151. Corollary.-All radii of the same circumference are equal.

152. Corollary. In the same circumference, a diameter is double the radius, and all diameters are equal.

153. Corollary.-Every point of the plane at greater distance from the center than the length of the radius, is outside of the circumference. Every point at a less distance from the center, is within the circumference. Every point whose distance from the center is equal to the radius, is on the circumference.

154. Theorem.-Circumferences which have equal radii are equal.

Let the center of one be placed on that of the other. Then the circumferences will coincide. For if it were otherwise, then some points would be unequally distant from the common center, which is impossible when the radii are equal. Therefore, the circumferences are equal.

155. Corollary.-A circumference may revolve upon, or slide along its equal.

156. Corollary.-Two arcs of the same or of equal circles may coincide so far as both extend.

157. Theorem.-Every diameter bisects the circumference and the circle.

For that part upon one side of the diameter may be turned upon that line as its axis. When the two parts thus meet, they will coincide; for if they did not, some points of the circumference would be unequally distant from the center.

158. A line which divides any figure in this manner, is said to divide it symmetrically; and a figure which can be so divided is symmetrical.

159. Theorem.-A diameter is greater than any other chord of the same circumference.

To be demonstrated by the student.

160. Problem.-Arcs of equal radii may be added together, or one may be subtracted from another.

For an arc may be produced till it becomes an entire circumference, or it may be diminished at will (35 and 145).

Therefore, the length of an arc may be increased or decreased by the length of another arc of the same radius; and the result, that is, the sum or difference, will be an arc of the same radius.

161. Corollary.—Arcs of equal radii may be multiplied or divided in the same manner as straight lines. 162. The sum of several arcs may be greater than a circumference.

163. Two arcs not having the same radius may be joined together, and the result may be called their sum; but it is not one arc, for it is not a part of one circumference.

APPLICATIONS.

164. The circumference is the only line which can move along itself, around a center, without suffering any change. For any line that can do this must, therefore, have all its points equally distant from the center of revolution; that is, it must be a circumference.

It is in virtue of this property that the axles of wheels, shafts, and other solid bodies which are required to revolve within a hollow mold or casing of their own form, must be circular. If they were of any other form, they would be incapable of revolving without carrying the mold or casing around with them.

165. Wheels which are intended to maintain a carriage always at the same hight above the road on which they roll, must be circular, with the axle in the center.

166. The art of turning consists in the production of the circular form by mechanical means. The substance to be turned is placed in a machine called a lathe, which gives it a rotary motion. The edge of a cutting tool is placed at a distance from the axis of revolution equal to the radius of the intended circle. As the substance revolves, the tool removes every part that is further from the axis than the radius, and thus gives a circular form to what remains.

PROBLEMS IN DRAWING.

16. The compasses enable us to draw a circumference, or an arc of a given radius and given center.

Open the instrument till the points are on the two ends of the given radius. Then fix one point on the given center, and the other point may be made to revolve around in contact with the surface, thus tracing out the circumference.

The revolving leg may have a pen or pencil at the point. In the operation, care should be taken not to vary the opening of the compasses.

168. It is evident that with the ruler and compasses (69),

1. A straight line can be drawn through two given points. 2. A given straight line can be produced any length.

3. A circumference can be described from any center, with any radius.

169. The foregoing are the three postulates of Euclid. Since the straight line and the circumference are the only lines treated of in elementary geometry, these Euclidian postulates are a sufficient basis for all problems. Hence, the rule that no instruments shall be used except the ruler and the compasses (68).

170. In the Elements of Euclid, which, for many ages, was the only text-book on elementary geometry, the problems in drawing occupy the place of problems in geometry. At present, the mathematicians of Germany, France, and America put them aside as not forming a necessary part of the theory of the science. English writers, however, generally adhere to Euclid.

171. Problem.-To bisect a given straight line.

With A and B as centers, and with a radius greater than the half of AB, describe arcs which intersect in the two points D

and E. The straight line joining these two points will bisect AB at C.

Let the demonstration be given by the student (109 and 151).

172. Problem.—To erect a perpendicular on a given straight line at a given point.

Take two points in the line, one on each side of the given point, at equal distances from it. Describe arcs as in the last prob

A

B

lem, and their intersection gives one point of the perpendicular. Demonstration to be given by the student.

173. Problem.—To let fall a perpendicular from a given point on a given straight line.

With the given point as a center, and a radius long enough, describe an arc cutting the given line BC in the points D and E. The line may be produced, if necessary, to be cut by the arc in two places. With D and E as centers, and with a radius greater than the half of DE, describe

arcs cutting each other in F. The

B

straight line joining A and F is perpendicular to DE. Let the student show why.

174. Problem.-To draw a line through a given point parallel to a given line.

Let a perpendicular fall from the point on the line. Then, at the given point, erect a perpendicular to this last. It will be parallel to the given line.

Let the student explain why (129).

175. Problem.—To describe a circumference through three given points.

The solution of this problem is evident, from Article 149.

176. Problem.-To find the center of a given arc or circumference.

Take any three points of the arc, and proceed as in the last problem.

177. The student is advised to make a drawing of every problem. First draw the parts given, then the construction requisite for solution. Afterward demonstrate its correctness.

Endeavor to make the drawing as exact as possible. Let the lines be fine and even, as they better represent the abstract lines of geometry. A geometrical principle is more easily understood by the student, when he makes a neat diagram, than when his drawing is careless.

TANGENT.

178. Theorem.-A straight line which is perpendicular to a radius at its extremity, touches the circumference in only one point.

Let AD be perpendicular to the radius BC at its extremity B. Then it is to be

proved that AD touches the circumference at B, and at no other point.

If the center C be joined by straight lines with any points of AD, the perpendicular BC will be shorter than any such oblique line (104). Therefore (153), every point of the line AD, except B, is outside of the circumference.

B

179. A TANGENT is a line touching a circumference in only one point. The circumference is also said to be tangent to the straight line.

called the point of contact.

The common point is