NATURE 



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THURSDAY, JANUARY 17, 1889, 



THE HISTORY OF MATHEMATICS. 

 A Short Account of the History of Mathematics. By 

 W. W. Rouse Ball. (London and New York : 

 Macmillan and Co., 1888.) 



THE quaint words addressed "to the great variety of 

 readers" by the editors of the folio Shakespeare 

 of 1623 are equally applicable to the useful compendium 

 of mathematical history which is the subject of our re- 

 view. " It is now public ; and you will stand for your 

 privileges, we know — to read and censure. Do so, but 

 buy it first : that doth best commend a book, the stationer 

 says. Then how odd soever your brains be or your wis- 

 doms, make your licence the same, spare not." But, as 

 goods are usually " bought by judgment of the eye, not 

 uttered by base sale of chapmen's tongues," we produce 

 our samples in the open market by making a few extracts 

 from Mr. Ball's book. 



In the opening chapter, on Egyptian and Phoenician 

 inathematics, we become acquainted with an old Egyp- 

 tian, " a priest named Ahmes," who, " somewhere between 

 ihe years 1700 B.C. and 1 100 B.C.," wrote, on imperishable 

 papyrus, a book entitled " Directions for Knowing all 

 Dark Things," which "is believed to be itself a copy 

 with emendations, of an older treatise of about the time 

 3400 B.C." Remembering that this work was written 

 certainly five hundred, and probably more than a thous- 

 and, years before the time of Thales, the first of the 

 Greek mathematicians, and founder of the Ionian school, 

 it must be regarded as a most remarkable production ; 

 for Profs. Cantor and Eisenlohr have shown that Ahmes 

 had some notion of trigonometry. In his problems on 

 pyramids, " Ahmes desires to find the ratio of certain 

 lines, which is equivalent to determining the trigonometri- 

 cal ratios of certain angles. The data and the results given 

 agree closely with the measurements of some of the ex- 

 isting pyramids." But perhaps the most interesting 

 feature of this ancient treatise is the algebraic notation 

 employed in it, which our author describes in these 

 words: — "The unknown quantity is always represented 

 by the symbol which means a heap ; addition is repre- 

 sented by a pair of legs walking forwards ; subtraction 

 by a pair of legs walking backwards, or by a flight of 

 arrows ; and equality by the sign //_." Our own -}- and 

 - first appeared in Widman's " Mercantile Arithmetic " 

 (published at Leipzig in 1489): with him (see p. 186, 

 Ch. XII.) they "are only abbreviations, and not symbols 

 of operation ; he attached little or no importance to them, 

 and would no doubt have been amazed if he had been 

 told that their introduction was preparing the way for a 

 complete revolution of the processes used in algebra." 

 The philosophic conception of the nature of algebra 

 (symbolized by the legs walking forwards and backwards ; 

 a notion closely related to, if not identical with, Sir W. 

 R. Hamilton's definition of algebra as the science of pure 

 time) perished with its author : the mere abbreviations 

 (+ and - ) lived and flourished — but then Widman was 

 able to print his book. 



The first date that can be assigned with absolute pre- 

 VoL. xxxix.— No. 1003. 



cision is that of Thales. " It is well known that he pre- 

 dicted a solar eclipse which took place at or about the 

 time he foretold : the actual date was May 28, 585 B.C." 

 It marks the real commencement of the history of mathe- 

 matics ; for the science, now revived in Greece, was at 

 this time neglected and completely forgotten by the 

 Egyptians. When we read that Thales, to the utter 

 amazement of the King and all who were present, showed 

 them how to find the height of a pyramid, by a simple 

 application of the theorem that the sides of equiangular 

 triangles are proportionals, we may well wonder why 

 Ahmes did not burst his mummy-case and appear in their 

 midst with his book opened at the problems on pyramids. 



From the time of Thales to that of Euclid, the know- 

 ledge of mathematical facts acquired in one generation 

 was transmitted to the next, almost exclusively by means 

 of oral tradition. That such was the case is mainly due 

 to the Pythagorean secret Society. " Pythagoras him- 

 self did not allow the use of text-books, and the assump- 

 tion of his school was, not only that all their knowledge 

 was held in common, and secret from the outside world, 

 but that the glory of any fresh discovery must be referred 

 back to their founder : thus Hippasus {circ. 470 B.C.) is 

 said to have been drowned for violating his oath by pub- 

 licly boasting that he had added the dodecahedron to 

 the number of regular solids enumerated by Pythagoras. 

 Gradually, as the Society became more scattered, it was 

 found convenient to alter this rule, and treatises contain- 

 ing the substance of their teaching and doctrines were 

 written. The first book of the kind was composed by 

 Philolaus {circ. 410 B.C.), and we are told that Plato 

 contrived to buy a copy of it." 



Now Anaximander, the immediate successor of Thales- 

 as head of the Ionian school, had the honour of teaching 

 Pythagoras ; while Eudoxus, Philolaus, and Plato, all 

 of them received their mathematical training from. 

 Archytas of Tarentum, who was one of the most cele- 

 brated of the Pythagoreans ; and " Menaschmus, who 

 was a pupil of Plato and Eudoxus," was alive as late as 

 325 B.C., which brings us down to about the time of 

 Euclid. Thus the chain of tradition connecting Thales 

 with Euclid is complete. Its successive links can be 

 traced in the second and third chapters of the work 

 before us. 



Among the contemporaries of Plato, Eudoxus of Cnidus 

 deserves special notice. His biography is to be found in, 

 Diogenes Laertius, who speaks of him as an astrono- 

 mer, geometer, physician, and statesman ; mentions his 

 great works on astronomy and geometry, and his minor 

 treatises on other subjects ; and refers to the fact that 

 he discovered curved lines. Modern research has found*, 

 out what the curves of Eudoxus were, though all his. 

 writings are lost : in our author's words, " he discussed! 

 some plane sections of the anchor ring," among themi 

 the curve which ought in future to be named after him,, 

 but is " generally called Bernouilli's lemniscate." Thus, 

 Eudoxus (who died in 355 B.C.) anticipated James 

 Bernouilli (d. 1705 A D.) by more than 2000 years ! 



The foundation of Alexandria by Ptolemy marks an 

 epoch in the history of mathematics. Alexander himself 

 did little more than choose the site, and it was entirely 

 due to Ptolemy that the city did not share the fate of 

 at least two others of the same name whose foundation 



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