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PROPOSITION IX.

To divide a Right Line given into as many equal Parts as you please.

Plate 2. Fig.

9.

Let AB be the Line proposed to be divided into fix equal

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A and B

AC and BD,

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Then from the Points

And upon the Line

Make fix equal Divifions, viz.

upon the Line AC, and

ROPONM upon the Line BD.

Then draw the Line EN, FO, GP, HQ, IR.

And the Line AB, will be divided into fix equal Parts by the Sections STUXY.

PROPOSITION X.

From a given Point, to draw a Right Line Plate 2. Fig. which fhall touch a propofed Circle.

10.

Let A be the Point from whence a Line is to be drawn that fhall touch the Circle DOP.

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Thus the Right Line AE will be the Tangent Line re

CA

ADB

D.

A

AE

D.

quired,

PRO

Plate 2. Fig.

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PROPOSITION XI.

To draw a Right Line which shall touch a Circle at a given Point.

Let ABC be the Circle, in the Circumference of which is the given Point A.

PRACTICE.

From the Point or Center
Draw the Line

D

DF

Prop. 1.

Through the given Point
Then to the given Point
And upon the Line

Draw the Perpendicular
Prolonged towards

A.

A

DF

AH

I.

Thus this Tangent Line HI will touch the Circle at the given Point A, which is what the Propofition required.

Plate 2. Fig.

PROPOSITION XII.

A Circle and a Right Line touching it being given, to find the Point where the said Right Line touches the faid Circle.

12.

Let ABC be the Circle touched by the Line GH.

We are to find the Point where the Line touches the Circle.

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The Section C will be the touching Point required.

Plate 2. Fig.

13.

PROPOSITION XIII.

To defcribe a Spiral Line upon a Right Line given.

Let IL be the Line upon which a Spiral Line is to be defcribed.

PRACTICE.

Divide half of the Line IL into as many equal Parts as you would defcribe Revolutions upon the faid Line.

EXAMPLE.

EXAMPLE.

Suppose you would defcribe four upon it.

Divide the half of the Line

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From the Point

BI

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Describe the Semi-Circles BC, DE, FG, HI.

B

Defcribe the Semi-Circles CD, EF, GH, IL.
and you will have the Spiral Line required.

PROPOSITION XIV.

Between two given Points, to find two others Plate 2. Fig. directly between them.

14.

Let AB be the given Points, between which two others are to be found directly even with them, and by means of which a Right Line may be drawn from the Point A to the Point B, with a fhor Ruler.

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These Points G and H will be the Points required; by the Affiftance of which one may, at three times, draw a Right Line from the Point A to the Point B, which could not be done at once with a Ruler fhorter than the Space between A and B.

BOOK

BOOK II.

Of the Conftruction of PLANE Figures.

Plate 3. Fig.

I.

PROPOSITION I.

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O conftruct an Equilateral Triangle upon a Right Line given of a determined Length. Let AB be the Line upon which an Equilateral Triangle is to be formed.

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Plate 3. Fig.

2.

PROPOSITION II.

To make a Triangle of three Right Lines, equal to three Right Lines given.

Let A, B, C, be the three Lines given, equal to which a Triangle of three Right Lines is to be made.

PRAC

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