(85 



SCIENCE. 



[Vol. XVII 1. No. 452 



branch of scieace, when a few men are working here and 

 there in comparative isolation. This variety continued to a 

 considerable extent throughout the seventeenth century. 



In this century we arrive at a new era in mathematical 

 development. This was brought about by the application of 

 algebra to geometry by Descartes in the early part, and the 

 discoverj' of the differential calculus by Newton and Leib- 

 nitz independently in the latter part of the century. Alge- 

 bra had been used in connection with geometry before Des- 

 cartes, but to him was due the discovery of the fact, that, if 

 the position of a point be given by co-ordinates, then any 

 equation involving those co-ordinates will represent some locus 

 all of whose properties are contained implicitly in the equa- 

 tion, and may be deduced therefrom by ordinary algebraic 

 operations. 



Descartes initiated the custom, which has become fixed, 

 of using the first letters of the alphabet for known and the 

 last for unknown quantities. He also appears to have been 

 the first to perceive that one general proof is sufficient for 

 any proposition algebraically treated, the different cases 

 which might arise by different arrangements of the equations 

 being covered by the possibility of any letter representing a 

 negative as well as a positive quantity, i.e., he distinguished 

 the intrinsic sign of a quantity or symbol. Hitherto it had 

 been considered .necessary to treat separately the forms of 

 the quadratic ax^ -\- hx ^= c, ax^ ^ bx -\- c, etc., which was 

 a natural result of the geometric method of arriving at the 

 solution. Descartes also introduced our present notation for 

 powei's, taking his exponents, however, only as positive and 

 integral. 



Contemporaneously with Descartes, Cavalieri, in Italy, 

 applied the so-called "method of indivisibles" to the com- 

 putation of areas, volumes, etc., a process which gave way 

 early in the eighteenth century to the integral calculus. At 

 this time, also, the beginnings of the mathematical theory of 

 probabilities were made by Pascal and Fermat in the solu- 

 tion of a certain problem which had been proposed. 



A tremendous impulse was given to all branches of math- 

 ematics in the latter part of the seventeenth century by the 

 genius of Newton. Besides his epoch-making discovery of 

 the " theory of fluxions," or differential calculus, he contrib- 

 uted to algebraic science the idea of the general exponent or 

 nVn power {n being positive, negative, integral, or fractional), 

 the binomial theorem, and a considerable part of the theory 

 of equations. 



To Leibnitz we owe the present notation of the differential 

 calculus, the introduction of the terms "co-ordinates" and 

 "axes of co-ordinates," and suggestions as to the use of in- 

 determinate coefficients and determinants, which, though 

 not developed by him, led, in the hands of others, to impor- 

 tant results. 



Jacob Bernoulli developed the fundamental principles of 

 the calculus of probabilities, and made the first systematic at- 

 tempts to construct an integral calculus. His brother John 

 dev^eloped the exponential calculus, and treated trigonometry 

 independently as a branch of analysis, it having been previ- 

 ously regarded as an adjunct of astronomy. The possibility 

 of a calculus of operations was first recognized by Brook 

 Taylor, after whom "Taylor's theorem" is named. De 

 Moivre contributed to the discussion of imaginaries the im- 

 portant theorem which bears his name. In 1748 MacLaurin 

 published an algebra which contained the results of some 

 earlier papers published by him, among others one on the 

 number of imaginary roots of an equation, and one on the 

 determination of equal roots by means of the first derivative. 



In the latter part of the eighteenth and beginning of the 

 nineteenth centuries mathematical advancement was rapid 

 under the powerful hands of Euler, Lagrange, Laplace, and 

 Legend re. To these great men we owe the calculus of vari- 

 ations, the initial discussion of the calculus of imaginaries 

 (which was afterw^irds systematized and developed by Gauss, 

 Cauchy, and others), the treatment of determinants, contri- 

 butions to the theory of equations, a large part of the inte- 

 gral calculus and differential equations, the development of 

 the theory of probabilities, the treatment of elliptic func- 

 tions, the method of least squares, and the specially algebraic 

 treatment of the theory of numbers. In this list are 

 included only those things which are of an algebraic 

 nature. 



We have now reached the beginning of our own century, 

 in which the advance has been so rapid in all directions as 

 to preclude more than a mere indication of some of the lines 

 along which this has taken place, without any attempt at an 

 enumeration of the illustrious names of those who have so 

 magnificently carried forward the work. 



The theory of equations has been perfected by the full use 

 of the complex unit a -j- hi, forming thus, in the words of 

 Cayley, a " universe complete in itself, such that, starting 

 in it, we are never led out of it." We have, in fact, a dou- 

 ble algebra as the instrument for the complete treatment of 

 all higher analysis, except that in which one of higher mul- 

 tiplicity is used. The field of quantios has been brilliantly 

 cultivated by Cayley, Sylvester, and others. The theory of 

 matrices has been developed by Cayley, and it was shown 

 by Professor J. Willard Gibbs, in his vice-presidential ad- 

 dress before this section at the Buffalo meeting in 1886, that 

 the simple and natural expression of this theory is in the 

 language of multiple algebra. The cp of Hamilton is a 

 matrix of the third order, and the Q of Grassmann a matrix 

 of the nth order. 



In the treatment of differential equations we have an al- 

 gebra of operations, due primarily to George Boole, carried 

 to a high degree of perfection', in which the symbol of differ- 

 entiation is treated precisely as if it were a real quantity. 

 In fact, we have come to regard scalar multiplication simply 

 as a particular case in the calculus of operations which cov- 

 ers every possible case of the effect of one symbol upon an- 

 other in producing some change in it. A further extension 

 of this same idea we have in the algebra of logic, invented 

 by the same author, and cultivated and extended by others 

 since his time. 



In conclusion, I propose to sketch briefly the development 

 of the idea of a multiplicity of fundamental units, which is 

 pervading more and more the mathematical thought of the 

 day. This proceeded along two distinct lines, one arising 

 from the interpi'etation of the imaginary, \/ — 1, and the 

 other entirely independent of this symbol or operation. 



The first attempt to give a geometric meaning to the ex- 

 pression a -\- hi appears to be due to Wallis in 1685, who 

 proposed to construct the imaginary roots of a quadratic by 

 going out of the line on which they would have been laid 

 off if real. In 1804 the Abbe Buee devised the now accepted 

 representation by laying off the terms containing i as a fac- 

 tor, at right angles to the others, and showed how to add 

 and subtract such expressions as a -|- hi. At about the same 

 time Argand published independently the same idea, and 

 still further developed it. The concept of a directed quan- 

 tity as represented by an algebraic symbol was thus necessa- 

 rily arrived at. Gauss, Cauchy, and others have elaborated 

 the complex unit more especially in the theory of numbers. 



