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2. A point is that which has no parts, being of itself indivisible, as A.

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3. A line has length but no breadth, as AB.. figures 1 and 2.

4 The extremities of a line are points, as the extremities of the line AB are the points A and B. figures 1 and 2.

5. A right line is the shortest that can be drawn between any two points, as the line AB. fig. 1. but if it be not the shortest, it is then called a curve line, as AB. fig. 2.

Plate 1.

6. A superficies or surface is considered only as having lengih and breadth, without thickness, as ABCD. fig. 3.

7. The extremities of a superficies are lines.

8. The inclination of two lines meeting one another (provided they do not make one continued line) or the opening between them, is called an angle. Thus fig. 4. the inclination of the line 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.

10. 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 curyed it is called a mix'd angle, as B. tig. 5. If both of them be curved it is called a curved-lined or spherical angle, as C. fig. 6.

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

12. 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.

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Plate 1.

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

14. 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.

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

16. The diameter of a circle is a right line drawn through the centre, and terminating on either side 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.

18. 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.

19. 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

Thus the right line HG, is the measure of the arc HBG. fig. 8.

20. 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

of the arc.

Plate 1.

and the arc HBG, or that which is comprehended between the said chord HG and the arc HDAEG are called segments. Whence 'tis 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.

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

22. 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 diaineter (by def. 22.) therefore producing DC to E the two diameters AB and DE cross one another at right angles, and thus the

Plate I.

periphery is divided into four equal parts, as BD, DA, AE, and, EB; (by def. 11.) 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.

23. 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.

24. 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 thro' the other end : thus BK is the tangent of the arc HB. fig. 8.

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

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

27. And what an arc wants of a semicircle is called the supplement thereof: Thus AH is the sup. plement of the arc HB. fig. 8.

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