# An introduction to linear drawing/Chapter 8

​﻿PART SECOND.
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Although several authors have written excellent trea- tises on the Art of Perspective, it is to be feared that they have presupposed such an acquaintance with ge- ometry, as is seldom attained by youth for whom this work is designed. A complete treatise is not intended, but merely such a familiar illustration of the first princi- ples of perspective as a common mind, acquainted with the former part of this work, may be able to compre- hend. All that has been attempted is the laying of a good foundation for future progress, should these few pages excite in the pupil a desire to know more of this useful and elegant art.
There is probably no one who has not remarked, that objects at a distance appear much smaller than their real size. The cause of this is to be found in the structure of* the eye, and in the laws of vision. To il- lustrate these laws, draw a right line of any length with a pen, then place the rule parallel to the line ; mark with the thumbs the length of the line upon the rule, and holding the thumbs there, stretch the arms at full length. Then begin gradually to draw the rule towards ​﻿the eyes, and it will be perceived that the space be- tween the two thumbs appears longer than the black line, and before the rule reaches the eyes, the differ- ence will be nearly as three to one.
This will lead the pupil to conclude, that objects of the same size appear the smaller the farther they are from the eye, and the larger the nearer they are to it, though the real size of the object is unchanged.
fig. 1. plate ii.
The apparent size of an object depends upon the an- gle formed by lines drawn from the extremities of an object to the eye, the eye forming the apex of the an- gle. Thus in fig. 1, let the circle represent the eye. Three arrows of equal length are placed at different distances from it. Lines are drawn from both extremi- ties of each arrow, so as to cross each other at the same point in the eyes, and continue on till they touch the back of the eye. The pupil has been informed (p. 4.) that it is not the length of the sides, but the opening of the sides which determines the size of the angle. If the pupil will measure the angle, which the lines from the extremities of the nearest arrow to the eye form, (see page 28,) he will find it to consist of about 92 de- grees, which are more than quarter of a circle. If he will then measure the angle formed by the lines from the second arrow, he will find it to consist of about 27 degrees. Finally, if he measure the angle of the third arrow, the farthest from the eye, he will find it to consist of only 14 degrees.
The lines of any angle if continued beyond the apex, as in this case, will form a second angle of the same size as the first. Of course the linds of the first angle, being extended beyond the apex to the back of the eye, form there a corresponding angle of 92 degrees, and ​﻿the other lines, extended in the same manner, form angles of 27 and 14 degrees.
The relative size of the angles is best represented on the back of the eye, where they are cut by a single circle, and where the distance between the lines gives the apparent size of the object as seen by the spectator. Thus, the distance between the dotted lines on the back of the eye is the apparent length of the farthest arrow, and so of the other lines, which may be easily traced to their proper arrows.
If the distance of the third arrow reduces the angle from 92 to 14 degrees, it must be evident that, if the arrow or object be much farther removed, it will form no angle at all with the eye, but will become a mere point, and then disappear or become invisible.
This point, at which the object ceases to form any angle with the eye, is called the Vanishing Point of the object; and that point in the eye where the lines cross each other is called the Point of Sight.
In figure 1, the arrows are represented of their real size, to show why they appear smaller as they recede. In figure 2, they are represented at their apparent size, to show the vanishing point to better advantage.
FIG. 2. PLATE II.
In the above figure the real size of the object is distinguished from the apparent size by dots, and as the vanishing point falls short of the last arrow, the conclusion is that this arrow is invisible to the eye of the observer.
The vanishing point is familiarly represented by the sides of a long and straight road, which seem to approach each other at a distance; or by the lamps of a long bridge, which, although in parallel lines, apparently meet when seen from either end of the bridge.
​﻿Perspective, then, is the art of drawing or represent- ing any object on a plane, or flat surface, as the object appears to the eye of the spectator.
The simplest way to represent an object in perspec- tive is to stand before a window, and, holding the head still, to draw on a pane of glass the outline of any ob- ject seen through it. But a more convenient apparatus is generally used.
This, and some of the more important terms used in Perspective Drawing, the pupil must endeavour to un- derstand.
1. The Perspective Plane is an upright square of glass, usually framed like a picture, with a base, so that it can stand up alone. This is placed between the eye of the spectator and the object to be drawn, and as the drawing is sometimes made directly upon it, it is sometimes called the Picture or the Plane of the Pic- ture.
2. Visual Rays are lines drawn from every part of the object, through the Perspective Plane to a point in the spectator's eye, which point, it has been said, is called the Point of Sight.
3. The Horizontal Line is a line drawn di- rectly across the Perspective plane at the height of the spectator's eye, and this may be of any height, although it is customary to draw the line about one third from the base or bottom of the Perspective Plane.
4. The Centre of the Picture is that point on the Perspective Plane, where a line, drawn from the spec- tator's eye to the Perspective Plane, would strike the Horizontal Line. This point is generally placed as near the centre of the Perspective Plane or Picture, as possible, and hence its name.
5. The Prime Vertical Line is a perpendicular line raised on the base of the picture, and passing through that point which is called the Centre of the Picture. ​﻿It is of course perpendicular to the Horizontal Line also.
6. The Points of Distance. One, called the dis- tance of the picture, is the distance of the eye from the centre of the Perspective Plane. The other called the Vanishing distance, is the distance from the eye to the Vanishing point on the Perspective Plane.
7. The Ground Plane is the table or any flat horizon- tal surface on which the Perspective Plane is supposed to stand.
8. The Base Line is a line necessarily marked across the Ground Plane by the base or bottom of the Per- spective Plane.
fig. 3. plate iu
This figure represents a road with passengers at differ- ent distances on it.
The Base Line is at the bottom of the figure.
The Horizontal Line cuts the four passengers in the middle, and is always parallel to the Base Line.
The Vanishing Point and Centre of the Picture are in this case the same point, where the lines converge on the Horizontal Line.
The Prime Vertical Line is the perpendicular raised on the Base Line, and passing through the Centre of the Picture.
Now to give each figure its relative proportion ac- cording to its distance, draw vanishing lines to the van- ishing point from the extremities of the nearest figure, and in the angle thus formed will be found the heights of all the other figures. To find the height of a figure on any part of the road, only lead off a horizontal line, parallel to the base line, for the foot of the object, and the height is found by raising a perpendicular from the horizontal until it strikes the other converging or van​﻿ishing line. Tbe dotted lines in the cut will illustrate this better than any description.
fig. 4. plate ii.
Suppose it be required to find the perspective dis- tance of nine trees equidistant from each other. Draw the Base Line just the real length of the row of trees, or longer, at pleasure. Then draw the Horizontal Line longer than the base line. From the left end of the base line draw a Vanishing Line to any point on the horizontal line. Thep taking the length of this van- ishing line, take a portion of the Horizontal Line to the left of the Vanishing Point, equal to the vanishing line in length. On the Base Line measure the exact length of the row of trees, and divide the Base Line into exact portions according to the number of trees, and place a dot at each distance. Then draw a line from each of these dots to the point on the left of the Horizontal Line, and the points where these lines intersect or cross the Vanishing Line are the perspective positions of the trees. If you wish to regulate the height of the trees, draw another vanishing line from the top of the tree on the BaseLine to the vanishing point, and the distance be- tween the two vanishing lines will be the height of the trees, as is shown by the dotted line in the figure.
fig. 5. plate ii.
Figure 5 represents the perspective of a square, one of whose sides is parallel to the base line. First draw the base and horizontal lines, then from the two angles of the square which touch the base line carry the two vanishing lines to thé vanishing point on the horizontal line. From the same angles draw two lines, one to the right and the other to the left of the vanishing point, and at equal distances on each side of it. Then draw a line ​﻿to connect the two points where these two lines inter- sect the vanishing lines, and the square is finished.
To draw the perspective of a second square, place one foot of the dividers at those angles of the first square which touch the base line ; draw the dotted quadrants or quarter circles ; and, from the points where they strike the base line, draw oblique lines to the points of distance on each side of the vanishing point, and the four points where they intersect the vanishing lines will be the four corners of the next perspective square; and you have only to connect these points by lines parallel to the base line. Pursue the same course in drawing the third square, and so on.
PIG. 6. PLATE II.
To draw the perspective of a solid body as a Cube or Prism. First, by directions under fig. 5, draw the perspective base, then raise perpendiculars to the base line, and by uniting their tops form that face of the cube or prism which is parallel to the Perspective Plane, and is the nearest face to the spectator's eye or point of sight. From the four corners of this face of the cube draw lines to the vanishing point, following the direc- tion of the right and left sides of the perspective base. Raisfc perpendiculars from the two distant points of the base until they touch the two upper vanishing lines, and connect these two points by a right line, and the per- spective cube is formed.
To draw a second Cube, draw a second base as in fig. 5, raise perpendiculars, and proceed as with the first cube.
In the directions for drawing a Cone (Part 1, Class 4) it was remarked that although the base was an exact circle, the laws of perspective required that it should be represented as an ellipsis or oval. The mode of representing a square in perspective has been shown ; ​﻿the next figure illustrates the rules for drawing a circle in perspective.
tig. 7. plate ii.
Draw the Circle and enclose it in a square. Draw parallels to the sides of the square where the diagonals cut the circle, then a parallel to the sides passing through the centre of the circle. From the points, where these parallels strike the base line, lead off vanishing lines to the vanishing point. Follow the directions given in fig. 5, for completing the squares ; but the circles must be drawn by the eye through the eight points formed by the intersection of the diagonals and vanishing lines. Of course, more vanishing lines may be made if the circle be large, and they are necessary.
It will be noticed,that in this figure one half of the lines is dotted. Thejobject of this is to show that, in drawing the arches of a bridge in perspective, the easiest way is to draw circles, the arch being half of a circle, and represented in the figure by the lines not dotted, the prime vertical line representing the surface of the water.
This figure may serve also to show the perspective of a Rhomb or Lozenge, represented by the diagonals of the squares. The Piers of the bridge deform the second rhomb, but, if no space be left between the squares or circles, the rhomb will be perfectly formed.
fig. 8. plate ii.
From the five angles of the Polygon, draw as many perpendiculars to the base line, and let the left hand perpendicular extend far enough down to intersect hor- izontal lines from the five angles abovementioned. Then from the points where the five perpendiculars strike the base line, lead off five vanishing lines to the vanishing ​﻿point in the horizontal line. From those points where the horizontal lines strike the left hand perpendicular, describe arcs of circles to the base line. From the points where these arcs strike the base line,lead off lines directed towards some one point on the horizontal line, but let them stop at the first vanishing line they strike, and from the point where they strike this vanishing line, draw parallels to the base lines, and by the figure it will be seen that they will strike the other vanishing lines in the points which form the angles of the perspective polygon. Draw lines to these points, and the figure is formed.
fig. 9. plate ii.
To draw the perspective of a house of which one side is parallel to the perspective plane, and in all simi- lar drawings, it is necessary to have the exact measure- ment of every part of the building which is to be rep- resented.
Having drawn the base line C D H and the hori- zontal line A B at the height of a man's head above it, that is, about 5 feet 6 inches, you may fix the centre of the picture at G on the horizontal line. Then raise per- pendiculars on the points C D, equal to the actual height of the building as previously laid down on a proportional scale of parts, and connect these perpendiculars by the horizontal line F E, parallel to the base line. This will complete the/row* of the building, or that part facing the spectator's eye.
To represent the end of the building, draw the van- ishing lines E B and D B. Ascertain the real width of the end, and mark it on the base line, say from D to H. Draw a line from H to the centre of the picture G, and the point where it cuts the vanishing line D B at I is the perspective width of the end. On the point I raise a perpendicular till it strikes the other vanishing ​﻿line at K, and you have the other corner of the building.
Draw the diagonals D K and E I, and the point L where they intersect will be the centre of the gable end.
Having measured the actual height of the gable end, continue the perpendicular D E till it reaches M, the actual height. Then draw the line M B. Erect a perpendicular on the centre of the gable end ? and the the point N, where it intersects the line M B, will be the point of the gable end. Then draw the lines N E and N K, and the end is completed.
To draw the Chimney, find its actual height above M, the actual height of the gable end, and continue the perpendicular D E M to O. Draw the line O B, then, oneach side of the centre P L, lay off on the base line two spaces Pa and Pb each equal to half the real breadth of the chimney, and raise the perpendiculars ac and bd.
To find the thickness of the chimney, lead off a line from O to Q equal to its real thickness. Draw the line Q B. Lead a horizontal from c till It strikes the line Q B at e; do the same from g to f; connect e and f by a perpendicular, and the perspective of the chimney is finished.
To draw the other gable end, you must suppose the house to be transparent, and proceed exactly as you did with the first. Then connect the points of the gables, and the line N R will form the ridge of the house.
The door and windows of the side parallel to the perspective plane must be drawn according to their actual dimensions, the rules of perspective only affecting the thickness of their edges.

#### FIG. 10. PLATE II.

To draw a house which stands oblique to the picture, that is, one which has no side parallel to the picture or ​﻿Perspective plane, begin with the base line C D and the horizontal line A B. Take Ethe corner of the building nearest to the spectator's eye, and draw the line E B for the bottom of one side of the building.
Then, to find the Vanishing point of the lines of the other side, from the Centre of the picture, which you may fix at F, draw the perpendicular F G equal to the Distance of the picture. Then draw the line G A at right angles to G B, and the point A, in which it cuts the horizontal line, will be the vanishing point of the other side of the building. To this point, therefore, draw the line E A for the bottom of this side of the building.
In order to find the apparent width of each side, it is necessary to have a distance point for each side. Take the space from A to I, equal in length to the line A G, the point I being the distance point of the side E A. In like manner with the distance B G mark the space B H on the horizontal line, the point H being the distance point of the side E B. On the base line measure the space E C equal to the real width of the side E A, and from the point C draw a line to the distance point I, which, cutting the line E A at L, will give the space L E, the perspective width of that side. In like manner measure off E D the real width of the other side, draw the line D H, and the space E N is the perspective width of this side.
From the corner E erect the perpendicular E M, the actual height of the house ; draw the line M A for the top of the building; raise a perpendicular on the point L till it strikes the line M A at K, and you have one front of the building completed. Do the same by the other side and you have M E O N, the other front.
Cross two diagonals to find the centre of the gable end near I. Carry the perpendicular E M to P, the real height of the point of the gable end. Draw the line P B, raise a perpendicular on the centre of the gable ​end near I, and the point $a$ where it strikes the line P B will be the perspective point of the gable end. Draw the lines M $a$ and O $a$, and the gable end is completed.
Having found the point Q of the other gable end, by the rules given under fig. 9, draw the line K Q for the slanting side of the roof, and the line $a$ Q will form the ridge of the house.
Mark the actual height and length of the door or window upon the perpendicular line E M, and draw lines from these points to A. The perspective height of the door and window will be found between these lines. To find the relative distance of the door and window from the end of the building, mark off the real distance on the base line and draw a line from that point to the point of distance I or H, as the case may be, and the point where this line strikes the lower vanishing line of the door or window will be the perspective distance of the side of the door or window.
FIG. 11. PLATE II.
To draw fig. 11, first draw the front of the arcade D Q E G H and A R C. Find the common centre O on the centre of Q G, at the height of the base of the arch. From the centre O draw a line towards the point of view V until you strike the point N, the centre of the perspective arch L M, which will terminate the proposed arcade.
If the thickness X H of the first arch (if the arcade is composed of several arches) project, as often happens, then, on the perspective depths X and 3, raise the perpendiculars X Y and 3, 2, to the perspective line of the top of the walls. On the point U, the centre of the opening of the arcade, raise the perpendicular U E. To