OS MAXilEMATiCAL TAULES. 57 



20 powers of r^ to 28 figures. It may be mentioned that the logarithms of 



10,000 primes were calcuhitod to 19 places, and tlie natural sines for every 

 minute (centesimal) to 22 places. This account of the ' Tables du Cadastre ' 

 has been abridged from a memoir by M. Lefort, in t. iv. (pp. [123]-[150]) of 

 tlie ' Annales de I'Observatoire do Paris ' (1858), where an explanation of the 

 methods of calculation, with the formulas &c., is given. The printing of the 

 table of natural sines was once begun. M. Lefort says that he has seen six 

 copies, all incomplete, although including the last page. De Morgan also men- 

 tions that he had seen some of the proofs. Babbage compared his table with 

 the ' Tables du Cadastre ;' and M. Lefort has given, by means of them, most 

 important lists of errors in Vlacq and Beiggs ; but these are almost the only 

 uses that have been made of tables the calculation of which required so great 

 an expenditure of time and money. " In 1820," says De Morgan, " a dis- 

 tinguished member of the Board of Longitude, London, was instructed by our 

 Uovernment to propose to the Board of Longitude of Paris to print an abridg- 

 ment of these tables, at the joint expense of the two countries. £5000 was 

 named as the sum which our Government was willing to advance for this 

 purpose ; but the proposal was declined " (Peuny Cyclopaedia, Article 

 " Prony "). The value of the logarithms of numbers is now materially 

 lessened by Mr. Sang's seven-figure table from 20,000 to 200,000 (see 

 Saxg, 1871, in this Article). 



Hogg (p. 241) gives the title " Notice sur Ics grandes tables logarithm, et 

 trigonom. calculees au Bureau du Cadastre," Paris, an IX. (=1801\ and 

 on the subject gives a reference to Bcnzeuberg's ' Angewandte Geora.' iii. 

 p. 557. 



Hill, 1799. Pive-figure logaritlims from 1 to 100 and from 1000 to 

 10,000, printed at full length, and with characteristics- — no difiercnces 

 (pp. 23-^8). The author was an accountant; and the table was intended 

 for commercial purposes, its use in which is explained in the book, 



Reishammer, 1800. These are commercial logarithms, intended for 

 merchants &c. When the number is less than unity, the logarithm of its 

 reciprocal (which the author calls the Jor/aritJime nef/afif)is tabulated; if 

 greater than unit}', its own logarithm (lor/ariihme 2^ositif). The first table 

 (which only occupies 2 pages) gives the locjarWimes neijatifs of the frac- 

 tions from ^i-y to 1, at intervals of -j-J^y ^o ^ places (the characteristics are 

 given, and not separated from the other figures). This is followed by the 

 principal table, which occupies 117 pages. On the first page are given the 

 lo(jarilhmcs nc</atlfs of 128 fractions, viz. of all proper fractions whose deno- 

 minators arc 60, 48, 40, or 32, arranged in order thus : — ^'^, -L, J^, ^L, J-y , 

 • • • -as' ou' ■§&• '^^^ r^st of ^^c logarithms are j^ositifs ; and the argu- 

 ments proceed from 1 to 111, with the 128 fractions just described inter- 

 mediate to each integer. Thus we have l^^j, l-J^, &c., 2-j,\, 2-Jyr, &c., as 

 arguments. The arguments then proceed from 111 to 207 at intervals of 

 •.,15^, from 207 to 327 at intervals of ^, thence to 807 at intervals of §, and 

 from 808 to 10,400 at intervals of unity, — all to 5 places. The characteristics 

 are given throughout. A page of proportional parts is added. 



There arc besides several small tables, to facilitate the calculations, only 

 one of which requires notice. It gives on a folding sheet the 128 fractions 

 previously described, expressed as fractious with denominators 100 and 10, 

 and also (when the numerator is integral) expressed as fractions with de- 

 nominators GO, 48, 40, 32, 30, 24, 20, IG, 15, 12, 8, 6, 5, 4, 3, 2. Thus -^ 

 = 10y''^-^100, and=l77'.j-rl0; as it cannot be expressed in lower terms 



