CASE 2. When the Point is too near the End of the Line.

With the center c, and any radius, defcribe the arc mns.-From the point m, with the fame radius, turn the compaffes twice over on the arc at n and s.-Again, with the centers n and s, defcribe arcs interThen draw Ar, and fecting in r. it will be perpendicular as required.

Another Method.

From any point m as a center, with the radius or diftance ma, defcribe an arc cutting the given line in n and A.-Through n and m draw a right line cutting the arc in 1.-Laftly, draw Ar, and it will be the perpendicular as required.

Another Method.

From any plane fcale of equal parts fet off am equal to 4 parts.With center A, and radius of 3 parts, defcribe an arc.-And with center m, and radius of 5 parts, crofs it at n.-Draw An for the perpendicular required.

Or any other numbers in the fame proportion as 3,

4, 5, will do the fame.

PRO

PROBLEM VI.

From a given Point A, out of a given Line BC, to let fall a Perpendicular.

CASE 1. When the Point is nearly oppofite the Middle of the Line.

With the center A, and any radius, defcribe an arc cutting BC in m and n.-With the centers m and n, and the fame, or any other radius, describe arcs interfecting in r.-Draw A Dr, for the perpendicular required.

B

A

D

C

m.

r

CASE 2. When the Point is nearly oppofite the End of the Line.

From A draw any line Am to meet BC, in any point m.-Bifect Am at n, and with the center n, and radius An or mn, describe an arc, cutting BC in D.-Draw A D the perpendicular as required.

Another Method.

From в or any point in вc, as a center, describe an arc through the point A. From any other center m in BC, defcribe another arc through A, and cutting the former arc again in n.-Through A and n draw the line ADn; and AD will be the perpendicular as quired.

re

B

A

N. B.

N. B. Perpendiculars may be more readily raised. and let fall, in practice, by means of a fquare, or other fit inftrument.

To divide a given Line A B into any propofed Number of Equal Parts.

From A draw any line AC at random, and from в draw BD parallel to it.-On each of thefe lines, beginning at a and B, fet off as many equal parts, of any length, as AB is to be divided into. Join the oppofite points of divifion by the lines A 5, 14, 23, &c; and they will divide A as required.

A

...

2345

C

D

B

5432

To divide a given Line A B in the fame Proportion as another Line CD is divided.

From A draw any line A E equal to CD, and upon it transfer the divifions of the line CD. -Join BE, and parallel to it draw the lines 1 1, 22, 33, &c; and they will divide Aв as required.

C

I 234

D

E

A

B

I 234

PRO

PROBLEM IX.

At a given Point A, in a given Line AB, to make an Angle equal to a given Angle c.

With the center c, and any radius, defcribe an arc mn.-With the center A, and the fame radius, defcribe the arc rs.-Take the distance mn between the compaffes, and apply it from r to s.-Then a line drawn through A and s, will make the angle A equal to the angle c as required.

PROBLEM X.

A

n

AV

B C

At a given Point A, in a given Line AB, to make an Angle of any propofed Number of Degrees.

With the center A, and radius equal to 60 degrees taken from a fcale of chords, defcribe an arc, cutting AB in m.-Then take between the compaffes, the proposed number of degrees from the fame fcale of chords, and apply them from m to n.-Through the point n draw an, and it will make the angle A of the number of degrees propofed.

A

m B

Note. Angles of more than 90 degrees are ufually taken off at twice.

Or the angle may be made with any divided arch or inftrument, by laying the center to the point A, and its radius along AB; then make a mark n at the propofed number of degrees, through which draw the line An as before.

PROBLEM XI.

To measure a given Angle A. Describe the arc mn with the chord of 60 degrees, as in the last Problem. Take the arc mn between the compaffes, and that extent, applied to the chords, will fhew the degrees in the given angle.

A

m B

Note. When the distance mn exceeds 90 degrees, it must be taken off at twice, as before.

Or the angle may be measured by applying the radius of a graduated arc, of any inftrument, to AB, as in the last problem; and then noting the degrees cut off by the other leg an of the angle.

PROBLEM XII.

To find the Center of a Circle. Draw any chord AB; and. bifect it perpendicularly with CD, which will be a diameter.-Bifect cp in the point o, which will be the

center.

C

B

D

PROBLEM XIII.

To defcribe the Circumference of a Circle, through three given Points, A, B, C.

From the middle point в draw chords to the other two points.Bifect thefe chords perpendicularly by lines meeting in o, which will be the center.-Then from the center o, at the distance oA, or oв, or oc, defcribe the circle.

A

B

Note.