AB to the line BC meeting each other in the point B, or the opening of the two lines BA and BC, is called an angle, as ABC.

Note, When an angle is expressed by three letters, the middle one is that at the angular point.

9. When the lines that form the angle are right ones, it is then called a right-lined angle, as ABC, fig. 4. If one of them be right and the other curved, it is called a mixed angle, as B. fig. 5. If both of them be curved, it is called a curved-lined or spherical angle, as C. fig. 6.

10. If a right line, CD (fig. 7.) fall upon another right line, AB, so as to incline to neither side, but make the angles ADC, CDB on each side equal to each other, then those angles are called right angles, and the line CD a perpendicular.

11. An obtuse angle is that which is wider or greater than a right one, as the angle ADE. fig. 7. and an acute angle is less than a right one, as EDB. fig. 7.

12. Acute and obtuse angles in general are called oblique angles.

13. If a right line CB. (fig. 8.) be fastened at the end C, and the other end B, be carried quite round, then the space comprehended is called a circle; and the curve line described by the point B, is called the circumference or the periphery of the circle; the fixed point C, is called its centre.

14. The describing line CB. (fig. 8.) is called the semidiameter or radius, so is any line from the centre to the circumference; whence all radii of the same or of equal circles are equal.

15. The diameter of a circle is a right line drawn thro' the centre, and terminating in opposite points of the circumference: and it divides the circle and circumference into two equal parts, called semicircles; and is double the radius, as AB or DE. fig. 8.

16. 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, fourths, &c. these parts being greater or less as the radius is.

17. A chord is a right line drawn from one end of an arc or arch (that is, any part of the circumference of a circle) to the other; and is the measure of the arc. Thus the right line HG, is the measure of the arc HBG. fig. 8.

18. The segment of a circle is any part thereof, which is cut off by a chord thus the space which is comprehended between the chord HG and the arc HBG, or that which is comprehended between the said chord HG and the arc HDAEG are called segments. Whence it is plain, fig. 8.

1. That any chord will divide the circle into two segments.

2. The less the chord is, the more unequal are the segments.

3. When the chord is greatest it becomes a diameter, and then the segments are equal; and each segment is a semicircle.*

19. A sector of a circle is a part thereof less

For the demonstration of this consult Prop. 15, Book HI. Simpson's Euclid.

than a semicircle, which is contained between two radii and an arc: thus the space contained between the two radii CH, CB, and the arc HB is a sector, fig. 8.

20. The right sine of an arc, is a perpendicular line let fall from one end thereof, to a diameter drawn to the other end: thus HL is the right sine of the arc HB.

The sines on the same diameter increase till they come to the centre, and so become the radius; hence it is plain that the radius CD is the greatest possible sine, and thence is called the whole sine.

Since the whole sine CD (fig. 8.) must be perpendicular to the diameter (by def. 20.) therefore producing DC to E, the two diameters AB and DE cross one another at right angles, and thus the periphery is divided into four equal parts, as BD, DA, AE and EB; (by def. 10.) and so BD becomes a quadrant or the fourth part of the periphery; therefore the radius DC is always the sine of a quadrant, or of the fourth part of the circle BD.

Sines are said to be of as many degrees as the arc contains parts of 360: so the radius being the sine of a quadrant, becomes the sine of 90 degrees, or the fourth part of the circle, which is 360 degrees.

21. The versed sine of an arc is that part of the diameter that lies between the right sine and the circumference: thus LB is the versed sine of the arc HB. fig. 8.

22. The tangent of an arc is a right line touching the periphery, being perpendicular to the end

of the diameter, and is terminated by a line drawn from the centre through the other end: thus BK is the tangent of the arc HB. fig. 8.

23. And the line which terminates the tangent, that is, CK, is called the secant of the arc HB. fig. 8.

24. What an arc wants of a quadrant is called the complement thereof: Thus DH is the complement of the arc HB. fig. 8.

25. And what an arc wants of a semicircle is called the supplement thereof: thus AH is the supplement of the arc HB. fig. 8.

26. The sine, tangent, or secant of the complement of any arc, is called the co-sine, co-tangent, or co-secant of the arc itself: thus FH is the sine, DI the tangent, and CI the secant of the arc DH: or they are the co-sine, co-tangent, or co-secant of the arc HB. fig. 8.

27. The sine of the supplement of an arc, is the same with the sine of the arc itself; for drawing them according to def. 20, there results the self-same line; thus HL is the sine of the arc HB, or of its supplement ADH. fig. 8.

28. The measure of a right-lined angle, is the arc of a circle swept from the angular point, and contained between the two lines that form the angle: thus the angle HCB (fig. 8.) is measured by the arc HB, and is said to contain so many degrees as the arc HB does; so if the arc HB is 60 degrees, the angle HCB is an angle of 60 degrees.

Hence angles are greater or less according as the arc described about the angular point, and

terminated by the two sides, contains a greater or less number of degrees of the whole circle.

29. The sine, tangent, and secant of an arc, is also the sine, tangent, and secant of an angle whose measure the arc is; thus because the arc HB is the measure of the angle HCB, and since HL is the sine, BK the tangent, and CK the secant, BL the versed sine, HF the co-sine, DI the co-tangent, and CI the co-secant, &c. of the arc BH; then HL is called the sine, BK the tangent, CK the secant, &c. of the angle HCB, whose measure is the arc HB. fig. 8.

30. Parallel lines are such as are equi-distant from each other, as AB, CD. fig. 9.

31. A figure is a space bounded by a line or lines. If the lines be right it is called a rectilineal figure, if curved it is called a curvilineal figure; but if they be partly right and partly curved lines, it is called a mixed figure.

32. The most simple rectilineal figure is a triangle, being composed of three right lines, and is considered in a double capacity; 1st, with respect to its sides; and 2d, to its angles.

33. In respect to its sides it is either equilateral, having the three sides equal, as A. fig. 10. 34. Or isosceles, having two equal sides, as B. fig. 11.

35. Or scalene, having the three sides unequal, as C. fig. 12.

36. In respect to its angles, it is either rightangled, having one right angle, as D. fig. 13.