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PROBLEM XIII.

Two unequal Right-lines being given, to form or make of them a Rightangled Parallelogram.

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lel, and of the fame Length, to AB; viz. make DC AB: Join DA with a Right-line, and it will form the Oblong or Parallelogram required.

As for Rhombus's and Rhomboides's, to wit, Oblique-angled Parallelograms, they are made, or defcribed, after the fame Manner with the two laft Figures; only instead of erecting the Perpendiculars, you must fet off their given Angles, and then proceed to draw their Sides parallel, &c. as before.

PROBLEM XIV.

In any given Circle, to infcribe or make a Triangle, whofe Angles fhalt be equal to the Angles of a given Triangle; as the Triangle FDG. (2. e. 4.)

Note, Any Right-lin'd Figure is faid to be infcrib'd in a Circle, when all the Angular Points of that Figure do juft touch the Circle's Periphery.

Draw any Right-line (as HK) fo as just to touch the Circle,

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PROBLEM XV.

In any given Triangle, as A BD, to defcribe a Circle that shall touch all its Sides. (4. e. 4.)

Bifect any two Angles of the Triangle, as A and B, and where the bifecting Lines meet (as at C) will be the Center of the Circle required; and its Radius will be the nearest Distance to the Sides of the Triangle.

PROBLEM XVI.

B

To defcribe a Circle about any given Triangle. (5. e. 4.)

D

This Problem is perform'd in all refpects like the Ninth, viz. by bifecting any Two Sides of the given Triangle; the Point, where those bifecting Lines meet, will be the Center of the Circle required.

PROBLEM XVII.

To defcribe a Square about any given Circle, (7. e. 4.)

Draw two Diameters in the given Circle (as DA and E B) croffing at Right Angles in the Center C; and, with the Circle's Radius C A, defcribe

from the extream Points of those Diameters, viz. A, B, D, E, crofs Arches, as at F, G, H, K; then join those Points where the Arches cross with Right Lines, and they will form the Square required.

F

D

E

G

A

H

B

K

PROBLÉM XVIII.

In any given Circle, to defcribe the largest Square it can
contain. (6. e. 4.)

Having drawn the Diameters, as D A and E B, bifecting each other at Right-angles in the Center C, (as in the laft Scheme); then join the Points A, B, D, and E, with Right lines, viz. A B, BD, DE, E A, and they will be Sides of the Square re quired.

PRO.

PROBLEM XIX.,

Upon any given Right-line, as AB, to defcribe a regular Pentagon, or Five-fided Polygon.

.K.

C

F

Make the given Line Radius, and upon each End of it de fcribe a Circle; and through thofe Points where the Circles crofs each other (as at G x) draw the Rightline Gex: Upon the Point G with the fame Radius defcribe the Arch HAB D, and laying a Ruler upon the Points D, e, mark where it crosses the other Circle, as at F. Again, lay the Ruler upon the Points H, e, and mark where it craffes the other Circle, as at C: Then from

H

D

the Points F and C (with the fame Radius as before) defcribe crofs Arches, as at K: Join the Points AF, FK, KC, and CB, with Right lines, and they will form the Pentagon required, viz. AF FKKC=CB=AB; and the Angles at A, B, C, K, F will be equal.

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PROBLEM XX.

In any given Circle, to defcribe a regular Pentagon.
(11. e. 4. & 10. e. 3.)

Or, in general Terms, to defcribe any regular Polygon in a

Circle.

Draw the Circle's Diameter D A, and divide it into as many

equal Parts as the proposed Polygon hath Sides; then make the whole Diameter a Radius, and defcribe the two Arches CA and CD. If a Right-line be drawn from the Point C, through the Second of thofe equal Parts in the Diameter; as at 2, it will affigna Point in the oppofite Semicircle's Periphery, as at B. Join DB with a Right-line, and it will be the Side of the Pentagon required.

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D

C

A

I

2 3 4

Thefe

Thefe Twenty Problems are fufficient to exercise the young Practitioner, and bring his Hand to the right Management of a Ruler and Compaffes, wherein I would advife him to be very ready

and exact.

As to the Reason why fuch Lines must be fo drawn as directed at each Problem, that, I prefume, will fully and clearly appear from the following Theorems; and therefore I have (for Brevity's Sake) omitted giving any Demonftrations of them in this Chapter, defiring the Learner to be fatisfied with the bare Knowledge of doing them only, until he hath fully confidered the Contents of the next Chapter; and then I doubt not but all will appear very plain and eafy.

CHA P. III.

A Collection of most useful Theorems in plain Geometry Demonstrated.

Note, In order to shorten feveral of the following Demonfirations, it will be neceffary to premife, that

1. THE Periphery (or Circumference) of every Circle (whether great or fmall) is fuppos'd to be divided into 360 equal Parts, called Degrees; and every one of thofe Degrees are divided into 60 equal Parts, call'd Minutes, &c.

2. All Angles are measured by the Arch of a Circle defcrib'd upon the Angular Point (See Defin. 9. Page 287.) and are esteem'd greater or lefs, according to the Number of Degrees contain'd in that Arch.

3. A Quadrant, or Quarter part of any Circle, is always 90 Degrees, being the Measure of a Right-angle (Defin. 6. P. 287.) and a Semicircle is 180 Degrees, being the Measure of two Right-angles.

4. Equal Arches of a Circle, or of equal Circles, measure equal Angles.

To thofe five general Axioms already laid down in Page 146, (which I here fuppofe the Reader to be very well acquainted with) it will be convenient to understand thefe following, which begin their Number where the other ended.

Arioms.

Arioms.

6. Every whole Thing is Greater than its Part. That is, the whole Line AB is greater than its Part A c, &c. SA

:is}

C

B

The fame is to be understood of Superficies's and Solids.

d

7. Every Whole is Equal to all its Parts taken together. That is, the whole Line A B is equal to its Parts AC + c d + de + e B. Š A—|—|—|—B. The fame is also true in Superficies's and Solids.

-

-B

8. Thofe Things which, being laid one upon another, do agres or meet in all their Parts, are equal one to the other.

But the Converfe of this Axiom, to wit, that equal Things being laid one upon the other will meet, is only true in Lines and Angles, but not in Superficies's, unless they be alike, viz. of the fame Figure or Form: As for Inftance, a Circle may be equal in Area to a Square; but if they are laid one upon the other, 'tis plain they cannot meet in all their Parts, because they are unlike Figures. Alfo, a Parallelogram and a Triangle may be equal in their Area's one to another, and both of them may be equal to a Square; but if they are laid one upon the other, they will not meet in all their Parts, &c.

Note, Befides the Characters already explain'd in Part I, and in other Places of this Tract, thefe following are added. Viz. denotes an Angle in general, and fignifies Angles; A fignifies a Triangle; fignifies a Square, and denotes a Parallelogram. And when an Angle is denoted by any three Letters (as, A B C) the middle Letter (as B) always denotes the Angular Point; and the other two Letters (as A B, and B C) denote the Lines or Sides of the Triangle which includes that Angie.

Thefe Things being premifed, the young Geometer may proceed to the Demonftrations of the following Theorems; wherein he may perceive an abfolute Neceflity of being well verfed in feveral Things that have been already deliver'd: And alfo it will be very advantageous to ftore up feveral ufeful Corollaries and Lemma's, as they become difcover'd Truths: For it often happens, that a Propofition cannot be clearly demonftrated a priori, or of itself, without a great Deal of Trouble; therefore it will be ufeful to have Recourfe to thofe Truths that may be affifting in the Demonftration then in Hand.

T H E O

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