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Set one Foot of the Compaffes in the Point a, extend the other fo as to reach beyond the Line AB, and draw an Arch to cut the Line A B in C and D ; put one Foot of the Compaffes in C, and, with any Extent above Half CD, draw an Arch, b; keep

ing the Compaffes at the fame

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in

Extent, put one Foot in D, and interfect the Arch b; through which Interfection, and the Point a, draw a E, the Perpendicular required.

A

D

a

E

B

Cafe 2. To let fall a Perpendicular from a Point, a, being nearly over the End of the Line A B: From any Part of the given Line, as D, draw a Line to the Point a; divide the Line D a into two equal Parts in E, by Proh. 1; fet one Foot of the Compaffes in E, and with the Distance D E defcribe an Arch from D to a, which will interfect the given Line A B in e; draw the Line a e, which will be perpendicular to the given Line A B.

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Prob. 5. To draw a Line parallel to a given Line, A B. Set one Foot of the Compaffes in E any Part of the Line, as at a; extend the Compaffes at Pleafure, unless a Distance be affigned, and deferibe an Arch, b; withthe fame Extent, in some other Part of the Line A B, as at e, describe the Arch c, lay a Ruler to the Extremities of the Arches, and draw the Line EF, which will be parallel to the given Line A B.

A a

св

Prob. 6. To draw a Line parallel to a given Line, A B, and

to pass through a given Point, a.

Set one Foot of the Compaffes in the given Point ; extend the other fo

as it may touch the Line A B, and not

pass it, in defcribing the Arch H; with

a

I

H

B

the fame Extent, fet one Foot in the A. Line A B, and defcribe the Arch I;

then draw a Line through the Point a to the Extremity of

Сс

the

the Arch I, which will be parallel to the given Line A B, afs through the Point a.

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Prob. 7. To make an Angle equal to a given Angle, A BC.

E A

B

T

Draw the Line PT at Pleasure ; fet one Foot of the Compaffes in B, and with any Diftance, B D, draw the Arch DE; keep the Compaffes at the fame Extent, and, placing one Foot in P, draw the Arch de at Pleafure: Set one Foot of the Compaffes in D, extend the other to E, fet this Extent from d towards, draw the Line P S through the Point b in the Arch de; and fo will SPT be an Angle equal to A BC.

P

Prob. 8. To make an Angle to any Number of Degrees.

This Problem has two Cafes, as the Angle required is lefs or more than 90 Degrees.

A

D

B

Cafe 1. To make an acute Angle of 30° on a given Line AB: Take 60 Degrees from a Line of Chords in the Compaffes, [The Defcription and Use of the Line of Chords will be given after the Definitions in Trigonometry.] Set one Foot of the Compaffes in the Point A, and defcribe an Arch, CD, at Plea fure; then fet one Foot of the Compaffes in the Brass Center in the Beginning of the Line of Chords, and bring the other to 30 on the Line; with this Extent, fet one Foot in C, with the other interfect the Arch CD in E, and through E draw the Line A E; fo will E A B be an Angle of 30 Degrees.

Cafe 2. To draw an obtufe Angle of 120° on a given Line, A B: Take 60 Degrees in the Compaffes from the Line of Chords, fet one Foot in A, and with the other defcribe a large Arch CD; take 90 Degrees from the Line

a

DZ 4

A

-B

of Chords, and fet one Foot in C, and interfect the Arch in a then take 30 Degrees from the fame Line of Chords; and fet them from a to b; a Line drawn from A to b will make an Angle, DBA, of 120 Degrees.

Prob. 9.

Prob. 9. To measure an Angle.

This Problem has two Cafes, the Angle being either lefs o more than 90°.

C

Ъ

B

a

Cafe 1. To measure an acute Angle, ABC: Take 60 Degrees from the Line of Chords, fet one Foot in A, and with the other describe the Arch a b from one Leg to the other, then fet one Foot of the Compaffes in a, and extend the other to b; with this Extent, fet one Foot on the Brafs Center at the Beginning of the Line, the other will reach to 54 on the Line of Chords, which is the Number of Degrees contained in the Angle BAC.

Cafe 2. To measure an obtufe Angle, A BC: Take 60 Degrees in the Compaffes from the Line of C Chords, fet one Foot in the angular Point A, and defcribe the Arch a c from one Leg to the other, as before; then take 90 Degrees from

Ъ

-B a

the Line of Chords, and fet it from a to b; then take the Extent from b to c in the Compaffes, as in the last Problem, and set one Foot in the Beginning of the Line of Chords, the other will reach to 41 Degrees, which, added to 90, is 131 Degrees, the Measure of the Angle BAC.

Prob. 10. To divide a given Line, BAC, into two equal Parts.

A

B

·D

Set one Foot of the Compaffes in the angular Point A, extend the Compaffes to any convenient Distance, and interfect each Leg, as at a and b, with the fame, or any other Extent above Half the Distance ba, fet one Foot in a, and draw the Arch D; then, keeping the Compaffes to the fame Extent, fet one Foot in b and interfect the Arch D in D; a Line drawn from the an gular Point A through the Interfection of the Arches will divide the Angle A into two equal Parts.

a c

Prob. II

f

Prob. 11. To divide a given Line, A B, into any Number of equal Parts.

A

123456789 C

987654321

B

From the End of the Line, A, draw the Line A C, fo as to make any Angle at A; from the other End of the Line, B, by Prob. 7, draw a Line BD, making an Angle at B equal to the Angle at A; which Lines AC and BD D will be parallel to each other: On each of thefe Lines, with any Extent of the Compaffes, from the Points A B, refpectively, fet off as many Parts equal to one another, fave one, as the given Line is to be divided into; as fuppofe 10, fet off 9 equal Parts, numbering them 1, 2, 3, &c. on each Line; draw Lines from 1, 2, 3, &c. in the Line A C, to 9, 8, 7, &c. in the Line BD; then will the Line A B be divided into ten equal Parts.

Prob. 12. From a Point, B, affigned out of a Circle, to draw a Tangent to the Circle, or a Line that shall touch the Circumference, but not interfect it.

From the given Point B, draw the Line B to the Center of the Circle, which is called the Secant; [Sines, Tangents, and Secants will be defined in the Beginning of the Trigonometry]. Divide this Line into two equal Parts in e, by Prob. 1; then, with one Foot of the Compaffes in the Point c, with the Extent B or co, draw the Semicircle કી AB, cutting the Circle in A; laftly,

B

draw a Line from B through the Interfection at A; this will be a Tangent to the given Circle.

Prob. 13. To make a Triangle, each of whofe Sides shall be equal to a given right Line.

A

Let A B be the given Line; draw A

B

a Line at Pleafure, larger than the given Line; then take the given Line A B in the Compafles, fet one Foot in A, the

Line laft drawn, with the other make a Mark at B, and with the fame Extent of the Compaffes draw the Arch e; keeping the Compaffes to the fame Extent, put one Foot in B, and draw the A.ch f, interfecting the former Arch; from the Interfection of the Arches draw

A

e

B

Lines to A and B, and the Triangle is completed, having each Side equal to the given Line A B.

C

A

B

--

Prob. 14. To draw a Triangle whofe Sides fhall be equal to three given Lines, if any two of them be longer than the third. Let A, B, C, be three given Lines: Draw a Line, A B, at Pleafure; take the Line C in the Compaffes, fet one Foot in A, and with the other make a Mark at B; then take the given Line B in the Compaffes, fet one Foot in B and defcribe the Arch C; then take the given Line A in the Compaffes, and fetting one Foot in A, interfect the Arch CA in C; laftly, draw the Lines A C

C

B

and B C, and the Triangle is completed, having three Sides,

each equal to one of the given Lines.

Prob. 15. To defcribe Ovals mechanically.

Draw two parallel Lines, as L and M, at a moderate Diftance, by Prob.5; then draw two others at the fame Diftance, crofs the former, as N and O; by the Croffing of thefe Lines will be made a Figure, ABCD, of four Sides: Extend the Compaffes at Pleafure, and fetting one Foot in D, defcribe the

Arch cde; with the fame Extent, fet

L h
M

B

D

one Foot in B, and describe the Arch f g h; then set one Foot of the Compaffes in C, and contract them fo as to reach the Point e, and defcribe the Arch Im; with the fame Extent, and one Foot in A, defcribe the Arch i k, and the Oval is completed: In the fame Manner, with a greater or lefs Extent

of

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