ON MATHEMATICAL TABLES. (13 



Dodson, 1742 (Antilogarithmic Canon). Numbers to elcvcu places 

 corresponding to logarithms from -00000 to 1-00000, at intervals of -00001, 

 arranged like a seven-figure logarithmic table, -with interscript differences, 

 and proportional parts at the bottom of the page. The changes in the fourth 

 figure in the middle of the column, both in the numbers and the differences, 

 are marked by points and commas, but not very clearly. Tliere is an intro- 

 duction of 84 pages ; and the tables occupy about 250 pages. 



In page ix of the Introduction an extract is given from Wallis, who states 

 that Harriot began, and Warner completed, a table of antilogarithms, which 

 was ready for press fifty years before. This was told Wallis by Dr. Pell, who 

 had assisted Warner in the calculation ; and Wallis mentions that he had 

 himself seen the calculation thirty years before, among Harriot's or Warner's 

 papers. Dr. Pell subsequently informed WaUis that the papers were in the 

 hands of Dr. Busby, and that he (Dr. Pell) hoped to publish them shortly. 

 Dr. Pell died in 1685 ; and at the time Wallis wrote Dr. Busby was 

 also dead, and the printing had not been begun. Speaking of this manu- 

 script De Morgan remarks : — " All our efforts to trace it, by help of published 

 letters <fcc., lead to the conclusion that, if existing, it must be among Lord 

 Macclesfield's unexamined manuscripts at Sbireburn Castle : this is by no 

 means improbable." See, however, some additional information and im- 

 portant remarks by De Morgan, ' Budget of Paradoxes ' (1872), pp. 457, 458. 



A list of thirty-six errors affecting the first eight figures in Dodson's 

 canon is given by Filipoavski in the preface to his 'Antilogarithms' (1849). 

 Mr. Peter Gray (' Insurance Record,' June 9, 1871) says that in 1847 he had 

 collected a list of 125 errors in Dodsou ; these he communicated to Shorteede, 

 and they were corrected in the plates of his tables (1849). Dodson's work 

 is unique of its kind, and it remained the only antilogarithmic canon for 

 more than a century after its completion, till in 1844 Shorteede published 

 the first edition of his tables ; in 1849 he published his second edition ; and 

 in the same year Filipowski's tables appeared. 



For hyperbolic antilogarithms (viz. e^ and e~*) see under miscellaneous 

 tables (§ 3, art. 25). 



The following are antilogarithmic tables described In § 4 : — 



AntUogarithmic Tables. — Gaediner, 1742, and (Avignon) 1770 [T. VI.] 

 (20 places); Dodson, 1747, T. XXXIII.; [Sheepshanks, 1844] [T. VII.]; 

 Shorteede (Comp. Log. Tab.), 1844 ; Shoetrede (tables), 1844, T. II., and 

 1849, T. II. ; FiLiPowsKi, 1849, T. I. ; Caelet, 1853 [T. II.], III. ; Stegmank, 

 1855, T. IL; HoIjel, 1858, T. VL ; Hxitton, 1858, T. IV.; Anontmohs 

 [1860 ?] (on a card) ; Paekhtjest, 1871, T. XXVIL, XXVIIL, and XXXV. 



Art. 15. Talks of (Briggian) Loganthmic Trigonometrical Functions, 



A general account of the introduction of Briggian logarithms is given in 

 § 3, art. 13 ; and Napier's ' Canon Mirificus' (1614), containing a Napierian 

 logarithmic canon, is described in § 3, art. 17. The first table of decimal 

 logarithms of numbers was published by Brtggs in 1617, and the first 

 (decimal) logarithmic canon by Gunter in 1620 (see below), giving the 

 results to 7 places. The next calculation was by Vlacq, who appended to 

 his ' Centum Chiliades ' in the ' Arithmetica ' of 1628 a minute logarithmic 

 canon to 10 places, obtained by calculating the logarithms of the sines &c. 

 of Ehetictjs. After the publication of his ' Ai-ithmetica ' in 1624, Beiggs 

 devoted himself to the calculation of logarithmic sines &c., and at his death 

 in 1631 had all but completed a ten-decimal canon to every hundredth of a 



