through A draw AC parallel to DE; then EC will be the fourth proportional required. Plate 2. fig. 27.

PROBLEM XI.

To find a mean proportional between two given lines, AB, BC.

MAKE AC equal ABXBC; bifect the line AC in the point D, with the centre D, and radius DA, or DC, describe the fe micircle AEC; erect the perpendicular BE, and it will be the mean proportional required. Plate 2. fig: 28.

PROBLEM XII.

To make a triangle with three given lines, AB, BC, CA.

TAKE any line AB for the bafe line; on the centre A, with the radius AC, describe an arch; on the centre B, with the radius BC, describe another arch, cutting the former in C; join CA and CB, and ABC is the triangle required. Plate 2. fig. 29.

PROBLEM XIII.

To measure any given angle from a line of chords.

FROM the angular point A, with the chord of 60° for a radius, describe an arch cutting the containing fides, produced, if neceffary, in the points D,E; take the diftance DE in your compaffes, and apply it to the line of chords. Thus the Thus the quantity of any angle is obtained.

Note, When the angle to be measured is obtufe, it must be taken off at twice. Thus, let the angle be 120°; first take 90° and 30°, or 60° and 60°, either of which will do.

PRO

PROBLEM XIV.

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

WITH the centre A, and radius 65° defcribe an arch, cutting AB in C; then take the proposed number of degrees in your compaffes, 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.

PROBLEM XV.

Upon a given line AB, to defcribe a fquare.

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

PROBLEM XVI.

To defcribe a parallelogram of a given length and breadth.

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

PROBLEM XVII.

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

PROBLEM XVIII.

About any given triangle to defcribe 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.

PROBLEM XIX.

To defcribe a circle in or about a given fquare.

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

PROBLEM XX.

To defcribe a fquare 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 infcribed fquare draw tangents, and they will form the circumfcribed fquare, abc d. Plate 3. fig. 37.

PROBLEM XXI.

To defcribe a circle through three given points, A,B,C, which are not in the fame ftraight line.

JOIN the middle point to the other two; bisect their distan ces 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,C1 Plate 3. fig. 38.

PROBLEM XXII.

A fegment of a circle being given, to defcribe the circle of which it is the fegment.

Draw AC the chord, and bifect 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, defcribe the circle, and it is done. Plate 3+ fig. 39. Or, take any three points in the fegment, and bifect their distances, and the bifecting lines will interfect each other in the centre, as in prob. 21.

PROBLEM XXIII.

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

ABC.

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

PRO

PROBLEM XXIV.

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.

PROBLEM XXV.

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; fo fhall the triangle FCG be equal to the given figure ABCDE, Plate 3. fig. 42.

PROBLEM XXVI.

To divide the area of a given circle into any number of equal parts, by concentric circles, fuppofe into three equal parts.

Divide the femidiameter AC into three equal parts, in the points a,b; alfo bifect AC in the point x; and upon x as centre, with the radius Ax, or xC, defcribe the femicircle AabC; and through the points of divifion a,b, erect perpendiculars to meet the femicircle in a, and b; then, on C as centre, with the diftances b,a, defcribe circles, and it is done. Plate 3. fig. 43.

PROBLEM XXVII.

The fundamental projection of the diagonal fcale.

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