504 



SCIENCE. 



[Vol. II., No. 30. 



yented by Edward Wright about the year 1600. Each 

 of tbese, as a particular case of the orthomorphic pro- 

 jection, belongs to the theory of the geometrical rep- 

 resentation of an imaginary variable. I have spoken 

 also of perspective, and (in an omitted paragraph) of 

 the representation of solid figures employed in 

 Monge's descriptive geometry. Monge, it is well 

 known, 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 unifprm 

 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 I'acacl., 1781). The normals of a 

 surface are, again, a particular case of a doubly infi- 

 nite system of lines, and are so connected with the 

 modern theories of congruences and complexes. 



The undulatory theory of light led to Fresnel's 

 wave-surface, — a surface of the fourth order, by far 

 the mo>t interesting one which had then presented it- 

 self. 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. E. Hamilton 

 the theory of conical refraction. The wave-surface 

 is now regarded in geometry as a particular case of 

 Kummer's quartic surface, with sixteen conical points 

 and sixteen singular tangent planes. 



My imperfect acquaintance, as weH with the mathe- 

 matics as the physics, prevents me from speaking of 

 the benefits which the theory of partial differential 

 equations has received from the hydrodynamical the- 

 ory 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 de- 

 tail, — 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 

 hillside and valley, stream, rock, wood, and flower. 

 But as for anything else, so for a mathematical the- 

 ory, — beauty can be perceived, but not explained. 

 As for mere extent, I might illustrate this by speak- 

 ing of the dates at which some of the great ex- 

 tensions have been made in several branches of 

 mathematical science. 



And, in fact, in the address as written, I 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. 

 These subjects are the general theory of the function 

 of an imaginary variable; the leading known func- 

 tions, 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 Argaud, War- 



ren, and Peacock, in regard to imaginaries in plane 

 geometry; Sir W. R. Hamilton's quaternions; Clif- 

 ford's biquaternions; the theories developed in Grass- 

 mann's ' Ausdehnungslehre,' with recent extensions 

 thereof to non-Euclidian space by Mr.^Homershara 

 Cox; also Boole's ' Mathematical logic,' and a work 

 connected with logic, but primarily mathematical and 

 of the highest importance, Shubert's ' Abzahlende 

 geometrie ' ( 1S78). I remark that all this in regard to 

 theories outside of ordinary matliematics 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 geome- 

 ters. Nothing is lost or wasted. The achievements 

 of Euclid, Archimedes, and Apollonius, are as admir- 

 able now as they were in their own days. Descartes' 

 method of co-ordinates is a possession forever. But 

 mathematics has never been cultivated more zealous- 

 ly and diligently, 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 llie process of the 



THE ENDOWMENT OF BIOLOGICAL 

 RESEARCH.^ 



It has become the custom for the presidents of the 

 various sections of this association to open the pro- 

 ceedings of the departments with the chairmanship 

 of which they are charged by formal addresses. In 

 reflecting on the topics which it might be desirable 

 for me to bring under your notice, as your president, 

 on tlie present occasion, it has occurred to me that I 

 might use this opportunity most fitly by departing 

 somewhat from the prevailing custom of reviewing 

 _ the progress of science in some special direction dur- 

 ing the past year, and that, instead of placing before 

 you a summary of the results recently obtained by 

 the investigations of biologists in this or that line of 

 inquiry, I might ask your attention, and that of the 

 external public (who are wont to give some kindly 

 consideration to the opinions expressed on these oc- 

 casions) to a matter which is even more directly con- 

 nected with the avowed object of our association; 

 namely, ' the advancement of science.' I propose to 

 place before you a few observations upon the pro- 

 vision which exists in this country for the advance- 

 ment of that branch of science to which section D 

 is dedicated; namely, biology. 



I am aware that it is usual for those who speak of 

 men of science and their pursuits to ignore altogether 

 such sordid topics as the one which I have chosen to 

 bring forward. A certain pride, on the one hand, and 

 a willing acquiescence, on the other hand, usually 

 prevent those who are professionally concerned with 



1 An address to the biological section of the British associa. 

 tion. By Prof. E. Uay I.anliester, M.A., F.R.S., F.L.S., presi- 

 dent of the section. From advance copy kindly furnished by 

 the editor of iVaiwre. 



