parallel sides of a regular hexagon, to construct that hexagon. 306. Fit a segment of a circle in a rectangle whose length is 3 and breadth 1. 307. Can you fit a segment of a circle in a rectangle whose length is 3 and breadth 2? 308. Can you place a circle in a quadrant ? 309. Give a figure of a symmetrical trapezoid whose parallel sides are 40 and 20, and the perpendicular distance between them 60; measure its angles by the line of sines, and calculate the area. 310. Show by a figure what the area of a rectangle is, whose length is 2 and breadth 11, and prove it by calculation. 311. Given, from a line of chords, the chord of 90°, it is required to find the radius of that line of chords. You have drawn one triangle similar to another, and one rhomboid similar to another; can you draw one trapezium similar to another? 312. Make of one piece of card a hollow em bossed tetrahedron; show how you arrange the surfaces to fit, and give a sketch of it when completed; and say if you can so arrange the surfaces on a plane as to have no reëntrant angles. 313. Can you make one triangle similar to another, and twice the size? 314. Can you make an irregular polygon similar to another, and twice the size? 315. Can you make an irregular polygon similar to another, and half the size? 316. Can you change a square to an obtuseangled isosceles triangle? 317. Can you show by a figure how much more is than ? 318. Can you make an isosceles triangle, each of whose sides shall be half the base? 319. Can you determine the size of an ob tuse angle by the line of sines? 320. Can you show by a figure that 2 is con. tained in 3 1 time? 321. Can you show that the sine of an arc is half the chord of double the arc? 322. Take an inch to represent a foot, and make a scale of feet and inches. 323. From the theorem, that triangles on the same base, and between the same parallels, are equal in surface, can you change a trapezi um into a triangle? 324. Can you change a triangle into a rectangle? 325. Make of a piece of card a hexahedron, embossed with semi-octahedrons; give a plan of the method by which you arrange the surfaces to fit, and give a sketch of the figure when made. 326. Can you convert a common trapezium into a symmetrical trapezium? 327. Can you construct a square, whose diagonal shall be 3 inches, and find the area of it? That portion of the radius of an arc which is intercepted between the sine and the extrem ity of the arc is called the versed sine of that arc. 328. Give an example of the versed sine of an arc. 329. Beginning at a point in a line, can you arrange the versed sines of all the degrees from 1° to 90°? i. e., can you make a line of versed sines? 330. Show when the versed sine of an arc is equal to the sine of the arc. 331. Show when the versed sine of an arc is equal to half the chord of the arc. 332. Say what versed sine is equal to the radius of the quadrant to which the line of versed sines belongs. 333. Given the versed sine of an arc equal to half the radius of that arc, to determine the number of degrees in that arc. 334. Can you reduce a figure of five sides to a triangle and to a rectangle? When lines or curves, or both, are symmet rically grouped about a point for effect, they take the name of star. 335. Invent and construct as beautiful a star as you can. When a body has 20 surfaces, whose sides and angles are respectively equal, it is called an icosahedron. 336. Make of one piece of card a hollow icosahedron;' represent on paper the method by which you arrange the surfaces to fit, and give a sketch of the icosahedron when made. 337. Describe an arc; let it be less than that of a quadrant, and draw to it the chord, the tangent, and co-tangent, the sine, and co-sine, and the versed sine. 338. Given the sine of an arc, exactly onefourth of the radius of that arc; it is required, by the protractor, to determine in degrees the length of such arc. 1 The tetrahedron, the hexahedron, the octahedron, the dodecahedron, and the icosahedron, take the name of regular bodies. These five regular bodies are also called Platonic bodies; and along with these Platonic bodies some place the sphere, as the most regular of all bodies. |