hended 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 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 sup

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

plement 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. 30, 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 equidistant 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 right-angled, having one right angle, as D, fig. 13,

37. Or obtuse-angled, having one obtuse angle, as E, fig. 14.

38. Or acute-angled, having all the angles acute, as F, fig. 15.

39. Acute and obtuse-angled triangles are in general called oblique-angled triangles, in all which any side may be called the base, and the other two the sides.

40. The perpendicular height of a triangle is a line drawn from the vertex to the base perpendicularly: thus if the triangle ABC be proposed, and BC be made its base, then if from the vertex A the perpendicular AD be drawn to BC, the line AD will be the height of the triangle ABC, standing on BC as its base, fig. 16.

Hence all triangles between the same parallels have the same height, since all the perpendiculars are equal from the nature of parallels.

41. Any figure of four sides is called a quadrilateral figure. 42. Quadrilateral figures, whose opposite sides are parallel, are called parallelograms: thus ABCD is a parallelogram, fig. 3, 17, and AB, fig. 18, 19.

43. A parallelogram whose sides are all equal and angles right is called a square, as ABCD, fig. 17.

44. A parallelogram whose opposite sides are equal and angles right is called a rectangle, or an oblong, as ABCD, fig. 3.

45. A rhombus is a parallelogram of equal sides, and has its angles oblique, as A, fig. 18, and is an inclined square.

46. A rhomboides is a parallelogram whose opposite sides are equal and angles oblique; as B, fig. 19, and may be conceived as an inclined rectangle.

47. Any quadrilateral figure that is not a parallelogram is called a trapezium. Plate 7, fig. 3.

48. Figures which consist of more than four sides are called polygons; if the sides are all equal to each other, they are called regular polygons. They sometimes are named from the number of their sides, as a five-sided figure is called a pentagon, one of six sides a hexagon, &c.; but if their sides are not equal to each other, then they are called irregular polygons, as an irregular pentagon, hexagon, &c.

49. Four quantities are said to be in proportion when the product of the extremes is equal to that of the means: thus if A multiplied by D be equal to B multiplied by C, then A is Isaid to be to B as C is to D.

POSTULATES, OR PETITIONS.

1. That a right line may be drawn from any one given point

to another.

2. That a right line may be produced or continued at pleasure. 3. That from any centre and with any radius the circumference of a circle may be described.

4. It is also required that the equality of lines and angles to others given, be granted as possible: that it is possible for one right line to be perpendicular to another at a given point or distance; and that every magnitude has its half, third, fourth, &c. part.

Note. Though these postulates are not always quoted, the reader will easily perceive where and in what sense they are to be understood.

AXIOMS, OR SELF-EVIDENT TRUTHS.

1. Things that are equal to one and the same thing are equal to each other.

2. Every whole is greater than its part.

3. Every-whole is equal to all its parts taken together. 4. If to equal things equal things be added, the whole will be equal.

5. If from equal things equal things be deducted, the remain ders will be equal.

6. If to or from unequal things equal things be added or taken, the sums or remainders will be unequal.

7. All right angles are equal to one another.

8. If two right lines not parallel be produced towards their nearest distance, they will intersect each other.

9. Things which mutually agree with each other are equal. NOTES.

A theorem is a proposition wherein something is proposed to be demonstrated.

A problem is a proposition wherein something is to be done or effected.

A lemma is some demonstration previous and necessary, to render what follows the more easy.

A corollary is a consequent truth, deduced from a foregoing demonstration.

A scholium is a remark or observation made upon something going before.

1

GEOMETRICAL THEOREMS.

THEOREM, I.

PL. 1. fig. 20.

If a right line falls on another, as AB, or EB, does on CD, it either makes with it two right angles, or two angles equal to two right angles.

1. If AB be perpendicular to CD, then (by def. 10) the angles CBA and ABD will be each a right angle.

2. But if the line fall slantwise, as EB, and let AB be perpendicular to CD; then the DBA=DBE+EBA: add ABC to each; then, DBA+ABC=DBE+EBA+ABC; but CBE=EBA+ABC, therefore the angles DBE+EBC=DBA+ABC, or two right angles. Q. E. D.

Corollary 1. Whence if any number of right lines were drawn from one point, on the same side of a right line, all the angles made by these lines will be equal to two right angles. 2. And all the angles which can be made about a point will be equal to four right angles.

THEOREM II.

PL. 1. fig. 21.

If one right line cross another (as AC does BD), the opposite angles made by those lines will be equal to each other: that is, AEB to CED, and BEC to AED:

By theorem 1, BEC÷CED= two right angles.

and CED+DEA= two right angles.

Therefore (by axiom 1) BEC+CED=CED+DEA; take CED from both, and there remains BEC=DEA (by axiom 5). Q. E. D.

After the same manner CED+AED= two right angles; and AED+AEB=two right angles; wherefore taking AED from both, there remains CED-AEB. Q. E. D.

THEOREM III.
PL. 1. fig. 22.

If a right line cross two parallels, as GH docs AB and CD, then, 1. Their external angles are equal to each other, that is, GEB=CFH. 2. The alternate angles will be equal, that is, AEF=EFD and BEF =CFE.

3. The external angle will be equal to the internal and opposite one on the same side, that is, GEB EFD and AEG=CFE.

4. And the sum of the internal angles on the same side are equal to two right angles; that is, BEF+DFE are equal to two right angles, and AEF CFE are equal to two right angles.