in any order we please, provided these products are equal.

Prop. 3. If four numbers are in proportion, they will be in proportion when taken alternately; that is, so that the first term shall be to the third, as the second is to the fourth.

If 12:9:36:27, then 12: 36 :: 9:27.

Prop. 4. If there be four numbers in proportion, and four other numbers in proportion, having the antecedents the same in both proportions, the consequents will be proportional.

Cor. 1. If an antecedent and consequent of one set, be equal to an antecedent and consequent of the other, the remaining terms will be proportional.

Cor. 2. If the consequents are the same, the antecedents will be proportional.

Prop. 5. If four numbers are in proportion, they will be in proportion when taken inversely; that is, when con- sequents are put for antecedents, and antecedents for consequents.

If

5:7:10:14, then 7:5:: 14:10.

Prop. 6. If several numbers are in proportion, any one antecedent will be to its consequent, as the sum of all the antecedents is to the sum of all the consequents.

If

2:43:6:5:10::7: 14,

then is 24 (2+3+5+7):(4+6+10+14),

Prop. 7. If four numbers are proportional, their squares are also proportional.

Prop. 8. If the two antecedents of a proportion, or one antecedent and one consequent, contain an equal part, that part may be omitted from the proportion.

here the antecedent and consequent both contain 7, omitting which, we have

2:36: 9.

Cor. If, in a proportion, an antecedent and consequent or two antecedents are equal, the other antecedent and consequent, or the two consequents, will be equal.

Prop. 9. Equimultiples of two numbers have the same ratio as the numbers themselves.

Take the numbers 5 and 2, their ratio is; now if both be multiplied by the same number, as 6, their equimultiples, 30 and 12, have a ratio 12.

Prop. 10. If there be two sets of proportional numbers, the products of the corresponding terms will be proportional.

BOOK V.

DEFINITIONS.

1. Similar figures are those, which have the angles of the one respectively equal to the angles of the other, and the sides containing the equal angles proportional.

2. The homologous sides of similar figures are those, which are each lying between two angles respectively equal.

3. In different circles, similar arcs, sectors, and segments are those whose arcs subtend equal angles at the

centre.

4. The altitude of a triangle is the perpendicular let fall from the vertex of an angle upon the base, or upon the base produced.

In the triangle ABC, if AB is taken as the base, CF is the altitude. In the triangle DHR, the base is DH, and the al

titude RI.

5. The altitude of a parallelogram is the perpendicular

between two opposite sides

taken as bases. Thus, EF,

C

or IH, is the altitude of the parallelogram ABDC, according as we assume the

A

sides AB, CD, or AC, BD, as the bases.

8. A linear unit is a line adopted as the measure of length; as an inch, a foot, a yard, &c.

9. A superficial unit is a surface, adopted as the measure of surface; as a square inch, square yard, acre, &c. 10. The area of a figure is the measure of its surface. 11. Figures with equal areas are called equivalent. 12. The square of a line, as AB, is expressed thus, AB2, read, "AB square."

13. The rectangle of two lines, as AB, CD, is written AB.CD, read, "AB multiplied into CD," or, shortly, "AB into CD."

PROP. I. THEOREM.

The area of a rectangle is obtained by multiplying the number of linear units in the base, by the number of the same linear units in the altitude.

Let the base of the rectangle ABCD contain a certain number of linear units, say eight, it is evident that the rectangle will contain as many times eight superficial units, as there are linear units in the altitude. For from the points of division K, O, &c. draw lines parallel to AD; and from the points of division E, G, &c. draw lines parallel to AB, and there will be as many squares formed, as the product of the number of linear units in the base, by the number of the same units in the altitude. In the case before us, the area contains thirtytwo times the superficial unit, which is a square whose side is the assumed linear unit.

Scholium. Whenever the base and altitude are commensurable, we may take for the linear unit the greatest common measure of them; and the superficial unit will be a square having the linear unit for a side. For example, if the base of a rectangle was 4 inches, and its altitude 3 inches, then inch would be a common measure; hence the base would contain 18, and the altitude 13 linear units, each inch in length. The superficial unit would be a square, whose side was inch; and hence would be of a square inch; therefore the supposed rectangle would contain 18.13 or 234 sixteenths of a square inch, or 234 square inches=1418 square inches.

If the base and altitude are incommensurable, we may still take the common measure so small (agreeably to Prop. 13 of B. IV.) that the part neglected will be practically unimportant.

(See Appendix, Prob. II.)