PLEASURES OF SCIENCE. y 



this in any one case, as in the number mentioned, for 3 added to 4 and 

 that to 8 make 15, which is plainly divisible by 3 ; and if you divide 

 348 by 3, you find the quotient to be 116, and nothing over. But this 

 does not at all prove that any other number, the sum of whose figures 

 is divisible by 3, will itself also be found divisible by 3, as 741 ; for 

 you must actually perform the division here, and in every other case, 

 before you can know that it leaves nothing over. Algebra, on the 

 contrary, both enables you to discover such general properties, and to 

 prove them in all their generality.* 



By means of this science, and its various applications, the most extra- 

 ordinary calculations may be performed. We shall give, as an example, 

 the method of Logarithms, which proceeds upon this principle. Take 

 a set of numbers going on by equal differences ; that is to say, the third 

 being as much greater than the second, as the second is greater than 

 the first ; thus, 1, 2, 3, 4, 5, 6, and so on, in which the common differ- 

 ence is 1 ; then take another set of numbers, such that each is equal to 

 twice or three times the one before it, or any number of times the one 

 before it ; thus, 2, 4, 8, 16, 32, 64, 128 ; write this second set of 

 numbers under the first, or side by side, so that the numbers shall stand 

 opposite to one another thus, 



1234567 

 2 4 8 16 32 64 128 



you will find, that if you add together any two of the upper or first set, 

 and go to the number opposite their sum, in the lower or second set, 

 you will have in this last set the number arising from multiplying toge- 

 ther the numbers of the lower set corresponding to the numbers added 

 together. Thus, add 2 to 4, you have 6 in the upper set, opposite to which 

 in the lower set is 64, and multiplying the numbers 4 and 16 opposite to 

 2 and 4, the product is 64. In like manner, if you subtract the upper 

 numbers, and look for the lower numbers opposite to their difference, 

 you obtain the quotient of the lower numbers opposite the number 

 subtracted. Thus, take 4 from 6 and 2 remains, opposite to which you 

 have in the lower line 4 ; and if you divide 64, the number opposite to 

 6, by 16, the number opposite to 4, the quotient is 4. The upper set 

 are called the logarithms of the lower set, which are called natural 

 inimbers : and tables may, with a little trouble, be constructed, giving 

 the logarithms of all numbers from 1 to 10,000 and more ; so that, 

 instead of multiplying or dividing one number by another, you have 

 only to add or subtract their logarithms, and then you at once find the 

 product or the quotient in the tables. These are made applicable to 

 numbers far higher than any actually in them, by a very simple process ; 

 so that you may at once perceive the prodigious saving of time and 

 labour which is thus made. If you had, for instance, to multiply 



* Another class of numbers divisible by 3 is discovered in like manner by algebra. 

 Every number of 3 places, the figures (or digits) composing which are in arithmetical 

 progression, (or ri?e above each other by equal differences,) is divisible by 3 : as, 123, 

 789, 357, 159, and so on. The same is true of numbers of any amount of places, 

 provided they are composed of 3, 6, 9, &c. numbers rising above each other by equal 

 differences, as 289, 299, 309, or 148, 214, 280, 346, or 30714208534564827(5198756, 

 which number of 24 places is divisible by 3, being composed of 6 numbers in a series 

 \vhose common difference is 1137. 



