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PROPOSITION XVI. THEOREM.
The surface of a sphere is to the whole surface of the circumscribed cylinder (including its bases), as 2 is to 3. lidities of these two bodies are to each other in the same ratio.
Let MNPQ be a great circle of the sphere; ABCD the circumscribed D square: if the semicircle PMQ and the half square PADQ are at the same time made to revolve about the diameter PQ, the semicircle will generate the sphere, while the half-square will generate the cylinder circumscribed about that sphere.
The altitude AD of that cylinder is equal to the diameter PQ; the base of
the cylinder is equal to the great circle, its diameter AB being equal to MN; hence (4. VIII.), the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter. This measure (10. VIII.) is the same as that of the surface of the sphere: hence the surface of the sphere is equal to the convex surface of the circumscribed cyTinder.
But the surface of the sphere is equal to four great circles; hence the convex surface of the cylinder is also equal to four great circles and adding the two bases, each equal to a great circle, the total surface of the circumscribed cylinder will be equal to six great circles; hence the surface of the sphere is to the total surface of the circumscribed cylinder as 4 is to 6, or as 2 is to 3; which was the first branch of our Proposition.
In the next place, since the base of the circumscribed cylinder is equal to a great circle, and its altitude to the diameter, the solidity of the cylinder (1. VIII.) will be equal to a great circle multiplied by its diameter. But (15. VIII), the solidity of the sphere is equal to four great circles multiplied by a third of the radius, in other terms, to one great circle multiplied by of the radius, or by of the diameter; hence the sphere is to the circumscribed cylinder as 2 to 3, and consequently the solidities of these two bodies are as their surfaces.
Scholium. Conceive a polyedron, all of whose faces touch the sphere; this polyedron may be considered as formed of pyramids, each having for its vertex the centre of the sphere, and for its base one of the polyedron's faces. Now it is evident that all these pyramids will have the radius of the sphere for their common altitude; so that each pyramid will be equal to one face of the polyedron multiplied by a third of the radius:
hence the whole polyedron will be equal to its surface multiplied by a third of the inscribed sphere's radius.
It is therefore manifest, that, the solidities of polyedrons circumscribed about the sphere are to each other as the surfaces of those polyedrons. Thus the property, which we have shewn to be true with regard to the circumscribed cylinder, is also true with regard to an infinite number of other bodies.
We might likewise have observed, that the surfaces of polygons, circumscribed about the circle, are to each other as their perimeters.
PROPOSITION XVII. PROBLEM.
The circular segment BMD being supposed to make a revolution about a diameter exterior to it, to find the value of the solid so produced.
On the axis, let fall the perpendiculars BE, DF; from the centre C, draw CI perpendicular to the chord BD; also draw the radii CB, CD.
The solid described by the sector BCA is equal to . CB.2AE (15. VIII.); the solid described by the sector DCA=
. CB2. AF; hence the difference of these two solids, or the solid described by the
sector DCB=3. CB2. (AF—AE)=. CB2. EF. But the solid described by the isosceles triangle DCB (14. VIII.) has for its measure . CI2. EF; hence the solid described by the segment BMD π. EF. (CB2—CI2). Now, in the right-angled triangle CBI, we have CB2-CI-BI2=1BD2; hence the solid described by the segment BMD will have for its measure T. EF. BD2, or 17. BD2. EF.
Scholium. The solid described by the segment BMD is to the sphere, which has BD for its diameter, as . BD. EF is to . BD3, or as EF to BD.
PROPOSITION XVIII. THEOREM.
Every segment of a sphere, included between two parallel planes,
is measured by the half-sum of its bases multiplied by its altitude, plus the solidity of a sphere whose diameter is that same altitude.
Let BE, DF (see the preceding figure), be the radii of the segment's two bases, EF its altitude, our segment thus being the one produced by the revolution of the circular space BMĎFE about the axis FE. The solid decribed by the segment BMD is equal to 17.BD2.EF (17. VIII.); and (6. VIII.) the truncated
cone described by the trapezium BDFE is equal to %.EF.(BE2+ DF2+ BE. DF); hence the segment of the sphere, which is the sum of those two solids, must be equal to . EF. (2BE2+2 DF2+2BE.BF+BD2). But, drawing DO parallel to EF, we shall have DO=DF-BE, hence (9. III.) DO2= DF2-2 DF. BE+ BE2; and consequently BD2-BO2+ DO2-EF2+ DF2-2 DF. BE + BE2. Put this value in place of BD2 in the expression for the value of the segment, omitting the parts which destroy each other; we shall obtain for the solidity of the segment,
* EF. (3 BE2+3 DF2+EF2), an expression which may be decomposed into two parts; the T. BE2+. DF2 one, ¿. EF. (3 BE2+3 DF2), or EF.
the half-sum of the bases multiplied by the altitude; while the other. EF represents (15. VIII. Schol.) the sphere of which EF is the diameter: hence every segment of a sphere, &c.
Cor. If either of the bases is nothing, the segment in question becomes a spherical segment with a single base; hence any spherical segment, with a single base, is equivalent to half the linder having the same base and the same altitude, plus the sphere of which this altitude is the diameter.
Let R be the radius of a cylinder's base, H its altitude: the solidity of the cylinder will be R2 × H, or πR2H.
Let R be the radius of a cone's base, H its altitude: the so
lidity of the cone will be R2x H, or
Let A and B be the radii of the bases of a truncated cone, H its altitude: the solidity of the truncated cone will be .H. (A2+B2+AB).
Let R be the radius of a sphere; its solidity will be R3. Let R be the radius of a spherical sector, H the altitude of the zone, which forms its base: the solidity of the sector will be R2H.
Let P and Q be the two bases of a spherical segment, Hits
altitude: the solidity of the segment will be
If the spherical segment has but one base, the other being nothing, its solidity will be PH+zH3.
END OF THE ELEMENTS OF GEOMETRY.
ELEMENTS OF GEOMETRY.
On some Names and Definitions.
SOME new expressions and definitions have been employed in this Work, where they seemed likely to give more accuracy and precision to geometrical language. We mean here to give some account of those changes, and to propose a few others, which might accomplish the same purpose more completely.
In the ordinary definition of the rectangular parallelogram and of the square, it is usual to say, that the angles of those figures are right; it would be more correct to say, that their angles are equal. For, to suppose that the four angles of a quadrilateral can be right, and even that the right angles are equal to each other, is to assume two propositions which require demonstration. This inconvenience, and several others of the same sort, might be avoided, if, instead of placing the definitions, according to the common practice, at the head of each Book, we were to disperse them over the course of the Book, each at the place where all it assumes is already proved.
The word parallelogram, according to its etymology, signifies parallel lines; it no more suits the figure of four sides, than it does that of six, of eight, &c. which have their opposite sides parallel. In like manner, the word parallelepipedon signifies parallel planes; it no more designates the solid with six faces, than the solid with eight, ten, &c. of which the opposite faces are parallel. The names, parallelogram and parallelepipedon, have the additional inconvenience of being very long. Perhaps, therefore, it would be advantageous to banish them altogether from geometry; and to substitute in their stead, the names rhombus and rhomboid, retaining the term lozenge, for quadrilaterals whose sides are all equal.
It might also be useful to extend the meaning of the word inclination, so as to make it synonymous with angle: both of them indicate a particular relation of two lines, or of two planes, which meet together, or would meet if produced. The inclination of two lines is nothing, when their angle is nothing; in other words, when the lines coincide, or lie parallel to each other. The inclination is greater when the angle is greater, or when the two lines form together a very obtuse angle. The quality of sloping has a different meaning; a line slopes the more towards another, the more it deviates from the perpendicular to that other.