2a by transposing or a, to the other side of the equation, we shall have 2' 9 &c. Or, by making q=2a, the expression for the simple radical will P 2a+ Ρ 2a &c. And in nearly the same way may any other expression of this kind be transformed to a quadratic, or a surd; to which they are always reducible, whether the periodic part consists of one, two, or more terms; or whether it commences in a regular or irregular manner. A similar mode of solution may also be applied to continued surds, or expressions of the form {a+√(a+√(a+))}, &c. the value of which, though apparently infinite, is always determinable by means of a certain equation; and, in some cases, it is a real integral or fractional quantity; for putting a√ {a+√(a+ (a+))} &c. we shall have, by squaring each side of the equation, 2a+a+(a+)} &c. the latter part of which is evidently equal to the original surd. Whence 22a+x, or x2—x. =a; and consequently, x=√(4+a), where, if a be now put equal to 2, the expression will become √{2+√(2+√(2+))} &c. = 1±√(+2)=2, or -1. = Again, let there be taken, as another instance of this kind, x=√{a+√(+√(a+√(b+)))} &c. Then, by squaring each side of the equation, as before, we shall have 2a+16+ √(a+√(b+))} &c., and x2-a = √{b+√(a+√(b+))} &c. And by again squaring each side of this last equation, 24-2a2 +a2=b+√{a+√(b+)} &c. or, x*—2ax2-x=b-a2, which equation, when solved in the usual way, will evidently give the value of x. To this we may add, that the square root of any quadratic number may be converted into a continued fraction, of the periodic kind, as follows.* *It has been long known, that any given continued periodic fraction could be reduced to a quadratic equation, and thence to a simple surd; but Lagrange appears to have been the first who has proved the reverse of this proposition, by showing that the square root of any whole number, or the root of a quadratic equation, can always be expressed by a continued periodic fraction.-See his work entitled De la Resolution des Equation Numeriques, p. 65. RULE. Find such an integral value a, of the given quantity x, that x-a shall be less than unity, and make x = also, such an integral value a' of x', that x'—«' shall be less than 1 unity, and make x'='+ Proceed in this manner with ", by putting its greatest integral value equal to a";, and so on. Then, by successively substituting the value of x', x", x''', &c. for their equals in the first of these expressions, we shall have + 1 '+ a""&c. for the continued fraction required. Where it is to be observed that the denominators a', a", a"", &c. are the partial quotients before mentioned; and the quantities x', x', x'", &c. resulting from the development of x, and of which the integers a', a", a"", &c. form the greater part, are the complete quotients. And if the partial quotients, thus found, be placed in a right line, as usual, and their corresponding converging fractions be determined according to the method before used, they will give the approximate values of the root required.* 1, Thus, if it be required to convert the square root of 19 into a continued fraction, we shall have, by following the above rule, 19-419-4 -=4+ 1 3 1 19+4 5 }(No19+4) =2+ =2+ + 3 19+2 ¿(√19+2) 5 5 √19+3 (19+3) 1 XIII =3+2 =3+ 3+ 2 2 19+3 (No19+3) XIV =1+ 1+ 5 5 3 19+4 1(19+4) 1 XVI= =8+ 1 1 19+4 }(√19+4) *It is not for the sake of the extraction of the square root, that this method has been devised, but on account of its application to indeterminate equations of the second degree; which admit of no other general method of solution; as was first shown by Lagrange, in the Memoirs of Berlin, for 1767 and 1768. Where xvII being the same as x, it is plain that, omitting the 4, which is the greatest integral part of 19,) the quotients 2, 1, 3, 1, 2, 8, already found, will always return again in the same order to infinity. 1 1 Whence 19=1+2+1 + 3 + 1 + 2 + 8 And if it should be required to convert the square root of 19 into a series of converging fractions, without first reducing it to the continued form, they may be obtained in the way before used, from the integral parts of the above results only. Thus, ... 48 61 TT 4T 5979 326 } .4, 2, 1, 3, 1, 2 8, 2, &c. Converging fractions...,,, 4, 11, II, 40°, 142, each of which fractions expresses the square root of 19 nearer than any of the preceding ones; and it is manifest, by setting down the quotients of the next and following periods, that they may be continued at pleasure to any degree of accuracy required. 2. Also, if it were required to convert the square root of 13 into a series of converging fractions, we shall have x=√13=3+ √13-3 13. 1 3 1 (No13+1) 13+1 1 XII: 3 3 =1+ =1+: No13 — 1 1+ 13+1 3 XIV =1+ =1+ =1+· 4 13+3 4 13-3 13+3 (No13+3) XV= =6+ =6+ 1 13+3 6+ (13+3) √13+3=1+~ where xvi being the same as x1, the integral parts of the process, 1, 1, 1, 1, 6, as in the former example, will return again contiuually, in the same order. Whence, for the remaining part of the operation, we shall have Quotients... ......3, 1, 1, 1, 1, 6, 1 &c. Converging fractions...,,,, 1, 18, 19 &c. each of =1+ √13+1 3 which will be the nearest square root of 13 that can be expressed in small numbers. 170. The operation made use of above for converting the square root of 19, and other nonquadrate numbers, into converging frac tions, may be rendered much more simple by observing the following law of the formation of the successive quotients, viz. let 19+m n =u&c. and 19+m' n =u&c. represent any two consecutive fractions in the examples there given, where m, m', m" &c. are the numbers that are to be added to the root; n, n', n" the divisors; and u, u', u', &c. the integral parts of the respective quotients. 19-m'2 n Then, from the obvious nature of the operation, we shall have in this case m'-nu-m, and n'— ; so that each value of m', n', and u, may be readily deduced from those of m, n, and u in the preceding fraction. Hence the operation above referred to will stand, by means of this law, as follows: 1 1 Where it will be found, by continuing the process, that the pe riod of quotients 2, 1, 3, 1, 2, 8, will recur again in the same order, ad infinitum. Hence, placing these quotients as in the former part of the work, and following a similar mode of operation, we shall have the following series of converging fractions: ( 4, 2, 1, 3, 1, 2, 8, b. 1, 1⁄2, 4, 44, 14, 40, &c.} each of which expresses 19 more nearly than any preceding one; and it is evident that they may be continued at pleasure to any degree of accuracy required. In the same manner also may the square root of any nonquadrate number N, be extracted, by supposing a to be the greatest integer contained in N, m, m', m'", &c. the numbers to be added to N, n, n', n', &c. the divisors, and u, u', u', &c. the integral parts of the several fractions, as before; viz. Where, by continuing the extraction, as in the former case, ✔N+m n or the sec we shall always arrive at a fraction equal to ond in the series; after which the quotients will constantly recur again in the same order. It is also evident from the practical example before given, and may be demonstrated generally, that the last quotient of every complete period of quotients, in this mode of extracting the square root of any number, is always equal to twice the greatest integer contained in N.* The Differential Method of Series. 171. The Differential Method is the method of finding the successive differences of the terms of a series, and thence any intermediate term, or the sum of the whole series. Problem I. To find the first term of any order of differences. Let a, b, c, d, e, &c. represent any series; then if the successive differences of the terms be taken, these differences will form a new series, which is called the first order of differences; in like manner, if the successive differences of the terms of this last series be taken, a new series, called the 2d order of differences, will be obtained. *The reader who may wish to see a full account of this subject is referred to the Theory of Numbers, by Legendre, so often mentioned; the Memoirs of the Academy of Berlin, an. 1767 and |