Euclid Euclid
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EUCLID EUCLID of hydrogen at low temperatures, Debye's law for gave are not wholly convincing, but the reference is specific heats of solids at low temperatures, the range certainly contrary to ancient practice and is not un- of applicability of Nernst's heat theorem, the differ- fairly characterized as naive ; and although it was ence (predicted by quantum theory) between the already in the text in the time of Proclus, it looks specific heats of ortho- and para-hydrogen, and Ein- like a marginal gloss which has crept in .2 Although stein's theory of specific heats . In the context of his it is no longer possible to rely on this reference, 3 a later research on reaction kinetics, Eucken studied general consideration of Euclid's works such as that contact catalysis and the exchange of vibrational presented here still shows that he must have written energy between gas molecules. after such pupils of Plato as Eudoxus and before Eucken was eager to discourage overspecialization Archimedes . in science . For example, he felt that physical chemis- Euclid's residence in Alexandria is known from try should be studied in terms of physics and mathe- Pappus, who records that Apollonius spent a long matics. (He believed, however, that for satisfactory time with the disciples of Euclid in that city .4 This teaching, physical chemistry should remain a separate passage is also attributed to an interpolator by subject because it is not at all obvious how to apply Pappus' editor, Friedrich Hultsch, but only for stylis- physics to chemical phenomena .) Similarly, Eucken tic reasons (and these not very convincing) ; and even hoped to increase the interaction between science and if the Alexandrian residence rested only on the au- technology; some of the research in his laboratory thority of an interpolator, it would still be credible was chosen for the sake of its practical application, in the light of general probabilities. Since Alexander and Eucken was chairman of a committee to promote ordered the foundation of the town in 332 B .C . and theoretical study of industrial processes . another ten years elapsed before it began to take shape, we get as a first approximation that Euclid's Alexandrian activities lay somewhere between 320 BIBLIOGRAPHY and 260 B .C . Apollonius was active at Alexandria under Ptolemy III Euergetes (acceded 246) and Ptol- I. ORIGINAL WORKS . Eucken wrote many physical emy IV Philopator (acceded 221) and must have is Fundamen- chemistry texts . An early English version received his education about the middle of the cen- tals of Physical Chemistry, trans. and adapted by E . R. tury. It is likely, therefore, that Euclid's life over- Jette and V . K. LaMer (New York, 1925). For a listing lapped that of Archimedes . of Eucken's papers, consult Chemical Abstracts for the This agrees with what Proclus says about Euclid period 1900-1950. II . SECONDARY LITERATURE . See E. Bartholome, in in his commentary on the first book of the Elements. Die Naturwissenschaften, 37 (1950), 481-483 ; and R . The passage, which is contained in Proclus' summary Oesper, "Arnold Eucken," in Journal of Chemical Educa- of the history of geometry, 5 opens : tion, 27 (1950), 540-541 . Not much younger than these [Hermotimus of Colo- SIGALIA DOSTROVSKY phon and Philippus of Medma, two disciples of Plato] is Euclid, who put together the elements, arranging in EUCLID (fl. Alexandria [and Athens?], ca. 295 B.C.), order many of Eudoxus' theorems, perfecting many of mathematics. The following article is in two parts: Life Theaetetus', and also bringing to irrefutable demon- and Works ; Transmission of the Elements . stration the things which had been only loosely proved Life and Works. by his predecessors. This man lived' in the time of the Although Euclid (Latinized as Euclides) is the most first Ptolemy; 7 for Archimedes, who followed closely celebrated mathematician of all time, whose name upon the first [Ptolemy], makes mention of Euclid, 8 and became a synonym for geometry until the twentieth further they say that Ptolemy once asked him if there century,' only two facts of his life are known, and were a shorter way to the study of geometry than the even these are not beyond dispute . One is that he Elements, to which he replied that there was no royal road to geometry . He is therefore younger than Plato's was intermediate in date between the pupils of Plato circle, but older than Eratosthenes and Archimedes ; for and Archimedes (b. ca. 287 B .c.) ; the (d. 347 B.c.) these were contemporaries, as Eratosthenes somewhere other is that he taught in Alexandria. says.9 In his aim he was a Platonist, being in sympathy Until recently most scholars would have been con- with this philosophy, whence he made the end of the tent to say that Euclid was older than Archimedes whole Elements the construction of the so-called Pla- on the ground that Euclid, Elements 1 .2, is cited in tonic figures . Archimedes, On the Sphere and the Cylinder 1 .2 ; but in 1950 Johannes Hjelmslev asserted that this refer- Since Plato died in 347, Ptolemy I ruled from 323 ence was a naive interpolation . The reasons that he and reigned from 304 to 285, and Archimedes was 4 1 4 EUCLID EUCLID born in 287, the chronology of this passage is self- from Athens, where he had been the governor, would consistent but allows a wide margin according to have known Euclid's reputation . It is possible that whether Euclid flourished in Ptolemy's rule or reign . this had already been established by one or more It is clear, however, that Proclus, writing over six books, but the only piece of internal or external evi- centuries later, had no independent knowledge, obvi- dence about the order in which Euclid wrote his works ously relying upon Archimedes for his lower date . The is that the Optics preceded the Phaenomena because story about the royal road is similar to a tale that it is cited in the preface of the latter . Euclid must Stobaeus tells about Menaechmus and Alexander .'° be regarded as the founder of the great school of Euclid may very well have been a Platonist, for mathematics at Alexandria, which was unrivaled in mathematics received an immense impetus from antiquity . Pappus or an interpolator 16 pays tribute to Plato's encouragement ; what Proclus says about his him as "most fair and well disposed toward all who relationship to Plato's associates, Eudoxus and The- were able in any measure to advance mathematics, aetetus, is borne out by his own works ; and if he were careful in no way to give offense, and although an a Platonist, he would have derived pleasure from exact scholar not vaunting himself," as Apollonius making the Elements end with the construction of the was alleged to do ; and although the object of the five regular solids . The testimony of so zealous a passage is to denigrate Apollonius, there is no reason Neoplatonist as Proclus is not, however, necessarily to reject the assessment of Euclid's character. It was conclusive on this point . presumably at Alexandria, according to a story by Confirmation of Proclus' upper date comes from Stobaeus,17 that someone who had begun to learn the relationship of Euclid to Aristotle, who died in geometry with Euclid asked him, after the first theo- 322 B.C . Euclid's postulates and axioms or "common rem, what he got out of such things . Summoning a notions" undoubtedly show the influence of Aris- slave, Euclid said, "Give him three obols, since he totle's elaborate discussion of these topics ." Aristotle, must needs make gain out of what he learns ." The on the other hand, shows no awareness of Euclid, and place of his death is not recorded-although the nat- he gives a proof of the proposition that the angles ural assumption is that it was Alexandria-and the at the base of an isosceles triangle are equal which date of his death can only be conjectured . A date is pre-Euclidean and would hardly have been cited about 270 B.C. would accord with the fact that about if Elements 1.5 had been at hand. 12 the middle of the century Apollonius studied with his If exact dates could be assigned to Autolycus of pupils. Pitane, greater precision would be possible, for in his Arabic authors profess to know a great deal more Phaenomena Euclid quotes (but without naming his about Euclid's parentage and life, but what they write source) propositions from Autolycus' On the Moving is either free invention or based on the assumption Sphere. Autolycus was the teacher of Arcesilaus, who that the so-called book XIV of the Elements, written was born about 315 B .C. It would be reasonable to by Hypsicles, is a genuine work of Euclid . suppose that Autolycus was at the height of his activ- Geometry: Elements (EToiXEia). Euclid's fame rests ities about 300 B.C., and the date that would best fit preeminently upon the Elements, which he wrote in the middle point of Euclid's active career is about thirteen books 18 and which has exercised an influence 295 n.c. ; but the uncertainties are so great that no upon the human mind greater than that of any other quarrel can be taken with the conventional round work except the Bible . For this reason he became date of 300 B.C . 13 known in antiquity as `0 Y_ToLxELWT11S, "the Writer of He is therefore a totally different person from the Elements," and sometimes simply as '0 FEW L Tp17s, Euclid of Megara, the disciple of Plato, who lived "the Geometer." Proclus explains that the "elements" about a hundred years earlier.14 His birthplace is are leading theorems having to those which follow unknown,15 and the date of his birth can only be the character of an all-pervading principle ; he likens guessed.