387 



VIETA, FRANCIS. 



VIETA, FRANCIS. 



(R. M.) Effectionum Geometricarum Canonica Recensio and Supple- 

 mcntum Geometries. The secoud of these works was first published 

 at Tours in 15$3. The former of these treatises is a collection of 

 problems in common geometry, intended to facilitate the solution of 

 problems of the second degree. The second treatise assumes the 

 construction of the conchoid of Nicomedes ; the finding of two mean 

 proportionals, the trisection of an angle, the inscription of a regular 

 heptagon in a circle, and the solution of the irreducible case of cubic 

 equations are made to follow. The last of these is contained in the 

 following proposition : " If there be two isosceles triangles, having 

 the equal sides of oue equal to those of the other, and the equal 

 angles of the second triple of those of the first, the cube of the base 

 of the first diminished by three times the parallelepiped under 

 the base of the first, and the square of the common side, is equal to 

 the parallelepiped under the base of the second and the square of the 

 common side." 



Pseudo-mesolabum. The term mesolabum was applied to any process 

 by which two mean proportionals could be found between two given 

 straight lines. By Paewdo-mesolabum Vieta means a process which, 

 though not limiting itself to Euclidean geometry, nevertheless is 

 effective on its own suppositions. A chord of a circle cuts a diameter, 

 and a perpendicular from one extremity of the chord cuts the 

 diameter produced, so that the part produced is equal to the chord. 

 This being the case, the segments of the chord are mean proportionals 

 between those of the diameter. When Vieta has finished his 

 pseudo-solution (merely ungeometrical), he then is ambitious of showing 

 how well he can reason falsely, and ends with a psettdo-theorema 

 (meaning one which is avowedly untrue, and given to be afterwards 

 exposed). Now, if a man will write a pseudo-method, which he 

 himself defines to mean no more than unallowed by Euclid, and 

 makes his treatise to end in nothing but a pseudo-theorems, (intended 

 to be false), not even the closest examination will prevent every one 

 from supposing that his pseudo-theorema is the finis atque corona of 

 his pseudo-method. 



(11. M., in which it is called Analytica Angularium Sectionum in tres 

 paries distributa). Ad Angulares Sectiones Theoremata /coOoXt/ccoTepo. 

 This is really Alexander Anderson's publication. Vieta sent him the 

 theorems, he found out the demonstrations, and published them, in 

 1615, at Paris, with a dedication to Charles, prince of Wales. Among 

 many trigonometrical theorems are here given some of the class of 

 which we shall presently speak with respect to Van Roomeu's problem. 

 The chord of an arc being given, the chords of its multiples and of 

 their supplements arc found. 



Ad Problem quod omnibus mathematicis totius orbis construendum 

 proposuit Adrianus Romanus Responsum. The circumstances under 

 which Vieta first saw this problem have been already stated from 

 Tallemant. It amounts to this : given the chord of an arc, to 

 express algebraically the chord of the 45th part of that arc ; but it is 

 given in the form of a proposed equation of the 45th degree. If 

 Vieta sat down at a window and solved several cases while Henry IV. 

 and the Belgian ambassador were talking in the room, it must have 

 been because he was then in full possession of his theory of angular 

 sections, and saw at once that Van Roomen's problem was a particular 

 case of it. But it must not be forgotten that the latter must also have 

 been in possession of the same or of cases of it. This answer of Vieta 

 is a full one, and appears to have been drawn up deliberately : he 

 gives the complete reduction of the problem, with a good deal of 

 what he must have supposed to be fun, but of a very ponderous and 

 sober character. He ends by proposing, in his turn, a problem, evi- 

 dently directed at Van Roomen, and by way of hit at his fearful 

 equation and enormous coefficients, he says, " Porro ad exerceudum 

 non cruciandum studiosorum ingenia, problema hujus modi con- 

 struendum subjicio." The problem is one of Apollonius, of which the 

 solution had been lost, Given three circles, to find a fourth touching 

 them all. 



Apollonius Gallus, seu exsuscitata Apollonii Pergcei irtpl eTrcKpuv 

 Geometria, first published by Vieta at Paris, in 1600, and addressed to 

 Van Roomen. It has, in the beginning, a Greek epistle, anonymously 

 addressed (perhaps by Van Roomen himself) $payi<i(TKCf Ovifry, which 

 is a presumption that the true pronunciation is Vieta. Van Roomen, as 

 appears by the introduction, solved the preceding problem by the 

 help of the hyperbola, on which Vieta rallies him in his manner, and 

 proceeds to a geometrical solution. He then gives geometrical solu- 

 tions of some problems which Regiomontanus had solved algebraically, 

 but professed himself unable to solve geometrically. He calls him- 

 self Apollonius Gallus, and Van Roomen, Apollonius Belga ; and from 

 that time it became a fashion for those who had done anything after 

 the manner of a particular Greek, to adopt the name of that Greek, 

 with an adjective of country annexed. Thus Snell, after his measure 

 of the earth, called himself Eratosthenes Batavus. 



Variorum de Rebus Mathematicis Responsarum liber octavus. This 

 book, first published at Tours in 1593, is preceded by an epistle from 

 Pet. Da., whoever he may be, which explains why it appeared. It 

 seems (at least it is so asserted) that there was at that time a great 

 excitement at Tours, not only among the educated, but even down to 

 the lowest of the people, about the quadrature of the circle, the prob- 

 lem of two mean proportionals, &c. ; and Pet. Da., who had seen 

 Vieta, and knew that he had a book on the subject lying by him, 



solicited and procured its publication. We have already spoken of the 

 first seven books, which, if they were ever written, are lost. This 

 book contains the history of, and remarks on, the method of finding 

 two mean proportionals, various modes of applying mechanical curves 

 to the quadrature of the circle, approximate solutions of the same 

 problem, and a collection of formulae for the solution of triangles, 

 with a short chapter on the calendar. 



Munimen adversus Nova Cydometrica. This was a refutation of 

 Joseph Scaliger's asserted quadrature of the circle, though the name 

 of Scaliger is not mentioned in it. This eminent scholar was exceed- 

 ingly angry, and attacked Vieta with much bitterness. But he 

 afterwards, according to De Thou, changed his tone, admitted his 

 error, and did justice to his opponent. Vieta himself had a high 

 respect for Scah'ger, as might be inferred from his suppression of the 

 name. If Isaac Casaubon is to be trusted, he thought most highly 

 even of the mathematical knowledge of Scaliger. In one of Casaubon's 

 letters to De Thou (p. 307 of the collection), he says, that on one 

 occasion he and a friend paid a visit to Vieta, and that, Scaliger's name 

 coming up in conversation, Vieta said, "I have so great an admiration 

 of that astounding genius, that I should think he alone perfectly 

 understands all mathematical writers, particularly those of the Greeks." 

 And he added, that he thought more highly of Scaliger when wrong 

 than of many others when right. 



Relatio Calendarii verb Greyoriani (Paris, 1600); Kalendarium Ore- 

 gorianum perpeluum, and Adversus Christophorum Clavium Expostu- 

 latio (Paris, 1602). We have said enough of these unfortunate works 

 in the preceding part of this article. The expostulation is preceded 

 by Greek verses addressed to Clavius. 



All the preceding works are contained, in the order in which we 

 have mentioned them, in the collected edition of Vieta's works, edited 

 by Schooten, and printed by the Elzevirs at Leyden in 1646. It 

 seems that Vieta's papers had either been almost entirely destroyed or 

 else exhausted; for though the Elzevirs, in 1640, advertised their 

 intention of printing such an edition (in the first number of the 

 ' Catalogus Universalis,' an annual book-list, printed at Amsterdam), 

 requesting those who had anything unpublished of Vieta's to com- 

 municate it, and giving the names (without dates, unfortunately) of all 

 that had been published, yet they could not print, six years after this 

 advertisement, one single treatise which did not appear in their own 

 advertisement as already known. We have yet to speak of two otber 

 works, both remarkable in their way, which are not in Schooten 's 

 collection. 



Harmonicon Caleste. This work has only been recovered in our 

 own day. Schooten's reason for not giving it was, that he could only 

 find an incomplete and inaccurate copy to print from ; but he says 

 that he had reason to suppose he should obtain a more complete copy, 

 which he promised to publish with other writings of Vieta : no such 

 work ever was produced. The very year before this preface of 

 Schooten appeared, Bouillaud, in the prolegomena to his ' Astronomia 

 Philolaica' (1645), says that Peter Dupuis (Petrus Puteanus) had lent 

 the manuscript to Merseune, and that some borrower, or more pro- 

 fessed thief (but which is not said^, had obtained it from Merseune, 

 and had never returned it. Some particular person is evidently 

 pointed at; Bouillaud says this borrower would neither restore it nor 

 a copy of it, and suspects that he meant to publish it as his own. 

 Bouillaud was a good authority in this matter : he was known to De 

 Thou, Schooten, &c., and Peter Dupuis was one of his colleagues in 

 the formation of the catalogue of De Thou's library, and perhaps, if 

 the story be true, got the manuscript out of that library to lend it to 

 Mersenue. This story has been repeated in many English writers on 

 this subject, from Sherburne down to Hutton, and always in the same 

 words. Some inquiries which the writer of this article made some 

 years ago at Paris through a most competent investigator, ended in 

 the assurance that it was in Bouillaud's handwriting in the Royal 

 Library at Paris, that he (Bouillaud) had himself lent the manuscript 

 to Cosmo de' Medici of Tuscany, which must have been after it was 

 recovered from Mersenne's honest friend, and of course after the publi- 

 cation of the 'Astronomia Philolaica.' Lately M. Libri (' Hist, des Sci. 

 Math, en Italie,' voL iv. p. 22) announces that there is an imperfect 

 manuscript in the Royal Library at Paris, and that the original manu- 

 script of Vieta (and an old copy, which however is mislaid) is in the 

 Magliabecchian Library at Florence (which confirms the last statement 

 of Bouillaud). He gives a short account of the contents of the Paris 

 manuscript, which contains various modifications of Ptolemy's theory, 

 and sufficient proof that Vieta well knew both the writings of Coper- 

 nicus and Tycho Brahe. Of the former he says that the excellence of 

 his system, if any, is destroyed by the badness of the geometry by 

 which it is explained ; and M. Libri states that he avows his oppo- 

 sition to the heliocentric system still more plainly in other places. 

 There is one conjecture which is worthy of some attention : we have 

 seen how imperfect is the evidence for attributing to APOLLONIUS the 

 opinion afterwards maintained by Copernicus; Vieta asserts that this 

 opinion was called Apollonian, not because Apollonius promulgated 

 it, but because the sun (Apollo) is in the centre of the system. 



It was said that the ' Harmouicon Cceleste ' was to be published, 

 but it has not yet appeared. 



Canon Matliematicus, seu ad Trianyula, cum adpendicibus, Lutetian, 

 apud Johunnem Mettayer, &c., 1579 ; to which is annexed, with a new 



