the sphere (158) is equal to four great circles multiplied by of the radius or of the diameter, which is the same as a great circle multiplied by or of the diameter. Therefore the two solidities are to each other as 2 to 3. 161. THEOREM.-The solidity of a spherical seg ment of one base, is found by taking the difference of the solidities of a spherical sector and cone, when the segment is less than a hemisphere; and the sum, when the segment is greater than a hemisphere. DEM.-This will be evident from a mere inspection of the figure. 1. The spherical segment generated by P F G (fig. 111) F 111 is equal to the sector generated by H F G minus the cone generated by HF P; or more briefly, segment P F G= sector HF G-cone HF P. 2. The spherical segment generated by O E A is equal to the spherical sector generated by H E A plus the cone generated by HEO; or more briefly, segment O E A=sector HEA+cone H E O. 162. THEOREM.-The solidity of a spherical segment of two bases is found by taking the difference between the solidities of two spherical segments, which have for their respective single bases, the two bases of the segment to be measured. DEM.-This also will be evident from inspection of the figure. Thus the segment generated by O E F P (fig. 111) is equal to the F 111 difference between the segment generated by OEG and the segment generated by PF G. Now by the preceding proposition, segment O E G=sector HEG -cone HE O; and segment P F G=sector HF G— cone HF P. Therefore segment O E F P=sector HE G-sector H FG+cone HF P-cone H E O. COMPARISON OF SOLIDS. 163. SCHO.-It is easy to compare solids after having ascertained the measures of their solidity; since for this purpose it is only necessary to compare those measures. Moreover if, in comparing two solidities, there be a common factor it may be omitted. Nor is the comparison limited to solids of the same kind. prism may be compared with a sphere, or a cone with the frustum of a pyramid, for their ratio must be the A same as that of their solidities. The following propositions, therefore, may be received as corollaries of the preceding demonstrations. COR.-Two prisms, two pyramids, two cylinders, or two cones are to each other as the products of their bases by their altitudes. If the altitudes are the same, they are as their bases. If the bases are the same, they are as their altitudes. 164. THEOREM.-The surfaces of two spheres are to each other as the squares of their radii, and the solidities are as the cubes of their radii. DEM. 1.Let S be the surface of one sphere, Ca great circle of that and (66) Cc: R2: r 2 that is, the surfaces are as the squares of their radii. 2.--The solidities of the two spheres are to each other as their surfaces multiplied by one third of their radii (158); that is, as SXR is to s×+ r. But since Ss: R we have (65, 66,) SXR : r2, : † r3 :: R3 : r3 ; that is, the solidities of the two spheres are as the cubes of their radii. SIMILAR SOLIDS. 165. DEF.-Two polyedrons of the same number of faces are similar, when their homologous solid angles are equal, and their homologous faces are similar polygons Also two cones or two cylinders are similar when their altitudes are to each other as the radii of their bases. The following are corollaries of the first definition. COR.-The homologous sides or edges of similar polyedrons are proportional; and the homelogous faces are to each other as the squares of their homologous sides. 166. THEOREM.-Two similar pyramids are to each other as the cubes of their homologous sides. DEM. Since by the definition the homologous solid angles are equal, the less pyramid may be placed in the F 93 greater so that the solid angles at A (fig. 93) shall coin cide. Moreover since the base G HIKL is, by the definition, similar to B C D E F, and is at the same time a section of the greater pyramid, the two bases are parallel (144). Now calling A F and A L the altitudes, (163) A-B C D E F : A-G H I K L :: B C D E FX AFGHIKLXAL. But BCDEFGHIKL: FE: LK2 and (131) Multiplying these two proportions term by term (67) we have LK 3 BCDEFXAFGHIK LXAL:: FE 3 Therefore, substituting this last ratio for its equal in the first proportion, we have A-B C D E F A-G HIKL:: FE : LK 9 which was to be demonstrated. 167. THEOREM.-Any two similar polyedrons are to each other as the cubes of their homologous sides. DEM.--No diagram is necessary for this demonstration. Let A be a solid angle of one polyedron, and a the homologous solid angle of the other. Since these solid angles are equal, we may suppose the less polyedron placed in the greater, so that the solid angles A and a shall coincide. Then if the greater polyedron be divided into pyramids, having their vertex in A, the planes which make these divisions, must evidently make corresponding divisions in the smaller. Thus the two similar polyedrons will be divided into the same number of similar pyramids, which by the preceding proposi tion, will be to each other as the cubes of their homologous sides (166). Hence a continued proportion might be formed, having the greater pyramids for its antecedents, the smaller pyramids for its consequents, and for its last ratio the cubes of two homologous sides of the two polyedrons. Then by adding the antecedents and consequents, excepting the fast, we should have the greater polyedron to the less as the cubes of their homologous sides. 168 THEOREM.-Two similar cones or cylinders are to each other as the cubes of the radii of their bases. DEM.-No diagram is necessary for this demonstration. Let C be one cone or cylinder, A its altitude, and R the radius of its base; and let c, a, r be corresponding expressions for the other cone or cylinder. Then (163) Cc: XR1×A : «×ra×a :: R1×A : r2xa. But by the definition (165) AaR: r Multiplying this, term by term, by the identical proportion R: r2:: R2 : r2, we have R1×A : r2xa :: R3 : r3. Substituting this last ratio for its equal in the first proportion, we have Cc: R3 : r3, which was to be demonstrated. APPENDIX. CONTAINING AN ACCOUNT OF THE PRACTICAL APPLICATION OF 169. We begin with the proposition of art. 17.Angles are measured by arcs of circles described from their vertices as centres. Upon this proposition depend the construction and use of all the instruments, which have been invented for the measurement of angles in space, as well as for tracing them upon paper. The protractor and its use, we have already mentioned, (18). The quadrant is an instrument used for measuring angles in a vertical plane. Its essential parts are represented in fig. 112. ED is a graduated arc of F 112 90° beginning at E. A C is a plumb-line attached to the vertex A. Near A and D are two sight-holes for determining accurately the direction of objects. The direction of a plumb-line A C, suspended freely, is called vertical; and the line F G, to which the vertical is perpendicular, is called horizontal. The angle HA G contained between the horizontal line and a line drawn to an object above it, is called the angle of eleva tion of the object: and the angle F A B, contained between the horizontal line and a line drawn to an object below it, is called the angle of depression of the object. Both these angles are readily measured by the quadrant. To find the angle of elevation H A G of an object H, the quadrant, kept always in a vertical plane by means of the plumb-line, is so placed that the object can be seen through the two sight-holes by the |