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


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

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


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



line AB for the base 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.


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 necessary, in the points D,E ; take the distance DE in your compafles, and apply it to the line of chords. Thus the

quantity of any angle is obtained. Noie, When the angle to be measured is obtuse, 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.

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

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 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.


Upon a given line AB, to describe a square.

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.

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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 ADBC, 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 tridngle 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 shall the triangle FCG be equal 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 femidiameter AC into three equal parts, in the points agb; also bisect AC in the point x; and upon x as centre, with the radius Ax, or xC, describe the semicircle AabC; and through the points of divifion a,b, erect perpendiculars to meet the semicircle in a, and b; then, on C as centre, 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

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