AD, it will produce the surface of the sphere, and each of the small arcs, as KL, will produce the surface of a truncated cone, of which KE will be half the upper base, and LF half the lower, (such as is represented in Fig. 1, Plate 7,) forming an integral part of the surface of the sphere; and the surface of this truncated cone will be equal to the product of KM or EF multiplied by the circumference whose radius is IC or AC. The triangle KML being similar to the triangle IHC, for the three sides of the one are perpendicular to the corresponding sides of the other, we have this proportion, KL: KM:: IC: IH; or, circumferences being to one another in the proportion of their radii, KL: KM:: the circumference described upon IC: the circumference described upon IH, consequently the product of KL multiplied into the circumference on IH is equal to the product of KM multiplied into the circumference on IC, or to the product of EF multiplied into the circumference on AC. Now the first of these products is the surface of the truncated cone formed by the motion of KL, which is therefore equal to the product of EF multiplied into the circumference on AC, or, in other words, to the product of the altitude of the truncated cone, multiplied into the circumference of any great circle of the sphere. In the same way, if we divide the circumference into a number of small arcs, similar to that represented by KL, the same result will be produced, and consequently the whole of these small truncated cones, composing by supposition the surface of the sphere, must be equal to the circumference of a great circle multiplied by the sum of the altitudes of these cones, which will of course form the whole of AD, which is the diameter of the sphere and hence the surfase, or superficial contents of a sphere, is equal to the pro duct, duct of its diameter multiplied into the circumference of a great circle on it, that is, of a circle whose diameter is that of the sphere. If we imagine a cylinder circumscribing and in contact with a sphere, that is, a cylinder whose altitude and the diameter of whose base are both equal to the diameter of the sphere, it will follow from what has just been shown, that the surface of the sphere is equal to the interior surface of the circumscribing cylinder; for the surface of a cylin der is equal to the product of the area of its circular base multiplied by its altitude; and in this case the circumference and diameter of the base must be precisely equal to those of the inscribed sphere. Since, therefore, in order to have the superficial area of a circle we multiply the circumference by one fourth part of the diameter, (Example 13 of Mensuration of Surfaces,) and since, in order to obtain the superficial area of a sphere,we multiply the circumference by the whole diameter, it follows that the superfi cial area of a sphere is equal to four times the area of one of its great circles. From the foregoing demonstrations it will also follow, that in order to obtain the superficial area of a segment or portion cut off from a sphere, we must multiply the circumference of a cirele whose diameter is that of the sphere, by the altitude of the segment; and that to obtain the superficial area of a portion of a sphere cut off by two parallel planes, we must multiply the circumference of a great circle by the altitude of the portion included between the two parallel planes. The rule, therefore, for finding the solidity of a sphere is, to multiply one third part of the radius, which is one sixth of the diameter, by four times the superficial contents of a great circle on a sphere, or four times the third part of the radius by the superficial contents of such a circle, or, lastly, two third parts of the diameter by the same superficial area. It was already shown, (Example 5 of Mensuration of Solids,) that the solidity of a cylinder is the product of the area of its base multiplied into its altitude; hence the solidity of a sphere will be two third parts of that of a circumscribing cylinder, that is, of one whose diameter and altitude are both equal to the diameter of the inscribed sphere. In comparing solid bodies in general, we inquire how often the number of parts, of a certain determinate magnitude, of which the one of the bodies is composed, may be contained in the number of the same parts of which the other body is composed; or we inquire the proportion between the number of equal parts contained in each of the solid bodies. Thus two prisms and two cylinders, or a prism and a cylinder, are to one another as the product of their respective bases multiplied by their altitudes; consequently prisms and cylinders on equal bases are in proportion as their altitudes, and prisms and cylinders of equal altitudes are in proportion as their bases. In the same way pyramids and cones, each being one third part of their corresponding prisms and cylinders, (Example 6 of Mensuration of Solids,) are to each other in the proportion of their respective bases and altitudes. The areas of similar surfaces are to one another in the proportion of the squares of their homologous or corresponding sides or lines: thus the area of an equilateral triangle 10 inches a side is 43,3 square inches; and the area of another equilateral triangle of 20 inches a side is 173,2 square inches: now if we take the square of the side 10, which is 100, and the square of the side 20, which is 400, we shall find that the area 43,3 is to the area 173,2 in the proportion of the square 100 to the square 400. J The same thing holds true, as the student may convince himself by a trial, with respect to the areas of all the similar figures and by analogy, the contents of all similar solid figures are to one another in the proportion of the cubes of their corresponding or homologous sides or lines, Thus the solid contents of a sphere of 10 inches diameter will be 261,773 nearly, and the solidity of another sphere of 20 inches diameter will be 2094,1829 nearly, then the cube of the diameter 10 being 1000, and that of the dia meter 20 being 8000, the solid contents 261,773 will be to the solid contents 2094,1829 as the cube 1000 is to the cube 8000. If it were required to form a solid body similar to a given solid, and whose contents were to be in a given proportion to the contents of the given solid, we must discover the line or side which, when cubed, would bear to the homologous line or side of the given solid the same proportion with that of their proposed solidities: thus if, for example, it were required to determine the diameter of a sphere whose solidity would be 8 times greater than that of a sphere whose diameter is 10 inches, we would state this proportion; as the solidity of the given sphere which may be expressed by 1 to the solidity of the sphere required, which is 8 times greater, or 8, so is the cube of the given diameter 10 = 1000 to a fourth proportional 8000, which will be the cube of the diameter of the required sphere; and extracting the cube root 20, (Arith. page 266,) this will be the diameter of a sphere whose solidity will be 8 times greater than that of one whose diameter is 10 inches. As in bodies consisting of the same substance, the weights are in proportion to the quantities of matter in each body, so by knowing the weight of any regular figure, as a ball for sphere of a determinate diameter, if we wish to learn the weight of another ball or sphere of the same sub stance, stance, but of a different diameter, we state this proportion; as the cube of the diameter of the first sphere whose weight is known to the cube of the diameter of the second, so is the weight of the first sphere to the weight of the second. SURVEYING OR LAND-MEASURING. By Surveying is generally understood the art of measur ing the contents of a field, an estate, a parish, a county, &c. and it comprehends three several operations, viz. 1st. making the survey, or measuring the dimensions of the ground; edly, delineating or laying down upon paper a correct representation of the ground surveyed, every part being exhibited in its due situation and proportion with respect to the others; and, 3dly, calculating the area or superficial contents of the several portions and of the whole. Of these operations the first is what is properly called surveying, the second is called plotting, protracting, or mapping, and the third is termed casting up, or computing the contents. 1st. The first operation, or surveying, consists of two parts, the making of observations for the angles, and the taking of lineal measures for the sides: the former operation being performed with one or other of the following instruments, viz. the theodolite, the circumferentor, the semicircle, the graphometer, the plain-table, the compass, or by the measuring chain alone: the latter operation is performed by the chain or the perambulator. The theodolite is made in various ways, according to the taste and skill of the artist, in order to render it the more simple and manageable, or the more accurate and convenient in use; but in general it consists of a brass circle of 6, 9, or 12 inches diameter, strengthened by four cross bars meet |