2. Draw the line CB, to touch those arcs without cutting them, and it will be parallel to AB as was required.

PROBLEM VIII. PLATE 7.

To divide a given line AB into any proposed number of equal parts.

1. From A, one end of the line, draw Ac, making any angle with AB; and from B, the other end, draw вd, making the angle ABd equal to BAC.

2. In each of the lines Ac, вd, beginning at A and B, set off as many equal parts, of any length, as AB is to be divided

into.

3. Join the points A 5, 14, 2 3, &c. and AB will be divided as was required.

PROBLEM IX. PLATE 7.

To find the centre of a given circle, or one already described. 1. Draw any chord AB, and bisect it with the perpendicular CD.

2. Bisect CD with the diameter Ef, and the intersection o will be the centre required.

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

1. From the centre o, draw the radius oA.

2. Through the point A draw DE perpendicular to os, and it will be the tangent required.

PROBLEM XI. PLATE 8.

To draw a tangent to a circle, or any segment of a circle ABC, through a given point B, without using the centre of the circle.

1. Take any two equal divisions upon the circle, from the given point B, towards d and e, draw the chord eв.

2. Upon в, as a centre, with the distance вd, describe the arc fdg, cutting the chord eв in f.

3. Make dg equal to df, through g draw gB, and it will be the tangent required.

PRACTICAL GEOMETRY.

PROBLEM XII. PLATE 8.

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A circle ABC being given, and a tangent DH to that circle, to find the point of contact.

1. Take any point e, in the tangent DH; from e, to the centre of the circle G, draw eG.

2. Bisect ea in f, and with the radius fe, or fe, describe the semicircle ecG, cutting the tangent and the circle in c, then will c be the point required.

Given three points, A, B, C, not in a right line, to find a number of points lying between them, so that they shall all be in the circumference of a circle without drawing any part of the circle, or finding its centre.

1. From A, through в and c, draw AB and Aƒ.

2. On A, as a centre, with any radius af, describe an arc fed, cutting AB in d, and ac in ƒ.

3. Bisect the arc df in e, through e draw ach.

4. Join CB, bisect it in g, draw gh perpendicular, cutting Ach at h, then h will be in the same circumference with A, C, B. In the same manner may a point be found between

ch and hв.

PROBLEM XIV.

PLATE 9.

Given three points, A, B, C, not in a right line, to find another point without those points, so that the four points shall all be in the circumference of a circle, without drawing any part of the circle, or finding its centre.

1. Draw Ae and AC, from A, through в and c.

2. On A, as a centre, with any radius Ae, draw the arc efg, cutting AB in e, and ac in ƒ.

3. Make fg equal to fe, through a and g draw ag indefinitely towards d.

4. Upon c, with the distance CB, cross the line Ag at d, it will be the point required.

If a fifth point, or any other number of points are required, the process will be the same.

PROBLEM XV. PLATE 9.

Given three points, A, B, C, not in a straight line, to draw a circle through them.

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

2. Upon d, with the distance da, dв, or dc, describe ABC, which will be the circle required.

PROBLEM XVI. PLATE 9.

To describe the segment of a circle to any length AB, and breadth CD.

1. Bisect AB by the perpendicular Dg, cutting AB in c. 2. From c, make CD on the perpendicular, equal to CD. 3. Bisect AD by a perpendicular ef, cutting pg in g.

4. Upon g, the centre, describe ADB, which will be the segment required.

PROBLEM XVII. PLATE 10.

To describe the segment of a circle, by means of two rules, to any length AB, and perpendicular height CD, in the middle of AB, without making use of the centre.

It will be most convenient for practice to make the rules CE and CF each equal to AB, as room is sometimes required.

1. Place the rules to the height at c, bring the edges close to A and B, tack them together at c, and fix a rod across them to keep them tight.

2. Put in pins at A and B, then move the rules round these pins, holding a pencil to the angular point at c, which will describe the segment required.

FIG. 2.-By means of a triangle, let AB be the length of the segment, and CD the perpendicular height in the middle.

B

1. Through the points D and в draw DB.

2. Draw DE parallel to AB for conveniency, then DE equal to DB, and join EB.

3. Make a triangle E, D, B, place pins at the points A, D, B, then move the triangle round the points D and B, and the angular point will describe half the segment; the other half will be described in the same manner, which will complete the whole segment, as required.

FIG. 3. PLATE 11.-Another method by means of points. Let AB be the length, and CD, bisecting AB perpendicular, the height.

1. Through D draw GH parallel to AB.

2. Draw DB, the half chord.

3. From в make вH perpendicular to DB, cutting GH in H, and make DG equal to DH.

PRACTICAL GEOMETRY.

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4. Draw AF and BE each perpendicular to AB, cutting GH in F and E.

5. Divide DG DH, CA CB, and AF BE, each into a like number of equal parts, as five.

6. Draw the cross lines 4 4, 3 3, 2 2, 1 1, &c.

7. From the division on AF and BE, draw lines to D, cutting the other cross lines at d, e, f, g, &c.

8. Placing pins in these points, bend a slip round them, and draw the curve by it, and it will be the segment required.

FIG. 4. PLATE 11.—Another method, by points (nearly true) when the segment is very flat. Let AB be the length, and CD bisecting AB, the perpendicular height.

1. Draw AE and BF perpendicular to AB, each equal to CD. 2. Divide CB and CA each into the same number of equal parts, as five.

3. From the points 4, 3, 2, 1, &c. on AB, draw the perpendicular 4 4, 3 3, 2 2, 1 1, &c. to AB.

4. Divide AF and BE each into five equal parts.

5. Draw lines from the points 1, 2, 3, 4, at each end, to D, and complete the segment in the same manner as Fig. 3.

PROBLEM XVIII. PLATE 11.

In a given triangle A, B, C, to inscribe a circle.

1. Bisect any two angles a and c, with the lines AD and CD. 2. From D, the point of intersection, let fall the perpendicular DE, it will be the radius of the circle required.

PROBLEM XIX. PLATE 12.

In a given square ABCD, to inscribe a regular octagon. 1. Draw the diagonals AC and BD, intersecting at e.

2. Upon the points A, B, C, D, as centres, with a radius ec, describe arcs hel, ken, meg, fei.

3. Join fn, mh, ki, lg, which will form the octagon required.

PROBLEM XX. PLATE 12.

In a given circle to inscribe an equilateral triangle, an hexagon, or a dodecagon.

For the Equilateral Triangle.

1. Upon any point A, in the 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 Hexagon.

Carry the radius AG six times round the circumference, the figure ABCDEF will be the hexagon.

For the Dodecagon.

Bisect the arc AB in h, and Ah, being carried twelve times round the circumference, will also form the dodecagon.

PROBLEM XXI. PLATE 12.

In a given circle to inscribe a square or an octagon. 1. Draw the diameters AC and BD, at right angles. 2. Join AB, BC, CD, DA, and ABCD will be the square. For the Octagon.

Bisect the arc AB in E, and AE, being carried eight times round, will form the octagon.

PROBLEM XXII. PLATE 13.

In a given circle to inscribe a pentagon, or a decagon.

For a Pentagon.

1. Draw the diameters AC and BD, at right angles.

2. Bisect Ec in ƒ, upon f, with the distance fD describe the arc Dg upon D, with the distance Dg, and describe the arc gu cutting the circle in н.

3. Join DH, and carry it round the circle five times, and the pentagon will be formed.

For the Decagon.

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

PROBLEM XXIII. PLATE 13.

In a given circle to inscribe any regular polygon.

1. Draw the diameter AB; from E, the centre, erect the perpendicular EFC, cutting the circle at F.

2. Divide Er into four equal parts, and set three parts from F to c.

3. Divide the diameter AB into as many equal parts as the polygon is required to have sides.

4. From c, through the second division in the diameter, draw CD.

5. Join AD, which will be a side of the polygon required.