NA rURE 



409 



THURSDAY, MARCH 3, iJ 



ARCHIMEDES. 

 The Works of Archimedes. Edited in Modern Notation, 

 with introductory Chapters by T. L. Heath, ScD., 

 sometime Fellow of Trinity College, Cambridge. Pp. 

 clxxxvi + 326. (Cambridge : at the University Press, 

 1897.) 



THIS is a companion volume to Dr. T. L. Heath's 

 valuable edition of the " Treatise on Conic Sec- 

 tions" by ApoUonius of Perga, and the same patience, 

 learning and skill which have turned the latter book 

 into a delightful guide to early Greek geometry have 

 been here applied to present in a most readable form the 

 extant works of perhaps the greatest mathematical genius 

 that the world has ever seen. The same general plan of 

 editing has been followed in this book as in that of 

 ApoUonius, such condensation and modernisation having 

 been introduced as is possible without alteration of the 

 methods employed. We consequently now possess for 

 the first time in English dress a reproduction of Archi- 

 medes' work, without addition or essential omission, from 

 which have been removed the thorns and briars that have 

 hitherto beset the path of the English student who 

 would make himself acquainted with the methods in- 

 vented and employed by the extraordinary genius of 

 Archimedes in order to make those measurements which 

 now-a-days are rendered easy only by the use of the 

 integral calculus. Not the least deterrent of these 

 obstacles have been the Doric dialect of parts of the 

 original and the abbreviations and corruptions in which 

 the text of the accessible editions abounds, so that hitherto 

 the French translation by Peyrard, or that in German 

 by Nizze, has been preferable for the purposes of study. 



The volume itself is a handsome example of mathe- 

 matical printing such as we are wont to expect from the 

 Cambridge University Press, and it is most fitting that it 

 should have been issued by that Press, since Torelli's 

 standard edition (in Greek with a Latin version) was 

 published by and at the expense of the sister University 

 under the editorship of Prof. Abram Robertson, D.D., of 

 Christ Church. 



The book will do much to restore Archimedes to his 

 proper place in the estimation of mathematicians ; for, 

 in spite of the admirable histories of mathematics that 

 lie to our hand, Archimedes is scarcely known at all 

 except as the discoverer of an important principle in 

 hydrostatics, or the constructor of a spiral, or the in- 

 ventor of a screw, or the destroyer of ships by the use of 

 a mirror. Indeed it is but little known that in his books 

 on Equilibrium he laid the foundation of theoretical 

 mechanics, and that in his treatises on Floating Bodies he 

 created the science of hydrostatics ; that he it was who 

 discovered 3! as a superior limit of the ratio of the 

 circumference of a circle to its diameter, and 3^9 as a 

 still closer lower limit ; that to him we owe the quadra- 

 ture of the conic sections and- of the surfaces generated 

 by their revolution, as well as the cubature of the volumes 

 so formed ; or that he was the author of a system of 

 representing numbers up to that which would now be 

 expressed by i with eighty thousand million millions of 



NO. 1479, VOL. 57 J 



cyphers following it. What else he may have done we 

 do not know for certain, <as our information rests on vague 

 accounts by later authors ; but eight at least of his pub- 

 lished works are lost, and these cannot but have shown 

 the same originality and been marked by the same pre- 

 sentation of new truths as those which have survived, 

 while none of them, or of those that we have, relate to 

 his numerous mechanical inventions, of which in com- 

 parison with his mathematical speculations he seems to 

 have thought meanly since, in Plutarch's words, he 

 would not deign to leave behind him any written work on 

 such subjects. 



Not the least important part of this book is the intro- 

 duction of 186 pages which precedes the 326 pages of 

 text. This contains a very valuable discussion of the 

 problems that attracted the attention of the early Greek 

 geometers, and also a most interesting account of the 

 various suggestions that have been put forward as to the 

 mode in which Archimedes obtained certain results which 

 he states without entering into details respecting them. 



Aft'er a preliminary recital of the stories told of 

 Archimedes by ancient writers, and a Short account of 

 the ingenious mechanical inventions they have attributed 

 to him, Dr. Heath discusses the MSS. sources of the 

 text that survive, and enumerates with slight details the 

 principal editions of their text that have been published. 

 From this we learn all that is really needful, though the 

 Latin version with commentary made by Abbot Ma ir- 

 olycus in 1534-49, and published at Palermo in 1685 by 

 Cyllenius Hesperius, also deserves notice. One might 

 further remark that if the titles of the different editions 

 are given at all in anything like fullness, they should be 

 given with absolute exactness, and this is not done in 

 any of the Latin versions save those of Torelli and 

 Heiberg. 



Then follows an excellent chapter upon the relation of 

 Archimedes to his predecessors, and this is full of interest 

 not only historically, but also as contrasting Archimedes' 

 methods with those of others, and illustrating his extra- 

 ordinary facility in the manipulation of proportions, a 

 special instance of which is, in modern notation, the 

 elimination of b, c, d from the conditions afb = djc 

 = cld>i, 2,d{a - c) = yv{a - d), {a - c){2a + 4b + 6c 

 + 3^ = 5/('^ + 2b + 2c + d). Next comes a clear 

 account of the Greek system of numerals and the Greek 

 modes of performing arithmetical operations, more 

 especially with reference to the approximations to the 

 square roots of numerics which are not squares ; but in 

 this last matter there is plenty of room for conjecture, as 

 not much direct information is available, and Dr. Heath 

 gives a fairly exhaustive account of what has been written 

 on the subject. The fifth chapter treats of the problems 

 known as vtvirtii, which deal with the straight lines that 

 have to verge towards a given point, and fulfil some 

 other condition too ; these have not much to do with 

 Archimedes, but they are exceedingly interesting, and 

 without their discussion a study of Greek geometry 

 would be incomplete. The Greek methods of geometric- 

 ally solving problems that practically involve cubic equa- 

 tions are next unfolded at length, and a discussion of the 

 classification of problems and loci as plane, solid and 

 (cufvi-) linear completes a most interesting chapter. 

 Finally Archimedes' anticipations of the integral calculus 



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