For suppose we wish to make the series differ from and similarly for any other assigned quantity. Therefore we say that ultimately when the number of terms is indefinitely increased, the series == 'i. IV. The Form 15. Suppose we have to find the value of the fraction a2-b2 in the limit, when b continually increases, and α- -b ultimately becomes equal to a. If we take the limit of a2-62 when b becomes equal to a, we find this to be 0; and also the limit of a-b, when b becomes equal to a, will be 0, and we shall have a2-b2 0 α- b =a+b =2a, when b becomes ultimately equal to a; and this is the limit required. Now, it must be borne in mind, that what is meant by the value of a fraction in the limit is not the value obtained by dividing the limit of the numerator by the limit of the denominator; but the value of the quotient, actually obtained by division, in the limit, or the value of the ratio of the numerator to the denominator, as the numerator and denominator approach the limit, and ultimately arrive at it. 16. The value of a ratio is not altered if we divide its two terms by the same quantity, or, which is the same thing, the value of a fraction is not altered if we divide both the numerator and denominator by the same quantity. However small the two terms of the ratio may be made, by division by another quantity, they still retain the same ratio, no matter how insignificant they may be in themselves. 17. We must regard the relation existing between two quantities, not as expressed by the difference between them, or how much one is larger than the other, but as how many times and parts of a time the one is contained in the other, or what multiple one is of the other. This is, in fact, the manner in which we regard matters of every-day life. We compare them with others of a like nature, and so pronounce them small or great. The quantities may be either great or small in themselves; but it is their relative value which gives us a notion of them as great or small. Thus, if there were 300 men in one assembly and 3000 in another, we should say, as a rule, that there were ten times as many in the latter as there were in the former, and not that there were 2700 more; and, again, the actual number 1000 may vary through any values, from very great to very small-it is all a matter of comparison. If it were stated that 1000 horses started in a race, we should say that it was simply ridiculous, the number was too large; if that 1000 men lived in one hamlet, that it was very large; if that there were 1000 men in one regiment, that it was large or beyond the average; and if that the 1000 men composed an invading army, that it was insignificant. Let us take an improper fraction 10000 this is equal to 1000 or 10 and the decimal point followed by a million zeros and 1 a million and 4 zeros and 1 = = 1000. Therefore, it follows, that we may make the numerator and denominator differ by less than any assignable quantity, and the ratio of the numerator to the denominator still remain equal to 1000. It will be seen then that it does not matter how small the terms of a ratio are, the value of the ratio remains unaltered. 18. Let us now revert to the limit of a2-b2 a Let ABCD be a square whose side is a, and QRSD a square whose side is b. Produce SR and QR to T and V. Then therefore AB-a and QR=b=AT, AVS=a2-b2, TB-a-b. AVS=2AR+TV, =2A T.TB+TB2; and the gnomon and Now Now suppose b or DQ to become larger and be represented by DQ' or AT". Then it will be seen that a2-b2 becomes smaller, and is now represented by AV'S', and that the rectangle, which was originally AR, has become longer and narrower, and is now represented by AR', and also that AV'S' is more nearly equal to 2AR' than AVS was to 2AR, since the square, which was originally TV, has become T" V'. Suppose, now, that b becomes still larger, and let it be represented by AT". Then the rectangle will have become still narrower, and the square TV" very small, and the gnomon A V"S" is more nearly equal to 2AR" than AV'S' was to 2AR'. By proceeding in this way, it will be seen that, eventually, when T moves up to B, that is, when b becomes equal to a, the rectangle will have become indefinitely narrow, and the square TV will have vanished altogether; that is, the gnomon will be represented by twice the line AB, since it will be represented by AB and BC; or, from (1), the limit of the ratio of a2-62 to a-b, when b ultimately becomes a, is equal to 24T or 2a. 19. This result might have been obtained thus: Let Then, when b=a-h. h=0, b=a. Substituting this value for b, we have a2-b2 a2-( a − h)2 = a-(a-h) a2-a2+2ah-h2 a-a+h 2ah-h2 = h =2a-h =2a, when h=0 or b=a. 20. We will illustrate the truth of these remarks by numerical examples. and this differs from 2a or 2× 10 by part of itself. and this differs from 2a or 2×100 by 1 part of |