# Popular Science Monthly/Volume 18/March 1881/Sketch of Professor Benjamin Peirce

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BENJAMIN PEIRCE.
* SKETCH OF PROFESSOR BENJAMIN PEIRCE.
THIS illustrious American mathematician and astronomer died in Boston, October 6, 1880, in the seventy-second year of his age. He was born in Salem, Massachusetts, April 4, 1809. He was graduated at Harvard College at the age of twenty. His father was a graduate of the same institution, and died its librarian. He was appointed tutor in 1831, University Professor of Mathematics and Natural Philosophy in 1833, and Perkins Professor of Astronomy and Mathematics in 1842, and was directly connected with the faculty of the college for forty-nine years. He was a member of all the learned societies in this country and in Europe, was elected Fellow of the Royal Society of England in 1857, and in 1867 received the degree of LL.D. from Harvard University.
We take the liberty of quoting from that excellent periodical, "The Harvard Register," for May, 1880 (to the courtesy of whose editor we are also indebted for the excellent likeness herewith presented), the following account of Professor Peirce's character and work, written, it will be observed, before his death, by Dr. Thomas Hill, ex-President of Harvard University:
"From 1836 to 1846 he issued a series of text-books on geometry, trigonometry, algebra, and 'curves, functions, and forces.' They were so full of novelties that they never became widely popular, except, perhaps, the trigonometry; but they have had a permanent influence upon mathematical teaching in this country; most of their novelties have now become commonplaces in all text-books. The introduction ​of infinitesimals or of limits into elementary books; the recognition of direction as a fundamental idea; the use of Hassler's definition of a sine as an arithmetical quotient, free from entangling alliance with the size of the triangle; the similar deliverance of the expression of derivative functions and differential coefficients from the superfluous introduction of infinitesimals; the fearless and avowed introduction of new axioms, when confinement to Euclid's made a demonstration long and tedious—in one or two of these points European writers moved simultaneously with Peirce, but in all he was an independent inventor, and nearly all are now generally adopted.
"All his writings are characterized by singular directness and conciseness, and particularly by a happy choice of notation—a point of great importance to the mathematician, lessening not only his mechanical labor in writing, but also his intellectual labor in grasping and handling the difficult conceptions of his science.
"His text-books were also complained of for their condensation, as being therefore obscure; but, under competent teachers, their brevity was the cause of their superior lucidity. In the Waltham High School his books were used for many years, and the graduates attained thereby a clearer and more useful applicable knowledge of mathematics than was given at any other high school in this country; nor did they find any difficulty in mastering even the demonstration of Arbogast's Polynomial Theorem, as presented by Peirce. The latter half of the volume on the Integral Calculus, full of marks of a great analytical genius, is the only part of all his text-books really too difficult for students of average ability.
"Gill's 'Mathematical Miscellany' contained many contributions which showed in a singular light the Harvard professor's power. For example, in the issues for May and November, 1839, he solved, by a system of coordinates of his own devising, several problems concerning the involutes and evolutes of curves, which would probably have proved impregnable by any other mode of approach.
"During the year 1842, Professors Peirce and Lovering published a 'Cambridge Miscellany of Mathematics and Physics,' in which Peirce gave an analytical solution of the motion of a top, a criticism of Espy's theory of storms, etc. About the same time he adapted the epicycles of Hipparchus to the analytical forms of modern science; and the method was used by Lovering in meteorological discussions communicated to the American Academy.
"The comet of 1843 gave Professor Peirce the opportunity, by a few strikinor lectures in Boston, to arouse an interest which led to the foundation of the observatory at Cambridge; and, by his discussions of the orbit with Sears C. Walker, he and that remarkable computer were brought to mutual acquaintance, and prepared for the still more important services to astronomy which they rendered after the discovery of Neptune. This planet was discovered in September, 1846, in ​consequence of the request of Leverrier to Galle that he should search the zodiac, in the neighborhood of longitude 325,° for a theoretical cause of certain perturbations of Uranus. But Peirce showed that the discovery was a happy accident; not that Leverrier's calculations had not been exact, and wonderfully laborious, and deserving of the highest honor; but because there were, in fact, two very different solutions of the perturbations of Uranus possible: Leverrier had correctly calculated one, but the actual planet in the sky solved the other; and the actual planet and Leverrier's ideal one lay in the same direction from the earth only in 1846. Peirce's labors upon this problem, while showing him to be the peer of any astronomer, were in no way directed against Leverrier's fame as a mathematician; on the contrary, he testified in the strongest manner that he had examined and verified Leverrier's labors sufficiently to establish their marvelous accuracy and minuteness, as well as their herculean amount.
"A few years later, 1851 to 1855, Peirce published the remarkable results of his labors upon Saturn's rings. Professor G. P, Bond had seen the ring divide itself and reunite, and had thereby been led to show by computation from Laplace's formulae that the ring could not be solid. Upon this Peirce investigated the problem anew, and showed that the ring, if fluid, could not be sustained by the planet; that satellites could not sustain a solid ring, but that sufficiently large and numerous satellites could sustain a fluid ring, and that the actual satellites of Saturn are sufficient.
"In 1849 he was appointed consulting astronomer to the 'American Ephemeris and Nautical Almanac,' and rendered efficient service in bringing that publication to its condition of honorable authority, particularly in the lunar tables which he furnished, in his treatment of Neptune, and various methods of computation. He also assisted Professor Bache in the Coast Survey, and was, for many years, of great service in that important national work before he was himself appointed superintendent in 1867. His calculations of the occultations of the Pleiades were very laborious and exact, and furnished an accurate means of studying the form, both of the earth and her satellite. His criterion for rejecting doubtful observations is an ingenious and valuable extension of the law of probabilities to its own correction. His detection of the mental error of lurking personal preferences for individual digits is a curious specimen of that acuteness of observation which characterizes his own mind.
"He held the office of Superintendent of the Coast Survey from 1867 to 1874. Coming after such able men as Hassler and Bache, to an office which required not only familiarity with mathematics and physics, but also great knowledge of men and executive ability, he was not found wanting, but showed that the theory of the Stoics will sometimes hold good to-day—the really great man shows himself great by any and every standard. The Coast Survey has, since the year 1845, ​steadily advanced in public favor, and its work commands the highest respect among all men competent to judge throughout the world, as being not only of direct service to the nation, but as making constant valuable additions to science.
"Many monographs, bearing the marks of Peirce's individuality and peculiar power, have been read by him before various academies, societies, and institutions; but only the results of most of them have ever been furnished for publication. Among these may be mentioned an investigation of the forms of stable equilibrium for a fluid in an extensible sack floating in another fluid, being an a priori embryology. Also, the motions of a billiard-ball, an instance in nature of discontinuity, when the ball leaves its curve, and goes on a tangent; another, the motion of a sling, curious from the immense variety of forms comprised under exceedingly simple uniform conditions.
"In 1857 he published a volume, summing up the most valuable and most brilliant results of analytical mechanics, interspersing them with original results of his own labor. A year or two later an American student in Germany asked one of the most eminent professors there, what books he would recommend on analytical mechanics: the answer was instantaneous, 'There is nothing fresher and nothing more valuable than your own Peirce's recent quarto.' In this volume occurs a singular instance of a characteristic which I have already mentioned. Peirce assumes as self-evident that a line which is wholly contained upon a limited surface, but which has neither beginning nor end on that surface, must be a curve reentering upon itself. By means of this hyper-Euclidean axiom he reduces a demonstration, which would otherwise occupy half a dozen pages, to a dozen lines.
"In 1870, through the 'labors of love' of persons engaged on the Coast Survey, an edition of a hundred lithographed copies was published, of certain communications to the National Academy upon 'Linear Associative Algebra.' In 1852 Hamilton, of Dublin, had published his wonderful volume on quaternions; and this had been followed by various other attempts to create an algebra more useful in geometrical and physical research than the coördinates of Descartes. Ordinary algebra deals only with quantitative relations, and the object of the arithmetic of lines and of Cartesian coördinates had been to reduce distances and directions to a comparison of quantity. But Hamilton introduced quality also; and his algebra employed the dimensions of space, unchanged and essentially diverse, in computation. His imitators and followers had not succeeded in improving or in really adding to his methods. But Peirce, in these communications to the Academy, attacks the problem, according to his wont, with astonishing breadth of view and boldness of plan. He begins with a definition of mathematics, shows the variety of processes included in his definition, passes then to its symbols, shows the nature of qualitative and of quantitative algebras, and of those which combine the two, ​and says he will investigate the general subject of algebra. First, he limits himself in this volume to algebras handling less than seven distinct qualities—that is, not exceeding six. The notation is then discussed, and the necessary enlargements and modifications of the algebraic signs and symbols are clearly defined. The distributive and associative principles in multiplication are adopted but not the commutative; and he confines himself to linear algebras—that is, to those in which every expression is reducible to an algebraic sum of terms each expressive of a single quality. After a full discussion of the general results which must be found in all algebras under these conditions, he begins with single algebras, then double, then triple, and so on up to sextuple, making nearly a hundred algebras which he shows to be possible, and of which he gives the great features. There are almost no comments upon them; and it is only by a patient examination for himself that the reader discovers that, of all these numerous algebras, only three have ever been heard of before. First, of the two single algebras we have one, which is the common algebra, including its simpler form of arithmetic. Secondly, of the three double algebras, we have one, viz., the calculus of Leibnitz and Newton. Thirdly, of over twenty quadruple algebras, only one has been used, the quaternions of Hamilton. Such is a brief abstract of this book of marvelous prophecy. The most noteworthy things which he has done since its publication are a course of Lowell lectures, given about a year ago, on 'Ideality in Science,' and a series of communications to the American Academy, which, it is understood, is still to be continued. In the Lowell lectures he embodies many of his views on philosophy and religion which are peculiarly dear to him, and are always listened to with profound interest, even by those of less religious nature. In the communications to the Academy he is discussing, with all his wonted power, questions of cosmical physics, and particularly theories concerning the source and supply of the sun's heat.
"While Professor Peirce has the tenacity of grasp and power of endurance which enable him to make the most intricate and tedious numerical computations, he is still more distinguished by intensity and fervor of action in every part of his nature, an enthusiasm for whatever is noble and beautiful in the world or in art, in fiction or real life; an exalted moral strength and purity; a glowing imagination which soars into the seventh heavens; an insight and a keenness of external observation which make the atom as grand to him as a planet; a depth of reverence which exalts him while he abases himself."