Let r=5, to find the numbers. 4. To divide a given number, which is the sum of two known squares, into two other squares. Let a2+b2=the number given, rx-a=the side of the first required square, sx-b-the side of the second, where rs. 2 Then will rx-a 2 + sx—b}2 = (r2x2—2 arx+a2+s2x2 - 2 bsx +b2=) r2 +s2.x2 -2 ar+2 bs.x+a2+b2=a2+b2 ··· r2+s2.x2 -2 ar+2 bs.x=0, or r2+s2.x2=2 ar+2 bs.x ; ·.· dividing by x, we have r+s2.x=2 ar+2 bs, ··· x= 2.ar+bs ; consequently rx— 2 s.ar+bs -a==side of the first square, and sx- ·b=: —b=side of the second. 2 EXAMPLES.-Let a=6, b=4, r=5, s=3; then will x= 42 17 Let a=4, b=3, r=2, and s=1, be given. 5. To find two numbers, of which the sum is equal to the square of the least. Ans. 6 and 3. 6. To divide the number 30 into two parts, such that their product will be a square number. Ans. 27 and 3. 7. To divide the number 129 into two parts, the difference of which will be a square number. Ans. 105 and 24. 8. What two numbers are those, whose product added to the sum of their squares, will make a square? Ans. 5 and 3. 9. To find two squares, such that their sum added to their 16 product may likewise make a square. Ans. and 9 10. To find two numbers, one of which being taken from their product, the remainder will be a cube. Ans. 3 and 108. 11. To find two numbers, such that either of them being added to the square of the other, the sum will be a square. An 12. To find three numbers, such that their sum, and likewise the sum of every two of them, will each be a square number. Ans. 42, 684, and 22. PART VII. ALGEBRA. INFINITE SERIES". A SERIES is a rank of quantities, which usually proceed ccording to some given law, increasing or decreasing succesively; the simple quantities which constitute the series are called its terms. 2. An increasing or diverging series is that in which the terms successively increase, as 1, 2, 3, 4, &c. a+3a+7a, &c. 3. A decreasing or converging series is that in which the terms successively decrease, as 5, 3, 1, &c. 10 a—7 a—2a, &c. a The doctrine and application of infinite series, justly considered as the greatest improvements in analysis which modern times can boast, were introduced about the year 1668, by Nicholas Mercator, who is supposed to have taken the first hint of such a method from Dr. Wallis's Arithmetic of Infinites; but it was the genius of Newton that first gave it a body and form. The principal use of infinite series, is to approximate to the values and sums of such fractional and radical quantities, as cannot be determined by any finite expressions; to find the fluents of fluxions, and thence the length and quadrature of curves, &c. Its application to astronomy and physics is very extensive, and has supplied the means whereby the modern improvements in those sciences have been made. The intricacy of this branch of science has exercised the abilities of some of the most learned mathematicians of Europe, and its usefulness has induced many to direct their chief attention to its improvement: among those authors who have written on the subject, the following are the principal; D'Alembert, Barrow, Briggs, the Bernoullis, Lord Brouncker, Bonnycastle, Des Cartes, Clairaut, Colson, Cotes, Cramer, Condorcet, Dodson, Euler, Emerson, Fermat, Fagnanus, Goldbach, Gravesande, Gregory, Halley, De l'Hôpital, Harriot, Huddens, Huygens, Horsley, Hutton, Jones, Kepler, Keill, Kirkby, Landen, De Lagni, Leibnitz, Lorgna, Manfredi, Monmort, De Moivre, Maclaurin, Montano, Nichole, Newton, Oughtred, Riccati, Regnald, Saunderson, Slusius, Sterling, Stuart, Simpson, Taylor, Varignon, Vieta, Wallis, Waring, &c. &c. 4. A neutral series is that in which the terms neither increase nor decrease, as 1, 1, 1, 1, &c. a+a+a+a, &c. 5. An arithmetical series is that in which the terms increase or decrease by an equal difference, as 1, 3, 5, 7, &c. 9, 6, 3, 0, &c. a+2a+3 a, &c. 6. A geometrical series is that in which the terms increase by constant multiplication, or decrease by constant division, as 1, 3 3, 9, 27, &c. 12, 6, 3, -, &c. a +2a+4a+8 a, &c. 7. An infinite series is that in which the terms are supposed to be continued without end; or such a series, as from the nature of the law of increase or decrease of its terms requires an infinite number of terms to express it. 8. On the contrary, a series which can be completely expressed by a finite number of terms, is called a finite or terminate series. 9. Infinite series usually arise from the division of the numerator by the denominator of such fractions as do not give a terminate quotient, or by extracting the root of a surd quantity. 10. To reduce fractions to infinite series. RULE I. Divide the numerator by the denominator, until a sufficient number of terms in the quotient be obtained to shew the law of the series. II. Having discovered the law of continuation, the series may be carried on to any length, without the necessity of farther division. 2. an-bn =an — 1 —all — ? b + aa — 3 b2 —, &c, which terminates in—¿a—¡y when n is an even number, but goes on indefinitely when n is odd. 3. an + bn - a+b = a" — 1 — a11 — 2 b + aa — 3 b 2 —, &c. which series terminates in + ba — 1, when n is an odd number, but goes on indefinitely when a is even, OPERATION. * 1+x) 1 (1-x+x2-x3+, &c. the series required. 4. The difference aa —¿a is not measured by the sum a+b. Hence, first, the difference of the nth powers of any two numbers is measured by the difference of the numbers, whether n be even or odd. Secondly, it is measured by the sum of the numbers, when n is even, but not when nis odd. Thirdly, the sum of the nth powers is measured by the sum of the numbers when n is odd, but not when n is even. In each of the quotients which terminate, the number of terms is equal to the index n. See an ingenious application of these conclusions in the Rev. Mr. Bridge's Lectures on Algebra, P. 248. 11. When any quantity is common to every term, the series may be simplified by dividing every term by that quantity, putting the quotients under the vinculum, and placing that quantity before the vinculum, with the sign x between. ent put under the vinculum and connected with the divisor x x1+ + + +, &c. which is a simpler form than that in the example. to an infinite series. Ans. 1+x+x2+x3+, &c. 3. Reduce 1 az x+z ns. — 12. To reduce compound quadratic surds to infinite series.. RULE. Extract the square root, (Art. 57. Part 3. Vol. I.) and continue the work until the law of the series be discovered; after -+, &c. =16 4 8 +, &c. 23 |