ON MATHEMATICAL TABLES. 75 



* 



1827, T. X. ; Bagay, 1829, T. XXII. ; Colemak, 1846, T. XXIV. ; Ixma^ 

 1871 [T. II.] 



(To 4 places) (viz. T. 74 of Eapeh) Croswell, 1791, T. V. ; Maskeltne 

 (Kequisite Tables), 1802, T. XV. ; Bowditch, 1802, T. XV. ; Andrew, 1805, 

 T. XIV.; Mackat, 1810, T. LI. ; Moore, 1814, T. XXV. ; Ditcom, 1820, 

 T. VII. ; Kerigan, 1821, T. XII. ; Stansburt, 1822 [T. II.] ; Kiddle, 

 1824, T. XXIX.; J. Taylor, 1833, T. XXXVL ; Beverley (1833?), T. 

 XVIII. ; Norte, 1836, T. XXXIV.; Gregory &c., 1843, T. VIII. ; Griffin, 

 1843, T. 41 ; J. Taylor, 1843, T. 35 ; Eumker, 1844, T. XXIV. ; Gordon, 

 1849, T. X. ; DoMKE, 1852, T. XL. ; Thomson, 1852, T. XIX. ; Eaper. 

 1857, T. 74. 



Proportional logarithms for every minute to 24'*, viz. log 1440 — log x. — ■ 

 (To 5 places) Galbraith, 1827, T. IX. 



(To 4 places) Stansbury, 1822, T. G ; Lynn, 1827, T. E; Gregory &c. 

 1843, T. XII. ; Gordon, 1849, T. XIX. ; Thomson, 1852, T. X. ; Kaper, 

 1857, T, 21A. 



Art. 19. Tables of Gaussian Logarithms. 



Gaussian logarithms have for their object to facilitate the finding of the 

 logarithms of the sum and difference of two numbers whose logarithms are 

 known, the numbers being themselves unknown ; on this account they are 

 often called Addition and Subtraction logarithms. The problem is therefore • 

 given log a and log 6, find log (a ± b) by the taking out of only one logarithm. 

 The utility of such logarithms was first pointed out by Leonelli, in a very 

 scarce book printed at Bordeaux in the year XI. (1802 or 1803), under the 

 title " Supplement logarithmique ;" but it met with no success. Leonelli's idea 

 was to construct a table to 14 places — an extravagant extent, as Gauss has re- 

 marked. The first table constructed was calculated by Gauss, and published 

 by him in vol. xxvi. (p. 498 et seq.) of Zach's ' Monatliche Correspondenz ' 



(1812) : it gives B and C for argument A, where A = log a?, B = log 1 1 + - ) 



C = log (1 + x), so that C = A + B ; and the use is as follows. We have 

 identically — 



log (a + b) = log a + log (l + ^ 



= log rt -f- B I for argument log - j. 



The rule therefore is, to take log a, the larger of the two logarithms, 

 and to enter the table with log a — logb as argument, we then have 

 log (a + b) = log n + B, or, if we please, = log 6 + C. For the difference, 

 the formula is log (a — b) =log b + A (argument sought in column C) if 

 log rt — log 6 is greater than -30103, and = log 6 — A (argument sought in 

 column B) if log a — log 6 is less than -30103 ; there are also other forms. 

 Gauss remarks that a complete seven-figure table of this kind would be very 

 useful. Such a table was formed by Matthiessen ; but the arrangement is 

 such that very little is gained by the use of it. This Gauss has pointed out 

 in No. 474 of the ' Astronomische Nachrichten,' 1843, and in a letter (1846) 

 to Schumacher, quoted by De Morgan. Gauss's papers on logarithms and 

 reviews of logarithmic tables from the ' Gottingische gelehrte Anzeigen,' 

 * Astronomische Nachrichten,' &e., are reprinted together on pp. 241-264 of 

 t. iii. of his ' Werke,' 1866. Of these pp. 244-252 have reference to Gaussian 

 logarithms and contain reviews of Pasquich, 1817 (§. 4), and Matihiessen, 



