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1% book12.tex --- Book XII of Euclid's Elements: Method of Exhaustion.
2%
3% All 18 propositions encoded. Book XII develops the Eudoxean method of
4% exhaustion (X.1) to compute curvilinear and 3D volumes: circles
5% (XII.2), pyramids (XII.5-XII.9), cones (XII.10-XII.15), and spheres
6% (XII.16-XII.18).
7%
8% Wording follows Heath (1908).
9
10\section{Book XII --- Method of Exhaustion}
11\label{sec:book-XII}
12
13\begin{claim}[Proposition XII.1: Similar inscribed polygons in circles are as squares on diameters]
14\label{prop:XII.1}
15Similar polygons inscribed in circles are to one another as the
16squares on the diameters.
17\end{claim}
18\begin{evidence}[Proof of XII.1]
19\label{ev:XII.1}
20Decompose each polygon into similar triangles by joining vertices to
21the centres; each pair of corresponding triangles is similar (VI.20)
22with side ratio equal to the diameter ratio; sum and combine via
23V.12.
24\dependson{XII.1}{VI.20}
25\dependson{XII.1}{V.12}
26\end{evidence}
27
28\begin{claim}[Proposition XII.2: Circles are as squares on diameters]
29\label{prop:XII.2}
30Circles are to one another as the squares on the diameters.
31\end{claim}
32\begin{evidence}[Proof of XII.2]
33\label{ev:XII.2}
34Apply the method of exhaustion: inscribe similar polygons in the two
35circles; by XII.1 they are in the ratio of squares on the diameters.
36Any deviation from that ratio at the level of the circles leads, via
37X.1, to a contradiction by choosing inscribed polygons close enough
38to fill the circle.
39\dependson{XII.2}{X.1}
40\dependson{XII.2}{XII.1}
41\end{evidence}
42
43\begin{claim}[Proposition XII.3: A pyramid on a triangular base is split into two equal smaller pyramids and two equal prisms]
44\label{prop:XII.3}
45Any pyramid which has a triangular base is divided into two pyramids
46equal and similar to one another, similar to the whole, and having
47triangular bases, and into two equal prisms; and the two prisms are
48greater than the half of the whole pyramid.
49\end{claim}
50\begin{evidence}[Proof of XII.3]
51\label{ev:XII.3}
52Bisect each edge (I.10); the midpoint cuts split the pyramid into
53two corner pyramids and two prisms. The two corner pyramids are
54similar to the original (their edges halved), and by I.34 the two
55prisms are congruent.
56\dependson{XII.3}{I.10}
57\dependson{XII.3}{I.34}
58\dependson{XII.3}{XI.39}
59\end{evidence}
60
61\begin{claim}[Proposition XII.4: Pyramids of equal height are as their bases (special case)]
62\label{prop:XII.4}
63If there be two pyramids of the same height which have triangular
64bases, and each of them be divided into two pyramids equal to one
65another and similar to the whole, and into two equal prisms, then,
66as the base of the one pyramid is to the base of the other pyramid,
67so will all the prisms in the one pyramid be to all the prisms in
68the other pyramid.
69\end{claim}
70\begin{evidence}[Proof of XII.4]
71\label{ev:XII.4}
72Each iteration of XII.3 doubles the number of prisms; by
73proportionality of bases (VI.1) the prism-sums maintain the same
74ratio as the original bases.
75\dependson{XII.4}{VI.1}
76\dependson{XII.4}{XII.3}
77\end{evidence}
78
79\begin{claim}[Proposition XII.5: Pyramids of equal height are as their bases]
80\label{prop:XII.5}
81Pyramids which are of the same height and have triangular bases are
82to one another as their bases.
83\end{claim}
84\begin{evidence}[Proof of XII.5]
85\label{ev:XII.5}
86Apply XII.4 in the limit of XII.3 iterations; the prism-sums
87exhaust the pyramids (X.1), so the base-ratio is the pyramid-ratio.
88\dependson{XII.5}{X.1}
89\dependson{XII.5}{XII.3}
90\dependson{XII.5}{XII.4}
91\end{evidence}
92
93\begin{claim}[Proposition XII.6: Pyramids of equal height on polygonal bases are as their bases]
94\label{prop:XII.6}
95Pyramids which are of the same height and have polygonal bases are
96to one another as the bases.
97\end{claim}
98\begin{evidence}[Proof of XII.6]
99\label{ev:XII.6}
100Triangulate each polygonal base; the polygon-pyramid is the sum of
101the triangular sub-pyramids; apply XII.5 and V.12.
102\dependson{XII.6}{V.12}
103\dependson{XII.6}{XII.5}
104\end{evidence}
105
106\begin{claim}[Proposition XII.7: Triangular prism is three equal pyramids]
107\label{prop:XII.7}
108Any prism which has a triangular base is divided into three pyramids
109equal to one another which have triangular bases.
110\end{claim}
111\begin{evidence}[Proof of XII.7]
112\label{ev:XII.7}
113Cut the prism by two planes through opposite edge-pairs; the three
114resulting pyramids share a common apex and have congruent base
115triangles, so by XII.5 they have equal volume.
116\dependson{XII.7}{XI.39}
117\dependson{XII.7}{XII.5}
118\end{evidence}
119
120\begin{claim}[Proposition XII.8: Similar pyramids are as the cubes on corresponding edges]
121\label{prop:XII.8}
122Similar pyramids which have triangular bases are in the triplicate
123ratio of their corresponding sides.
124\end{claim}
125\begin{evidence}[Proof of XII.8]
126\label{ev:XII.8}
127By XII.7 a prism is three equal pyramids; by XI.33 similar
128parallelepipeds (and hence prisms) are in the triplicate ratio of
129edges; transfer to pyramids by XII.5.
130\dependson{XII.8}{XI.33}
131\dependson{XII.8}{XII.5}
132\dependson{XII.8}{XII.7}
133\end{evidence}
134
135\begin{claim}[Proposition XII.9: Equal pyramids have reciprocally proportional bases and heights]
136\label{prop:XII.9}
137In equal pyramids which have triangular bases the bases are
138reciprocally proportional to the heights; and those pyramids which
139have triangular bases in which the bases are reciprocally
140proportional to the heights are equal.
141\end{claim}
142\begin{evidence}[Proof of XII.9]
143\label{ev:XII.9}
1443D analogue of VI.15 for pyramids; via XII.5 / XII.6 the area-times-
145height proportion factors into the reciprocal proportion of bases
146and heights.
147\dependson{XII.9}{VI.14}
148\dependson{XII.9}{XII.5}
149\dependson{XII.9}{XII.6}
150\end{evidence}
151
152\begin{claim}[Proposition XII.10: A cone is one-third of the cylinder on the same base and height]
153\label{prop:XII.10}
154Any cone is a third part of the cylinder which has the same base
155with it and equal height.
156\end{claim}
157\begin{evidence}[Proof of XII.10]
158\label{ev:XII.10}
159Inscribe a pyramid on a polygonal base in both cone and cylinder;
160XII.7 makes the pyramid one-third the prism; apply X.1 to refine the
161inscribed polygon to fill the circle (XII.2); the limit gives the
162cone-to-cylinder ratio.
163\dependson{XII.10}{X.1}
164\dependson{XII.10}{XII.2}
165\dependson{XII.10}{XII.7}
166\end{evidence}
167
168\begin{claim}[Proposition XII.11: Cones and cylinders of equal height are as their bases]
169\label{prop:XII.11}
170Cones and cylinders which are of the same height are to one another
171as their bases.
172\end{claim}
173\begin{evidence}[Proof of XII.11]
174\label{ev:XII.11}
175By XII.2 the bases (circles) are in the squared-diameter ratio; by
176the formula in XII.10, the volumes follow the same ratio.
177\dependson{XII.11}{XII.2}
178\dependson{XII.11}{XII.10}
179\end{evidence}
180
181\begin{claim}[Proposition XII.12: Similar cones and cylinders are as cubes on diameters]
182\label{prop:XII.12}
183Similar cones and cylinders are to one another in the triplicate
184ratio of the diameters in their bases.
185\end{claim}
186\begin{evidence}[Proof of XII.12]
187\label{ev:XII.12}
188Analogue of XII.8 for cones / cylinders: by similarity the height
189scales proportionally with the diameter; cube of the linear ratio
190gives the volume ratio.
191\dependson{XII.12}{XII.8}
192\dependson{XII.12}{XII.10}
193\dependson{XII.12}{XII.11}
194\end{evidence}
195
196\begin{claim}[Proposition XII.13: Parallel sections of a cylinder are in ratio of distances]
197\label{prop:XII.13}
198If a cylinder be cut by a plane which is parallel to its opposite
199planes, then, as the cylinder is to the cylinder, so will the axis
200be to the axis.
201\end{claim}
202\begin{evidence}[Proof of XII.13]
203\label{ev:XII.13}
204Parallel cross-sections give equal circles (XI.16 implies parallel
205diameters); the volume scales linearly with axial length by XII.11.
206\dependson{XII.13}{XI.16}
207\dependson{XII.13}{XII.11}
208\end{evidence}
209
210\begin{claim}[Proposition XII.14: Cylinders on equal bases are as their heights]
211\label{prop:XII.14}
212Cones and cylinders which are on equal bases are to one another as
213their heights.
214\end{claim}
215\begin{evidence}[Proof of XII.14]
216\label{ev:XII.14}
217By XII.13 the volume is proportional to the axis when the base is
218fixed.
219\dependson{XII.14}{XII.11}
220\dependson{XII.14}{XII.13}
221\end{evidence}
222
223\begin{claim}[Proposition XII.15: Equal cones / cylinders have reciprocal proportions]
224\label{prop:XII.15}
225In equal cones and cylinders the bases are reciprocally proportional
226to the heights; and those cones and cylinders in which the bases
227are reciprocally proportional to the heights are equal.
228\end{claim}
229\begin{evidence}[Proof of XII.15]
230\label{ev:XII.15}
231Analogue of XII.9 / VI.15 for cones and cylinders.
232\dependson{XII.15}{XII.9}
233\dependson{XII.15}{XII.11}
234\dependson{XII.15}{XII.14}
235\end{evidence}
236
237\begin{claim}[Proposition XII.16: Inscribe in the larger of two concentric circles a polygon not touching the smaller]
238\label{prop:XII.16}
239Given two circles about the same centre, to inscribe in the greater
240circle an equilateral polygon with an even number of sides which
241does not touch the lesser circle.
242\end{claim}
243\begin{evidence}[Proof of XII.16]
244\label{ev:XII.16}
245Bisect arcs repeatedly (III.30) until the chord-to-arc gap is
246smaller than the difference of radii; this guarantees that the
247inscribed polygon avoids touching the smaller circle.
248\dependson{XII.16}{III.30}
249\dependson{XII.16}{X.1}
250\end{evidence}
251
252\begin{claim}[Proposition XII.17: Inscribe in the larger of two concentric spheres a polyhedron not touching the smaller]
253\label{prop:XII.17}
254Given two spheres about the same centre, to inscribe in the greater
255sphere a polyhedral solid which does not touch the lesser sphere at
256its surface.
257\end{claim}
258\begin{evidence}[Proof of XII.17]
259\label{ev:XII.17}
2603D analogue of XII.16: apply XII.16 in great-circle cross-sections,
261then triangulate the sphere using XI.27 to assemble a polyhedron
262strictly inside the larger sphere and outside the smaller.
263\dependson{XII.17}{XI.27}
264\dependson{XII.17}{XII.16}
265\end{evidence}
266
267\begin{claim}[Proposition XII.18: Spheres are in triplicate ratio of diameters]
268\label{prop:XII.18}
269Spheres are to one another in the triplicate ratio of their
270respective diameters.
271\end{claim}
272\begin{evidence}[Proof of XII.18]
273\label{ev:XII.18}
274Apply the method of exhaustion: inscribe similar polyhedra (XII.17);
275by XII.12 (similar cones) and the polyhedron's similar-pyramid
276decomposition, the inscribed solids are in the triplicate ratio of
277diameters. By X.1 the inscribed solids approach the spheres in
278volume; the limit gives the result.
279\dependson{XII.18}{X.1}
280\dependson{XII.18}{XII.8}
281\dependson{XII.18}{XII.12}
282\dependson{XII.18}{XII.17}
283\end{evidence}
284