10. A plane triangle is a space included by three right lines, 2nd has three angles. 11. A right angled triangle has one right angle, as A BC. The side A C, opposite the right angle, is called the hypothenuse; the sides A B and B C are respectively called the B base and perpendicular. B 12. An obtuse angled triangle has one obtuse angle, as the angle at B. D 13. An acute angled triangle has all its é three angles acute, as D. E 14. An equilateral triangle has three equal sides, and three equal angles, as E. A 15. An isosceles triangle has two equal sides, and the third side eater or less than each of cc the equal sides as F. C 16. A quadrilateral figure is a space bounded by four right lines, and has four angles; when its opposite sides are equal, it is called a parallelogram. ili G 17. A square has all its sides equal, and all its angles right angles, as G. 18. A rectangle is a right angled parallelogram, whose length exceeds its breadth, as B, (see figure to definition 2). I 19. A rhombus is a parallelogram having all its sides and each pair of its opposite angles equal, as I. be 20. A rhomboia is a parallelogram having its opposite sides and angles equal, as K. K B 21. A trapezium "is bounded by four straight lines, no two of which are parallel to each other, as L A line connecting any two of its angles is called the diagonal, A. as B. 22. A trapezoid is a quadrilateral, having two of its opposite sides parallel, and the re M maining two not, as M. 23. Polygons have more than four sides, and receive particular names, according to the number of their sides. Thus, a pentagon has five sides; a hexagon, six; a heptagon, seven ; an octagon, eight ; &c. They are called regular polygons, when all their sides and angles are equal, otherwise irregular blue polygons. B equa T A ch D right osite а 7. 24. A circle is a plain figure, bounded by a curve line, called the circumference, which is ill is everywhere equidistant from a point C within, called the centre. 25. An arc of a circle is a part of the circumference, as A B. 26. The diameter of a circle is a I equal parts, each of which is called a 27. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees, and each degree into 60 minutes, each minute into 60 seconds, &c. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees. Fing 28. The measure of an angle is an arc of gles any circle, contained between the two lines А which form the angle, the angular point being the centre; and it is estimated by the ing number of degrees contained in that arc : thus the arc A B, the centre of which is C, dal it ic thus written 42° 29' 48". Eing Oui llel care E B m PROBLEMS IN PRACTICAL GEOMETRY. (In solving the five following problems only a pair of common compasses and a straight edge are required; the problems beyond the fifth require a scale of equal parts; and the two last a line of chords: all of which will be found in a common case of instruments.) PROBLEM I. To divide a given straight line A B into two equal parts. From the centres A and B, with any radius, or opening of the compasses, greater than half A B, describe two arcs, cutting each other in C and D; draw CD, and it will cut A B in the middle point E. PROBLEM II. At a given distance E, to draw a straight line CD, parallel to a given straight line A B. E From any two points m and r, in the line A B, with a distance equal to с D E, describe the arcs n and s :-draw CD to touch these arcs, without cutA B ting them, and it will be the parallel required. Note. This problem, as well as the following one, is usually performed by an instrument called the parallel ruler. PROBLEM III. Through a given point r, to draw a straight line CD parallel to a given straight line A B. From any point n in the line A B, with SD the distance nr, describe the arc rm:—from centre r, with the same radius, describe the arc ns:-take the arc mr in the compasses, Am and apply it from n to s:--through r and s draw C D, which is the parallel required. PROBLEM IV. From a given point P in a straight line A B to erect a per. pendicular. I When the point is in or near the middle of the line. On each side of the point P take any two equal distances, Pm, Pn; from the points m and n, as centres, with any radius greater than P m, describe two arcs cutting each other in C; through C, draw CP, and it will A be the perpendicular required. P B P 2. When the point P is at the end of the line. With the centre P, and any radius, describe the arc nr 8;—then with the same radius, and taking n and r as centres, cut off the equal arcs n r and rs:—again, with centres r and s, describe arcs interBecting in 0:-draw CP, and it will be the perpendicular required. Note. This problem and the following one are usually done with an instrument called the square. PROBLEM V. From a given point C to let fall a perpendicular to a given line. 1 When the point is nearly opposite the middle of the line. From C, as a centre, describe an arc to cut A B in m and n ;-with centres m and n, and the same or any other radius, describe arcs intersecting in o: through Cand o draw Co, the perpendicular required. с 2. When the point is nearly opposite the end of the line. From C draw any line C m to meet B A, in any point m;-bisect C m in n, and with the centre n, and radius Cn, or mn, describe an arc cutting B A in P. Draw CP for the perpendicular required. PROBLEM VI. To construct a triangle with three given right lines, any two of which must be greater than the third. (Euc. I. 22.) Let the three given lines be 5, 4 and 3 C yards. From any scale of equal parts lay off the base A B = 5 yards; with the centre A and radius AČ = 4 yards, describe an arc; with centre B and radius Á CB = 3 yards, describe another arc cutting the former arc in C: draw A C and CB; then A B C is the triangle required. PROBLEM VII. Given the base and perpendicular, with the place of the latter on the base, to con- с struct the triangle. Let the base AB А B В 7, the per A A perpendicular DC, which make = 3:-draw A C and CB; then A B C is the triangle required. PROBLEM VIII. To describe a syuare, whose side shall be of a given length. Let the given line A B be three chains. At Di the end B of the given line erect the perpendicular B C, (by Prob. IV. 2.) which make AB:—with A and C as centres, and radius A B, describe arcs cutting each other in D: draw A D, DC and the square will be comB pleted. PROBLEM IX. To describe a rectangled parallelogram having a given length and breadth. Let the length A B = 5 chains, and the breadth BC 2. At B erect the perpendicular BC, and make it = 2: with the centre A and radius B C deB scribe an arc; and with centre C and radius A B, describe another arc, cutting the former in D:join A D, DC to complete the rectangle. PROBLEM X. The base and two perpendiculars being given to construct a trapezoid Let the base AB = 6, and the D 0 perpendiculars A D and BC, 2 and 3 chains respectively. Draw the perpendiculars A D, DC, as given B above, and join D C, thus complet ing the trapezoid. PROBLEM XI. Draw the diagonal D B, and C E pa- ABCD. PROBLEM XII. A E |