. > 



TABLK. 





(how, Montucla does not state ; probably by private communication, 

 or perhaps Montucla ought to have cited the ' Fortsetzung der Rechen- 

 kuust,' 1783) that thi.s passage of Bramer had led him to look at some 

 old tables which he had bought, and which had lain by neglected. 

 And in these old tables he says he found the above work of Byrgius. 

 This occurs in the second edition of Montucla 's History, vol. ii. p. 10 ; 

 see also Kastner's History, vol. ii. p. 375, and vol. iii. p. 14; and 

 Delambre, ' Hist, de 1'Ast. Hod.,' vol. i. pp. 560-566. It will be noticed 

 that Byrgius did not publish till six years after Napier ; so that in all 

 probability Kapler is first in point of invention as well as publication. 

 Byrgius's system begins with as a logarithm and 10" as a number ; for 

 every increase of the logarithm by 10, the number is multiplied by 

 1-0001; so that 10m has for its number 10 (1-0001)"-'. This is 

 undoubtedly a rude table of logarithms, or rather of numbers to loga- 

 rithms ; and since Byrgius carried it up to 230270, the number to 

 which is 9DXO; 1 ' '!'!'. lie certainly secured the main advantages of 



hmic calculation. 



Delambre, who has in general treated Napier with fairness, has in 

 one instance formed a conclusion on premises so strange, that we 

 hardly remember the like in any historian. (' Astr. Mod.,' i. 287-291.) 

 Ursxis Dithmarsus, the pupil of Byrgius, in his work of 1583 already 

 mentioned, hints at some method by which he can calculate sines 

 even in numbers, and arithmetically ; and afterwards he talks of 

 doing this in common numbers, by inscription, and in logistic num- 

 bers, by section of the angle. What he means Delambre cannot 

 understand, neither, we should suppose, can any one else : but, 

 seeing that he is a pupil of Byrgius, who afterwards made an 

 antilogarithmic table. Delambre interprets him as possibly eon- 

 vtyini; the ideas of Byrgius to Napier. That is to say, certain unin- 

 telligible professions of Ursus, who does not even attribute them to 

 Byrgius, and in which DeUuubre himself, with all his knowledge of 

 logarithms, can neither see logarithms nor anything else, may have 

 given the first idea of logarithms to Napier, or may furnish presump- 

 tion that Byrgius gave that idea in some other way. One would 

 really almost suppose that Delambre had been misled by the 

 antithesis of comm<m and Iw/istic numbers, the usual terms of the 

 day for integers and fractions. Thus, Kepler begins the Kudolphin 

 table* by a chapter on the loyiitici he means to use, and warns the 

 l-yitta that he will express the distances of the planets by dividing 

 that of the earth and sun into 100,000 parts. 



The following is a summary of the ground of presumption that 

 N ipi'-r's system was well advanced, in thought at least, if not in actual 

 calculation, before 15S8. Kepler testifies that Napier gave Tycho 

 Brahe strong hint* of what wag coming in 1594. Now in 1593 Napier 

 published the tirst edition of his interpretation of the Apocalypse ; and 

 there is no reason to doubt his declaration that he considered this as 

 the main business of his life, and mathematics as only secondary. 

 This heavy work shows what was his main employment in the years 

 preceding 15U4, in which year his system was well advanced : which 

 we take to make probable, all things put together, that it was well 

 advanced at least four or five years earlier. Napier returned from 

 travel and settled down to study in 1571, and probably he soon began 

 his researches. It is to be noticed that his system of logarithms did 

 not stand alone. He informs us that he tried many plans to facilitate 

 calculation, some of which might perhaps be published : thi- 

 iti thu preface to his canon of logarithms. Accordingly, in 1617, the 

 'Kabdologia' appeared [NAPIER'S BONKS]; and this was only <'iio. of 

 three plans, of which the others are but named. Napier was not in 

 haste to publish. His son informs us that the second tract, the 

 ' Constructio,' was written years before the name logarithm was 

 invented ; and this description must have followed the actual calculation 

 of much, if not all, of the canon. 



1B24. Briggs, ' Arithmetica Logarithmica," London. Logarithms 

 (decimal) to fifteen places, from to 20,000, and from 90,000 to 

 Hi), with interscript differences. After his death, in 1631, a 

 nt was, it is said, made by one George Miller ; the Latin title and 

 explanatory parts were replaced by English ones ; ' Logarithmicall 

 Arithmetike, &c. Wo must doubt the reprint of the tables, and think 

 that they were Briggs's own tables, with an English explanation pre- 

 fixed in place of the Latin one. Wilson (in his History of Navigation, 

 prefixed to the third edition of Robertson) says that some copies ol 

 Vlacq, of 1628, were purchased by our booksellers, and published at 

 London with an English explanation premised, dated 1631. Mr. Babbage 

 (to whose large and rare collection of tables we were much indebted 

 in the original article) has one of these copies ; and the English explana- 

 tion and title is the same as that which was in the same year attached to 

 the asserted reprint of Briggs. We have no doubt that Briggs anc 

 Vlacq were served exactly in the same manner. Some copies of Briggs 

 hare, after the ' Finis,' another chiliad of logarithms, headed ' Chiliaf 

 centenma prirua,' and arranged like the preceding ones ; also, a page o: 

 square roots, to eleven decimals, from 101 to 200. In some copies 

 the page of errata follows the additions. 



It24. Benjamin Ureinus, 'Magnus Canon Triangulorum Logarith 

 micus ex voto et consilio Illustr. Neperi p.m. novissimo,' Cologne (a! 

 the end, Berlin), Ito. This is an extension of the urvjinal Naperian 

 logarithms to eight figures, and to every ten seconds : the last places 

 are much more correct. The arrangement is entirely that of Napier 

 with the addition of the tabular differences, hea<lr<l ,/;/,>: ,.r I)., th> 



\KT* AMI Ml. HIV. V"t. Ml. 



TABLE. 991 



leading of the logarithmic tangents being different, (contraction of 

 differentia). There is no preface ; but the ' Trigonometria cum magno 

 jogarithinoruin cauone,' published in the next year at Cologne by 

 Jrsinus, contains the necessary explanations. It is bound up with the 

 canon in the copy we have seen ; and probably the canon was not 

 ssued without it. 



(1624.) John Kepler, ' Chilias Logarithmorum,' Marpurg, and (1625) 

 Supplemeutum . . . continens Prsecepta de eorum Usu.' These were 

 reprinted by Maseres, in vol. i. of the ' Scriptores Logarithmic!.' See 

 a very full account of them also in the first volume of Delambre's 

 rlistory of Modern Astronomy. The logarithms are strictly Naperian, 

 0( 1 )1000, but four ciphers are put to the end of each number, to make 

 ,he radius ten millions. There are five columns, of which this is a 

 pecimen : 



44 30' 26" | 7010000 | 16 U 49 26 s | 3552474 | 42 4' 



The number here is 701, and the sine being 7010000, the angle is 

 44" 30' 26". The logarithm is 3552474. And if 1000 represent 24>>, 

 then 701 represents 16 h 49 26 s ; while if 1000 represent 60, 701 

 represents 42 4'. There are also interscript differences. And thus 

 Kepler originated the species of table now called logistic. 



1625. Wingate, ' Arithmetique Logarithmique,' Paris, (reprinted at 

 Gouda, in 1628, according to Murhard). Wingate was an Englishman 

 who first carried Briggs's logarithms iuto France. The work was 

 reprinted in England, in the same year. Dodson, Hutton, Ward, &c., 

 say the year of the French publication was 1624 ; but Lalande and 

 Delambre knew of none previous to lb'26, and a copy of the last date 

 which we have examined bears no mark of being a second edition, and 

 refers to nothing as published before, except a tract on the rule of 

 proportion (Guuter's Scale). The logarithms are from Gunter. 



But we have found a copy dated 1625, and we are satisfied, from the 

 date of the "privilege" and other things, that this was the first edition. 

 That date is November 4, 1624, and the printing is stated as having 

 been finished April 4, 1625. This edition and that of 1626 are from 

 the same types, except in their title-pages and a page or two of the 

 postfixed explanations. The latter has also a further appendix of 

 differences and some points of explanation. It has also additional 

 (perhaps, for the same thing may have been torn out of our copy of 

 1625) a folding sheet of mean proportionals between 10 and 1. The 

 contents are, seven-decimal logarithms of numbers 0(1)1000 with 

 interscript differences ; and 0(1')45 logarithms of sines and tangents, 

 with the complemental parts on opposite pages. These logarithms are 

 from Gunter. This is the introduction of Briggs's logarithms into 

 France ; that of Napier's was made, as noted, by B. Vincent. 



(1626.) Henrion's 'Logarithms,' Paris. (Dodson, followed by 

 Hutton.) Lalande knew nothing of this work, nor Delambre. All we 

 can learn is from Dechales, who states that Henrion wrote on the pro- 

 portional compasses in (1623), reprinted in (1681), and on the rule of 

 proportion (which we take to be Gunter's scale) in (1626) ; and that 

 this last work contains logarithms of numbers up to 2000. 



1626. 'Tables des Logarithmes pour les nombres d'una 10000, com- 

 posdes par Henry Brigge. A. Goude. Par Pierre Rammaseyn.' The negli- 

 gence of a bookbinder enables us to clear up some confusion, in rather a 

 singular manner. Sherwin states that he examined his table by one of 

 Vlacq's, in large* octavo, printed at Gouda in 1626, of which table we 

 find no other mention. The table before us corresponds in every respect, 

 except that there is no author's name ; but no one except Vlacq can be 

 mentioned, who was in the least likely to have printed logarithms at 

 (rvuda in or about 1626. At one time we thought that this table was 

 the original of the long series of small tables called after Vlacq ; but 

 this was a mistake (see 1625, Gellibrand), and the mistake was partly 

 due to the following circumstances. This table, Gouda, 1626, having 

 the title, when not cut away, above described, and which we have also 

 seen with a Dutch title and preface, is the table which is always bound 

 up at the end of ' Sciographia, or art of Shadowes .... by T. W[ells], 

 Esq.,' London, 1635, large octavo. It has a preface by Gellibrand, 

 who was thus accessory to the introduction of one small table by Vlacq 

 in the very year in which he (Gellibrand) published another small table, 

 the reprints of which were destined to be called by Vlacq's name. 

 That the book was intended to have these logarithms bound at the end 

 is evident from every page of it. Now the fact stands as follows : 

 A sufficient number of copies of the logarithms having been procured 

 from abroad, the binder was directed to caucel the title-page of the 

 logarithms, and to append them to the work. Accordingly, most 

 copies have no title to the logarithms, which look quite like part of 

 the work. But in some copies the binder has not cancelled as required ; 

 we have obtained two (since our first article was written), and there is 

 another in the library of the Royal Society. But in all three copies 

 the title of the logarithms is cut half way up with knife or scissors, as 

 a direction to the binder to caiicel it. One of our copies has this 

 Dutch title-page to the table, ' Henrici Briggii Tafel van Logarithm! 

 voor de Ghetallen van een tot 10000. Ter Goude . . . 1626.' And 

 the work (though the same impression as beforej has a different title- 



* The work we shall describe would not now be called large octavo ; but 

 may have been so in Sherwin's eyes. The octavo sizes (and indeed all the sizes) 

 varied as much U3 they do in our own day, when between post, demy, royal, 

 >Vr., we hardlv know what is and what is not octavo. 



3s 



