496 



NA TURE 



{Sept. 20, 1883 



prominent way. Tables were calculated of the logarithms of 

 numbers, and of those of the trigonometrical functions. 



The circular function and the logarithm were thus invented 

 each for a practical purpose, separately and without any proper 

 connection with each other. The functions are connected 

 through the theory of imaginaries, and form together a group of 

 the utmost importance throughout mathematics : but this is 

 mathematical theory ; the obligation of mathematics is for the 

 discovery of the functions. 



Forms of spirals presented themselves in Greek architecture, 

 and the curves were considered mathematically by Archimedes ; 

 the Greek geometers invented some other curves, more or less 

 interesting, but recondite enough in their origin. A curve which 

 mi^ht have presented itself to anybody, that described by a 

 pjint in the circumference of a rolling carriage wheel, was first 

 noticed by Mersenne in 1615, and is the curve afterwards con- 

 sidered by Roberval, Pascal, and others, under the name of the 

 Koulette, otherwi-e the Cycloid. Pascal (1623-1662) wrote at 

 the age of seventeen his " Essais pour les Coniques," in seven 

 short pag:s, full of new views on these curves, and in which he 

 give^, in a paragraph of eight lines, his theory of the inscribed 

 hexagon. 



Kepler (1571-1630) by his empirical determination of the laws 

 of planetary m ition, brought into connection with astronomy one 

 of the forms of conic, the ellipse, and established a foundation 

 for the theory of gravitation. Contemporaiy with him, for 

 most of his life, we have Galileo (1564- 1642), the founder of 

 the science of dynamics ; and closely following upon Galileo, we 

 have Isaac Newton (1643-1727) : the "Philosophise naturalis 

 Principia Mathematica," known as the " Principia," was first 

 published in 1687. 



The physical, statical, or dynamical questions which presented 

 themselves before the publication of the " Principia" were of no 

 particular mathematical difficulty, but it is quite otherwise with 

 the crowd cf interesting questions arising out of the theory of 

 gravitation, and which, in becoming the subject of mathematical 

 investigation, have c intributed very much to the advance of 

 mathematics. We have the problem of two bodies, or what is 

 the same thing, that of the motion of a particle abDut a fixed 

 centre of force, for any law of force ; we have also the (mathe- 

 matically vtry interesting) problem of the motion of a body 

 attracted to two or more fixed centres of force ; then, next 

 preceding that of the actual solar system — the problem of three 

 bodies; this has ever been and is far beyond the power of 

 mathematics, and it is in the lunar and planetary theories 

 replaced by what is mathematically a different problem, that of 

 the motion of a bo.ly under the action of a principal central 

 force and a disturbing force ; or (in one mode of treat- 

 ment) by the problem of disturbed elliptic motion. I would 

 remark that we have here an instance in which an astro- 

 nomical fact, the observed slow variation of the orbit of 

 a planet, has directly suggested a mathematical method, applied 

 to other dynamical problems, and which is the basis of very 

 extensive modern investigations in regard to systems of differ- 

 ential equations. Again, immediately arising out of the theory 

 of gravitation, we have the problem of finding the attraction of 

 a solid body of any given form upon a particle, solved by Newton 

 in the case of a homogeneous sphere, but which is far more 

 difficult in the next succeeding cases of the spheroid of revolution 

 (very ably treated by Maclaurin) and of the ellipsoid of three 

 unequal axes : there is perhaps no problem of mathematics which 

 has been treated by as great a variety of methods, or has given 

 rise to so much interesting investigation as this last problem of 

 the attraction of an ellipsoid upon an interior or exterior point. 

 It was a dynamical problem, that of vibrating strings, by which 

 Lagrange was led to the theory of the representation of a function 

 as tne sum of a series of multiple sines and cosines ; and con- 

 nected with this we have the expansions in terms of Legendre's 

 functions P„, suggested to him by the question just referred to of 

 the attraction of an elli soid ; the subsequent investigations of 

 Laplace on the attractions of bodies differing slightly from the 

 sphere led to the functions of two variables called Laplace's 

 functions. I have been speaking of ellipsoids, but the general 

 theory is that of attractions, which has become a very wide 

 branch of modern mathematics ; associated with it we have in 

 particular the names of Gauss, Lejeune-Uirichlet, and Green ; and 

 I must not omit to mention that the theory is now one relating to 

 ((-dimensional space. Another great problem of celestial me- 

 chanics, that of the motion of the earth about its centre of gravity, 

 ■ n the most simple case, that of a body not acted upon by any 

 ,orces, is a very interesting one in the mathematical point of view. 



I may mention a few other instances where a practical or 

 physical question has connected itself with the development of 

 mathematical theory. I have spoken of two map projections — 

 the stereographic, dating from Ptolemy ; and Mercator's pro- 

 jection, invented by Edward Wright about the year 1600 : each 

 of these, as a particular case of the orthomorphic projection, 

 belongs to the theory of the geometrical representation of an 

 imaginary variable. I have spoken also of perspective, and (in 

 an omitted paragraph) of the representation of solid figures em- 

 ployed in Monge's descriptive geometry. Monge, it is well 

 ki own, is the author of the geometrical theory of the curvature 

 of surfaces and of curves of curvature : he was led to this theory 

 by a problem of earthwork — from a given area, covered with 

 earth of uniform thickness, to carry the earth and distribute it 

 over an equal given area, with the least amount of cartage. For 

 the solution of the corresponding problem in solid geometry he 

 had to consider the intersecting normals of a surface, and so 

 arrived at the curves of curvature (see his " Memoire sur les 

 Deblais et les Remblais," Mem. de /' "Acad, , 1781). The normals 

 of a surface are, again, a particular case of a doubly infinite 

 system of lines, and are so connected with the modern theories of 

 congruences and complexes. 



The undulalory theory of light led to Fresnel's wave-surface, a 

 surface of the fourth order, by far the most interesting one which 

 had then presented itself. A geometrical property of this surface, 

 that of having tangent planes each touching it along a plane 

 curve (in fact, a circle), gave to Sir W. R. Hamilton the theory 

 of conical refraction. The wave-surface is now regarded in 

 geometry as a particular case of Rummer's quartic surface, with 

 sixteen conical points and sixteen singular tangent planes. 



My imperfect acquaintance as well with the mathematics as 

 the physics prevents me from speaking of the benefits which the 

 theory of Partial Differential Equations has received from the 

 hydrodynamical theory of vortex motion, and from the great 

 physical theories of electricity, magnetism, and energy. 



It is difficult to give an idea of the vast extent of modern 

 mathematics. This word "extent" is not the right one: I 

 mean extent crowded with beautiful detail — not an extent of 

 mere uniformity, such as an objectless plain, but of a tract of 

 beautiful country seen at first in the distance, but which will 

 bear to be rambled through and studied in every detail of hill- 

 side and valley, stream, rock, wood, and flower. But, as for 

 anything else, so for a mathematical theory — beauty can be per- 

 ceived, but not explained. As for mere extent, I might illus- 

 trate this by speaking of the dates at which some of the great 

 extensions have been made in several branches of mathematical 

 science. 



And in fact, in the Address as written, 1 speak at considerable 

 length of the extensions in geometry since the time of Descartes, 

 and in other specified subjects since the commencement of the 

 century : the.-e subjects are the general theory of the function of 

 an imaginary variable ; the leading known functions, viz. the 

 elliptic and single theta-functions and the Abelian and multiple 

 theta-functions ; the Theory of Equations and the Theory of 

 Numbers. I refer also to some theories outside of ordinary 

 mathematics : the multiple algebra or linear associative algebra 

 of the late Benjamin Peirce ; the theory of Argand, Warren, 

 and Peacock in regard to imaginaries in plane geometry; Sir W. 

 R. Hamilton's quaternions, Clifford's biquaternions, the theories 

 developed in Grassmann's " Ausdehnungslehre," with recent 

 extensions thereof to non-Euclidian space by Mr. Homersham 

 Cox; also Boole's "Mathematical Logic," and a work con- 

 nected with logic, but primarily mathematical and of the highest 

 importance, Schubert's "Abzahlende Geometrie " (1S78). I 

 remark that all this in regard to theories outside of ordinary 

 mathematics is still on the text of the vast extent of modern 

 mathematics. 



In conclusion I would say that mathematics have steadily 

 advanced from the time of the Greek geometers. Nothing is 

 lost or wasted ; the achievements of Euclid, Archimedes, and 

 Apollonius are as admirable now as they were in their own days. 

 Descartes' method of coordinates is a possession for ever. But 

 mathematics have never been cultivated more zealously and dili- 

 gently, or with greater success, than in this century — in the last 

 half of it, or at the present time : the advances made have been 

 enormous, the actual field is boundless, the future full of hope. 

 In regard to pure mathematics we may most confidently say : — 



*' Yet I doubt not through the ages one increasing purpose runs. 

 And the thoughts of men are widened with the process of the suns." 



