532. Mensuration treats of the measurement of magnitudes.

In reference to the kinds of magnitude, mensuration is of four kinds, namely, Lines, Angles, Surfaces, and Solids.

A Straight Line is one that does not change its direction. Lines and Angles, and Surfaces and Solids in part, have been somewhat fully discussed in Arts. 202-224, and also 247-258. It is proposed in this Chapter simply to complete the subject of mensuration, so far as may be proper in a work of this kind, by adding some Definitions, Explanations, and Rules, which the pupil is now prepared to comprehend, with reference to surfaces and solids.

SQUARE MEASURE.

533. A Surface is that which has length and breadth without thickness (204).

534. An Area is a definite amount of surface.

535. A Triangle is a plane surface bounded by three straight lines. (Fig. 1.)

Triangles are equilateral, isosceles, and scalene, when the three sides are equal, when two sides are equal, and when no two sides are equal, respectively.

They are also right-angled, obtuse-angled, acute-angled, and equiangular, when they have one right-angle, when they have one obtuseangle, when all the angles are acute, and when all the angles are equal, respectively.

FIG. 4.

A B C, Fig. 1, is a right-angled triangle, and also a scalene triangle.

A D C, Fig. 2, and D C B, Fig. 4, are obtuse-angled-triangles, and also scalene tri

angles.

A Quadrilateral is a surface bounded by four straight lines, Figs. 2, 3, and 4.

Quadrilaterals are subdivided into parallelograms, trapezoids, and trapeziums, that is, four-sided figures having their opposite sides parallel, A BEC, Fig. 3; having two sides parallel and two in

clined to each other, Fig. 4; having no two sides parallel, Fig. 2, respectively.

Lines are said to be parallel when they will not meet, however far they may be produced, as C E and A B, Fig. 3; and D C and A B, Fig. 4.

E

B

A surface bounded by five sides is called a pentagon; one

by six sides,

A

D

a hexagon, by seven, a heptagon; by eight, an octagon; by ten, a decagon; by eleven, an undecagon; by twelve, a dodecagon.

536. The Base of a plane figure is the side on which it is supposed to rest, as B C, Fig. 1; A B, Figs. 2, 3, and 4.

537. The Altitude is the perpendicular distance between the base and the vertex of the opposite angle, oř between the base and the opposite side; as A B, Fig. 1, and C D, Fig. 3, and D E, Fig. 4.

538. The Diagonal of a plane figure is the straight line joining the vertices of two angles not connected by one of the sides; as A C, Fig. 2, and D B, Fig. 4.

In Fig. 3, if we cut off the part A D C and place it to the right of the figure C D BE, so that A C will coincide with B E, we see that ABEC will be changed to a rectangle (207), which is equivalent to A B E C, and whose base is equal to A B, and altitude C D; and we find the area by multiplying together the two dimensions A B and CD (209). Therefore, to find the area of any parallelogram we have this

539. RULE.-Multiply the base by the altitude.

And since a triangle is one-half of the parallelogram having the same base and altitude, (See A B C, Fig. 3,) we find its area by this 540. RULE.-A. Multiply the base by one-half the· altitude.

And since a trapezoid is only two triangles, B C D and B A D, Fig. 4, and they have the same altitude, D E, and the base of one is A B, and D C may be regarded as the base of the other, we may take the sum of the areas of the two triangles as the area of the trapezoid. And as we have the area of one of the triangles equal to one-half its base multiplied by its altitude, and the area of the other triangle equal to one-half its base by the same altitude, we must have the area of the trapezoid equal to

541. One-half the sum of its two sides, multiplied by its altitude.

The area of a triangle is also found by this

542. RULE.-B. From the half sum of the three sides subtract each side separately. Then extract the square root of the continued product of these remainders and the half sum of the sides..

The area of any plane figure may be found

543. By dividing it into triangles, calculating the area of each triangle separately, and then finding the sum of these areas.

If a figure is regular, that is, has its sides and angles equal, each to each, its area may be found

544. By multiplying the distance around it (the perimeter), by one-half the perpendicular distance from its cente. to one of its sides.

If a regular figure has its sides exceedingly small and an exceedingly large number of them, the sides taken together form the circumference (223) of a circle, (222), and the perpendicular distance from the centre to one of its sides is the radius (223); and, therefore, we say that

545. The area of a circle equals the product of the circumference, by one-half the radius.

It is proven in Geometry, that the ratio of the circumference of a circle to its diameter is 3.14159, nearly. By means of this truth we deduce various rules for determining different parts of a circle when other parts are known.

Thus, since the circumference is 3.14159 times the diameter,

To find the circumference when the diameter is known,

546. RULE.-Multiply the diameter, or twice the radius, by 3.14159.

To find the diameter when the circumference is known,

547. RULE.-1. Divide the circumference by 3.14159.