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à 1. From the points AB, as cer.tres, with any distance greater than half AB, describe arcs cutting each other in c and d.

2. Draw the line cd and the point E, where it cuts AB, will 1. Take any point d above the be the middle of the line required. line, and with the radius or dis

tance dC, describe the arc eCf,

cutting AB in e and C. Prob. 2.-From a given point 2. Through the centre d and C, in a given right line AB, to the point e, draw the line edf, erect a perpendicular.

cutting the arc eCf, in f. When the point is near the middle 3. Through the points f and C, of the line.

draw the line fc, and it will be the perpendicular required.

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radius, describe the arc de, cutting 1. From the point B, with any AB in e and d.

radius, describe the arc AC. 2. From the points e,d with the 2. From A and C, with the same, or any other radius, describe same or any other radius, describe two arcs cutting each other, in f. arcs cutting each other in d.

3. Through the points C,f, draw 3. Draw the line Bd, and it will the line CDs, and CD will be the bisect the angle ABC, as was reperpendicular required.

quired. Prob. 4.-At a given point D, upon the right line DE, to make an

Prob. 6.- To trisect, or divide, angle equal to a given angle aBl.

a right angle ABC into three equal angles.

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1. From the point B, with any 1. From the point B, with any radius BA, describe the arc AC, radius, describe the arc ab, cutting cutting the legs BA, and BC, in the legs Ba, Bb, in the points a and b. A and C. 2. Draw the line De, and from

2. From the point A, and C, the point D, with the same radius with the radius AB, or BC, cross as before, describe the arc ef, cut the arc AC in d, and e. ting DE in e.

3. Through the points e,d, draw 3. Take the distance la, and the lines Be, Bd, and they will apply it to the arc ef, from e to f. trisect the angles as was required.

4. Through the points Df, draw the line Df, and the angle eDf, will be equal to the angle iBa, as

Prol. 7. - Through a given was required.

point C, to draw a line parallel to

a given line AB. Prob. 5.—To divide a given an.

с gle ABC into two equal angles.


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B 1. Take any point d, in AB, upon d and C, with the distance Cd, describe two arcs eC, and df, cutting the line AB, in e and d.

2. Make df equal to eC; join Cs, and it will be parallel to'AB as required. LI

When the parallel is to be at a given de, upon the circle, from the given distance EF from AB. point B, and draw the chord eB.

2. Upon B, as a centre, with E -F

the distance Bd, describe the arc D fdg, cutting the chord eB in f.

3. Make dg equal to df, through 8 draw gB, and it will be the tan

gent required. А

B 1. From any two points c and d, in the line AB, with a radius equal

Prob. 10.- A circle ABC being to EF, describe the arcs e and f. given, and a tangent CH to that

2. Draw the line CD, to touch circle, to find the point of contact. those arcs without cutting them, and it will be parallel to AB as was required.

Prob. 8.-To draw a tangent to a given circle, that shall pass through a given point A.





1. Take any point e, in the tangent CH, and join eG.

2. Bisect eG in f, and with the radius fe, or fG, describe the semicircle eCG, cutting the tangent and

the circle in C, it will be the point 1. From the centre O, draw the required. radius OA.

2. Through the point A, draw DE perpendicular to OA, and it Prob. 11.-Given three points will be the tangent required.

A, B, C, not in a straight line, to

draw a circle through them. Prob. 9.-To draw a tangent to a ciicle, or any segment of a circle

B ABC, through a given point B, without making use of the centre of the circle.


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1. Bisect the lines AB, and BC, by perpendiculars, meeting at d.

2. Upon do with the distance

da, dB, or dC, describe ABC, L 1 Take any two equal arcs Bd, will be the circle required.

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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 D

the equilateral triangle required.

For the heragon.

Carry the radius AG six times E

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

gon. 2. From D, the point of inter

For the dodecagon. section, 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. inscribe a square or an octagon.

Prob. 15.-—In a given circle to


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11 C

D r 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 trapeBisect 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


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1. Divide the given trapezium 1. Upon the points A and B,

ABCD into two triangles, by a with a radius equal to AB, describe 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.

În the same manner may any irregular polygon be made equal and similar to a given irregular

polygon, by dividing the given A

polygon into triangles, and conF

structing the triangles in the same E

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

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