September 23, 1904.] 



SCIENCE. 



389 



time, the development of abstract ideas 

 among the Greeks preceded the concrete 

 knowledge of natural phenomena. When 

 we reflect that in the geometry of Euclid 

 the science of space was brought to such 

 logical perfection that even to-day its teach- 

 ers are not agreed as to the practicability 

 of any radical improvement upon it, we 

 can not avoid the feeling that a very slight 

 change in the direction of the intellectual 

 activity of these people would have led to 

 the beginning of natural science. But it 

 would seem that the very purity and per- 

 fection which was aimed at in their system 

 of geometry stood in the way of any ex- 

 tension or application of its methods and 

 spirit to the field of nature. One example 

 of this is worthy of attention. In modern 

 teaching the idea of magnitude as gener- 

 ated by motion is freely introduced. A 

 line is described by a moving point ; a plane 

 by a moving line; a solid by a moving 

 plane. It may, at first sight, seem singu- 

 lar that this conception finds no place in the 

 Euclidian system. But we may regard 

 the omission as a mark of logical purity 

 and rigor. Had the real or supposed ad- 

 vantages of introducing motion into geo- 

 metrical conceptions been suggested to 

 Euclid, we may suppose him to have replied 

 that the theorems of space are independent 

 of time ; that the idea of motion necessarily 

 implies time, and that, in consequence, to 

 avail ourselves of it would be to introduce 

 an extraneous element into geometry. The 

 result was that, in keeping this science pure 

 from ideas which did not belong to it, it 

 failed to form what might otherwise have 

 been the basis of physical science. Its 

 founders missed the discovery that the 

 methods of geometric demonstration could 

 be extended into other and wider fields 

 than that of space. Thus not only the 

 development of applied geometry, but the 

 reduction of other conceptions to a rigorous 



mathematical form was indefinitely post- 

 poned. 



The idea of continuous increase in time is 

 that by which the conceptions of the 

 infinitesimal calculus can most easily find 

 root in the mind of the beginner. It is 

 quite possible that the contempt of the 

 ancient philosophers for the practical ap- 

 plication of their science, which has con- 

 tinued in some form to our own time, and 

 which is not altogether unwholesome, was 

 a powerful factor in preventing the de- 

 velopment of this idea. 



Astronomy is necessarily a science of ob- 

 servation pure and simple, in which experi- 

 ment can have no place except as an aux- 

 iliary. The vague accounts of striking 

 celestial phenomena, handed down by the 

 priests and astrologers of antiquity, were 

 followed in the times of the Greeks by ob- 

 servations having, in form at least, a rude 

 approach to precision, though nothing like 

 the degree of precision that the astronomer 

 of to-day would reach with the naked eye, 

 aided by such rude instruments as he could 

 fashion from the tools at command of the 

 ancients. 



The rude observations commenced by the 

 Babylonians were continued with gradually 

 improving instruments, first by the Greeks 

 and then by the Arabians; but the results 

 failed to afford any insight into the true 

 relation of the earth to the heavens. What 

 was most remarkable in this failure is that 

 to take a first step forward, which would 

 have led on to success, no more was neces- 

 sary than a course of abstract thinking 

 vastly easier than that required for work- 

 ing out the problems of geometry. That 

 space is infinite is an unexpressed axiom, 

 tacitly assumed by Euclid and his succes- 

 sors. Combining this with the most ele- 

 mentary consideration of the properties of 

 the triangle, it would be seen that a given 

 body of any size could be placed at such a 

 distance in space as to appear to us like a 



