however the negative value as insignificant or 96 96 useless, since in the equation = -1, the x+8 negative value of x, viz. -32, affords a solution to the same question negatively expressed, as follows:-A boy sold 2 shillings' worth of apples. If he had sold 8 more apples for the same money, he would have gained a farthing a piece more for them; how many had he? And it should be observed that in algebraical operations, since + multiplied by +y is equal to -x multiplied by y; in the multiplication and involution of quantities, new values of the unknown quantities are introduced, which, if not again excluded by the nature of the question, will appear in the final equation. (Art. 74.) Ex. 2. Divide a line 20 inches in length into two such parts, that the product, or rectangle under the whole and one part, may be equal to the of the other part. square Let x=one part, then 20-x=the other by the question x2=20 × (20 −x) x2+20x=400 x2+20x+100=500 ... x+10=/500 x=-10/500 that is x=500-10 or -500-10 Here the positive value answers the given conditions; and the negative value shows, that if the line be produced 500+10 inches, the square of the part produced is equal to the rectangle under the line given and the line made up of the whole and part produced. When the problem or question under consideration requires the determination of two or more unknown quantities by means of as many equations, from which a final quadratic equation can be obtained; the same rules will apply as in simple equations, observing however that when the given equations take a particular form, there are certain modes of resolving them that can be often conveniently adopted, and which experience alone can teach, by which means many other equations, which, although not quadratic equations, may by such artifices be reduced to forms, to which the mode of solution that has been pointed out is applicable. Ex. 3. Given x+y=a find x and y From 1st equatn. x2 + 2xy + y2=a2 From 2nd equation Subtract 4xy=4b Subtract x2-2xy+y2=a2 — (2a2 — 2b)=2b — a2 Extr. square root x-y=±√2b-a2 And since x+y=a by the 1st equation; .. x and y are found by the last example. Cube 1st eq. 2+3x2y+3xy2+y3=a3 and y can be determined as in the 3rd Example. 6. Divide the number 40 into 2 parts, that their product may be 375. Ans 15 and 25. 7. The sum of 2 numbers is 34, and the sum of their squares is 650; find the numbers. Ans. 11 and 23. 8. The product of 2 numbers is 273, and the sum of their squares is 610; find the numbers. Ans. 13 and 21. 9. Find a number such, that of it taken from its square will make 30. 10. Ans. 8. The difference between 2 numbers is 9, and their product 52; find the numbers. Ans. 4 and 13. 11. Two steam-vessels start together; one of them goes at the rate of one knot per hour faster, and runs 144 miles in two hours less time, than the other. How many knots per hour did each go? Ans. 8 and 9 knots respectively. 12. A regiment of 1025 men was formed into 2 solid squares. One square had 5 men more in a side than the other. What number of men were in each square? Ans. 400 and 625. RATIOS. 79. Def.-Ratio is the relation which two quantities of the same kind have to one another, in respect to magnitude; the comparison being made by considering how often one is contained in the other. If a and b represented the numerical values of any two quantities, the ratio of a to b is expressed by the quotient or fraction or by a:b. α Ъ ; In this ratio a is called the antecedent, and b the consequent. 80. Ratios are compared with each other by means of the fractions which express them. Thus |