ON MATIIEM^.TICAL TABLES. 19 



figure primes by letters, to save room, and, in fact, to use letters tliroiigli- 

 out — and further to simplify the printing by writing «■* as 4a, &c., which 

 would do equally well ; we then have Dilling's tables, which have not the 

 smallest connexion with logarithms. Such a table might once have been 

 found useful ; but the slightest consideration shows that (except as a factor 

 table) it would be all but valueless now. The space a large table of the kind 

 would occupy, the impossibility of arranging the antifactor table so as to 

 admit of easy entry, and the great convenience of existing tables (both 

 Pythagorean and logarithmic) are alone sufficient to prove this. 



Crelle, 1836. This table occupies 1000 pages, and gives the product of 

 a number of seven figures by 1, 2, ... , 9, by a double operation, very much 

 in the same manner as Bretschneider's does for a number of five : viz., each 

 page is divided into two tables ; thus, to multiply 9382477 by 7, we turn to 

 page 825, and enter the right-hand table at line 77, column 7, where we find 

 77339 ; we then enter the left-hand table on the same page, at line 93, 

 column 7, and find 050, so that the product required is 05077339.^ We think 

 for numbers seven figures long the table effects a considerable saving of time, 

 as it is as easy to use as BRExsciiNEinEK s for five figures. It would take some 

 little practice to use the table rapidly in all cases, as of course the mode of 

 entry, &e. must be varied according as the number consists of seven, six, 

 five, &c. figures ; but the value of a table is measured not by the trouble 

 lequired to learn to use it, but by the time saved by means of it after the 

 computer has learnt its use. 



Bretschneider, 1841. This table is for the multiplication of any 

 number up to 100,000 by a single digit. On each page there are two tables, 

 the upper of which occupies ten lines, and the lower fifty. An example will 

 show the method of using the table. Suppose it required to multiply 

 50878 by 7, then the table is entered on the page headed 0800 (the headings 

 run from to 99, with two ciphers added to each). Facing 78 in the lower 

 table we find *146 ; and in the upper table facing 568, in the column for 7, 

 we find 397; the product required is therefore 398146, the third figure 

 being increased because the 146 was marked by an asterisk. The arguments 

 in the upper table, on the page headed 0800, are 08,168,208 . . . 908 (twice 

 repeated for the two cases when succeeding numbers are less and greater 

 than 50), and also 1, 2 ... 9, as the table is of double entry. 



The arrangement of the table is thus very ingenious ; but, as De Morgan 

 has remarked, multiplication by a single digit is so simple an operation that 

 it is questionable how far a table is serviceable when its use requires three 

 distinct points to be attended to. 



The introduction (10 pages) gives a complete explanation of how the table 

 can be used when the number of figures is greater than five. Having made 

 some use of the table for this purpose, we do not think any time is saved by 

 it ; at all events, not imtil the computer has had much practice in using it. 



Crelle, 1804. This magnificent table gives products up to 1000 x 1000, 

 arranged in a most convenient and elegant manner, one consequence of which 

 is that all the multiples of any number appear on the same page. It is also 

 very easy to get used to the arrangement of the table, which is as useful for 

 divisions as multiplications. It can be used for multiplying numbers which 

 contain more than three figures, by performing the operation, three figures 

 at a time ; but it requires some practice to do this readily ; and a similar 

 remark applies to the extraction of square roots. 



There is one great ineouvenieuco that every computer must feel in using 

 the work, viz. that the multiples of numbers ending in arc omitted, so that, 



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