Chapter XXVIII: Book XII

S 85. In the last part of Book XI. we have learnt how to compare the
volumes of parallelepipeds and of prisms. In order to determine the
volume of any solid bounded by plane faces we must determine the
volume of pyramids, for every such solid may be decomposed into a
number of pyramids.

As every pyramid may again be decomposed into triangular pyramids, it
becomes only necessary to determine their volume. This is done by the

_Theorem._--Every triangular pyramid is equal in volume to one third
of a triangular prism having the same base and the same altitude as
the pyramid.

This is an immediate consequence of Euclid's

Prop. 7. _Every prism having a triangular base may be divided into
three pyramids that have triangular bases, and are equal to one
another._

The proof of this theorem is difficult, because the three triangular
pyramids into which the prism is divided are by no means equal in
shape, and cannot be made to coincide. It has first to be proved that
two triangular pyramids have equal volumes, if they have equal bases
and equal altitudes. This Euclid does in the following manner. He
first shows (Prop. 3) that a triangular pyramid may be divided into
four parts, of which two are equal triangular pyramids similar to the
whole pyramid, whilst the other two are equal triangular prisms, and
further, that these two prisms together are greater than the two
pyramids, hence more than half the given pyramid. He next shows (Prop.
4) that if two triangular pyramids are given, having equal bases and
equal altitudes, and if each be divided as above, then the two
triangular prisms in the one are equal to those in the other, and each
of the remaining pyramids in the one has its base and altitude equal
to the base and altitude of the remaining pyramids in the other. Hence
to these pyramids the same process is again applicable. We are thus
enabled to cut out of the two given pyramids equal parts, each greater
than half the original pyramid. Of the remainder we can again cut out
equal parts greater than half these remainders, and so on as far as we
like. This process may be continued till the last remainder is smaller
than any assignable quantity, however small. It follows, so we should
conclude at present, that the two volumes must be equal, for they
cannot differ by any assignable quantity.

To Greek mathematicians this conclusion offers far greater
difficulties. They prove elaborately, by a _reductio ad absurdum_,
that the volumes cannot be unequal. This proof must be read in the
_Elements._ We must, however, state that we have in the above not
proved Euclid's Prop. 5, but only a special case of it. Euclid does
not suppose that the bases of the two pyramids to be compared are
equal, and hence he proves that the volumes are as the bases. The
reasoning of the proof becomes clearer in the special case, from which
the general one may be easily deduced.

S 86. Prop. 6 extends the result to pyramids with polygonal bases.
From these results follow again the rules at present given for the
mensuration of solids, viz. a pyramid is the third part of a
triangular prism having the same base and the same altitude. But a
triangular prism is equal in volume to a parallelepiped which has the
same base and altitude. Hence if B is the base and h the altitude, we
have

Volume of prism = Bh,
Volume of pyramid = 1/3Bh,

statements which have to be taken in the sense that B means the number
of square units in the base, h the number of units of length in the
altitude, or that B and h denote the numerical values of base and
altitude.

S 87. A method similar to that used in proving Prop. 5 leads to the
following results relating to solids bounded by simple curved
surfaces:--

Prop. 10. _Every cone is the third part of a cylinder which has the
same base, and is of an equal altitude with it._

Prop. 11. _Cones or cylinders of the same altitude are to one another
as their bases._

Prop. 12. _Similar cones or cylinders have to one another the
triplicate ratio of that which the diameters of their bases have._

Prop. 13. _If a cylinder be cut by a plane parallel to its opposite
planes or bases, it divides the cylinder into two cylinders, one of
which is to the other as the axis of the first to the axis of the
other;_ which may also be stated thus:--

_Cylinders on the same base are proportional to their altitudes._

Prop. 14. _Cones or cylinders upon equal bases are to one another as
their altitudes._

Prop. 15. _The bases and altitudes of equal cones or cylinders are
reciprocally proportional, and if the bases and altitudes be
reciprocally proportional, the cones or cylinders are equal to one
another._

These theorems again lead to formulae in mensuration, if we compare a
cylinder with a prism having its base and altitude equal to the base
and altitude of the cylinder. This may be done by the method of
exhaustion. We get, then, the result that their bases are equal, and
have, if B denotes the numerical value of the base, and h that of the
altitude,

Volume of cylinder = Bh,
Volume of cone = 1/3Bh.

S 88. The remaining propositions relate to circles and spheres. Of the
sphere only one property is proved, viz.:--

Prop. 18. _Spheres have to one another the triplicate ratio of that
which their diameters have._ The mensuration of the sphere, like that
of the circle, the cylinder and the cone, had not been settled in the
time of Euclid. It was done by Archimedes.