27. A LEMA is an auxiliary proposition.

28. A COROLLARY is an obvious consequence of one or more propositions.

29. A SCHOLIUM is a remark made upon one or more propositions, with reference to their connection, their use, their extent, or their limitation.

30. An HYPOTHESIS is a supposition made, either in the statement of a proposition, or in the course of a demonstration.

31. Magnitudes are equal to each other, when each contains the same unit an equal number of times.

32. Magnitudes are equal in all their parts, when they may be so placed as to coincide throughout their whole extent.

Of Plane and Rectilineal Surfaces.

1. A superfices or surface has two dimensions; length and breadth REMARK.-The extremities of superfices are lines. A term or boundary is the extremity of anything. A figure is enclosed by one or more boundaries.

NOTE.-Rectilineal figures are contained by straight lines. Trilateral figures or triangles, by three straight lines; quadrilateral figures, by four straight lines; multilateral figures or polygons, by more than four straight lines.

2. An equilateral triangle has three equal sides, as A.
3. An isoceles triangle has two equal sides, as B.
4. A right-angled triangle has one right angle, as C.

REMARK.-The side opposite the right angle is called the hypothenuse.

5. A scalene triangle has three unequal sides, as D.
6. An obtuse-angled triangle has one obtuse angle as O.
7. An acute-angled triangle has three acute angles, as A.

8. A square has all its sides equal, and all its angles right angles, as E

9. A rhombus has all its sides equal, but its angles are not all right angles, as H.

10. A rhomboid has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles, as FG.

11. A trapezoid has only two of its sides parallel, as IK.

12. All other four-sided figures beside these are called trapeziums. NOTE. The terms oblong and rhomboid are not often used; practically the following definitions are used: Any four-sided figure is called a quadrilateral. A line joining two opposite angles of a quadrilateral is called a diagonal. A quadrilateral which has its opposite sides parallel is called a parallelogram. The words square and rhombus are used in the same as defined by Euclid, and the word rectangle is used instead of the word oblong.

Some writers propose to restrict the word trapezium to a quadrilateral which has two of its sides parallel; and it would certainly be convenient if this restriction were universally adopted.

Of the Circle, see Plate 3.

1. A circle is a plain figure bounded by a curved line called the circumference, every point of which is equally distant from a point within. And this point within is called the center A, F, C, B, H, D, G is the circumference and E the center of the circle. Fig. 2 Plate 3.

2. A radius is a straight line drawn from the center to any point of the circumference AE, CE, BE, DE. Fig. 2 Plate 3.

3. A diameter is a straight line drawn through the center and terminating in the circumference, as AB, CD. Fig. 2 Plate 3

REMARK.—All radii of the same circle are equal. All diameters are equal, and each is double of the radius.

4. An arc is any part of the circumference, as AF. Fsg. 1 Plate 3. 5. A chord is a straight line joining the extremities of an arc, as FH. Fig. 1 Plate 3.

6. A segment is that part of a circle between an arc and its chord AG, HB, and GHD; Fig. 2 Plate 3 are segments of the circle ABCD. 7. A sector is that part of a circle included within an arc and the radii drawn to its extremities FEC; Fig. 2 Plate 3 is a sector of the circle ABCD, also AEF, CEB.

REMARK.—AEF is also called a sextant because it is one-sixth part of the circle, and CEB is called a quadrant, because it is one-fourth part of the circle.

TRIGONOMETRY.

Trigonometry is the science which teaches how to determine the several parts of a triangle from having certain parts given.

Plane trigonometry treats of plane triangles; spherical trigonometry treats of spherical triangles.

The circumference of a circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 minutes, and each minute into 60 seconds. Degrees, minutes, and seconds are designated by the characters Thus 23° 14′ 35′′ is read 23 degrees, 14

minutes, and 35 seconds.

Since an angle at the center of a circle is measured by the arc intercepted by its sides, a right angle is measured by 90°, two right angles by 180°, and four right angles are measured by 360°.

The complement of an arc is what remains after subtracting the are from 90°. Thus the are DF, Fig. 1 Plate 3, is the complement of AF. The complement of 25° 15' is 64° 45'.

In general, if we represent any arc by A, its complement is 90°—A. Hence, if an arc is greater than 90°, its complement must be negative. Thus the complement of 100° 15′ is-10° 15'. Since the two acute angles of a right-angled triangle are together equal to a right angle each of them must be the complement of the other.

The supplement of an arc is what remains after subtracting the are from 180°. Thus the arc BDF is the supplement of the arc AF. The supplement of 25° 15′ is 154° 45'. In general, if we represent any arc by A, its supplement is 180°-A. Hence, if an arc is greater than 180°, its supplement must be negative. Thus the supplement of 200° is—20°. Since in every triangle the sum of the three angles is 180°, either angle is the supplement of the sum of the other two. The sine of an arc is the perpendicular let fall from one extremity of the arc on the radius passing through the other extremity. Thus FG is the sine of the arc AF, or of the angle ACF.

Every sine is half the chord of double the arc. Thus the sine FG is the half of FH, which is the chord of the arc FAH, double of FA. The chord which subtends the sixth part of the circumference, or the chord of 60°, is equal to the radius (Loomis' Geom., Prop. IV., Book VI.); hence the sine of 30° is equal to half of the radius.

The versed sine of an arc is tween the sine and the arc. AF.

that part of the diameter intercepted beThus GA is the versed sine of the arc

The tangent of an arc is the line which touches it at one extremity, and is terminated by a line drawn from the center through the other extremity. Thus AI is the tangent of the arc AF, or the angle ACF. The secant of an arc is the line drawn from the center of the circle through one extremity of the arc, and is limited by the tangent drawn through the other extremity. Thus CI is the secant of the arc AF, or of the angle ACF

Thus

The cosine of an arc is the sine of the complement of that arc. the arc DF, being the complement of AF, FK is the sine of the are DF, or the cosine of the arc AF.

The cotangent of an arc is the tangent of the complement of that are. Thus, DL is the tangent of the arc DF, or the cotangent of the arc AF.

The cosecant of an arc is the secant of the complement of that arc. Thus CL is the secant of the arc DF, or the cosecant of the arc AF.