AFB very nearly; that is, the above number will differ from the truth by a very small decimal, whose highest place is 17 places below unity. Whence .0000000243269992842=the length of the arc AFB or of the part of the whole circumference extremely near. Now if the length of the arc AFB as above determined be multiplied into the denominator of this fraction, the product will be 6.2831853061898472=the circumference of a circle whose diameter is 2, very nearly.

253. Having found the circumference of a circle, we can readily find the area, if not with strict accuracy, at least sufficiently near the truth for any practical purpose, in order to which, let us suppose an indefinite number of straight lines drawn from the centre to the circumference, these will divide the circle into as many sectors, the bases of which will be indefinitely small arcs, and their common altitude the radius of the circle; now since these small arcs coincide indefinitely near with the sides of a circumscribed or inscribed polygon of the same number of sides as there are sectors, these sectors may evidently be considered as triangles, the bases of which are the above small arcs, and their common altitude the radius; but half the base of a triangle, multiplied into the altitude, will give the area (42. 1.) wherefore, (half the sum of the bases, that is) half the circumference of the circle, multiplied into the radius, will give the area of the triangles, that is, the area of the circle; thus 6.2831853, &c. x 1

2

diameter is 2.

=3.1415926, &c.=the area of a circle, whose

254. Having found the circumference of a circle, whose diameter is 2, we are by means of it enabled to find the circumference of any other circle, whatever its diameter may be; for let the inscribed polygon (whose sides coincide indefinitely near with the circumference) have n sides, the length of each being r; and let a similar polygon be inscribed in any other circle having the length of its sides, then will nr=the periphery of the first polygon, and ns= that of the second. Let the radius of the former circle, t=that of the latter; then if lines be drawn from each centre to the points of division in the respective circumferences, we shall have 1:r::t:s, (4.6.) whence (16.5.) 1: t::r: s, and consequently (15.5.) 1: t:: nr

ns, that is, the peripheries of the similar polygons are to each

other as the radii of their circumscribed cricles; but these polygons coincide indefinitely near with their circumferences : wherefore the circumferences of circles are as their radii.

255. The area of one circle being known, that of another circle having a given diameter, may be found; let D=the diameter of a circle, A=its area, and d=the diameter of another circle, whose area x is required; then (2. 12.) D2 : d2 : : A : x, de A whence r the area required.

D2'

PRACTICAL GEOMETRY.

255. Practical Geometry teaches the application of theoretical Geometry, as delivered by Euclid and other writers, to practical uses".

256. To draw a straight line from a given point A, to represent any length; in yards, feet, inches, &c.

RULE. I. Let each of the divisions on any convenient scale of equal parts represent a yard, foot, inch, or other unit of the measure proposed.

II. Extend the compasses on that scale until the number of units proposed be included exactly between the points.

III. With this distance in the compasses, and one foot on A, describe a small arc at B; lay the edge of a straight scale or ruler from A to B, and draw the line AB with a pen or pencil, and it will be the line required.

EXAMPLES.-1. To draw a straight line from the point A to represent 12 inches.

The following problems are intended as an introduction to the practical application of some of the principal propositions in the Elements of Euclid, and likewise to assist the student in acquiring a knowledge of the use of a case of mathematical instruments. From a great variety of problems usually given by writers on Practical Geometry, we have selected such as appear most necessary, and likewise such methods of solving them as appear most simple and obvious; to a learner well acquainted with Euclid, other methods will occur, and he should be encouraged to exercise his ingenuity in discovering and applying them. The best elementary treatises on Practical Geometry and Mensuration, are those of Mr. Bonnycastle and Dr. Hutton.

With one foot on ✪ extend the other to the 12th division on the scale you choose to adopt, and apply that distance from A as above directed, and it will give the length proposed.

2. To draw a line that shall represent 35 yards.

Let each primary division be considered as 10 yards, then will each subdiv sion represent 1 yard; apply the compasses from 3 backwards (to the left) to the 5th subdivision, and 35 subdivisions will be included between the points; apply this from the given point and draw the line as before.

3. To draw a line equal to 263.

On the diagonal scale, let each primary division represent 100, then will each subdivision represent 10, and the distance which each diagonal slopes or the first parallel will be 1, on the second 2, on the third 3, and so on; therefore for 263 extend from the number 2 backwards to the sixth subdivision, on the third parallel, (viz. the 4th line downwards) and it will be the distance. required.

257. To measure any straight line'.

RULE. Extend the compasses from one extremity of the given line to the other, and apply this distance to any convenient scale of equal parts, the number of parts intercepted between the points, will be the length required.

Note. If the sides of a rectilineal figure are to be measured, the same scale must be used for them all; and one scale must be used for each of two or more lines, when their relative length is required to be ascertained.

258. To bisect a given straight line AB.

By the word measure is meant the relative measure of a line, that is, the length of that line compared with the length of another line, both being measured from the same scale; if we call the subdivisions of the scale feet or yards, the line will represent a line of as many feet or yards as it contains such subdivisions; to find the absolute measure of a line in yards or feet, we must evidently apply a scale of actual yards or feet to it.

• Any scale of equal parts may be employed for this purpose, but it will be proper to choose one that will bring the proposed figure within the limits you intend it to occupy; every part (viz. every line) of the figure must be measured by one scale, and not one line of the figure by one scale, and another line by another.

RULE I. With any distance in the compasses greater than half the given line, let arcs be described from the centres A and B, cutting each other in C and D.

II. Draw a straight line from C to D, and it will bisect the given straight line, as was required.

A

-B

259. From a given point D, in a given straight line AB, to erect

a perpendicular FD.

RULE I. From any point C

Lay the centre of the protractor on D, and let the 90 on its circumference exactly coincide with the given line; draw the line FD along the radius, and it will be the perpendicular required.

259. B. From a given point F, to let fall a perpendicular to a given straight line AB. See the preceding figure.

RULE I. In AB take any point E, join FE, and bisect it in C, (Art. 258.)

II. From C as a centre with the distance CF or CE, describe

If the points AC and BC be joined, this rule may be proved by Euclid 8.1.

"The proof of this rule depends on Euclid 31.3. Of the various methods for erecting a perpendicular, given by writers on Practical Geometry, this is the most simple and easy.

the circle EDF; join FD, and it will be the perpendicular required *.

260. Through a given point E to draw a straight line parallel to a given straight line AB.

RULE I. Take any point Fin AB, and from E and F as centres, with the distance EF, describe the arcs EG, FH.

II. Take the distance EG in the compasses, and apply it from F to H on the arc FH.

III. Through E and H draw

C

E

the straight line CD, and it A
will be parallel to AB as was required".

BY THE PARALLEL RULer.

H

D

B

Lay the ruler so, that the edge of one of its parallels may exactly coincide with the line AB. Holding it steady in that position, move the other parallel up or down until it cut the point E, through which draw a line CED, and it will be parallel to AB.

If E be too near, or too distant for the extent of the ruler, first draw a line parallel to, and at any convenient distance from AB, to which draw a parallel through E as before, and it will be parallel to AB.

261. At a given point A, in a given straight line AB, to make an angle BAC, which shall measure any given number of degrees. RULE I. Extend the compasses from the beginning of the scale of chords (mark

ed C,) to the 60th degree, and from the given point A, with this distance, describe an arc cutting AB (produced if necessary) in E.

F

E B

II. Extend the compasses from the beginning of the scale of

* This depends on Euclid 31.3.

▾ Since the arcs EG, HF are equal, the angles EFG, FEH at the centres are equal, (Euclid 27.3.) and therefore AB is parallel to CD, Euclid 27.1.