TABLE. 



TABLE. 



988 





tractatur canon mathematicus, seu ad triangula : item Canonion, c., 

 &c., &c., Londini, apud Franciscum Bouvier, 1589.' This publisher is 

 not mentioned by Ames. 



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

 with a third title-page, as follows : ' Fran. Vietaei Libellorum Suppli- 

 cuin in Regia magistri, insignia que Mathematici, varia opera mathe- 

 matics : in quibus tractatur Canon Mathematicus, seu ad triangula ; 

 item Canonion, &c., Parisiis, apud Bartholomseum Macaeum,' &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 mis-spelt, and letters and lines which are broken, in 

 all three : also by the fact that the second title-page, ' Francisci Vietsei,' 

 4c., exists, 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 ita extreme scarceness, a bibliographical curiosity ; we 

 have repeatedly examined 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 the 

 printing ; and the same stamping is apparent both in the fourth and 

 fifth. The canon matltematicia 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 frac- 

 tions 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 remarkable that in 

 the cosines Vieta does use a species of decimal notation, leaving a blank 

 pace instead of using a decimal point ; for, to an hypothenuse 100,000, 

 the base to an angle of 2-T 2' is what we should now write 91330'9. 

 There ia 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 calculations, for mention of which see Button's 

 ' History of Trigonometrical Tables,' prefixed to his logarithms, aud 

 inserted in his tracts. A short preface by Mettayer, prefixed to the 

 ' L'uiversalium Inspectionuin,' &c., states that Vieta found great diffi- 

 culties 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 in/tliciter editia, and hopes 

 that a second edition will be of better authority. 



Besides the three title-pages above mentioned, there must have been 

 a fourth ; for in the title of that which Delambre examined was the 

 motto l>ura el quittee, which certainly was not in either of the three 

 seen by us. The work has well obeyed the direction given : it has 

 lasted in silence, having never been described in catalogues or histories 

 till modem times. Copies seem to have been rare in Germany ; neither 

 Weidltr, Heilbronner, nor Kastner mentions it. Button never saw 

 but his own copy ; Montucla (in France !) never saw more than two 

 one in the royal library, and one sold at the Soubise sale (but it is not 

 in the catalogue of that sale), which the historian would fain have 

 bought, had nut a curieujc bid too high. And this was only by the 

 time Moutucla's second edition was written, for by the mention of it 

 made in the first edition it is clear that the author had never seen it. We 

 have examined in London at least eight copies. We have mentioned 

 the complaints which the author had to make against the printers : 

 M..ntucU states that Vieta bought in as many copies as he could. 

 There are several signs of something odd having taken place in the 

 printing ; and the following is worth mention : To one of the copies 

 we have seen (as well as to one of those in the Museum) is appended 

 one folio sheet, in correction of a mass of errors in one sheet of the 

 collection of formula: : this sheet is a separate publication, with the 

 date, 1679, and printer's name on it (J. Mettayer). 



i imitates Kheticus in his method of heading the tables, but in 

 addition u>tes the word sine, and calls the table of tangents Jacumla, 

 and that of secants fircunduama. He complains that elegant names 

 hare not been found, and states that he gets his denominations from 

 certain Rhapvidi (as he calls them ; it is not often that mathematical 

 tabulators are called rhapsodists), whom he does not name. In a later 

 work, the 'Reaponsa,' Ac. [ VIETA, in Bioo. Div.], published in 1593, 

 he names and objects to the words tangent and secant, which by that 

 time he bad seen. And he proposes to call the tangents proeinci or 

 aauint*, and the secants tranttinuiiui lines. 



As to the matter of Vieta's tables, it in worth notice that they must 

 have been made by independent calculation. They do not exhibit the 

 errors in the last tangents and secants which appear in all writings 

 to the more correct publication of Rheticus by Pitiscus. On the 

 additions made by Vieta to the theory of trigonometry we have not 

 here to speak : but we may simply say that they made the computa- 

 tion of a trigonometrical canon a much easier thing than it had 

 theretofore been. Delambre is quite right when he observes that the 

 ' Trigonometria firitannica ' of Briggs is altogether French in all that 

 relates to the non-logarithmic part of it. Had he known a little more 

 of Vieta, he could have reinforced his assertion. For Briggs's method 

 of solving equations which Uelambre (evidently not understanding it) 

 describes as an obscure mixture of division and extraction of roots, was 

 the slightly amended form of Vieta's numerical exegesis, to which we 



have referred in INVOLUTION AUD EVOLUTION. And as, by Gellibraud's 

 account, we trace the commencement of Briggs's labours to shortly 

 after the time when Vieta first published this exegesis, it is by no means 

 an unlikely conclusion that the power of trisection and quinquisection 

 given by this mode of solving equations first put it into his head to 

 construct the table. [INVOLUTION AND EVOLUTION.] 



Purbach and Regiomontanus had seen the advantage of adopting 

 decimal tables, though their use of the radius 600, &c., was a remnant 

 of sexagesimalism. It was rtserved for Maurice Bressius to show 

 himself a century behind his time, by publishing in his ' Metrices 

 Astronomies; Libri Quatuor," Paris, 1581, folio, sexagesimal tables to 

 every minute of sines, tangents, and secants, or as he calls them, sines, 

 adscripts, and hypothenuses. Thus, the radius being 60", the sine of 

 57 20' is given as 50 30' 34" ; and the adscript and hypothenuse as 

 1 sex. 33 34' 40", and 1 sex. 51 9' 44" ; 1 sex. meaning 60. Accus- 

 tomed as we are to look upon sexagesimal division as sacred to angular 

 and horary measure, we are apt to forget that the time was when other 

 subdivisions were rarely used in Europe. 



As yet we do not find the modern names o tanyent and secant. 

 These were introduced in 1583 by a young man of twenty-two years, 

 Thomas Finck, of Flensborg in Denmark (who died in 1656, aged 95), 

 in his ' Geometric Rotundi [sic] Liber X4V.,' Basle, 4to. His part in 

 the matter was quite forgotten, and has been recently revived (see Phil. 

 Mag., May, 1845). He introduces the words with expressions which 

 cannot * be interpreted otherwise than as a proposal of his own. to 

 which it must be added that no earlier use of these words has ever 

 been brought forward. The tables of sines, tangents, and secants, so 

 called, which Finck has introduced in his work, are to every minnte, 

 and to a radius of 10,000,000. Finck deserves a much higher name ~ 

 than he has got, for the contents of this work alone : there are other 

 writings of his, which we have not seen. He calculated his own 

 secants by a theorem which answers to the formulas 







Sec B = tan 6 + tan (45 ). 



In (1585-6) Clavius published at Rome, in quarto, his edition of 

 Theodosius, to which is appended a treatise on triangles, and a table of 

 sines, tangents, and secants, under those names to a radius of 10,000,000. 

 They were reprinted in the folio collection of his works, Mayence, 

 1611 ; and the table of sines only in his ' Astrolabium,' Home, 1593, 

 4to. It is clear, on inspection, that these tables are, so far as taugents 

 and secants are concerned, a reprint of those of Finck, in their pre- 

 liminary theorems, in their arrangement, in their omissions, and in 

 their errors, as well as in the new terms with which they are headed. 

 The name of Finck is suppressed as well as that of Rheticus ; both of 

 them were Protestants, and Clavius was a Jesuit, high in favour at 

 Rome. Delambre expresses his astonishment that Clavius, in recapitu- 

 lating the names of celebrated writers on dialling, should have omitted 

 Sebastian Munster. The fact was that Minister followed Luther. We 

 are not quite certain that a greater than Clavius was altogether exempt 

 from this laughable weakness. When Vieta suppresses the names of 

 his authorities, as above noted, calling them merely rhapsodists, we 

 may almost suspect that he wanted to avoid speaking of Rheticus and 

 Reiuhold ; for he was very intolerant. 



In (1591) Philip Lansberg published ' Triangulorum Geometriaj 

 Libri Quatuor,' Leyden, 4to, the first work known to Hutton in which 

 the words tangent and secant are used; and in 1592 Magini published 

 ' De Planis Triangulis Liber Unicus," Venice, 4to. Both these contain 

 full tables, takpn from Clavius ; and Magini is said to have repeated 

 them in his ' Primutn Mobile,' Bologna and Venice (1609). Magini, 

 who goes beyond Clavius in historical reference, wilfully suppresses 

 the name of Finck. 



We at first thought ourselves unable to give a date to the tables of 

 Stevinus, except within a few years, and conjecturally. That he pub- 

 lished his Arithmetic in 1585, and that Snell t collected many of his 

 works in Latin in 1605-8, are the facts which are supposed to mark 

 out the known limits of his career. The tables must have been pub- 

 lished after 1593, since Vieta's names for the tangent and secant are 

 mentioned ; probably long after, for Vieta's works were of very slow 

 travel. We ourselves believe fully that the Cosmographia, which 

 contains the tables, was never published until it appeared in what is 

 called Snell's collection (in 1608). These tables are to every minute, 

 to a radius of 10,000,000, and they are copies of Finck, Clavius, &c. 

 Recent researches in Belgium have made it appear that Stevinus was 



* " Erit AI tangens .... Sic vocare placuit [i. ., nobis] quia .... Damus 

 aliquid .... Regiomontano .... damus ctiam aliquid receptlB consuetudini. 

 Vcrum id non facile damus ut vcrba ea in usu rctineamus quibus elegantiora, 

 breviora, significantioro, veriora habeamus." And again ; " Sequitur .... qua) 

 vulgo canon tecundus, nobis canon tangentiuin dicitur : et canon hypotenusarum 

 Rhetico, nobis canon secantium vocatur." 



t It can be made very obvious that Stevinus was alive throughout the whole 

 of the printing of these two volumes (or five volumes bound in two). In the 

 very last page of the last volume (index excepted), the author excuses himself 

 for not fulfilling certain announcements, because he had not made up his mind 

 about the subjects of them, and the printer could not wait. And this after 

 referring to the places of the several matters in the very volumes which are 

 supposed to be the collection of the editor. Besides, Snell, the reputed editor, 

 was only seventeen years old when the work was published. 



