PROPOSITION XXVIII. THEOREM.

In every parallelogram, the opposite sides and angles are equal each to each.

B

C

Let ABCD be a parallelogram: then will AB=DC, AD=BC, the angle A=C, and the angle ADC=ABC. For, draw the diagonal BD, dividing the parallelogram into the two triangles, ABD, DBC. Now, since AD, BC, are parallel, the angle ADB DBC (P. 20, c. 2); and since AB, CD, are parallel, the angle ABD=BDC: and since the

side DB is common, the two triangles are equal (P. 6); therefore, the side AB, opposite the angle ADB, is equal to the side DC, opposite the equal angle DBC (p. 10, s.), and the third sides AD, BC, are equal: hence, the opposite sides of a parallelogram are equal.

Again, since the triangles are equal, the angle A is equal to the angle C (P. 10, s.) Also, the angle ADC cornposed of the two angles, ADB, BDC, is equal to ABC, composed of the corresponding equal angles DBC, ABD (A. 2): hence, the opposite angles of a parallelogram are equal.

Cor. 1. Two parallels AB, CD, included between two other parallels AD, BC, are equal; and the diagonal DB divides the parallelogram into two equal triangles.

Cor. 2. Two parallelograms which have two sides and the included angle in the one equal to two sides and the included angle in the other, each to each, are equal.

Let the parallelogram ABCD, have the sides AB, AD, and the included angle BAD equal to the sides AB, AD, and the included angle BAD, in the next figure; then will they be equal.

A

D

B

C

For, in each figure, draw the diagonal DB. By the last corollary, the diagonal divides each parallelogram into two equal triangles: but the triangle BAD in one parallelogram, is equal to the triangle BAD in the other (P.5): hence, the parallelograms are equal (A. 6).

PROPOSITION XXIX. THEOREM.

If the opposite sides of a quadrilateral are equal, each to each, the equal sides are parallel, and the figure is a parallelogram.

D

Let ABCD be a quadrilateral, having its opposite sides respectively equal, viz.: AB=DC, and AD=BC; then will these sides be parallel, and the figure a parallelogram. For, having drawn the diagonal BD the two triangles ABD, BDC, have all the sides of the one equal to the corresponding sides of the other; therefore they are equal, and the angle ADB, opposite the side AB, is equal to DBC, opposite CD (P. 10, s.); therefore the side AD is parallel to BC (p. 19, c. 1) For a like reason AB is parallel to CD: therefore, the quadrilateral ABCD is a parallelogram.

A

B

PROPOSITION XXX. THEOREM.

If two opposite sides of a quadrilateral are equal and parallel, the other sides are equal and parallel, and the figure is a parallelogram.

Let ABCD be a quadrilateral, having the sides AB, CD, equal and parallel; then will the figure be a parallel

ogram.

For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, since AB is parallel to DC, the alternate angles ABD, BDC are equal (P. 20, c. 2); moreover, the

side DB is common, and the_side_AB=DC; hence, the triangle ABD is equal to the triangle DBC (P. 5); there fore, the side AD is equal to BC, the angle ADB=DBC, and consequently AD is parallel to BC (P. 19, c. 1); hence, the figure ABCD is a parallelogram.

PROPOSITION XXXI. THEOREM.

The two diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other.

Let ADCB be a parallelogram, AC and DB its diago nals, intersecting at E; then will AE=EC, and DE= EB.

Comparing the triangles AED, BEC, we find the side AD=CB (P. 28), the angle ADB CBE, and the angle DAE=ECB (p. 20, c. 2); hence, these triangles are equal (P. 6); consequently,

=

B

E

A

AE, the side opposite the angle ADE, is equal to EC opposite CBE, and DE opposite DAE is equal to EB opposite ECB.

Scholium. In the case of the rhombus, the sides AB, BC being equal, the triangles AEB, EBC, have all the sides of the one equal to the corresponding sides of the other, and are therefore equal: whence, it follows, that the angles AEB, BEC, are equal, and therefore, the two diago nals of a rhombus bisect each other at right angles.

BOOK II.

OF RATIOS AND PROPORTIONS.

DEFINITIONS.

1. PROPORTION is the relation which one magnitude bears to another magnitude of the same kind, with respect to its being greater or less.*

2. RATIO is the measure of the proportion which one magnitude bears to another; and is the quotient which arises from dividing the second by the first. Thus, if A and B represent magnitudes of the same kind, the ratio of A to B is expressed by

B

A

A and B are called the terms of the ratio; the first 18 called the antecedent, and the second, the consequent.

3. The ratio of magnitudes may be expressed by num bers, either exactly or approximatively; and in the latter case, the approximation may be brought nearer to the true ratio than any assignable difference.

Thus, of two magnitudes, one may be considered to be divided into some number of equal parts, each of the same kind as the whole, and regarding one of these parts as a unit of measure, the magnitude may be expressed by the number of units it contains. If the other magnitude contain an exact number of these units, it also may

* See Davies' Logic of Mathematics: Proportion, § 267.

be expressed by the number of its units, and the two magnitudes are then said to be commensurable.

If the second magnitude do not contain the measuring unit an exact number of times, there may perhaps be a smaller unit which will be contained an exact number of times in each of the magnitudes. But if there is no unit of an assignable value, which is contained an exact number of times in each of the magnitudes, the magnitudes are said to be incommensurable.

It is plain, however, that if the unit of measure be repeated as many times as it is contained in the second magnitude, the result will differ from the second magnitude by a quantity less than the unit of measure, since the remainder is always less than the divisor. Now, since the unit of measure may be made as small as we please, it follows, that magnitudes may be represented by num bers to any degree of exactness, or they will differ from their numerical representatives by less than any assignable magnitude.

4. We will illustrate these principles by finding the ratio between the straight lines CD and AB, which we will suppose commensurable.

From the greater line AB, cut off a part equal A C to the less CD, as many times as possible; for example, twice, with the remainder BE

From the line CD, cut off a part, CF, equal to the remainder BE, as many times as possible; once, for example, with the remainder DF.

From the first remainder BE, cut off a part equal to the second, DF, as many times as possible; once, for example, with the remainder BG.

From the second remainder DF, cut off a part equal to BG, the third remainder, as many times. as possible.

-E

-G

Continue this process, till a remainder occurs, which is contained exactly, a certain number of times, in the preceding one.

Then, this last remainder will be the common measure of the proposed lines. Regarding this as unity, we shall