PART VII.

APPLICATION OF ARITHMETIC TO GEOMETRY, OR MENSURATION.

387. Generally speaking a Body is anything that can be seen, touched, or weighed.

Bodies are either solid, liquid, or gaseous; metals, stones, wood are solid bodies; water, wine, &c., are liquid bodies, and lastly, the atmosphere, lighting gas, &c., are gaseous bodies.

A body cannot exist without occupying a part of space; or a body has three dimensions, length, breadth, and thickness or depth. Thus, E in the accompanying diagram, A B is the length, B E the breadth, and A C the thick

ness.

388. The content, volume or solidity of a body is the part of space which it occupies; the boundaries or limits of a body are its faces, or its surface.

Surfaces may be plane like the front of a house, the top of a table, a mirror, &c., and curved as in a ball, a turnip, &c., there are also mixed surfaces, composed of plane and curved surfaces. It must be observed that no account is taken of thickness when speaking of surfaces.

The boundaries or limits of surfaces are lines, the right or straight line is represented by the edge of a well made ruler, the direction of a thread at the end of which is hung a heavy body; the curved line, of which we have representations in the circum ference of a circle, the edge of a basin, &c., when a line is partly straight and partly curved it is said to be mixed; then we may consider lines without having regard to the surface. The place of intersection or of meeting of two or more lines, is called a point, thus we may speak of a point independently of the lines.

389. The distance between two points is measured by the straight line which connects them, for it is the shortest path between them.

390. An angle is the inclination of two lines, which meet, the point of meeting is called the rexter of the angle.

When one straight line meeting another straight line. makes with it two angles, which may be equal or unequal, in the first case, the lines are vertical to one another, and the equal angles are said to be right, in the second, when the lines are oblique to one another, the angle which is less than a right angle is called acute, and that which is greater obtuse.

391. The distance from a point to a line is measured by the vertical line drawn from the point to the line.

Two lines are parallel when they are everywhere equidistant. The vertical line which joins the parallels is their true or shortest distance.

392. By polygon is meant a plain surface, enclosed by right lines, which are called the sides of the polygon.

The simplest of all polygons is the triangle, or trigon, having three angles and three sides. A triangle is right-angled, obtuseangled, or acute-angled, as one of its angles is right, obtuse, or all three acute. It is also equilateral, when the three sides are equal to one another; isosceles, when only two sides are equal; and scalene, when all three are unequal.

A quadrilateral, or tetragon, is a polygon of four sides and four angles.

A pentagon, a figure of five sides and five angles.

A hexagon, a figure of six sides and six angles.

A heptagon, a figure of seven sides and seven angles.

An octagon, a figure of eight sides and eight angles, &c.

If the sides of a polygon be equal, as also the angles, or if a polygon be equilateral and equiangular, it is called a regular polygon.

A quadrilateral, with the opposite sides parallel, is called parallelogram. A parallelogram, which has four equal sides, and its angles right, is called a square.

A parallelogram, which has four equal sides, and its angles oblique, is called a rhombus.

A parallelogram, with two pairs of equal sides, and its angles right, is called a rectangle.

A parallelogram with two pairs of equal sides and its angles oblique is called a rhomboid.

A four-sided figure with only one pair of parallel sides is called a trapezoid.

A four-sided figure with no parallel sides is called a trapezium.

393. The simplest of curves is the circumference, of which every point is equally distant from the centre. The circle is the surface contained within the circumference.

Any line drawn from the centre to the circumference is called radius, and a line drawn through the centre and terminated at the circumference is called the diameter, therefore a diameter is composed of two radii. All diameters of the same circle are equal to one another.

Every circumference of the circle is supposed divided into 360 parts, called degrees, the degree into 60 minutes, the minute into 60 seconds, &c.

An arc is a part of the circumference, and the right line joining its extremities is the chord.

394. To measure the length of a line, we fix upon some unit of measure, as an inch, a foot, &c., and this unit is repeated till it makes up the line, and the number of times it is contained gives the units of length of that line.

The perimeter of a figure is the sum of its sides.

395 Every circumference is 3.14159 times its diameter, or the ratio of the circumference to the diameter is 22 nearly, which ratio is generally expressed by π.

Therefore to find the circumference multiply the diameter by 3.14159, or by 22 in ordinary cases.

And to find the diameter, divide the circumference by 3.14159 or by 22.

396. The unit used to measure angles is the angle of one degree. From the vertex of an angle, with any radius, suppose an arc described, the number of degrees contained in the arc between its sides, indicates the measure of the angle.

The instrument employed to find this number of degrees is called a protractor.

An angle of 36 degrees, 40 minutes, and 30 seconds, is expressed thus: 36° 40′ 30′′.

The sum of the angle in every triangle is the same as that of two right angles, or of 180°.

397. EXERCISES.

1. What is the perimeter of a triangle, the sides of which are 35 feet, 28 feet, and 44 feet 7 inches respectively?

2. A rectangular piece of ground, the adjacent sides of which are 116 feet and 104 feet, is to be surrounded by a ditch; its length is required?

3. At what rate, per hour, does a horse go, that runs three times round a field of a rectangular shape, the adjacent sides of which are 960 and 1118 yards, in 8 minutes and 20 seconds? 4. What is the circumference of a circular well the diameter of which is 5 yards, 2 feet, 8 inches?

5. What will be the expense of planting with box a circular garden, whose diameter is 7 yards 10 inches, at 2d. per yard?

6. What is the diameter of a piece of water, the circumference of which is 124 yards ?

7. The radius of the equator is 3962.824 miles, find at what rate per hour is any object on the equator carried round during the earth's rotation in 24 hours?

8. The mean distance of the earth from the sun is 95 millions of miles, at what rate per hour is the earth carried round the sun during its revolution in 365 days?

9. The radius of the equator is 3962:824 miles. What is the distance on the equator of two places which are 36° 24′ 16′′ apart?

10. The diameter of a circle is 7.4 feet. Required the number of degrees in an arc, whose length is 5.6 feet.

MENSURATION OF SURFACES.

398. The unit of superficial measure is a square surface,

P

the length of the side of which is the lineal unit; thus if x y be the lineal unit, the square x y p q is

the superficial unit.

399. The surface or area of a rectangle is found by multiplying together two adjacent sides; if one side is 7 feet, and the other 5 feet, then 7×5, or 35, expresses that the area of the rectangle contains 35 times the surface of the square foot, or is 35 square feet. This is easily shown by dividing one side of a rectangle into 7 equal parts, and the other into 5; and by drawing from the points of division, lines parallel to the sides, the area contains 7 × 5 equal

squares.

400. The two contiguous sides of a rectangle are its length and breadth, and are called its dimensions. The area of a square is found by multiplying one side by itself; thus if the side is 12 yards, the square contains 12 x 12, or 144 square yards. This is evident, for a square is a rectangle whose contiguous sides are equal.

401. To find the area of a parallelogram multiply its length by its vertical breadth, or its base by its height. For a parallelogram has the same area as a rectangle of the same base and the same height.

402. To find the area of a triangle, multiply the base by the vertical height, and half the product, will be the area; any side may be taken as the base of the triangle, and its altitude or height is the vertical line drawn from the vertex of the opposite angle to the side taken as a base, This is evident, since the area of a triangle is the half of that of a parallelogram, having the same base and the same altitude.

The area of a triangle is also found as follows:

From half the sum of the three sides, subtract each side separately, multiply this half sum and the three remainders continually together, and extract the square root of the product.

Ex. 1. The base of a triangle is 32.5 feet, and its vertical height 18.8 feet. Required the area.

Here area.

32-5 x 18.8
2

305.5 square feet.

Ex. 2. If the sides of a triangle are 15.4, 20, and 24 yards, find its area.