339. Given the versed sine of an arc, exactly one-fourth of the radius of that arc; it is required, by the protractor, to determine the degrees in that arc. 340. How would you prove the correctness of a straight-edge, of a parallel ruler, of a set square, of a drawing-board, of a protractor, and of a line of chords? 341. Reduce an irregular hexagon with a reentrant angle to a triangle. 342. Reduce an irregular octagon with two reëntrant angles to a triangle. 25 It has been agreed upon by arithmeticians that fractions whose denominators are either 10, or some multiple of 10, as, %, 1856, etc., may be expressed without their denominators, by placing a dot at the left hand of the numerator: thus, may be expressed .5; thus, .25; 125 thus, .125; and thus, .05. Such expressions are called decimals. 25 Like other fractions, decimals may be illustrated either by a line and parts of that line, or by a surface and parts of that surface. 343. By dividing a line, supposed to represent a unit of length, illustrate the value of .5, .25, and .125, etc. 344. By means of a square representing a unit of surface, exhibit the value of .5, .25, and .125. 345. Out of an apple, or a turnip, or a potato, cut a cube: call each of its linear dimensions 2, and determine its solid content, and prove by arithmetic. 346. Show by means of a cube, and prove by arithmetic, what the cube of 1 is. 347. Can you place nine trees in ten rows of three in a row? 348. With 10 divisions of a diagonal scale for its side, construct an equilateral triangle, and call such side 1; and determine the length of its perpendicular to three decimal places, and prove its truth by calculation. 349. Can you calculate the area of an equilateral triangle whose side is 1? 350. Illustrate by geometry the respective values of .9, .99, .999, .9999. A circle may be supposed to consist of an indefinite number of equal isosceles triangles, having their bases placed along the circumference of the circle, and their vertices all meeting in the centre of the circle. And as the areas of together would be all these triangles added equal to the area of the circle: To find the area of a circle-multiply the radius which is the perpendicular common to all these imaginary triangles, by the circumference which is the sum of all their bases, and divide the product by 2. Reckoning the circumference of a circle as 3 times its diameter: 351. Find the area of a circle whose diameter is 1. Reckoning the circumference of a circle to be 3.1416 times the diameter: 352. Find the area of a circle whose diameter is 1. Circles being similar figures, the areas of circles are to each other as the squares of their radii, their diameters, or their circumferences. 353. Find the area of a circle whose radius is 5, and find the area of another circle whose radius is 7, and see whether their respective quantities agree with the rule. 354. A circular grass-plot has a diameter of 300 feet, and a walk of 3 yards wide round it; find the area of the grass-plot, and also the area of the walk. 355. Can you find the area of a sector whose radial boundaries are each 20 yards, and whose arc contains 35°? 356. The largest pyramid in the world stands upon a square base, whose side is 700 feet long. The pyramid has four equilateral triangles for its surfaces. Calculate what number of square feet, square yards, and acres, the base of such pyramid stands upon, and the number of square feet on each of its triangular surfaces; calculate also its perpendicular height, and prove its correctness by geometry; give in card a model of the pyramid; say what solid it is a part of, and give a sketch of the model. 357. There is a rhomboid of such a form that its area may be found by means of one of its sides, and one of its diagonals. Give a plan of it. 358. Can you convert a square whose side is 1 into a rhombus whose long diagonal is twice as much as the short one; and can you find, both by geometry and by calculation, the length of the side of that rhombus? 359. Can you convert an equilateral triangle into an irregular pentagon? 360. Point out upon a tetrahedron two lines that are in the same plane, and two that are not in the same plane. 361. Make of card a truncated octahedron, and give a plan of it, and a sketch of the figure. 362. Show how many cubes may be made to touch at one point. 363. Show by a figure how many cubes may be made to touch one cube. |