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364. You have calculated the perpendicular height of an equilateral triangle whose side is 1; can you say how far up that perpendicular it is from the base to the centre of the triangle?
A solid formed by revolving a rectangle about one of its sides takes the name of cylinder,' and it may be called a circular prism.
365. Can you find the surface of the cylinder whose length is 1, and whose diameter is 1 ?
A sphere may be formed by revolving a semicircle about the diameter as an axis.
The surface of a sphere whose diameter is 1, is equal to the surface of a cylinder whose diameter is 1 and height 1. Give a figure in illustration of what is meant.
366. Find the surface of a sphere whose diameter is 1, and also the surface of a sphere whose diameter is 2. Compare the two surfaces together, and say whether the ratio the less has to the greater accords with the law,
1 When a cylinder is long it takes the name of rod, as a rod of iron.
“ The areas of similar figures are to each other as the squares of their homologous sides.”
367. Can you erect an hexagonal pyramid whose slant sides shall be equilateral triangles ?
368. Make a box of strong pasteboard, and let the length be five inches, breadth four, and depth three, and let it have a lid that shall not only cover the box, but have edges clasping it when shut, and hanging over the top of the box three-eighths of an inch.
369. Can you plant 19 trees in 9 rows of 5 in a row ?
370. Can you convert a scalene triangle into a symmetrical trapezium?
371. Place a hexagon inside an equilateral triangle, so that three of its sides may touch it, and show the ratio the hexagon bears to the tringle.
372. A philosopher had a window a yard square, and it let in too much light; he blocked up one half of it, and still had a
window a yard high and a yard wide. Say how he did it.
373. Can you divide an equilateral triangle into two equal parts by a line drawn parallel to one of its sides?
374. Given the chord of an arc 50, and the sine of the arc 40; required the versed sine by calculation, and point out on the figure that it is equal to radius minus co-sine.
375. Can you divide a common triangle into two equal parts by a line parallel to one of its sides?
376. Can you divide a triangle into two equal parts by a line from any point in any one of its sides?
377. Show how many solid feet there are in a solid yard.
378. Make an oblique square prism with two rectangular sides and two rhomboidal sides.
379. Make an oblique square prism with all its sides equally rhomboidal.
380. Can you place an equilateral triangle in a square so that one angular point of the equilateral triangle may coincide with one angular point of the square, and the other two angular points of the triangle may touch, at equal distances from the angle of the square, two of the sides of the square ?
381. Can you divide a line into 5 equal parts ?
382. Can you divide a line into 3} parts?
333. Can you divide a line as any other line is divided ?
384. Can you answer by geometry the question, When three yards of cloth cost 12s., what will five yards cost ?
The rule by which areas are found, when the dimensions are given in feet and inches, takes the name of duodecimals; such areas being always expressed in feet, twelfths of feet, or parts, twelfths of parts, or square inches. (See Young's “Mensuration.")
Duodeciinals are used chiefly by artisans for the purpose of determining the quantity of work they have done, or the quantity of materials they have used.
385. Give a plan of a duodecimal part, that is, of a twelfth of a foot.
386. Give a plan of a duodecimal inch, that is, of one-twelfth of a part, and show that its size is the same as the square inch, although the forms may differ.
387. Give a plan that shall show the area of a square whose side is 1 ft. 1 in., and prove by duodecimals.
388. Give by a scale of an inch to a foot the plan of a board, 3 ft. 4 in. long, and 2 ft. 2 in. wide, and prove by duodecimals the
389. Ascertain by geometry how many inches there are in the diagonal of a square foot, and how many in the diagonal of a cubic foot, and prove by calculation.
390. Can you make an octagon which shall have its alternate sides one-half of the others, and that shall still be symmetrical ?
391. Can you place in a pentagon a rhombus that shall touch with its angular points