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Prob. 20.-To make a triangle 1. Bisect the altitude AF in G, equal to a given trapezium ABCD. through G draw ED parallel to BC. D
2. C draw CD perpendicular to
BC, and BE parallel to CD, then
Prob. 23.-To make a rect
angle, having a side equal to a E
given line AB, and equal to a given
rectangle CDFF. 1. Draw the diagonal BD, make
D the triangle.
E Prob. 21.–To make a triangle equal to any given right-lined figure ABCDE.
G IL B 1. Produce the sides of the rectangle CF, DE, FE, and CD.
2. Make EG equal to AB, C
through G draw LH parallel to E
DE, cutting CD produced at L.
3. Draw the diagonal LE, and
produce it till it cut CF at K.
4. Draw KH, parallel to EG, 1. Produce the side AB both then will EGH be the rectang e ways at pleasure.
required. 2. Draw the diagonals AD and BD, and make EF and CG paral
Prob. 24.–To make a square lel to them.
equal to a given rectangle ABCD. 3. Join DF, DG, then DFG
H will be the triangle required.
Much after the same manner may any other right-line figure be reduced to a triangle.
2. Bisect AE in I, on I, as a
centre with the radius IE or IA E G
describe the semicircle AHE.
3. Produce the side CB to cut the circle in H, on BH describe
the square BHGF, and it w I bo BE
the square required.
Prob. 25.-To make a square 1. Make an angle HEI at pleaequal to two given squares A and sure, from E, make EF equal to B.
CD, and EG equal to AB, join
3. Make EH equal to EG, and draw HI parallel to FG, then EI
will be the third proportional reI
quired, that is, EF: EG::EH:EI, T. A
or CD : AB :: AB : EI.
Prob. 28.--Three right lines D E
AB, CD, EX, being given, to find 1. Make DE equal to the side
a fourth proportional. or the square A, and DF perpen. dicular to DE, equal to the side of the
B. 2. Draw the hypothenuse FE ; on it describe the square EFGH, it will be the square required. Prob. 26.-To make a square
Ti H equal to three given squares A, 1. Make the angle HGI at B, C.
pleasure ; from G make GH equal
tc AB; GI equal to CD; and с A
2. Make GK equal to EF, draw KL through K parallel to HI,
then GL will be the fourth propor. B
tional required ; that is, GH : GI
:: GK: GL, or AB : CD :: EF E
; GL. 1. Make DE equal to the side of the square A, and DF perpen Prob. 29.-To divide a given dicular to DE, equal to the side line AB, in the same proportion of the square B.
as another, CD is divided, 2. Join FE, draw FG perpendicular to it.
LD 3. Make FG equal to the side
I of the square C; join GE, then GE will be the side of the square required.
22 i k 2 K к Prol. 27.-Two right lines AB, and CD, being given, to find a
1. Make any angle KHI; and third proportional.
make HI equal to AB; then apply
the several parts of CD froin H A
to K, and join KI.
2. Draw the lines le, if, kg, parallel to IK, and the line HÌ, will be divided in h, i, k, l, as was, required.
Definitions. Def. 1. Linear perspective is the art of describing accurately, on a plane surface, the representation of any given object.
In order to give the reader a clear and comprehensive idea of what is meant by the art of Perspective, it must be remarked, that a picture, if ex. ecated in a high degree of perfection, would so far deceive the spectator as to be taken for the original objects, or to appear exactly as the original ol). jects would, if occupying the same situation in reality, in the picture where it is but a representation of them only.
In order to produce this effect, it is necessary that the rays of light should come from the several parts of the picture to the spectator's eye, just as they wonld from the real objects themselves, in every respect whether it regards the direction of the visual lines, strength of light and shadow, or colour with reference te every corresponding part.
Thus, in fig. 1, suppose the spectator's eye to be at 0, viewing the representation of a cube abede : the original cube being ABCDE in its real situation. The light from any point of the picture ought to come to O, by the ray 10, in the same direction as it would from the corresponding point A of the original cube by the ray AO. The circumstances above alluded to, make the executive part of Painting to consist of three parts, viz. Drawing, Colouring, aud Light and Shadow ; drawing, which is what we have here principally in view, has solely to do with the position, and forms of the object to be delincated, and, when this is accurately done, according to mathematical roles, and not by guess or acquired liabit of the hand and eye, it is called Pero spective.
We shall, therefore, in our following demonstrations, continually recur to this general foundation, by showing, by exanıple, that the rays of light will come in the same directions from the several points in the picture, as they would from the corresponding points of the original objects.
Def. 2. When lines, drawn according to a certain law, from several points of any object, are cut by a plane, the figure formed by the intersections of the rays on that plane, is termed the projection of that object.
The lines generating that projection, taken altogether, are termed the system of rays. And, when these rays pass all through one point, they are denominated a pyramid of rays, and when this point is in the spectator's eye, they are called the optic pyramid.
Def. 3. When the system of rays are all parallel to each other, and perpendicular to the horizon, and the projection is made on a plane parallel to the horizon, it is called the ichnography of the object proposed.
Def. 4. When the system of rays are all parallel to each other, and perpendicular to the horizon, and the projection is made on a plane perpendicular to the horizon, it is then called the orthography of the object.
The ternis ichnography and orthograplıy may be illustrated thus, Let GHIK, fig. 2, be a plane parallel to the horizon, and the rays Aa, Bb, Ce, &c, from the several points of the octahedron, ABCDEF being perpendicular to, and parallel to each other, the projection abcd made by them, is called the ichino. graphy of the object.
Again, in fig. 2, let the rays Aa, Bb, Cc, &c. be parallel to each other, and perpendicular to the plane GHLM, on which the projection is made, and let the plane GHLM be perpendicular to the plane GHIK; then will the figure abcdef be the orthography of the object ABCDEF.
These are the common definitions of the terms ichnography and orthography, but we shall use them to signify any two projections that are made by systems of parallel rays, when the rays in those systems are perpendicular to each other, and to the plane on which the projections are made, as in the present figure, without any regard to their situation with respect to the horizon.
In this kind of projection, the projection of any particular point, or line, is usually called the seat of that point, or line, on the plane of projection. Thus, a is the seat of the point A on the plane GHIK, and af is the seat of line AF the plane GHLM.
Def. 5. When the projection is made by a pyramid of rays, it is the schenography,
Thus, (fig. 1) the figure ahcde projected on the plane FGHI by the rays AO, BO, CO, &c. coming from the several parts of the cube ABCDE 10 the point 0, is the schenography of the object ABCDE.
We shall, hereafter, see that this projection is the true perspective representation of the object ABCDE, as it would appear to the eye of the spectator at 0.
It is also evident, that the shadows of objects are thus generated, when the light is supposed to come from a single point. Though in case of the light coming from the Sun or Moon, they being at so great a distance, that the rays may be practically considered as parallel to each other.
Def. 6. The point of sight is where the spectator's eye should be placed to view the picture.
This point is the same with the vortex of the optic pyramid, as is evident from Theorem %, where we will show, that the representation of any object is no other than the schenographic representation of the object on the plane of the picture.
Def. 7. If from the point of sight a line be drawn perpendicular to the picture, the point where that line intersects the picture is termed the centre of the picture, and the distance between this centre and the point of sight is termed the distance of the picture.
Def. 8. If a plane be imagined to pass through the point of sight parallel to the picture, this plane is termed the directing plane.
Def. 9. By original object is meant, whether it be the point, line, surface, or solid, the real object in its own situation, and which is represented in the picture.
Def. 10. By original planes, we mean the plane in which any original, point, line, or plane figure is situated.
Def. u. The point where any original line cuts the picture, is
called simply the intersection of that line, or the intersecting point of that line.
Def. 12. The line produced by an original plane cutting the picture, is simply termed the intersection of the original plane.
Dif. 13. The point where any original line cuts the directing plane, is termed the directing point of that original line. And a line drawn through that directing point, and the point of sight, is termed the director of that original line.
Def. 14. The line, whereby any original plane cuts the directing plane, is termed the directing line of that criginal plane.
Def. 15. A line drawn through the point sight parallel to any original line, is termed the parallel of that original line.
Def. 16. A plane passing through the point of sight parallel to any original plane, is termed the parallel of that original plane.
Dif. 17. The point where the parallel of any original line cuts the picture, is called the vanishing point of that line, and the distance between the vanishing point and the point of sight, is called the distance of that vanishing point.
Cor. 1. Hence it is clear, that original lines, which are parallel to each other, have the same vanishing point, for a line passing through the spectator's eye parallel to them all, generates their common vanishing point by this definition.
Cor. 2. Those lines which are parallel to the picture have no vanishing point. Because the lines, which should produce the vanishing points, are, in this case, themselves parallel to the picture, and, therefore, never can cut or intersect it.
Cor. 3. The lines that generate the vanishing points of any two original lines, make the same angle at the spectator's eye, as the original lines do with each other.
Dif. 18. The line, whereby the parallel of any original plane cuts the picture, is termed the vanishing line of that plane.
And if from the point of sight there be drawn a line cutting that vanishing line at right angles, the point where it is so cut is called the centre of the picture. And the distance between the centre and the point of sight is called the distance of the vanishing line.
Cor. 1. Hence, original planes, that are parallel, have the same vanishing line. For a plane parallel to them all, passing through the spectator's eye, produces that vanishing line.
Cor. 2. The vanishing points of lines in parallel planes, are in the vanishing line of those planes, for the lines that generate these vanishing points, are all in the plane that produces the vanishing line.
Cor.3. The planes wbich produce the vanishing lines of two original planes, have their common intersection passing through the spectator's eye, parallel to the intersection of the original planes, and are inclined to each other in the same angle as the original planes are to each other.
Cor. 4. The vanishing point of the common intersection of two planes, is the intersection of the vanishing lines of those planes.
Cor. 5. The vanishing line, and intersection of the same original