18 REPORT 1873. 



by 2, 3, or 5) to 10,500. A short aud graudiloqucnt dedication to the 

 French Institute is prefixed. 



Eogg gives also a German title, ' Pinacothek, oder Sammlung allgemein- 

 niitzlicher Tafeln fiir Jedermann' &c. 



Gruson, 1799. A table of products to 9 x 10,000. The pages, which 

 arc very large (containing 125 lines), are divided into two by a vertical line, 

 each half page containing ten columns, giving the numbers and their first 

 nine multiples : the first half of the first page thus ends at 9 x 124, the 

 second half at 9 x 249 ; and there are 1992 tabular results to the page. The 

 table has only one tenth of the range of Bketschneider's ; but the result is 

 given at once ; however, the large size of the page (almost, if not quite, the 

 largest we have seen for a table) is a great disadvantage. There are two 

 pages of explanation &c. 



The title describes the table as extending to 100,000, the above being only 

 the first part, "We do not know whether any more was published, but think 

 probably not. Eogg mentions no more. At the end of the introduction 

 three errors occurring in some copies are given. 



Martin, 1801. This is a large collection of tables on money-changing, 

 rentes, weights and measures, &c. The only part of the book that needs 

 notice here is Chapter XI., which contains a multiplication table giving the 

 first nine multiples of the numbers from 101 to 1052 (19 pp.). 



Dilling, 1820. In the use of a table of logarithms to multiply numbers 

 together, the logarithms used are of no vahie in themselves, being got rid of . 

 before the final result. If, therefore, letters a, 6, c, ... be used instead, we 

 have no occasion to know the values of any one of them, but only the way in 

 which they are related to one another. The present table is constructed for 

 numbers up to 1000 on this principle ; within this range there are about 170 

 l^rimes, the logarithms of which have to be denoted by separate symbols, 

 a,h, . .. , z, ffj, 6j, . . . , &c. ; the powers of 2 are denoted by numbers ; thus 

 log (2^)=2, log (2')=3, &c. ; and the logarithms of any number to 1000 can 

 be easily expressed in not more than four terms; thus log 84=2 + rt-|-c. 

 There is also a table of antilogarithms arranged according to the last letter 

 involved; thus log 21=«,-|-o, log 15=a-|-6, the sum =2« + Z»4-c ; and 

 entering the antilogarithmic table at c, we find 315 the product. We can 

 thus only multiply numbers whose pi'oduct is less than 1000 ; and a table of 

 products of the same size would certainly have been more useful. The table 

 can of course be used for division, square roots, &c., but only if the result is 

 integral, so that it is little more than a matter of curiosity. Tliis table was 

 intended, however, only as a specimen, to be followed by a larger one to 

 10,000. We believe the continuation was not published ; and Eogg refers to 

 no Other work of Dilling, 



The work, although nominally a table of logarithms, is included in this 

 article, as it is reaUy a multiplication table. It is the only table we have met 

 with involving a principle which at one time would have been of value with 

 respect to multiplication, viz. to resolve the numbers into their prime factors, 

 and multiply them by adding their factors. Thus 21 =3 X 7, 15 = 3x5, and 

 their product 315=3' X 5 X 7 ; if therefore we had a table giving the prime 

 factors of all numbers from 1 to 1000, arranged in order, and another table 

 of like extent giving the numbers corresponding to the same products of 

 factors, arranged with the largest factor first, and the others in descending 

 order, so as to facilitate the entry, we could perform multiplication (where 

 the product does not exceed 1000) by addition only. In the construction of 

 such a table it would soon be found convenient to replace the two and throe 



