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square, but greater than A's. Now if A should purchase all the front of B's lot, so as to extend the rear boundary line of his own through B's lot, parallel to the street, the two neighbors would possess equal quantities of land. Find the length of one side of B's lot.

Ans. 6(1+1/2) rods. 45. There are three numerical quantities having the following relations to each other;—the sum of the squares of the first and second, added to the first and second, is 32; the sum of the squares of the first and third, added to the first and third, is 42; and the sum of the squares of the second and third, added to the second and third, is 50. Required the quantities.

1st, 3 or 4; Ans. 2d, 4 or-5;

3d, 5 or -6. 46. What is the side of that cube which contains as many solid units as there are linear units in the diagonal through its opposite

Ans. V3.

corners.

47. It is required to find two quantities such that their sum, their product, and the sum of their squares, shall all be equal to each other.

Ans. }(3+V-3), and {(3:13). 48. Find those two numbers whose sum, product, and difference of their squares, are all equal to each other.

Ans. 1(3+/5), and 1(1+75). 49. Find two numbers, such that their product shall be equal to the difference of their squares, and the sum of their squares

shall be equal to the difference of their cubes.

Ans. Ešv5, and 1(5+/5).

SECTION VI.

PROPORTION, AND THE THEORY OF PERMUTATIONS

AND COMBINATIONS

PROPORTION.

315. Two quantities of the same kind may be compared, and their numerical relation determined, by finding how many times one contains the other. This mode of comparison gives rise to ratio and proportion.

316. Ratio is the quotient of one quantity divided by another of the same kind regarded as a standard of comparison.

There are two methods of indicating the ratio of two quantities.

1st. By writing the divisor before the dividend, with two dots between them; thus,

a: 6

indicates the ratio of a to b, where a is the divisor and b the dividend. 2d. In the form of a fraction; thus, the ratio of a to b may

be written

a

317. A Compound Ratio is the product of two or more ratios. Thus,

Simple ratios, Compound ratio, 318. The Duplicate Ratio of two quantities is the ratio of

a:6
c:d
ac : bd

their squares.

319. The Triplicate Ratio of two quantities is the ratio of their cubes.

320. Proportion is an equality of ratios. Thus, if two quantities, a and b, have the me ratio as two other quantities, c and d,

the four quantities, a, b, c, d, taken in their order, are said to be proportional. Proportion may be written in two ways; thus,

a:b::c:d, which is read, a is to b as c is to d; or thus,

a:b=c:d, which may be read as the other, or, the ratio of a to 6 is equal to the ratio of c to d. The second method of writing proportion is recommended as the more appropriate.

321. A Couplet is the two quantities which form a ratio.

322. The Terms of a proportion are the four quantities which are compared

323. The Antecedents in a proportion are the first terms of the two couplets; or the first and third terms of the proportion.

324. The Consequents in a proportion are the second terms of the two couplets; or the second and fourth terms of the proportion.

325. The Extremes in a proportion are the first and fourth terms.

326. The Means in a proportion are the second and third terms.

327. When the first of a series of quantities has the same ratio to the second, as the second has to the third, as the third to the fourth, and so on, the several quantities are said to be in continued proportion, and any one of them is a mean proportional between the two adjacent ones. Thus, if

a : b = b;c=cid=d:e, then a, b, c, d, and e are in continued proportion, and b is a mean proportional between a and c, c a mean proportional between b and d; and so on.

328. One quantity is said to vary directly as another when the two quantities, by reason of their mutual dependence, have always a constant ratio, so that if one be changed the other will be changed in the same proportion.

Thus, for illustration, suppose, in the purchase of a commodity, a certain quantity, A, costs a certain sum, B. Now if the price of unity remain the same, it is evident that 2 A will cost 2B; 3A will

cost 3B; and in general, m A will cost . In this case the cost is said to vary directly as the quantity.

329. One quantity is said to vary inversely as another when the first has a constant ratio to the reciprocal of the other.

330. One quantity is said to vary as two others jointly, when it has a constant ratio to the product of the two.

331. The Sign of Variation is the symbol oc; thus, the expression, A oc B, signifies that A varies as B.

From the definition of variation, it is evident that the expression, A o B, is equivalent to the proportion,

A:B=m:1, where m is a constant. This proportion gives

A = mB. Hence the general truth,

If A vary as B, then A is equal to B multiplied by some constant quantity.

PROPOSITIONS IN PROPORTION.

332. A Proposition is the statement of a truth to be demonstrated, or of a problem to be solved.

333, A Scholium is a remark showing the application or limitation of a preceding proposition. 334. If in the proportion

a:b=cid, the second method of indicating ratio be employed, we have

6 d

(A)

a

which is the fundamental equation of proportion; and any proposition relating to proportion will be proved, when shown to be consistent with this equation.

PROPOSITION I.-In every proportion, the product of the extremes is equal to the product of the means.

a

с

Let

a:b=c:d represent any proportion ;

6 d then by formula (A), clearing of fractions, bc = ad.

That is, the product of b and c, the means, is equal to the product of a and d, the extremes. SCHOLIUM. From the last equation, we have

ad The first mean,

b

(1)

ad The second mean,

7

с

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.

Hence,

1st. Either mean is equal to the product of the extremes divided by the other mean. (1)

22. Either extreme is equal to the product of the means divided by the other extreme. (2)

PROPOSITION II.—Conversely :-If the product of two quantities is equal to the product of two others, then two of them may be taken for the means, and the other two for the extremes of a proportion. Let

bc ad.

b d Dividing by ac,

; hence by formula (A), a: b=c:d, in which the factors of the first product, bc, are the means, and the factors of the second product, ad, are the extremes.

PROPOSITION III.-If four quantities be in proportion, they will be in proportion by ALTERNATION; that is, the antecedents will be to each other as the consequents. Let

a:b=c:d;

b d then by formula (A),

a

с

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(1)

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