15. If 640 acres go to a square mile, what is the length of each side of a square piece of land which contains 100 acres?

16. Find the length of the side of a square field which contains 7 acres 3 roods 15 perches.

17. A room 21 feet long required 49 yards of carpet of a yard broad. Find the breadth of the room.

18. The number of yards of paper required to cover the four walls of a room 54 feet wide and 30 feet high is 880, and the breadth of the paper is of a yard. Required the length of the room.

19. The first of two pictures is 1 ft. 6 in. by 2 ft., the second 2 ft. by 2 ft. 6 in., and they are to be framed in the same way; if the glass and frame of the former cost 7s. 6d. and that of the latter 11s. 2d., what is the price of the glass per square foot, and of the frame per foot of length?

20. A merchant buys cotton (27 inches in width) at 5 cents per square yard. He pays a duty of 2 cents per square yard, and 15 per cent. ad valorem. For what price per yard should he sell it in order to gain 25 per cent. on his outlay? 21. Of two squares of carpet, one measures 44 feet more round than the other, and 187 square feet in area. What are their sizes?

22. A square field of grain containing ten acres is to be cut down by a reaper working round and round; the cut of the reaper is 6 feet. How many rounds must the reaper take before the field is half cut?

23. A square plot of ground, 21 yards in the side, is sold for the greatest num. ber of sovereigns which can be placed flat upon it. Find the price, the diameter of the sovereign being seven eighths of an inch.

24. What length of fence is required to enclose 100 acres; first, when the land is in the form of a square; second, when in the form of a rectangle, having length 2 times breadth?

B

SECTION XII.-SURFACE, Continued.

ART. 82.-Area of the Sector of a Circle. Consider any

Fig.7.

A

sector of a circle ABC; draw the chord BC (Fig. 7). The area of the triangle ABC' is a first approximation to the area of the sector ABC. Bisect the sector by the line AD; join BD, DC; the sum C of the areas of the triangles ABD and ADC is a second approximation to the area of the sector.

Bisect each of these again; the sum of the areas of the four triangles will be a third approximation to the area of the sector. When the bisection has been continued a large number of times, the altitude of each of the triangles will not differ sensibly from the radius of the sector, and the sum of the lengths of the bases will not differ sensibly from the arc of the sector. Hence the area is given by the rate

then, by multiplying these two equivalences together, L arc appears common to both sides, and may therefore be eliminated, and we obtain

ART. 84.—Area of an Ellipse. The circle described with the major axis AA' for diameter is called the auxiliary circle (Fig. 8). The area of that circle is given by

The breadth of the ellipse is everywhere, as at PM, derived from the corresponding breadth of the circle, as QM, according to the ratio by which the minor axis BB' is derived from the diameter

of the circle DD'. Hence the area of the ellipse itself is given by

or

SL semi-major axis by L semi-minor axis.

ART. 85.-Surface of the Common Cylinder. The flat portion of the surface consists of two equal and parallel circles (Fig. 9). If the curved surface were cut parallel to the axis, unrolled, and flattened out, it would form a rectangle having the circumference of the cylinder for length and the axis for breadth. Hence the area of the curved surface is given by

1 SL circumference by L axis.

The dependence of the circumference on the radius of the cylinder is given by

2 L circumference = L radius ;

.'. 2 S by L circumf. = L circumf. by L axis by L radius, i.e., 2π SL axis by L radius.

ART. 86. Surface of the Common Cone. The flat portion of the surface is a circle (Fig. 10). If the curved surface were cut along the slant height, unrolled, and flattened out, it would form a sector of a circle, having the slant height for radius and the circumference of the base for arc. Hence (Art. 82) the area of its curved surface is given by

ART. 87.-Surface of a Sphere. The area of a spherical zone (Fig. 11) is given by

2 SL radius of sphere by L height of zone.

Hence, as the height of a hemispherical zone is equal to the radius of the sphere, we have for a hemispherical surface

2 S (L radius of sphere)2.

=

Hence, for the area of a spherical surface,

4 S (L radius)2.

ART. 88. Solid Angle. Just as a linear angle can be specified by means of the rate connecting the circumference of a circle with its radius (Art. 67), so a solid angle can be specified by means of the rate connecting the area of a surface of a sphere with the square of its radius. Thus a solid angle is specified in terms of S per (L radius)2;

and the unit-rate S per (L radius)2 is sometimes called a steradian, that is, a solid radian.

To deduce the dependence of the area of a spherical lune on its spherical angle, 4 S per (L radius)= 360 degrees;

ART. 89.-Change of Surface. Suppose that the length of a plate expands according to the rate

a L increment per L original length,

(1)

and its breadth according to the rate

BL increment per L original breadth,

(2)

then the ratios of expansion are—

1 + a L expanded length = L original length,

(3)

1 + ẞ L expanded breadth = L original breadth.

(4)

Hence, if the included angle remain constant,

(1 + a)(1 + ẞ)L2 expanded surface = L2 original surface,

(5)

and

{(1+a)(1 + ẞ) −1}L2 increm. of surface = L2 orig. surface. (6) The reciprocal of (5) is

1

(1 + a)(1 + ẞ)

and the rate of diminution,

(1

1

L' original surface = L2 expanded surface,

(7)

(8)

(1 + a)(1 + B))L2 decrement = La expanded surface,

When a and B are both small, the value of (5) is 1+ a + ß; of (6), a + ẞ; of (7), 1-a-ẞ; and of (8), a + B.

When ẞ is equal to a, the value of (5) is (1+a)2, and its approximation is 1+2a; the value of (6) is 2a + a2, and of its approximation 2a; and so on.

EXAMPLES.

Ex. 1. The sides of a rectangle are 16 feet and 10 feet respectively. Find, to four places of decimals, the length of the diagonal of a square, whose area equals that of the rectangle.

16 feet long by 10 feet broad,

1 (foot side)2=foot long by foot broad,

2 (foot diag.)2=1 (foot side)2;

16 × 10 × 2 (foot diag.)2,

8/5 feet diag.;