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Prob. 12.-In a given triangle 1. Upon any point A, in the A, B, C, to inscribe a circle. circumference with the radius AG,

describe the arc BGF.

2. Draw BF, make BD equal to BF.

3. Join DF, and BDF will be the equilateral triangle required.

For the heragon.

Carry the radius AG six times A

round ihe circumference, the fi 1. Bisect any two angles A and gure ABCDEF will be the hexa C, witls the lines AD, and CD.


For the dodecagon. 2. From D, the point of intersection, let fall the perpendicular

Bisect the arc AB in h, and Ah DE, it will be the radius of the being carried twelve times round circle required.

the circumference, will also form

the dodecagon. Prob. 13.-In a given square ABCD, toinscribe a regular octagon.

Prob. 15.-In a given circle to

inscribe a square or an octagon. 7 D


E 2

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D I 1. Draw the diagonals AC, and 1. Draw the diameters AC and D, intersecting at e.

BD, at right angles.' 2. Upon the points A, B, C, D, 2. Join AB, BC, CD, DA, and as centres, with a radius eC, de- ABCD will be the square. scribe arcs, hel, ken, meg, fei.

For the octagon. 3. Join fn, mh, ki, lg, it will be Bisect the arc AB in 'E, and the octagon required.

AE being carried eight times

round, will also form the octagon. Prob. 14.-In a given circle to inscribe an equilateral triangle, an Prob. 16.-In a given circle to hexagon, or a dodecagon. inscribe a pentagon, or a decagon. For the equilateral triangle.

For a pentagon.

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1. Draw the dianieters AF and 1. Draw AB equal to the line GH, at right angles, cutting each D. other in I.

2. Upon A, with the distance 2. Bisect GI in f, upon f, with E, describe an arc at C. the distance of fa, describe the 3. Upon A, with the distance arc Ag: upon A, with the dis. F, describe another arc, intersecttance Ag, describe the arc gE ing the former at C. cutting the circle in E.

4. Draw AC and CB, and ABC 3. Join AE, and carry it round will be the triangle required. the circle five times, then will ABCDE be the pentagon required. For the decagon.

Prob. 19.-To make a trape. Bisect the arc AE in i, and Di zium equal, and similar to a given being carried ten times round, will trapezium ABCD. also form the decagon.

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



1. Divide the given trapezium 1. Upon the points A and B, with a radius equal to AB, de

ABCD into two triangles, by a scribe arcs, cutting each other at

diagonal AC.

2. Make EF equal to AB upon C. 2. Draw AC, and BC, it will whose three sides will be re

EF, construct the triangle EFG, be the triangle required.

pectively equal to the triangle

ABC. Prob. 18.–To make a triangle, whose three sides shall be equal ac, construct the triangle EGH,

3. Upon EG, which is equal to 10 three given lines D, E, F, if any whose two sides EH, and GH, two are greater than the third.

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 conI

structing the triangles in the same E

manner in the required polygen, as is shown by figures.


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 BCDE will be the rectangle required.

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Prob. 23.-To niake a rect.

angle, having a side equal to a B

given line AB, and equal to a given 1. Draw the diagonal BD, make rectangle CDEF. CE parallel to it, meeting the side

F K AB, produced in E. 2. Join DE, and ADE will be

D the triangle.


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


1. Produce the sides of the rect. D

angle CF, DE, FE, and CD.

2. Make EG equal to AB, С

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.


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

1. Produce the side AB, make
BE equal to BC.

2. Bisect AE in I, on I, as a

centre with the radius IE or IA 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 B I

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 reU

quired, that is, EF: EG ::EH: EI, IA

or CD : AB :: AB: EI. A

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 perpendicular to DE, equal to the side of

I the square B.

2. Draw the hypothenuse FE ; on it describe the square EFGH,

I. it will be the square required.




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: GL.

Prob. 26.- To make a square

I II 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

join HI.

2. Make GK equal to EF, draw

KL through K parallel to HI, F

then GL will be the fourth proporB

tional required ; that is, GH : GI

:: GK: GL, or AB : CD:: EF

I 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.


D 3. Make FG equal to the side

CH of the square C; join GE, then GE will be the side of the square required.

h i k ZK Prob. 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 A

the several parts of CD froin H

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.





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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 hy 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 oh. 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 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 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 40, in the same direction as it would from the corresponding point A of the original cube by the ray A0. 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 roles, and not by guess or acquired habit of the hand and eye, it is called Per.. 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

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