in such a position that every other side of the octagon may coincide with a side of the square?

194. Fit an equilateral triangle inside a circle, and another outside, in such positions with regard to each other as shall show the ratio the inner one has to the outer.

195. Can you place four octagons in a group to touch at their angles?

196. Can you fit a hexagon outside a circle?

197. Can you place four octagons to meet in one point, and to overlap each other to an equal extent ?

198. Can you let fall a perpendicular to a line from a point given above that line?

Those instruments by which an angle can be constructed so as to contain a certain number of degrees, or by which we can measure an angle, and determine how many degrees it contains, as also by which we can make an arc of a circle that shall subtend a certain number of degrees, or can measure an arc and determine how many degrees it subtends, are called protractors.

Protractors commonly extend to 180°; though there are protractors that include the whole circle, that is, which extend to 360°.

199. Make of a piece of card as accurate a protractor as you can.

200. Make by a protractor an angle of 45°, and prove by geometry whether it is accurate.

or not.

201. Can you contrive to divide a square into two equal but dissimilar parts?

202. Make with a protractor an angle of 60°, and prove by geometry whether it is cor

rect or not.

203. Make an angle, and determine by the protractor the number of degrees it contains.

204. Make by geometry the arc of a quad

rant, and determine by the protractor the number of degrees that arc subtends.

205. Show how many hexagons may be made to touch one hexagon at the sides.

That which an angle lacks of a right angle, that is, of 90°, is called its complement.

206. Make a few angles, and say which their complements are.

207. Make an angle of 70°, and measure its complement.

That which an angle lacks of 180° is called its supplement.

208. Make a few angles, and their supplements, and measure them by the protractor.

209. Make by geometry an angle of 30°, and its supplement, and measure by the protractor the correctness of each.

210. Can you make a semicircle equal to a circle?

211. Make a few triangles of different forms, and measure by the protractor the angles of each, and see if you can find a triangle whose angles

added together amount to more than the angles of any other triangle added together.

212. Can you make a pentagon in a circle by means of the protractor?

213. Make of one piece of card a hollow square pyramid, and let the slant height be twice the diagonal of the base. Give a plan of your method, and a sketch of the pyramid, when completed.

214. Can you make a pentagon outside a circle by means of a protractor?

215. Can you, by means of a protractor, make a pentagon without using a circle at all?

It has already been said that the chord of an arc is a line joining the extremities of that arc.

216. With the assistance of a semicircular protractor, can you contrive to place on one line the chords of all the degrees from 1° to 90° or, in other words, can you make a line of chords?

217. Can you say why the line of chords should not extend as far as 180°?

There is one chord which is equal in length to the radius of the quadrant to which all the chords belong; that is, which is equal to the radius of the line of chords.

218. Say which chord is equal to the radius of the line of chords.

219. Make, by the line of chords, angles of 26°, 32°, 75o, and prove, by the protractor, whether they are correct or not.

220. How, by the line of chords, will you make an obtuse angle, say one of 115°?

221. Can you make, with the assistance of a line of chords, a triangle whose angles at the base shall each be double of the angle at the vertex?

222. Make a triangle, whose sides shall be 21, 15, and 12, and measure its angles by the line of chords and by the protractor.

223. There is one side of a right-angled triangle that is longer than either of the other two. Give its name, and show from such fact that the chord of 45° is longer than half the chord of 90°.