436. In an ascending geometrical series, each succeeding term is found by multiplying the ratio into the preceding term. * If the first term is a, and the ratio r, Then axr=ar, the second term, ar xr=ar3, the fourth, arxar, the third, ar3 xr=ar*, the fifth, &c. And the series is a, ar, ar2, ar3, ar1, ar3, &c. 437. If the first term and the ratio are the gression is simply a series of powers. same, If the first term and the ratio are each equal to r, Then rxr=r2, the second term, rrr, the third; the pro 3xrr, the fourth, rxrr, the fifth. And the series is r, r2, r3, r^, r3, ro, &c. 438. In a descending series, each succeeding term is found by dividing the preceding term by the ratio. If the first term is ar, and the ratio r, The series is aro, ar3, ara, ar3, ar2, ar, a, &è. If the first term is a and the ratio r, ä a α The series is a,773,&c,or(Art.207.)a, a^~1‚a ̃ ̃3‚&¤, If the first term is 1, and the ratio 2, The series is 1, 1, 1, 1, 76, 77, ¿'; &c. 1 2 3 By attending to the series a, ar, ar2, ar3, ar, ars, &c. it will be seen that, in each term, the exponent of the power of the ratio is one less, than the number of the term. If then a=the least term, z=the greatest, r=the ratio n=the number of terms; we have the equation z=ar"-1, that is, 439. In geometrical progression, the greatest term is equal to the product of the least, into the power of the ratio whose index is one less than the number of terms. When the least term and the ratio are the same, the equation becomes z=rr”-1=r”. See art. 437. 440. Of the four quantities a, z, r, and n, any three being given, the other may be found.* *See Note K. Dd 3. Divid. the 1st by a, and extracting the root, (Art. 297.) 441. The next thing to be attended to is the rule for finding the sum of all the terms. If any term, in an increasing geometrical series, be multiplied by the ratio, the product will be the succeeding term. (Art. 436.) Of course, if each of the terms be multiplied by the ratio, a new series will be produced, in which all the terms except the last will be the same, as all except the first in the other series. To make this plain, let the new series be written under the other, in such a manner, that each term shall be removed one step to the right of that from which it is produced in the line above. Take, for instance, the series Here it will be seen, at once, that the 2, 4, 8, 16, 32 4, 8, 16, 32, 64. four last terms in first in the lower the upper line are the same, as the four line. The only terms which are not in both, are the first of the one series, and the last of the other. So that when we subtract the one series from the other, all the terms except these two will disappear, by balancing each other. a, ar, ar2, ars, arn-I arn-1, arn. If the given series is In this equation, ar" is the last term in the new series, and is therefore the product of the ratio into the last term in the given series. Hence, 442. The sum of a series in geometrical progression is found, by multiplying the greatest term into the ratio, subtracting the least term, and dividing the remainder by the ratio less one. Prob. If in a series of numbers in geometrical progression, the first term is 6, the last term 1458, and the ratio 3, what is the sum of all the terms? Let the series be a, ar, ar2, ar3, art, &c. By the nature of geometrical progression, a:ar::ar: ar2 :: ar2 : ar3 :: ar3 : ar*, &c. In each couplet let the antecedent be subtracted from the. consequent according to art. 389. 6. Then a: ar::ar-a: ar2-ar:: ar2 -ar: ar3 — ar2, &c. That is, the first term is to the second, as the difference between the first and second, to the difference between the second and third; and as the difference between the second and third, to the difference between the third and fourth, &c. Cor. If quantities are in geometrical progression, their differences are also in geometrical progression. Thus the numbers And their differences geometrical progression. 3, 9, 27, 61, 243, &c. 6, 18, 54, 162, &c. are in 444. Several quantities are said to be in harmonical progression, when, of any three which are contiguous in the series, the first is to the last, as the difference between the two first, to the difference between the two last. See art. 400. Thus the numbers 60, 30, 20, 15, 12, 10, are in harmonical progression. For 60: 20:60-30:30-20, And 20:12::20-15:15-12, And 30:15::30-20:20-15, And 15:10::15–12:12–10. SECTION XV. INFINITES AND INFINITESIMALS.* ART. 445. THE word infinite is used in different senses, The ambiguity of the term has been the occasion of much perplexity. It has even led to the absurd supposition, that propositions directly contradictory to each other may be mathematically demonstrated These appar ent contradictions are owing to the fact, that what is proved of infinity, when understood in one particular manner, is often thought to be true also, whep the term has a very different signification. The two meanings are insensibly shifted, the one for the other, so that the proposition which is really demonstrated, is exchanged for another which is false and absurd. To prevent mistakes of this nature, it is important that the different meanings be carefully distinguished from each other. 446. INFINITE, in the highest, and perhaps the most proper sense of the word, is that which is so great, that nothing can be added to it, or supposed to be added. Infi In this sense, it is frequently used, in speaking of moral and metaphysical subjects. Thus, by infinite wisdom is meant that which will not admit of the least addition. nite power is that which cannot possibly be increased, even in supposition. This meaning of infinity is not applicable to the mathematics. That which is the subject of the mathematics is quantity; (Art. 1.) such quantity as may be conceived by the human mind. But no idea can be formed of a quantity so great that nothing can be supposed to be added to it. In this sense, an infinite number is inconceivable. We may increase a number by continual addition, till we obtain one that shall exceed any limits which we please to assign. By this, however, we do not arrive at a number to which *Locke's Essays, Book 2. Chap. 17. Berkley's Analyst. Preface to Maclaurin's Fluxions. Newton's Princip. Saunderson's Algebra, Art. 336. Mansfield's Essays, Emerson's Algebra, Prob. 73, nothing can be added; but only at one that is beyond any limits which we have hitherto set. Farther additions may be made to it, with the same ease, as those by which it has already been increased so far. It is therefore not infinite, in the sense in which the term has now been explained. It is absurd to speak of the greatest possible number. No number can be imagined so great, as not to admit of being made greater. We must therefore look for another meaning of infinity, before we can apply it, with propriety, to the mathe matics. 447. A mathematical quantity is said to be infinite, when it is supposed to be increased beyond any determinate limits. By determinate limits are meant such as can be distinctly stated.* In this sense, the natural series of numbers 1, 2, 3, 4, 5, &c. may be said to be infinite. For, if any number be mentioned ever so great, another may be supposed still greater. The two significations of the word infinite are liable to be confounded, because they are in several points of view the same. The higher meaning includes the lower. That which is so great as to admit of no addition, must be beyond any determinate limits. But the lower does not necessarily imply the higher. Though number is capable of being increased beyond any specified limits; it will not follow, that a number can be found to which no farther additions can be made. The two infinites agree in this, that, according to each, the things spoken of are great beyond calculation. But they differ widely in another respect. To the one, nothing can be added. To the other, additions can be made at pleas ure. 448. In the mathematical sense of the term, there is no absurdity in supposing one infinite greater than another. We may conceive the numbers and 2 2 2 2 2 2 2 &c. to be each extended so far as to reach round the globe, or to the most distant visible star, or beyond any greater boundary which can be mentioned. But, if the two series be equally extended, the amount of the one will be twice as great as the other, though both be infinite. So, if the series a+ a2 + a3 + a1 + a3 &c. 9a+9a2 + 9a3 +9a1+9a3 &c. and * See Note L. |