tal line, is considerably above it: the lines at k show the manner of finding the height of the roof. The point of sight, s in fig. 108., may be in any place relative to the plan, so that the distance from the picture is not less than the length of the picture. The custom is to choose a point from which the object will look best, and drawings viewed at small angles are more pleasant than those viewed at great angles, with the point of sight too near. When the fronts of objects are parallel to the plane and above the eye, the representation is called parallel perspective. When the objects are parallel to the plane, but below the eye (as fig. 107.) it is called parallel bird's-eye perspective. When objects are oblique to the plane (as fig. 110.), it is angular perspective; and when oblique to the plane, but below the eye (as fig. 109.), it is angular bird's-eye perspective. These are merely conditional terms, for there are, strictly speaking, only two kinds of perspective, linear and aerial. Linear perspective is the perspective of lines which we have been describing; and aerial perspective is the art of giving the due tone of light, shadow, and colour of objects according to their distance and the medium through which they are seen.

L

To draw a Circle in Perspective. Let fig. 111. a be the circle; draw a square about it, then draw the two diagonals of the square, and where they cut the circle let those be the angles of an inner square. Draw also the two diameters of the circle. Let HL be the horizontal line, c the

Fig. 111.

centre, and the

point at H the point of distance; then put all the straight

lines in perspective, which will give eight points of the circle, when the curve is completed by hand.

Shadows and Tinting. The light may be in front, at either side, or even at the back of a picture; but for perspective views of buildings one side is preferable. Of course, all shadows are produced by opaque objects intercepting the light; which may either be natural light from the sun, &c., or artificial light from a candle or other luminous body. The rays of natural light, in consequence of the immense distance of the sun, are parallel; but the rays of artificial light diverge from a point; consequently, shadows thrown by objects intercepting the sun's light are of the same breadth, while shadows thrown by objects intercepting the rays of artificial light increase in width as it leaves the objects. In geometrical drawing the rays of light fall upon the object at an angle of 45°, as we have seen, but the shadows in a picture or perspective drawing are found thus: - We first determine upon the direction in which we wish the light to come into the picture; then find the altitude of the light, which may be high or low according to what we wish the drawing to appear. From the altitude we drop a perpendicular to the base, and produce the latter: we then draw a line from the altitude to the top of the object, and that line continued to the base determines the length of the shadow, which is the co-tangent of the angle that the light makes with the surface on which the shadow is thrown. In plainer terms, the shadow is found by producing parallel lines from the top of the object in the direction of the light until they meet the surface on which the shadows fall. The shadows caused by artificial light are found by producing lines from the luminous point touching the angles of the different projections until they meet the surfaces on which the shadows are thrown, and the length of the shadow is determined. In shading perspective drawings, the shadows and shades must be darkest where they are nearest the foreground; and each individual part must not have its own light and shadow, but must be blended with the whole. This is one distinction between the manner of treating geometrical and perspective

drawings. In the foreground of the latter, bright lights are opposed to dark shadows, while the back-ground is blended into one less definite mass. The same holds good as to tinting — bright and distinct colours are opposed to each other in the foreground, but in the distance they are subdued, as if a mist or haze had come over the objects, which in nature is caused by the intervening body of air. The eye calculates more readily the size of objects by this gradation of tint, than by the magnitude of the objects in the representation. The proper distribution of light and shadow in a picture can only be properly managed by one who has the eye of an artist; but any one who can shade polygons, when geometrically represented, will be able to shade and tint a building in perspective.

Reflection and Refraction. All objects in nature reflect light through the medium of the air, even when the sun is not shining on them; but it is with the reflection of objects in water that we have to do at present. It is an universal law of Optics that the angles of incidence and reflection are equal. The angle of incidence is the angle at which objects are presented to a reflecting surface, and the angle of reflection is the angle at which the image is reflected; both are equal, but in opposite directions. Let bac, fig. 112., be the angle of incidence, then da e is the angle of reflection, and d a in this case would be the line in which the eye of an observer was situated; but images in water will be better understood

e

d

Fig. 112.

by a figure. Let f, g, h (fig. 113.) be three trees at different distances from the edge of a piece of water, and at different distances from the eye of an observer on the opposite bank of the water at i; then k would be the reflection of f; 7 the reflection of g; and m the reflection of h, as seen by this observer. When objects stand on the brink of water, their whole height is reflected; but when they stand at a distance from it, the height of the ground, or rather the height which the ground subtends at the eye, must be deducted from the

Fig. 113.

m

height of the object for the depth of the reflection. Little need be said of refraction; it simply means the turning away of rays of light from their direct course. If light falls perpendicularly on a piece of water, it passes through it perpendicularly; but if it falls obliquely, then it does not continue the same line of obliquity through the water, but in a direction approaching more to the perpendicular, because water is more dense than air; and if passing through glass, it would be still more refracted from its original direction, as glass is more dense than water; but when it has passed through either body it resumes its original direction in passing through its original medium.

To varnish water-coloured Drawings. -Prepare the drawing with a strong coating of isinglass in a liquid state, laid on with a large flat camel's-hair brush, taking care not to go over the same place twice while wet, for fear of raising the colours. The isinglass must float in a body during the operation, which must be rapid, but with a light touch, lest the colours should run. The drawing thus prepared to receive, when perfectly dry, two or three coats of Masters's patent varnish, each coat being perfectly dry before the succeeding coat is laid on.

219

CHAP. XIII.

MISCELLANEOUS TABLES.

SECTION I.

INTEREST AND ANNUITY TABLES.

Explanation of the Tables referring to Interest and Annuities. TABLES I. and II. The present worth of a sum of money, as 17., to be received at a future period, is that which, laid out and improved at a given rate of interest during that period, will amount to the proposed sum by the time it becomes due. The difference between the present worth and the sum itself is called the discount. Hence, to find the present worth of a sum of money improved at a given rate for a fixed time, divide the proposed sum by the amount of 11. improved at the assigned rate for the given time, and the quotient will be the present worth: subtract the present worth from the sum proposed and the remainder will be the discount.

Ex. If 17. be divided by 1.047. the quotient is 9615387. which is the present value of 17. due one year hence.; and, if from 17. we take that quotient, the difference 0384627. is the discount: for, 961538+038462=17. Hence the use and application of TABLES I. and II., and particularly the latter, as shown in example 2. page 43.

TABLE III. This table is fully explained in pages 35. and 36., article Compound Interest.

TABLE IV. This table is explained, and its construction fully shown, in pages 37. and 38.

TABLE V. The construction of this table is shown in pages 38. and 39.; and example 1. page 39. and example 5, page 40. show its application and use. Thus, in column 1. the term of years is found, and the reader traces from 21 years across the line to the third column or 5 per cent., and