53. An Angle on a Segment, or an Arc, is that which is contained by two lines, drawn from any point in the opposite part of the circumference to the extremities of the arc, and containing the arc between them. Thus A and D (in the last figure) are both angles upon the arc BEC.

54. An Angle at the Center is one whose vertex is at the center of the circle. An Eccentric Angle is one whose vertex is not at the center. An Angle at the Circumference is one whose vertex is in the circumference. This last is also called an Inscribed angle.

55. Similar arcs, in different circles, are those which subtend equal angles at the center.

56. A right line is a Tangent to a circle, or touches it, when it has but one point in common with the circle.

57. Two circles Touch each other when they have but one point common, or when they have a common tangent.

58. A right-lined figure is Inscribed in a circle, or the circle Circumscribes the figure, when all the angular points of the figure are in the circumference of the circle.

59. A right-lined figure Circumscribes a circle, or the circle is Inscribed in the figure, when all the sides of the figure touch the circumference of the circle.

60. One right-lined figure is inscribed in another, or the latter circumscribes the former, when all the angular points of the former are placed in the sides of the lat

ter.

61. A Secant is a line that cuts a circle, lying partly within and partly without it.

62. The Altitude of a triangle is a perpendicular let fall from the vertex of either angle upon the opposite side, called the base.

63. In a right-angled triangle, the side opposite the right angle is called the Hypothenuse; and the other two sides are called the Legs, and sometimes the Base and Perpendicular.

64. The altitude of a parallelogram

or trapezoid is the perpendicular distance between the parallel sides.

The bases of a trapezoid are the parallel sides.

65. Two triangles, or other right-lined figures, are said to be mutually equilateral when all the sides of the one are equal to the corresponding sides of the other, each to each; and they are said to be mutually equiangular when the angles of the one are respectively equal to those of the other.

66. Identical polygons are such as are both mutually equilateral and equiangular, or that have all the sides and all the angles of the one respectively equal to all the sides and all the angles of the other, each to each; so that if the one figure were applied to, or laid upon the other, all the sides of the one would exactly fall upon and cover all the sides of the other; the two becoming, as it were, but one and the same figure.

67. Similar polygons are of the same shape, but not the same size; they have all the angles of the one equal to all the angles of the other, each to each, and the corresponding or homologous sides, as they are called, proportional.* The homologous sides are those

Perhaps it will be a little plainer to say that the homologous sides in the two figures have the same ratio. Thus, if the first side in the one figure (beginning in both at the sides adjacent equal angles) be three times as great as the first side in the other, the second side in the first figure will be three times as great as the second side in the other figure, and so on.

The ratio of the corresponding sides of the polygon is called the ratio of similitude.

similarly situated, or those adjacent equal angles, or, in triangles, those opposite equal angles.

68. A Proposition is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem.

69. A Problem is something proposed to be done. 70. A Theorem is a truth proposed to be demonstrated.

71. A Hypothesis is a supposition made in the enunciation of a proposition, or in the course of a demon

stration.

72. A Lemma is something which is premised, or demonstrated, in order to render what follows more

easy.

73. A Corollary is a consequent truth, gained immediately from some preceding truth or demonstration.

74. A Scholium is a remark or observation made upon something going before it, and may require a demonstration or may not.

Axioms.

1. Things which are equal to the same thing are equal to one another.

2. When equals are added to equals, the wholes are equal.

3. When equals are taken from equals, the remainders are equal.

4. When equals are added to unequals, the wholes are unequal.

5. When equals are taken from unequals, the remainders are unequal.

6. Things which are double the same thing, or equal things, are equal to each other.

7. Things which are halves of the same thing are equal.

8. The whole is greater than its part.

9. Every whole is equal to all its parts taken together.

10. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts. 11. All right angles are equal.

12. Angles that have equal measures, or arcs, are equal.

13. A straight line is the shortest distance between two points. Corollary.-One side of a triangle is less than the sum of the other two.

14. But one straight line can be drawn between two points.*

EXERCISE WITH RULE AND DIVIDERS UPON THE RIGHT LINE AND

ANGLE.

1. Make a line equal to the sum of two given lines. Of four. 2. Make a line equal to the difference of two given lines.

3. Make a line equal to five times one given line and six times another.

4. Find how many times one given line is contained in another. 5. Find a common measure of two given lines.t

6. Make a straight line equal in length to a broken line.

7. Make a straight line equal in length approximately to a curve.‡ 8. With several given points as centers, to describe circles with given lines as radii.

9. To find a point which shall be at given distances from two given points.

10. Draw the radius of a circle as a chord of the same.

11. Make an angle double a given angle. Triple.

12. Measure the number of degrees in a given angle by means of a brass or paper circle or semicircle, divided into degrees, called a pro

tractor.

13. Make an angle equal to the sum of several given angles. 14. Draw a line through a given point parallel to a given line. 15. Draw through given points several parallels to a given line.

* A straight line joining two points is the direction of the one from the other. Two points are said to determine a line. Two points of a line being given, the line is given; for it is the line joining them.

This is done in a manner analogous to the corresponding operation in Arithmetic and Algebra, by applying the smaller line to the larger as many times as it will go; and the remainder to the smaller given line, and so on.

This may be done by taking such small portions of the curve as are nearly straight.

16. Draw through a given point, without a given line, a line forming with it a given angle.

17. To make an angle with two given lines for sides.

18. In how many points may 20 lines cut each other, no two of which are parallel?

19. In how many when twelve of them are parallel?

20. In how many when 4 are parallel in one direction, 5 in another, and 6 in another?

21. In how many points will 36 lines intersect, 24 of which pass through the same point?

THEOREM I.

If two triangles have two sides and the included angle in the one equal to two sides and the included angle in the other, the triangles will be identical, or equal in all respects.

A

D

In the two triangles ABC, DEF, if the side AB be equal to the side. DE, the side AC equal to the side DF, and the angle A equal to the angle D, then will the two triangles be identical, or equal in all respects.

B

CE

F

For, conceive the triangle ABC to be applied to, or placed on, the triangle DEF, in such a manner that the point A may coincide with the point D, and the side AB with the side DE, which is equal to it.

Then, since the angle A is equal to the angle D (by hyp.),† the side AC must differ in direction from the side AB by the same amount that the side DF does from DE; hence AC must take the same direc

*The student will do well, at first, to cut two triangles out of pasteboard or paper, and place one upon the other; or imagine the first of the above triangles to be cut out of the page and placed upon the other; or conceive the sides to be fine wires, so that the triangle can be taken off the page.

Hyp. stands for hypothesis. This term is much used, and signifies generally that what is stated is given or supposed true at the outset.