Commentary on Proposition XII.12¶
Back to Proposition XII.12.
The method of proof is precisely that of the previous proposition. The only addition is caused by the necessity of proving that, if similar equilateral polygons be inscribed in the bases of two similar cones, and pyramids be erected on them with the same vertices as those of the cones, the pyramids (are similar and) are to one another in the triplicate ratio of corresponding edges.
Let KL, MN be the axes of the cones, L, N the vertices, and let BT, FP be sides of similar polygons inscribed in the bases. Join BK, TK, BL, TL, PM, FM, PN, FN.
Now BKL, FMN are right-angled triangles, and, since the cones are similar, BK : KL = FM : MN. [XI. Def. 24]
Therefore (1) $\Delta$s BKL, FMN are similar. [VI. 6] Similarly (2) $\Delta$s TKL, PMN are similar.
Next, in $\Delta$s BKT, FMP, the angles BKT, FMP are equal, since each is the same fraction of four right angles; and the sides about the equal angles are proportional; therefore (3) $\Delta$s BKT, FMP are similar.
Again, since from the similar $\triangle\text{s}$ BKL, FMN, and the similar $\triangle\text{s}$ BKT, FMP respectively, LB : BK = NF : FM, BK : BT = MF : FP, ex aequali, LB : BT = NF : FP. Similarly LT : TB = NP : PF. Inverting the latter ratio and compounding it with the preceding one, we have, ex aequali, LB : LT = NF : NP.
Thus in $\triangle\text{s}$ LTB, NPF the sides are proportional in pairs ; therefore (4) $\triangle\text{s}$ LTB, NPF are similar.
Thus the partial pyramids L-BKT, N-FMP are similar. In exactly the same way it is proved that all the other partial pyramids are similar.
Now (pyramid L-BKT) : (pyramid N-FMP) = ratio triplicate of (BK : FM). The other partial pyramids are to one another in the same triplicate ratio. The sum of the antecedents is therefore to the sum of the consequents in the same triplicate ratio, i.e. (pyramid L-ATBU…) : (pyramid N-EPFQ…) = ratio triplicate of ratio (BK : FM) = ratio triplicate of ratio (BD : FH).
[The fact that Euclid makes this transition from the partial pyramids to the whole pyramids in the body of this proposition seems to me to suggest grave doubts as to the genuineness of the Porism to XII. 8, which contains a similar but rather more general extension from the case of triangular pyramids to pyramids with polygonal bases. Were that Porism genuine, Euclid would have been more likely to refer to it than to repeat here the same arguments which it contains.]
Now we are in a position to apply the method of exhaustion. If $X$, $X'$ be the volumes of the cones, $d$, $d'$ the diameters of their bases, and if
$$(\text{ratio triplicate of } d : d') \ne X : X',$$
then must
$$(\text{ratio triplicate of } d : d') = X : O,$$
where $O$ is either less or greater than $X'$.
Suppose that $O$ is less than $X'$. Construct in the way described a pyramid ($\Pi'$) in $X'$ leaving over portions of $X'$ together less than $(X' - O)$, so that $X' > \Pi' > O$, and construct in $X$ a pyramid ($\Pi$), with the same vertex as $X$ has, on a polygon inscribed in its base similar to the base of $\Pi'$.
Then, by what has just been proved,
$$\begin{aligned} \Pi : \Pi' &= (\text{ratio triplicate of } d : d') \\ &= X : O\text{, by hypothesis,} \end{aligned}$$
and, alternately,
$$\Pi : X = \Pi' : O.$$
But $X$ includes, and is therefore greater than, $\Pi$; therefore
$$O > \Pi'.$$
But, by construction,
$$O < \Pi' :$$
which is impossible.
Therefore
$$O\text{ cannot be less than } X'.$$
Suppose, if possible, that
$$(\text{ratio triplicate of } d : d') = X : O,$$
where $O$ is greater than $X'$; then
$$(\text{ratio triplicate of } d : d') = Z : X',$$
or, inversely,
$$(\text{ratio triplicate of } d' : d) = X' : Z,$$
where $Z$ is some solid less than $X$.
This is proved impossible by the exact method of Part I. Hence $O$ cannot be either greater or less than $X'$, and
$$X : X' = (\text{ratio triplicate of ratio } d : d').$$