12. What two numbers are those, whose sum, multiplied by the greater, is equal to 77, and whose difference, multiplied by the less, is equal to 12. Ans. 4 and 7 13. A grazier bought as many sheep as cost him 601., and after reserving 15 out of the number, sold the remainder for 541., and gained 2s. a head by them: how many sheep did he buy? Ans. 75 14. It is required to find two numbers, such that their product shall be equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes. Ans. 5 and (5+√5) 15. The difference of two numbers is 8, and the difference of their fourth powers is 14560; required the numbers. Ans. 3 and 11 16. A company at a tavern had 8l. 15s. to pay for their reckoning; but, before the bill was settled, two of them went away; in consequence of which those who remained had 10s. apiece more to pay than before: how many were there in company? Ans. 7 17. A person ordered 71. 4s. to be distributed among some poor people; but, before the money was divided, there came in, unexpectedly, two claimants more, by which means the former received a shilling a piece less than they would otherwise have done; what was their number at first? Ans. 16 persons 18. It is required to find four numbers in geometrical progression such, that their sum shall be 15, and the sum of their squares 85. Ans. 1, 2, 4, and 8 19. The sum of two numbers is 11, and the sum of their fifth powers is 17831; required the numbers ? Ans. 4 and 7 20. It is required to find four numbers in arithmetical progression such, that their common difference shall be 4, and their continued product 176985. Ans. 15, 19, 23, and 27 21. Two detachments of foot being ordered to a station at the distance of 39 miles from their present quarters, begin their march at the same time; but one party, by travelling of a mile an hour faster than the other, arrive there an hour sooner; required their rates of marching? Ans. 31 and 3 miles per hour 22. It is required to find two numbers, such that the square of the first plus their product, shall be 140, and the square of the second minus their product 78. Ans. 7 and 13 OF CUBIC EQUATIONS. A cubic equation is that in which the unknown quantity rises to three dimensions; and like quadratics, or those of the higher orders, is either simple or compound. A simple cubic equation is of the form x3+ax=b, x3+ax2=b, or x3+ax2 + bx=c, in each of which, the known quantities a, b, c, may be either or —. Or, either of the two latter of these equations may be reduced to the same form as the first, by taking away its second term; which is done as follows: RULE. Take some new unknown quantity, and subjoin to it a third part of the coefficient of the second term of the equation with its sign changed; then if this sum. or dif ference, as it may happen to be, he substituted for the original unknown quantity and its powers, in the pro posed equation, there will arise an equation wanting its second term. Note. The second term of any of the higher orders of equations may also be exterminated in a similar manner, by substituting for the unknown quantity some other unknown quantity, and the 4th, 5th, &c. part of the coefficient of its second term, with the sign changed, according as the equation is of the 4th, 5th, &c. power. EXAMPLES. 1. It is required to exterminate the second term of the equation x3+3ax2=b, or x3+3ax2 —b=0. in which equation the second power (22), of the unknown quantity, is wanting. 2. Let the equation x3-12x2+3x=-16, be transformed into another, that shall want the second term. Here x=x+4, (+4) 3=23+12z2+48z+64 Then — 12(x+4)2=—— - 1222 - 96z — 192 +3(x+4) Whence 23-45z-116=-16 Or 23-452=100 which is an equation where z2, or the second term, is wanting, as before. 3. Let the equation x3-6x2-10, be transformed into another, that shall want the second term. Ans. y3 12y=26 4. Let y3-15y2+81y=243, be transformed into an equation that shall want the second term. 3 Ans. x3+6x=88 7 9 5. Let the equation 3+ 2x2 + 2x-1/8 x3 16 =0, be transformed into another, that shall want the second term. 11 3 Ans. y3+ 16 4 6. Let the equation 2x3-3x2+4x-5-0, be transformed into another, that shall want its second term. OF THE SOLUTION OF CUBIC EQUATIONS. RULE. Take away the second term of the equation when necessary, as directed in the preceding rule. Then, if the numeral coefficients of the given equation, or of that arising from the reduction above mentioned, be substituted for a and b in either of the following formulæ, the result will give one of the roots, as required. 4 27 Where it is to be observed, that when the coefficient a, of the second term of the above equation, is negative, a3 27' α as also, in the formula, will be negative; and if 3' b the absolute b be negative, in the formula, will, also, be 2 b2 negative; but will be positive. (e) It may, likewise, be remarked, that when the equation is of the form x3. аx +b (e) This method of solving cubic equations is usually ascribed to Cardan, a celebrated Italian analyst of the 16th century; but the authors of it were Scipio Ferreus, and Nicolas Tartalea, who discovered it about the same time, independently of each other, as is proved by Montucla, in his Historire des Mathematiques, Vol. I. p. 568, and more at large in Hutton's Mathematical Dictionary, Art. Algebra. The rule above given, which is similar to that of Cardan, may be demonstrated as follows: Let the equation, whose root is required, be x3 +ax=b. And assume y+zx, and 3yz-a. Then, by substituting these values in the given equation, we shall have y3+3y2z+3yz2 +z3+aX(y+z)=y3+z3+3yz×(y+8)+ax (y+2)=y3+23-ax(y+2)+ax(y+2)=b, or y3+23=b. And if, from the square of this last equation, there be taken 4 times the cube of the equation yza, we shall have y6-2y320 +z6b2+2α3, or But the sum of this equation and y3+z3=b, is 2y3=b+√(62± 3) and their difference is 2x3=b-✔(62+4a3); whence y=3/36+√({b2+7a3), and z=36—(762+3). From which it appears, that y+z, or its equal x, is = 336+√(762+7a3)+36−√(163+7a3), which is the theo |