the side BD included between these equal angles is common to the two triangles, they are equal (Prop. VI.); hence the side A B opposite the angle ADB is equal to the side DC opposite A

the angle DBC (Prop. VI. Cor.); and, in like manner, the side AD is equal to the side BC; hence the opposite sides of a parallelogram are equal.

Again, since the triangles are equal, the angle A is equal to the angle C (Prop. VI. Cor.); and since the two angles DBC, ABD are respectively equal to the two angles AD B, BD C, the angle ABC is equal to the angle AD C.

109. Cor. 1. The diagonal divides a parallelogram into two equal triangles.

110. Cor. 2. The two parallels A D, B C, included between two other parallels, A B, CD, are equal.

PROPOSITION XXXII.-THEOREM.

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

Let ABCD be a quadrilateral having its opposite sides equal; then will the equal sides be parallel, and the figure be a parallelogram.

A

D

B

C

For, having drawn the diagonal BD, the triangles A BD, BDC have all the sides of the one equal to the corresponding sides of the other; therefore they are equal, and the angle A D B opposite the side A B is equal to D B C opposite CD (Prop. XVIII. Sch.); hence the side A D is parallel to BC (Prop. XX.). For a like reason, AB is parallel to CD; therefore the quadrilateral A B C D is a parallelogram.

PROPOSITION XXXIII.-THEOREM.

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

Let ABCD be a quadrilateral, having the sides A B, CD equal and parallel; then will the other sides also be equal and parallel.

Draw the diagonal BD; then, since A

D

B

C

A B is parallel to CD, and BD meets them, the alternate angles A BD, BDC are equal (Prop. XXII.); moreover, in the two triangles A BD, DB C, the side BD is common; therefore, two sides and the included angle in the one are equal to two sides and the included angle in the other, each to each; hence these triangles are equal (Prop. V.), and the side A D is equal to BC. Hence the angle A D B is equal to D B C, and consequently A D is parallel to BC (Prop. XX.); therefore the figure ABCD is a parallelogram.

PROPOSITION XXXIV. - THEOREM.

113. The diagonals of every parallelogram bisect each other.

Let ABCD be a parallelogram,

and A C, DB its diagonals, intersecting at E; then will A E equal E C, and BE equal ED.

A

D

C

E

B

For, since AB, CD are parallel, and BD meets them, the alternate angles CDE, A BE are equal (Prop. XXII.); and since AC meets the same parallels, the alternate angles BAE, ECD are also equal; and the sides AB, CD are equal (Prop. XXXI.). Hence the triangles ABE, CDE have two angles and the in

cluded side in the one equal to two angles and the included side in the other, each to each; hence the two triangles are equal (Prop. VI.); therefore the side A E opposite the angle A B E is equal to CE opposite CDE; hence, also, the sides BE, DE opposite the other equal angles are equal.

114. Scholium. In the case of a rhombus, the sides AB, BC being equal, the triangles A E B, EBC have all the sides of the one equal to the corresponding sides of the other, and are, therefore,

D

C

E

B

equal; whence it follows that the angles AEB, BEC are equal. Therefore the diagonals of a rhombus bisect each other at right angles.

PROPOSITION XXXV.-THEOREM.

115. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

Let ABCD be a quadrilateral, and D

AC, DB its diagonals intersecting at E;

then will the figure be a parallelogram.

C

E

B

For, in the two triangles ABE, CDE, the two sides AE, E B and the included A angle in the one are equal to the two sides CE, ED and the included angle in the other; hence the triangles are equal, and the side AB is equal to the side CD (Prop. V. Cor.). For a like reason, A D is equal to C B; therefore the quadrilateral is a parallelogram (Prop. XXXII.).

BOOK II.

RATIO AND PROPORTION.

DEFINITIONS.

116. RATIO is the relation, in respect to quantity, which one magnitude bears to another of the same kind; and is the quotient arising from dividing the first by the second.

A ratio may be written in the form of a fraction, or with the sign: .

A

B'

Thus the ratio of A to B may be expressed either by or by A: B.

117. The two magnitudes necessary to form a ratio are called the TERMS of the ratio.

The first term is called the

ANTECEDENT, and the last, the cONSEQUENT.

118. Ratios of magnitudes may be expressed by numbers, either exactly, or approximately.

This may be illustrated by the operation of finding the numerical ratio of two straight lines, A B, CD.

From the line CD cut off a part 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 D F, as many times as possible; once, for example, with the remainder BG.

Proceed thus till a remainder arises, which is exactly contained a certain number of times in the preceding one. Then this last remainder will be the common measure of the proposed lines; and, regarding it as unity, we shall easily find the values of the preceding remainders; and, at last, those of the two proposed lines, and hence their ratio in numbers.

Suppose, for instance, we find GB to be contained exactly twice in FD; BG will be the common measure of the two proposed lines. Let BG equal 1; then will FD equal 2. But EB contains FD once, plus GB; therefore we have E B equal to 3. CD contains E B once, plus FD; therefore we have CD equal to 5. AB contains CD twice, plus EB; therefore we have A B equal to 13. Hence the ratio of the two lines is that of 13 to 5. If the line CD were taken for unity, the line AB would be; if A B were taken for unity, CD would be.

It is possible that, however far the operation be continued, no remainder may be found which shall be contained an exact number of times in the preceding one. In that case there can be obtained only an approximate ratio, expressed in numbers, more or less exact, according as the operation is more or less extended.

119. When the greater of two magnitudes contains the less a certain number of times without having a remainder, it is called a MULTIPLE of the less; and the less is then called a SUBMULTIPLE, or measure of the greater.

Thus, 6 is a multiple of 2; 2 and 3 are submultiples, or measures, of 6.

120. EQUIMULTIPLES, or LIKE MULTIPLES, are those which contain their respective submultiples the same number of