PROB. XIV. PL. 3. fig. 2. To divide a right line AB, in the point E, so that AE shall have the same proportion to EB, as two given lines C and D have. Draw an indefinite blank line, AF, to the extremity of the line AB, to make with it any angle; lay the line C, from A to C; and D, from C to D; and join the points B and D by the line BD ; through Cdraw CE parallel to BĎ (by prob. 8.) so is E the point of division. For, by cor. 1. theo. 20. AC:AD::AE : AB. Or, C:D:: AE: EB. To describe a circle about a triangle ABC, or f which is the game thing ) through any three points, A, B, C, which are not situated in a right line. By prob. 4. Bisect the line AC by the perpendicular DE, and also CB, by the perpendiculár FG, the point of intersection H, of these perpendiculars, is the centre of the circle required, from which take the distance to any of the three points A, B, C, and describe the circle ABC, and it is done. For, hy cor. to theo. 8. The lines DE and FG, must each pass through the centre, therefore, their point of intersection H, must be the centre. SCHOLIUM. By this method the centre of a circle may be found, by having only a segment of it given. PROB. XVI. PL. 3. fig. 4. To make an angle of any number of degrees, at the point 1, of the line AB, suppose of 45 degrees. a From a scale of chords take 60 degrees, for 600 is equal to the radius (by cor. theo. 15.) and with that distance from A, as a centre, describe a circle from the line AB; take 45 degrees, the quantity of the given angle, from the same scale of chords, and lay it on that circle from a to b, through A and be draw the line AbC; and the angle A will be an angle of 45 degrees, as required. If the given angle be more than 900, take its half (or divide it into any two parts less than 90) and lay them after each other on the arc, which is described with the chord of 60 degrees; through the extremity of which, and the centre, let a line be drawn, and that will form the angle required, with the given line. PROB. XVII. PL. 3. fig. 5. To measure a given angle, ABC. If the lines which include the angle, be not as long as the chord of 600 on your scale, produce them to that or a greater length, and between them so produced, with the chord of 60° from B, describe the arc e d; which distance e d, measured on the same line of chords, gives the quantity of the angle BAC, as required; this is plain from def. 17. PROB. XVIII. Pl. 3. fig. 6. To make a triangle BCE equal to a given quadrilateral figure ABCD. Draw the diagonal AC, and parallel to it (by prob. 8.) DE, meeting AB produced in E ; then draw CÉ, and ECB will be the triangle required. For the trianglès ADC, AEC, being upon the same base AC, and under the same parallel ED, (hy cor. to theo. 13.) will be equal, therefore if ABC be added to each, then ABCD=BEC. PROB. XIX. PL. 3. fig. 7. To make a triangle DFH, equal to a given five-sided figure ABCDE. Draw DA and DB, and also EH and CF, parallel to them (by prob. 8.) meeting AB produced in Hand F; then draw DH, DF, and the triangle HDF is the one required. For the triangle DEA=DHA, and DBC= DFB (by cor. to then. 13.) therefore by adding these equations, DEA + DBC=DHA + DFB if to each of these ADB be added ; then DEA + ADB+DBC=ABCDE=(DHA+ABD+DFB, DHE PROB. XX. PL. 3. fig. 8: To project the lines of chords, sines, tangents and secants, with any radius. L On the line AB, let a semicircle ADB be described ; let CDF be drawn perpendicular to this line from the centre C ; and the tangent BE perpendicular to the end of the diameter; let the quadrants, AD, DB, be each divided into 9 equal parts, every one of which will be 10 degrees; if i hen from the centre C, lines be drawn through 10, 20, 30, 40, &c. the divisions of the quadrant BD, and continued to BE, we shal there have the tangents of 10, 20, 30, 40, &c. and the secants ( 10, 20, C 30, &c. are transferred to the line CF, by describing the arcs 10, 10: 20, 20 : 30, 30, &c. If from 10, 20,30,&c. the divisions of the quadrant BD, there be let fall perpendiculars, let these be trarisferred to the radius CB, and we shall have the sines of 10, 20, 30, &c. and if from A we describe the arcs 10, 10: 20, 20 : 30, 30, &c. from every division of the arc AD; we shall have a line of chords. The same way we may have the sine, tangent, &c. to every single degree on the quadrant, by subdividing each of the 9 former divisions into 10 equal parts. By this method the sines, tangents, Sc. may be drawn to any radius ; and then, after they are transferred to lines on a rule, we shall have the scales of sines, tangents, &c. ready for use. MATHEMATICAL DRAWING INSTRUMENTS. a The strictness of germetrical demonstration admits of no other instruments, than a rule and a pair of compasses. But, in proportion as the practice of geometry was extended to the different arts, either connected with, or dependent upon it, new instruments became necessary, some to answer peculiar purposes, some to facilitate operation, and others to promote accuracy. As almost every artist, whose operations are connected with mathematical designing, furnishes himself with a case of drawing instruments suited to his peculiar purposes, they are fitted upin various modes, some containing more, others, lewer instruments. The smallest collection put into a case, consists of a plane scale, a pair of conipasses with a moveable leg, and twospare points, which may be applied occasionally to the compasses; one of these points is to hold ink; the other, a porte crayon, for holding a piece of black-lead pencil. What is called a full pocket case, contains the following instruments. A pair of large compasses with a moveable point, an ink point, a pencil point, and one for dotting; either of those points may be inserted in the compasses, instead of the moveable leg. A pair of plain compasses somewhat smaller than those with the moveable leg. A pair of bow compasses. A drawing pen with a protracting pin in the upper part. A sector. a Large collections are called, magazine cases of instruments ; these generally contain A pair of six inch compasses with a moveable leg, an ink point, a dotting point, the crayon point, so contrived as to hold a whole pencil, two additional pieces to lengthen occasionally one leg of the compasses, and thereby enable them to measure greater extents, and describe circles of a larger radius. A pair of hair compasses. |