March 27, 1903.] 



SCIENCE. 



497 



some of the answers. It appears that the 

 Egyptian did not make any additions to 

 the work of Ahmes for more than a thou- 

 sand years. 



About the seventh century b.c. the 

 Greefe showed such a deep interest in 

 learning that they began to go to foreign 

 countries (especially Egypt) in quest of 

 knowledge. They soon excelled their 

 teachers, and inaugurated a golden period 

 of mathematical progress which has had 

 no equal until recent times. Hence we 

 naturally look to the Greeks for funda- 

 mental discoveries whose beauty and gen- 

 erality have engaged the admiration and 

 interest of all who became acquainted with 

 them. 



One of the earliest of these is the proof 

 that there are lines which do not have a 

 common measure. Pythagoras observed 

 that it is impossible to divide the side of 

 a square into such a number of equal parts 

 that one of these parts is contained an 

 integral number of times in the diagonal. 

 This fact made a very deep impression on 

 the Greek mind. It is one of those great 

 truths which can never be fully established 

 by experiment, and yet does not rest on 

 postulates or axioms which appear some- 

 what arbitrary. It thus stands in sharp 

 contrast with the older discovery that the 

 sum of the angles of a plane triangle is 

 equal to two right angles, and deserves to 

 be placed in a higher category of mathe- 

 matical truths. 



A scholium of Euclid's 'Elements,' 

 which is supposed to be due to Proelus, 

 bears evidence of the high regard Avhich the 

 Greeks had for the discovery of the incom- 

 mensurable or the irrational. It reads as 

 follows : " It is said that the man who first 

 made the theory of the irrational public 

 died in a shipwreck because the unspeak- 

 able and invisible should always be kept 

 secret, and that he who by chance first 

 touched and uncovered this symbol of life 



was removed to the origin of things where 

 the eternal waves wash around him. Such 

 is the reverence in which these men held 

 the theory of the irrational quantities." 



Aristotle frequently speaks of the fact 

 that the diagonal of a square whose side 

 is unity is irrational, and in one instance 

 states that otherwise an even number would 

 be equal to an odd number. The meaning 

 of this is made clear by Euclid's proof of 

 the fact that the V2, which is the value of 

 the given diagonal, is irrational. Euclid 

 says, in substance, if we assume that 

 V2 = m/n a rational number, it follows 

 that 2n^ = m^. The fraction m/n may be 

 supposed to be reduced to its lowest terms, 

 and hence at least one of the two numbers 

 m, 11 must be odd. This, however, makes 

 the equation 2n^ = m? impossible, since the 

 square of an even number is always divis- 

 ible by 4 and the square of an odd number 

 is odd. By dividing both members of the 

 equation 2?i^ = m^ by 2 an odd number 

 would be equal to an even number, as 

 Aristotle says. It appears very probable- 

 that this elegant proof is due to the Pythag- 

 oreans, possibly to Pythagoras himself. 



Another fundamental discovery of the 

 Greeks is the use of infinite convergent 

 series. Aristotle observed that the sum of 

 an infinite number of small things may 

 be finite and Archimedes frequently finds 

 the sum of an infinite series in the solution 

 of a problem. For instance, in fiinding the 

 area of a portion of the parabola he ob- 

 serves that it is equal to the area of a 

 given triangle multiplied by the infinite 

 geometric series 1 + i +(i)^ +(i)^ +•"• 

 and he proves that the sum of this series 

 cannot be greater or less than 4/3. His 

 proof is practically the same as that found 

 in our elementary algebras. 



In finding the sum of such infinite series 

 Archimedes answered in a very explicit 

 and definite manner some of the difficult 

 questions raised by Zeno two hundred years 



