The Charles S. Peirce-Simon Newcomb Correspondence

Carolyn Eisele

Proceedings of the American Philosophical Society, Vol. 101, No. 5. (Oct. 31, 1957), pp. 409-433.

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http://www.jstor.org Tue Mar 11 07:51:41 2008 THE CHARLES S. PElRCE-SIMON NEWCOMB CORRESPONDENCE CAROLYN EISELE Hunter College

CAREFULLYtucked away in the files of the was Peirce's father, the renowned Benjamin. Manuscript Division of the Library of Congress However, while Newcomb the elder moved fre- and in the archives of Widener Library and quently from place to place in Nova Scotia to eke Houghton Library at Harvard University are out a subsistence living, Benjamin Peirce was the two ends of a correspondence that stirs the firmly established in Cambridge as one of the imagination and quickens the pulse of the scien- adornments of the Harvard faculty in the Profes- tist or historian interested in scieritific Americana sorship of Natural Philosophy and Mathematics. of the late nineteenth century. The correspond- Charles was born into comfortable, upper middle ents are two of the greatest intellects ever pro- class surroundings with the silver spoon of the in- duced in America and their exchange of opinion tellectual aristocracy planted firmly in his mouth. regarding matters scientific and personal serves Both boys were avid readers of whatever came as an interesting personalized documentation of to hand. Neither needed much help in the con- the scientific thought of their period. The re- quest of elementary mathematics. Xewcomb nowned astronomer, Simon Newcomb, consid- tells of his state of rapture when he stumbled, at ered the sheaf of letters from Charles S. Peirce the aee of fifteen. on the beauties of Euclid in his important enough to file away with those of grandfather's copq. of the Simpson editi~n.~At other important men in science whom he knew. the age of thirteen, Peirce came upon his Peirce, America's belatedly recognized giant in brother's copy of Whately's Logic and at once logic and philosophy, preserved a number of absorbed the principles set forth in tile text. Xewcomb's letters as well as the drafts of some Newcomb was o.lly thirteen years old when he of his letters to Kewcomb of which we have no was hired out by his father as a manual laborer other record . The correspondence is being pub- to ease the financial strains at home. After lished herein for the first time.' It is not all three years, the boy had the opportunity to ap- continuous although most of it lies in the period prentice himself to a "man of science," a doctor 1889-1894. who turned out to be a quack. Flight from what The fact that the two men were compatriots then proved to be an intolerable bondage brought and contemporaries in science almost year for him to the shelter of this country which his fore- year in a time of unprecedented scientific inquiry bears had left for the deeper wilderness of the and discovery suggests an inevitable personal and north. Bv the time he was nineteen, Newcomb professional contact in the circumscribed society reached kiaryland, and accepted a teaching post of American scientific life in those davs. A in a country school. In 1856 he was employed short sketch of their lives will account for this ex- as a tutor in a family on a plantation whose change of opinion and criticism in their later proximity to Washington at last brought him vears. into a world where men "wrote books" and Although he was born a Canadian in the re- people "knew the Inen who wrote book^."^ The mote town of Wallace, Nova Scotia, Newcomb's hunger for books took him to the Smithsonian heritage was as thoroughly New England as that of Peirce. Xewcomb's father was a teacher. as Reminiscences of an astronomer, by Simon Newcomb, 14, 17, 18. The correspondence was discovered by the writer while An excellent short account of Newcomb's life and work doing research for a book on the activities of Charles S. is found in the Memoirs of the National Academy of Peirce as a historian of science under a grant from the Science 17, 1924, by \i7.W. Campbell. Three additional American Philosophical Society. It is being published papers by R. C. Archibald are of much value: Simon with the permission of the heirs of both men and of the Newcomb (1835-1909). Bibliography of his life and work, Philosophy Department of Harvard University. The Science, n. s. 44 (1147): 871-878, Dec. 22, 1916; an earlier writer is indebted to Dr. Elizabeth McPherson of the Li- version of the foregoing in The Proceedings and Transactions brary of Congress who was first to suggest the possible of the Royai Society of Canada, 1905, 2nd ser., 11 Sect. 111: existence of Peirce materials in the Newcomb Collection in 79 ; biographical note in the Semicentennial history of the the Manuscript Division. American Mathematical Society, 124-139. 410 CAROLYN EISELE [PROC. AMER. PHIL.SOC.

Institution where he met Joseph Henry and J. E. comb was marked by mutual respect and esteem. Hilgard. These two men, recognizing the young When Charles boasts in his writings, as he often man's ability, sought to place him at first in the does, of the scientific Clite who regularly visited Coast Survey. But nothing came of their hope. his renowned father at home, Newcomb's name Later their sponsorship led him at the age of is invariably listed among the callers. More- twenty-two "into the world of sweetness and over, a warm friendship seems to have existed light on one frosty morning in January, 1857," between Newcomb and Charles Henry Davis,? when he took his seat as a computer between Benjamin's brother-in-law and Charles' uncle. Joseph Winlock and John Runkle "before a Davis was Superintendent of the Nautical Alma- blazing fire in the office of the 'Nautical Almanac' nac Office and in 1865 became Superintendent of at Cambridge, Mass."4 the Naval Observatory, where Newcomb had The official routine of the Almanac office per- been assigned also as a Professor of Mathematics mitted great flexibility in the arrangement of in the Navy.8 working hours and thus made it possible for In 1870 the United States Government sent a Newcomb to join the Lawrence Scientific School solar eclipse expedition to the Mediterranean "early in '57 for the purpose of studying mathe- under the direction of Benjamin Peirce who took matics under Professor Benjamin Peir~e."~He with him, among others, the two young men. was graduated Bachelor of Science in 1858 and Newcomb was in the party stationed at Gibraltar for the three ensuing years was classified as a while Charles did his observing in Sicily. resident graduate. The friendship between Benjamin Peirce and Charles Peirce, in his youth, had been under Newcomb must have been strengthened by the the personal tutelage of his father, and was pre- overlap in their work in the Nautical Almanac pared by the age of sixteen to enter Harvard Col- Office, for Benjamin had been Consulting Astroll- lege.6 He was graduated without any particular omer to this agency since its organization in 1849. distinction by the College in 1859 and was given Later, in 1875, when Benjamin learned that an assignment as aid in the Coast Survey to Charles would not be considered as a candidate which he was formally appointed in 1861. He, for the post of Director of the Harvard Observa- too, found it possible to do this work and attend tory, he threw his considerable scientific weight classes simultaneously at the Lawrence Scientific to the support of Newcomb for the coveted posi- School. With a specialization in chemistry, tion. He was unable to persuade Newcomb to Peirce was graduated a Bachelor of Science in consider an offer which had been made by Presi- 1863, the first summa cum laude from that in- dent Eliot. Newcomb wanted the Observatory stitution; he had been graduated a Master of post no more than that of the Superintendency Arts from Harvard Universitv the vear before. of the Coast Survey which Sylvester had urged Perhaps the fact that peke ;as four years him to seek after Patterson's resignation in 1881.9 younger than Newcomb and the fact that New- Two years later, on September 15, 1877, New- comb lacked social roots in Cambridgeu accounted comb became head of the Nautical Almanac to a large extent for the surprising lack of evi- Office. dence of a personal friendship such as Peirce enjoyed at the time with Alexander Agassiz or Davis' paternal attitude toward Newcomb is revealed with William James or even, in a more formal in "Formative influences," by Simon Newcomb, Forum 11: 183-191, Mar.-Aug. 1891. See also "Professor Benjamin way, with Chauncey Wright. However, a gen- Peirce," by Newcomb, Proceedings of the Royal Society of uine friendliness between Peirce pbre and New- Edinburgh 11: 739-742, Nov. 1880-July 1882. Newcomb was appointed by President Lincoln in 1861. Reminiscences of an astronomer, 1. 9 Osiris 1 : 85-154. "Unpublished letters of James Letter to Rev. J. Walker from Newcomb dated March Joseph Sylvester and other new information concerning 25, 1857, re conditions for prizes. Sent from Nautical his life and work" by Raymond Clare Archibald. In a Almanac Office. Now in the archives of Widener Library. letter to Newcomb dated Oct. 20, 1881, Sylvester writes, Mrs. Elaine Trehub, then of the library staff, brought it "Who is to be the new superintendent of the Coast Sur- to the attention of the writer. The writer wishes to express vey? Why should you not allow it to be known that her gratitude to Mr. Kimball Elkins, Senior Assistant in you would accept the appointment supposing you would the Archives, and to Mr. Clifford Shipton, Custodian of the be willing to do so!" Sylvester was the eminent British Archives, for securing the permission of Harvard Univer- mathematician who served as the first chairman of the sity to use this material. Department of Mathematics at the Johns Hopkins Uni- No definitive biography of Peirce has as yet been versity (1876-1883). He returned to England in 1884 written. There is a short biographical account by Paul to occupy the chair of Savilian Professor of Geometry at Weiss in the American Dictionary of Biography. Oxford. VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE 41 1

Just as the official work of Benjamin Peirce attempt with Michelson to measure the velocity and Newcomb had overlapped, so did that of of light.15 Charles and Newcomb although they were affili- Their scientific re~utationbrought - both men ated with two different government agencies. membership in the National Academy of Science, For example, after Charles had begun his pendu- Newcomb in 1869 and Peirce in 1877, where both lum-swinging experiments as the Assistant in the later became members of the important Academy Coast Survev in charge- of the measurement of Committee on Weights and Measures. gravity, he mentioned in a reportlo that New- Other similarities in interests and activities are comb had seemed to discover a possible new numerous, but only a few of the more significant factor adding further difficulty to the pendulum- will be mentioned. In 1871, after fulfilling the swinging problem. Again, when Peirce needed duties of his solar eclipse expedition assignment, the calculations and map of the eclipse of June Newcomb "holed-in" at the Paris Observatory 29. 1878." Newcomb was asked to furnish co~ies library, within ear-shot of the besieging nation- of 'the same. When, much later, Peirce was' on alist forces, to make a study of the records of the defensive against Superintendent Menden- earlier astronomers, beginning with 1675, of the hall's attack on the "backwardness of his work," occultations of brightu stars bv the moon. He he countered with, "Now anvbodv who has ever later considered this the most important "find" done such work in such a way,-ask such men as he ever made, for he was able, as a result of these Langley or Newcomb,-will tell you that it is im- studies, to confirm his suspicions that Hansen's possible to make any reliable estimate of how tables. then in -general use, were unreliable. much time it will take."12 Charles Peirce, driven similarly by an interest in While Newcomb was building his scientific past scientific achievement and a need to utilize career in the Nautical Almanac Office and at the historical documentarv evidence in current scien- Naval Observatory, Charles Peirce was acquiring tific researches, made a thorough study of a a not inconsiderable scientific reputation in the manuscript of a thirteenth-century work16 at the Coast Survey on three counts. His photometric Bibliothhque Nationale while on an official Coast researched3 at the Harvard College Observatory Survev mission in Paris. brought him recognition as a first-rate astrono- Both men had teaching- experience,- and in mer; his pendulum work in Europe as well as in both cases it was severely curtailed. In New- America brought him international recognition ; comb's case the pressure of other responsibilities and his measurement of the meter from the wave made it impossible for him to devote full time to length of light was applauded at the time and an academic career.'' In Peirce's case personal proved to be the forerunner of similar work by eccentricities and an inability to work in har- Michelson and Morley.14 Newcomb was later to mony and "in harness" with others was to frus-

lo From a letter dated July 1, 1873, in the Coast Survey trate continually his ardent desire for a formal files in the National Archives in Washington. Peirce academic connection. writes, "Newcomb, in a paper in the last Ast. Nachrichten, Although creative mathematical skill was in- says he finds that pendulums hung by springs twist and untwist as they oscillate and says this will affect the time cidental to the work of both men, Peirce regarded of oscillation." l1 Ibid. A letter dated April 26, 1878. l6 Under a Congressional appropriation, Newcomb l2 National Archives. From a letter dated March 30, worked in Washington while Michelson experimented in 1888. When Chas. A. Schott, Assistant in charge of the Cleveland (1880-1 882). Computing Division of the Coast Survey, was asked to The Efiistle of Petrus Perenrinus- on the lodestone. Paris investigate a certain aspect of Peirce's procedure, he ap- MS. NO. 7j78. - proved of it and added, "vide Prof. Newcomb's investiga- l7 Newcomb's ~rofessorshi~of mathematics in the Navv tion to free the so-called standard R. A.'s from periodic has already been mentioned. He later became Professor errors; Wash. Obs. 1870." This remark is from a footnote of Mathematics and Astronomy at the Johns Hopkins to a letter to Superintendent Patterson dated Jan. 14, 1879. University (1884-1893 ; 1898-1900. Lecturer from 1876 l3 Photometric researches, in the Annals of the Harvard to 1883). Peirce had lectured on the philosophy of science College Observatory 9, by Charles S. Peirce. at Harvard (1864-1865) ; on philosophy (1869-1870) ; and The writer read a paper entitled: Charles S. Peirce, on logic (1870-1871). He lectured at the Lowell Institute nineteenth century man of science, to the Met. N. Y. on the logic of science in 1866. He became a Lecturer on Section of the History of Science Society on November 27, Logic under the auspices of the Mathematics Department 1956. A similar paper has been accepted for publication under Sylvester at the Johns Hopkins University in the in Scripta .Wathematica in the near future. period 1879-1884. The official phrasing of the records l4 Michelson and Morley suggested the wave length of veils, perhaps forever, the mystery of Peirce's enforced de- sodium light as the standard of measure. tachment from the University at a critical period of his life. 412 CAROLYN EISELE [PROC.AMER. PHIL.SOC.

Newcomb as essentially a practical astr~nomer.'~tion in the last vears of his life. For the files of And yet Newcomb was later to be elected for the the Assistants' correspondence in the National years 1897 and 1898 to the office of President of Archives, written in the decade after Benjamin's the American Mathematical Society, one of the death in 1880, reveal the tragic story of the many positions of high honor which he held in gradual decline in value to the Survey of Peirce's his lifetime.lg Peirce's self-admitted primary service as a member of that organization. His interest and creative talent lay in the area of increasing inability to participate harmoniously logic, especially the logic of scientific method. in close teamwork with his colleagues led to Because both men developed an unusually wide personal skirmishes with Thorn, the "lawyer" range of interests, both were asked to write ex- Superintendent (1885-1889). Peirce's contemp- tensively for newspapers and periodicals and tuous attitude, as revealed in his letters, toward both were members of editorial boards of dic- what he considered the scientific ineptitude of tionaries and encyclopedias. Newcomb wrote Mendenhall, Thorn's successor, rendered inevi- numerous books of popular interest in a facile table his own resignation from the ranks of the style. Peirce, however, was to find it impossible Survey in 1891. to produce the manuscripts that would have Furthermore, despite Peirce's pretensions to made his work known in the longer book form.23 sound business acumen. he found himself bv the Later in life their careers again diverged. mid-nineties divested of family inheritance as Newcomb, having retired at the mandatory age well as the regular source of income which the of sixty-two from the Nautical Almanac Office, Survev attachment had assured him. Beset bv was able to continue many of his professional legal difficulties and faced with the permanent activities. This was especially true of his im- loss of his valuable library as well as the tempo- portant lunar studies which enjoyed the patron- rary loss of his Milford estate, "Arisbe," he found age of the Carilegie Institution after his retire- survival for himself and his wife barely<. ~ossible ment." This support was first granted in 1903 by accepting numerous odd writing assignme~lts, and Newcomb continued with the work almost some of which will be referred to in the course of to the end of his life. He was highly revered this paper. After negotiations had been con- and honored, decorated by foreign governments, cluded for him to use "Arisbe" as his permanent a member of every important honorary scientific residence, he spent the remainder of his life work- organization in the western world, and president ing part of his land, writing and rewriting inces- of many learned societies. He seems indeed to santly whenever the opportunity offered itself, have been a notable product of the studious and sending his manuscripts, whenever he was application of his favorite motto, "Whatsoever so employed, to the various publishers by mail. thy hand findeth to do, do it with all thy might." Occasionally he went farther afield to attend Peirce's unhappy star, on the other hand, meetings of the National Academy of Science, to brought him, despite the recognition of his valu- visit Langley on business at the Smiths~nian,~~ able contributions to logic and his earlier cele- or to lecture at Cambridge- or in Boston. brated work in science, such insecurity and And at the very end, Peirce became a victim of misery that only the generous solicitude of a that dread malady, cancer. Strangely enough small group of sympathizers, headed by the noble this disease was to fell Newcomb also. Both William James, rescued him from actual destitu- struggled to the very end to leave behind a pre- cious heritage of ideas gleaned from a lifetime of

l8 Newcomb enjoyed a reputation as a political economist also. 22 Mr. H. by. Dorsey, who had been the Administrative l9 He became Vice-President of the American Philoso- Assistant to Secretary Langley in those days, recalled in a phical Society on January 1, 1909. He had been elected to telephone conversation with the writer early in July, 1952, membership in the Society on January 18, 1878. Pierce's ill-groomed appearance on such visits and the bit- The correspondence of Peirce with various publishers, terness between Peirce and Langley created by the contro- now to be found in the Peirce Collection in the archives of versy over an article entitled by Peirce, "The laws of Widener Library, reveals that Peirce had numerous op- nature and Hume's arguments against miracles," which portunities to publish but either could or would not meet Peirce had prepared at Langley's request for publication the conditions set by the publishers. References to details by the Smithsonian Institution. The correspondence be- in correspondence other than that between Peirce and tween Peirce and Langley on this subject has been pub- Newcomb in the Peirce Collection, are being made with lished by Philip P. Wiener in Proc. Amer. Philos. Soc. 91 the permission of the Philosophy Department at Harvard (2): 201-228, 1947. Further details appear in a paper by University. the writer in Jour. Hist. Ideas, Oct., 1957. It is entitled: 21 Established in 1902. The scientist-philosopher C. S. Peirce at the Smithsonian. VOL. 101, NO. 5, 19.571 PEIRCE-NEWCOMB CORRESPONDENCE 413 rich experience. Each man was just past sev- discrepancy between the calculated and observed enty-four at the time of his death.23 positions of Mercury," and that, "Soon after- In closing this brief outline of the careers of ward the theory of relativity seemed to be this these men, it may be well to quote some of new law." Einstein concluded his letter with Peirce's opinion of Newcomb's work. What the sentence, "It was thus that the theory of rela- Peirce's opinion of Newcomb's contributions to tivity completed the work of the calculus of the science of his time had been is obvious from perturbations and brought about a full agree- the opening paragraph of his review24 in the ment between theory and experience." Without Nation of Newcomb's The Reminiscences of an the foreknowledge of the scientific edifice which Astronomer. It is expressed in the words, Newcomb's labors were capable of supporting, Peirce had judged his man fairly and well. Newcomb is quite the most distinguished man of Newcomb's estimate of Peirce's general capa- science in this country to-day, as well as one of the bilities as a logician and scientist is difficult to most eminent in the whole world. His name will judge, for materials with such references seem to remain upon the page of scientific history, and be no longer extant. The correspondence which eventually take its place high in the second rank, distinctly above Leverrier's or even Hansen's be- follows reflects Newcomb's judgment of Peirce's cause of the breadth of his work. thought as regards numerous specific matters, mathematical and otherwise. Earlier, in 1901, Peirce had written in the The first item27in the correspondence, chrono- opening article for the Post, in its special issue logically speaking, reflects not only Newcomb's dedicated to a Review of the XIXth Cent~ry,~~interest in political economy but Peirce's ap- proach to questions in that field too. In the Thus it happens that we have a magnificent group last of the series of five papers written for the of great astronomers living among us to-day. We Monist in 1893,28 the paper entitled "Evolu- stand too close to them to take in their true pro- tionary Love," Peirce takes full advantage of an portions. But it is certain that the names of opportunity to castigate the author of a "hand- Chandler, Langley, Newcomb, Pickering, and several book of political economy-the most typical others are indelibly inscribed upon the heavens. and middling one I have at handM-because the author had conferred the title "love of self" on It is remarkable that Peirce could have so clearly anticipated what the judgment of New- what Peirce called pure greed. The handbook comb's work by the next generation was to be. was the Principles of Political Economy (1886) For his opinion was confirmed a quarter of a and the author was Simon Newcomb. Writing- century later when Albert Einstein wrote to a in a similar vein in a manuscript of about 1906,z9 daughterz6of Newcomb, "Your father's life-work Peirce recalls and reaffirms at this late date his is of monumental importance to astronomy." earlier reactions to Newcomb's work. He mentioned especially her father's contribu- tions to the calculus of perturbations. It was I remember two passages in my writings in which I noted in the "Sketch of Simon Newcomb" that made as much fun as politeness would allow of Newcomb "had at times remarked that a new writers who undertook to tell us what was "conducive to our welfare." Once it was Simon Newcomb law of nature must be discovered to explain the who was talking like that in his book on Political Economy; and I remarked that an economist, far 23Simon Newcomb (March 12, 1835-July 11, 1909). from having any qualifications for exploring this Charles S. Peirce (Sept. 10, 1839-April 19, 1914). most occult of all matters, was particularly unfit 24 78 (2021) : 237. March 24, 1904. for the task owing to his habit of taking it for 26 The Century's great men in science, Sat. Jan. 12, 1901. Peirce contributed the leading article to the Post's Review granted that wealth was desirable. of the XIXth century. On the second page the leading article was by Newcomb and was entitled: Advance in 27 From the Newcomb Collection, MS. Division, Library astronomical science. of Congress. The writer has made an effort to clarify all 26 Science, n. s. No. 69: 248-249, March 1, 1929. Dr. allusions in each letter. Limitations of space do not permit Einstein sent this letter from Berlin to Mrs. Josepha a mathematical analysis of detail in this paper. The Whitney on July 15, 1926. It is found also in a pamphlet writer has in preparation a critical review of Peirce's mathe- entitled "A brief sketch of Simon Newcomb" which was matical contributions. prepared by this daughter and submitted by her and her 28 The collected papers of Charles S. Peirce, edited by sisters Anita McGee and Emily N. Wilson to the Electors Charles Hartshorne and Paul Weiss (6.291). Also found of the Hall of Fame of New York University. Copy in in Chance, looe, and logic by Morris R: ohe en. Butler Library at Columbia University. rJ Ibid., 6.517. CAROLYN EISELE

One of the earliest attempts at an exchange of turned to England and Newcomb had succeeded opinion on this subject has been preserved in the him in the position of Editor-in-Chief of the first item. American Journal of Mathematics for the periods Dec. 17, 1871 1885-1893 and 1899-1900. There seems to have U. S. Coast Survey Office been considerable friction regarding the material Washington Peirce wished to have published in that Journal, for echoes of his complaints about Newcomb's Dear Sir lack of cooperation are heard in many places. What I meant by saying that the law of Supply For example he writes in one place,31 and Demand only holds for unlimited competition is this. I take the law to be, that the price of an I wrote out in 1885 a full systematic exposition of article will be such that the amount the producers the notation and its use: but I did not print it. can supply at that price with the greatest total Indeed, it was, in effect, refused by the editor of profit, is equal to what the consumers will take at the Am. J. Math., Simon Newcomb, who said he that price. This is the case with unlimited com- would print it, if I would declare it was a mathe- petition because nothing that any individual pro- matical paper. That I could not do. ducer does will have an appreciable effect on the price; therefore he simply produces as much as he can profitably. But when production is not thus Milford, Pa. 1889 Jan. 17 stimulated, the price will be higher and at that My dear Newcomb : higher price a greater amount might be profitably If the following proposition be not too well-known supplied. or obvious, possibly the proof I give of it may be To state this algebraically :- worth filling up some corner of your journal with The amount that can be profitably produced at a sometime. certain price X is the value of y which makes Through a point in space let 4 right lines be (X, - z) a maximum so that X - Duz = 0. But drawn. Let 4 points, assumed one on each of these the price x which the producer will set will be that lines, be taken as the vertices of a tetrahedron; and which will make (xy - z) a maximum so that let a second tetrahedron be formed in the same way. y + D,y(x - Duz) = 0. Then the lines of intersection of corresponding Clearly x>X because D,ymathematicians were who had not read or had not with "the best mathematical thought of the day" studied Lobatchewsky, in reference to that subject. to regard the value infinity as a mere point through If you were to say that my definitions mentioned which a variable passes without coming to any are not in accord with the prevalent mathematical abrupt termination there or anywhere. Of course, thought, perhaps you could prove your point; but the real thing which the mathematical object I would rather be in accord with views which are in simulates may do so, but pure mathematical lines my opinion surely destined to prevail in a few years. join themselves though (as in hyperbolic geometry) 2.-You say you cannot comprehend what I they have to pass through a region of imaginary mean by "the point at which a variable upon which values before they do so. And in pure mathematics a function depends passes through infinity." there is no difference between a line and a linear I will explain. First, observe that I do not say quantity, unless a mere point of view. So that when there is such a variable when limit means simply it is said that a variable "never" reaches its limit, boundary. Thus, where time passes from one day that will not prevent our conceiving it as getting to the next, I do not say that at Oh of the second day there and passing through it, all the same. any variable considered passes through infinity. 3.-You say you object to my definition of limit, Second, I am not speaking of a variable connected "totally." Now let us see what it is. I give first a with a function in a strictly definite manner. But generic definition, and then a specific one under it. when it is said that a limit is a value which a variable (After which I criticize another definition which I is conceived to approach indefinitely but never to do not assent to.) The general definition is: "the reach, then I say there is an express reference to precise boundary between two continuous regions of another variable. The words of approach and magnitude or quantity." Thus, a tangent is the never are words of time and relate to some vaguely precise boundary between the region of neighboring conceived variable. The word never indicated the right-handed secants and neighboring left-handed point where that variable, assimilated to time in secants. I then give a specific definition under the language used, passes through infinity. I that: "the point at which a variable upon which imagine (see figure) in a plane a line rotating about some function depends passes through infinitv." I have shown abdve wh; I give the-first as ;he universal meaning, and why and how I hold that as used by writers on the differential calculus it really has the second meaning, though they fail to make themselves perfectly clear about it, even to themselves. 4.-The definition then given and criticized is not "intended to be the sense in which the word is always used." It is intended to be a statement of an attempt frequently made to analyze the usual 418 CAROLYN EISELE [PROC. AMER. PHIL. SOC. meaning in the differential calculus. You say I am and infinitesimals are non-existent in thought" you wrong in describing this definition as tacitly assum- are altogether mistaken in supposing mathematicians ing that the variable depends upon another. I did agreed upon. This simply shows as I said you have not intend quite to say that, for I do not think it not read the papers. In the next place, even if quite true. It would be true, if for "tacitly" mathematicians were agreed on this psychological we read "expressly." But that is not what I nonsense, it would not prove anything at all about intended to say, for that is a statement to which I the proposition A + i = A. should not object at all. What I object to about the 9.-You say you never heard of anyone claiming definition is, in the first place, that it assumes this that we cannot reason correctly about infinite vague variable, which I fully admit, to increase by quantities. What? about infinite quantities strictly finite steps. The variable it expressly and rightly speaking, or about what the method of limits sub- asserts. The increment by finite steps it tacitly and stitutes for them and calls by that name? What unnecessarily assumes. This criticism I shall not was the motive of Lagrange's Theorie des fonctions? defend because you do not attack it. The first book I pull out is Jordan's treatise where 5.-You say you do not understand what I mean I read: "Quant A. I'infini, il kchappe A. toute mesure by saying that that definition's "overlooking the et ne sauvait entrer dans un calcul." essential cond[it]ions of continuity." But that is not what I say. I say it overlooks the essential Very faithfully element of continuity. That is to say, it defines a C. S. Peirce limit by that very conception embodied by the word infinitesimal (which word I forgive because Milford, Pa. 1890 (. . .) everybody overlooks or forgets its etymological My dear Newcomb : meaning) namely as if it were nothing more than that Merry Christmas (. . . .) As you signify in your toward which a series converges,-the sum of an last that you have got through with your criticisms infinite series. Here there is no continuity; while a on my definitions of infinitesimal, limit, and Doctrine limit, according to my notion of what people mean of limits, I will summarize them and my answers. by it, is the boundary between two continuous You "object to my definition of limit totally and regions. of infinitesimal partially, as non-accordant with 6.-You say I do not define Method or doctrine the best mathematical thought of the day, and of limits. I profess to do so. I do not confine myself savoring too much of notions current a century ago, to stating what I suppose to be its logical basis, nor and current still among men whose ideas are im- do I state at all what I suppose to be its logical properly co-ordinated." basis. I state what I conceive to be its essence; Let us see by a review of the arguments whether and that is as much as the dictionary can find room you have succeeded in justifying this judgment for in regard to any doctrine or method. Perhaps, I or not. might have a little specimen of it. Against my definition of infinitesimal, you object 7.-I don't understand what your proposition is that I do not define it in terms of the doctrine of about a + i = a. If a person adopts the concep- limits. I do refer to that mode of treatment, tion of infinitesimals, this certainly seems a funda- stating that it avoids sundry difficulties. I do not mental proposition of the calculus. If you adopt adopt as giving the (. . .) and accurate meaning of the method of limits an infinitesimal is regarded infinitesimal, (. . .) view which one of its chief as the limit of a quantity that vanishes. That defenders characterizes thus: "The method of A + lim B = A when lim B = 0 cannot well be limits is, in reality nothing more than one way of denied, and this is all the meaning that an adherent evading the use of the word infinite in its absolute to the doctrine of limits can see in A + i = A. But sense." But this does not satisfy you. You perhaps from that point of view it may not be a thought "that for 50 years, at least, everyone who fundamental proposition. understood the subject accepted the view that an 8.-You say you think I am wrong as to mathe- infinitesimal is simply a quantity approaching zero maticians differing in regard to the interpretation as its limit. In reply, I instanced Georg Cantor, of A + i = A. I think that shows you have not when you remarked that the papers of Cantor read recent discussions of infinitesimals etc. To "came before" you some years ago, but you could support this you say all would agree that "all see nothing of value in them. Your expression reasoning about infinitesimals and infinites" admits suggests that you rejected something offered for the of being reduced to the forms of Euclid. If for the Am. J. Math. The editor of the Acta Mathematica, words in quotations you put "the results of the which I may without disrespect for your journal calculus, in regard to finite quantities," I admit it. characterize as without doubt the most profound Otherwise, it is quite indefensible. As for Euclid's journal of mathematics published, saw so much in axioms, I consider them as innocent as they are Cantor's papers that he gave up one entire number impotent,--except the ridiculous 9th, which has to them,-although most of them had already nothing to do with the subject. That "infinities been published. The (. . .) excited a discussion in VOL.101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE 419 which some of the (. . .) mathematical heads of our shown to be nothing but a reductio ad absurdum. time have taken part. Not to be acquainted with (. . .) the logic of the doctrine of limits is as stupid them is to be unprepared to enter into a modern and narrow as that of the Stoics, its consequences discussion of the conception of infinity. are not so bad, since it happens that the most In order to show the difficulties with which the important results really can be established by the doctrine of infinitesimals is beset, I had remarked method of limits. that it was necessary, i being infinitesimal, to put On the whole, I have nothing to modify in my A + i = A, an equation "representing" a funda- definition of infinitesimal, though I could have made mental proposition of the calculus. the matter clearer in much greater space. It is to You think this statement "illogical or inexact." be remembered that it is no purpose of a dictionary It "simply astonishes" you. Your "objection" is to inculcate sound doctrine; it is simply to explain simply that of the school-boy. If A + i = A, then, words, ideas, etc. Moreover it would not be substracting A, it follows by Euclid's axiom, i = 0." expected that under infinitesimal, I should repeat all In reply, I pointed out that I had indicated in the that has been put under infinite. Dictionary two ways which had been resorted to to To my definition of the word limit, you "object escape this. One is to say i = 0. That zero is totally." You "cannot comprehend what is meant subject to all the rules of algebra without exception by the point at which a variable upon which some will, I take it, hardly be disputed at this time of function depends, passes through infinity." It day. It is true (. . .) as you say, with an assured would seem then that you are not au fait of what air that taken in connection with its accompanying has been done of recent years concerning the unconsciousness of what has been going on in the foundations of the theory of functions. The fashion- logical world is charming, that "zero cannot be able definition of to-day, if I may use the expression, treated as a term of a ratio under any circumstances." of a limit among the chief upholders of the doctrine You do not say how you would prove this proposi- of limits is the following. The limit of a variable, tion. The other way of escaping the difficulty x, is a value, c, such that for every positive quantity, about A + i = A, I said was, to regard the sign E, sufficiently small, a value N can be assigned to of equality as meaning measurable equality. In a number n, making mod (x, - c) < E for every order to make my meaning clear to you, I supposed value of n exceeding N. Here in n you have the a telepathic communication with infinitesimal beings. vague (variable) to which I refer becoming infinite We can measure A but not i; they can measure at the limit. You will perceive therefore that the i but not A. Neither of us can measure any fact that you could not comprehend what I meant difference between A + i and A, as a difference; did not suffice to relegate me to the class of those though they can measure i. whose ideas are imperfectly coordinated. In order to have successfully attacked my defini- My definition departs from the fashionable one tions, the one thing you would have had to show in two ways. First, it removes the unnecessary was that the idea of a quantity immeasurably small restriction that n must vary by discrete steps; involves contradiction. This you have utterly since I regard the conception of limit, when under- failed to do; and have shown you were not properly stood in the way it must be understood to make it (equipped) for the discussion. serve its purposes, as well as in the thought of However, you return to the equation A + i = A most of those who employ it, as involving or, at in your last, and undertake to prove that it does not least, not excluding the idea of continuity. In the represent a fundamental proposition of the calculus. second place, in the interest of co-ordination of You propose to make this out by showing that the ideas, I regard infinity, just as in projective geometry, application of it is equivalent to the "passage to the as a value to be "passed through." But you must limit." This, however, even if established would not suppose that when I thus vaguely describe my leave your proof incomplete. It would still remain reason for treating the matter as I have done that to be shown that the passage to the limit is not a my reason itself is vague. On the contrary, it fundamental part of the doctrine of limits; a premise rests on exact algebraic discussion of the reasoning required by your reasoning, though it would have concerning finite and infinite quantities which can- a double back action if granted. not be successfully controverted. To say of views But you do not show that the application of so founded that they "savor of notions current a A + i = A is equivalent to the passage to the century ago" suffices to demolish them just as much limit. You only show, what was well known, that as kicking the great pyramid would bring that down. there is a close analogy and correspondence between Having given my own definition of limit, I state the two ways of reasoning. There is an almost the common one which I reject. I have to put this precisely similar analogy between scientific induc- into popular language, and I use an expression tion and the reductio ad absurdum. There was an which has often been used by the advocates of the ancient dispute between the Epicureans and Stoics, way of limits. I admit that in the bare statement, the former advocating the use of induction, the without seeing how it is used, it is not evident to a latter only allowing it in cases where it could be person not very well up in the subject that the CAROLYN EISELE statement in popular language is open to the infinitesimals can be correctly reasoned about charge of representing the vague variable as varying mathematically. To that you will not reply, because by discrete steps. Such however is the peculiar it is an argumentum ad hominem. To show, import of the word never. however, that I assume that dx is not infinitesimal I admit this definition is somewhat wanting in (the very opposite of what I do assume) you say perspicuity. The vague variable requires explana- that I make it "subject to laws of multiplication." tion and the nature of the objection to the common Now, this argument has no validity, unless I deny definition needs elucidation. that infinitesimals are subject to the laws of algebra. I am unable to see however that you have shown So it appears that in your view those who hold the my views to be antiquated or confused. It is you doctrine of limits as well as everybody else you ever who do not seem to be aware of modern ideas. heard of admit that infinitesimals are subject to \Ye come now to the definition of the doctrine of mathematical reasoning, while those who adhere to limits. You say "I suppose there is a method of the method of infinitesimals do not regard them as reasoning in mathematics called the method of the subjects of such reasoning! You are all mixed limits . . . which method was employed by Euclid." up here. Those who consider that infinitesimals can This involves an inaccuracy in regard to the phrase be reasoned about mathematically have no need of which is serious as regards any criticism of my resorting, and do not resort, to any adjurant theory definition. Euclid uses in the 6th book a method for the differential calculus such as the doctrine of of reasoning which Archimedes developed into the limits. method of exhaustions. Supposing this method of You say that I "object to the way in which" the reasoning were the essence of the method of limits, doctrine of limits "has been defined in such a way it would be far from true that Euclid used the as would lead the reader to suppose that the whole method of limits. The difference is this: the method was fallacious." This is a singularly hasty method of exhaustions is a method of direct geo- statement which will not bear examination, as you metrical reasoning; the method of limits is a method shall see. In the first place, I have raised no of establishing the first propositions of the differen- objection in the dictionary to any definition of the tial calculus. This is the language generally doctrine of limits. I have objected to a definition recognized by careful writers. The method of frequently given to "limit." But that does not limits is a method of establishing the calculus lead any rational person to suppose the whole without the use of true infinitesimals nor series by method of limits fallacious. To the doctrine of reasoning concerning the limits of finites. (. . .) limits, I have made two criticisms. The first is, such, it has several varieties. The one which you that the notion that we (cannot) reason directly use, and which you would have me exclusively about infinitesimals is unfounded. No sane per- recognize, is not generally used, and is the least son could conclude from that remark that indirect demonstrative of any. reasoning about them was fallacious. The second You endeavor to make a distinction between the remark is that the method of infinitesimals "har- method and doctrine of limits, which I have treated monizes better with recent advances in mathe- as synonymous, by making the doctrine the "princi- matics." If a reader is so thoughtless as to suppose ples" of the method. But the distinction is not this means the method of limits is in downright tenable. In establishing the propositions of the conflict with recent discoveries in mathematics, he calculus it is necessary to enumerate the principles will fall into the error you mention. I don't think and there is none but a hair-splitting distinction there is much danger of that, but I will change to between the method and the doctrine of limits. "is more in the spirit of modern mathematical You also think the latter "somewhat incorrectly" philosophy," in another edition. called the doctrine. I don't know what you have You say you "cannot gather whether" I "admit got in your head. It is like saying Ptolemy's the validity of the method of limits or deny it." Almagest discusses no "problems." If I know any- This I regard as high praise. I hope I may have thing of the terminology of logic, the definition of treated all controverted doctrines in such a way doctrine in the Century Dictionary is correct: "the that it is not easy to gather on which side of the principles of any branch of knowledge; anything controversy I range myself. held for true." But I will mention that, in fact, if a limit is In undertaking to state the essential characters correctly defined, so that the approach to it may of the doctrine of limits, I begin by saying that it be continuous, and the great (distin . . .) of the rests on the idea that infinitesimals in the strict discrete and continuous infinite be not lost from sense "cannot be reasoned about mathematically." sight, the reasoning being like that concerning the To this you object, because you never heard of line at infinity in projective geometry, then I see anybody's holding such an opinion. It is, never- no objection to the method of limits except its theless, the general opinion of those who adhere to unnecessary circumbendibus. the doctrine of limits. I have twice asked you In the form which you would give to it, the error whether you really mean to say that you hold that is not serious, but I do not think it is perfectly VOL.101, NO.5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE 421

demonstrative. To say that the error of a state- to my attempt to make out a negative curvature ment is less than any assignable magnitude is of space. certainly not to say there is no error: Euclid 1.-The small number of negative parallaxes, con- himself would admit that. sidering the errors to which such work is subject, is You say, I give no adequate definition of what an argument that their real values are not very the method is. But the truth is you are mistaken small. I think infinitely distant stars may for as to what the method of limits is. It does not aught we know have parallaxes of 0".10. consist in reasoning in a special way, and were I to 2.-\\.'hen we look at the numbers of stars of different select two forms of the method as illustrations, magnitudes, we are struck first with the fact that thus greatly lengthening the definition, neither they increase with the magnitude in a pretty would be that which you wish me to give exclusively. regular way, and second that they do not increase On the whole, I do not think your original judg- nearly so fast as equable distribution requires, ment of the three definitions has been shown to be especially for the fainter stars. manifestly just. Now the surface of a sphere (or the reciprocal of Yours very faithfully the light) is proportional to [sin h(r/2k)I2. C. S. Peirce The volume of a sphere is proportional to [sin h(r/k) - (r/k)]. Peirce refers constantly in his writings to the In the D M the number of stars of magnitude influence of Lobachewsky, Clifford, Cayley, Can- 4.8 and brighter is 468 = (2.670) of magnitude tor, and Klein on the direction of his mathema- 8.8 and brighter is 70197 = (4.846) tical thought. Historically, the idea that our This gives the ratio of increase per magnitude (0.544)instead of (0.600)according to the Euclidean space might be non-Euclidean with a scientifi- Theory. I cannot now go further, as I want the cally verifiable curvature had been developed latest photometric measures of DM stars. along with the non-Euclidean geometries shortly 3.-It is well-known that the proper motions of before Peirce penned the next letter~.4~It is faint stars are not much smaller than those of curious that he should in these letters cling to the brighter ones. Some years ago I compared the theory of negative curvature when the computa- photometric measures of Seidel with Madler's proper tions gave him the "reverse of what" he wished, motions and found as follows for he was an old hand at the theory and practice of testing scientific hypotheses. He was most Distance from Square Root Proper Distance from certainly acquainted with Riemann's celebrated Mag Motion Brightness paper, which in Clifford's translation becomes, On the Hypotheses which Lie at the Bases of Geom- etry. Apparently the strong Lobachewskian in- fluence, which is so evident in his mathematical manuscripts, prevailed at this time.

My dear Newcomb: Now a proper motion has two parts, a parallactic part, inversely proportional to tan h(r/2k) and a want to get into circumstances in which can I I stellar part, inversely proportional to sin h(r/2k). pursue certain researches. I want you to do certain The component directed toward the apex of the things to aid me, and to that end, I want you first to sun's motion is in the mean entirely parallactic, the remark how encouraging the figures look in regard component perpendicular to that is entirely stellar. These two are in the mean not far from eaual. 46 Xo date, Xewcomb Collection, Library of Congress. Hence, the total proper motion is in the mean The first page of a draft is in the Peirce Collection, Box VB2b, at \t7idener Library. proportional to Dec. 21, 1891, Newcomb Collection, Library of Congress. (The number sheet mentioned is not in the folder in the 1 A files.) sin h"7/2k) + tan h2(7/2k) Dec. 21, 1891, draft in Peirce Collection, Box VB2b, Widener Library. Dec. 23, 1891, ibid. where A is such that the terms are generally nearly No date, ibid. equal. Now tan h(r/2k)is nearly constant for large The writer has found in the archives of the Smithsonian values of r;so that this is not very remote from Institution a complete manuscript by Peirce entitled: On two map-projections of the Lobatschewskian plane. It reached Langley too late to be read at the Academy of Science meeting for which it had been intended. CAROLYN EISELE

To show this, the ratio of Distance from brightness should like to have a whack at it myself; and as I to Distance from proper motion is (multiplied by a have found out what I have, I think I am entitled constant) to that whack. month or six weeks' work might show how the Observed thing was promising. But to discuss the matter as Mag Ratio 41 + 2 cos h2(r/2k) Where 2k = Dist it should be discussed, from six to nine months of 5th mag stars. would not be too long. The question is, can some The reason I mul- appropriation be made, or some millionaire be found, 1.78 tiply cos h2 (r/2k) to pay $3000 for this, $2000 for six months of my 1.85 by 2 is to make work and $1000 for an assistant? Will you and 2.02 the two terms other men, say Langley, King, Powell, Rood, John 2.40 about their actual Fiske, be willing to express the opinion that it is a 3.05 relative values. piece of work most desirable to have done? There were only a few 6th magnitude stars in In my mind, this is part of a general theory of the Seidel's list. universe, of which I have traced many consequences, 4.-But the question is whether the component of -some true and others undiscovered,-and of which many more can be deduced; and with one striking the proper motion perpendicular to the great circle trust there would be little difficulty in from the star to the apex of the sun's motion really success, I getting other deductions tested. It is certain that diminishes (as r/2k increases) relatively to the com- the theory if true is of great moment. ponent in the line to the apex. mThatinterest would you take in the matter? According to preliminary tests it really does so. Thus, I take all the stars of your catalogue from Very faithfully Sh 30m to gh 30m and assume the meridian through C. S. Peirce each star to be the line through the sun's apex (a rather violent assumption). There are 53 stars. I wonder whether a professorship of logic in Of these, 7 are moving towards the apex. These I Stanford's University would be beyond hope for me. reject and the 7 that are moving most rapidly away to balance the first. I then find for each star the My dear Newcomb : ratio of motion in arc of a great circle perpendicu- lar to the meridian to its motion in the meridian. Since writing you I have taken 63 faint stars from I divide them in the stars brighter than 6 and your catalogue and as near each as I could a bright fainter than 6 mag. The lists are nearly equal. one and for these have calculated the proper motions From each list I reject 3 sporadic very high values away from the apex and in the perpendicular direc- of the ratio, and the three lowest values to balance tion. The result is most decidedly the reverse of them. For the rest I find the mean ratio 0.92 for what I wished; and this shows itself in several ways. the bright stars and 0.68 for the faint ones. The I enclose some numbers. difference is, no doubt, excessive. My computations are without check; but errors I try the same for lgh. But here the stars do not to affect the result are impossible, I think. The happen to be so favorable. The same treatment apex was taken at a = 270' 6 = +30° for 1850. gives Yours very faithfully, for 10 bright stars 1.12 C. S. Peirce for 6 faint stars 0.91 It still seems to me the subject should be pursued. All these results are favorable to the hypothesis; Milford Pa 1891 Dec. 21 and can only be otherwise explained by four different suppositions. The argument appears to me strong enough to call for the closest examination. Milford Pa 1891 Dec. 21 The hypothesis is capable of being tested in many My dear Newcomb : ways. It seems to me that an exhaustive discussion One thing is clear. It is that those figures I sent of it is called for. you lend no support whatever to the idea of a The discovery that space has a curvature would positive curvature, but are rather against it. The be more than a striking one; it would be epoch- making. It would do more than anything to break striking non decrement of the peculiar motion with up the belief in the immutable character of mechani- the brightness needs to be investigated. But the cal law, and would thus lead to a conception of the formula for the surface of the sphere and for the universe in which mechanical law should not be the peculiar motion equally involve sin h(rj2k) or head and centre of the whole. sin (r/2k). It would contribute to the improving respect paid Very faithfully to American science, were this made out here. I C. S. Peirce VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE

I have given a whole day to this. I cannot any Two of these letters from Peirce drew from longer afford the luxury of unremunerative work but Newcomb a reply46which, in its discussion of the very little. possible curvature of space, anticipates the sub- Milford 1891 Dec. 23 ject of his Presidential address before the Ameri- My dear Newcomb : can Mathematical Society on December 29, 1897.47 That address was entitled "The Philo- I have for the present given up the idea that sophy of Hyper-Space." Although it cannot be anything can be concluded with considerable proba- bility concerning the curvature of space. The best reproduced in toto here, a few excerpts will serve argument I could make was as follows. By long to clarify Newcomb's position and to highlight motion, I mean the component of proper motion his scientific conservatism in refusing to entertain directly away from the apex of the sun's motion. a hypothesis which had not yet come completely By cross motion the motion at right angles to that. unscathed through the acid test of experiment. Now I found the cross motion and brightness had His address reads, in part, remarkably little to do with one another. Separat- ing the stars (I mean the 126 I wrote to you of) into I cannot but fear that some confusion on this subject two equal sets according to the magnitude of either is caused by the tendency among both geometers of these quantities, I find the other quantity nearly and psychologists to talk of space as an entity in equal for both sets. In point of fact, the intrinsically itself. . . . For us the limits of space are simply the brighter stars have in the mean probably a greater limits to which we can suppose a body to move. mass and therefore by the law of action and reaction Hence when space itself is spoken of as having a smaller absolute motion. Consequently, I said, possible curvatures, hills and hollows, it seems to the geometric mean of the distance as deduced from me that this should be regarded only as a curvature, the brightness and from the cross motion will if I may use the term, of the laws of position of probably be remarkably near the . . . . material bodies in space. Clifford has set forth, incomplete with acuteness and great plausibility, that the minute spaces occupied by the ultimate atoms of New York 7 \Vest 43 St. matter may, in this respect, have properties different My dear Newcomb from the larger space which alone makes itself known to our conceptions. If so, we should only regard The following question recently put to me seems this as expressive of some different law of motion, or, to involve fundamental points relating to infinity. since motion is only a change of position, of some I should like to see how you would answer it. different law of position among the molecules of bodies. Newcomb continues with the observation that "This consideration leads us to a possible form of space relations distinct from those of our Euclidean geometry, and from the hypothesis of space of more than three dimensions. I refer to what is commonly known as 'curved space.' " He speaks of the two independent substitutes which had now been made for the Euclidean axiom and the need of testing any curved space hypothesis by experience, and of the narrow Upon the straight line OX erect an infinite series of limits of the two extremes of the earth's orbit equidistant perpendiculars OA, BC, DE, FG, etc. within which "the measures of stellar parallax and draw the oblique line OS. Then the space give no indication that the sum of the angles of a included between AOX and the circle at infinity triangle in stellar space differs from two right bears some finite ratio to the space included in angles." Moreover, he adds, AOS and the circle at infinity. But it bears an infinite ratio to the space included between two The wise man is one who admits an infinity of successive perpendiculars, the base line, and the possibilities outside the range of his experience, but circle at infinity. Hence the diagonal OS must cross all the perpendiculars, contrary to Lobatchew- 46 Ibid. sky. Where is the fallacy? 47 Science, n.s., 7: 1-7, Jan. 7, 1898. See also address delivered at meeting of the N. Y. Math. Society, Dec. 28, Yours very truly, 1893, (N. Y. Math. Soc. Bull., 95-107), entitled: Modern C. S. Peirce mathematical thought. 424 CAROLYN EISELE [PROC..4MER. PHIL.SOC.

who in considering actualities is not decoyed by the among us. I do not know where to look for funds temptation to strain the facts of experience in order to do this with. to make them accord with glittering possibilities. . . . As for the Stanford University, I have never We are justified by experience in saying that the been in any way consulted respecting it, and in fact space relations which we gather from observation know nothing about it, except what I have seen in around us are valid for the greatest- distances which print. I do not therefore feel able to do anything separate us from the most distant stars. We have no in that direction. right to extend the conclusion further than this, We Yours very truly, must leave it to our posterity to determine whether, Simon Newcomb in either way, the hypothesis of hyperspace can be Prof. Chas. S. Peirce used as an explanation of observed phenomena. Milford Penna. Newcomb's reply to Peirce now follows. Newcomb's extreme conservatism with regard Nautical Almanac Ofice to mathematical matters in the first of the follow- ing letters48 is very surprising. Peirce was, Bureau of Equipment and Recruiting without doubt, the more daring intellectual of Navy Department the two. For example, he later corresponded U'ashington Dec. 24, 1891. enthusiastically with Langley on the develop- Mv dear Peirce : ment of the aeroplane49while Newcomb believed not at all in its potential, practical value. I received your two letters, each in due time, but Discussing the power of the am so crowded with work that I cannot give the ap- subject the study which it deserves. I can therefore proach to the calculus50Peirce gives an illustra- only refer to the different points in a general way. tion from the work of Fermaty5' and First, does not the fact that all recent determina- with the sentence, "The method of indivisibles tions of parallax are relative, prevent us drawing had recognized that infinitely large numbers may an" conclusion as to a limit ultimate Darallax? have definite ratios, SO that the division is appli- It seems to me it does. That is to say the relative cable to them." In a footnote to this sentence, parallaxes of distant stars will converge towards Peirce remarks that "Newcomb errs in saying zero on either hypothesis. (Johnson's Cyclopedia, 1894, IV, 576) this Second, in drawing conclusions from statistics of method is 'medieval,' and his descri~tionof it is stellar magnitudes, the number of disposable hy- not very characteristic. He is also wrong potheses seems to me so great that we can scarcely (Funk's Dictionary, indivisible) in calling it an test them even on the supposition of homaloidal application of the method of limits." space ;much less, then, can we draw a conclusion as to The second and third 1ette1-s~~are self-explana- the curvature of space. The fact is, it seems to me doubtful whether we can derive any law of relation tory. between an absolute magnitude of a star and the Nautical Almanac Office number of stars having that magnitude. Navy Dept.--Wash. D. C. Third, the proposition that the proper motions of Mar. 9th. 1892 faint stars are not much smaller than those of bright My dear Peirce: ones, seems to me not established, except in this sense; that, given a proper motion of two or three Your last letter seems decisive in favor of a seconds or more in a century, the number of stars proposition which I have often been inclined to having it range through a wide degree of brightness. maintain, to wit, that all philosophical and logical But you would thus exhaust the bright stars much discussion is useless. If there is any one question quicker than you would the faint ones. which illustrates the correctness of the doctrine of infinities, always maintained by me, it is the very As to getting a grant of money for the purpose one suggested by the demonstration you and Hal- you mention, it seems to me the difficulties are stead sent me. I have always held that infinity, insuperable. In the first place, the task of getting the scientific world to accept any proof now possible 48 Peirce Collection, Widener Library. Box V B2b. that space is not homaloidal, is hopeless, and you 49 Peirce-Langley Correspondence at the Smithsonian could have no other satisfaction than that of doing Institution. a work for posterity. In the next place, it is, I 53 The collected Papers of Charles S. Peirce, 4.15 1. 51 Peirce could have referred to his father's approach to believe, unusual if not unprecedented, to pay an the derivative in B. Peirce's book entitled An elementary investigator to do a work of his own out of trust treatise on curves, functions and forces. funds for the advancement of science, at least " Peirce Collection. Widener Library. Box VB2b. VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE considered in itself, could not be treated as a During the period in which the next letter53 mathematical quantity, and that it is pure nonsense was penned, Peirce was preparing the lectures on to talk about one infinity being greater or less than the history of science which he delivered at the another. The ground for this view I think I Lowell Institute that year. His familiarity with mentioned in our correspondence a year ago; the the old star names was to be expected after his very meaning of the word infinity is something notable work two decades earlier in the prepara- without bounds. But we can compare two magni- tudes only by comparing their bounds. Therefore tion of the historical portions of the Photometric I say the reasoning in question is baseless. What Researches. more can I say? Milford Pa. 1892 July 28 Yours very truly, S. Newcomb Simon Newcomb LL.D. Mr. Charles S. Peirce Sup't. Nautical Almanac The Century Club Dear Sir: New York, N. Y. In observing for time, I find the old star-names, Washington, D. C. where they exist more convenient than Bayer's April 23, 1895 designations. No doubt, many other observers My dear Peirce: would be of my mind. I suggest these names be I was puzzled by something in the Century inserted in the Almanac list, or at least, the following Dictionary, which can, I think, be explained by no of them. one but yourself. I refer to the definition of an a Andromedae Alpheratz oddly odd number. As no odd number can have /3 Cassiepeae Chaaph any except odd factors, I do not see what the a N Shedir definition amounts to. It is I know a very old one, fl Ceti Diphda but did it originate in anything but a blunder of p Andromedae Mirach some ancient editor? Yours very sincerely I do not think the second word in the name of a S. Newcomb constellation should begin with a capital. Mr. Charles S. Peirce Yours respectfully C. S. Peirce Nautical Almanac Office, Navy Department (Only five of the ninety-eight names given by Washington May 11, 1895. Peirce are listed here) My dear Peirce : The enclosed paper has been offered to the Amer. Without academic or government affiliation Journal of Math. Before deciding what to do with after 1891, Peirce decided to bring his materials it I would be much pleased to have you examine it, together for publication in a set of twelve vol- and point out any unsoundness that you may find. umes to be entitled "The Principles of Philos- I have not yet looked at it carefully, as I can better ophy: Or, Logic, Physics, and Psychics, con- do so after having your examination of the subject sidered as a unity, in the Light of the Nineteenth before me. Century." A prospectuss4 was prepared by You may address your remarks to Dr. Craig, who Peirce and subscriptions at two dollars and fifty is now responsible editor, especially as I may be away when they arrive. cents per volume were solicited in advance. Yours very truly, Peirce's synopsis reads as follows: S. Newcomb Vol. I-Review of the Leading Ideas of the Professor C. S. Peirce Nineteenth Century. Milford, Pa. Vol. 11-The Theory of Demonstrative Reasoning. Thanks for your reply to Vol. 111-The Philosophy of Probability. my question about "oddly odd." Vol. IV-Plato's World: An Elucidation of the Ideas of Modern Mathematics. The mathematical content of a number of the Vol. V-Scientific Metaphysics. letters already quoted reflects Peirce's need for tenacity in the pursuit of a clarification of the 63 Peirce Collection. Widener Library. Draft. Box meaning of each term. For the mathematical VB2b. concepts of infinity, continuity, and probability 64 There is an extant copy of the synopsis in the Harvard University Archives section of the LVidener Memorial as well, were to be found among the deepest building. The writer is indebted to Mr. Robert Haynes, foundation stones of his system of philosophy. Assistant Librarian, for securing permission to publish it. CAROLYN EISELE

Vol. VI-Soul and Body. Mr. C. S. Peirce Vol. VII-Evolutionary Chemistry. (The title Milford may probably be changed.) Penna Vol. VIII-Continuity in the Psychical and Moral Sciences. The Thomas Bayes mentioned in the first of Vol. IX-Studies in Comparative Biography. the next two 1ette1-s~~developed a basic theorem Vol. X-The Regeneration of the Church. in the subject of inverse probabilities which was Vol. XI-A Philosophical Encyclopedia. published posthumously in the London Philo- Vol. XII-Index raisonn.6 of Ideas and Words. sophical Transactions, vols. 53 and 54 for the years 1763 and 1764. According to R. A. Peirce could not find a publisher to support Fis~her,~'Bayes was the first to use mathematical the project. Henry Holt rejected it in a letter probability inductively, "that is, for arguing dated December 2, 1893.66 from the particular to the general, or from the In the letter55a to Peirce about the proposed sample to the population." With his abiding volumes, Newcomb's remark about "inverse interest in all possible forms of logical inference probability" is surprising since he had written and his own logic grounded in the trilogy of in- papers on the theory of probabilities for Runkle's 1859-1861. ductive, deductive, and abductive inference, Mathematical Monthly in the period Peirce would of necessity be deeply concerned UTashingtonD. C. with the validity of conclusions so drawn. Jan. 3rd, 1894 Dear Sir: Washington, D. C. January 16, 1894. I am persuaded that whatever you might write My dear Peirce : on the subject of scientific philosophy would be provocative of thought and discussion, and therefore There seems no immediate occasion for me to do interesting, whether one accepted your conclusions more than merely acknowledge yours of Jan. 6. I or not. You may therefore put me down as a sub- quite coincide with your expression of the spirit in scriber to your proposed volumes. which you treat the subject, although I fear my Your 2nd, 3rd, and 4th vols. are those which I philosophy would diverge a good deal from yours. should think would have the most scientific value, Your prospectus is well fitted to excite curiosity, and which I would therefore rather see come out first. and yet I fancy that the last paragraph, and possibly I am curious to know what the doctrine of inverse several of the preceding paragraphs, will not attract probabilities is, as I see you propose to refute it. an audience. I do not mean by this to imply that I am sorry to see that you repeat the implication there is anything objectionable or open to criticism that somebody holds the dogma that we cannot in the paragraphs above referred to. But experience reason mathematically about infinity. That we can- has taught me that there are some subjects on which not correctly reason about infinity as if it were a nobody wants to be really informed. magnitude is a proposition which I think no one As I do not know exactly what Bayes' theorem is, ought to dispute; but if you do dispute it, I am sure I am still in the dark as to your objection to inverse you, as a logician, ought to put the proposition into probabilities. the shape in which your opponents uphold it. In your letter you say, "You begin by finding I am sorry to say that you greatly overestimate fault with a sentence quoted verbatim from your- the value of any expression from me on your subject. self." I am curious to know what that sentence is, My experience leads me to believe that people have and where it was uttered. very little confidence in my views on subjects out- Yours very truly, Simon Newcomb side of mathematics and astronomy. The general subject of the greater number of your volumes is Mr. C. S. Peirce one on which people already have their minds Milford, Penna made up. I could make a number of criticisms both on the Washington, D. C. expressions in your circular, and the descriptions of February 7, 1894. your volumes, but as it seems to be printed in its My dear Peirce :- final form it is not worth while to do so. Excuse my inattention to your letters. I am Yours very sincerely, overrun with work, the result of having ten years' S. ~ ~ computations~ ~of about six~ or eight~ computorsb to get .

55 Peirce Collection. Publishers' Correspondence. 56 Peirce Collection. Widener Library. Box V B2b. Widener Library. 67 Statistical methods for research workers, Edinburgh, 668 Peirce Collection. Widener Library. Box VB2b. 1938. Also E. T. Bell, Development of mathematics. VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE into shape and weave in with astronomical theory. However, my purpose is not to tackle you but only But for this you would find me paying much more to explain that what may look like a reply to you attention to your projects. was really written long before your address. I am greatly obliged for your explanation about the inverse ~robabilities. The ~roblemvou allude Very truly to is of course one in which an element is lacking. C. S. Peirce But your programme gave the idea that you meant Their overlap in interests would account for to subvert one of the best grounded theories of mathematics. many of the mutual scientific associations of Your second letter puzzles me. In the first place Peirce and Newcomb. Among the most eminent if you do not agree to my conclusions by all means of these was the British astronomer Sir Joseph say so in the plainest English you can use. I would Norman Lockyer. He had headed the British a good deal rather be killed by a rattling attack Eclipse Expedition to the Mediterranean in 1870 than "todgeschweigen," as the Germans say. But and was closely associated with the American I am utterly unconscious of having said that there team in Sicily. Peirce's first wife, Melusina Fay, has been no great advance in mathematics since was a working member of the American group Euler and La Grange. On the contrary I thought and in her official report to the Coast Survey she my whole address was devoted to showing the spirit refers to Lockyer's general helpfulne~s.~~ of that advance. Peirce himself often expressed his great respect Yours very truly, S. Newcomb. for Lockyer's work and the quality of his reason- ing. In later years he reviewed Lockyer's Inor- One of Peirce's meager sources of income dur- ganic Evolution as Studied by Spectrum Analysis ing these years was as a reviewer of books for and paid him the following tribute60at that time. Garrison's Nation. The passage in the review He said. referred to in the next letter58 is the opening paragraph which reads as follows: That the relations among the chemical elements are to be explained by some sort of evolutionary process Many good people fancy that the advances of is the only idea we can at present entertain. LVe mathematics, like those of jurisprudence, become ought to begin then, with trying how the hypothesis manifest only when the state of things in one of the simplest kind of evolution that could answer generation is compared with that in another; and the purpose will fit the facts, and adhere to that that they are merely in the nature of extensions of until it is refuted. Lockyer's seems to be that old methods to new cases. . . . \Ve are speaking of simplest hypothesis. . . . pure mathematics, not celestial mechanics. In the Newcomb letter61which follows, Peirce's Dear Newcomb, approach to many of the problems in the history Some months ago I wrote for the Nation a notice of science stands revealed. Using the tools of of some books on the Theory of Functions. The the modern astronomer he checked mathemati- Nation is not exactly ravenously hungry for that cally the assumptions on which Lockyer had sort of thing and the proofs only reached me this based his historical theories in the Dawn of week. I then saw that you might fancy I meant to Astronomy. Peirce's review62was based on these attack your address, which of course I had not seen computations and exposed Lockyer's incompe- when I wrote what I did. I added a sentence to tence in this field. He questioned Lockyer's soften the contradiction. But really I think you claim that "Egyptian temples were generally are wrong. You might as well say that Chemistry oriented to the risings and settings of stars." has made no progress since Geber; because we can no more transmute the metals now than we could Some of the argument against the hypothesis is then. given in the second letteP3 following. You must take mathematics to mean pure mathe- U.S. Coast and Survey expedition to the Mediterranean matics and conceive its problems as they are con- for observing the eclipse of 1870. Appendix No. 16 of ceived today. On that point of view, the advance U. S. C. S. Report for 1870 (Dec. 22, 1870). of mathematics seems to me wonderfully rapid 60 The Nation 70 (1819), May 10, 1900. and accelerating. Peirce Collection. LVidener Library. Box V B26. 62 The Nation 58 (1500) : 234, March 29, 1894. 68 Newcomb Collection. Library of Congress. The 63 Newcomb Collection, Library of Congress. Peirce's books reviewed were Theory of functions of a complex approach to problems in the history of science was em- variable by Forsyth ;A treatise on the theory of functions by phasized by the writer in a paper which will appear in the Harkness and Morley ; and Trait6 d'analyse by E. Picard, Proceedings of the Eighth International Congress for the The Nation 58 (1498) : 197, Mar. 15, 1894. History of Science held at Florence, Italy, September, CAROLYN EISELE [PROC.AMER.PHIL. SOC.

Nautical Almanac Office, Naval OB with a polar projection made about 700 B. c. and Georgetown Heights, D. C. Feb 13, 1894 then radii vectores will not suit any projection I can think of but one from the vertex of the cone tangent Dear Mr. Peirce: to the sphere on the circle of perpetual occultation Mr. Garrison desires me to send you certain data at 35' to 40'. Nothing in Egypt will answer at all. respecting the obliquity of the ecliptic and the Now 719 B. C.Egypt was defeated in a great battle positions of the fixed stars for epochs of ten thousand by Sargon whose capital was Nineveh, lat 36', and years back, which will be within a few minutes of about that time Assyria began to be more influential arc. I do not know of any formulae which will in intellectually in Egypt. I am therefore inclined themselves serve your purpose. Of course the strongly to think that the original from which after ordinary formulae for precession do not apply, and many copies and changes this "zodiac," or rather the data for the rigorous trigonometric reduction planisphere was taken was made on such a projection have not, so far as I am aware, ever been carefully (or some projection practically equivalent) at computed. So the best I can do is to send you the Nineveh about that time. For if made in Egypt data for making the computation. how could it fail to show Canopus, and in short all Having the path of the pole of the ecliptic during stars, (but one, apparently) of lower declination the past 10000 years, as shown by the numbers on than -48' which at Denderat have an altitude at the diagram, the motion of the pole of the earth can culmination of 16'? Now since the planisphere was be computed by mechanical quadratures, within put up about A. D. 14, it follows that the Egyptians one or two minutes of arc. It would seem from the had made no observations of the stars to speak of diagram that the obliquity has been diminishing for 700 years; and that quite accords with my opinion during almost the entire period, or, more exactly, of their scientific nullity. The idea of expecting to about eight thousand years, so that it was probably find such a people orienting their temples with any nearly one degree greater ten thousand years ago sufficient accuracy to settle chronology, I think is than it is now. U'ith the former position of the difficult. Doubtless if we could find temples of the pole the places of the stars can be computed by the pyramid epoch, or thereabout, that could be shown trigonometric reduction. by independent evidence to be oriented to stars, I am very glad you are ready to take up that book then the case might be very different. Perhaps of Lockyer's. If he has really produced a sound Lockyer may do so in the future. I should be glad scientific work, by all means let Sir Joseph have the to see him go to work. So far his evidence is slender. credit of it. But if it is of a piece with much of his Very truly other work, let him be shown up. C. S. Peirce Yours very sincerely, Peirce made a last great effort in 1899to return S. Newcomb Peirce to the Coast Survey as Inspector, Bureau of Milford Penna Weights and Measures. His earlier Coast Sur- vey work had given him a particular competence Milford Pa 1894 Feb. 20 for this position as the first of the next two My dear Newcomb : letters64 shows. He was not successful, however, in his quest for the job. Among his sponsors I enclose you a circular showing what use I am were Asaph Hall, Seth Chandler, and George A. putting your letter to. Plimpton. The second letter65 was apparently In regard to Lockyer's Theory, I think, in view of written by Newcomb while on one of his favorite its possible importance if true, in setting early Egyptian chronology, which I think has a bearing walking trips in Switzerland. on our whole conception of man's development, that he has made out sufficient case to warrant the Milford Pa 1899 June 10 expenditure of time and money to collect further My dear Newcomb: facts. At the same time, I think there is much I am an applicant for Civil Service Examination against the theory. for the place of "Inspector of Standards," Office of As for the circular zodiac of Denderat, it is possible Weights and Measures, U.S.C. and G. Survey. to locate upon it with some probable accuracy,- Salary $3000, the same I formerly had. When I aside from any measurements made upon it,-the shall have been accepted as a candidate for examina- positions of about 40 stars. Then measuring the tion, the first question put to me, I am informed, places of those stars, we find they agree in R. A. will be, Name five persons who will answer questions concerning your "scientific administrative qualifica- 1956. A more general statement about his activities in that area was made in a report on his work which appeared 6' Ibid. in Year Book Amer. Philos. Soc. for 1954: 353-358, 1955. 65 Peirce Collection. Widener Archives. Box V b5. VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE tions and experience." I naturally apply to my difficult. So I have postponed answering it and colleagues on the Academy Committee on Weights other letters till I could settle down. and Measures of which you are a member. I do not see how I could say anything of real Allow me to remind you that you have known value about your "scientific administrative qualifica- about the following work of mine pertinent to the tions & experience." Your work was done for the question : Coast Survey and its records are there, and tell their own story. The Superintendent has at his command lst, that in 1874 I first undertook to "weigh the all the data for reaching a conclusion and nothing earth" by setting up a balance over a deep shaft that I could say could add to his knowledge of the and comparing the weight of 10 kilos above and subject. It is very clear to me that the persons to below. I did not succeed because I could only whom the C. C. Comm. should apply are those who work when the machinery at the Hoosac tunnel have made a more careful study than I have of your was stopped and that could only be stopped for work or have had occasion to examine it. These are two hours on successive Sundays. Now there but to be found among the C. S. and other pendulum were difficulties about oscillations of the wire and experts. about moisture which could not be overcome in Sincerely yours, that short time. But the method, which I first S. Newcomb proposed, was soon after applied successfully by others. Early in 1904, Garri~on~~requested Peirce to 2nd, that my determinations of the absolute give priority in the selection of his reviewing as- value of gravity were superior to those which immediately preceded them; if not to all previous signments to Newcomb's Reminiscences of an ones. Astronomer. Peirce's review67was highly com- 3rd, that that work involved my bringing to plimentary to Newcomb as a person and as a America the first authoritative line-metre. scientist. Garrison observed in a letter to Peirce 4th, that my work of comparing the length of a dated March 10, 1904, that it would please bar with the mean ruling of a glitter-plate, with Newcomb. a view to obtaining a check on secular changes in Apparently Peirce had included in the original the lengths of bars, had many merits. My version of the review an extended description of measurements of the deviation of the ray were the succession to the seat to which Newcomb had more accurate perhaps than any measurements of been elected in the French Academy of Sciences a large angle ever made. The peculiar comparator I invented and used enabled me to build up from in 1895.68 The next letteP9reveals one possible a double centimeter with a probable error of less source of Peirce's information regarding it. than a millionth part, which was quite a feat. Since the original review was too long, Garrison True, Rowland found a break in my glitter which deleted the discussion of the succession and, at vitiated my value of the wave length of light. Peirce's suggestion, submitted it to the Evening But that does not, I believe, affect my main Post. The Post printed that part of the f'dis- purpose, as long as that plate can be used again. embowelled Newcomb" as the leading article in Sth, that I proposed and used the method of two the Saturday Supplement on March 5 of the reversible pendulums in order to compare the same year. lengths of the yard and the metre, a new idea. Peirce called the article "French Academv of 6th, that I was for some months in charge of the Office of Weights and Measures under Hilgard as Sciences" and in it he traced the history of the Superintendent; and only left because his physical eight seats of the associe's e'trangers created in condition was such as to cause me embarrassment 1699. Since five of these places had not existed which I thought required me to quit Washington. previously, they were filled at the time by the election of Hartsoeker, Newton, Jacob and John May I name you as one of the five persons who Bernoulli, and Viviani, all more or less mathe- will answer questions as desired, and may I hope maticians. Peirce mentions the fact that Frank- that those answers will be favorable? lin, Rumford, and Newcomb were the only Yours very truly Americans honored by election and then, man by C. S. Peirce man, shows the succession of Newcomb to New-

Maloja, Switzerland Ibid. Garrison file of letters. Jan. 8. July 21, 1899 67 The Nation 78 (2021): 237, March 24, 1904. My dear Peirce: 68 Peirce knew something about the Academy at first hand. He had lectured there on June 14, 1880 on the When your letter of June 10 reached me I was at a value of gravity. mountain recess where correspondence was very Og Kewcomb Collection. Library of Congress. CAROLYN EISELE ton's seat. Indeed, this must have pleased New- Hill. Newcomb praised his work on every pos- comb very much. sible occasion and paid particular tribute to him in his Reminiscences of an Astron~rner'~with a Milford Pa 1904 Jan. 15. statement opening with the words, "Perhaps the My dear Newcomb : most eminent and interesting man associated Living here so far away from libraries, when I do with me during this period was Mr. George W. not know what book to borrow in order to get a Hill, who will easily rank as the greatest master given piece of information, I have to ask the good of mathematical astronomy during the last quar- offices of some person up in the subject. ter of the nineteenth century." Now when you received the extraordinary honor Peirce wrote a note on "Mr. G. W. Hill's of election as Foreign Associate of the French Moon Theory" in the Nation71 on October 19, Academy of Sciences, I think you must have had 1905. He lauded Hill's achievements in celestial the curiosity to know, lst, when the rule that there should be five was made and what previous number mechanics. He speaks of Hill's "staggering con- there had been, if any; and the whole history of the ception of an infinite determinant" and of his institution of five members. 2nd, I would like to success in "virtually solving a differential equa- see the complete list of all there ever have been tion of an infinite order." In doing so Hill down to today. 3rd any curious anecdotes con- probably did not "perceive that he was applying nected with the matter. Baconian reasoning to mathematics." Peirce I remember a book by one of the de Candolles speaks of Vol. I of the Collected Mathematical which I think gave the names of most of them. Works of George William Hill as bringing the But I may be mistaken and at any rate the book is "oldtime glow of exultant American feeling." thirty years old. By October 17, 1907, in a review72of Vol. IV If you know of such a book, I wish you would of the Peirce speaks less glow- kindly write to Dr. Herbert Putnam and tell him Collected Works, that is the book I am after. I will write to him ingly of Hill's work and begins the second para- that I would like him to send me De Candolle[']s graph with the statement that book and also a book or book[s] for the title or titles of which I have asked you to send to him. Then This science . . . is reduced to an art of performing you will have no responsibility in the matter but excessively intricate calculations. It must be a will merely furnish a piece of information to me to peculiar mind that can devote a lifetime to it; and whom everything relating to the history of science with less devotion there is no chance of being able is interesting. to improve it. I enclose some pages of a Syllabus that may possibly interest you. It certainly would had I This review brought a fiery retort from New- been able to print the whole. But though I limited comb in a letter written on October 17 to the the copies to 100, the money gave out when the Editor of the Nation and which appeared in the printer got so far. The whole would have been issue of October 31, 1907.73 Newcomb goes on about a hundred and fifty such pages. Logic is a to say, subject which does not amount to a row of pins unless it is treated systematically and it is necessary Now if you will slightly change your wording, and to push through a mass of stuff like the greater part say that through the labors of a series of investiga- of what I send and then through matter of extreme tors from the time of Newton to that of Hill the subtlety and difficulty of comprehension of another theory in question is being reduced to an art of kind, in the way of logical analysis, before one can performing intricate calculations, you will hit the treat in a solid way of the part of the matter that is truth. What gives significance to the work of Hill generally interesting. I have a great quantity and those in the same field is not their patience in ready for the press which I think, more keenly the performing calculations, but their ability to show more experience I gain, is of really great importance. how it is possible, by calculations within the power But there is no hope of its ever being printed. It of one man, to reach results which would have seems a fearful piece of egotism to study so deeply required the labor of many lives if the methods had for myself alone. But it is not my fault. not been invented. Any good computer, under Very truly capable supervision, can make the intricate calcula- C. S. Peirce tions. It is the method that costs.

70 Pp. 218-223. Among the computers on Newcomb's staff in ,, 81 (2103): 321. the Nautical Almanac Office, none was so highly 72 85 (2207): 355. prized by Newcomb as the mathematician G. W. 73 85 (2209) : 396. VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE

Appended to the above is a note which reads men do not like to tell you to your face that you are as follows: wrong. I don't like to do so myself; though it is not because offending you would inconvenience me By calculations, we did not mean numerical com- otherwise than by sadness. You may be sure that putations. Professor Newcomb expresses, as his nobody outside your group entertains a greater own dictum, what we intended to say. We have intelligent admiration for Hill and you than I do. already done justice to Dr. Hill's mathematical invention; but there is little of that in the fourth C. S. Peirce volume, which we have had under examination. The last letter75 in this collection emphasizes The Reviewer. the psychological as well as the optical factors of which the astronomer must be cognizant when The following 1ette1-74needs no further com- - drawing conclusions from telescopic observa- ment. tions. The problem is as old as the telescope P. 0. Milford Pa. 1907 Oct. 31 itself and became a major issue in the contro- versies following the work of Lowell76 and of My dear Newcomb : Schiaparelli on the canals of Mars. The Nation, which comes to me today, brings to Newcomb expressed his opinion77that, despite my mind your anger at my notice of the fourth Lowell's investigations of these basic factors as volume of Hill's works. I am sure I like you no they related to-his researches on Mars, further less for being angered and think it very natural. research was necessary. He mentions especially At the same time,-well, it is the facts of the the "psychology of vision, that branch of the situation and my stating those facts which irritate subject which relates to accuracy of conception you. You cannot put your finger upon anything I and estimate" and which is "an almost virgin said that is not perfectly true, and this is shown by field." Newcomb had devised certain experi- your saying that if I had said so and so, I should have been right. Now that is just what I did say in ments on visibilitv and visual intermetation substance. That the science of celestial mechanics which, in their approach to the probiem, were by its own perfectionment is now reduced to calcu- quite different from those of Lowell. He did lation. Of course, I did not mean numerical com- not question the subjective reality of the canal putation, but just that sort of art that there is in system but he felt that the proof of its objective Delaunay's method. It is not a method for finding reality would be incomplete until further research out any substantially new truth, but is a method on the process of visual inference had been made. for calculating a result according to well known These remarks were refuted point by point by principles. Lowell in a later issue of the journal78which had I never cast any slur on the men who do this sort publicized Newcomb's criticisms. Lowell refers of thing. I said they must have peculiar minds; but in some measure, this is true of any specialist. to the results of investigations made by H. That you and Hill and other theoretical astronomers Dennis Taylor which were "entirely opposite in find in the afternoon of life that their own successes corlclusion from what is supposed to exist by have rendered their science uninteresting to most Professor Newcomb." He points to Newcomb's people,-even to most mathematicians is distressing; use of transmitted light rather than reflected but it is a fact. Meantime, for those who keep on light in his experiments and holds rigidly to his I for one have an especial admiration, and the less own original position. interesting from any broad standpoint their work A one-page "Note on Preceding PaperU79by has been rendered, the more they deserve to be Newcomb follows the Lowell article, and a one- applauded; especially, since there is ground for hope page "Reply to Professor Newcomb's Note,"so that something important may eventually come of the work. gives Lowell the last thrust in this joust. In short, if you will reread what I said coolly, you In the letter which follows, Peirce refers to the will see, that I was neither mistaken nor was I results of the psychological experiments he per- wanting in esteem for the theoretical astronomers, formed in collaboration with J. Jastrow and and that that which my article contained that was disagreeable was due to my expressing truths that 76 Ibid. may be unpleasant to a man like you, but are truths 76 Mars and its canals by Lowell, Annals of the Lowell just the same for all that. Observatory 3 : 268-277. In short you read into my article a tone which 77 Astrophysical Journal 26 (1): 1-17, July, 1907. l8 26 really was not there. Your position is such that Ibid. (3): 131-141, Oct., 1907. Ibid. 26 (3): 141, Oct., 1907. 74 Newcomb Collection. Library of Congress. na Ibid. 26 (3) : 142, Oct., 1907. CAROLYN EISELE which he reported in a paper entitled "On Small but were stopped by the utter impossibility of getting Differences in Sensation."B1 a piece of Bristol board containing a square inch of uniform luminosity. No doubt this might have P. 0. Milford Pa 1908 Jan. 7 been overcome. But Jastrow and I were severally pressed with other work and we dropped the investi- My dear Newcomb : gation-contenting ourselves with what we had Needless to say that I have read your paper on done. I hope you will scrutinize the tables of com- the canals of Mars with great interest. For though parisons and the precautions which were more care- I have always thought that granting the reality of ful and studied and elaborate than the memoir what Lowell has observed it is far from proving the states. work of intelligent inhabitants, yet I have worked I became perfectly satisfied by the run of the enough with a longish equatorial, that of Cambridge, curve that it is merely a question of a sufficient for near 3 years constantly, and have worked on molition (my term for a volition prescinded from stellar photometry still more, and on other photo- all purposiveness) of direct attention, voluntary or metry and chromatics very much more and on allied involuntary, to enable a person to perceive the psychological problems to have great interest in difference between any two feelings, or sensations, anything such a man as you puts forward on the of different intensities or of different energies of subject. excitation under the same internal conditions. Our I have no doubt that the utterance of these words tables will enable you to form your own opinion as of caution from a man of such authority will do to this. But whatever conclusion you come to, let good to Lowell himself, not to speak of such men as me tell you that Jastrow without any previous Morse and Story who on the basis of no more than training except about two months experimentation a few weeks in Lowell's observatory, put forward with me upon pressure sensations, as soon as he opinions and arguments of no weight at all. Yet I took up photometry got as his average perceptible do not think your investigation is up to your ratio of luminosity (with no great exertion of atten- standard; and I will mention one or two points tion, though not in the fashion of the psychologist,) which seem to me weak, besides the general objection about 301/300 as well as I remember now. Cer- that it is all too narrow and does not bring light tainly far less than had been said. Now if this upon the psychological laws involved. principle,-a very broad psychological principle it In the first place I note that you accept as is, with a thousand important consequences,-be established the dictum of Gustav Theodor Fechner correct, there is a fundamental weakness in your that the least sensible ratio of light is 101/100. If work in that you give the reader no assurance that you will look in volume I11 Mem. of the U. S. Nat. you were worked up to the pitch of attention that Acad. of Sci. you will find a paper by me and my then Lowell and his assistants without doubt have been. student in logic Joseph Jastrow devoted to the Another small criticism, pointing the opposite question whether there is or is not such a thing as way is that you admit without qualification that a "Differenz-Schwelle" or least perceptible difference some canals there are; and for no better reason than of sensation; and of course, our conclusions being that they have been photographed. But what negative, can only be founded on that weakest form proof is there that the leading causes of illusion in of induction, which consists in inferring that a given telescopic observation are absent in examining the phenomenon does not exist because it has not doubtless nearly invisible lines on the photograph? appeared in a certain run of experience,-a form of In the present indispensableness of large expendi- inference that though per se very weak we are tures for astronomy we ought to give a little extra obliged to resort to in almost or quite every science. praise, to be understood as a tactful recognition of One does not believe in ghosts, or in meteorites, the force of character involved in the conquest of because critical investigation has never found them. needles1-eyedifficulty, to the work of a young man of Now you know that I am au fait at experimental great wealth who shows a real devotion to the science, psychology; and I have not an exalted opinion of beyond what we should to the young fellow who is psychologists as scientific observers. Jastrow and I presumably working for an assistant-professorship in began with sensations of pressure and for a reason some college; and we cannot help admiring Lowell's I will shortly mention we ended there. At once, work. He and all should be made to feel that it is using such precautions as any astronomer would use bound to advance astronomy and all science not a in observing faint nebulas, without any practice we little, whether his observations turn out to be obser- found that if there were any least perceptible ratio vations of the real Mars, or whether they turn out of pressure, it was twenty or thirty times nearer to be observations of illusions partly based on instru- unity than the psychologists had made it to be. mental imperfections; in the former case infusing into We afterward tried to do the same thing for light; human veins a new motive for pursuing astronomical study, and in the latter case enforcing the lesson that 81 National Academy of Science Memoirs 3 :75-83. Read percepts are not by any means as Karl Pearson calls on Oct. 17, 1884. them, the "first impressions of sense," but are VOL. 101, NO. 5, 19571 PEIRCE-NEWCOMB CORRESPONDENCE results of "Schluss-Verfahrungen" of our deeper might do me and those to whom I may be of any lying consciousness, closely and in much detail, use more service than you would think. analogous to the different varieties of critical induc- Very faithfully tion, yet having certain general characters of their C. S. Peirce own, which must be brought to light before we can make the best use of the finer kinds of observations. I confess that your reasoning about the visibility These letters are not only a valuable addition of a line appears to me to involve a momentary to scientific Americana of their period but serve forgetfulness of the fact that not merely what we also to deepen our acquaintance with two out- call "illusions," that is, perceptual inferences that standing personalities of that epoch. That are refuted by others, but also all results of percep- these men left an indelible imprint on the intel- tion, have processes behind them which have all the lectual life of America cannot be denied. They characters of reasoning except that of being conscious came by their intellectual powers and earned and, thereby self-controlled, and thereby being critical. You seem to me tacitly to assume that the personal distinction in such strikingly different process of perceiving a line is considerably simpler ways that one is reminded again of the criterion than my observations have led me to surmise that it by which Peirce would judge a man's greatness. is, (and in some measure to test my surmise). How- It was formulated to read, ever, I must not put much stress on that, since I have no leisure to expound my theory. Who, for instance, shall we say are the great men of By the way, since Garrison's death, an important science? . . . Some hold that they are fashioned of item of bread and butter has been almost wholly the most ordinary clay, and that only their rearing cut off from me by my no longer being invited to and environment, conjoined with fortunate oppor- write for the Nation. There is a person on the staff tunities, make them what they are. The heaviest who I believe thinks himself a very superior philoso- weight, intellectually, among these writers, main- pher,-as most students of the subject severally tains, on the other hand, that circumstances are as rate themselves. It is very singular and seems to powerless to suppress the great man as they would show that their methods of determining their valua- be to subject a human being to a nation of dogs. tions are at fault, since there is no room for half of But it was only the blundering Malvolio who got them to be right. I thought it likely, too, that the notion that some are born great; the sentence of your letter about my remarks on Celestial Dynamics the astute Maria was: "Some are become great, increased their indisposition to trust to me. The some atcheeves greatnesse, and some have greatnesse whole difference between us seems to me to have con- thrust uppon em." . . . My opinion will, I fear, be sisted in my using the word "calculation" in a well- set down by some intellectual men as foolishness, established sense considerably broader than your though it has not been lightly formed, nor without interpretation of what I said. I meant that there long years of experimentation-that the way to was no present prospect of important general judge of whether a man was great or not is to put physical discoveries, such as those of Keppler and aside all analysis, to contemplate attentively his Newton flowing from the work (unless perhaps life and works, and then to look into one's heart centuries hence) nor even of any mathematical and estimate the impression one finds to have theory comparable for instance with Laplace's func- been made. . . . The great man is the impressive tions in importance. If anything of the kind is to personality; and the question whether he is great is come from your study of the movement of Mars, a question of impression.82 tant mieux! LVell, I only introduce the matter to say that if you are disposed to do me such a good The century's great men in science, The Evening Post, turn, a word dropped in conversation with Lamont Jan. 12, 1901.