GEOMETRY.

GEOMETRY is a Science which treats of the properties of Magnitude.

PART I.

Geometrical Definitions.

1. A Point is a small Dot; or, Mathematically considered, is that which has no parts, being of itself indivisible.

2. A Line has length but no breadth.

3. A Superficies or Surface, called also Area, has length and breadth, but no thickness.

4. A Solid has length, breadth, and thickness.

5. A Right Line is the shortest that can be drawn between two Points.

6. The inclination of two Lines meeting one another, or the opening between them, is called an Angle. Thus at B. Fig. 1. is an Angle, formed by the meeting of the Lines AB and BC.

Fig. 1.

A

7. If a right Line CD. Fig. 2. fall upon another Right Line AB, so as to incline to neither side, but make the Angles on each side equal, then those Angles are called Right Angles; and the Line CD is said to be Perpendicular to the other Line.

B

Fig. 2.

D

B

Fig. 3.

8. An Obtuse Angle is greater than a Right

Angle; as ADE. Fig. 3.

9. An Acute Angle is less than a Right Angle; as EDB. Fig. 3.

Note. When three letters are used to express an Angle, the middle letter denotes the angular Point.

10. A Circle is a round Figure, bounded by a Line equally distant from some Point, which is called the Centre. Fig. 4.

11. The Circumference or Periphery of a Circle is the bounding Line; as ADEB. Fig. 4.

12. The Radius of a Circle is a Line drawn from the Centre to the Circumference; as CB. Fig. 4. Therefore all Radii of the same Circle are equal.

13. The Diameter of a Circle is a Right Line drawn from one side of the Circumference to the other, passing through the Centre; and it divides the Circle into two equal parts, called Semicircles; as AB or DE. Fig. 5.

14. The Circumference of every Circle is supposed to be divided into 360 equal parts, called Degrees; and each Degree into 60 equal parts, called Minutes; and each Minute

into 60 equal parts, called Seconds; and these into Thirds, &c.

Note. Since all Circles are divided into the same number of Degrees, a Degree is not to be accounted a quantity of any determinate length, as so many inches or feet, &c. but is always to be reckoned as being the 360th part of the Circumference of any Circle, without regarding the bignessof the Circle.

15. An Arch or Arc of a Circle is any part of the Circumference; as BF or FD. Fig. 5; and is said to be an Arch of so many Degrees as it contains parts of 360 into which the whole

16. A Chord is a Right Line drawn from one end of an Arch to the other, and is the measure of the Arch; as HG is the Chord of the Arch HIG. Fig. 6.

A

Note. The Chord of an Arch of 60 degrees is equal in length to the Radius of the Circle of which the Arch is a part.

17. The Segment of a Circle is a part of a Circle, cut off by a Chord; thus the space comprehended between the Arch HIG and the Chord HG is called a Segment. Fig. 6.

18. A Quadrant is one quarter of a Circle; as ACB. Fig. 6. 19. A Sector of a Circle is a space contained between two Radii and an Arch less than a Semicircle; as BCD or ACD. Fig. 6.

20. The Sine of an Arch is a Line drawn from one end of the Arch, perpendicular to the Radius or Diameter drawn through the other end: Or, it is half the Chord of double the Arch; thus HL is the Sine of the Arch HB. Fig. 7.

21. The Sines on the same Diameter in- A crease in length till they come to the Centre, and so become the Radius. Hence it is plain that the Radius CD Fig. 7. is the greatest possible Sine, or Sine of 90 Degrees.

Fig. 7.

F

K

D

22. The Versed Sine of an Arch is that part of the Diameter or Radius which is between the Sine and the Circumference thus LB is the Versed Sine of the Arch HB. Fig. 7.

23. The Tangent of an Arch is a Right Line touching the Circumference, and drawn perpendicular to the Diameter; and is terminated by a Line drawn from the Centre through the other end of the Arch; thus BK is the Tangent of the Arch BH. Fig. 7.

Note. The Tangent of an Arch of 45 Degrees is equal in length to the Radius of the Circle of which the Arch is a part.

through one end of the Arch till it meets the Tangent; thus CK is the Secant of the Arch BH. Fig. 7.

25. The Complement of an Arch is what the Arch wants of 90 Degrees, or a Quadrant; thus HD is the Complement of the Arch BH. Fig. 7.

26. The Supplement of an Arch is what the Arch wants of 180 Degrees, or a Semicircle; thus ADH is the Supplement of the Arch BH. Fig. 7.

27. The Sine, Tangent or Segant 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 Co-Sine, Co-Tangent, and CoSecant 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 A. distant from each other; as AB and CD.

Fig. 8.

Fig. 8.

B

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.

A

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

Fig. 10.

A

Fig. 11.

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.

Fig. 12.

Fig. 13.

37. An Acute Angled Triangle has all its Angles Acute. Fig. 9, or 10.

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.