the assigning to an inanimate object the sentiments and language of a living being. Perspective is the art of drawing upon a plane surface true resemblances or pictures of objects, as the objects themselves appear to the eye from any distance and situation, real or imaginary.

It was in the sixteenth century that perspective was revived, or rather re-invented. It owes its birth to painting, and particularly to that branch of it which was employed in the decorations of the theatre, where landscapes were properly introduced, and which would have looked unnatural and horrid if the size of the objects had not been pretty nearly proportioned to their distance from the eye. We learn from Vitruvius, that Agatharchus, instructed by Eucyclus, was the first who wrote upon this subject; and that afterwards the principles of the art were more distinctly taught by Democritus and Anaxagoras, the disciples of Agatharchus. Of the theory of this art, as described by them, we know nothing, since none of their writings have escaped the general wreck that was made of ancient literature in the dark ages of Europe. However, the revival of painting in Italy was accompanied with a revival of this art.

The first person who attempted to lay down the rules of perspective was Pietro del Borgo, an Italian. He supposed objects to be placed beyond a transparent tablet, and endeavoured to trace the images which rays of light emitted from them would make upon it. But we do not know what success he had in this attempt, because the book which he wrote upon the subject is not now extant. It was, however, very much commended by the famous Egnazio Dante; and, upon the principles of Borgo, Albert Durer constructed a machine, by which he could trace the perspective appearance of objects.

Balthazar Perussi studied the writings of Borgo, and endeavoured to render them more intelligible. To him we owe the discovery of points of distance, to which all lines that make an angle of forty-five degrees with the ground line are drawn. A little time after, Guido Ubaldi, another Italian, found that all the lines that are parallel to one another, if they be inclined to the ground line, converge to some point in the horizontal line; and that through this point also, a line drawn from the eye, parallel to them, will pass. These principles, put together, enabled him to make out a pretty complete theory of perspective.

Great improvements were made in the rules of perspective by subsequent geometricians; particularly by Professor Gravesande, and still more by Dr Brooke Taylor, whose principles are in a great measure new, and far more general than any that had been made before his time.

In order to understand the principles of perspective, it will be proper to consider the plane on which the representation is to be made as transparent, and interposed between the eye of the spectator and the object to be represented. Thus, suppose a person at a window looks through an upright pane of glass at any object beyond it, and, keeping his head steady, draws the figure of the object upon the glass with a black-lead pencil, as if the point of the pencil touched the object itself; he would then have a true representation of the object in perspective, as it appears to his eye.

In order to this, two things are necessary; first, that the glass be laid over with strong gum-water, which, when dry, will be fit for drawing upon, and will retain the traces of the pencil; and, secondly, that he look through a small hole in a thin plate of metal, fixed about a foot from the glass, between it and his eye, and that he keep his eye close to the hole, otherwise he might shift the position of his head, and consequently make a false delineation of the object.

Having traced the figure or outline of the object, he may go over it again with pen and ink, and when the ink is dry, put a sheet of paper upon it, and trace it thereon with a pencil; then taking away the paper and laying it upon a table, he may finish the picture by giving it the colours, lights, and shades, as he sees them in the object itself. By these means he will have a true resemblance of the object.

To every person who has a general knowledge of the principles of optics, this must be self-evident. For, as vision is occasioned by pencils of rays coming in straight lines to the eye from every point of the visible object, it is plain that, by joining the points in the transparent plane, through which all those pencils respectively pass, an exact representation must be formed of the object, as it appears to the eye in that particular position, and at that determined distance; and were pictures of things to be always first drawn on transparent planes, this simple operation, with the principle upon which it is founded, would comprehend the whole theory and practice of perspective. As this, however, is far from being the case, rules must be deduced from the sciences of optics and geometry, for drawing representations of visible objects upon opaque planes; and the application of these rules constitutes what is properly called the art of perspective.

Previously to our laying down the fundamental principles of this art, it may not be improper to observe, that when a person stands right against the middle of one end of a long avenue or walk, which is straight and equally broad throughout, the sides thereof appear to approach nearer and nearer to each other as they are further and further from his eye; or the angles under which their different parts are seen become less and less according as the distance from his eye increases; and if the avenue be very long, the sides of it at the farthest end will seem to meet, and there an object that would cover the whole breadth of the avenue, and be of a height equal to that breadth, would appear to be only a mere point.

Having made these preliminary observations, we shall now proceed to the practice of perspective, which is built upon the following:

**THEOREM I.**

Let $abcd$ (fig. 1) represent the ground plan of the figure to be thrown into perspective, and $efgh$ the transparent plane through which it is viewed by the eye at $E$. Let these planes intersect in the straight line $KL$. Let $B$ be any point in the ground-plan, and BE a straight line, the path of a ray of light from that point to the eye. This will pass through the plane efgh in some point b; or B will be seen through that point, and b will be the picture, image, or representation of B.

If BA be drawn in the ground-plan, making any angle BAK with the common intersection, and EV be drawn parallel to it, meeting the picture-plane or perspective plane in V, and VA be drawn, the point b is in the line VA so situated that BA is to EV as bA to bV.

For since EV and BA are parallel, the figure BA6VEBB is in one plane, cutting the perspective plane in the straight line VA; the triangles BA6, EV6, are similar, and BA : EV = bA : bV.

Cor. 1. If B be beyond the picture, its picture b is above the intersection KL; but if B be between the eye and the picture, as at B', its picture b' is below KL.

2. If two other parallel lines BA', ES, be drawn, and A', S, be joined, the picture of B is in the intersection of the lines AV and AS.

3. The line BA is represented by bA, or bA is the picture of BA; and if AB be infinitely extended, it will be represented by AV. V is therefore called the vanishing point of the line AB.

4. All lines parallel to AB are represented by lines converging to V from the points where these lines intersect the perspective-plane; and therefore V is the vanishing point of all such parallel lines.

5. The pictures of all lines parallel to the perspective-plane are parallel to the lines themselves (fig. 8).

6. If through V (fig. 1) be drawn HVO parallel to KL, the angle EVH is equal to BAK.

Remark. The proposition now demonstrated is not limited to any inclination of the picture-plane to the ground-plane; but it is usual to consider them as perpendicular to each other, and the ground-plane as horizontal. Hence the line KL is called the ground-line, and OH the horizon-line; and OK, perpendicular to both, is called the height of the eye.

If ES be drawn perpendicular to the picture-plane, it will cut it in a point S of the horizon-line directly opposite to the eye. This is called the point of sight, or principal point.

7. The pictures of all vertical lines are vertical, and the pictures of horizontal lines are horizontal, because these lines are parallel to the perspective plane.

8. The point of sight S is the vanishing point of all lines perpendicular to the perspective plane.

The above proposition is a sufficient foundation for the whole practice of perspective, whether on direct or inclined pictures, and serves to suggest all the various practical constructions, each of which has advantages which suit particular purposes. Writers on the subject have either confined themselves to one construction, from an affection of simplicity or fondness for system; or have multiplied precepts, by giving every construction for every example, in order to make a great book, and give the subject an appearance of importance and difficulty. An ingenious practitioner will avoid both extremes, and avail himself of the advantage of each construction as it happens to suit his purpose. We shall now proceed to the practical rules, which require no consideration of intersecting planes, and are all performed on the perspective plane by means of certain substitutions for the place of the eye and the original figure. The general substitution is as follows:

Let the plane of the paper be first supposed to be the ground-plan, and the spectator to stand at F (fig. 2). Let it be supposed that the ground-plan is to be represented on a plane surface, standing perpendicularly on the line GKL of the plan, and that the point K is immediately opposite to the spectator, or that FK is perpendicular to GL; then FK is equal to the distance of the spectator's eye from the picture.

Now suppose a piece of paper laid on the plan with its straight edge lying on the line GL; draw on this paper KS perpendicular to GL, and make it equal to the height of the eye above the ground-plan. This may be much greater than the height of a man, because the spectator may be standing on a place much raised above the ground-plan. Observe also that KS must be measured on the same scale on which the ground-plan and the distance FK were measured. Then draw HSO parallel to GL. This will be a horizontal line, and (when the picture is set upright on GL) will be on a level with the spectator's eye, and the point S will be directly opposite to his eye. It is therefore called the principal point, or point of sight. The distance of his eye from this point will be equal to FK. Therefore make SP (in the line SK) equal to FK, and P is the projecting point or substitute for the place of the eye. It is sometimes convenient to place P above S, sometimes to one side of it on the horizontal line, and in various other situations; and writers ignorant of, or inattentive to, the principles of the theory, have given it different denominations, such as point of distance, point of view, &c. It is merely a substitute for the point F in fig. 1, and its most natural situation is below, as in this figure.

The art of perspective is conveniently divided into Ichography, which teaches how to make a perspective draught of figures on a plane, commonly called the ground-plan; and Scenography, which teaches how to draw solid figures, or such figures as are raised above this plan.

Prob. 1. To put into perspective any given point of the ground plan.

First general construction.

From B and P (fig. 2) draw any two parallel lines BA, PV, cutting the ground-line and horizon-line in A and V, and draw BP, AV, cutting each other in b; b is the picture of B.

For it is evident that BA, PV of this figure are analogous to BA and EV of fig. 1, and that BA : PV = bA : bV. If BA' be drawn perpendicular to GL, PV will fall on PS, and need not be drawn. A'V will be A'S. This is the most easy construction, and nearly the same with Ferguson's.

Second general construction.

Draw two lines BA, BA', and two lines PV, PD, parallel to them, and draw AV, A'D, cutting each other in b; b is the picture of B by Cor. 2. This construction is the foundation of all the rules of perspective that are to be found in the books on this subject. They appear in a variety of forms, owing to the ignorance or inattention of the authors to the principles. The rule most generally adhered to is as follows:

Draw BA perpendicular to the ground-line, and AS to the point of sight, and set off Aβ equal to BA. Set off SD equal to the distance of the eye in the opposite direction from S that β is from A, where B and E of fig. 1 are on opposite sides of the picture; otherwise set them the same way. D is called the point of distance. Draw BD, cutting AS in b.

This is evidently equivalent to drawing BA' and PS perpendicular to the ground-line and horizon-line, and BA" and PD (fig. 2) making an angle of 45° with these lines, with the additional puzzle about the way of setting off A'A" and SD, which is avoided in the construction here given.

This usual construction, however, by a perpendicular and the point of distance, is extremely simple and convenient; and two points of distance, one on each side of S, serve for all points of the ground-plan. But the first general construction requires still fewer lines, if BA be drawn perpendicular to GL, because PV will then coincide with PS.

Third general construction.

Draw BA (fig. 3) from the given point B perpendicular to the ground-line, and AS to the point of sight. From the point of distance D set off Dd equal to BA, on the same or the contrary sides as S, according as B is on the same or the contrary side of the picture as the eye. Join dA, and draw Dd parallel to dA. b is the picture of B. For SD, Dd are equal to the distances of the eye and given point from the picture, and SD : Dd = bS : bA.

This construction does not naturally arise from the original lines, but is a geometrical consequence from their position and magnitude; and it is of all others the most generally convenient, as the perpendicular distances of any number of points may be arranged along SD without confusion, and their direct situations transferred to the ground-line by perpendiculars such as BA; and nothing is easier than drawing parallels, either by a parallel ruler or a bevel square, used by all who practise drawing.

Prob. 2. To put any straight line BC of the ground-plan in perspective.

Find the pictures b, c, of its extreme points by any of the foregoing constructions, and join them by the straight line bc.

Perhaps the following construction will be found very generally convenient.

Produce CB till it meet the ground-line in A, and draw PV parallel to it; join AV, and draw PB, PC, cutting AV in b, c. V is its vanishing point, by Cor. 3 of the fundamental theorem.

It must be left to the experience and sagacity of the drawer to select such circumstances as are most suitable to the multiplicity of the figures to be drawn.

Prob. 3. To put any rectilineal figure of the ground-plan in perspective.

Put the bounding lines in perspective, and the problem is solved.

The variety of constructions of this problem is very great, and it would fill a volume to give them all. The most generally convenient is to find the vanishing points of the bounding lines, and connect these with the points of their intersection with the ground line. For example, to put the square ABCD (fig. 5) into perspective.

Draw from the projecting point PV, PW, parallel to AB, BC, and let AB, BC, CD, DA, meet the ground line in α, z, δ, β, and draw αV, δV, zW, βW, cutting each other in abcd, the picture of the square ABCD. The demonstration is evident.

This construction, however, runs the figure to great distances on each side of the middle line, when any of the lines of the original figure are nearly parallel to the ground-line.

The following construction avoids this inconvenience.

Let D be the point of distance. Draw the perpendiculars Aα, Bβ, Cz, Dδ, and the lines Ae, Bf, Cg, Dh, parallel to PD. Draw Sα, Sβ, Sz, Sδ, and De, Df, Dg, Dh, cutting the former in a, b, c, d, the angles of the picture.

It is not necessary that D be the point of distance; only the lines Ae, Bf, &c. must be parallel to PD.

Remark. In all the foregoing constructions the necessary lines (and even the finished picture) are frequently confounded with the original figure. To avoid this great inconvenience, the writers on perspective direct us to transpose the figure; that is, to transfer it to the other side of the ground-line, by producing the perpendiculars Aα, Bβ, Cz, Dδ, till αA', βB', &c. are respectively equal to Aα, Bβ, &c.; or, instead of the original figure, to use only its transposed substitute A'B'C'D'.

This is an extremely proper method. But in this case the point P must also be transposed to P' above S, in order to retain the first or most natural and simple construction, as in fig. 7, where it is evident, that when BA = AB', and SP = SP', and BP' is drawn, cutting AS in b, we have bA : bS = BA : PS = BA : PS, and b is the picture of B; whence follows the truth of all the subsequent constructions with the transposed figure. **THEOREM II.**

The length of any vertical line standing on the ground plane is to that of its picture as the height of the eye to the distance of the horizon-line from the picture of its foot.

Let BC (fig. 8) be the vertical line standing on B, and let EF be a vertical line through the eye. Make BD equal to EF, and draw DE, CE, BE. It is evident that DE will cut the horizon-line in some point d, CE will cut the picture-plane in c, and BE will cut it in b, and that bc will be the picture of BC, and is vertical, and that BC is to bc as BD to bd, or as EF to bd.

Cor. The picture of a vertical line is divided in the same ratio as the line itself. For BC : BM = bc : bm.

**PROB. 5. To put a vertical line of a given length in perspective standing on a given point of the picture.**

Through the given point b (fig. 9) of the picture draw SbA from the point of sight, and draw the vertical line AD, and make AE equal to the length or height of the given line. Join ES, and draw be parallel to AD, producing bc, when necessary, till it cut the horizontal line in d, and we have bc : bd = AE : AD, that is, as the length of the given line to the height of the eye, and bd is the distance of the horizon-line from the point b, which is the picture of the foot of the line. Therefore (Theor. 2) bc is the required picture of the vertical line.

This problem occurs frequently in views of architecture, and a compendious method of solving it would be peculiarly convenient. For this purpose, draw a vertical line XZ at the margin of the picture, or on a separate paper, and through any point V of the horizon-line draw VX. Set off XY, the height of the vertical line, and draw VY. Then from any points b, r, on which it is required to have the pictures of lines equal to XY, draw bs, rt, parallel to the horizon-line, and draw the verticals su, tv; these have the lengths required, which may be transferred to b and r. This, with the third general construction for the base points, will save all the confusion of lines which would arise from constructing each line apart.

**PROB. 6. To put any sloping line in perspective.**

From the extremities of this line, suppose perpendiculars meeting the ground-plane in two points, which we shall call the base points of the sloping line. Put these base points in perspective, and draw, by last problem, the perpendiculars from the extremities. Join these by a straight line. It will be the picture required.

**PROB. 7. To put a square in perspective, as seen by a person not standing right against the middle of either of its sides, but rather nearly even with one of its corners.**

In fig. 10, let ABCD be a true square, viewed by an observer, not standing at o, directly against the middle of its sides AD, but at O almost even with its corner D, and viewing the side AD under the angle AOD; the angle AOD (under which he would have seen AD from o) being 60 degrees.

Make AD in fig. 11 equal to AD in fig. 10, and draw SP and Oo parallel to AD. Then, in fig. 11, let O be the place of the observer's eye, and SO be perpendicular to SP; then S shall be the point of sight in the horizon SP.

Take SO in your compasses, and set that extent from S to P; then P shall be the true point of distance, taken according to the foregoing rules.

From A and D draw the straight lines AS and DS; draw also the straight line AP, intersecting DS in C.

Lastly, through the point of intersection C draw BC parallel to AD; and ABCD in fig. 11 will be a true perspective representation of the square ABCD in fig. 10. The point M is the centre of each square, and AMC and BMD are the diagonals.

**PROB. 8. To put a reticulated square in perspective, as seen by a person standing opposite to the middle of one of its sides.**

A reticulated square is one that is di- vided into several little squares, like net-work, as fig. 12, each side of which is divided into four equal parts, and the whole surface into four times four (or sixteen) equal squares.

Having divided this square into the given number of lesser squares, draw the two diagonals AxC and BxD.

Make AD in fig. 13 equal to AD in fig. 12, and divide it into four equal parts, as Ae, eg, gi, and iD.

Draw SP for the horizon, parallel to AD, and through the middle point g of AD, draw OS perpendicular to AD and SP. Make S the point of sight, and O the place of the observer's eye.

Take SP equal to SO, and P shall be the true point of distance. Draw AS and DS to the point of sight, and AP to the point of distance, intersecting DS in C; then draw BC parallel to AD, and the outlines of the reticulated square ABCD will be finished.

From the division points e, g, i, draw the straight lines ef, gh, ik, tending towards the point of sight S; and draw BD for one of the diagonals of the square, the other diagonal AC being already drawn.

Through the points r and s, where these diagonals cut ef and ik, draw lm parallel to AD. Through the centre point x, where the diagonals cut gh, draw no parallel to AD. Lastly, through the points v and w, where the diagonals cut ef and ik, draw pq parallel to AD; and the reticulated perspective square will be finished.

This square is truly represented, as if seen by an observer standing at O, and having his eye above the horizontal plane ABCD on which it is drawn, as if OS was the height of his eye above that plane; and the lines which form the small squares within it have the same letters of reference with those in fig. 12, which is drawn, as it would appear, to an eye placed perpendicularly above its centre x.

**Prob. 9. To put a circle in perspective.**

If a circle be viewed by an eye placed directly over its centre, it appears perfectly round; but if it be obliquely viewed, it appears of an elliptical shape. This is plain by looking at a common wine-glass set upright on a table.

Make a true reticulated square, as fig. 12, of the same diameter as you would have the circle; and setting one foot of your compasses in the centre x, describe as large a circle as the sides of the square will contain. Then, having put this reticulated square into perspective, as in fig. 13, observe through what points of the cross lines and diagonals of fig. 12 the circle passes; and through the like points in fig. 13 draw the ellipsis, which will be as true a perspective representation of the circle as the square in fig. 13 is of the square in fig. 12.

This is Mr Ferguson's rule for putting a circle in perspective; but the following rules by Wolf are perhaps more universal.

If the circle to be put in perspective be small, describe a square about it. Draw first the diagonals of the square, and then the diameters ha and de (fig. 14), cutting one another at right angles; draw the straight lines fg and bc parallel to the diameter de. Through b and f, and likewise e and g, draw straight lines meeting DE, the ground line of the picture, in the points 3 and 4. To the principal point V draw the straight lines IV, 3V, 4V, 2V, and to the points of distance L and K, 2L and 1K. Lastly, join the points of intersection a, b, d, f, h, g, e, c, by the arcs ab, bd, df, and abfghijeca will be the circle in perspective.

If the circle be large, so as to make the foregoing practice inconvenient, bisect the ground line AB, describing, from the point of bisection as a centre, the semicircle AGB (fig. 15), and from any number of points in the circumference C, F, G, H, I, &c., draw to the ground line the perpendiculars C1, F2, G3, H4, I5, &c. From the points A, 1, 2, 3, 4, 5, B, draw straight lines to the principal point or point of sight V, likewise straight lines from B and A to the points of distance L and K. Through the common intersections draw straight lines as in the preceding case; and you will have the points a, c, f, g, h, i, b, representatives of A, C, F, G, H, I, B. Then join the points a, c, f, &c., as formerly directed, and you have the perspective circle acfghibihjeca.

Hence it is apparent how we may put not only a circle, but also a pavement laid with stones of any form, in perspective. It is likewise apparent how useful the square is in perspective; for, as in the second case a true square was described round the circle to be put in perspective, and divided into several smaller squares, so in this third case, we make use of the semicircle only for the sake of brevity, instead of that square and circle.

**Prob. 10. To put a reticulated square in perspective, as seen by a person not standing right against the middle of either of its sides, but rather nearly even with one of its corners.**

In fig. 16, let O be the place of an observer, viewing the square ABCD almost even with its corner D. Draw at pleasure SP for the horizon, parallel to AD; and make SO perpendicular to SP; then S shall be the point of sight, and P the true point of distance, if SP be made equal to SO.

Draw AS and DS to the point of sight, and AP to the point of distance, intersecting DS in the point C; then draw BC parallel to AD, and the outlines of the perspective square will be finished. This done, draw the lines which form the lesser squares, as taught in Prob. 8, and the work will be completed. You may put a perspective circle in this square by the same rule as it was done in fig. 13.

**Prob. 11. To put a cube in perspective, as if viewed by a person standing almost even with one of its edges, and seeing three of its sides.**

In fig. 17, let AB be the breadth of either of the six equal square sides of the cube AG; O the place of the observer, almost even with the edge CD of the cube, S the point of sight, SP the horizon parallel to AD, and P the point of distance taken as before.

Make ABCD a true square; draw BS and CS to the point of sight, and BP to the point of distance, intersecting CS in G. Then draw FG parallel to BC, and the uppermost perspective square side BFGC of the cube will be finished.

Draw DS to the point of sight, and AP to the point of distance, intersecting DS in the point I; then draw GI parallel to CD; and, if the cube be an opaque one, as of wood or metal, all the outlines of it will be finished; and then it may be shaded as in the figure.

But if you want a perspective view of a transparent glass cube, all the sides of which will be seen, draw AH toward the point of sight, FH parallel to BA, and HI parallel to AD; then AHID will be the square base of the cube, perspectively parallel to the top BFGC; ABFH will be the square side of the cube parallel to CGID, and FGHI will be the square side parallel to ABCD.

As to the shading part of the work, it is such mere children's play, in comparison of drawing the lines which form the shape of any object, that no rules need be given for it. Let a person sit with his left side toward a window, and he knows full well that if any solid body be placed on a table before him, the light will fall on the left-hand side of the body, and the right-hand side will be in the shade.

**Prob. 12. To put any solid in perspective.**

Put the base of the solid, whatever it be, in perspective by the preceding rules. From each bounding point of the base, raise lines representing in perspective the altitude of the object; by joining these lines and shading the figure according to the directions in the preceding problem, you will have a scenographic representation of the object. This rule is general; but as its application to particular cases may not be apparent, it will be proper to give the following example of it.

**Prob. 13. To put a cube in perspective as seen from one of its angles.**

Since the base of a cube standing on a geometrical plane, and seen from one of its angles, is a square seen from one of its angles, draw first such a perspective square; then raise from any point of the ground-line DE (fig. 18) the perpendicular HI equal to the side of the square, and draw to any point V in the horizontal line HR the straight lines VI and VH. From the angles d, b, and c, draw the dotted lines d2 and c1 parallel to the ground line DE. Perpendicular to those dotted lines, and from the points 1 and 2, draw the straight lines L1 and M2. Lastly, since HI is the altitude of the intended cube in a, L1 in c and b, M2 in d, draw from the point a the straight line fa perpendicular to abe, and from the points b and e, bg and ec, perpendicular to bcl, and abde being according to rule, make af = HI, bg = ec = L1, and hd = M2. Then, if the points g, h, e, f be joined, the whole cube will be in perspective.

**Prob. 14. To put a square pyramid in perspective, as standing upright on its base, and viewed obliquely.**

In fig. 19, let AD be the breadth of either of the four sides of the pyramid ATCD at its base ABCD; and MT its perpendicular height. Let O be the place of the observer, S his point of sight, SE his horizon, parallel to AD and perpendicular to OS; and let the proper point of distance be taken in SE produced toward the left hand, as far from S as O is from S.

Draw AS and DS to the point of sight, and DL to the point of distance, intersecting AS in the point B. Then from B draw BC parallel to AD; and ABCD shall be the perspective square base of the pyramid.

Draw the diagonal AC, intersecting the other diagonal BD at M, and this point of intersection shall be the centre of the square base.

Draw MT perpendicular to AD, and of a length equal to the intended height of the pyramid; then draw the straight outlines AT, CT, and DT; and the outlines of the pyramid (as viewed from O) will be finished; which being done, the whole may be so shaded as to give it the appearance of a solid body.

If the observer had stood at o, he could have only seen the side ATD of the pyramid; and two is the greatest number of sides that he could see from any other place of the ground. But if he were at any height above the pyramid, and had his eye directly over its top, it would then appear as in fig. 20, and he would see all its four sides, E, F, G, H, with its top t just over the centre of its square base ABCD; which would be a true geometrical, and not a perspective square.

**Prob. 15. To put two equal squares in perspective, one of which shall be directly over the other, at any given distance from it, and both of them parallel to the plane of the horizon.**

In fig. 15, Plate CCCCVIII., let ABCD be a perspective square on a horizontal plane, drawn according to the foregoing rules, S being the point of sight, SP the horizon (parallel to AD), and P the point of distance.

Suppose AD, the breadth of this square, to be three feet; and that it is required to place just such another square, EFGH, directly above it, parallel to it, and two feet from it.

Make AE and DH perpendicular to AD, and two thirds of its length; draw EH, which will be equal and parallel to AD; then draw ES and HS to the point of sight S, and EP to the point of distance P, intersecting HS in the point G: this done, draw FG parallel to EH; and you will have two perspective squares, ABCD and EFGH, equal and parallel to one another, the latter directly above the former, and two feet distant from it, as was required.

By this method shelves may be drawn parallel to one another, at any distance from each other in proportion to their length.

**Prob. 16. To put a truncated pyramid in perspective.**

Let the pyramid to be put in perspective be quinquangular. If from each angle of the surface whence the top is cut off, a perpendicular be supposed to fall upon the base, these perpendiculars will mark the bounding points of a pentagon, of which the sides will be parallel to the sides of the base of the pyramid within which it is inscribed. Join these points, and the interior pentagon will be formed with its longest side parallel to the longest side of the base of the pyramid. From the ground line EH (fig. 21) raise the perpendicular HI, and make it equal to the altitude of the intended pyramid. To any point V draw the straight lines IV and HV, and by a process similar to that in Prob. 16, determine the scenographical altitudes \(a\), \(b\), \(c\), \(d\), \(e\). Connect the upper points \(f\), \(g\), \(h\), \(i\), \(k\) by straight lines; and draw \(lk\), \(fm\), \(gn\), and the perspective of the truncated pyramid will be completed.

Cor. If in a geometrical plane two concentric circles be described, a truncated cone may be put in perspective in the same manner as a truncated pyramid.

**Prob. 17. To put in perspective a hollow prism lying on one of its sides.**

Let ABDEC (fig. 22) be a section of such a prism. Draw HI parallel to AB, and distant from it the breadth of the side on which the prism rests; and from each angle internal and external of the prism let fall perpendiculars to HI. The parallelogram will be thus divided by the ichnographic process below the ground-line, so as that the side AB of the real prism will be parallel to the corresponding side of the scenographic view of it. To determine the altitude of the internal and external angles: from H (fig. 23) raise HI perpendicular to the ground-line, and on it mark off the true altitudes HI, H2, H3, H4, and H5. Then, if from any point V in the horizon be drawn the straight lines VH, V1, V2, V3, V4, V5, or V1; by a process similar to that of the preceding problem, will be determined the height of the internal angles, viz. \(1 = aa\), \(2 = bb\), \(4 = dd\); and of the external angles, \(3 = cc\) and \(5 = ee\); and when these angles are formed and put in their proper places, the scenograph of the prism is complete.

**Prob. 18. To put a square table in perspective standing on four upright square legs, of any given length with respect to the breadth of the table.**

In fig. 1, Plate CCCCVIII., let ABCD be the square part of the floor on which the table is to stand, and EFGH the surface of the square table, parallel to the floor.

Suppose the table to be three feet in breadth, and its height from the floor to be two feet; then two thirds of AD or EH will be the length of the legs \(i\) and \(k\); the other two (\(l\) and \(m\)) being of the same length in perspective.

Having drawn the two equal and parallel squares ABCD and EFGH, as shown in Prob. 15, let the legs be square in form, and fixed into the table at a distance from its edges equal to their thickness. Take Aa and Dd equal to the intended thickness of the legs, and ab and dc also equal thereto. Draw the diagonals AC and BD, and draw straight lines from the points \(a\), \(b\), \(c\), \(d\), towards the point of sight S, and terminating at the side BC. Then, through the points where these lines cut the diagonals, draw the straight lines \(n\) and \(o\), \(p\) and \(q\), parallel to AD; and you will have formed four perspective squares (like ABCD in fig. 19) for the bases of the four legs of the table: and then it is easy to draw the four upright legs by parallel lines, all perpendicular to AD, and to shade them as in the figure.

To represent the intended thickness of the table-board, draw eh parallel to EH, and HG toward the point of sight S: then shade the spaces between these lines, and the perspective figure of the table will be finished.

**Prob. 19. To put five square pyramids in perspective, standing upright on a square pavement composed of the surfaces of eighty-one cubes.**

In fig. 1, Plate CCCCVI., let ABCD be a perspective square drawn according to the foregoing rules, S the point of sight, P the point of distance in the horizon PS, and AC and BD the two diagonals of the square.

Divide the side AD into nine equal parts (because 9 times 9 is 81) as Aa, ab, bc, &c., and from these points of division \(a\), \(b\), \(c\), \(d\), &c., draw lines toward the point of sight S, terminating at the farthermost side BC of the square. Then, through the points where these lines cut the diagonals, draw straight lines parallel to AD, and the perspective square ABCD will be subdivided into eighty-one lesser squares, representing the upper surfaces of eighty-one cubes, laid close to one another's sides in a square form.

Draw AK and DL, each equal to Aa, and perpendicular to AD; and draw LN toward the point of sight S: then draw KL parallel to AD, and its distance from AD will be equal to Aa. This done, draw al, bm, cn, do, ep, fq, gr, and hs, all parallel to AK; and the space ADKL will be subdivided into nine equal squares, which are the outer upright surfaces of the nine cubes in the side AD of the square ABCD.

From the points where the lines which are parallel to AD in this square meet the side CD thereof, draw short lines to LN, all parallel to DL, and they will divide that side into the outer upright surfaces of the nine cubes which compose it; and then the outsides of all the cubes that can be visible to an observer, placed at a proper distance from the corner D of the square, will be finished.

As taught in Prob. 14, place the pyramid AE upright on its square base Atea, making it as high as you please; and the pyramid DH on its square base hawD, of equal height with AE.

Draw EH from the top of one of these pyramids to the top of the other; and EH will be parallel to AD. Draw ES and HS to the point of sight S, and HP to the point of distance P, intersecting ES in F.

From the point F draw FG parallel to EH; then draw EG, and you will have a perspective square EFGH (parallel to ABCD) with its two diagonals EG and FH intersecting one another in the centre of the square at I. The four corners of this square, E, F, G, H, give the perspective heights of the four pyramids AE, BF, CG, and DH; and the intersection I of the diagonals gives the height of the pyramid MI, the centre of whose base is the centre of the perspective square ABCD.

Lastly, place the three pyramids BF, CG, MI, upright on their perspective bases at B, C, and M; and the required perspective representation will be finished as in the figure.

**Prob. 20. To put upright pyramids in perspective, on the sides of an oblong square or parallelogram, so that their distances from one another shall be equal to the breadth of the parallelogram.**

In most of the foregoing operations we have considered the observer to be so placed as to have an oblique view of the perspective objects: in this we shall suppose him to have a direct view of fig. 2, Plate CCCCVI., that is, standing right against the middle of the end AD which is nearest to his eye, and viewing AD under an angle of 60 degrees.

Having cut AD in the middle, by the perpendicular line SS, take S therein at pleasure for the point of sight, and draw ES for the horizon, parallel to AD. Here SS must be supposed to be produced downward, below the limits of the plate, to the place of the observer; and SE to be produced towards the left hand beyond E, far enough to take a proper point of distance therein, according to the foregoing rules.

Take Ad at pleasure, and Dg equal to Ad, for the breadths of the square bases of the two pyramids AE and DF next the eye: then draw AS and dS, and likewise DS and gS, to the point of sight S; and DG on to the point of distance, intersecting AS in G: then from G draw GI parallel to AD; you will have the first perspective square AGID of the parallelogram ABCD.

From I draw IH to (or toward) the point of distance, intersecting AS in H: then, from H draw HK parallel to AD, and you will have the second perspective square GHKI of the parallelogram. Go on in this manner till you have drawn as many perspective squares up toward S as you please.

Through the point e, where DG intersects gS, draw bf parallel to AD; and you will have formed the two perspective square bases Abed and efDg of the two pyramids at A and D.

From the point f (the upper outward corner of efDg) draw fh toward the point of distance, till it meets AS in h; then from this point of meeting draw hm parallel to GI, and you will have formed the two perspective squares Ghih and ImIn, for the square bases of the two pyramids at G and I.

Proceed in the same manner to find the bases of all the other pyramids, at the corners of the rest of the perspective squares in the parallelogram ABCD, as shown by the figure. Then,

Having placed the first two pyramids at A and D upright on their square bases, as shown in Prob. 9, and made them of any equal heights at pleasure, draw ES and FS from the tops of these pyramids to the point of sight S; place all the rest of the pyramids upright on their respective bases, making their tops touch the straight lines ES and FS; and all the work, except the shading part, will be finished.

**Prob. 21. To put a square pyramid of equal-sized cubes in perspective.**

Fig. 3, Plate CCCCVI., represents a pyramid of this kind, consisting, as it were, of square tables of cubes, one table above another; 81 in the lowest, 49 in the next, 25 in the third, 9 in the fourth, and 1 in the fifth or uppermost. These are the square numbers of 9, 7, 5, 3, and 1.

If the artist is already master of all the preceding operations, he will find less difficulty in this than in attending to the following description of it; for it cannot be described in a few words, but may be executed in a very short time.

In fig. 4, Plate CCCCVI., having drawn PS for the horizon, and taken S for the point of sight therein (the observer being at O), draw AD parallel to PS for the side (next the eye) of the first or lowermost table of cubes. Draw AS and DS to the point of sight S, and DP to the point of distance P, intersecting AS in the point B. Then, from B draw BC parallel to AD, and you will have the surface ABCD of the first table.

Divide AD into nine equal parts, as Aa, ab, bc, cd, &c., then make AK and DL equal to Aa, and perpendicular to AD. Draw KL parallel to AD, and from the points of equal division at a, b, c, &c. draw lines to KL, all parallel to AK. Then draw AS to the point of sight S, and from the division points a, b, c, &c. draw lines with a black-lead pencil, all tending towards the point of sight, till they meet the diagonal BD of the square.

From these points of meeting draw black-lead lines to DC, all parallel to AD; then draw the parts of these lines with black ink which are marked 1, 2, 3, 4, &c. between hE and DC.

Having drawn the first of these lines bg with black ink, draw the parts ai, bh, cl, &c. (of the former lines which met the diagonal BD) with black ink also; and rub out the rest of the black-lead lines, which would otherwise confuse the following part of the work. Then draw LF toward the point of sight S; and from the points where the lines l, 2, 3, 4, &c. meet the line DC, draw lines down to LF, all parallel to DL; and all the visible lines between the cubes in the first table will be finished.

Make iG equal and perpendicular to βi, and qM equal and parallel to iG; then draw GM, which will be equal and parallel to iq. From the points k, l, m, n, &c. draw kn, lo, mp, &c. all parallel to iG, and the outsides of the seven cubes in the side Gq of the second table will be finished.

Draw GS and MS to the point of sight S, and MP to the point of distance P, intersecting GS in H; then, from the point of intersection H, draw HI parallel to AD; and you will have the surface GHIM of the second table of cubes.

From the points n, o, p, q, &c. draw black-lead lines toward the point of sight S, till they meet the diagonal MH of the perspective square surface GHIM; and draw SM, with black ink, toward the point of sight.

From those points where the lines drawn from n, o, p, q, &c. meet the diagonal MH, draw black-lead lines to MI, all parallel to AD; only draw the whole first line γl with black ink, and the parts 2, 3, 4, &c. and nt, ou, pe, &c. of the other lines between γN and MI, and GM and γl, with the same; and rub out all the rest of the black-lead lines, to avoid further confusion. Then from the points where the short lines 1, 2, 3, &c. meet the line MI, draw lines down to qL, all parallel to Mg, and the outer surfaces of the seven cubes in the side ME will be finished; and all these last lines will meet the former parallels 2, 3, 4, &c. in the line qE.

Make iO equal and perpendicular to γl, and yP equal and parallel to iO; then draw OP, which will be equal and parallel to ty. This done, draw OS and PS to the point of sight S, and PP to the point of distance P in the horizon. Lastly, from the point Q, where PP intersects OS, draw QR parallel to OP; and you will have the outlines OQRP of the surface of the third perspective table of cubes.

From the points u, v, w, x, draw upright lines to OP, all parallel to tO, and you will have the outer surfaces of the five cubes in the sides Oy of this third table.

From the points where these upright lines meet OP, draw lines toward the point of sight S, till they meet the diagonal PQ; and from these points of meeting draw lines to PR, all parallel to OP, making the parts 2, 3, 4, 5, of these lines with black ink which lie between ZY and PR. Then, from the points where these lines meet PR, draw lines down to yN, which will bound the outer surfaces of the five cubes in the side PN of the third table.

Draw the line δl with black ink; and, at a fourth part of its length between δ and Z, draw an upright line to S, equal in length to that fourth part, and another equal and parallel thereto from Z to V; then draw SV parallel to δZ, and draw the two upright and equidistant lines between δZ and SV, and you will have the outer surfaces of the three cubes in the side SZ of the fourth table.

Draw SS and VS to the point of sight S in the horizon, and VP to the point of distance therein, intersecting SS in T; then draw TU parallel to SV, and you have STUV, the surface of the fourth table, which being reticulated or divided into nine perspective small squares, and the uppermost cube W placed on the middlemost of the squares, all the outlines will be finished; and when the whole is properly shaded, as in Plate CCCCVI. fig. 3, the work will be done.

**Prob. 22. To represent a double cross in perspective.**

In fig. 5, Plate CCCCVI., let ABCD and EFGH be the two perspective squares, equal and parallel to one another, the uppermost directly above the lowermost, drawn by the rules already laid down, and as far asunder as is equal to the given height of the upright part of the cross; S being the point of sight, and P the point of distance, in the horizon PS taken parallel to AD.

Draw AE, DH, and CG; then AEHD and DHGC shall be the two visible sides of the upright part of the cross; of which the length AE is here made equal to three times the breadth EH.

Divide DH into three equal parts, HI, IK, and KD. Through these points of division, at I and K, draw MO and PR parallel to AD; and make the parts MN, IO, PQ, KR, each equal to HI; then draw MP and OR parallel to DH.

From M and O draw MS and OS to the point of sight S; and from the point of distance P draw PN, cutting MS in T; from T draw TU parallel to MO, and meeting OS in U; and you will have the uppermost surface MTUO of one of the cross pieces of the figure. From R draw RS to the point of sight S; and from U draw UV parallel to OR; and OUVR shall be the perspective square end next the eye of that cross part.

Draw PMX (as long as you please) from the point of distance P, through the corner M; lay a ruler to N and S, and draw XN from the line PX; then lay the ruler to I and S, and draw YZS. Draw XY parallel to MO, and make XW and YB equal and perpendicular to XY; then draw WB parallel to XY, and WXYYB shall be the square visible end of the other cross part of the figure.

Draw BK toward the point of sight S; and from U draw UP to the point of distance P, intersecting YS in Z; then, from the intersection Z draw Za parallel to MO, and Zb parallel to HD, and the whole delineation will be finished.

This done, shade the whole, as in fig 6, Plate CCCCVI., and you will have a true perspective representation of a double cross.

**Prob. 23. To put three rows of upright square objects in perspective, equal in size, and at equal distances from each other, on an oblong square plane, the breadth of which shall be of any assigned proportion to the length thereof.**

Fig. 7, Plate CCCCVI., is a perspective representation of an oblong square plane, three times as long as it is broad, having a row of nine upright square objects on each side, and one of the same number in the middle; all equally high, and at equal distances from one another, both longwise and crosswise, on the same plane.

In fig. 8, Plate CCCCVI., PS is the horizon, S the point of sight, P the point of distance, and AD (parallel to PS) the breadth of the plane.

Draw AS, NS, and DS, to the point of sight S; the point N being in the middle of the line AD; and draw DP to the point of distance P, intersecting at AS in the point B; then, from B draw BC parallel to AD, and you have the perspective square ABCD.

Through the point i, where DB intersects NS, draw ae parallel to AD; and you will have subdivided the perspective square ABCD into four lesser squares, as aiN, NieD, aBai, and ikCe.

From the point C (at the top of the perspective square ABCD) draw CP to the point of distance P, intersecting AS in E; then from the point E draw EF parallel to AD; and you will have the second perspective square BEFC.

Through the point l, where CE intersects NS, draw bf parallel to AD; and you will have subdivided the square BEFC into the four squares Blk, klfC, bEmI, and ImEf.

From the point F (at the top of the perspective square BEFC) draw FP to the point of distance P, intersecting AS in I; then from the point I draw IK parallel to AD; and you will have the third perspective square EIKF.

Through the point n, where FI intersects NS, draw eg parallel to AD; and you will have subdivided the square EIKF into four lesser squares, Ecnm, mngF, elon, and nokg.

From the point K (at the top of the third perspective square EIKF) draw KP to the point of distance P, intersecting AS in L; then from the point L draw LM parallel to AD; and you will have the fourth perspective square ILMK.

Through the point p, where KL intersects NS, draw dh parallel to AD; and you will have subdivided the square ILMK into the four lesser squares Idpo, ophK, dLqp, and pqMh.

Thus we have formed an oblong square ALMD, whose perspective length is equal to four times its breadth, and it contains sixteen equal perspective squares. If greater length was still wanted, we might proceed farther on toward S.

Take A3, equal to the intended breadth of the side of the upright square object AQ (all the other sides being of the same breadth), and AO for the intended height. Draw O18 parallel to AD, and make D8 and 47 equal to A3; then draw 3S, 4S, 7S, and 8S to the point of sight S; and among them we shall have the perspective square bases of all the twenty-seven upright objects on the plane.

Through the point 9, where DB intersects 8S, draw 110 parallel to AD; and you have the three perspective bases A1 2 3, 4 5 6 7, 8 9 10 D, of the three upright square objects at A, N, and D.

Through the point 21, where eb intersects 8S, draw 14, 11 parallel to AD; and you will have the three perspective squares a14 15 16 17 18 19 20, and 21 11 e22, for the bases of the second cross row of objects; namely, the next beyond the first three at A, N, and D.

Through the point w, where CE intersects 8S, draw a line parallel to BC; and you will have three perspective squares; at B, h, and C, for the bases of the third row of objects; one of which is set up at B.

Through the point x, where fc intersects S, draw a line parallel to bf; and you will have three perspective squares, at b, l, and x, for the bases of the fourth cross row of objects.

Go on in this manner, as you see in the figure, to find the rest of the square bases, up to LM; and you will have 27 upon the whole oblong square plane, on which you are to place the like number of objects, as in fig. 7, Plate CCCCVI.

Having assumed AO for the perspective height of the three objects at A, N, and D (Plate CCCCVI, fig. 8) next the observer's eye, and drawn O 18 parallel to AD, in order to make the objects at N and D of the same height as that at O; and having drawn the upright lines 4-15, 7 W, 8 X, and D 22, for the heights at N and D; draw OS and RS, 15 S and WS, XS and 22 S, all to the point of sight S; and these lines will determine the perspectively equal heights of all the rest of the upright objects, as shown by the two placed at a and B.

To draw the square tops of these objects, equal and parallel to their bases, we need only give one example, which will serve for all.

Draw 3 R and 2 Q parallel to AO, and up to the line RS; then draw PQ parallel to OR, and OPQR shall be the top of the object at A, equal and parallel to its square base A 1 2 3. In the same easy way the tops of all the other objects are formed.

When all the rest of the objects are delineated, shade them properly, and the whole perspective scheme will have the appearance of fig. 7, Plate CCCCVI.

**Prob. 24. To put a square box in perspective, containing a given number of lesser square boxes of a depth equal to their width.**

Let the given number of little square boxes or cells be sixteen; then four of them make the length of each side of the four outer sides ab, bc, cd, da, as in Plate CCCCVII.fig. 9, and the depth ef is equal to the width ae. Whoever can draw the reticulated square by the rules laid down towards the beginning of this article, will be at no loss about putting this perspective scheme in practice.

**Prob. 25. To put stairs with equal and parallel steps in perspective.**

In fig. 10, Plate CCCCVII., let ab be the given breadth of each step, and ai the height thereof. Make be, cd, de, &c. each equal to ab, and draw all the upright lines ai, bl, cn, dp, &c. perpendicular to ah (to which the horizon ssS is parallel); and from the points i, l, n, p, r, &c. draw the equidistant lines IB, JC, ND, &c. parallel to ah; these distances being equal to that of IB from ah.

Draw xi touching all the corner-points l, n, p, r, t, v; and draw 2 16 parallel to xi, as far from it as you want the length of the steps to be.

Toward the point of sight S draw the lines a1, i2, k3, l4, &c. and draw 16 15, 14 13, 12 11, 10 9, 8 7, 6 5, 4 3, and 2 1, all parallel to Ah, and meeting the lines w 15, u 13, s 11, &c. in the points 15, 13, 11, 9, 7, 5, 3, and 1: then from these points draw 15 14, 13 12, 11 10, 9 8, 7 6, 5 4, and 3 2, all parallel to ha; and the outlines of the steps will be finished. From the point 16 draw 16 A parallel to ha, and Az 16 will be the part of the flat at the top of the uppermost step. This done, shade the work as in fig. 11, Plate CCCCVII., and the whole will be finished.

**Prob. 26. To put stairs with flats and opening in perspective, standing on a horizontal pavement of squares.**

In fig. 12, Plate CCCCVII., having made S the point of sight, and drawn a reticulated pavement AB with black-lead lines, which may be rubbed out again; at any distance from the side AB of the pavement which is nearest to the eye, and at any point where you choose to begin the stair at that distance, as a, draw Ga parallel to BA, and take ab at pleasure for the height of each step.

Take ab in your compasses, and set that extent as many times upward from F to E as is equal to the first required number of steps O, N, M, L, K; and from these points of division in EF draw 1b, 2d, 3f, 4h, and Ek, all equidistant from one another, and parallel to Fa; then draw the equidistant upright lines ab, td, nf, vh, uk, and Im, all perpendicular to Fa; then draw mb, touching the outer corners of these steps at m, h, f, d, and b; and draw ns parallel to mb, as far from it as you want the length of the steps K, L, M, N, O to be.

Towards the point of sight S draw mn, l5, ka, i6, hp, fq, dr, and bs. Then (parallel to the bottom-line BA) through the points n, o, p, q, r, s, draw n8; 5, 14; 6, 15; 7, 16; 1, 17; and 2s; which done, draw n5 and o6 parallel to lm, and the outlines of the steps K, L, M, N, O will be finished.

At equal distances with that between the lines marked 8 and 14, draw the parallel lines above, marked 9, 10, 11, 12, and 13; and draw perpendicular lines upwards from the points n, o, p, q, r, s, as in the figure.

Make Hm equal to the intended breadth of the flat above the square opening at the left hand, and draw HW toward the point of sight S, equal to the intended length of the flat; then draw WP parallel to Hm, and the outlines of the flat will be finished.

Take the width of the opening at pleasure, as from F to C, and draw CD equal and parallel to FE. Draw GH parallel to CD, and the short lines marked 33, 34, &c. just even with the parallel lines 1, 2, &c. From the points where these short lines meet CD, draw lines toward the point of sight S till they meet DE; then from the points where the lines 38, 39, 40, &c. of the pavement meet Cg, draw upright lines parallel to CD; and the lines which form the opening will be finished.

The steps P, Q, R, S, T, and the flat U above the arch V, are done in the same manner with those in fig. 10, as taught in Prob. 25, and the equidistant parallel lines marked 18, 19, &c. are directly even with those on the left-hand side of the arch V, and the upright lines on the right-hand side are equidistant with those on the left.

From the points where the lines 18, 19, 20, &c. meet the right-hand side of the arch, draw lines toward the point of sight S; and from the points where the pavement lines 29, 30, 31, 32, meet the line drawn from A towards the point of sight, draw upright lines toward the top of the arch.

Having done the top of the arch, as in the figure, and the few steps to the right hand thereof, shade the whole, as in fig. 13, Plate CCCCVII., and the work will be finished.

**Prob. 27. To put upright conical objects in perspective, as if standing on the sides of an oblong square, at distances from one another equal to the breadth of the oblong.**

In fig. 14, Plate CCCCVII., the bases of the upright cones are perspective circles inscribed in squares of the same diameter; and the cones are set upright on their bases by the same rules as are given for pyramids, which we need not repeat here.

In most of the foregoing operations we have considered the observer's eye to be above the level of the tops of all the objects, as if he viewed them when standing on high ground. In this figure, and in fig. 17 and fig. 18, Plate CCCCVIII., we shall suppose him to be standing on low ground, and the tops of the objects to be above the level of his eye.

In fig. 14, Plate CCCCVII., let AD be the perspective breadth of the oblong square ABCD; and let Aa and Dd (equal to Aa) be taken for the diameters of the circular bases of the two cones next the eye, whose intended equal heights shall be AE and DF.

Having made S the point of sight in the horizon parallel to AD, and found the proper point of distance therein, draw AS and dS to contain the bases of the cones on the left-hand side, and DS and dS for those on the right.

Having made the two first cones at A and D of equal height at pleasure, draw ES and FS from their tops to the point of sight, for limiting the perspective heights of all the rest of the cones. Then divide the parallelogram ABCD into as many equal perspective squares as you please; find the bases of the cones at the corners of these squares, and make the cones thereon, as in the figure.

If you would represent a ceiling equal and parallel to ABCD, supported on the tops of these cones, draw EF, then EFGH shall be the ceiling; and by drawing ef parallel to EF, you will have the thickness of the floor-boards and beams, which may be what you please.

This shows how any number of equidistant pillars may be drawn of equal heights to support the ceiling of a long room, and how the walls of such a room may be represented in perspective at the backs of these pillars. It also shows how a street of houses may be drawn in perspective.

**Prob. 28. To put a square hollow in perspective, the depth of which shall bear any assigned proportion to its width.**

Fig. 17, Plate CCCCVIII., is the representation of a square hollow, of which the depth AG is equal to three times its width AD; and S is the point of sight over which the observer's eye is supposed to be placed, looking perpendicularly down into it, but not directly over the middle.

Draw AS and DS to the point of sight S; make ST the horizon parallel to AD, and produce it to such a length beyond T that you may find a point of distance therein not nearer S than if AD were seen under an angle of sixty degrees.

Draw DU to the point of distance, intersecting AS in B; then from the point B draw BC parallel to AD; and you will have the first perspective square ABCD, equal to a third part of the intended depth.

Draw CV to the point of distance, intersecting AS in E; then from the point E draw EF parallel to AD; and you will have the second perspective square BEFC, which, added to the former one, makes two thirds of the intended depth.

Draw FW to the point of distance, intersecting AS in G; then from the point G draw GH parallel to AD; and you will have the third perspective square EGHF, which, with the former two, makes the whole depth AGHD three times as great as the width AD, in a perspective view.

Divide AD into any number of equal parts, as suppose S; and from the division points a, b, c, d, &c. draw lines toward the point of sight S, and ending at GH; then through the points where the diagonals BD, EC, GF, cut these lines, draw lines parallel to AD; and you will have the parallelogram AGHD reticulated, or divided into 192 small and equal perspective squares.

Make AI and DM equal and perpendicular to AD; then draw IM, which will be equal and parallel to AD; and draw IS and MS to the point of sight S.

Divide AI, IM, and MD into the same number of equal parts as AD is divided; and from these points of division draw lines toward the point of sight S, ending respectively at GK, KL, and LH.

From those points where the lines parallel to AD meet AG and DH, draw upright lines parallel to AI and DM; and from the points where these lines meet IK and LM draw lines parallel to IM; then shade the work, as in the figure.

**Prob. 29. To represent a semicircular arch in perspective, as if it were standing on two upright walls, equal in height to the height of the observer's eye.**

After having gone through the preceding operation, this will be more easy by a bare view of fig. 18, Plate CCCCVIII., than it could be made by any description; the method being so much like that of drawing and shading the square hollow. We need only mention, that aTbEA and DFcdl are the upright walls on which the semicircular arch is built; that S is the point of sight in the horizon Tt, taken in the centre of the arch; and d in fig. 17 is the point of distance; and that the two perspective squares ABCD and BEFC make the parallelogram AEFD of a length equal to twice its breadth AD.

**Prob. 30. To represent a square in perspective, as viewed by an observer standing directly even with one of its corners.**

In fig. 19, Plate CCCCVIII., let A9BC be a true square, viewed by an observer standing at some distance from the corner C, and just even with the diagonal C9.

Let pSP be the horizon, parallel to the diagonal AB; and S the point of sight, even with the diagonal C9. Here it will be proper to have two points of distance, p and P, equidistant from the point of sight S.

Draw the straight line l 17 parallel to AB, and draw A8 and B10 parallel to CS. Take the distance between 8 and 9 in your compasses, and set it off all the way in equal parts from 8 to 1, and from 10 to 17. The line l 17 should be produced a good way farther both to right and left hand from 9, and divided all the way in the same manner.

From these points of equal division, 8, 9, 10, &c. draw lines to the point of sight S, and also to the two points of distance p and P, as in the figure.

Now it is plain that aeb9 is the perspective representation of A9BC, viewed by an observer even with the corner C and diagonal C9. But if there are other such squares lying even with this, and having the same position with respect to the line l 17, it is evident that the observer, who stands directly even with the corner C of the first square, will not be even with the like corners G and K of the others; but will have an oblique view of them, over the sides FG and IK, which are nearest his eye: and their perspective representations will be egf6 and hki3, drawn among the lines in the figure; of which the spaces taken up by each side lie between three of the lines drawn toward the point of distance p, and three drawn to the other point of distance P.

**Prob. 31. To represent a common chair, in an oblique perspective view.**

The original lines to the point of sight S, and points of distance p and P (Plate CCCCVIII., fig. 19), being drawn as in the preceding operation, choose any part of the plane, as Iani13, on which you would have the chair L to stand. There are just as many lines (namely two) between l and m or 13 and n, drawn toward the point of distance p, at the left hand, as between l and 13, or m and n, drawn to the point of distance P on the right: so that lm, mn, n13, and 13l, form a perspective square.

From the four corners l, m, n, 13, of this square raise the four legs of the chair to the perspective perpendicular height you would have them; then make the seat of the chair a square equal and parallel to lam13, as taught in Prob. 15, which will make the two sides of the seat in the direction of the lines drawn toward the point of distance p, and the fore and back part of the seat in direction of the lines drawn to the other point of distance P. This done, draw the back of the chair leaning a little backward, and the cross bars therein tending toward the point of distance P. Then shade the work as in the figure, and the perspective chair will be finished.

**Prob. 32. To present an oblong square table in an oblique perspective view.**

In fig. 19, Plate CCCCVIII., M is an oblong square table, as seen by an observer standing directly even with C9 (see Prob. 30), the side next the eye being perspective-ly parallel to the side ae of the square abe9. The foremen-tioned lines drawn from the line 1 17 to the two points of distance p and P, form equal perspective squares on the ground plane.

Choose any part of this plane of squares for the feet of the table to stand upon; as at p, q, r, and s, in direction of the lines op and rs for the two long sides, and ts and qr for the two ends; and you will have the oblong square or parallelogram qrst for the part of the floor or ground-plane whereon the table is to stand: and the breadth of this plane is here taken in proportion to the length as 6 to 10; so that, if the length of the table be ten feet, its breadth will be six.

On the four little perspective squares at q, r, s, and t, place the four upright legs of the table, of what height you please, so that the height of the two next the eye, at o and p, shall be terminated by a straight line drawn to the point of distance P. This done, make the leaf M of the table an oblong square, perspectively equal and parallel to the oblong square qrst on which the feet of the table stands. Then shade the whole, as in the figure, and the work will be finished.

If the line 1 17 were prolonged to the right and left hand, and equally divided throughout (as it is from 1 to 17), and if the lines which are drawn from p and P to the right and left hand sides of the plate were prolonged till they came to the extended line 1 17, they would meet it in the equal points of division. In forming large plans of this sort, the ends of slips of paper may be pasted to the right and left edges of the sheet on which the plan is to be formed.

**Of the Anamorphosis, or reformation of distorted images.**

By this means, pictures that are so mishapen as to exhibit no regular appearance of any thing to the naked eye, shall, when viewed by reflection, present a regular and beautiful image. The inventor of this ingenious device is not known. Simon Stevinus, who was the first that wrote upon it, does not inform us from whom he learned it. The principles of it are laid down by S. Vanzelard, in his Perspective Conique et Cylindrique; and Gaspar Schott professes to copy Marius Bettinus in his description of this piece of artificial magic.

It will be sufficient for our purpose to copy one of the simplest figures of this writer, as by this means the mystery of this art will be sufficiently unfolded. Upon the cylinder of paper, or pasteboard, ABCD, fig. 16, Plate CCCCVIII., draw whatever is intended to be exhibited, as the letters IHS. Then with a needle make perforations along the whole outline; and placing a candle G behind this cylinder, mark upon the ground-plane the shadow of them, which will be distorted more or less, according to the position of the candle or the plane, &c. This being done, let the picture be an exact copy of this distorted image, let a metallic speculum be substituted in the place of the cylinder, and let the eye of the spectator have the same position before the cylinder that the candle had behind it. Then looking upon the speculum, he will see the distorted image restored to its proper shape. The reformation of the image, he says, will not easily be made exact in this method, but it will be sufficiently so to answer the purpose.

Other methods, more exact and geometrical than this, were found out afterwards: so that these pictures could be drawn by certain rules, without the use of a candle. Schott quotes one of these methods from Bettinus, another from Herigonius, and another from Kircher, which may be seen in his Magia, vol. i. p. 162, &c. He also gives an account of the methods of reforming pictures by speculums of conical and other figures.

Instead of copying any of these methods from Schott or Bettinus, we shall present our readers with that which Dr Smith hath given us in his Optics, vol. i. p. 250, as, no doubt, the best, and from which any person may easily make a drawing of this kind. The same description answers to two mirrors, one of which, fig. 24, is convex, and the other, fig. 25, is concave.

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**Fig. 24.**


**Fig. 25.**

 taking the eye, raised above R, to the given height RO, somewhat greater than that of the cylinder, for a luminous point, describe the shadow acdf (of a square, fig. 24, or parallelogram standing upright upon ae as a base, and containing the picture required) anywhere behind the arch lap. Let the lines drawn from R to the extremities and divisions of the base a, b, c, d, e, cut the remotest part of the shadow in the points f, g, h, i, k; and the arch of the base in l, m, n, o, p; from which points draw the lines laf, mbg, ncH, oDI, pEK, as if they were rays of light that came from the focus R, and were reflected from the base lap; so that each couple, la, lb, produced, may cut off equal segments from the circle. Lastly, transfer the lines laf, mbg, &c., and all their parts in the same order, upon the respective lines laf, mbg, &c., and having drawn regular curves, by estimation, through the points A, B, C, D, E, through F, G, H, I, K, and through every intermediate order of points; the figure ACEKHF, so divided, will be the deformed copy of the square, drawn and divided upon the original picture, and will appear similar to it, when seen in the polished cylinder, placed upon the base lap, by the eye in its given place O.

The practical methods of drawing these images seem to have been carried to the greatest perfection by J. Leopold, who, in the Acta Lipsiensia for the year 1712, has described two machines, one for the images to be viewed with a cylindrical and the other with a conical mirror. The person possessed of this instrument has nothing to do but to take any print he pleases, and while he goes over the outlines of it with one pen, another traces the anamorphosis.

By methods of this kind, groves of trees may be cut, so as to represent the appearance of men, horses, and other objects, from some one point of view, which are not at all discernible in any other. This might easily be effected by one person placing himself in any particular situation, and giving directions to other persons what trees to lop, and in what manner. In the same method it has been contrived, that buildings of circular and other forms, and also whole groups of buildings consisting of walls at different distances, and with different positions to one another, should be painted so as to exhibit the exact representation of particular objects, which could only be perceived in one situation. Bettinus has illustrated this method by drawings in his Apiaria.

It may appear a bold assertion to say, that the very short sketch now given of the art of perspective is a sufficient foundation for the whole practice, and includes all the expeditious rules peculiar to the problems which most generally occur. It is, however, true; and the intelligent reader will perceive, that the two theorems on which the whole rests include every possible case, and apply with equal facility to pictures and originals in any position, although the examples are selected of perpendicular pictures, and of originals referred to horizontal planes, as being the most frequent. The scientific foundation being so simple, the structure need not be complex, nor swell into such volumes as have been published on the subject; which by their size deter from the perusal, and give the simple art the appearance of intricate mystery, whilst, by their prices, they defeat the design of their authors to promote the dissemination of knowledge amongst practitioners. The treatises on perspective acquire their bulk by long and tedious discourses, minute explanations of common things, or by great numbers of examples; which indeed render some of these books valuable from the variety of curious cuts, but do not at all instruct the reader by any improvements made in the art itself. For it is evident that most of those who have treated this subject have been more conversant in the practice of designing than in the principles of geometry; and therefore when, in their practice, the cases which have occurred put them upon trying particular expedients, they have thought them worth communicating to the public as improvements in the art; and each author, fond of his own little expedient, which by a scientific person would have been known as an easy corollary from the general theorem, has made it the principle of a practical system, thus narrowing instead of enlarging the knowledge of the art; and the practitioner, tired of the bulk of the volume, in which a single maxim is tediously spread out, and the principle on which it is founded kept out of his sight, contents himself with a remembrance of a maxim perhaps not understood, and keeps it slightly in his eye to avoid gross errors. We can appeal to the whole body of painters and draughtsmen for the truth of this assertion; and it must not be considered as an imputation of remissness or negligence on their part, but as a necessary consequence of the ignorance of the authors from whom they have taken their information. This is a strong statement, but it is not the less just. Several mathematicians of eminence have written on perspective, treating it as the subject of pure geometry, as it really is; and the performances of Dr Brooke Taylor, Gravesande, Wolf, De la Caille, and Emerson, are truly valuable, by presenting the art in all its perspicuous simplicity and universality. The works of Taylor and Emerson are more valuable, on account of the very ingenious and expeditious constructions which they have given, suited to every possible case. The merit of the former has been universally acknowledged by all the British writers on the subject, who never fail to declare that their own works are composed on the principle of Dr Brooke Taylor; but any man of science will see that these authors have either not understood them, or aimed at pleasing the public by fine cuts and uncommon cases; for, without exception, they have omitted his favourite constructions, which had gained his predilection by their universality, and attached themselves to inferior methods, or perhaps to inventions which they were pleased to regard as their own. What has been laid down in this article is not professed to be according to the principles of Dr Brooke Taylor, because the principles are not peculiar to him, but the necessary results of the theory itself, and inculcated by every mathematician who has taken the trouble to consider the subject. They are sufficient not only for directing the ordinary practice, but likewise for suggesting modes of construction in every case out of the common track. A person of ingenuity will have a laudable enjoyment in inventing rules for himself; and will be better pleased with such fruits of his own ingenuity, than in reading the tedious explanation of examples devised by another; and with this view we would, with Dr Taylor, "advise all our readers not to be contented with the scheme they find here; but, on every occasion, to draw new ones of their own, in all the variety of circumstances they can think of." This will take up more time at first, but they will find the vast benefit and pleasure of it by the extensive notions it will give them of the nature of the principles."

The art of perspective is necessary to those arts where there is any occasion for designing, as architecture, fortification, carving, and generally to all the mechanical arts; but it is more particularly necessary to the art of painting, in which nothing can be done without it. A figure in a picture, which is not drawn according to the rules of perspective, does not represent what is intended, but something else. Indeed, we hesitate not to say, that a picture which is faulty in this particular is as blameable as any composition in writing which is defective in point of orthography or grammar. It is generally thought very ridiculous to pretend to write a heroic poem, or a fine discourse, upon any subject, without understanding the propriety of the language in which we write; and to us it seems no less ridiculous for one to pretend to paint a good picture without understanding per- Perspective. Yet how many pictures are there to be seen, which are highly valuable in other respects, and yet are entirely faulty in this point? Indeed, this fault is so very general, that we cannot remember ever having seen a picture which is entirely free from it; and, what is more to be regretted, the greatest masters have been the most guilty of it. Such examples cause it to be overlooked; but the fault is not the less, but the more to be lamented, and deserves the greater care in avoiding it for the future. The great occasion of this fault is certainly the bad method which is generally used in educating persons in this art; for the young people are generally put immediately to drawing, and when they have acquired a facility in that, they are put to colouring. And these things they learn by rote, or by practice only, and are not at all instructed in any rules of art; by which means, when they come to make any designs of their own, though they are very expert at drawing and colouring every thing which offers itself to their fancy, yet for want of being instructed in the strict rules of art, they do not know how to govern their inventions with judgment, and become guilty of many gross mistakes, which prevent themselves, as well as others, from finding the satisfaction which they otherwise would do in their performances. To correct this defect, we would recommend it to masters of the art of painting to consider whether it would not be necessary to establish a better method for the education of their scholars, and to begin their instructions with the technical parts of painting, before they let them loose to follow the inventions of their own uncultivated imaginations.

The art of painting, taken in its full extent, consists of two parts; the inventive and the executive. The inventive part is common to poetry, and belongs more properly and immediately to the original design (which it conceives and disposes in the most proper and agreeable manner) than to the picture, which is only a copy of that design already formed in the imagination of the artist. The perfection of this art depends upon the thorough knowledge the artist has of all the parts of his subject; and the beauty of it consists in the happy choice and disposition which he makes of it. It is in this that the genius of the artist discovers and shows itself; whilst he indulges and humours his fancy, which here is not confined. But the executive part of painting is wholly confined and restricted to the rules of art, which cannot upon any account be dispensed with; and therefore in this the artist ought to govern himself entirely by these rules, and not to take any liberties whatsoever. For any thing that is not truly drawn according to the rules of perspective, or not truly coloured or truly shaded, does not appear to be what the artist intended, but something else. Wherefore, if at any time the artist happen to imagine that his picture would look better if he swerved a little from these rules, he may assure himself, that the fault belongs to his original design, and not to the strictness of the rules; for what is perfectly agreeable and just in the real original objects themselves, can never appear defective in a picture where these objects are accurately copied.

Therefore, to offer a short hint of the thoughts we have some time had as to the method which ought to be followed in instructing a scholar in the executive part of painting, we would first have him to learn the most common principles of practical geometry, and the elements of plain geometry and of common arithmetic. When he is sufficiently perfect in these, we would next have him to learn perspective; and when he has made considerable progress in the latter, so as to prepare his judgment with right notions of the alterations which figures must undergo, when they come to be drawn upon a flat surface, he may then be put to drawing by view, and be exercised in this alone with perspective, till he comes to be sufficiently perfect in both. Nothing should be more familiar to a painter than perspective; for it is the only thing which can make the judgment correct, and help the fancy to invent with ten times the ease that it could do without it.

We earnestly recommend to our readers the careful perusal of Dr Taylor's Treatise, as published by Colson in 1749, and Emerson's, published along with his Optics, as still the best scientific works on the subject.

(B.B.)