Perspective teaches how to represent objects on a plane surface, such as they would appear at a certain distance and height, upon a transparent plane perpendicular to the horizon, placed between the objects and the eye. In order to understand this subject, a general knowledge of the principles of Optics is absolutely necessary. The foundation The business of perspective, therefore, is to lay down geometrical rules for finding the points \(a, b, c, d, e, f\) upon the plane; and hence also we have a mechanical method of delineating any object very accurately.

Perspective is either employed in representing the ichnographies or ground plots of objects; or the scenographies, or representations of the objects themselves.

But before we give any examples of either, it will be proper to explain some technical terms in regard to perspective in general; and first, the horizontal line is that supposed to be drawn parallel to the horizon through the eye of the spectator; or rather it is a line which separates the heaven from the earth, and which limits the sight. Thus, A, B (ibid. fig. 2.) are two pillars below the horizontal line CD, by reason the line is elevated above them; in fig. 3. they are said to be equal with it, and in fig. 4. raised above it. Thus, according to the different points in view, the objects will be either higher or lower than the horizontal line. The point of sight A (ibid. fig. 5) is that which makes the central ray on the horizontal line \(ab\); or, it is the point where all the other visual rays DD unite. The points of distance, C, C, are points set off in the horizontal line at equal distances on each side of the point of sight A; and in the same figure BB represents the base line, or fundamental line: EE is the abridgment of the square; of which D, D, are the sides; F, F, the diagonal lines, which go to the points of distance C, C. Accidental points are those where the objects end: these may be cast negligently; because neither drawn to the point or sight, nor to those of distance, but meeting each other in the horizontal line. For example, two pieces of square timber G and H (ibid. fig. 6.) make the points I, I, I on the horizontal line; but go not to the point of sight K, nor to the points of distance C, C; these accidental points serve likewise for casements, doors, windows, tables, chairs, &c. The point of direct view, or of the front, is when we have the object directly before us; in which case, it shews only the foreside; and, if below the horizon, a little of the top; but nothing of the sides, unless the object be polygonous.

Thus the plan ABCD. (ibid. fig. 7.) is all in front, and if it were raised we should not see anything of the sides AB or CD, but only the front AD: the reason is, that the point of view E being directly opposite thereto, causes a diminution on each side; which however is only to be understood where an elevation is the object; for if it be a plan, it shews the whole, as ABCD.

The point of oblique view, is when we see an object aside of us, and as it were afloat, or with the corner of the eye; the eye, however, being all the while opposite to the point of sight; in which case, we see the object laterally, and it presents to us two sides or faces.

For instance, if the point of sight be in F, (ibid. fig. 8.) the object GHIK will appear athwart, and show two faces GK and GH, in which case it will be a side point.

We shall now give some examples, by which it will appear, that the whole practice of perspective is built upon the foundation already laid down. Thus, to find the perspective appearance or a triangle ABC (ibid. fig. 9.) between the eye and the triangle draw the line DE, which is called the fundamental line; from 2 draw 2 V, representing the perpendicular distance of the eye above the fundamental line, be it what it will; and through V draw, at right angles to 2 V, HK parallel to DE: then will the plane DHKE represent the transparent plane on which the perspective representation is to be made. Next to find the perspective points of the angles of the triangle, let fall perpendiculars A 1, C 2, B 3, from the angles to the fundamental DE: set off these perpendiculars upon the fundamental opposite to the point of distance K, to B, A, C; from 1, 2, 3, draw lines to the principal point V; and from the points A, B, and C, on the fundamental line, draw the right lines AK, BK, CK, to the point of distance K, which is so called, because the spectator ought to be so far removed from the figure or painting, as it is distant from the principal point V. The points \(a, b, c\), where the visual lines V 1, V 2, V 3 intersect the lines of distance AK, BK, CK, will be the angular points of the angle \(abc\), the true representation of ABC.

To draw a square pavement in perspective. See fig. 10. and 11. of Plate CXLIII.

Suppose your piece of pavement to consist of 64 pieces of marble, each a foot square. Your first business is, to draw an ichnographic plan or ground-plot of it, which is thus performed. Having made an exact square of the size you intend your plan, divide the base and horizon into eight equal parts; and from every division in the base to its opposite point in the horizon, rule perpendicular lines: then divide the sides into the same number, ruling parallel lines across from point to point: so will your pavement be divided into 64 square feet; because the eight feet in length, multiplied by the same in breadth, give the number of square feet or pieces of marble contained in the whole: then rule diagonals from corner to corner; and thus will your ground-plot appear as in fig. 10.

Now, to lay this in perspective, draw another square to your intended size, and divide the base line AB into eight equal parts, as before; then fix your point of sight C in the middle of the horizon DE, and from the same point rule lines to every division in the base AB; after which, rule diagonal lines from D to B, and from E to A, answerable to those in the ground-plot, and your square will be reduced to the triangle ABC; then from the point F, where the diagonal DB intersects the line AC, to the opposite intersection G, where the diagonal EA crosses the line CB, rule a parallel line, which is the abridgment of the square.

Then through the points where the diagonals cross the rest of the lines which go from the base to the point of sight, rule parallel lines, and your square pavement will be laid in perspective, as in fig. 11.

To diminish a square viewed by the angle D. See Plate CXLIII. fig. 12.

Having described the plane ABCD, draw a line to touch or raise the angle B, and falling perpendicularly on BD.

This being continued as a base line, lay your ruler on the side of the square AD and DC, and where the ruler cuts the terrestrial line make the points H, I.

Then from H and B draw lines to the point of distance P, and from I draw a line to the other point of distance G; and in the intersection of those lines, make points, which will give you the square KLMB.

To do without the plan: set off the diameter each way from the middle point B, as to H and I. But in either case no line is to be drawn to the point of sight Q.

To diminish a Circle. See Plate CXLIII. fig. 13.

Draw a square ABCD about it, and from the angles AD and CB draw diagonals, dividing the circle into eight parts, and through the points where they cut it OO, draw lines from the base line perpendicular to DEF.

Then draw two diagonals QR, SP, intersecting each other at right angles in the centre G.

Having thus disposed the plan, draw lines from all the perpendiculars to the point of sight H; and where they are intersected by the diagonals AK and BI, make points; the two last of which M, N give the square, which is to be divided into four by diagonals, intersecting each other in the point P.

In the last place, from the extremes of this cross, draw curve lines through the said points, which will give the form of the circle in perspective.

Of the measures upon the base in perspective, Pl. CXLIV.

By the base line alone any depth may be given, and in any place at pleasure, without the use of squares; which is a very expeditious way.

As for example, suppose the base line BS, (fig. 1.) the point of view A, and the points of distance DE; if now you would make a plan of a cube BC, draw two occult or dotted lines from the extremes BC to the point of sight; then to give the breadth, take the same measure BC, and set it off on the terrestrial line CF, and from F draw a line to the point of distance D; and where this line intersects the first ray C in the point G, will be the diminution of the plan of the cube BHGC.

If you would have an object farther towards the middle, take the breadth, and the distance of the base line, as IK; and to have the depth, set it as you would have it on the same base as LM, and its width both on LM. Then from L and M draw occult lines to the point of distance D, and from the points NO, where those lines intersect the ray K, draw parallels to the terrestrial line, and you will have the square QPON.

After the same manner you may set off the other side of the square which should be on the base, as BHGC is here transferred to V. The points M and T, which are only two feet from the point S, afford a very narrow figure in K, as being very near.

Of the base line, and a single point of distance.

Since the depths and widths may be had by the means of this base line, there is no need of any further trouble in making of squares; as shall be shewn in this example.

Suppose a row of trees or columns is to be made on each side; on the base line lay down the place, and the distance between them, with their breadth or diameters, as ABCDEFG; then laying a ruler from the point of distance O to each of the points ABCDEFG, the intersections it makes on the visual ray AH will be the bounds of the objects desired. Fig. 2.

To set them off on the other side upon the ray GH, set one foot of the compasses upon the point of the eye H, and with the other strike an arch; the point wherein this cuts the ray GH, will be the corresponding bound.

Thus M will be the same with N, and so of the rest; through which drawing parallels, you will have the breadths.

And as for the length, you may make it at pleasure: setting it off from A, as for instance, to P, and then from P drawing a line to H; and where this cuts the other parallels, will be formed the plan required; which you may make either round or square.

To find the height and proportion of any objects, as they appear above the horizon on a supposed plane. See Plate CXLIV. fig. 3.

First rule your horizontal line NO, and fix your point of sight, as at M; then mark the place of your nearest pillar, by making a dot for the base or bottom, as at A; and another for the summit or top, as at B; rule a line from A to the point of sight M, and another from B to M, and these two lines will give the height of any number of pillars. As for example: Suppose you would have a pillar at C, fix your dot for the base, and rule from thence a parallel line to meet the diagonal AM at D; then rule the perpendicular DE to the diagonal BM: which perpendicular is the height of your figure required at C. Or, if you would place pillars at F and I, observe the same method, ruling the parallels FG and IK, and the perpendiculars GH and KL will give their heights at the distances required.

To find the diameter or thickness of pillars at any particular distances, you are also to be guided by that nearest the base. For instance: Suppose your nearest pillar AB to be ten feet high, and one foot in diameter; divide it from top to bottom into ten equal parts, and set off one of them upon the base of the pillar: then rule a line from the point of sight M to the diameter P, and you will have the thickness of all your pillars on their respective parallels or bases.

The same rule exemplified in objects below the horizon. See Plate CXLIV. fig. 4.

If you would know the heights of a number of figures below the horizon, rule your horizontal line QR, and fix your point of sight, as at P: then place your nearest figure, or mark the dots for the head and feet, by the points A and B, which answers the same purpose; and rule from these dots to the point of sight the lines AP and BP: and if you would find the height of a figure to be drawn at c, rule from thence the parallel cd to the diagonal BP, and the perpendicular de will give the height required. The same directions will shew the height of a figure at any other distance you have a mind to place it, as at f, i, and m, by ruling the parallels fg, ik, and mn; and from each of these their respective perpendiculars gb, kl, and no; which perpendiculars will shew the heights of the figures at f, i, and m.

4. To draw a direct view. See Plate CXLIV. fig. 4.

To illustrate this example, suppose you were to draw the inside of a church, as represented in this figure: First take your station at the point A, in the centre of the base line. BC: from which you have a front view of the whole body of the church, with all the pillars, &c. on each side: then fix your horizon at any height you think proper, as at DE; inset it by the perpendicular EA: and where these two lines intersect, is the point of sight F. This perpendicular will pass through the centres of all the arches in the dome or cupola: which centres may be found by any three given points. Next divide your base line into any given number of feet; and the visual lines, ruled from these divisions to the point of sight, will reduce all your objects to their just proportion, by setting off their height upon a perpendicular raised at their respective distances. The base, in the example here given, is divided into twelve equal parts of five feet each; from which (supposing your front column to be 35 feet high) take seven divisions from the base line of your drawing, and set them off upon the perpendicular GH; then (supposing this column to be five feet thick at the base) set off one of those divisions upon the parallel IK, which is the breadth required. So that, by proportioning this scale to any distance by the foregoing directions, you may not only find the dimensions of all your columns, but also of every distinct part of them, as well as of all the doors, windows, and other objects that occur.

For instance: Having found the height and breadth of your first or nearest column G, draw from the top and bottom of the said column to the point of sight the lines HF and KF; after which, rule the line IF from the base of the column to the point of sight, and you have the height and breadth of all the rest of the columns, as has been already shewn in fig. 3.

By ruling lines from the points a, b, c, d, &c. to the point of sight, you will see that all the summits and bases of your columns, doors, windows, &c. must tend immediately to that point; and by lines drawn from the points 1, 2, 3, 4, &c. on each side, to the correspondent points on the opposite side, may be seen all the parts of your building lying upon the same parallel.

To draw an oblique view. See fig. 6. of Plate CXLIV.

First draw your horizontal line AB; then, if your favourite object be on the right hand, as at C, place yourself on the left hand upon the base line, as at D; then from that station erect a perpendicular DE, which will pass through the horizon at the point of sight F; to which rule the diagonals GF and HF, which will shew the roof and base of your principal building C, and will also, as before directed, serve as a standard for all the rest.

Observe also, either in direct or oblique views, whether the prospect before you make a curve; for if it does, you must be careful to make the same curve in your drawing.

To draw a perspective view, wherein are accidental points. See fig. 7. of Plate CXLIV.

Rule your horizontal line ab, and on one part of it fix your point of sight, as at c; from which rule the diagonals cd and ce on the one side, and cf and cg on the other; which will shew the roofs and bases of all the houses in the street directly facing you; (supposing yourself placed at A in the centre of the base line:) Then fix your accidental points g and h upon the horizontal line, and rule from them to the angles ik and lm, (where the streets on each side take a different direction, towards the accidental points g and h) and the lines gi and gk give the roofs and bases of all the buildings on one side, as li and mb do on the other.

Accidental points seldom intervene where the distance is small, as in noblemen's seats, groves, canals, &c. which may be drawn by the strict rules of perspective: but where the prospect is extensive and varied, including mountains, bridges, castles, rivers, precipices, woods, cities, &c. it will require such an infinite number of accidental points, that it will be better to do them as nature shall dictate, and your ripened judgment approve.

To find the centre for the roof of a house, in an oblique view. See fig. 8. of Plate CXLIV.

Suppose from the point of sight A, the visual lines AB and AC be drawn, BC being one perpendicular given, and DE the other, rule the diagonals from D to C, and from E to E, and the perpendicular FG, raised through the point of their intersection, will shew the true centre of the roof, as will appear by ruling the lines GE and GC.

For want of being acquainted with this necessary rule, many who have been well versed in other parts of perspective, have spoiled the look of their picture, by drawing the roofs of their houses out of their true perpendicular.

We shall conclude by giving a few practical rules. 1. Let every line, which in the object, or geometrical figure, is straight, perpendicular, or parallel to its base, be so also in its scenographic delineation. 2. Let the lines, which in the object return at right angles from the fore-right side, be drawn scenographically from the visual point. 3. Let all straight lines, which in the object return from the fore-right side, run in a scenographic figure into the horizontal line. 4. Let the object you intend to delineate, standing on your right-hand, be placed also on the right hand of the visual point; and that on the left-hand, on the left-hand of the same point; and that which is just before, in the middle of it. 5. Let those lines which are (in the object) equidistant to the returning line be drawn in the scenographic figure, from that point found in the horizon. 6. In setting off the altitude of columns, pedestals, and the like, measure the height from the base line upwards, in the front or fore-right side; and a visual ray down that point in the front shall limit the altitude of the column or pillar, all the way behind the fore-right side, or orthographic appearance, even to the visual point. This rule you must observe in all figures, as well where there is a front or fore-right side, as where there is none. 7. In delineating ovals, circles, arches, croffes, spirals, and crofs-arches, or any other figure in the roof of any room, first draw ichnographically, and so with perpendiculars from the most eminent points thereof, carry it up unto the ceiling; from which several points, carry on the figure. 8. The centre in any scenographic regular figure is found by drawing lines from opposite angles: for the point where the diagonals cross, is the centre. 9. A ground-plane of squares is alike, both above and below the horizontal line; only the more it is distant above or beneath the horizon, the squares will be so much the larger or wider. 10. In drawing a perspective figure, where many lines come together, you may, for the directing of your eye, draw the diagonals in red; the visual lines in black; the perpendiculars in green, or other different colour, from that which you intend the figure shall be of. 11. Having considered the height, distance, and position of the figure, and drawn it accordingly, with side or angle against the base; raise perpendiculars from the several angles, or designed points, from the figure to the base, and transfer the length of each perpendicular, from the place where it touches the base, to the base on the side opposite to the point of distance; so will the diametral drawn to the perpendiculars in the base, by intersection with the diagonals, drawn to the several transferred distances, give the angles of the figures, and so lines drawn from point to point will circumscribe the scenographic figure. 12. If in a landscape there be any standing-waters, as rivers, ponds, and the like, place the horizontal line level with the farthest sight or appearance of it. 13. If there be any house, or the like, in the picture, consider their position, that you may find from what point in the horizontal lines to draw the front and sides thereof. 14. In describing things at a great distance, observe the proportion, both in magnitude and distance, in draught, which appears from the object to the eye. 15. In colouring and shadowing of every thing, you must do the same in your picture, which you observe with your eye, especially in objects lying near; but, according as the distance grows greater and greater, so the colours must be fainter and fainter, till at last they lose themselves in a darkish sky-colour. 16. The catoptrics are best seen in a common looking-glass, or other polished matter; where, if the glass be exactly flat, the object is exactly like its original; but, if the glass be not flat, the resemblance alters from the original; and that more or less, according as the glass differs from an exact plane. 17. In drawing catoptric figures, the surface of the glass is to be considered, upon which you mean to have the reflection; for which you must make a particular ichnographical draught, or projection; which on the glass must appear to be a plane full of squares, on which projection transfer what shall be drawn on a plane, divided into the same number of like squares; where though the draught may appear very confused, yet the reflection of it on the glass will be very regular, proportional, and regularly composed. 18. The dioptric, or broken beam, may be seen in a tube through a crystal or glass, which hath its surface cut into many others, whereby the rays of the object are broken. For to the flat of the crystal, or water, the rays run straight; but then they break and make an angle, which also by the refracted beams is made and continued on the other side of the same flat. 19. When these faces on a crystal are returned towards a plane placed directly before it, they separate themselves at a good distance on the plane; because they are all directed to various far distant places of the same. See Optics.