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1. Draw the diameters AF and GH, at right angles, cutting each other in I.

2. Bisect GI in f, upon f, with the distance of fA, describe the arc Ag upon A, with the distance Ag, describe the arc gE cutting the circle in E.

3. Join AE, and carry it round the circle five times, then will ABCDE be the pentagon required. For the decagon.

Bisect the arc AE in i, and Di being carried ten times round, will also form the decagon.

Prob. 17.-Upon a given line AB, to describe an equilateral triargle.

D.

1. Draw AB equal to the line

2. Upon A, with the distance E, describe an arc at C.

3. Upon A, with the distance F, describe another arc, intersecting the former at C.

4. Draw AC and CB, and ABC will be the triangle required.

Prob. 19. To make a trapezium equal, and similar to a given trapezium ABCD.

D

B

G

H

1. Upon the points A and B, with a radius equal to AB, describe arcs, cutting each other at C.

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ABCD into two triangles, by a 1. Divide the given trapezium diagonal AC.

2. Make EF equal to AB upon 2. Draw AC, and BC, it will whose three sides will be reEF, construct the triangle EFG, be the triangle required.

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pectively equal to the triangle ABC.

AC, construct the triangle EGH, 3. Upon EG, which is equal to whose two sides EH, and GH, are respectively equal to AD and CD, then EFGH will be the trapezium required.

In the same manner may any irregular polygon be made equal and similar to a given irregular polygon, by dividing the given polygon into triangles, and constructing the triangles in the same manner in the required polygon, as is shown by figures.

Prob. 20.-To make a triangle equal to a given trapezium ABCD.

D

1. Draw the diagonal BD, make CE parallel to it, meeting the side AB, produced in E.

2. Join DE, and ADE will be the triangle.

Prob. 21. To make a triangle equal to any given right-lined figure ABCDE.

E

F A

C

B G

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1. Produce the sides of the rectangle CF, DE, FE, and CD.

2. Make EG equal to AB, through G draw LH parallel to 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.

2. Draw the diagonals AD and BD, and make EF and CG parallel to them.

3. Join DF, DG, then DFG will be the triangle required.

Much after the same manner may any other right-line figure be reduced to a triangle.

Prob. 22.-To reduce a triangle ABC to a rectangle.

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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 execnted 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 objects 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 would 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 to every corresponding part.

Thus, in fig. 1, suppose the spectator's eye to be at O, viewing the represen tation of a cube abcde: 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 aO, 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 delineated, and, when this is accurately done, according to mathematical rules, and not by guess or acquired habit of the hand and eye, it is called Perspective.

We shall, therefore, in our following demonstrations, continually recur to this general foundation, by showing, by example, 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.

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