COMMENTARY ABBREVIATIONS OF WORKS FREQUENTLY CITED

A Astronomia instaurata (Amsterdam, 1617), the 3rd ed. of the RB'Oolutions B The 2nd ed. of the Reoolutions (Basel, 1566) GV Valla, Giorgio, De expetendis et fugiendis rebus (Venice, 1501) Me Menzzer, C. L., Uber die Kreisbewegungen der Weltkorper (Leipzig, 1939; reprint of Thom 1879 ed.) MK Birkenmajer, L.A., Mikolaj Kopernik (Cracow, 1900). An abridged English version was published by Xerox University Microfilms, Ann Arbor, Michigan, in 1976, too late for use herein. Mu Nikolaus Kopernikus Gesamtausgabe, vol. II (Munich, 1949) N Copernicus, De reoolutionibus orbium coelestium (Nuremberg, 1543) NCCW Nicholas Copernicus Complete Works: vol. I (London/Warsaw, 1972) p Prowe, Leopold, Nicolaus Coppernicus (Berlin, 1883-1884; reprinted, Osnabriick: Zeller, 1967): PI: vol. I, part I PI1: vol. I, part II PII: vol. II P-R Peurbach, George and Johannes Regiomontanus, Epitome (Venice, 1496) PS Ptolemy, Syntaxis PS 1515 Ptolemy, Syntaxis; Latin translation issued in Venice, 10 January 1515 SC Birkenmajer, L. A., Stromata Copernicana (Cracow, 1924) T Copernicus, De r8fJolutionibus orbium caelestium (Thom, 1873) 3CT Rosen, Edward, Three Copernican Treatises, 3rd ed. (New York: Octagon, 1971) W Copernicus, De reoolutionibus orbium coelestium (Warsaw, 1854) Z Zinner, Emst, Entstehung und Ausbreitung der coppernicanischen Lehre (Erlangen, 1943) ZGAE Zeitschrift jar die Geschichte und Altertumskunds Ermlands

332 NOTBS ON PP. VII-XV

NOTES ON TilE FRONT MATTER

P. VII-xn. No Table of Contents appears in Copernicus' autograph. What appears in N was prepared by the editors of N not by Copernicus. P. XV. This title page was designed in Nuremberg by the persons responsible for the first edition of the RBf!olutions (cited hereafter as N). They surely did not consult Copernicus, who was at that time gravely ill in distant Frombork. Whether he as the author had supplied his own formal title page cannot be determined from his autograph in its present condition. For at .some undetermined time (as was indicated in NCCW, I, 6, 11) the first leaf of the first quire was carefully cut away and the loose edge glued firmly in place to avoid damage to the rest of the autograph. On what is now the recto or upper side of the first folio, in the lower right comer, the letter a as the designation of the first quire was written by someone other than Copernicus, whereas he himself had written all the other quire designations. In the absence of folio 0 (zero, as the missing leaf has been labeled) nobody can say with any degree of certainty whether or not it was the title page as conceived by Copernicus, if indeed he did conceive one. A dark cloud therefore hangs over the title Six Books on the Revolutions of the Hetl'08nly Spheres (De r8'Do­ lutionibus orbium coslsstium libri VI). In particular, the words orbium coelestium were deleted in a number of copies of N. The unauthorized insertion therein of Andreas Osiander's Foreword was bitterly attacked by George Joachim Rheticus (1514-1574), Copernicus' only disciple and his loyal supporter. Rheticus' thoroughly justified protest against the interpolation of Osiander's Foreword may somehow have given rise to the notion that the Nuremberg preacher also without warrant tacked on the two words orbium coelestium (on the assumption that De rBflolutionibus, used by Copernicus for convenience as a short title, was likewise his choice for the full title). Yet in themselves these two words orbium coslsstium are completely unobjectionable. For (like Rheticus) Copernicus believed that the visible celestial bodies, that is, the stars and planets, were imbedded in invisible heavenly spheres ( orbes coslestss), which moved the visible bodies in accordance with the cosmological thinking accepted since Greek antiquity. Hence, although we cannot be absolutely sure what formal title, if any, Coper­ nicus had in mind for his masterpiece, on conceptual grounds no fault can be found with the title as printed in N. For at the head of Book I, Chapter 10, in the autograph Copernicus himself wrote "The Order of the Heavenly Spheres" (De ordine coelestium orbium; NCCW, I, fol. sr, line 1), and in the Preface he referred to the "revolution of the celestial spheres" (revolutions orbium coslestium). Thus, the two contested words not only expressed one of his basic ideas but also formed !p1 integral part of his working vocabulary. If the deletion of orbium coelestium occurred before these supposed celestial spheres were forever banished from the heavens by Tycho Brahe, that great admirer of Copernicus, then perhaps the opposition was based on our astronomer's choice of words in the opening sentence of his Preface. There he refers to the six books which he has written "about the revolutions of the spheres of the universe" (de revolutionibus sphaerarum mundi). Perhaps some poorly informed person may have failed to realize the semantic equivalence of the expressions sphaerarum mundi and orbium coelestium. As a perceptive stylist, Copernicus studiously avoided the excessive repetition of the same locution. When sphaerarum mundi and orbium coelestium flowed from his pen, they were entirely interchangeable and the occurrence of the first expression in the Preface guarantees the validity of the second expression in the title. Although sphaera and orbis in this cosmographical context are in general synonymous, as precise mathematical terms they refer to two quite different bodies: a solid sphere, by contrast with a hollow spherical shell or ring. This distinction between sphaera and orbis was set forth with complete clarity by the celebrated philologist, astronomer, and geographer, Sebastian Miinster (1488-1552), in his textbook of elementary mathematics, Rudi­ menta mathematica (Basel, 1551), p. 60:

333 NOTBS ON PP. XV-XVI

Among solid figures, the most important is the sphere, [which is] the most regular of all. It is a regular solid body, bounded by a single surface ... We conceive the sphere to be generated by the complete rotation of a semicircle: while the diameter of the semicircle remains fixed, the plane sur­ face of that circle is rotated ... An orb (orbis) is also a solid figure. It is bounded, however, by two round spherical surfaces, namely, an interior [surface], which is called concave, and an exterior [surface], which is labeled convex. If these surfaces have the same center, the orb will be uniform, that is, of equal thickness throughout. But if the surfaces have different centers, these will make the orb's thickness nonuniform and irregular. This is the kind which the heavens of all the planets have. Copernicus' conceptions of sphaera and orbis agree with those of his somewhat younger contemporary, Mtinster. The blurb on the tide page was obviously not composed by Copernicus. As a piece of advertising copy, it was evidendy contributed by the printer-publisher himself, for Johannes Petreius (Hans Peter) was an author in his own right. Two years later, in 1545, when he issued Girolamo Cardano's Ars magna (translated into English by T. Richard Witmer as The Great Art or the Rules of Algebra, Cambridge, Mass., 1968), Petreius placed a sim­ ilar blurb on the tide page of that foundation of the theory of equations. Then, when he published Cardano's De subtilitate in 1550, Petreius attributed the blurb to himself. By the same token, when Petreius published Algorithmus demonstratus in 1534, his blurb ended: Quare eme, lege, & iwaberis (Therefore buy, read, and enjoy yourself). Nine years later, his blurb for Copernicus closed with the same sentiment in slighdy different language: Igitur eme, lege, jruere. All four of these promotional passages unmistakably emanate from the same hand. The warning on the tide page to nonmathematical readers to stay away from the R800lutions (Copernicus' short tide is equally serviceable in English) in all likelihood was due to Osiander, who edited for Petreius not only Copernicus' R~olutions but also Cardano's Ars magna. Although Osiander became famous for his militant theological views and stirring sermons, his favorite hobby was the mathematical sciences. He left the warning to nonmathematicians on the tide page of the R800lutions in Greek. This warning was then commonly believed to have been inscribed over the entrance to Plato's Academy. As long as this famous school existed, nobody made any reference to any such inscription. But after the emperor Justinian ordered all pagan schools, includ­ ing Plato's Academy, to be closed in 529, the first reference to this supposed inscription was written by Joannes Philoponus in his Commentary on Aristotle's Soul (Commentaria in Aristotelem graeca, XV (Berlin, 1897), p. 117, lines 26-27). When Fran~is Viete (1540-1603) savagely attacked Copernicus as incompetent, the French math­ ematician turned this very warning against Copernicus himself. In denouncing Copernicus' ''ungeometrical procedure," Viete, although writing in Latin, used a Greek word (ageometresia) related to the first word in the warning, evidendy on the (mistaken) assumption that it was Copernicus who had been responsible for placing it on the tide page. That mistake did not die with Viete. In his Astronomical R~olution (Paris, 1973), p. 73, § 9, Alexandre Koyre said: "no doubt with the approval of his master [Copernicus], he [Rheticus] put on the tide page of De Revolutionibus the famous adage which, [at least] according to tradition, was placed above the portals of the [Platonic] Academy." The printing of the R~olutions was finished a few days before 21 March 1543, when Sebastian Kurz, an employee of the Fugger banking firm, sent a copy from Nuremberg to Emperor Charles V. On the other hand, Copernicus in Frombork did not receive a copy until 24 May 1543, the day he died. See Marcel Bataillon, "Charles-Quintet Copemic: documents inedits," Bulletin hispanigue, 1923,25:256-258, or La R~ de Pologne, 1923, 2nd series, 1: 131-134, and AtJant, awe, apris Copernic (Paris, 1975), p. 184. P. XVI. This forceful presentation of the fictionalist philosophy of science was inserted by Osiander after Rheticus, the first editor of the RetJolutions, had left Nuremberg for Leipzig University, which had just appointed him professor of mathematics. Petreius then entrusted the editing of the R~olutions to Osiander, a prolific author, some of whose writings had been printed by Petreius. Osiander had presumably first met Rheticus when the latter, as a young professor at Wittenberg University, in 1538 obtained a leave of absence for the purpose of visiting the German astronomers. Then, in 1540 when Rheticus' First Report about the Copernican astronomy was published in Gdansk, a copy was sent to Osiander. The Lutheran preacher was shocked by the new system's claim to be true, since he regarded divine revelation as the sole source of truth. On 20 April 1541 he wrote Copernicus a letter, the only surviving part of which reads as follows:

I have always felt about hypotheses that they are not articles of faith but the basis of computa­ tion; so that even if they are false it does not matter, provided that they reproduce exacdy the phenomena of the motions. For if we follow Ptolemy's hypotheses, who will inform us whether the sun's nonuniform motion occurs on account of an epicycle or on account of the eccentricity,

334 NOTBS ON P. XVI

since either arrangement can explain the phenomena? It would therefore appear to be desirable for you to touch upon this matter somewhat in your introduction. For in this way you would mollify the peripatetics and theologians, whose opposition you fear. On the same day Osiander addressed a companion letter to Rheticus, who was then also in Frombork waiting for Copernicus to put the final touches on the RefJolutions. Osiander's second letter continued along the lines laid down in the first: The peripatetics and theologians will be readily placated if they hear that there can be different hypotheses for the same apparent motion; that the present hypotheses are brought forward, not because they are in reality true, but because they regulate the computation of the apparent and combined motion as conveniently as may be; that it is possible for someone else to devise different hypotheses; that one man may conceive a suitable system, and another a more suitable, while both systems produce the same phenomena of motion; that each and every man is at liberty to devise more convenient hypotheses; and that if he succeeds, he is to be congratulated. In this way they will be diverted from stem defense and attracted by the charm of inquiry; first their antagonism will disappear, then they will seek the truth in vain by their own devices, and go over to the opinion of the author. Unfortunately today we no longer have Copernicus' reply. But Johannes Kepler (1571-1630), who still had access to it, reported that "Copernicus, strengthened by a stoical firmness of mind, believed that he should publish his convictions openly, even though the science should be damaged" (3CT, p. 23). Hence Osiander knew that his own fictionalist philosophy was rejected by Copernicus, who regarded the secular human reason as fully capable of attaining the truth about the physical universe, some of whose secrets he had himself unveiled. But by one of those strange twists of fate, of which human history is all too full, control over the printing of the RefJolutions passed into the hands of an editor whose fundamental outlook was diamet­ rically opposed to that of the author, who had already resolutely resisted the (future) editor's efforts to persuade him to mask his thoughts. Having failed in this attempt, Osiander slipped his own fictionalist pronouncement, from which he carefully withheld his name, in with the authentic front matter so surreptitiously that Petreius did not notice the intru­ sion. Although Osiander successfully concealed from Petreius his authorship of the interpolated Foreword, he later openly confessed his ruse. Whereas Rheticus promptly detected it and Giordano Bruno denounced its author as a jackass, the interpolated Foreword fooled many readers, including the great nineteenth-century historian of astronomy J. B. J. Delambre, who did not suspect its spuriousness. Neither did Bernardino Baldi (1553-1617), author of the earliest surviving substantial biography of Copernicus, which was completed on 7 October 1588. After summarizing some key features of the new astronomy (the heavens stationary, with the earth moving around the sun, at rest in the center of the universe), Baldi remarked that Copernicus "nevertheless excuses himself by saying he did that, not because he believes it to be true that this is the nature of things, but because he was induced to do so by thinking that in this way he could with greater convenience accomplish what he had undertaken" (translated from the Italian text reprinted in Bilmski, La 'Oita di Copernico di B. Baldi, 1973, pp. 21-22, lines 58-60). Baldi's misattribution of Osiander's Foreword to Copernicus is matched by his misstatements about Copernicus and Bologna. Baldi (line 23) puts Copernicus' age when he reached Bo­ logna "about 21," although the law student and future astronomer was already 23 years, 8 months. Baldi says "about 21" because he is not sure whether Copernicus was born in 1472 or 1473, yet he is sure that Copernicus entered Bologna in 1494 (line 25). Actually he matriculated in 1496. In the history of ideas Osiander was neither the first not the last to espouse his own brand of philosophy of science. In the 19th century the distinguished organic chemist Kekule declared: The question whether atoms exist or not has but little significance in a chemical point of view: its discussion belongs rather to metaphysics. In chemistry we have only to decide whether the assumption of atoms is an hypothesis adapted to the explanation of chemical phenomena. More especially have we to consider the question, whether a further development of the atomic hy­ pothesis promises to advance our knowledge of the mechanism of chemical phenomena. I have no hesitation in saying that, from a philosophical point of view, I do not believe in the actual existence of atoms, taking the word in its literal signification of indivisible particles of matter. I rather expect that we shall some day find, for what we now call atoms, a mathemat­ ico-mechanical explanation, which will render an account of atomic weight, of atomicity, and of numerous other properties of the so-called atoms. As a chemist, however, I regard the assump­ tion of atoms, not only as advisable, but as absolutely necessary in chemistry. I will even go

335 NOTES ON PP. XVI-XVII

further, and declare my belief that chemical atoms exist [italics in original), provided the term be understood to denote those particles of matter which undergo no further division in chemical metamorphoses ... We may, in fact, adopt the view of Dumas and of Faraday, "that whether matter be atomic or not, thus much is certain, that, granting it to be atomic, it would appear as it now does" (Richard Anschutz, August KekuU, Berlin, 1929, II, 366). In the Ptolemaic system the elongation of Venus from the sun was explained by making that planet revolve around the circumference of an epicycle whose center traveled with the sun's mean motion. Con­ sequently the radius of Venus' epicycle had to be long enough to produce that planet's greatest elongation from the sun of more than 40°, say, 45°. In the resulting right isosceles triangle, set the radius of Venus' epicycle= 1. Then the planet's perigeal distance from the earth at the center of the Ptolemaic universe ;;;;; t, and its apogeal distance ;;;;; 2!-, "more than four times" greater. Hence, the apparent diameter of Venus, if Ptolemy were right, would appear more than four times greater at perigee than at apogee, and the body of the planet more than sixteen times greater. No such variation in the brightness of Venus had ever been reported, as Osiander correctly remarks. Osiander's emphasis on this defect in the Ptolemaic system has been misattributed to Copernicus: "In one glaring case he [Copernicus] points out Ptolemy's error in failing to account for the variation in brightness of Venus" (Derek Price, "Contra-Copernicus," in Critical Problems in the History of Science, University of Wis­ consin Press, 1959, p. 198). At no time does Copernicus call attention to Ptolemy's failure to account for the absence of variation in the naked-eye brightness of Venus. Perhaps the glare of this case so blinded Price that he confused Osiander with Copernicus. The latter's objection to the huge size of Venus' epicycle in the Ptole­ maic system is based, not on the absence of the concomitant variations in the planet's brightness, but on the prin­ ciple of plenitude: the universe is full, and contains no such large unused spaces as would be contained within the epicycle of Venus, if Ptolemy were right (Revolutions, I, 10). P. XVII. Nicholas Schonberg (1472-1537) went to Varmia in 1518 as a papal emissary in an effort to make peace between the Knights of the Teutonic Order and the kingdom of Poland. At that time SchOnberg did not meet Copernicus and probably did not even hear about him. That situation changed, however, after the death of Pope Clement VII on 25 September 1534, for his secretary, Johann Albrecht Widmanstetter (1506- 1577), then entered Schonberg's service. In 1533, between 6 June and 9 September, Widmanstetter had de­ livered an "explanation of Copernicus' opinion about the earth's motion" in the Vatican gardens, for which he was handsomely rewarded by the pope, who presented him with a rare Greek manuscript (3CT, p. 387). Wid­ manstetter's interest in the Copernican astronomy continued after his change of employers, and he may well have drafted this letter to Copernicus, which was signed by SchOnberg on 1 November 1536. Although Coper­ nicus kept Schonberg's letter in his files and later released it so that it could be printed in the front matter of the Revolutions in 1543, during the intervening half-dozen years Copernicus did not accept Schonberg's invita­ tion to send his writings to Rome or to have them copied in Frombork at the cardinal's expense. Copernicus' inaction and ·silence were presumably motivated by his characteristic prudence. To fill this lull, an utterly unhistorical scenario was concocted by Baldi. He imagined that Schonberg had Copernicus' work; recognized its perfection and excellence; showed it to the pope, by whose judgment it was approved. The said Cardinal [Schonberg] addressed himself to Copernicus to ask him for many reasons to be willing to publish it ... Copernicus dedicated it to Pope Paul III, by whose judgment, as has been said, it had been approved. What reward Co­ pernicus obtained for it and what happened in the said affair, I would not know (Bilinski, Vita, pp. 22-23, lines 103-106, 109-111). The cause of historical accuracy would have been far better served had Baldi not pretended to know that a copy of the manuscript of the Revolutions was possessed by Schonberg, shown by him to the pope, and approved by the pope; and that SchOnberg addressed himself to Copernicus a second time. No such communication from Schonberg to Copernicus after 1 November 1536 has ever been documented, nor has any papal approval of the Revolutions. In the shorter version of his biography of Copernicus (Cronica de' matematici, Urbino, 1707, p. 121) Baldi restricts himself to stating that "Copernicus dedicated his great work on the Revolutions to Paul III," while keeping silent about any supposed approval by the pope. The contrasting case of Girolamo Fracastoro, who was born about five years later than Copernicus and died ten years after him, is instructive. Like Copernicus, Fracastoro dedicated his work on astronomy (Homocentrics, Venice, 1538) to Pope Paul III. But unlike self-effacing, taciturn Copernicus, Fracastoro tells us quite plainly in his dedication how he was induced to direct it to the pope. Fracastoro felt deeply indebted for his leisure time to his generous patron, Gian Matteo Giberti, the bishop of Verona, his native city. Hence in the first in­ stance Fracastoro desired to dedicate his Homocentrics to Giberti.

336 NOTES ON P. XVII

However, the bishop replied: ''You will publish not under mine, but under mightier auspices. For if this gift is mine, it is my pleasure that the new work should be dedicated to the new pope," Paul III having just been elevated to the papal throne. On the one hand, Bishop Giberti had Fracastoro's work and instructed that author to dedicate it to Pope Paul III. On the other hand, despite Baldi, Cardinal Schonberg never had Coper­ nicus' work and never showed it to the pope, who never approved it. In fact, the favorite theologian of Pope Paul III, his Master of the Sacred and Apostolic Palace, Bartolomeo Spina "had planned to condemn his book [Copernicus' Revolutions], but being prevented at first by illness and then by death, he was not able to carry out this" plan. This hostile attitude on the part of Paul Ill's closest theological adviser toward the Revolutions is recorded by Spina's intimate friend, Giovanni Maria Tolosani, who adds: "After [Spina's death in 1546] I undertook to perform this task in this little work for the purpose of safeguarding the truth for the common advantage of Holy Church" (Studia CopernicatuJ, VI, 42). The "little work" in question was appended by Tolosani to his long treatise On the Truth of Holy Scripture. This tract, in manuscript, was later utilized by the Dominican preacher who initiated the attack on that great Copernican, GalUeo, which culminated in the sentencing of the eminent Italian scientist to life imprisonment (Studia Coper­ nicatuJ, VI, 31). Had Baldi perhaps heard about Pope Clement VII's actual approval in 1533 of Widmanstetter's "explana­ tion of Copernicus' opinion about the earth's motion"? Did Baldi then somehow mistakenly transmogrify that real happening into an imaginary approval by Pope Paul III of Copernicus' Revolutions? Had Copernicus ac­ tually received Paul III's permission to print the Revolutions, what earthly reason would have deterred him from making a public proclamation to that effect at some prominent point in his Preface? See Edward Rosen, "Was Copernicus' Revolutions Approved by the Pope?", JourtuJl of the History of Ideas, 1975, 36: 531-542. Schonberg referred to the "eighth heaven." This term designated the sphere of the fixed stars, below which the traditional astronomy located each of the seven planets in its own heaven, while the eighth heaven was the sphere of the fixed stars. Of these eight heavens or spheres, six were somehow overlooked by Thomas S. Kuhn, whose Copernican Revolution (Harvard University Press, 1957, 1966) repeatedly refers to "the ancient two-sphere universe." Schonberg's agent, Theodoric of Reden, was the representative of the Varmia Chapter in Rome. Presumably it was he who had brought the astronomy of Copernicus, his fellow-canon, to the attention ofWidmanstetter. After his return to Frombork, Theodoric served as one of the four executors of Copernicus' last will and testament (which has not survived; 3CT, p. 404).

337 NOTES ON P. 3

NOTES ON THE REVOLUTIONS

P. 3, line 3. In September 1543 Petreius presented to Rheticus' friend Achilles Pirmin Gasser (1505-1577) a copy of the Revolutions which is preserved in the Vatican Library (Enrico Stevenson, Jr., Inventario dei libri stampati palatino-vaticani, Rome, 1886-1889, vol. 1, part 2, page 161, no. 2250; a facsimile of the title page in Karl Heinz Burmeister, Achilles Pirmin Gasser, Wiesbaden, 1970, I, 77). On folio 2v of the first signature of his copy Gasser wrote that Copernicus' Preface was "composed at Frombork in Prussia in the month of June 1542" (Z, p. 451). Presumably Gasser was so informed by Petreius or by Rheticus. In either case it would appear that when Rheticus left Frombork in the autumn of 1541 to return to his post as professor of mathematics at Wit­ tenberg University, the manuscript of the Revolutions which he took with him did not contain this Preface (because it had not yet been written). Then at the end of the winter semester on 1 May 1542 Rheticus went on leave from Wittenberg University to Nuremberg, where Petreius began to print the Revolutions. Was it there and then that the plan was devised to have Copernicus write this Preface, dedicating the Revolutions to the reigning pope? If so, we can understand why the autograph of the Preface has not survived. For if Copernicus, after writing the Preface in June 1542, sent it directly to Nuremberg, then it must have shared the fate of that manuscript copy of the Revolutions from which Petreius printed N. For that printer's copy has utterly vanished, together with the autograph of the Preface. What has survived, on the other hand, is the autograph of the Revolutions which remained behind at Frombork when Rheticus left some nine months before Copernicus wrote the Preface. Since a typesetter could easily be confused by the twists and turns indicated by the numerous deletions and insertions in Copernicus' autograph, a fair copy of it had to be provided for Petreius' workshop by Rheticus. None of this intricate story (if it has been correctly disentangled above) was familiar to Galileo, who knew only the first two printed editions of the Revolutions (1543, 1566). What has since been gleaned from the letters and marginal notes written by Copernicus' contemporaries was still undiscovered in the time of Galileo. By drawing some wholly unwarranted inferences, that great Italian scientist made the dreadful mistake, in his Letter to the Grand Duchess Christina, of saying that Copernicus, "having assumed his laborious enterprise by order of the supreme pontiff, ... dedicated his book to Paul Ill." Of course no supreme pontiff ever issued any order that the Revolutions should be written by Copernicus, who needed no orders from the supreme pontiff or from anybody else to compose the work to which he devoted the whole of his mature life. When Galileo's valiant supporter Tommaso Campanella wrote his Defense of Galileo (Frankfurt, 1622) while serving a sentence of life imprisonment in a Neapolitan jail as a condemned heretic, Campanella went even farther astray than Galileo by saying that "Pope Paul III ... to whom Copernicus dedicated the book ..• approved it" and gave his "permission that the book should be printed." Actually, Campanella in jail knew no more than Galileo (and Baldi) outside jail about what had happened in Frombork and Nuremberg, three-quar­ ters of a century before. As a matter of sober historical fact, there is not the slightest shred of evidence that Paul III was apprised in advance of Copernicus' intention to publish the Revolutions and dedicate it to him, nor is there any indication whatever that the pope looked with anything but disfavor on the Revolutions and its dedica­ tion. Nevertheless, lack of evidence has not prevented numerous writers from uncritically repeating Baldi's, Galileo's, and Campanella's unhistorical assertions in a bewildering variety of forms. P. 3:19. This letter is translated below at pp. 25-26. The addressee Hipparchus is not to be identified with the great astronomer of that name, who lived in the 2nd century B.C. and had nothing whatever to do with the Pythagorean&. P. 3:31. For Schonberg's letter to Copernicus, see page XVII. Copernicus' description of Schonberg as "re­ nowned in every field of learning" would seem to be polite flattery rather than a candid evaluation of the car­ dinal, whose pitifully small intellectual output is catalogued in Jacques Quetif and Jacques Echard, Scriptores ordinis praedicatorum, Paris, 1719-1723, II, 103-104 (reprinted, Burt Franklin, New York, 1959). P. 3:33. Giese (1480-1550), a fellow-canon of Copernicus since 1504 and one of his closest friends, became bishop of Chehnno (Kulm, in German) on 22 September 1537. His knowledge of theology had been demon­ strated in a polemical work which he published (in two editions) at Cracow in 1525 at the urging of Copernicus. Six years after the astronomer's death, Giese became bishop of Varmia on 20 May 1549. P. 3:37. In his Art of Poetry (lines 388-389) the Roman writer Horace advised budding authors not to publish their work as soon as it was finished but to hold it back "until the ninth year" thereafter. Copernicus' quadruplication of this incubation period has sometimes been misunderstood to mean that he hatched the Rev­ olutions for thirty-six whole years. That computation would have him start (or even, in an alternative version, finish) the Revolutions in 1507 (or 1506). Actually, his first glimpse of the geokinetic universe did not occur to him before 1508 (3CT, p. 339), and it was at least some years later before he began to write the Revolutions.

338 NOTES ON P. 3

By saying in June 1542 that the work was then in its fourth novennium, Copernicus implied that he had started it at some time before 1515. P. 3:38. Copernicus' failure to mention Rheticus by name here has recently been branded a "scandal" and "betrayal of Rheticus." But Rheticus himself did not complain that he had been slighted by Copernicus, nor did he feel or express any resentment toward his teacher. On the contrary, Rheticus later declared publicly that Copernicus "is never praised enough," and avowed that he had "always cherished, esteemed, and honored Copernicus not only as a teacher but also as a father." His own father having been beheaded as a supposed sorcerer, Rheticus, a Protestant and professor of mathematics at Wittenberg University, the militant think-tank of the anti-papal Lutheran heresy, realized full well that his name would be out of place in a Preface addressed to Pope Paul III and honoring the cardinal of Capua and the bishop of Chelmno. Although we do not have any direct information about Rheticus' reaction to N, we know that he promptly sent two copies to Giese, together with a letter that has unfortunately been lost. However, we do have Giese's reply to Rheticus, which he wrote from Lubawa, the see of his diocese of Chelmno, on 26 July 1543. Because this document is of prime historical importance, an English translation of it follows.

On my return from the royal wedding in Cracow [of Prince Sigismund Augustus of Poland with Elisabeth, archduchess of Austria], in Lubawa [Lobau] I found the two copies, which you had sent, of the recently printed treatise of our Copernicus. I had not heard about his death before I reached Prussia. I could have balanced out my grief at the loss of that very great man, our brother, by reading his book, which seemed to bring him back to life for me. However, at the very threshold I perceived the bad faith and, as you correctly label it, the wickedness of Petreius, which produced in me an indignation more intense than my previous sorrow. For who will not be anguished by so disgraceful an act, committed under the cover of good faith? Yet I am not sure whether [this misconduct] should be attributed to this printer, who depends on the labor of others, rather than to some jealous person. Grieving that he would have to abandon the previous beliefs if this book achieved fame, perhaps he took advantage of that [printer's] ingenuousness to diminish faith in the treatise. However, lest the man should escape scot-free who permitted himself to be misled by someone else's deception, I have written to the City Council of Nuremberg, indicating what I thought had to be done in order to restore faith in the author. I am sending you the letter together with a copy of it, to enable you to decide how the affair should be managed on the basis of what has been started. For I see nobody better equipped or more eager than you to take this matter up with that City Council. It was you who played the leading part in the enactment of the drama, so that now the author's interest seems to be no greater than yours in the restoration of that which has been distorted. Provided that this interests you at all, I ardently implore you to pursue this matter with the utmost earnestness. If the first sheets are going to be printed again, it seems that you should add a brief introduction, which would cleanse the stain of chicanery also from those copies which have already been dis­ tributed. I should like in the front matter also the biography of the author tastefully written by you, which I once read. I believe that your narrative lacks nothing but his death. This was caused by a hemorrhage and subsequent paralysis of the right side on 24 May, his memory and mental alertness having been lost many days before. He saw his treatise only at his last breath on his dying day. The distribution of the published treatise before his death will not be an obstacle, since the year agrees, and the day when the printing was finished was not indicated by the pub­ lisher. I should like also the addition of your little tract, in which you entirely correctly de­ fended the earth's motion from being in conflict with the Holy Scriptures. In this way you will fill the volume out to a proper size and you will also repair the injury that your teacher failed to mention you in his Preface to the treatise. I explain this oversight not by his disrespect for you, but by a certain apathy and indifference (he was inattentive to everything which was nonscien­ tific) especially when he began to grow weak. I am not unaware how much he used to value your ac­ tivity and eagerness in helping him. As for the copies of the treatise which you sent to me, I am deeply grateful to the donor. These copies will serve me as a permanent reminder to preserve the memory not only of the author, whom I always cherished, but also of you. Just as you proved yourself to be an energetic assistant to him in his labors, so now you have helped us with your effort and care lest we be deprived of the enjoyment of the finished work. It is no secret how much we all owe you for this zeal.

339 NOTES ON PP. 3-4

Please let me know whether the book has been sent to the pope; for if this was not done, I would like to carry out this obligation for the deceased. Farewell (Latin text in PII, 419-421). Rheticus must have acted with great alacrity in adopting Giese's suggestion that his complaint against Petreius should be forwarded to the City Council of Nuremberg. That body in tum transmitted the complaint to Petreius, who submitted a draft reply to the Council. In the Council's manuscript archives for 1543 Hier­ onymus Baumgartner, the Council's secretary, recorded the following decision taken on Wednesday, 29 August: Send to Tiedemann [Giese], bishop of Chehnno in Prussia, the answer written by Johannes Pe­ treius to the bishop's communication. The acerbities in Petreius' answer should be omitted and sweetened. Add to it: no punishment can be infiicted on Petreius in this matter on the basis of his answer (MK, p. 403). Unfortunately, Petreius' draft and the Council's official reply to Giese have not survived. However, the essential content of Petreius' self-defense can be inferred from a note written at the top of fol. 2r in the front matter of the copy of N owned by Michael Miistlin (1550-1631), the Tiibingen professor of astronomy, to whom mankind is deeply indebted because it was he who converted the great Johannes Kepler to Copemicanism. Miistlin's copy of N is now preserved in the municipal library of Schaffhausen, Switzerland. In his own tiny handwriting Miistlin, who was endowed with exceptionally acute vision, wrote: With reference to this Foreword [by Osiander], I, Michael Miistlin, found the following words written somewhere among the books of Philip Apian (which I bought from his widow). Although the writer did not append his name, nevertheless I was very easily able to recognize from the formation of the letters that the handwriting was that of Philip Apian. Hence I conjecture that these words had been copied by him from some source, doubtless for the purpose of preserving them. "On account of this Foreword George Joachim Rheticus, the Leipzig professor and disciple of Copernicus, became embroiled in a very bitter wrangle with the printer [Petreius]. The latter asserted that the Foreword had been turned over to him with the rest of the treatise. Rheticus, however, suspected that Osiander had put it in the front matter of the work. If Rheticus knew this to be a fact, he declared, he would so maul the fellow that he would mind his own business and not dare to mutilate astronomers any more in the future." Nevertheless, Apian told me that Osiander had openly admitted to him that he had added this [Foreword] as his own idea (Z, p. 453). Unfortunately we do not know exactly when Osiander made this open admission to Philip Apian (1531-1589). The latter lived in Ingolstadt, where his father Peter Apian was professor of mathematics at the local university. Osiander was compelled to leave Nuremberg for religious reasons about 18 November 1548 and look for a post elsewhere. At this time he may have traveled the fifty miles south to lngolstadt, where he had been admitted to the university as a young man. If so, Philip Apian may have heard Osiander's statement regarding the Fore­ word in Ingolstadt in November 1548. The next stage in the transmission of Osiander's confession is more definite. For on 3 December 1568 Miistlin matriculated at Tiibingen University, where Philip Apian became the professor of mathematics on 1 March 1570. Miistlin bought his copy of N on 6 July 1570. It must have been then, or not long thereafter, that Apian repeated to Miistlin Osiander's admission that it had been his own idea to interpolate his Foreword inN. He did so surreptitiously. Hence Petreius first found out that the Foreword had been written by somebody other than Copernicus when the City Council of Nuremberg forwarded Giese's complaint to the printer, with a request for his reply. Since the City Council accepted Petreius' explanation that he had been deceived by Osiander's sleight of hand, and decided to drop the whole matter forthwith, Giese's proposal that the front matter of the Re'Oo­ Iutions should be revised was not put into practice. Hence Rheticus' biography of Copernicus and his defense of the compatibility of the new astronomy with the Bible were not printed (and have since disappeared from sight), while his name remained excluded from the Re'Oolutions, which in all probability would never have been published had he not intervened. On the other hand, a cordial working relation continued between Petreius and Osiander, as is shown by the production, two years after the Reflolutions, of Cardano's Ars magna, which was dedicated to Osiander, edited by him, and printed by Petreius. P. 4:12. For Copernicus' definition of the "tropical year'' as distinguished from the "sidereal year", see III, 1. P. 4:15. Homocentric spheres, all turning around the same center- the earth at rest in the middle of the universe - were the only cosmological units admitted by Aristotle and the Aristotelians. This principle

340 NOTES ON P. 4 of homocentricity having been found incapable of accounting for all the known celestial phenomena, some were explained by moving the center of the heavenly spheres away from the central earth or, alternatively, by mount­ ing the heavenly body on an epicycle whose center was carried around by a deferent, either homocentric or eccentric. The astronomical system employing eccentrics and epicycles culminated in Claudius Ptolemy's Syntaxis (miscalled the "Almagest") in the second century of the Christian era. The Greek text of the Syntaxis (cited hereafter as PS) was first printed at Basel in 1538, and a copy of that first edition was presented to Coper­ nicus by Rheticus in 1539, too late for any extensive effect on the composition of the Revolutions (see note to p. 89 :13). During the preceding quarter-century Copernicus used two Latin translations of PS. Gerard of Cremona's (Venice, 1515; cited hereafter asPS 1515) had been made from an Arabic version, whereas George of Trebizond's (Venice, 1528) was based directly on Ptolemy's Greek text. Copernicus also had access to the Epitome of Ptolemy's Almagest by George Peurbach and Johannes Regiomontanus (Venice, 1496; cited hereafter as P-R). The Varmia Chapter's copy of PS 1515 is now in Uppsala (MK, pp. 242-292), but its copy of P-R has not been tracked down (ZGAE, 5: 374-375). Copernicus' use of the 1528·translation is established by internal evidence (see notes to pp. 14:34 and 15:17). P. 4:22. "The first principles of uniform motion" require a rotating circle to traverse in equal times equal arcs as measured from the circle's own center. As Ptolemy formulated this principle (PS, III, 3), "all the east­ ward motions of the planets ... are uniform and circular by nature; that is, the straight lines which are conceived of as causing the rotation of the planets or their circles in all cases in equal times uniformly traverse equal angles at the center of each of the circles." Then in PS, IX, 2, Ptolemy explained why it is "our task to derive all the ob­ served nonuniformities of the five planets ... as resulting from perfectly uniform and circular motions." For, only such uniform circular motions are "in conformity with the nature of the divine [heavenly bodies], which are remote from disorder and irregularity." Having declared that any departure from uniform circular motion is disorderly, irregular, and incompatible with the nature of celestial bodies and heavenly motions, Ptolemy proceeds in PS, IX, 5, to introduce just such a departure: "The epicycles cannot have their own centers carried about on thGse eccentrics around whose centers the epicycles form equal angles in equal times in uniform east­ ward motions ... The epicycles' centers revolve around circles (a) equal to the eccentrics which produce the a­ nomaly but (b) not described about the same centers." The distance from the epicycle's center to the eccentric's center remains constant, but the angle formed by the epicycle's center at the eccentric's center in equal times is unequal. This is the contradiction of "the first principles of uniform motion" which Copernicus urges here against "those who devised the eccentrics." P. 4:25. Just as Copernicus previously recalled Horace's advice to aspiring authors (note to p. 3:37), here he echoes the first five lines of the Art of Poetry. P. 4:39. Copernicus' conception of the universe as a world machine (machina mundi) is to be linked with his insistence on absolute adherence to the principle of uniform motion. A mechanical circle, or wheel, when it executes a motion of rotation, must tum uniformly about its own center, if the rotation is destined to last indefinitely (in accordance with the traditional beliefs of the astronomers). Uniform circular motion, consequently, is a mechanical necessity. Any smoothly running machine, and above all the smoothly running world machine, may also appear beautiful to an admirer. But his outlook is not, for that reason, exclusively esthetic. Cicero is cited a few lines below this reference to the world machine. Copernicus may perhaps have been familiar with that passage in the Roman philosopher's Nature of the Gods (I, 10) where a speaker recalls "Plato's denial that any shape is more beautiful than" the sphere, "whereas to me the figure of the cylinder or cube or cone or pyramid seems to be lovelier." Judgments concerning beauty may vary. But if a mechanical wheel, on the earth or in the astronomer's heaven, is constrained to tum with uniform speed around some center other than its own, it will not last as long as Copernicus was confident that the heavens would. Hence, his world machine could run only on absolutely uniform motions. P. 4:45. "According to Theophrastus, Hicetas of Syracuse believes that the sky, sun, moon, stars, in short, all the heavenly bodies stand still, and that nothing in the universe moves except the earth. As this rotates and turns around its axis with the greatest speed, all the same phenomena occur as would appear if the heavens moved and the earth stood still" (Cicero, Academic Questions, II, 39, 123). A manuscript containing Cicero's Academic Questions was available to Copernicus in the library of the Frombork Chapter (ZGAE, 5: 377, n. 56). Mter finding this passage in Cicero's Academic Questions, Copernicus copied it out on the lower half of sig. a2v in Pliny the Elder's Natural History (Venice, 1487); a photofacsimile of his handwritten excerpt in the copy of Pliny now in the library of Uppsala University, Sweden, was published in MK, facing p. 567. Like the early printed editions, the Cicero manuscript to which Copernicus had access mistakenly gave "N" as the initial letter of the Syracusan geokineticist's name. Cicero is cited here but not quoted, unlike the next ancient source. That made sense astronomically, whereas in Cicero the denial of the moon's motion as a result of the earth's rotation must have convinced Copernicus that quotation was inadvisable.

341 NOTES ON P. 5

P. 5:1. The Opinions of the Philosophers, III, 13. This work was ascribed to Plutarch in Copernicus' time, but is now regarded as falsely attributed to him. Since Copernicus quotes pseudo-Plutarch in the original language, he must have had access to the Greek text, which was printed for the first time in the Aldine edition of Plutarchi opuscula LXXXXII (Venice, 1509), with the quoted passage at p. 328. It omits two key words (alia treptikos), which were omitted by Copernicus too. Although Copernicus' astronomy resembled the Pythagorean only to a limited extent, his quotation of pseudo­ Plutarch's references to the Pythagorean& Philolaus and Ecphantus led some superficial readers and writers to call his system Pythagorean. This tendency was strengthened when, on 5 March 1616, the Roman Catholic Sacred Congregation of the Index, in a decree dealing with a number of books, suspended the Revolutions until corrected. Our treatise was condemned in the following terms: It has come to the attention of the aforesaid Sacred Congregation that the Pythagorean doctrine of the earth's mobility and the sun's immobility, a doctrine which is false and completely contrary to divine Scripture and which Nicholas Copernicus teaches in his De revolutionibus orbium caeles­ tium . . . is already widely known and is accepted by many persons ... Therefore, lest an opinion of this sort spread any farther to the detriment of Catholic truth, the Congregation has resolved that the said Nicholas Copernicus' De revolutionibus orbium ... shall be suspended until corrected.

In 1620 the following Warning (Monitum) was announced: A Warning to the Reader of Nicholas Copernicus and a Correction of that [Author] Although the Fathers of the Sacred Congregation of the Index resolved that the book on the revolutions of the universe by Nicholas Copernicus, the well-known astronomer, should be completely prohibited because the author has no hesitation in asserting, not as hypothesis but as solid truth, principles concerning the position and motion of the terrestrial globe which are incompatible with Holy Script and its true and catholic interpre­ tation (conduct which is intolerable in a Christian), nevertheless, because his writings contain much that is highly useful to the Church, the Congregation has unanimously reached the conclusion that the works of Copernicus, which have been printed heretofore, should be permitted, and the Congregation has granted its permission, provided that, in accordance with the subjoined emendation, those passages are corrected in which the author discusses, not in terms of hypothesis but of flat assertion, the position and motion of the earth. Future editions are permitted, except that the aforesaid passages shall be emended as follows and this correction prefixed to the Preface by Copernicus. The "emendation of the passages in Copernicus' book which were deemed to deserve correction" will be indicated at the appropriate places. The condemnation of the Revolutions took place in the interval between two official revisions of the Index of Prohibited Books, its latest issue having been authorized by Pope Clement VIII in 1596. To a reprint (Rome, 1624) of this Clementine Index there were appended All the Decrees hitherto promulgated regarding Books Pro­ hibited since the Index of Clement VIII, including the suspension decree of 1616 and the correction decree of 1620. In the alphabetical list (Elenchus) of proscribed authors, which was published at Rome in 1632 by Franciscus Magdalenus Capiferreus, Secretary of the Congregation, Copernicus' name was entered. He was in due course taken up into the next official revision of the Index in 1664, the Index of Prohibited Books issued by order of Pope Alexander VII, which reprinted the suspension md correction decrees. The Revolutions remained on the Index more than two centuries, during which time no edition of the work appeared. For, the third edition was brought out in 1617 before the suspension decree was printed; and the fourth edition was published after the work had been removed from the Index published in 1835. The third edition of the Revolutions (Amsterdam, 1617) was entitled Astronomia instaurata and will be cited hereafter as A; the fourth edition (Warsaw, 1854) will be cited hereafter as W. P. 5 :36. This proverb is quoted by Copernicus in Latin, although it does not appear in any classical Roman writer. It was first translated into Latin by Erasmus, who had found it in Aristophanes and incorporated it in his Chiliades adagiorum (Venice: Aldus, 1508), where it appears as no. 29 in the sixth hundred of the second thousand. There is no evidence that Copernicus was familiar with Erasmus' collection of proverbs at first hand. But his friend Giese corresponded with Erasmus. The latter in 1526-1527 published a polemical tract as the Shieldbearer (Hyperaspistes) for his Diatribe against Luther. Hyperaspistes, the first word in Erasmus' tide, was used by his admirer Giese to form the tide of his own treatise upholding the compatibility of Copernicanism with the Bible, if read properly. Giese's (lost) Hyperaspisticon quoted Erasmus' "quite favorable" judgment of Copernicus, which is otherwise unknown. In the absence of any direct contact between Copernicus and Erasmus or the Dutch scholar's collection of proverbs, Giese is the most likely channel by which Erasmus' Latin rendering of Aristophanes' proverbial utterance became known to Copernicus.

342 NOTES ON PP. 5-7

P. 5 :37. As its first emendation, the Congregation of the Index demanded the deletion of the material from the beginning of this paragraph through the famous utterance "AstronomJ is written for astronomers" (from Si fortasse to hi nostri labores). P. 5:38. Expanding on Copernicus' defense of the Revolutions as entirely compatible with the Bible, when properly read, Rheticus wrote a little tract along the very same lines. In the judgment of Bishop Giese, as we saw above (note to p. 3 :38), Rheticus "entirely correctly defended the earth's motion from being in conflict with the Holy Scriptures." But like Giese's Hyperaspisticon, Rheticus' reconciliation of Copernicanism with the Bible failed to survive its passage into the clutches of the Catholic Counter-Reformation. P. 5:41. Lactantius, Dimne Institutes, Ill, 24. P. 5:47. While the Fifth Lateran Council (1512-1517) was in session, Pope Leo X announced that he had "consulted the greatest experts in theology and astronomy," whom he had "advised and encouraged to think about remedying and suitably correcting " the calendar, which was out of joint. The pope added that the experts "have conscientiously heeded me and my instructions, some of them in writing, others orally." But when these written and oral discussions produced no suitable corrections, Leo X issued a general appeal. To the Holy Roman Emperor, for example, on 21 July 1514 he dispatched a message urging that "of all the theologians and astron­ omers whom you have in your empire and domains, you should order... every single one of high renown ... to come to this sacred Lateran Council ... But if there be any who for a legitimate reason cannot come to the Coun­ cil, Your Majesty will please instruct them ... to send me their opinions carefully written." A similar notice, dated three days later, was distributed in printed form to the heads of other governments and of all universities. This general invitation was repeated on 1 June 1515 and 8 July 1516. Paul of Middelburg (1445-1533), bishop of Fossombrone, published a report to Leo X about the outcome of that pope's efforts to stimulate projected correc­ tions of the defects in the current calendar. In that report, entitled Secundum compendium correctionis calendarii (Rome, 1516), Paul of Middelburg listed Copernicus among those who wrote, not among those who traveled to the Eternal City (sig. b 1 ). Unfortunately, what Copernicus wrote has not been found. But the surviving evidence disproves Galileo's statement "that when the Lateran Council under Leo X took up the correction of the church calendar, Copernicus was called to Rome from the most remote parts of Germany to undertake its reform." Because of Galileo's enormous prestige, this misstatement has been often repeated, as has also his related error that the Gregorian calendar of 1582 was "regulated by Copernicus' teachings." P. 6:4. Baldi mistakenly imagined that, prior to the Oost) letter written by Paul of Middelburg to Co­ pernicus around 1516, these two men had known each other personally. For, when speaking of Copernicus' arrival in Italy in 1496, Baldi makes the unsupported assertion that "on that occasion he likewise bec:ame friendly with, and made the acquaintance of, all the intellectuals active at that time in Italy, among them Paul of Mid­ delburg, who was then in the service of Duke Guido I of Urbino" (Bilinski, Vita, p. 21, lines 28-31). There is, however, absolutely no evidence that Copernicus ever so much as set foot in Urbino, or that he ever met Paul of Middelburg while he was in Italy, from 1496 to 1503. Had Paul of Middelburg ever been a friend of Coper­ nicus, would not the latter have so indicated here, instead of referring formally and stiffly to "that most dis­ tinguished man, Paul, bishop of Fossombrone, who was then in charge of this matter"? How utterly different is Copernicus' mention of "a man who loves me dearly, Tiedemann Giese, bishop of Chehnno"! As for Baldi's lively imagination, in his Cronica de' matematici (Urbino, 1707), p. 81, he attributed to Henry Bate of Malines "a treatise on the star of the Magi," which Bate never wrote, according to Aleksander Birken­ majer, whose doctoral dissertation dealt with Bate (Studia Copernicana, I, 110). P. 6:7. Like the description of Cardinal SchOnberg as "renowned in every field of learning," Copernicus' implication that Paul Ill was a learned astronomer is to be regarded as polite flattery. P. 7:3. Copernicus' Introduction to Book I of the Revolutions is preserved in the autograph (fol. 1r-2r). But this brief statement was not printed inN, presumably because it was superseded by the lengthier Preface to the entire work. The Preface, as we saw above (note to p. 3 :3), was written by Copernicus in June 1542 as his last addition to the Revolutions. The omission of the Introduction to Book I was the act of Rheticus, pre­ sumably in agreement with Copernicus, since both Rheticus and Giese did not complain about this omission when they later found fault with the Nuremberg edition as it left the press (note to p. 3 :38). Copernicus' autograph was not known to those who edited or published the second and third editions of the Revolutions (Basel, 1566,; Amsterdam, 1617). By the time of the fourth edition, however, the autograph was recovered, and from it the Introduction to Book I was printed for the first time in W. Since then every subsequent edition (Thoro, 1873,; Munich, 1949,; Warsaw, 1972) has of course included the Introduction to Book I. Hereafter the Thoro edition will be cited as T, and the Munich edition (Nikolaus Kopernikus Gesamtausgabe, vol. II) as Mu. P. 7:11. Copernicus found such etymologies in Pliny, Natural History, II, 3, 8: "What the Greeks called the 'cosmos' to label it an ornament, we [Romans] term mundus on account of its perfect and absolute elegance. We also say caelum, undoubtedly with reference to a carving." Besides the aforementioned Venice 1487 edition

343 NOTES ON P. 7 of the Natural History (note to p. 4 :45), Copernicus had access also to his Chapter's copy of the Rome 1473 edition. By connecting the noun mundus ( = universe) with the adjective mundus ( = pure), Copernicus harks back to an authority older than Pliny. Modem philologists regard the derivation of caelum and mundus as uncertain. P. 7:13. One example would be Plato, whose Timaeus ends with a reference to heaven as "visible god." Marsilio Ficino's translation of Plato's Works into Latin (Florence, about 1485) was available to Copernicus in his Chapter's library (SC, pp. 306-307). P. 7:15. Copernicus here remarks that the science of astronomy was formerly also called "astrology." The latter term, when he wrote, was not restricted, as it is nowadays, to the immensely popular delusion that human affairs on this planet are governed in some inscrutable way by the other planets and the stars. Even as late as 17 September 1676 John Evelyn wrote: "There dined with me Mr. Flamested the learned Astrologer & Mathematitian, whom now, his Majestie had established in the new Observatorie in Greenewich Park ... '' (Diary, ed. B.S. de Beer, Oxford: Clarendon Press, 1955, IV, 98). Fortune-telling astrology received absolutely no support from Copernicus. In this respect he differed markedly from Brahe, Galileo, and Kepler, to mention only a few of the celebrated astronomers who believed in astrology and practiced it for one reason or another. In particular, the contrast between Copernicus and his disciple Rheticus in this regard is complete. Nowhere in the Revolutions nor anywhere else in the unquestion­ ably authentic writings of Copernicus can the slightest trace of belief in astrology be found. On the other hand, Rheticus' addiction to astrology is notorious. In this connection the fate of a poem written by Johannes Dantiscus (1485-1548), bishop of Varmia, is illuminating. On 9 June 1541 Dantiscus invited Copernicus to dinner, and shortly thereafter sent the astronomer a "very gracious and quite friendly" letter, accompanied by an "elegant epigram." These quotations are taken from Copernicus' reply of 27 June to his bishop, who was widely recognized by contemporaries as one of Eu­ rope's leading Neo-Latin poets. In this reply Copernicus indicated that Dantiscus' poem was directed "to the readers of my [six] books" about the Revolutions, and also characterized the poem as "relevant" (ad rem). Fur­ thermore, he promised that he would display the bishop's "tide in the forefront" of his work. Nevertheless, the tide of Bishop Dantiscus did not appear in the front matter of the Reoolutions, although Pope Paul III, Cardinal Schonberg, and Bishop Tiedemann Giese are very prominent in those opening pages. Bishop Dantiscus' name was completely absent from the Revolutions in 1543, and likewise from the Trig­ onometry in 1542. Yet Dantiscus' "elegant epigram" was printed by Rheticus at the end of his Dedication of Copernicus' Trigonometry. Rheticus was careful, however, not to mention the name ofDantiscus, a Roman Catholic bishop, in Copernicus' Trigonometry, which was published in Wittenberg, the innermost citadel of the anti-papal movement headed by Luther. Concealing the identity of the author, Rheticus nevertheless printed the poem, which was kept away from the reader of the Revolutions for whom it was originally intended. Addressing the pro­ spective student, it asserts that "these writings show you the way to the sky," a thoroughly "relevant" descrip­ tion of the Revolutions, but hardly of the Trigonometry. Nevertheless, Dantiscus' poem was printed by Rheticus, who had full control over the contents of the Trigonometry, presumably because four of its lines announce to the student that You must first master the doctrine which these principles Set briefly before you, if you wish to know What fates govern future events, what disasters Are brought to people by hostile stars. This professed ability of astrology to penetrate the datk veil covering human destiny was dear to the heart of Rheticus but utterly foreign to the thinking of Copernicus. P. 7:32. Copernicus read Plato's Laws in Ficino's aforementioned translation (note top. 7 :13). The passages utilized here are 809 C-D, 818 C-D. P. 7:38. Copernicus' phraseology "abesse ... ut ... difJinus effici ... possit, qui nee so lis nee lunae nee re­ liquorum siderum necessariam habeat cognitionem" echoes Cardinal Bessarion's In calumniatorem Platonis (Venice, 1503), of which Copernicus possessed his own copy. The corresponding passage may be found in Ludwig Mohler, Kardinal Bessarion, II, 595: 30-34. P. 7:41. For Copernicus a hypothesis was a fundamental proposition, underlying a sustained process of reasoning. In his vocabulary, a hypothesis was not a tentative or uncertain suggestion, which he rather called a "coniectura." Before Newton uttered his famous cry "Hypotheses non jingo" in 1713, he too, like Copernicus, labeled his basic ideas "hypotheses" in 1687. For example, in the first edition of his Mathematical Principles of Natural Philosophy, the proposition that the center of the universe was at rest was Hypothesis IV. See "New­ ton's Use of the Word Hypothesis" at pp. 575-589 in I. Bernard Cohen, Franklin and Newton (Philadelphia, 1956).

344 NOTES ON PP. 8-9

P. 8:9. One example would be Ptolemy's lunar theory, according to which at its closest approach to the earth the moon would be about half as far away as when it is at its farthest distance from the earth. In that case its "diameter would similarly look twice as large and half as large ... The contrary is self-evident" (IV, 2). P. 8:12. The last three words (mathematicorum peritiam vincit} were taken from John Peter of Lucca's frequently printed Latin translation of Plutarch's Roman Questions. In this instance Copernicus did not quote Plutarch's Greek text, which was available to him at p. 240 in the volume from which he quoted pseudo-Plutarch in his Preface. Since Plutarch's Roman Question 24 discusses the moon and the month, not "the sun's tropical year," evidently here Copernicus was relying on his sometimes unreliable memory. P. 8:22. This modest claim to originality, uttered almost apologetically as though to confess a reluctant introduction of novel ideas, and coupled with a generous acknowledgment of indebtedness to earlier investi­ gators, both in thought and language closely follows Pliny, Natural History, II, 13, 62. P. 8 :26. The view that the universe is spherical was propounded earl} in the history of Greek astronomy and remained the dominant conception in ancient and medieval thought. Of the four causes advanced by Copernicus to explain the sphericity of the universe, the first, the perfec­ tion of the spherical figure, was a mathematico-esthetic judgment familiar to Greek antiquity and succeeding ages, when it was commonly argued that this geometrical form, by reason of its perfection, was the appropriate shape for the universe. The second cause, based on the theorem that of all solids with equal surfaces the sphere encloses the greatest volume, was utilized by Ptolemy (PS, I, 3) and his imitators. The doctrine that the heavenly bodies are spherical, having prevailed over rival views, was incorporated in the astronomical tradition, and that drops of water assume a spherical shape was a commonplace of ancient and medieval science. P. 8:27. Copernicus' expression "needing no joint" (nulla indigens compagine) was modeled after Pliny's nullarum egens compagium (Natural History, II, 2, 5). P. 8:34. In speaking of the "divine" bodies Copernicus' autograph (fol. 2r, lines 5-6) was simply following a long-established custom. But his Nuremberg editors, evidently fearful of religious censure, altered the ex­ pression to the "celestial" bodies (N, fol 1•). In his First Report Rheticus too had called the planets "these divine bodies" (3CT, p. 145: divinis his corporibus). Did he change Copernicus' "divine" to "celestial" when it was decided to dedicate the Re'Dolutions to the pope? P. 8:36. Copernicus' first argument for the sphericity of the earth is derived from Aristotle's theory of centripetal impulse: "It is inherent in the nature of earth to move from all sides to the center" (Hea'Dens, II, 14). Aristotle's writings loomed large in the curriculum of the universities attended by Copernicus. P. 8:37. That the earth's surface is not perfectly spherical and that the irregularities are, nevertheless, of negligible proportions were familiar propositions long before Copernicus' time. P. 8:42. Copernicus' proof of the sphericity of the earth in the north-south direction is patterned after Aristotle's Hea'Dens, II, 14, and PS, I, 4. P. 8:43. In Copernicus' Star Catalog, after II, 14, Canopus, a star of the first magnitude, is the next to the last star in the southern constellation Argo or Ship. In saying that "Italy does not see Canopus" (Canopum non cernit Italia}, Copernicus echoes Pliny's "non cernit ... Canopum Italia" (Natural History, II, 70, 178). But in adding that Canopus "is visible in Egypt," Copernicus does not find it necessary to repeat Pliny's detailed state­ ment that "to observers in Alexandria Canopus appears to reach nearly one-fourth of a sign [ = 71/ 2 °] above the earth." P. 8:44. The River's last star is another first-magnitude star in another southern constellation (p. 110:35). P. 9:6. The last sentence of this paragraph repeats Pliny, Natural History, II, 72, 180. But in the two editions of Pliny available to Copernicus, the close of Pliny's sentence, as printed, was unintelligible and there­ fore had to be revised by Copernicus. P. 9:15. Up to this point this paragraph paraphrases Pliny, Natural History, II, 65, 164-165. According to Aristotle, Generation and Corruption, II, 3, "Earth and water form that which moves toward the center." P. 9:17. In his Geography, a work cited by Copernicus here in I, 3, Ptolemy asserted the unity of the ter­ raqueous sphere: "From the mathematical disciplines we obtain the proposition that the continuous surface of land and water, taken as a whole, is spherical" (1, 2, 7). It had been shown by Archimedes that "the surface of any fluid at rest is spherical and has the same center as that of the earth" (Floating Bodies, I, 2). P. 9:23. According to the teleological point of view, which was accepted by Copernicus and his contem­ poraries, the universe had been created for the sake of living creatures, particularly mankind. P. 9:24. Here Copernicus departs from Ptolemy, who rejected the view that the entire known earth is completely surrounded by water (Geography, VII, 7, 4; VIII, 1, 4). P. 9:27. One example would be Ristoro d'Arezzo, whose Della composizione del mondo of 1282 said: "Then water will be ten times greater than earth, and air ten times greater than water, and fire ten times greater than air" (IV, 3).

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P. 9:29. According to Campanus of Novara's Computus maior, Ch. 3, "The elements arise from and pass into one another ... From one measure of earth ten measures of water are generated" (fol. 159v, Sphaera mundi 7UJ'Oiter recognita, Venice 1518; for the copy presented by Copernicus to Rheticus, see SC, pp. 320-321). P. 9:31. In the aforementioned Sphaera mundi noviter recognita, Capuanus of Manfredonia's revised Commentary on Sacrobosco's Sphere (fol. 37v) stated that "the earth is not of uniform weight throughout, but is heavier in one part than in another. The reason for this is that in one part, where it has no hollows and cavities it is denser and more compact, whereas elsewhere it is porous and full of cavities. Therefore its center of mag­ nitude is not its center of gravity." P. 9:32. Copernicus' compressed argument may be more easily understood when expanded as follows.

If the volume of water were seven times greater than the volume of land, the volume (V1) of the terraqueous sphere would be 7+1 = 8, and the volume (V2) of land would be 1. Since V1 :V2 = d13 :d23, 8:1 = d13 :dh then 2:1 = d1 :d2 or d 2 = id1 ; that is, the diameter (d2) of the land would equal the radius (id1) of the ter­ raqueous sphere or the distance from its center to the circumference of the waters. Then, since earth as the heavier element must occupy the center, no land could protrude above the surface of the waters. P. 10:1. According to Ptolemy's Geography, which Copernicus cites in his next sentence, the inhabited portion of the earth extended over approximately 80° of latitude, from 63° north to about 17° south. Hence Egypt, at about 23° north, would be "almost in the middle of the inhabited lands." P. 10:2. Here Copernicus intended to repeat Pliny, Natural History, II, 68, 173: centum quindecim milibus passuum Arabicus sinus distet ab Aegyptio mari. But in saying inter Aegyptium mare Arabicumque sinum vix quin­ decim superesse stadia, Copernicus not merely reduced Pliny's miles to stadia, which are only eighths of a mile, but he also omitted Pliny's centum (hundred). By this combination of extraordinary errors, Copernicus nar­ rowed the Isthmus of Suez from 115 miles (Pliny's statement) to less than 2 miles. P. 10:3. "The eastern limit of the known earth is marked by the meridian drawn through the metropolis of the Sinae ... the western limit, by the meridian drawn through the Fortunate Islands, its distance .. . from the easternmost meridian being the 180° of a semicircle" (Ptolemy, Geography, VII, 5, 13-14). Ptolemy drew his prime meridian through the Canary Islands (then called the Fortunate Islands) because they were the western­ most land known in his time: "We divide the equator into ... 180° and distribute the numbers, beginning with the westernmost meridian" (I, 24, 8). Copernicus referred to Ptolemy's geographical treatise as the Cosmography because that was the title given in the U1m 1486 edition, which he consulted in his Chapter's library (MK, pp. 337-341). P. 10:4. "The portion of the earth which is inhabited by us is boundedontheeastbyunknown land, border­ ing on the eastern peoples of Greater Asia, the Sinae and those in Sera" (Ptolemy, Geography, VII, 5, 2). P. 10:5. Copernicus' "Cathagia" may denote only North China, although this term was also applied to the whole of China, which is still known as "Kitai" in Russian. P. 10:11. Here Copernicus undoubtedly had in mind Augustine's City of God, XVI, 9: The legend that there are antipodes, that is, human beings on the earth's opposite side, where the sun rises when it sets for us, making footprints facing our feet, must not be believed in any way. No assertion is put forward that this was learned through any historical information. On the contrary, by a sort of reasoning, it is conjectured that the earth hangs within the heaven's convexity, and the universe has the same place as its center and as its lowest level. On this basis the opinion is formed that the earth's other side, which is below us, cannot lack human inhab­ itants. Those who hold this view fail to notice that even if the universe is believed, or by some argument is proved, to be round and spherical, it still does not follow that in that part the earth emerges from the mass of water; secondly, even if it does so emerge, it does not follow at once that it is inhabited. Scripture does not lie in any way. Narrating what has happened, it establishes its credibility by the fulfillment of its predictions. It is too ridiculous to say that some men from this part of the earth could have crossed the immense ocean, sailed to that part of the earth, and established themselves so that there too the human race should be propagated from that one first man (Corpus Christianorum, series latina, 48 (1955), 510: 1-19). P. 10:13. Since Copernicus' "America ... is diametrically opposite the Ganges district of India," obviously he did not apply the term to the whole of the recently discovered hemisphere. Therefore, in saying that America was "named after the ship's captain who found it," he did not ignore Columbus nor lack historical perspective, as he has been charged. Although none of the literature of discovery can be shown to have passed through his hands, internal evi­ dence points to Martin Waldseemiiller's Cosmographiae introductio (St. Die, 1507) as his chief source. This little book, which has attained renown because it coined the word America, contained an Introduction to Cosmog-

346 NOTES ON P. 10

raphy, the Four Voyages of Amerigo Vespucci, and a map of the world. A comparison of our passage with the data in the Cosmographiae introductio will reveal Copernicus' dependence on Waldseemiiller: (1) Copernicus gives 60° as the extent to which travelers in Cathay have penetrated Ptolemy's unknown land, which began at 180°; and Waldseemiiller's map shows 240° as the eastern boundary of Cathay. (2) Copernicus speaks only of Spanish and Portuguese expeditions; and the Cosmographiae introductio does likewise. (3) Copernicus regards America as the principal island discovered during these voyages; and America is de­ clared to be an island (Cosm .. intro., pp. XXX, 70) and is so depicted on Waldseemiiller's map. (4) Copernicus affirms that America is named after its discoverer, a ship's captain (ab inventore ... navium prae­ fecto) ; and the name is derived from the discoverer ( ab . . . inventore), who is described as a ship's cap­ tain (uno ex naucleris naviumque praefectis; Cosm. intro., pp. XLV, 88, and map). (5) Copernicus says that the size of America is undisclosed; and its size is not yet completely known, accord­ ing to a legend on the map. (6) Copernicus refers to many other islands, previously unknown; and there is announced a description of various islands, not mentioned by ancient authors (Cosm. intro., pp. XLV, 88, and map). (7) Copernicus concludes that there is little reason to marvel at the existence of antipodes or antichthones; and it has been proved that in the far south there are antipodes (Cosm. intro., pp. VIII, 41), who are also called antichthones (pp. XXV, 63). (8) According to Copernicus, America is diametrically opposite the Ganges district of India; and the mouths of the Ganges are located at 145° just below the Tropic of Cancer, while America is placed at 325° just above the Tropic of Capricorn. It will be observed that Copernicus departs from Waldseemiiller in only one pronouncement, namely, that America is thought to be a second orbis terrarum. A reason for this divergence may be proposed. Waldsee­ miiller called America the fourth part of the earth, because he viewed Europe, Africa, and Asia as three separate continents (pp. XXV, XXVIII-XXIX, 63, 68-69). Copernicus, on the other hand, conceived these land masses to be a single continent and orbis terrarum, as we saw in the first paragraph of I, 3. For him, therefore, America could not constitute a fourth part of the earth; instead, he accepted the current designation of it as a second orbis terrarum. If the foregoing analysis has demonstrated Copernicus' close dependence on Waldseemiiller, then the mean­ ing which the latter attached to the geographical term America is of supreme importance. At one point in the text of the Cosmographiae introductio he uses America as the name of the entire newly discovered region (pp. XXX, 70); in another passage he restricts it to the zone between the Tropics of Cancer and Capricorn (pp. XVIII, 54); and in his only other reference to its position, he puts it in the southern hemisphere near the Tropic of Capricorn (pp. XXV, 62-63). It is in the last of these locations that he prints the legend America on his map, which shows no single all-embracing appellation for the whole of the recendy discovered territory. Which of these meanings of the term America did Copernicus adopt? The answer is provided by the eighth point of our earlier comparison of his statements with Waldseemiiller's. By the assertion of its diametrical op­ position to the Ganges district of India, Copernicus placed America in the vicinity, of Capricorn. Hence, when he wrote that America was named after its discoverer, he did not mean that the first man to set foot upon trans­ atlantic soil was Vespucci; what he had in mind was that Vespucci discovered an important area in the south­ ern hemisphere. Even if he knew none of the literature of discovery save Waldseemuller, he could not have been unaware of Columbus' achievements, since the latter was named on the map as the discoverer of certain islands. Nor could he have been deceived regarding Columbus' priority, for in Vespucci's Second Voyage reference was made to an island discovered a few years before by Columbus (pp. LXXXV, 132); and on Waldseemiiller's map a legend mentioning the two discoverers put Columbus first and Vespucci second. P. 10:19. This observational proof of the earth's sphericity was familiar to Copernicus from Aristotle's Heavens, II, 14: "In eclipses of the moon the boundary line is always convex. Hence, since the eclipses are due to the interposition of the earth, the shape of the line must be caused by the surface of the earth, which is therefore spherical." P. 10:25. Copernicus based these statements on pseudo-Plutarch's Opinions of the Philosophers (III, 9-11), from which he departed in three respects. (1) Like everybody else, pseudo-Plutarch said nothing about Empedocles' conception of the shape of the earth. But Empedocles regarded the moon as fiat, according to pseudo-Plutarch (II, 27). This characterization was transferred to the earth by Copernicus, perhaps on Aristotle's principle that "what is true of one heav­ enly body is true of all" (Heavens, II, 11). (2) By the same token, Heraclitus' description of the sun and moon as bowl-shaped (pseudo-Plutarch, II, 22,

347 NOTBS ON PP. 10-11

27) was applied to the earth by Copernicus, although Heraclitus "gives no account of the nature of the earth," according to Diogenes Laertius (IX, 11). (3) Pseudo-Plutarch's report that, according to Xenophanes, the earth extends infinitely downward was embel­ lished by Copernicus, who made Xenophanes' earth diminish in thickness toward the bottom. In so doing, Copernicus was misled by an error in Giorgio Valla's translation of the Opinions of the Philosophers into Latin. Valla introduced the term "thickness," which corresponds to nothing in pseudo-Plutarch's Greek, and left its meaning in the context unclear. Copernicus apparently attempted to remove the obscurity by substituting submissa for Valla's immissam. Valla used his translation of the Opinions of the Philosophers as Books XX-XXI in his De expetendis et fu­ giendis rebus (Venice, 1501, 2 volumes). A copy of Valla (cited hereafter as GV) formed part of the library of Copernicus' Chapter (ZGAB, 5: 375). Copernicus' remark that the earth "is perfectly round, as the philosophers hold" may seem strange to those readers who have been deluded into imagining that in that age everybody thought the earth was flat, and had to be shown otherwise by Columbus. "Of all the vulgar errors connected with Columbus, the most persistent and the most absurd is that he had to convince people 'the world was round.' Every educated man in his day believed the world to be a sphere, every European university so taught geography" (Samuel Eliot Morison, Admiral of the Ocean Sea, Boston, 1942, p. 33). Columbus himself regarded the earth as having the shape of a pear (Morison, p. 557). P. 10:29. Ancient Greek theoretical astronomy was dominated by the conviction that the heavenly motions must be circular. This quasi-theological dogma held sway until the ellipticity of the planetary orbits was dem­ onstrated by Kepler in his New Astronomy of 1609. After Copernicus leaves the broad cosmological generali­ zations of Book I behind him, and in his later Books confronts the technical details of astronomy, at certain times he finds it convenient to introduce a heavenly motion that is not circular, but rectilinear and reciprocat­ ing, like a piston's. However, in these cases he at once hastens to demonstrate that this kind of straight back­ and-forth oscillation can be produced by a combination of "two circular motions acting conjointly" (Ill, 4; V, 25). If these circular motions are unequal rather than equal, they will describe an ellipse, as Copernicus pointed out in a deleted passage (Ill, 4). P. 10:30. This single proposition states Copernicus' entire conception of celestial mechanics. For him the universe is an all-embracing sphere, containing lesser spheres; and the sphere, as a geometrical form, is endowed with the property of circular motion. This is the sum total of his answer to the question, why do the heavens go round? Satisfied with this explanation of the cause of motion, he envisages the task of the astronomer as the endeavor to trace the patterns of motion, to solve the problem, how do the heavens go round? Copernicus derived from Aristotle the idea that "by nature the spherical body moves forever in a circle" (Heavens, II, 3). But the Stagirite could not impute circular motion to every sphere, since he taught that the earth was stationary (II, 14). Differentiating it from the upper bodies, he asserted that they were composed of a fifth element, aither, to which he ascribed natural circular motion (1, 2, 3; II, 7). Thus he accounted for the rotation of the celestial spheres and the stability of the earth. While accepting Aristotle's theory of motion in the main, Copernicus was constrained to accommodate it to his geokinetic astronomy. Accordingly, he dropped the aither from his cosmology, and he attributed natural circular motion to the sphere as a geometrical form. Nevertheless, the innermost and outermost spheres of Copernicus' cosmos have no natural circular motion, since his sun and stars are stationary. Although it cannot be shown that Copernicus was acquainted with the writings of Nicholas of Cusa (c. 1401-1464), that German cardinal had said in his De Judo globi of 1463, Book 1:

The spherical figure, then, is most suitable for eternal motion. If it acquires motion naturally, it will never come to rest. Therefore, if it moves about itself in such a way that it is the center of its own motion, it moves forever. And this is the natural motion with which the outermost sphere moves, a motion without violence or weariness, a motion shared by all bodies that have a natural motion. According to Cusa, God had set the spheres in motion once and for all at the time of the creation, providing them with the initial impulse which keeps them running forever. Thus he dispensed with Aristotle's unmoved Prime Mover, which operated as the unceasing original cause of all motion (Physics, VIII, 6, 10; Metaphysics, XII, 7, 8). Copernicus, on the other hand, by attributing eternal circular motion to the sphere as a geometrical form, had no need for an initial impetus or a Prime Mover. Nor did he have any use for "intelligences" as "movers of the heavens," in the manner of Theodoric of Freiberg (P. Duhem, Le Systime du monde, Ill, repr. 1958, 388). P. 11 :16. This pronouncement makes unmistakably clear Copernicus' adherence to a form of the traditional doctrine of the spheres. As the Italian natural philolopher Francesco Patrizi put it in his N(Jf)(J de umf}6f'sis phi-

348 NOTES ON PP. 11-13 losophia, Book XVII (Venice, 1593): Copernicus "thought that the planets, like the other heavenly bodies, were carried by spheres, to which they were attached." P. 11 :17. Here Copernicus repeats Aristotle's argument (Heavens, II, 6) that the perfection of the heavenly bodies requires their motion to be uniform and free from irregularities: "Since everything that is in motion is moved by something, any irregularity of the motion must be caused by either the mover or the moved or both. For if the mover fails to exert a constant force, or if the moved is altered instead of remaining the same, or both change, there is nothing to prevent the moved from moving irregularly. But none of these things can take place in the heavens." P. 11 :28. Euclid, Optics, Proposition 5 : "Equal magnitudes at unequal distances look unequal, and the one nearer to the eye always seems larger." Since the Greek text of Euclid's Optics was first published in 1557, after the death of Copernicus, he must have used a Latin translation. In the present instance this could not have been Valla's, whose Book XV, Ch. 3, omitted Prop. 5. This proposition was included, however, in Bartolomeo Zam­ berti's translation of Euclid's works (Venice, 1505). Zamberti's version of Prop. 5 provided the model for Co­ pernicus' restatement of this optical principle in IV, 1. P. 11:29. This is a special case of the optical principle that "Of bodies moving equally fast, those farther away seem to travel more slowly." The theorem is enunciated in its general form at the beginning of I, 10. There Copernicus tells us that his source is Euclid's Optics (his only citation of the work by name); and his language is identical with Zamberti's Theorem LVI, Prop. 57. This proposition of Euclid's Optics was omit­ ted by GV. P. 11:30. Copernicus' salutary warning that the earth's motion affects our observations of the motions of the other heavenly bodies bas been misunderstood to mean that he still retained the traditional dichotomy be­ tween celestial and terrestrial phenomena. P. 11 :42. The Sacred Congregation of the Index demanded that this sentence should be corrected to read: "Nevertheless, if we examine the matter carefully, we shall see that, as far as saving the appearances of the ce­ lestial motions is concerned, it makes no difference whether the earth is in the middle of the universe or outside the middle." P. 11 :45. The principle of relative motion had small importance for astronomers as long as the earth was considered stationary. But a geokinetic system must try to understand the effects of the earth's motion upon the observed phenomena. P. 12:1. Since Copernicus bas already shown familiarity with Euclid's Optics, he was presumably acquaint­ ed with its Proposition 51: "If several objects are moving at different speeds in the same direction as that in which the eye is also being carried, any object moving at equal speed with the eye seems to be stationary." P. 12:16. In his eagerness to avoid being accused of originality (or, in the language of the time, of introduc­ ing new ideas), Copernicus searched for ancient supporters of geokineticism. Having found a suitable passage in pseudo-Plutarch, he displayed it prominently in his Preface, and relies on it here to show that Heraclides and Ecphantus accepted a rotation of the earth about its axis, although they had no further thought of the earth's revolving through space in a progressive motion. For Hicetas a like conclusion follows from the passage in Cicero which was quoted above (note to p. 4:45). Was it by oversight or deliberate intent that Copernicus linked Heraclides with the Pytbagoreans? The connection between Heraclides and the brotherhood is slight (Diogenes Laertius, V, 86), and was probably un­ known to Copernicus. P. 12:33. "That the earth is one of the heavenly bodies, that it revolves in a circle about the center, and that it produces night and day" were beliefs ascribed by Aristotle (Heavens, II, 13) to the Pytbagoreans as a body. Copernicus without warrant transferred all these same beliefs to Philolaus as an individual. Yet the daily rota­ tion is not assigned to the earth by Philolaus, according to pseudo-Plutarch as quoted by Copernicus in his Pref­ ace. Nor is there any evidence elsewhere that Philolaus toward the end of the fifth century B.C. perceived the earth's rotation. P. 12:36. Copernicus took this indication of Plato's esteem for Philolaus from Cardinal Bessarion's In calumniatorem Platonis, I, 5, 1 (Venice, 1503). Copernicus' copy of this vigorous defense of Plato, the first com­ prehensive study of that philosopher, was bound behind another work, on the title page of which he wrote his name as owner of the composite volume. On fol. 8v of Bessarion, Copernicus wrote the marginal note "Plato's travels" (MK, p. 131). P. 12:38. Here Copernicus rather closely follows PS, I, 6. P. 13:8. Copernicus utilizes Book I, Proposition 6, of Theodosius' Spherics: "Of the circles in a sphere, those that pass through the center of the sphere are great circles." While the Greek text of this work was not printed until after Copernicus' death, he bad two Latin translations, one in GV (XIII, 5), and the other in Sphaera mundi noviter recognita (note to p. 9 :29).

349 NOTBS ON PP. 13-14

P. 13:12. "The dioptra was invented by the makers of instruments to direct the sight along a straight line by conducting it through a narrow space and to prevent it from straying in any direction" (Olympiodorus, Com­ mentary on Aristotle's Meteorology, III, 6). One use of the dioptra was familiar to Copernicus from Pliny's Nat­ ural History, II, 69, 176, where this instrument confirms the equality of day and night "when sunrise and sunset are seen along the same line at the time of the equinox." P. 13:12. The water level (chorobates) was known to Copernicus from Vitruvius, Architecture, VIII, 5, 1, available to Copernicus in his Chapter's library (ZGAE, 5: 375, 377). He copied a short passage out of Vi­ truvius into his copy of Witelo's Optics (PI2, 410). P. 13:16. "Six of the twelve signs are always visible above the earth to everybody, while the other six are invisible. Then again, when all of the latter are simultaneously visible above the earth, the others are at the same time invisible. Thus, from the fact that the same semicircles are completely cut off sometimes above the earth and sometimes below the earth, it becomes clear that the ecliptic too is bisected by the horizon" (PS, I, 5). P. 13:23. Here Copernicus' reasoning is patterned upon Euclid's in the Preface to the Phenomena: "The horizon is also one of the great circles. For it always bisects the ecliptic ... which is a great circle, since it always keeps six of the twelve signs above the earth ... But if on a sphere ... a circle cuts any great circle in half ... the cutting circle is itself a great circle. Consequently, the horizon is one of the great circles." From Zamberti's translation Copernicus borrowed one expression (semper bijariam dispescit), but for the better part of this pas­ sage he adhered to GV (XVI, 1). P. 13:27. Copernicus does not pause to make the obvious exception for the case where all three points (the point in the firmament, the point on the earth's surface, the center of the earth) are on the same straight line. P. 13 :40. These properties of a rigid sphere undergoing rotation were pointed out by Aristotle: "When a body rotates in a circle, some part of it must remain stationary, namely, that which is at the center"; "That the circles should have their velocities proportional to their sizes is not absurd but inevitable ... For that which is carried on the larger circle must be swifter" (Heavens, II, 3, 8). P. 13:43. In Copernicus' Star Catalog, the Little Bear is the first of the northern constellations, while the Eagle in the northern region and the Little Dog in the southern region are constellations relatively close to the ecliptic. P. 14:1. As Euclid indicated in the Preface to his Phenomena, "If a sphere rotates uniformly about its axis, all the points on the surface of the sphere in an equal period of time describe similar arcs of the parallel circles on which they are carried." P. 14:11. The principle governing the motion of parts of a rotating sphere obviously does not apply to the planets, because they do not swing round the heavens in equal times. The difference in their periods gave rise to the rule which Aristotle formulated as follows: "It is reasonable that the body nearest to the simple and pri­ mary revolution should complete its circle in the longest time and the one farthest away in the shortest, and so with the others, the nearer in a longer time, the farther in a shorter" (Heavens, II, 10). P. 14:19. In the autograph (fol. 5•) the last seven lines were marked for deletion, presumably by someone other than Copernicus himself. For when he decided to cancel a passage of comparable length, he drew vigorous horizontal, vertical, diagonal, or crisscross strokes all through the doomed passage. Here, on the other hand, a small inconspicuous open loop indicates the omission, which affects also the top line of fol. 5v, although there no deletion sign appears. Both of these traits are contrary to Copernicus' usual practice and cannot be matched elsewhere in his autograph. Why was this admirable section at the end of I, 6 removed? Was its reference to atoms the reason for its deletion? As uncreated and imperishable entities, atoms did not appeal to those who believed that the universe had been created at some time in the past and would dissolve at some time in the future. P. 14:34. The argument was stated by Aristotle as follows: "That heavy objects are carried toward the center of the earth is indicated by the fact that they move toward the earth not along parallel lines but at the same angles. Hence they are carried toward a single center, which is the center of the earth. It is therefore clear that the earth must be at rest in the middle" (Heavens, II, 14). But Copernicus presents Ptolemy's amplification of the argument: "All heavy objects are carried toward the earth... Since we have shown that it is ... spherical, in all its parts without exception the direction and motion of bodies having weight, I mean their proper motion, are always and everywhere at right angles to the uninclined plane which is drawn through the point where the falling body makes contact ... If it were not stopped by the surface of the earth, it would drop clear down to the very center, since the straight line which leads to the center is always perpendicular to the plane which is tangent to the sphere at the point of intersection where the contact is made" (PS, I, 7). Similarities of expression suggest that here Copernicus was using George of Trebizond's Latin translation of PS. P. 14:38. This is Aristotle's doctrine of natural place: "Whither the earth is carried by nature, there it also

350 NOTES ON PP. 14-16 rests by nature." In its generalized form the proposition reads: "The elements remain by nature where they are borne by nature" (HeafJens, II, 13; III, 2). P. 14:40. Here again, Copernicus' language unmistakably resembles the corresponding passage in George of Trebizond's translation of PS. P. 15:17. Ptolemy did indeed hold that "living creatures and discrete heavy bodies would be left behind dangling in the air, while the earth itself would finally at great speed fall out of the very heavens. Yet such con­ sequences need merely be conceived to appear entirely ridiculous" (PS, I, 7). But the Greek astronomer did not envisage these horrific consequences as resulting from a rotation of the earth. According to his analysis, they would ensue if the entire earth had the same downward motion as any particle of earth or heavy body. It seems likely, then, that Copernicus is reconstructing Ptolemy's position from memory. Even so, he echoes words from George of Trebizond's translation of PS. P. 15:19. This argument is not adduced by Ptolemy but forms part of Aristotle's rejection of geokineticism: "Heavy objects thrown forcibly upward return perpendicularly to their starting place" (Heavens, II, 14). P. 15:21. Copernicus closes this review of ancient opinion by resuming Ptolemy's rebuttal of the earth's rotation: "No cloud would ever be seen drifting eastward, nor any other flying or thrown thing" (PS, I, 7). P. 15:22. No portion of Copernicus' work disturbed the Sacred Congregation of the Index more deeply than I, 8, which outlines a physical theory consonant with the earth's motion. In its correction decree the Congre­ gation declared: "This whole chapter could be eliminated because the author openly discusses the reality of the earth's motion while refuting the arguments put forward by the ancients to prove that it remains at rest. Nev­ ertheless, since he always seems to be speaking problematically, for the purpose of satisfying students and pre­ serving intact the sequence and arrangement of the book, let the chapter be emended as follows." The decree then ordered three specific changes, which will be noted below. P. 15:27. According to Aristotle, "everything that is in motion moves either naturally or unnaturally and under compulsion." "It is apparent that ... the unnatural is destroyed most quickly," whereas the natural con­ tinues forever (Physics, VIII, 4; HeafJens, I, 2). P. 15:44. Again Copernicus accepts Aristotle's dicta: "The infinite cannot be traversed"; "the infinite cannot be moved in any way" (Hea'Oens, I, 5, 7). P. 15:46. "It has been shown that beyond the heavens there neither is nor can be body. Hence it is clear that neither space nor void ... is out there." "There is nothing outside the universe" (Aristotle, Heavens, I, 9; Physics, IV, 5). P. 16:3. The "inner concavity" of the heavens is the lower surface of the eighth sphere or sphere of the fixed stars. Such a finite inner concavity might conceivably coexist with a finite outer convexity or an infinite expanse upward. P. 16:6. "It is impossible that the infinite should move at all" (Aristotle, Hea'Oens, I, 7; 274b30). P. 16:9. In seeking to overcome the opposition to the earth's rotation, Copernicus starts by countering the thesis that any rotation on the part of the earth must let loose disruptive centrifugal forces. To this conten­ tion he enters a twofold rejoinder. He maintains, first, that the rotation is natural and therefore perpetual, a con­ cept which compels him, as we shall presently see, to revise Aristotle's theory of motion. His second retort is of the "you too" variety and leads him into the problem of the infinite. Those who deny that the earth rotates must attribute the daily rotation to a turning of the heavens. But if a rotation of the earth would cause it to dis­ integrate, a rotation of the heavens would burst them asunder. Copernicus' imaginary opponent now accepts the centrifugal consequences of a rotation of the universe, but utilizes them to account for the enormous expanse of the heavens. According to this view, the earth cannot rotate, for if it did, it would fly apart; whereas the universe does rotate, and its immense size is due to that fact. This position Copernicus demolishes by showing that it involves a contradiction: if the heavens move outward, they must become infinite; but if infinite, they cannot move. The opponent tries to break this logical chain by arguing that, although the heavens grow, they stop short of infinity; for they cannot cross their finite limit, beyond which there is no space to be entered. This modification, Copernicus replies, assumes that the nothing which is beyond the heavens can prevent them from expanding. If there is to be nothing outside the heavens, they must be all-inclusive or infinite, but in that case they will be stationary. For, the only universe that can move is a finite universe. Now Copernicus' geokinetic system required the heavens to be motionless. He might have been expected, then, to conclude that the universe is infinite. But an infinite universe can have no center, and without a center Copernicus' spherical astronomy would be wholly unhinged. Hence he avoids the unqualified proposition, and restricts himself to declaring that ''the universe is . . . similar to the infinite," "the heavens ... present the aspect of an infinite magnitude" and "on the testimony of the senses the earth is related to the heavens ... as a ... finite to an infinite magnitude" (I, 6; after I, 11). As a close student of Aristotle's writings he was perhaps de-

351 NOTES ON P. 16 terred by the philosophical difficulties attending the categorical assertion that an infinite body actually exists. In any case, he was patently unwilling to accept either half of the vexed antinomy - the universe is finite, the universe is infinite. His final attitude may be elicited from these two judgments: "How far this immensity [of the heavens] extends is not at all clear"; the universe's "limit is unknown and unknowable" (I, 6, 8). Hence he ended by tossing the problem into the lap of the natural philosophers, from whom as a professional astronomer he dissociated himself. In Paradise Lost, VIII, 76-77, John Milton said that God "his Fabric of the Heavens/ Hath left to their disputes." Milton's expression, like Copernicus' "Si'De ... finitus sit mundus ... disputationi phys­ iologorum dimittamus," is modeled on "mundum tradidit disputationi eorum" (Vulgate, Ecclesiastes, III, 11). So is D'Alembert's reply to certain theologians: "Although religion is solely intended to govern our morals and our faith, they thought that it was made to enlighten us also about the system of the universe, that is to say, about those matters which the Almighty explicitly left to be discussed by us" (Encyclopedie, Vol. I, Paris, 1751, Discours preliminaire, p. XXIV). P. 16:10. Of the three specific changes ordered by the Sacred Congregation to be made in I, 8, the first would alter this passage to read as follows: "Why then can we not grant it the motion due to its form, in pref­ erence to attributing a movement to the entire universe, the limit of which is unknown and unknowable, and regard the celestial phenomena as resembling the utterance of Vergil's Aeneas?" P. 16:21. Copernicus now turns to the topic of atmospheric phenomena, since they constituted Ptolemy's chief argument against the rotation of the earth. The leading astronomical spokesman for a stationary earth was perfectly willing to admit that the celestial phenomena, taken by themselves, were satisfactorily explained by assigning the daily rotation to the earth instead of to the heavens: "Now some people propose what they regard as a more plausible explanation, even though they have no arguments against our position; and they think that no contrary evidence will arise against them if, for example, they assume that the heavens are motionless and that the earth executes from west to east about the same axis approximately one rotation each day ... So far as the phe­ nomena of the celestial bodies are concerned, nothing perhaps would prevent the situation from being as indi­ cated by this theory in its simpler form; but in view of what takes place about us and in the air, such an assump­ tion would appear absolutely ridiculous" (PS, I, 7). P. 16:24. Ptolemy was aware that his arguments based on atmospheric phenomena could be met by affirm­ ing that the air participates in the earth's rotation: "Those who contend that the earth rotates may take the posi­ tion that the air too is carried around the earth in the same direction and at equal speed" (PS, I, 7). P. 16:29. Aristotle, for example, held that "a great part of the air is carried about the earth by the motion of the celestial rotation" (Meteorology, I, 3, 7). P. 16:31. Copernicus' use of the terms repentina, cometae, and pogoniae recalls Pliny, Natural History, II, 22, 89. P. 16:32. Aristotle held that comets and bearded stars are generated in the upper air (Meteorology, I, 7). P. 16:35. This is Copernicus' answer to Ptolemy's objections to a joint rotation of earth and air: "The compound bodies in the air would always seem to fall behind the common motion of both air and earth. Or if these bodies were borne along in unison with the air, they would no longer appear to move either forward or backward; but they would always appear stationary and, whether flying or thrown, undergoing no shift or change of position" (PS, I, 7). P. 16:36. "The wind is understood to be nothing but a wave of air" (Pliny, Natural History, II, 44, 114). Edmond Halley agreed that "wind is most properly defined to be the Stream or Current of the Air." Halley then continued: "where such Current is perpetual and fixt in its course, 'tis necessary that it proceed from a permanent unintermitting Cause. Wherefore some have been enclined to propose the diurnal Rotation of the Earth upon its Axis, by which, as the Globe turns Eastwards, the loose and fluid particles of the Air, being so exceeding light as they be, are left behind, so that in respect of the Earths surface they move Westwards, and become a Constant Easterly Wind. This opinion seems confirmed, for that these Winds are found only near the Equinoctial, in those Parallels of Latitude where the diurnal Motion is swiftest." On account of "the insufficiency of that Hy­ pothesis," Halley called attention to "the Action of the Suns Beams upon the Air and Water, as he passes every day over the Oceans." Yet this is merely traditional phraseology coming from the pen of Halley, who was a con­ vinced Copernican. Hence, when he published "An Historical Account of the Trade Winds ... with an attempt to assign the Phisical cause of the said Winds" (Royal Society of London, Philosophical Transactions, 1686- 1687, 16: 153-168; quoted passage at pp. 164-165), Halley explained the trade winds as a result of the earth's daily rotation (and annual orbital revolution). Long afterwards, in 1902, Henri Poincare (1854-1912), the distinguished French mathematician and philos­ opher of science, in the course of his onslaught against Isaac Newton's concept of absolute space, asked whether "it would make any sense to say that the earth rotates? If there is no absolute space, can a thing tum without turning with respect to something?" (in English translation, Science and Hypothesis, reprint, Dover, New York,

352 NOTES ON PP. 16-17

1952, p. 114). Returning to this theme four years later in his Value of Science (English translation, reprint, Dover, 1958, pp. 140-141), Poincare remarked: the statement that the earth rotates has no meaning ... or, rather, these two statements, the earth rotates, and it is more convenient to suppose that the earth rotates, have one and the same mean­ ing. These words have given rise to the strangest interpretations. Some have thought that they saw therein the rehabilitation of Ptolemy's system ... Behold the apparent daily motion of the stars and the daily motion of the other heavenly bodies, and in addition, the flattening of the earth, the rotation of Foucault's pendulum, the twist­ ing of the cyclones, the trade winds, and what else besides? For the Ptolemaist, all these phenomena have no link between them; for the Copernican, they are produced by one and the same cause. In saying that the earth rotates, I declare that all these phenomena have an intimate connection, and that is true [italics in original], and that remains true even though there is not and cannot be absolute space. So much for the rotation of the earth about itself; what about its revolution around the sun? Here again we have three phenomena which, for the Ptolemaist, are absolutely independent and, for the Copernican, are related to the same origin. These are the apparent displacements of the planets on the celestial sphere, the aberration of the fixed stars, the parallax of these same stars. Is it by accident that all the planets display a nonuniformity with a period of one year, and that this period is exactly equal to that of the aberration, exactly equal also to that of the parallax? To accept Ptol­ emy's system is to answer, yes; to accept Copernicus' system is to answer, no; this is to declare that there is a bond linking the three phenomena, and that is still true even though there is no absolute space. In Ptolemy's system, the movements of the heavenly bodies cannot be explained by the action of central forces: celestial mechanics is impossible. The intimate relations which celestial me­ chanics reveals to us between all the celestial phenomena are true relations; to assert the immo­ bility of the earth would be to deny these relations, and that would therefore be to deceive ourselves. Despite Poincare's vigorous and categorical denial that his conventionalist philosophy of science provided any basis whatever for the resuscitation of Ptolemy, we have lately been solemnly assured that "today we cannot say that the Copernican theory is 'right' and the Ptolemaic theory 'wrong' in any meaningful physical sense. The two theories ... are physically equivalent to one another" (Fred Hoyle, Nicolaus Copernicus, London, 1973, p. 79). P. 16:40. Copernicus is compelled to depart from Aristotle's theory of motion, since geokineticism cannot be reconciled with the Stagirite's view that the natural motion of earth is downward. For Copernicus, the earth as a whole has a natural circular motion. But individual bits of earth undeniably drop downward. Hence he formulates the doctrine that the earth, as a spherical whole, moves naturally in a circle, while parts of the earth, in addition to sharing in this circular motion, have a rectilinear motion of their own. P. 16:44. The definition is Aristotle's (Generation and Corruption, II, 4; Meteorology, IV, 9). Thomas Aquinas' commentary on the latter work was left unfinished, and his anonymous continuator used the very phraseology repeated here by Copernicus (Aquinas' Opera omnia, Leonine ed., III, Appendix, p. CXXXIX). P. 16:45. The recently introduced firearms and cannon were personally familiar to Copernicus, who served as a military commander in his Chapter's war against the Knights of the Teutonic Order. P. 17:2. Aristotle, Heavens, I, 7. P. 17:4. According to Aristotle, "a rotating sphere, although it is in motion, is yet in a sense at rest, since it continues to occupy the same place" (Physics, VIII, 9). P. 17:16. Here Copernicus discards one of the four traditional elements of which the universe was believed to be composed. For while Copernicus saw and felt earth, water, and air all around him, he had no material evi­ dence for the existence of the fourth element, a supposed invisible sphere of fire encircling the air and lying just below the region of the heavens. Copernicus of course recognizes the presence of earthly fire, but is dubious about the elemental fire in the traditional cosmology. He was not alone in this skeptical attitude, as a result of which the metaphysical poet John Donne, in An Anatomy of the World (London, 1611), lines 205-206, exclaimed: And new Philosophy calls all in doubt, The mement of fire is quite put out. P. 17:18. Copernicus' discussion of acceleration and deceleration does not depart from Aristotle's ideas. According to the Stagirite, "only circular motion can be uniform, because things that travel in a straight line exhibit different velocities at the outset and near their goal. For they all move faster, the farther they recede

353 NOTES ON P. 17

from their initial stationary position." This rule applies to fire as an element and simple body: "Earth moves faster as it approaches the middle, but fire, as it nears the upper limit." However, since the fires within the range of our observation consume earthy fuel, the principle that violent motions decelerate comes into operation: "A thing that is approaching its natural place of rest seems to move ever faster, whereas a thing that is set vio­ lently in motion does the opposite"; "everything slows down as it recedes from the source of its enforced mo­ tion." Finally, motion ceases when the moving object arrives at its natural destination: "everything stops moving when it reaches its proper place" (Physics, V, 6; VIII, 9; Heavens, I, 8, 9). P. 17 :22. In order to construct a theory of motion consistent with geokineticism, Copernicus had to abandon Aristotle's principle that the movements of whole and part are identical. The Stagirite argued, for example, that "The earth does not have a circular motion. If it did, each of its parts would have that same motion, whereas in fact they all move in a straight line toward the middle" (Heavens, II, 14). Copernicus emerges, then, with the generalization that wholes move differently from their parts. Circqlar motion may exist dissociated from recti­ linear motion, just as a living creature may be free from sickness. But a rectilinear motion may be superimposed upon a body undergoing rotation, just as sickness may befall a healthy living creature. Copernicus' modification of the traditional laws of motion deeply impressed the Portuguese mathematician Pedro Nunes (1502-1578). Although he was entirely unsympathetic to the new astronomy, in his Rules and Instruments for the Art of Navngation, Ch. 11 (Opera, Basel, 1566, pp. 105-106) Nunes wrote:

It is a question for philosophers to discuss whether Copernicus, by means of the arguments used by Ptolemy to show that the earth has no circular motion at all, reasons soundly when he says that not only the earth but also earthy things and all heavy things, wherever they may be located, are carried in a natural motion from west to east, while they undergo an additional rectilinear [motion] when they depart in any way whatever from their natural places, and that circular [mo­ tion] subsists with rectilinear [motion] not otherwise than "being alive" with "being sick." For, nothing will be said to move either away from the middle or toward the middle which does not also move around the same middle. Copernicus devised these [principles] in order to be able to explain why, if the earth travels in a circle, heavy bodies hurled upward nevertheless return ver­ tically to the places lying below them.

Only an anti-Copernican like Nunes would characterize the novel reasoning in I, 8, as "the arguments used by Ptolemy to show that the earth has no circular motion at all." In the following generation another anti-Copernican, Christopher Clavius (1537-1612), in the fourth edition of his Commentary on the Sphere of Sacrobosco (Lyon, 1593; reissued, 1594), disapproved of Copernicus' departure from Aristotle's doctrine that earth is a simple body and, as such, bas only one simple motion: Many absurdities and errors are contained in Copernicus' position, for instance, that the earth ... moves with a triple motion. I hardly understand how this could happen, since according to phi­ losophers [only) one motion is linked with one simple body (p. 520). P. 17:30. The second change in I, 8, demanded by the Sacred Congregation would transform this sentence as follows: "It is no more difficult, moreover, to attribute motion to that which is enclosed and occupies some space, namely, the earth, than to the framework of space." P. 17:36. In Aristotle's theory every circular motion or motion around the middle had to be performed about the center of the universe. The simplicity of this homocentric system is shattered by Copernicus' extension of the meaning of motion around the middle. In his more complex world there are many centers. Every spherical whole, by reason of its form, has a circular motion about its own center, which need not be identical with any other center. Circular motion, moreover, is no longer on a par with rectilinear motion, since the latter affects only parts, that is, incomplete bodies, which discard this temporary characteristic as soon as they join their spheri­ cal whole. P. 17:38. The Sacred Collil'egation ordered the deletion of this conclusion. P. 17:40. The question referred to here was propounded in the title of I, 5, in two parts. The first part asked "Does circular motion suit the earth?" and Copernicus has now answered it affirmatively. The second part, inquiring about the position of the earth, will be answered in I, 9. This closing sentence of I, 8 (fol. 7", line 8) was omitted from N, probably because it sounded exactly like a familiar structural formula in a scholastic disputation. That was the kind of academic training to which Coper­ nicus had been subjected as a student for a dozen years at three universities, but his editors wanted to break completely away from that hidebound tradition. P. 17:44. The Congregation ordered the first sentence of I, 9 changed to read as follows: "Accordingly, since I have assumed that the earth moves, I suggest that we should now consider also whether several motions

354 NOTES ON PP. 17-19 can suit it." The Congregation's change eliminates Copernicus' proposed conclusion that the earth "can be regarded as one of the planets." P. 18:5. According to Aristotle, "the earth and the universe happen to have the same center; a heavy body moves also toward the center of the earth, but it does so only incidentally, because the earth has its center at the center of the universe." "If someone removed the earth to where the moon is now, each of the [earth's separate] parts would move not toward it but toward the place where it is now" (Heavens, II, 14; IV, 3). P. 18:9. Here Copernicus veers sharply away from Aristotle. For instead of Aristotle's single center of heaviness or gravity in the entire universe, Copernicus correctly saw that there are multiple centers. Just as heavy terrestrial objects tend toward the center of the earth, so heavy lunar objects tend toward the center of the moon, and so on for the other heavenly bodies. The careful reader will note, however, that this tendency is for Copernicus a yearning for togetherness innate in all the parts that belong together. There is in Copernicus not the slightest hint of the later idea of gravitation as a mutual attraction of physical bodies, however that at­ traction is to be explained. P. 18:22. The harmoniousness of the entire universe was a commonplace theme emphasized by various schools of ancient Greek thought. In his Preface, it will be remembered, Copernicus called "the structure of the universe and the true symmetry of its parts" the "principal consideration." P. 18:23. In closing I, 9 with the expression "if only we look at the matter, as the saying goes, with both eyes," or in Latin, ambobus (ut aiunt) oculis, fol. 7v, last line, Copernicus transformed a phrase which he had noticed in GV, XV, 3, sig. aa8r. There, in translating Prop. 25 of Euclid's Optics, GV used ambobus ... oculis in the literal, physiological sense of looking at the visible object "with both eyes" rather than with only one eye. Acknowledging parenthetically by ut aiunt that he was resorting to a quotation, Copernicus expanded the meaning of ambobus oculis to signify unimpeded intellectual insight. The Greek sources used by GV were enumerated by J. L. Heiberg, "Philologische Studien zu griechischen Mathematikern: III. Die Handschriften Georg Vallas von griechischen Mathematikern," Jahrbilcher filr classische Philologie, suppl. 12 (1881), 377-402. P. 18:26. Aristotle, Heavens, II, 10 (quoted in note to p. 14:11). P. 18:29. As was pointed out above (note top. 11 :29), here Copernicus repeats the language of Zamberti's translation of Euclid's Optics, Prop. 56-57, which became no. 53 in Zamberti's translation of Euclid's works. P. 18:37. Al-Bitruji (Alpetragius, as he was called in Latin) climaxed the Islamic attack against Ptolemy in returning to the homocentric theory. Shortly after 1185 Al-Bitruji wrote his Book on the Sphere in Arabic. This was translated into Latin by Michael Scot in 1217 and, from Moses ben Tibbon's Hebrew version of 1259, again into Latin in 1529 by Calo Calonymus ben David, a Neapolitan Jew. This translation was published two years later in Venice as part of a composite volume, Sphaerae tractatus, too late to affect Copernicus' writing of I, 10. By the same token Michael Scot's translation was not printed in the lifetime of Copernicus, nor did he see any manuscript copy of it. Al-Bitruji's original text would have been useless to Copernicus, since he did not know Arabic. Without access to Al-Bitruji himself and to his translators, Hebrew and Latin, Copernicus learned about this Spanish Muslim's peculiar arrangement of Venus and Mercury from P-R, Book IX, Prop. 1. Unlike Copernicus, Regiomontanus possessed his own personal copy of Michael Scot's translation .of Al­ Bitruji. P. 18:39. In the Timaeus (39B) Plato spoke of the sun's light as "shining throughout the entire heaven." From this statement the Platonists whom Copernicus has in mind here drew the inference that the planets had no light of their own. P. 18:40. Through a misunderstanding of this passage it has often been said that Copernicus predicted the discovery of the phases of Venus and Mercury when means of improving human vision were found. The inven­ tion of the telescope about half a century after Copernicus' death permitted the phases of the inner planets to be seen for the first time. It was only thereafter that this fable about "Copernicus' prophecy" came into existence. But in his age of naked-eye observation the absence of moonlike phases from Venus and Mercury was used by Platonists as an argument against the infrasolar position assigned to those two planets by the Ptolemaists. P. 18:44. No transit of Venus or Mercury was observed until after the death of Copernicus. P. 19:3. It was Ptolemy's contention that Venus and Mercury must be located between the sun and moon, otherwise "this great space would remain empty, as though it had been forgotten and ignored by nature." Ptol­ emy propounded this argument in his Planetary Hypotheses, Book II, the original Greek text of which has not yet been found. It was not translated into Latin, and the Arabic and Hebrew versions would have been useless to Copernicus. But the Greek text was still available to the fifth-century neoplatonic philosopher Proclus, whose Hypotyposis was partially translated into Latin in GV, Book XVIII, to which Copernicus had access, as we saw above (note to p. 10:25). P. 19:7. 641/ 8 x 18 = 1155 ~ 1160. P. 19:8. Copernicus obtained most of the numerical values in this passage from GV (Book XVIII, Ch.

355 NOTBS ON P. 19

23, sig. gg 6V). Moreover, this dependence of Copernicus on GV is confirmed by his repetition of some of GV's expressions (!1mdicant rationem; minimum solis interoallum; inams; comperiunt; compleri numeros; succedat). Since a copy of GV is hard to find nowadays, it may be useful to insert here the substance of the relevant passage in GV: With regard to the order of the planets ... some [astronomers] find a conjectural [arrangement) by operating with the perigees and apogees: that is to say, the moon's apogee is immediately followed by Mercury's perigee, and in turn Mercury's apogee by the perigee of Venus, and the latter's apogee by the sun's perigee. Thus, by this kind of reasoning the relative order has been obtained. For, they put the moon's [greatest) distance ... [from the center of the universe] ... = 64P10', with the earth radius= 1P, but the sun's least distance= 116QP, ... the difference between them being 1096P... Since... there is no vacant space in the arrangement of the universe, and the distances are crammed full with what belongs within them, these astronomers think it proper to examine the ratios of the apogees and perigees of Mercury and Venus, and thereby to determine whether these values can fill out the numbers mentioned above. Thus these astronomers find that from the apogee of Mercury's epicycle to the center of the ecliptic ... ~ 177P33', which is Mercury's greatest distance. Once more, because there is a great gap between this 177P 33' and the sun's perigee= 1160P, they consider it proper, in order to avoid an empty space, to insert another sphere, namely, Venus' ...

Although Copernicus follows GV quite closely here, he veers off sharply from his source in interpreting the significance of the number 177P 33', which Copernicus rounds off as 177!. For GV, this number means the distance "from the apogee of Mercury's epicycle to the center of the ecliptic... which is Mercury's greatest distance" ( Mercurii ab apogio epicycli ad centrum usque signiferi ... quantum est Mercurii maximum intenJallum). On the other hand, for Copernicus 177! becomes Mercury's interapsidal distance, that is, the distance between its apsides (inter absides Mercurii, fol. sr, line 5 up). Only so could Copernicus allot 91QP to Venus. This value is the approximate result of subtracting from 1096P ( = the interval between the moon's apogee and the sun's perigee) the figure of 177!, interpreted in Copernicus' manner as Mercury's interapsidal distance. The ultimate source of this discussion is a passage in Ptolemy's Planetary Hypotheses, which was inacces­ sible to Copernicus but was recently recovered by Bernard R. Goldstein (Transactions of the American Philo­ sophical Society, 1967, 57, pt. 4, p. 7). What reached Copernicus was GV's somewhat less than faithful transla­ tion of Proclus, whose treatment of the question is translated and discussed by Willy Hartnet:, Oriens-Occidens (Hildesheim: Olms, 1968), pp. 323-326.

P. 19:13. As Copernicus understood the computations of the Ptolemaists,Mercury's apogee= 641/ 8 +177! = 2411/ 8• Then, 2411/ 8 +910 = 11511/ 1 ~ 1160. P. 19:16. Copernicus' autograph (fol. 8r,line 3 up) says Non ... fatentur, "they do not admit," that is, the Ptolemaists do not admit. By a typographical error N (fol. sr, line 13) printed Non ... fatlur, "we do not admit." One influential reader, namely, Galileo, failed to recognize the misprint and was therefore responsible for ini­ tiating the historical error that the opacity of the planets was denied by Copernicus, whereas he attributed that denial to the Ptolemaists. P. 19:18. "There may be planets beneath the sun, but they need not be in any plane passing through the sun and our eye. Rather, they may lie in another plane and for that reason produce no visible passage before the sun, just as no eclipses occur for the most part when the moon at the time of conjunction passes below the sun" (PS, IX, 1). P. 19:22. In the autograph (fol. 8v, line 5) Copernicus wrote "Albategnius," the usual Latin designation of Al-Battani. Later he deleted "Albategnius" and in the left margin wrote "Machometus," Al-Battani's forename. Copernicus had access neither to the Arabic original of Al-Battani's great astronomical treatise nor to the Latin translation thereof (Nuremberg, 1537). Copernicus' source for Al-Battani's statement was rather P-R (IX, 1). P-R, however, said that it was the ancients who made the sun's apparent diameter ten times larger than Venus', according to Al-Battani. This was transformed by Copernicus into the opinion of Al-Battani. He was assigned to the place where he observed by the use of the assimilated Arabic article ar-Raqqa, which was latinized as Araccensis or, as here, Aratensis. Al-Battani's private observatory was discussed by Aydin Sayili, The Observ­ atory in Islam (Ankara, 1960), pp. 96-98. P. 19 :24. Ibn Rushd (or Averroes, as Islam's greatest philosopher was called in Latin) in the twelfth century wrote in Arabic a Paraphrase of Ptolemy's Syntaxis, which was translated into Hebrew by Jacob Anatoli of Naples in 1231. That excellent student of Hebrew, Giovanni Pico della Mirandola, in his posthumous Disputations against Predicti'Oe Astrology (Bologna, 1495-1496) stated: "In his Paraphrase of Pto'Umy's Syntaxis Ibn Rushd says that he once observed what looked like two blackish spots on the sun. When he examined the tables for

356 NOTBS ON P. 19 that time, he found Mercury located in the sun's rays" (X, 4; reprinted, Florence, 1946-1952, II, 374: 14-17). This episode was mentioo.ed by Kepler in his Optics, where he raised no question about the propriety of Averroes' name (Gesammelte Wer.ke, II, 265 :7'). Later, in a letter dated 7 April 1607 and addressed to his former teacher Mastlin, Kepler asked "whether Averroes' Paraphrase of Ptolemy, cited by Copernicus, is extant" (ibid., XV, 418: 48-49). At about the same time Kepler inquired of a wealthy patron where he could obtain "Averroes' Paraphrase of Ptolemy, referred to by Copernicus" (ibid., XV, 462: 354). Then on 28 May 1607 occurred a Singular Phenomenon, in reporting which Kepler quoted the aforementioned passage of his Optics and inserted a poem; in neither of these places did he express any doubt about Averroes (ibid., IV, 83: 20, 96 :36). The Kepler-Mastlin correspondence was interrupted for over three years, and when it was resumed a more urgent topic pushed the Averroes question into the background (ibid., XVI, no. 592). Early in 1612, however, Kepler tripped himself up. Addressing another correspondent, he wrote about "Averroes (or, as I conjecture, Avenrodan, as Pico testifies in his work against the astrologers" (ibid., XVII, 9: 83-84). By describing this utterance as a conjecture (conijcio), Kepler confesses his uncertainty. Evidently he did not take the time to track down the passage in Pico. Had he done so, he would have seen that Pico spec­ ifies Averroes, not Avenrodan (Ali ibn Ridwan, 998-1061). However, Kepler was commenting with consider­ able urgency on an important book that had just been published. Stored away in the back of Kepler's mind from an earlier perusal of Pico, a general recollection may have persisted that this famous opponent of astrology often mentions Avenrodan by name (more than thirty times), but Averroes far less frequently (seven times, including our passage). After all, the principal purpose of Pico's treatise was to attack astrologers. Avenrodan wrote a commentary on an astrological tract by (pseudo-) Ptolemy, whereas Averroes paraphrased Ptolemy's Synta»s, a purely astronomical volume absolutely unstained by astrology. In view of the greater prominence of Avenrodan in Pico, and of Kepler's relegation of this matter to the periphery of his attention at that time, his conjectural substitution of Avenrodan for Averroes seems entirely understandable. Indeed, its occurrence in a private letter may also make it appear quite harmless. Yet harm was done not long afterwards when Kepler converted his private conjecture of 1612 into a public certainty: "Avenrodan saw two spots in the sun (as Pico della Mirandola states in his book written against astrology)" (Ephemerides, Preface, p. 17; Kepleri opera omnia, ed. Frisch, II, 786: 13-14). By this time Kepler was so sure he was right that he launched an offensive: "AlthOugh this [passage) was copied from Pico by Coper­ nicus, nevertheless he changed the name of Avenrodan to that of Averroes" (ed. Frisch, ibid., lines 15-17). But of course Copernicus did no such thing. For, Pico cites Averroes' Paraphrase of Ptolemy's Syntaxis, and no such work was ever written by or attributed to Avenrodan. Kepler's former teacher must have been involved somehow in this erroneous accusation against Copernicus. For, Kepler goes on to say that Copernicus "supplied Masdin with the vain task oflookingfor the passage through all of Averroes' commentaries" (ed. Frisch, ibid., lines 17-18). These numerous and voluminous writings had been published in Latin translation, together with Aristotle's works (Venice, 1562-1574; reprinted, Frankfurt/ Main, 1962). What Pico cited, however, was not a commentary on Aristotle by Averroes, but his Paraphrase ofJ?tolemy. No wonder Mastlin hunted in vain "through all of Averroes' commentaries." These had been trans­ lated into Latin, whereas the Ptolemy Paraphrase had not. In fact, no Arabic manuscript of the original survives, and it is known only because it had been translated into Hebrew, a language which Pico had mastered (Moritz Steinschneider, Die hebraeischen iJbersetzungen des Mittelalters und die Juden als Dolmstscher (Graz, 1956, reprint of 1893 ed., pp. 546-549). Moreover, he possessed a manuscript of Jacob Anatoli's Hebrew translation (Pearl Kibre, The Library of Pico della Mirantlola, 2nd printing, New York, 1966, pp. 203-204). Yet, despite the correctness of Copernicus' citation and the unsoundness of Kepler's charge that Coper- nicus erred, as Z's example of Copernicus' "mistakes in citing older writings" Z adduces his ascription to Averroes of the observation of a dark spot on the sun. Milsdin searched in vain through all of Averroes' commentaries without finding the passage. Actually, this was a matter concerning Aven Rodan. Copernicus had indeed taken the passage from Pico della Mirandola's treatise against astrology and miswritten the name Aven Rodan as Averroes (p. 510). Z's arrant nonsense was repeated in 1951 by Max Caspar: "Copernicus mistakenly wrote Averroes instead of Aven Rodan" (in Kepler, Gesammelte Werke, XV, 549, note to line 45). Since Copernicus as a student at Cracow University was subjected to the normal dose of astrology, his later indifference to the supposed art of foretelling the future may have been produced by his familiarity with Pico's Disputations, which was published shortly before his arrival in Italy and from which he extracted this reference to Averroes. P. 19:30. In the autograph (fol. 8v, line 14) Copernicus put the moon's perigeal distance at "more than 49" {plusguam iL}, "as will be made clear below." But below, in IV, 17, 22, 24, he made that distance more than

357 NOTBS ON PP. 19-20

52, to be exact, 52P 17', and N printed 52 here in I, 10, in order to make Copernicus consistent with himself. He himself, however, after revising upward his estimate of the moon's perigeal distance, failed to change this passage in the autograph. This number 49, therefore, sheds light on the still incompletely solved question when Copernicus composed the various parts of his Revolutions. Obviously, I, 10, and fol. 8 were written before Co­ pernicus made the observations and computations which induced him to change from 49 to 52. It is of course true that 52 is more than 49, and perhaps someone may think that 52P 17' at three places in Book IV carries out the promise in I, 10 to make "clear below" that the moon's perigeal distance is "more than 49, according to a more accurate determination." However, had Copernicus already possessed the 52P 17' result when he was writing I, 10, surely he would then have said "more than 52" instead of "more than 49." For, the larger number would have strengthened his objection to the reasoning of the Ptolemaists. The fact that he used 49 proves that he did not yet have the 52 result. As a by-product of this analysis, it becomes evident that "more than" (plusquam) connotes a fraction less than unity, so that "more than 49" implies "less than 50." It seems clear that the 52P 17' result was derived from the calculations based on the two observations dis­ cussed in IV, 16. Since these are dated 27 September 1522 and 7 August 1524, we may safely conclude that Copernicus wrote I, 10, and fol. 8 before 27 September 1522. Folio 8 in quire a was written on paper C, the first of the four kinds of paper used for the autograph. Although it has not yet been possible to assign a precise date to paper C on the basis of its watermark, other considerations confirm this conclusion that fol. 8 was written in or before 1522 (see NCCW,, I, p. 3). P. 19:32. Here again, as in I, 8 (note top. 17:16). Copernicus implies that he doubts the existence of the sphere of "elemental" fire. P. 19:35. Once more Copernicus makes a promise that he fails to carry out, but this time N allowed it to stand, perhaps feeling that it was justified by the implications of V, 21, 22. This passage in I, 10, may have suggested to Osiander (note top. XVI) the Venus argument with which he thought he could prove the perma­ nent unreliability of astronomy, which has instead shown itself to be a self-correcting discipline, like the other branches of natural science. P. 20:6. These "certain other Latin writers" may have included Vitruvius, Architecture, IX, 6: "The planets Mercury and Venus perform their retrogradations and stations by winding a wreath as they travel around the sun's rays as center." P. 20:7. Martianus Capella's encyclopedia, often c:alled The Marriage of Philology and Mercury, VIII, 857: "Venus and Mercury... locate the center of their circles in the sun." P. 20:10. Since Copernicus' words "terram non ambiunt" echo Capella's "terras ... non ambiunt," he may have had access to the 1499 Vicenza or 1500 Modena edition of that once popular encyclopedia. P. 20:11. Copernicus' expression "absidas corroersas habent" repeats "confJBrsas habent ... apsidas" from Pliny's Natural History, II, 14, 72. Apparently Copernicus derived from Pliny's obscure discussion the impression that the author of the Natural History ascribed a heliocentric orbit to Venus and Mercury. At II, 13, 63, Pliny explains that he uses the Greek word apsidts as meaning "circles." P. 20:35. The Congregation ordered "Hence I feel no shame in asserting ... " changed to "Hence I feel no shame in assuming ...". P. 20:36. Here Copernicus introduces for the first time the expression orbis magnus to denote the earth's annual revolution about the sun. This technical term became, as it were, the trademark of the Copernican astron­ omy down to the time of Newton. It might more naturally be translated into English as "great circle," which is preempted, however, as an established label in spherical geometry. P. 20 :37. Copernicus here says that the center of the universe is "near" (circa) the sun. He chose this ex­ pression deliberately, since he knew that his orbits could not be concentric, but had to be somewhat eccentric. Some ill-informed critics have insisted that Copernicus' astronomy was not really heliocentric because the center of his universe was ·outside the sun. In I, 9, he said that ''the sun occupies the middle of the universe," at variance with his statement here. For his conscious use of this ambiguity, or "amphibology" as he calls it, see III, 25. P. 20:39. The Congregation ordered the replacement of "rather" (potius) by "therefore" (conseguenter). It is not immediately obvious how the change from "rather" to "therefore" served the Congregation's purposes. Perhaps the Congregation felt that potius implied a state of actual fact, whereas conseguenter suggested merely a logical inference. P. 20:44. Although Copernicus carefully refrained from naming the authors he has in mind, one possi­ bility is the Muslim astronomer Nasir a1 Din (1201-1274), director of the Maragha Observatory, who was known as "al Tusi" from his birthplace Tus in Persia, and whose "Tusi couple" was adopted in a modified form by Copernicus (III, 4). To the ample number of spheres used by Ptolemy, Tusi added 33 in a section of his Kitab al-tatlhkira, translated from Arabic into French by Carra de Vaux in Paul Tannery, Recherches sur l'histoire de l'astronomie ancienne (Mbnoires ds Ia Socilt4 des sciences ... de Bordeaux, 1893, pp. 351, 358-359).

358 NOTBS ON PP. 20-22

P. 20:46. These familiar maxims were quoted from Galen by Copernicus' disciple Rheticus: ''Nature does nothing in vain"; "Our Creator is so skillful that each part he has made has not only one usefulness but two or three or frequently four" (On the Usefulness of the Parts of the Body, tr. M. T. May, Ithaca, 1968, pp. 501-502). P. 21. Copernicus' diagram was not printed in N exactly as he drew it (fol. 9"). For example, his Arabic numerals 1-7 were replaced by Roman numerals I-VII (N, fol. 9v). His diagram merely mentioned the moon by name, but that body's orbit was indicated by N, which also included its symbol, the only such symbol in the entire diagram. Copernicus wrote the legend for the sphere of the stars below that circle, evidently for the sake of avoiding confusion with the words of his text, which hug two-thirds of that circumference quite tightly. By contrast, the printed page is designed much more legibly, with the lines of the text set distinctly apart from the legend for the sphere of the stars, even though that is placed above the outermost circle. Following that pattern, N placed the legend for each of the three outer planets just above its circle. In carrying out this plan, N inadvertently obscured Copernicus' intention, with the result that some rather wild conjectures have arisen about his conception of the cosmos. His drawing in the autograph allotted to each planet two circles, the inner one for its perihelian distance and the outer one for its aphelian distance. This outer circle did double duty as the inner circle of the next higher planet, in accordance with Copernicus' theory of contiguous spheres. Thus, the innermost circle around the central sun marks Mercury's perihelian distance. Then, as we move away from the sun, the next circle is Mercury's aphelion, which is also Venus' perihelion. In turn, Venus' aphelion is the perihelion of the earth in the autograph, and of the moon in N. Here, however, in addition to the moon's perihelion and aphelion, there is a separate circle for the earth's center, so that the earth-moon system has three circles instead of the usual two. Just above the earth-moon system inN the reader sees what looks like a most un-Copernican empty space in the region of the planets, because the legend for Mars was placed above its aphelion, instead of between its aphelion and perihelion, as in the cases of the earth-moon, Venus, and Mercury. This Mars placement may have been nothing more than a convenience for the typesetter, and no cosmological significance should be at­ tached to it. The same should be said about the region of the sun, which has in its middle something that has been misinterpreted as a small circle but is actually the mark left unintentionally by the fixed, blunt foot of the compass with which Copernicus drew the circles in this diagram. It will be observed that in drawing the circle for Jupiter, Copernicus inadvertently permitted the moving foot of the compass to wiggle a little in the 5 o'clock region. P. 21:4. This "first principle" was explained in the opening paragraph of I, 10. P. 21:9. In asserting that the immovable sphere of the stars is the place (locus) of the universe, to which the motions of all the other heavenly bodies are compared, Copernicus is applying Aristotle's generalization that "without place ... motion is impossible" (Physics, III, 1). P. 22:6. The sun was called the mind of the universe (mundi ... mentem) and ruler of the heavens (caeli ... rector) by Pliny, Natural History, II, 4, 13. Cicero is another writer in whom Copernicus found the sun called the "mind of the universe" (mens mundi, Republic, VI: Scipio's Dream, Ch. 17). If Copernicus attributed the de­ scription of the sun as the lantern of the universe (lucernam mundi) to a specific author, that writer has not yet been identified by students of the subject. P. 22:7. From this remark some overenthusiastic writers have concluded that Copernicus is to be asso­ ciated with Renaissance Hermetic magic or neoplatonic mysticism. But in this unique reference to that collec­ tion of spurious theological treatises which disguised its true nature by invoking the name of the famous Greek divinity Hermes, Copernicus does not so much as mention that god and calls the (supposed) author "Trime­ gistus" (fol. 10r, iine 6), although Copernicus knew Greek well enough to realize that Thrice Greatest should be Trismegistus. Moreover, nowhere in that sprawling body of writings ascribed to Hermes Trismegistus is the sun called a visible god, as Copernicus asserts that it was. Evidently Copernicus had no first-hand acquaint­ ance with the Hermetic corpus, and was relying once more on his imperfect memory, perhaps of a university lecture he had once heard. That professor may have had access to a manuscript of the Epitome of the Di'Dine Institutes by that same Lactantius who was chided by Copernicus near the end of his Preface to the Revolutions. In his Epitome Lactantius uses the expression 'Disibilem deum (quoted here in I, 10, by Copernicus) in making his own Latin translation of a passage from a Hermetic treatise entitled Asclepius. There, however, the expression "visible god" was attached to the perceptible universe, as Copernicus correctly indicated in his Introduction to Book I (first paragraph). Here in I, 10, on the other hand, his faulty memory misapplied "visible god" to the sun. P. 22:8. Sophocles calls the sun all-seeing, not in his Electra, but in his Oedipus at Co/onus, line 869. Co­ pernicus does not quote the original Greek here, as he did in his Preface, but instead repeats Pliny's expre88ion omnia intuens (Natural History, II, 4, 13).

359 NOTBS ON PP. 22-25

P. 22:11. "In a book on animals Aristode says that the moon's nature is more closely akin to that of the earth than to that of the other heavenly bodies" (Averroes, De substantia orbis, Ch. 2, published in Venice in the fifteenth century together with a Latin translation of some of Aristode's works). Copernicus repeated this mis­ statement by the greatest Muslim commentator on Aristotle without being aware that Aristode's Generation of Animals, IV, 10, regards the moon, not as akin to the earth, but as a "second and lesser sun." P. 22:30. Euclid, Optics, Prop. 3. P. 22:33. The contrast between the twinkling of the stars and the steady light of the planets was used by Copernicus in his Letter against Werner as a handy illustration of how astronomers draw inferences from obser­ vations: "The science of the stars is one of those subjects which we learn in the order opposite to the natural order. For example, in the natural order it is first known that the planets are nearer than the fixed stars to the earth, and then aa a consequence that the planets do not twinkle. We, on the contrary, first see that they do not twinkle, and then we know that they are nearer to the earth." Here Copernicus is echoing Aristode's HetlfJens II, 8, and Posterior Analytics, I, 13. P. 22:36. The Congregation ordered the deletion of the closing sc;ntence of I, 10, presumably because it portended the tremendous expansion of the relatively tiny cosmos in which the traditionally minded theolo­ gians felt more at home. P. 22:38. The Congregation ordered the title of I, 11, changed to "The Hypothesis of the Earth's Triple Motion, and the Proof of That" Hypothesis. P. 23:9. Copernicus appears to be thinking of I, 6. There, however, he used the Goat and Crab to prove the identity of the centers of the horizon and ecliptic, and not as here, the diametrical opposition of the earth's real position in the zodiac and the sun's apparent position. P. 23:17. This third motion of the earth seemed indispensable to Copernicus because he regarded our planet as firmly attached to an invisible sphere. But after all this imperceptible celestial furniture had been swept out of the heavens, and the earth had become an unattached body moving freely in space, Copernicus' third terrestrial motion was discarded as unnecessary. Thus, in his N«» Astronomy, Chapter 57, Kepler wrote: While the earth's axis remains almost exacdy parallel to itself in all its positions throughout the annual revolution of the [earth's] center, summer and winter are produced. To the extent, however, that very long periods of time alter the inclination of the axis, the fixed stars are believed to advance and the equinoxes to precess ..• Copernicus was mistaken in thinking that a special prin­ ciple was needed to make the earth oscillate annually from north to south and back again in such a way that summer and winter result, and that the operation of this [libration], in phase with the revolution [of the center], will produce the uniform recurrence of the tropical and sidereal year (since they are nearly equal). On the other hand, by itself a fixed direction of the earth's axis, about which the daily motion occurs, accomplishes all those [results], except only for the very slow precession of the equinoxes (Gesammelte Werke, III, 350: 22-25, 30-37). P. 24:13. In the autograph (fol. nr, line 17) his followed by se (deleted). In this instance Copernicus wrote a long J and allowed the short deletion stroke to cross it, so that a careless reader could have misread the deleted seas an f. This is exacdy what happened inN (fol. nv, line 7), which mistakenly printed HF, although H without F is required by the context. The misprint HF was repeated by the next five editions. Undoubtedly contributing to this error is Copernicus' repetition of se in line 18 as well as the nonexistent form "convertentens" instead of convertens. P. 24:16. In the autograph (fol. nr, line 22) this circle perpendicular to the ecliptic is lettered abc. But this lettering was not included by Copernicus in the accompanying diagram which he drew at the foot of fol. ur, and was therefore wisely omitted by N. The center of the right-hand circle was originally lettered b, over which Copernicus later wrote c. P. 24:25. In the autograph (fol. nr, line 8 up) by a slip Copernicus wrote ac, which was corrected in N to AB, since B is the position of the supposed observer. Moreover, just below, C is called the "opposite point," so that the previous observation could not have been made from AC. P. 25 :20. For the opinions of Phllolaus, as attested by antiquity and as exalted by Copernicus, see his Pref­ ace and note to p. 12 :33. P. 25:21. Prior to the recovery and publication of this deleted passage, it was maintained that Copernicus was entirely unaware of Aristarchus' geokineticism. Kepler, for example, asked: "Will anyone deny that the system which makes the earth one of the moving planets was discovered by Copernicus, who did not have the slightest inkling of Aristarchus' theory?" (Appendix to his Hyperaspistes, in The Controoersy on the Comets of 1618, Philadelphia, 1960, p. 344). Copernicus in fact derived litde more than an inkling from GV, XXI, 24. For, that unreliable encyclopedist

360 NOTES ON P. 25

distorted the ancient Greek statement that "According to Aristarchus, the sun and the fixed stars are stationary, while the earth revolves around the ecliptic." GV's mistranslation had "Aristarchus locate the sun beyond the fixed stars." This in itself may have been enough to persuade Copernicus to delete the passage containing his reference to Aristarchus as a geokineticist. But Copernicus may have had an additional motive for discarding Aristarchus the earth-mover. For in Plutarch's Face in the Moon Copernicus may have read about the ancient philosopher who "thought that the Greeks should indict Aristarchus of Samos on the charge of impiety for setting in motion the hearth of the universe, the evidence being that he tried to save the phenomena by assuming that the heavens are stationary and that the earth revolves in an oblique circle, while at the same time rotating about its axis." If Copernicus noticed this statement on page 932 of Plutarch's Opuscula LXXXXII, from page 328 of which he excerpted the Greek passage in his Preface, he may have decided to dissociate himself from Aristarchus. For if it is true that Aristarchus was the Copernicus of antiquity, Copernicus had no desire to become a modem Aristarchus and be indicted for impiety with the customary unpleasant consequences. Our most important ancient source of information about Aristar!=hus' astronomical system is Archimedes' Sand-Reckoner, with which, however, Copernicus was unacquainted. P. 25:25. Perhaps Copernicus has in mind Cardinal Bessarion's comment that Plato ''urges a very few to most complete mastery of these disciplines," including astronomy (In calumniatorem Platonis,IV, 12; ed. Mohler, II, 593: 4-5). P. 25:33. The Greek text of Lysis' letter was available to Copernicus at sig. r6v-7v in Bpistolae difJBI'sorum philosoplwrum (Venice, 1499). A copy of this collection of letters written by twenty-six Greek philosophers, orators, and rhetoricians belonged to the library of Copernicus' Chapter (ZGAH, 5: 376). In addition, Cardinal Bessarion's translation of Lysis' letter into Latin was also accessible to Copernicus in his copy of Bessarion's In calumniatorem Platonis, fol. 2V-3r. This passage is specially marked and underlined in Copernicus' copy of Bessarion's attack on George of Trebizond as a vilifier of Plato (MK, p. 131). The two translations of Lysis' letter into Latin - Bessarion's and Copernicus' - were printed side by side in MK, pp. 132-134. The merest glance shows at once that Copernicus' translation followed Bessarion's quite closely. Why did Copernicus depart at all from Bessarion's translation? Although the cardinal's native language was Greek, an outstanding contemporary philologist praised him as the "best Latinist among the Greeks" (ed. Mohler, I, 251). Nevertheless, he did not learn Latin until he was about forty years old. Perhaps Copernicus flattered himself that he could improve on Bessarion's Latin style. He did adhere more faithfully to the Greek as, for instance, when he specified that the initiation into the Pythagorean brotherhood lasted five years, a re­ quirement omitted by Bessarion. The Greek text of Lysis' letter, as printed in the 1499 collection of epistolographers and translated by Bessarion and Copernicus, is somewhat longer than the version of Lysis' letter that is preserved in Iamblichus' Life of Pythagoras, chapter 17, sections 75--78 (ed. Ludwig Deubner, Leipzig, 1937). Whoever first produced the abbreviated text that Iamblichus (c. 250-330 A.D.) later incorporated in his biography of Pythagoras omitted the longer version's opening reference to the dissolution of the Pythagorean brotherhood and the dispersion of its former members. Neither at the start (nor anywhere else) did the abbreviator admit that the Pythagoreans had suffered so severe a blow. Instead, he began by taking from the longer version's fourth paragraph (as trans­ lated here) Lysis' rebuke of his addressee's un-Pythagorean conduct as a private individual. But just as one swallow does not make a spring, so one backslider's misbehavior does not entail the end of the entire brotherhood. That confraternity goes on, even if Lysis' correspondent fails to return to the fold in good faith. At this point the abbreviator repeated the powerful closing sentence of the longer version. Instead of stopping here, however, the abbreviator continues with the parenthetical insertion ''he says." This intrusion is of course an implicit reference to Lysis as the author of the longer version, which from this point to the end of our third paragraph provides the rest of the abbreviated version. The longer version's subsequent mention of Pythagoras' daughter was quoted by Diogenes Laertius, LifJBs of Eminent Philosophers, VIII, 42. Two of the three oldest manuscripts of Diogenes Laertius identify Lysis' addressee as Hippasus (not Hipparchus; Armand Delatte, La fJie de Pythagore de Diogme Lalrce, Brussels, 1922; Academie royale de Belgique, Classe des Lettres, Memoires, Collection in 8°, 2f! serie, t. 17, fasc. 2, p. 138, line 11). According to Iamblichus' Life of Pythagoras (ch. 18, section 88), "Hippasus ... who was one of the Pythagorean& .•. was the first to write down and divulge" a Pythagorean secret. Later on, concerning ''the first to reveal the nature of the commensurables and incommensurable&" Iamblichus remarks that this renegade

was not only excluded from the common life and company, but a tombstone was erected for him as though he, who had once been a member, had actually departed from the life of mankind (ch. 34, section 246).

361 NOTES ON P. 2S

Here, although no renegade is mentioned by name, presumably Hippasus is intended. Hence, the designation of Hippasus as the recipient of Lysis' letter in two of the three oldest manuscripts of Diogenes Laertius is without question historically more acceptable than is an otherwise unknown Hipparchus. For there is no Hipparchus in the list of 218 male Pythagoreans known by name. This list was repeated by Iamblichus (Life of Pythagoras, ch. 36, section 267) from a compilation made long before his time. To be sure, the list may not be complete. Nevertheless, the absence of Hipparchus from it may have been responsible for Iamblichus' evident hesitation when he said that Lysis' letter was addressed "to a certain Hipparchus" (ed. Deubner, p. 42, line 23). In the aforementioned passage of Diogenes Laertius, where two of the three oldest manuscripts identify Hippasus as Lysis' correspondent, the third manuscript has a gap which a later hand filled with "Hipparchus." This defective reading was adopted early in the nineteenth centurY by an influential edition of Diogenes Laertius, although in a note it cited manuscript and editorial preference for Hippasus (ed. Hiibner, Leipzig, 1828-1831, II, 275, note "1"). Through another authoritative nineteenth-centurY edition of Diogenes Laertius (Paris, 1850, ed. Cobet, p. 214, line 23) the erroneous substitution of Hipparchus for Hippasus became more firmly established. This blunder began in connection with Aristotle's statement (Metaphysics 984a7) that according to Hippasus the most fundamental element was fire. But when a Christian polemicist, for whom philosophy was the mother of heresy, wrote a treatise On the Soul about 210, he misattributed to Hipparchus, instead of to Hippasus, the doctrine that the soul was made of fire (Tertullian, De anima, ch. 5, ed. J.H. Waszink, Amsterdam, 1947, p. 6, line 6). Some two centuries later, Macrobius (ed. Willis, Leipzig, 1970, II, 59, line 8) echoed this misattribution of the fiery soul to Hipparchus (William H. Stahl, Macrobius, Commentary on the Dream of Scipio, New York, 1966, p. 146). Although a late sixteenth-century annotator called attention to Tertullian's error, it was recently repeated by Timothy David Barnes, Tertullian (OXford, 1971), p. 207. Whether Tertullian was misled by an earlier doxographer was discussed by Werner W. Jaeger, Nemesios von Emesa (Berlin, 1914), pp. 94-96. In any case and in a different context Clement of Alexandria said in his Miscellanies, Book V, ch. 9, section 57, that Hipparchus the Pythagorean, being guilty of writing out Pythagoras' teachings in plain language, was expelled from the brotherhood, and a tombstone was erected for him, as though he were dead (Clemens Alexandrinus, Stromata I-VI, 3rd ed. by Stiihlin and Friichtel, Berlin, 1960, p. 364, lines 27-29).

The foregoing statements by Clement forcibly recall what we have already quoted from section 246 in Iamblichus' Life of Pythagoras. Indeed the resemblance is close enough to suggest that both these passages were derived from a common source. If so, that unknown author may have refrained from mentioning any miscreant by name. Whereas this reticence was continued by Iamblichus, Clement rashly supplied the wrong name, Hipparchus. Some two centuries later, Hipparchus was unhesitatingly designated as Lysis' addressee by Synesius (c. 365-c. 414; Synesii ... opera, Paris, 1612, p. 279B; Letters of Synesius of Cyrene, tr. Augustine FitzGerald, London, 1926, p. 237). Another turncoat Pythagorean, according to Iamblichus' Life of Pythagoras (ch. 31, section 199), was Phi­ lolaus:

Reduced to great and grinding poverty, Philolaus was the first to release those three famous books, which are said ... to have been bought for a hundred minas at the instigation of Plato.

In the Phaedo (61E) Plato implies that Philolaus spent some time in Thebes. This was where Lysis settled down after he escaped, with only one other Pythagorean, from their meeting-house in Crotona to which the anti­ Pythagorean& had set fire. But the Commentary on Plato's Phaedo attributed to Olympiodorus, the sixth-century Neoplatonist, altered the names of the two successful escapees to Hipparchus and Philolaus (ed. William Norvin, Hildersheim, 1968; reprint of 1913 ed., p. 9, lines 17-18). This pair of names was repeated by an anonymous scholiast on Plato's Phaedo, writing at some unknown time (Scholia platonica, ed. W.C. Greene, Haverford, 1938, p. 9). In Iamblichus' account of the Crotona catastrophe, however, the two who escaped were Lysis and Archippus (ch. 35, section 249). Lysis' companion was gratuitously transformed into Hipparchus (Rheinisches Museumfilr Philologie, 1879,34: 262) by Erwin Rohde, whose unsupported equation, Archippus = Hipparchus, has often been uncritically repeated. If, with Iamblichus, we retain Archippus as Lysis' coescapee, we have no reason to regard Archippus as Lysis' addressee, since Archippus betrayed no Pythagorean secrets. But Hippasus did, and thereby established his eligibility to be Lysis' addressee. The chronological uncertainties surrounding Lysis and Hippasus have strengthened the feeling of skepticism about the authenticity of the Lysis letter, which is frequently branded "spurious." Such a condemnation, however, means only that the historical Lysis was not the true author of our letter. That document undoubtedly formed

362 NOTES ON PP. 25-31

part of a substantial pseudepigraphical pythagoreanizing literature, which circulated under the names of the famous Pythagorean& of olden times. As a highly conspicuous element in this philosophical movement, the longer version of our document was written by a later Pythagorean, and then abbreviated by a still later Pytha­ gorean. These two ghost-writers concealed their identities by pretending that Lysis was the author of both versions of our document. In either form it was a more recent product (perhaps of the 3rd or 2nd century B.C.), deliberately and falsely attributed to a well-known Pythagorean of the 5th and 4th centuries B.C. This category of pretended Pythagorean writings was unfamiliar to Copernicus. In that same spirit of innocence he accepted the Opinions of the Philosophers as a genuine work of Plutarch, although it is now recognized as a post-Plutarchan production, which was promulgated under the name of the celebrated biographer. Why was the pseudo-Lysis letter concocted in the first instance? Its initial fabricator may have wanted to wrap a layer of authentication around the apocryphal "notes" of Pythagoras which, having passed from him to his daughter and granddaughter, were suddenly discovered and published as an integral part of the Pythagorean revival (Walter Burkert, Weisheit und Wissenschaft, Nuremberg, 1962, p. 436, n. 86). This primitive form of the pseudo-Lysis letter was subsequently shortened for the purpose of deleting any reference to the catastrophe that overwhelmed the Pythagorean order, and to fortify the pretense that the brotherhood was still in a flourishing state. The names of Pythagoras' daughter and granddaughter were given in the 1499 edition of the Greek epistol­ ographers as Damo and Bistalia (the latter should have been Bitale). These genuine Greek names of women were infelicitously latinized as Dama and Vitalia by Bessarion, whom Copernicus followed. When Copernicus contrasted male faithlessness with female fidelity, he omitted Bessarion's comment that Damo remained loyal "even though she was a woman." That slur may have looked to Copernicus like an anti-feminine insult inserted by Bessarion without any warrant in the 1499 text of the Greek epistolographers. Perhaps Copernicus was unaware that Diogenes Laertius' quotation from the pseudo-Lysis letter ends with the remark that Damo behaved nobly "even though she was a woman." Bessarion gave his reader no hint that he was blending Diogenes Laerti'us with the 1499 Greek text. P. 27:26. In the terminology of ancient Greek geometry, a problem differed from a theorem in that the former was concerned with a construction, in this instance, of the Table of the Straight Lines Subtended in a Circle, which Copernicus placed immediately after the problem. For the sake of brevity this Table will be cited hereafter as "Lines Subtended." P. 27:33. Copernicus owned a copy of the first printed edition of Euclid's Elements (Venice, 1482), a trans­ lation into Latin which was essentially based on a previous Arabic version rather than on the original Greek text. That was first printed at Basel in 1533, and a copy was presented to Copernicus by Rheticus in 1539 too late to affect the composition of the Revolutions. P. 27:40. Copernicus cites Euclid's Elements according to the 1482 edition, whose arrangement in general differs from that in the Greek manuscripts. P. 31:8. The corresponding Table in PS, I, 11, extended from 0° to 180°. P. 31 :10. The corresponding Table in PS, I, 11, progresses by halves of a degree. P. 31 :12. The corresponding Table in PS, I, 11, gives the length of the chord in sexagesimal parts of a di­ ameter assumed"= 120P. On the other hand, Copernicus expresses the length of the chord in decimal parts of a diameter assumed = 200,000. This shift from the sexagesimal to the decimal basis, accompanied by the re­ duction of the scope of the Table from the semicircle to the quadrant, and by the reduction of the argument from 30' to 10', is not found in P-R, GV, PS 1515. Copernican scholars have not yet discovered what model, if any, Copernicus followed in transforming Ptolemy's sexagesimal Table of Chords into an early form of the modem Table of Natural Sines. For, Copernicus' half-line is nothing but the sine.

Thus, with Copernicus' diameter= 200,000 the radius= 100,000. Copernicus' half-line AB = sin~C, for values of ~C ranging between 0° and 90°, is expressed as a five-place decimal part of 100,000.

363 NOTBS ON PP. 31-40

Although Copernicus saw the advantage of the decimal over the sexagesimal arrangement, he studiously avoided the term "sine" as a neologism not found in ancient writers. "Copernicus shuns the word 'sine,' as I was told by Rheticus"; this annotation was written in the copy of Copernicus' Revolutions which came into the possession of Johannes Praetorius (1537-1616), who studied with Rheticus at Cracow in 1569 (Z, p. 454, reading "Abhorret"). P. 31:16. In the Preface to his Rudolphine Tables (Ulm, 1627), Kepler said: Even though this work [Copernicus' Re'l)olutions] has tables added to its explanations of the proofs, nevertheless, as far as I know, there is nobody nowadays who uses those tables for the purpose of computation... On the one hand, tables should be convenient lists, easy to use. This handy usefulness had even been enhanced in the case of the Alfonsine [Tables] and the other authors of tables by the book's format, the numerical tables being presented in a single series, with very brief instructions up front at the beginning. On the other hand, Copernicus' book scatters the tables throughout the text of the proofs, like Ptolemy's Syntaxis. As a result, those who want the theory have their concentration split apart by the text, those who want the use have their concentration split apart by the tables, and the work deprives itself of its principal usefulness (Kepler, Gesammelte Werke, X, 39: 40-41; 40: 4-11). Had Kepler lived two years longer than he did and had he known Italian well enough to read Galileo's Dialogue of 1632, he would have seen that when a Table of Arcs and Chords had to be consulted for a particular entry, a copy of Copernicus' Re'l)olutions was right there at hand with the desired information in this Table of the Straight Lines Subtended in a Circle (Galileo Galilei, Opere, national edition, reprinted, Bologna, 1968, VII, 207, lines 34-35). P. 32. The care with which Copernicus' Table was scrutinized may still be seen in the copy of the Rev­ olutions which was acquired by Mastlin in 1570. At eight places in this Table Mastlin corrected mistakes, five of which are to be found in Copernicus' autograph, while the remaining three are typographical errors inN. P. 32:7. In the 3rd column of his Table of Lines Subtended (fol. 1SV) Copernicus made the proportional difference of 291 extend from 0° 0' to 2° 40'. By actual computation, however, th-:: difference in the 2nd column is only 290 at 3 places: 0° 40', 1° 30', and 2° 20'. Then 290 in the 3rd column extends from 2° 50' through 5° 40', with 291 in the 2nd column at 3° 0' and 3° 30'. In like manner Copernicus records the diminishing proportional differences in the 3rd column until he reaches the vicinity of 90°, where the sine= 100,000. At 12° 20' Copernicus' 21350 was misprinted inN as 12350, the modern value being 21360. At 13° 50' Copernicus wrote 23900, which was repeated inN, although the 3rd column requires 23910. At 20° 50' and 21° 0' Copernicus' last digits should be 5 and 7. At 22° 10' the 5th digit should be 0. At 25° 10' the 3rd digit should be 5. At 25° 30' the 3rd digit is 3 by dittography, and should be 0; although Copernicus wrote 43351, he actually used 43051 in computing 25° 40', where he miswrote 43393 for 43313. Again, presumably by dit­ tography, Copernicus wrote 43555 for 25° 50', where the 4th digit should be 7. By anticipatory dittography, at 37° 40', instead of 61107, Copernicus wrote 61177. To this erroneous figure he added a difference of 200, instead of the required 230, and reached the false result for 37° 50' of 61377, instead of 61337. At 40° 10', by dittography Copernicus repeated 2 for the 3rd digit, which should be 5. Failing to notice this oversight, for 40° 20' he wrote 4 for the 3rd digit, which should be 7. At 72° 40' by dittography Copernicus wrote 9 for the 4th digit instead of 5. Once more by dittography, for 72° 50' he wrote a third 5 for the 4th digit. For 73° 0' he wrote 95600 dittographically, since to 95630, the correct value, he added the difference of 85 to obtain 95715, corresponding to 73° 10'. Once again by dittography, for 76° 10' Copernicus wrote 97009, where 97030+69 = 97099. At 82° 10', in adding to 99027 the proportional difference of 40, Copernicus repeats the 4 as the 4th digit. These dittographical errors have been recorded here in the hope that an analysis of them will help to throw light on the origin of Copernicus' Table of Lines Subtended. His other tables suffer from similar defects. In general, the tables in the autograph (NCCW, I) differ some­ what from those inN. To what extent were these changes introduced by Rheticus in his capacity as editor of N? Rheticus' independent career in science after leaving Copernicus in 1541, was concerned mainly with the con­ struction and publication of mathematical tables. But his fair copy or transcript of the Re'l)olutions for the printer's use has not survived. With it has perished the possibility of ascertaining what changes Rheticus made in the autograph's tables for N. In printing the tables, N made typographical errors. Later editions tried to straighten out the errors, not always with success. The complete elucidation of this aspect of the Re'l)olutions re­ quires further research. P. 40:16. In the autograph (fol. 19v, line 12) by a slip Copernicus wrote et si, where the sense requires a second aut, as was recognized by N (fol. 19v). P. 40:36. Originally (fol. 19v, lines 11-10 up) Copernicus wrote " ... degrees, whereof the circumference

364 NOTES ON PP. 40-46 of a circle [is) 360° ." Later he deleted this formulation, which he replaced by " ... degrees, whereof 180 are equal to two right angles." N (fol. 2or, lines 6-7) confusingly combined parts of these two different, although equiv­ alent, expressions by printing, not 180, but 360 as equal to two right angles. N was followed by the next three editions, but T (p. 54, lines 11-12) restored the later correct reading of Copernicus' autograph. P. 41:15. In the autograph (fol. 20', line 12) by a slip of the pen Copernicus wrote ab, which was quietly corrected to BC by Me, p. 44, no. 5. P. 41:27. An oversight in Copernicus' enunciation of Theorem liE (or VI, according to the numeration inN, fol. 2Qf) was pounced upon by Nunes, who called attention to what is now known as the "ambiguous case": In Rectilinear Triangles, Theorem VI [our liE] Copernicus went astray in the same way [as later on in Spherical Triangles, Theorem XI]. For if two sides of a triangle are given together with only one angle at the base, the remaining side together with the remaining angles cannot be known, unless the given angle is either right or obtuse, or if it is acute, unless it is opposite the longer of the given sides. For if other conditions are proposed [the given acute angle being placed opposite the shorter of the two given sides], it will not be clear from the assumptions whether the remaining angle at the base is acute or its obtuse supplement, and therefore the base too will remain unknown (Rules and Instruments, ed. Basel, 1566, p. 105).

P. 42:37. The numbering of Copernicus' Theorems on Spherical Triangles underwent a series of changes before Rheticus gave it its final form: the Roman numerals from I to XV. Originally Copernicus numbered his Theorems by means of the Arabic numerals from 1 to 12. P. 43:32. In Miistlin's copy of N (fol. 22•, line 12) another hand inserted parallelae in the right margin. P. 43:46. Christopher Clavius in his treatise on Spherical Triangles (Opera mathematica, Mainz, 1611-1612, I, 179), pointed out that What Copernicus enunciates in Theorem IV is not always true ... For even though right angleD and angle ABD are known, together with side AD [reading latere instead of latera], because [the known side] is opposite the known angle ABD and not the right angle, for that reason we shall not arrive at a knowledge of the remaining angle and the remaining sides. For, the remaining sides may be either AB and BD or AC and CD, and the remaining angle may be either BAD or CAD, as is evident. Therefore, something else will have to be known in addition, before the remaining angle and the remaining sides are determined.

A

P. 45:1. Theorems I through V were numbered by Copernicus not only with Arabic numerals but also with the letters of the Greek alphabet, from alpha through epsilon. However, beginning with the next Theorem the correspondence between the Arabic numerals and the Greek letters stops. The Greek letters themselves were not used by Rheticus on fol. 24-25 in connection with Theorems XIII-XV. P. 45:12. Beginning with this Theorem, the numbering was changed more than once. What actually became Theorem VI, likewise indicated by the Greek letter digamma = 6, had previously been numbered 7 by means of the Greek letter zeta and the Latin letter G, both deleted, and 11 by means of the Latin letter L, also deleted. Another deleted sign is difficult to interpret. P. 45:15. The corresponding equal sides must be opposite the right angles. By widening this condi­ tion to "either of the equal angles," Copernicus incorrectly includes the corresponding angles which are not right angles, as was pointed out by Clavius, Spherical Triangles, Opera mathematica, I, 179. P. 46:3. This Theorem was originally numbered 10 by means of the Arabic numeral, later deleted. Then the number was reduced to 8, as indicated by the Greek letter eta, also deleted. In the final stage this Theorem was called 7, as indicated by the Roman numeral VII and the Greek letter zeta. However, the correct Roman letter G was deleted, while H was allowed to remain in the left margin although the correspond-

365 NOTES ON PP. 46-47 ing Greek letter eta was deleted in the right margin. Could somebody have miscounted the letters in the Roman alphabet? P. 4:6:25. This Theorem is numbered 8 by means of the Roman numeral VIII as well as the Greek letter eta. Once more, as in the case of the preceding Theorem, the correct Roman letter H was deleted and replaced by I, perhaps as part of the miscounting suggested in the preceding note. P. 46:26. By permitting an ambiguity in this enunciation Copernicus left himself open to the following criticism by Nunes: This relation of the sides and angles of a triangle was not heeded enough [not only by Menelaus, but] also by Nicholas Copernicus of Torut1. The latter was principally concerned with the question how, by means of Ptolemy's method, epochs, and proofs, he might again bring to public attention the ancient and almost forgotten astronomy of Aristarchus of Samos with the earth in motion while the sun and eighth sphere are at rest, as mentioned by Archimedes in his work on the Sand­ reckoner. For in [Copernicus'] Re-oolutions, I, 14, which deals with Spherical Triangles, Theorem VIII reads as follows: "If two triangles have two sides equal to the two corresponding sides, as well as an angle equal to an angle, whether it be the angle included by the equal sides, or an angle at the base, the base will also be equal to the base, and the remaining angles to the remaining angles." I shall show, however, by a simple proof that the last part is not true. For in the spherical triangle ABC, let the two sides AB and AC be equal. But we shall extend the base BC to D; how­ ever, let arc CD be less than a semicircle. Through the points A and D we shall draw AD as the arc of a great circle. In the two spherical triangles ABD and ACD, therefore, the two sides AB and AD of triangle ABD are equal to the two sides AC and AD of triangle ACD, while ADB is a common angle, being at the base of both triangles. Therefore the base BD of triangle ABD will be equal to the base CD of triangle ACD, according to Nicholas Copernicus' Theorem VIII, the part [BD] being equal to the whole [CD], an impossibility. The same absurdity follows with regard to the two angles BAD and CAD, for one is a part of the other. Also, angle DBA will always be unequal to angle DCA, unless sides AB and AC, which were assumed to be equal, are quadrants. Accordingly, let us make them smaller than quadrants, and therefore angle DCA will be acute, DBA obtuse, and ADB acute. Hence, what is enunciated in Theorem XI, that every triangle having two sides and an angle given becomes a triangle of given angles and sides is unsound (Rules and Instruments, ed. Basel, 1566, pp. 104-105). A D{_ -.... ____.c B Nunes' criticism of I, 14, VIII, was repeated by Clavius, who said: This [theorem] is not true, unless it is specified that each of the remaining angles at the base is either larger or smaller than a right angle (Spherical Triangles, in Opera mathematica, I, 181). P. 4:7:5. This Theorem is numbered 9 by means of the Roman numeral IX as well as the Greek letter theta. By the miscounting already mentioned, the Roman letter K was placed in the right margin. However, in an earlier numeration this Theorem had been called 11, as indicated by the Arabic numeral, deleted in the right margin. P. 47:20. This Theorem is numbered 10 by means of the Roman numeral X. The correct Roman letter K in the left margin was deleted, while the wrong letter L was allowed to remain in the right margin. Deleted in that margin is the Arabic numeral12, indicating the last Theorem originally written by Copernicus, as is shown by the last two (deleted) lines on fol. 26r. P. 47:34. This Theorem is numbered 11 by means of the Roman numeral XI as well as the Greek letters iota alpha (deleted in the left margin). Also deleted in the left margin is the correct Roman letter L, whereas in the right margin the wrong letter M was allowed to remain. In an earlier numeration, the general enuncia­ tion "Every triangle having two sides and an angle given becomes a triangle of given angles and sides" was numbered 6 by means of the Arabic numeral, later deleted. Similarly, the Arabic numeral 7 had been assigned to the special case, inserted in the right margin, wherein the given sides are equal.

366 NOTBS ON PP. 47-51

P. 47:35. Nunes' criticism of I, 14, XI, was amplified by Clavius as follows: For even if sides AD and AB, as well as angle D, are known, nevertheless we shall not proceed therefrom to a determination of the other side and the other angles. For, the remaining side may be either DB or DC, and so on. Therefore, something else in addition must be known before the remaining side as well as the remaining angles become known (Spherical Triangles, in Opera mathematica, I, 181). P. 48:19. This Theorem is numbered 12 by means of the Roman numeral XII as well as by the Greek letters iota beta in the right margin, although in the left margin the Arabic numeral 12 was deleted. Once more, the correct Roman letter M was deleted to make way for the wrong letter N. P. 48:20. In his Rules and Instruments (ed. Basel, 1566, p. 105) Nunes commented as follows: Copernicus made no less an error in Theorem XII, which reads as follows: "Furthermore, if any two angles and a side are given, the same results will follow" [that is, all the angles and sides will be known]. For, construct a spherical triangle BCG, in which two sides BC and CG, taken together, are equal to a semicircle. Prolong side BG [reading latere instead of latera] to A, and draw a great circle through A and C. In triangle ABC let two angles CAB and CBA be known, together with side AC, which is opposite angle CBA. Through what is assumed to be known, the remaining angle [ACB] and the remaining sides [AB, BC] will not yet -be known. For since the two sides CB and CG, taken together, are equal to a semicircle, therefore angle ABC will be equal to angle BGC. Accordingly, also in triangle ACG, two angles CAG and AGC are assumed to be known, and side AC, opposite angle AGC, is assumed to be known. For in both triangles ABC and AGC, there are the same assumptions. Hence, from what is assumed to be known, it cannot yet be determined whether the remaining angle, which was not known, is ACB or ACG [not ABG, as in the printed edition], and whether the remaining sides which were not known are CB and AB or CG and AG. c

A G B

Nunes' criticism of I, 14, XII, was repeated by Clavius, Spherical Triangles, Opera mathematica, I, 179 and 180 (in the latter case, two sides are given as well as two angles). P. 48 :44. This Theorem is numbered 13 by means of the Roman numeral XIII. Again, the wrong letter 0 stands in the right margin. The last three theorems, XIII-XV, were written on paper F, the last batch of paper used by Copernicus. Folios 24 and 25 were inserted by him in quire c after the arrival of Rheticus with a copy of Regiomontanus' treatise On all Kinds of Triangles (English translation, Madison, 1967). The study of this book persuaded Copernicus to delete on fol. 22v his original draft of Theorem 13, then numbered 8. When he expanded this draft, the designation 13 was permitted to remain on fol. 22v, right margin, in the form of the Greek letters iota gamma (or perhaps Roman "c" in place of its equivalent Greek gamma). P. 49:33. This Theorem is numbered 14 by means of the Roman numeral Xliii. Its position on fol. 25r-v clearly indicates that it was intended to be the last of the three added Theorems. In the left margin of fol. 251' there is the Roman letter f, the significance of which is unclear. P. 50:14. This Theorem is numbered 15 by means of the Roman numeral XV. In the left margin of fol. 24v the Roman letter f is deleted, and the letter g remains; the significance of both letters has not been satis­ factorily explained. P. 50:42. In the absence of any indication on fol. 2SV that Copernicus brought Book I to an end there, that indication was supplied by N, which was followed by the later editions. P. 51:11. This is Aristotle's definition of time (Physics, IV, 11). P. 51:17. In saying that he takes "the opposite position," Copernicus rejects the Ptolemaic teaching that the earth is motionless. Nevertheless it is maintained by Fred Hoyle, Nicolaus Copernicus (London, 1973), p. 79, that "Today we cannot say that the Copernican theory is 'right' and the Ptolemaic theory 'wrong' in any meaningful sense. The two theories ... are physically equivalent to one another." But the two theories are ob­ viously not physically equivalent, since the Copernican theory has the earth perform the daily rotation, which

367 NOTES ON PP. 51-53

produces the flattening of the earth at its poles, whereas the motionless Ptolemaic earth would be a sphere, not an oblate spheroid. P. 51:19. Copernicus' forthright declaration of his intention to continue to use the everyday language of his geostatic predecessors was overlooked by some of his critics, who accused him of wavering in loyalty to his own professed principles. However, wherever his moving planet earth had to be contrasted with the pre­ Copernican stationary earth, he emphasized that issue quite bluntly. On the other hand, wherever this contrast was of minimal importance, he felt no hesitation in falling back on the traditional terminology. In II, 1, for example, he says "the sun reaches the meridian." Such language is of course impermissible in the Copernican astronomy, strictly interpreted, since the Copernican sun is absolutely motionless. But for the purposes of Book II in general, the customary phraseology is convenient. It would have been far more cumbersome to have to say that "the imaginary great circle through the zenith of a place and the poles of the celestial equator, in the course of the earth's daily rotation, passes directly through the center of the sun." P. 51:23. Copernicus does not indicate who wrote this couplet. Yet these two dactylic hexameters are clearly modeled after the line quoted above (1, 8) from Vergil's Aeneid. Both there and here recedunt is used to terminate a hexameter. Moreover, Vergil's Provehimur is partially echoed here in 'IJehimur, which implies Copernicus' authorship of this couplet. To be sure, the ancient Roman and Renaissance Latin poets were quite fond of refer­ ring to the passage of sun and moon, and the recurring comings and goings of the stars. Yet all of these writers without exception were committed to the view that the earth is motionless, and therefore none of them would have described us humans as being "borne (fJehimur) by the earth." No wonder that the learned investigators who patiently searched for this couplet in the poetical works of antiquity and the Renaissance have failed to find it. These scholars hesitated to identify Copernicus as the author of this distich because he wrote no other poetry. Yet his characteristic reluctance to put his own name forward prominently should be recalled when we seek to understand why he failed to identify the author of this couplet, as he cited Vergil by name in I, 8. Incidentally, when this distich was quoted by Baldi (Bilinski, Vita, p. 22, line 66), for cices in Copernicus' second line, Baldi mistakenly substituted obitus, a term which he encountered much more frequently in the Latin poets whom he read in preparation for writing his own Latin poetry. But Baldi's obitus (settings) is far less suitable as the subject of recedunt (drop out of sight) than is Copernicus' cices (rounds, vicissitudes, alternations): P. 51:40. In Mastlin's copy of N (fol. 28r, line 13, right margin).another hand supplied quae oriuntur, et occidere as the sort of words which Copernicus somehow omitted (fol. 26v, last line). P. 52:10. "Antarctic" occurs in Proclus' Sphaera, the Greek text of which was published together with a translation into Latin at Venice in 1499. Proclus' brief introduction to spherical astronomy formed the last section of a composite volume which was owned by Copernicus. His reference to other Greeks was probably based on the Opinions of the Philosophers, III, 14, and on the woodcut in P-R (sig. a3v). P. 52:22. Why did Copernicus (fol. 27r, lines 19-20) at first mention Eratosthenes' and Posidonius' con­ nection with the size of the earth, and then later delete their names? Our best source for their estimates is Cleo­ medes, who was translated into Latin by Giorgio Valla (Venice, 1498). Valla's volume was published on 30 September 1498, when Copernicus was working closely with the professor of astronomy in nearby Bologna. Cleomedes (1, 10) treats in considerable detail Posidonius' and Eratosthenes' efforts to estimate the size of the earth. In I, 2 (opening sentence) Cleomedes says: "Five parallel circles are drawn in the heavens". Shortly there­ after he adds: "Beneath these five sections of the heavens, which are marked off by the aforementioned circles, lie five parts of the earth." However, Cleomedes does not regard the terrestrial circles as primary, and as pro­ jected imaginatively into the heavens. This was Copernicus' view. Perhaps he realized, on later reflection, that his view was not present in Cleomedes, and in any case was not linked with Eratosthenes and Posidonius. These second thoughts may have persuaded Copernicus to delete those names. Valla's translation displays no genuine title. His title page begins: Giorgio Valla Placentino lnterprete, and then lists the Greek works translated by him, beginning with Nicephorus' Logic and including Cleomedes (sig, h¥-lY). P. 52:38. Hence the instrument itself is called a "quadrant." Copernicus' description of its construction is based on PS, I, 12, second device. P. 53:13. In view of Copernicus' awareness of the desirability of the maximum precision in this measure­ ment, it is significant that he follows Ptolemy in putting an unspecified object on the graduated quadrant to mark the shadow and its midpoint. Copernicus must have known and declined to adopt the alternative device described and illustrated in P-R, I, 17. To the center of that quadrant there was attached a movable pointer bearing two peepholes through which sunlight passed, instead of being forced to cast a shadow by being inter­ cepted by an opaque pin or cylinder. P. 53:23. In the autograph (fol. 27V, line 16 up) by a slip Copernicus wrote 23° 52' 20", which was corrected in N (fol. 2gr). P. 53:28. Three such contemporaries or near-contemporaries are named below in III, 6: Peurbach (who

368 NOTES ON PP. 53-70 died a dozen years before Copernicus was born); Peurbach's pupil, Regiomontanus; and Domenico Maria No­ vara, Copernicus' teacher (note to p. 129 :44). P. 53:32. Hence, when Copernicus wrote II, 2, he had already made up his mind about the upper and lower limits between which the obliquity of the ecliptic oscillated, according to his (mistaken) theory. P. 54:22. According to Lines Subtended, for 23° 30': 39875; for 23° 20': 39608; as partial difference, hence, for 23° 28': 39822, instead of which Copernicus by a slip wrote 3822 (fol. 28r, line 18 up). When he noticed this error, he did not insert the missing 9, but instead placed a dot below the 8 to call attention to the omission. This did not affect his calculation, since just below (line 15 up) he gives 19911 (corrected from 19905) as half of the number in question. P. 54:24. According to Lines Subtended, 19937 for 11° 30'; 19652 for 11° 20'; hence, 19,909 ~ 19,911 for 11° 29' (19905, according to fol. zsr, line 15 up). P. 54:32. In the autograph (fol. 28r, line 7 up) Copernicus by a slip put AF = 64° 30'. But he has assumed .t;.GEH = 23° 28'; hence, AF = BF-BA = 90°-23° 28' = 66° 32', the correct value which he uses just below at fol. 28v, line 2. His error was corrected in N (fol. 3or). P. 54:37. In the extant autograph this has not been said previously by Copernicus. Perhaps he did so in a preliminary draft that was not incorporated in the autograph. P. 54:45. In the autograph (fol. 28v, line 8, and top drawing in the left margin) Copernicus incorrectly used the letter K for the south pole as well as for the north pole. The mistake in the text was corrected from arc KMG to HGM on fol. 30r in N, whose correction sheet called attention to the mislabeling in the diagram. P. 55:6. By a slip of the pen Copernicus wrote DB in the autograph (fol. 28v, line 16), instead of DC, as in his third figure in the left margin of fol. 28v. This error was corrected in the 1964 Russian edition, p. 77. P. 55:29. After "I did not mind adding it too," Copernicus wrote an unimportant sentence which he later decided to delete (fol. 28v, lines 5-4 up). P. 55:30. After "the same results will be clear in any other obliquity of the ecliptic," Copernicus originally deleted the remainder of II, 3 (fol. 28v, 30V). Later he cancelled the deletion by writing at the foot of fol. 28v: "This material, up to the next chapter, should not be deleted." P. 61 :4. Euclid's introduction to astronomy was available to Copernicus in Zamberti's translation of 1505. P. 61:11. The latitude of a place on the earth may be defined by its angular distance from the equator, in the modern manner, or, in the ancient manner, by the ratio of the year's longest day to its shortest day at the place in question. In the northern hemisphere an ancient zone was bounded by two parallels of latitude between which the length of the longest day increased by !h, or by !h near 60° north latitude (PS, II, 6). In actual prac­ tice, however, usually only seven "climes" were recognized. Copernicus evidently took his list of places from GV, XVI, 1 (sig. bb 4v). P. 61:22. Copernicus' insistence that "the latitudes of the places ... agree with the recorded ancient obser­ vations" implies his rejection of the thesis put forward by his teacher at Bologna, Domenico Maria Novara (whose name he courteously refrains from mentioning here), to the effect that Mediterranean latitudes had increased by 1° 10' since the time of Ptolemy. P. 62 :30. By a slip of the pen in the autograph (fol. 32v, line 13 up) Copernicus forgot to equate EH with half the difference between the longest day and the equinoctial day, as he had done seven lines above. P. 63:21. Posidonius' definition of parallel lines is preserved in Proclus' Commentary on the First Book of Euclid's Elements (tr. Glenn R. Morrow, Princeton, 1970, p. 138). But at this stage of the composition of the Re-oolutions Copernicus must have known about Posidonius' definition from some other source, since he acquired his copy of Proclus as a gift from Rheticus in 1539 (PJZ, p. 407). P. 64:10. Under the heading "The Rising and [Setting] of the [Zodiacal] Signs and Degrees of the Ecliptic as well as of the Stars" Copernicus originally started a new chapter, and then he deleted these 19 lines (fol. 33v). They may have continued on a folio which he removed, since this quire has been reduced from the normal quinternion to a quaternion (NCCW, I, 11-12). These 19 deleted lines (and perhaps also their continuation) were not discarded by Copernicus, but rather postponed. For after the Tables on fol. 34r-36•, followed by II, 8, on fol. 36v, Copernicus substantially repeated the deleted material under a modified title in II, 9, on fol. 36V-37r. P. 70:20. The general adoption of equal hours, displacing seasonal hours, was undoubtedly hastened by the introduction and spread of the mechanical clock. Driven at first by the regulated drop of a descending weight, this ingenious device measures "equal hours, common to daytime and nighttime." Copernicus is understandably rather vague regarding the period when equal hours were generally adopted, since he had no precise knowledge about the origin and growing popular acceptance of this extremely influential invention. Even now our zealous

369 NOTES ON PP. 70-78 and talented historians of medieval technology are still uncertain regarding the date of the construction of the first weight-driven clock. The earliest mechanisms utilizing the steady downward pull of a weight for measuring the passage of time are conveniently explained by Carlo M. Cipolla, Clocks and Culture 1300-1700 (London: Collins, 1967), pp. 111-114. P. 70:27. At the end of II, 8 (fol. 36v), Copernicus started to write what is now the beginning of II, 10. But after he had put down its title and first four lines, he suddenly remembered that he had to insert the material he had dropped at the end of II, 7. This he proceeded to do (fol. 36v-37~, as was indicated just above, and then at the close of what is now II, 9, he repeated with slight modifications on fol. 37r the title and opening lines of II, 10, which he had previously deleted on fol. 36v, so that we have a postponement or rearrangement rather than a deletion. P. 71:14. By a slip of the pen (fol. 37•, line 15 up) Copernicus wrote "AFH, the angle of the obliquity AHF." N (fol. 39•) correctly deleted "AFH." P. 71:46. Copernicus forgot to say that the meridian altitude AB is also given, although that is an indis­ pensable part of his argument. His omission was repaired by N, which correctly inserted cum AB altitudine meridiana (fol. 39v). P. 72:7. Copernicus' instruction to "complete the quadrants EAG and EBH" (fol. 37v, line 17 up) is incompatible with the accompanying diagram in the left margin of fol. 37v. This merely reproduces the situa­ tion in the diagram on fol. 28•, as was recognized by N, which supplied a diagram appropriate for II, 10 (fol. 39V), P. 73. This "Table of the Ascensions of the [Zodiacal] Signs in the Revolution of the Right Sphere" is an expansion of the "Table of Right Ascensions" after II, 3. That earlier Table was limited to the first three signs, but gave the right ascension for every degree in the first quadrant. Here, on the other hand, Copernicus proceeds by jumps of 6° for all twelve signs. Immediately thereafter, on fol. 38v, Copernicus started another table, which he did not finish and cancelled by drawing a diagonal line through it. He drew this line in red ink, which he ordinarily used for the rubrics of his tables. In this instance he deleted the table before writing the rubrics. The vertical framework of this un­ finished table for the oblique sphere consists of 7 columns, for north latitudes 39° to 57° at intervals of 3°, Ten horizontal columns each contain 3 entries, the bottom entry for every latitude being 30° of the Ram, whose symbol appears above each of the 7 vertical columns. After recording the computations for latitudes 39° to 48°, Copernicus abandoned the table halfway through the first entry for latitude 51°, with none of the ascensions entered in the vertical column at the extreme left, which remained completely empty. P. 76. In Mastlin's copy of N (fol. 42•), the Table of the Angles Formed by the Ecliptic with the Horizon has been extended in the left margin to north latitudes 31° and 36°, while some of the printed values have been altered. P. 78:6. Folio 46, like folio 47, consists of paper C, the only such sheet in quire e, which was otherwise made up of the later papers D and E (NCCW, I, 7, 12). P. 78:9. This is the only place in the autograph in which Copernicus used Greek letters to designate points in a diagram. P. 78:11. By a slip Copernicus refers to Theorem Von Spherical Triangles, which appears on fol. 22• and deals with a right triangle. Just below it, however, is the Theorem which Copernicus intended to cite, since it discusses his case of a given angle included between two given sides. P. 78:14. The word here translated as "intercept" is schoenus, which Copernicus used instead of "sine" (see note to p. 31 :12). In Pliny's Natural History (VI, 30, 124; XII, 30, 53) schoenus occurs as a Persian (and Greek) measure of distance. P. 78:17. It was these "others," presumably post-Ptolemaic authors, from whom Copernicus excerpted the sine (schcenus). These "others" are eliminated, together with their newfangled sine, in the final version of II, 12, which Copernicus wrote on fol. 41r, made of the later paper E. P. 78:20. The title of II, 13, repeats the heading of GV, XV, 3. GV, at sig. cc 7•, line 13, explains that he has been expounding mainly Autolycus' On Risings and Settings, one of the earliest Greek astronomical works to survive intact. Since this Greek text was not printed until 1885, GV had to work from a manuscript. That Greek manuscript of Autolycus formed part of GV's extensive private collection and, still bearing his ownership mark, is now in Barbarinianus graecus 186 in the Vatican Library (Giovanni Mercati, Codici Iatini ... e i codici greci Pio. Studi e testi 75, Citta del Vaticano, 1938, p. 204, n. 3). P. 78:28. In referring to the "ancient mathematicians" in general, Copernicus does not mention Autolycus by name. Nor, for that matter, does he ever mention GV. P. 78:30. Instead of merely repeating GV here, Copernicus introduces two stylistic improvements. First, where GV used cum twice, in two different senses (cum una cum sole astrum oritur}, Copernicus replaces the first

370 NOTES ON PP. 7S-83 cum by quando. Secondly, GV's oritur, echoing ortus, becomes emergit in Copernicus. These alterations are commended to the consideration of those who mistakenly suppose that Copernicus was insensitive to literary niceties. P. 78:30. GV's definition of a star's true morning setting is defective, since he has this event occur "when the star sets equally with the sun" (cum pariter cum sole astrum occidit; XV, 3, line 3). But of course the sun does not set in the morning. GV forgot to match Autolycus' "rising" sun, as Copernicus realized full well. Hence he replaced GV's pariter cum sole by oriente sole. Copernicus' improvement of GV's Latin style and his correc­ tion of GV's defective definition show that at some places in II, 13, he does not merely follow Autolycus word for word (despite Studia Copernicana, IV, 693). P. 78:36. See D. R. Dicks, Early Greek Astronomy to Aristotle (London: Thames and Hudson, 1970), p. 13. P. 79:8. Throughout its apparent yearly revolution from west to east against the background of the zo­ diacal signs, the sun always moves faster than the outer planets, Mars, Jupiter, and Saturn. Hence, the relative position of the sun and an outer planet changes in much the same way as the relative position of the sun and the starS, so that the phenomena of the first rising and the first setting are similar for the outer planets and the stars. For the inner planets, Mercury and Venus, however, the situation is entirely different. Between the time of their greatest western elongation and greatest eastern elongation, these two planets overtake the sun by travelling eastward in the firmament faster than the sun. For that reason their apparent evening rising occurs later than the true evening rising at the time of their superior conjunction with the sun. Like Ptolemy, Copernicus uses the terms "evening rising" and "morning setting" with respect to the inner planets to identify their first visibil­ ity in the evening sky, and their last visibility in the morning sky. P. 79:21. These values for the first visibility of these five planets as a function of the angle of the sun's de­ scent below the horizon were taken by Copernicus from PS, XIII, 7. P. 80:4. In keeping with his usual practice at the start of a new Book, Copernicus left space for a large ornamental initial letter on fol. 46v, 1. 1-4. He did the same for the definitive version of II, 14, on fol. 42r, 1. 9-13. P. 80:6. This earlier draft was written on paper C, whereas the definitive version on fol. 42r-4¥ was written on the later papers D and E (NCCW, I, 7). Moreover, in the earlier draft (fol. 4tr, lines 13 down and 6 up) Copernicus used the Arabic numerals 2! and 360, whereas in the definitive version he switched to the corre­ sponding Roman numerals (fol. 42v, line 15; fol. 4Y, line 11). In general, in the final form of the RBfJolutions Copernicus confined the use of Arabic numerals to his Tables and the related computations. P. 80:23. PS, VII, 3, reports two observations in Rome by Menelaus in which stars were located as being in conjunction with the moon. P. 81:7. For Ptolemy, the fundamental fact was the length of the solar year, and since the motion of precession affected the position of the stars, he postponed his discussion of them toPS, Books VII-VIII. For Copernicus, on the other hand, the stars were motionless in space. Hence he placed his Star Catalog in II, 14, ahead of his discussion of the sun's (apparent) motions (Book III). At this place in the autograph (fol. 42v, line 5) Copernicus wrote the caption "Observing the Place of the Sun by the Use of Instruments," but later deleted it. P. 81 :18. At this place in the autograph (fol. 42v, line 17) Copernicus wrote the caption "Locating the Moon and the Stars by the Same Method," but later deleted it. P. 83:19. Sun at Fishes 30°-3° 2' 30" = 26° 57' 30" Ram 30° Bull 30° = 5° 10'

92° 7' 30" = 921/8 °. P. 83:35. With the moon at 5° 24' within the Twins, and Basiliscus' distance from the moon= 571f10°, Twins 24° 36' Crab 30° Basiliscus at Lion 2° 30' 57° 6'. P. 83:37. Basiliscus' distance from summer solstice: Crab 30 Lion 2° 30' 32° 30'. P. 83:40. Here Copernicus dates Ptolemy's observation on 24 February 139 A.D. In his Letter Against Werner, however, he made the day 22, not 24. There he merely repeated Werner's 22 February, because he was so intent on correcting the latter's gross error of eleven years in the date of this same observation that he ignored the minor discrepancy of two days (3CT, p. 97). Tycho Brahe, in his copy of B, which was reproduced at Prague in 1971, wrote in the right margin of fol. 46r: "Instead of the 24th day, we must read 23. For otherwise the

371 NOTES ON PP. 86-90

P. 86:51. In the Dragon, for star 20, which is described as being "Of the two small stars west of the triangle, the one to the east," Copernicus gave as the longitude 200°0' (fol. 53•, line 11). In this instance his source was apparently PS 1515 (fol. 78v, line 21). Using 3 columns (zodiacal sign, degrees, minutes), PS 1515 reported for this star "3 26 40" ( = 116°40'). From this value Copernicus had to subtract 6°40'. Instead of this mental arithmetic (116°40'-6°40' = 110°0'), Copernicus operated with the so-called "physical" sign ( = 60°) and thereby arrived at the strange result 200°0'. Mastlin in his copy of N (fol. 47v), noted that SchOner, Reinhold, and the Alfonsine Tables all agreed in assigning to this star the longitude of 110°0', as Copernicus also should have done. He was evidently aware that something was wrong here, since in the right margin of fol. 53r he put a mark alongside this line to call attention to the need for a correction. Elsewhere (fol. 88r, 114r, 143r, 153r, 173v), when he used this same mark ( +) in the margins, he carried out the required emendation. Here, however, he failed to introduce any modification. This neglect may be taken as a measure of the smaller importance he attached to his Star Catalog as compared with the rest of the Revolutions. P. 87:5. Here again, as in the case ofthe preceding star (20 in the Dragon) and presumably through the same error, Copernicus emerged with longitude 195°0' (fol. 53r, line 12). His source (PS 1515, fol. 78v, line 22) gave 3 21 40 (= 111°40'). PS 1515's value minus 6°40' = 105°0' would be valid for PS, SchOner, Reinhold, and the Alfonsine Tables, as Mastlin noted in his copy of N (fol. 47v). P. 87:18. To this star described as being "to the east, on the tail," Copernicus assigned longitude 192°30' (fol. 53r, line 21). His source (PS 1515, fol. 78v, line 31) gave "3 19 10" ( = 109°10'). By subtracting 6°40', Copernicus should have arrived at 102°30'. But again he mistakenly used the "physical" sign = 60° in reaching his false result 192°30'. P.87:19. Once more, in the case of the last star in the Dragon, Copernicus' longitude is too large by 90° (fol. 53r, line 22). He was in excess by 3 signs in all 4 instances of stars in the Dragon for which PS 1515's first component of the longitude was 3 signs. P. 87:21. For the first three constellations Copernicus used the standard designations, which he could have found in PS 1515 (fol. 78r_v). For the fourth constellation, however, he did not follow PS 1515 ("Cheichius"), but adopted "Cepheus" instead. In general, Copernicus did not derive his Star Catalog exclusively from PS 1515 but utilized other sources as well, particularly GV. P. 88:43. As the longitude of star 5 in the Kneeler, Copernicus wrote at first 190°0', but he later corrected the number of degrees to 220 (fol. 54r, line 11). Presumably he was using 7 16 40 ( =226°40') in PS 1515 (fol. 79r, line 22 up). In subtracting 6°40', he arrived at the partial result 10°0', which he wrote down in the appro­ priate columns. But, in shifting to the circumferential notation, he must have looked at the preceding line in PS 1515, which has 6 in the column for the number of signs. This gave Copernicus the sum 190. But, noticing his error in this instance, he deleted 190 and replaced it by 220. P. 89:13. In the Kneeler, for star 20, described as "the one to the west, of the three stars in the left foot," Copernicus gave as the longitude 188°40' (fol. 54r, line 8 up). Here again, without any question, he relied on PS 1515. For in the column for the zodiacal signs, by a typographical error PS 1515 had 6 instead of 7 (fol. 79r 7 up). From PS 1515's value 6 15 20 ( = 195°20'), Copernicus subtracted 6°40', and thus arrived at his 188°40'. But PS's actual value was 7 15 20 = 225°20'. Had Copernicus been familiar with this correct figure from any source other than PS 1515, he would have arrived at the result 218°40', in agreement with Ptolemy, the Alfonsine Tables, and Schoner, as was noted by Mastlin in his copy of N (fol. 48v, line 17). Here we may recall (as was mentioned, note on p. 4 :15) that Copernicus received from Rheticus a copy of the first edition of the Greek text of Ptolemy's Syntaxis in 1539, the year after it was published in Basel. Had Copernicus consulted its Star Catalog, he would have seen that it put Kneeler 20 at 15 20 within the Scorpion(= 225°20'; p. 176, misnumbered 174, line 9). Evidently Copernicus did not use PS 1538 to verify his own Star Catalog. P. 90:11. In the Swan, to star 9, described as "the last of the three, at the tip of the wing," Copernicus assigned longitude 310°0' (fol. 54v, line 7 up). Here again he followed as his source PS 1515's 10 16 40 (fol. 79v, 9th constellation, line 9) = 316°40'. From this value Copernicus subtracted 6°40' and arrived at his result 310°0'. However, in this constellation of the Swan, in the column for the sign, PS 1515 increased the number from 9 to 10 one line too soon. Hence Copernicus' longitude 310° is 30° too high, as was noted by Mastlin in his copy of N (fol. 49r, line 12), where he cited 280° as the equivalent value given by Ptolemy, the Alfonsine Tables, and Schoner. Had Copernicus examined his copy of PS 1538 (p. 176, misnumbered 174, line 2 up), he would have recognized that PS 1515 put Swan 9 in the wrong sign. P. 90:14. In the Swan, the star described by Copernicils as being "at the tip of the left wing" is assigned to celestial latitude 74°0' (fol. 54v, line 4 up). Here Copernicus unmistakably followed as his source PS 1515 (fol. 79v, 9th constellation, line 12). This gross northward displacement was changed by Kepler from 74° to 47° in his copy ofN, which was recently reprinted (New York/London: Johnson, 1965, fol. 49r, line 15). Not only did Kepler emend this latitude but he also numbered the stars in this constellation, the only one of the forty-eight

373 NOTES ON P'P. !10-92 constellations which he treated in this special manner. He did so because a previously unnoticed star had been observed in the Swan in 1600. When this exciting information reached Kepler, he took an active part in the lively debate over the question whether this third-magnitude star was or was not a true nova (it is now known to be a variable). Kepler published his account of this matter in his Astronomical Report on the Star of the Third Magnitude in the Swan Which Was Unknown unti/1600 and is Not Yet Extinct (Prague, 1606), which he inserted in his New Star (also Prague, 1606). Kepler's Report is now available in his Gesammelte Werke, I, 293-311. Had Copernicus looked up Swan 12's latitude in GV (sig. dd4r, line 25), he would have seen there PS's value 44, and then perhaps surmised that in PS 1515, 74 as Swan 12's latitude was typographically a dittography of Swan 9's 74. P. 90:39. For the longitude of the last star in Cassiopeia (fol. 55r), Copernicus originally wrote 27 in the zodiacal column. Then when he was engaged in the process of changing the number of degrees to the circum­ ferential basis, he forgot to do so in this instance, and by dittography he repeated 27. This was corrected to 357 by an intercolumnal note in Miistlin's copy of N, while in Brahe's copy of B 357 was written heavily over the printed number 27 (fol. 49r, line 2 up). Their attention was directed to Cassiopeia in 1572 by the sudden appear­ ance in that constellation of the immensely bright new star which became the subject of important discussions by both of them. P. 91:37. In PS 1515 (fol. 80r, line 15 up) the first-magnitude star Capella was misplaced in the wrong sign, so that once more Copernicus emerged with a longitude too great by 30° (78°20' instead of 48°20'; fol. 55v, line 9 up). This discrepancy was noted by Miistlin in his copy of N (fol. 5or, line 6). P. 92:5. As the longitude of the last star in the Reinsman, described as being "the small star in the left foot," Copernicus found in PS 1515, fol. 80r, line 4 up, "1 0 40" ( = 30°40'). From this value he subtracted 6°40' and emerged with 24° (fol. 56r, line 5). However, had he consulted his copy of PS 1538 for Reinsman 14 (p. 178, line 16 up), he would have seen that PS's longitude for this last star in this constellation was 50°40'. Thereupon he would surely have recognized that merely by a typographical error PS 1515 had omitted the first digit in the 2nd column, thereby reducing the longitude of Reiii.sman 14 by 20°. This discrepancy was noted by Miistlin in his copy of N (fol. 50r, line 18), where he attributed the corresponding longitude of 44 o to Ptolemy. P. 92:20. For the longitude of Serpent Carrier 9, described as being "in the right elbow," Copernicus found in PS 1515, fol. 80v, line 12, "7 26 40." From this value, at first he subtracted 6°40' and emerged with 20°0' within the appropriate sign. Later, when he was shifting to the circumferential notation, by dittography he repeat­ ed 20°, instead of adding 20° to 7 signs(= 210°). Hence, his final erroneous result was 220°0' (fol. 56r, line 16). This deficiency of 10° was noted by Miistlin in his copy ofN (fol. 5or, line 10 up), where he ascribed the corre­ sponding longitude of 230° to the Alfonsine Tables, Ptolemy, and SchOner. P. 92:21. As the longitude of Serpent Carrier 10, described as "the one to the west in the right hand," Copernicus found in PS 1515, fol. 80v, line 13, "7 2 20" = 212°20'. From this value, he subtracted 6°40' and emerged with 205°40' (fol. 56r, line 17). However, had he consulted his copy of PS 1538 (p. 178, line 4 up), he would have recognized that PS 1515located Serpent Carrier 10 in the wrong sign. Hence Copernicus' longitude for this star was too low by 30°. This deficiency was noted by Miistlin in his copy of N (fol. 5or, line 9 up), where he ascribed the corresponding longitude of 235°40' to Ptolemy, Schoner, and the Alfonsine Tables. P. 92:22. In the case of Serpent Carrier 11, PS 1515 (fol. 80v, line 14) repeated the error of 1 sign too few, discussed in the preceding Note with regard to Serpent Carrier 10. Here, however, PS 1515 made the additional error of adding 2°, instead of 1°, to the longitude of the preceding star, as Copernicus would have recognized instantly, had he compared lines 4 up and 3 up on p. 178 in his copy of PS 1538. This excess of 1o was found also in the Alfonsine Tables and Schoner, as was noted by Miistlin in his copy of N (fol. 50r, line 8 up). P. 92:30. As the longitude of Serpent Carrier 18, described as "touching the heel," in PS 1515 (fol. 80v, line 21) Copernicus found "7 26 10" = 236°10'. From this value he subtracted 6°40' and emerged with 229°30' (fol. 56v, line 11 up). However, had he consulted his copy of PS 1538 (p. 179, line 6), he would have seen that somehow PS's Greek symbol ~ for 7 had been reduced by 1. This deficiency was noted by Miistlin in his copy of N (fol. 5or, last line). GV (sig. dd5r, line 18) also has Scorpion 27 1/6 ( = 237°10'), from which Copernicus would have derived 230°30'. P. 92:49. As the longitude of Serpent 2, described as "touching the nostrils," in PS 1515 (fol. 80v, line 21 up) Copernicus found "6 27 40" = 207°40'. From this value he subtracted 6°40' and emerged with 201°0' (fol. 56v, line 8). However, had he consulted his copy of PS 1538 (p. 179, line 23), he would have seen that 7 had mistakenly replaced 1. This confusion, which was not very common when Greek numerals were used, may have occurred while the still somewhat unfamiliar numerals in the Arabic version of PS were being taken over into the Latin translation, or a copy thereof, underlying PS 1515. In any case, Copernicus' longitude for Serpent 2 exceeded by 6° the corresponding values in the Alfonsine Tables and SchOner, as was noted by Miistlin in his copy of N (fol. 5ov, line 18 up). GV gave Balance 21 1/2 ( = 201 °30'; sig. dd5r, line 13 up).

374 NOTES ON PP. 93-94

P. 93:9. As the longitude of Serpent 6, described as being "north of the head," in PS 1515 (fol. sov, line 17 up) Copernicus found "6 2S 10" = 20S0l0'. From this value he subtracted 6°40' and emerged with 201°30' (fol. 56v, line 12). However, had he consulted his copy ofPS 153S (p. 179, line 21 up), he would have seen that there the number of degrees was 23, not 28. Hence, for the longitude of Serpent 6, Copernicus exceeded PS 153S by 5°,as was noted by Miistlin in his copyofN (fol. sov, line 14up). GV also gave Balance 23 1/6 ( = 203°10'; sig. dd5'", line 9 up). P. 93:47. For the longitude of the 1st star outside the constellation of the Eagle, described as being "south of the head, the one to the west," in PS 1515 (fol. Slr, line 20) Copernicus found "9 S 40" = 278°40'. From this value he subtracted 6°40' and emerged with 272°0' (fol. 57r, line 12). However, had he consulted his copy of PS 1538 (p. 1S2, line 14), he would have seen that the number of degrees was 3, not S. Hence, in the longitude of this star Copernicus exceeded PS 1538 by 5°, as was noted by Miistlin in his copy ofN (fol. Sir, line 19 up). GV also gave the number of degrees as 3 (sig. ddSV, line 23 up). P. 94:30. As the longitude of Horse Segment 2, described as "the one to the east," in PS 1515 (fol. s1r, line 18 up), in addition to the number of signs, Copernicus found "2S 0." Performing the subtraction 2S00'- 6040', he emerged with 21 °20' (fol. 57r, line 2 up). But when he shifted to the circumferential notation, in adding the number of degrees to 9 signs = 270°, somehow he substituted 22 for 21 and reached a result 1° greater than the corresponding values of Reinhold, Ptolemy, the Alfonsine Tables, and Schaner, as was noted by Miistlin in his copy of N (fol. 51v, line 5). P. 94:35. As was pointed out in NCCW, I, 12, Copernicus shifted from the ancient to the modem method of indicating the celestial longitudes of the stars when he reached this constellation of the Winged Horse or Pegasus. In the traditional order of the 20 stars in Pegasus, the first 3 were in the sign of the Fishes, the 4th was near the end of the preceding sign of the Water Bearer, the 5th and 6th toward the beginning of the Fishes, and all the rest back again in the Water Bearer. This zigzag arrangement was evidently not disturbing to Ptolemy and his followers, who were satisfied to locate each star within a given zodiacal sign. But "the twelve zodiacal signs ... are derived from the equinoxes and solstices" and "have shifted a considerable distance away from those constellations of the fixed stars which originally agreed with them in name and position" (Copernicus, R8fJolu­ tions, II, 14; III, 1). Since, according to Copernicus, "the equinoxes and solstices seem to move forward" by contrast with his stars, which remain forever motionless, he counted all celest~ longitudes from the star to which he assigned the permanent and invariant longitude 0°0'. Hence, in the case of Pegasus, the Ptolemaic pattern of the 20 component stars became less suitable in Copernicus' eyes than an alternative arrangement which would list these stars more nearly in the order of increas­ ing celestial longitude. Accordingly, at the head of his Pegasus list Copernicus put the star with the lowest longitude in that constellation, namely, the traditional Pegasus 17. Copernicus took its description, not from PS 1515 (fol. s1v, line 7: Quae est in muscida), but from GV ( sig. dd6r, line 21: Quae in rictu). From its longitude (Water Bearer 51/3 = 305°20') Copernicus subtracted 6°40' and emerged with 29S 0 40' (fol. 51V, line 6). For his Pegasus 2, Copernicus chose GV's Pegasus 15 (Water Bearer 9 1/6 = 309°10'). From this value Copernicus subtracted 6°40' and emerged with 302°30' (fol. 57v, line 7). Then he must have noticed that, for the minutes, PS 1515 (fol. Slv, line 5) gave 20 rather than GV's 10. He therefore erased the 3 in the column for the minutes and wrote a 4 over it. For his Pegasus 5, Copernicus took GV's Pegasus 14. But he recognized that its entry for the degrees (Water Bearer 2 1/2 = 302°30') was typographically defective. Hence, from PS 1515 (fol. s1v, line 4) he accepted instead 10 20 30 ( = 320°30'). From this value he subtracted 6°40' and emerged with 313°50' (fol. 57V, line 10). For his Pegasus 6, Copernicus took GV's Pegasus 11 (Water Bearer IS 50 = 3lS050'[ -6°401 = 312°10'). Copernicus put a dot, however, over his second digit to indicate that he wrote the 1 over an earlier 2. This oblit­ erated 2 shows that Copernicus had previously accepted PS 1515's longitude (fol. Sir, line 3 up: 10 28 50= 328°50'). For his Pegasus S, Copernicus took GV's Pegasus 20 (Water Bearer 12 1/3 = 312°20'; 312°20'-6°40' = 305°40'). He wrote the 3rd digit very heavily over a previous entry. If the digit thus obliterated was a 6, its presence was probably due to nothing more than a miscalculation, which was quickly corrected. As the description of Pegasus S, Copernicus at first followed GV's description of his Pegasus 20: (Quae) in sinistra sura. Later he re­ placed sura by subfragine, writing bf heavily over the earlier ra (fol. 57v, line 13). For his Pegasus 9, Copernicus took GV's Pegasus 19, but not its longitude (Water Bearer 171/2 =- 317°30'; 317°30'-6°40' = 310°50'). Since Copernicus emerged with 311°0' (fol. 57v, line 14), he evidently adopted PS 1515's longitude (fol. Slv, line 9: 10 17 40 = 317°40'). For his Pegasus 10, Copernicus took GV's Pegasus IS but as in the preceding case, not its longitude (Water Bearer 231/2 = 323°30'; 323°30'-6°40' = 316°50'). Again Copernicus preferred PS 1515's longitude (fol. s1v, line 8: 10 23 40 = 323°40'), since he emerged with 317°0' (fol. 57V, line 15).

375 NOTES ON P. 94

For his Pegasus 17, Copernicus took GV's Pegasus 4. But he rejected its longitude (Fishes 261/2 = 356°30'), preferring PS 1515's 11 26 40 = 356°40' (fol. 81v, line 10 up). From PS 1515's 356°40' he subtracted 6°40' and emerged with 350°0'. In all the remaining cases, Copernicus used GV. Thus, his 3 was GV's 16; his 4, GV's 13; his 7, GV's 12; his 11, GV's 9; his 12, GV's 10; his 13, GV's 7; his 14, GV's 8; his 15, GV's 5; his 16, GV's 6; his 18, GV's 3; his 19, GV's 2; and his 20, GV's 1. For these 12 stars, he subtracted 6°40' from GV's longitude and emerged with the values which he entered on fol. 57v in the column for the longitude. Then, when he proceeded to record the latitudes of the stars in Pegasus, he must somehow have forgotten that he had rearranged the order within that constellation. Evidently he began by simply copying out the latitudes as he found them in GV. Thus, his initial latitude entry must have been 26°0', as in GV. Only after he reached his own Pegasus 12 did he realize that his first 11 latitude entries were wrong. At that point he turned back to his own Pegasus 1 ( = PS's Pegasus 17). But he took its latitude 21 °30' from PS 1515 rather than from GV's entry, where by an unusual typographical error a dot had replaced the 1 (2. 1/2). Over his original 2nd latitude entry (12°30') Copernicus wrote heavily 16°50' (for PS's Pegasus 15). His 3rd latitude entry had been 31 °0', which he deleted and replaced by 16°0' (for PS's Pegasus 16), while altering the magnitude from 2 to 4. In the case of his Pegasus 4, the magnitude had to be changed from 2 to 5 (for PS's Pegasus 13). Hence he could not write over the original figure, as in the preceding instance, but instead he had to strike out the 2 and substitute a 5 in the right margin. His 4th latitude had originally been 19°40', as in PS 1515 (rather than GV's 19 1/2). This figure had to be changed to 15°0' (as for PS's Pegasus 13). To do so, Coper­ nicus wrote a 5 heavily over the 9, while the erased 4 is still visible. For his Pegasus 5, in like manner Copernicus had to alter 25°30' to 16°0'. Over the original 2, he wrote a 1, surmounted by a dot, rewrote the 5 as 6, and erased the 3, leaving its trace easily visible. In the column for the magnitudes he deleted the 4 and wrote a 5 in the right margin. For his Pegasus 6, Copernicus again deleted the 4 in the column for the magnitude and wrote a 3 in the right margin (for PS's Pegasus 11). The original latitude (25°0') was conveniently reduced to 18°0' by drawing a line through the number for the degrees and writing the new number in the vacant column to the left. For his Pegasus 7 (PS's Pegasus 12) Copernicus wrote a 4 over the 3 in the column for the magnitude, while for the latitude he had to alter 35°0'. This he did by writing a 1 heavily over the previous digit and rewriting the 5 as an 8. Then he realized that he had taken the latitude of PS's Pegasus 11 instead of 12. Consequently he drew a line through the reconstructed 18 in the column for the degrees and wrote 19 in the vacant column to the left. Copernicus took his 8th latitude entry from PS 1515 (24°30'), not from GV (34 1/2). Then in shifting to the value for his own Pegasus 8 ( = PS's Pegasus 20), he again preferred PS 1515's 36°30' to GV's 36 1/2 1/3 ( = 36°50'). As a result no change was necessary in the column for the minutes, and he simply drew a line through the 24 and wrote 36 in the vacant column to the left. Copernicus' 9th latitude entry was 29°0', which had to be changed to 34°15' (for PS's Pegasus 19). In this instance he drew a line through the 29, wrote 34 in the vacant column to the left, inserted a 1, and wrote a 5 over the 0. Copernicus' lOth latitude entry was 29°30', which had to be changed to 41 °10' (for PS's Pegasus 18). Hence he drew a line through the 29, wrote 41 in the vacant column to the left, and in the column for the minutes he wrote a 1 heavily over the 3, with a dot above to call attention to the change. Copernicus' 11th latitude entry was 18°0', which had to be changed to 29°0' (for PS's Pegasus 9). Accordingly he drew a line through the 18 and wrote 29 in the vacant column to the left. While no change was required in the column for the minutes, the magnitude had to be modified, and this Copernicus did by writing a 4 heavily over the previous 3. It was here, in connection with his Pegasus 12, that Copernicus stopped taking over the latitude entries in PS's order, which was quite different from his own. Beginning with his own Pegasus 12, Copernicus was atten­ tive to the necessity of looking up the appropriate latitude in his two main sources, GV and PS 1515. Where they differed, as in the case of Copernicus' Pegasus 14 ( = PS's Pegasus 8), Copernicus followed PS 1515's 24°30' rather than GV's 34°30'. Again, for his Pegasus 17 (= PS's Pegasus 4), Copernicus followed PS 1515's 19°40' rather than GV's 19°30'. In one instance, Copernicus' Pegasus 15 (= PS's Pegasus 5), where both GV and PS 1515 gave 25°30', Copernicus wrote 25°40', probably inadvertently. Copernicus' rearrangement of the stars in Pegasus was noticed by Miistlin. In his copy of N (fol. 51v, left margin) alongside Copernicus' description of each star in Pegasus, Miistlin wrote the appropriate number accord­ ing to PS's order. Yet, despite his profound and minute study of the Reuolutions, Miistlin failed to discern Co­ pernicus' reason for reordering the stars in Pegasus, as he would surely have done had he had access to Coper­ nicus' autograph.

376 NOTES ON PP. 95-99

P. 95:50. The longitude in the last star in Andromeda differs by a sign in PS 1515 as compared with GV. In this instance Copernicus' adherence to the circumferential notation may have induced him to follow PS 1515's 0 11 40 ( = 11 °40'), from which he subtracted 6°40' and emerged with 5°0' (fol. 5'8r, line 17). But had he consulted his copy of PS 1538 (p. 183, line 10 up), he would have seen that it placed Andromeda 23 in the sign of the Fishes, as did GV (sig. dd6r, line 5 up). GV's longitude (Fishes 11 1/2 = 341 °30'), however, fell10' short ofPS's (341 °40'). Hence Copernicus would have emerged, not with 5°0', but with 33SOO', as was noted by Miisdin in his copy of N (fol. 52r, line 18). P. 97:7. The source of Copernicus' longitudes is unmistakably indicated by the case of Ram 1. Its longitude was given by PS 1515' (fol. 81v, line 10 up) as 0 6 40, from which Copernicus subtracted 6°40' to make Ram 1 the ''first of all" the stars at longitude 0°0' (fol. 5'8•, line 3 up). But according to GV (sig. dd6v, line 8), Ram 1's longitude was Ram 6 30 ( = 6°30'), and according to PS 1538 (p. 184, line 3), Ram 6 20 ( = 6°20'). Copernicus' decision to assign 0°0' celestial longitude to Ram 1 departed from Ptolemy's practice in the Syntaxis, but agreed with the procedure in his Handy Tables, where, however, he used the Litde King or Regulus as his zero point. In that work, which was later than the Syntaxis, the celestial longitude of a star was obtained by adding its longitude as recorded in the Catalog to the mean motion of Regulus (J .B.J. Delambre, Histoire de l'astronomie ancienne, II, 623; New York, 1965, reprint of the Paris 1817 edition). The Greek text cited here by Delambre is found in Heiberg's edition of Ptolemy's Minor Astronomical Works (Opera Astronomica Minora), Leipzig, 1907, II, 167. Furthermore, in his Hypotheses of the Planets (Heiberg, II, 80: 25-27), Ptolemy gave the longitude of the Litde King at the beginning of his Alexander era, and the same procedure was followed in the Canobic inscription (ed. Heiberg, II, 152: 3). P. 97:13. For the longitude of Ram 5, PS 1515's value exceeded GV's by 10'. Here too Copernicus chose to follow PS 1515's 16°30' because it conformed to a pattern of increasing longitude, rather than GV's 6 1/2 (sig. dd6v, line 12). Yet had Copernicus consulted his copy of PS 1538 (p. 184, line 6), he would have seen that it made Ram 5's longitude smaller than that of all but one of the preceding stars in the Ram. Hence, in his copy of N (fol. 52•, line 5' up), alongside Copernicus' longitude of 9°50' for Ram 5, Miisdin wrote "359°5'0', according to Ptolemy and Reinhold." P. 98:13. In the right margin of fol. 59• Copernicus placed ''Venus' apogee at 48°20'." PS (X, 1) located this apogee at 55°, from which value Copernicus obtained his own simply by subtracting 6°40' =the longitude in PS of Copernicus' zero point. This entry concerning the location of Venus' apogee does not appear in N, which was printed from the copy of Copernicus' autograph that was made by Rheticus before his departure from Frombork. Presumably an item of such importance as the place of a planetary apogee would not have been omitted by Rheticus from the copy which he was preparing for the printer, nor would the printer have excluded it from N. Hence, the Venus entry may have been written in the margin of fol. 59r by Copernicus after Rheticus' return to Wittenberg. If this rea­ soning is historically sound, it is equally applicable to the entries concerning the other planets on the margins of fol. 60•, 61r-v, as well as to the longitude change in the margin of fol. 59v, P. 98:14. For Bull20 and 21, PS 1515' (fol. 82•, lines 24-23 up) gave the same longitude (1 25 40 = 55°40'). From this value Copernicus subtracted 6°40' and emerged with 49° (fol. 59•, lines 8-9). In GV (sig. dd6v, lines 6-5 up), however, these longitudes differed by 10° (Bull 15 1/2, 25 1/2). In this instance Copernicus' copy of PS 1538 would not have helped him at all, since by a typographical oversight (p. 184, line 3 up) it omitted Bull 21's longitude, latitude, and magnitude. In his copy of N (fol. 52v, line 4 up) Miisdin noted that for Ptolemy, Sch6ner, and the Alfonsine Tables the corresponding longitude for Bull 20 would have been 39°0'. A northern latitude was assigned by PS 1515 to both Bull 20 and 21, whereas GV put Bull 20 in a southern latitude. In this respect Copernicus followed GV as well as in its magnitude 3 for Bull 21 (rather than PS 1515's 4). P. 99:11. For Twins 7, Copernicus as usual followed GV's description. In this instance, however, Coper­ nicus corrected a typographical error in GV (sig. dd7•, line 19 up), where instead of "the left (sinistro) shoulder of the eastern (sequentis) Twin," by dittography the printed text read "The eastern (sequenti) shoulder of the eastern (sequentis) Twin." By an additional anticipatory dittography GV assigned the longitude of Twins 8 also to Twins 7. For this reason Coperni<.us took the longitude of Twins 7 from PS 1515 (fol. 82v, line 14: 2 26 40 = 86°40'), from Which he SUbtracted 6°40' and emerged With 80°0' (fol. 59v, line 10), P. 99:13. For the longitude of Twins 9 there was a difference of 3° between GV (sig. dd7r, line 17 up: Twins 26 1/6 = 86°10') and PS 1515' (fol. 82v, line 16: 2 23 10 = 83°10'). As was his usual practice, Copernicus accepted PS 1515's numerical value 83°10', from which he subtracted 6°40' and emerged with 76°30' (fol. 59v, line 12). In his copy of N (fol. 5'3•, last line) Miisdin noted that for Ptolemy the corresponding value would have been 791/2°, In fact PS 1538 (p. 185, misnumbered 192, line 12 up) did give 261/6°, in agreement with GV's value, equivalent to 79 1/2° for Copernicus. Miisdin also noted that magnitude 3 inN (fol. 5'3r, last line) is wrong and should be 5', as indeed Copernicus intended (fol. 59v, line 12). GV's latitude 1/3°, instead of .3°, exemplifies

377 NOTES ON PP. 99-102 the frequent confusion of a fraction with the corresponding integer, the difference being indicated in the Greek numerical notation by the presence or absence of a superlinear stroke, like our sign for minutes. P. 99:14. To Copernicus' description of Twins 10 (fol. 59v, line 13), N (fol. 53v, line 4) added "bright­ er" (maior), which was repeated by B, A, W. P. 99:15. For Twins 11, Copernicus accepted GV's longitude, Twins 18 1/4 ( = 78°15'; sig. dd7•, line 15 up). From this value he subtracted 6°40' and emerged with 71 °35' (fol. 59v, line 14). Yet inN (fol. 53v, line 5), for the minutes, in addition to 1/2 = 30', 1/6 was printed = 10', making a total of 40' instead of 35'. In Kepler's copy of N, 1/6 was deleted and replaced by 1/12 ( = 5') in the left margin. In his copy of N, Miistlin also noted that for Ptolemy the corresponding value would be 71 1/2 1/12 = 71°35'. P. 99:16. For Twins 12, Copernicus apparently accepted PS 1515's longitude for Twins 11: 2 21 40 ( =81 • 40'), from which he subtracted 6°40' and emerged with 75°0' (fol. 59v, line 15). For he may have recognized that PS 1515 interchanged the longitudes of Twins 11 and 12. His clue may have been the latitudes. Here he followed GV's 2°30' and 0°30', perhaps inferring that PS 1515's latitude (2°30') for Twins 12 belongs with Twins 11. In his copy of N, Miistlin noted the interchange of Twins 11 and 12 in Schoner and the Alfonsine Tables. P. 99:21. For Twins 16, Copernicus took PS 1515's longitude 2 10 10 ( = 70°10'; fol. 82v, line 24). From this figure he subtracted 6°40' and emerged with 63°30' (fol. 59v, line 19). Later, however, in the left margin he wrote 63 36, and afterwards he deleted the 36 and replaced it by 20. This change in the minutes from 30 to 36 to 20 did not affect N, which printed 63 1/2 (fol. 53v, line 10). Was this change in longitude, like the planet­ ary apogees, inserted by Copernicus in the autograph after Rheticus' departure from Frombork? P. 99:36. For the last of the 7 stars outside the constellation of the Twins, GV seems to have no entries for the longitude, latitude, and magnitude (sig. dd7v, line 4). Copernicus recognized, however, that part of the lengthy description of No. 4 in this group had somehow become detached from the rest, and printed as though it denoted a separate star. In other words, In recta linea borea (sig. dd7v, line 2) may look to the unpracticed eye like an independent description, but Copernicus understood full well that this phrase actually belonged with No. 4 in the last line on sig. dd7•. By transferring this misplaced phrase to its proper position, Copernicus obtained the values apparently missing in GV for No. 7. But now he found a discrepancy of more than 5° in the longitudes given by GV (Crab 1/2 = 90°30') and PS 1515 (3 5 40 = 95°40'; fol. 82v, line 24 up). Had he followed GV exactly, he would have emerged with 83°50' ( = 90°30' -6°40') instead of 84°0' (fol. 59v, line 4 up). Had he instead chosen to follow PS 1515, he would have emerged with 89°0', as was noted by Miistlin in his copy ofN (fol 53v line 21). P. 99:44. For Crab 4, Copernicus found a discrepancy in the longitude as given by GV (sig. dd7v, line 13: Crab 13 = 103°0') and PS 1515 (fol. 82v, line 18 up: 3 10 20 = 100°20'). By following the latter (100°20'- 6°40') Copernicus emerged with 93°40' (fol. 60•, line 4). Had he followed GV (103°0'-6°40'), he would have emerged with 96°20', as was noted by Mastlin in his copy of N (fol. 53v, line 10 up). P. 100:20. On fol. 60• Copernicus started to write the position of Mars' apogee in the right margin at the top. Then on second thought he decided to put the entry lower down in the margin, so that it would appear alongside the neighboring stars in the Lion. What he actually wrote was "Mars' apogee at 109°50'." But this should be 108°50' (as in V, 15, on fol. 165v, left margin). For, PS (X, 7) put Mars' apogee at 115°30', and 115°30'- 6°40' = 108°50'. P. 100:28. For the longitude of Lion 12, Copernicus found a gross discrepancy between GV (Crab 1/6 = 90°10'; sig. dd7v, line 13 up) and PS 1515 (3 24 10 = 114°10'; fol. 83•, line 11). As a careful student, Coper­ nicus may have realized that the white space in GV between the name of the constellation and the fraction 1/6 was exactly wide enough for a number of degrees in 2 digits, and he may therefore have surmised that 24 had somehow dropped out of GV. With this number 24 uppermost in his mind, he performed the usual subtraction of 6°40', and emerged with 117°30' (fol. 60•, line 6 up). This result shows that his minuend was 124°10' instead ofPS 1515's 114°10'. The correct value, 107°30', was attributed to Ptolemy, Reinhold, Schoner, and the Alfonsine Tables by Miistlin in his copy of N (fol. 54•, Lion, line 12). P. 101:35. In the autograph (fol. 61•, right margin) Copernicus placed "Jupiter's apogee at 154°20'." This is 6°40' less than PS' value of 161 o (XI, 1). P. 101 :51. In the autograph (fol. 61 •, right margin) Copernicus placed Mercury's apogee at a longitude which he rewrote. At first he simply subtracted 6°40' from PS' preliminary value of 186° and thus put Mercury's apogee at 179°20'. However, after continuing to read to the end of PS' discussion (IX, 7), which advanced Mercury's apogee by 4°, Copernicus transformed the 7 into an 8 and deleted the 9, over which he wrote a 3, so that his longitude for Mercury's apogee once more is 6°40' less than PS' true value of 190°. P. 102:21. For the longitude of Claws 4, Copernicus found a considerable discrepancy between PS 1515 (fol. 83v, line 16 up: 6 27 40 = 207°40') and GV (sig. dd8v, line 11: Balance 17 1/2 = 197°30'). Apparently he began to subtract 6°40' from the value in PS 1515 since he wrote a 2 in the place for the hundreds in the

378 NOTES ON PP. 102-110 column for the degrees (fol. 61', line 3 up). Then he shifted to GV's value, for he put a dot over the 2, which he proceeded to cover by a heavy 1. However, in subtracting 6°40' from GV's 197°30', he emerged with 191 °0' instead of 190°50'. P. 103:22. In the autograph (fol. 61v, left margin), Copernicus placed "Saturn's apogee at 226°30' ." Presum­ ably he intended to subtract 6°40' from PS' value of 233° (XI, 5) but missed by 10'. P. 104:23. The longitudes of these two stars were changed by Copernicus (fol. 62v, lines 7-8). But the readings printed in NCCW, vol. II, Latin version, at p. 101, lines 7-8, assign to these two stars the same longitude, latitude, and magnitude, so that they would be identical or indistinguishable. Copernicus, however, clearly says that the first one is to the west, and the second one to the east. Moreover, according to the critical apparatus at the foot of p. 101 in the Latin version, the reading in line 8 was corrected by Copernicus from 261 to 262. Yet the text in line 8 reads 261, not 262. The number of degrees should indeed be 261 for both these stars. But for the western one, Copernicus partially erased the first digit in the minutes column, thereby indicating that his final preference for these two longitudes was 261 °0' and 261° 10'. P. 105:6. Although Copernicus assigned to this star the latitude 0°0', he inadvertently labeled its latitude "southern" (fol. 62v, line 4 up). He was more alert in the similar case (p. 100, line 27), where tllis column is left blank, although Copernicus himself wrote "0" to indicate that the star in question had neither northern nor southern latitude (fol. 60', line 7 up). P. 105:13. For the longitude of this star (fol. 63', line 2) Copernicus at first inadvertently wrote 289°40', the correct numbers for the following star. He noticed his error when he wrote 289°40' the second time. Then he went back to the previous entry and rather unsuccessfully modified the original 9 to an 8 by writing over it. Above the 4 he placed our present sign for minutes, in order to substitute a 1 for the previous digit, as he had done on fol. 61 ', line 3 up. His replacement of 4 by 1 was not noticed by any edition earlier than NCCW, vol. II, Latin version, p. 101, line 35, and p. 399. P. 106:39. In the constellation of the Fishes, GV put "the one to the north, of the two stars in the back of the head of the western fish" in tile 3rd place, while the 4th place was assigned by GV to "the one to the west, of the two stars in the back of the western fish" (sig. ee2', lines 25-24 up). On the other hand, PS 1515 reversed this order. The longitude of Fishes 3 in PS 1515's order exceeds that of Fishes 4 by 2°10'. In this respect Coper­ nicus follows PS 1515: the longitude of his Fishes 3 is 321 °30', while that of his Fishes 4 is 319°20' (fol. 63v,lines 5-4 up). However, his nomenclature follows, not PS 1515's order, but GV's. Consequently the longitude of Copernicus' Fishes 3 belongs with his Fishes 4, and vice versa. This interchange suggests that in compiling his star catalog, he based his descriptions on GV (somewhat modified), but his numerical values on PS 1515, and did not meticulously check to make sure that both his lists meshed properly. P. 107:48. Here (fol. 64", lines 6 up - 5 up) Copernicus says: "I remarked above that Berenice's Hais was named by the astronomer Conon." Evidently Copernicus did not take the trouble to glance at his previoug reference (fol. 60v, lines 2Q-21), where he listed the stars "which they call Berenice's Hair" witllout mentionino the name of Conon. The latter flourished about the middle of the tllird century B. C. The statement that "T'' gratify [Berenice's husband, King] Ptolemy, tile matllematician Conon put Berenice's Hair among the starsy was made by Theon the Grammarian in his commentary on Aratus' Phenomena (line 146). In Copernicus' cop6 of Aratus-Theon, a marginal note called attention to Conon's naming of Berenice's Hair (sig. 02v, lines 15-16; cf. MK, p. 135). P. 110:26. By an oversight PS 1515 (fol. 86v, lines 16-17) placed River 27 and 28 in the same sign as the 17 stars before them and tile 6 stars after tllem. On tile oilier hand, GV (sig. ee3v, lines 18-19) correctly located River 27 and 28 in the following sign. In the case of these 2 stars Copernicus followed GV's longitudes (Bull 4 1/6, Bull 5 = 34°10', 35°0'). By subtracting 6°40', he emerged with 27°30' and 28°20' (fol. 65v, lines 7-6 up). P. 110:43. Copernicus (fol. 66', lines 9-10) described Hare 5 "In the chin" as "dimmer," and Hare 6 "At tile end of the left forefoot" as "brighter." These descriptions were interchanged by N (fol. 59•, lines 19-18 up), followed by Band A. Copernicus' two main sources, however, agreed in classifying Hare 5 and 6 as both "brighter" than the 4th magnitude. Copernicus was brought into alignment witll GV and PS 1515 by T (p. 147, lines 26-27) by ignoring minor in Copernicus' autograph. P. 110:51. For the Hare's last star, described as being "at the tip of the tail," Copernicus found divergent longitudes in his two main sources. According to PS 1515 (fol. 86v, line 22 up), the longitude in question was 2 11 40 ( = 71 °40'). Had Copernicus subtracted 6°40' from this figure, he would have emerged with 65°0', the value assigned to Ptolemy and the Alfonsine Tables by Miisdin in his copy of N (fol. 59•, line 12 up). On the other hand, had Copernicus followed GV meticulously in this instance (sig. ee3v, line 12 up: Twins 2 1/2 = 62°30'), he would have emerged with 55°50'. Instead, he writes 5'6°0' (fol. 66•, line 16). This difference of 10' is, as usual, attributable to the difference between tile longitudes of Ram 1 in PS 1515' and GV (see note top. 97:7).

379 NOTES ON PP. 111-115

P. 111:27. For the Dog's next to the last star, described as being "at the tip of the right foot," Copernicus found a difference of only 10' in the longitude. GV (sig. ee4r, line 8) gave Twins 9 1/2 = 69°30', as compared with 69°40' in PS 1515 (fol. 86v, line 3 up: 2 9 40). Had Copernicus subtracted 6°40' from this latter figure, he would have emerged with 63°0', the value attributed to Ptolemy, Schoner, and the Alfonsine Tables by Mast­ lin in his copy of N (fol. 59v, line 11). And in fact the first digit in Copernicus' longitude entry was a 6 (fol. 66v, line 3). But he later wrote a 7 quite heavily over the 6, and turned the longitude entry into 77°0'. The reason for this wide departure from both his main sources would appear to be that as he was performing his customary subtraction of 6°40', his glance fell on the entry two lines higher in PS 1515. There he read 2 23 40 ( = 83°40'), from which he subtracted 6°40' and emerged with 77°0'. Hence he wrote a 7 heavily over his previous 6. His latitude entry, 55°40', likewise differs from both his sources, which agree on the value 53°45'. Copernicus' depar­ ture from this figure is perhaps likewise to be explained as a momentary glance at the lines just above and below the proper line in PS 1515. This suggestion is strengthened by the fact that originally he wrote the magnitude belonging to the line above, and then wrote 3 correctly over the previous 4. P. Ill :28. For the longitude of the Dog's last star, described as being "at the tip of the tail," Copernicus found a difference of a sign (less 10') between his two main sources. For, PS 1515 put all 18 stars of the Dog without any exception in the Twins. On the other hand, GV correctly moved Dog 18 into the Crab (sig. ee4r, line 9: Crab 2 = 92°0'). Interestingly enough, Copernicus combined the best features of his two sources. From GV, he took the sign, but not the degrees and minutes, in which he followed PS 1515's 2 10 (fol. 86v, line 2 up). Thus he put the older longitude at 92°10', from which he subtracted 6°40' and emerged with 85°30' (fol. 66v, line 3). A sign less, 55°30' was attributed to the translator of Ptolemy, namely PS 1515, and to the Alfonsine Tables by Mlistlin in his copy of N (fol. 59v, line 12). P. 111:41. Longitude 52°20' requires sequens ("to the east"), as in PS 1515, fol. 87r, line 12 and in GV, sig. ee4r, line 20). Yet Copernicus wrotepraecedens ("to the west," fol. 66v, line 15). Presumably he did so because he was already thinking about the description of the next star as antecedens ("to the west") at longitude 49°20'. N failed to correct this slip. P. 112:47. With regard to the latitude of Argo 29, Copernicus found that his two main sources disagreed by 10°. For GV (sig. ee4v, line 6) gave 43 1/3, as compared with 53 20 in PS 1515 (fol. 87r, line 9 up). Whereas in such cases Copernicus usually accepted the numerical values in PS 1515, in this instance he correctly followed GV. For according to the description of Argo 29 and 30, their latitudes should not differ by much, and Argo 30 was placed at 43°30' by both GV and PS 1515. P. 113:17. The first-magnitude star Canopus (Argo 44) was placed at south latitude 69°0' by PS 1515 (fol. 87v, line 10), whereas GV put it 6° farther south (sig. ee4v, line 21). Here again Copernicus followed GV's correct value 75°0' (fol. 67r, last line). P. 113:19. With regard to the latitude of the last star in Argo, Copernicus found a difference of nearly 10° in his two main sources. PS 1515 (fol. 87v, line 11) placed Argo 45 at 61 °50' as compared with 711/2 1/4 ( = 71 °45') in GV (sig. ee4v, line 22). Copernicus took the degrees from GV, but the minutes from PS 1515 (71 °50'; fol. 67v, line 1). His description of this star as "brighter" than the 3rd magnitude (maior: fol. 67v, line 2) was omitted by N, and therefore also by B, A, and W, but was restored by T (p. 150, line 38) from the autograph. P. 113:41. Copernicus' description of this star as lying "to the east and to the south" (fol. 67v, line 16) was curtailed by N (fol. 6ov, line 12), perhaps because the words et Borea had to be brought down from the line above and left too little room for et australis. These words were restored from the autograph by T (p. 151, line 20). P. 114:21.ForCup 4and 7, PS 1515 gave the obviously erroneous longitudes 5 30 0 and5 50 40(fol. 87v,lines 11 and 8 up). Since the numbers 30 and 50 are unacceptable in the column for degrees of longitude, Copernicus must have recognized that they were merely anticipatory dittographic displacements of the number of minutes of latitude of these two stars. Hence, for Cup 4, he relied on GV (sig. ee5r, line 11: Virgin 7 = 157°0'), from which value he subtracted 6°40' and emerged with 150°20' (fol. 68r, line 9). In the case of Cup 7, however, he must have seen that GV's value for the minutes (Virgin 11/3 1/6) was typographically defective, since 1/3+1/6 = 1/2°. Hence, he must have assumed that GV intended to say 1/2 1/6 = 40', in agreement with PS 1515. Thus Coper­ nicus emerged with 151 °40'-6°40' = 145°0' (fol. 68r, line 12). P. 115:7. As the latitude of Centaur 11, Copernicus found 20 50 in both his sources (PS 1515, fol. 88r, line 18; GV, sig. ee5r, line 16 up). But when he was copying these figures into his own catalog, by dittography he wrote 20 20. Then, when he noticed his error, he impressed a 5 quite heavily over the second 2 (fol. 68r, last line). This 5 must have been regarded as just a smudge, and therefore N printed 20°0' as the latitude of Centaur 11. This error was repeated in the next three editions and was first corrected by T (p. 153, line 7). In his copy of N, however, Mlistlin noted that Reinhold, Ptolemy, SchOner, and the Alfonsine Tables all gave 20°50' (fol. 6tr, line 18 up).

380 NOTES ON PP. 115-120

P. 115:29. As the longitude of Centaur 29, GV (sig. ee5v, line 3) gave Balance 16 followed by a poorly printed fraction. If this was read by Copernicus as 1/2, then he had as the longitude 196°30'. By subtracting 6°40' from this figure, he should have emerged with 189°50', instead of which he wrote 179°50' (fol. 68v, line 16 up). This computational error cannot be imputed toPS 1515 (fol. 88r, line 23 up), which gave as Centaur 29's longitude 6 16 20 ( = 196°20'). Subtraction of 6°40' from this figure would have yielded 189°40', the value ascribed to Ptolemy, Schoner, and the Alfonsine Tables by Miistlin in his copy of N (fol. 61v, line 4). P. 115:31. To obtain the longitude of Centaur 30, Copernicus added to his (erroneous) longitude of Centaur 29 the increase of 1°10' given by GV (sig. ee5v, lines 4-5: Balance 161/3, Balance 171/2 = 196°20', 197°30'). By adding 1°10' to 179°50', Copernicus emerged with 181°0' (fol. 68v, line 15 up). On the other hand, 191°0' would have been the equivalent value for Ptolemy, Reinhold, Schoner, and the Alfonsine Tables, as was noted by Miistlin in his copy of N (fol. 61v, line 5). A was the first edition to replace 181 °0' by 191 °0'. P. 115:34. For the longitude of Centaur 33, Copernicus chose to follow PS 1515 (fol. 88r, line 19 up: 6 15 20 = 195°20'). From this value Copernicus subtracted 6°40' and emerged with 188°40', identical with the longitude of Centaur 32 (fol. 68v, lines 13-12 up), as in PS 1515. On the other hand, GV's longitude for Centaur 33 was Balance 6 1/3 ( = 186°20') or go less than its longitude for Centaur 32 (Balance 15 1/3= 195°20'; sig. ee5v, lines 6-7). Had Copernicus followed GV rather than PS 1515, he would have emerged with 179°40', the equivalent value ascribed to Ptolemy by Miistlin in his copy of N (fol. 61v, line 8), on the basis of PS 1538 (p. 200, line 3). In short, Copernicus failed to recognize in PS 1515's longitudes for Centaur 32 and 33 the sort of dittography which we have already seen so often in Copernicus himself. P. 115:36. For the longitude of Centaur 35 (the first-magnitude star later called Alpha Centauri) Copernicus followed PS 1515 (fol. 88r, line 17 up: 6 8 20 = 188°20'). From this value he subtracted 6°40' and emerged with 181 °40' (fol. 68v, line 10 up). On the other hand had he followed GV (sig. eeSV, line 9), Copernicus would have placed Alpha Centauri within the sign of the Scorpion at 81/3 ( =218°20'). By subtracting 6°40' from this value, he would have emerged with 211 °40', the equivalent value ascribed to unnamed "others" by Miistlin in his copy of N (fol. 61v, line 10). All37 stars in the Centaur without exception were placed in the same sign by PS 1515, whereas GV correctly removed Alpha Centauri from that sign and put it in the following sign. P. 116:22. With regard to the longitude of Hearth 2, Copernicus found a curious disagreement between his principal sources. On the one hand, GV (sig. ee5v line 16 up) placed Hearth 2 at 3 o within the Archer ( = 243 °0'). Had Copernicus subtracted 6°40' from GV's value, he would have emerged with 236°20'. Instead, he wrote 233°40' (fol. 69r, line 15 up). This result implies that he started from 240°20' as Hearth 2's longitude. PS 1515 did give 0 20 for the degrees and minutes, but misplaced Hearth 2 in the previous sign (fol. 88v, line 14). Appar­ ently, then, Copernicus took GV's sign but PS 1515's degrees and minutes. His greater reliance on PS 1515 prevented him from inferring that its 0°20' was probably a conversion of 3° into 1/3°. P. 117:20. With regard to the longitude of the last star in the Southern Fish, Copernicus found his principal sources in complete agreement: 9 26 0 ( = 296°0') in PS 1515, fol. 88v, line 12 up; Goat 26 ( = 296° 0') in GV sig. ee6r, line 19. Had Copernicus consulted his copy of PS 1538 (page 201, line 21), there too he would have seen Goat 26. Thus he had no way of knowing that in the course of the transmission of PS's star catalog, in this instance 1/6° had been transformed into 6°, so that the correct reading 20°10' had become 26°. P. 119:15. The Olympic games were held in honor of the pagan Greek god Zeus every fourth year. This four-year interval, known as an Olympiad, was assigned a number in a consecutive series beginning in 776 B.C., the first year of the first Olympiad. Every year thereafter was given its appropriate number within its appropriate Olympiad. In the absence of an era commanding general recognition among the ancient Greeks, the historian Polybius (XII, 11) tells us, one of his predecessors "made comparisons from earliest times of the ephors with the kings in Lacedaemon, and he set the archons in Athens and the priestesses in Argos alongside the victors in the Olympic games." On this basis Polybius proceeded to use the Olympiads as his chronological framework. Thus he says (I, 3; III, 1): "My History begins in the !40th Olympiad." This Olympiad scheme was followed by other Greek historians and geographers until the "Olympic games ceased ... to be celebrated" near the close of the reign of the Roman emperor Theodosius I, who died in 395 and whom orthodox Christians glorified as the scourge of the pagans (George Cedrenus, Compendium of History, in Migne, Patrologia graeca, vol. 121, column 622, 573). With the cessation of the Olympic games, the Olympiad chronology fell into disuse. P. 119:35. For this early theory that the sphere of the stars swings back and forth with an amplitude of 8°, which was intended to explain the phenomena of precession, see J. L. E. Dreyer, A History of Astronomy from Thales to Kepler (New York, 1953), pp. 203-204. P. 120:4. Near the end of I, 11, Copernicus said that "more recent writers now add on a tenth sphere." When he wrote that statement, he had not yet seen Johann Werner's Motion of the Eighth Sphere (Nuremberg, 1522), which postulated the eleventh sphere to which Copernicus here refers. Since Copernicus sent off his

381 NOTES ON PP. 120-121

Letter against Werner on 3 June 1524, we may reasonably conclude that he finished I, 11, before that date, and wrote III, 1, after that date. P. 120:33. Taking this observation from PS (VII, 3), Copernicus equated I, 36, Callippus, with 30 Alex· ander, an era not mentioned by PS in connection with this observation by Timocharis, nor with the next one by the same observer in the same year. P. 120:41. Here (fol. nr, line 4) Copernicus locates Regulus in the Lion's chest (pectore), as he did also in his Star Catalog (fol. 6Ql). But there he deleted pectore and replaced it in the left margin by corde (heart). Again, in dating Hipparchus' observation, PS (VII, 2) made no reference to Alexander's era, which was intro­ duced by Copernicus here. P. 120:43. In reporting these observations, PS (VII, 3) mentioned neither the Christian era nor Alexander's era, both of which were introduced by Copernicus here. P. 120:45. In the autograph (fol. nr, lines 8-9) Copernicus mistakenly wrote "from the autumnal equinox," which was corrected to "from the solstice" by Rheticus in the right margin (NCCW, I, 17). P. 121:1. "Away from the autumnal equinox" was inserted by Rheticus in the right margin, line 10, fol. nr. P. 121:2. In reporting this observation, PS (VII, 2) did not refer to Alexander's era, which was introduced by Rheticus (fol. nr, left margin). P. 121:5. The expression "from the autumnal equinox" was added by Rheticus in the right margin, fol. nr, line 13. Copernicus took the longitudes of the Spike and the first star in the Scorpion from PS' catalog. How­ ever, in assigning to the Spike the longitude of 86 ° 30', Copernicus here followed GV's erroneous value in Book XVII (sig. dd sr). But in his own Star Catalog (fol. 61 r, line 6) Copernicus used PS' value of 26° 40' within the Virgin for the Spike, since the Spike's longitude of 170° in Copernicus' Catalog consists of 5 signs (5x 30° = 150°)+26° 40'-6° 40'. In the left margin of fol. W of his copy of B, Brahe wrote: "Only Valla has this value [86° 30'], whereas others [have] 86° 40'." P. 121:9. These two observations by Al-Battani were available to Copernicus in P-R (VI, 7). P. 121:16. The Egyptian year is explained by Copernicus at the end of III, 6, where he also explains why he uses it. P. 121:18. The Spike's meridian altitude, when corrected for refraction (which was completely neglected by Copernicus), is now put at 27° 2'. P. 121:19. In his Astronomiae instauratae mechanica (Wandsbek, 1598) Tycho Brahe reported that in the year 1584 I dispatched one of my student assistants in astronomical research with a sex­ tant ... to measure very precisely with the help of this [instrument] the altitude of the pole at From­ bork. For I surmised that Copernicus' determination of this quantity was very nearly 3' too low. This was indicated to me by the fact that the sun's motion and the greatest obliquity of the ecliptic differed from the numerical values furnished by him. Experience itself also confirmed that this was so. For by means of my instrument in very many observations of the fixed stars and the sun, the altitude of the pole there was found to be 54° 22f ... But on the basis of his own observation Copernicus assumed the latitude of that place to be 54° 19!'. Therefore his value is 23/ 4'less than the correct figure. Previously I drew that conclusion entirely from his own data and the derived calculations regarding the motion of the sun (Brahe, Opera omnia, V, 45: 11-25). In his effort to correct Copernicus' defective value for the latitude of Frombork, Brahe went nearly as far above the true value 54° 21' 6" as Copernicus had been below it. P. 121:20. Round off 54° 19t' as 54° 20' +27- 81° 20' +840 90° P. 121:45. According to Lines Subtended, for 25° 30': 43,051; for 25°20': 42,788; hence, for 25° 28f: 43,010. P. 121:46. The declination = 8° 40', for which Lines Subtended give 15,069. P. 121:47. t chord 2AB:BE = t chord 2AH:HIK 39,832:100,000 = 43,010:HIK HIK = 107,978 OP: OK= t chord 2AH:HIK OP = MA = 15,069 15,069:0K = 43,010:107,978 107,978X 15,069 = 1,627,120,482+43,010 = 37,831.

382 NOTBS ON PP. 121-122

P. 121:47. HIK-OK=HO HIK 107,978 OK- 37,831 HO 70,147 P. 121:48. HGL = BGD-2(BH= 2°) = 176° HG = l(176°) = 88° for 88°, in Lines Subtended: 99~939. P. 122:1. 01 = HOI-HO = 99,939-70,147 = 29,792. At first Copernicus wrote 29,892 (fol. 72v, line 12 up), but later, having noticed the error, he wrote a 7 over the 8. P. 122:2. 99,939:29,792 = 100,000:29,810. P. 122:3. According to Lines Subtended, 30,071, for 17° 30'; 29,793 for 17"20'; hence, for 29,810, 17° 21'. P. 122:5. In the autograph (fol. nv, line 7 up) Copernicus wrote 1515 over an earlier number. This started with MDX. These three numerals(= 1510) were followed by vertical strokes over which stood dots, signifying unities (perhaps four of them). Then Copernicus erased them and put a V ( = 5) to the left of the X. P. 122:11. Ptolemy: 462 Alexander Timocharis: 30 Alexander 432 It was for the purpose of facilitating the computation of this interval that Copernicus introduced the Alexander era in reporting Timocharis' first observation of the Spike (note to p. 120 :33). Then he forgot to do the same for Ptolemy's observation, so that Rheticus had to insert the Alexander year in the margin (note to p. 121:2). P. 122:14. 432 years ~ 41/ 8 centuries, for 41/ 8 °. P. 122:15. Ptolemy: 462 Alexander 32° 30' Hipparchus: 196 Alexander 29° 50' 266 years 2° 40' 21/a centuries 21/ 8 ° P. 122:18. Since Copernicus makes this interval 782 years (fol. 73•, line 10), he is operating with 1204 Alexander, for Al-Battani, and - 422 Alexander, for Menelaus 782 years Copernicus' Alexander year for Al-Battani (fol. 12•, line 19) now reads "Mccllii," with the smudge oblit­ erating 2 i's. When Copernicus reduced this number from 1204 to 1202 on foJ. 12•, he forgot to make the cor­ responding adjustment here and elsewhere.

P. 122:20. With 11° 55' = 715' in 782 years, 60' in 653/ 6 years ~ 66 years. P. 122:20. Presumably Copernicus is operating with 1204 as Al-Battani's Alexander year. Then 463 was his Alexander year for Ptolemy, not 462 as supplied by Rheticus (note top. 121 :2; 1204-463 = 741). How­ ever, the interlineation of uni and anni (fol. 73', line 13), as well as the misplacement of the dot over x instead of the i in Dccxli, are indications of great haste on the part of Copernicus. P. 122:21. Al-Battani: Regulus 44° 05'; Scorpion 47° 50' Ptolemy: 32 30; 36 20 11° 35' 11° 30' With 11 o 30' = 690' in 741 yeara, 60' in 64'/6 years ~ 65 years. P. 122:22. Copernicus (1525): 1849 Alexander Al-Battani: 1204 645 yeara. P. 122:23. Copernicus forgets to disclose how he determined this difference of 9° 11', since his comparison star is the Spike, of which he cites no observation by Al-Battani. 9° 11' = 551':645 years= 60' for 70! years~ 71 years. P. 122:24. The precession of the equinoxes is in fact uniform, being approximately 50" a year, 1° in 72 years, and 360° in 26,000 years. The misconception that the precession was nonuniform arose, in part, because Ptolemy had understated the rate of precession for 100 years as 1 o, although the actual precession was about 1° 24'. By way of compensation Al-Battani had overstated the rate for 66 years as 1°, instead of only 56'. As a result of these opposite misses, to some of his predecessors and to Copernicus, the precession during the course

383 NOTES ON PP. 122-126

of many centuries looked nonuniform, slowly shifting from deficiency to excess. This wholly imaginary non­ uniformity in the precession of the equinoxes was discarded by Tycho Brahe. In his Astronomiae instauratae mechanica (Wandsbek, 1598; Opera omnia, Copenhagen, 1913-1929, V, 113, lines 9-17) the great Danish as­ tronomer remarked: I have also noticed that in the longitudes [of the stars] the intricacy of the nonuniformity is not as great as Copernicus believed. For, what he imagined in this regard insinuated itself through defects in the observations, both ancient and recent. Hence the precession of the equinox in these times is also not as slow as he indicated. For at present the fixed stars traverse 1°, not in 100 years in accordance with his computation, but in only 71! [years]. In the past they always uniformly completed very nearly this [motion], if the observations of [our] predecessors are properly delimited, with only a trivial irregularity, arising accidentally from another source. In his Astronomiae instauratae progymnasmata (Opera omnia, II, 256: 17-19) Brahe concluded: I do not yet want to announce a final judgment on this matter, in the belief that it is more prudent to withhold it for several years and reserve it for [my] general treatment of astronomy [which Brahe did not live long enough to write]. P. 122:27. Here Copernicus commits a historical blunder. In II, 2, he had correctly recalled that Ptolemy's determination of the obliquity as 23 ° 51 '20" agreed with Eratosthenes' and Hipparchus'. But the exhaustive study of Aristarchus of Samos by Thomas L. Heath (reprint, Oxford University Press, 1959) evidences no such determination by Copernicus' ancient predecessor. P. 122:29. The source of _Copernicus' statements about the values of the obliquity as propounded by Al­ Battani, Al-Zarkali, and Profatius has not yet been identified. Here (fol. 73•, lines 22-23) Copernicus attri­ buted to Al-Battani the value of 23°36', but at fol. 79•, line 23, he switched after some hesitation to 23°35'. P. 122:30. After "23° 28!"' Copernicus originally wrote "or 29', according to some" [authorities]. Later he deleted this alternative (fol. 73', line 13 up). P. 122 :32. Evep. if these determinations of the obliquity of the ecliptic had been historically accurate, they would ha~e exhibited only a steady diminution, from Ptolemy's 23° 51' 20" to Copernicus' 23° 28' 30". But Copernicus linked the change in the obliquity with the precession, and turned what is in fact a unidirectional decrease into a cyclical phenomenon, oscillating between a maximum of 23 ° 52' and a minimum of 23 ° 28' in the course of 3434 years (III, 6). P. 124:20. This line was drawn by Copernicus in the autograph (fol. 74') as a slightly squashed figure 8. Its two loops were mistakenly separated inN (fol. 66v). A far more serious error was committed in T (p. 164), which replaced Copernicus' nearly elliptical loops by osculating circles. This misrepresentation was repeated in Me (p. 136), in the 1964 Russian translation (p. 164), and also by Otto Neugebauer, Vistas in Astronomy, 1968, 10: 96 (" ... a figure eight curve made by two small contacting circles.") P. 124:22. In the 4th edition of his Commentary on the Sphere of Sacrobosco (p.168) Clavius introduced the follow­ ing remarks regarding III, 3: Copernicus "speaks in a confused way and explains himself and expresses himself with such great difficulty that he can hardly be understood, since he seems to me, with regard to the last two motions, to write statements completely inconsistent with one another. For he wants the first motion, by which the greatest declination of the sun is changed, to be brought about by an approach and withdrawal of the pole of the universe from the pole of the ecliptic through 24' in the solstitial colure. But the process of the last motions, which produces nonuniformity in the motion of the fixed stars, called by him the precession of the equinoxes, is caused by the departure of the same pole of the universe to either side of the colure by so great a distance that the equator described by it, when it is at its greatest distance from the colure, intersects the ecliptic in two points which are 1 o 10' away from the equinoctial points of the first movable, both to the east and to the west. As a result, by this motion the pole of the equator describes a sort of figure like a twisted crown, as he himself says. This is cut in two by the colure so that two ellipses [eclipses, in the printed text!], mutually tangent in latitude, are formed in such a way that their minor axes produce an almost straight line and cut off 24' of the colure. But who does not see that these statements are completely inconsistent? For if the pole creeps, as it were, up and down along the colure, how could the same pole be understood to be able to move at the same time outside the colure? Or if it does travel to either side [of the colure], how could the same (pole] at the same time move up and down along the colure? I for one sincerely admit that I never could perfectly understand this contradiction." In line 7 of this note Clavius implies that the now standard term "precession of the equinoxes" was introduced by Copernicus. P. 126:9. Unfortunately Copernicus did not identify those whom he called "some people" (aliqui, fol.

384 NOTES ON PP. 126-129

75r, line 11 up, and again fol. 75v, line 12). Whoever these unnamed persons were, evidently they were familiar with the theorem that a rectilinear oscillation could be generated by a suitable combination of circular motions. By calling attention to this prior knowledge of the theorem, Copernicus implicitly disclaims any originality on his own part. On the other hand, he gives no indication of being aware that the theorem had been enunciated some three centuries earlier by Tusi. In a work critical of Ptolemy and intended to improve him, Tusi introduced a lemma with the remark that "On this point I have received nothing from my predecessors, and I have myself invented what I am going to present here" (Carra de Vaux in Tannery, p. 348). In its original form, the Tusi couple consisted of a pair of circles, corresponding to Copernicus' ADB and GHD. But Copernicus' third circle CDE has been added to Tusi's own arrangement. How the Tusi couple, either in its original or in a modified version, reached Copernicus is still unknown. No pre-Copernican translation of Tusi's Kitab al-tadhkira has been found, nor could Copernicus read Arabic. Whatever the route by which Tus1's couple came to Copernicus may eventually tum out to have been, for him the Persian astronomer's ingenious invention was immensely valuable. For, in order to produce the (sup­ posedly recurring) variation (a) in the rate of the precession of the equinoxes and (b) in the obliquity of the e­ cliptic, Copernicus could have a point, itself the center of a rotating ball, slide back and forth along a segment of a straight line at a changing rate of speed. In Copernicus' mechanical cosmos, however, everything has a spheri­ cal shape and a circular motion. But the Tusi couple possesses the great virtue of generating a rectilinear motion by the interaction of rotating circles or spheres. That is why Copernicus welcomed the opportunity of introduc­ ing the Tusi couple into the precession and obliquity mechanism. Thereby he struck an additional blow at the strict Aristotelian dichotomy of heaven and earth, wherein rectilinear motion was confined to the lowly earth and circular motion was the distinguishing characteristic of the lofty heavenly bodies. By contrast, in Copernicus' non-Aristotelian cosmos the earth is a heavenly body, and therefore there is no longer any reason why rectilinear motion, so familiar on the earth, should not be equally at home elsewhere in the heavens. Galileo, in his early essay On Motion, took the Tusi couple from Copernicus (Galileo, Opere, national ed., I, 326 : 4-9; English translation in Galileo Galilei on Motion and on Mechanics, Madison, Wisconsin, 1960, p. 97). Although Galileo's essay On Motion cannot be dated precisely, he must have written it between 1589 and 1592. Hence he was quite familiar with the Revolutions when he wrote to Kepler, on 4 August 1597, that he had "many years ago gone over to the doctrine of Copernicus ... our teacher" (Galileo, Opere, X, 68 : 17-18, 22). P. 126:13. This deleted passage was printed for the first time in T, which mistakenly described it as "of the greatest importance in the history of astronomy" (p. 166). Stimulated by this misconception, Me exclaimed that Copernicus dimly foresaw "the elliptical shape of the planetary orbits" (Notes, p. 22). But in his genera­ tion of the ellipse, Copernicus is not in the least thinking about planetary orbits. Since Copernicus deleted this reference to the ellipse, he did not carry out his promise to discuss the topic elsewhere. P. 127:6. The word translated as "let it be" ends fol. 75v, the sentence being continued on fol. 78r. In be­ tween, Copernicus inserted a sheet of paper E (fol. 76-77), thereby making a sexternion of quire h which con­ sisted originally of five sheets of paper C (NCCW, I, 7, 13). The reason for the insertion was that originally Co­ pernicus ended III, 5, halfway down fol. 78r, and started III, 6, immediately below III, 5. Mterwards he decided to make an addition to III, 5, and, having no room on fol. 78•, he inserted a sheet of paper E that came to be numbered fol. 76-77. On fol. 76r he wrote the addition to III, 5, leaving the lower half of the page blank. P. 128 :37. Copernicus wrote "Aristyllus" as a correction for "Aristarchus" (fol. 78v, line 2). In PS 1515 (fol. 73r, 75v) he saw the name of an ancient Greek astronomer in the garbled form "Arsatilis." This he changed to Aristarchus in his personal copy (fol. 7SV), which is now in the library of Uppsala University, Sweden. As late as 3 June 1524, in his Letter against Werner, he still clung to the mistaken identification of "Arsatilis" with Aristarchus. It was only thereafter that he deleted "Aristarchus" here and replaced it in the margin with the cor­ rect name, Aristyllus. He undoubtedly should have made the same substitution toward the end of II, 2, where he left "Aristarchus of Samos" and "Aristarchus" unchanged (fol. 73r, lines 20 up and 12 up). Had he inter­ changed the names there, he would not have extricated himself from the historical error involved, since a de­ termination of the obliquity as 23 a 51' 20" cannot be attributed to Aristyllus any more than it can to Aristarchus (note to p. 122:27). P. 128:37. Agrippa is mentioned only once in PS (VII, 3), as an observer contemporary with Menelaus. P. 129:4. According to III, 2, Copernicus: 1849 Alexander Timocharis: 30 1819 years P. 129:5. This period of 432 years is the interval from Timocharis (30 Alexander) to Ptolemy (462 Alex­ ander; note to p. 122:11).

385 NOTES ON PP. 129-135

P. 129:5. This period of 742 years is the interval from Ptolemy (462 Alexander) to Al-Battani, whose Alex­ ander year is therefore 1204 here (note to p. 122:18). P. 129:6. Copernicus: 1849 Alexander AI-Battani: 1204 645 years. P. 129:11. 1819 years: 360°+21°24'=381°24' 381°24':1819=360°:1716.9, for which Copernicus writes 1717 years. P. 129:13. 85° 30' +146° 51' +127° 39' = 360°; 90° 35' +155° 34' +113° 51'= 360°. P. 129:18. 1819 years 645 -1717 -102 102 543 years. P. 129:21. From Timocharis to Copernicus, 1819 years (note top. 129:4). Spike, Copernicus, 17° 21' from the first point of the Balance (III, 2); Timocharis, 82° 20' from the first point of the Crab= 22° 20' within the Virgin; from Virgin 22° 20' to Balance 17° 21' = 25° 1'. But with 25° 1' (= 1501') for 1819 years, for 1717 years Copernicus should have found 23° 37' (= 1417'), not 23° 57'(= 1437'; fol. 78v, line 4 up). When this number was repeated (fol. 79r, line 9), Copernicus wrote 57' over an erased number which it is now hard to decipher. As the value for the apparent motion, 23° 57' may have bad a disturbing effect on the computation of the mean motion. P. 129:25. Since 23° 57'(= 1437') are traversed in 1717 years, then 360°(= 21600') would be completed in 25809 years, not 25816 years (as in fol. 79r, line ll); 25816+1717 = 151/ 28 • P. 129:30. Here again Copernicus forgot to change Aristarchus to Aristyllus (note to p. 128:37). P. 129:35. At first Copernicus wrote the minutes of Al-Battani's value mistakenly as 27 (xmj), fol. 7gr, line 23. When he noticed this error, he erased the 2 i's and prefixed a third x in the left margin. In doing so, however, he forgot that he had previously given the number of minutes as 36 (fol. 7¥, line 17 up). P. 129:38.SinceCopernicus' earliest recorded observation is dated 9 March 1497 (V, 27), we may reasonably conclude that he wrote Ill, 6, about 1527. His reference to his "frequent observations" bas been overlooked by those who mistakenly say that he observed infrequently, and who fail to realize that he discusses, not all his observations, but only a selected few. P. 129:43. P-R, Book I, Prop. 17. P. 129:44. Domenico Maria Novara (1454-1504) was the professor of astronomy at the University of Bologna when Copernicus was enrolled as a student there. Copernicus "was not so much the pupil as the assistant and witness of observations of the learned Domenico Maria," as he himself informed Rheticus (3CT, p. 111). In 3434 years, Copernicus believed, the obliquity of the ecliptic would complete its cycle and return to its former maximum value of 23u52'. Thereafter, the obliquity would enter upon a fresh cycle of 3434 years, during which it would once more descend to its minimum value of 23°28'. In proposing this cycle, Copernicus contra­ vened the available evidence, which indicated only a steady decrease of the obliquity from 23°51'20" to about 23°281/ 1'. In opposition to Copernicus' conception that the obliquity would increase and decrease alternately, Cam­ panella insisted that, since only a decrease was historically known, only a continuation of that decrease should be expected. Therefore the distance between the sun and the earth would diminish until the luminary's intense heat destroyed our abode in a final conflagration, thus fulfilling after a fashion the grim vision in Revelation 20. This eschatological bonfire was of course incompatible with Copernicus' implicit anticipation of the unending repetition of his 3434-year cycle for the obliquity of the ecliptic. See Michel-Pierre Lerner, "Campanella et Co­ pernic," in AfJant, afJec, apris Copernic, pp. 220, 227, 229. P. 131. Originally Copernicus placed these four Tables immediately after his Star Catalog, which ends on fol. 69v. Later, when he realized that these Tables would be better placed if they followed his historical and theoretical discussion of precession in III, 1-6, he cut away the two Tables of the Uniform Precession in Years and in Days, leaving fol. 69bis a stub in quire g (NCCW, I, 5, 12). He did not similarly remove fol. 70, the last sheet in quire g, out of fear that the folio might fall apart if he did so. Instead, by means of diagonal lines, he cancelled the two Tables of the Nonuniform Motion in Years and Days. Then he rewrote all four Tables on fol. 8or-s1v, changing many of the values in the process. P. 135:10. According to the Table of the Uniform Motion of the Precession of the Equinoxes in Years, after Ill, 6, 420 years= 7X60 years : 5° 51' 24" 12 10 2 25"' 432 years : 6° 1' 26"25"', for which Copernicus writes 6°.

386 NOTES ON PP. 135-139

P. 135:11. The star in Scorpion was located (III, 2) by Ptolemy at 36° 20' Timocharis 32 difference 4° 20'; 6°-4° 20' = 1° 40'. P. 135:14. This passage, beginning in the autograph with the last line on fol. 82r, is marked for postponement by a vertical stroke in the left margin, joined to a horizontal line running across the bottom of the page. This indication of postponement is continued on fol. 82v, where the left margin shows a long vertical wavy stroke ending just above the line containing the material to be inserted (our p. 136, 2nd paragraph). Op­ posite it, in the left margin of fol. 82v, Copernicus rewrote the first three words of the postponed passage, and then on second thought he deleted them. P. 135:34. Here begins the postponed passage, which was discussed in the preceding note. P. 136:2. Since

387 NOTES ON P. 139

edges to arise from his own ungeometrical procedure. Therefore the Apollonius of France [Viete] will stir up the astronomers too with his Second Appendix. As a geometer, Copernicus was surely less skillful than as an unskilled calculator. Hence he left out what had been omitted by Ptolemy, and moreover he committed very many errors. But I shall supply the missing material and correct the blunders in my "Francelinis" [honoring Fran.;oise de Rohan]. There I shall also set forth the Prussian [Tables'] computation of the celestial motions by means of the so-called Apollonian hypotheses, if there is dissatisfaction with the Ptolemaic hypotheses, freed from motion about an extraneous center and sub-centers or inclinations of epicycles. The impact of Viete's onslaught against Copernicus was temporarily checked by the limited circulation of his works. "His writings, although not sparse, were rare because he had them printed at his own expense, and kept the copies in his own hands. As a man far removed from all desire for profit, he generously gave copies to his friends and to experts in these matters." This account of Viete's management of his publications was print­ ed not very long after the mathematician's death by his friend Jacques-Auguste de Thou (1553-1617) in the fourth edition (Paris, 1618) of his celebrated study of contemporary history, Historiarum sui temporis libri, Book 129, year 1603. But when Viete's mathematical works were collected and reissued (Opera mathematica, Leiden, 1646), his Apollonius Gallus was included at pp. 325-346, with the Second Appendix at pp. 343-346. Viete's Opera mathematica were recently reprinted (Hildesheim, 1970). When his Apollonius Gallus was first published, it came into the hands of a patron of science, who promptly· sent a copy to Tycho Brahe. At that time Tycho's team included Kepler, who saw Viete's book without having the opportunity to examine it carefully (Kepler, Gesammelte Werke, XIV, 134, lines 276-277). Hence on 12 July 1600 Kepler wrote to the patron: I am sending you a problem in geometry, which you will transmit to Viete if you wish astronomy to profit.... Heretofore I have used it but without any proof.... For I had to use a double fiction or, so to speak, a fiction squared: quite rightly, then, a gambler's unscientific procedure, to use the term employed by Viete in demonstrating the Copernican problem of three observations of such opposi­ tions. This demonstration by Viete gave me hope that my question too can be solved by him. tf the proof occurs to me first, I shall impart it to him. Heretofore I have sought this solution in vain, I suppose because I have had little practice in this field (ibid., XIV, 132: lines 174-175, 184-194). Whether Kepler's problem was transmitted to Viete is not known. For his part Kepler failed to find a neat solu­ tion, and after prolonged and painful wrestling with the question he came to the conclusion that There will be keen geometers like Viete, who will consider it to be something great to prove that this method is unprofessional. For in this matter this objection has been raised by Viete• against Ptolemy, Copernicus, and Regiomontanus. Let those geometers therefore go ahead and solve the problem geometrically and they will be great divinities in my judgment. As for me, for the purpose of deducing four or five conclusions from a single argument (comprising four observations and two hypotheses), that is, for the purpose of getting back to the right road out of the labyrinth, it is sufficient to have shown, instead of the geometrical light, the unscientific thread (by means of which you will nevertheless be conducted to the solution). If the method is hard to understand, much harder is the investigation of the matter without any method (ibid., III, 156: lines 9-18). Viete insisted on precision and contributed nothing to the advancement of astronomy. That science was raised to higher levels by Copernicus and Kepler, who resorted to approximations in the absence of rigorous solutions. "Having only contempt for the mathematical abilities of astronomers, especially Copernicus, he [Viete] set out to show how a real mathematician can develop models far more elegant than any thought up by astronomers .... However, he stops at geometry, an.d so does not consider it his province to inquire into questions ... that cannot be answered by geometry alone," such as whether his "equations correctly describe the motion of a planet" and whether "observations ... confirm ... the accuracy of these equations" (Journal for the History of Astronomy, 1975, 6: 206-207). The author of this recent article imagines that Viete planned to compile "French tables" (ibid., pp. 185, 188, 189, 196, 207). This unhistorical notion rests on a misunderstanding of a neologism introduced by Viete in the Second Appendix to his Apollonius Gallus. Viete had dedicated his Introduction to the Analytical Art (Tours, 1591) to a noble lady whom he addressed, in his characteristically effusive manner, as "Melusinis." Her "dearest sister, Fran.;oise de Rohan," whom Viete had served as a legal adviser in a breach of promise suit, had given him a safe place of refuge after the crushing defeat of the Huguenots early in 1589. It was from the home of Fran~oise that Viete dated the dedication of his Introduction to the Analytical Art. After the death of Fran~oise in December 1591, Viete intended to commemorate her in the title of his projected work on astronomy. Since this was to be written in Latin, he had to recast her name in a Latin form that would not suggest the various

388 NOTES ON PP. 139-140

extraneous connotations connected with Francisca, the customary Latin equivalent of Fran~ise. In his effon to avoid such confusions, for the fanciful name of Melusinis' sister, Viete coined "Francelina." His title accordingly became "Francelinis," in the tradition of Aeneis for Aeneas and Achilleis for Achilles. Viete's "Francelinis" was devised to honor Fran~ise de Rohan, not France. He planned no "French tables," which exist only in the imag­ ination of our recent author, who writes: "Viete never completed the French tables, but he did begin writing a large astronomical treatise" (ibid., p. 185). The title of that treatise was to have been "Francelinis" which later gave way to Harmonicum coeleste. In that work, which Viete never completed, he proposed to use "the so-called Apollonian hypotheses," as we saw above. The Apollonian hypotheses were so called because they placed the sun in the middle of the planetary motions. Viete, who loved Greek, often substituted the name of Apollo, the Greek god of the sun, for the name of that body. Thus, Viete's Apollonian hypotheses followed Copernicus in making the planets revolve around the sun. But Viete, the self-styled Apollonius Gallus or French Apollonius, did not accept Coper­ nicus' epoch-making recognition of the earth as a moving planet. Like Tycho, Viete continued to keep the earth stationary at the center of the universe. He indicated, as we saw above, that he would reson to the "Apollonian hypotheses, if there is dissatisfaction with the Ptolemaic hypotheses" as modified by himself. Our recent author (p. 185), failing to understand Viete's simple Latin, transfers the dissatisfaction from the Ptolemaic to the Apollo­ nian hypotheses. Besides the Ptolemaic and Apollonian hypotheses, in the Harmonicum coeleste Viete considers his own hypotheses francilinideae and harmonia francilinidea (G. Libri, Histoire des sciences mathbnatiques etc., Paris, 1838-1841; 2nd ed., Halle, 1865'; IV, 298, 299, 301). Such uses of francilinidea by Viete, which are absolutely incompatible with the equation Francelinidis = French tables, are not mentioned by our recent author, who promulgated this utterly outlandish equation without explaining it or justifying it in any way whatever. An earlier misunderstanding of Viete's neologism "Francilinidean" declared that he so named his theories "after himself" (British Journal for the History of Science 1964-1965, 2: 295). But Viete, who publicly and proudly proclaimed himself to be the French Apollonius, "Apollonius Gallus," after one of the most famous Greek mathe­ maticians of antiquity, would hardly have deigned to perpetuate his fame by incorporating "Fran~ois," his undistinguished given name, in so obscure a title as "Francelinis," which has been misunderstood in two different ways in the recent past. P. 139:47. Copernicus diminishes DG = 45° 17!', by 2° 47!' = 42° 30'. He also increases DF = 45° 17!' by 2° 47!' = 48° 5'. P. 140:3. DGCEPAF = DG+GCEP+PAF = 42° 30' +155 34 +113 51 311° 55' P. 140:4. DGCEP = DG+GCEP = 42° 30' +155 34 198° 4' P. 140:6. According to the Table of Prosthaphaereses, after III, 8, fox 311°55':+52'; for 42°30': -47!' (for 42°: -47'); for 198°4':+21'. P. 140:11. 1st interval: j-(311 o 55')= 155° 57!'

2nd interval: j-(42!0 ) = 21° 15' 3rd interval: !(198° 4') = 99° 2' P. 140:12. At this point in the autograph (fol. 85r, line 20) Copernicus proceeded directly from III, 9, to III, 11. This indicates that he had already written III, 10, on the insetted sheet, fol. 76v, with the last two lines of III, 10, at the top of fol. 77r. Later, a German hand transcribed them at the bottom of fol. 76v. P. 140:20. Copernicus says "about" 1387 years, because in III, 6, he reponed his determinations of the obliquity as extending over 30 years. P. 140:21. According to the Table of the Nonuniform Motion of the Equinoxes in Years, the 3rd Table after III, 6, for 1380 years= 23X 60 years: 2x 60° +24° 40' 15" +7 44 1 49"' 1387 years 145° 24' 16" 49"', instead of which Copernicus writes 144° 4' (fol. 76v, line 10) ~ 1374 years. Since he has just made the interval from Ptolemy to himself "about 1387 years" rather than exactly 1387 years, perhaps he determined the simple anomaly in that period around 1512, and retained that determination thereafter. N, however, replaced 144° 4' by 145° 24' (fol. 7fr, line 11).

389 NOTES ON PP. 140-141

P. 140:29. Here N intended to replace 75° 19' by the altered value required by the change discussed in the preceding note. Instead of 76° 39', N misprinted 76° 29' (fol. 76r, line 13 up). P. 140:34. GK = GB+KB = 932 967 1899 P. 140:36. 1899:2000 = 22' 56":24'2", for which Copernicus writes 24'. P. 141:10. Nebuchadnezzar, the second ruler of that name, reigned from 604 to 562 B.C. He belonged to the Chaldean dynasty, properly so called, whereas Nabonassar, whom Copernicus labeled a Chaldean and whom we today regard as a Babylonian, ruled nearly a century and a half earlier than Nebuchadnezzar II. The latter was incorrectly substituted for Nabonassar by PS 1515 (fol. 33v: Nabuch.) and by P-R (Bk. III, Prop. 21). P. 141 :11. PS (III, 7) computed "424 Egyptian years from the reign of Nabonassar to the death of Alex­ ander" the Great. Since the latter event was associated by Copernicus in his Letter against Werner with 323 B.C. (3CT, pp. 94-95), Copernicus was aware that Nebuchadnezzar II, who conquered Jerusalem in 586 B.C., lived much later than Nabonassar, the beginning of whose reign (on 26 February 747 B.C.) was taken by Ptolemy as the earliest of his eras. P. 141 :12. Shalmaneser, the fifth ruler of that name, was king of the Assyrians, not of the Chaldeans, from 726 to 722. Hence, Shalmaneser V did not ascend the Assyrian throne immediately after the death of the Ba­ bylonian King Nabonassar (747-734 B.C.). P. 141:14:. This round number, 28 years, would make the 1st Olympiad begin one year too late: 747 +28 = 775 B.C., instead of 776 B.C. Whereas the start of Nabonassar's reign had long been 11ssociated with 747 B.C. (3CT, p. 94), the beginning of the Olympiad chronology was (and is) less well known. The Olympiads were ignored by Ptolemy and post-Ptolemaic astronomers, being essentially a time scheme used by political and military historians. Unfortunately, Copernicus does not tell us by whom "the first Olympiad is found to have preceded Nabonassar by 28 years." The reintroduction of the Olympiad chronology is an additional in­ dication of Copernicus' humanistic attachment to Greek antiquity. P. 141:16. Censorinus dedicated his book On Birthdays (De die natali) in 238 A.D. In Chapter 21 Cen­ sorious said only that "the Olympic games are celebrated ... on summer days." Copernican scholars have not yet identified the "other recognized authorities," from whom Copernicus took the statement that the Olympic games started at the time of the summer solstice (rather than at the first full moon after that solstice). P. 141 :20. Hecatombaeon was the first month of the Athenian calendar. Since other Greek communities used their own calendars, which began at different times of the year and used other names for the months, there was no calendar in general use among the ancient Greeks. In his copy of N (fol. 76v, left margin) Miistlin wrote: "This calculation by Copernicus [of the interval] from the beginning of the Olympiads to Nabonassar falls one whole year short of the truth, which is 28Y247d." P. 141:23. Copernicus' words Kalendas Ianuarii, unde Julius Caesar anni a se constituti fecit principium were taken directly from Censorinus, Ch. 21, 7. P. 141:24. Copernicus' words pontijex maximus suo tertio et M. Aemilii Lepidi consulatu are quoted directly from Censorious, Ch. 20, 10. P. 141:27. Copernicus' words Ex hoc anno ita a Julio Caesare ordinato caeteri ... Iuliani are quoted directly from Censorious, Ch. 20, 11. For stylistic reasons, however, Copernicus replaced Censorious' ad nostram memo­ riam by deinceps, and appellantur by sunt appellati. Copernicus' expression ex quarto Caesaris consulatu is likewise repeated from Censorinus, Ch. 20, 11. P. 141:28. Copernicus' words quamfJis ante diem XVI Kalendas Februarii ... difJi filius ... sententia Munati Planci a senatu caeterisque civibus appellatus ... se septimo et M. Vipsanio consulibus. Sed Aegypti, quod biennio ante in potestatem fJenerint ... were quoted directly from Censorinus, Ch. 21, 8-9. But Copernicus deemed it to be advisable to explain to his readers that it was Julius Caesar who had been deified, whereas Censorinus, writing while the Roman Empire was still a living political institution, felt no need to do so. P. 141:30. At fol. 8SV, line 9, Copernicus wrote the name of the gens of Lucius Munatius Plancus incor­ rectly as "Numatius." This erroneous form is found in the edition of Censorinus which was published on 12 May 1497 in Bologna, while Copernicus was studying at the university in that city. P. 14:1 :38. In Copernicus' time the question had not yet been raised to what extent Ptolemy's star catalog incorporated the (lost) catalog of his great predecessor Hipparchus. P. 14:1:39. Since the Egyptian year consisted of exactly 365 days and disregarded the leap year, every fourth year it fell one day behind the Roman year of 365! days. Hence, from the beginning of the Christian era to the epoch of Ptolemy's star catalog on 24 February 139 ( = "138 Roman years, 55 days"), the Egyptian year slipped 34 (= 136+4) days behind the Roman calendar. P. 141:41. To compute the time elapsed from the 1st Olympiad to the epoch of Ptolemy's star catalog,

390 NOTBS ON PP. 141-143

Copernicus added the component intervals: from 1st Olympiad to Nabonassar 27' 247d Alexander 424 Julius Caesar 278 118! Augustus 15 ~ Christ 29 130! Ptolemy 138 89 [= 55+34] 831! 2-730 913Y 10ltd Copernicus has silently dropped !d from this total, because in passing from Christ to Ptolemy the era shifts only 12h from midnight in the Roman calendar to noon in the Egyptian calendar. P. 141:42. According to the Tables of the Uniform Motion of the Precession in Years and in Days, after III, 6, for900Y=15X60 12°33' 1" 13Y 10 52 37"' 6od 8 15 41d 5 38 12° 44' 7" 30'", for which Copernicus writes 12° 44'. According to the Tables of the Nonuniform Motion of the Equinoxes in Years and in Days, after III, 6, for 900Y 60°+34° 21' 2" 13Y 1 21 46 13'" 6od 1 2 2 41d 42 23 95° 44' 32" 38'", for which Copernicus writes 95° 44'. P. 141:45. According to the Table of the Prosthaphaereses of the Equinoxes, after III, 8, fot"42°: 47'. P. 142:5. Originally Copernicus wrote the number of minutes as 44 (fol. 85v, last line), as he had done just above (fol. 85v, line 15 up; note top. 141 :42). Then in the last line of fol. 85v he deleted 44 and replaced it by 45 in the bottom margin. P. 142:6. 360°+21° 15' = 381.15' -95 45 285° 30' P. 142:11. This epoch, 5° 32' for the Christian era, was written by Copernicus quite prominently in an otherwise vacant central column in his Table of the Uniform Motion of the Precession of the Equinoxes (after III, 6; fol. 8or). Nand B, however, omitted this epoch, which was first reinstated by A. Had Copernicus placed his epochs at the head or foot of his columns of figures or in some other conspicuous position, his Tables would have been more convenient to consult and use. Had he lived in a place where scientific activity was widely pur­ sued in an environment resembling a research center, he might have recognized the utility of such readily com­ prehended captions. Actually he spent the most productive years of his life far from personal contact with fellow­ scientists. Moreover, he was equally remote from the universities, where the needs of elementary students might have induced him to introduce such valuable rubrics. An illuminating contrast is provided by that experienced and effective teacher Miistlin who, in his copy of N (fol. 70v, below the left-hand column), listed all the relevant epochs. P. 143:14. 20° 55' 2" 20 55 16 5 32 26° 48' 13", for which Copernicus writes 26° 48'. P. 143:19. 120° 37 15' 3" 2 37 15 2 4 2 645

391 NOTES ON PP. 143-145

P. 143:23. 2X 166° 40' = 333° 20' = 5X 60°+33° 20'. P. 143:26. But 32' +26° 48' = 27° 20'. For the number of the minutes, Copernicus originally wrote 22, which he replaced first by 19 and finally by 21 (fol. 87r, line 3). In making this decision, he may have been in­ fluenced somewhat by the previously neglected 13" in excess of 26° 48' for the mean precession (note top. 143:14). Another consideration is discussed in the following comment. P. 143:29. The agreement with the 21' in III, 2, undoubtedly affected Copernicus' final decision at fol. 87r, line 3. But those who accuse Copernicus of fudging his figures should bear in mind that in III, 2, he reported the minutes as "approximately" 21 (proxime, fol. nv, line 9 up). P. 143:39. According to the Table of the Nonuniform Motion of the Equinoxes in Years, after III, 6, for 880Y = 14 x 60Y = 840Y 60° + 28° 3' 38" 40Y 4 11 36 6"' 92° 15' 14" 6'" epoch of Christ 6 45 990 P. 143:40. According to the Table of the Prosthaphaereses, after III, 8, for 99°, 25'. P. 144:19. Aristarchus' determination of the length of the year was found by Copernicus in Censorinus, Ch. 19. P. 144:33. Ptolemy 462Y 68d 191hh Hipparchus 176 363 12 285Y 70d 71f.h. P. 144:34. 285+4 = 71d 6h. P. 144:35. 71d ~-70d7 1f6h = 224/r.h 22.8:24 = 19:20. P. 144 :43. A day was divided either into 24 hours, each 60 hour-minutes long ( = 24h X 6()Ill), or Into 60 day-minutes, each 60 day-seconds long ( = 6odm x 60ds = 3600d5). According to these methods of dividing a day, the tropical year consisted, in addition to 365d, either of 6h - 1/ 300d, or of 15dm - 1/ 800d. According to the second reckoning, 1/ 300d = 12ds, and the tropical year= 365d 14dm 48d8, P. 145:1. Copernicus took Al-Battani's observation from P-R (Bk. III, Prop. 2), which timed the equinox at "43/ 4h before sunrise," rather than 43/ 6 (as in Copernicus, fol. 88r, line 8). P. 145:4. In IV, 29, Copernicus explains how to reduce the time of an observation made on one meridian to the local time of another meridian. 445m P. 145:10. 7d 2/.h = 168.4h = 10104ffi+743 = 13m(+ -- ~ 365). u 743 - P. 145:14. At first Copernicus wrote "Varmia" (fol. 88r, line 15 up). Then, deleting"Varmia," in the right margin he wrote Frauenburg, the German place name meaning "the Fortress of Our Lady." For this he devised a Greek counterpart "Gynautia," which he then replaced by "Gynopolis," an exact Greek equivalent of German Frauenburg. Previously, in III, 2 (fol. nr, line 10 up), Copernicus had called his place of observation "Hermia," perhaps with the idea of recalling to any reader familiar with Greek that this was the location of the cathedral of whose chapter Copernicus was a member. At fol. nr, line 8 up, he even constructed a derived form of Hermia, but he then deleted both it and "Hermia" two lines above. Thereafter he dropped "Hermia" altogether, since he preferred "Gynopolis" as the Greek name of his place of residence. A recent would-be belittler of Copernicus professes to see some sort of mystification in Copernicus' liking of this Greek name. But our poorly informed belittler has no understanding of Copernicus' sincere love of Greek antiquity and of his valiant ·efforts to pro­ mote Greek studies in his native land. P. 145:16. Before finally timing the moment of this equinox at "! hour after sunrise," Copernicus had previously written "before" sunrise, and then in the margin "1 hour before" sunrise (fol. 88r, line 12 up). The reason for these changes is not clear. The moment of this autumnal equinox at Frombork was 8:31A.M., ac­ cording to Z, p. 204. P. 145:21. Copernicus: 1840, 6 Phaophi = 1839Y 36d = 1838Y 401d Ptolemy: 463, 9 Athyr 462 69 1376Y 332d Copernicus' th after sunrise at Frombork ;;;; 1!h after sunrise at Alexandria Ptolemy's local time ;;;; 1h after sunrise at Alexandria th

392 NOTES ON PP. 145-147

P. 145:23. 158d 6h -153 63/4h 4d 23;fh, rather than 4d 223/,h. 4d223f4h:633Y= 1d:127.9Y, for which Copernicus writes 128 years. P. 145:24. 1376 -7- 4 = 344d -332 th --1-1-23!- :;;; 12d.

P. 145:25. 1376 -7- 12 = 1142/3, for which Copernicus writes 115 years. P. 145:28. Originally Copernicus timed the moment of this equinox at "3;fh before sunrise" (fol. 88v, lines 2-3) = 2:45 A.M., instead of 4:20 A.M. The true moment was 1:05 A.M., mean time, according to Z, p. 204. P. 145 :29. The interval between the observations of the vernal equinox by Ptolemy and Copernicus is again 1376Y 332d, because both astronomers observed the vernal equinox following the autumnal equinox just dis­ cussed. In each case the interval between equinoxes is 178d. Ptolemy's year is 463 Alexander; 9 Athyr = 69th day+178d = 247th day= 7 Pachon. For Copernicus, the interval extends from 14 September 1515 to 11 March 1516 = 178d (16 in September 1515, 31 in October, 30 in November, 31 in December, 31 in January 1516, 29 in February, 10 in March).

P. 145:30. Copernicus: 41/3h after midnight Ptolemy: 1h after noon 15lfah + 1 difference between Frombork and Alexandria. 161/ah P. 145:42. Copernicus obtained this information about Thabit from P-R (Bk. III, Prop. 2). In Thabit's treatise On the Solar Year, as printed in Latin translation in Francis J. Carmody, The Astronomical Works of Thabit b. Qurra (Berkeley, 1960), p. 74, section 108, the solar year is given as 365d 15' 22" 47"' 30"". P. 145:44. 15dm = 1-d = 6h 1d = 6odm = 360ods = 24h = 1440m = 86,4008 1ds =216m 23ds = 91/om = 9m 12s P. 146:27. In his brief essay On the Measurement of a Circle, Prop. 1, in order to find the area of a circle Archimedes inscribed and circumscribed a square. Within the circumscribed square and around the inscribed square he constructed a series of polygons whose area progressively approached the area of the given circle. This quantity could be considered analogous to the sun's mean motion, while its nonuniform motion was com­ parable to the varying area of the polygons. After drawing this analogy (fol. 89r, lines 18-17 up), Copernicus deleted it, perhaps because he realized that his reader would have little, if any, access to the writings of Archi­ medes.

P. 147:3. Originally the difference was given as only 1ds (fol. 89v, line 9). Later Copernicus added 10/60ds in the left margin, because he had increased by that amount his determination of the length of the sidereal year: Copernicus 365d 15dm 24ds wdt - Thabit 365d 15dm 23ds 1ds lOdt P. 147:4. Originally Copernicus' determination of the length of the sidereal year (fol. 89v, lines 10-11) was 365d 15dm 24ds, to which he later added 10f60ds in the left margin. P. 147:5. This value was originally 6h 9m 36 24/ 608 (fol. 89v, line 11). Later Copernicus deleted the 24/808, erased the j at the end of xxxvj and wrote over the v, making it a fourth x. Consequently he had to delete the two columns for seconds and sixtieths of seconds in his Table of the Sun's Simple Uniform Motion in Years, after III, 14, and he wrote two new columns of figures to the right of the ruled spaces containing the deleted numbers. 15dm = 6h

24ds 93/5m = 9m 368 10dt = 4" 6h 9m 408 P. 147:15. Annual simple uniform: 5X 60°+59° 44' 49" 7"' 4"" +precession: 50 12 5 composite uniform:

393 NOTES ON PP. 147-157

daily simple uniform: 59' 8" 11"' 22"" +precession: 8 15 composite uniform: 59' 8" 19"' 37"" P. 154:1. Copernicus started to write this Chapter on fol. 94'". But no sooner had he put down the tide and number of the Chapter than he realized that his Tables of the Sun's Simple Uniform Motion in Years and in Days (fol. gor-v) and his Tables of the Sun's Uniform Composite Motion in Years and in Days (fol. 931'-") should be accompanied by his Tables of the Sun's Uniform Motion in Anomaly in Years. Hence he deleted this Chapter's tide and number, and putting his Table of the Sun's Uniform Motion in Anomaly in Years on fol. 94'", on fol. 94v he recommenced this Chapter. Later on, having written III, 23, on the Solar Anomaly, he recomputed his original figures, with which he was now dissatisfied (fol. 102v, left margin). Hence he crossed out the Table for the solar anomaly on fol. 94'", and wrote a fresh Table of the Sun's Uniform Motion in Anomaly in Years as well as the accompanying Table in Days. These two Tables he wrote on a sheet of paper E, which he inserted in quire i, previously composed of papers C and D. This is the reason why his Anomaly Tables, now numbered fol. 91r-v, are out of their proper place, since logically they should follow the Uniform Composite Motion on fol. 931'-V. Moreover, having no use for fol. 92r-v, he left it blank (NCCW, I, 8, 13). P. 155:10. Copernicus' employment of eccentrics elicited the following reflections from Nunes: Since Copernicus adopts eccentric orbs, he will therefore have to assume others, in order to fill out the planetary spheres concentric with the universe. Hence, in my judgment, he should have sought only this one objective, how [reading quonam instead of quoniam] he could make the tables of the celestial motions more accurate on the basis of his own observations and those of others. He could have done so by having the eighth sphere move, with the sun also in motion, but the earth remaining stationary in the middle of the universe, as in the conventional astronomy (Rules and Instrumsnts, in Opera, Basel, 1566, p. 106).

Nunes' belated advice to Copernicus to confine his efforts to improving the tables, and to forget about revising the basic astronomical concepts springs from the Portuguese mathematician's unwavering adherence to the traditional cosmology. Nunes' true attitude toward Copernicus was masked for a long time by a false reading in Baldi's biography of Copernicus as published by Guido Zaccagnini in the second edition of his biography Bernardino Baldi (Pistoia, 1908), p. 331. There the unsuspecting reader was told that "Pedro Nunes praises .•. Copernicus and calls him an astronomer not only worthy of being compared with the ancients but absolutely marvelous in astronomical matters." When Bronislaw Bilinski reprinted Zaccagnini's defective version of Baldi's biography of Copernicus in Studia Copernicana, IX (1973), BiliD.ski pointed out that this lavish praise of Copernicus emanated, not from Nunes, but from Peter Ramus (p. 76). Shortly thereafter Bilinski gained access to Baldi's autograph manuscript, and thereby confirmed that the presence of Nunes' name in connection with the praise of Copernicus was noth­ ing more than a faulty reading (Bilinski, La Vita di· Copernico di B. Baldi, 1973, p. 23). P. 155:13. Had Copernicus not been content to retain the traditional geocentric terminology (note to p. 51 :19), he might have reflected that "apogee," meaning literally "farthest from the earth," was utterly inap­ propriate for a position of the earth, and should therefore have been replaced by "aphelion," our present term. By the same token, he might have replaced ''perigee," two lines below, by ''perihelion." P. 155:24. CFD > (CBD = ABB) > AFB. P. 155:29. Euclid, Optics, Prop. 5 (note top. 11:28). P. 156:45. As between the epicycle and the eccentric, Copernicus hesitates in this particular case. But he has no hesitation in believing that one or the other exists in the heavens ( e:cistat in caelo). P. 157:18. The essence of this proof, which is modeled after PS (III, 3) may be restated as followa: GDF >DGF BDG=BGD BDF = GDF+BDG; BGF = DGF+(BGD = BDG) .·.BDF >BGF P. 157:39. According to the Table of the Sun's Simple Uniform Motion in Days, after III, 14, for 6Qd 59° 8' 11" 22"' 34 33 30 38 26 t 29 34 6 94ld 93° 8' 23" 54"', for which Copernicus writes 93° 9'

394 NOTES ON PP. 157-159

for 6od 59° 8' 11" 22'" 32 31 32 22 3 ! 29 34 6 92ld 91° 10' 7" 31"', for which Copernicus writes 91° 11'. P. 158:10. According to Lines Subtended, for 2° 10', 3781, with the radius = 100,000, and 378 with the radius = 10,000. P. 158:11. According to Lines Subtended, for 1°: 1745; for 50': 1454; hence for 59'= BH: 1716 with the radius= 100,000, and 172 with the radius= 10,000. P. 158:13. (378)9 = 142,884 (172)2 = 29,584 172,468 ~ (415)2 Originally Copernicus wrote 415 (fol. 97•, line 3). Then he erased the last digit and replaced it by a 7, which he subsequently converted into a 4. P. 158:14. 414X 24 = 9936 ~ 10,000. P. 158:15. EF:EL = NE:t chord (2NH) 414:172 = 10,000:4154.6 According to Lines Subtended, for 24° 30': 41,469 ~ 41,546. P. 158:23. According to the Table of the Sun's Simple Uniform Motion in Days, after III, 14, 59° 8' 11" 22"' for 60d 59° 8' 11" 22'" for 6od 27 35 49 18 28d 29 34 5 41 30d 7 23 31 1/8d 7 23 31 1/8d

86° 51' 24" 11'" S81/8d 88° 49' 40" 34'" 901/8d. P. 158:35. Copernicus obtained this information about Al-Battani and Al-Zarkali from P-R (Bk. III, Prop.l3). P. 158:40. Here Copernicus pretty plainly indicates that his attention was drawn to the problem of the length of the year by the inquiries connected with the attempt at calendar reform during the Fifth Lateran Council (note top. 5 :47). Hence his concern with this problem did not lead him to the geokinetic astronomy, a thesis which has been advanced recently. Copernicus' dissatisfaction with the Ptolemaic system and his search for a more acceptable alternative preceded his preoccupation with the problem of calendar reform. P. 158:41. Beneath this number for the interval between the vernal and the autumnal equinoxes, Copernicus originally wrote a different number (fol. 97r, line 3 up). At first, followingPS' 94td+92!d, he had 187d ( clxxX'Oij). Then, putting a dot below the c to call attention to the faultiness of this number, at its end he erased the 2 i's, whose surmounting dots are still faintly visible, and he prolonged the final digit, over which he put a fresh dot. The number of day-minutes was at first 20! ( xxs), consistent with 178d 53!dm (fol. 97v, line 5) as the interval from the autumnal equinox to the following vernal equinox, and making the tropical year 365d 14dm. Finally, the double x was erased and replaced by the present v. P. 158:46. Emphasizing the difficulty of precise observation of the solstices, P-R, III, 14, recommended instead concentration on the midpoints of 4 constellations a quadrant apart: Bull, Lion, Scorpion, Water Bearer. When Copernicus (fol. 97v, lines 2-3) started to list these constellations, he began badly with the Ram and Virgin. Striking these out, he switched to the Bull, and again mistakenly repeated the Virgin, before going on to the right track with the Lion, Scorpion, and Water Bearer. P. 159:1. According to the Table of the Sun's Simple Uniform Motion in Days, after III, 14, for 45d 44° 21' 8" 31"' (1 d 59' 8'1 11"') 16dm 16' 45d 16dm 44° 37' for 120d = 2X 60d 60° +58° 16' 22" 58d 57 9 54 59"' 53ldm "'53 30 178d 53fdm 176° 19' 46" 59"' P. 159:2. At fol. 97v, line 8, Copernicus says "Reproduce the circle ABCD," without anticipating that he will promptly proceed to arrange these letters in the order ADBC. P. 159:7. With B at the autumnal equinox, and Cat mid-Scorpion, <;:.BFC = 45°.

395 NOTES ON PP. 159-162

P. 159:13. 131 o 42' +45 23 177° 5' Evidently recalling the 46" belonging with 176° 19' for 178d 53-!dm (note top. 159 :1), Copernicus at first made the sum CAD= 177° 6' (fol. 97v, line 20). Later he deleted the 6', which he replaced by 5!' in the right margin. P. 159:23. Here Copernicus operates with CAD= 177° 6', the value which he originally used (note to p. 159:13). P. 159:26. Originally Copernicus wrote 322. Later he altered the second 2 to a 3 (fol. 97v, line 3 up). He must have shifted from 322 to 323 after he had written IV, 21, where he left 322 unchanged (fol. 130v, line 7 up). P. 159:30. EL:EF = 10000:323 = 60P:1P 56' 17", for which Copernicus writes 1P 56'. P. 159:32. 10000 -;- 323 = 30.96; 323 X 31 = 10013. P. 160:34. The lower diagram on this page, drawn to accompany the reasoning in the second paragraph of III, 18, is not found in the holograph. The same is true of the diagram on p. 161, accompanying the reasoning at the end of III, 18. These two diagrams were supplied by N, in place of the incipient diagram begun by Coper­ nicus on fol. 98v. P. 161:7. Again Copernicus originally wrote 322 (fol. 99•, line 3), which he changed to 323 after he had composed IV, 21 (note to p. 159:26). P. 161:18. Copernicus: 1840 Alexander, 6 Phaophi, th after sunrise 1839 complete years, 35 complete days, 18th + (time difference between Frombork and Alexandria) 1 1838Y 400d 19th Hipparchus: 177 Alexander, 3rd intercalary day, midnight 176Y 363d 12h 183SY 40oct 19ih - 176 363 12 1662Y 37d+(7th = 18dm 45ds) P. 161:25. Here Copernicus writes 176Y 362d 27idm = llh, reduced to Frombork local time. On the other hand, in III, 18, he had implicitly used 363d (note top. 161 :18). There, however, he erred by 1d, since midnight on the 3rd intercalary day = 2d 12h. P. 161 :26. According to the Table of the Sun's Simple Uniform Motion in Years and in Days, after III, 14, for 120Y = 2X 60Y 59 X 60° = 3540° -3240 300°+29° 38' 14" 300 +45 49 50 35"' 5X60° = 300 +54 49 8 1 58 16 22 ,..., 27 6 1032° 42' 35" - 720 312° 43' P. 161:42. In III, 13, Copernicus had informed his reader that, in antiquity, 1 Hecatombaeon fell on the day of the summer solstice, so that here he is allowing for the number of days by which the Roman or Julian calendar has fallen behind in his own time. P. 162:8. These "others" were identified as the authors of the Alfonsine Tables by Miistlin in his copy of N (fol. 9ov, lines 4-5, interlineation). P. 162:16. P-R, III, 13: "Ptolemy concluded that the solar apogee was stationary and fixed with respect to the vernal and autumnal equinoctial points. Al-Battani found ... the arc BH [from the solar apogee to the summer solstice] = 7°43'. Yet Al-Zarqali ... found ... the arc BH = 12°10'. This certainly seems remarkable, since Al-Zarqali lived later than Al-Battani .... Al-Zarqali, 193 years after Al-Battani, ... having found BH = 12°10', was therefore compelled to say that the center of the sun's eccentric moved on a certain small circle." P. 162:29. This passage is commended to the attention of those who believe that Copernicus uncritically accepted all the observations of all his predecessors: "he shows a blind respect for their slightest observations" (Delambre, Histoire de l'astronomie moderne, 1969 reprint of 1821 edition, p. 105).

396 NOTES ON PP. 162-165

P. 162:37. By a slip of the pen (fol. 1oor, line 16) Copernicus wrote "6°1/2 1/3" =~50', a value 10' higher than his conclusion at the end of III, 16 (~40'). N's correction of the second fraction to 1/6 was followed by the later editions. P. 162 :42. Ptolemy, having found the solar apogee in his own time exactly where Hipparchus had located it some three centuries earlier, concluded that its position was fixed forever at 65°30' from the vernal equinox. But Thabit ibn Qurra put the solar apogee in his time at 82°45'. The displacement of the apogee, since Hipparchus' deter­ mination, would therefore have been about 18° in about 12 centuries ~ 1o in two-thirds of a century. Since this was e-~.ual to Thabit's rate of precession, he concluded "that the apogee of the sun remains fixed once and for all in regard to the fixed stars" (Dictionary of Scientific Biography, I, 510). Al-Battani's location of the solar apogee, half a century after Thabit, at 82°17' gives him "no special claim to the discovery of the motion of the solar apogee" (ibid., I,. 510-511). On the other hand, "the first who actually made a clear (and very correct) numerical statement concerning the proper motion •.. was al-Zarqali" (ibid., I, 511). His rate for the solar apogee's motion was about 12" a year, some 8 times faster than the modem value. But, like Thabit, al-Zarqali believed that the apogee moved forward and backward alternately. Hence Copernicus' "continuous, regular, and progressive advance" of the solar apogee may be regarded as the earliest such statement in the history of astronomy. P. 164:28. Again Copernicus has no grounds for choosing among kinematically equivalent arrangements, but he is certain that one of them takes place (locum habeat). P. 164:39. In the autograph (fol. 101r, line 2 up) Copernicus originally wrote 416, which he deleted and replaced by 417 in the right margin. In a previous reference to this eccentricity (III, 16; fol. 97r, line 3) he used 4 digits, with the radius = 100,000, as he noted in the right margin. Then, deciding to employ only 3 digits, he obliterated the last figure so that it can barely be deciphered, but he forgot to make the corresponding reduction in the radius from 100,000 to 10,000. Whatever the 3rd digit may originally have been, he wrote over it to make the eccentricity 414. In a second previous reference (III, 18: fol. 98v, line 16 up) he made the eccentricity 416, although the lower part of the 6 is smudged and seems to have been written over a 7. By the same token, in three later references to this eccentricity in III, 21 (fol. 101v, lines 6, 17 up, 14 up), he wrote 416 with a dot below the 6: ~· Perhaps he used this device of a dot below the 6 to indicate the change to 417 which he made explicit in the right margin of fol. 101r. If the successive stages of his thinRing about this eccentricity have been correctly disentangled, at first he vacillated between 416 and 417, and later finally decided on 414. P. 164:40. Originally 322 (fol. 101v, line 1). For the time when he changed to 323, see note top. 159:26. P. 165:6. For the minutes of ~CAD, Copernicus originally wrote 55, which he later changed to 24 (fol. 101v, line 15). For 14° 24', according to Lines Subtended, the corresponding chord is 2486, which Copernicus wrote in the left margin, replacing an erroneous number 2596 in line 16. for 14 o 30' 25,038 14 20 24,756 10 282 1 28.2 4 112.8 14 24 2486, with radius = 10,000. Copernicus put a dot below the 8 in (2486), perhaps when (in the left margin) he changed the number of min- utes from 24 to 21, for which the chord would be 2478. This shift to 21' became necessary when Copernicus altered the anomaly to 165° 39' (fol. 101r, right margin; 101v, line 5). P. 165:10. According to Lines Subtended, for 4° 20': 7555; for 4°10': 7265; hence, for 4° 13': 7352, or with radius = 10,000, 735. P. 165:12. AB:AC = 3225:735 = 416:94.8, for which Copernicus wrote "about 94," with a 5 written over the 4 (fol. 101v, line 17 up). P. 165:14. Since Copernicus makes this difference 321 at the outset (fol. 101v, line 14 up), the dot below the 6 in 416 in this line and three lines above indicates the shift to 417, discussed in note top. 164:39. P. 165:15. For circumferential ~CBD, Copernicus wrote 4° 23' (fol. 101v, line 12 up), as he had originally done 7 lines above. But there he erased the first x, reducing the minutes to 13'. Hence here in line 12 up, after originally making the central angle's minutes 12, he deleted that number and replaced it by 6! in the right margin. P. 165:21. FDB:EF = 369:48 = 10,000:1300. P. 165:22. According to Lines Subtended, with radius= 10,000, 1305 for 7° 30'; 1276 for 7°20'; and 1300 for 7°28'.

397 NOTES ON PP. 165-172

P. 165:27. These values will be found in Column 3 of the Table of the Solar Prosthaphaereses, after III, 24. p, 165:40. Copernicus repeated (fol. 1W, line 13 up) the slip he had made at fol. 10or, line 16. But this time his error was not corrected in N and the later editions (see note to p. 162:37). P. 165:45. Here Copernicus put the mean solar apogee at 71 o 37' (fol. 102v,line 10). This number of minutes was inconsistent with 32' in III, 22 (fol. 102r, right margin, replacing deleted 13' in the text, line 8 up). Hence, N made Copernicus self-consistent by reading 32 here (fol. 93V). This alteration entailed another. For on fol. 102v, in the left margin and also in lines 13-14, Copernicus put the mean solar distance from the apogee at 82° 58', which, when added to his 71° 37', yielded 154° 35', as in III, 18. Consequently, N felt constrained to raise Co­ pernicus' 82° 58' to 83° 3', in order to compensate for the 5' lost in reducing 71° 37' to 71° 32'. P. 166:1. Copernicus' value for the mean annual displacement of the apogee of the apparent sun or earth = 87614"". When this shift continues for 1580 years, the cumulative effect= 10° 40' 53" ~ 10° 41', Coper­ nicus' figure. The current value is about 2! times greater (Astronomical Journal, 1974, 79: 58). Copernicus himself made no great fuss over his discovery of this phenomenon, preferring to permit the informed reader to draw the appropriate conclusion from his scattered statements. B:ut his disciple Rheticus bluntly asserted that Copernicus, "carefully investigating the motions of the apsides of the sun and of the other planets, ... team­ ed ... that the apsides have independent motions in the sphere of the fixed stars" (3CT, p. 120). The way had been prepared for Copernicus' discovery by conflicting post-Ptolemaic determinations of the place of the (ap­ parent) sun's apogee. P. 166:14. 572 complete Olympiads= 4X 572Y = 2288Y leapdays 572d 1 complete year in 573rd Olympiad 1 July-August 1515 62 365d transferred from column of days 1 September 1515 12 2290Y 646 -365 281d 18th from noon to the time of Copernicus' observation 1sh = 45dm th = 1 dm( + 15da, which are ignored) 46dm interval from 1st Olympiad to Copernicus' observation: 2290Y 281cl46dm. P. 166:16. The number of minutes was changed from 33' to 49' by N (fol. 93v). By subtracting 42° 49' from 83° 3', N's value for the mean solar distance from the apogee, that is, the anomaly, in 1515, N arrived at 40° 14' for the 1st Olympiad, for which Copernicus' own figure was 40° 25' (replacing 29° 4'; fol. 102v,line 14). According to the Tables of the Sun's Uniform Motion in Anomaly in Years and in Days, after III, 14, 2290Y = 38X 60Y+10Y:5X 60° = 300°+57° 24' 7" 48'" 281d = 4x60d+41d: 40 24 33 2 46dm ~ s/,d 45 2290Y 281d 46dm 398° 33' 40" 50"'= 38° 33', or exactly 4° (~ 4d) less than 42° 33' in Copernicus' autograph (fol. 102v, line 13). P. 169:33. In his Rules and Instruments (Opera, Basel, 1566, p. 106) Nunes remarked: As far as astronomy is concerned, Copernicus interchanges the places of the sun and earth. In order to make the sun and fixed stars motionless, he attributes to the earth a triple motion on an eccentric orb, together with two librations, so that the observations of the fixed stars in every age may be able to be consistent with one another. Being opposed to the new astronomy, Nunes turned Copernicus' reasoning topsy-turvy. Copernicus did not start out with the intention of immobilizing the sun and stars. On the contrary, Copernicus began by recognizing the earth's true status as a planet. Its real daily rotation about its axis transformed the corresponding stellar motion into nothing more than an optical illusion. Hence the motionlessness of the stars was, for Copernicus, the result of one of the earth's motions, and not a purpose, as Nunes mistakenly asserted. By the same token, the sun's motionlessness was a consequence of two of the earth's motions, the annual orbital revolution as well as the diurnal rotation, so that again Nunes confused an outcome of Copernicus' thought with his motive. P. 172:9. Yet in Ill, 19 (fol. 99v, line 7) this distance was 96° 16' (= 0° 16' within the Crab). Here, before writing "0° 36' within the Crab" (fol. 106", right margin), Copernicus had put in the text "29° 57' within the Crab," the sign being changed to the Twins before both of these indications were deleted.

398 NOTES ON PP. 172-176

P. 172:14. 15 time-degrees= 1h = 6om

P. 173:11. Copernicus' emphasis on the close kinship between the earth and the moon became a distin­ guishing feature of his cosmology as contrasted with Aristotle's and Ptolemy's. For the Greeks, the moon was a ce­ lestial body, whereas the earth was not. With Copernicus, the earth became a celestial body, closely akin to the moon. This kinship provided a great follower of Copernicus, Kepler, with the foundation of his theory of mutual gravitational attraction between earth and moon. The generalization of this attraction by that other great Coper­ nican, Newton, gave physical astronomy one of its basic principles, universal gravitation. P. 175:9. This Prop. 5 of Euclid's Optics has already been cited in notes to pp. 11 :28 and 155:29. P. 176:2. "The motion of the heavenly bodies is uniform, eternal, and circular" (1, 4). P. 176:14. Here Copernicus carefully refrains from calling this "certain other point" by its newfangled name "equant." P. 176:20. This is one of the passages which must have impelled Kepler to explain to the readers of his N61JJ Astronomy why Copernicus rejected the equant. In Chapter 4 of that work Kepler wrote: With its center at B, describe the eccentric DE. Let its eccentricity be BA, so that A is the place of the observer [literally, eye]. The line drawn through BA will show the apogee in D and the perigee in F. On this line, above B, lay off another distance BC, equal to BA. C will be the equant point, that is, the point from which the planet traverses equal angles in equal times, although it disposes its circle not around C but around B. This device is denounced by Copernicus, V, 4, and IV, 7 [actually IV, 2] for other reasons but also on the charge that it violates the principles of physics by declaring the motions in the heavens to be nonuniform. For, choose a point Eon the circle traversed by the body of the planet, and connect E with C, B, and A. Now let DCE, like ECF, be a right angle. Hence these angles are equal, being traversed in equal times, and DCE, the exterior angle, is equal to CBE+CEB, the interior angles. Therefore, when the part CEB is subtracted, the remainder CBE or DBE will be smaller than DCE. Consequently FBE is larger than DCE or FCE. But the arc DE measures the angle DBE, and the arc EF measures the angle EBF. Accordingly, DE is smaller than EF, and the planet traverses them in equal times.

D

F

Therefore, the same solid sphere (Copernicus believes in solid spheres) to which the planet is attached is slow while the planet, as it is carried by the sphere, proceeds from D to E, and fast when it goes from E to F. As a result, the whole solid sphere is sometimes fast, sometimes slow. This is rejected by Copernicus as ridiculous (Gesammelte Werke, III, 73: 9-31). P. 176:33. PS, V, 13: lunar perigee= 33P 33'; V, 15: lunar apogee= 64P 10'. P. 176:41. Copernicus' reasoning at this point was assailed by James Reid of Edinburgh in 1618. Reid couched his assault in the following four propositions: (1) The moon's greatest distance [from the earth] is twice its least distance. (2) Although the moon does not appear twice as large at perigee as at apogee, there is a fallacy in Copernicus' reasoning by which he concludes that an object looks twice as big when it is twice as near. (3) Ptolemy's distances can be correct while, nevertheless, at both distances the [moon's apparent] diameter does not vary greatly. (4) And for this reason the science of optics will decide that if [two bodies of] equal size are observed at different

399 NOTES ON PP. 176-177

distances, the ratio of their apparent sizes is smaller than the ratio of their distances, since apparent sizes differing imperceptibly from each other can have quite a significantly [different] distance [from the observer] (Journal for the History of Astronomy, 1974, 5: 126, 131). Four years later, however, Reid receded so far from his defense of Ptolemy and opposition to Copernicus that he concluded: "we have no true knowledge of astronomical matters, especially those which depend on the pla­ netary motions" (ibid.). P. 176:43. This defect in the Ptolemaic theory had previously been pointed out in P-R (Book V, Prop. 22), whence Copernicus took it for use in his earlier Commentariolus, written between 1508 and 1514. P. 176:48. Ptolemy (PS, V, 14) mentions Hipparchus' (lost) explanation of this instrument, and his own use of it for measuring the apparent diameter of the sun and moon. P. 177:12. Because Ptolemy's equant violated the principle of uniform circular motion, it was not suited to be a component in the mechanical universe of Copernicus. Hence in his lunar theory he replaced the equant by this second or outer epicycle, on whose circumference he located the moon, while the center of this small second epicycle was carried around on the circumference of the first, larger, and inner epicycle. One result of this arrangement would have to be a periodic penetration by the moon into the space of the first epicycle. Hence this could not be a solid orb. For if it were, then the solid body of the moon would periodically crash into the solid body of the first lunar epicycle. Consequently Copernicus found himself in an acute dilemma. On the one hand, his mechanical universe could not accommodate the equant. On the other hand, the second epicycle, his own substitute for the equant, could not be a solid orb. In that case, what was this epicyclet? With regard to this perplexing and fundamental question, Copernicus preferred to maintain a discreet silence. He neither asserted nor denied the reality of the solid orbs. Their physical existence was later refuted by Tycho Brahe. In a discussion of the supposed sphere of Sat­ urn, Brahe conditionally supposed that "this is solid and real, as Copernicus too seems to have thought in agree­ ment with the opinion which had been accepted for a long time" (Astronomiae instauratae progymnasmata, Part II; Opera, II, 398: 32-34). But in Part III of the same work Brahe became less hesitant, saying "if the spheres were real, as Copernicus too admitted" (Opera, III, 173: 7-8). Kepler likewise categorically declared that Co­ pernicus "believed in the reality of the spheres" (Opera, ed. Frisch, I, 282: I, 10, line 3 up). Another firm believer in the reality of the spheres was Pedro Nunes. He approved of Copernicus' second lunar epicycle because it overcame the unacceptably large variation in the size of the moon's apparent diameter:

Not without reason does [Copernicus] place the moon on an epicycle of an epicycle, with the center of the smaller [epicycle] on the circumference of the larger [epicycle]. I point out, how­ ever, that the entire smaller [epicycle] must be enclosed within the larger [epicycle] to avoid shattering the heaven, if [Copernicus]. deems that [second epicycle] serviceable (Rules and In­ struments of the Art of Navigation, Basel, 1566, p. 106, lines 7-10).

Had Copernicus shared Nunes' view concerning the necessity of avoiding heavenly collisions and therefore placed the lunar epicyclet wholly within the epicycle, he would not have reduced Ptolemy's excessive variation of the apparent lunar diameter. For Copernicus, therefore, agreement with the observations was palpably more important than strict consistency with a theory, which need not be propounded in explicit terms. This overriding concern for agreement with the obserVations may explain why Copernicus ignored Tusi's lunar theory (on the assumption that he was somehow familiar with it), although he wholeheartedly adopted a modified form of the Tusi couple (III, 4). Tusi "expressly states that it is his aim to devise a model preserving the Ptolemaic extteme values for the moon's distance from the center of the earth" and "accepts these extteme values as undebatable" (Physis, 1969, 11: 291-292). Since any such Tusi model would retain Ptolemy's coun­ terfactual variation in the lunar apparent diameter, this part of the great Persian astronomer's theory was abso­ lutely worthless to Copernicus. Even more worthless is the recent remark that in Copernicus' lunar theory "the second epicyclic sphere is entirely contained within the first ... The intersection of spheres is not permitted" (Proceedings of the American Philosophical Society, 1973, 117: 467). The categorical assertion that "the intersection of spheres is not permitted" in Copernicus' astronomy is not supported by any reference to anything that Copernicus ever said or implied, is itself the merest product of a self-inflated historical dogmatism, and is in flagrant contradiction with Coper­ nicus' explicit statement here in IV, 3, that the moon on the epicyclet periodically penetrates into the region of the first epicycle, as shown by the diagram which Copernicus drew with his own hand (NCCW, I, fol. 109) and by our dogmatist's own Figure 14 (p. 468). P. 177:41. Here Copernicus' language (caetera mundi pura sint et diurnae lucis plena) echoes supra lunam

400 NOTES ON PP. 177-186

pura omnia ac diurnae lucis plena in Pliny, Natural History, II, 10, 48, while Copernicus' noctem non aliud esse ••• guam terrae umbram repeats Pliny's neque aliud esse noctem quam terrae umbram. Pliny's hebetari recurs in Co­ pernicus, but Pliny's talis figura semper mucrone deficiat is replaced by Copernicus' in conicam figuram nititur desinitque in mucronem (fol. 109r, line 2 up-109v, line 2). P. 178:15. By a gross error Copernicus wrote "thirty-seventh" (trigesima septima, fol. 109v, lines 12-11 up). Before expressing this number in letters, he had originally used Roman numerals ( xxX'Oij), which he deleted. This deleted numeral suggests how Copernicus may have come to make this mistake. If his computations on a separate work sheet pointed to Meton's Olympiad as the 87th (lxxxvij), in transcribing this result in his auto­ graph Copernicus may have lost sight of the initial "l". His blunder was overlooked inN (fol. 101r) and was first publicly corrected in W (p. 255). In his copy of N, however, Miistlin replaced trigesima by octoge­ sima. P. 178:16. Copernicus found this expression in Censorinus' discussion of Meton (ch. 18). However, Coper­ nicus reversed the order of the components in Censorinus' 1;1umeral dekaenneateris (ed. Venice, 1498, sig. d3V). P. 178:23. Copernicus' reference to 304Y shows that he was relying on Censorinus rather than on Ptolemy, who always uses the round number "about 300Y" (PS, III, 1). P. 178:25. By an oversight Copernicus wrote 760 (Dcclx, fol. nor, line 5), instead of 3760. His error was overlooked in N (fol. 101 v) and was first corrected in W (p. 255). P. 186:33. By a slip Copernicus wrote 135 (cxxXfJ; fol. 114r, line 7 up). In the next paragraph he dates the 3rd eclipse "1Y 137d 5h" after the 2nd eclipse, which occurred on 20 October 134. Copernicus' erroneous date 135 was printed inN (fol. 105v, 106~, and first corrected in A (p. 247, misnumbered 239).

P. 186:39. Bull: 30°-13!0 16° 45' Balance: 30°-251/ 8 o 4° 50' 4 whole signs 120 4 whole signs 120 Balance 25 10 Fishes 14 5 161° 55' 138° 55' By a slip, as the number of degrees Copernicus wrote 137 (cxxX'Oij, omitting ani, fol. 114v,line 3), an error not previously corrected. P. 186:40. 1st observation: 17 Hadrian 19 Pauni nh 1sm remainder in Pauni lQd 12 45 Epiphi, Mesori, 5 intercalary days 65 18 Hadrian 1Y 19 Hadrian: Thoth, Phaophi, Athyr 90 2nd observation: Choiach 1 11 interval: P. 186:42. 2nd observation: 19 Hadrian 1 Choiach in 19 Hadrian: Choiach 8 months+5 intercalary days in 20 Hadrian: 7 months Pharmuthi 18 16 502 5 -365 interval: lY 137d 5h P. 186:44. Sun Moon 1Y 359° 44' 129° 37' 120d = 2x60d (60+58) 118 16 (60X 24) 22 53 46d 45 20 (360)+ 180 20 46 58 12 3 524° 18' 365° 19' -360 -360 164° 18' 5° 19' 5 19 combined uniform motion of sun and moon 169° 37'

401 NOTES ON PP. 186-188

P. 186:45. lY 88° 43' 120d = 2 X 60d (1560 = 1440 +) 120 7 47 (360+) 240 59 12 52 470° 21' -360 lunar motion in anomaly 110° 21' P. 186:46. Sun Moon 1Y 359° 44' 129° 37' 120d = 2X60d (60+58) 118 16 (60X24) 22 53 17d 16 45 (180+27) 207 14 5th 14 2 48 494° 59' 362° 32' -360 -360 ---- 134° 59' 2° 32' 2 32 combined uniform motion of sun and moon 137° 31'

The deficiency of 3', in comparison with 137° 34', Copernicus' figure, results from neglecting the sixtieths of a minute and a second. P. 187:1. 1Y 60° 28 43' 2X60d 26X60° = (1500-1440)+120 7 47 17d 180 42 6 5th 3 1Y 137d 5}h 441° 36' -360 81° 36' P. 187:2. 169° 37' -161 55 7° 42 P. 187:3. 138" 55' -137 34 1° 21'

P. 187:38. By an arithmetical error Copernicus wrote 1220460 (fol. 115r, line 10 up). The faulty digit 6 was printed in N (fol. 106v), and was first corrected in A (p. 249). P. 187:45. LM=2KM DM+KM = DK DM+2KM = LD (DM+KM)2 = DK2 DM(DM+2KM) = LDXDM (1) DW+IDMX~+~2 =~ W DW+IDMX~=illX~ (1-2) ~2 = DK2 - LDXDM LDxDM+~2 = DK2

P. 188:18. By a slip (fol. llSV, line 12 up), as the number of the minutes Copernicus wrote 49 ( iL), although he actually operated with 59, the correct figure, a few lines below, where he calculated the moon's mean place in the third eclipse. The correction from 49 to 59 is being made here for the first time publicly, but it was made privately by Brahe in his copy of B (fol. 107v, line 1).

402 NOTES ON P. 188

P. 188:25. 1st eclipse: from Scorpion 9° 53', to Bull 12° 21' Scorpion 9° 53' 5 whole signs 150 Bull (30°-12°21') ~ 17 40 177° 33' 2nd eclipse: from Ram 29!0 to Balance 26° 43' Ram 29° 30' 5 whole signs 150 Balance (30°-26° 43') 3 17 182° 47' 3rd eclipse: from Virgin 17° 4' to Fishes 11° 44' Virgin 17° 4' 5 whole signs 150 (30°-11 ° 44') 18 16 185° 20' P. 188:32. Eclipse's end 1~2om beginning 10 52 308 duration 3 27 30 half-duration 1 43 45 7 mid-eclipse 12 36 15 ~ 12h 35m = f12h after midnight

Copernicus wrote "th+1h2" in the right margin of fol. 116r, and forgot to delete "plus the twelfth part of an hour" in line 8. P. 188:37. By means of the fifth note in the right margin of fol. 116r, Copernicus originally timed the be­ ginning of this second eclipse at "two-fifths and the twentieth part of a uniform hour" before midnight. When he deleted the twentieth part (vigesima parte), he removed the smallest subdivision of time connected with an observation made by himself and reported in the Revolutions. In like manner, he used 1/ 20h in the seventh note in the right margin of fol. 116r to help mark the beginning of his third lunar eclipse, but again he deleted et vigesima. Copernicus' introduction of an interval as short as 3m in the timing of an astronomical observation suggests that he may have occasionally resorted to a mechanical clock. However, he makes no mention of any such time-measuring instrument anywhere in the Revolutions, and his double deletion of 1/ 20h in IV, 5, may imply his lack of confidence in that relatively recent invention. Thus, when he reports his observations of planets, he is usually content with the even hour: 5 uniform hours after midnight (V, 9); 11 hours after midnight, 3 hours after midnight, 19 hours after midnight (V, 11); 1 hour after midnight, 8 hours after noon, 7 hours before noon, 1 hour after sunset, and the beginning of the 8th hour after noon (V, 23). These indications of the time of an observation suggest that Copernicus may have been using an hourglass, in which the sand ran out of the upper container into the lower in an hour, although he makes no explicit mention of such a device. The smaller model, which emptied out in half an hour, may have been the basis of Copernicus' timing of the autumnal equinox on 14 September 1515 at "th after sunrise" (III, 13, 18; fol. ssr, right margin, 3rd entry; fol. 98v, left margin). But Copernicus evidently did not rely exclusively on sand clocks, whether of the hourly or half-hourly variety. Thus, he times two oppositions of Saturn (V, 6) at "11/ 5h before midnight" and "62/ 5h after midnight" (fol. l5¥, right margin). Here too in IV, 5, as we saw at the beginning of this Note, " 2/sh before midnight" marked the beginning of an eclipse. Does this fractionizing of an hour indicate resort to a mechanical clock?

The first lunar eclipse in IV, 5, is said to have begun "P/8h before midnight" and to have ended "21/ 3h after midnight" (fol. 116r, lines 4, 6). These parts of an hour were changed by Copernicus, from 1/ 8 to 1/ 3 and back again to 1/ 8, and from 1/ 3 to 1/ 8 to 1/ 3 to 1/ 8 and finally to 1/ 3, perhaps because his notes did not make it absolutely clear which fraction belonged with which observation. Unsympathetic critics have suspected Copernicus of fudg­ ing his figures. Did such persons really understand how people kept track of time in the sixteenth century, before the invention of the pendulum clock? In this connection a letter written to Rheticus by Matthias Lauterwalt early in 1545 is very instructive (Bur­ meister, Rhetikus, III, 59-60). Rheticus had timed the beginning of the lunar eclipse of 28 December 1544 at Leipzig at 3: 30 A.M. "But," Lauterwalt reproaches him, "you did not add whether that happened according to an accurately corrected clock or according to an uncorrected ordinary clock. But it is certain, if this was really your observation, that that clock was completely wrong and was slow ... Copernicus ... observes in accordance with an accurately corrected clock ... When I observed the eclipse, according to the clock on the church at Wit-

403 NOTES ON PP. 188-189 tenberg it was seen about half of the quarter-hour before 4 [ =3: 52! A.M.]. At the same time by means of the sand clock I also observed the hours of the [church] clock so that they would not be so nonuniform only to per­ ception, and I observed that sunrise occurred 41Il before 8 [o'clock]. Hence the error of the clock was made clear. For if it had struck correctly, then the sun would have risen at 8:07. Therefore it was slow by 11m." Lauter­ walt's use of the sand clock to check the mechanical clock in 1544 and his conclusion that the latter was off by 11m may be analogous to Copernicus' introduction of an interval as short as ~. presumably obtained from a me­ chanical clock, and his subsequent discarding of that interval as too uncertain to be used. P. 188:47. Balance (30°-22° 25') 7° 35' 10 whole signs 300 Virgin 22 12 329° 47' P. 188:48. Virgin (30°-22° 12') 7°48' 11 whole signs 330 Virgin 11 21 349° 9' P. 189:1. From 12: 35 A.M., 7 October 1511 to 1: 20 A.M., 6 September 1522: 1511 October 24d 23h 2sm November 30 December 31 10 whole years 1522 January through August 243 September 5 1 20 leapdays (1512, 1516, 1520) 3 10Y337d ~ P. 189:3. From 1:20 A.M., 6 September 1522 to 4:25 A.M., 26 August 1523: 1522 September 24" 22h 40m October through December 92 1523 January through July 212 August 25 4 25 354"- ~ sm P. 189:5. Sun Moon 180° 357° 28' 36 13' 295 40 57 13 36 28 (7 X 60 = 420-360) 60 31 3 2 24 689° 38' 364°53' -360 -360

lOY 337d 'J6h 329° 38' 4° 53' Forgetting that he was operating with the combined mean motion of the sun and moon, at first Copernicus wrote the figure for the sun alone, 329° (fol. 1161', line 2 up). Then he remembered that he had to add to 329° the figure for the moon's motion. Consequently he deleted 329° in the text, and wrote 334° in the right margin. For the minutes, he had originally written 43 (xliij), which he changed to 47. The sum would be 31', if the fractions of a minute are disregarded, as above. P. 189:6. lOY (2X60°) 120° 47 11' 300 19 29 (480°-360°) 120 324 26 610° 30' -360 250° 30'

404 NOTES ON PP. 189-190

For the number of minutes Copernicus originally wrote 33 (xxxiij; fol. 1161', line 2 up), which he changed to xxX'Oj in the text, confirmed by Arabic 36 in the right margin. The foregoing computation is lower by a few minutes because fractions of a minute were ignored. P. 189:8. Sun Moon 30od = 5x6od 2400 57° 13' 55 40' 54d 53 13 (600°-360°) 240 58 18 3hgm 8 1 :35 354d3" gm 3490 1' +357 6 706 7 -360 346° 7' Forgetting that he was operating with the combined motion of the sun and moon, at first Copernicus wrote the number of degrees for the sun alone, 349° (cccxlix; fol. 116v, line 1). The number of minutes given above is some­ what lower than his, because again fractions of a minute have been disregarded here. P. 189:9. 30oc' = 5x 60d 300° 19 29' 54d (11 X 60° = 660°-360°) :300 .e:30 1 :38 666° :37' -:360 :3060 :37' This result is 6' lower than Copernicus' because fractions of a minute were disregarded. P. 189:15. BAC = :306° 4:3' .·.CB(= :360°-BAC) = 5:3 17 ACB=250 36 -CB = 5:3 17 AC = 197° 19' P. 189:16. 5° -2 59' 2° 1' P. 189:24. In writing the number of minutes, Copernicus apparently forgot to include the o in xoiiij (fol. 116v, line 17 up). To rectify the omission, he wrote a o heavily over the first i, so that N mistakenly printed this number as 18 (fol. 10sr). The error was corrected in W (p. 267), relying on the four dots clearly visible over the numeral, although it shows only the last 3 i's. P. 189:31. DE:AE = 19,865:702 = 8024:28:3.6, for which Copernicus writes 28:3. P. 189:47. DF:FG = 116,226:10,000 = 100,000:8603.9, for which Copernicus writes 8604. P. 189:48. These post-Ptolemaic and pre-Copernican astronomers have not yet been identified. P. 190:22. Here Copernicus uses the new geokinetic terminology ("the earth's annual" [motion]), whereas a few lines below, in the beginning of IV, 6, he reverts in the same context to the traditional pre-Copernican phraseology, "moon's distance from the sun," "moon's motion away from the sun." P. 190:23. 1st eclipse, from Balance 24° 13' to Ram 22° 3' Balance 5° 17' 5 whole signs 150 Ram 22 :3 177° 20', or 31' less than Copernicus' figure. Originally he had put the moon's mean motion in the 1st eclipse at 22° within the Ram plus 13' (fol. 111v, line 7). 2nd eclipse, from Virgin 23° 59' to Fishes 26° 50' Virgin 6° 1' 5 whole signs 150 Fishes 26· 50 182° 51'

405 NOTES ON PP. 190--191

3rd eclipse, from Virgin 13° 2' to Fishes 13° Virgin 16° 28' 5 whole signs 150 Fishes 13 179° 28', or 30' less than Copernicus' figure. Originally he had put the moon's mean motion in the 3rd eclipse at 13° within the Fishes plus a certain (now undecipherable) number of minutes; fol. 117", line 11. P. 190:37. From 19 Hadrian, 2 Choiach = 20 October 134 (IV, 5) to 5 September 1522: 134 October 11 d 2h November- December 61 1387 complete years 1522 January- August 243 September s 1 2om leapdays (136-1520) 347 667d -365

P. 191:9. 19 Hadrian, 2 Choiach = 20 October 134 (IV, 5) 133Y leapdays (4-132) 33d 134 January through September273 October 19 22h 133Y

P. 191:11. 120Y = 2X60Y (4X60X60° = 14400°) (19X60° = 1140°-1080° =) 60° 14 45' l3Y 240 5 5

57 13 300 4 46 11 692° 49' -360

P. 191:12. 120Y = 2X60Y (2X60X60° = 7200°) (57 X 60° = 3420°-3240° =) 180° 26 18' 13Y 60 13 20 3ood = 5x60d (60x60° = 3600°) (5X60° =) 300 19 29 300 26 37 11 42 937° 26' -720 217° 26', with a slight discrepancy from Copernicus' figure for the minutes, because here the fractions of minutes were disregarded.

406 NOTES ON P. 191

P. 191:14. 182° 47' 64° 38' +360 +360 542° 47' 424° 38' -332 49 -217 32 209° 58' 207° 6' With regard to the number of minutes for this lunar anomaly, Copernicus' figure has been read as 7 (fol. 11sr, line 11 up). However, the first i is prolonged downward as a terminal j and is also surmounted by a dot. The vertical stroke to its right has been read as an additional i, despite the absence of a dot and a swerving toward the right, whereas Copernicus' final j normally turns toward the left. P. 191 :16. Copernicus arrives at the total of 19-ij-d by reckoning from noon on 21 June, the day of the sum­ mer solstice, to midnight preceding 1 January: June 1od July through December 184 from noon to midnight ! 194!-d Then he equates 193 Olympiads, 2Y, 194fd, with Egyptian years: 193 Olympiads= 4YX 193 = 772Y + 2 774Y in 772Y, leapdays 193d +194! 387!-d -365 775Y 22!-d However, as the number of days, Copernicus gives 12!-d, as he did above (III, 19). But there he equated 1 He­ catombaeon with 1 July, thereby correcting for the lOd by which the Julian calendar had slipped backward in his own time (nunc, "now," as he says there). Here, on the other hand, he implicitly equates 1 Hecatombaeon with 21 June by making the total of days 194!, over and above complete years, in the Olympic-Christian interval. Forgetting that the total of 775Y 12!-d for that interval in III, 19, was based on his Julian correction, he repeats that total here, although 193 Olympiads, 2Y 194fd = 775Y 22}d, not 12!-d. P. 191:19. In III, 11, instead of giving the undivided period from Alexander to Christ as 323Y 130td, Copernicus inserted Julius Caesar and Aui~Jstus: from Alexander to Caesar 278Y 118!-d Caesar Augustus 15 246! Augustus Christ 29 130! 495}d -365 1Y 323Y 130!-d P. 191:20. As the sum of the two sub-intervals in the preceding note: from Caesar to Augustus 15Y 246!-d Augustus Christ 29 130! 377d -365 1Y 45Y 12d P. 191:30. In saying that "Frombork... is located at the mouths of the Vistula River," Copernicus was expressing the geographical conception of his own time as well as of ours. This was an "apparently senseless mystification," according to one of our poorly informed contemporaries. For a complete refutation of his ridic­ ulous accusation against Copernicus, see 3CT, pp. 290-291. As the name of the river, Copernicus uses the Latin form Istula (fol. 11sv, line 13), which was one of a number of variants used by the geographers and historians of antiquity and also of his own time.

407 NOTES ON PP. 191-199

P. 191:32. This information about eclipses observed simultaneously in Frombork and Cracow was ob­ tained by Copernicus from letters he received from Cracow mathematicians. These letters, now lost, were still available in the 17th century, as is clear from the revised biography of Copernicus in the 2nd edition of Szymon Starowolski's Scriptorum polonicorum Bxat't'ovrtit; (Venice, 1627). P. 191:33. The city Epidamnus was founded on the peninsula called Dyrrachium, which later gave its name to the city too. Today it is situated in Albania, where it is known as Durri!s, although better known as Durazzo, its appellation in Italian. P. 192:13. According to Lines Subtended, for 7° 40': 13341, with radius= 100,000, or 1334, with radius = 10,000. P. 193:14. By a slip Copernicus wrote (fol. l19r, last line) instead of jl, el (not previously corrected). P. 193:15. CE:EL = 1097:237 = 10,000:2160.4, for which Copernicus writes 2160. P. 193:17. According to Lines Subtended, 21,644 for 12°30'; and 21,360 for l2°20'(by a slip, 21,350, fol. 1~); hence, 21,587 with radius= 100,000, or 2159 ~ 2160, with radius= 10,000, for 12° 28'. P. 193:32. Copernicus says "91/8 hours of the day" (horis diei 110'0em et triente transactis, fol. 119v, Ch. 10, lines 7-8), following PS 1515, fol. 49v, line 13 up: 110'0Bm horis et tertia horae diei praeteritis. These hours of the day are reckoned from sunrise= 6 A.M. Brahe in his copy of B (fol. 111v, line 7) deleted no'Oem = 9, and replaced it in the margins by 3 P.M. P. 193:33. Sun at Crab 10° 54' Crab 19° 6' moon at Lion 29 distance moon-sun 48° 6' P. 193:38. At Lion 29°, the moon was 3 signs= 90° from Scorpion 29°, then rising. P. 193:41. Here 31/ 3h in the afternoon (a meridie) is equivalent to 91/ 8b. of the day (diei), reckoned from sunrise = 6 A.M., near the beginning of this paragraph (see note to p. 193 :32). P. 193:42. Ptolemy mistakenly believed that Rhodes lay on the same meridian as Alexandria (PS, V, 3). How the displacement of Rhodes 1/8h = 2!0 west of Alexandria was found by Copernicus is still unknown. According to Copernicus, 41/8h Alexandria= 4'1 Rhodes= 31/8h Cracow. Hence, Copernicus put the 4ifference 5 0 in longitude between Rhodes and Cracow= /8h = 12! • This exceeds modern values by about 5°= 1fah· P. 193:43. 197 Alexander, 17 Pauni = 196Y 9x3Qd = 27od Pauni 16 196Y P. 193:45. As the number of minutes, Copernicus originally wrote, following Ptolemy, 5 (t~; fol. 1~, line 6). Putting a dot over the "• he converted it into an i, to which he added 2 more i's. P. 194:2. For the number of minutes, Copernicus originally wrote 9 (ix), which he deleted and replaced by 5 (v; fol. 12or, line 10). P. 194:9. Forgetting that he had already lowered this moon-sun distance to 45° 5' (see the preceding note), here Copernicus left the number of minutes at 9 (ix; fol. 120r, line 15 up). Yet in the line immediately above, he diminished arc FG from 90° 18' to 90° 10' (= 2X45° 5'). P. 194:22. True sun at Crab 10° 40': Crab 19° 20' apparent moon at Lion 28°37 distance moon-sun 47° 57' P. 194:43. The expression "head of the Dragon" for the moon's ascending node was found by Coper­ nicus in GV, Book XVIII, Ch. 4 (sig. gglr). But whereas GV ascribed this term to the "barbarians" (barbaros), Copernicus altered this harsh description to the "moderns" (neoterici; fol. 12ov, line 6 up). The identification of the lunar nodes with the extremities of a fictitious dragon is examined historically by Hartner, Oriens-Oc­ cidens, pp. 359-377. P. 194:44. The "tail of the Dragon" is the "modern" term for the moon's descending node. P. 195:31. 2° 44': 1° 33' = 164:93 = 60:341/ 41, for which Copernicus writes 34. P. 195:33. For the number of minutes, here Copernicus wrote and retained 18 (Xtliij; fol. 121r, line 6 up). When he later returned to IV, 10, and there lowered the given arc from 90° 18' to 90° 10' (note to p. 194:9), he forgot to make the corresponding reduction here. P. 198:29. By the common scribal error called "dittography," Copernicus repeated the word "latitude" (fol. 122v, line 5 up), although "longitude" is clearly required by the context, as was first pointed out in Me (p. 222, n. 313). P. 199:28. Copernicus puts the midtime of the eclipse at 2 seasonal hours after midnight (fol. 123v, line

408 NOTES ON P. 199

15). On the other hand, PS 1515 (fol. 63") correctly repeats Ptolemy's statement that the midtime occurred at zth, as P-R (Book VI, Prop. 6) also did. P. 199:30. Copernicus puts the sun's longitude at 6° within the Bull (fol. 123v, line 17), whereas PS 1515 said 7° (fol. 63v), as did also P-R (Book VI, Prop. 6), both correctly.

P. 199:33. By a slip Copernicus wrote 3/ 5h (quintis; fol. 123v, line 14 up), inconsistent with 45m = 3/ 4h (fol. 1241", line 1: scrup xlv). The correction of quintis to quartis was made privately in Mastlin's copy of N as well as in Brahe's copy of B (fol. 11SV, line 10 up), but not publicly in any previous edition or transla­ tion. P. 199:37. 150 Alexander, 27 Phamenoth, 2:20 A.M.= 6x3od = 18od Phamenoth 26 149Y P. 199:38. When he first reduced this eclipse to Cracow time, th behind Alexandria time, Copernicus changed 21/ 3h to 11/ 3h, without labeling it uniform time (fol. 123v, line 18). Here, when he repeats 141/ 3h = 21/ 3h, Alexandria time, he mistakenly thinks this is local time, although he has called it "21/ 3 uniform hours" (horas aequinoctiales duas cum triente; fol. 123v, line 16). Hence, when he subtracts Ih and obtains 13 1/ 3h, he names this the local Cracow time, which he subjects to a correction of 1om in order to obtain the uniform time at Cra­ cow. But the Cracow uniform time would have been 131/ 3h, not 13jh. P. 199:41. In the Table of the Lunar Prosthaphaereses, after IV, 11, 1° 23' is the first epicycle's pros­ thaphaeresis for 165° ~ 163° 33'. P. 199:43. From Alexander to Christ 323Y 130d 12h 16m (IV, 7) from Christ to 2nd eclipse, complete years 1508 1509: January through May 151 June 1 11 45 1 1 0 1 leapdays (4-1508) 377 660 1-365 1832Y 295d Oh 1m= 11h 5~ less than Co­ pernicus' interval, 1832Y 295d 11h 5sm. Copernicus evidently stumbled over this computation. Initially, his total for the days was 3, then 88, and for the hours 22 plus a fraction (fol. 123v, last 2 lines). P. 199:45. According to the Tables of the Moon's Motion in Years and in Days, after IV, 4, 1800Y = 30X60Y (4X60X60° = 14400°) (48X60° = 2880°) 41° 18' 12" 0'" 32Y 3X60° 180 7 56 3 25 z4Qd = 4 x 60d (48X60° = 2880°) 45 46 46 55d (11X60° = 660°-360°) 300 10 29 28 2 nh 55m ~ 6 591° 30' 29" 27"' -360- 231° 30' Alexander's era (IV, 7) +310 44 542° 14' -360 182° 14' 29" 27"' Initially, Copernicus' number for the minutes was apparently 15 (xv), but later he squeezed in 3 i's (fol. 124r, line 3).

409 NOTES ON PP 199-200

P. 199:45. According to the Tables of the Moon's Motion in Anomaly in Years and in Days, after IV, 4, 1800Y = 30X60Y (2X60X60° = 7200°) (21X60° = 1260°-1080° 180° 34 33' 37" 32Y 300 19 0 51 52'" 24Qd = 4X 6()d :52 X 60° = 3120°-2880° = 240 15 35 46 300 58 34 26 47 6 1153° 44' 41" 39'" -1080 73°45' Alexander's era (IV, 7) + 85 41 159° 26' For the number of minutes, Copernicus left 55 (lfl), after deleting 1 or 2 i's (fol. 12¥, line 3). For the number of normalized degrees, initially he wrote 141 (cxlj), which he corrected to 161. P. 199:46. In the Table of the Lunar Prosthaphaereses, after IV, 11, the first epicycle's prosthaphaeresis for 159° : 1° 55'; for 162° : 1° 39'; hence, 1° 43' for 161° 13'. P. 200:1. At the time of the 1st eclipse the solar apogee preceded the summer solstice(= Crab 0°) by 24!0 = Twins5!0 (III, 16); hence, the sun at Bull6° preceded the apogee by 29!0 .At the time of the 2nd eclipse the apogee followed the summer solstice by 68/ 8 ° ( = Crab 61/ 3 °; III, 16); hence, the sun at Twins 21° preceded the apogee by 151/ 8°, P. 200:2. 1st eclipse: 7ha of the moon's diameter= 7 digits 2nd eclipse: 8/11 = 8 digits. P. 200:5. 2nd eclipse: t 0 farther from the node, because 1 additional digit was darkened; ascending node, because the moon's southern side was darkened. 1st eclipse: descending node, because the moon's northern side was darkened. P. 200:7. In passing from one node to the other, the moon moves 180° in latitude; in this instance it was ! 0 away from completing such a passage. P •. 200~12. 2nd eclipse: 1832Y 295d nh 45m (local) ssm (uniform) -1st eclipse: 149 206 13 20 30 1683Y ~ 22h 25m 25m For the number of minutes Copernicus wrote 1 x too many (XXXfJ; fol. 124r, line 13 up}. This correction was made in Miistlin's copy of N (fol. 116f, Ch. 13, line 5 up). P. 200:13. The moon's latitudinal motion completes 13 revolutions a year plus 148° 42' of the 14th revolution (IV, 4): 1683Y X 13 = 21,879 revolutions 1683X 148° 691 +324° 1683X42' 3 + 98°6' 1 3 22,577 revolutions P. 200 :32. Initially Copernicus located Cracow 6 ° east of Rome (fol. 124v, line 15). By lowering this distance to 5°, he moved farther away from the true value, 7 1/2°. Whatever Copernicus' source for the distance Cracow­ Rome may have been, it certsinly was not the Alfonsine Tables, 1492 edition. For its Table of Longitudes and Latitudes of the Principal Cities and Regions of Europe (sig. el") placed Cracow at 2h2om and Rome at 1h40m east of the prime meridian. The time difference between these two cities would therefore have been 40m = 10°. It has been mistakenly asserted (Proceedings of the American Philosophical Society, 1973, 117:426) that the Alfonsine Tables and one other book "together provide the working library for an astronomer of Copernicus' time". In the Reuolutions Copernicus never cites the Alfonsine Tables. Their value of 10° for the distance Rome-Cracow is exactly as far above the true value as Copernicus' final value was below it.

P. 200:33. With 15° = lh, at 5° east of Rome, Cracow's local time precedes Rome's by 1/ 8h,

410 NOTES ON PP. 200-201

P. 200:35. From Alexander to Christ: from Christ to eclipse: complete years 1499 1500, January through October 304 November 5 2 2om leapdays (4-1500) 375 814 -730 + 2 84"- 1824Y 84d 14't 2om P. 200:37. Anomaly, 2nd eclipse: 294° 44' = 65° 16' from the higher apse Anomaly, 1st eclipse: 64° 38' P. 200:40. 1st eclipse 2nd eclipse Higher apse Twins 5° 30' Crab a middle apse Virgin 5 30 Balance 640 24 30 23 20 sun at Balance 25 10 Scorpion 23 16 sun's distance from a middle apse 49° 40' 46° 36' P. 200:42. As the basis of his inference that the latitude was southern, Copernicus might have recalled that the shadows were in the north. P. 200:45. apparent uniform 2nd eclipse: 1824Y 84d = 1823Y 449d 14't 2om 14't161D = 13h 761D 1st eclipse (IV, 14, below) - 457 91 10 9 54 --1~3~66~Y~3~5-8d,---4't.-2~0~m------4't~22_m_

Initially Copernicus wrote 22, the correct difference for the uniform minutes (fol. 124v,last line). Then he noticed that he had omitted the number for the hours. Hence he deleted the 22, inserted 4 as the correct number of hours, and wrote 24 for the apparent minutes. Catching this error, he deleted the 4 i's, leaving the correct number 20. Proceeding to the uniform minutes, initially he wrote probably 26, over which he wrote 24 (fol. 125', line 1). An indication of this transformation is the presence of 2 final long-stroke i's. These alterations of the nuinbers for the minutes are connected with the changes in the figure for the minutes from Alexander to Copernicus' observation. There his final number was 20, superseding 2 previous figures (perhaps 24, then 12; fol. 124Y). P. 200:46. According to the Tables of the Moon's Motion in Latitude in Years and in Days, after IV, 4, 1366Y = 46Y +(1320Y = 22X 60Y) 22 X 60Y: 31 X 60° = 1860°-1800° 60° 40 36' 46Y 46 358d = 58d+(30od = 5X6od) 5x60d: 60x60° = 3600° 6X60° = 360° 848

47 18 2 27 1366Y 358d 159° 55' P. 201:12. From Alexander to Christ 323Y 130d 12h from Christ to Ptolemy's observation at 10 P.M. on 20 October 134; complete years 133 134 January through September 273 October 19 22 leapdays (4-132) 33 1 1oh 456d 1 -365 457Y 91d 1oh

411 NOTES ON PP. 201-202

P. 201:20. This interval from the 1st Olympiad to Alexander was given above, in III, 11, as 27Y 247d from 1st Olympiad to Nabonassar +424 from Nabonassar to Alexander 451Y 247d from 1st Olympiad to Alexander. P. 201:22. The interval from Alexander to Caesar was given above, in III, 11 as 278Y 118id +451 247 from 1st Olympiad to Alexander 1 365ld

730Y 12h from 1st Olympiad to Caesar.

P. 202:10. The title of PS, V, 12, does indeed refer to the construction of the parallactic instrument (organon parallaktikon). This term, however, was not used in P-R, V, 13, which calls this instrument "Ptolemy's rulers' (regulae Ptolemei, sig. f 2v). By the same token, PS 1515, V, 12, does not use the expression "parallactic instrument," speaking instead of the "instrument by which the amount of the lunar parallax is determined" (fol. 53r). Since Copernicus wrote Revolutions, IV, 15, on paper D in quire n, a normal quinternion, he could not have based his statement (fol. 125v, line 17) that Ptolemy called this device the parallactic instrument on the Greek text of PS, to which he had no access before the summer of 1539. We may therefore conclude that Copernicus learned about Ptolemy's nomenclature from some source other than the first printed edition of the Greek te"xt of PS, where organon parallaktikon occurs on pp. 107, 121. This term was employed also by Proclus' Hypotyposis, IV, 49, but there it was derived from the use of the instrument to detect parallaxes. Hence this Proclus passage, even if it were known to Copernicus, would hardly have induced him to state that Ptolemy calls this device the parallactic instrument. In any case, Copernicus' familiarity with Proclus' Hypotyposis came to him through GV. The latter's translation of Proclus' parallaktikon organon is commutatile ... instrumentum (fol. ff 4v, line 10), a far cry from Copernicus' terminology. It seems clear, then, that Copernicus' awareness of Ptolemy's name for the parallactic instrument was not obtained from P-R, PS 1515, GV, nor from the Greek texts of PS and Proclus' Hypotyposis. If further investigation should succeed in disclosing this source, perhaps we shall then also understand why, in his account of the construction of the parallactic instrument, Copernicus omits Ptolemy's plumb line, supports his own device on a vertical pole around which it can swing, unlike Ptolemy's instrument which was fixed in the plane of the meridian, and changes the markings on his ruler from Ptolemy's sexagesimal system of 60 and subdivisions thereof to 1414 = 1000 X .,fi as the side of the square inscribed in a circle of unit radius. Copernicus' parallactic instrument was later acquired by Tycho Brahe, who reported in his Astronomiae instauratae mechanica that I have acquired [an instrument] of this kind made entirely of wood alone. It had once belonged to that peerless man, Copernicus. It had even been made with his own hands (as was said). It was sent to me as a gift by Johannes Hanow, a canon of Frombork, where Copernicus used to live ... [In 1584 Brahe sent a student to Frombork, as we saw above in note top. 121 :19. This Canon Johannes Hanow was a nephew of the Canon Johannes Hanow who died on 23 January 1575; ZGAE, 1929, 23: 755, no. 44]. When my student came back to me, he not only returned in perfect condition the sextant which I had entrusted to him but he also brought this second [device], the Copernican parallactic instrument, sent as a gift to me from that canon whom I mentioned. As soon as I saw it, even though it was made of wood and was awkward to use, nevertheless because it was a reminder of so great a master, who was said to have made it, I was so happy that I could not stop myself from ... composing at once a poem in epic meter.

When Brahe published this poem of 34 hexameters in his Letters (1596; Opera omnia, VI, 266-267), he dated its composition on 23 July 1584. He pointed out that Copernicus had used black ink for the dividing lines (VI, 253: 28, 265: 38), had tried to overcome the warping of the wood, which Brahe identified as fir (VI, 104: 1), and had designed inconvenient eyepieces. These were in the Copernican instrument ... holes, through which the stars are seen with great difficulty. There is this additional disadvantage that the forward hole ... must be wider than the other, if the stars are to be observed properly through the former. As a result, this must take up a frac­ tional part of 1°, that is, at least 1/8 ° or 1/10 °. But during the observation it is not known whether or not the star is seen exactly in the center of that hole. Thus, an error of several minutes can

412 NOTES ON PP. 202-205

occur. Hence it is a wonder how not only Copernicus but also the ancients, who used such eyepieces, could have attained any precision, even if everything else was in perfect order (Brahe, Opera omnia, v, 46: 9-18). Copernicus' parallactic instrument shared the fate of Tycho's other instruments, which were destroyed during the Thirty Years' War. See John L. E. Dreyer, Tycho Brahe (New York: Dover, 1963; reprint of the Edinburgh, 1890 ed.), pp. 125, 365-366. P. 203:5. In giving the sun's position as 5° 28' within the Balance (fol. 126r, last 2 lines), Copernicus neglected to specify that, according to Ptolemy, this was the true sun, whereas the mean sun was at 7° 31' within the Balance. Hence, the moon's elongation from the (mean) sun being 78° 13', Balance 22° 29' Scorpion 30 Archer 25 44 = mean moon 78° 13' Archer 25° 44' prosthaphaeresis 7 26 Goat 3° 10', as given in PS 1515 (fol. 541'), not 3° 9', as in Copernicus (fol. 126v, line 4), who mis­ takenly followed P-R (Book V, Prop. 15). P. 203:11. Moon's zenith distance 50° 55' moon's declination 23° 49' -1 7 latitude of Alexandria 30 58 ®o- ~o4T moon's latitude -4 59 49° 48' P. 203:32. From the beginning of the Christian era to- 5: 40 P. M., 27 September 1522: 1521 complete years 1522 January through August 243d September 26 leapdays (1520-;...4) 380 649 1 -365 1522Y P. 203:40. Copernicus' determination of the latitude ofFrombork as 54° 19' was found by Brahe's assistant to be somewhat too low (Dreyer, Brahe, p. 124). P. 203:47. From the beginning of the Christian era to 6 P.M., 7 August 1524: 1523 complete years 1524 January through July 212d August 6 leapdays (4-1524) 381 599 1 -365 1524Y P. 204:30. According to Lines Subtended, for 50', 1454; AC:CE = 1454:99,219 = 1:68.2, for which Co­ pernicus writes 68. P. 204:34. Initially Copernicus put the moon's apparent zenith distance in the 2nd observation of IV, 16, at 81° 42!-' (fol. 127r, line 12 up). Later he changed this value to 82° by adding a 2nd terminalj to the number of degrees, deleting the reference to minutes, and erasing the s (=!').In like manner he raised 81° 42!' to 82° at fol. 127v, lines 2-3. With the moon's mean zenith distance computed as 80° 55' (originally 42'; fol. 127v, line 2), the parallax= 1° 5' (fol. 127v, lines 3 down, 5 up). To 1° 5', according to Lines Subtended, the corre­ sponding chord is 1891, as at fol. 127v, line 4 up (1° 10':2036; 1°:1745; 10':291; 5':146; 1° 5':1745+146 = 1891). P. 205:1. This ratio, with which Copernicus operates later (fol. 128r, line 15 up), is derived from the values 98,953:1745, used initially by Copernicus forCE and AC (fol. 127v, line 4 up). When he there replaced 98,953 by 99,027, and 1745 by 1891, he did not recompute the ratio, which would be CE:AC = 52P 22': 1P, instead of 56P 42':1P, With AC = 1745, the parallax= i=AEC = 1°. Hence, N (fol. 119r), retaining 80° 55' for the

413 NOTES ON PP. 205-208

moon's mean zenith distance in the 2nd observation of IV, 16, made the apparent zenith distance 81° 55', although Copernicus' autograph reads 82°, replacing 81° 42f. By the same token, the autograph's value for ~ABC (65'; fol. 127v, line 5 up) was reduced to 60' (fol. 119v). P. 205:7. According to the Table of the Lunar Prosthaphaereses, after IV, 11, for 195°: 2° 39'; for 192°: 2° 7'; hence, for 194° 10': 2° 30'. P. 205:9. ~KDB = 59° 43'. According to Lines Subtended, for 59° 50': 86,457; for 59°40': 86,310; hence, for 59° 43': 86,354. P. 205:15. DE:EK = 91,856:86,354 = 100,000:94,010.2, for which Copernicus writes 94,010. P. 205:17. KE:DE = 94,010:100,000 = 56P 42':60P 18.8', for which Copernicus writes 18'. KE:DF = 94,010:8600 = 56P 42':5P 11' KE:DFG = 94,010:13,340 = 56P 42':8P 2.7', for which Copernicus writes 2'. P. 205:22. Where we might have expected 52P 16', initially Copernicus wrote 52P and the abbreviation for minutes (fol. 128•, line 9 up). Then, thinking of a convenient fraction, he switched to tp, while forgetting to delete the abbreviation for minutes. Finally, he struck out the ·h over which he placed 17 as the number of minutes. P. 205:23. In arriving at these values of 65!P and 55P 8', perhaps Copernicus bore in mind that DE= 6QP 18.8' ~ 6QP 19'. P. 207:12. By a slip Copernicus altered 56 to 58 (Lvjij; fol. 129v, line 12 up). This error inN was corrected in Miistlin's copy (fol. 12IV, line 14). P. 207:14. KL:KD = 3'11":60' = 64P 10':1209.4P, rounded off by Ptolemy as 1210P. P. 207:17. KM:KMS = 14' 22":60' = 64P 10':267.98P ~ 268P. P. 207:26. 29f' X 13h = 1° 16' 42" ~ 1 o 163/ 4'. P. 207:29. These findings were attributed to Al-Battani by P-R, V, 21: "Al-Battani, however, finds the eclipses he observed different in extent and duration from what was indicated by Ptolemy's computations .... When the moon was in the apogee of its epicycle during eclipses, Al-Battani found that the diameter of the moon in the apogee of its epicycle was 291/ 2' .... But he kept the ratio of the moon's radius to the shadow's radius which Ptolemy gave, that is, 5: 13 or 1:2 3/5.... Al-Battani also announces a variation in the sun's [apparent] diameter. For when the sun is at its greatest distance [from the earth], he says that [the sun's diameter]= 311/3', in agree­ ment with Ptolemy .... The moon's greatest distance [from the earth] = 64P10' .... At its apogee the sun's dis­ tance = 1146P, of which the earth's radius = 1P .... At that time the length of the shadow's axis = 254P of the same units." P. 207:32. Having previously reduced the eccentricity of the annual orbit from Ptolemy's 1/ 24 to his own value of 1/ 31 (III, 16, end), Copernicus now correspondingly enlarges the sun's apparent diameter from Ptolemy's 31' 20" to 31' 40". P. 208:1. The Congregation ordered the deletion of the words "these three heavenly bodies, because the earth is not a heavenly body, as Copernicus makes it out to be." P. 208:4. KL:KD = 65!P:1179P = 1:18. P. 208:5. With LO = 17' 8" (IV, 19), 18XLO = 5P 8' 24" =1- SP 27' (fol. 130v, line 5). However, Co­ pernicus' original value for LO was 18' 11" (fol. 130•, line 6 up), and 18X 18' 11" ~ SP 27'. Hence, when Coper­ nicus reduced LO from 18' 11" to 17' 9" and finally to 17' 8" in IV, 19 (fol. 130•, line 5 up), he forgot to alter "18XLO ~ SP 27"' in IV, 20 (fol. 130v, lines 4-5). Matthias Lauterwalt of Wittenberg, writing to Rheticus in 1545, said: "It is hard to attribute to the author [Copernicus] the beginning of [IV,] 20, in my judgment, and I am greatly puzzled why you did not correct such errors, since you reviewed the author's computations in Books I-IV" (Burmeister, Rhetikus, III, 63: lines 10 up-8 up). Having no access to Copernicus' autograph, Lauterwalt perforce based his opinion exclusively on N. Like many another careful student ofN, Lauterwalt received the impression from N that Copernicus had made numerous slips in elementary arithmetic. Because such readers were unaware of the intricate events preced­ ing the publication of N, they failed to realize that computational errors in N were often due, not to Copernicus' ineptness as a calculator, but rather to the unfinished state of his manuscript when he finally consented to its publication. His remark in the Preface that his manuscript had been buried among his papers and had lain concealed for many years gave rise to the inference that the withheld manuscript had long been in a state ready for publication. The painful truth is that it was not really ready for publication even when it was released to the printer. Lauterwalt's statement that Rheticus "reviewed the author's computations in Books I-IV" throws addi­ tional light on events in Nuremberg. Lauterwalt's reliability is guaranteed by the closeness of his contact with Rheticus, even after the latter had begun to teach mathematics at Leipzig University. For, Rheticus' handwritten

414 NOTES ON PP. 208-212

notes on his observation of the lunar eclipse at Leipzig on 28 December 1544 were in Lauterwalt's possession early in 1545. Hence, there is every reason to take at face value Lauterwalt's assertion that Rheticus reviewed Copernicus' "computations in Books I-IV." If that is the correct reading, we may conclude that in his editing of the Reoolutions Rheticus had not gone beyond Book IV when he had to leave Nuremberg in order to begin teaching at Leipzig University in October 1542. Was it then that Osiander succeeded Rheticus as editor of the Reoolutions, Books V-VI? Lauterwalt was admitted to Wittenberg University on 11 June 1540 (Album academiae Vitebergensis, I [Leip· zig, 1841], 180). The misreading of his surname as "Lauterbach" by MK (pp. 591,593, 606-608) was corrected on pp. 682, 698. Nevertheless Z continued to talk about "Lauterbach." Z (pp. 257, 270) called "Lauterbach" a student in 1545, presumably because in closing his letter to Rheticus he signed himself tui studiosus (Yours faithfully). Since Lauterwalt hailed from Elbillg, he labeled Copernicus his compatriot (conte"aneus). P. 208:6. SKD = SK+KLD = 265+1179 = 1444. P. 208:7. 265X 5P 27' = 1444}P. 8 7 P. 208:9. (5P 27') = 161.879 ~ 161 / 8• P. 208:10. Here Copernicus put the radius of the moon= 17' 9" (fol. 130v, line 14) before he lowered that value by 1' (fol. 130r, line 5 up). When he made that reduction in IV, 19, he forgot to make it here. 7 7 P. 208:14. 42 /8 X 161 / 8 = 6940, for which Copernicus writes 7000-63 = 6937. P. 208:17. Again Copernicus cites Euclid, Optics, Prop. 5 (see notes to pp. 11:28, 155:29 and 175:9). P. 208:22. Here Copernicus left 322 unchanged (fol. 130v, line 7 up), when he later returned to III, 16, 18, where he raised 322 to 323 (notes to pp. 159:26 and 161:7). P. 208:26. 10,322:9678 = 1179:1105.4, for which Copernicus writes 1105.1179-1105 = 74; 74+2 = 37; 37 + 1105 = 1142. P. 208:27. 1,000,000+1179 = 848.18, for which Copernicus writes 848. P. 208:29. 1,000,000+1105 = 904.98, for which Copernicus writes 905. P. 208:32. Actually, in IV, 19, no value for the sun's apparent diameter at apogee was shown (ostensum est). On the contrary, Copernicus simply put it (posuerimus) = 31' 40" (fol. 13or, lines 16-17). This was also his figure here at first. But when his computations led to the result 31' 48", he squeezed in "viij" in two places (fol. 131r, lines 9, 11), and forgot to go back to make the same change in IV, 19. P. 208:39. Copernicus here retains the exaggerated traditional value. For the reduction in post-Coper­ nican times, see A. Pannekoek, A History of Astronomy (London: Allen & Unwin, 1961), pp. 283-284. P. 208:41. Copernican scholars have not yet identified those astronomers who inferred the sun's apparent diameter's mean length from the sun's apparent hourly motion. P. 208:42. By an arithmetical slip Copernicus wrote (fol. 131r,line 12 up) 141/ 5• The fourth I was de­ leted in Miistlin's copy of N (fol. 123r, line 9). P. 209:8. The half moon's apogee is given here= 68° 21' (fol. 131r, line 3 up). Previously, in IV, 17, Copernicus put this value= 681/ 8P = 68P 20' (fol. 128r, line '11 up). P. 209:26. Here Copernicus again calls attention to the defect in the Ptolemaic lunar theory that under· estimated the moon's perigeal distance (note to p. 176 :43). P. 209:43. CZ:ZB = BK:KS 4P 27':1105P = 1P:248P 18.9', for which Copernicus writes 19'. P. 210:2. SK:KB = SM:MR 248P 19': lP = 186P 19':45' 1.9", for which Copernicus writes 45' 1". P. 210:7. Since the maximum variation in the diameter of the shadow is 57", the heading over the last column in the second table following IV, 24, should be seconds, not minutes as inN, fol. 126v. This typographical error in N was pointed out by Lauterwalt (Burmeister, Rhetikus, III, 63, lines 7 up - 5 up). P. 211:30. Here the diameter of the moon's 2nd epicycle is lettered BF. In the second diagram accompany­ ing IV, 17, that diameter= DFG-DF = 89/60-511/60 = 251/60 earth-radii. P. 211 :31. GA = }(BF = 2P 51') ~ lP 25' AC = AB+BC = lP 25'+5P 11' = 6P 36'. P. 211:37. BF:BL = 2P 37':46' = 60':17.6', for which Copernicus writes 18'. P. 211 :38. By a slip Copernicus wrote 7th column (fol. 13Y, last line). P. 211:41. Although Copernicus constructed angle MBN = 60°, in his diagram of fol. 13Y he omitted line BN, which was supplied by N. P. 211:44. 3P 7':55' = 60:17.6, for which Copernicus writes "~ 18." P. 212:8. In IV, 17 (fol. 128r, line 13 up) Copernicus had put the distance from the center of the earth to the center of the moon's 1st epicycle at 60P plus 18' instead of 19' as here (fol. 133v, line 11 up). P. 212:11. lOP 22':2P 27' = 60:14.2, for which Copernicus writes 14.

415 NOTES ON PP. 215-227

P. 215:33. By dittography Copernicus wrote "second" limit immediately after giving the number of seconds (fol. 13sr, line 3 up). P. 215:36. In the last column, the proportional minutes for g6o are 32, and 35 for 102°, so that 34 belongs with 100°. P. 217:10. Although Copernicus (fol. 136r, lines 6-5 up) calls these arcs KM and LG, his drawing shows a somewhat different arrangement, which was modified by N. P. 218:11. This is the 15th star in the constellation of the Bull, in Copernicus' Star Catalog, after II, 14. P. 218:18. Since Aldebaran at 5° 10' south latitude was 1/ 3 of the moon's apparent diameter(~ 32') closer to the southern hom than to the northern hom, the star was some 5' south of the moon's center, which was therefore at about 5° 6' south latitude. P. 218:1g, 14g6 complete years 1497 January through February sgd March 8 23'1 leapdays (4-1496) = 374d = lY gd 1 g 14g7Y P. 218:20. In saying that "Cracow is nearly go east of Bologna," Copernicus relied on some source other than his edition of the Alfonsine Tables. Those Tables (sig. eF) did not list Bologna, but placed both Venice and Florence at 1h34"' east of the prinle meridian, with Cracow at zhzom; the difference, 46"1, = 111/2°, Hence, Copernicus and his source were much closer to the true value (~81/2°) than were the Alfonsine Tables. P. 218:21. With 15° = lh, go= 36m. P. 218:27. At first, thinking of a number< 60, Copernicus wrote scr (minutes; fol. 137v, line 3). Then he deleted the abbreviation, which he replaced by 1° (pars una), because he momentarily had in mind a value slightly greater than 60'. When he accepted 51' as the definitive value, however, he forgot to strike out pars una. That was done in Miistlin's copy of N (fol. 12gr, Ch. 27, line 4 up). P. 221 :44. Copernican scholars have not yet identified those astronomers who relied exclusively on the moon's hourly motion to find the tinle of the true syzygy. Are these astronomers also those who, from the sun's apparent hourly motion, inferred the mean length of the sun's apparent diameter (note to p. 208:41)? P. 222:8. 15° = lh = 6QID; 1° = 4"'; 1' = 48.

P. 224:14. In zh = 12om the moon traverses 1 o = 60' (IV, zg), Hence it covers 2' in 4"' = 1fi6h. P. 224:23. Copernican scholars have not yet identified the many astronomers who determined the por­ tions of the luminaries in partial eclipse by reference to the surfaces eclipsed rather than to the diame­ ters.

P. 225:g. PS (VI, 7) repeated Archinledes' famous delimitation of n as< 31/ 7 but > 310/71' and put the ratio circumference: diameter = 3P 8' 30": 1. But, although Ptolemy mentioned Archinledes by name, he did not associate him with Syracuse, nor did he cite his treatise's title as Measurement of the Circle. Copernicus must have obtained this additional information about Archimedes elsewhere. P. 226:4. Copernican scholars have not yet identified the astronomers who treated eclipses more fully. Presumably they, or some of them, are referred to in notes to pp. 208:41, 221 :44, and 224:23. P. 226 :7. When Copernicus reduced the number of Books in the Revolutions to 6, he forgot to change his entry here (fol. 14lv), which he left as "Here ends the 5th Book of the Revolutions." P. 227:14. In the autograph Copernicus originally ended the introduction to Book Vat fol. 142r, line 12, and passed inlmediately to Ch. 1. Mter writing its title and first two sentences, he struck out these nine lines and went back again to the introduction in order to recall the names of the planets allegedly used by Plato in the Timaeus. Yet in the Timaeus Plato did not call the five planets by these names, which were introduced long after his death. How is this misattribution to be explained? Chalcidius' Commentary on Plato's Timaeus (first edition: Paris, 1520) gave the names of the planets in the form repeated by Copernicus. If Chalcidius was Copernicus' source for this passage, three interesting inferences suggest themselves. First, Copernicus failed to separate what the commentator Chalcidius said about the Timaeus from what Plato hinlself had said in the Timaeus more than seven centuries earlier. In other words, Copernicus did not go back to the Timaeus itself to verify whether Plato actually used these planetary names. Secondly, Copernicus was aware that the bare names themselves might appear strange and even unintelligible to his readers, most of whom could not be assumed to be familiar with Greek. Hence, instead of merely repeating his source, Co­ pernicus provided explanations of the meaning of these planetary names, and in the case of Venus he added

416 NOTES ON PP. 227-244 two fairly familar names. Lastly, if Chalcidius was Copernicus' source here, we would have an additional reason to think that he started to write Book V after 1520. It is of course true that there were several Chalcidius manu­ scripts at Cracow when Copernicus was a student there. But was he permitted to delve into precious manuscripts when he was in his late teens and early twenties? It is also true that the planetary names in question appear in other ancient authors (for instance, Cicero, pseudo-Plutarch, Martianus Capella). Although Copernicus knew their works, none of them connected these planetary names specifically with Plato's Timaeus. In Chalcidius, on the other hand, the planetary names appear in the context of the Timaeus, as they do also in Copernicus. P. 227:40. A planet "always moves forward with its own motion." Yet at times it seems to stop and reverse its direction. These departures from the planet's proper motion are not real, but only apparent. These appear­ ances are due to our motion as observers on planet earth orbiting the sun. An (imaginary) observer watching the planets from the sun (regarded as motionless) would see the planets perform only their own proper, direct motion. He would observe no halts and no changes of direction. This insight constitutes Copernicus' mightiest contribution to our understanding of the planets' behavior. At the same time it provides a striking proof of the earth's annual revolution around the sun. That revolution, it is often said, remained unproved until the effect of the earth's annual motion on a star was detected in the form of an annual stellar parallax perceptible only with a vastly improved telescope, unavailable in the time of Copernicus. But his discovery of planetary parallax (which he called their "motion in commutation," motus commutationis, back-and-forth motion) is just as powerful a proof of the earth's orbital motion as is annual stellar parallax. The minuteness of that phenomenon delayed its dis­ covery until technological progress caught up with it. But planetary parallax, a large-scale phenomenon, was discovered by Copernicus with the naked eye. See Jean-Claude Peeker, "Retour sur Copernic, Kepler, Bessel et les parallaxes," L'Astronomie, 1974, 88. P. 228:14. Here once again, as he previously did in the Introduction to Book II, Copernicus reminds his readers that it is sometimes convenient and harmless to use the ordinary heliokinetic terminology (note to p. 51 :19). P. 228:42. By anticipatory dittography Copernicus wrote "six times" (sexies, instead of "sixty times," sexagies; fol. 143r, line 13 up), presumably because he was already thinking about Jupiter's six sidereal revolu­ tions, mentioned three lines below. P. 229:22. In the seconds column for Venus' daily motion, by a slip Copernicus wrote 49 (iL; fol. 143v, line 14 up). This number should be 59, as in the Table (fol. 147V, line 3) and as is required for consistency with Venus' annual motion. P. 240:27. In his Republic, Book VI, Cicero incorporated a section called "Scipio's Dream," in which the planets were said "to pursue their circular and spherical" paths (§ 15). P. 240:32. The admission that "a circular motion can be uniform with respect to an extraneous center not its own" was one of the situations that gave Copernicus "the occasion to consider the motion of the earth and other ways of preserving uniform motion and the principles of the science." At this point he still refrains from using the term "equant" for the device which helped to induce him to conceive of the earth as a moving celestial body. Yet in his earlier Commentariolus he did refer to "certain equants" (aequantes quosdam circulos) as impelling him to find "a more reasonable arrangement ... in which everything would move uniformly about its proper center, as the rule of absolute motion requires" (3CT, pp. 57-58). Half a millennium before Copernicus, the great Muslim scientist Ibn Al-Haytham (965-c. 1040) had like­ wise rejected the equant because it violated the principle of uniform motion. He did so in his Doubts concerning Ptolemy, the Arabic text of which was recently published (Al-Shukuk 'ala Batlamyus, edd. A.I. Sabra and N. Shehaby, Cairo: National Library Press, 1971). But the Doubts were never translated into Latin, and were there­ fore inaccessible to Copernicus. Did he nevertheless hear some faint echo of the objection to the equant on the part of Ibn Al-Haytham? If so, it is noteworthy that the Muslim's rejection of the equant did not lead him to geokineticism. The relevant passage was translated and discussed by Salomon Pines, "Ibn Al-Haytham's Critique of Ptolemy," Proceedings of the Tenth International Congress of the History of Science (Paris, 1964), pp. 548-549. P. 242:32. According to Copernicus, the ancient astronomers believed that the planets deviate from per­ fectly circular orbits. The mistaken notion that, according to him, their planetary orbits were absolutely circular and that he intended "to disprove an opinion of the ancients" is based on the printed editions, all of which failed to reproduce the colon in fol. 151 r, line 7. They all thereby obscured Copernicus' meaning. This would of course have been clear to Otto Neugebauer ("On the Planetary Theory of Copernicus," Vistas in Astronomy, 10: 94), who would not have charged Copernicus with so serious an error had he consulted the autograph instead of rely­ ing exclusively on the printed editions. P. 244:13. In his analysis of the longitudinal motions of the five planets, Copernicus begins with Saturn,

417 NOTES ON P. 244

which is followed by the other two outer planets, Jupiter and Mars, in that order, and then Venus and Mer­ cury. Ptolemy, on the other hand, proceeds in the opposite direction, from Mercury outward to Saturn (PS, IX, 7-XI, 8). In his tables (PS, IX, 4; XI, 11; XII, 8) and in his treatment of the arcs of retrogradation (XII, 2-6), however, Ptolemy shifts to the order which was adopted by Copernicus. P. 244:15. In PS 1515 (fol. 122v) the name of this Egyptian month was distorted to "machur," which Co­ pernicus interpreted as Mechyr (fol. 152r, line 4), the 6th Egyptian month, whereas Ptolemy meant the 9th month Pachon. This correction was made by someone other than Miistlin in his copy of N (fol. 143r, right margin), by Brahe in his copy of B, and publicly for the first time by A. P. 244:19. In the first opposition Ptolemy reported Saturn's longitude as 1° 13' within the Balance= 181° 13'. Applying to this value a correction for precession of about 6° 33', Copernicus obtained 174° 40' as a round number (jere). On the other hand, in the 2nd and 3rd oppositions, his precession correction was precisely 6° 37'. Why then did he use an approximate correction for the 1st opposition? The explanation is surely not a "computational error," as Z mistakenly thought (p. 510). Z failed to notice that Copernicus labeled Saturn's position in the first opposition as only approximate. P. 244:23. By a slip, instead of quindecim (fifteen), Copernicus wrote undecim (eleven), although 3 lines below he correctly wrote the Roman numeral xv. In his copy of B, Brahe corrected 11 to 15 (fol. 143v, line 4). P. 244:25. Ptolemy located Saturn in the 2nd oppposition at 9° 40' within the Archer= 249° 40'. Hence in this instance Copernicus' correction for precession was precisely 6° 37' +243° 3' = 249° 40'. P. 244:30. In the 3rd opposition Ptolemy located Saturn at 14° 14' within the Goat= 284° 14'. Then once more Copernicus' correction for precession was precisely 6° 37' +277° 37' = 284° 14'. P. 244:32. From 5 P.M., 26 March 127 to 3 P.M., 3 June 133: 121 March 5d 7h April through December 275 5 complete years 128-132 133 January through May 151 June 2 15 leapdays (128-132) 2 435 1 -365 6Y P. 244:35. According to the Tables of Saturn's Parallactic Motion in Years and in Days, after V, 1, for 6Y 240° 45 12' 18" 58"' 57 7 44 5 9 31 17 20

22h 352° 43' 20" 23"', for which Copernicus writes 352° 44.' P. 244:37. From 3 P.M., 3 June 133 to 11 P.M., 8 July 136: 133 June 27d gh July through December 184 2 complete years, 134-135 136 January through June 181 July 7 11 leapday (136) 1 400 1 -365 3Y P. 244:39. According to the Tables of Saturn's Parallactic Motion in Years and in Days, after V, 1, for 3Y 300° 22 36' 33 19 48 356° 43'

418 NOTES ON PP. 244-248

P. 244:46. Copernicus' proof requires A, B, and C to be joined to E. But the resulting intersections of AE, BE, and CE with the epicyclet's circumference are not used in this proof, and therefore Copernicus does not letter these intersections in his diagram (fol. 152"). K, L, and M mark the intersections of the epicyclet's circumference with AD, BD, CD, and not with AE, BE, CE. P. 245:19. For Archimedes' indirect attack on the problem of finding a square whose area is (nearly) equal to the area of a circle see note to p. 146:27. P. 245:22. Copernicus' rejection of this overelaborate treatment by Ptolemy should be carefully con­ sidered by those who are fond of accusing Copernicus of merely parroting Ptolemy. P. 246:2. Actually Ptolemy's 1st arc= 57° 5' (PS 1515, fol. 124V), For the 2nd arc, Ptolemy gave 18° 38' (not 18° 37', as in fol. 153r, lines 9-10). P. 246:3. DF:DE = 6oP:6P 50'= 10,000:1139. By a strange slip, as though he had put DE= 6P 5' 45" 36"', Copernicus wrote, instead of 1139, 1016 (fol. 153r, line 12). Yet in the very next line he ac­ tually operated with 1139, since he equated 8/, of the number in question with 854 and 1/, with 285. The correc­ tion from 1016 to 1139 was made in Mastlin's copy of N and in Brahe's copy of B (fol. 14¥, line 3 up) but publicly for the first time in A. P. 246:19. Initially Copernicus put BDE = 161° 23' (fol. 15Y, lines 4 up- 3 up). Hence at that time he was still operating with FB = 18° 37' (see above, note top. 246 :2). Later he reduced the number of minutes n BDE to 22 by erasing the first i. In so doing, he returned to Ptolemy's value for the 2nd arc = 18° 38'. P. 246:23. For the number of minutes in OBL, by a slip Copernicus wrote 36 (fol. 153v, line 2). But he actually operates with 38 (see the preceding note). P. 246:27. Copernicus mistakenly described (fol. 153v, line 6) BED as the remainder, although it is ac­ tually the minuend, as was first noticed in Mu (p. 298, n. on line 31). P. 246:29. Copernicus mistakenly equated angle ODE with 56° 30', instead of with the supplement of that value, an error first noticed in the 1952 English translation (p. 747). For the number of minutes Copernicus initially wrote 30 (xxx; fol. 153v, line 9). He reduced this number to 29 by inserting ani above the line and before the las~ x. Actually, however, he operated also with 30' (p. 247:2) and repeated that number in the form of the fraction 1/ 1 toward the end of V, 5 (fol. 1541', line 18). P. 246:43. For the number of minutes Copernicus wrote, instead of 37, 14 by anticipatory dittography, since he was already thinking of the 14 at the end of this same line (fol. 153v, line 2 up). This correction was made in Mastlin's copy of N (fol. 14¥, line 4). P. 246:43. Instead of the required PEF, by a slip Copernicus wrote PDF (fol. 153v, last line), an error which was corrected in Mastlin's copy of N (fol. 14¥, line 5) and publicly for the first time in Mu (p. 299, n. on line 12). P. 247:21. Initially Copernicus timed this opposition at "nearly gh after midnight" (fol. 154r, line 2 up). Then, erasing the numeral (which can now be read only with difficulty), Copernicus shifted in the right margin to "2h before sunrise." Finally, he replaced this 2nd .indication by "61f5h after midnight," repeated at fol. 156v, lines 7 up - 6 up. Hence, from beginning to end he changed from "nearly 9" to 4 to 6: 24, all three of these determinations clustering around the early morning. Making the astounding error of transposing this opposi­ tion to the evening, Z (p. 209) argued that Copernicus could not have used the moon, then new, as an interme­ diary and that Copernicus blundered in this opposition, which Z proposed to postpone by a month to 10 No­ vember! But Copernicus' final determination (6:24A.M.) was made before he computed the interval between the 2nd and 3rd oppositions 5lines later (fol. 154v, line 4; see below, note to p. 248:3). That interval is of course completely incompatible with Z's captious criticism.

P.248:1. From 10:48 P.M., 5 May 1514 to noon, 13 July 1520: 1514 May 26d lh 12m June through December 214 5 complete years 1515-1519 1520 January through June 181 July 12 12 leapdays (1516, 1520) 2 435 1 -365 6Y

419 NOTES ON PP. 248-252

P. 248:3. From noon, 13 July 1520 to 6: 24 A.M., 10 October 1527: 1520 July 18d 12h August through December 153 6 complete years 1521-1526 1527 January through September 273 October 9 6 24111 leapday (1526) 1 454 1 -365 7Y P. 248:5. For Saturn's mean proper motion, it will be recalled (V, 1), Copernicus felt it to be unnecessary to provide tables. For Saturn's proper motion he gave 12° 12' 46" for 1Y; hence, 73° 16' 36" for 6Y. The rest of Saturn's proper motion is obtained by subtracting the appropriate entries in the Tables of Saturn's Parallactic Motion in Years and in Days, after V, 1, from the corresponding entries in the Tables of the Sun's Simple Uniform Motion in Years and in Days, after III, 14. Thus, for 70d 33dm 60d Sun 59° 8' Saturn 57° 7' 10 9 51 9 31 7od 68° 59' 66° 38' -66 38 2° 21' 1 2° 22' 6Y 73 16 36" 75° 38' 36", for which Copernicus writes 75° 39'. P. 248:28. DB:AE = 19090:8542 = 13501:6041. Instead of 6041, Copernicus writes 6043 (fol. 154v, last line). He does so because he originally put DB= 13506 (fol. 154v, line 12 up, where he erased the 6 and wrote 1 over it; and line 2 up, where the 6 is still plainer than the 1). With DE = 13506, AE = 6043. When Copernicus changed DB from 13506 to 13501, he forgot to make the corresponding change in AE, which he left= 6043. P. 248:44. Fol. 155r line 16 up: In formulating this subtraction, Copernicus interchanged the subtrahend and minuend. His error was corrected for the first time by W. P. 248:46. FG:FD = 10,000:1200 = 60P:7P 12'. P. 249:3. FD:DK = 1200:650 = 10,000:54161/ 8• Yet the last digit, although partially erased, was not a 7 and what is left looks like a 1 (fol. 15¥, line 7 up). Now, according to Lines Subtended, for 32° 50': 54220; for 32° 40': 53975; hence, for 32° 45': 54098, or 5410 with the radius = 10,000. Th'l: angle corres­ ponding to 54167 would be about 32° 48'. P. 250:47. Previously Copernicus had written 7 (vij; fol. 15-¥, last line). Here, however, he says 8 (octo; fol. 156v, line 12). Had he retained the 7 here, Saturn's lower apse would have been at 601/ 8 ° without the "about" (jere; fol. 156v, line 16). But he definitely adopted 8 when, at the end of V, 6, he located Saturn's higher apse at 240° 21' (fol. 156v, line 3 up). P. 252:1. Initially Copernicus timed this observation at "2h before sunrise" ( = 4 A.M., fol. 157r, line 11). When he shifted to a somewhat later time, he replaced 2 by 6, without any fraction, and did not replace "before sunrise" by "after midnight," as he had previously (note to p. 247 :21). P. 252:2. From 11 A.M., 8 July 136 to 6 A.M., 10 October 1527: 136 July 23d 13h August through December 153 1390 complete years 137-1526 1527 January through September 273 October 9 6 leapdays (140-1524) 347 805 2 -730 1392Y

420 NOTES ON PP. 252-255

Copernicus writes 48dm, perhaps because he remembers that his final determination of the time of his 3rd opposi­ tion was some minutes after 6 o'clock. P. 252:8. Here (fol. 157•, line 20), for the minutes Copernicus wrote 45, in agreement with his original computation 6 lines above. There, however, he later squeezed in 3 i's to raise the number of minutes to 48. But he made the change from 45 to 48 after he had already performed this subtraction involving the lower num­ ber, which remains unaltered in the autograph. This is, then, another instance in which Copernicus modified a numerical result without making the changes required elsewhere. P. 252:13. The corresponding modem value would be about 1l in 100 years (Astronomical Journal, 1974, 79: 58). P. 252:17. From the beginning of the Christian era to 20 Hadrian, 24 Mesori = 8 July 136: 135 complete years 136 January through June 181d July 7 uh leapdays (4-136) ------34 135Y 222d uh = 27ldm for which Copernicus writes 27dm (fol. 157•, line 5 up). P. 253:5. From the beginning of the Christian era to 5 A.M., 24 February 1514: 1513 complete years 1514 January 31d February 23 sh leapdays (4-1512) 378 432 1 -365 1514Y 67d 5h = 12ldm, for which Copernicus writes 13dm (fol. 157", lines 4 up- 3 up), where Copernicus' error of 77d (lxxvij, with one x too many) was corrected in A (p. 355). P. 253:8. Here Copernicus places Saturn's higher apse at "about 2401/ 3°" (Jere; fol. 158•, line 3). At the end of V, 6, Copernicus had said more precisely "240° 21'" (fol. 156v, line 3 up). P. 253:19. Instead of 41 for the number of degrees, by a slip (fol. 15S•,line 17) Copernicus wrote 40 (xl, without an i), which was corrected in A (p. 355). P. 254:7. Although Copernicus had given the number of minutes as 31 just above (fol. 15S•,line 1), here by a slip of the pen he wx:ote 33, an error corrected by A. P. 254: 10. Here Copernicus wrote the number (31'), which he should have written in the line above. Perhaps because he was conscious of the error, his attention was diverted from 35', the number required here. This cor­ rection too was made by A. Copernicus was evidently not as careful as he might have been in transferring the results of his computations from his work sheets to the autograph. P. 254:15. BD:EL = 10,000:1090 = 60P:6P 321/5', for which Copernicus writes 32' (fol. 158v, line 8). P. 254:15. In Ptolemy's theory for Saturn, the radius of the epicycle (analogous to the radius of the earth's annual orbit in Copernicus) = 6P 30' (PS, XI, 6). P. 254:21. 1090:10,569 = 1P:9P 41.8', for which Copernicus writes 42' (fol. 158v, line 16). P. 254:22. 1090:9431 = 1P:8P 39'. P. 254:42. By a slip (fol. 15gt,line 3), for the number of degrees Copernicus wrote 6 (vj), an error which was corrected in A (p. 357). P. 255:1. From 11 P.M., 1 Epiphi, 17 Hadrian to 10 P.M., 14 Phaophi, 21 Hadrian: 17 Hadrian Epiphi 28d 13h Mesori 30 intercalary 5 3 complete years 18-20 Hadrian 21 Hadrian Thoth 30 Phaophi 13 10 3Y P. 255:2. From 10 P.M., 14 Phaophi, 21 Hadrian to 5 A.M., 21 Athyr, 1 Antoninus Pius:

421 NOTES ON PP. 255-257

21 Hadrian Phaophi 16d 1~ 10 complete months (Athyr-Mesori) 300 intercalary 5 1 Antoninus Pius 2 complete months (Thoth-Phaophi) 60 Athyr 20 17 17h 402 1Y -365 ~

P. 255:9. 60P:5tp = 10,000P:9161/ 3P, for which Copernicus writes 917 (fol. 159r, line 9 up). P. 255:22. By a slip (fol. 159v, lines 10-11) Copernicus interchanged BAD and·DBA, errors which were corrected in W (p. 371). P. 256:11. For the sake of conciseness Copernicus omits some steps in his geometrical reasoning. Thus, if the intersection of DB and BL is called Y, then LBB+DBB (= 4'+12') =DYE= 16', and FDB = 177° 10' =(DYE= 16')+(FBL = 176° 54'). P. 256:18. BCM = (DCB = 2° 8')+DCM By construction, DOM = FDC But FDC = 180°(-GDC = 30° 36') = 149° 24' • •• ECM = 2° 8' +149° 24' = 151 o 32', not 147° 44', as Copernicus thought (fol. 16or, line 7, where he originally had a different number, which he partially erased so that it is no longer legible). The correction 151° 32' was made in Miistlin's copy of N (fol. 15ov, line 5 up) and publicly for the first time in W. P. 256:23. LEM = (GEM= 33° 23')+(LBG = 3° 6') = 36° 29' LEG= 180°-(FBL = 176° 54')= 3° 6' P. 256:26. Here, by a slip, for the number of minutes Copernicus wrote 30 (xxx; fol. 16or, line 18). A little later on, however, he used the correct value, 22 (xxij; fol. 16or, line 4 up). This correct value 22 was substituted for 30 in Miistlin's copy of N (fol. 151r, line 6) and for the first time publicly in W. P. 257:3. From 11 A.M., 30 April 1520 to 3 A.M., 28 November 1526: 1520 April 13h May through December 245d 5 complete years, 1521-1525 1526 January through October 304 November 27 3 leapday (1524) 1 577 1 -365 6Y

P. 257:6. From 3 A.M., 28 November 1526 to 7 P.M., 1 February 1529: 1526 November 2d 21 h December 31 2 complete years 1527-1528 1529 January 31 February 19 leapday (1528) 1 1 --:-(24-+)-1(;h-;- 66d Here, in place of 1(;11, Copernicus writes 3gdm (fol. 160v, line 12), whereas in the interval between his first two oppositions he correctly equated 1(;11 with its precise counterpart 4Qdm. Did he perhaps originally time his 3rd opposition at 6: 36 P.M. rather than 7 P.M. on 1 February 1529? P. 257:22. ED= 10918P (fol. 160v, line 3 up) would be the correct length of the side opposite <~:OED= 66° 10' as an inscribed angle = 33° 5', as a central angle, according to Lines Subtended, where for 33° 10': 54,708; and for 33° 0': 54,464; hence, for 33° 5', 54,586 or 5459 with radius = 10,000. 2X 5459 = 10,918, which is Copernicus' mistaken value for ED. Had he not confused 66° 10' = <~:OED with 64° 10' = <~:DOE,

422 NOTES ON PP. 257-260

he would have arrived at a value for ED in fair agreement with Ptolemy's result. Copernicus' error was first pointed out publicly by Antonie Pannekoek, "A Remarkable Place in Copernicus' De revolutionibus," in Bul­ letin of the Astronomical Institutes of the Netherlands, 1945, 10: 68-69. In his copy of B, however, Brahe cor­ rected Copernicus' error as well as its consequences (fol. 151v-152v). P. 257:32. ED:AE = 18992:9420 = 10,918:5415.3, for which Copernicus writes 5415 (fol. 161•, line 9). P. 257:40. CE:DE = 18150:10918 = 17727:10663.5, for which Copernicus originally wrote 10663, but later he wrote over the 3 to make it a 5 (fol. 161r, line 17). P. 257:46. Instead of EDxDB, by reverse dittography (fol. 161r, line 14 up), Copernicus wrote EDxDE, an error which was corrected in N (fol. 152r). P. 257:48. By a slip Copernicus inverted the two terms in this subtraction by writing that (FDH) 2 was to be subtracted from rectangle GDxDH (fol. 16F, line 11 up). This error was corrected in T (p. 347, lines 15-17, and note). P. 258:2. FG:E.D = 10;000:1193 = 60P:7P 9.48', for which Copernicus writes 9' (fol. 161r, line 8 up). P. 258:21. Yet in V, 5, Copernicus reported Saturn's eccentricity in Ptolemy= 6P 50'= 1139 = 854+285, and in V, 6, Copernicus increased this eccentricity to 7P 12' = 1200 = 900+300, only "slightly different" (parum distant; fol. 155', line 11 up). P. 258:22. In V, 10, Copernicus reported Jupiter's eccentricity in Ptolemy= 5¥' = 917, "agreeing almost exactly with the observations" (observatis propemodum respondebant; fol. 159r, lines 8 up - 7 up). Here, in V, 11, Copernicus' own eccentricity for Jupiter = 7P 9' = 1193. P. 258:23. Had not Copernicus assumed the wrong arc for -t.DCE, the error mentioned in note to p. 257:22, his value for Jupiter's eccentricity would not have differed so widely from Ptolemy's. P. 258:40. For the number of minutes in -t,EAK, by a slip (fol. 162r, line 5), Copernicus wrote 34 (xxxiiij), which was corrected to 41 in Miistlin's copy of N (fol. 152v, line 5 up) and for the first time publicly in T (p. 349, line 4, and note). P. 258:44. Here again, for brevity's sake, Copernicus omits some steps in his geometrical reasoning. If the intersection of AD and KE is called X, then (AEK = 57')+(DAE = 2° 39') = 180°-(AXE = 176° 24') But AXE= (ADE = 180°-45° 2' = 134° 58') +(KED = 41 o 26'). P. 259:3. DEL= DEB-(BEL = 1° 10') But DEB= 180°-(BDE = 64° 42')-(DBE = 3° 40') = 111 o 38' Then DEL = 111 o 38' -1 o 10' = 110° 28'. P. 259:3. By a slip (fol. 162r, line 12 up) Copernicus mislettered this angle AED, im error not noticed in any previous edition or translation. P. 259:11. By a slip (fol. 162•, line 6 up) Copernicus mislettered this angle FCD, an error which was silently corrected in T (p. 349, line 20). P. 259:12. By a slip (fol. 162r, line 5 up) Copernicus mislettered the side DE, an error which was corrected in Mu (p. 315, line 9, and note). P. 259:18. In the diagram on fol. 162r Copernicus did not designate the intersection of DC and EM by any letter. Call this intersection X. Then, EXC = 180°-(XEC = 1°)-(DCE = 2° 51')= 176° 9'. But EXC = EDX ( = 180°-[FDX = 49° 8'] = 130° 52') +(DEX = DEM). Then 176° 9' = 130° 52'+DEM, and DEM = 45° 17'. P. 259:20. For the number of minutes in -t.LEM, Copernicus writes 10 (x; fol. 162v, line 5). The reason is that in his original value for -t.DEM the number of minutes was 18 (fol. 162v, line 3). Mter performing the subtraction with 18 and obtaining the result 10', he went back to line 3 and erased the third i, but left the outcome of the subtraction unchanged. P. 260:16. This discrepancy of 1' in Jupiter's parallactic motion may be due to Copernicus' operating with 52' in the left margin of fol. 162v, instead of 51' in the adjoining line 20. It is these 51' which belong with the parallax, whereas the 52' do not. P. 260:18. Copernicus misplaces "about" (Jere; fol. 162v, line 6 up), which belongs with 1° 5' =Jupiter's parallactic motion, not with 104° 54'= Jupiter's mean motion. P. 260:21. With Ptolemy's observation timed at 5 A.M. and his own at 7 P.M., how did Copernicus arrive at a difference of 37dm (= 1~ 48m; fol. 163', line 1)? To make the rest of the interval= 1392Y 99d, he must have equated the Egyptian date of Ptolemy's observation with 7 October 137:

423 NOTES ON PP. 260-263

137 October 24d November-December 61 1391 complete years 138-1528 1529 January 31 leapdays 347 1 from the hour column 464 1 1392Y P. 260:27. The corresponding modem value would be about 1!' in 300 years (Astronomical Journal, 1974 79: 58). P. 260:34. This reckoning (fol. 163r, line 14) would put Ptolemy's observation on 11 November 137, a date irreconcilable with the computation in V, 12 (note top. 260 :21). for the hour Copernicus writes lodm = 4'1, which he puts in the right margin, replacing 5, the number he had used previously (fol. 159l, line 6; 162v, line 3 up). P. 261:6. Copernicus specifies the 12th day before the Calends of March (fol. 16Y, line 4 up). His com­ putation at fol. 163v, line 3, shows that he did not take into account 29 February in 1520, a leap year. P. 261:10. From the beginning of the Christian era to 6 A.M., 18 February 1520: 1519 complete years 1520 January 31d February 17 fJl leapdays (4-1516) 379 427 1 -365 1520Y P. 262:25. PB:BS = 9698:1791 = 10373:1915.7, for which Copernicus writes 1916 (fol. 164", line 13). P. 262:26. 6QP:UP 30' = 10,000:19162/ 8• P. 262:29. RET:ADC = 2X 1916:2x 10,000 = 3832:20,000 = 1P:5P 13' 9", which Copernicus writes at fol. 164", line 20, although here at lines 18-19 he contents himself with 5P 13'. P. 262:30. AD:DB = 10,000:687 = 5P 13':21' 30", for which Copernicus wrote 21' 29" (fol. 16¥, line 15 up). This discrepancy is negligible, however, since he put BP = 1/ 8 DE= 7' 10" (same line). Yet he actually operated with BP = 7' 9" in computing Jupiter's distance at aphelion and perihelion. P. 263:8. From 1 A.M., 26 Tybi, 15 Hadrian to 9 P.M., 6 Pharmuthi, 19 Hadrian: 15 Hadrian Tybi 3d uh 7 complete months 210 intercalary 5 3 complete years 16-18 Hadrian 19 Hadrian 7 complete months 210 Pharmuthi 6 9 434 1 -365 4Y 6¢ 2oh = 5odm P. 263:11. From 9 P.M., 6 Pharmuthi, 19 Hadrian to 10 P.M., 12 Epiphi, 2 Antoninus Pius: 19 Hadrian Pharmuthi 23d 1sh 4 complete months 120 intercalary 5 3 complete years: 20-21 Hadrian, 1 Antoninus Pius 2 Antoninus Pius 10 complete months 300 Epiphi 12 10 1 (24+)1h 461 1 4Y

424 NOTES ON PP. 264-268

P. 264:5. In giving the size of ..J::ADE, Copernicus forgot to include the number of minutes (27') in ad­ dition to 138°, For, as he points out, ..J::ADE = 180° - (<}:FDA = 41 °33'); fol. 164v, line 3 up. P. 264:12. By a slip (fol. 165l, line 20) Copernicus mislettered the second angle AED, instead of LED, an error corrected iii T (p. 355, line 32, and note). Then, in the following line, to be consistent with himself, Copernicus wrote DEA, where DEL is required by the context, a correction made silently in T. For if the in­ tersection of DA and EL is called X, then AEL+DAE = 1 o 56' +5° 7' = 7° 3' = DXE. But DXE = 7° 3' = ADF-DEL = 41° 33'-34° 30'. Copernicus later puts DEL (not DEA) = 34° 30' when he computes the whole of MEL toward the end of his discussion of Mars' 2nd opposition. On the other hand, DBA= DEL+AEL = 34° 30' +1 ° 56'= 36° 26'. P. 264:18. By a slip (fol. 165l, line 8 up), for the number of minutes in <}:EBM Copernicus wrote 13 (xiij), where 9 is required. P. 264:27. By an extraordinary error (fol. 165v, line 3) Copernicus put CED = 37° 39', a result which he evidently obtained by the subtraction CDE-DCE = 44° 21'-6° 42' = 37° 39'. Instead, he should of course have used the subtraction 180°-(DCE+CDE = 6° 42' +44° 21' =51 ° 3') = 128° 57'. Yet, in obtaining NED in line 7, just below, he implicitly used CED = 128° 57' -# 37° 39'. The correction from 37° 39' to 128° 57' was made in Miistlin's copy of N (fol. 156', line 4 up). P. 265:15. From 1 A.M., 5 June 1512 to 8 P.M., 12 December 1518: 6Y 19ld, For 19h = 47fdm, Coper­ nicus writes 45dm (= ISh; fol. 166r, line 15). P. 265:16. From 8 P.M., 12 December 1518 to 5 A.M., 22 February 1523: 4Y 72d. For gh = 22!-dm, Copernicus writes 23dm (fol. 166r, line 16). P. 265: diagram. This diagram was supplied by N (fol. 157V) in place of the diagram begun and abandoned by Copernicus in the autograph (fol. 166v). P. 266:13. For the number of minutes in BF, by a slip (fol. 166v, line 5) Copernicus wrote 18 (xviij}, although he operated with 25 in computing Mars' 2nd opposition. P. 266:30. Copernicus inserted the discussion of triangle BDE in the left margin of fol. 166v. There, for the number of minutes in BDE, he finally wrote 35 ( xxxv), consistent with 25 for BF, as was pointed out in the preceding Note. Yet he did not go back to line 5, where he left 18 unchanged. On the other hand, in the margin his original number for the minutes seems to have been 37, although 42 would have been required to be con­ sistent with 18 for BF. These inconsistencies are not surprising, in view of the fact that Copernicus drew the circle and only four of the lines needed in the diagram, which he failed to complete and letter. P. 266:34. For the number of minutes in EBM, Copernicus originally wrote 18 (xviij; fol. 166v, last line), which would seem to be nothing more than a careless repetition of the 18 for BF (see the 2 preceding Notes). However, Copernicus deleted the 18 and replaced it by 36 (xxxvj), which he obtained by using 25 for the number of minutes in BF. Yet it is characteristic that he did not return to line 5 to change the 18 there to 25. P. 267:3. If the intersection of NE and CD is called X, then C~N +DCE = 50' +2° 6' = 2° 56' = DXE = FDC-DEN = 16° 36'-13° 40'. P. 267:12. Copernicus' figure;;:; 47' per century is again far greater than the modem value ;;:; 27' per century (Astronomical Journal, 197 4, 79: 58). P. 267:41. In reducing Ptolemy's 3rd observation to Cracow local time, for the purpose of comparing it with his own observation, Copernicus changes from 10 P.M., Alexandria time (V, 15) to 9 P.M., Cracow time, since he operates on the assumption that Cracow lies Jh = 15° east of Alexandria (III, 18). P. 267:45. This Egyptian date (9 P.M., 12 Epiphi, 2 Antoninus Pius) is equated by Copernicus (V, 18) with 8:48P.M., 27 May 139. From that time to 5 A.M., 22 February 1523:1384Y 251d 1oh 12m (=25!dm), instead of which Copernicus writes 19dm ( = 7h 36m; fol. 168r, line 3). P. 268:8. From the beginning of the Christian era to 9 P.M., 12 Epiphi, 2 Antoninus Pius, Copernicus reckons 138Y 18Qd 52dm: 138 complete years leapdays (4-136) 34d 139 A.D. January through April 120 May 26 52dm (= 8:48 P.M.;;:; 9 P.M.), 138Y 18od, 9 P.M., 27 May 139 (see preceding note). P. 268:9. For the number of minutes in this parallactic motion, in the right margin of fol. 168r Coper­ nicus finally wrote 22 (xxij). Since he actually operated with 4 in computing the position for the beginning of the Christian era= 238° 22' (in line 7 of this chapter), it seems clear that 22 in the right margin opposite line 4 of this Chapter is another example of Copernicus' anticipatory dittography. The correction from 22 to 4 was made in Brahe's copy of B (fol. 158v, Ch. 18, line 5) and for the first time publicly in W.

425 NOTES ON PP. 268-270

P. 268:26. Mars

1J8 P. 269:20. Since FSB = 70° 32' according to Lines Subtended, for 70° 40' :94,361 and for 70° 30': 94,264; hence, for 70° 32' : 94,283.4, or with radius = 10,000, 9428. P. 269:20. Since BFS = 35° 9', according to Lines Subtended, for 35° 10': 57,596 and for 35° 0' : 57,358; hence, for 35° 9': 57,572.2, or with radius= 10,000, 5757. P. 269:22. BF:BS = 9428:5757=10,776:6580.1, for which Copernicus writes "about 6580" (Jere; fol. 169", line 6 up). P. 270:5. ES:ADE= 6580:10,960 = 1P:1P39'56"21'", for which Copernicus finally wrote 57" (lvjj), after deleting a now illegible fraction above 56 (fol. 169v, line 4). P. 270:6. ES:BC = 6580:9040 = 1P: 1P 22' 25" 54'", for which Copernicus finally wrote 26" by squeez­ ing vj in, between the lines (fol. 169v, line 4). This change from 20" to 26" is connected with the deleted passage in VI, 3, where Copernicus let 20 stand, after erasing something (perhaps vj; fol. 1931', line 1). P. 270:27. This Theon was mistakenly identified with Alexandria by Copernicus (fol. 169v, line 6 up). But Theon of Alexandria lived in the fourth century, long after the death of Ptolemy. It was another Theon, whose observations were used by his somewhat younger contemporary, Ptolemy. The latter, however, did not connect this Theon with Smyrna or with any other Greek community. But, after saying that for the present purpose he did not find a satisfactory observation made by the old astronomers, Ptolemy continues: "On the other hand, I did find what I need for this investigation among the observations which were made in our own time. And I have found among the observations which have come to us from the observations of Theon the mathematician ... " (PS 1515, fol. 109", Ch. 1, lines 11-14). Ptolemy's reference to Theon "the mathematician" as his contemporary, and his remark that he had access to Theon's observations may have given Copernicus the misimpression that this Theon was connected with Alexandria. In the summer of 1539 Copernicus received from Rheticus a copy of the first Greek edition of Ptolemy's Syntaxis (Basel, 1538), which was accompanied by Theon's commentary thereon. This commentator was prom­ inently identified on the title page, in both Greek and Laqp, as "Theon of Alexandria." Had Copernicus merely glanced at this title page before writing V, 20, he would instantly have realized that the Theon of Alexandria who wrote the commentary on the Syntaxis, and therefore lived later than Ptolemy, could not possibly be the Theon from whom Ptolemy, while still alive and composing the Syntaxis, borrowed an observation of Venus. Hence we may conclude that the commentary of Theon effectively came to Copernicus' attention only after he had finished writing V, 20. On the other hand, by the time Copernicus wrote V, 35 (fol. 197V), he adopted an expression known to him only from the 1538 edition of the Syntaxis. We may therefore date the effective reception by Copernicus ofRheti­ cus' presentation copy of the Syntaxis with Theon after the composition of V, 20, and before the composition of V, 35. P. 270:31. Ptolemy put the mean sun at 141/,0 within the Fishes= 344° 15', from which Copernicus sub- tracted 6° 34' for precession, in order to obtain 337° 41'. P. 270:34. At first Copernicus equated 4 Antoninus Pius with 144 A.D., then with 142 A.D. (fol. 17or, line 5). This is merely a slip on his part, as can be seen from his detailed discussion in the Letter against Werner (3CT, pp. 94-97) and in V, 26, where he matches 4 Antoninus Pius with 140 A.D. (fol. 176v, last 3lines). Here in V, 20, the erroneous counterpart 142 was correct!=d to 140 in Miistlin's copy of N (fol. 160v, line 5) and for the first time publicly in Me, Notes, p. 56, n. 417. P. 270:36. Ptolemy put the mean sun at 53/ 4° within the Lion= 125° 45', from which Copernicus sub­ tracted about 6° 45' for precession, in order to obtain about 119° (fol. 11or, line 8). P. 270:45. Copernicus mistakenly dated this observation in 4 Hadrian (fol. 17or, line 18) because he fol­ lowed P-R (Book X, Prop. 1; sig. 12V). The year is given incorrectly also in PS 1515 as 2 Hadrian (fol. 109f). In PS (X, 1) the year was 12 Hadrian. According to the Egyptian reckoning, 1 Hadrian began on 25 July 116, when Hadrian was regarded as starting to share the highest power with his predecessor Trajan; hence, 4 Hadrian commenced on 25 July 119. Fortunately, Copernicus used neither 4 Hadrian nor 119 as a basis for further com­ putation. In Miistlin's copy of N (fol. 160v, lines 16, 18) 4 Hadrian was corrected to 12, and 119 to 127.

426 NOTES ON PP. 271-273

P. 271:1. Ptolemy put the sun's mean place at 17° 52' within the Balance= 197° 52', from which Co­ pernicus subtracted 6° 39' for precession, in order to obtain 191° 13' (fol. 17or, line 13 up). P. 271:5. Ptolemy put the sun's mean place at 21/16 ° within the Goat= 272° 4', from which Copernicus subtracted 7° 4' for precession, in order to obtain 265° (fol. 11or, line 8 up). P. 271:14. Ptolemy put the sun's mean place at 252/ 6 ° within the Bull= 55° 24', from which Copernicus subtracted 6° 34' for precession, in order to obtain 48° 50' (fol. 110v, line 5). P. 271:17. The correct equivalent in the Roman calendar is 18 November, which Copernicus actually started to write (fol. 110v, line 9: xiiij Cal). But instead of continuing with the proper month (December), he shifted to January, and then changed the number of the day from 14 to 5, perhaps by confusion with another observation of Venus by Ptolemy in 21 Hadrian (PS, X, 1, last observation). The correction to 18 November = 14 K.alends December was made in Mastlin's copy of N (fol. 161r, line 2) and for the first time publicly in A. P. 271:18. Ptolemy put the sun's mean place at 25° 30' within the Scorpion = 235° 30', from which Copernicus subtracted 6° 36' for precession, in order to obtain 228° 54' (fol. 170v, line 10). P. 271:37. According to Lines Subtended, for 44° 50': 70,505; and for 44° 40': 70,298; hence, for 44° 48': 70 ,463.6, or 7046, with radius = 10,000. P. 271 :39. Here (fol. 111r, line 2) Copernicus put the greatest elongation in Ptolemy's 3rd observation in v, 20, = 471/s 0 • This was also the value which he originally used in V, 20 (fol. 170v, line 12). But there he later deleted "one-third" and replaced it by 16' in the right margin. Here in V, 21, however, he did not make the corresponding change but allowed "one-third" to remain. Yet he operated, not with 47° 20', but with 47° 16' to obtain the length of DF. For, according to Lines Subtended, for 47° 20': 73,531; and for 47° 10': 73,333; hence, for 47°16': 73,451.8, for which Copernicus writes 7346, with radius= 10,000. On the other hand, with DBF = 47° 20', DF = 7353. In Brahe's copy of B, the fraction to be added to 47° for DBF was changed from 1/3 to!(= 15', fol. 161r, line 5 up). P. 271:40. DF:BD = 7346:10,000 = 7046:9591.6, for which Copernicus mistakenly writes 9582 (fol. 111r, line 5), which would make DF = 7353. P. 271:43. AC:DE = 9791:7046 = 1P:431/6'; AC:CD = 9791:209 = lP:l' 16'' 51'", for which Coper­ nicus writes "about 1!'" (fol. 171 r, line 8). P. 271:44. AC:DE = 1P:431/6' = 10,000:71941/3, for which Copernicus ultimately writes 7193, although originallyhehad somefractionalvalue,whichheobliterated(fol.171r,line9).AG:GD = 1P:1!' = 10,000:2081/3, for which Copernicus writes "about 208." P. 272:6. Ptolemy put the mean sun at 25!0 within the Water Bearer= 325° 30', from which Coper­ nicus subtracted 6° 40' for precession, in order to obtain 318° 50' (fol. 111r, line 7 in this chapter). p. 272 :6. Ptolemy located Venus at 11° 55' within the Goat = 281° 55', from which Copernicus subtracted 6° 40' for precession, in order to obtain 275° 15' (fol. 111r, line 8 in this chapter). P. 272:12. By a slip (fol. 111r, line 7 up) Copernicus wrote, instead of sextante = 1/60 dextante = 6/ 6, an error which was corrected in Mastlin's copy of N (fol. 161v, Ch. 22, line 15) and for the first time publicly in A (p. 385). P. 272:19. By a slip (fol. 111v, line 1) Copernicus wrote EGD, which was corrected to EGO in Mastlin's copy of N and Brahe's copy of B (fol. 161v, line 10 up) and for the first time publicly in T (p. 368, line 6). P. 272:25. According to Lines Subtended, for 2° 30': 4362;andfor2° 20': 4071;hence,for2° 23': 4158.3, or 416 with radius= 10,000. P. 273:9. By a slip (fol. 112r, line 8) Copernicus wrote 20th, although PS 1515 said correctly "29th" (fol. nov). P. 273:16. In putting 33f4h = gdm 32ds (fol. 112r, line 13 up), Copernicus' pen slipped, substituting xxxij for xxiij. For, 3~m = gdm 22td•, which he often rounds off as 23. P. 273:17. Ptolemy located the sun at 22° 9' within the Archer= 262° 9', from which Copernicus sub­ tracted 6° 39' for precession, in order to obtain 255° 30' (fol. 112r, line 17 in the deleted passage). P. 273:26. With the moon at 209° 55' and the star at 209 40, the intervening distance is 15', to be divided 1!:1 = 9:6, so that Venus = 209° 46' = 209° 55' -9' = 209° 40' +6'. By the same token, with regard to the latitude, with the moon at 4° 42' and the star at 1 20 the intervening distance is 3°22', to be divided 1!:1;;:;; 2°:1°20', so that Venus= 2° 40' = 4° 42'-2° ;;:;; 2° 42' = 1° 20' +1 o 20' = 2° 40'.

427 NOTES ON PP. 274-276

P. 274:6. Instead of BCE, by a slip Copernicus wrote BDE (fol. 172v, line 15 up). At fol. 172v, last line, he equated CDFwith 54° 20' = 2X 27° 10', and at fol. 172v, line 7 up, he put DCE (= BCE) = 27° 10', so that CDF = 2 x BCE. On the other hand, BDB will soon emerge = 28° (see note to p. 274:13). P. 274:9. Copernicus drew this diagram (fol. 172V) on such a scale and placed it so that he had no room to complete Venus' orbit and show the actual intersection with EF at Land with FK at K. From this diagram the reader can easily see the great dispatity between the perigeal and apogeal distances of Venus. As a conse­ quence, the naked-eye brilliance of Venus should vary considerably. In fact, it does not. Since Copernicus could not explain the absence of this vatiation, he followed his usual practice in such matters: a prudent silence. He never "points out Ptolemy's error in failing to account for the vatiation in brightness of Venus," despite Price's unsupported attack on Copernicus in this regatd (see note top. XVI). That great Copernican, Galileo, understood this situation much better. In his magnificent Dialogue of 1632 Galileo pointed out in his Third Day that "with regatd to the slight Vatiation in her [Venus'] size, Copernicus said nothing ... I suppose because he could not explain to his own satisfaction a phenomenon so irreconcilable with this thesis" (Opere, national edition, VII, 362: 12-15). In this connection GalUeo made the unfortunate mistake of attributing to Copernicus the idea that Venus was either self-luminous or transparent, a conception actually ascribed by Copernicus to others. Galileo was misled by a misprint inN, and his magisterial prestige induced many later writers to follow his example; see Rosen, "Copernicus on the Phases and the Light of the Planets," Organon, 1965, 2: 69-74. P. 274:13. Therefore, BDE = DCB+CED = 27° 10'+50' = 28°. SinceBDE= 28°andCDF = 54° 20', CDF :p 2x BDE, the slip corrected above in note top. 274:6. P. 274:16. The whole of FDE = FDB+BDB = 125° 40'+28° = 153° 40' :P 152° 50'. Copernicus made the sum = 152° 50' (fol. 17:¥, line 2) because he mistook BCB for BDE. But BDE = BCB + CED = 27°10'+50' = 28°. P. 274:30. Copernicus' date (xiij Cal Januarii; fol. 17:¥, line 10 up)= 3 A.M., 20 December 138 (origi­ nally 139) is inconsistent with his interval from the beginning of the Christian era to Ptolemy's observation (fol. 172r, lines 16-19), where he gave 138Y lad 33/,h = 137 complete yeats 138 January through November 334"- December 15 38f4h leapdays (4-136) 34 383 1 -365 138Y

P. 275:5. Here (fol. 174v, line 17) Copernicus gives this stat's longitude= 15lt0 • This was also Coper­ nicus' value in his Catalog of Fixed Stats (fol. 6ov, line 5 up), and it so appears inN (fol. 54V). Thereafter, some one other than Copernicus blotted out the zero and wrote a poorly formed 5 over it; for the sake of clatity, he then placed 35 in the adjoining vacant column. Presumably he atrived at 151° 35' by subtracting from Ptolemy's longitude 8° 15' within the Virgin = 158° 15' Copernicus' usual correction for precession 6 40 151° 35'. P. 275:18. Instead of the correct sum 147° 4', by anticipatory dittography Copernicus wrote 144° 4' (fol. 175r, line 1). P. 275:25. Arc KLG =semicircle KL+(LG =

428 NOTES ON PP. Z77-279

P. 277:26. Copernicus evidently equated dawn, 18 Mesori, 13 Ptolemy Philadelphus with 3: 30 A.M., 12 October 272 B.C. 272 October 20th November-December 271 complete years leapdays 68 271Y 148d 20th +1529 87 19! 1800Y 1 (24+)1()h(= 4Qdm) 236d For 1529Y 87d 19jh as the interval from the beginning of the Christian era to Copernicus' observation, see the preceding Note. P. 277:36.From3: 45 A.M., 16(not 20;seenotetop. 274:30) December 138 to 7:30P.M., 12March 1529: 138 December 15d 2Q!h 1390 complete years 139-1528 1529 January through February 59 March 11 19-l leapdays (140-1528) 348 1 (24+) 434 1 -365 1391Y ~ (fol. 174", line 5 up). Copernicus' computation of this interval shows that he actually reckoned from 16 Decem­ ber, so that he obviously intended to write X'Oij Cal., and xiij Cal. is a mere slip of the pen at fol. 17:¥, line 10 up. P. 277:46. Here (fol. 174v, line 3) Copernicus writes 9-ldm~ 3"/,h (fol. 112•, line 19); 9tdm = 3h481D. P. 278:8. Here Copernicus evidently dated the beginning of the 1st Olympiad at noon on 1 July 775 B.C.: 775 184d 502 complete years 774-273 272 January through September 273 October 11 1(jh leapdays (772-276) 125 593 1 -365 503Y For 18 Mesori, 13 Ptolemy Philadelphus = 12 October 272, see note to p. 277:26. P. 278:39. Although Ptolemy used the equant model, he did not employ any such term. Arabic writers, however, introduced a designation which, when converted into Latin, was reduced by stages to "equant." This name appears in John of Seville's twelfth-century Latin translation of Al-Farghani, but not in Gerard of Cre­ mona's rendering of Ptolemy's Syntaxis into Latin. This was done in 1175, printed in 1515, and read by Coper­ nicus. He encountered "equant" in P-R (for example, Book IX, Prop. 7). This work was completed before 28 April 1463, but not printed until 31 August 1496. See Francis S. Benjamin and G. J. Toomer, Campanus of NOfJara (University of Wisconsin Press, 1971), p. 405, n. 3, for a discussion of the emergence of the term "equant." P. 279:9. According to Proclus' Commentary, a straight line can be "generated by [the combiQ.ation of] multiple motions" (tr. Morrow, p. 86). This reference to Proclus was written by Copernicus in the right margin of fol. 176• after he had received a copy of the first edition of the Greek text of Proclus' Commentary, together with the first edition of the Greek text of Euclid, as a gift from Rheticus in the spring of 1539. Hence, Copernicus could not have written this marginal note before 1539. Had he known about this method of generating a straight line from Albert of Brudzewo's Commentary on Peurbach (Milan, 1495), why would he have waited until 153~ and cited Proclus then? Copernicus' previous failure to cite Albert may indicate an unfamiliarity on his part with Brudzewo's Commentary on Peurbach. On the other hand, Copernicus' reference to the ellipse in the deleted passage in III, 4, seems indirectly reminiscent of this very passage in Proclus. For here that commentator on Euclid deals with a straight line at­ tached at both ends to the sides of a right angle. As an example, let one end of a ladder lean against a vertical wall, while the ladder's other end rests on the horizontal ground. Now let the ladder slide, with the upper end

429 NO TBS ON PP. 279-281

slipping down along the wall while the lower end moves away from the wall. If at the same time the midpoint of the ladder generates an arc of a circle, all the other points on the ladder (except the extremities) generate ellipses. Proclus' construction has been erroneously identified with Tusi's couple (Journal for the History of Astron­ omy, 1973, 4: 129). Somehow Proclus' Greek was there mistranslated as follows: "If we imagine a straight line resting on the sides of a right angle, its centre will describe a circle." But if a straight line rests anywhere, its center will describe nothing. The mistranslation continues with the straight line's "ends moving rectilinearly," whereas the transmitted Greek text says "uniformly," although that has perforce been emended to "nonuni­ formly" (=unequally; Paul Tannery, Memoires scientifiques, II, 36). The mistranslation continues by having "the centre moving curvilinearly," whereas the Greek text says "nonuniformly." Moreover, we are told that "Proclus's aim was to demonstrate how a cyclic movement is obtained from two rectilinear ones." Actually, Proclus is wrestling with an asserted distinction between a simple line and a mixed line; although the circle is a simple line, it can be generated by a nonuniform motion, under the given conditions. P. 280:13. By a slip Copernicus mislettered this diameter HK (fol. 176v, left margin, note 3, line 4). The diagram to which he here refers appears on fol. 176r. His error was corrected in T (p. 378, line 13). P. 280:18. In the Table, which Copernicus actually called "Mercury's Parallactic Motion in Days," the entry for 58d = 3X60° = 180°, so that a complete revolution of 360° = 2X58d = 116d. P. 280:35. 137 complete years leapdays (4-136) 34d 138 January through May 151 June 3 42!dm = 17h 188d Hence, Copernicus dated Ptolemy's observation 5 P.M., 4 June 138, Cracow time. P. 280:37. Ptolemy located the mean sun at 10-lo within the Twins= 70° 30', from which Copernicus subtracted 6 40 to obtain 63° 50' for his mean place of the sun. P. 280:43. 140 complete years leapdays (4-136) 34d 141 January 31 February 2 12dm = 411 48m 67d Hence, Copernicus dated Ptolemy's observation 4:48 A.M., 3 February 141. P. 280:44. Ptolemy located the mean sun at 10° within the Water Bearer= 310°, from which Copernicus subtracted 6° 41' for precession, in order to obtain 303° 19' for the place of the mean sun. P. 281:4. Instead of 276° 49' = Mercury's place in the 2nd observation, Copernicus wrote the sun's place in the 1st observation= 63° 50' (fol. 177f, line 8). Yet in the following computations he correctly used 276° 49'. P. 281:10. Ptolemy put the mean sun at 9° 15' within the Balance= 189° 15', from which Copernicus subtracted 6° 37' for precession, to obtain 182° 38' for his own position of the mean sun (fol. 177f, line 14). P. 281:12. Ptolemy located Mercury at 20° 12' within the Virgin = 170° 12', from which Copernicus subtracted 6° 37' for precession, so that he should have obtained 163° 35' for his own position of Mercury. By a transpositional slip, however, for the number of degrees, instead of clxiij, he wrote cxliij (fol. 177f, line 16), an error which was corrected in Miistlin's copy of N (fol. 166f, line 8) and for the first time publicly in A (p. 396). P. 281 :13. In equating "the same year of Hadrian," that is, 19 Hadrian, with a particular year in the Chris­ tian calendar, Copernicus does not mean that the two years are coextensive. Thus, Ptolemy made his 1st obser­ vation in Athyr, the 3rd Egyptian month, but his 2nd observation in Pachon, the lOth month. Since Hadrian's regnal year started in the summer, Ptolemy's 1st observation was made in 134, and his 2nd observation in 135. For this last numeral Copernicus mistakenly wrote Mccct~ (= 1305; fol. 177f, line 18). Had he used the Roman numeral CXXXfJ = 135 in his preliminary notes for this passage, he would not have shifted to Mccco at fol. 177r, line 18. On the other hand, the Hindu-Arabic numeral 135 in the preliminary notes could conceivably have been erroneously converted into Mccctl at fol. 177f, line 18. If this passage exemplifies Copernicus' usual prac­ tice, then he actually operated privately with the Hindu-Arabic numerals, but later transformed them into the traditional Roman numerals for purposes of publication. P. 281:14. Ptolemy located Mercury at 4° 20' within the Bull = 34° 20', from which Copernicus subtracted 6° 37' for precession, to obtain 27° 43' for his own position of Mercury (fol. 177f, line 20).

430 NOTES ON PP. 281-282

P. 281:15. Ptolemy located the mean sun within the Ram at 11° 5', from which Copernicus subtracted 6° 37' for precession, to obtain 4° 28' for his own position of the mean sun (fol. 177l, line 21). P. 281:31. According to Lines Subtended, for 19° 10': 32,832, and for 19°0':32,557; hence, for 19°3': 32,639.5, for which Copernicus writes 32639 (originally 32649; fol. 177v, line 7). P. 281:32. According to Lines Subtended, with DBF = 23° 15', for 23° 20': 39,608, and for 23°10': 39,341; hence, for 23° 15': 39,474.5, for which Copernicus writes 39,474 (fol. 177v, line 8). P. 281:33. Segment AD was mislettered by Copernicus "ADC" (fol. 177Y, line 10). Just above, in line 6, he made the same error, but there he corrected it by deleting the C. Here, however, he let the mislettering ·remain.

P. 281:34. FD = ED:AD = 32,639:100,000; F'D = AD ( 32,639 ) 100,000

F D : D B = 39474: 100,000; FD =DB (39,474) 100,000

AD ( 32,639 ) = DB ( 39,474) 100,000 100,000

AD:DB = 39,474:32,639 = 100,000:82,684.8forwhichCopernicus writes 82,685 (fol. 177v, left margin, replacing an error in line 10). P. 281:38. AC:DE = 91,342:32,639 = 60':21' 26" AC:CD = 91,342:8658 = 60':5' 41" P. 281:39. AC:DF = 91,342:32,639 = 100,000:35,732.9, for which Copernicus writes 35,733 (fol. 177Y, line 15). AC:CD = 91,342:8658 = 100,000:9478.7, for which Copernicus writes 9479 (fol. 177v, left margin). P. 281 :45. -129 complete years leapdays (4-128) 32d 130 January through June 181 July 3 45Cim= 18h 216d Hence, Copernicus dated Theon's observation at 6 P.M., 4 July 130. P. 281:46. Ptolemy located the mean sun at 10° 5' within the Crab = 100° 5', from which Copernicus subtracted 6° 35' for precession, in order to obtain 93° 30' for his own position of the mean sun. P. 282:1. By a slip Copernicus wrote "west of" (praecedere; fol. 177v, line 7 up). This error was corrected by Z, p. 448. P. 282:2. The fraction should be 1/ 8, not 8/ 4 (dodrans; fol. 177v, line 5 up; Z, p. 448). P. 282:4. Ptolemy assigned this observation to 24 Mesori. The number of the day was mistakenly changed to 21 in PS 1515 (fol. 101r, line 15 up), which was undoubtedly the source of Copernicus' error. P. 282:5. 138 complete years leapdays (4-136) 34d 139 January through June 181 July 4 12dm = 4'/,h 21¢ Hence, Copernicus misdated Ptolemy's observation at 4:48A.M., 5 July 139, the correct day being 8 July, in accordance with the preceding note. P. 282:6. Ptolemy located the mean sun at 10° 20' within the Crab= 100° 20', from which Copernicus subtracted 6° 41' for precession, in order to obtain 93° 39' for his own position of the mean sun. P. 282:8. Ptolemy placed Mercury at 20° 5' within the Twins= 80° 5', from which Copernicus subtracted 6° 41' for precession, in order to obtain 73° 24' for his own position of Mercury. P. 282:19. According to Lines Subtended, for 3° 0': 5234, f9r which, with the radius= 10000, Coper­ nicus writes 524 (fol. 178r, line 17). P. 282:24. Originally (fol. 178r, right margin) Copernicus put CFI = 634, because he was operating with DF=422( +IF= 212). Then he realized that, to obtain CFI, he must add to IF= 212, not DF, but CF = 524. Hence, deleting 634, he replaced it by 737, for which he later substituted 736, followed by the upper two-thirds of his sign fort. He made this change from 737 to 736!- after he had written 737 at fol. 18or, line 3. There he wrote

431 NOTES ON PP. 282-285 a 6 heavily over the 2nd 7 and squeezed! into the line, after doing the same thing at fol. 178v,line 5 above the deleted passage. P. 282:25. See above, note to p. 281:32. P. 282:27. EF:FH = 10,000:3947 = 10,014:3952.53, for which Copernicus writes 3953 (fol. 178r, line 6 up). P. 282:44. The Ptolemaic equivalent of this Copernican formulation occurs in PS, IX, 8. P. 283:30. EF:FG = 9540:3858 = 10,000:4044, instead of which Copernicus wrote 4054 (fol. 179r, line 6 up). He obtained this result by operating with FG = 3868; beneath the heavily written 5 at fol. 179r, line 11 up, presumably a 6 lies obliterated. This value (3868) should be the sum of 3573 (fol. 178v, line 4) plus

3/ 4 X 380. However, for 3/ 4 X 380 Copernicus mistakenly wrote 295, presumably by dittography because he had. obtained 1/ 4 X 380 = 95 (fol. 179r, line 14 up). When he noticed this error, he replaced 295 by 285, and 3868 by 3858. Then, moving down to 4054, he deleted it. But, instead of substituting the correct value 4044, he re­ peated 4054 (by dittography ?) in the right margin. While Copernicus put FG = 4054, he made the corresponding angle= 23° 55' (fol. 179r, line 5 up). Ac­ cording to Lines Subtended, 40674 for 24° 0' -40408 -23 50 266 10 133 5 +40408 +23 50 40541 But he lowered the number of minutes from 55 to 52!, so that he evidently intended to reduce 4054, al­ though, as we just saw, he failed to carry out that intention. An angle of 23° 52!' would correspond to 4047, with the radius = 10,000. P. 283 :41. Copernicus interchanges the distances in longitude and latitude. He makes the longitudinal distance "2 lunar diameters" and the latitudinal distance "1 lunar diameter" (fol. 179v, lines 11-12), whereas Ptolemy had put the longitudinal separation at 1lunar diameter and the latitudinal separation at 2 lunar diame­ ters. P. 283:45. Ordinarily Copernicus did not write fractions in the Egyptian manner, which required the nu­ merator to be unity, with rare exceptions. Here, however, he writes "a half and a third," which he equates with "five-sixths" (fol. 179v, lines 16-17). However, in his own Star Catalog (fol. 61v, line 19) as in Ptolemy's, this star's latitude = 1° 40', not 1° 50'. Hence we have here still another example of Copernicus' anticipatory dit­ tography, since 2 lines below he gives 1° 50' as the latitude of Mercury, not of the star. P. 284:3. Misled by a confusing rendering in PS 1515 (fol. 108r, line 9), Copernicus said "during the next 4 days" (subsequentibus iiij diebus; fol. 179v, line 11 up), whereas Ptolemy meant "on the 4th day thereafter." P. 284:13. Originally (fol. 180r, line 3) Copernicus wrote 737. Later he wrote a 6 heavily over the 2nd 7, and squeezed ! into the line, in conformity with the changes he made on fol. 178r, right margin, and fol. 178v, line 13 up. This reduction in the length of CI from 737 to 736! accompanies the corresponding reduction of IF from 212 (fol. 178r, line 13 up; fol. 179r, line 4) to 211! (fol. 180r, line 12). P. 284:23. Copernicus failed to draw this diagram in the autograph. He puts diameter LM = 380P, the greatest variation in Mercury's distance from the center of its orbit [V, 27]. P. 285 :2. This passage has sometimes been misunderstood to mean that Copernicus never saw the planet Mercury. Thus, Le Verrier said that Copernicus "never could manage to see Mercury" (Annales de l'Obser­ vatoire de Paris, V, 1859, 1-2). Presumably Le Verrier was misled by Delambre's mistaken remark that "Coper­ nicus had never been able to see Mercury on account of the mists of the Vistula" (Histoire de l'astronomie moderne, I, 134). P. 285:11. Five years after Walther died in 1504, his house in Nuremberg was sold by his heirs to the fa­ mous artist Di.irer (Albrecht Darer's Wohnhaus und seine Geschichte, Nuremberg, 1896, pp. 4-6). Walther's handwritten observations passed into the possession of Johann Schoner, who printed them at Nuremberg four decades later. This was the year following the death of Copernicus and the publication of his Revolutions in that same city. Walther's observations were appended to Schoner's edition of some writings and observations of Walther's teacher, Regiomontanus (Scripta clarissimi mathematici M. Joannis Regiomontani, Nuremberg, 1544). Walther's observations were printed a second time in Willebrord Snel's edition of the observations of the Landgrave of Hesse, Coeli et siderum in eo errantium observationes Hassiacae (Leiden, 1618). The third edition appeared as part of the Historia coelestis (Augsburg, 1666), with the editor's name disguised as Barettus (not "Barethis," as in Donald de B. Beaver, "Bernard Walther," Journal for the History of Astronomy, 1970, 1: 40).

432 NOTES ON P. 285

P. 285:13. Walther omits this reference to the instrument in his observation of 9 September (Scripta ..• Regiomontani, fol. 55r, misnumbered 59). However, in his first Mercury observation of 1491, dated 26 August, he says that his instrument was directed toward Aldebaran ("Armillis rectificatis per Aldebaran"); under 31 Au­ gust, he repeats "rectificatis Armillis ut prius"; and under 2 September he again remarks "Armillis rectificatis iterum per Aldebaran." In this connection it should be noted that Walther has no hesitation in using the Arabic star-name Aldebaran. In its place Copernicus, true to his humanistic harking back to classical antiquity, sub­ stitutes Palilicium. In his Star Catalog, it will be remembered, when Copernicus listed the first-magnitude star in the southern eye of the Bull, he said that it was "called 'Palilicium' by the Romans" (fol. 58v, last line). P. 285:14. Walther says "I found Mercury at 13° 23' within the Virgin." At first Copernicus made the fraction "about two-fifths" :;;;;: 24', which he deleted and replaced by "ith of a sign" = 15' (fol. 180v, lines 8-7 up). Copernicus' final decision to use l 0 appears there in the left margin. In Copernicus' successive shifts from 23' to "about 24"' to 15' to 30', certain scholars have imagined that they detected some sinister or dishonest motive. P. 285:16. Walther's observations of Mercury in 1491 began on 26 August and ended on 11 September. In the observation selected by Copernicus, that of 9 September, Walther remarked "Mercury appeared quite faint," and two days later Walther reported: "11 September, Mercury was still visible but very dimly." P. 285:17. From the beginning of the Christian era to 5 A.M., 9 September 1491: 1491Y 258d 5h = 12tdm. P. 285:19. To 149° 48' =sun's mean place, Copernicus adds 26°59' for precession, in order to obtain 176° 47' as the sun's longitude. In V, 23, he put the precession= 27°24' in 1529 (p. 276:12). P. 285:21. Johann Schoner (1477-1547) was an ordained Roman Catholic priest until rebellious peasants threatened to put all such clergymen to death. Thereafter Schoner taught mathematics at the Melanchthon High School in Nuremberg. It was he who edited Regiomontanus and Walther in 1544. P. 285:22. Whereas Copernicus' source (Scripta ... Regiomontani, fol. 58•) says this observation was made on "9. Januarii," Copernicus the humanist dates it in the Roman style "5 days before the Ides of January." This observation was performed by Walther, but it was misattributed to Schoner by Copernicus (fol. 181r, line 3). Why did Copernicus make this misattribution? Zinner stated, as though it were a historical fact, that "Copernicus had three determinations of Mercury's position reported to him by Schoner" (Leben und Wirken des Johannes Muller '/Jon Konigsberg genannt Regiomon­ tanus, Munich, 1938, p. 173; second edition, Osnabriick, 1968, p. 231). For any direct contact between Copernicus and Schoner, there is not the slightest shred of historical evidence. Nevertheless, we have recently been assured that Copernicus "knew, through Johann Schoner ... that Walther had an extensive series of Mer­ cury observations ... Perhaps there was a closer connection between Copernicus and Walther, although it is only a matter of conjecture. In the fall of 1496, spring of 1501 and spring of, 1503, Copernicus may have passed through Nuremberg, and thus might have visited with Walther, who was there at the time. It would seem that Copernicus would hardly have avoided the famous pupil of Regiomontanus, yet it is not known whether the two astronomers met" (Beaver, p. 42). The evidence for any personal contact between Copernicus and Walther is even less visible than for any contact between Copernicus and SchOner. Somehow Z deluded himself into thinking that the statement made by Copernicus at the beginning of the preceding paragraph "proves... that he [Copernicus] received Walther's observations through Schoner" (p. 212). Nevertheless, Copernicus' statement proves no such thing and offers not the slightest clue to the process by which Copernicus obtained the Nurem­ berg observations of Mercury. By misconstruing the meaning of primum in this context (fol. 180v, line 12 up), Z devised a false explana­ tion of Copernicus' misattribution. Whereas Copernicus' primum means merely that the "first" of the three Mercury observations was made by Walther, Z mistranslated primum by "at first" (zuerst}, with the implication that later the observations were made by someone other than Walther. According to Z, Copernicus "bethought himself that Walther died in the year 1504 and therefore did not carry out all [three] observations." But in 1504, Walther died on 19 June, and the two 1504 observations were made on 9 January and 18 March. Therefore, if Copernicus knew the time of Walther's death, he had no reason to withdraw the two 1504 observations from him. Why, then, did Copernicus make this mistake, and how did he obtain Walther's observations? Before Rheticus came to confer with Copernicus in Frombork, he visited Schoner at Nuremberg. It may well have been Schoner who suggested to Rheticus that he should learn about the new astronomy directly from Coper­ nicus. Is that not the reason why about halfway to his destination, at Poznan, on 14 May 1539 Rheticus wrote SchOner a (lost) letter informing him that the trip had been undertaken (3CT, p. 109)? In V, 35, Copernicus introduces an unusual term, with which he became acquainted through a book brought to him by Rheticus in 1539. Is it not entirely plausible that here in V, 30, Copernicus utilizes the Nuremberg

433 NOTES ON P. 285

observations brought to him by Rheticus from Schoner? Is the presence of Schoner's name, as the forthcoming editor of Walther's observations, the explanation of Copernicus' misattribution? That was corrected by someone other than Miistlin in his copy of N (fol. 169v, line 10, left margin). P. 285:25. At fol. 181•, line 6, Copernicus mistakenly wrote "Water Bearer," where the context clearly requires "Goat." For in lines 6-7 he located Mercury 23° 42' west of the sun, with the planet at a fraction over 3° within the Goat (lines 3-4). Hence the sun, at 27° 7', must be within the Goat, not the Water Bearer. Coper­ nicus was evidently thinking about some other Nuremberg observation of Mercury. The fraction over 3° for Mercury's position within the Goat was changed by Copernicus from! (line 4, like his source) to 1/ 3 (right margin). Hence, with the sun at 27° 7' within the Goat (not the Water Bearer), and Mercury at 3° 20' within the same sign, the planet's westward elongation was 23° 47'. Yet, for the minutes, Copernicus mistakenly wrote 42 (xlij, line 7). Nevertheless, his value for this interval was actually 23° 47', as is shown by his statement later in V, 30 (fol. 195•, lines 5-7) that the computed distance 23° 46' "takes only a little away" (parum demunt) from the observed elongation, which must therefore be 23° 47'. This correction to 23° 47', and the replacement of Water Bearer by Goat, were indicated by a hand other than Miistlin's in his copy of N (fol. 169v, lines 13-14). In Brahe's copy of B, the Water Bearer was deleted and above it was written the sign for the Goat. P. 285:26. Where his source said "18 Martii" (Scripta ... Regiomontani, fol. 60'), Copernicus again fell back on the Roman equivalent "15 days before the Calends of April" (fol. 181•, line 8). P. 285:27. Copernicus' source put Mercury at 26° 30' within the Ram. At first Copernicus omitted the fraction, apparently inadvertently; then in the right margin he inserted "cum deunce unius gradus" (with 11/12 ° = 55'; fol. lSI•). But N (fol. 169v, line 16) misread deunce as decima (1/10° = 6'). Miistlin accepted decima as the correct reading. He realized, however, that something was amiss here, and wrote in the left margin of fol. 169v of his copy of N: I think we must read "at 27° minus 1/10 °" [ = 26°54']. Brahe too sensed an error, and wrote "26° 48"' in the margin of his copy of B. P. 285:33. From 5 A.M., 9 September 1491 to 6:30 A.M., 9 January 1504: 1491 September 21d 19h October through December 92 12 complete years 1492-1503 1504 January 8 6 3om leapdays (1492-1500) 3 1 (24+ )1h 3om = 3dm 45d• 12Y 125d P. 285:34. According to the Tables of the Sun's Simple Uniform Motion in Years and in Days, after III, 14, for 12Y 5X 60° = 300° 56 57' 49" 24'" 120d = 2X 60d 60 58 16 22 5d 4 55 40 56 3dm 45d• ~ 4 --::---,----,.,-- 4800 13' 52" 20'" -360 120° 13' 52" 20"', for which Copernicus writes 120° 14' (fol. 181•, line 16). P. 285:35. According to the Tables of Mercury's Parallactic Motion in Years and in Days, after V, 1, for 12Y 4 X 60° = 240° 47 28' 37" 18"' 12od = 2x6od 6X60° = 360 12 48 27 5d 15 32 1 8 3dm45d8 ~ 12 676° 1' 5" 26"' -360 316°, for which Copernicus writes 316° 1' (fol. 181•, line 17) after correcting an error. P. 285:36. From 6: 30 A.M., 9 January 1504 to 7:30 P.M., 18 March 1504:

434 NOTES ON PP. 285-288

1504 January 22d 17h 3om February 29 March 17 19 30 1 (24+) 13h = 32!-dm 69d

Instead of 32!-dm, Copernicus writes 31dm45d8 (= 12h42m; fol. 181r, line 18). P. 285:37. According to the Table of the Sun's Uniform Motion in Days, after III, 14, for 6Qd 59° 8' 9d 8 52 3ldm 45ds ~ 32 68° 32' = sun's mean motion, for which Copernicus should have written motus, not locus ( = place, fol. 181 r, line 18). P. 285:44. Copernicus writes 28!- (xXYJiij s; fol. 181r, line 7 up). P. 285:47. With Mercury's apogee at 211° 30' and the sun's mean position in the 1st observation at 149° 48', the sun's distance from the planet's apogee= 360°-211° 30' = 148° 30' +149 48 298° 18', for which Copernicus writes 298° 15' (fol. 181r, line 3 up). P. 285:48. In the 2nd observation the mean sun was at 297° 7', corrected for precession of 27° 8', = 269° 59' -211 30 58° 29'. P. 285:48. In the 3rd observation the mean sun was at 5° 39', corrected for precession of 27° 8', = 365° 39' - 27 8 338° 31' -211 30 127° 1'.

P. 286:5. Copeq:~.icus began to designate this angle by some letters which he deleted (fol. 181v, line 3 and replaced (incorrectly) by IEC, an error which was corrected in T (p. 389, line 19, and note). P. 286:17. Copernicus (fol. 18F, line 17) mislettered this angle POM, an error which was corrected in Me (Notes, p. 62, n. 455). P. 286:18. To arrive at the ratio 190:105, Copernicus should have operated with 5519 = sin 33° 30' =

435 NOTES ON PP. 288-302

323Y I3Qd I2h In V, 29, he dates this ancient observation 59 I7 Ish(= 45dm) after Alexander's death 264Y 112d ISh The 3rd recent observation of Mercury was made at 7:30 P.M., IS March I501: I503 complete years I504 January through February 59d March I7 leapdays (4-I504) 376 452 I -365 I504Y S7d I9h 3()1D + 264 112 IS I (24+) I~ 3om= 333/,dm, I76SY 20od for which Copernicus writes 33dm (fol. I95", line I6 up). P. 2SS:I2. In V, I, Copernicus gives Mercury's annual motion as 3 complete revolutions plus about 54° a value suitable also for the slightly different figures in the Table after V, 1. Now, 20X 54° = 10S0° = 3 complete revolutions +(20x3) = 60 63 P. 2SS:I4. Here (fol. I95", line 11 up) Copernicus omitted "200," an oversight which was corrected in T (p. 393, line 22, and note). P. 2SS:15. Here (fol. I95v, line 10 up) Copernicus omitted the word for "years", and instead of the correct number I76S (MDCClxviij) he wrote 576S (VDCCbroiij), the first numeral of which appears by anticipatory dittography, VDlxx being the number of the revolutions in the next line. This error was corrected in Mastlin's copy of N (fol. I72r, line 3) and for the first time publicly in T (p. 393, line 23, and note).

P. 2SS:20. The corresponding modem value would be about 6' 21/ 3" for 63 years (Astronomical Journal, I974, 79:5S). A discrepancy between the theoretical and observed motions of the perihelion of Mercury was first pointed out by Le Verrier. The existence of this discrepancy was confirmed by Simon Newcomb, who arrived at a figure of 43" a century. This anomaly became fundamental in relativity theory, which calls for a pre­ cession in Mercury's perihelion of 43" a century over and above the value demanded by previous theory. P. 2SS:24. Above, in p.ote top. 2SS:S, we saw that from the beginning of the Christian era to 7:30P.M., 1S March I504, there were I504Y S7d I9h 30m= 483/,dm, for which Copernicus writes 4Sdm (fol. I96f, line 2). P. 2SS :43. Mercury's daily motion is given in V, I, as well as in the Table of Mercury's Parallactic Motion in Days, after V, I, as 3° 6' 24", from which subtract 59' S" as the daily motion of the Sun's (that is, Earth's) 2° 7' 16" Simple Uniform Motion in Days, after III, I4. P. 302:32. Here (fol. I97v, line 2 up) Copernicus used the Greek word "lemmation," which does not occur in the Latin sources consulted by him, but which he read in the first edition of the Greek text of the Syntaxis, a copy of which he received from Rheticus in I539. He started to write V, 35, on fol. Issr, where he set down only the title of the chapter, and then indicated by a special mark that the chapter itself would be found later, out of its proper place. This is fol. 197v, a sheet of paper E, which he had begun to use only a few years before Rheticus' arrival in Frombork (NCCW, I, 4). Hence, Copernicus' composition of V, 35, may be dated no earlier than the summer of I539, as was suggested in the Russian translation of I964. GV (Bk. XVIII, Ch. 4, sig. ff SV) refers to Apollonius' theorem as in'lJentum, and Copernicus had previously used demonstrata (V, 3; fol. I49v, last line). Since Copernicus was still busy writing the later chapters of Book V in 1539, we can understand why Rheticus stated in his First Report, completed on 23 September in that year, that Copernicus "has in large measure already accomplished" his labors (3CT, p. I62). On 15 April 1541 a Wittenberg friend of Rheticus passed along to Mel­ anchthon the information that Rheticus "has written from Prussia that he is waiting for his teacher [Coper­ nicus] to finish his work" (the Revolutions; Rosen, "Rheticus' Earliest Extant Letter to Paul Eber," Commentary by Karl Heinz Burmeister, Isis, I970, 61: 3S5-3S6). As late as 2 June I54I Rheticus still reports that his "teacher ... is writing a great deal" (Burmeister, Rhetikus, III, 27). P. 302:47. Copernicus originally wrote "AEG sectorem" (fol. I9sr, line I7). Then he realized that "sector

436 NOTES ON PP. 302-307

ABG" belonged in the second ratio, not in the first. Hence he deleted sectorem, but forgot to change the letter­ ing for the sector (ABG) to the lettering required for the triangle (ABC). This error was corrected in T (p. 405, line 1, and note). P. 303:35. In the 2nd inequality, line GB was mislettered GF by Copernicus (fol. 199r, line 1), an error which was corrected in Mu (p. 369, note to line 32). P. 303:41. Instead of CM, N printed CL (fol. 180v, line 15 up). This is no ordinary typographical error. For, N also put the second station in L, not M; it drew line ELM, not BMN; it compared half of LM, not MN, with LB, not ME; it put point L, not M, at the second station; and it referred to arc FCL, not FCM. These readings in N do occur in the autograph (fol. 199r, lines 9-13). In every case, however, these readings which appeared in N were deleted in the autograph and replaced by those in the present translation. How is this sit­ uation to be explained? In all likelihood, when Rheticus copied the autograph for the printer, these changes had not yet been made. In other words, Copernicus continued to improve his autograph after Rheticus' depar­ ture from Frombork early in the fall of 1541. From Rheticus' penultimate copy, as we may call it, N was printed. In Miistlin's copy of N, these errors were corrected (fol. 180v). P. 304:28. Copernicus says "draw... DG perpendicular to BFB." But on 'fol. 199v, after having drawn line DG, he indicated that it should be deleted by placing five cross strokes through it. Presumably he did so because in the earlier version he had used triangle DFG, which he did not use in the printed version. Thinking that he had no need for line DG, he deleted it. But in so doing, he forgot his earlier instruction to "draw DG." P. 305:17. This product 63963984 (fol. 20or, line 5 up) is another example of Copernicus' occasional dit- tography, since the fourth and fifth digits should be 51, not a repetition of 63. P. 305:21. DB:DA = 10,000:6580 = 60P:39P 28.8', for which Copernicus writes 29' (fol. 200v, line 3). P. 305:23. 99P 29'X 20P 31' = 2041P 3.9', for which Copernicus writes 4' (fol. 20ov, line 5). P. 305:25. 2041P 4' +3P 16' 14" = 7347840" + 11774" = 624P 4.4', for which Copernicus writes 624P 4' (fol. 200v, line 8). P. 305:27. DB:BF = 60P:24P 58' 52"= 216000":89932" = 10000P:4163P 31', for which Copernicus writes 5' (fol. 2oov, left margin). P. 305:34. At fol. 200v, line 6 up, Copernicus started to write the customary word in abbreviated form par (degrees). This he deleted and replaced by the highly unusual term sectionibus. P. 305:46. DB:DA = 10,960:6580 = 216,000":129,678" = 60P:36P 1' 18" 50"', for which Copernicus writes 20" (fol. 201r, line 13). P. 305:48. ABXBC = 96P 1' 20"X 23P 58' 40" = 345,680"X 86,320 = 29,839,097,600"+12,960,000= 2302P 23' 56", for which Copernicus writes 58" (fol. 201r, line 15). P. 305:51. DB:DF = 6QP: 30P 4' 51"= 216,000": 108,291" = 100,000:50,134.7, for which Coper­ nicus writes 50,135 (fol. 201r, line 15 up). P. 306:9. DB: AD= 9040:6580; 60P = 216,000"X6580 = 1,421,280,000"+9040 = 157,221" = 43P 40' 21". P. 306:14. DB:BF = 6QP:18P 59' 58"= 216,000":68,398" = 100,000:31,665.7, for which Copernicus writes 31,665 (fol. 201v, line 7). P. 306:39. Book V in the autograph is in such a disorganized state that Copernicus failed to indicate where it came to an end. That indication was supplied by N, which was followed by the later editions. P. 307:29. By linking the motions of the other planets in latitude to the earth's orbital motion, Copernicus believed that he was strengthening his argument for a moving earth. Instead, he was singling out one particular planet, the earth, to which he was assigning a privileged status. For if the earth really was merely one of a number of planets revolving around the sun, why should the latitudinal motions of the other planets be governed by the earth's longitudinal revolution? In adopting this principle, Copernicus clung too closely to his Ptolemaic model and failed to perceive that his conception of the earth as a moving planet required a thoroughgoing reconstruc­ tion of the traditional theory of planetary latitudes. As things stand, then, the moving earth was Copernicus' main theme throughout the Revolutions. Yet a detractor recently said that Copernicus' synopsis of the theory occupies less than twenty pages at the beginning of the book, or about five per cent of the whole. The remaining ninety-five per cent consists of the application of it. And when that is completed, there is hardly anything left of the original doctrine. It has, so to speak, destroyed itself in the process. This may be the reason why no summary, conclusions, or winding-up of any kind is found at the end of the book, although we are repeatedly promised it in the text. It is characteristic of this detractor that, although he claims a summary is "repeatedly promised," he fails to cite a single instance of this promise, which Copernicus never made.

437 NOTES ON PP. 308-310

P. 308:10. At fol. 1891", line 11 up, Copernicus wrote aliis (other), where the context clearly requires mediis (midway), as was suggested in T (p. 413, Note to line 24). P. 311:27. As Copernicus drew the diagram (fol. 192•), F would be the perigee, and G the apogee, in con­ trast to the text (fol. 191v, line 13 up). P. 313:2. At fol. 192v, line 12, a smudge in the number of minutes for Jupiter may indicate that Coper­ nicus origmally wrote 8 ('Oiij), in agreement with Ptolemy. As for Mars, Copernicus wrote "7 minutes" and then deleted the expression. P. 313:4. In agreement with Ptolemy, originally (fol. 192v, line 15) Copernicus wrote 4 (iiij), which he later deleted 8-!ld replaced by 5 ('D) for Mars. P. 313:17. As the number of seconds Copernicus originally wrote 26 (xX'Di; fol. 193•, line 1). Later he erased 'Oi. In V, 19, he gave lP 22' 26" as Mars' perigeal, not apogeal, distance. P. 313:32. Instead of ED, Copernicus wrote FD (fol. 19~, line 18). P. 313:33. Instead of GED, Copernicus wrote DFE (fol. 193•, line 14 up), an error which was corrected in T (p. 421, line 3, and note). P. 314:11. Inadvertently omitting an x from XX'Oiij (fol. 193v, line 6 up), Copernicus wrote 18, a slip which was corrected in Mu (p. 383, note on line 22). The erroneous reading 19 inN (fol. 186v, line 4) was corrected in Mastlin's copy to 28. P. 314:13. Saturn: 2° 16'-(!X28' = 14') ~ 2° 3' Jupiter: 1° 18'-(!X24' = 12') = 1°6' P. 315:2. Here (fol. 194v, line 4), for the minutes Copernicus writes 50 (L) instead of "about 51" (Lj jere; fol. 193v, line 1). P. 315:4. According to Lines Subtended, for 45°, 70711, with radius= 100,000. P. 315:8. DE:GE = 10,000:2929 = 50!:14.79 ~ 15. P. 315:12. ED:FG = 10,000:7071 = 6580:4652.7, for which Copernicus writes 4653 (fol. 194v, line 17). P. 315:33. By a slip (fol. 203•, line 8), for the number of degrees Copernicus wrote 8 ('Viii), an error which was corrected in N (fol. 187Y). P. 316:23. See above, note to p. 315:4. P. 316:28. BE:EK = 10,000:7071 = 7193:5086 (= HK). P. 316:28. BE:KL = 10,000:308 = 7193:221.5, for which Copernicus writes 221 (fol. 203v, line 13 up). P. 316:28. BE:BL = 10,000:7064 = 7193:5081. P. 316:40. By a slip this angle of 45° 57' was mislettered by Copernicus ALM (fol. 204•, line 1). But he has already shown ALM to be a right angle (fol. 203v, line 11 up). The proper lettering, MAL, was introduced by T (p. 426, line 11, and note). P. 316:47. BH:BK = 10,000:7071 = 3953:2795. P. 317:10. By a slip (fol. 20¥,line 9 up) this diagonal was mislettered by Copernicus LK, an error which was rectified in W (p. 466). P. 317:30. Here (fol. 204v, lines 8-9) Copernicus doubles Ptolemy's values, 1/8° for Venus and 3{4. 0 for Mercury, which he cites correctly, however, when he returns to this topic in VI; 8 (fol. 207Y, lines 14-12 up). P. 318:46. At fol. 20SV, line 14, Copernicus wrote secundum, presumably by dittography of the same word in the preceding line. This error inN (fol. 190•, line 8) was corrected in Mastlin's copy. The emendation sunt in Mu (p. 389, note on line 34) is undoubtedly right. P. 319:13. BA:AD = BD:DF 10,000:6947 = 7193:4996.97, for which Copernicus writes 4997 (corrected from 4994; fol. 205v, line 6 up). P. 319:23. AG:FG = 6940:4988 = 10,000:7187.3, for which Copernicus writes 7187 (fol. 206•, line 10). According to Lines Subtended, 71,934 for 46° 0', and 71,732 for 45°50'; hence, 71,873.4, or with radius = 10,000, 7187 for 45° 57'. P. 319:24. AD:DF = 6947:4997 = 10,000:7193 (see note to p. 319:13). According to Lines Subtended, 71,934, or with radius= 10,000, 7193 for 46°. P. 319:33. AB:AD = BD:DF 10,000:9340 = 3573:3337.2, for which Copernicus writes 3337 (fol. 206•, line 11 up). P. 320:7. AB:AD = BD:DF (by a slip Copernicus wrote BF for the last line). 10,208:7238 = 7193:5100.2, instead of which Copernicus, presumably by an error in transcription, writes 5102 (fol. 206v, line 17). P. 320:9. AD:DG = 7238:309 = 10,000:426.9, for which Copernicus writes 427 (fol. 206•, line 14 up).

438 NOTES ON PP. 320-330

P. 320:10. According to Lines Subtended, 4362 for 2° 30'; and 4071 for 2°20'; hence, 4274.7, or with radius = 10,000, 428 for 2°27'. P. 320:13. AB:AD = BD:DF 9792:6644 = 7193:4880.5, instead of which Copernicus writes 4883 (fol. 206v, line 9 up). P. 321:29. See note top. 317:30. P. 321:41. Copernicus shared the commonly held belief in the instantaneous transmission of light, which was first decisively refuted more than a century after his death. P. 321:42. The word tempori, indispensable to Copernicus' argument (fol. 207v, last line), was misprinted ipsi in N (fol. 192r, lines 6-7), where it was replaced by tempori in Mlistlin's copy. P. 322:15. According to Lines Subtended, for 10', 291, or 29 with radius = 10,000. P. 322:26. According to Lines Subtended, for 50': 1454 and for 40': 1163; hence, for 45': 1308.5, or 131, with radius 10,000. P. 327-328. In the autograph (fol. 211r-v) Copernicus put the deviations for Venus and Mercury in columns 7-8. N, however, transferred the deviation for Venus to column 5, presumably in order to group the 3 columns for Venus together. But N failed to make the corresponding change in the captions, so that what was intended to be Venus' deviation is there mistakenly headed Mercury's declination (fol. 194v). This cluster of errors was corrected in Mlistlin's copy of N. P. 330:8. Copernicus' autograph simply stops at fol. 212v. A suitable closing formula was supplied by N, which was followed by the later editions.

439 INDEX OF PERSONS

Prepared by Brna Hilfstein

(Numbers in italics refer to the commentary)

Agrippa (the astronomer) 128, 385 Battani, al- 19, 121, 122, 128, 129, 139, 144, 145, Agrippa, Marcus 141 158, 160, 162, 207, 356, 382-384, 386, 387, 392, Albategnius see Battani, at- 397, 414 Albert of Brudzewo 429 Baumgartner, Hieronymus 340 Alembert, D' 352 Bessarion 344, 349, 361, 363 Alexander VII 342 Bilinski 335, 336, 343, 368, 394 Alexander the Great 120, 141, 390 Birkenmajer, Aleksander 343 Alfraganus see Farghani, al­ Bitale 26, 363 Alhazen see Haytham, Ibn at- Bitruji, al- 18, 355 Ali ibn Ridwan 357 Brahe Alpetragius see Bitruji, al- annotated Revolutions 371, 372, 374, 382, 402, Anatoli, Jacob 356, 357 408, 409, 418, 419, 423, 427, 434 Anaximander 10 cited 333, 344, 388, 413 Anaximenes 10 Mechanica 382, 384, 412 Antoninus Pius 83 Progymnasmata 384, 400 Antony 141 Bruno, Giordano 335 Apian, Peter 340 Philip 340 Caesar, Julius 141, 390 Apollonius 241, 302, 306, 387, 388, 436 Callippus 120, 144 Aquinas 353 Calo Calonymus 355 Aratus 84, 372, 379 Campanella 338, 386 Archimedes 144, 146, 225, 345, 361, 366, 393, 419 Campanus of Novara 346 Archippus 362 Capuanus of Manfredonia 346 Aristarchus 25, 122, 144, 360, 361, 366, 384-386 Cardano 334, 340 Aristophanes 342 Censorinus 141, 390, 392, 401 Aristotle Chalcidius 416,417 cited 14, 17, 22, 340, 357, 385, 399 Charles V 334 Generation and Corruption 345, 353 Cicero 4, 12, 240, 341, 349, 359, 417 Generation of Animals 360 Clavius 354, 365-367, 384 HeafJens 15, 25, 345, 347-351, 353-355, 360 Clement VII 336, 337 Metaphysics 348, 362 Clement VIII 342 Meteorology 352 Clement of Alexandria 362 Physics 348, 351, 353, 354, 359, 367 Cleomedes368 Posterior Analytics 360 Cleopatra 141 Aristyllus 128, 385, 386 Columbus 346, 347 Arzachel see Zarkali, al­ Conon 107, 379 Augustine 346 Copernicus Augustus 141 Commentariolus 400, 417 Autolycus 370, 371 Letter against Werner 360,371,382,385,390,426 Avenrodan see Ali ibn Ridwan Star Catalog 85-117, 372, 373 Averroes see Rushd, Ibn Trigonometry 344

Damo 26, 363 Baldi 335, 338, 343, 368, 394 Dantiscus 344 Barettus 432 Delambre 335, 432 Bate, Henry 343 Democritus 10

441 INDEX OF PERSONS

Diogenes Laertius 348, 349, 361, 362 Job 84, 372 Donne 353 John Peter of Lucca 345 Durer 432 John of Seville 429 Justinian 334

Ecphantus 5, 12, 342, 349 Kekule 335 Empedocles 10, 347 Kepler Erasmus 342 cited 344 Eratosthenes 52, 53, 368, 384 correspondence 385, 388 Euclid Ephemerides 357 cited 29, 40-42, 44, 49, 156, 163, 164, 178, 207, his copy of the Revolutions 373 372, 429 Hyperaspistes 360 Elements 26, 27, 43, 45, 63, 121, 126, 127, 154, New Astronomy 348, 360, 399 155, 283, 363 New Star 374 Optics 18, 349, 355, 360, 394, 399 Optics 357 Phenomena 61, 350 quoted 335 Evelyn, John 344 Rudolphine Tables 364 Singular Phenomenon 357 Farghani, al- 429 Kurz, Sebastian 334 Ficino 344 Firmicus 372 Lactantius 5, 343, 359 Flamsteed 344 Lauterwalt, Matthias 403, 404, 414, 415 Foucault 353 Leo X 5, 343 Fracastoro 336, 337 Lepidus 141 Fugger 334 Leucippus 10 Le Verrier 432, 436 Luther 342, 344 Galen 359 Lysis 3, 25, 361-363 Galileo 337, 338, 343, 344, 356, 364, 385, 428 Gasser 338 Macrobius 362 George of Trebizond 341, 350, 351, 361 Martianus Capella 20, 358, 417 Gerard of Cremona 341, 429 Miistlin (Maestlin) Giberti, Gian Matteo 336, 337 annotated Revolutions 340 Giese 3, 339, 340, 342, 343 corrected Revolutions' Guido I 343 (a) Star Catalog 372-381 (b) tables 364, 370, 391 Hadrian 426, 427 (c) text 372, 390, 401 Halley 352 correspondence 357 Hanow, Johannes 412 his copy of the Revolutions 365, 368, 409, 418, Haytham, Ibn al- 417 419, 422, 423, 426-428, 430, 434-438 Heraclides 5, 12, 349 Melanchthon 436 Heraclitus 10, 347, 348 Menelaus 80, 120, 122, 128, 177, 366, 371, 385 Hermes, Thrice Greatest (Trismegistus) 22, 359 Menzzer 332 Hesiod 84, 372 Meton 178, 401 Hicetas 4, 12, 341, 349 Milton, John 352 Hipparchus (the astronomer) 53, 119, 120, 122, 128, Moses ben Tibbon 355 Munster, Sebastian 333, 334 1~M~1~1~-~~1~1~1~1~~~ 195, 205, 206, 228, 338, 372, 382, 384, 390, 397, 400 Nabonassar 141, 390 Hipparchus (the Pythagorean) 3, 25, 26, 338, 361,362 Nasir al-Din al-Tusi 358, 385, 400, 430 Hippasus 361, 362 Nebuchadnezzar II 141, 390 Homer 84, 372 Newcomb, Simon 436 Horace 338, 341 Newton, Isaac 344, 352, 358, 399 Hoyle, Fred 353, 367 Nicephorus 368 Nicholas of Cusa 348 Novara, Domenico Maria 129, 369, 386 Iamblichus 361, 362 Nunes 354, 365-367, 394, 398, 400

442 INDEX OF PERSONS

Olympiodorus 350, 362 Ramus, Peter 394 Osiander XVI, 333-336, 340, 358, 435 Regiomontanus 129, 285, 332, 341, 355, 369, 388, 432, 433 Patrizi 348 Reid, James 399 Paul III 3, 336-338, 343, 344 Reinhold 372, 373, 375, 377, 378, 380, 381 Paul of Middelburg 6, 343 Rheticus Peter, Hans see Petreius and Osiander 333-335,340 Petreius XV, 334, 338, 340 brought books to Copernicus 341, 363, 367, Peurbach 129, 332, 341, 368 369, 373, 426, 429, 433, 436 Philolaus 5, 12, 25, 342, 349, 360, 362 cited 338, 359, 378, 435 Philoponus 334 correspondence 339, 403, 433, 436 Pico della Mirandola, Giovanni 356, 357 defended Copernicus 343 Plato 12, 19, 25 edited Copernicus' Academy 334 (a) Revolutions 343, 345, 365, 377, 382, 383, 437 cited 341, 349, 361 (b) Trigonometry 344 Laws 7, 344 First Report 334, 345, 436 Phaedo 362 quoted 386, 398 Timaeus 18, 227, 344, 355, 416, 417 Ristoro d'Arezzo 345 Pliny the Elder 341, 343-346, 350, 352, 358, 359, Rohan, Fran~ise de 388, 389 370, 401 Rushd, Ibn 19, 356, 357, 360 Plutarch 8, 342, 345, 361, 363 Plutarch (pseudo-) 4, 342, 345, 347-349, 417 Poincare, Henri 352, 353 Schonberg XVII, 3, 336-338, 343 Polybius 381 Schaner 285, 372-375, 377, 378, 380, 381, 432, 433 Posidonius 52, 63, 368, 369 Scot, Michael 355 Praetorius, J obannes 364 Shalmaneser V 141, 390 Proclus 52, 279, 355, 368, 369, 412, 429, 430 Sigismund Augustus 339 Profatius 122, 129, 384 Snel 432 Ptolemy Sophocles 22, 359 cited 8, 25, 78, 84, 121, 128-130, 140, 155, 159, Spina, Bartolomeo 337 179, 189, 190, 201, 204, 207, 209, 225, 245, Starowolski 408 247, 250-252, 255, 258, 260, 265, 282, 285, Stevin 387 288, 358, 366, 369, 384, 385, 387, 399, 401, Sto:ffler 372 408, 409, 414, 418, 423, 438 Synesius 362 Geography 10, 345, 346 Handy Tables 377 hypotheses 334, 388, 389 Tertullian 362 instruments 368, 400, 412 Thabit ibn Qurra 145, 147, 393, 397 lunar theory 345, 400 Theodoric of Freiberg 348 observations 135, 139, 145, 268, 277, 371, 372, Theodoric of Reden XVII, 337 383, 423, 425, 43o-432 Theodosius I 381 opposed 367, 419 Theodosius of Bithynia 349 Planetary Hypotheses 355, 356, 377 Theon of Alexandria 372, 426 quoted 341, 350, 352, 356 Theon (of Smyrna?) 270, 271, 281 solar apogee 162, 397 Theon the Grammarian 372, 379 Star Catalog 371, 372, 374, 377, 390, 432 Theon, the younger 84, 372, 431 Syntaxis 15, 18, 19, 27, 30, 43, 53, 64, 80, 81, Theophrastus 341 83, 119, 122, 141, 144, 146, 156-158, 160, Thou, De 388 170, 171, 178, 186, 188, 191, 193, 194, 199, Timocharis 120, 122, 128, 129, 135, 139, 140, 177, 275, 200, 202, 203, 206, 208, 228, 240, 244, 246, 277, 382, 387 248, 254, 262, 263, 267, 269-273, 278, 280, Tolosani, Giovanni Maria 337 281, 283, 302, 307, 308, 313, 315-317, 319, Trajan 120 321, 332, 341, 357, 364, 373, 377, 426, 429, Tusi see Nasir al-Din 436 table of chords 363 Pythagoras 25, 26, 361-363 Valla, Giorgio 332, 348, 349, 355, 368, 382 Pythagoreans 3, 25, 338, 342, 349, 361-363 Vergil 16, 352, 368

443 INDEX OF PERSONS

V espucci 34 7 Widmanstetter 336, 337 Viete 334, 387-389 Witelo 350 Vitruvius 350, 358 Xenophanes 10, 348 Waldseemilller 346, 347 Walther, Bernard 285, 286, 432-434 Zamberti 349, 350, 355, 369 Werner, Johann 371, 381 Zarkali, al- 122, 129, 158, 160, 162, 384, 395, 397

444 INDEX OF PLACES

Prepared by Erna Hilfstein

Africa 347 Leipzig 334, 340, 403 Albania 408 Lubawa (Lobau) 339 Alexandria 61, 83, 144, 145, 160, 186, 193, 199, 200, 202, 203, 244, 273, 345, 408, 409 Macedonia 191 America 10, 346, 347 Maragha 358 Asia 347 Mediterranean 10, 369 Athens 178 Meroe 61

Black Sea 61 Naples 338 Bologna 218, 335, 368, 369, 386, 390, 416 Nile 284 Nuremberg XV, 285, 334, 338-340, 432-434 Canary Islands 346 Cathay 10, 347 Chehnno 3, 338, 340, 343 Persia 358 China 346 Constantinople 61 Poland 336, 339 Cracow 160-162, 186, 191, 193, 199-201, 218, 222, Portugal 10 244, 267, 273, 274, 277, 280, 338, 357, 364, 408- Poznan 433 410, 416, 425 Prussia 121, 338-340, 436 Crotona 362 Raqqa 19, 144, 145, 207, 356 Dnieper 61 Red Sea 10 Durazzo (Durres) 408 Reden XVII, 337 Dyrrhachium 191, 408 Rhodes 61, 84, 144, 193, 408 Rome xvn, 200, 337, 342, 343, 371, 410 Egypt 8, 345, 346 Elblu (Elbing) 415 Schaft'hausen 340 Epidamnus 191, 408 Sera 346 Europe 347 Sinae 346 Spain 10 Florence 416 Suez 346 Fortunate Islands 346 Switzerland 340 Fossombrone 6, 343 Syene 61 Frauenburg see Frombork Syracuse 416 Frombork 145, 160, 161, 186, 191, 276, 333-335, 337, Syria 144 338, 341, 377, 378, 392, 396, 407, 412, 433, 436, 437 Thebes 362 latitude of 121, 203, 382, 413 Toruli XV, 366 Tiibingen 340 Ganges 10, 346, 347 Tus 358 GdaDsk (Danzig) 334 Germany 343 Uppsala 341, 372, 385 Greenwich 344 Urbino 343

Hellespont 61 Varmia 336-338, 341, 344, 392 Vatican 336, 338, 370 India 10, 346, 347 Venice 416 Ingolstadt 340 Verona 336 Italy 8, 12, 343, 345, 357 Vistula 191, 284, 407, 432

Kulm ses Chelmno Wittenberg 334, 338, 339, 344, 377, 403

445 INDEX OF SUBJECTS

Prepared by Erna Hilfstein

acceleration 17, 353 [clock] aether (aither) 19, 348 sand 403, 404 air 15, 16, 19, 345, 352, 353 cloud 15, 16, 351 Almagest 341 colure 54, 127, 128, 384 amphibology 358 comet 16, 352 antichthones 10, 347 commutation 176 antipodes 10, 346, 347 conic section 126 aphelion 394 conjunction 173, 178 apogee 11, 155, 394 constellation 83, 85-117, 119, 372 apse 358 conventionalism 353 higher 155 lower 155 day 60, 350, 392 motion of 398 beginning of 79 ascension equinoctial 61, 62, 70, 71, 350 oblique 70, 72, 74, 77 intercalary 130 right 54, 57, 59, 70, 72, 77 longest 62, 171, 369 astrolabe 81-83, 146, 177, 192, 193, 244, 285 natural 170, 171, 194 astrology 7, 344, 357 shortest 62, 171, 369 astronomy XVI, 5, 7, 343, 358, 361 solstitial 61 atom 14, 335, 350 uniform 170 attraction 355, 399 deceleration 353, 354 axiom 15, 176 decimal system 363, 387 declination 54, 56, 59 calendar latitudinal 309, 312, 315, 317 Athenian 390 deferent 155, 341 Egyptian 390 degree 55, 364 Gregorian 343 deviation 309, 317, 321 Julian 396, 407 digit 200, 206 reform of 6, 343, 395 dioptra 13, 82, 176, 202, 205, 206, 350 Roman 161, 390, 391, 396 dodecatemories 70 cannon 353 Dragon Canobic Inscription 377 head of the 194, 408 centrifugalism 351 tail of the 194, 408 centripetalism 345, 350, 354, 355 chemistry 335 earth circle as a planet 12, 17, 355, 360, 368, 385, 389, 399, arctic 52 437 antarctic 52, 368 as an element 9, 14, 15, 346, 353, 354 grand (orbis magnus) 20, 302, 304, 307-309, 311, at rest or in motion XVI, XVII, 3-5, 11-18, 25, 358 26, 51, 173, 227, 228, 240, 256, 265, 270, great 13, 349, 358 278, 302, 307, 335, 336, 340-343, 348, osculating 384 349, 351-354, 366-368, 389, 394 ratio of circumference to diameter 225 axis of 23, 25, 122, 144, 360 squaring of 245 center of gravity of 9, 10, 18 clime 61, 84, 369 equator of 23 clock motion of, anti-Biblical? 339, 342, 343 mechanical 369, 370, 403 place of 3, 11-15, 17, 20, 349, 350, 355, pendulum 403 389

446 INDEX OF SUBJECTS

[earth] horizon 13, 51-53, 59-61, 71, 76, 77, 350, 360, rotation of 14, 170, 341, 349, 351-353 371 shadow of 177, 178, 198,205-207,209, 210, 414 hour shape of5, 8-11,345,347, 348,350,353,368 equal 70, 369 size of 13, 14, 208, 368 seasonal 70, 369 third motion of 360 hourglass 403 triple motion of 22-25, 51, 354, 360, 398 hydrascope 52 122, 265, 334, 335, eccentric XVI, 4, 155, 156, 174, 334, 341, 394, 399 hypothesis XV, XVI, 4, 7, 25, 26, eclipse 9, 162, 173, 207, 215, 217, 218, 356, 408 342, 344, 352, 360 lunar 10, 177-179, 186, 191, 198, 200, 205, 222-226, 347, 403, 415 impetus 17 solar 77, 177, 178, 191, 223 impiety 361 ecliptic 13, 23, 51, 52, 54, 60, 61, 71, 76, 120, 122, incommensurable 361 173, 307, 308, 350, 356, 360 Index of Prohibited Books 342 obliquity of 23-25, 51, 53, 61, 72, 119, 120, Index, Sacred Congregation of 342, 343, 349, 352, 122-124, 126, 128-130, 136, 140, 194, 369, 354, 355, 358, 360 382, 384-386 infinite 13, 15, 16, 26, 351 pole of 384 intelligences 348 elements fifth 348 Jupiter 18, 2Q-22, 78, 79, 146, 227-229, 232, 233, four 15, 351, 353 240, 241, 243, 254-262, 265, 270, 293, 294, 301, transmutation of 9, 345 306, 307, 371, 418, 423 ellipse 126, 348, 385, 429 apogee of 101, 256, 261, 378 as orbit 348, 385 latitude of 309, 312-315, 325, 326, 329 epicycle XVI, 4, 155, 156, 174, 334, 341, 394 epoch 141 Lateran Council 5, 343, 395 equant 278, 399, 400, 417, 429 latitude equation 136 parallels of 23, 51, 60, 61, 64, 352, 369 equator (equinoctial) 22, 51-54, 59, 60, 71, 120, planet's motion in 437 122, 352 terrestrial 61, 369 mean 123, 124 libration 123, 125, 128, 308, 321, 360, 398 equinoctial point 25, 80, 396 light 321, 439 equinox 80, 84, 145, 350, 375 longitude autumn 24 celestial 372, 375, 377 mean 123 terrestrial 222 precession of 25, 83, 119, 120, 122, 126, 128, 129, 131-135, 139, 141-144, 146, 186, 309, 360, magic 359 371, 381, 383-385 Mars 18, 20, 23, 78, 79, 227-229, 234, 235, 240, 241, sprihg 24, 54, 70, 71, 81, 83, 372 243, 262-270, 295, 296, 301, 304-307, 418 era 142, 381 apogee of 100, 267, 378 Christian 382 color of 22 of Alexander 382, 383 latitude of 309, 312-315, 325, 326, 329 of Nabonassar 390 place of 21 rising and setting of 79, 371 fictionalism 335 mechanics, celestial 348, 353 fire 15-17, 19, 345, 354, 358, 362 Mercury 22, 227-229, 238-242, 278-290, 299-304, flame 16 306, 356, 418 356, 378 force, centrifugal 15 apogee of 19, 101, 281, 285, 287, 288, fractions 27, 387, 432 center of orbit of 20, 358 latitude of 308-312, 315-324, 327-330 geokineticism 349, 351--354, 360, 417 perigee of 19, 356 gravitation 355, 399 perihelion of 436 gravity 9, 10, 18, 346, 355 period of 22 phases of 355 heresy 362 place of 18-20, 355 homocentric (concentric) 4, 18, 336, 340, 341, 354, rising and setting of 78, 79, 371 355 size of 19

447 INDEX OF SUBJECTS

[Mercury] opposition 20, 173 transit of 19, 355 optics 7, 11, 13, 22, 155, 321, 349, 399 visibility of 285, 432 meridian 52-54, 58, 71, 77, 368, 392 parallactic instrument 202, 203, 412, 413 prime 346, 416 parallax metaphysics 335 planetary 227-229, 244, 252, 280, 302, 309, method 4 417 month 6, 10, 14, 173, 345 stellar 353, 417 Egyptian 130 parallel lines 63, 369 moon 14, 82, 83, 173, 345, 419 pendulum 353 apogee of 19, 356 perigee 11, 155, 394 apparent diameter of 177, 200, 205, 206, 209, perihelion 394 214, 345, 399, 400, 414 peripatetics 9, 334 conjunction of, with sun 218-222 piston 348 distance of, from the earth 18-20, 176, 203, place, natural 17, 350, 354 204-207, 209, 345, 356, 358, 399, 414 planet 11 elpngation of, from the sun 19 latitude of 119, 123 hourly motion of 83 light of 360 kinship of, with earth 22, 173, 360, 399 non-self-luminous 18, 355 latitude of 179, 184, 185, 198, 202, 203, 205, 321, retrograde motion of 18 410 self-luminous? 19, 355 nature of 360, 399 shape of 18 opposition of, to sun 218, 220-222 size of 19 parallax of 83, 176, 177, 192, 193, 202-205, 209, stations of 18 210, 215-218, 252 plenitude, principle of 356, 359 perigee of 19, 357, 358, 399 pole, altitude of 61, 62, 70, 382 size of 208 precession see equinox motion 51, 80 prime mover 348 cause of 10, 17, 348 problem 363 circular 14, 15, 17 prosthaphaeresis 136-138, 142, 167, 168, 195-197, first 10, 11, 384 290-300 natural 15, 348, 351, 354 perpetual 348 quadrant 368 rectilinear 14, 15, 17 relative 11, 349 refl.exion 312 simple 14, 15, 17 refraction 382 uniform 4, 341, 349, 353, 400 revelation XVI, 334 unnatural 351 rising 307 mysticism 359 acronycal 244 evening 78 morning 78 night 60, 79, 177, 349 true vs visible 78 node 173, 308, 408 rotation, daily 10, 12-17, 22, 23, 51, 59, 60, 64, 78, nova 374 79, 120, 173, 352, 353, 368, 398 number 51 common 137 Saturn 14, 18, 2o-22, 78, 79, 146, 227-231, 240, numeral 241, 243-254, 258, 265, 270, 291, 292, 301, 306, Arabic 371, 430 307, 371, 400, 417, 418, 423 Greek 27, 374 apogee of 103, 252, 379, 420, 421 Hindu 27, 430 latitude of 309, 312-315, 325, 326, 329 Roman 27, 371, 401, 430 setting 307 evening 78 obliquation 309, 312, 318 morning 78, 371 observatory 344, 356 true vs visible 78 ocean 9 sexagesimal system 363, 387, 412 Olympiad 141, 142, 161, 381, 390, 391, 401, 429 sextant 382, 412 Olympic games 141, 381, 390 shadow at noon 60, 61

448 INDEX OF SUBJECTS sign Tables physical 373 Alfonsine 364, 372-375, 377-381, 410, 416 zodiacal18, 60, 70, 73, 83, 84, 345, 350, 371- ''French" 388, 389 373, 375 Prussian 388 sine 363, 364, 370 teleology 345 solstice 60, 61, 80, 84, 119, 145, 158, 375 telescope 355 summer 24, 51, 53, 54, 62, 141, 144, 396 terminology, geostatic 51 winter 23, 51, 53, 54, 62 Teutonic Order, Knights of the 336, 353 solstitial point 25, 80 theologians 335, 343, 352, 360 sorcerer 339 theology 338, 343, 359 soul 362 theorem 363 space 12, 17, 351, 354, 360 Thirty Years' War 413 absolute 352 time 10, 51, 367 empty 356 apparent 171 sphere 8, 345, 360 uniform 171 beautiful? 341, 345 time-degree 171, 172 contiguous 359 times of the equator 55 eighth XVII, 337, 351, 366, 394 tropic 51-53, 61, 173 eleventh 120, 381 of Cancer 347 heavenly XV, 333, 334 of Capricorn 24, 347 homocentric 340 motion of 10, 13, 348, 350 universe ninth 25, 120 center of 17, 18, 20, 23, 154, 169, 173, 346, oblique 59, 60, 71, 72 355, 358 right 59, 71-73 harmony of 18 solid 333, 334, 399 place of 21, 359 tenth 25, 120, 381 shape of 8, 26, 345, 346, 348 terraqueous 346 size of 20, 26 stars symmetry of 22 distanceof22 latitude of 121, 122 new 374 Venus22,23,227-229,236,237, 240,241,243,270- sphere of 21, 22, 25, 26, 80, 81, 244, 337, 351, 281, 297-298, 301-304, 306, 416-418, 427 359 apogee of 19, 98, 356, 377 stationary? 26, 348, 359, 361, 371, 375, 381, 398 apparent diameter of XVI, 336, 356 twinkling of 22, 360 brightness of 336, 428 variable 374 center of orbit of 20, 358 summer 360 elongation of 336, 371 sun epicycle of XVI, 19, 336 altitude of 61, 79, 81 latitude of 308-312, 315-324, 327-330 apogee of 159, 162, 219, 397 perigee of 19, 356 apparent diameter of 206-208, 214, 356, 400, period of 21, 22 414 phases of 355 at rest XVI, 18, 20, 22, 342, 348, 361, 366, 368, place of 18-20, 23, 355 398 rising and setting of 78, 79, 371 declination of 81, 384 size of 19, 336 distance of, from earth 19, 20, 23, 154, 206-208, transit of 355 386, 4!4 visibility 79 function of 359 first 307, 308, 371 location of XVII, 18-20, 22, 154, 358 last 371 parallax of 206, 208, 210, 213, 215 void 15, 351 perigee of 19, 356 Vulgate 352, 372 size of 19, 208 transit of 18, 19 water-clock 70 sundial 60, 61 water level 13, 52, 350 system, Ptolemaic 336, 353, 395 wind 16, 352 syzygy 218, 221, 222 trade 352, 353

449 INDEX OF SUBJECTS winter 360 year seasonal 144 world machine 4, 341 sidereal 119, 144, 146, 340, 360, 393 solar 80, 144, 145, 228, 371, 393 year 6, 10, 173 tropical 4, 8, 119, 144, 146, 147, 340, 345, 360, Egyptian 130, 142, 382, 390 392 Julian 141 leap 390 zenith 52, 60, 77 natural 119, 144, 145 zodiac 11, 97, 360 Olympic 119 signs of 13, 70, 119 Roman 142, 390 zone 369

450