called the altitude of the parallelogram, or triangle. Thus AD is ihe base of the parallelogram ABEC, or triangle ABC, and CD is the altitude. Fig. 15. 32. All plane figures contained under more than four sides, are called polygons; of which those having five sides, are called pentagons; those having six sides, hexagons, and so on. 33. A regular polygon is one whose angles, as well as sides, are all equal. 34. A circle is a plane figure, bounded by one curve line called the circumference or periphery, every part of which is equally distant from a certain point within the circle; and this point is called the centre, Fig. 16. 35. The radius of a circle is a straight line drawn from the centre to the circumference, as CB, Fig. 17. a 36. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference, as AE, Fig. 17. It divides the circle into two equal parts, called semicircles. 37. A quadrant is one quarter of a circle, as ACB, Fig. 17. a Note. The fourth part of the circumference of a circle is also called a quadrant. 38. A segment of a circle is the figure contained by a right line, and the part of the circumference it cuts off: thus AEBA and AEDA are segments of the circle ABED, Fig. 16. 59. An arc of a circle is any part of the circumference; 40. Ratio is a mutual relation between two quantities of the same kind with respect to magnitude. Note.-A ratio is generally expressed, either by two numbers or by two right lines. 41. When two quantities have the same ratio as two other quantities, the four quantities taken in order are called proportionals; and the last is said to be a fourth proportional to the other three. 42. When three quantities of the same kind are such that the first has to the second the same ratio which the second has to the third, the third is called a third proportional to the first and second, and the second is called a mean proportional between the first and third. a GEOMETRICAL PROBLEMS. PROBLEM I. To bisect a right line, AB, Fig. 18. Open the dividers to any distance more than half the line AB, 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 per. pendicular, Fig. 19. From the point A, lay off on each side, the equal distances AC, AD; from C and D, as centres, with any radius 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 III. To raise a 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 AB 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 AE, 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 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 AE, describe the arc DE, and with the same distance and centre B describe the arc FG; make HG equal to DE, and through B and H draw the line BH; then will the angle HBG be equal to the angle A. PROBLEM VII. To bisect any right lined angle BAC, Fig. 24. In the lines AB and AC, from the point A, set off equal distances AD and AE; with the centres D and E and any distance more than half DE, describe two arcs cutting each other in F; from A through F draw the line AG, and it will bisect the angle BAC. PROBLEM VIII. To describe a triangle that shall have its sides respectively equal to three right lines D, E and F, of which any two must be together greater than the third, Fig. 25. Make AB 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 AB or BC describe two arcs cutting each other in D; draw AD and CD, then will ABCD be the square required. a 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 l, 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 AP, making an 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 plain 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 |