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(6) If two pyramids upon equal bases and between the same parallel planes be cut by a third plane parallel to the two former, the sections will be equal to one another.
Thus, if ABCS and A'B'C'S' be upon equal bases and between the same parallel planes (that is, having ABC, A'B'C' in one plane, and the vertices S, S' in a plane parallel to it), and abc, a'b'c' be the sections made in these pyramids by a plane parallel to ABC, A'B'C', they will have equal areas. (c) If the parallel sections abc, a'b'd' be made by different planes,
their areas will be in the duplicate ratio of their respective distances from the vertices S and S', reckoned on any convenient parallel lines. That is, for example,
area abc : area a'b'c':: Sa: S'a'. (d) If two pyramids upon equal bases and between the same parallel
planes be cut by two other planes also parallel to those planes ; and on the sections most remote from the vertices of the prisms as bases prisms be described limited by the other cutting plane; then these prisms will be equal.
Thus if (P) and (P') be the bases, ABC, A'B'C' and DEF, D'E'F' be the two other planes, the prisms upon the bases ABC, A'B'C' between the same two planes parallel to the bases (P) (P') will be equal.*
* In the figures the edges of the prisms are taken coincident with, and parallel to, the edges SA, S'A' of the pyramids, as described in the note on (a) preceding.
(e) If a pyramid between parallel planes be cut by another plane
parallel to and equidistant from them, and prisms be erected on the base of the pyramid, and on the section, each limited by the adjacent planes : these two prisms will be less than that erected on the base of the pyramid, and limited by the extreme planes.
Let SABC be the pyramid, ABC its base, DSE a plane parallel to ABC through the vertex S, and D'E'c a plane equidistant from these two; and let ABCDES be the prism on the base ABC limited by the extreme planes, and ABCD'E'c, abcdes the other two, limited each by the plane adjacent to its base: then these two together will be less than the first prism limited by the extreme planes.
Note. The prisms ABCDES and abcdes are similar, and the side of the former is double that of the latter. Their volumes are therefore as 23 : 18 or as 8:1; or the smaller one is one-eighth of the larger. Also, the prism ABCD'E'c is one-half of the larger one ABCDES.
Both together, then, are 1 1 5 2
of the larger prism. This ratio is constantly the same from 8 8 step to step, if we successively bisect as in the next paragraph. (f) If two pyramids between the same parallel planes, and upon equal
bases be cut by planes parallel to these, however numerous and close to each other; and if prisms be constituted on each of these in succession, each prism terminating in the adjacent cutting plane : then the whole of the prisms about one of the pyramids are equal to the whole of those about the other, both separately and collectively.
This is only a generalized statement of the preceding. It is clear that as we increase the number, and consequently the minuteness of the divisions, these series of prisms will collectively become less and less different in volume from the two pyramids; and it is easy to infer that we may, by the multiplicity of our divisions, render that difference as small as we please. The equality remains the same throughout; and there is no limit that can be set to the closeness of our approximation of the volume in this system of prisms to the volume of the two pyramids.
Here, however, the equality of the pyramids and their systems of dependent prisms is, after all, only approximate and not exact for any finite number of deviating planes. We are, therefore, here obliged to introduce a new principle :
“Every property which is true up to the limit is true at the limit.”
The proposition is true of the prisms up to their limiting pyramids ; and therefore, true of the pyramids, which are the limits of the systems of prisms.
This is precisely the mode in which all such inquiries are conducted, differing only in formal details, in all that relates to figures which do not admit of comparison by superposition, transpositions, or ratios. See
several parts of the Mensuration in the first volume of this course, pp. 271, 277.
If the successive divisions be those of continual bisection of the distances of the original planes, it will furnish an interesting exercise of calculation to ascertain the ratio between the prism on the original base and the sums of the successive series of those which result from subdivision. It will be found that they converge towards the ratio of 3 : 1, closer and closer at each successive step; but that they never in any stage actually attain to that ratio exactly. The bisection is only chosen for the ease of calculation; any other mode would have given the same tendent ratio 3:1, but the fractions would have been different.
PROPOSITION XIII. Every pyramid is one-third of the volume of a prism of the same base
and altitude, Let ABCDEF be a prism ; draw AE, DC, EC and let planes pass through these lines two and two; thus dividing the prism into three triangular pyramids, whose bases are AEB, AED and DFE having the common vertex C. These are to be shown equal to one another, and hence each of them one-third of the prism itself.
(a) Since the face ABED of the prism is a parallelogram, the triangle ABE is equal to the triangle ADE; and hence (Prop. XII.) the pyramid ABEC is equal to the pyramid ADEC.
(6) Since ABC is equal to DEF, and the planes are parallel, the pyramid ABCE on the base ACB and vertex E, is equal to the pyramid DEFC on the base DEF and vertex C; they being on equal bases and between the same parallel planes.
(c) Hence the three pyramids, each of the other two being equal to ABCE, are all equal; and therefore each equal to one-third of the prism of the same base ABC and between the parallels, viz., the altitude or distance of the planes ABC and DEF.
(d) What has been proved in reference to the figures in this and the preceding Proposition to figures which are sketched for simplicity as triangular pyramids and prisms, is equally true whatever rectilinear figures constitute their bases; for all prisms can either be transformed to triangular ones as in (Prop. vii.) or divided into a series of triangular ones, and reasoned upon collectively,
[In a manner somewhat analogous, but also involving the principle that the circle is the limit of its circuniscribing polygons made by successively doubling the number of its sides, the relative volumes of cones, cylinders, and spheres are investigated. Thus, are deduced such theorems as the following:
(a) Cylinders upon equal bases and between the same (or equidistant pairs of) parallel planes are equal in volume.
(6) Cones, the same.
(c) If a cone and cylinder hare equal bases and be between the same (or equidistant pairs of) parallel plones, the volume of the cone is one-third of the volume of the cylinder,
(d) Similar cones, similar cylinders, and spheres hare their surfaces as the squares of their homologous lines ; and their volumes as the cubes of those lines. In the spheres the diameters are usually taken, but not of necessity, as the homologous lines.
(e) The volumes of two cylinders, or of two cones are in the ratio compounded of the ratio of their bases and the ratio of their altitudes.
(f) The surface of the sphere is equal to four times the area of its great circle.
(9) Every sphere is in volume equal to two-thirds of its circumscribing cylinder. Hence arises the property of the one, sphere, and cylinder, attributed* to Archimedes, and said to have been modelled on his tomb.
Let AKBCLD be a right cylinder whose height AD is equal to the diameter AB of its base; let also EGFHM be its inscribed sphere, and AKBE a cone on the same base AKB having its vertex in E the centre of the end DLC of the cylinder; then the volume of the cone and sphere together is equal to the volume of the cylinder. For other properties see the Treatise of Archimedes himself in an edition of his works.]
THE CONE, CYLINDER, AND SPHERE.
PROPOSITION I. Every section of a sphere made by a plane is a circle. 1. Let the plane pass through the centre 0 of the sphere: then all lines drawn from the centre in this plane to the line of intersection being radii of the sphere, they are equal; and hence (Euc. I. Def. 15) the line of intersection is a circle, of which O is the centre.
2. Let the plane not pass through the centre, but cut the sphere in the figure ABC: then ABC will be a circle.
For from O draw OP perpendicular to the plane ABC, and join PA, PB, PC, etc., and likewise OA, OB, OC, etc.
Then in the right-angled triangles OPA, * We say “ attributed,” because, though Archimedes is the earliest writer in whose works these properties are found, it does not necessarily follow that he was the discoverer of every proposition in his Treatise on the Cone and Cylinder, any more than that Euclid was the discoverer of every proposition in his Elements.
OPB, the two sides OA, OB are equal, and OP common; and hence, also, AP, BP are equal. In the same manner PC, etc., are each equal to AP. Whence ABC is a circle, of which P is the centre (Euc. I. Def. 15).
CoR. 1. If a right cone have its vertex at the centre of a sphere, the sections of the cone and sphere will be circles.
Cor. 2. If a cone have its base coincident with a plane section of the sphere, and its vertex at the centre, it will be a right cone.
PROPOSITION II. If a line or plane be drawn through a point in the sphere, perpendi
cular to the radius of that point, it will touch the sphere.
1. Let OA be a radius of the sphere, and BC any line perpendicular to OA drawn through A: then AB will touch the sphere.
For let the plane AOBC be drawn, and any other line OP in this plane to meet AB in P, and the circular section made by that plane in G.
Then (as in Euc, I. 17.) P is without the circle AGH, and hence AB is a tangent to it.
But AB is in the plane AGH, and hence a if it meet the sphere in any other point it must be in this plane; which has been shown to be impossible.
2. Let the plane CEDB be perpendiculor to OA and pass through A: it will touch the sphere.
For since the line OA is perpendicular to the plane CEDB, it is perpendicular to every straight line in it through A ; and it has been shown that these straight lines touch the sphere. The plane, therefore, which contains them touches the sphere, since there is no point in that plane through which one of these straight lines does not pass.
PROPOSITION III. If two spheres intersect each other, their intersection will be a circle
the centre of which is in the line joining the centres of the spheres, and whose plane is perpendicular to that line.
Let A, B be the centres of two spheres which cut one another ; then the section CDH is a circle. For, draw any two planes through AB to cut the spheres in the circles KCNG, MCLG, and KDNH, MHLD. Join CG meeting AB in E; and join AC, CB, AG, GB, AD, DB, AH, HB, DE, EH.
Then, since CA, CB, are equal to AD, DB, and AB common to the two triangles ACB, ADB, the angles ACB, ADB, are equal ; as are likewise the angles CAB, DAB, and the angles CBA, DBA.
Again, since the sides CA, AE are equal to the sides DA, AE, and the included angles equal, the angle DEA is equal to the angle CEA, and the side DE to the side CE.