ON MATHJiMATlCAL TABLES. 53 



No further calculation of logarithms of numhers took pluco till the end of 

 the last century, when the great French manuscript tables (the ' Tables 

 Du Cadastre' — see description of them below) were comiKited under the 

 direction of Prony. These, as is well known, hare never been published. 



In 1794 Vega published his ' Thesaurus Logarithmorum Completus/ which 

 contains a complete ten-figure table from 1000 to 101,000. It was not, how- 

 ever, the result of a fresh calculation, but was cojued from Ylacq, after ex- 

 amination and correction of many errors (see Vega's ' Thesaurus,' § 4). 



In 1871 Mr. Sang published his seven-figure table of logarithms of numbers 

 to 200,000, the second half of which was obtained by a new calculation. It is 

 thus seen that, with the exception of the Tables du Cadastre, and the second 

 half of Mr. Sang's table, every one of the hundreds of the tables that have 

 appeared has been copied from Eriggs or Vlacq ; and considering the enor- 

 mous number of calculations in which logarithms have been employed, 

 and the vast saving they have eftected of labour, it must be admitted that 

 (apart from the fact that the great tables of Ertggs and Ylacq remain 

 still unsuperseded) great historical interest attaches to the original com- 

 putation. 



Ylacq's ten-figure table contains about 300 errors (leaving out of consi- 

 deration errors atfecting only the last figure by a unit). The greater number 

 of these were found cither by Vega, or by Lefort from comparison with the 

 Tables dtt Cadastre : complete references and a small subsidiary list are 

 given in the ' Monthly Notices of the Royal Astronomical Society ' for May 

 and June 1872. While speaking of ten-figure logarithms, we may men- 

 tion PiNETo's table described below ; but Vlacq (1628 or 1631) and Vega 

 (1794) are far preferable : they are unfortunately so rare, however, that not 

 many besides those Avho have access to a good library can make use of 

 them, and, except to a few, the employment of ten-figure logarithms in their 

 most convenient form is denied : we much prefer Vlacq to Vega for use, as 

 the arrangement is more convenient. 



To return to the history of logarithmic tables to a less number of figures. 

 In 1625 Wingatc published at Paris his ' Arithmetique Logarithmetique,' con- 

 taining seven-figure logarithms to 1000, and logarithmic sines and tangents 

 from GuNTER (see De Morgan ; the full title of the Gouda edition of Wingate 

 (1628) is given by Hogg, p. 408), thus introducing Briggian logarithms into 

 France ; and in 1626 appeared both Henrion's 'Traicte' (§ 4) at Paris, con- 

 taining 20,000 logarithms from Briggs and Gunters logarithmic sines and 

 tangents, and De Decker's ' Nieuwe Telkonst ' (§ 4) at Gouda, giving also 

 logarithms from Briggs and Gunter; then Vlacq began to calculate logarithms, 

 and brought them in 1628 to the state in which they now are. There is a table' 

 of logarithms in Norwood's 'Trigonometric' (1031) ; and in 1633 appeared 

 Eoe's table (§ 4), in which the first four figures of the logarithm are printed 

 at the top of the column. This was an advance halfway to the modern arrange- 

 ment, which was introduced in its present form in John Newton's eight-figure 

 table (1658). On Fatohaber, 1631, and Oughtred, 1657, see § 4. 



Tables of seven- and five-figure logarithms are too numerovis to notice 

 here separately. The cliief line of descent is Briggs, Vlacq, IIoe, perhaps 

 Newton, the editions of Sherwin, Gardiner; and then both Hutton and 

 Callet bring down the succession to the present day. A very fair account 

 of several logarithmic tables is given by Hogg in section iv. " Elementar- 

 Gcometrie " (B) of his ' Ilandbuch,' who added to the books described in this 

 part of his bibliography a description of the contents. But the reader must 

 be warned against trusting his accounts, except where he is clearly describing 



