

TABLE. 





The person who ia used to accurate descriptions of books 

 might possibly, without this warning, throw away the thin folio we 

 are speaking of, under the idea that it could not be in any sense an 

 edition of the Opua Palatinum : which in fact it is not, though it is an 

 elition of all that was corrected. The 86 pages of reprint are easily 

 distinguishable by the inferiority of paper and type.* The title-page 

 of the thin book is a sort of fly-title, without date, &e., on the first 

 page, as follows : ' Georgia Joachiini Rhetici magnus cauon doctrinse 

 triangulorum ad decades secundorum scrupulorum, et ad partes 



'0 00000. Recens emendatus a Bartholomaeo Pitisco Silesio. . . . 

 Addita eat brevis commonefactio de fabrica et usu hujus Canonis 



i hie, una cum brevi coinmonefactione etiam separatim ab 



opere Palatino venditur. In bibliopoleio Harniscbiano." And the 

 'ia has a title page of its own, as follows : ' Bartholomsei 

 Pitisci (iriinbergensis Silesii Brevis et Perspicua commonefactio de 

 fabrica et usu magni canonis doctrina; triangulorura Georgii Joachiini 

 Klu-tici. Xeoatadii Typis Nicolai Schrarnmii MDCVII.' It thus 

 appears that the date is 1607, which no one has yet noted, except 

 Kastner, copying an older description, apparently without any distinct 

 separate knowledge of what he was describing. 2. Pitiscus published, 

 Frankfort, 1613 (misprinted on two of the titles 1513, by omission 



>, folio, the tables of Rheticua by which himself was enabled to 

 make the preceding corrections, under a long descriptive title begin- 

 ning with ' Thesaunis Matheinaticus.'* The contents, described in 

 modern language, are : sines to every ten seconds and to fifteen 

 decimals, with first, second, third, and sometimes more differences ; 

 those of the first and last degrees, also to fifteen places, and to every 

 second ; the fundamental sines, from which the rest were calculated, 



nty places : the sines of every 10th, 30th, and 50th second in 



-t 35 minutes to 22 places (this last table was done by Pitiscus 

 1 in July, 1U1 3, very shortly after the publica- 



' the Thesaurus. 

 \Viii-n v..'come to reflect, we find that the tables of liheticus did 



ike such an epoch in the history of these things as might have 

 been expected. The ten-minute canon, 135 1 , which we have described, 

 and of which the memory was almost lost, introduced the secants, 

 completed the system, and suggested to Vieta both the extension and 



in. II. ul Kheticm published his own large table before his 

 death, in 1 -lit have been otherwise: but deferred as this 



publication was, partly till 1 596, seventeen years after Vieta's Canon 

 had appeared, and partly till 1613, the year before the publication of 

 logarithm*, it turned out that the impulse had already been given from 

 oilier quarters. The next great tables of sines which were produced 

 were the work of Briggs, who was, as we have seen, exclusively the 

 follower of Vieta in this part of the matter. The labours of Rheticus 

 became little more than a tradition, though Vlacq used the last halt' of 

 his quadrant in the construction of logarithmic sines. Vossius, 1650, 

 knew nothing definite of the tables except the Thesaurus, and that 

 only in time to insert it in the additions to his work. Sherburne, 



lias not a word of tables. Briggs hardly mentions Rheticus; his 

 biographers not at all. The Je.niit Blancanus omits him as a con- 

 demned writer; and it is to be noticed that he was, as to this 

 matter, worse off than Copernicus himself; for he was ilamnatua 



ind the absolute prohibi- 



imst all his writings must have tended to the oblivion into 

 which his name fell. AVeidler, 1711, writing in the University of 



ubrrg, in which Hheticm taught, had not seen the Opus Pala- 

 tinmn, and knew nothing of what 1'itwcus had done. In the Berlin 



irs for 1786, John Bernoulli (the younger) revived the know- 

 ledge of the ' Opus Palatinum ' and the ' Thesaurus ; ' and Lalaude had 



usly come at some statement to the effect that I'itiscus had 



ceived instructions to correct the former. But Bernoulli knew 



i% of these corrections, and nothing was known until chance 



threw a copy of the corrected Opus Palatinum into the hands of 



iii'd it in a paper printed in the fifth volume of the 



the Institute, 18H4-. Uelambre gave an accurate account 



iMing of the second volume of the' Histoire de 1'Astronomie 



u'la had given nothing but mistakes. Mutton knew 



alii. Kastner (1796, who would have got much more 



The eorrected copies of the Work, thick or thin, may he distinguished from 

 the uncorrected ones in a moment, as follows* Look ut the bottom of 

 at the running title* of the columns. The uncorreclcd copy will have, ai it 

 ought to hare, 



Basii Differentia Hypolhenusa. 



Hut the corrected copy (qni> ctutotlict ipioi custoilti!) will have, as it ought not 



to bare, 



Ilypotheuusa Differentia Basis. 



t The copies of the two works, the ' Opus PalatinuoV and the ' Thesaurus,' 

 which belonged to Deiamhre, were bought at the sale of his books by Mr. Uabhagv. 

 The copy of the ' Thesaurus ' is curious : it once belonged to De Thou, and was 

 lK>quealhed to Dclambre by Liilande. It sold at the sale for 2 16 francs; the 

 Opus Palatinum ' for CO francs. Mr. Uabbutre has aho a copy of the corrected 

 >h thm volume) He informed us that, in 1828, Kcuss, the librarian at 

 on, and the indefatigable editor of the ' Hepcrtorium Commentationum,' 

 *c., the most complete digest of scientific transactions which exists, was 

 alt ,'gcther ignorant of tlie existence of any corrections of the Opus Palatimun. 

 This is * truly singular Instance of the slowness with which bibliographical 

 information spread*. 



TABLE. (Kjy 



credit if he had given a proper name to his valuable work of biblio- 

 graphy, instead of calling it a history of mathematics) has a detailed 

 account of all the matter, except the corrections of the Opus Palatinum, 

 on which he could only quote from a periodical of 1789. 



In (1599) Pitiscus published his own work on Trigonometry, with 

 tables, generally to seven places, and having intervals which may be 

 described, as presently noted, by (1") 1' (2") 10' (10") 1 (!') 45". 

 The edition of 1608, now before us, has of course the corrected tangents 

 and secants. It was reprinted again in (1612), and Dechales mentions 

 a reprint, by Henrion, in (1623). 



Pitiscus will always be remarkable as the priest who wished that all 

 his brethren were mathematicians,* to make them manageable and 

 benevolent. 



Among the non-logarithmic tables, which were published after the 

 invention of Napier turned all the calculators another way, was that of 

 Schooten, ' Tabulaj Sinuum,' &c., Amsterdam, 1627, a complete canon 

 to seven places, in a pocket volume with pages of two inches by four. 

 It is often said to contain no error ; but we believe the author's own 

 assertion in the preface is the source of this opinion : Mutton found 

 many errors in the last figures. There were two principal editions, 

 one with explanations in Latin, the other in Dutch. Lipenius says * 

 this was reprinted in (1638 ?) and we know there is an edition of 

 (1672) at Rouen, and of 1683 at Brussels. Editions are mentioned of 

 (164ii) and (1664), and also a Spanish edition, Brussels (1683), and of 

 1688 a\ Amsterdam, from different type. Joh. Meyer's tables, 

 Strasburg (1619), contain sines, tangents and secants, squares, and 

 cubes. Those of Adrian Metius (1633) give a complete canon, to 

 minutes, to seven decimals. In (1627) Snell published his ' Doctrina 

 triangulorum canonica,' Leyden, containing a complete canon to every 

 minute, and to seven places. Cruger's ' Synopsis Trigonometric,' 

 Danzig (1612), gave a five-decimal canon to minutes. Albert Girard's 

 ' Tables des sinus,' &c., Hague 1627, are to five decimals; there was 

 a Dutch reprint in (1629). Adrian Romanus gave tables (Delambre, 

 'Astr. Mod.,' vol. ii. p. 35) in (1609); they were taken from Clavius. 

 The greater part of the contents of this paragraph are taken from 

 different sources, and not from the books themselves. We might 

 mention some anonymous tables from various catalogues, but anony- 

 mous works of this kind are so rare that we always suspect them. 

 One, however, now before us, deserves mention. It is a thin quarto, 

 Wurtzburg (Herbipolis), 1625, announced as intended for the students 

 of the University. It contains a semi-quadrantal table of sines (only), 

 and ia entitled ' Canon sinuum ad decempedam accomodatus.' The 

 table is to a radius of 10,000, but the four places of each sine are 

 severally called radii, pedes, digiti, grana; and are so headed in 

 every column. The degree is divided into 12 parts, each of which is 

 called a minute. 



Alsted's ' Encyclopaedia,' 1649, the earliest work which has bulk 

 enough to be compared with modem works of the same name, gives 

 nothing more than a canon to degrees and seven decimals, with another 

 to ten miuutes and five decimals. The name only of logarithms is 

 mentioned, and an insufficient definition given. 



j; 6. Tables of Lvyarttkma, Before we enter on this subject we 

 shall give a hint which may be worth the attention of future com- 

 pilers, though in joining together two articles of older date than 

 the additions we have not been able to take advantage of it to any 

 great extent. Systems of tables may be arranged and spoken of either 

 By their K'jtn-ces or by their forms. Thus in thirfking of old books of 

 logarithms we may have to ask whether they came from Vlacq or direct 

 : iggs ; we also want to know the number of figures, the arrange- 

 ment, &c. The tables of logarithms might with some trouble be divided 

 into sets, those in each set being lineal descendants of their predecessors. 



Our mention of different works will be found, as to length and 

 minuteness, much out of proportion to their celebrity, in many cases : 

 this cannot be avoided when we have information to give which is nut 

 commonly found. 



1614. Napier, ' Mirifici Logarithmorum Canonis Descriptio, Ejusque 

 usus, in utraque Trigonometria, ut etiam in omui Logistica Mathematica, 

 Amplissimi, Facillimi, et expeditissimi explicatio. Authore ae Inveu- 

 tore, loanne Nepero, Barone Merchistonii, &c. Scoto. Edinburgi, Ex 

 ofticina Andreic Hart, Bibliopola), CIO.DC.XIV.' 4to. Sints and Naperian 

 logarithms of sines and tangents, to every minute and seven decimals. 



It must be specially noted that the logarithms which Napier himself 

 published are not precisely those which are now called Naperian ; 

 that is, they are not the simple logarithms to the base e = 2'7182818. . 

 As the sines increase, his logarithms decrease. As he uses no decimal 

 point, both his sines and logarithms are integers, the former to a radius 

 of 10 millions. And if N be a sine and L the logarithm of it, as they 

 stand in Napier, the equation connecting them ia 



N = 10,000,000 f U1 .""".""" 



* In his preface he says, " Mansuetudo autein, bone Deus, quantum ft quam 

 rarum est Theoiogorum ornamentum ! Et quam optandum esset hoc siculo, 

 omncs Theologos ease mathematicos, hoc est, homines tructabilea et mansuetos." 

 lYrhaps the union of the characters of divine and mathematician gives a 

 peculiar right to speak well of the latter ; for Barrow says, " Tenerrimu: frontis 

 et stomachi robustissimi, aut si mavis, pudentissimum tit'dmjue patientissimum 

 genus hominum sunt mathematici." We accept the si macis, for there is no 

 saying how the moderns might translate the first epithets. 



