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To make an angle of any proposed number of degrees, with a given

line AB.

WITH the centre A, and radius 69° describe an arch, cutting AB in C; then take the proposed number of degrees in your compafies, and with this for a radius and centre C, defcribe another arch, cutting the former in D; join AD, and the thing is done. See fig. 30. plate 2.


Upon a given line AB, to describe a square.

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UPON the point AB erect a perpendicular AD, equal to AB; from the centre B, with the radius AB, describe an arch; and on D as centre, with the same radius describe another arch, cutting the former in the point C; join DC and BC, and it is done. See fig. 32. plate 2.



To describe a parallelogram of a given length and breadth.

Make BC perpendicular to AB; upon A, as centre and radius BC, describe an archi; with the centre C, and radius AB, describe another arch, cutting the former in D; then join DC and DA and it is done. See fig. 33. plate 2.


To describe a circle in a given triangle, ABC.

BISECT any two of the angles with the lines AD and BD; from D drop a perpendicular DE, upon any one of the three

fides :

fides; then upon D for a centre, and radius DE, describe the circle, and it is done. Plate 2. fig. 34.


About any given triangle to describe a circle.

BISECT any two fides, BA, BC, by perpendiculars, DE, DF, with the centre D, and radius equal to the distance of any one of the angles, describe a circle. Plate 3. fig. 35.


To describe a circle in or about a given square.

DRAW two diagonals to the given square; at the intersection D drop a perpendicular DE; on D as centre, with the radius DE, describe a circle for the inscribed circle; on D as centre, with half the diagonal for the radius, describe another for the circumscribed circle. Plate 3. fig. 36.


To describe a square in or about a given circle.

DRAW two diameters, AB, CD, at right angles to each other ; join their extremities for the inscribed square ADBG, and, at the angular points of the inscribed square draw tangents, and they will form the circumscribed square, abcd. Plate 3.

fig. 37

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To describe a circle through three given points, A,B,C, which art

not in the fame Araight line.

JOIN the middle point to the other two ; bisect their distances perpendicularly by straight lines meeting in D; then with the centre D, and distance of either of the three given points as radius, describe a circle, and it shall pass through A,B,C. Plate 3. fig. 38.


A segment of a circle being given, to describe the circle of which it

is the segment.

Draw AC the chord, and bisect it at right angles by BD ; then join AB, and make the angle BAD equal to the angle ABD; draw AD, then with the point D as centre, and radius DA, DB or DC, describe the circle, and it is done. Plate 3. fig. 39. Or, take any three points in the segment, and bisect their distances, and the bisecting lines will interfect each other in the centre, as in prob. 21.


To describe a parallelogram that shall be equal to a given triangle,


Biscet BC in E; join AE, and draw CD equal and parallel to AE; then join AD, and AECD is the parallelogram required. Plate 3. fig. 40.


To make a triangle equal to a given trapezium, ABCD.

Draw the diagonal BD, and through C draw CE parallel to BD, and meeting AD produced in E, join BE; and the triangle ABE is equal to the trapezium ABCD. Plate 3. fig. 41.


To make a triangle equal to an irregular polygon, ABCDE.

Draw the diagonals, CA, CE, through B,D; draw DG and BF parallel to them, meeting the base AE, produced both ways in F and G; join CF and CG; so Thall the triangle FCG be cqual to the given figure ABCDE, Plate 3. fig. 42.


To divide the area of a given circle into any number of equel parts,

by concentric circles, suppose into three equal parts.

Divide the semidiameter AC into three equal parts, in the points ayb; also bisect AC in the point x; and upon x as centre, with the radius Ax, or xC, describe the femicircle AabC; and through the points of division a,b, erect perpendiculars to meet the femicircle in a, and b; then, on C as cen-, tre, with the distances b,a, describe circles, and it is done. Plate 3. fig. 43.


The fundamental projection of the diagonal scale.

Draw a line AE, of any convenient length; divide it into 12 equal parts; complete these into parallelograms of a conveni


B. 2

ent height, by drawing parallel lines; divide the altitude of these rectangles into ten equal parts, and, through each of these parts, draw parallel lines the whole length of the scale. Divide the first division AB into ten equal parts, also CD into as many, and connect these points of division by diagonal lines, and the scale is finished.

In taking measures from the diagonal scale--- If the large divisions be reckoned units, the small divisions from A to B will be decimals. If the great divisions be 10, each of the small divisions is an unit; and if the great divisions be 100, then each of the small divisions is 10, and each division in the altitude is.

an unit.

If it were required to take off 456 from the scale ; with one foot of the compasses on 4, extend the compafles till you have 4 of the great divifions and 5 of the lefler ; then flide up your compafles with a parallel motion till you come to 6 on the pas rallel lines, and you have the extent required.

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The construction of the line of chords, fines, tangents, and secants,

About the centre C, with any convenient radius *, describe the semicircle ADB; erect the perpendicular CF, which will divide the semicircle into two quadrants, viz. AD, BD : divide the quadrant DB into nine equal parts, and upon the point B erect a perpendicular BT, then draw AD and BD.

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On B as centre, transfer each of these divisions in the quadrant DB, to the straight line BD; then is BD a line of chords.

From the points 10, 20, 30, &c. in the quadrant BD, drop perpendiculars upon the diameter AB; transfer the perpendi

cụlais The degrees are numbered from B to D.

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