And when paid ev'ry way 'twill admit, the amount 8. DIOPHANTINE PROBLEMS. Unlimited problems relating to square and cube numbers, right angled triangles, &c. were first and chiefly treated of by Diophantus of Alexandria, and from that circumstance they are usually named Diophantine Problems. These problems, if not duly ordered, will frequently bring out answers in irrational quantities; but with proper management this inconvenience may in many cases be avoided, and the answers obtained in commensurable numbers. The intricate nature and almost endless variety of problems of this kind, render it impossible to lay down a general rule for their solution, or to give rules for an innumerable variety of particular cases which may occur. The following rules will, perhaps, be found among the best and most generally applicable of any that have been proposed. RULE I. Substitute one or more letters for the required root of the given square, cube, &c. so that, when involved, either the given number, or the highest power of the unknown quantity, may be exterminated from the given equation. f Diophantus has been considered by some writers as the inventor of Algebra; others have ascribed to him the invention of unlimited problems: but the difficult nature of the latter, and the masterly and elegant solutions he has given to most of them, plainly indicate that both opinions are erroneous. Diophantus flourished, according to some, before the Christian æra; some place him in the second century after Christ, others in the fourth, and others in the eighth or ninth. His Arithmetics, (out of which have been extracted most of the curious problems of the kind at present extant,) consisted origi nally of thirteen books, six of which, with the imperfect seventh, were pub lished at Basil in 1575, by Xylander; this fragment is the only work on Algebra, which has descended to us from the ancients: the remaining books have never been discovered. See Vol. I. p. 327. Of those who have written on, and successfully cultivated, the Diophantine Algebra, the chief are, Bachet de Mezeriac, Brancker, Bernoulli, Bonnycastle, De Billy, Euler, Fermat, Kersey, Ozanam, Prestet, Saunderson, Vieta, and Wolfius. II. If, after this operation, the unknown quantity be of but e dimension, reduce the equation, and the answer will be and. III. But if the unknown quantity be still a square, cube, &c. bstitute some new letter or letters for the root, and proceed before directed. IV. Repeat the operation until the unknown quantity is reced to one dimension; its value will then readily be found, om whence the values of all the other quantities will likewise e known. 1. To divide a given square number into two parts, so that square number. ach may be a ANALYSIS. Let a2=the given square number, x2=one of the arts; then will a2-x2=the other part, which, by the problem, must likewise be a square. Let rx-a=the side of the latter quare, then will (rx-a2=) r2x2-2 arx+a2=a2x2, whence x= =the side of the first square, and rx-a=(; the side of the second square; wherefore 2 are `r2+1 are the parts required; where a and r may be taken at pleasure, provided r be greater or less than unity. Q. E. I. is the second condition. Q. E. D. EXAMPLES.-Let the square number 100 be proposed to be divided into two parts, which will be squares. Mr. Bonnycastle, in his solution of the problem, (Algebra, third Edit. P. 143.) has omitted this restriction, which is evidently necessary; for if ✈ be supposed =1, then will the numerator of the fraction solution become nugatory. ar2-a 2+1 vanish, and the Here a2=100, and a=10. First, assume r=2, then will 40 2 ar 2 =)8=the side of the first square, and rx—a=6= the side of the second square; for 8)2+6)=(64+36=)100, as was required. Secondly, assume r=3, then will x=(==)6, and rx—a=8, 60 10 as before. Thirdly, assume r=4, then x= and rx-a=( 80 320 -10=) 17' 17 17 289 289 Divide 36 into two square numbers. Here a2=36, a=6; assume r=2, then x=— To divide 25 into two square numbers. Ans. 16 and 9. 2. To find two square numbers having a given difference. Let d the given difference, axb=d, whereof ab, and let x=the side of the less square, and x+b=the side of the greater; then will x+b — x2= (x2 + 2 bx+b2x2=) 2 bx+b2=ab; divide the side of the less square; by hypothesis ab=d. Q. E. D. EXAMPLES.-To find two square numbers, whereof the greater exceeds the less by 11. a-b And 2 2 (1=1=)5=side of the less square. Whence 6)2=36, and 5)2=25, are the squares required. To find two squares, whose difference is 15. nd 49. To find two squares differing by 24. Ans. 64 3. To find two numbers, whose sum and difference will be both squares. Let xone of the numbers, x2-x=the other; then will their _sum (x+x2 —x=) x2, evidently be a square number. And since (x2-X · —x=) x2 — 2x=their difference, must likewise be a square; let its side be assumed=x―r, then will (x—r} =)x2-2 xr+r2=x2—2 x, or 2 xr—2 x=r2, ··· x: 2 and 2r-2' =2; If 2 be substituted in this example for r, both numbers will come out = that is, their sum will be 4, and difference 0; wherefore r must not only be greater than 1, (as is asserted in Bonnycastle's Algebra, p. 146.) but greater than 2, 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 2 3, 9, 27, &c. 12, 6, 3, —, &c. a+2à+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 nume rator 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. 1. a"-fn a-b ́=a" — 1 + a" — 2b+an — 3b2+, &c. to + ba —1, which series evidently terminates. 2. an-bn ·=a1 — 1 —a11 — 2b + aa — 3b2 —, &c. which terminates in-b¤ — 1 y when n is an even number, but goes on indefinitely when n is odd. 3. a" + ba · 1 — a1 — 2 b + aa — 3 b2 —, &c. which series terminates in + ba — 1, when n is an odd number, but goes on indefinitely when ʼn is even, |