Source — Euclid's Elements, encoded as an rrxiv paper

1% book08.tex --- Book VIII of Euclid's Elements: Continued Proportions.
2%
3% All 27 propositions encoded. Book VIII studies geometric progressions
4% of integers, especially their reduction to a least sequence in the
5% given ratio.  It is the integer counterpart to Book V's continuous
6% theory and feeds directly into Book IX's number-theoretic applications.
7%
8% Wording follows Heath (1908).
9
10\section{Book VIII --- Continued Proportions}
11\label{sec:book-VIII}
12
13\begin{claim}[Proposition VIII.1: Least continued proportion has coprime extremes]
14\label{prop:VIII.1}
15If there be as many numbers as we please in continued proportion, and
16the extremes of them be prime to one another, the numbers are the
17least of those which have the same ratio with them.
18\end{claim}
19\begin{evidence}[Proof of VIII.1]
20\label{ev:VIII.1}
21Suppose a smaller set $b_1, \dots, b_n$ in the same ratio existed.
22By ex aequali (VII.14) the ratio of extremes $b_1 : b_n$ equals $a_1
23: a_n$, so by VII.21 (least in a ratio are coprime) the original
24$a_1$, $a_n$ would not be coprime --- contradiction.
25\dependson{VIII.1}{VII.14}
26\dependson{VIII.1}{VII.20}
27\dependson{VIII.1}{VII.21}
28\end{evidence}
29
30\begin{claim}[Proposition VIII.2: Construct a continued proportion with a given common ratio]
31\label{prop:VIII.2}
32To find numbers in continued proportion, as many as may be prescribed,
33and the least that are in a given ratio.
34\end{claim}
35\begin{evidence}[Proof of VIII.2]
36\label{ev:VIII.2}
37For $a : b$ with $\gcd(a, b) = 1$, the sequence $a^{n-1}, a^{n-2}
38b, \dots, b^{n-1}$ is in continued proportion with ratio $a : b$; by
39VII.27 the extremes are coprime, so by VIII.1 the sequence is least
40in its ratio.
41\dependson{VIII.2}{VII.27}
42\dependson{VIII.2}{VIII.1}
43\end{evidence}
44
45\begin{claim}[Proposition VIII.3: Least continued proportion has coprime extremes (converse setup)]
46\label{prop:VIII.3}
47If as many numbers as we please in continued proportion be the least
48of those which have the same ratio with them, the extremes of them
49are prime to one another.
50\end{claim}
51\begin{evidence}[Proof of VIII.3]
52\label{ev:VIII.3}
53Suppose the extremes shared a common divisor $d > 1$.  By VII.20
54each term would be divisible by some power of $d$, producing a
55smaller sequence in the same ratio --- contradicting minimality.
56\dependson{VIII.3}{VII.20}
57\dependson{VIII.3}{VII.21}
58\dependson{VIII.3}{VIII.1}
59\end{evidence}
60
61\begin{claim}[Proposition VIII.4: Common ratio across multiple chains]
62\label{prop:VIII.4}
63Given as many ratios as we please in least numbers, to find numbers
64in continued proportion which are the least in the given ratios.
65\end{claim}
66\begin{evidence}[Proof of VIII.4]
67\label{ev:VIII.4}
68Reduce each ratio to lowest terms by VII.33.  Compound them by
69multiplying numerators and denominators across; the resulting
70sequence is in continued proportion with the prescribed ratios.
71\dependson{VIII.4}{VII.33}
72\dependson{VIII.4}{VIII.2}
73\end{evidence}
74
75\begin{claim}[Proposition VIII.5: Plane numbers have a compound ratio of their sides]
76\label{prop:VIII.5}
77Plane numbers have to one another the ratio compounded of the ratios
78of their sides.
79\end{claim}
80\begin{evidence}[Proof of VIII.5]
81\label{ev:VIII.5}
82For plane numbers $ab$ and $cd$: $ab : cd = (a : c) \cdot (b : d)$
83in the language of compound ratios.  Verified by direct computation
84using VII.17 / VII.18.
85\dependson{VIII.5}{VII.17}
86\dependson{VIII.5}{VII.18}
87\dependson{VIII.5}{def:VII.16}
88\end{evidence}
89
90\begin{claim}[Proposition VIII.6: First measures last iff first measures second]
91\label{prop:VIII.6}
92If there be as many numbers as we please in continued proportion, and
93the first do not measure the second, neither will any other measure
94any other.
95\end{claim}
96\begin{evidence}[Proof of VIII.6]
97\label{ev:VIII.6}
98Contrapositive of VIII.7: divisibility propagates through the
99sequence, so failure at the first step prevents any later
100divisibility relation.
101\dependson{VIII.6}{VII.20}
102\dependson{VIII.6}{VIII.1}
103\end{evidence}
104
105\begin{claim}[Proposition VIII.7: First measures last implies first measures second]
106\label{prop:VIII.7}
107If there be as many numbers as we please in continued proportion, and
108the first measure the last, it will measure the second also.
109\end{claim}
110\begin{evidence}[Proof of VIII.7]
111\label{ev:VIII.7}
112If $a_1 \mid a_n$, reduce $a_1, \dots, a_n$ to lowest terms (VIII.3);
113since the lowest extremes are coprime but $a_1$ divides $a_n$, the
114ratio $a_1 : a_n$ must be 1:1 in lowest terms, forcing $a_1 \mid
115a_2$ via VII.20.
116\dependson{VIII.7}{VII.20}
117\dependson{VIII.7}{VIII.3}
118\end{evidence}
119
120\begin{claim}[Proposition VIII.8: Intermediate numbers in a proportion]
121\label{prop:VIII.8}
122If between two numbers there fall numbers in continued proportion
123with them, then, however many numbers fall between them in continued
124proportion, so many will also fall in continued proportion between
125the numbers which have the same ratio with the original numbers.
126\end{claim}
127\begin{evidence}[Proof of VIII.8]
128\label{ev:VIII.8}
129The number of geometric means between two numbers depends only on
130their ratio; scaling by a common factor changes the magnitudes but
131not the ratio, so the same number of means fall between the scaled
132pair.
133\dependson{VIII.8}{VII.13}
134\dependson{VIII.8}{VIII.2}
135\end{evidence}
136
137\begin{claim}[Proposition VIII.9: Coprimality between unit and a sequence]
138\label{prop:VIII.9}
139If two numbers be prime to one another, and numbers fall between them
140in continued proportion, then, however many numbers fall between them
141in continued proportion, so many will also fall in continued
142proportion between each of them and a unit.
143\end{claim}
144\begin{evidence}[Proof of VIII.9]
145\label{ev:VIII.9}
146Coprime extremes correspond to a least continued proportion (VIII.1);
147the unit extends the proportion at both ends, and VIII.2 gives the
148matching extension on each side.
149\dependson{VIII.9}{VIII.1}
150\dependson{VIII.9}{VIII.2}
151\end{evidence}
152
153\begin{claim}[Proposition VIII.10: Counting means between unit and number]
154\label{prop:VIII.10}
155If numbers fall between each of two numbers and a unit in continued
156proportion, however many numbers fall between each of them and a
157unit in continued proportion, so many also will fall between them in
158continued proportion.
159\end{claim}
160\begin{evidence}[Proof of VIII.10]
161\label{ev:VIII.10}
162Concatenate two unit-anchored continued proportions; VII.14 (ex
163aequali) confirms the joined sequence remains in continued
164proportion.
165\dependson{VIII.10}{VII.14}
166\dependson{VIII.10}{VIII.9}
167\end{evidence}
168
169\begin{claim}[Proposition VIII.11: Squares have one mean proportional]
170\label{prop:VIII.11}
171Between two square numbers there is one mean proportional number, and
172the square has to the square the ratio duplicate of that which the
173side has to the side.
174\end{claim}
175\begin{evidence}[Proof of VIII.11]
176\label{ev:VIII.11}
177For squares $a^2$, $b^2$: the mean proportional is $ab$ (since $a^2 :
178ab = ab : b^2 = a : b$), and $a^2 : b^2$ is the duplicate of $a : b$.
179\dependson{VIII.11}{VII.17}
180\dependson{VIII.11}{VII.18}
181\dependson{VIII.11}{def:VII.18}
182\end{evidence}
183
184\begin{claim}[Proposition VIII.12: Cubes have two mean proportionals]
185\label{prop:VIII.12}
186Between two cube numbers there are two mean proportional numbers,
187and the cube has to the cube the ratio triplicate of that which the
188side has to the side.
189\end{claim}
190\begin{evidence}[Proof of VIII.12]
191\label{ev:VIII.12}
192For cubes $a^3$, $b^3$: the two means are $a^2 b$ and $a b^2$, and
193$a^3 : b^3$ is the triplicate of $a : b$.
194\dependson{VIII.12}{VII.17}
195\dependson{VIII.12}{VII.18}
196\dependson{VIII.12}{def:VII.19}
197\dependson{VIII.12}{def:V.10}
198\end{evidence}
199
200\begin{claim}[Proposition VIII.13: Powers of a continued proportion are in continued proportion]
201\label{prop:VIII.13}
202If there be as many numbers as we please in continued proportion, and
203each by multiplying itself make some number, the products will be
204proportional; and if the original numbers by multiplying the products
205make certain numbers, the latter will also be proportional.
206\end{claim}
207\begin{evidence}[Proof of VIII.13]
208\label{ev:VIII.13}
209Squares (and cubes) of terms in continued proportion are themselves
210in continued proportion, by VII.27 and VIII.2.
211\dependson{VIII.13}{VII.27}
212\dependson{VIII.13}{VIII.2}
213\end{evidence}
214
215\begin{claim}[Proposition VIII.14: Square measures square iff side measures side]
216\label{prop:VIII.14}
217If a square measure a square, the side will also measure the side;
218and if the side measure the side, the square will also measure the
219square.
220\end{claim}
221\begin{evidence}[Proof of VIII.14]
222\label{ev:VIII.14}
223$a^2 \mid b^2 \iff a \mid b$, by Euclid's lemma (VII.30) applied
224prime-by-prime.
225\dependson{VIII.14}{VII.30}
226\dependson{VIII.14}{VIII.11}
227\end{evidence}
228
229\begin{claim}[Proposition VIII.15: Cube measures cube iff side measures side]
230\label{prop:VIII.15}
231If a cube number measure a cube number, the side will also measure
232the side; and if the side measure the side, the cube will also
233measure the cube.
234\end{claim}
235\begin{evidence}[Proof of VIII.15]
236\label{ev:VIII.15}
237Same prime-by-prime argument as VIII.14 for cubes.
238\dependson{VIII.15}{VIII.12}
239\dependson{VIII.15}{VIII.14}
240\end{evidence}
241
242\begin{claim}[Proposition VIII.16: Squares non-measuring]
243\label{prop:VIII.16}
244If a square measure not a square, neither will the side measure the
245side; and if the side measure not the side, neither will the square
246measure the square.
247\end{claim}
248\begin{evidence}[Proof of VIII.16]
249\label{ev:VIII.16}
250Contrapositive of VIII.14.
251\dependson{VIII.16}{VIII.14}
252\end{evidence}
253
254\begin{claim}[Proposition VIII.17: Cubes non-measuring]
255\label{prop:VIII.17}
256If a cube number measure not a cube number, neither will the side
257measure the side; and if the side measure not the side, neither will
258the cube measure the cube.
259\end{claim}
260\begin{evidence}[Proof of VIII.17]
261\label{ev:VIII.17}
262Contrapositive of VIII.15.
263\dependson{VIII.17}{VIII.15}
264\end{evidence}
265
266\begin{claim}[Proposition VIII.18: Mean proportional between similar plane numbers]
267\label{prop:VIII.18}
268Between two similar plane numbers there is one mean proportional
269number, and the plane number has to the plane number the ratio
270duplicate of that which the corresponding side has to the
271corresponding side.
272\end{claim}
273\begin{evidence}[Proof of VIII.18]
274\label{ev:VIII.18}
275For similar plane numbers $ab$ and $cd$ with $a : b = c : d$, the
276mean proportional is the geometric mean of $ab$ and $cd$, which by
277VII.19 / VIII.2 equals $ad$ (or $bc$, equal by VII.19).
278\dependson{VIII.18}{VII.19}
279\dependson{VIII.18}{VIII.5}
280\dependson{VIII.18}{def:VII.21}
281\end{evidence}
282
283\begin{claim}[Proposition VIII.19: Mean proportionals between similar solid numbers]
284\label{prop:VIII.19}
285Between two similar solid numbers there fall two mean proportional
286numbers, and the solid number has to the solid number the ratio
287triplicate of that which the corresponding side has to the
288corresponding side.
289\end{claim}
290\begin{evidence}[Proof of VIII.19]
291\label{ev:VIII.19}
292For similar solid numbers $abc$ and $def$: the two means are $abf$
293and $aef$ (or symmetric variants); together they give a continued
294proportion in triplicate ratio.
295\dependson{VIII.19}{VIII.12}
296\dependson{VIII.19}{VIII.18}
297\end{evidence}
298
299\begin{claim}[Proposition VIII.20: Mean proportional characterises similar plane numbers]
300\label{prop:VIII.20}
301If one mean proportional number fall between two numbers, the numbers
302will be similar plane numbers.
303\end{claim}
304\begin{evidence}[Proof of VIII.20]
305\label{ev:VIII.20}
306Converse of VIII.18: if $a : m = m : b$ then $a$ and $b$ admit
307factorisations as similar plane numbers via VII.19.
308\dependson{VIII.20}{VII.19}
309\dependson{VIII.20}{VIII.18}
310\end{evidence}
311
312\begin{claim}[Proposition VIII.21: Two mean proportionals characterise similar solid numbers]
313\label{prop:VIII.21}
314If two mean proportional numbers fall between two numbers, the
315numbers are similar solid numbers.
316\end{claim}
317\begin{evidence}[Proof of VIII.21]
318\label{ev:VIII.21}
319Converse of VIII.19.
320\dependson{VIII.21}{VIII.19}
321\dependson{VIII.21}{VIII.20}
322\end{evidence}
323
324\begin{claim}[Proposition VIII.22: Squares in continued proportion]
325\label{prop:VIII.22}
326If three numbers be in continued proportion, and the first be square,
327the third will also be square.
328\end{claim}
329\begin{evidence}[Proof of VIII.22]
330\label{ev:VIII.22}
331If $a^2 : m = m : c$, then $m^2 = a^2 c$ so $c = (m/a)^2$, hence $c$
332is square.  VII.19 ensures the division produces an integer.
333\dependson{VIII.22}{VII.19}
334\dependson{VIII.22}{VIII.11}
335\end{evidence}
336
337\begin{claim}[Proposition VIII.23: Cubes in continued proportion]
338\label{prop:VIII.23}
339If four numbers be in continued proportion, and the first be cube,
340the fourth will also be cube.
341\end{claim}
342\begin{evidence}[Proof of VIII.23]
343\label{ev:VIII.23}
344Same scheme as VIII.22 with two means; the fourth term is the cube
345of the ratio's denominator scaled appropriately.
346\dependson{VIII.23}{VIII.12}
347\dependson{VIII.23}{VIII.22}
348\end{evidence}
349
350\begin{claim}[Proposition VIII.24: Square ratio implies square ratio]
351\label{prop:VIII.24}
352If two numbers have to one another the ratio which a square number
353has to a square number, and the first be square, the second will
354also be square.
355\end{claim}
356\begin{evidence}[Proof of VIII.24]
357\label{ev:VIII.24}
358By VIII.11 the ratio of squares has a mean proportional; transferring
359that mean to $a^2 : b$ forces $b$ to be square by VIII.22.
360\dependson{VIII.24}{VIII.11}
361\dependson{VIII.24}{VIII.22}
362\end{evidence}
363
364\begin{claim}[Proposition VIII.25: Cube ratio implies cube ratio]
365\label{prop:VIII.25}
366If two numbers have to one another the ratio which a cube number has
367to a cube number, and the first be cube, the second will also be
368cube.
369\end{claim}
370\begin{evidence}[Proof of VIII.25]
371\label{ev:VIII.25}
372Same argument as VIII.24 for cubes via VIII.12 and VIII.23.
373\dependson{VIII.25}{VIII.12}
374\dependson{VIII.25}{VIII.23}
375\end{evidence}
376
377\begin{claim}[Proposition VIII.26: Similar plane numbers have square ratio]
378\label{prop:VIII.26}
379Similar plane numbers have to one another the ratio which a square
380number has to a square number.
381\end{claim}
382\begin{evidence}[Proof of VIII.26]
383\label{ev:VIII.26}
384By VIII.18 similar plane numbers admit a mean proportional, and the
385ratio (squares of corresponding sides) is a square-to-square ratio.
386\dependson{VIII.26}{VIII.18}
387\dependson{VIII.26}{def:VII.21}
388\end{evidence}
389
390\begin{claim}[Proposition VIII.27: Similar solid numbers have cube ratio]
391\label{prop:VIII.27}
392Similar solid numbers have to one another the ratio which a cube
393number has to a cube number.
394\end{claim}
395\begin{evidence}[Proof of VIII.27]
396\label{ev:VIII.27}
397By VIII.19 similar solid numbers admit two mean proportionals, and
398the ratio is in the triplicate (cube-to-cube) ratio of corresponding
399sides.
400\dependson{VIII.27}{VIII.19}
401\dependson{VIII.27}{def:VII.21}
402\end{evidence}
403