October 12, 188;^.] 



SCIENCE. 



503 



Copernicus, Reinholc), Miiurolycus, Victa, and many 

 others. The tabulations of the functions tangent 

 and secant are due to Reinliold and Maurolycus re- 

 spectively. 



Logarithms were invented, not exclusively with 

 reference to the calculation of trigononielrical tables, 

 but in order to facilitate numerical calculations gen- 

 erally. The invention is due to John Napier of Mer- 

 chiston, who died in 1618, at sixty-seven years of age. 

 The notion was based upon refined mathematical rea- 

 soning on the comparison of the spaces descriljed by 

 two points ; the one moving with a uniform velocity, 

 the other with a velocity varying according to a given 

 law. It is to be observed that Xapier's logarithms 

 were nearly, but not exactly, those which are now 

 called, sometimes Napierian, but more usually hy- 

 perbolic logarithms, those to the basee; and that 

 the change to the Kise 10 (the great step by which 

 the invention was perfected for the object in view) 

 was indicated by Xapier, but actually made by Henry 

 Briggs. afterwards Savilian professor at Oxford (d. 

 1630). But it is the hyperbolic logarithm which is 

 mathematically important. The direct function e^', 

 or exp. X, which has for its inverse the hyperbolic log- 

 arithm, presented itself, -but not in a 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 im- 

 aginaries, and form together a group of the utmost 

 importance throughout mathematics : but this is math- 

 . emalical 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 mathe- 

 matically by Archimedes. The Greek geometers in- 

 vented some other curves more or less interesting, 

 but recondite enough in their origin. A curve which 

 might have presented itself to anybody, that described 

 by a point in the circumference of a rolling carriage- 

 wheel, was first noticed by Merseniie in 1015, and is 

 the curve afterwards considered by Roberval, Pascal, 

 and others, under the name of the roulette, other- 

 wise the cycloid. Pascal (1623-62) wrote, at the age 

 of seventeen, his ' Essais pour les coniques ' in seven 

 short pages, full of new views on these curves, and 

 in which he gives, in a paragraph of eight lines, his 

 theory of the inscribed hexagon. 



Kepler (1.571-1630), by his empirical determination 

 of the laws of planetary motion, brought into con- 

 nection with astronomy one of the forms of conic, 

 the ellipse, and established a foundation for the theo- 

 ry of gravitation. Contemporary with him for most 

 of his life, we have Galileo (1.504-1042), the founder 

 of the science of dynamics ; and closely following 

 upon Galileo, we have Isaac Xewton (1643-1727). 

 The ' Philosophiae naturalis principia mathemalica,' 

 known as the ' Principia,' was first published in 1087. 



The physical, statical, or dynamical questions 

 which presented themselves before the publication 

 of the 'Principia' were of no particular mathemati- 



cal difficulty ; but it is quite oil erwise with the crowd 

 of interesting questions arising out of the theory of 

 gravitation, and which, in becoming the subject of 

 mathematical investigation, have contributed 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 about a fixed centre 

 of force, for any law of force ; we have also the 

 problem (mathematically very interesting) of the 

 motion of a body attracted to two or more fixed cen- 

 tres 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 mathemat- 

 ics ; and it is in the lunar and planetary theories re- 

 placed by what is mathematically a different problem, 

 — that of the motion of a body luider the action of a 

 principal central force and a disturbing force, — or, in 

 one mode of treatment, by the problem of disturbed 

 elliptic motion. I would remark that we have here 

 an instance in which an astronomical fact, the ob- 

 served 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 differential equations. Again : immedi- 

 ately 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 Maclau- 

 rin), and of the ellipsoid of three unequal axes. There 

 is, perhaps, no problem of mathematics which has 

 been treated by so 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 dynam- 

 ical problem, that of vibrating strings, by which 

 Lagrange was led to the theory of the representation 

 of a function as the sum of a series of multiple sines 

 and cosines ; and connected with this we have the 

 expansions in terms of Legendre"s functions P„, sug- 

 gested to him by the question, just referred to, of the 

 attraction of an ellipsoid. The subsequent investiga- 

 tions 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-Dirichlet, 

 and Green ; and I must not omit to mention that 

 the theory is now one relating to )i-dimensional 

 space. Another great problem of celestial mechan- 

 ics, that of the motion of the earth abput its centre 

 of gravity (in the most simple case, that of a body 

 not acted upon by any forces), is a very interesting 

 one in the mathematical point of view. 



I may mention a few other instances where a prac- 

 tical 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 projection, in- 



