2 and suppose that : represents the sum of it. Putting +1 in place of z, o (z+1) will in like manner be the sum of the series, Subtract the one of these series from the other, term by term; we shall have : 2 4:(+1) for the sum of the remainder, which, in its expanded form, will be a + + + &c. 2.2+1 2.2+1.2+2 2 2.2+1.2+2.2+3 But this remainder may be put under the form Divide this equation by : (≈ + 1), and to simplify the result, let a : z be a new function of z, such that := 2.9: (+1); we may ↓ : s = x + ↓ : ( ~ + 1 )' But by successively inserting ≈ + 1, ≈ + 2, &c., in place of z in this equation, there will result from it ; &c. ≈+2+4:(≈ + 3) Hence the value of :z, may be expressed by the continued fraction, Reciprocally this continued fraction, produced to infinity, has for its Now suppose z=; the continued fraction will become in which all the numerators, except the first, are equal to 4 a, and the denominators form the series of odd numbers, 1, 3, 5, 7, &c. The value of this continued fraction may, therefore, be expressed by But these series have a relation to some admitted formulas; and it is well known that, putting e for the number whose hyperbolic logarithm is 1, the foregoing expression becomes eNa eNa '+e√ √ a; so that we shall have, generally, From this, two principal formulas are derived, according as a is positive or negative. First, let 4 a = x2; we shall have This formula will serve as the basis of our demonstration. Before proceeding to it, however, we must prove the two following Lemmas. to be a continued fraction prolonged to infinity, in which all the numbers m, n, m', n' are positive or negative integers; if the component fractions m m' m" n n" &c. are all less than unity, then will the total value of the continued fraction be of necessity an irrational number. In the first place, this value is less than unity. For, without interfering with the general applicability of the continued fraction, we are at liberty to suppose all the denominators, n, n', n", &c. to be positive; in which case, taking a single term of the proposed series, we shall, than n; and since they are both integers, m will also be less than n + Hence the value which results from the two terms m n + m2 is less than unity. Calculate three terms of the proposed continued fraction; and in the first place, as we have just seen, the value of the part will still be less than unity: hence the value which results from the three terms m " + m' n' + m" is less than unity. By continuing the same process, it will appear, that whatever number of terms in the proposed continued fraction be calculated, the value resulting from them is less than unity; hence the total value of the fraction prolonged to infinity, is also less than unity. It cannot be equal to unity except in the single case, when the proposed fraction had the form This being proved, if the value of the continued fraction is not admitted to be an irrational number, suppose it to be a rational number, for example, B and A being any integers; we shall then have Let C, D, E, &c. be indeterminate quantities, such that and so on to infinity. These different continued fractions having all their terms less than unity, their values or sums will be less than unity, as we have proved above; and thus we shall have BA, C ≤ B, D ≤ C, &c. ; so that the series A, B, C, D, E, &c. goes on decreasing to infinity. But the combination of the continued fractions we are treating of gives And since the two first numbers A and B are integers by hypothesis, it follows that all the others C, D, E, &c. which were hitherto undetermined, are also integers. Now it implies a contradiction to suppose that an infinite series A, B, C, D, E, &c. can at once be decreasing and composed of integer numbers; for, besides, no one of the numbers A, B, C, D, E, &c. can be zero, since the proposed continued fraction B C D extends to infinity, and therefore the sums represented by A'B'C -' &c. must always be something. Hence our hypothesis, that the sum of the proposed continued fraction was equal to a rational quantity, cannot stand; hence that sum is of necessity a rational number. LEMMA II. The same suppositions continuing as in the former Lemma, m m' m" if the component fractions &c. are of any magnitude what n n'' " ever at the beginning of the series, provided after a certain interval they become less than unit; we assert, that the proposed continued fraction, if it still extends to infinity, will have an irrational value. For if, reckoning from "V &c. to infinity, are less than unit, then by Lemma I, the V it is evident that being irrational, all the quantities w, w", "", must be so likewise. ய But "", the last of these, is equal to the proposed con tinued fraction, hence the value of this fraction is irrational. We are now in a condition to resume our subject, and demonstrate this general proposition. THEOREM. If an arc is commensurable with the radius, its tangent will be incommensurable with that radius. m Put the radius 1, and the arc a m and in being whole num n |