chapter xv.

t Fermat's Diophanttts, note on p. 339; or Brassinne's Precis, p. 127. + Euler's Algebra (English trans. 1797), vol. n, chap, xv, p. 2-17.
42 ARITHMETICAL RECREATIONS [CH. II
interest and elegance, but its conclusions have little practical importance, and since his time it has been discussed by only a few mathematicians, while even of them not many have made it their chief study. This is the explanation of the fact that it took more than a century before some of the simpler results which Fermat had enunciated were proved, and thus it is not surprising that a proof of the theorem which he succeeded in establishing only towards the close of his life should involve great difficulties.
In 1823 Legendre* obtained a proof for the case of n = 5; in 1832 Lejeune Dirichletf gave one for n=14; and in 1840 Lame and Lebesgue J gave proofs for w = 7.
The proposition appears to be true universally, and in 1849 Kummer§, by means of ideal primes, proved it to be so for all numbers except those (if any) which satisfy three conditions. The proof is complicated and difficult, and there can be little doubt is based on considerations unknown to Fermat. It is not known whether any number can be found to satisfy these con- ditions. It was shown a considerable time ago, that there is no number less than 100 which does so. Recently the method has been developed by L. E. Dickson ||. His calculations show that Fermat's result is true if n is less than 6857. But mere numerical verifications have little value ; no one doubts the truth of the theorem, and its interest lies in the fact that we have not yet succeeded in obtaining a rigorous general demon- stration of it. The general problem was also attacked on other lines by Sophie Germain, who showed that it was true for all numbers except those (if any) which satisfied certain defined con- ditions. I may add that to prove the truth of the proposition when n is greater than 4, obviously it is sufficient to confine
* Reprinted in his Theorie des Nombi'es, Paris, 1830, vol. n, pp. 361 — 368 : see also pp. 5, 6.
t Crelle's Journal, 1832, vol. ix, pp. 390—393.
X Liouville's Journal, 1841, vol. v, pp. 195—215, 276—279, 348—349.
§ References to Kummer's Memoirs are given in Smith's Report to the British Association on the Theory of Numbers, London, 1860.
II See L'Intermediaire des Mathematiciens, Paris, 1908, vol. xv, pp. 247 — 248 ; Messenger of Mathematics, Cambridge, 1908, vol. xxxviii, pp. 14 — 32 ; and Quarterly Journal of Mathematics, Cambridge, 1908, vol. xl, pp. 27 — 45.
CH. Il] ARITHMETICAL RECREATIONS 43
ourselves to cases where « is a prime. A prize * of 100;000 marks has been offered for a general proof, to be given bet ore 2U07.
Naturally there has been much speculation as to how Fer- mat arrived at the result. The modern treatment of higher arithmetic is founded on the special notation and processes introduced by Gauss, who pointed out that the theory of discrete magnitude is essentially different from that of con- tinuous mngnitude, but until the end of the last century the theory of numbers was treated as a branch of algebra, and such proofs by Fermat as are extant involve nothing more than elementary geometry and algebra, and indeed some of his arguments do not involve any symbols. This has led some writers to think that Fermat used none but elementary algebraic methods. This may be so, but the following remark, which I believe is not generally known, rather points to the opposite conclusion. He had proposed, as a problem to the English mathematicians, to show that there was only one integral solution of the equation a?'^ + 2 = 3/' : the solution evidently being x = b, 2/ = 3- On this he has a notef to the effect that there was no difficulty in finding a solution in rational fractions, but that he had discovered an entirely new method — sane pulcherrima et subtilissima — which enabled him to solve such questions in integers. It was his intention to write a work \ on his researches in the theory of numbers, but it was never completed, and we know but little of his methods of analysis. I venture however to add my private suspicion that continued fractions played a not unimportant part in his researches, and as strengthening this conjecture I may note that some of his more recondite results — such as the theorem that a prime of the form 4n -f 1 is expressible as the sum of two squares — may be established with comparative ease by properties of such fractions.
♦ Ulntermidiaire des Mathematiciens, vol. xv, pp. 217—218, for references
and details.
t Fermat's Diophantus, bk. vi, prop. 19, p. 320; or Brassinne's Pricis,
p. 122.
I Fermat's Diophantus, bk. iv, prop. 31, p. 181 ; or Brassinne's Precis, p. 82.
•i^
CHAPTER TIL
GEOMETRICAL KECREATIONS.
In this chapter and the next one I propose to enumerate certain geometrical questions the discussion of which will not involve necessarily any considerable use of algebra or arithmetic. Unluckily no writer like Bachet has collected and classified problems of this kind, and I take the following instances from my note-books with the feeling that they represent the subject but imperfectly. Most of this chapter is devoted to questions which are of the nature of formal propositions: the next chapter contains a description of various trivial puzzles and games, which the older writers would have termed geometrical.
In accordance with the rule I laid down for myself in the preface, I exclude the detailed discussion of theorems which involve advanced mathematics. Moreover (with one or two exceptions) I exclude any mention of the numerous geomet- rical paradoxes which depend merely on the inability of the eye to compare correctly the dimensions of figures when their relative position is changed. This apparent deception does not involve the conscious reasoning powers, but rests on the inaccurate interpretation by the mind of the sensations derived through the eyes, and I do not consider such paradoxes as coming within the domain of mathematics.
Geometrical Fallacies. Most educated Englishmen are acquainted with the series of logical propositions in geometry associated with the name of Euclid, but it is not known so generally that these propositions were supplemented originally by certain exercises. Of such exercises Euclid issued three
CH. Ill]
GEOMETRICAL RECREATIONS
45
series: two containing easy theorems or problems, and the third consisting of geometrical fallacies, the errors in which the student was required to find.
The collection of fallacies prepared by Euclid is lost, and tradition has not preserved any record as to the nature of the erroneous reasoning or conclusions; but, as an illustration of such questions, I append a few demonstrations, leading to obviously impossible results. Perhaps they may amuse any one to whom they are new. I leave the discovery of the errors to the ingenuity of my readers.
First Fallacy*. To prove that a right angle is equal to an angle which is greater than a right angle. Let A BCD be a rectangle. From A draw a line AE outside the rectanole, equal to AB or DC and making an acute angle with AB, as
indicated in the diagram. Bisect CB in H, and through H draw HO at right angles to CB. Bisect CE in K, and through K draw KO at right angles to CE. Since CB and CE are not parallel the lines HO and KO will meet (say) at 0. Join OA, OE, OC, and OD.
The triangles ODC and OAE are equal in all respects. For, since KO bisects CE and is perpendicular to it, we have
* I believe that this and the fourth of these fallacies were first published in this book. They particularly interested Mr C. L. Dodgson ; see the Lewis Carroll Picture Book, London, 1899, pp. 20-4, 2Gu, where they appear in the form in which I originally gave them.
46
GEOMETRICAL RECREATIONS
[CH. ill
00= OE. Similarly, since HO bisects GB and DA and is per- pendicular to them, we have OD = OA. Also, by construction, DC = AE. Therefore the three sides of the triangle ODC are equal respectively to the three sides of the triangle OAE. Hence, by Euc. I. 8, the triangles are equal ; and therefore the angle ODG is equal to the angle OAE.
Again, since HO bisects DA and is perpendicular to it, we have the angle ODA equal to the angle OAD.
Hence the angle ADC (which is the difference of ODC and ODA) is equal to the angle DAE (which is the difference of OAE and OAD). But ADC is a right angle, and DAE is necessarily greater than a right angle. Thus the result is impossible.
Second Fallacy*. To prove that a part of a line is equal to the whole line. Let ABC be a triangle; and, to fix our ideas, let us suppose that the triangle is scalene, that the angle B is
acute, and that the angle A is greater than the angle G. From A draw AD making the angle BAD equal to the angle (7, and cutting BG in D. From A draw AE perpendicular to BG.
The triangles ABG, ABD are equiangular; hence, by Euc.