Note. In the fame manner may the center of an arc of a circle be found.

PROBLEM XIV.

Through a given Point A, to draw a Tangent to a given Circle.

CASE 1. When A is in the Circumference of the Circle.

From the given point A, draw Ao to the center of the circle. Then through a draw BC perpendicular to Ao, and it will be the tangent as required.

B

A C

CASE 2. When A is out of the Circumference.

From the given point A, draw AO to the center, which bifect in the point m.—With the center m, and radius mA or mo, describe an arc cutting the given circle in n.-Through the points A and n, draw the tangent BC.

B

m

11

To find a Third Proportional to two given Lines A B, Ac.

Place the two given lines, AB, AC, making any angle at A, and join BC.-In AB take AD equal to AC, and draw DE parallel to BC. So fhall A E be the third proportional to AB and AC.

That is, AB AC :: AC : AE.

A

A

B

C

C

E

Ꭰ Ᏼ

PROBLEM XVI.

To find a Fourth Proportional to three given Lines, AB,

To find a Mean Proportional between two given Lines,

To divide a Line AB in Extreme-and-Mean Ratio.

Raife Bс perpendicular to AB, and equal to half AB. Join AC. With center c, and radius CB, Crofs AC in D. Laftly, with center A, and radius AD, Crofs AB in E, which will divide the line AB in extreme and mean ratio, namely, fo that the whole line is to the greater part, as the greater part is to the lefs part. That is, AB AE :: AE : EB.

A

D

E

B

PRO

PROBLEM XIX.

To make an Equilateral Triangle on a given Line A B.

From the centers A and B, with the radius AB, defcribe arcs, interfecting in c.-Draw Ac and BC, and it is done.

Note. An ifofceles triangle may be made in the fame manner, taking for the radius the given length of one of the equal fides.

PROBLEM XX.

C

B

To make a Triangle with three given Lines AB, AC, BC.

With the center A and radius AC, defcribe an arc.-With the center B, and radius Bc, defcribe another arc, cutting the former in c.-Draw AC and BC, and ABC is the triangle A required.

PROBLEM XXI.

C

A

B

B.

To make a Square upon a given Line A B.

Draw BC perpendicular and equal

to AB.

From A and c with the radius AB, defcribe arcs interfecting in D.-Draw AD and CD, and it is done.

Another Way.

On the centers A and B, with the radius AB, defcribe arcs croffing at o.-Bifect AO in n.-With center o, and radius on, crofs the two arcs in c and D.-Then draw AC, BD,

CD.

C

D

C

口

PROBLEM XXII.

To defcribe a Rectangle, or a Parallelogram, of a given
Length and Breadth.

Place Bc perpendicular to AB.-With center A, and radius AC, defcribe an arc.With center c, and radius AB, defcribe another arc, cutting the former in D.-Draw AD and CD, and it is done.

D

B C

B

Note. In the fame manner is defcribed any oblique parallelogram, only drawing BC, making the given oblique angle with AB, instead of perpendicular to it.

PROBLEM XXIII.

To make a regular Pentagon on a given Line AB.

Another Method.

Make Bm perpendicular and equal to AB.-Bifect AB in n; then with the center n, and radius nm, crofs AB produced in o.With the centers A and B, and radius A o, describe arcs interfecting in D, the oppofite angle of the pentagon.

Laftly, with center D, and radius AB, crofs thofe arcs again in c and E, the other two angles of the figure. Then draw the lines from angle to angle, to complete the figure.

A third Method, nearly true.

On the centers a and B, with the radius AB, describe two circles, interfecting in m and n. -With the fame radius, and the center m, defcribe гAOBS, and draw mn cutting it in 0.-Draw roc and SOB, which will give two angles of the pen

tagon. Lastly, with radius AB, and centers c and E, describe arcs interfecting in D, the other angle of the pentagon nearly.