GEOMETRICAL PROBLEMS. PROBLEM I. To bisect a right line, A B, Fig. 18. Open the dividers to any distance more than half the line A B, and with one foot in A, describe the arc CFD; with the same opening, and one foot in B, describe the arc CGD, meeting the first arc in C and D; from C to D draw the right line CD, cutting AB in E, which will be equally distant from A and B. PROBLEM II. At a given point A, in a right line EF, to erect a perpendicular, Fig. 19. From the point A, lay off on each side, the equal distances AC, AD; from C and D, as centres, with any interval greater than AC or AD, describe two arcs intersecting each other in B; from A to B, draw the line ; AB, which will be the perpendicular required. PROBLEM IIT. To raise the perpendicular on the end B of a right line AB, Fig. 20. Take any point D not in the line AB, and with the distance from D to B, describe a circle cutting A B in E; from E through D draw the right line EDC, cutting the periphery in C, and join CB, which will be perpendicular to AB. PROBLEM IV. " To let fall a perpendicular upon a given line BC, from a given point A, without it, Fig. 21. In the line BC take any point D, and with it as a centre and distance DA describe an arc AGE, cutting BC in G, with G as a centre, and distance GA, describe an arc cutting AGE in E, and from A to E draw the line AFE; then AF will be perpendicular to AB. PROBLEM V. Through a given point A to draw a right line AB, parallel to a given right line CD, Fig. 22. From the point A to any point F, in the line CD, draw the right line AF; with F as a centre and distance FA, describe the arc A E, and with the same distance and centre A describe the arc FG; make FB equal to AE, and through A and B draw the line AB, and it will be parallel to CD. PROBLEM VI. At a given point B, in a given right line LG, to make an angle equal to a given angle A, Fig. 23. With the centre A and any distance A E, describe the arc DE, and with the same distance and centre B describe the arc FG; make HG equal to D E, and through B and H draw the line BH; then will the angle HBG be equal to the angle A. PROBLEM VII. . To bişect any right lined angle BAC, Fig. 24. In the lines AB and AC, from the point A set off equal distances A D and AE; with the centres D and E and any distance more than half D E describe two arcs cutting each other in F; from A through F draw the line A G, and it will bisect the angle BAC. PROBLEM VIII. To make a triangle of any three right lines D, E and F, of which any two together must be greater than the third, Fig. 25. Make A B equal to D; with the centre A and distance equal to E, describe an arc, and with the centre B and distance equal to F describe another arc, cutting the former in C; draw AC and BC, and ABC is the triangle required. PROBLEM IX. Upon a given line AB to describe a square, Fig. 26. At the end B of the line AB, by problem 3, erect the perpendicular BC, and make it equal to AB; with A and C as centres, and distance A B or BC describe two arcs cutting each other in D; draw AD and CD, then will ABCD be the square required. PROBLEM X. To describe a circle that shall pass through the angular points A, B and C, of a triangle ABC, Fig. 27. By problem 1, bisect any two of the sides, as AC, BC, by the perpendiculars DE and FG; the point H where they intersect each other will be the centre of the circle; with this centre and the distance from it to either of the points A, B, or C, describe the circle. PROBLEM XI. To divide a given right line AB into any number of equal parts, Fig. 28. Draw the indefinite right line A P, making any angle with AB, also draw BQ parallel to AP, in each of which, take as many equal parts AM, MN, &c. Bo, on, &c. as the line AB is to be divided into; then draw Mm, Nn, &c. intersecting AB in E, F, &c. and it is done. PROBLEM XII. To make a plane diagonal scale, Fig. 29. Draw eleven lines parallel to, and equidistant from each other; cut them at right angles by the equidistant lines BC; EF; 1, 9; 2, 7 ; &c. then will BC, &c. be divided into ten equal parts; divide the lines EB, and FC, each into ten equal parts, and from the points of division on the line EB, draw diagonals to the points of division on the line FC: thus join E and the first division on FC, the first division on EB and the second on FC, &c. Note.-Diagonal scales serve to take off dimensions or numbers of three figures. If the first large divisions be units, the second set of divisions, along EB, will be will be 100th parts. If HE be tens, EB will be units, and BC will be 10th parts. If HE be hundreds, BE will be tens, and BC units. And so on, each set of divisions being tenth parts of the former ones. For example, suppose it were required to take off 242 from the scale. Extend the dividers from E to 2 towards H; and with one leg fixed in the point 2, extend the other till it reaches 4 in the line EB; move one leg of the dividers along the line 2, 7, and the other along the line 4, till they come to the line marked 2, in the line BC, and that will give the extent required. |