Again, the two triangles ABC, DEC, are equiangular: for the angles BAC, BDC, are equal, standing on the same arc BC; and the angle DCE is equal to the angle BCA, by adding the common angle ACE to the two equal angles DCA, BCE; therefore the third angles E and ABC are also equal: but AC, DC, and AB, DE, are the like sides: therefore the rectangle Ac. DE is the rectangle AB. DC (th. 62).

Hence, by equal additions, the sum of the rectangles AC. BE+ AC. DE is AD BC + AB DC. But the former sum of the rectangles AC BE + AC DE is the rectangle AC. BD (th. 30): therefore the same rectangle ac BD is equal to the latter sum, the rect. AD. BC + the rect. AB. DC (ax. 1). Q. E. D.

Corol. Hence, if ABD be an equilateral triangle, and c any point in the arc BCD of the circumscribing circle, we have For AC BD being =AD. BC + AB. DC; dividing by BD = AB = AD, there results AC BC + DC.

AC BC DC.

OF RATIOS AND PROPORTIONS.

DEFINITIONS.

DEF. 76. RATIO is the proportion or relation which one magnitude bears to another magnitude of the same kind, with respect to quantity.

Note. The measure, or quantity, of a ratio, is conceived, by considering what part or parts the leading quantity, called the Antecedent, is of the other, called the Consequent ; or what part or parts the number expressing the quantity of the former, is of the number denoting in like manner the latter. So, the ratio of a quantity expressed by the number 2, to a like quantity expressed by the number 6, is denoted by 2 divided by 6, or or the number 2 being 3 times contained in 6, or the third part of it. In like manner, the ratio of the quantity 3 to 6, is measured by or; the ratio of 4 to 6 is or; that of 6 to 4 is for ; &c.

77. Proportion is an equality of ratios. Thus,

78. Three quantities are said to be proportional, when the ratio of the first to the second is equal to the ratio of the

second to the third. As of the three quantities a (2), b (4), c (8), where, both the same ratio.

79. Four quantities are said to be proportional, when the ratio of the first to the second, is the same as the ratio of the third to the fourth. As of the four, a (4), в (2), c (10), d (5), where == 2, both the same ratio.

Note. To denote that four quantities, A, B, C, D, are proportional, they are usually stated or placed thus, A : B :: C: D; and read thus, a is to в as c is to D. But when three quantities are proportional, the middle one is repeated, and they are written thus, A: B:: B: C.

The proportionality of quantities may also be expressed very generally by the equality of fractions, as at pa. 118..

80. Of three proportional quantities, the middle one is said to be a Mean Proportional between the other two; and the last, a Third Proportional to the first and second.

81. Of four proportional quantities, the last is said to be a Fourth Proportional to the other three, taken in order.

82. Quantities are said to be Continually Proportional, or in Continued Proportion, when the ratio is the same between every two adjacent terms, viz. when the first is to the second, as the second to the third, as the third to the fourth, as the fourth to the fifth, and so on, all in the same common ratio.

As in the quantities 1, 2, 4, 8, 16, &c.; where the common ratio is equal to 2.

83. Of any number of quantities, A, B, C, D, the ratio of the first A, to the last D, is said to be Compounded of the ratios of the first to the second, of the second to the third, and so on to the last.

84. Inverse ratio is, when the antecedent is made the consequent, and the consequent the antecedent. Thus, if 1:2::3:6; then inversely, 2:1::6:3.

85. Alternate proportion is, when antecedent is compared with antecedent, and consequent with consequent.-As, if 1:23:6; then, by alternation, or permutation, it will be 1:3 : : 2:6.

86. Compound ratio is, when the sum of the antecedent and consequent is compared, either with the consequent, or

with the antecedent. Thus, if 1:2 position, 1+2 : 1 :: 3 + 6: 3, and 6: 6.

3:6, then by com. 1+2 : 2 :: 3+

87. Divided ratio, is when the difference of the antecedent and consequent is compared, either with the antecedent or with the consequent. Thus, if 1 2 3 6, then, by division, 2-1:1::6–3; 3, and 2 — 1 : 2 :: 6 - 3: 6.

Note. The term Divided, or Division, here means subtract. ing, or parting; being used in the sense opposed to compounding, or adding, in def. 86.

THEOREM LXVI.

EQUIMULTIPLES of any two quantities have the same ratio as the quantities themselves.

Let A and B be any two quantities, and ma, mв, any equimultiples of them, m being any number whatever: then will ma and mв have the same ratio as a and в, or a : B :: ma:

Corol. Hence, like parts of quantities have the same ratio as the wholes; because the wholes are equimultiples of the like parts, or a and в are like parts of ma and mв.

THEOREM LXVII.

IF four quantities, of the same kind, be proportionals; they will be in proportion by alternation or permutation, or the antecedents will have the same ratio as the consequents*.

The author's object in these propositions was to simplify the doctrine of ratios and proportions, by imagining that the antecedents and consequences may always be divided into parts that are commensurable. But it is known to mathematicians that there are certain quantities or magnitudes, such as the side and the diagonal of a square, which cannot possibly be divided in that manner by means of a common measure. The theorems themselves are true, nevertheless, when applied to these incommensurables; since no two quantities of the same kind can possibly be assigned, whose ratio cannot be expressed by that of two numbers, so near, that the difference shall be less than the least number that can be named. From the greater of two unequal magnitudes we may take, or suppose taken, its half, from the remaining half, its half,

Let A : B :: Ma: mB; then will a : ma :: B : MB.

and so on, by continual bisections, until there shall at length be left a magnitude less than the least of two magnitudes; or, indeed, less than the least magnitude that can be assigned; and this principle furnishes a ground of reasoning.

Or, somewhat differently, let A and B be two constant quantities, a and b two variable quantities, which we can render as small as we please, if we have an equality between A+ a, and в + b, or, in other words, if the equation ▲ + a = =B+b holds good whatever are the values of a and b, it may be divided into two others, a = B, between the constant quantities, and, ab, between the variable quantities, and which latter must obtain for all their states of magnitude. For if, on the contrary, we suppose ▲ = BQ, we shall have A -B= =b-a = Q, an absurd result; since the quantities a and b being susceptible of diminishing indefinitely, their difference cannot always be Q. This is the principle which constitutes the method of limits. In general, one magnitude is called a limit of another, when we can make this latter approach so near to the former, that their difference shall be less than any given magnitude, and yet so that the two magnitudes shall never become strictly equal.

=

=

Let us here apply the principle to the demonstration of this proposition, that the ratio of two angles ACB, NOP, is equal to that of the arcs, ab, np, comprised between their sides, and drawn from their respective summits as centres with equal radii.

If the 'arcs pn, ba, are commensurable, their common measure bm will be contained n times in pn, r times in ba; so that we shall have the equal ratios

ba T

Through each

point of division, m, n', &c. draw the lines mc, n'c, &c. to the summits c, and o, the angles proposed will be divided into n, and r, equal angles,

bcm, mcn', poq, qor, &c. We shall, therefore, have

PON n

= Hence BCA ↑

If the arcs are incommensurable, divide one of them, ba, into a number r of equal parts, bm, mn', &c. and set off equal parts pq, qr, &c. upon the other arc pn; and let s be the point of division that falls nearest to n. Draw oss. Then, by the preceding, ba, ps, being commensurable, we

Here Nos and ns are susceptible of indefinite variation, according as we change the common measure, bm, of ba; they may, therefore, be VOL. I.

42

Otherwise. Let A: B:: C: D; then shall B: A :: C):

In a similar manner may most of the other theorems be demonstrated.

THEOREM LXVIII.

Ir four quantities be proportional; they will be in proportion by inversion, or inversely.

Let A B MA: mB; then will в: A :: mb : ma.

Ir four quantities be proportional; they will be in proportion by composition and division.

Corol. It appears from hence, that the sum of the greatest and least of four proportional quantities, of the same kind, exceeds the sum of the other two. For, since

A: A+B ma ma + mв, where A is the least, and mamв the greatest; then m + 1. A + m3, the sum of the greatest and least, exceeds m + 1 . A + B, the sum of the two other quantities.

THEOREM LXX.

IF, of four proportional quantities, there be taken any equimultiples whatever of the two antecedents, and any equi

rendered as small as we please, while the other quantities remain the same. Consequently, by the nature of limits, as above explained, we have the equal ratios

PON

BCA

pn

ba

, or PON: BAC:: pn: ba.