It is pretty evident, from this most interesting statement, that Newton, having simply observed the succession of the numbers, tried various formulæ until he found one which agreed with them all. He was so little satisfied with this process, however, that he verified particular results of his new theorem by comparison with the results of common 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 conclusive 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 probably thus discovered, by the ancient Greek geometers; and we have pretty clear evidence of this in the Commentaries of Proclus.[140] 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.[141] But discovery in such cases means nothing more than suggestion, and it is always by pure deduction that the general law is really established. As Proclus puts it, *we must pass from sense to consideration*. [140] Bk. ii. chap. iv. [141] *Philosophical Transactions* (1866), vol. 146, p. 334. [Illustration]