2π radians. If, then, the length of the circle is c L, its curvature is 2π/c radian per L curve.

But if we are given the radius of the circle as r L, then its circumference is 2πr L, and the rate of curvature is expressed as

1/r radian per L curve.

A portion of any curve, throughout which the curvature does not change sensibly, coincides with a portion of the arc of some circle. Hence, if the radius of that circle is r L, the curvature for that portion of the curve will be

1/r radian per L curve.

The radius of the equivalent circular arc is called the radius of

curvature.

EXAMPLES.

Ex. 1. Express 12° 34′ 56′′ in terms of radian; and 3 radians in terms of deg. min. sec.

Ex. 2. Calculate from the primary definition of the metre, and the relation of the metre to the mile, the length of the diameter of the earth in miles.

40,000,000 metres circumf.,

113 metres diam. = 355 metres circumf.,

Ex. 3. How much must a rail 6 ft. long be bent in order to

fit into a curve of half a mile radius ?

360 degrees = × 1760 yards arc,

Ex. 4. The radius of the earth's orbit subtends an angle of four-tenths of a second at 61 Cygni ; how far is the star from the

sun?

E

Fig.4.

S

C

Here the dependence of ES upon SC is given by the tangent of the angle ECS, that is, of '4". For so small an angle the value of the radian is equal to the value of the tangent with sufficient accuracy for the present calculation.

01745 mile arc per mile radius = 1 degree,

1 degree = 3600 seconds,

Observation.—It is more common to put the point after the first figure, so that the index of 10 may be the mantissa of the logarithm of the number.

1. Express

EXERCISE X.

of a right angle in degrees and also in grades.

2. The angle which is subtended by an arc equal to the radius is 206,264 8 seconds. Reduce this to grades and decimals, and thence deduce the value of 1/π.

3. A circle has a circumference of 123 yards.

What is the length of its radius?

4. How many radians are there in an arc of 114° 35' 30" ?

5. Express 45 degrees in terms of radian; and 4.5 radians in terms of degree. 6. Express NW. by N. and SW. by W. in degrees, reckoning east from north. 7. How many points and degrees are there between N. by E. and SSE.? 8. If the length of of the earth's circumference be 69 miles, find its diameter, taking diameter to circumference as 7 to 22.

9. From the primary definition of the metre and its relation to the statute mile deduce the average number of miles in a degree of latitude.

10. The cotangent of the angle of a roof is 15; what is the ratio of rise to span? 11. A railway line has a gradient of 1 in 80 for a distance of 3 miles; what is the total rise or fall?

12. The sines of two angles of a triangle are and, and the side opposite to the former is 10 yards; required the side opposite the latter, and the third side.

13. The base of the Great Pyramid is a square 764 feet in the side, and the perpendicular height is 486 feet; required the length of the slant side of one of the triangular faces.

14. A man 5 feet 4 inches in height standing at a distance of 52 feet from the base of an electric-light tower casts a shadow 8 feet long on the pavement. What is the height of the tower?

15. The difference of longitude between two places is 5 degrees, and the latitude of both is 45 degrees; find the distance between them along the parallel of latitude. (Take the radius of the earth to be 4,000 miles.)

16. The length of a railway curve which has a uniform curvature is one mile, and the change of direction is 30 degrees. Find the value of the curvature and of the radius of curvature.

17. Express the sea-mile in terms of the kilometre.

18. Compare the nautical-fathom with the common fathom.

SECTION XI.-SURFACE.

ART. 75.-General Unit of Surface. The general unit of surface is appropriately denoted by S. It can be defined by means of the unit of length, and the mode commonly adopted is as follows: -The unit of surface is the area of the square which is formed with the unit of length as the side. When S is so defined,

we have

1S = L2,

where L2 denotes square L. Such a unit of surface is called a systematic unit. When S is not so defined we have

k S = L2,

where k denotes some number, integral or fractional.

In what follows S is commonly restricted to being systematic.

ART. 76.-Imperial Units of Surface. In the imperial system the principal unit of surface is the area of the square formed by the yard, and denominated the square yard. Of the other units of surface some are derived systematically from the corresponding linear unit; and two, the acre and the rood, used in the measurement of land, are defined in terms of the square yard.

i.e.,

sq.yd.

or

sq.ft.

ft.

yd.

yd. broad,

32 feet21 yard2;

9 square feet 1 square yard. This is also evident from an inspection of the figure, where the longer side represents a yard and the shorter side a foot.

In science and the arts the square foot and the square inch are commonly used.

ART. 77.-Metric Units of Surface. In the metric system the unit of surface is derived from the area of the square formed by a unit of length. The natural unit is the area of the square formed by the metre, but it is an inconveniently small unit for the purpose of measuring land. Hence the special name of are was given, not to the area of the square formed by the metre, but to the area of the square formed by the dekametre, which is 100 times the former. From the are are derived, as will be seen from the table appended, the usual decimal multiples and submulti