4. Find the present value of £48. 10s. due two years hence, compound interest being reckoned at 4 per cent.

5. Find the proportion between the interest of £100 for 4 years, at 6 per cent., and the discount of the same sum payable at the end of 4 years, at the same rate of interest.

6. Find the present value of a bill for £174. Is. 104d. due in 128 days, interest being at the rate of 5 per cent.

7. Find the discount on £663. 17s. due 6 months hence at 4 per cent.

8. What ready money will discharge a debt of £1056. 11s. 10d. due 8 months hence at 43 per cent. per annum?

9. What ready money will discharge a debt of £1056. 18s. due 4 months hence at 43 per cent. simple interest?

10. Find the true discount on £142. 1s. 9d. due 18 months hence at 3 per cent. per annum.

11. Find the discount on £51. 15s. 10d. due 4 years hence at 3 per cent. simple interest.

Summary.

Arithmetical Progression.—Quantities which increase or diminish by the same amount at each step are said to be in Arithmetical Progression.

The last term is given by l=a + (n − 1) d.

n

Sum of n terms, 8=5(a+1)

=

n

{2a + (n − 1)d}.

Geometrical Progression.—In a Geometrical Progression the quantities increase or decrease by a constant factor.

The last term is given by l=arn-1.

The sum of n terms by s='

a (m − 1)_a(1 − pm)

r-1

=

1-r

The Arithmetic Mean of two numbers is one-half the sum of the numbers.

The Geometric Mean is the square root of the product of the numbers.

Interest.-Money paid for the loan of a sum borrowed at a fixed rate is called interest, and is called Simple Interest when the principal remains unaltered, but Compound Interest when the principal continually alters.

Pxrxt
100

Amount:

=

P(1+100)".

Discount. The abatement made when a sum of money is paid before it is due is called discount.

CHAPTER XIV.

ORTHOGRAPHIC PROJECTION. TRUE LENGTHS AND INCLINATIONS OF LINES REFERRED TO TWO AND THREE CO-ORDINATE PLANES.

Lines. - Lines may be straight or curved, or straight in one part of their length and curved in another.

Straight Line.-A straight line may be defined for practical purposes as the shortest distance between two points; or as that line which lies evenly between its extreme points.

Planes. A plane is a surface such that the straight line joining any two points on it lies wholly in that surface.

с

A

B

FIG. 75.-Plane surface.

Perhaps a clear notion of what this definition implies may be obtained by using a flat sheet of paper, as in Fig. 75. If any two points, A and B, on the surface of the paper be selected, it will be seen that the line joining them lies in the surface. Next proceed to bend or crease the paper between the points, D as by bending along CD. The surface no longer remains in one plane, and the shortest, or straight line, joining the two points A and B, does not lie in the surface.

The intersection of two planes is a straight line, because the straight line joining any two points in their line of intersection must lie in both planes.

From a sheet of cartridge paper or thin cardboard cut out two rectangular strips. Cut one narrower than the other, as

shown at A (Fig. 76). Το pass A through B, without bending or creasing, it is necessary to cut a slit S in B.

The two planes A and B are now easily seen to intersect in the straight line S.

B

A

d

S

FIG. 76.

Vertical
Plane

Instead of using one strip narrower than the other a useful model of what are called the vertical and horizontal planes can be made by using two equal strips of paper or cardboard, along the centre of one of which a slit is made as shown at S (Fig. 76). In the other strip the two slits, indicated by the lines ab, cd, are so cut that when folded over the slip can be passed through the slit S. When the two pieces are opened out, a model of the vertical and horizontal planes of projection, as shown in Fig. 77, is formed. 'The two planes are supposed to rotate about the line xy

Horizontal
Plane.

Vertical
Plane

FIG. 77.- Model of the co-ordinate planes of projection.

as a hinge, until one continuous surface or plane is obtained, corresponding to a sheet of

paper.

Descriptive Geometry-By the methods of descriptive geometry the form of any figure may be determined by constructions drawn to scale. These constructions are made on one

plane only, and it is possible by this means to represent a solid having the three dimensions, length, breadth, and thickness, on a plane having but two dimensions, length and breadth, such as a sheet of paper.

This is effected by what are called projections.

Projections of a Point.-The position of a point in space is determined when its projections on two intersecting planes are given.

Thus, if a point A (Fig. 78) be given, the projection a of the point on the horizontal plane is the intersection of a perpendicular let fall from the point on to that plane. In a similar manner, if a perpendicular be drawn from A to the vertical plane the intersection of the perpendicular with this plane is the projection of the point a' on the vertical plane.

A

a

FIG. 78.-Projection of a point.

The two planes of projection, intersecting in a line xy, when in position are usually at right angles to each other and are always referred to as the horizontal plane of projection, or shortly, H.P., and the vertical plane of projection, or v.P. The point a on the H.P. is called the plan of the point, and the point a' on the V.P. the elevation of the point A.

These two projections determine definitely the position of the point A in space, because if from a and a' perpendiculars are drawn to the H.P. and v.P. respectively, the point A will lie at some point in the first, and also at some point in the second perpendicular, and therefore must be at the point of their intersection.

It is essential to note that the given point and its projections

must lie in one plane. Hence, if the perpendiculars do not intersect, the two projections a and a' are not the projections of the same point.

The distance Aa is equal to the distance of a from xy and the given point from the H.P.; and Aa' is equal to the distance of a' and the point from the v.P.

from xy

Thus, the distance of the plan of a point from the ground line (xy) indicates the distance of the point from the V.P.; and the distance of the elevation of a point from the ground line indicates the distance of the point from the H.P.

Other Planes of Projection.-As other planes of projection besides those named may be used, the elevation of a point is marked by an italic letter with a dash at the upper right-hand corner of the letter, thus a'. The plan of a point is marked by the same italic letter, but without a dash, as a.

The capital letter A is only used to specify the point itself, and is not shown except when a perspective view of the planes is made, as in Fig. 78.

Ex. 1. A point A is 5" in front of the V.P. and '7′′ above the H.P. Another point B is at a distance of 6" from the H.P. and 4" from the V.P. Show the projections of the two points.

Commence by drawing the ground line xy (Fig. 78); draw a line aa' perpendicular to xy.

Make the distance from xy to a' equal to 7" and from xy to a equal to 5" as shown.

Then a', a are the required projections of the point A.

In a similar manner measuring distances 6" above and 4" below xy, on a line b'b, drawn perpendicular to xy, the two projections b'b of a point B can be obtained.

Projections of a Line.--The projection of a line AB on a plane MN (Fig. 79) is obtained as follows:

From A and B let fall perpendiculars (as shown by the dotted lines), on the plane MN. The line joining the points where these dotted lines meet the plane is the projection required.

Angle between a line and plane, or the inclination of a line to a plane, is the angle between the line and its projection on the plane; thus, if BA produced meets the plane as shown (Fig. 79), the inclination of the line to the plane is the angle between the line and its projection on the plane.