76 . REPORT — 1873. 



« 

 1818 (below). The largest tables are Zech (reprinted from Hulsse's edition 

 of Vega) and Wittstein, which answers the purpose Gauss had in view the 

 best of all : there is also a good introduction to the latter (in French and 

 German), explaining the use and objects of the tables. 



Whenever in this Eeport the letters A, B, C are used in the description 

 of Gaussian logarithms, they are always supposed to have the meanings 

 assigned to them by Gauss (which are explained above), unless the con- 

 trary is expressly stated. Of course all Gaussian tables have reference to 

 Briggian (not hyperbolic) logarithms. 



Leonelli, 1806. This is the German translation of Leonelli's work, and 

 suggested to Gauss the construction of his table in Zach's ' Correspondenz.' 

 The book consists of two parts : in the first there are 9 pages of tables &c. 

 wanted in the construction of logarithms, viz. log cc, log 1-x, log (1-Oa-), . . . . 



log (l-OOOOOOOOOO.r), for x = 1, 2, 9, to 20 places, and the same for 



hyperbolic logarithms; also log -1, -2 (9-9), and log l-Oo:, log l-OOOo--, 



log 1-OOOOO.r, and log 1-OOOOOOO.r, for x = 01, 02, ... . 99. 



The second part is headed " Theorie der Ergiinzungs- und Verminderungs- 

 Logarithmen zur Berechnung der Logarithmen der Summen und Differenzen 

 yon Zahlen aus ihren Logarithmen," and on pp. 52-54 the specimen table is 



given ; log x being the argument, it gives log j 1 + - J and log (1 + x) as 



tabular results to 14 places, for arguments from -00000 to '00104 at 

 intervals of -00001. [It wiU be noticed that the above are the same as 

 Gauss's A, B, and C] The middle page of this table (p. 53) is nearly an 

 inch longer than any of the other pages of the book. The original work, 

 according to Houel, 1858, ^ Avertissement,' p. vi, was published at Bordeaux, 

 An XI., under the title " Supplement logarithmique," (fee. 



Gauss, 1812. b/^ = log fl + -\\, and C (= log (1 + x)) are given for 



argument A(= log x) from A = -000 to 2-000 at intervals of -001, thenco 

 to 3-40 at intervals of -01, and to 5-0 at intervals of -1, all to 5 places, with 

 differences. The table occupies 27 small octavo pages. Gauss's paper is re- 

 printed from the ' Correspondenz ' in t. iii. pp. 244—246 of his ' Werke,' 

 1866 ; but the table is not reproduced there. 



Matthiessen, 1818. B and C are given to 7 places for argument A, 

 from A = -0000 to 2-0000 at intervals of -0001, thence to 3-000 at intervals 

 of -001, to 4-00 at intervals of -01 and to 5-0 at intervals of -1 ; also for 

 A = 6 and 7, with proportional parts. 



As C = A 4- B, the last three figures are the same for B and C, so that 

 the arrangement is, column of A, column of first four figures of B, column of 

 first four figures of C, column of last three figures of B and C, proportional 

 parts ; the eye has therefore to look in two different columns to take out a 

 logarithm. There is also another disadvantage, viz. that as there are only 

 four figures of argument, if it is to be used as a seven-figure table three more 

 must be interpolated for. 



The introduction is both in German and Latin. 



Mr. Gray, who recalculated a considerable portion of this table, found that 

 it contained numerous errors (see Gray, 1849, below). See also the intro- 

 ductoi-y remarks to this article. 



Weidenbach, 1829. Modified Gaussian logarithms. Log x (= A) is 



the argument, and log '^' ^ (= B) is the tabular result. A and B are thus 

 X -~ 1 ^ 



