INTRODUCTORY EXPLANATIONS. 15 



limits, as an amount ; and thus, within those limits, we 

 reduce all questions of multiplication and division to 

 addition and subtraction, by reference to the tables. 



We thus perceive a simple principle applied with 

 much labour, bnt such as is performed once for all. 

 The notion above elucidated was the first on which 

 logarithms were constructed; in time came more easy 

 methods. We now take another abbreviation which is 

 perpetually occurring in our subject. It is the multi- 

 plication of all the successive numbers from 1 up to 

 some high number ; that is, the continuation of the pro- 

 cess following. Let [10], for instance, represent the 

 product of all the numbers, from 1 up to 10, both in- 

 clusive, or let 



{10] stand forlx2x3x4x5x6x7x8x9x 10 = 3628800 



[1] is 1 ; [2] is 2; [3] is 6; [4] is 24; [5] is 120; [6] 

 is 720 ; [7] is 5040 ; [8] is 40,320 ; and so on. This 

 labour becomes absolutely unbearable when the numbers 

 become larger; thus, [30] contains 33 places of figures, 

 and [1000] contains 2568 figures. But, nevertheless, 

 we cannot deal with problems in which there are 1000 

 possible cases without knowing, either nearly or exactly 9 

 the value of [1000]. It will, however, be sufficient 

 to know this value very nearly ; within, say, a thou- 

 sandth part of the whole ; that is, as nearly as when, the 

 answer of a problem being 1 000, we find something be- 

 tween 999 and 1001. We now put before the reader 

 who can use logarithms a rule for this approximation, 

 with an example ; intending thereby to show the reader 

 who does not comprehend the process how mathematics 

 enter this subject in the abbreviation of tedious comput- 

 ations. 



RULE. To find very nearly the value of [a given 

 number], from the logarithm of that number, subtract 

 4342945, and multiply the difference by the given 

 number, for a first step. Again, to the logarithm of the 

 given number add '7981799^ and take half the sum, for 

 a second step. Add together the results of the first and 



