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Unto a given Right Line P Q, to draw another Right Line S T, which shall be Parallel to it; and at the Diftance of the gi ven Line M.

Fig. XV.J Practice. Flft, Take in your Compaffes the Length of the given

Line of Distance M.

2. Set one Foot in O, (near to one end of the give Line PQ,) and with the other defcribe the Arch a a: Alfo, upon the Point L, (near the other end of the given Line) defcribe the Arch bb; then,

3. By the Convexity (or Tops) of thofe Two Arches, if you draw a Right Line ST, it will be Parallel to the given Line PQ, and at the Distance of the Line M.


To a given Line V W, to draw another Right Line Y A, which
Shall pass through a given Point R, and Parallel to V W.

Fig. XVI. Practice. First, Set one Foot of the Compaffes in the given Point

R, and with the other defcribe the Arch cc, fo that

it may only touch the given Line V W; and with the fame Diftance, fet one Foot in any Point of the Line V W, as at X, and with the other defcribe the Arch dd.

2. A Line drawn through the given Point R, by the Convexity of the Arch dd, as the Line YR A, that Line fhall be a Parallel to the Line V W.


A fecond Way to draw a Line Parallel to a given Line A B, which
Shall pass through a given Point P.




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Irft, With the Distance P A, upon the Point B, defcribe the Arch e e.

2. With the Distance AB, upon P, describe the Arch ƒ ƒ, crof fing the Arche e in the Point C.

3. A Right Line drawn from the given Point P, through C, fhall be Parallel to the given Line A B.



To make an Angle DFE, equal to a given Angle A B C

Practice. First, Upon the Angular Point B, at any Distance, defcribe an Arch gg. Then,

2. Having drawn another Line, as F E, upon the end F (with the fame Distance) defcribe the Arch bb.

3. Take the Distance g g in your Compaffes, and fet it from b to b: Then,

4. A Line drawn from F through b, as F b D, fhall make the Angle DF E equal to the Angle AB C.

...PRO B. X.

To divide an Angle G H K into Two Equal Parts.



Practice. Fr the Compaffes) defcribe an Arch cutting the

Irft, Upon the Angular Point H (with any Distance Fig.XIX. of

Two Sides, containing the Angle, in the Points h and s.

2. The Compaffes opened to the fame (or any other) Distance, fet one Foot in b, and with the other defcribe the Arch k k; and (with the fame Distance) one Foot fet in s, describe the Arch i i, croffing k k in the Point L.

3. Join H L, and fo is the Angle G H K divided into Two Equal Parts, by the Line H L.

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How to divide a given Right Line M N, into any Number of
Equal Parts Suppofe&Five.

Practice. Fine NO at Pleafute, making the Angle Ở NM
Irft, From one end of the given Line, as N, draw the Fig. XX.


2. Upon the Point M (by Prob. IX.) make the Angle N M P equal to the Angle ONM; or (by Prob. VIII.) through the Point M, draw the Line M P Parallel, to the Line O N

3. With any fmall Diftance of the Compaffes, one Foot being fet in N, run over Four of thofe Distances upon the Line NO, at the Points 1, 2, 3, 4-Do the like upon the Line MP: Then,

C 2

4. If

4. If you draw the obfcure Lines 1 4, 23, 3 2, and 4 I, they will cross the given Line M N in the Points d, c, b, a, dividing it into Five Equal Parts, as was required.


To make an Equilateral Triangle ABC, whofe Sides Shall be Equal to a given Line O.

Fig. Practice. M


Fig. XXII.

Ake the Side B C equal to the given Line O, and with the fame Diftance of the Compaffes, fetting one Foot in B, with the other defcribe the Arch mm; and fet alfo on C, defcribe the Arch 17, croffing the former Arches in A: Then join A B and A C, fo have you conftituted a Triangle A B C, whofe Three Sides are feverally equal to the given Line O.


To make a Triangle G HK, whofe Three Sides fhall be Equal to the Three given Right Lines D, E, F.

Practice. Flr,

Make the Line G H equal to the given Line D.


2. Take the given Line E in
your Compaffes, and fetting one E
Foot in H, with the other de-
fcribe the Arch n n.


2. Take the Line F in your Compaffes; and fetting one Foot in G, with the other defcribe an Arch o o, croffing the former Arch in the Point K.

4. Join G K and H K; fo fhall you have conftituted a Triangle; whofe Three Sides G H, HK, and G K, are equal to the Three given Lines D E and F.

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To make a Geometrical Square B CDE, whofe Sides Shall be equal to the given Line A.

Fig. Practice. XXIII.

Ift, Make the Line BC for one Side of the Square, equal to the given Line A, and on one End thereof,

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Fig. XXI.



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