OF SOLIDS. the whole circumference of the sphere by the height of the part required. Ex. 1. Required the convex superficies of a sphere, whose diameter is 7, and circumference 22. Ans. 154. Ex. 2. Required the superficies of a globe, whose diameter Ans. 1809-5616. is 24 inches. Ex. 3. Required the area of the whole surface of the earth, its diameter being 79573 miles, and its circumference 25000 miles. Ans. 198943750 sq. miles. Ex. 4. The axis of a sphere being 42 inches, what is the convex superficies of the segment whose height is 9 inches? Ans. 1187-5248 inches. Ex. 5. Required the convex surface of a spherical zone, whose breadth or height is 2 feet, and cut from a sphere of Ans. 78-54 feet. 12 feet diameter. I. PROBLEM VIII. To find the Solidity of a Sphere or Globe. RULE 1. Multiply the surface by the diameter, and take of the product for the content *. Or, which is the same thing, multiply the square of the diameter by the circum ference, and take of the product. RULE II. Take the cube of the diameter, and multiply it by the decimal ·5236, for the content. RULE III. Cube the circumference, and multiply by ⚫01688. Ex. 1. To find the solid content of the globe of the earth, supposing its circumference to be 25000 miles. Ex. 2. Supposing that a cubic •269 of a lb. avoird. what is the 5.04 inches diameter ? Ans. 263750000000 miles. inch of cast iron weighs weight of an irou ball of = the diameter, c * For, put d the circumference, and s surface of the sphere, or of its circumscribing cylinder; also, a == number 3.1416. the the Then, is is = the base of the cylinder, or one great circle of the sphere; and d is the height of the cylinder: therefore 4ds is the content of the cylinder. But of the cylinder is the sphere, by th. 117, Geom. that is, of ads, or ds is the sphere; which is the first rule. Again, because the surfaces is ad2; therefore ds = Lad1 = ·5236d3, is the content, as in the 2d rule. Also, d being = c÷a, therefore fad3 = fc3 ÷ a3 = •01688, the 3d rule for the content. PROBLEM IX. To find the Solid Content of a Spherical Segment. * RULE 1. From 3 times the diameter of the sphere take double the height of the segment; then multiply the remainder by the square of the height, and the product by the decimal 5236, for the content. RULE II. To 3 times the square of the radius of the segment's base, add the square of its height; then multiply the sum by the height, and the product by 5236, for the content. Ex. 1. To find the content of a spherical segment, of 2 feet in height, cut from a sphere of 8 feet diameter. Ans. 41-888. By corol. 3, of theor. 117, Geom. it ap pears that the spheric segment PFN, is equal to the difference between the cylinder ABLO, and the conic frustum ABMQ. E G But, putting d = AB or FH the diameter of the sphere or cylinder, h = FK the height of the segment, r = PK the radius of its base, and a = 3-1416; then the content of the cone ABI is = · fad1× fri = ad; and by D the similar cones ABI, QMI, as FI: KI' :: ad: ad3× ( પુત h. H = the cone QMI; therefore the cone ABI the cone qui = {ad›—¿{ad› × (3d77")1 = jad:h— jadh3 + jah3 is the conic frustum of ABMQ. And dad his = the cylinder ABLO. Then the difference of these two is fadh? — Zah1 — fah3 × (3d —2h), for the spheric segment PFN; which is the first rule. Again, because PK FK X KH (cor. to theor. 87, Geom.) or r2 = h 3r+ha which being substituted in the former rule, it becomes fah2 X =fah × (3r? + h), which is the 2d rule. Note. By subtracting a segnent from a half sphere, or from another segment, the content of any frustum or zone may be found. OF SOLIDS. "Ex. 2. What is the solidity of the segment of a sphere, its height being 9, and the diameter of its base 20? Ans. 1795-4244. Note. The general rules for measuring the most useful figures having been now delivered, we may proceed to apply them to the several practical uses in life, as follows. LAND SURVEYING. SECTION I. DESCRIPTION AND USE OF THE INSTRUMENTS. 1. OF THE CHAIN. LAND is measured with a chain, called Gunter's Chain, from its inventor, the length of which is 4 poles, or 22 yards, or 66 feet. It consists of 100 equal links; and the length of each link is therefore of a yard, or of a foot, or 7.92 inches. 66 Land is estimated in acres, roods, and perches. An acre is equal to 10 square chains, or as much as 10 chains in length and 1 chain in breadth. Or, in yards, it is 220 × 22 = 4840 square yards. Or, in poles, it is 40 X 4 = 160 square poles. Or, in links, it is 1000 × 100 100000 square links: these being all the same quantity. Also, an acre is divided into 4 parts called roods, and a rood into 40 parts called perches, which are square poles, or the square of a pole of 5 yards long, or the square of of a chain, or of 25 links, which is 625 square links. So that the divisions of land measure will be thus: 625 sq. links = 1 pole or perch The lengths of lines measured with a chain, are best set down in links as integers, every chain in length being 100 links; and not in chains and decimals. Therefore, after the content is found, it will be in square links; then cut off five of the figures on the right hand for decimals, and the rest will be acres. These decimals are then multiplied by 4 for roods, and the decimals of these again by 40 for perches. EXAM. Suppose the length of a rectangular piece of ground be 792 links, and its breadth 385; to find the area in acres, roods, and perches. 792 385 3960 6336 2376 3(4920 3.04920 4 •19680 40 7.87,200 Ans. 3 acres, 0 roods, 7 perches. 2. OF THE PLAIN TABLE. THIS instrument consists of a plain rectangular board, of any convenient size: the centre of which, when used, is fixed by means of screws to a three-legged stand, having a ball and socket, or other joint, at the top, by means of which, when the legs are fixed on the ground, the table is inclined in any direction. To the table belong various parts, as follow. 1. A frame of wood, made to fit round its edges, and to be taken off, for the convenience of putting a sheet of paper on the table. One side of this frame is usually divided into equal parts, for drawing lines across the table, parallel or perpendicular to the sides; and the other side of the frame is divided into 360 degrees, to a centre in the middle of the table; by means of which the table may be used as a theo. dolite, &c. 2. A magnetic needle and compass, either screwed into the side of the table, or fixed beneath its centre, to point out the directions, and to be a check on the sights. 3. An index, which is a brass two-foot scale, with either a small telescope, or open sights set perpendicularly on the ends. These sights and one edge of the index are in the same plane, and that is called the fiducial edge of the index. To use this instrument, take a sheet of paper which will cover it, and wet it to make it expand; then spread it flat on the table, pressing down the frame on the edges, to stretch it and keep it fixed there; and when the paper is become dry, it will, by contracting again, stretch itself smooth and flat from any cramps and unevenness. On this paper is to be drawn the plan or form of the thing measured. Thus, begin at any proper part of the ground, and make a point on a convenient part of the paper or table, to represent that place on the ground; then fix in that point one |