113. Cor. 1. If two proportions have an antecedent end its consequent the same in both, the remaining terms will be in proportion.

114. Cor. 2. Therefore, by alternation (Theo. V.), if two proportions have the two antecedents or the two consequents the same in both, the remaining terms will be in proportion.

THEOREM X.

115. If any number of magnitudes are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents.

Let AB::C:D::E: F; then will

A: BA+C+E: B+D+F.

For, from the given proportion, we have

A×

AX D= BX C, and AX F = B × E.

By adding A × B to the sum of the corresponding sides of these equations, we have

A× B+ A× D+A × F = A× B+B × C + B × E. Therefore,

A× (B+D+F) = B× (A+C+E).

Hence, by Theo. II.,

A: B:: A+ C+E:B+D+F.

THEOREM XI.

116. If there be two sets of proportional magnitudes, the products of the corresponding terms will be proportionals.

Let A B C : D, and E: F::G: H; then will

:

AXE:BXF:: CXG: DX H.

For, from the first of the given proportions, by Theo. I., we

have

AX DBX C;

and from the second of the given proportions, by Theo. I., we

have

EXH-FX G.

Multiplying together the corresponding members of these equations, we have

AX DXEX H= BX CXFX G.

Hence, by Theo. II.,

AXE:BX F:: CXG: DX H.

THEOREM XII.

117. If four magnitudes are proportionals, their like powers and roots will also be proportional.

Let A B C : D; then will

A”

:

A: B" :: C: D", and A: B:: CA: DA.

For, from the given proportion, we have

Raising both members of this equation to the nth power, we have

and extracting the nth root of each member, we have

Hence, by Theo. II., the last two equations give

FOR ORIGINAL THOUGHT, ON REVIEW.

118. 1. If three magnitudes are in proportion, the product of the two extremes is equal to the square of the mean.

2. If four magnitudes are proportionals, the first and second may be multiplied or divided by the same magnitude, and also the third and fourth by the same magnitude, and the resulting magnitudes will be proportionals.

3. If four magnitudes are proportionals, the first and third may be multiplied or divided by the same magnitude, and also the second and fourth by the same magnitude, and the resulting magnitudes will be proportionals.

4. If there be two sets of proportional magnitudes, the quotients of the corresponding terms will be proportionals.

120. The Circumference or periphery of a circle is its entire bounding line; or it is a curved line, all points of which are equally distant from a point within called the center.

121. A Radius of a circle is any straight line drawn from the center to the circumference; as the line CA, CD, or CB.

122. A Diameter of a circle is any straight line drawn through the center and terminating in both directions in the circumference; as the line A B.

123. All the radii of a circle are equal; all the diameters are also equal, and each is double the radius.

arc, and the two radii drawn to the extremities of the arc; as the surface included between the arc AD, and the two radii CA, CD.

129. A Tangent to a circle is a straight line which, how far so ever produced, meets the circumference in but one point; as the line CD. The point of meeting is called the point of contact; as the point M.

130. Two circumferences touch each other, when they have a point of contact without cutting one another; thus two circumferences touch each other at the point A, and two at the point B.

131. A straight line is inscribed in a circle when its extremities are in the circumference; as the line AB, or BC.

C

A

A

B

B

132. An Inscribed Angle is one which has its vertex in the circumference, and is formed by two chords; as the angle ABC.