Broblem 12. To make a Square whose Sides shall be equal to a given Right Line. Let F be the Right Line, with which it is required to make the Square A B CD. Construction. First, take the Line F in the Compaffes, and set it from A to B. Next, on the Point B erect the Perpendicular BD, by Problem 3, equal to the Line F. With the fame Wideness of the Compasses, setting one Foot in D, describe the Arch 00; and with the fame Extent of the Compaffes, fetting one Foot in A, describe the Arch e e, cutting the former in C. Lastly, from C draw Lines to A and D, which will complete the Square required. Architects, Surveyors, &c. make Ufe of this Problem in laying out Ground for Buildings, or for Pleasure. Problem 13. To make a Parallelogram, or long Square, whose Length and Breadth shall be equal to two Right Lines given. Let the two Lines given be A and B; the former the Length, and the latter the Breadth of the Parallelogram required to be made. Construction. First, take with the Compasses the Length of the Line A, and set it from C to D; upon the Point D erect a Perpendicular DE, equal to the Line B. Then take the Line A again in the Compasses, and setting one Foot in E, describe the Arch 00; next, take the Line B in the Compasses, set one Foot in C, and describe the Arch e e, interfecting the other in F. Lastly, draw the Lines FC and FE; and CDEF will be the Parallelogram re quired. This Problem, like the last, is equally useful to Surveyors, Architects, and Mechanics. Problem 14. To make a Rhombus, each of whose Sides shall be equal to a given Right Line; and whose acute Angle shall also be equal to an Angle given. Let the Line given be A B, and the Acute Angle CA B, to delineate a Rhombus, whose Sides and Acute Angles fhall be equal thereto. Construction. First, make the Angle at D equal to the Angle A, by Problem 8. Then take the Line AB in the Compasses, and fetting one Foot in D, describe the Arch a b, cutting the Side D Fin F. Next, on the Points F and E, with the fame Extent A B, describe the two Arches oo, and ee, interfecting each other at G. Lastly, draw the Lines F G and E G, fo will the Rhombus DEGF be formed, corresponding with the Line and Angle given. Note. In like Manner any Rhomboides, whose Length, Breadth, and Acute Angle are given, may be easily conftructed. Problem 15. To divide the Circumference of a Circle into any Number of Parts not exceeding ten. Let AFCG be a Circle given to be so divided. Construction. First, draw the Diameter AEC, which will divide the Circumference into two equal Parts. Take the Semi-diameter AE or CE in the Compasses, and setting one Foot in A, with the other make in the Circumference the Points BD, which Line BD will divide the Circle into 3 equal Parts. Divide the Diameter AC into two equal Parts by Problem 1. at right Angles with FG, and draw A F, which will divide the Circle into 4 equal Parts. Next, set one Foot of the Compaffes in H, where the third Part BD cuts the Diameter A C, and extend the other to F, and defcribe the Arch F1; then draw the Line FI, which will divide the Circle into 5 equal Parts. The Semidiameter A E or CE will always divide the Circle into 6 equal Parts. Half the third Part, i. e. BH or HD, will divide the Circle into 7 equal Parts. Divide the Angle AEF, by Problem 7, into two equal Parts with the Line EK, cutting the Limb in K; then draw AK, and that Line will divide the Circle into 8 equal Parts. Take one third of the Arch BAD, as DL, and that Space will divide the Circle into 9 equal Parts. Lastly, take the Line EI in the Compasses, and that Distance will divide he Circle into 10 equal Parts. |