232 THE PRINCIPLES OF SCIENCE. [CHAF. 



multiplication, and the rule for the extraction of the 

 square root. Newton, in fact, gave no demonstration 

 of his theorem ; and the greatest mathematicians of the 

 last century, James Bernoulli, Maclaurin, Landen, Euler, 

 Lagrange, &c., occupied themselves with discovering a con- 

 clusive method of deductive proof. 



There can be no doubt that in geometry also discoveries 

 have been suggested by direct observation. Many of the 

 now trivial propositions of Euclid's Elements were pro- 

 bably thus discovered, by the ancient Greek geometers ; 

 and we have pretty clear evidence of this in the Commen- 

 taries of Proclus. 1 Galileo was the first to examine the 

 remarkable properties of the cycloid, the curve described by 

 a point in the circumference of a wheel rolling on a plane. 

 By direct observation he ascertained that the area of the 

 curve is apparently three times that of the generating circle 

 or wheel, but he was unable to prove this exactly, or to 

 verify it by strict geometrical reasoning. Sir George Airy 

 has recorded a curious case, in which he fell accidentally by 

 trial on a new geometrical property of the sphere. 2 But 

 discovery in such cases means nothing more than sugges- 

 tion, and it is always by pure deduction that the general 

 law is really established. As Proclus puts it, *.ve mutt 

 pass from sense to consideration. 



Given, for instance, the series of figures in the accom- 

 panying diagram, measurement will show that the curved 

 lines approxim te to semicircles, and the rectilinear figures 

 to right-angled triangles. These figures may seem to 

 suggest to the mind the general law that angles inscribed 



1 Bk. ii. chap. iv. 



* Philosophical Transactions (1866), vol. 146, p. 334. 



