OF RATIOS AND PROPORTIONS WHICH RESPECT
MAGNITUDES.

DEFINITIONS.

85. THE following Definition of Ratio is usually given in the 5th. Book of Euclid's Elements.

"Ratio is a mutual relation of two magntiudes of the same kind to one another in respect of quantity."

scure.

This definition is frequently objected to as imperfect and obAnd it seems difficult to acquire a correct idea of the ratio of two magnitudes from the definition, if we are limited to the consideration of magnitudes abstractedly. By the help of numbers however, it becomes perfectly intelligible. Thus, if we divide the line or magnitude AB into 3 equal parts, and the magnitude CD contains 4 of those parts, the relation of AB to CD is the same as that of 3 to 4, which in numbers, is the ratio of the magnitudes AB and CD in respect of quantity.

Let GH be any other line or magnitude divided into 6 equal parts, and suppose IK contains 8 of those parts.

I

H
K

Then the relation or ratio of GH to IK is the same as that of AB to CD, because GH is contained or can be taken in IK as often as AB is contained or can be taken in CD, for the same reason that 6 is contained in 8 as often as 3 is contained in 4, that is, because.

Those four lines or magnitudes are proportional; viz. AB is to CD, as GH is to IK; and are set down in the manner of proportional numbers, thus AB CD :: GH: IK. And the proportion must evidently hold good whether AB and CD

are commensurable or incommensurable when compared with GH and IK.

86. Quantities of the same kind which are commensurable or can be divided into like parts, or parts of the same magnitude, may be compared in the same manner as we compare numbers in geometrical proportion (133, 134, arith.). Thus if AB contains 2; CD, 3; GH, 4; and IK, 6 equal parts, those lines or magnitudes will evidently have the same proportion as the number of equal parts into which they are respectively divided;

AHB

CHHD

GHHH

Or suppose the equal parts are again divided into a like number of equal parts, as 10 for example; then AB will contain 20; CD, 30; GH, 40; and IK, 60; therefore the quantities or lines will be in the proportion of 20, 30, 40, and 60; or as 2, 3, 4, and 6, the same as before.

Hence it is evident (if we make use of a common measure, as in Practical Geometry) that commensurable magnitudes may be represented by numbers, and their properties, as far as relates to proportion, demonstrated arithmetically. In the following theorems therefore, we shall sometimes refer to the articles in arithmetic which treat of proportion, in order to abridge the operations.

THEOREMS.

87. Parallelograms AC, GK between the same parallels, or having the same altitude, are to one another in the same ratio as their bases AD, GR.

For suppose AD is divided into 3 equal B parts, and that GR contains 2 of those parts. Then if lines are drawn from the

points of division parallel to the sides, the

parallelogram AC will be divided into 3, and the parallelogram GK into 2 equal parallelograms, because they stand upon equal bases (82a corol.)

Therefore

3 is to 2, as the paral. AC is to the paral. GK. Or AD GR paral. C: parl. GK.

::

B

CH

And if the bases AD, GR are incommensurable, the like proportion must evidently hold good. Suppose the base GR is the side of a square, and the base AD its diagonal (83, corol.). Let AN GR, and draw NO parallel to DC: and take NP so that AN and NP are commensurable.

A

NPD G

R

Then, paral. BN : paral. BP :: AN : AP.

And by continually taking commensurable parts in the remainder PD, we may at last, approximate nearer to D than any assignable distance. Consequently the parallelogram BD will ultimately be to the parallelogram BN (or HR) as AD to GR.

Corol. 1. Since triangles are the halves of their parallelograms (824. corol. 1.), therefore triangles having the same, or equal altitudes, are to one another as their bases.

Corol. 2. If RK, and DC be taken for the equal bases of the parallelograms RH and DB, then RG and DA will be their altitudes: Therefore parallelograms, or triangles, on equal bases, are respectively in the same ratio as their heights.

88. Parallelograms CADB, ORQP, having unequal bases and altitudes, are as the rectangles of the bases and altitudes.

Make BG, CH, and PS, ON, perpendicular to CB, OP, respectively; then the rectangle HB is equal to the parallelogram AB, and the rectangle NP equal to the parallelogram RP (82).

HAGD

Then, because equals must have equal ratios, As rectangle to rectangle, so is parallelogram to parallelogram.

Scholium. The parallelograms are also said to be in the compound ratio of their bases and altitudes. For if CB: OP, and BG: PS denote the ratio of the bases, and altitudes, respectively, the rectangles of the corresponding terms or CB x BG: OP × PS will denote the compound ratio or the ratio of their rectangles. (141, Arith.)

Suppose CB2, BG = 5, OP = 4, PS = 3; then denotes the ratio of CB to OP; and that of BG to PS; and their product × 3 (or 42) is the compounded ratio or that of the parallelograms, namely, as 10 to 12.

89. If four right lines AB, DC, PD, BR are proportional (AB DC :: PD: BR, or AB : PD :: DC: BR); the rectangle PC made with the two means DC, PD, is equal to the rectangle AR made with the two extremes AB, BR.

Let CO BR, and RQ = DC. Then the rectangles AR, DO having equal altitudes, will be as their bases (87); and for the same reason the rectangles PC, BQ will also be as their bases;

AB DC rectang. AR: rectang. DO;
PD: BR :: rectang. PC: rectang. BQ;

But AB DC: PD: BR, therefore by equality of ratios rectang. AR: rectang. DO :: rectang. PC: rectang. BQ:

Now the surfaces or rectangles DO, BQ contained under the same or equal lines (DC, BR) must be equal; therefore the consequents being equal, the antecedents or rectangles AR, PC will also be equal.

Or thus: Since the rectangle of two lines is analogous to the product of two numbers, if AB : DC :: PD : BR, then AB × BR = DC x PD *. (93, Arith.)

Corol. 1. When DC and PD are equal, the rectangle PC becomes a square; and its side is a mean proportional between the other two lines AB and BR (151, Arith.).

Corol. 2. Hence also, the product of the base and perpendicular gives the area or surface of a parallelogram.

=

*Here the surfaces of the rectanoles or parallelograms AR and PC are denoted by AB BR, and DC x PD. And if AB= 8, BR = 3, DC : 6, and FD = 4 (inches, for example); then 8 5 and 6 x 4 are the surfaces or areas of those rectangles in square inches.