PERSPECTIVE teaches how to represent objects on a plane superficies, 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.

It was in the 16th 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 Æschylus, was the first who wrote upon this subject; and that afterwards the principles of this 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 Perspective 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 this subject is not now extant. It is, 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 Peruzzi studied the writings of Borgo, and endeavoured to make them more intelligible. To him we owe the discovery of points of distance, to which all lines that make an angle of 45 degrees with the ground-line are drawn. A little time after, Guido Ulbaldi, 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 geometers; particularly by professor Gravefendt, and still more by Dr Brook Taylor, whose principles are, in a great measure new, and far more general than any before him.

In order to understand this subject, a general knowledge of the principles of Optics is absolutely necessary. The foundation of perspective may be understood, by supposing the pentagon ABDEF (fig. 1.) were to be represented by the rules of perspective on the transparent plane VP, placed perpendicularly on the horizontal plane HR; dotted lines are imagined to pass from the eye C to each point of the pentagon, as CA, CB, CD, &c. which are supposed, in their passage through the plane, PV to leave their traces or vestigia in the points a, b, d, &c. on the plane, and thereby to delineate the pentagon abdef; which, as it strikes the eye by the same rays that the original pentagon ABDEF does, will be a true perspective representation of it.

The business of perspective, therefore, is to lay down geometrical rules for finding the points a, b, 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, 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 plane ABCD, (ibid. fig. 7.) is all in front; and if it were raised we should not see any thing 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 aslant, 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 shew 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 of 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 Perspective from the figure or painting, as it is distant from the principal point V. The points a, b, and c, where the visual lines V1, V2, V3, intersect the lines of distance AK, BK, CK, will be the angular points of the angle a b c, the true representation of ABC.

To draw a Square Pavement in Perspective.

Suppose your piece of pavement to consist of 64 pieces of marble, each a foot square. Your first business is to draw an ichnographical 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, 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 O.

To diminish a Circle. See 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 Cube in Perspective.

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

As, for example, suppose the base line BS (fig. 1.) Plate CXXXVI. fig. 10. 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 X, 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.

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. Plate CCXXXVIII

Fig. 1.

PHENICOPTERUS Ruber The Flamingo.

Fig. 2.

Fig. 35.

Fig. 34.

Fig. 36. To find the Height and Proportion of any Object, as they appear above the Horizon on a Supposed Plane.

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.

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 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 gh, kl, and no; which perpendiculars will shew the heights of the figures at f, i, and m.

To draw a Direct View.

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 a front view of the whole body of the church, with all the pillars, &c. on each side: then fix your horizon on any height you think proper, as at DE: bisect 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 these 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 KP; 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.

First draw your horizontal line AB; then, if your Fig. 6; 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 thro' 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.

Rule your horizontal line ab, and on one part of it Fig. 7: 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 streets on each side take a different direction, towards the accidental points g and h), and the lines gi and gh give the roofs and bases of all the buildings on one side, as lh and mh do on the other.

Accidental points seldom intervene where the distance is small, as in noblemens 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.

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 is in the object, or geometrical figure, be 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, crosses, spirals, and cross-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 into 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 the side or angle against the base; raise perpendiculars from the several angles, or designed points, from the figure of 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 catoptries 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 distant places of the same. See Optics.

Of the Anamorphosis, or Reformation of Distorted Images.

By this means pictures that are so misshapen, 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. Vauzelard in his Perspective Conique et Cyindrique; 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 palteboard, ABCD, Plate draw whatever is intended to be exhibited, as the let. ters IHS. Then with a needle make perforations along the whole out-line; 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. 34, is convex, and the other, fig. 35, is concave.

In order to paint upon a plane a deformed copy ABCDEKIHGF of an original picture, which shall appear regular, when seen from a given point O, elevated above the plane, by rays reflected from a polished cylinder, placed upon the circle lnp, equal to its given base; from the point R, which must be supposed to lie perpendicularly under O, the place of the eye, draw two lines RaRe; which shall either touch the base of the cylinder, or else cut off two small equal segments from the sides of it, according as the copy is intended to be more or less deformed. Then, 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 aekf (of a square axz, fig. 36, or parallelogram standing upright upon its base ae, and containing the picture required) anywhere behind the arch lnp. 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 a focus R, and were reflected from the base lnp; so that each couple, as lA, lR, 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 lnp, 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 Lipiensia, 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 out-lines 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.

Perspiration

Perspiration, or Graphical Perspective, in optics. See there, p. 5584.

PERSPIRATION, in medicine, the evacuation of the juices of the body through the pores of the skin. Perspiration is distinguished into sensible and insensible; and here sensible perspiration is the same with sweating, and insensible perspiration that which escapes the notice of the senses; and this last is the idea affixed to the word perspiration when used alone.