or end is a, and radius r, moving in the direction of its axis; because then s = 1, and a = pr2, where p = 3.1416; then pnv22 pnv22 the retarding 402. Corol. 5. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of motion. But were its face an elliptic section, or a conical surface, or any other figure everywhere equally inclined to the axis, or direction of motion, the sine or inclination being : then, the number of particles of the fluid striking the face being still the same, but the force of each, opposed to the direction of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resistpar222 ing force & would be 4g PROPOSITION LXXIX. 403. The Resistance to a Sphere moving through a Fluid, is but Half the Resistance to its Great Circle, or to the End of a Cylinder of the same Diameter, moving with an Equal Velocity. H LET AFEB be half the sphere, moving in the direction CEG. Describe the paraboloid AIEKB on the same base. Let any particle of the medium meet the semicircle in F, to which draw the tangent FG, the radius FC, and the ordinate FIH. Then the force of any particle on the surface at r, is to its force on the base at H, as the square of the sine of the angle G, or its equal the angle FCH, to the square of radius, that is, as HF to CF. Therefore the force of all the particles, or the whole fluid, on the whole surface, is to its force on the circle of the base, as all the HF to as many times cr2. But Cris CA2 = AC CB, and HF2 = AH HB by the nature of the circle: also, AH. HB: AC . CB HI CE by the nature of the parabola; consequently the force on the spherical surface, is to the force on its circular base, as all the HI's to as many CE's, that is, as the content of the paraboloid to the content of its circumscribed cylinder, namely, as 1 to 2. 1 404. Corol. Hence, the resistance to the sphere is R = pnq22 , being the half of that of a cylinder of the same diameter. diameter. For example, a 9lb iron ball, whose diameter is 4 inches, when moving through the air with a velocity of 1600 feet per second, would meet a resistance which is equal to a weight of 1323lb, over and above the pressure of the atmosphere, for want of the counterpoise behind the ball. PRACTICAL EXERCISES IN MENSURATION. QUEST. 1. WHAT difference is there between a floor 28 feet long by 20 broad, and two others, each of half the dimensions; and what do all three come to at 45s. per square, or 100 square feet? Ans. dif. 280 sq. feet. Amount 18 guineas. QUEST. 2. An elm plank is 14 feet 3 inches long, and I would have just a square yard slit off it; at what distance from the edge must the line be struck? Ans. 7 inches. QUEST. 3. A cieling contains 114 yards 6 feet of plastering, and the room 28 feet broad; what is the length of it? Ans. 369 feet. QUEST. 4. A common joist is 7 inches deep, and 25 thick; but wanting a scantling just as big again, that shall be 3 inches thick; what will the other dimension be? Ans. 113 inches. QUEST. 5. A wooden cistern cost me 3s. 2d. painting within, at 6d. per yard; the length of it was 102 inches, and the depth 21 inches; what was the width? Ans. 27 inches. QUEST. 6. If my court-yard be 47 feet 9 inches square, and I have laid a foot-path with Purbeck stone, of 4 feet wide, along one side of it; what will paving the rest with flints come to, at 6d. per square yard? Ans. 5. 165. Ožd. QUEST. 7. A ladder, 263 feet long, may be so planted, that it shall reach a window 22 feet from the ground on one side of the street; and, by only turning it over, without moving the foot out of its place, it will do the same by a window 14 feet high on the other side; what is the breadth of the street? Ans. 37 feet 9 inches. QUEST. 8. The paving of a triangular court, at 18d. per foot, came to 100.; the longest of the three sides was 88 feet; required the sum of the other two equal sides? QUEST. 9. There are two columns in the ruins of Perse polis left standing upright: the one is 64 feet above the plain, and the other 50: in a straight line between these stands an ancient small statue, the head of which is 97 feet from the summit of the higher, and 86 feet from the top of the lower column, the base of which measures just 76 feet to the centre of the figure's base. Required the distance between the tops of the two columns? Ans. 157 feet nearly. QUEST. 10. The perambulator, or surveying wheel, is so contrived, as to`turn just twice in the length of 1 pole, or 16 feet; required the diameter ? Ans. 2.626 feet. QUEST. 11. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner made but one: the wheels were both 4 feet high; and supposing them fixed at the distance of 5 feet asunder on the axletree, what was the circumference of the track described by the outer wheel? Ans. 62.83 feet. QUEST. 12. What is the side of that equilateral triangle, whose area cost as much paving at 8d. a foot, as the pallisading the three sides did at a guinea a yard? Ans. 72.746 feet: QUEST. 13. In the trapezium ABCD, are given, AB = 13, BC 31, CD = 24, and DA 18, also в a right angle; required the area ? Ans. 410 122. QUEST. 14. A roof which is 24 feet 8 inches by 14 feet 6 inches, is to be covered with lead at 8lb. per square foot: what will it come to at 18s per cwt. ? Ans. 221. 19s. 10d. QUEST. 15. Having a rectangular marble slab, 58 inches by 27, I would have a square foot cut off parallel to the shorter edge; I would then have the like quantity divided from the remainder parallel to the longer side; and this alternately repeated, till there shall not be the quantity of a foot left: what will be the dimensions of the remaining piece? Ans. 20.7 inches by 6.086. QUEST. 16. Given two sides of an obtuse-angled triangle, which are 20 and 40 poles; required the third side, that the triangle may contain just an acre of land? Ans. 58.876 or 23.099. QUEST. 17. The end wall of a house is 24 feet 6 inches in breadth, and 40 feet to the eaves; of which is 2 bricks thick, more is 14 brick thick, and the rest I brick thick. Now the triangular gable rises 38 courses of bricks, 4 of which usually make a foot in depth, and this is but 44 inches, or or half a brick thick: what will this piece of work come to at 57. 10s. per statute rod? Ans. 20. 11s. 74d. QUEST. 18. How many bricks will it take to build a wall, 10 feet high, and 500 feet long, of a brick and half thick; reckoning the brick 10 inches long, and 4 courses to the foot in height? Ans. 72000. QUEST. 19. How many bricks will build a square pyramid of 100 feet on each side at the base, and also 100 feet perpendicular height: the dimensions of a brick being supposed 10 inches long, 5 inches broad, and 3 inches thick?" Ans. 3840000. QUEST. 20. If, from a right-angled triangle, whose base is 12, and perpendicular 16 feet, a line be drawn parallel to the perpendicular, cutting off a triangle whose area is 24 square feet; required the sides of this triangle? Ans. 6, 8, and 10. QUEST. 21. The ellipse in Grosvenor-square measures $40 links across the longest way, and 612 the shortest,within the rails: now the walls being 14 inches thick, what ground do they enclose, and what do they stand upon? Ans. Senclose 4 ac. Or. 6 p. stand on 1760 sq. feet. QUEST. 22. If a round pillar, 7 inches over, have 4 feet of stone in it: of what diameter is the column, of equal length, that contains 10 times as much? Ans. 22.136 inches. QUEST. 23. A circular fish-pond is to be made in a garden, that shall take up just half an acre; what must be the length of the chord that strikes the circle? Ans. 274 yards. QUEST. 24. When a roof is of a true pitch, or making a right angle at the ridge, the rafters are nearly of the breadth of the building: now supposing the eaves-boards to project 10 inches on a side, what will the new ripping a house cost, that measures 32 feet 9 inches long, by 22 feet 9 inches broad on the flat, at 15s. per square? Ans. 87. 15s. 91⁄2d. QUEST. 25. A cable, which is 3 feet long, and 9 inches in compass, weighs 22lb; what will a fathom of that cable. weigh, which measures a foot about ? Ans. 783 lb. QUEST. 26. My plumber has put 281b. per square foot into a cistern, 74 inches and twice the thickness of the lead long, 26 inches broad, and 40 deep: he has also put three stays across it within, of the same strength, and 16 inches deep, deep, and reckons 22s. per cwt. for work and materials. I, being a mason, have paved him a workshop, 22 feet 10 inches broad, with Purbeck stone, at 7d. per foot; and on the balance I find there is 3s. 6d. due to him; what was the length of the workshop, supposing sheet lead of inch thick to weigh 5.899lb the square foot? of an Ans. 32 feet, 02 inch. QUEST. 27. The distance of the centres of two circles, whose diameters are each 50, being given, equal to 30; what is the area of the space enclosed by their circumferences? Ans. 559-119. QUEST. 28. If 20 feet of iron railing weigh half a ton, when the bars are an inch and quarter square; what will 50 feet come to at 34d. per lb, the bars being of an inch square? Ans. 201. Os. 2d. QUEST. 29. The area of an equilateral triangle, whose base falls on the diameter, and its vertex in the middle of the arc of a semicircle, is equal to 100: what is the diameter of the semicircle? Ans. 26-32148. QUEST. 30. It is required to find the thickness of the lead in a pipe, of an inch and quarter bore, which weighs 14lb. per yard in length; the cubic foot of lead weighing 11325 ounces? Ans. 20737 inches. QUEST. 31. Supposing the expense of paving a semicircular plot, at 2s. 4d. per foot, come to 10.; what is the diameter of it? Ans. 14.7737 feet. QUEST. 32. What is the length of a chord which cuts off of the area from a circle whose diameter is 289 ? Ans. 278.6716. QUEST. 33. My plumber has set me up a cistern, and, his shop-book being burnt, he has no means of bringing in the charge, and I do not choose to take it down to have it weighed; but by measure he finds it contains 64 square feet, and that it is precisely of an inch in thickness. Lead was then wrought at 21. per fother of 194 cwt. It is required from these items to make out the bill, allowing 65 oz. for the weight of a cubic inch of lead? Ans. 4. 11s. 2d. QUEST. 34. What will the diameter of a globe be, when the solidity and superficial content are expressed by the same number? Ans. 6. QUEST. 35. A sack, that would hold 3 bushels of corn, is 224 inches broad when empty; what will another sack ....contain, |