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27. The Sine, Tangent or Secant of the Complement of any Arch is called the Co-Sine, Co-Tangent or Co-Secant of the Arch; thus FH is the Sine, DI the Tangent and CI the Secant of the Arch DH; or they are the CoSine, Co-Tangent and Co-Secant of the Arch BH. Fig. 7. 28. The measure of an Angle is the Arch of a Circle contained between the two Lines which form the Angle, the angular Point being the Centre; thus the Angle HCB. Fig. 7. is measured by the Arch BH: and is said to contain so many Degrees as the Arch does.

Note. An Angle is esteemed greater or less according to the opening of the Lines which form it, or as the Arch intercepted by those Lines contains more or fewer Degrees. Hence it may be observed, that the bigness of an Angle does not depend at all upon the length of the including Lines; for all Arches described on the same Point, and intercepted by the same Right Lines, contain exactly the same number of Degrees, whether the Radius be longer or shorter.

29. The Sine, Tangent or Secant of an Arch is also the Sine, Tangent or Secant of the Angle whose measure the Arch is.

30. Parallel Lines are such as are equally distant from each other; as AB and CD. Fig. 8.

31. A Triangle is a Figure bounded by three Lines; as ABC. Fig. 9.

32. An Equilateral Triangle has its three sides equal in length to each other. Fig. 9.

33. An Isocles Triangle has two of its sides equal, and the other longer or shorter. Fig. 10.

34. A Scalene Triangle has three unequal Sides. Fig. 11.

35. A Right Angled Triangle has one Right Angle. Fig. 12.

36. An Obtuse Angled Triangle has one Obtuse Angle. Fig. 13.

37. An Acute Angled Triangle has all its Angles

38. Acute and Obtuse Angled Triangles are called Oblique Angled Triangles, or simply Oblique Triangles; in which the bottom Side is generally called the Base and the other two, Legs.

39. In a Right Angled Triangle the longest Side is called the Hypothenuse, and the other two, Legs, or Base and Perpendicular.

Note. The three Angles of every Triangle being added together will amount to 180 Degrees; consequently the two Acute Angles of a Right Angled Triangle amount to 90 Degrees, the Right Angle being also 90.

40. The perpendicular height of a Triangle is a Line drawn from one of the Angles to its opposite Side; thus the dotted Line AD. Fig. 14. is the perpendicular height of the Triangle ABC.

Note. This Perpendicular may be drawn from either of the Angles; and whether it falls within the Triangle, or on one of the Lines continued beyond the Triangle, is immaterial.

41. A Square is a Figure bounded by four equal Sides, and containing four Right Angles. Fig. 15.

42. A Parallelogram, or Oblong Square, is a Figure bounded by four Sides, the opposite ones being equal and the Angles Right. Fig. 16.

43. A Rhombus is a Figure bounded by four equal Sides, but has its Angles Oblique. Fig. 17.

44. A Rhomboides is a Figure bounded by four Sides, the opposite ones being equal, but the Angles Oblique. Fig. 18.

45. The perpendicular height of a Rhombus or Rhomboides is a Line drawn from one of the Angles to its opposite Side; thus the dotted Lines AB. Fig. 17. and Fig. 18. represent the perpendicular height of the Rhombus and Rhomboides.

46. A Trapezoid is a Figure bounded by four Sides, two of which are parallel though of unequal lengths. Fig. 19. and Fig. 20.

Note. Fig. 19. is sometime called a Right Angled

47. A Trapezium is a figure bounded by four unequal Sides. Fig. 21.

48. A Diagonal is a Line drawn between two opposite Angles; as the Line AB. Fig. 21.

49. Figures which consist of more than four Sides are called Polygons; if the sides are equal to each other they are called regular Polygons, and are sometimes named from the number of their sides, as Pentagon, or Hexagon, a Figure of five or six Sides, &c; if the Sides are unequal they are called irregular Polygons.

PART II.

Geometrical Problems.

PROBLEM I. To draw a Line parallel to another Line at any given distance; as at the Point D, to make a Line, parallel to the Line AB. PLATE 1. Fig. 22.

With the Dividers take the nearest distance between the Point D and the given Line AB; with that distance set one foot of the Dividers any where on the Line AB, as at E, and draw the Arch C; through the Point D draw a Line so as just to touch the top of the Arch C.

A more convenient way to draw parallel Lines is with a parallel Rule.

PROBLEM II. To bisect a given Line; or to find the middle of it. Fig. 23.

Open the Dividers to any convenient distance, more than half the given Line AB, and with one foot in A describe an Arch above and below the Line, as at C and D; with the same distance, and one foot in B describe Arches to cross the former; lay a Rule from C to D, and where the Rule crosses the Line, as at E, will be

PROBLEM III. To erect a Perpendicular from the end, or any part of a given Line. Fig. 24.

Open the Dividers to any convenient distance, as from D to A, and with one foot on the Point D, from which the Perpendicular is to be erected, describe an Arch, as AEG; set off the same distance from A to E and from E to G; upon E and G describe two Arches to intersect each other at H; draw a Line from H to D, and one Line will be perpendicular to the other.

Note. There are other methods of erecting a Perpendicular, but this is the most simple.

PROBLEM IV. From a given Point, as at C, to drop a Perpendicular on a given Line AB. Fig. 25.

With one foot of the Dividers in C describe an Arch to cut the given Line in two places, as at F and G; upon F and G describe two Arches to intersect each other below the Line as at D; lay a Rule from C to D and draw a Line from C to the given Line.

Perpendiculars may be more readily raised and let fall, by a small Square made of Brass, Ivory or Wood.

PROBLEM V. To make an Angle at E, equal to a given Angle ABC. Fig. 26.

Open the Dividers to any convenient distance, and with one foot in B describe the Arch FG; with the same distance and one foot in E, describe an Arch from H; measure the Arch FG, and lay off the same distance on the Arch from H to I; draw a Line through I to E, and the Angles will be equal.

PROBLEM VI. To make an Acute Angle equal to a given number of Degrees, suppose 36. Fig. 27.

Draw the Line AB to any convenient length; from a Scale of Chords take 60 Degrees with the Dividers, and with one foot in B describe an Arch from the Line AB; from the same Scale take the given number of Degrees, 36, and lay it on the Arch from C to D; draw a line from B through D, and the Angle at B will be an Angle of 36

PROBLEM VII. To make an Obtuse Angle, suppose of 110 Degrees. Fig. 28.

Take a Chord of 60 Degrees as before, and describe an Arch greater than a Quadrant; set off 90 Degrees from B to C, and from C to E set off the excess above 90, which is 20; draw a Line from G through E and the Angle will contain 110 Degrees.

Note. In a similar manner Angles may be measured ; that is, with a Chord of 60 Degrees describe an Arch on the angular Point, and on a Scale of Chords measure the Arch intercepted by the Lines forming the angle.

A more convenient method of making and measuring Angles is to use a Protractor instead of a Scale and Dividers.

PROBLEM VIII. To make a Triangle of three given Lines, as BO, BL, LO. Fig. 29.

Draw the Line BL from B to L; from B, with the length of the Line BO, describe an Arch as at 0; from L, with the length of the Line LO, describe another Arch to intersect the former; from O draw the Lines OB and OL, and BOL will be the Triangle required.

PROBLEM IX. To make a Right Angled Triangle, the Hypothenuse and Angles being given. Fig. 30.

Suppose the Hypothenuse CA 25 Rods or Chains, the angle at C 35o 30' and consequently the Angle at A 54° 30'. See Note after the 39th Geometrical Definition.

Note. When Degrees and Minutes are expressed, they are distinguished from each other by a small Cypher at the right hand of the Degrees, and a Dash at the right hand of the Minutes; thus 35° 30' is 35 Degrees and 30 Minutes.

Draw the Line CB an indefinite length; at C make an Angle of 35° 30'; through where that number of Degrees cuts the Arch draw the Line CA 25 Rods, which must be taken from some Scale of equal parts;

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