ON MATHEMATICAL TABLES. 77 



" reciprocal," the relation between them being, infact, 10 * + " = 10^ + 10" + 1, 

 so that either A or B may be regarded as the argument. The table gives E to 

 five places with differences, from A = -382 to A = 2-002 at intervals of -001, 

 from A = 2-00 to A = 3-60 at intervals of -01, and then to 5-5 at intervals 

 of -1. The corrimencement of the table being at A = '382 does not render it 

 incomplete, by reason of the reciprocitj' referred to above, since for arguments 

 less than '382 we can take B as the argument. Thus, at the beginning of 

 the table A and B are very nearly equal, viz. A = -382, B = 0-38355 ; 

 A = -383, B = '38255. There is an introduction of 2 pp. by Gauss. 



The use of the table in the solution of triangles is very apparent, e. g. in 



the formula cot - = -^^, tan — , in Napier's analogies, &c. 



2 a — 2 



Gray, 1849. Modified Gaussian logarithms. T. I. Log (1 + x) is the 

 tabular result for log x as argument ; and the range is from log .^• = -0000 

 to 2-0000 at intervals of -0001, to 6 places, with proportional parts to 

 hundredths (viz. 100 proportional parts of each difference). 



T. ir. Log (1 — x) is the tabular result for log x as argument; and the 

 range is from log x = 3-000 to 1-000 at intervals of -001, and from 1-0000 

 to i-9000 at intervals of -0001, to 6 places, with complete proportional parts. 

 The first table might have been copied from ITatthiessen by contracting the 

 7 places of the latter to 6 ; but it was recalculated by Mr. Gray, and many 

 errors were thereby found in Matthiessen's table (Introduction, p. vi) ; the 

 second t<able was also tke result of an original calculation. Some remarks 

 and references on the subject of Gaussian logarithms &c. will be found in 

 the Introduction to the work. 



Since writing the above account, Mr. Gray has sent us a copy of his 

 * Addendum to Tables and Formulae for the computation of Life Contin- 

 gencies .... Second Issue, comprising a large extension of the principal 

 table . . . . ' London, 1870, 8vo (26 pp. of tables and an introduction), which is 

 a continuation of the work under notice, and is intended to be bound up with it, 

 a new title having reference to the whole work when so augmented being added. 

 The ' Addendum ' contains a table of log (1 + x) to 6 places for argument 

 log X, from log x = 3-000 to I-OOO at intervals of -001, and from 1-0000 to 

 0-0500 at intervals of -0001, the latter portion having proportional parts for 

 every hundredth of the differences added : the whole of course the result of 

 an original calculation. Mr. Peter Gray was the first to perceive the utility 

 of Gaussian logarithms in the calculation of life contingencies, and to him is 

 due their introduction as well as the calculation of the necessary tables, which 

 it is evident are valuable mathematically, apart from the particular subject 

 for which they were undertaken. 



Zech, 1849. Table of seven-figure Gaussian logarithms. Denoting, 



done by Gauss, log x, log [ 1 + - j, and log (1 -j- x), by A, B, C 



respectively, then the table gives B to seven places, from A = -0000 to 

 A = 2-0000 at intervals of -0001, from A = 2-000 to A = 4-000 at intervals of 

 •001, and thence to 6-00 at intervals of -01, with proportional parts through- 

 out ; the whole arranged as an ordinary seven-figure logarithm table, and 

 headed Addition table. 



The Subtraction table gives C to 7 places, from B = -0000000 to -0003000 

 at intervals of -0000001, thence to -050000 at intervals of -000001, and 

 thence to -30300 at intervals of -00001 to seven places, with proportional 

 parts. 



as was 



