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
