ESSAYS 129 



that the margin of his book was too small to contain it. 1 The enunciation of 

 Eisenstein's results occupied part of one page of Crelle's Journal, but their demon- 

 stration by Smith and Minkowski required a small volume. This proof, as well as 

 the proof of Kummer's results, are examples of some of the most delicate and 

 intricate demonstrations to be found in the whole range of mathematical analysis. 



As the integers are the fundamental elements of the Theory of Numbers, it is 

 not surprising that questions are easily suggested and theorems discovered 

 inductively, though it does seem strange that their demonstrations should so 

 frequently be very difficult and indeed occasionally baffle all mathematical analysis. 

 The law of quadratic reciprocity, which is one of the most important and beautiful 

 theorems, was discovered by Euler nearly half a century before the first rigid 

 demonstration was given by Gauss in complete ignorance of the work of his pre- 

 decessors. Kummer discovered the general law of reciprocity about ten years 

 before he was able to give a proof of it. Stern found by induction that the value 

 of x in the equation {p is a prime) p = Sn + 3 = x* + ly* could be found from a 

 simple congruence, though it remained for Eisenstein to demonstrate and generalise 

 this result. Waring's conjecture in 1787 that every positive integer could be 

 expressed as a sum of m positive n th powers, where m is independent of the 

 integer, was not proved until 1909 by Hilbert. Goldbach's Theorem, that every 

 even number can be expressed as the sum of two primes, has neither been proved 

 nor disproved. Similarly, while our readers will have no difficulty in finding a 

 few solutions in integers of the indeterminate equation ofy 2 = x z + 17, the general 

 solution is still unknown ; although Fermat stated that he had discovered an 

 entirely new method — sane pulcherrima et subtilissima — which enabled him to solve 

 such questions in integers. 2 



In 1890, at the meeting of the British Association, Dr. Glaisher said : " I am 

 sure that no subject loses mere than mathematics by any attempt to dissociate it 

 from its history." This applies with great force to the Theory of Numbers, as can 

 be seen from any history of mathematics, but more especially from the many 

 volumes by Bachman and from the original memoirs. The Higher Arithmetic 

 seems to include most of the romance of mathematics. No other study requires 

 such steadfast devotion or gives more pleasure to its investigators. As Gauss 

 wrote to Sophie Germain, " The enchanting beauties of this sublime study are 

 revealed in their full charm only to those who have the courage to pursue it ! " and 

 we are reminded of the folk-tales, current amongst all peoples, of the Prince 

 Charming who can assume his proper form as a handsome prince only because of 

 the devotedness of the faithful heroine. 



The original memoirs frequently suggest the toilsome labours of a mountaineer- 

 ing party attempting to ascend hitherto unexplored summits. We realise at once 

 that the authors are grappling with extraordinary difficulties. Their paths abound 

 with innumerable obstacles, and new methods must be devised to overcome them. 

 Progress may be slow, and it may be many years before their efforts are crowned 

 with success and an imperishable monument erected as a testimony to their genius 

 Who can fail to sympathise with Eisenstein when we read his account of his 

 efforts to find the number of classes of ternary quadratics (during which hede- 

 veloped a tolerably complete account of the arithmetic properties of the ternary 

 quadratic and very likely of the general quadratic), and to imagine his joy, when, 

 after ten years of arduous labours, as he himself tell us, the goal was in sight ? It 



1 Cajori's History of Mathematics, p. 179. For a full account see Ball's 

 Mathematical Recreations, p. 40, or his History of Mathetnatics, p. 295. 

 3 Ball, Mathematical Recreations, p. 43. 



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