OBJECTS, PLEASURES, AND ADVANTAGES OF SCIENCE. 



which are unknown, and of which we only know that they stand in certain relations to known numbers, 

 or to one another. The instance now taken is an easy one ; and you could, by considering the question a 

 little, answer it readily enough ; that is, by trying different numbers, and seeing which suited the conditions ; 

 for you plainly see that 5 and 4 are the two numbers sought ; but you see this by no certain or general rule 

 applicable to all cases, and therefore you could never work more difficult questions in the same way ; and 

 even questions of a moderate degree of difficulty would take an endless number of trials or guesses to 

 answer. Thus a shepherd sold his flock for 801. ; and if he had sold four sheep more for the same money, 

 he would have received one pound less for each sheep. To find out from this, how many the flock consisted 

 of, is a very easy question in algebra, but would require a vast many guesses, and a long time to hit upon 

 by common arithmetic.* And questions infinitely more difficult can easily be solved by the rules of algebra. 

 In like manner, by arithmetic you can tell the properties of particular numbers ; as, for instance, that the 

 number 348 is divided by 3 exactly, so as to leave nothing over : but algebra teaches us that it is only one 

 of an infinite variety of numbers, all divisible by 3, and any one of which you can tell the moment you see 

 it ; for they all have the remarkable property, that if you add together the figures they consist of, the sum 

 total is divisible by 3. You can easily perceive 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, with 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 .f 



By means of this science, and its various applications, the most extraordinary calculations may be per- 

 formed. 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, and the common difference being the number you begin with ; 

 thus, 1, 2, 3, 4, 5, 6, and so on, in which the common difference 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, 

 but the common multiplier being the number you begin with: 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 together 

 the numbers of the lower set corresponding or opposite 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 one of the upper numbers 

 from another, and opposite to their difference in the upper line, you look to the lower number, it is the 

 quotient found from dividing one of the lower numbers by the other opposite the subtracted ones. 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 numbers; 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 iu 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 7,543,283 by itself, and that pro- 

 duct again by the original number, you would have to multiply a number of 7 places of figures by an equally 

 large number, and then a number of 14 places of figures by one of 7 places, till at last you had a product of 

 21 places of figures a very tedious operation; but, working by logarithms, you would only have to take 

 three times the logarithm of the original number, and that gives the logarithm of the last product of 21 

 places of figures, without any further multiplication. So much for the time and trouble saved, which is 

 still greater in questions of division ; but by means of logarithms many questions can be worked, and of 

 the most important kind, which no time or labour would otherwise enable us to resolve. 



Qecunetry teaches the properties of figure, or particular portions of space, and distances of points from 

 each other. Thus, when you see a triangle, or three-sided figure, one of whose sides is perpendicular to 



* It u 16. 



t Another claw of numbers divisible by 3 U discovered in like manner by algebra. Every number of 3 places, the figures 

 (or digits) composing which are in arithmetical progression, (or rise above each other by equal differences,) is divisible by 3: 

 u 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, 

 Ac., numbers rising above each other by equal differences, as 289, 299, 309, or 148, 214, 280, or 307142085345648276198756, 

 which number of 24 places is divisible by 3, being composed of 6 numbers in a series whose common difference is 1137. Thii 

 property, too, in only a particular case of a much more general one. 



