(1) from the upper edge, and (2) from the lower edge, of the trench to the top of the rampart.

(30) Find the expense of glazing a window 51 ft. by 3 ft., with diamond quarries, whose diagonals are 9 and 7 inches, at 2s. 9d. per dozen.

Ans. 8s. 3d.

(31) How much paper yd. wide, will be sufficient to paper a room 22 ft. 5 in. long, 12 ft. 1 in. broad, and 11 ft. 3 in. high? And how much will it cost at 44d. per yard? (1) Ans. 115 yds. (2) Ans. £2. 3s. 11⁄2d.

(32) How many square feet of board will be required to make a rectangular box with lid, of which the length, breadth, and depth are 3 ft., 21 ft., and 1 ft. 21 in. respectively? Ans. 29 sq. ft. 58 sq. in.

III. OF ANGLES, CIRCULAR LINES, AND CIRCULAR

AREAS.

232. As a line is measured by the ratio which it bears to another known line; and a surface, or area, by the ratio which it bears to another known surface or area; so an angle is measured by comparing it with another known angle, as the unit of measure. This unit is the right angle. And as the lineal foot is divided into parts called inches, which are again subdivided; and the square foot into square inches, &c., so the right angle is supposed to be divided into 90 equal angles or parts, called degrees; each degree into 60 equal parts called minutes; and each minute into 60 equal parts called seconds.

Hence a right angle is arithmetically expressed by 90 degrees, usually written thus, 90°; half a right angle is 45°; one-third of a right angle is 30°; two right angles are 180°; and four right angles are 360°. One-fourth of a right angle is 224, that is, 22° 30 min., usually written thus, 22° 30'.

Ánd an angle which is measured by degrees, minutes, and seconds, is usually denoted by the marks °, placed to the right of the digits expressing the number

of such degrees, minutes, and seconds, respectively. Thus, 10 degrees, 6 minutes, and 21 seconds would be written thus, 10° 6′ 21′′; and would measure an angle greater than the ninth part of a right angle by 6' 21".

If a certain angle cannot be exactly expressed in degrees, minutes, and seconds, the remainder after seconds is expressed in decimal parts of a second.

But although theoretically the right angle is the unit, or standard, of measure for angular magnitude, because it is an angle which meets us at every turn, and incapable of being misunderstood, yet it is plain from what has been already said, that practically the unit is the 90th part of a right angle, called a degree. Every angle we hear mentioned, or described, is said to be so many degrees, or degrees and fractions of a degree; the right angle as a unit disappears when we really come to work. Practically, therefore, we must consider the degree, or the 90th part of a right angle, as the unit of angular magnitude.

233. To measure a given* angle.

As the footrule, or some other material standard of measure, is required for measuring lines, and areas, so the unit or standard of angular measure being a certain angle called a degree, some convenient representative of this unit is required wherewith to measure a proposed angle. This we have in an instrument called

THE PROTRACTOR,

which is usually, for school purposes†, a thin plate of brass in the form of a semicircle, with a concentric segment cut out of the middle, as represented in the annexed diagram.

The semicircular band, called the limb of the instrument, is divided into 180 equal parts by straight lines, all of which, if produced, pass through the centre 0 of the semicircle; the outer edge of these is subdivided,

*

"Given," that is, by being presented to us traced on a plane surface, its arithmetical magnitude being unknown.

The Protractor used by sailors and surveyors in actual work is always a complete circle.

each into 10 equal parts, and then the main divisions are marked 10, 20, 30, 40......180, from right to left on the inner rim, and the same from left to right on the outer rim—the whole limb being divided for this purpose into two rims by concentric semi-circular arcs marked on it. (The reason for the two graduations in a reverse order will appear afterwards). Then, if C be the point in

either rim marked 90, and OC be joined, it is evident that AOC=BOC= a right angle: and if from the point O lines be drawn to each point of division in the outer edge of the limb, on the principle that in the same circle equal arcs subtend equal angles at the centre (59, Part 1.), it is plain, that the right angle is divided by those lines into 90 equal parts, and that, therefore, each of these parts is a degree (232). Also any number of such equal parts will together make an angle which is measured by that number of degrees. Hence, if OD be a straight line from O meeting the inner circumference, for example, at the point marked 50, then AOD=130°; and BOD=50o.

So, then, to measure G any proposed angle, as FEG, place the Protractor so that the centre O coincides with the vertex E, and the outer straight edge 40 with FE; then you have simply to observe where

F

E

the line EG meets the outer curved edge of the instrument; and the number of divisions of the limb from

PART III.

3

angles are required to be measured or laid down with great accuracy, will be described among other instruments in a future chapter.

237. To measure the circumference of a given circle.

Following the method before employed in measuring a curved line (219) by means of a string or tape, the circumference of a circle which is accessible at every point may be measured. And this is the method most commonly employed, when practicable.

But it is so usual to consider a circle given, when its radius or diameter only is given, that it becomes necessary to measure circumferences of circles by determining what proportion the circumference bears to the radius; and as this proportion is proved to be the same for all circles, (93, Part 1.), that is, a constant quantity, the number which represents it is an important number in many mathematical calculations.

It was shewn in (93) that if C, c, represent the circumferences of any two different circles, and D, d, their diameters, Cc:: D: d, or C: D::c: d, that is,

C c

=

D d

or the circumference of a circle bears an invariable ratio to its diameter, and therefore to its radius. This fixed number is commonly denoted by the Greek letter (read pi), the first letter of the Greek word periphery, or circumference. So that if

π

or the circumference of a circle is equal to its diameter multiplied by T.

But still the question remains, What is the numerical value of ? or, How many times is the diameter contained in the circumference?

π

Now, as this value, or number, is the same for every circle, it is obvious, that a single accurate measurement should be sufficient to determine it. It might appear at first sight, that nothing can be more easy than to take a perfectly constructed circle, and measure its circumference with a cord; and then measure the diameter in like manner, and find the ratio of two measurements, which will give the numerical value of π.

But, as in the case of the diagonal and side of a square before mentioned (228), so here also it is found, that the ratio of the circumference of a circle to its diameter cannot be exactly expressed in numbers. Each separately can be measured with perfect exactness; but both do not admit of being measured exactly by the same linear unit, however small that unit may be. This is another example of incommensurable lines.

Practically, or 3, is found to express for

22
7

many

purposes with sufficient exactness the value of π, the circular multiplier;

355
113

is still nearer; and it will

seldom be necessary to use a nearer approximate value than 3.1416.

Thus, for rough calculations,

circumference of circle =

22

x diameter;

7

using whichsoever of the two is the most convenient. (See NOTE at the end of this section, p. 255).

N.B. Although the value of cannot be expressed without some error; yet that error may be made as small as we please. For the value has been calculated to many places of decimals, and is found to be, up to 20 places, as follows:

3.14159265358979323846 &c.

If then for the value of π we use

22

7

that is,

3.142..., it is plain that this differs from the true value by a quantity less than 01; that is, supposing the diameter to be 100 inches, the circumference, with this value of π, will be 3142... inches; but the true value is 314-15... inches; therefore the error in this circumis less than one-tenth of an inch.

ference from using

22