305 



VIETA, FRANCIS. 



VIETA, FRANCIS. 



366 



seven first books of the ' Responsa,' which have not come down to us, 

 though tradition has preserved the name : and ' Ad logiaticen specio- 

 sam note posteriores,' of which even the very name has disappeared 

 from the history of algebra. We cannot help hoping that some old 

 library may yet be found to contain this collection. Other writers 

 take the words of the title in a sense between that of quotation and de- 

 scription. Thus Alexander Anderson says, " Restitutum Mathematicam 

 Analysin F. Vieta) debetia, (t>i\o/j.a0f7s." And Walter Warner (preface 

 to Harriot), "Artis Analyticce Restitution em, F. Vieta aggressus est."] 



We believe it will be shorter and clearer to leave the preceding 

 passage in brackets (for which we thought we had very fair evidence), 

 and to make a suspected correction, as another writer would do, in 

 preference to mixing up the mistake (if it be a mistake) and the cor- 

 rection. The first publication of the 'Isagoge,' &c. (1591), bears on 

 its title-page that it is ' Seorsim excussa ab Opere Restitute Mathe- 

 nmticso Analyseos, seu Algebra) Novic:' and on the reverse of the 

 title-page appears ' Opere Restitute Mathematics Analyseos, seu Alge- 

 bra Hovft, contiuentur : Operi autem Preposita est sequens 



epistola.' Ten works are given by title, which may, all but the seven 

 books and the nota; posteriores already noticed, be collected from the 

 indication (K. M.) in the following list; and the epistle is the dedication 

 to Catherine of Parthenai before alluded to. Blancanus (1615) places 

 ' Opus Restitute,' &c., in the list of Vieta's works ; and Morhof says 

 that Vieta wrote "Isagoge, &c., seu Algebra Nova." Can any evidence 

 be more positive to the fact that a work was published, or at least 

 written out for publication ? The absence of date or printer's name 

 tells nothing as to that period ; for books were then few, and did not 

 require the minute accuracy of description which is now necessary to 

 distinguish one work from another : moreover, whether this be the 

 reason or not, such accuracy of description was not usual. Why then 

 do we not continue to believe that such a work was published ? In 

 the first place it is entirely lost, and with it the Responsa and the notes 

 posteriores, which is not likely to have happened to a large collection 

 of Vieta's works ; in the second place, Anderson, in his publication 

 (which be gives us to understand was the first that was made) of the 

 treatise 'De Recognitione,' &c., tells us something about Vieta's habits, 

 which seem to explain the whole. " He was," says Anderson, " in the 

 habit of referring to as finished " (insignire solebat), and by their 

 names, works which, though undertaken in his own mind, and digested 

 in order, were not even so much as fairly written down, owing to the 

 interruption which his studies received from his public duties. This 

 then may be the whole secret : Vieta gave a list of the works which 

 he intended to publish, under the name which he intended to give 

 them collectively. The seven books of the ' Responsa ' and the notes 

 posteriores never, on this supposition, were published at all. And it 

 will afterwards appear that there was a reason why the eighth book of 

 the ' Responsa ' should have been published without the rest ; though 

 it is singular, if the list above named bo only of works intended, that 

 this eighth book, which must have been as finished as the rest, should 

 not have been mentioned. It is almost incredible moreover that 

 Alexander Anderson should have published a few of Vieta's theorems, 

 with his own demonstrations, as new, if Vieta had published them, 

 and more, twenty years before. 



(R. M.) In Artem Analyticam Isagoge, first published by Vieta him- 

 self, at Tours, in 1591. Here are laid down the principles of homo- 

 geneity before alluded to, and the common axioms used in the solution 

 of simple equations. Many new terms are introduced, of which only 

 two have lasted, namely, the distinction of equations into pure and 

 adfected. The law of homogeneity is a fanciful deduction from certain 

 well-known analogies between arithmetic and geometry, and the 

 manner in which it is applied renders this book of Vieta somewhat 

 obscure. The following is a specimen : " Lineam rectam curves non 

 comparat (probably corrupt, comparare non licet), quia angulus est 

 medium quiddam inter linearn rectam et planam figuram. Repugnare 

 itaque videtur homogcneorum lex." 



(R. M.) Ad logisticen speciosam notce priorcs. The notce posteriores, 

 as just mentioned, are lost. Logistice Speciosa is the literal algebra, as 

 distinguished from logistice numerosa, or common arithmetic. Here 

 are various questions in algebraical addition and multiplication : the 

 powers of a binomial are raised up to the sixth inclusive, and the law 

 of the exponents is given, but not that of the coefficients. Particular 

 notice is taken of the addition of powers of A-f B and A B, and, in 

 a few cases, of the composition of A" B". Various methods are 

 given of forming right-angled triangles whose sides shall be whole 

 numbers. 



(R. M.) Zeteticorum libri quinque. The first book contains problems 

 producing simple equations, of which the following are specimens : 

 Given x y, x + z, and the ratio of y to z, to find x ; given the sum 

 or difference of two numbers, and of given proportions of those num- 

 bers, to find the numbers. Here, as elsewhere, Vieta uses the capital 

 letters only, and represents the unknown quantities by vowels, and 

 the known quantities by consonants. The second book is full of those 

 problems of the second and third degree, which produce unadfected 

 equations, _ solved as in our modern works. The third book contains 

 the reduction into equations and solution of questions in proportion, 

 and also of right-angled triangles. The fourth and fifth books give 

 the solutions of various of those problems now called Diophantine, 

 mostly collected from Diophantus himself. We find here the first use 



of the vinculum connecting terms whose result is considered as a 

 whole. Blancanus says that Cataldi explained this work of Vieta in 

 what he calls " coutinuatio algebra? proportionalis," which cannot be 

 the "nova algebra proportionate," Bologna, 1619, published after 

 Blancanus wrote. 



(R. M. as to the first, not the second.) De Equationum Recognitione 

 et Emendatione libri duo. First put together by Alexander Anderson, 

 who obtained the materials from Alelmus or Aldaume (who had charge 

 of Vieta's papers), and published these books at Paris in 1615. The 

 first six chapters of the treatise I)e Itecoynitione are employed in 

 demonstrating that equations of the second and third degree spring 

 from questions upon three or four continued proportionals, except in 

 the irreducible case of the latter species, which is shown to depend on 

 the trisection of an angle. Where a cubic equation has one root only, 

 and that negative, the equation is deduced which has the corresponding 

 positive root. The two roots of an equation of which one is negative 

 are not considered, but the equation is deduced which has a positive 

 root corresponding to the negative root of the former, and this equa- 

 tion is called contradictory to the former. Various methods are found 

 by which an equation of a higher degree may be deduced from a given 

 one, a synthetical process, apparently introductory to the subsequent 

 depression of equations. In the treatise de Emendatione, Vieta lays 

 down rules for destroying the second term of an equation of the second 

 or third degree. He then shows, in a cubic equation which has the 

 highest term negative, how to avoid this by a transformation which is 

 in effect finding the equation whose roots are reciprocals to the roots 

 of the former equation. We have not space to enter minutely into 

 the various transformations ; we will only remark generally, that an 

 equation is considered unfit for use in which the highest power of the 

 unknown quantity is negative, or has a coefficient, and that the greater 

 part of the reductions employed would not be necessary to a modern 

 analyst. These books leave the reader in possession of the methods 

 then known for the depression or solution of equations of the second, 

 third, and fourth degrees. They are a luxuriant exercise of the power 

 newly derived from Vieta'a improvements in notation. He concludes 

 by showing how to construct an equation which shall have given 

 positive roots : which form the suggestive basis of the subsequent 

 discoveries of Harriot. On this he observes, " Atque hsec elegans et 

 perpulcrse speculationis sylloge, tractatui alioquin effuso, finem 

 aliquem et Coronida tandem imponito." Dr. Huttou mistranslates 

 when ('Hist. Alg. Tracts,' vol. ii.) he concludes from, these words that 

 Vieta only announces the theorem, " and for this strange reason, that 

 he might at length bring his work to a conclusion." Nevertheless, 

 Hutton's account is generally a very good one. 



(R. M.) De' Numerosa Potestatum pur arum, atqus adfectarum ad 

 exegesin resolutione tractatus. This work, first published, with Vieta's 

 consent, at Paris in 1600, has at the end a letter (herein before referred 

 to) from Ghetaldi to Michael Coignet, a Belgian mathematician, who 

 states that at his earnest entreaty Vieta had consented to allow the 

 work to be published, on condition that he (Ghetaldi) would take the 

 trouble of editing it. This letter mentions the seven books of the 

 Responsa, the Harmonicon Gceleste, &c. The numerose exegesis, as the 

 method herein explained was frequently denominated, passed through 

 the hands of Harriot, Oughtred, and Wallis, with some improvements, 

 but was so prolix, and required so much calculation, that when New- 

 ton's method appeared it gradually sank out of use. The late Mr. 

 Horner of Bath reproduced it, with a capital improvement in the 

 mode of making the successive computations, which will establish it 

 permanently. Recently, Mr. Thomas Weddle of Newcastle, author of 

 ' A New &c. Method of solving Numerical Equations,' has produced 

 the kindred method of finding the highest denomination of the root, 

 and correcting it by successive multiplications, instead of additions ; 

 a method which has considerable advantages when the degree of the 

 equation is high. To return to Vieta : when the root is irrational, 

 and any given degree of approximation is required, instead of using 

 fractions, the equation is found whose roots shall be ten, or a hundred, 

 &c., times the root of the given equation, which roots are then 

 extracted by the method within a unit. The introduction of our 

 notation for decimal fractions had not taken place at the time we are 

 speaking of, though wo should not be justified in drawing this con- 

 clusion from the mere fact of not finding it used by Vieta. From his 

 avocations perhaps, but more from the imperfect modes of communi- 

 cation (for there were then no scientific associations), he appears not 

 to have been perfectly aware of what was going on in other parts of 

 the mathematical world. So that it is impossible to say, at present, 

 whether some of the things which we know to have been discovered 

 before his time, may not have been, as far as he knew, the fruits of his 

 own investigation. " He neglects to avail himself of the negative 

 roots of Cardan " (but this however was done, on principle, and from 

 a determined refusal of all symbolical extension) ; " the numerical 

 exponents of Stifelius, instead of which he uses the names of the 

 powers themselves ; or the fractional exponents of Stevinus ; or the 

 commodious way of prefixing the coefficient before the quantity ; and 

 such like circumstances, the want of which gives his algebra the 

 appearance of an age much earlier than his own." (Hutton, ' Tracts,' 

 ii. 273.) He had however seen the exponents of Steviuus, and the 

 prefixed coefficients, for Van Roomen's problem, as given by himself, 

 contains both. 



