substituting for one unknown quantity the product of the other and a third unknown quantity, Examples. Assume x=vy. ..: vy + dye = (B). From the equation (A) yer vel And .. Take the equation (B) yo= utd e Hence, we bv = vt door av + ad=evo – bev ; beta ad Therefore, being an equation from wh. the value of v may be found. But having the value v, the value of y may be found from the equation yo= of a; and, consequently, the value of 3 (= vy) will be known. (89.) 2. What are those two nunibers, the sum of which multiplied by the greater is 154; and whose difference multiplied by the less is equal to 24? Solu.-Let x = the greater, and y= the less number ; Then will (x+y) x=r?+xy=77, 77 or, y'= Assume I =VY then voy? +vyo=77, 12 V-1 2 { 12 Hence, 77 or 12v2 +12v=770-77; 77 12 12 ; -72 Now either of these values of v answers the conditions of tie 7 12 12 question ; but let us take v = ; then will yo 4 VI =16; and y= +4, also x=vy=+*+4= +7. 3 From which it appears that the numbers are 4 and 7. 3. It is required to find two numbers, such, that the square of the greater minus the square of the less may be 56, and the square of the less plus one-third of their product may be 40. Ans. 9 and 5. 4. There are two numbers, such, that thrice the square of the greater plus twice the square of the less is 110; and half their product plus the square of the less is 4 ; it is required to find these two numbers. Ans. 6 and 1. Miscellaneous Examples for practice. 1. What two numbers are those whose sum is 2y, and their product 100 ? Ans. 25 and 4. 2. What two numbers are those wnose sum is 20, and their product 36? Ans. 2 and 13 3. To divide the number 33 into two such parts that their product shall be 162. Ans. 27 and 6. 4. To divide the number 60 into two such parts, that their product may be to the sum of their squares in the ratio of 2 to 5. Ans. 20 and 40. 5. The difference of two numbers is 5, and a fourth part of the. product is 26; required those two numbers. Ans. 13 and 8. 6. The difference of two numbers is 3, and the difference of their cubes is 117; what are those numbers ? Ans. 2 and 5 7. The difference of two numbers is 6 ; and if 47 be added to twice the square of the less, it will be equal to the square of the greater ; what are the numbers ? Ans. 11 and 17. 8. There are two numbers whose sum is 30; and one-third of their product plus 18 is equal to the square of the less number; it is required to find those numbers. Ans. 9 and 21. 9. What number is that which, when divided by the product of its two digits, the quotient is 3; and if 18 be added to it, the digits will be inverted : Ans. 24 10. There are two numbers whose product is 120: if 2 be added to the less, and 3 subtracted from the greater, the product of the sum and remainder will also be 120; what are these numbers ? Ans. 15 and 8. 11. What two numbers are those whose sum multiplied by the greater is equal to 77; and whose difference multiplied by the lesser is equal to 12? Ans. 4 and 7. 12. To resolve the number 6 into two such factors, that the sum of their cubes shall be 35. Ans. 3 and 2. 13. To divide the number 20 into two such parts, that their product shall be equal to 105 ? Ans. 10+N-5, and 10-N-5. 14. A grazier bought as many sheep as cost him 60l. and, after reserving fifteen, he sold the remainder for 541. and gained 2s. a head by them ; how many sheep did he buy? Ans. 75. 15. To find a number such, that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. Ans. 7 ur 3. 16. The sum of two numbers is 8, and the sum of their cubes is 152 ; what are the numbers ? Ans. 3 and 5. 17. Two partners, A and B, gained 140l. by trade; A.'s money was three months in trade, and his gain was 6ol. less than his stock; and B.'s money, which was 501. more than A.'s, was in trade five months; what was A.'s stock ? Ans. 100l. 18. To divide 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36. 19. A and B distribute 1200l. each among a certain number of persons; A relieves 40 persons more than his friend B, and B gives to each person 51. a piece more than A. Now, how many persons were relieved by A and B respectively? Ans. 120 by A, and 80 by B. 20. The sum of two numbers is 6, and the sum of their 5th power is 1056 ; what are the numbers ? Ans. 2 and 4. 21. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. Ans. 10 and 14. 22. A and B. set off at the same time to a place distant 300 miles : A travels at the rate of one mile an hour quicker than B, and gets to the end of his journey 10 hours before him. How many miles per hour did each person travel ? Ans. A travelled 6, and B 5 miles an hour. 23. The sum of two numbers is 7, and the sum of their 4th power is 641; what are the numbers ? Ans. 2 and 5. 24. A company at a tavern had sl. 15s. to pay for their reckoning ; but, before the bill was settled, two of them leti the room, and they who remained had jos a piece może to pay tlian before: how many were there in company? Ans. 7. 62) OF EQUATIONS OF ALL DEGREES. The following is the translation of a letter received by the Editor of this Course, Jul" .8th, 1820, from the French Listitute, on account of an Essay on Involution and Evolution, then published; which letter will show the degree of approbation attached to the following general method of depressing and auginenting the roots of equations, and the subsequent extraction of their roots, see Nicholson's Analytical and Arithmetical Essays, published November, 1820, where this interesting branch on equations is. FRENCH INSTITUTE, ROYAL ACADEMY OF SCIENCES. Paris, July 10, 1820. The Perpetual Secretary of the Academy to Mr. Nicholson. SIR,—The Academy has received with a lively degree of interest, the Essay that you obligingly addressed to it on Involution and Evolution, or a Method of determining the Numerical Value of any Function of an unknown Quantity. I am desired, in its name, to thank you for sending this interesting work, which has been honourably placed in the Library of the Academy; and to express the sense of obligation which the Institution entertains for your attention, Receive, Sir, I beg, the assurance of the most distinguished respect with which I am, Your's, &c. B. G. CUVIER. Definitions. 1. TRANSFORMATION of an equation, is the method of finding another equation which shall have all its roots greater or less than the roots of the original equation by a given quantity. 2. Extraction is the method of finding such a value of the unknown quantity in an equation as will make the numerical value of both sides equal when that value is substituted for the unknown quantity representing it. 3. The Root of an Equation is the value of the unknown quantity, 4. When an equation contains one or niore powers of an unknown quantity, it is said to be of such a degree or order as is indicated by the exponent of the highest power of the unknown quantity. 5. A Simple Equation is that in which the exponent of the upknown quantity is unity. 6. A Quadratic Equation is that in which the exponent of the highest power of the unknowu quantity is e. 1 7. A Cubic Equation is that in which the exponent of the highest power of the unknown quantity is 3. 8. A Biquadratic Equation is that in which the exponent of the bighest power of the unknown quantity is 4. 9. An Equation of the nth Degree is that in which the exponent of the highest power of the unknown quantity is n, where n may be any assigned value whatever. Scholium.-The solutiou of simple egnations is not the object of this tract, as the root is fouod by plain division. Here I shall only treat of the higher equations, and show a general method of extracting their roots. For those who are curious in the history of this important subjact, I shall refer them to my Essay on Jovolution and Evolution. Proposition I. Problem. To transform the equation Az" + B2 - +C%*~2...+Lx=N. into another, of which the root shall be less or greater thau the root of the equation proposed by a given quantity a. Let v be the remaining part of the root; then will x=v+a; also let u the exponent of any power of z or of v ta; therefore, by the binonial theorem, TT avut HIT 12119 z*=(v+a)u=vato Quvv-+ 13 Trabzon +, &c. But, by reversing the order of the factors, ulo (U-1):11 (u~2)311 1311 add 3+, &c. and this equation will be equivalent to the following: j'11 2-ili 34-11 44-311 ***"17"*+in-stiaveritis liaywas + in-stia vurms +, &c. By substituting n, n-1, 9--2, &c. for u, and multiplying the respective powers and their values by the co-efficients, we have the following value of each of the terms of the given equation, viz. No.'l. 20-111* Azo = Av"+Ti Aava-s+ 30-11 4"-311 3-311 Bava—+ EN mli Cumm? tii Cava-s +&c. 1o-3 TiBa’ya-3+ &c. Cson 3 + &c. &c But the values of „B, n-,C, n-D, &c. exhibited in Prop: 3, Figurate numbers, are identical to the co-efficients now exhibited of |