HK was taken along HG. Lastly, from each point of division of HB, extend a line perpendicular to HG.

These perpendiculars are parallel to each other and to CD (129). These parallels by construction intercept equal parts of HB. Therefore (135), they are equally distant from each other. Hence, HG is divided by them into equal segments (134); that is, each one passes through one of the previously ascertained points. of the line HG.

But the last of these points was beyond the line CD, and as the parallel can not cross CD (120), the corresponding point of HB is beyond CD. Therefore, HB and CD must cross each other.

ANGLES WITH PARALLEL SIDES.

138. Theorem.-When the sides of one angle are parallel to the sides of another, and have respectively the same directions from their vertices, the two angles are equal.

If the directions BA and DC are the same, and the directions DE and BF are the

same, then the angles ABF and CDE are equal.

For each of these angles is equal to the angle CGF (124).

139. Let the student demonstrate that when two of the parallel sides have opposite directions, and the other two have the same direction, then the angles are supplementary.

B

G

A

D

G

C

D

E

F

Let him also demonstrate that if both sides of one angle have directions respectively opposite to those of the other, then again the angles are equal.

Geom.-5

ANGLES WITH PERPENDICULAR SIDES.

140. Theorem.-Two angles which have their sides respectively perpendicular are equal or supplementary.

D

H

F

If AB is perpendicular to DG, and BC is perpendicular to EF, then the angle ABC is equal to one, and supplementary to the other of the angles formed by DG and EF (86).

G

BH parallel to EF.

Now, ABI and CBH

E

are right angles (127), and therefore equal (90). Subtracting the angle HBA from each, the remainders HBI and ABC are equal (7). But HBI is equal to FGD (138), and is the supplement of EGD (139). Therefore, the angle ABC is equal or supplementary to any angle formed by the lines DG and EF.

APPLICATIONS.

141. The instrument called the T square consists of two straight and flat rulers fixed at right angles to each

other, as in the figure. It is used to draw parallel lines.

Draw a straight line in a direction perpendicular to that in which it is required to draw parallel lines. Lay the cross-piece of the T ruler along this line. The other piece of the ruler gives the direction of one

of the parallels. The ruler being moved along the paper, keeping the cross-piece coincident with the line first described, any number of parallel lines may be drawn.

What is the principle of geometry involved in the use of this instrument?

142. The uniform distance of parallel lines is the principle upon which numerous instruments and processes in the arts are founded.

If two systems, each consisting of several parallel lines, cross each other at right angles, all the parts of one system included between any two lines of the other system will be equal. The ordinary framing of a window consists of two systems of lines of this kind; the shelves and upright standards of book-cases and the paneling of doors also afford similar examples.

143. The joiner's gauge is a tool with which a line may be drawn on a board parallel to its edge. It consists of a square piece of wood, with a sharp steel point near the end of one side, and a movable band, which may be fastened by a screw or key at any required distance from the point. The gauge is held perpendicular to the edge of the board, against which the band is pressed while the tool is moved along the board, the steel point tracing the parallel line.

144. It is frequently important in machinery that a body shall have what is called a parallel motion; that is, such that all its parts shall move in parallel lines, preserving the same relative position to each other.

The piston of a steam-engine, and the rod which it drives, receive such a motion; and any deviation from it would be attended with consequences injurious to the machinery. The whole mass of the piston and its rod must be so moved, that every point of it shall describe a line exactly parallel to the direction of the cylinder.

CHAPTER IV.

THE CIRCUMFERENCE.

145. Let the line AB revolve in a plane about the end A, which is fixed. Then the

point B will describe a line which returns upon itself, called a circumference of a circle. Hence, the following definitions:

A CIRCLE is a portion of a plane bounded by a line called a CIRCUMFERENCE, every point of

B

which is equally distant from a point within called the CENTER.

146. Theorem.-A circumference is curved throughout. For a straight line can not have more than two points equally distant from a given point (111).

147. Corollary.-A straight line can not cut a circumference in more than two points.

148. The circumference is the only curve considered in elementary geometry. Let us examine the properties of this line, and of the straight lines which may be combined with it.

HOW DETERMINED.

149. Theorem.-Three points not in the same straight line fix a circumference both as to position and extent. The three given points, as A, B, and C, determine

A

B

D

E

the position of a plane. Let the given points be joined by straight lines AB and BC. At D and E, the middle points of these lines, let perpendiculars be erected in the plane of the three points.

By the hypothesis, AB and BC make an angle at B. Therefore, GD is not perpendicular to BC, for

if it were, AB and BC would be parallel (129). Hence, DG and EH are not parallel (117), since one is perpendicular and the other is not perpendicular to BC. Therefore, DG and EH will meet (137) if produced. Let L be their point of intersection.

Since every point of DG is equidistant from A and B (108), and since every point of EH is equidistant from B and C, their common point L is equidistant from A, B, and C. Therefore, with this point as a center, a circumference may be described through A, B, and C. There can be no other circumference through these three points, for there is no other point besides L equally distant from all three (112).

Therefore, these three points fix the position and the extent of the circumference which passes through them.

ARCS AND RADII.

150. An ARC is a portion of a circumference.

A RADIUS is a straight line from the center to the circumference.

A DIAMETER is a straight line passing through the center, and limited at both ends by the circumference. A CHORD is a straight line joining the ends of an arc.