Sympathetic Vibratory Physics

Theory of Proportion

"A proportion is an expression of equality between two equal ratios; and is written in one of the following forms:

a:b = c:d; a:b::c:d; a/b=c/d.

This proportion is read, "a is to b as c is to d"; or "the ratio of a to b is equal to the ratio of c to d."

The terms of a proportion are the four quantities compared; the first and third terms are called the antecedents, the second and fourth terms, the consequents; the first and fourth terms, the extremes, the second and third terms, the means.

Thus, in the proportion a:b = c:d; a and c are antecedents, b and d the consequents, a and d the extremes, b and c the means.

The fourth proportional to three given quantities is the fourth term of the proportion which has for its first three terms the three given quantities taken in order.

Thus, d is the fourth proportional to a, b, and c in the proportion a:b = c:d.

The quantities a, b, c, d, e, are said to be in continued proportion, if a:b + b:c = c:d = d:e.

If three quantities are in continued proportion, the second is called the mean proportional between the other two, and the third is called the third proportional to the other two.

Thus, in the proportion a:b = b:c; b is the mean proportional between a and c; and c is the third proportional to a and b."

Propositions of Proportion

Proposition I
In every proportion the product of the extremes is equal to the product of the means.

Proposition II
The mean proportional between two quantities is equal to the square root of their products.

Proposition III
If the product of two quantities is equal to the product of two others, either two may be made the extremes of the proportion in which the other two are made the means.

Proposition IV
If four quantities are in proportion, they are in proportion by alternation; that is, the first term is to the third as the second is to the fourth.

Proposition V
If four quantities are in proportion, they are in proportion by inversion; that is, the second term is to the first as the fourth is to the third.

Proposition VI
If four quantities are in proportion, they are in proportion by composition; that is, the sum of the first two terms is to the second term as the sum of the last two terms is to the fourth term.

Proposition VII
If four quantities are in proportion, they are in proportion by division; that is, the difference of the first two terms is to the second term as the difference of the last two terms is to the fourth term.

Proposition VIII
If four quantities are in proportion, they are in proportion by composition and division; that is, the sum of the first two terms is to their difference as the sum of the last two terms is to their difference.

Proposition IX
In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to is consequent.

Proposition X
The products of the corresponding terms of two or more proportions are in proportion.
Corollary: If three quantities are in continued proportion, the first is to the third as the square of the first is to the square of the second.

Proposition XI
Like powers of the terms of a proportion are in proportion.

Proposition XII
Equimultiples of two quantities are in the same ratio as the quantities themselves.
[Equimultiples of two quantities are the products obtained by multiplying each of them by the same number. Thus, ma and mb are equimultiples of a and b.]

Proposition XIII
If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally.
Corollary 1: One side of a triangle is to either part cut off by a straight line parallel to the base as the other side is to the corresponding part.
Corollary 2: If two lines are cut by any number of parallels, the corresponding intercepts are proportional.

Proposition XIV
If a straight line divides two sides of a triangle proportionally, it is parallel to the third side.

Proposition XV
The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides.

Proposition XVI
The bisector of an exterior angle of a triangle divides the opposite side externally into segments which are proportional to the adjacent sides.
Corollary: The bisectors of an interior angle and an exterior angle at one vertex of a triangle meeting the opposite side divide that side harmonically.
[If a given straight line is divided internally and externally into segments having the same ratio, the line is said to be divided harmonically.]

Proposition XVII
Two mutually equiangular triangles are similar.
Corollary 1: Two triangles are similar if two angles of the one are equal, respectively, to two angles of the other.
Corollary 2: Two right triangles are similar if an acute angle of the one is equal to an acute angle of the other.

Proposition XVIII
If two triangles have an angle of the one equal to an angle of the other, and the including sides proportional, they are similar.

Proposition XIX
If two triangles have their sides respectively proportional, they are similar.

Proposition XX
Two triangles which have their sides respectively parallel, or respectively perpendicular, are similar.
Corollary: The parallel sides and the perpendicular sides are homologous sides of the triangle.

Proposition XXI
The homologous altitudes of two similar triangles have the same ratio as any two homologous sides.

Proposition XXII
If two parallels are cut by three or more transversals that pass through the same point, the corresponding segments are proportional.

Proposition XXIII
Conversely: If three or more non-parallel straight lines intercept proportional segments upon two parallels, they pass through a common point.

Proposition XXIV
The perimeters of two similar polygons have the same ratio as any two homologous sides.

Proposition XXV
If two polygons are similar, they are composed of the same number of triangles, similar each to each, and similarly placed.

Proposition XXVI
Conversely: If two polygons are composed of the same number of triangles, similar each to each, and similarly placed, the polygons are similar.

[Source: Plane and Solid Geometry; Book III; pages 135-158. George A. Wentworth, 1902. Ginn & Company, The Athenæum Press, Boston, Massachusetts]

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Propositions of Proportion