quantity of ground does it enclose, and how much does it occupy? Ans. The wall encloses 4a. Or. 6p. and occupies 1760.531 square feet. 7. The area of a right-angled triangle is 60, and the hypothenuse 17; required the two legs. Ans. 15 and 8. 8. Two sides of an obtuse-angled triangle are 5 and 10 chains; what must be the length of the third side, that the triangle may contain just 2 acres of ground? Ans. 8.06225 or 13.60147 chains. 9. Two travellers, A and B, departed from an inn, at the hour of eight in the morning; A proceeded north west, at the rate of 6 miles an hour, and B north east, at the rate of 8 miles an hour; how far were they distant from each other, at twelve o'clock of the same day? Ans. 40 miles. 10. Two boys amusing themselves at a game called snatch-apple, in a room 13 feet high, find that by standing 12 feet from each other, the apple, which is suspended from the ceiling by a string, and in a right line between them, when put in motion, just touches each of their mouths. Required the area of the sector described by the string and apple; the perpendicular height of each boy's mouth, from the ground, being 5 feet. Ans. 64.35726 feet. 11. What is the area of an isosceles triangle inscribed in a circle whose diameter is 24; the angle included by the equal sides of the triangle being 30 degrees? Ans. 134.3538. 12. A maltster has a kiln that is 18 feet square; but he intends to pull it down, and build a new one, that may dry three times as much at once as the old one; what must be its length, if its breadth be 24 feet? Ans. 40 feet. 13. The side AB of a triangular field is 40, BC 30, and CA 25 chains; required the sides of a triangle parted off by a division-fence made parallel to AB, and proceeding from a point in CA, at the distance of 9 chains from the angle A. Ans. 16, 19.2, and 25.6 chains. 14. A field in the form of a right-angled triangle is to be divided between two persons, by a fence made from the right angle meeting the hypothenuse perpendicularly, at the distance of 880 links from one end; required the area of each person's share, the length of the division-fence being 660 links. Ans. 2a. 3r. 241p. and 1a. 2r. 211p. 15. It is required to part from a triangular field whose three sides measure 1200, 1000, and 800 links respectively, 1 acre, 2 roods, and 16 perches, by a line parallel to the longest side. Ans. The sides of the remaining triangle are 927, 7724, and 618 links respectively. 16. The perambulator, or surveying wheel, is so contrived, as to turn just twice in the length of a pole, or 16 feet; what is its diameter ? Ans. 2.626 feet. 17. Required the side of an equilateral triangle whose area is just two acres. Ans. 6.79617 chains. 18. The sides of a triangle are 20, 30, and 40 respectively; what is the area of its inscribed circle? Ans. 130.8999. 19. In an isosceles triangle, two circles are inscribed touching each other and the sides of the triangle; the diameters of the circles are 9 and 25; required the sides of the triangle. Ans. 44.27083, 44.27083, and 41.66666. 20. The base of a field, in the form of a trapezoid, is 30, and the two perpendiculars are 28 and 16 chains respectively; it is required to divide it equally between two persons, by a fence parallel to the perpendiculars. Ans. The division-fence is 22.8035 chains, and it divides the base into two parts, whose lengths are 17.0087, and 12.9913 chains respectively. 21. A field in the form of an equilateral triangle, contains just half an acre; what must be the length of a tether, fixed at one of its angles, and to a horse's nose, to enable him to graze exactly half of it? Ans. 48.072 yards. 22. The diameter of a circular estate is 25 chains; what is the length of the chord which divides it into two segments whose areas are to each other as 2 to 1 ? Ans. 24.1062 chains. 23. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made three turns while the inner wheel made two. The wheels were of equal height; and supposing them fixed at the statutable distance of 5 feet asunder on the axle-tree; what was the circumference of the track described by each wheel? Ans. The length of the track described by the outer wheel is 94.248 feet, and that described by the inner wheel 62.832 feet. 24. If the frustrum of a cone whose diameters are 8 and 12 feet respectively, be made to revolve with its slant side upon a horizontal plane, until it returns to its first position; what will be the area of the space passed over by the slant side, the length of which is 10 feet? Ans. 1570.8 feet. 25. Being desirous of finding the height of a steeple, I placed a looking-glass at the distance of 100 feet from its base, on the horizontal plane, and walking backward 5 feet, I saw the top of the steeple appear in the centre of the glass; required the steeple's height, my eye being 5 feet 6 inches from the ground. Ans. 110 feet. 26. Three men bought a grinding-stone of 50 inches diameter, each paying part of the expense; what part of the diameter must each person grind down for his share? Ans. The first must grind down 9.1752, the second 11.9573, and the third 28.8675 inches. 27. Wanting to know the height of the cathedral at York, I measured the length of its shadow, and found it to be 200 feet. At the same time, a staff 5 feet long, cast a shadow of 4 feet; required hence the height of that elegant and magnificent structure. Ans. 250 feet. 28. At Matlock, near the Peak, in Derbyshire, where there are many surprising curiosities of nature, is a rock by the side of the river Derwent, rising perpen dicularly to a wonderful height, which, being inaccessible, I endeavoured to measure in the following manner: At the distance of 340 feet from the bottom of the rock, I fixed a staff, 8 feet in length, perpendicularly to the horizontal plane; and at a convenient distance from this, I fixed another, 3 feet long, so that by looking over both their tops, I could just see the summit of the rock. The distance between the staves I found to be 4 feet 6 inches. From these data I determined the perpendicular altitude of the rock; and you are requested to repeat the process. Ans. The height of the rock is 128.5925 yards. 29. There are two columns in the ruins of Persepolis, left standing upright; one of which is 64 feet above the horizontal plane, and the other 50. Between these, in a right line, stands a 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, whose base is just 76 feet from the centre of the figure's base; required hence the distance between the tops of the two columns. Ans. 157.03687 feet. 30. A gentleman a garden had, Tell us how wide the walk must be. Ans. 25.96876 feet, Note 1. Persepolis, mentioned in the 29th question, was the ancient capital of the Persian empire. This immense and renowned city was taken by Alexander the Great, about 300 years before the Christian æra, who, at the instigation of the depraved courtezan Thais, ordered it to be set on fire. Its magnificent ruins are about 50 miles N. E. of Schiras, the present capital of Persia. 2. The 30th question was first proposed in the Ladies' Diary for the year 1708. The poetry is not very elegant; but the question is more original in its present form than it would be in prose. PART IV. MENSURATION OF SOLIDS, THE DESCRIPTION AND USE OF THE CARPENTER'S RULE, TIMBER MEASURE, And Miscellaneous Questions concerning Solids. SECTION I. MENSURATION OF SOLIDS. DEFINITIONS. 1. A SOLID is a figure which generally consists of three dimensions; viz. length, breadth, and thickness. 2. The measurement of a solid is called its solidity, capacity, or content. 3. The contents of solids are estimated by a cube, whose side is one inch, one foot, one yard, &c.; hence the solidity of a body is said to be so many cubic inches, feet, yards, &c. as are contained in that body. 4. A cube is a solid having six equal square sides. 5. A parallelopipedon is a solid having six rectangular sides, every opposite two of which are equal and parallel. |