ACB. Hence the volume of the polygonal sector must be less than that of the spherical sector AC B, which is a part of it, which is absurd. Hence the product of the spherical surface by a third of its radius is not less than the volume of the spherical sector. Hence this product must be equal to the volume of the spherical sector. If the circular sector A CB be supposed to increase by the increase of the angle ACB, until it becomes equal to a semicircle, the corresponding spherical sector will become equal to the whole sphere. Hence the volume of the entire sphere is equal to the product of its surface by a third of its radius. (246) Cor. 1.—The volume of a spherical sector is equal to that of a cone whose altitude is the radius, and whose base is equal to the spherical base of the sector; for the volume of the cone is equal to its base multiplied by a third of its altitude. This may also appear from considering the spherical sector to be formed of an infinite number of cones having the same vertex, equal altitudes, and infinitely small bases, which bases may be conceived to form the spherical base of the sector. (247) Cor. 2.-The volume of a sphere is equal to that of a cone whose base is equal to the surface of the sphere, and whose altitude is equal to the radius. (248) Cor. 3.-The volumes of spheres are as the cubes of their radii or diameters. For their surfaces are as the squares of the radii, and these being multiplied by one third of the radii give products which are as the cubes of the radii. PROPOSITION XVI. (249) Let a square and an equilateral triangle be cir cumscribed round the same circle, the base of the equilateral triangle coinciding with a side of the square, and let the whole figure revolve round the altitude B A of the triangle. A sphere will thus be described circumscribed by a cylinder and cone; the entire surfaces of this sphere, cylinder, and cone, are in continued proportion, the common ratio being 2:3, and their volumes are also in continued proportion and in the same ratio. B The surface of the sphere is equal to four times one of its great circles (234). The cylindrical surface is equal to this; and as each of the bases of the cylinder is a great circle, the entire surface of the cylinder is equal to six times a great circle. Hence the ratio of the surface of the sphere to the entire surface of the cylinder is 4:6, or 2:3. If the radius CA of the circle be r, half the base of the equilateral triangle will be 3.r; hence the area of the base of the cone will be 3 pl 7. The circumference of the base will be 2.v3.ry, and the side of the cone is 2,~3.r; therefore the conical surface is 2.3.rén, or 6 y 7 (222); to which if the base be added, the entire surface of the cone will be 9 pén, or nine times the area of a great circle. Thus it appears, that the entire surfaces of the sphere, cylinder, and cone, are respectively equal to 4, 6, and 9 great circles, and are, therefore, in continued proportion, the common ratio being 2:3. The volume of the sphere is equal to 18 7 (245). The volume of the cylinder is 278 7. The base of the cone being 3 r27, and its altitude 3r, its volume is 3 pm. Hence the volumes of the three solids are as the numbers 4, 2, 3, or 4, 6, 9, which are in continued proportion, the common ratio being 2 : 3. (250) Cor.—The base of the cone is equal to three great circles, and its conical surface to six. Hence in such a cone the conical surface is double its base. APPENDIX No. I. GEOMETRICAL ANALYSIS. SECTION 1. Introduction. (1) ANALYSIS, or resolution, is a name given to a species of mathematical investigation, which commencing with the assumption of that which is sought as if it were given, a chain of relations is pursued which terminates in what is given (or may be obtained) as if it were sought. SYNTHESIS, or composition, is a process the very reverse of this ; being one in which the series of relations exhibited commences with what is given, and ends with what is sought. Consequently analysis is the instrument of invention, and synthesis that of instruction. The analysis of the ancients is distinguished from that of the moderns by being conducted without the aid of any calculus, or the use of any principles except those of Geometry, the latter being conducted entirely by the language and principles of Algebra. The ancient is, therefore, called the Geometrical Analysis. The interest which the Geometrical Analysis derives from its antiquity, and from having been the instrument by which the splendid results of the ancient Geometry were obtained, would alone be sufficient to render it an object of attention even after the discovery of the more powerful agency of Algebra. But this is not its only nor its principal claim upon our notice. Its inferiority, compared with the modern analysis, in power and facility, is balanced by its extreme purity and rigour; and though its value as an instrument of discovery be lost, yet it must ever be considered as a most useful exercise for the mind of a student; and it may be fairly questioned, whether it may not be more conducive to the improvement of the mental faculties than the modern analysis, unless the latter be pursued much farther than it usually is in the common course of academical education, in which the student acquires little more than a knowledge of its notation. Newton was fully aware of the advantages attending the cultivation of this branch of mathematical science, and in many parts of his works laments that the study of it has been so much abandoned. He considered, that, however inferior in power and despatch the ancient method might be, it had greatly the advantage in rigour and purity; and he feared, that by the premature and too frequent use of the modern analysis the mind would become debilitated and the taste vitiated. We must however confess, that the pretensions of the ancient method to superior rigour do not seem to us to be as well founded as they are sometimes considered. It would be no very difficult matter to expunge the algebraical symbols from a modern investigation, and substitute for them their meaning expressed in the language used in geometrical investigations; but would such a change confer upon them' greater rigour, or would it give to the conclusions greater validity? And yet this is precisely what Newton himself has done in many parts of his great work, the Principia. His theorems are, evidently, investigated algebraically; but in demonstrating them, the process is disguised by the substitution of lines and geometrical figures for the algebraical species and formulæ. It cannot but excite astonishment, that a man of his extraordinary sagacity could so far deceive himself, as to suppose that by such a proceeding his reasoning acquired greater rigour. But without reference to the modern analysis, we conceive that the ancient method has sufficient claims to our attention on the score of its own intrinsic beauty. It has this 'further advantage, that we can enter at once upon its most interesting discussions without the repelling task of learning any new language or system of notation. In the application of the Geometrical Analysis to the solution of problems, or the demonstration of theorems, no general rules nor invariable directions can be given which will apply in all cases. The previous construction to be used, and the preparatory steps to be taken, depend on the particular circumstances of the question, and must be determined by the sagacity of the analyst; and his skill and taste will be evinced in the selection of the properties or affections of the given or sought quantities on which he founds his analysis ; for the same question may frequently be investigated in many different ways. In submitting a problem to analysis, its solution, in the first instance, is assumed; and from this assumption a series of consequences are drawn, until at length something is found which may be done by established principles, and which if done will necessarily lead to the execution of what is required in the problem. Such is the analysis. In the synthesis, then, or the solution, we retrace our steps; beginning by the execution of the construction indicated by the final result of the analysis, and ending with the performance of what is required in the problem, and which constituted the first step of the analysis. When a theorem is submitted to analysis, the thing to be determined is, whether the statement expressed by it be true or not. In the analysis this statement is, in the first instance, assumed to be true; and a series of consequences is deduced from it until some result is obtained, which either is an established or admitted truth, or contra. dicts an established or admitted truth, If the final result be an esta. blished truth, the theorem proposed may be proved by retracing the steps of the investigation, commencing with that final result, and concluding with the proposed theorem. But if the final result contradict an established truth, the proposed theorem must be false, since it leads to a false conclusion. |