369 



VIETA, FRANCIS. 



VIETA, FRANCIS. 



370 



title-page, ' Francisci Vietsoi universalium Inspectionum ad Canonem 

 Mathematicum liber singularis, Lutetiao,' &c., as before. 



This same book, from the same types, is also found with another 

 title-page, us follows : ' Fraucisci Vietaoi opera mathematica, in qui- 

 bus tractatur canon mathematicus, seu ad triaugula : item Canonion, 

 &c. &c. &c., Londini, apud Franciacum Bouvier,' 1589 (but though 

 bearing the imprint London, it is evidently printed on the Continent). 



The same book, again from the same types, is in the British 

 Museum with a third title-page, as follows : ' Fran. Vietsei Libellorum 

 Supplicuin in Regia magistri, insignia que Mathematici, varia opera 

 mathematica: in quibus tractatur Canon Mathematicus, seu ad trian- 

 gula ; item Canoniou, &c., Parisiis, apud Bartholomseum Macceum,' 

 &c., 1609. 



That the second- and third are really the same book as the first, 

 with a new title-page, we have ascertained by carefully comparing 

 various words which are misspelt, and letters and lines which are 

 broken, in all three; also by the fact that the second title-page, 

 ' Francisci Vietsei,' &c., is the same, date and all, in the second. In 

 the third the second title-page is taken out, and Mettayer's address is 

 printed after the first. This book was, from its extreme scarceness, a 

 bibliographical curiosity: we have seen five copies, three with the first 

 title-page, one with the second, and one with the third : in two of the 

 first three, some figures which are not found in the third have been 

 stamped in after printing ; and the same stamping is apparent both in 

 the fourth and fifth. The canon mathematicus is the first table in 

 which sines and cosines, tangents and cotangents, secants and cosecants, 

 are completely given ; they are arranged in the modern form, in which 

 each number entered has a double appellation. But the notation of 

 decimal fractions not being invented, the mode of description is as 

 follows : to give the sine and cosine of 24 2', Vieta states that, the 

 hypothenuse being 100,000, the perpendicular and base are 40,727 and 

 91,330 9 ; and in a similar way for the others : and here it- is remark- 

 able that in the cosines Vieta does use a species of decimal notation, 

 leaving a blank space instead of using a decimal point ; for, to an 

 hypothenuse 100,000, the base to an angle of 24 2' is what we should 

 now write 91,330 - 9. There is also a large collection of rational- 

 sided right-angled triangles, which form a trigonometrical canon, but 

 not ascending by equal angles. The work concludes with a copious 

 collection of trigonometrical formulae and various numerical calcula- 

 tions, for mention of which see Hutton's ' History of Trigonometrical 

 Tables,' prefixed to his logarithms, and inserted in his tracts. A short 

 preface by Mettayer, prefixed to the ' Univeraalium Inspectionum,' &c., 

 states that Vieta found great difficulties in getting tables printed at all, 

 and also that plagiarists had printed and sold something of the kind, 

 but what is not stated. Vieta himself (Schooten, p. 323) calls this 

 book infeliciter editus, and hopes that a second edition will be of better 

 authority. 



Having now given, we believe, as complete an account of Vieta as 

 existing materials can furnish, in consideration of the very meagre 

 manner in which his biography is usually treated (the article in the 

 ' Biographic Universelle ' is very poor, considering that the work is 

 French, and Vieta the greatest French mathematician of the 16th 

 century), we may speak briefly upon the merit of his writings. Vieta 

 is a name to which it matters little that we have not dwelt on several 

 points which would have made a character for a less person, such as 

 his completion of the cases of solution of right-angled spherical 

 triangles, his expressions for the approximate quadrature of the circle, 

 his arithmetical extensions of the same approximation, and so on. 

 The two great pedestals on which his fame rests, are his improvements 

 in the form of algebra, which he first made to be a purely symbolical 

 science, and showed to be capable of wide and easy application in 

 ordinary hands; his application of his new algebra to the extension of 

 trigonometry, in which he first discovered the important relations of 

 multiple angles ; and his extension of the ancient rules for division 

 and extraction of the square and cube roots to the exegetic process 

 for the solution of all equations, which, with Mr. Horner's new mode 

 of conducting the calculation, is becoming daily of more importance. 

 He did not, as some of the French say, lay down the view of equa- 

 tions which was afterwards done by Harriot ; but he gave strong 

 suggestions towards it, stronger suggestions than the Italian algebraists 

 had furnished him with for his own new algebra : it is Harriot's praise 

 that he saw how to go on from where Vieta had stopped, as it is that 

 of Vieta to have proceeded from the point at which Cardan had 

 stopped. Neither did he, as some of the French again say (but not 

 from national feeling in this instance), first apply algebra to geometry; 

 for if by the application of algebra be meant the method of co-ordin- 

 ates, that .application is wholly due to Des Cartes, assisted, no doubt, 

 by the power which Vieta conferred on algebra. But if nothing more 

 be meant than the solution of geometrical problems by help of alge- 

 braical symbols and methods, many have claims before Vieta; for 

 instance, Regiomontanus, Cardan, and Bombelli. Nay, Vieta himself 

 points out that Regiomontanus had solved problems algebraically 

 which he complained of not being afterwards able to do geometrically; 

 and Vieta himself supplies the geometrical verification of Regiomon- 

 tanus's algebraical solutions. Neither did he, as some of the French 

 again _say, show how to form the coefficients of the powers of a 

 binomial : he saw, no doubt, the connection of them with the series, 

 1, 2, 3,&c. f 1, 3, 6, &c., 1, 4, 10, &c., as Tartaglia had done before 



BIOQ. DIV. VOL. VI. 



him ; but he did not show how to form them by any algebraical law, 

 as Newton afterwards did. If a Persian or an Hindoo, instructed 

 in the modern European algebra, were to ask, " Who, of all the indi- 

 vidual men, made the step which most distinctly marks the separation 

 of the science which you now return to us from that which we 

 delivered to you by the hands of Mohammed Ben Musa ? " the 

 answer must be Vieta. 



The earliest history of algebra is that contained in the mixed 

 treatise of Wallis (in English, 1685; in Latin, 1693). Wallis had a 

 partiality for Harriot, which not only blinded him to much of the 

 merit of Vieta, but furnished him with spectacles by which he could 

 sec most of the discoveries of the latter only in the writings of the 

 former. Montucla has fairly and properly exposed this tendency; 

 but that he may be disqualified to throw a stone at Wallis, he, in his 

 turn, gravely and seriously declares that he cannot see the merit of 

 the invention of aa, aaa, &c., to represent the powers of a, instead of 

 Vieta's mode. Montucla is not altogether fair to the Italian alge- 

 braists who preceded Vieta, as to which he has been severely criticised 

 by Cossali, and also by M. LibrL But these Italian historians have 

 a corresponding fault : they make a painful endeavour to show that 

 the peculiar discoveries of Vieta are to be found in the writings of 

 their own illustrious countrymen, and particularly of Cardan. Cossali 

 will even have it that Cardan has even something equivalent to, or 

 very nearly approaching to, Des Cartes's theorem on the roots of 

 equations; and constantly endeavours to show that Cardan might, 

 could, would, or should, or ought to have had something which he just 

 stops short of saying Cardan actually had. He wants to make his 

 countrymen a school of constructive discoverers ; if Cardan had only 

 carried the contents of page x farther than he did, and seen some- 

 thing at page y which he did not see, then he would have been able 

 at page z to do something which he did not do, but which Vieta did 

 do. M. Libri starts more fairly : " In France," he observes (voL iv., 

 p. 1), "Vieta made algebra approach nearer to perfection, and, per- 

 haps, caused the labours of his predecessors to fall into too much 

 neglect." This is perfectly true, and might have been more positively 

 expressed; but a little further on we find (p. 7), "In truth his dis- 

 coveries seem to be not comparable to those of Ferro or Ferrari." 

 This is truly strange ; for in the next sentence we find he " was an 

 eminently philosophical mind, and is more to be admired for hia 

 methods than for the results which he obtained from them." Can it 

 seriously be M. Libri's opinion that the inventor of an isolated result 

 is to be placed above one who increases the power of the human race 

 over every branch of science ? and is it not the surest test of the 

 greatness of a discovery, that it is a method, not a result, and that 

 the power which it gives to others makes succeeding results obtained 

 from it more remarkable than those of the inventor himself ? If ever 

 it lias been true that coming events have thrown their shadows before, 

 it has been in the progress of the mathematics : it never has happened, 

 in the case of any great discovery, that it was made upon quite a clear 

 field. No one can read the history of science without finding that 

 there was always, in the time immediately preceding the promulgation 

 of any new method, a constant tendency towards the invention of that 

 method, a series of efforts the results of which have speedily merged in 

 those of the man for whom the discovery was reserved. This leaves the 

 relative merit of investigators unaltered ; if it depress Vieta, it also de- 

 presses Tartaglia and Cardan. To us it raises all three ; for it points 

 out that they have severally succeeded where their predecessors have 

 failed, and relieves them from the consequences of the supposition that 

 it was merely their good fortune which led their thoughts to that which 

 another might as easily have attained if his thoughts had been turned 

 towards the subject. If sometimes too much Gallicism shows itself, 

 by way of exception, in the admirable history of Montucla, it is not 

 half so offensive as the constant and always recurring nationality of 

 the Italian historians, which renders it necessary to watch them so 

 closely, that the end of it will be a general conviction that they are not 

 to be safely read at all, without the original authorities at hand, on 

 any matter in which claims of country can enter. M. Libri, in finding 

 out, and with perfect correctness, that Cataldi used continued fractions 

 before Brounker, and infinite series (or at least an infinite series) before 

 Wallis, and in making a very just remark on the interest with which 

 the first dawnings of the doctrine of infinites should bo regarded, 

 forgets that Vieta had preceded Cataldi, to the extent of using a 

 combination of the infinite product and series united. It would be 

 difficult, we think, to produce an earlier germ of the doctrine just 

 alluded to than is seen in the celebrated expression given by Vieta 

 for the quadrature of a circle, which we should now express thus 

 2 



Va). A/{a+ V (a+ Va)\ &c. 



where a means half a unit. ('Resp. Math.,' Schooten, p. 400.) 



Both Vieta and Cossali endeavour to show that the Italian alge- 

 braists used letters for quantities, both known and unknown. So they 

 did, no doubt, and so did Euclid, and so (according to M. Libri him- 

 self) did Aristotle. But who combined the use of letters with that of 

 symbols of operation so as to produce algebraical formulae, and to give 

 to the operations of algebra that technical character which makes 

 them resemble the operations of arithmetic ? One look at any page 

 of the Italian algebraists will show the difference between their algebra 



2u 



