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the unit, the remainder, or undivided part of any quantity may be conceived to be so diminished, as to be of no assignable value compared with the other part, and as there are no conditions which limit this subdivision, the ratio of these quantities may be expressed

as before by the fraction

a

b

DEF. 2.-A proportion consists of 2 equal ratios. Or if, of four quantities; the first contains the second, or part of the second, as often as the third contains the fourth, or like part of the fourth; they are said to be proportionals. That is, if a, b, c, d be the numerical values of 4 geometrical quantities; the pro

portion is represented by the equation

a с

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bd

which is usually expressed by saying, a is to b as c is to d. In this proportion, a and c are called the antecedents, and b and d the consequents. The proportion is also expressed by a:b::c:d, or by a b c d as most convenient; in which a and d are called the extremes, and b and c the means. And in any proportion if the means are equal to each other, i. e. if a:b::b: c, thén b is said to be a mean proportional between a and b.

DEF. 3. Similar rectilineal figures are those which have their respective angles equal to each other, and the sides about the equal angles proportional.

Hyp.

Hyp.

PROP. LXV. THEOR.

16. 6 Eu.

If four geometrical quantities, numerically represented by a, b, c, d, be proportionals; that is, if ab::c:d; then will ad=bc.

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If there be four quantities represented as before by a, b, c, d; and if ad=bc, then will a: b::c: d.

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Prop. 65.

PROP. LXVII. THEOR. 16. 5 Eu.

If as before a:b::c:d; then alternately a:c::b: d.

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PROP. LXVIII. THEOR. 12. 5 Eu.

If there be any number of proportional quantities, represented numerically as a:b:: c:d::e:f, &c.; then will one antecedent be to its consequent, as the sum of all the antecedents to the sum of all the consequents; or a:b::a+c+e, &c.: b+d+f, &c.

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.. ab+ad+af, &c. = ba+be+be, &c.
That is, a(b+d+f)=b(a+c+e),
..aba+c+e, &c. : b+d+f, &c.

Prop. 66.

PROP. LXIX. THEOR.

If as before a:b::c:d, then will a2 : b3 : ; c2 : d'.

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PROP. LXX. THEOR. 15. 5 Eu.

Equimultiples of two quantities have the same ratio as the quantities themselves.

Let ma and mb be equimultiples of a and b; then will ab:: ma: mb.

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Triangles and parallelograms of the same altitude are to each other as their bases.

In str. line BD take any number of equal parts or units BG, GC, CH, &c. From pt. A without the line BD draw AB, AG, AC, AH, &c. also, draw AF, EC, FD respectively || BD, AB, AC; and the As ABC, ACD having the same altitude, viz. the perpendicular drawn from A to BD; then will base BC: base CD:: ABC: ACD::

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m

A E F

BGCHIK D

Prop. 35. Then As ABG, AGC, ACH, &c. are equal to

Cor. 1.

each other;

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If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides proportionally: and if the sides be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

1st. Draw DE || BC one of the sides of the ABC; then shall DB: AD :: EC: AE; also AB: AC:: AD: AE.

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Prop. 70.

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