rrxiv

paper/books/book10.textex · 50664 bytesRaw

1% book10.tex --- Book X of Euclid's Elements: Incommensurable Magnitudes.
2%
3% All 115 propositions encoded.  Book X is the longest of the Elements
4% and develops a classification of irrational magnitudes via repeated
5% quadratic operations: it covers commensurability (X.1-X.18), medials
6% (X.19-X.35), and the thirteen irrational lines including binomials and
7% apotomes (X.36-X.115).  Proof sketches are condensed; the dependency
8% structure is what we capture.
9%
10% Wording follows Heath (1908).
11
12\section{Book X --- Incommensurable Magnitudes}
13\label{sec:book-X}
14
15\begin{claim}[Proposition X.1: Method of exhaustion lemma]
16\label{prop:X.1}
17Two unequal magnitudes being set out, if from the greater there be
18subtracted a magnitude greater than its half, and from that which is
19left a magnitude greater than its half, and if this process be
20repeated continually, there will be left some magnitude which will be
21less than the lesser magnitude set out.
22\end{claim}
23\begin{evidence}[Proof of X.1]
24\label{ev:X.1}
25By the Archimedean property (Definition V.4), some multiple of the
26smaller magnitude exceeds the larger.  Iterated halving (or more)
27brings the remainder below the smaller in a finite number of steps.
28\dependson{X.1}{def:V.4}
29\end{evidence}
30
31\begin{claim}[Proposition X.2: Anthyphairesis detects incommensurability]
32\label{prop:X.2}
33If, when the lesser of two unequal magnitudes is continually
34subtracted in turn from the greater, that which is left never
35measures the one before it, the magnitudes will be incommensurable.
36\end{claim}
37\begin{evidence}[Proof of X.2]
38\label{ev:X.2}
39A common measure would persist through anthyphairesis (Common Notion
403); non-termination of the algorithm thus implies no common measure
41exists.
42\dependson{X.2}{X.1}
43\dependson{X.2}{cn:3}
44\dependson{X.2}{def:X.1}
45\end{evidence}
46
47\begin{claim}[Proposition X.3: Greatest common measure of commensurables]
48\label{prop:X.3}
49Given two commensurable magnitudes, to find their greatest common
50measure.
51\end{claim}
52\begin{evidence}[Proof of X.3]
53\label{ev:X.3}
54Apply anthyphairesis (the Euclidean algorithm on magnitudes); by X.2
55the algorithm terminates exactly when a common measure exists.
56\dependson{X.3}{X.2}
57\dependson{X.3}{def:X.1}
58\end{evidence}
59
60\begin{claim}[Proposition X.4: GCM of three commensurables]
61\label{prop:X.4}
62Given three commensurable magnitudes, to find their greatest common
63measure.
64\end{claim}
65\begin{evidence}[Proof of X.4]
66\label{ev:X.4}
67Apply X.3 to the first two; then to the result with the third.
68\dependson{X.4}{X.3}
69\end{evidence}
70
71\begin{claim}[Proposition X.5: Commensurables have a number ratio]
72\label{prop:X.5}
73Commensurable magnitudes have to one another the ratio which a number
74has to a number.
75\end{claim}
76\begin{evidence}[Proof of X.5]
77\label{ev:X.5}
78If $d$ is a common measure of $a$, $b$, then $a = md$, $b = nd$, so
79$a : b = m : n$ (a ratio of integers).
80\dependson{X.5}{def:V.5}
81\dependson{X.5}{def:X.1}
82\end{evidence}
83
84\begin{claim}[Proposition X.6: Number ratio implies commensurable]
85\label{prop:X.6}
86If two magnitudes have to one another the ratio which a number has
87to a number, the magnitudes will be commensurable.
88\end{claim}
89\begin{evidence}[Proof of X.6]
90\label{ev:X.6}
91Converse of X.5: if $a : b = m : n$, dividing $a$ by $m$ produces a
92common measure of $a$ and $b$.
93\dependson{X.6}{X.5}
94\end{evidence}
95
96\begin{claim}[Proposition X.7: Incommensurables have no number ratio]
97\label{prop:X.7}
98Incommensurable magnitudes have not to one another the ratio which
99a number has to a number.
100\end{claim}
101\begin{evidence}[Proof of X.7]
102\label{ev:X.7}
103Contrapositive of X.6.
104\dependson{X.7}{X.6}
105\end{evidence}
106
107\begin{claim}[Proposition X.8: No number ratio implies incommensurable]
108\label{prop:X.8}
109If two magnitudes have not to one another the ratio which a number
110has to a number, the magnitudes will be incommensurable.
111\end{claim}
112\begin{evidence}[Proof of X.8]
113\label{ev:X.8}
114Contrapositive of X.5.
115\dependson{X.8}{X.5}
116\end{evidence}
117
118\begin{claim}[Proposition X.9: Commensurability of squares from commensurability of sides]
119\label{prop:X.9}
120The squares on straight lines commensurable in length have to one
121another the ratio which a square number has to a square number; and
122squares which have to one another the ratio which a square number
123has to a square number will also have their sides commensurable in
124length.
125\end{claim}
126\begin{evidence}[Proof of X.9]
127\label{ev:X.9}
128If $a : b = m : n$ (integers) then $a^2 : b^2 = m^2 : n^2$; converse
129holds by VIII.14 applied to integers and VI.22 applied to magnitudes.
130\dependson{X.9}{VI.22}
131\dependson{X.9}{VIII.14}
132\dependson{X.9}{X.5}
133\dependson{X.9}{X.6}
134\end{evidence}
135
136\begin{claim}[Proposition X.10: Construct incommensurables]
137\label{prop:X.10}
138To find two straight lines incommensurable, the one in length only,
139the other in square also, with an assigned straight line.
140\end{claim}
141\begin{evidence}[Proof of X.10]
142\label{ev:X.10}
143Take a non-square integer ratio (e.g.\ $2 : 1$) and use it via X.9 to
144construct an incommensurable-in-length pair; build the
145incommensurable-in-square one similarly with a ratio that is neither
146square nor cube.
147\dependson{X.10}{VI.13}
148\dependson{X.10}{X.9}
149\end{evidence}
150
151\begin{claim}[Proposition X.11: Commensurability is transitive]
152\label{prop:X.11}
153If four magnitudes be proportional, and the first be commensurable
154with the second, the third also will be commensurable with the
155fourth; and if the first be incommensurable with the second, the
156third also will be incommensurable with the fourth.
157\end{claim}
158\begin{evidence}[Proof of X.11]
159\label{ev:X.11}
160Commensurability $\iff$ rational ratio (X.5 / X.6); rational ratios
161are preserved under equality of ratios.
162\dependson{X.11}{X.5}
163\dependson{X.11}{X.6}
164\end{evidence}
165
166\begin{claim}[Proposition X.12: Commensurability is transitive (three magnitudes)]
167\label{prop:X.12}
168Magnitudes commensurable with the same magnitude are commensurable
169with one another.
170\end{claim}
171\begin{evidence}[Proof of X.12]
172\label{ev:X.12}
173If $a \sim c$ and $b \sim c$ (commensurable), then $a \sim b$ by
174composing ratios (X.11).
175\dependson{X.12}{X.11}
176\end{evidence}
177
178\begin{claim}[Proposition X.13: Incommensurable preserved through transitivity]
179\label{prop:X.13}
180If two magnitudes be commensurable, and one of them be incommensurable
181with any magnitude, the remaining one will also be incommensurable
182with the same.
183\end{claim}
184\begin{evidence}[Proof of X.13]
185\label{ev:X.13}
186Contrapositive of X.12.
187\dependson{X.13}{X.12}
188\end{evidence}
189
190\begin{claim}[Proposition X.14: Squares preserve commensurability of sides]
191\label{prop:X.14}
192If four straight lines be proportional, and the square on the first
193be greater than the square on the second by the square on a straight
194line commensurable with the first, the square on the third will also
195be greater than the square on the fourth by the square on a straight
196line commensurable with the third.
197\end{claim}
198\begin{evidence}[Proof of X.14]
199\label{ev:X.14}
200Proportionality lifts to the squares; the deviation magnitude
201inherits the commensurability relation.
202\dependson{X.14}{VI.22}
203\dependson{X.14}{X.11}
204\end{evidence}
205
206\begin{claim}[Proposition X.15: Sum of commensurables is commensurable]
207\label{prop:X.15}
208If two commensurable magnitudes be added together, the whole will
209also be commensurable with each of them; and if the whole be
210commensurable with one of them, the original magnitudes will also be
211commensurable.
212\end{claim}
213\begin{evidence}[Proof of X.15]
214\label{ev:X.15}
215$a, b$ have a common measure $d$, so $a + b = (m + n) d$ shares $d$.
216Converse: if $a + b$ and $a$ share a measure, then so does $b$ by
217Common Notion 3.
218\dependson{X.15}{cn:3}
219\dependson{X.15}{def:X.1}
220\end{evidence}
221
222\begin{claim}[Proposition X.16: Sum with incommensurable]
223\label{prop:X.16}
224If two incommensurable magnitudes be added together, the whole will
225also be incommensurable with each of them; and if the whole be
226incommensurable with one of them, the original magnitudes will also
227be incommensurable.
228\end{claim}
229\begin{evidence}[Proof of X.16]
230\label{ev:X.16}
231Contrapositive of X.15.
232\dependson{X.16}{X.15}
233\end{evidence}
234
235\begin{claim}[Proposition X.17: Application of areas with commensurable difference]
236\label{prop:X.17}
237If there be two unequal straight lines, and to the greater there be
238applied a parallelogram equal to the fourth part of the square on the
239less and deficient by a square figure, and if it divide it into parts
240which are commensurable in length, then the square on the greater
241will be greater than the square on the less by the square on a
242straight line commensurable in length with the greater.
243\end{claim}
244\begin{evidence}[Proof of X.17]
245\label{ev:X.17}
246By VI.28 / VI.29 the application of areas with deficient/excess
247square corresponds to solving a quadratic; commensurability of the
248parts forces commensurability of the discriminant.
249\dependson{X.17}{VI.28}
250\dependson{X.17}{X.14}
251\dependson{X.17}{X.15}
252\end{evidence}
253
254\begin{claim}[Proposition X.18: Application of areas with incommensurable difference]
255\label{prop:X.18}
256If there be two unequal straight lines, and to the greater there be
257applied a parallelogram equal to the fourth part of the square on the
258less and deficient by a square figure, and if it divide it into parts
259incommensurable in length, then the square on the greater will be
260greater than the square on the less by the square on a straight line
261incommensurable in length with the greater.
262\end{claim}
263\begin{evidence}[Proof of X.18]
264\label{ev:X.18}
265Same construction as X.17 with the opposite hypothesis;
266incommensurability of the parts forces incommensurability of the
267discriminant.
268\dependson{X.18}{VI.28}
269\dependson{X.18}{X.16}
270\dependson{X.18}{X.17}
271\end{evidence}
272
273\begin{claim}[Proposition X.19: Rectangle on rationals is rational]
274\label{prop:X.19}
275The rectangle contained by rational straight lines commensurable in
276length is rational.
277\end{claim}
278\begin{evidence}[Proof of X.19]
279\label{ev:X.19}
280Rational sides commensurable in length have integer ratios; product
281of two such sides is in rational ratio to the assigned-square area.
282\dependson{X.19}{X.9}
283\dependson{X.19}{def:X.3}
284\dependson{X.19}{def:X.4}
285\end{evidence}
286
287\begin{claim}[Proposition X.20: Rational area divided by rational side is rational]
288\label{prop:X.20}
289If a rational area be applied to a rational straight line, it produces
290as breadth a straight line rational and commensurable in length with
291the straight line to which it is applied.
292\end{claim}
293\begin{evidence}[Proof of X.20]
294\label{ev:X.20}
295$A = \ell \cdot b$ with $A$ and $\ell$ rational forces $b$ rational by
296X.19 / X.9.
297\dependson{X.20}{X.19}
298\end{evidence}
299
300\begin{claim}[Proposition X.21: Medial rectangle]
301\label{prop:X.21}
302The rectangle contained by rational straight lines commensurable in
303square only is irrational, and the side of the square equal to it is
304irrational.  Let the latter be called medial.
305\end{claim}
306\begin{evidence}[Proof of X.21]
307\label{ev:X.21}
308Commensurable-in-square-only means the square on each is rational
309but the lengths are not in integer ratio.  The rectangle is then in
310a non-rational ratio to a rational area; its square root is the
311medial straight line (Definition XIII.3).
312\dependson{X.21}{X.9}
313\dependson{X.21}{def:X.3}
314\dependson{X.21}{def:XIII.3}
315\end{evidence}
316
317\begin{claim}[Proposition X.22: Square on a medial]
318\label{prop:X.22}
319The square on a medial straight line, if applied to a rational
320straight line, produces as breadth a straight line rational and
321incommensurable in length with that to which it is applied.
322\end{claim}
323\begin{evidence}[Proof of X.22]
324\label{ev:X.22}
325A medial squared is rational; applying it to a rational base, the
326breadth is rational; the incommensurability follows from the fact
327that the medial is incommensurable in length with the rational.
328\dependson{X.22}{X.21}
329\end{evidence}
330
331\begin{claim}[Proposition X.23: Magnitudes commensurable with medial are medial]
332\label{prop:X.23}
333A straight line commensurable with a medial straight line is medial.
334\end{claim}
335\begin{evidence}[Proof of X.23]
336\label{ev:X.23}
337Commensurability preserves the medial property: scaling a medial by
338a rational ratio leaves it medial.
339\dependson{X.23}{X.21}
340\dependson{X.23}{def:XIII.3}
341\end{evidence}
342
343\begin{claim}[Proposition X.24: Rectangle of commensurable medials]
344\label{prop:X.24}
345The rectangle contained by medial straight lines commensurable in
346length is medial.
347\end{claim}
348\begin{evidence}[Proof of X.24]
349\label{ev:X.24}
350Product of two medials in rational length-ratio is again the
351geometric mean of two rationals (Definition XIII.3).
352\dependson{X.24}{X.21}
353\dependson{X.24}{X.23}
354\end{evidence}
355
356\begin{claim}[Proposition X.25: Rectangle of medials commensurable in square only]
357\label{prop:X.25}
358The rectangle contained by medial straight lines commensurable in
359square only is either rational or medial.
360\end{claim}
361\begin{evidence}[Proof of X.25]
362\label{ev:X.25}
363Two cases depending on whether the rectangle has a rational
364square-root.  Both cases are realised by explicit constructions.
365\dependson{X.25}{X.21}
366\dependson{X.25}{X.24}
367\end{evidence}
368
369\begin{claim}[Proposition X.26: Medial difference is not rational]
370\label{prop:X.26}
371A medial area does not exceed a medial area by a rational area.
372\end{claim}
373\begin{evidence}[Proof of X.26]
374\label{ev:X.26}
375A difference $M_1 - M_2 = R$ with $R$ rational and $M_1$, $M_2$
376medial would force $M_1$, $M_2$ to be in a rational ratio,
377contradicting their being medial in distinct families.
378\dependson{X.26}{X.21}
379\dependson{X.26}{X.25}
380\end{evidence}
381
382\begin{claim}[Proposition X.27: Two medial lines commensurable in square]
383\label{prop:X.27}
384To find medial straight lines commensurable in square only which
385contain a rational rectangle.
386\end{claim}
387\begin{evidence}[Proof of X.27]
388\label{ev:X.27}
389Construct two such medials from a fixed rational by extracting two
390square-roots of ratios in lowest terms.
391\dependson{X.27}{X.21}
392\dependson{X.27}{X.22}
393\end{evidence}
394
395\begin{claim}[Proposition X.28: Medials enclosing a medial rectangle]
396\label{prop:X.28}
397To find medial straight lines commensurable in square only which
398contain a medial rectangle.
399\end{claim}
400\begin{evidence}[Proof of X.28]
401\label{ev:X.28}
402Same construction as X.27 with an extra medial step.
403\dependson{X.28}{X.21}
404\dependson{X.28}{X.27}
405\end{evidence}
406
407\begin{claim}[Proposition X.29: Two rationals commensurable in square, square-difference of commensurable kind, Lemma 1]
408\label{prop:X.29}
409To find two rational straight lines commensurable in square only such
410that the square on the greater is greater than the square on the
411less by the square on a straight line commensurable in length with
412the greater.
413\end{claim}
414\begin{evidence}[Proof of X.29]
415\label{ev:X.29}
416Take a rational line $a$ and apply X.17 with a deficient square; the
417construction gives the required pair.
418\dependson{X.29}{X.17}
419\end{evidence}
420
421\begin{claim}[Proposition X.30: Same as X.29 with the discriminant incommensurable]
422\label{prop:X.30}
423To find two rational straight lines commensurable in square only such
424that the square on the greater is greater than the square on the
425less by the square on a straight line incommensurable in length with
426the greater.
427\end{claim}
428\begin{evidence}[Proof of X.30]
429\label{ev:X.30}
430Apply X.18 (the incommensurable analogue of X.17).
431\dependson{X.30}{X.18}
432\end{evidence}
433
434\begin{claim}[Proposition X.31: Two medials with rational discriminant relation]
435\label{prop:X.31}
436To find two medial straight lines commensurable in square only,
437containing a rational rectangle, such that the square on the greater
438is greater than the square on the less by the square on a straight
439line commensurable in length with the greater.
440\end{claim}
441\begin{evidence}[Proof of X.31]
442\label{ev:X.31}
443Combine X.27 with the discriminant-control of X.29.
444\dependson{X.31}{X.27}
445\dependson{X.31}{X.29}
446\end{evidence}
447
448\begin{claim}[Proposition X.32: Two medials, medial rectangle, commensurable discriminant]
449\label{prop:X.32}
450To find two medial straight lines commensurable in square only,
451containing a medial rectangle, such that the square on the greater
452is greater than the square on the less by the square on a straight
453line commensurable in length with the greater.
454\end{claim}
455\begin{evidence}[Proof of X.32]
456\label{ev:X.32}
457Same as X.31 with X.28 in place of X.27.
458\dependson{X.32}{X.28}
459\dependson{X.32}{X.31}
460\end{evidence}
461
462\begin{claim}[Proposition X.33: Sum of squares rational, rectangle medial, sides incommensurable in square]
463\label{prop:X.33}
464To find two straight lines incommensurable in square which make the
465sum of the squares on them rational but the rectangle contained by
466them medial.
467\end{claim}
468\begin{evidence}[Proof of X.33]
469\label{ev:X.33}
470Apply X.30: a difference-of-squares construction with an
471incommensurable discriminant produces such a pair.
472\dependson{X.33}{X.30}
473\end{evidence}
474
475\begin{claim}[Proposition X.34: Sum medial, rectangle rational]
476\label{prop:X.34}
477To find two straight lines incommensurable in square which make the
478sum of the squares on them medial but the rectangle contained by
479them rational.
480\end{claim}
481\begin{evidence}[Proof of X.34]
482\label{ev:X.34}
483Variant of X.33 with the medial/rational roles swapped, via X.31.
484\dependson{X.34}{X.31}
485\dependson{X.34}{X.33}
486\end{evidence}
487
488\begin{claim}[Proposition X.35: Sum medial, rectangle medial, sum incommensurable with rectangle]
489\label{prop:X.35}
490To find two straight lines incommensurable in square which make the
491sum of the squares on them medial and the rectangle contained by
492them medial and moreover incommensurable with the sum of the squares
493on them.
494\end{claim}
495\begin{evidence}[Proof of X.35]
496\label{ev:X.35}
497Variant of X.34 with both quantities medial; use X.32.
498\dependson{X.35}{X.32}
499\dependson{X.35}{X.34}
500\end{evidence}
501
502\begin{claim}[Proposition X.36: Binomial straight line]
503\label{prop:X.36}
504If two rational straight lines commensurable in square only be added
505together, the whole is irrational; and let it be called binomial.
506\end{claim}
507\begin{evidence}[Proof of X.36]
508\label{ev:X.36}
509The sum $a + b$ with $a$, $b$ rational and incommensurable in length
510has square $a^2 + b^2 + 2ab$ where $2ab$ is medial (X.21), so the
511square on the sum is the sum of a rational and a medial: irrational.
512\dependson{X.36}{X.21}
513\dependson{X.36}{def:X.II.1}
514\end{evidence}
515
516\begin{claim}[Proposition X.37: First bimedial straight line]
517\label{prop:X.37}
518If two medial straight lines commensurable in square only and
519containing a rational rectangle be added together, the whole is
520irrational; and let it be called first bimedial.
521\end{claim}
522\begin{evidence}[Proof of X.37]
523\label{ev:X.37}
524Sum of two medials with rational rectangle; the square consists of
525two medials and a rational --- irrational.
526\dependson{X.37}{X.27}
527\dependson{X.37}{X.36}
528\end{evidence}
529
530\begin{claim}[Proposition X.38: Second bimedial straight line]
531\label{prop:X.38}
532If two medial straight lines commensurable in square only and
533containing a medial rectangle be added together, the whole is
534irrational; and let it be called second bimedial.
535\end{claim}
536\begin{evidence}[Proof of X.38]
537\label{ev:X.38}
538Same scheme as X.37 with the rectangle medial instead of rational.
539\dependson{X.38}{X.28}
540\dependson{X.38}{X.37}
541\end{evidence}
542
543\begin{claim}[Proposition X.39: Major straight line]
544\label{prop:X.39}
545If two straight lines incommensurable in square which make the sum
546of the squares on them rational, but the rectangle contained by them
547medial, be added together, the whole straight line is irrational;
548and let it be called major.
549\end{claim}
550\begin{evidence}[Proof of X.39]
551\label{ev:X.39}
552Sum of an X.33 pair has square = rational + medial: irrational.
553\dependson{X.39}{X.33}
554\dependson{X.39}{X.36}
555\end{evidence}
556
557\begin{claim}[Proposition X.40: Side of a rational plus medial area]
558\label{prop:X.40}
559If two straight lines incommensurable in square which make the sum
560of the squares on them medial, but the rectangle contained by them
561rational, be added together, the whole straight line is irrational;
562and let it be called the side of a rational plus a medial area.
563\end{claim}
564\begin{evidence}[Proof of X.40]
565\label{ev:X.40}
566Sum of an X.34 pair; same scheme as X.39.
567\dependson{X.40}{X.34}
568\dependson{X.40}{X.39}
569\end{evidence}
570
571\begin{claim}[Proposition X.41: Side of the sum of two medial areas]
572\label{prop:X.41}
573If two straight lines incommensurable in square which make the sum
574of the squares on them medial, and the rectangle contained by them
575medial and also incommensurable with the sum of the squares on them,
576be added together, the remaining straight line is irrational; and
577let it be called the side of the sum of two medial areas.
578\end{claim}
579\begin{evidence}[Proof of X.41]
580\label{ev:X.41}
581Sum of an X.35 pair; same scheme as X.39.
582\dependson{X.41}{X.35}
583\dependson{X.41}{X.40}
584\end{evidence}
585
586\begin{claim}[Proposition X.42: A binomial has unique decomposition]
587\label{prop:X.42}
588A binomial straight line is divided into its terms at one point only.
589\end{claim}
590\begin{evidence}[Proof of X.42]
591\label{ev:X.42}
592Suppose two decompositions $a_1 + b_1 = a_2 + b_2$ of the same
593binomial.  Comparing rationals and medials in the squares forces
594$a_1 = a_2$ and $b_1 = b_2$.
595\dependson{X.42}{X.36}
596\end{evidence}
597
598\begin{claim}[Proposition X.43: A first bimedial has unique decomposition]
599\label{prop:X.43}
600A first bimedial straight line is divided at one and the same point
601only.
602\end{claim}
603\begin{evidence}[Proof of X.43]
604\label{ev:X.43}
605Same uniqueness argument as X.42 applied to the first bimedial.
606\dependson{X.43}{X.37}
607\dependson{X.43}{X.42}
608\end{evidence}
609
610\begin{claim}[Proposition X.44: A second bimedial has unique decomposition]
611\label{prop:X.44}
612A second bimedial straight line is divided at one point only.
613\end{claim}
614\begin{evidence}[Proof of X.44]
615\label{ev:X.44}
616Same uniqueness argument applied to the second bimedial.
617\dependson{X.44}{X.38}
618\dependson{X.44}{X.43}
619\end{evidence}
620
621\begin{claim}[Proposition X.45: A major has unique decomposition]
622\label{prop:X.45}
623A major straight line is divided at one and the same point only.
624\end{claim}
625\begin{evidence}[Proof of X.45]
626\label{ev:X.45}
627Same uniqueness argument applied to the major.
628\dependson{X.45}{X.39}
629\dependson{X.45}{X.44}
630\end{evidence}
631
632\begin{claim}[Proposition X.46: Side of rational+medial has unique decomposition]
633\label{prop:X.46}
634The side of a rational plus a medial area is divided at one and the
635same point only.
636\end{claim}
637\begin{evidence}[Proof of X.46]
638\label{ev:X.46}
639Same uniqueness argument.
640\dependson{X.46}{X.40}
641\dependson{X.46}{X.45}
642\end{evidence}
643
644\begin{claim}[Proposition X.47: Side of two medial areas has unique decomposition]
645\label{prop:X.47}
646The side of the sum of two medial areas is divided at one and the
647same point only.
648\end{claim}
649\begin{evidence}[Proof of X.47]
650\label{ev:X.47}
651Same uniqueness argument.
652\dependson{X.47}{X.41}
653\dependson{X.47}{X.46}
654\end{evidence}
655
656\begin{claim}[Proposition X.48: First binomial straight line]
657\label{prop:X.48}
658To find the first binomial straight line.
659\end{claim}
660\begin{evidence}[Proof of X.48]
661\label{ev:X.48}
662Construct $a + b$ with $a$ commensurable in length with the
663assigned rational and the square-discriminant commensurable with the
664greater (X.29).
665\dependson{X.48}{X.29}
666\dependson{X.48}{X.36}
667\dependson{X.48}{def:X.II.1}
668\end{evidence}
669
670\begin{claim}[Proposition X.49: Second binomial straight line]
671\label{prop:X.49}
672To find the second binomial straight line.
673\end{claim}
674\begin{evidence}[Proof of X.49]
675\label{ev:X.49}
676Construct as in X.48 but with $b$ (rather than $a$) commensurable
677with the assigned rational.
678\dependson{X.49}{X.48}
679\dependson{X.49}{def:X.II.2}
680\end{evidence}
681
682\begin{claim}[Proposition X.50: Third binomial straight line]
683\label{prop:X.50}
684To find the third binomial straight line.
685\end{claim}
686\begin{evidence}[Proof of X.50]
687\label{ev:X.50}
688Neither term commensurable with the assigned rational, but the
689square-discriminant commensurable with the greater (X.29 variant).
690\dependson{X.50}{X.29}
691\dependson{X.50}{X.48}
692\dependson{X.50}{def:X.II.3}
693\end{evidence}
694
695\begin{claim}[Proposition X.51: Fourth binomial straight line]
696\label{prop:X.51}
697To find the fourth binomial straight line.
698\end{claim}
699\begin{evidence}[Proof of X.51]
700\label{ev:X.51}
701$a$ commensurable with the assigned rational, square-discriminant
702incommensurable with $a$ (X.30).
703\dependson{X.51}{X.30}
704\dependson{X.51}{X.48}
705\dependson{X.51}{def:X.II.4}
706\end{evidence}
707
708\begin{claim}[Proposition X.52: Fifth binomial straight line]
709\label{prop:X.52}
710To find the fifth binomial straight line.
711\end{claim}
712\begin{evidence}[Proof of X.52]
713\label{ev:X.52}
714$b$ commensurable, discriminant incommensurable with the greater.
715\dependson{X.52}{X.30}
716\dependson{X.52}{X.51}
717\dependson{X.52}{def:X.II.5}
718\end{evidence}
719
720\begin{claim}[Proposition X.53: Sixth binomial straight line]
721\label{prop:X.53}
722To find the sixth binomial straight line.
723\end{claim}
724\begin{evidence}[Proof of X.53]
725\label{ev:X.53}
726Neither term commensurable, discriminant incommensurable.
727\dependson{X.53}{X.30}
728\dependson{X.53}{X.52}
729\dependson{X.53}{def:X.II.6}
730\end{evidence}
731
732\begin{claim}[Proposition X.54: Rectangle on a first binomial is rational on rational]
733\label{prop:X.54}
734If an area be contained by a rational straight line and the first
735binomial, the side of the area is the irrational straight line which
736is called binomial.
737\end{claim}
738\begin{evidence}[Proof of X.54]
739\label{ev:X.54}
740$\sqrt{R \cdot \text{first binomial}}$ has the form of a binomial in
741the assigned rational base.
742\dependson{X.54}{X.36}
743\dependson{X.54}{X.48}
744\end{evidence}
745
746\begin{claim}[Proposition X.55: Rectangle on a second binomial yields a first bimedial]
747\label{prop:X.55}
748If an area be contained by a rational straight line and the second
749binomial, the side of the area is the irrational straight line which
750is called first bimedial.
751\end{claim}
752\begin{evidence}[Proof of X.55]
753\label{ev:X.55}
754Same pattern as X.54.
755\dependson{X.55}{X.37}
756\dependson{X.55}{X.49}
757\dependson{X.55}{X.54}
758\end{evidence}
759
760\begin{claim}[Proposition X.56: Rectangle on a third binomial yields a second bimedial]
761\label{prop:X.56}
762If an area be contained by a rational straight line and the third
763binomial, the side of the area is the irrational straight line which
764is called second bimedial.
765\end{claim}
766\begin{evidence}[Proof of X.56]
767\label{ev:X.56}
768Same pattern.
769\dependson{X.56}{X.38}
770\dependson{X.56}{X.50}
771\dependson{X.56}{X.55}
772\end{evidence}
773
774\begin{claim}[Proposition X.57: Rectangle on a fourth binomial yields a major]
775\label{prop:X.57}
776If an area be contained by a rational straight line and the fourth
777binomial, the side of the area is the irrational straight line which
778is called major.
779\end{claim}
780\begin{evidence}[Proof of X.57]
781\label{ev:X.57}
782Same pattern.
783\dependson{X.57}{X.39}
784\dependson{X.57}{X.51}
785\dependson{X.57}{X.56}
786\end{evidence}
787
788\begin{claim}[Proposition X.58: Rectangle on a fifth binomial yields a side of rational+medial]
789\label{prop:X.58}
790If an area be contained by a rational straight line and the fifth
791binomial, the side of the area is the irrational straight line which
792is the side of a rational plus a medial area.
793\end{claim}
794\begin{evidence}[Proof of X.58]
795\label{ev:X.58}
796Same pattern.
797\dependson{X.58}{X.40}
798\dependson{X.58}{X.52}
799\dependson{X.58}{X.57}
800\end{evidence}
801
802\begin{claim}[Proposition X.59: Rectangle on a sixth binomial yields side of two medials]
803\label{prop:X.59}
804If an area be contained by a rational straight line and the sixth
805binomial, the side of the area is the irrational straight line which
806is called the side of the sum of two medial areas.
807\end{claim}
808\begin{evidence}[Proof of X.59]
809\label{ev:X.59}
810Same pattern.
811\dependson{X.59}{X.41}
812\dependson{X.59}{X.53}
813\dependson{X.59}{X.58}
814\end{evidence}
815
816\begin{claim}[Proposition X.60: Square on a binomial yields a first binomial]
817\label{prop:X.60}
818The square on the binomial straight line applied to a rational
819straight line produces as breadth the first binomial.
820\end{claim}
821\begin{evidence}[Proof of X.60]
822\label{ev:X.60}
823Inverse of X.54: squaring and dividing by the assigned rational
824recovers the first binomial.
825\dependson{X.60}{X.48}
826\dependson{X.60}{X.54}
827\end{evidence}
828
829\begin{claim}[Proposition X.61: Square on a first bimedial yields a second binomial]
830\label{prop:X.61}
831The square on the first bimedial straight line applied to a rational
832straight line produces as breadth the second binomial.
833\end{claim}
834\begin{evidence}[Proof of X.61]
835\label{ev:X.61}
836Inverse of X.55.
837\dependson{X.61}{X.49}
838\dependson{X.61}{X.55}
839\dependson{X.61}{X.60}
840\end{evidence}
841
842\begin{claim}[Proposition X.62: Square on a second bimedial yields a third binomial]
843\label{prop:X.62}
844The square on the second bimedial straight line applied to a rational
845straight line produces as breadth the third binomial.
846\end{claim}
847\begin{evidence}[Proof of X.62]
848\label{ev:X.62}
849Inverse of X.56.
850\dependson{X.62}{X.50}
851\dependson{X.62}{X.56}
852\dependson{X.62}{X.61}
853\end{evidence}
854
855\begin{claim}[Proposition X.63: Square on a major yields a fourth binomial]
856\label{prop:X.63}
857The square on the major straight line applied to a rational straight
858line produces as breadth the fourth binomial.
859\end{claim}
860\begin{evidence}[Proof of X.63]
861\label{ev:X.63}
862Inverse of X.57.
863\dependson{X.63}{X.51}
864\dependson{X.63}{X.57}
865\dependson{X.63}{X.62}
866\end{evidence}
867
868\begin{claim}[Proposition X.64: Square on side-of-rational-plus-medial yields a fifth binomial]
869\label{prop:X.64}
870The square on the side of a rational plus a medial area applied to
871a rational straight line produces as breadth the fifth binomial.
872\end{claim}
873\begin{evidence}[Proof of X.64]
874\label{ev:X.64}
875Inverse of X.58.
876\dependson{X.64}{X.52}
877\dependson{X.64}{X.58}
878\dependson{X.64}{X.63}
879\end{evidence}
880
881\begin{claim}[Proposition X.65: Square on side-of-two-medials yields a sixth binomial]
882\label{prop:X.65}
883The square on the side of the sum of two medial areas applied to a
884rational straight line produces as breadth the sixth binomial.
885\end{claim}
886\begin{evidence}[Proof of X.65]
887\label{ev:X.65}
888Inverse of X.59.
889\dependson{X.65}{X.53}
890\dependson{X.65}{X.59}
891\dependson{X.65}{X.64}
892\end{evidence}
893
894\begin{claim}[Proposition X.66: Commensurable with binomial is binomial]
895\label{prop:X.66}
896A straight line commensurable in length with a binomial straight
897line is itself also binomial and the same in order.
898\end{claim}
899\begin{evidence}[Proof of X.66]
900\label{ev:X.66}
901Multiplication by a rational ratio preserves the binomial type and
902order.
903\dependson{X.66}{X.36}
904\dependson{X.66}{X.48}
905\end{evidence}
906
907\begin{claim}[Proposition X.67: Commensurable with bimedial is bimedial]
908\label{prop:X.67}
909A straight line commensurable in length with a bimedial straight
910line is itself bimedial and the same in order.
911\end{claim}
912\begin{evidence}[Proof of X.67]
913\label{ev:X.67}
914Same scheme as X.66.
915\dependson{X.67}{X.37}
916\dependson{X.67}{X.38}
917\dependson{X.67}{X.66}
918\end{evidence}
919
920\begin{claim}[Proposition X.68: Commensurable with major is major]
921\label{prop:X.68}
922A straight line commensurable with a major straight line is itself
923major.
924\end{claim}
925\begin{evidence}[Proof of X.68]
926\label{ev:X.68}
927Same scheme.
928\dependson{X.68}{X.39}
929\dependson{X.68}{X.67}
930\end{evidence}
931
932\begin{claim}[Proposition X.69: Commensurable with side of rational+medial is the same]
933\label{prop:X.69}
934A straight line commensurable with the side of a rational plus a
935medial area is itself such a side.
936\end{claim}
937\begin{evidence}[Proof of X.69]
938\label{ev:X.69}
939Same scheme.
940\dependson{X.69}{X.40}
941\dependson{X.69}{X.68}
942\end{evidence}
943
944\begin{claim}[Proposition X.70: Commensurable with side of two-medials is the same]
945\label{prop:X.70}
946A straight line commensurable with the side of the sum of two medial
947areas is itself such a side.
948\end{claim}
949\begin{evidence}[Proof of X.70]
950\label{ev:X.70}
951Same scheme.
952\dependson{X.70}{X.41}
953\dependson{X.70}{X.69}
954\end{evidence}
955
956\begin{claim}[Proposition X.71: Rational + medial sum is one of the four irrationals]
957\label{prop:X.71}
958If a rational and a medial area be added together, four irrational
959straight lines arise, namely either a binomial, a first bimedial, a
960major, or a side of a rational plus a medial area.
961\end{claim}
962\begin{evidence}[Proof of X.71]
963\label{ev:X.71}
964The square of any of the four classes (X.36, X.37, X.39, X.40) is
965the sum of a rational and a medial; conversely, every such sum
966arises in exactly one of these forms.
967\dependson{X.71}{X.36}
968\dependson{X.71}{X.37}
969\dependson{X.71}{X.39}
970\dependson{X.71}{X.40}
971\end{evidence}
972
973\begin{claim}[Proposition X.72: Medial + medial sum yields a bimedial or a side of two medials]
974\label{prop:X.72}
975If two medial areas incommensurable with one another be added
976together, the remaining two irrational straight lines arise, namely
977either a second bimedial or a side of the sum of two medial areas.
978\end{claim}
979\begin{evidence}[Proof of X.72]
980\label{ev:X.72}
981The square of X.38 or X.41 is a sum of two incommensurable medial
982areas; converse runs the same way.
983\dependson{X.72}{X.38}
984\dependson{X.72}{X.41}
985\dependson{X.72}{X.71}
986\end{evidence}
987
988\begin{claim}[Proposition X.73: Apotome straight line]
989\label{prop:X.73}
990If from a rational straight line there be subtracted a rational
991straight line commensurable with the whole in square only, the
992remainder is irrational; and let it be called apotome.
993\end{claim}
994\begin{evidence}[Proof of X.73]
995\label{ev:X.73}
996$a - b$ with $a$, $b$ commensurable in square only is the negation
997of the binomial case (X.36); the same argument shows it is
998irrational.
999\dependson{X.73}{X.36}
1000\end{evidence}
1001
1002\begin{claim}[Proposition X.74: First apotome of a medial]
1003\label{prop:X.74}
1004If from a medial straight line there be subtracted a medial straight
1005line commensurable with the whole in square only, and containing
1006with the whole a rational rectangle, the remainder is irrational;
1007and let it be called first apotome of a medial.
1008\end{claim}
1009\begin{evidence}[Proof of X.74]
1010\label{ev:X.74}
1011Negation of X.37.
1012\dependson{X.74}{X.37}
1013\dependson{X.74}{X.73}
1014\end{evidence}
1015
1016\begin{claim}[Proposition X.75: Second apotome of a medial]
1017\label{prop:X.75}
1018If from a medial straight line there be subtracted a medial straight
1019line commensurable with the whole in square only, and containing
1020with the whole a medial rectangle, the remainder is irrational; and
1021let it be called second apotome of a medial.
1022\end{claim}
1023\begin{evidence}[Proof of X.75]
1024\label{ev:X.75}
1025Negation of X.38.
1026\dependson{X.75}{X.38}
1027\dependson{X.75}{X.74}
1028\end{evidence}
1029
1030\begin{claim}[Proposition X.76: Minor straight line]
1031\label{prop:X.76}
1032If from a straight line there be subtracted a straight line
1033incommensurable in square with the whole, which with the whole makes
1034the squares on them added together rational, but the rectangle
1035contained by them medial, the remainder is irrational; and let it
1036be called minor.
1037\end{claim}
1038\begin{evidence}[Proof of X.76]
1039\label{ev:X.76}
1040Negation of X.39.  This is the "minor" line (Definition XIII.4).
1041\dependson{X.76}{X.39}
1042\dependson{X.76}{def:XIII.4}
1043\end{evidence}
1044
1045\begin{claim}[Proposition X.77: Line producing with rational area a medial whole]
1046\label{prop:X.77}
1047If from a straight line there be subtracted a straight line
1048incommensurable in square with the whole which with the whole makes
1049the sum of squares medial but twice the rectangle rational, the
1050remainder is irrational; let it be called that which produces with
1051a rational area a medial whole.
1052\end{claim}
1053\begin{evidence}[Proof of X.77]
1054\label{ev:X.77}
1055Negation of X.40.
1056\dependson{X.77}{X.40}
1057\dependson{X.77}{X.76}
1058\dependson{X.77}{def:XIII.5}
1059\end{evidence}
1060
1061\begin{claim}[Proposition X.78: Line producing with medial area a medial whole]
1062\label{prop:X.78}
1063If from a straight line there be subtracted a straight line
1064incommensurable in square with the whole which with the whole makes
1065both the sum of squares and twice the rectangle medial and the two
1066sums incommensurable with one another, the remainder is irrational;
1067let it be called that which produces with a medial area a medial
1068whole.
1069\end{claim}
1070\begin{evidence}[Proof of X.78]
1071\label{ev:X.78}
1072Negation of X.41.
1073\dependson{X.78}{X.41}
1074\dependson{X.78}{X.77}
1075\end{evidence}
1076
1077\begin{claim}[Proposition X.79: Apotome has unique annex]
1078\label{prop:X.79}
1079Only one rational straight line can be annexed to an apotome which
1080is commensurable with the whole in square only.
1081\end{claim}
1082\begin{evidence}[Proof of X.79]
1083\label{ev:X.79}
1084Uniqueness analogue of X.42 for apotomes.
1085\dependson{X.79}{X.42}
1086\dependson{X.79}{X.73}
1087\end{evidence}
1088
1089\begin{claim}[Proposition X.80: First-apotome-of-medial uniqueness]
1090\label{prop:X.80}
1091Only one medial straight line can be annexed to a first apotome of
1092a medial which is commensurable with the whole in square only and
1093forms with it a rational rectangle.
1094\end{claim}
1095\begin{evidence}[Proof of X.80]
1096\label{ev:X.80}
1097Same uniqueness pattern.
1098\dependson{X.80}{X.43}
1099\dependson{X.80}{X.74}
1100\end{evidence}
1101
1102\begin{claim}[Proposition X.81: Second-apotome-of-medial uniqueness]
1103\label{prop:X.81}
1104Only one medial straight line can be annexed to a second apotome of
1105a medial which is commensurable with the whole in square only and
1106forms with it a medial rectangle.
1107\end{claim}
1108\begin{evidence}[Proof of X.81]
1109\label{ev:X.81}
1110Same uniqueness pattern.
1111\dependson{X.81}{X.44}
1112\dependson{X.81}{X.75}
1113\end{evidence}
1114
1115\begin{claim}[Proposition X.82: Minor uniqueness]
1116\label{prop:X.82}
1117Only one straight line can be annexed to a minor.
1118\end{claim}
1119\begin{evidence}[Proof of X.82]
1120\label{ev:X.82}
1121Same uniqueness pattern.
1122\dependson{X.82}{X.45}
1123\dependson{X.82}{X.76}
1124\end{evidence}
1125
1126\begin{claim}[Proposition X.83: Uniqueness for X.77's line]
1127\label{prop:X.83}
1128Only one straight line can be annexed to the line producing with a
1129rational area a medial whole.
1130\end{claim}
1131\begin{evidence}[Proof of X.83]
1132\label{ev:X.83}
1133Same uniqueness pattern.
1134\dependson{X.83}{X.46}
1135\dependson{X.83}{X.77}
1136\end{evidence}
1137
1138\begin{claim}[Proposition X.84: Uniqueness for X.78's line]
1139\label{prop:X.84}
1140Only one straight line can be annexed to the line producing with a
1141medial area a medial whole.
1142\end{claim}
1143\begin{evidence}[Proof of X.84]
1144\label{ev:X.84}
1145Same uniqueness pattern.
1146\dependson{X.84}{X.47}
1147\dependson{X.84}{X.78}
1148\end{evidence}
1149
1150\begin{claim}[Proposition X.85: First apotome]
1151\label{prop:X.85}
1152To find the first apotome.
1153\end{claim}
1154\begin{evidence}[Proof of X.85]
1155\label{ev:X.85}
1156Take a first binomial $a + b$ (X.48); the difference $a - b$ is the
1157first apotome.
1158\dependson{X.85}{X.48}
1159\dependson{X.85}{X.73}
1160\dependson{X.85}{def:X.III.1}
1161\end{evidence}
1162
1163\begin{claim}[Proposition X.86: Second apotome]
1164\label{prop:X.86}
1165To find the second apotome.
1166\end{claim}
1167\begin{evidence}[Proof of X.86]
1168\label{ev:X.86}
1169Use the second binomial as the model (X.49).
1170\dependson{X.86}{X.49}
1171\dependson{X.86}{X.85}
1172\dependson{X.86}{def:X.III.2}
1173\end{evidence}
1174
1175\begin{claim}[Proposition X.87: Third apotome]
1176\label{prop:X.87}
1177To find the third apotome.
1178\end{claim}
1179\begin{evidence}[Proof of X.87]
1180\label{ev:X.87}
1181Use the third binomial (X.50).
1182\dependson{X.87}{X.50}
1183\dependson{X.87}{X.86}
1184\dependson{X.87}{def:X.III.3}
1185\end{evidence}
1186
1187\begin{claim}[Proposition X.88: Fourth apotome]
1188\label{prop:X.88}
1189To find the fourth apotome.
1190\end{claim}
1191\begin{evidence}[Proof of X.88]
1192\label{ev:X.88}
1193Use the fourth binomial (X.51).
1194\dependson{X.88}{X.51}
1195\dependson{X.88}{X.87}
1196\dependson{X.88}{def:X.III.4}
1197\end{evidence}
1198
1199\begin{claim}[Proposition X.89: Fifth apotome]
1200\label{prop:X.89}
1201To find the fifth apotome.
1202\end{claim}
1203\begin{evidence}[Proof of X.89]
1204\label{ev:X.89}
1205Use the fifth binomial (X.52).
1206\dependson{X.89}{X.52}
1207\dependson{X.89}{X.88}
1208\dependson{X.89}{def:X.III.5}
1209\end{evidence}
1210
1211\begin{claim}[Proposition X.90: Sixth apotome]
1212\label{prop:X.90}
1213To find the sixth apotome.
1214\end{claim}
1215\begin{evidence}[Proof of X.90]
1216\label{ev:X.90}
1217Use the sixth binomial (X.53).
1218\dependson{X.90}{X.53}
1219\dependson{X.90}{X.89}
1220\dependson{X.90}{def:X.III.6}
1221\end{evidence}
1222
1223\begin{claim}[Proposition X.91: Side of first-apotome area is an apotome]
1224\label{prop:X.91}
1225If an area be contained by a rational straight line and a first
1226apotome, the side of the area is an apotome.
1227\end{claim}
1228\begin{evidence}[Proof of X.91]
1229\label{ev:X.91}
1230Dual of X.54 for apotomes.
1231\dependson{X.91}{X.54}
1232\dependson{X.91}{X.85}
1233\end{evidence}
1234
1235\begin{claim}[Proposition X.92: Side of second-apotome area is a first apotome of medial]
1236\label{prop:X.92}
1237If an area be contained by a rational straight line and a second
1238apotome, the side of the area is a first apotome of a medial.
1239\end{claim}
1240\begin{evidence}[Proof of X.92]
1241\label{ev:X.92}
1242Dual of X.55.
1243\dependson{X.92}{X.55}
1244\dependson{X.92}{X.86}
1245\dependson{X.92}{X.91}
1246\end{evidence}
1247
1248\begin{claim}[Proposition X.93: Side of third-apotome area is a second apotome of medial]
1249\label{prop:X.93}
1250If an area be contained by a rational straight line and a third
1251apotome, the side of the area is a second apotome of a medial.
1252\end{claim}
1253\begin{evidence}[Proof of X.93]
1254\label{ev:X.93}
1255Dual of X.56.
1256\dependson{X.93}{X.56}
1257\dependson{X.93}{X.87}
1258\dependson{X.93}{X.92}
1259\end{evidence}
1260
1261\begin{claim}[Proposition X.94: Side of fourth-apotome area is a minor]
1262\label{prop:X.94}
1263If an area be contained by a rational straight line and a fourth
1264apotome, the side of the area is a minor.
1265\end{claim}
1266\begin{evidence}[Proof of X.94]
1267\label{ev:X.94}
1268Dual of X.57.
1269\dependson{X.94}{X.57}
1270\dependson{X.94}{X.88}
1271\dependson{X.94}{X.93}
1272\end{evidence}
1273
1274\begin{claim}[Proposition X.95: Side of fifth-apotome area produces rational-plus-medial complement]
1275\label{prop:X.95}
1276If an area be contained by a rational straight line and a fifth
1277apotome, the side of the area is the line producing with a rational
1278area a medial whole.
1279\end{claim}
1280\begin{evidence}[Proof of X.95]
1281\label{ev:X.95}
1282Dual of X.58.
1283\dependson{X.95}{X.58}
1284\dependson{X.95}{X.89}
1285\dependson{X.95}{X.94}
1286\end{evidence}
1287
1288\begin{claim}[Proposition X.96: Side of sixth-apotome area produces medial-plus-medial complement]
1289\label{prop:X.96}
1290If an area be contained by a rational straight line and a sixth
1291apotome, the side of the area is the line producing with a medial
1292area a medial whole.
1293\end{claim}
1294\begin{evidence}[Proof of X.96]
1295\label{ev:X.96}
1296Dual of X.59.
1297\dependson{X.96}{X.59}
1298\dependson{X.96}{X.90}
1299\dependson{X.96}{X.95}
1300\end{evidence}
1301
1302\begin{claim}[Proposition X.97: Square on apotome yields first apotome]
1303\label{prop:X.97}
1304The square on an apotome straight line applied to a rational
1305straight line produces as breadth a first apotome.
1306\end{claim}
1307\begin{evidence}[Proof of X.97]
1308\label{ev:X.97}
1309Inverse of X.91.
1310\dependson{X.97}{X.85}
1311\dependson{X.97}{X.91}
1312\end{evidence}
1313
1314\begin{claim}[Proposition X.98: Square on first apotome of medial yields second apotome]
1315\label{prop:X.98}
1316The square on a first apotome of a medial straight line applied to
1317a rational straight line produces as breadth a second apotome.
1318\end{claim}
1319\begin{evidence}[Proof of X.98]
1320\label{ev:X.98}
1321Inverse of X.92.
1322\dependson{X.98}{X.86}
1323\dependson{X.98}{X.92}
1324\dependson{X.98}{X.97}
1325\end{evidence}
1326
1327\begin{claim}[Proposition X.99: Square on second apotome of medial yields third apotome]
1328\label{prop:X.99}
1329The square on a second apotome of a medial straight line applied to
1330a rational straight line produces as breadth a third apotome.
1331\end{claim}
1332\begin{evidence}[Proof of X.99]
1333\label{ev:X.99}
1334Inverse of X.93.
1335\dependson{X.99}{X.87}
1336\dependson{X.99}{X.93}
1337\dependson{X.99}{X.98}
1338\end{evidence}
1339
1340\begin{claim}[Proposition X.100: Square on minor yields fourth apotome]
1341\label{prop:X.100}
1342The square on a minor applied to a rational straight line produces
1343as breadth a fourth apotome.
1344\end{claim}
1345\begin{evidence}[Proof of X.100]
1346\label{ev:X.100}
1347Inverse of X.94.
1348\dependson{X.100}{X.88}
1349\dependson{X.100}{X.94}
1350\dependson{X.100}{X.99}
1351\end{evidence}
1352
1353\begin{claim}[Proposition X.101: Square on rational-plus-medial producer yields fifth apotome]
1354\label{prop:X.101}
1355The square on the line producing with a rational area a medial whole
1356applied to a rational straight line produces as breadth a fifth
1357apotome.
1358\end{claim}
1359\begin{evidence}[Proof of X.101]
1360\label{ev:X.101}
1361Inverse of X.95.
1362\dependson{X.101}{X.89}
1363\dependson{X.101}{X.95}
1364\dependson{X.101}{X.100}
1365\end{evidence}
1366
1367\begin{claim}[Proposition X.102: Square on medial-plus-medial producer yields sixth apotome]
1368\label{prop:X.102}
1369The square on the line producing with a medial area a medial whole
1370applied to a rational straight line produces as breadth a sixth
1371apotome.
1372\end{claim}
1373\begin{evidence}[Proof of X.102]
1374\label{ev:X.102}
1375Inverse of X.96.
1376\dependson{X.102}{X.90}
1377\dependson{X.102}{X.96}
1378\dependson{X.102}{X.101}
1379\end{evidence}
1380
1381\begin{claim}[Proposition X.103: Commensurable with apotome is apotome]
1382\label{prop:X.103}
1383A straight line commensurable in length with an apotome is itself an
1384apotome and the same in order.
1385\end{claim}
1386\begin{evidence}[Proof of X.103]
1387\label{ev:X.103}
1388Dual of X.66.
1389\dependson{X.103}{X.66}
1390\dependson{X.103}{X.73}
1391\end{evidence}
1392
1393\begin{claim}[Proposition X.104: Commensurable with apotome of medial is the same]
1394\label{prop:X.104}
1395A straight line commensurable in length with an apotome of a medial
1396is itself such an apotome of the same order.
1397\end{claim}
1398\begin{evidence}[Proof of X.104]
1399\label{ev:X.104}
1400Dual of X.67.
1401\dependson{X.104}{X.67}
1402\dependson{X.104}{X.74}
1403\dependson{X.104}{X.103}
1404\end{evidence}
1405
1406\begin{claim}[Proposition X.105: Commensurable with minor is minor]
1407\label{prop:X.105}
1408A straight line commensurable with a minor is itself a minor.
1409\end{claim}
1410\begin{evidence}[Proof of X.105]
1411\label{ev:X.105}
1412Dual of X.68.
1413\dependson{X.105}{X.68}
1414\dependson{X.105}{X.76}
1415\dependson{X.105}{X.104}
1416\end{evidence}
1417
1418\begin{claim}[Proposition X.106: Commensurable with rational-medial producer is the same]
1419\label{prop:X.106}
1420A straight line commensurable with the line producing with a rational
1421area a medial whole is itself such a line.
1422\end{claim}
1423\begin{evidence}[Proof of X.106]
1424\label{ev:X.106}
1425Dual of X.69.
1426\dependson{X.106}{X.69}
1427\dependson{X.106}{X.77}
1428\dependson{X.106}{X.105}
1429\end{evidence}
1430
1431\begin{claim}[Proposition X.107: Commensurable with medial-medial producer is the same]
1432\label{prop:X.107}
1433A straight line commensurable with the line producing with a medial
1434area a medial whole is itself such a line.
1435\end{claim}
1436\begin{evidence}[Proof of X.107]
1437\label{ev:X.107}
1438Dual of X.70.
1439\dependson{X.107}{X.70}
1440\dependson{X.107}{X.78}
1441\dependson{X.107}{X.106}
1442\end{evidence}
1443
1444\begin{claim}[Proposition X.108: Side of rational minus medial is one of the four irrationals]
1445\label{prop:X.108}
1446If from a rational area a medial area be subtracted, the side of the
1447remaining area arises as one of four irrationals: an apotome, a
1448first apotome of a medial, a minor, or the line producing with a
1449rational area a medial whole.
1450\end{claim}
1451\begin{evidence}[Proof of X.108]
1452\label{ev:X.108}
1453Dual of X.71.
1454\dependson{X.108}{X.71}
1455\dependson{X.108}{X.73}
1456\dependson{X.108}{X.74}
1457\dependson{X.108}{X.76}
1458\dependson{X.108}{X.77}
1459\end{evidence}
1460
1461\begin{claim}[Proposition X.109: Medial minus rational yields apotome or producer-of-medial]
1462\label{prop:X.109}
1463If from a medial area a rational area be subtracted, two other
1464irrational straight lines arise, namely a first apotome of a medial
1465or the line producing with a rational area a medial whole.
1466\end{claim}
1467\begin{evidence}[Proof of X.109]
1468\label{ev:X.109}
1469Variant of X.108.
1470\dependson{X.109}{X.74}
1471\dependson{X.109}{X.77}
1472\dependson{X.109}{X.108}
1473\end{evidence}
1474
1475\begin{claim}[Proposition X.110: Medial minus medial yields second apotome of medial or medial-producer]
1476\label{prop:X.110}
1477If from a medial area there be subtracted a medial area
1478incommensurable with the whole, the remaining two irrational straight
1479lines arise: a second apotome of a medial or the line producing with
1480a medial area a medial whole.
1481\end{claim}
1482\begin{evidence}[Proof of X.110]
1483\label{ev:X.110}
1484Variant of X.108 / X.109.
1485\dependson{X.110}{X.75}
1486\dependson{X.110}{X.78}
1487\dependson{X.110}{X.109}
1488\end{evidence}
1489
1490\begin{claim}[Proposition X.111: Apotome and binomial are distinct]
1491\label{prop:X.111}
1492The apotome is not the same as the binomial.
1493\end{claim}
1494\begin{evidence}[Proof of X.111]
1495\label{ev:X.111}
1496A binomial has a rational sum of squares plus a medial rectangle; an
1497apotome has a rational difference of squares minus a medial
1498rectangle; if they coincided, the two combinations would coincide,
1499forcing the medial part to be rational --- contradiction.
1500\dependson{X.111}{X.36}
1501\dependson{X.111}{X.73}
1502\end{evidence}
1503
1504\begin{claim}[Proposition X.112: Square on rational divided by binomial is an apotome]
1505\label{prop:X.112}
1506The square on a rational straight line applied to the binomial
1507straight line produces as breadth an apotome the terms of which are
1508commensurable with the terms of the binomial and in the same ratio.
1509\end{claim}
1510\begin{evidence}[Proof of X.112]
1511\label{ev:X.112}
1512$R^2 / (a + b) = a' - b'$ with $a'$, $b'$ in the same ratio as $a$,
1513$b$.  Verified by direct manipulation of the square-on-binomial
1514identity.
1515\dependson{X.112}{X.91}
1516\dependson{X.112}{X.97}
1517\end{evidence}
1518
1519\begin{claim}[Proposition X.113: Square on rational divided by apotome is a binomial]
1520\label{prop:X.113}
1521The square on a rational straight line applied to an apotome produces
1522as breadth a binomial the terms of which are commensurable with the
1523terms of the apotome and in the same ratio.
1524\end{claim}
1525\begin{evidence}[Proof of X.113]
1526\label{ev:X.113}
1527Inverse of X.112.
1528\dependson{X.113}{X.54}
1529\dependson{X.113}{X.60}
1530\dependson{X.113}{X.112}
1531\end{evidence}
1532
1533\begin{claim}[Proposition X.114: Rectangle on binomial and apotome can be rational]
1534\label{prop:X.114}
1535If an area be contained by an apotome and the binomial the terms of
1536which are commensurable with the terms of the apotome and in the
1537same ratio, the side of the area is rational.
1538\end{claim}
1539\begin{evidence}[Proof of X.114]
1540\label{ev:X.114}
1541The rectangle on $(a - b)$ and $(a' + b')$ with $a' = ka$, $b' = kb$
1542equals $k(a^2 - b^2)$, which is rational.
1543\dependson{X.114}{X.112}
1544\dependson{X.114}{X.113}
1545\end{evidence}
1546
1547\begin{claim}[Proposition X.115: Medials yield infinitely many irrationals]
1548\label{prop:X.115}
1549From a medial straight line there arise irrational straight lines
1550infinite in number, and none of them is the same with any preceding.
1551\end{claim}
1552\begin{evidence}[Proof of X.115]
1553\label{ev:X.115}
1554By repeated mean-proportional construction (VI.13) on the medial,
1555each new line is irrational with respect to all earlier ones (using
1556the unique-decomposition results X.42--X.47, X.79--X.84).
1557\dependson{X.115}{VI.13}
1558\dependson{X.115}{X.21}
1559\dependson{X.115}{X.114}
1560\end{evidence}
1561

1% book10.tex --- Book X of Euclid's Elements: Incommensurable Magnitudes. 2% 3% All 115 propositions encoded. Book X is the longest of the Elements 4% and develops a classification of irrational magnitudes via repeated 5% quadratic operations: it covers commensurability (X.1-X.18), medials 6% (X.19-X.35), and the thirteen irrational lines including binomials and 7% apotomes (X.36-X.115). Proof sketches are condensed; the dependency 8% structure is what we capture. 9% 10% Wording follows Heath (1908). 11 12\section{Book X --- Incommensurable Magnitudes} 13\label{sec:book-X} 14 15\begin{claim}[Proposition X.1: Method of exhaustion lemma] 16\label{prop:X.1} 17Two unequal magnitudes being set out, if from the greater there be 18subtracted a magnitude greater than its half, and from that which is 19left a magnitude greater than its half, and if this process be 20repeated continually, there will be left some magnitude which will be 21less than the lesser magnitude set out. 22\end{claim} 23\begin{evidence}[Proof of X.1] 24\label{ev:X.1} 25By the Archimedean property (Definition V.4), some multiple of the 26smaller magnitude exceeds the larger. Iterated halving (or more) 27brings the remainder below the smaller in a finite number of steps. 28\dependson{X.1}{def:V.4} 29\end{evidence} 30 31\begin{claim}[Proposition X.2: Anthyphairesis detects incommensurability] 32\label{prop:X.2} 33If, when the lesser of two unequal magnitudes is continually 34subtracted in turn from the greater, that which is left never 35measures the one before it, the magnitudes will be incommensurable. 36\end{claim} 37\begin{evidence}[Proof of X.2] 38\label{ev:X.2} 39A common measure would persist through anthyphairesis (Common Notion 403); non-termination of the algorithm thus implies no common measure 41exists. 42\dependson{X.2}{X.1} 43\dependson{X.2}{cn:3} 44\dependson{X.2}{def:X.1} 45\end{evidence} 46 47\begin{claim}[Proposition X.3: Greatest common measure of commensurables] 48\label{prop:X.3} 49Given two commensurable magnitudes, to find their greatest common 50measure. 51\end{claim} 52\begin{evidence}[Proof of X.3] 53\label{ev:X.3} 54Apply anthyphairesis (the Euclidean algorithm on magnitudes); by X.2 55the algorithm terminates exactly when a common measure exists. 56\dependson{X.3}{X.2} 57\dependson{X.3}{def:X.1} 58\end{evidence} 59 60\begin{claim}[Proposition X.4: GCM of three commensurables] 61\label{prop:X.4} 62Given three commensurable magnitudes, to find their greatest common 63measure. 64\end{claim} 65\begin{evidence}[Proof of X.4] 66\label{ev:X.4} 67Apply X.3 to the first two; then to the result with the third. 68\dependson{X.4}{X.3} 69\end{evidence} 70 71\begin{claim}[Proposition X.5: Commensurables have a number ratio] 72\label{prop:X.5} 73Commensurable magnitudes have to one another the ratio which a number 74has to a number. 75\end{claim} 76\begin{evidence}[Proof of X.5] 77\label{ev:X.5} 78If $d$ is a common measure of $a$, $b$, then $a = md$, $b = nd$, so 79$a : b = m : n$ (a ratio of integers). 80\dependson{X.5}{def:V.5} 81\dependson{X.5}{def:X.1} 82\end{evidence} 83 84\begin{claim}[Proposition X.6: Number ratio implies commensurable] 85\label{prop:X.6} 86If two magnitudes have to one another the ratio which a number has 87to a number, the magnitudes will be commensurable. 88\end{claim} 89\begin{evidence}[Proof of X.6] 90\label{ev:X.6} 91Converse of X.5: if $a : b = m : n$, dividing $a$ by $m$ produces a 92common measure of $a$ and $b$. 93\dependson{X.6}{X.5} 94\end{evidence} 95 96\begin{claim}[Proposition X.7: Incommensurables have no number ratio] 97\label{prop:X.7} 98Incommensurable magnitudes have not to one another the ratio which 99a number has to a number. 100\end{claim} 101\begin{evidence}[Proof of X.7] 102\label{ev:X.7} 103Contrapositive of X.6. 104\dependson{X.7}{X.6} 105\end{evidence} 106 107\begin{claim}[Proposition X.8: No number ratio implies incommensurable] 108\label{prop:X.8} 109If two magnitudes have not to one another the ratio which a number 110has to a number, the magnitudes will be incommensurable. 111\end{claim} 112\begin{evidence}[Proof of X.8] 113\label{ev:X.8} 114Contrapositive of X.5. 115\dependson{X.8}{X.5} 116\end{evidence} 117 118\begin{claim}[Proposition X.9: Commensurability of squares from commensurability of sides] 119\label{prop:X.9} 120The squares on straight lines commensurable in length have to one 121another the ratio which a square number has to a square number; and 122squares which have to one another the ratio which a square number 123has to a square number will also have their sides commensurable in 124length. 125\end{claim} 126\begin{evidence}[Proof of X.9] 127\label{ev:X.9} 128If $a : b = m : n$ (integers) then $a^2 : b^2 = m^2 : n^2$; converse 129holds by VIII.14 applied to integers and VI.22 applied to magnitudes. 130\dependson{X.9}{VI.22} 131\dependson{X.9}{VIII.14} 132\dependson{X.9}{X.5} 133\dependson{X.9}{X.6} 134\end{evidence} 135 136\begin{claim}[Proposition X.10: Construct incommensurables] 137\label{prop:X.10} 138To find two straight lines incommensurable, the one in length only, 139the other in square also, with an assigned straight line. 140\end{claim} 141\begin{evidence}[Proof of X.10] 142\label{ev:X.10} 143Take a non-square integer ratio (e.g.\ $2 : 1$) and use it via X.9 to 144construct an incommensurable-in-length pair; build the 145incommensurable-in-square one similarly with a ratio that is neither 146square nor cube. 147\dependson{X.10}{VI.13} 148\dependson{X.10}{X.9} 149\end{evidence} 150 151\begin{claim}[Proposition X.11: Commensurability is transitive] 152\label{prop:X.11} 153If four magnitudes be proportional, and the first be commensurable 154with the second, the third also will be commensurable with the 155fourth; and if the first be incommensurable with the second, the 156third also will be incommensurable with the fourth. 157\end{claim} 158\begin{evidence}[Proof of X.11] 159\label{ev:X.11} 160Commensurability $\iff$ rational ratio (X.5 / X.6); rational ratios 161are preserved under equality of ratios. 162\dependson{X.11}{X.5} 163\dependson{X.11}{X.6} 164\end{evidence} 165 166\begin{claim}[Proposition X.12: Commensurability is transitive (three magnitudes)] 167\label{prop:X.12} 168Magnitudes commensurable with the same magnitude are commensurable 169with one another. 170\end{claim} 171\begin{evidence}[Proof of X.12] 172\label{ev:X.12} 173If $a \sim c$ and $b \sim c$ (commensurable), then $a \sim b$ by 174composing ratios (X.11). 175\dependson{X.12}{X.11} 176\end{evidence} 177 178\begin{claim}[Proposition X.13: Incommensurable preserved through transitivity] 179\label{prop:X.13} 180If two magnitudes be commensurable, and one of them be incommensurable 181with any magnitude, the remaining one will also be incommensurable 182with the same. 183\end{claim} 184\begin{evidence}[Proof of X.13] 185\label{ev:X.13} 186Contrapositive of X.12. 187\dependson{X.13}{X.12} 188\end{evidence} 189 190\begin{claim}[Proposition X.14: Squares preserve commensurability of sides] 191\label{prop:X.14} 192If four straight lines be proportional, and the square on the first 193be greater than the square on the second by the square on a straight 194line commensurable with the first, the square on the third will also 195be greater than the square on the fourth by the square on a straight 196line commensurable with the third. 197\end{claim} 198\begin{evidence}[Proof of X.14] 199\label{ev:X.14} 200Proportionality lifts to the squares; the deviation magnitude 201inherits the commensurability relation. 202\dependson{X.14}{VI.22} 203\dependson{X.14}{X.11} 204\end{evidence} 205 206\begin{claim}[Proposition X.15: Sum of commensurables is commensurable] 207\label{prop:X.15} 208If two commensurable magnitudes be added together, the whole will 209also be commensurable with each of them; and if the whole be 210commensurable with one of them, the original magnitudes will also be 211commensurable. 212\end{claim} 213\begin{evidence}[Proof of X.15] 214\label{ev:X.15} 215$a, b$ have a common measure $d$, so $a + b = (m + n) d$ shares $d$. 216Converse: if $a + b$ and $a$ share a measure, then so does $b$ by 217Common Notion 3. 218\dependson{X.15}{cn:3} 219\dependson{X.15}{def:X.1} 220\end{evidence} 221 222\begin{claim}[Proposition X.16: Sum with incommensurable] 223\label{prop:X.16} 224If two incommensurable magnitudes be added together, the whole will 225also be incommensurable with each of them; and if the whole be 226incommensurable with one of them, the original magnitudes will also 227be incommensurable. 228\end{claim} 229\begin{evidence}[Proof of X.16] 230\label{ev:X.16} 231Contrapositive of X.15. 232\dependson{X.16}{X.15} 233\end{evidence} 234 235\begin{claim}[Proposition X.17: Application of areas with commensurable difference] 236\label{prop:X.17} 237If there be two unequal straight lines, and to the greater there be 238applied a parallelogram equal to the fourth part of the square on the 239less and deficient by a square figure, and if it divide it into parts 240which are commensurable in length, then the square on the greater 241will be greater than the square on the less by the square on a 242straight line commensurable in length with the greater. 243\end{claim} 244\begin{evidence}[Proof of X.17] 245\label{ev:X.17} 246By VI.28 / VI.29 the application of areas with deficient/excess 247square corresponds to solving a quadratic; commensurability of the 248parts forces commensurability of the discriminant. 249\dependson{X.17}{VI.28} 250\dependson{X.17}{X.14} 251\dependson{X.17}{X.15} 252\end{evidence} 253 254\begin{claim}[Proposition X.18: Application of areas with incommensurable difference] 255\label{prop:X.18} 256If there be two unequal straight lines, and to the greater there be 257applied a parallelogram equal to the fourth part of the square on the 258less and deficient by a square figure, and if it divide it into parts 259incommensurable in length, then the square on the greater will be 260greater than the square on the less by the square on a straight line 261incommensurable in length with the greater. 262\end{claim} 263\begin{evidence}[Proof of X.18] 264\label{ev:X.18} 265Same construction as X.17 with the opposite hypothesis; 266incommensurability of the parts forces incommensurability of the 267discriminant. 268\dependson{X.18}{VI.28} 269\dependson{X.18}{X.16} 270\dependson{X.18}{X.17} 271\end{evidence} 272 273\begin{claim}[Proposition X.19: Rectangle on rationals is rational] 274\label{prop:X.19} 275The rectangle contained by rational straight lines commensurable in 276length is rational. 277\end{claim} 278\begin{evidence}[Proof of X.19] 279\label{ev:X.19} 280Rational sides commensurable in length have integer ratios; product 281of two such sides is in rational ratio to the assigned-square area. 282\dependson{X.19}{X.9} 283\dependson{X.19}{def:X.3} 284\dependson{X.19}{def:X.4} 285\end{evidence} 286 287\begin{claim}[Proposition X.20: Rational area divided by rational side is rational] 288\label{prop:X.20} 289If a rational area be applied to a rational straight line, it produces 290as breadth a straight line rational and commensurable in length with 291the straight line to which it is applied. 292\end{claim} 293\begin{evidence}[Proof of X.20] 294\label{ev:X.20} 295$A = \ell \cdot b$ with $A$ and $\ell$ rational forces $b$ rational by 296X.19 / X.9. 297\dependson{X.20}{X.19} 298\end{evidence} 299 300\begin{claim}[Proposition X.21: Medial rectangle] 301\label{prop:X.21} 302The rectangle contained by rational straight lines commensurable in 303square only is irrational, and the side of the square equal to it is 304irrational. Let the latter be called medial. 305\end{claim} 306\begin{evidence}[Proof of X.21] 307\label{ev:X.21} 308Commensurable-in-square-only means the square on each is rational 309but the lengths are not in integer ratio. The rectangle is then in 310a non-rational ratio to a rational area; its square root is the 311medial straight line (Definition XIII.3). 312\dependson{X.21}{X.9} 313\dependson{X.21}{def:X.3} 314\dependson{X.21}{def:XIII.3} 315\end{evidence} 316 317\begin{claim}[Proposition X.22: Square on a medial] 318\label{prop:X.22} 319The square on a medial straight line, if applied to a rational 320straight line, produces as breadth a straight line rational and 321incommensurable in length with that to which it is applied. 322\end{claim} 323\begin{evidence}[Proof of X.22] 324\label{ev:X.22} 325A medial squared is rational; applying it to a rational base, the 326breadth is rational; the incommensurability follows from the fact 327that the medial is incommensurable in length with the rational. 328\dependson{X.22}{X.21} 329\end{evidence} 330 331\begin{claim}[Proposition X.23: Magnitudes commensurable with medial are medial] 332\label{prop:X.23} 333A straight line commensurable with a medial straight line is medial. 334\end{claim} 335\begin{evidence}[Proof of X.23] 336\label{ev:X.23} 337Commensurability preserves the medial property: scaling a medial by 338a rational ratio leaves it medial. 339\dependson{X.23}{X.21} 340\dependson{X.23}{def:XIII.3} 341\end{evidence} 342 343\begin{claim}[Proposition X.24: Rectangle of commensurable medials] 344\label{prop:X.24} 345The rectangle contained by medial straight lines commensurable in 346length is medial. 347\end{claim} 348\begin{evidence}[Proof of X.24] 349\label{ev:X.24} 350Product of two medials in rational length-ratio is again the 351geometric mean of two rationals (Definition XIII.3). 352\dependson{X.24}{X.21} 353\dependson{X.24}{X.23} 354\end{evidence} 355 356\begin{claim}[Proposition X.25: Rectangle of medials commensurable in square only] 357\label{prop:X.25} 358The rectangle contained by medial straight lines commensurable in 359square only is either rational or medial. 360\end{claim} 361\begin{evidence}[Proof of X.25] 362\label{ev:X.25} 363Two cases depending on whether the rectangle has a rational 364square-root. Both cases are realised by explicit constructions. 365\dependson{X.25}{X.21} 366\dependson{X.25}{X.24} 367\end{evidence} 368 369\begin{claim}[Proposition X.26: Medial difference is not rational] 370\label{prop:X.26} 371A medial area does not exceed a medial area by a rational area. 372\end{claim} 373\begin{evidence}[Proof of X.26] 374\label{ev:X.26} 375A difference $M_1 - M_2 = R$ with $R$ rational and $M_1$, $M_2$ 376medial would force $M_1$, $M_2$ to be in a rational ratio, 377contradicting their being medial in distinct families. 378\dependson{X.26}{X.21} 379\dependson{X.26}{X.25} 380\end{evidence} 381 382\begin{claim}[Proposition X.27: Two medial lines commensurable in square] 383\label{prop:X.27} 384To find medial straight lines commensurable in square only which 385contain a rational rectangle. 386\end{claim} 387\begin{evidence}[Proof of X.27] 388\label{ev:X.27} 389Construct two such medials from a fixed rational by extracting two 390square-roots of ratios in lowest terms. 391\dependson{X.27}{X.21} 392\dependson{X.27}{X.22} 393\end{evidence} 394 395\begin{claim}[Proposition X.28: Medials enclosing a medial rectangle] 396\label{prop:X.28} 397To find medial straight lines commensurable in square only which 398contain a medial rectangle. 399\end{claim} 400\begin{evidence}[Proof of X.28] 401\label{ev:X.28} 402Same construction as X.27 with an extra medial step. 403\dependson{X.28}{X.21} 404\dependson{X.28}{X.27} 405\end{evidence} 406 407\begin{claim}[Proposition X.29: Two rationals commensurable in square, square-difference of commensurable kind, Lemma 1] 408\label{prop:X.29} 409To find two rational straight lines commensurable in square only such 410that the square on the greater is greater than the square on the 411less by the square on a straight line commensurable in length with 412the greater. 413\end{claim} 414\begin{evidence}[Proof of X.29] 415\label{ev:X.29} 416Take a rational line $a$ and apply X.17 with a deficient square; the 417construction gives the required pair. 418\dependson{X.29}{X.17} 419\end{evidence} 420 421\begin{claim}[Proposition X.30: Same as X.29 with the discriminant incommensurable] 422\label{prop:X.30} 423To find two rational straight lines commensurable in square only such 424that the square on the greater is greater than the square on the 425less by the square on a straight line incommensurable in length with 426the greater. 427\end{claim} 428\begin{evidence}[Proof of X.30] 429\label{ev:X.30} 430Apply X.18 (the incommensurable analogue of X.17). 431\dependson{X.30}{X.18} 432\end{evidence} 433 434\begin{claim}[Proposition X.31: Two medials with rational discriminant relation] 435\label{prop:X.31} 436To find two medial straight lines commensurable in square only, 437containing a rational rectangle, such that the square on the greater 438is greater than the square on the less by the square on a straight 439line commensurable in length with the greater. 440\end{claim} 441\begin{evidence}[Proof of X.31] 442\label{ev:X.31} 443Combine X.27 with the discriminant-control of X.29. 444\dependson{X.31}{X.27} 445\dependson{X.31}{X.29} 446\end{evidence} 447 448\begin{claim}[Proposition X.32: Two medials, medial rectangle, commensurable discriminant] 449\label{prop:X.32} 450To find two medial straight lines commensurable in square only, 451containing a medial rectangle, such that the square on the greater 452is greater than the square on the less by the square on a straight 453line commensurable in length with the greater. 454\end{claim} 455\begin{evidence}[Proof of X.32] 456\label{ev:X.32} 457Same as X.31 with X.28 in place of X.27. 458\dependson{X.32}{X.28} 459\dependson{X.32}{X.31} 460\end{evidence} 461 462\begin{claim}[Proposition X.33: Sum of squares rational, rectangle medial, sides incommensurable in square] 463\label{prop:X.33} 464To find two straight lines incommensurable in square which make the 465sum of the squares on them rational but the rectangle contained by 466them medial. 467\end{claim} 468\begin{evidence}[Proof of X.33] 469\label{ev:X.33} 470Apply X.30: a difference-of-squares construction with an 471incommensurable discriminant produces such a pair. 472\dependson{X.33}{X.30} 473\end{evidence} 474 475\begin{claim}[Proposition X.34: Sum medial, rectangle rational] 476\label{prop:X.34} 477To find two straight lines incommensurable in square which make the 478sum of the squares on them medial but the rectangle contained by 479them rational. 480\end{claim} 481\begin{evidence}[Proof of X.34] 482\label{ev:X.34} 483Variant of X.33 with the medial/rational roles swapped, via X.31. 484\dependson{X.34}{X.31} 485\dependson{X.34}{X.33} 486\end{evidence} 487 488\begin{claim}[Proposition X.35: Sum medial, rectangle medial, sum incommensurable with rectangle] 489\label{prop:X.35} 490To find two straight lines incommensurable in square which make the 491sum of the squares on them medial and the rectangle contained by 492them medial and moreover incommensurable with the sum of the squares 493on them. 494\end{claim} 495\begin{evidence}[Proof of X.35] 496\label{ev:X.35} 497Variant of X.34 with both quantities medial; use X.32. 498\dependson{X.35}{X.32} 499\dependson{X.35}{X.34} 500\end{evidence} 501 502\begin{claim}[Proposition X.36: Binomial straight line] 503\label{prop:X.36} 504If two rational straight lines commensurable in square only be added 505together, the whole is irrational; and let it be called binomial. 506\end{claim} 507\begin{evidence}[Proof of X.36] 508\label{ev:X.36} 509The sum $a + b$ with $a$, $b$ rational and incommensurable in length 510has square $a^2 + b^2 + 2ab$ where $2ab$ is medial (X.21), so the 511square on the sum is the sum of a rational and a medial: irrational. 512\dependson{X.36}{X.21} 513\dependson{X.36}{def:X.II.1} 514\end{evidence} 515 516\begin{claim}[Proposition X.37: First bimedial straight line] 517\label{prop:X.37} 518If two medial straight lines commensurable in square only and 519containing a rational rectangle be added together, the whole is 520irrational; and let it be called first bimedial. 521\end{claim} 522\begin{evidence}[Proof of X.37] 523\label{ev:X.37} 524Sum of two medials with rational rectangle; the square consists of 525two medials and a rational --- irrational. 526\dependson{X.37}{X.27} 527\dependson{X.37}{X.36} 528\end{evidence} 529 530\begin{claim}[Proposition X.38: Second bimedial straight line] 531\label{prop:X.38} 532If two medial straight lines commensurable in square only and 533containing a medial rectangle be added together, the whole is 534irrational; and let it be called second bimedial. 535\end{claim} 536\begin{evidence}[Proof of X.38] 537\label{ev:X.38} 538Same scheme as X.37 with the rectangle medial instead of rational. 539\dependson{X.38}{X.28} 540\dependson{X.38}{X.37} 541\end{evidence} 542 543\begin{claim}[Proposition X.39: Major straight line] 544\label{prop:X.39} 545If two straight lines incommensurable in square which make the sum 546of the squares on them rational, but the rectangle contained by them 547medial, be added together, the whole straight line is irrational; 548and let it be called major. 549\end{claim} 550\begin{evidence}[Proof of X.39] 551\label{ev:X.39} 552Sum of an X.33 pair has square = rational + medial: irrational. 553\dependson{X.39}{X.33} 554\dependson{X.39}{X.36} 555\end{evidence} 556 557\begin{claim}[Proposition X.40: Side of a rational plus medial area] 558\label{prop:X.40} 559If two straight lines incommensurable in square which make the sum 560of the squares on them medial, but the rectangle contained by them 561rational, be added together, the whole straight line is irrational; 562and let it be called the side of a rational plus a medial area. 563\end{claim} 564\begin{evidence}[Proof of X.40] 565\label{ev:X.40} 566Sum of an X.34 pair; same scheme as X.39. 567\dependson{X.40}{X.34} 568\dependson{X.40}{X.39} 569\end{evidence} 570 571\begin{claim}[Proposition X.41: Side of the sum of two medial areas] 572\label{prop:X.41} 573If two straight lines incommensurable in square which make the sum 574of the squares on them medial, and the rectangle contained by them 575medial and also incommensurable with the sum of the squares on them, 576be added together, the remaining straight line is irrational; and 577let it be called the side of the sum of two medial areas. 578\end{claim} 579\begin{evidence}[Proof of X.41] 580\label{ev:X.41} 581Sum of an X.35 pair; same scheme as X.39. 582\dependson{X.41}{X.35} 583\dependson{X.41}{X.40} 584\end{evidence} 585 586\begin{claim}[Proposition X.42: A binomial has unique decomposition] 587\label{prop:X.42} 588A binomial straight line is divided into its terms at one point only. 589\end{claim} 590\begin{evidence}[Proof of X.42] 591\label{ev:X.42} 592Suppose two decompositions $a_1 + b_1 = a_2 + b_2$ of the same 593binomial. Comparing rationals and medials in the squares forces 594$a_1 = a_2$ and $b_1 = b_2$. 595\dependson{X.42}{X.36} 596\end{evidence} 597 598\begin{claim}[Proposition X.43: A first bimedial has unique decomposition] 599\label{prop:X.43} 600A first bimedial straight line is divided at one and the same point 601only. 602\end{claim} 603\begin{evidence}[Proof of X.43] 604\label{ev:X.43} 605Same uniqueness argument as X.42 applied to the first bimedial. 606\dependson{X.43}{X.37} 607\dependson{X.43}{X.42} 608\end{evidence} 609 610\begin{claim}[Proposition X.44: A second bimedial has unique decomposition] 611\label{prop:X.44} 612A second bimedial straight line is divided at one point only. 613\end{claim} 614\begin{evidence}[Proof of X.44] 615\label{ev:X.44} 616Same uniqueness argument applied to the second bimedial. 617\dependson{X.44}{X.38} 618\dependson{X.44}{X.43} 619\end{evidence} 620 621\begin{claim}[Proposition X.45: A major has unique decomposition] 622\label{prop:X.45} 623A major straight line is divided at one and the same point only. 624\end{claim} 625\begin{evidence}[Proof of X.45] 626\label{ev:X.45} 627Same uniqueness argument applied to the major. 628\dependson{X.45}{X.39} 629\dependson{X.45}{X.44} 630\end{evidence} 631 632\begin{claim}[Proposition X.46: Side of rational+medial has unique decomposition] 633\label{prop:X.46} 634The side of a rational plus a medial area is divided at one and the 635same point only. 636\end{claim} 637\begin{evidence}[Proof of X.46] 638\label{ev:X.46} 639Same uniqueness argument. 640\dependson{X.46}{X.40} 641\dependson{X.46}{X.45} 642\end{evidence} 643 644\begin{claim}[Proposition X.47: Side of two medial areas has unique decomposition] 645\label{prop:X.47} 646The side of the sum of two medial areas is divided at one and the 647same point only. 648\end{claim} 649\begin{evidence}[Proof of X.47] 650\label{ev:X.47} 651Same uniqueness argument. 652\dependson{X.47}{X.41} 653\dependson{X.47}{X.46} 654\end{evidence} 655 656\begin{claim}[Proposition X.48: First binomial straight line] 657\label{prop:X.48} 658To find the first binomial straight line. 659\end{claim} 660\begin{evidence}[Proof of X.48] 661\label{ev:X.48} 662Construct $a + b$ with $a$ commensurable in length with the 663assigned rational and the square-discriminant commensurable with the 664greater (X.29). 665\dependson{X.48}{X.29} 666\dependson{X.48}{X.36} 667\dependson{X.48}{def:X.II.1} 668\end{evidence} 669 670\begin{claim}[Proposition X.49: Second binomial straight line] 671\label{prop:X.49} 672To find the second binomial straight line. 673\end{claim} 674\begin{evidence}[Proof of X.49] 675\label{ev:X.49} 676Construct as in X.48 but with $b$ (rather than $a$) commensurable 677with the assigned rational. 678\dependson{X.49}{X.48} 679\dependson{X.49}{def:X.II.2} 680\end{evidence} 681 682\begin{claim}[Proposition X.50: Third binomial straight line] 683\label{prop:X.50} 684To find the third binomial straight line. 685\end{claim} 686\begin{evidence}[Proof of X.50] 687\label{ev:X.50} 688Neither term commensurable with the assigned rational, but the 689square-discriminant commensurable with the greater (X.29 variant). 690\dependson{X.50}{X.29} 691\dependson{X.50}{X.48} 692\dependson{X.50}{def:X.II.3} 693\end{evidence} 694 695\begin{claim}[Proposition X.51: Fourth binomial straight line] 696\label{prop:X.51} 697To find the fourth binomial straight line. 698\end{claim} 699\begin{evidence}[Proof of X.51] 700\label{ev:X.51} 701$a$ commensurable with the assigned rational, square-discriminant 702incommensurable with $a$ (X.30). 703\dependson{X.51}{X.30} 704\dependson{X.51}{X.48} 705\dependson{X.51}{def:X.II.4} 706\end{evidence} 707 708\begin{claim}[Proposition X.52: Fifth binomial straight line] 709\label{prop:X.52} 710To find the fifth binomial straight line. 711\end{claim} 712\begin{evidence}[Proof of X.52] 713\label{ev:X.52} 714$b$ commensurable, discriminant incommensurable with the greater. 715\dependson{X.52}{X.30} 716\dependson{X.52}{X.51} 717\dependson{X.52}{def:X.II.5} 718\end{evidence} 719 720\begin{claim}[Proposition X.53: Sixth binomial straight line] 721\label{prop:X.53} 722To find the sixth binomial straight line. 723\end{claim} 724\begin{evidence}[Proof of X.53] 725\label{ev:X.53} 726Neither term commensurable, discriminant incommensurable. 727\dependson{X.53}{X.30} 728\dependson{X.53}{X.52} 729\dependson{X.53}{def:X.II.6} 730\end{evidence} 731 732\begin{claim}[Proposition X.54: Rectangle on a first binomial is rational on rational] 733\label{prop:X.54} 734If an area be contained by a rational straight line and the first 735binomial, the side of the area is the irrational straight line which 736is called binomial. 737\end{claim} 738\begin{evidence}[Proof of X.54] 739\label{ev:X.54} 740$\sqrt{R \cdot \text{first binomial}}$ has the form of a binomial in 741the assigned rational base. 742\dependson{X.54}{X.36} 743\dependson{X.54}{X.48} 744\end{evidence} 745 746\begin{claim}[Proposition X.55: Rectangle on a second binomial yields a first bimedial] 747\label{prop:X.55} 748If an area be contained by a rational straight line and the second 749binomial, the side of the area is the irrational straight line which 750is called first bimedial. 751\end{claim} 752\begin{evidence}[Proof of X.55] 753\label{ev:X.55} 754Same pattern as X.54. 755\dependson{X.55}{X.37} 756\dependson{X.55}{X.49} 757\dependson{X.55}{X.54} 758\end{evidence} 759 760\begin{claim}[Proposition X.56: Rectangle on a third binomial yields a second bimedial] 761\label{prop:X.56} 762If an area be contained by a rational straight line and the third 763binomial, the side of the area is the irrational straight line which 764is called second bimedial. 765\end{claim} 766\begin{evidence}[Proof of X.56] 767\label{ev:X.56} 768Same pattern. 769\dependson{X.56}{X.38} 770\dependson{X.56}{X.50} 771\dependson{X.56}{X.55} 772\end{evidence} 773 774\begin{claim}[Proposition X.57: Rectangle on a fourth binomial yields a major] 775\label{prop:X.57} 776If an area be contained by a rational straight line and the fourth 777binomial, the side of the area is the irrational straight line which 778is called major. 779\end{claim} 780\begin{evidence}[Proof of X.57] 781\label{ev:X.57} 782Same pattern. 783\dependson{X.57}{X.39} 784\dependson{X.57}{X.51} 785\dependson{X.57}{X.56} 786\end{evidence} 787 788\begin{claim}[Proposition X.58: Rectangle on a fifth binomial yields a side of rational+medial] 789\label{prop:X.58} 790If an area be contained by a rational straight line and the fifth 791binomial, the side of the area is the irrational straight line which 792is the side of a rational plus a medial area. 793\end{claim} 794\begin{evidence}[Proof of X.58] 795\label{ev:X.58} 796Same pattern. 797\dependson{X.58}{X.40} 798\dependson{X.58}{X.52} 799\dependson{X.58}{X.57} 800\end{evidence} 801 802\begin{claim}[Proposition X.59: Rectangle on a sixth binomial yields side of two medials] 803\label{prop:X.59} 804If an area be contained by a rational straight line and the sixth 805binomial, the side of the area is the irrational straight line which 806is called the side of the sum of two medial areas. 807\end{claim} 808\begin{evidence}[Proof of X.59] 809\label{ev:X.59} 810Same pattern. 811\dependson{X.59}{X.41} 812\dependson{X.59}{X.53} 813\dependson{X.59}{X.58} 814\end{evidence} 815 816\begin{claim}[Proposition X.60: Square on a binomial yields a first binomial] 817\label{prop:X.60} 818The square on the binomial straight line applied to a rational 819straight line produces as breadth the first binomial. 820\end{claim} 821\begin{evidence}[Proof of X.60] 822\label{ev:X.60} 823Inverse of X.54: squaring and dividing by the assigned rational 824recovers the first binomial. 825\dependson{X.60}{X.48} 826\dependson{X.60}{X.54} 827\end{evidence} 828 829\begin{claim}[Proposition X.61: Square on a first bimedial yields a second binomial] 830\label{prop:X.61} 831The square on the first bimedial straight line applied to a rational 832straight line produces as breadth the second binomial. 833\end{claim} 834\begin{evidence}[Proof of X.61] 835\label{ev:X.61} 836Inverse of X.55. 837\dependson{X.61}{X.49} 838\dependson{X.61}{X.55} 839\dependson{X.61}{X.60} 840\end{evidence} 841 842\begin{claim}[Proposition X.62: Square on a second bimedial yields a third binomial] 843\label{prop:X.62} 844The square on the second bimedial straight line applied to a rational 845straight line produces as breadth the third binomial. 846\end{claim} 847\begin{evidence}[Proof of X.62] 848\label{ev:X.62} 849Inverse of X.56. 850\dependson{X.62}{X.50} 851\dependson{X.62}{X.56} 852\dependson{X.62}{X.61} 853\end{evidence} 854 855\begin{claim}[Proposition X.63: Square on a major yields a fourth binomial] 856\label{prop:X.63} 857The square on the major straight line applied to a rational straight 858line produces as breadth the fourth binomial. 859\end{claim} 860\begin{evidence}[Proof of X.63] 861\label{ev:X.63} 862Inverse of X.57. 863\dependson{X.63}{X.51} 864\dependson{X.63}{X.57} 865\dependson{X.63}{X.62} 866\end{evidence} 867 868\begin{claim}[Proposition X.64: Square on side-of-rational-plus-medial yields a fifth binomial] 869\label{prop:X.64} 870The square on the side of a rational plus a medial area applied to 871a rational straight line produces as breadth the fifth binomial. 872\end{claim} 873\begin{evidence}[Proof of X.64] 874\label{ev:X.64} 875Inverse of X.58. 876\dependson{X.64}{X.52} 877\dependson{X.64}{X.58} 878\dependson{X.64}{X.63} 879\end{evidence} 880 881\begin{claim}[Proposition X.65: Square on side-of-two-medials yields a sixth binomial] 882\label{prop:X.65} 883The square on the side of the sum of two medial areas applied to a 884rational straight line produces as breadth the sixth binomial. 885\end{claim} 886\begin{evidence}[Proof of X.65] 887\label{ev:X.65} 888Inverse of X.59. 889\dependson{X.65}{X.53} 890\dependson{X.65}{X.59} 891\dependson{X.65}{X.64} 892\end{evidence} 893 894\begin{claim}[Proposition X.66: Commensurable with binomial is binomial] 895\label{prop:X.66} 896A straight line commensurable in length with a binomial straight 897line is itself also binomial and the same in order. 898\end{claim} 899\begin{evidence}[Proof of X.66] 900\label{ev:X.66} 901Multiplication by a rational ratio preserves the binomial type and 902order. 903\dependson{X.66}{X.36} 904\dependson{X.66}{X.48} 905\end{evidence} 906 907\begin{claim}[Proposition X.67: Commensurable with bimedial is bimedial] 908\label{prop:X.67} 909A straight line commensurable in length with a bimedial straight 910line is itself bimedial and the same in order. 911\end{claim} 912\begin{evidence}[Proof of X.67] 913\label{ev:X.67} 914Same scheme as X.66. 915\dependson{X.67}{X.37} 916\dependson{X.67}{X.38} 917\dependson{X.67}{X.66} 918\end{evidence} 919 920\begin{claim}[Proposition X.68: Commensurable with major is major] 921\label{prop:X.68} 922A straight line commensurable with a major straight line is itself 923major. 924\end{claim} 925\begin{evidence}[Proof of X.68] 926\label{ev:X.68} 927Same scheme. 928\dependson{X.68}{X.39} 929\dependson{X.68}{X.67} 930\end{evidence} 931 932\begin{claim}[Proposition X.69: Commensurable with side of rational+medial is the same] 933\label{prop:X.69} 934A straight line commensurable with the side of a rational plus a 935medial area is itself such a side. 936\end{claim} 937\begin{evidence}[Proof of X.69] 938\label{ev:X.69} 939Same scheme. 940\dependson{X.69}{X.40} 941\dependson{X.69}{X.68} 942\end{evidence} 943 944\begin{claim}[Proposition X.70: Commensurable with side of two-medials is the same] 945\label{prop:X.70} 946A straight line commensurable with the side of the sum of two medial 947areas is itself such a side. 948\end{claim} 949\begin{evidence}[Proof of X.70] 950\label{ev:X.70} 951Same scheme. 952\dependson{X.70}{X.41} 953\dependson{X.70}{X.69} 954\end{evidence} 955 956\begin{claim}[Proposition X.71: Rational + medial sum is one of the four irrationals] 957\label{prop:X.71} 958If a rational and a medial area be added together, four irrational 959straight lines arise, namely either a binomial, a first bimedial, a 960major, or a side of a rational plus a medial area. 961\end{claim} 962\begin{evidence}[Proof of X.71] 963\label{ev:X.71} 964The square of any of the four classes (X.36, X.37, X.39, X.40) is 965the sum of a rational and a medial; conversely, every such sum 966arises in exactly one of these forms. 967\dependson{X.71}{X.36} 968\dependson{X.71}{X.37} 969\dependson{X.71}{X.39} 970\dependson{X.71}{X.40} 971\end{evidence} 972 973\begin{claim}[Proposition X.72: Medial + medial sum yields a bimedial or a side of two medials] 974\label{prop:X.72} 975If two medial areas incommensurable with one another be added 976together, the remaining two irrational straight lines arise, namely 977either a second bimedial or a side of the sum of two medial areas. 978\end{claim} 979\begin{evidence}[Proof of X.72] 980\label{ev:X.72} 981The square of X.38 or X.41 is a sum of two incommensurable medial 982areas; converse runs the same way. 983\dependson{X.72}{X.38} 984\dependson{X.72}{X.41} 985\dependson{X.72}{X.71} 986\end{evidence} 987 988\begin{claim}[Proposition X.73: Apotome straight line] 989\label{prop:X.73} 990If from a rational straight line there be subtracted a rational 991straight line commensurable with the whole in square only, the 992remainder is irrational; and let it be called apotome. 993\end{claim} 994\begin{evidence}[Proof of X.73] 995\label{ev:X.73} 996$a - b$ with $a$, $b$ commensurable in square only is the negation 997of the binomial case (X.36); the same argument shows it is 998irrational. 999\dependson{X.73}{X.36} 1000\end{evidence} 1001 1002\begin{claim}[Proposition X.74: First apotome of a medial] 1003\label{prop:X.74} 1004If from a medial straight line there be subtracted a medial straight 1005line commensurable with the whole in square only, and containing 1006with the whole a rational rectangle, the remainder is irrational; 1007and let it be called first apotome of a medial. 1008\end{claim} 1009\begin{evidence}[Proof of X.74] 1010\label{ev:X.74} 1011Negation of X.37. 1012\dependson{X.74}{X.37} 1013\dependson{X.74}{X.73} 1014\end{evidence} 1015 1016\begin{claim}[Proposition X.75: Second apotome of a medial] 1017\label{prop:X.75} 1018If from a medial straight line there be subtracted a medial straight 1019line commensurable with the whole in square only, and containing 1020with the whole a medial rectangle, the remainder is irrational; and 1021let it be called second apotome of a medial. 1022\end{claim} 1023\begin{evidence}[Proof of X.75] 1024\label{ev:X.75} 1025Negation of X.38. 1026\dependson{X.75}{X.38} 1027\dependson{X.75}{X.74} 1028\end{evidence} 1029 1030\begin{claim}[Proposition X.76: Minor straight line] 1031\label{prop:X.76} 1032If from a straight line there be subtracted a straight line 1033incommensurable in square with the whole, which with the whole makes 1034the squares on them added together rational, but the rectangle 1035contained by them medial, the remainder is irrational; and let it 1036be called minor. 1037\end{claim} 1038\begin{evidence}[Proof of X.76] 1039\label{ev:X.76} 1040Negation of X.39. This is the "minor" line (Definition XIII.4). 1041\dependson{X.76}{X.39} 1042\dependson{X.76}{def:XIII.4} 1043\end{evidence} 1044 1045\begin{claim}[Proposition X.77: Line producing with rational area a medial whole] 1046\label{prop:X.77} 1047If from a straight line there be subtracted a straight line 1048incommensurable in square with the whole which with the whole makes 1049the sum of squares medial but twice the rectangle rational, the 1050remainder is irrational; let it be called that which produces with 1051a rational area a medial whole. 1052\end{claim} 1053\begin{evidence}[Proof of X.77] 1054\label{ev:X.77} 1055Negation of X.40. 1056\dependson{X.77}{X.40} 1057\dependson{X.77}{X.76} 1058\dependson{X.77}{def:XIII.5} 1059\end{evidence} 1060 1061\begin{claim}[Proposition X.78: Line producing with medial area a medial whole] 1062\label{prop:X.78} 1063If from a straight line there be subtracted a straight line 1064incommensurable in square with the whole which with the whole makes 1065both the sum of squares and twice the rectangle medial and the two 1066sums incommensurable with one another, the remainder is irrational; 1067let it be called that which produces with a medial area a medial 1068whole. 1069\end{claim} 1070\begin{evidence}[Proof of X.78] 1071\label{ev:X.78} 1072Negation of X.41. 1073\dependson{X.78}{X.41} 1074\dependson{X.78}{X.77} 1075\end{evidence} 1076 1077\begin{claim}[Proposition X.79: Apotome has unique annex] 1078\label{prop:X.79} 1079Only one rational straight line can be annexed to an apotome which 1080is commensurable with the whole in square only. 1081\end{claim} 1082\begin{evidence}[Proof of X.79] 1083\label{ev:X.79} 1084Uniqueness analogue of X.42 for apotomes. 1085\dependson{X.79}{X.42} 1086\dependson{X.79}{X.73} 1087\end{evidence} 1088 1089\begin{claim}[Proposition X.80: First-apotome-of-medial uniqueness] 1090\label{prop:X.80} 1091Only one medial straight line can be annexed to a first apotome of 1092a medial which is commensurable with the whole in square only and 1093forms with it a rational rectangle. 1094\end{claim} 1095\begin{evidence}[Proof of X.80] 1096\label{ev:X.80} 1097Same uniqueness pattern. 1098\dependson{X.80}{X.43} 1099\dependson{X.80}{X.74} 1100\end{evidence} 1101 1102\begin{claim}[Proposition X.81: Second-apotome-of-medial uniqueness] 1103\label{prop:X.81} 1104Only one medial straight line can be annexed to a second apotome of 1105a medial which is commensurable with the whole in square only and 1106forms with it a medial rectangle. 1107\end{claim} 1108\begin{evidence}[Proof of X.81] 1109\label{ev:X.81} 1110Same uniqueness pattern. 1111\dependson{X.81}{X.44} 1112\dependson{X.81}{X.75} 1113\end{evidence} 1114 1115\begin{claim}[Proposition X.82: Minor uniqueness] 1116\label{prop:X.82} 1117Only one straight line can be annexed to a minor. 1118\end{claim} 1119\begin{evidence}[Proof of X.82] 1120\label{ev:X.82} 1121Same uniqueness pattern. 1122\dependson{X.82}{X.45} 1123\dependson{X.82}{X.76} 1124\end{evidence} 1125 1126\begin{claim}[Proposition X.83: Uniqueness for X.77's line] 1127\label{prop:X.83} 1128Only one straight line can be annexed to the line producing with a 1129rational area a medial whole. 1130\end{claim} 1131\begin{evidence}[Proof of X.83] 1132\label{ev:X.83} 1133Same uniqueness pattern. 1134\dependson{X.83}{X.46} 1135\dependson{X.83}{X.77} 1136\end{evidence} 1137 1138\begin{claim}[Proposition X.84: Uniqueness for X.78's line] 1139\label{prop:X.84} 1140Only one straight line can be annexed to the line producing with a 1141medial area a medial whole. 1142\end{claim} 1143\begin{evidence}[Proof of X.84] 1144\label{ev:X.84} 1145Same uniqueness pattern. 1146\dependson{X.84}{X.47} 1147\dependson{X.84}{X.78} 1148\end{evidence} 1149 1150\begin{claim}[Proposition X.85: First apotome] 1151\label{prop:X.85} 1152To find the first apotome. 1153\end{claim} 1154\begin{evidence}[Proof of X.85] 1155\label{ev:X.85} 1156Take a first binomial $a + b$ (X.48); the difference $a - b$ is the 1157first apotome. 1158\dependson{X.85}{X.48} 1159\dependson{X.85}{X.73} 1160\dependson{X.85}{def:X.III.1} 1161\end{evidence} 1162 1163\begin{claim}[Proposition X.86: Second apotome] 1164\label{prop:X.86} 1165To find the second apotome. 1166\end{claim} 1167\begin{evidence}[Proof of X.86] 1168\label{ev:X.86} 1169Use the second binomial as the model (X.49). 1170\dependson{X.86}{X.49} 1171\dependson{X.86}{X.85} 1172\dependson{X.86}{def:X.III.2} 1173\end{evidence} 1174 1175\begin{claim}[Proposition X.87: Third apotome] 1176\label{prop:X.87} 1177To find the third apotome. 1178\end{claim} 1179\begin{evidence}[Proof of X.87] 1180\label{ev:X.87} 1181Use the third binomial (X.50). 1182\dependson{X.87}{X.50} 1183\dependson{X.87}{X.86} 1184\dependson{X.87}{def:X.III.3} 1185\end{evidence} 1186 1187\begin{claim}[Proposition X.88: Fourth apotome] 1188\label{prop:X.88} 1189To find the fourth apotome. 1190\end{claim} 1191\begin{evidence}[Proof of X.88] 1192\label{ev:X.88} 1193Use the fourth binomial (X.51). 1194\dependson{X.88}{X.51} 1195\dependson{X.88}{X.87} 1196\dependson{X.88}{def:X.III.4} 1197\end{evidence} 1198 1199\begin{claim}[Proposition X.89: Fifth apotome] 1200\label{prop:X.89} 1201To find the fifth apotome. 1202\end{claim} 1203\begin{evidence}[Proof of X.89] 1204\label{ev:X.89} 1205Use the fifth binomial (X.52). 1206\dependson{X.89}{X.52} 1207\dependson{X.89}{X.88} 1208\dependson{X.89}{def:X.III.5} 1209\end{evidence} 1210 1211\begin{claim}[Proposition X.90: Sixth apotome] 1212\label{prop:X.90} 1213To find the sixth apotome. 1214\end{claim} 1215\begin{evidence}[Proof of X.90] 1216\label{ev:X.90} 1217Use the sixth binomial (X.53). 1218\dependson{X.90}{X.53} 1219\dependson{X.90}{X.89} 1220\dependson{X.90}{def:X.III.6} 1221\end{evidence} 1222 1223\begin{claim}[Proposition X.91: Side of first-apotome area is an apotome] 1224\label{prop:X.91} 1225If an area be contained by a rational straight line and a first 1226apotome, the side of the area is an apotome. 1227\end{claim} 1228\begin{evidence}[Proof of X.91] 1229\label{ev:X.91} 1230Dual of X.54 for apotomes. 1231\dependson{X.91}{X.54} 1232\dependson{X.91}{X.85} 1233\end{evidence} 1234 1235\begin{claim}[Proposition X.92: Side of second-apotome area is a first apotome of medial] 1236\label{prop:X.92} 1237If an area be contained by a rational straight line and a second 1238apotome, the side of the area is a first apotome of a medial. 1239\end{claim} 1240\begin{evidence}[Proof of X.92] 1241\label{ev:X.92} 1242Dual of X.55. 1243\dependson{X.92}{X.55} 1244\dependson{X.92}{X.86} 1245\dependson{X.92}{X.91} 1246\end{evidence} 1247 1248\begin{claim}[Proposition X.93: Side of third-apotome area is a second apotome of medial] 1249\label{prop:X.93} 1250If an area be contained by a rational straight line and a third 1251apotome, the side of the area is a second apotome of a medial. 1252\end{claim} 1253\begin{evidence}[Proof of X.93] 1254\label{ev:X.93} 1255Dual of X.56. 1256\dependson{X.93}{X.56} 1257\dependson{X.93}{X.87} 1258\dependson{X.93}{X.92} 1259\end{evidence} 1260 1261\begin{claim}[Proposition X.94: Side of fourth-apotome area is a minor] 1262\label{prop:X.94} 1263If an area be contained by a rational straight line and a fourth 1264apotome, the side of the area is a minor. 1265\end{claim} 1266\begin{evidence}[Proof of X.94] 1267\label{ev:X.94} 1268Dual of X.57. 1269\dependson{X.94}{X.57} 1270\dependson{X.94}{X.88} 1271\dependson{X.94}{X.93} 1272\end{evidence} 1273 1274\begin{claim}[Proposition X.95: Side of fifth-apotome area produces rational-plus-medial complement] 1275\label{prop:X.95} 1276If an area be contained by a rational straight line and a fifth 1277apotome, the side of the area is the line producing with a rational 1278area a medial whole. 1279\end{claim} 1280\begin{evidence}[Proof of X.95] 1281\label{ev:X.95} 1282Dual of X.58. 1283\dependson{X.95}{X.58} 1284\dependson{X.95}{X.89} 1285\dependson{X.95}{X.94} 1286\end{evidence} 1287 1288\begin{claim}[Proposition X.96: Side of sixth-apotome area produces medial-plus-medial complement] 1289\label{prop:X.96} 1290If an area be contained by a rational straight line and a sixth 1291apotome, the side of the area is the line producing with a medial 1292area a medial whole. 1293\end{claim} 1294\begin{evidence}[Proof of X.96] 1295\label{ev:X.96} 1296Dual of X.59. 1297\dependson{X.96}{X.59} 1298\dependson{X.96}{X.90} 1299\dependson{X.96}{X.95} 1300\end{evidence} 1301 1302\begin{claim}[Proposition X.97: Square on apotome yields first apotome] 1303\label{prop:X.97} 1304The square on an apotome straight line applied to a rational 1305straight line produces as breadth a first apotome. 1306\end{claim} 1307\begin{evidence}[Proof of X.97] 1308\label{ev:X.97} 1309Inverse of X.91. 1310\dependson{X.97}{X.85} 1311\dependson{X.97}{X.91} 1312\end{evidence} 1313 1314\begin{claim}[Proposition X.98: Square on first apotome of medial yields second apotome] 1315\label{prop:X.98} 1316The square on a first apotome of a medial straight line applied to 1317a rational straight line produces as breadth a second apotome. 1318\end{claim} 1319\begin{evidence}[Proof of X.98] 1320\label{ev:X.98} 1321Inverse of X.92. 1322\dependson{X.98}{X.86} 1323\dependson{X.98}{X.92} 1324\dependson{X.98}{X.97} 1325\end{evidence} 1326 1327\begin{claim}[Proposition X.99: Square on second apotome of medial yields third apotome] 1328\label{prop:X.99} 1329The square on a second apotome of a medial straight line applied to 1330a rational straight line produces as breadth a third apotome. 1331\end{claim} 1332\begin{evidence}[Proof of X.99] 1333\label{ev:X.99} 1334Inverse of X.93. 1335\dependson{X.99}{X.87} 1336\dependson{X.99}{X.93} 1337\dependson{X.99}{X.98} 1338\end{evidence} 1339 1340\begin{claim}[Proposition X.100: Square on minor yields fourth apotome] 1341\label{prop:X.100} 1342The square on a minor applied to a rational straight line produces 1343as breadth a fourth apotome. 1344\end{claim} 1345\begin{evidence}[Proof of X.100] 1346\label{ev:X.100} 1347Inverse of X.94. 1348\dependson{X.100}{X.88} 1349\dependson{X.100}{X.94} 1350\dependson{X.100}{X.99} 1351\end{evidence} 1352 1353\begin{claim}[Proposition X.101: Square on rational-plus-medial producer yields fifth apotome] 1354\label{prop:X.101} 1355The square on the line producing with a rational area a medial whole 1356applied to a rational straight line produces as breadth a fifth 1357apotome. 1358\end{claim} 1359\begin{evidence}[Proof of X.101] 1360\label{ev:X.101} 1361Inverse of X.95. 1362\dependson{X.101}{X.89} 1363\dependson{X.101}{X.95} 1364\dependson{X.101}{X.100} 1365\end{evidence} 1366 1367\begin{claim}[Proposition X.102: Square on medial-plus-medial producer yields sixth apotome] 1368\label{prop:X.102} 1369The square on the line producing with a medial area a medial whole 1370applied to a rational straight line produces as breadth a sixth 1371apotome. 1372\end{claim} 1373\begin{evidence}[Proof of X.102] 1374\label{ev:X.102} 1375Inverse of X.96. 1376\dependson{X.102}{X.90} 1377\dependson{X.102}{X.96} 1378\dependson{X.102}{X.101} 1379\end{evidence} 1380 1381\begin{claim}[Proposition X.103: Commensurable with apotome is apotome] 1382\label{prop:X.103} 1383A straight line commensurable in length with an apotome is itself an 1384apotome and the same in order. 1385\end{claim} 1386\begin{evidence}[Proof of X.103] 1387\label{ev:X.103} 1388Dual of X.66. 1389\dependson{X.103}{X.66} 1390\dependson{X.103}{X.73} 1391\end{evidence} 1392 1393\begin{claim}[Proposition X.104: Commensurable with apotome of medial is the same] 1394\label{prop:X.104} 1395A straight line commensurable in length with an apotome of a medial 1396is itself such an apotome of the same order. 1397\end{claim} 1398\begin{evidence}[Proof of X.104] 1399\label{ev:X.104} 1400Dual of X.67. 1401\dependson{X.104}{X.67} 1402\dependson{X.104}{X.74} 1403\dependson{X.104}{X.103} 1404\end{evidence} 1405 1406\begin{claim}[Proposition X.105: Commensurable with minor is minor] 1407\label{prop:X.105} 1408A straight line commensurable with a minor is itself a minor. 1409\end{claim} 1410\begin{evidence}[Proof of X.105] 1411\label{ev:X.105} 1412Dual of X.68. 1413\dependson{X.105}{X.68} 1414\dependson{X.105}{X.76} 1415\dependson{X.105}{X.104} 1416\end{evidence} 1417 1418\begin{claim}[Proposition X.106: Commensurable with rational-medial producer is the same] 1419\label{prop:X.106} 1420A straight line commensurable with the line producing with a rational 1421area a medial whole is itself such a line. 1422\end{claim} 1423\begin{evidence}[Proof of X.106] 1424\label{ev:X.106} 1425Dual of X.69. 1426\dependson{X.106}{X.69} 1427\dependson{X.106}{X.77} 1428\dependson{X.106}{X.105} 1429\end{evidence} 1430 1431\begin{claim}[Proposition X.107: Commensurable with medial-medial producer is the same] 1432\label{prop:X.107} 1433A straight line commensurable with the line producing with a medial 1434area a medial whole is itself such a line. 1435\end{claim} 1436\begin{evidence}[Proof of X.107] 1437\label{ev:X.107} 1438Dual of X.70. 1439\dependson{X.107}{X.70} 1440\dependson{X.107}{X.78} 1441\dependson{X.107}{X.106} 1442\end{evidence} 1443 1444\begin{claim}[Proposition X.108: Side of rational minus medial is one of the four irrationals] 1445\label{prop:X.108} 1446If from a rational area a medial area be subtracted, the side of the 1447remaining area arises as one of four irrationals: an apotome, a 1448first apotome of a medial, a minor, or the line producing with a 1449rational area a medial whole. 1450\end{claim} 1451\begin{evidence}[Proof of X.108] 1452\label{ev:X.108} 1453Dual of X.71. 1454\dependson{X.108}{X.71} 1455\dependson{X.108}{X.73} 1456\dependson{X.108}{X.74} 1457\dependson{X.108}{X.76} 1458\dependson{X.108}{X.77} 1459\end{evidence} 1460 1461\begin{claim}[Proposition X.109: Medial minus rational yields apotome or producer-of-medial] 1462\label{prop:X.109} 1463If from a medial area a rational area be subtracted, two other 1464irrational straight lines arise, namely a first apotome of a medial 1465or the line producing with a rational area a medial whole. 1466\end{claim} 1467\begin{evidence}[Proof of X.109] 1468\label{ev:X.109} 1469Variant of X.108. 1470\dependson{X.109}{X.74} 1471\dependson{X.109}{X.77} 1472\dependson{X.109}{X.108} 1473\end{evidence} 1474 1475\begin{claim}[Proposition X.110: Medial minus medial yields second apotome of medial or medial-producer] 1476\label{prop:X.110} 1477If from a medial area there be subtracted a medial area 1478incommensurable with the whole, the remaining two irrational straight 1479lines arise: a second apotome of a medial or the line producing with 1480a medial area a medial whole. 1481\end{claim} 1482\begin{evidence}[Proof of X.110] 1483\label{ev:X.110} 1484Variant of X.108 / X.109. 1485\dependson{X.110}{X.75} 1486\dependson{X.110}{X.78} 1487\dependson{X.110}{X.109} 1488\end{evidence} 1489 1490\begin{claim}[Proposition X.111: Apotome and binomial are distinct] 1491\label{prop:X.111} 1492The apotome is not the same as the binomial. 1493\end{claim} 1494\begin{evidence}[Proof of X.111] 1495\label{ev:X.111} 1496A binomial has a rational sum of squares plus a medial rectangle; an 1497apotome has a rational difference of squares minus a medial 1498rectangle; if they coincided, the two combinations would coincide, 1499forcing the medial part to be rational --- contradiction. 1500\dependson{X.111}{X.36} 1501\dependson{X.111}{X.73} 1502\end{evidence} 1503 1504\begin{claim}[Proposition X.112: Square on rational divided by binomial is an apotome] 1505\label{prop:X.112} 1506The square on a rational straight line applied to the binomial 1507straight line produces as breadth an apotome the terms of which are 1508commensurable with the terms of the binomial and in the same ratio. 1509\end{claim} 1510\begin{evidence}[Proof of X.112] 1511\label{ev:X.112} 1512$R^2 / (a + b) = a' - b'$ with $a'$, $b'$ in the same ratio as $a$, 1513$b$. Verified by direct manipulation of the square-on-binomial 1514identity. 1515\dependson{X.112}{X.91} 1516\dependson{X.112}{X.97} 1517\end{evidence} 1518 1519\begin{claim}[Proposition X.113: Square on rational divided by apotome is a binomial] 1520\label{prop:X.113} 1521The square on a rational straight line applied to an apotome produces 1522as breadth a binomial the terms of which are commensurable with the 1523terms of the apotome and in the same ratio. 1524\end{claim} 1525\begin{evidence}[Proof of X.113] 1526\label{ev:X.113} 1527Inverse of X.112. 1528\dependson{X.113}{X.54} 1529\dependson{X.113}{X.60} 1530\dependson{X.113}{X.112} 1531\end{evidence} 1532 1533\begin{claim}[Proposition X.114: Rectangle on binomial and apotome can be rational] 1534\label{prop:X.114} 1535If an area be contained by an apotome and the binomial the terms of 1536which are commensurable with the terms of the apotome and in the 1537same ratio, the side of the area is rational. 1538\end{claim} 1539\begin{evidence}[Proof of X.114] 1540\label{ev:X.114} 1541The rectangle on $(a - b)$ and $(a' + b')$ with $a' = ka$, $b' = kb$ 1542equals $k(a^2 - b^2)$, which is rational. 1543\dependson{X.114}{X.112} 1544\dependson{X.114}{X.113} 1545\end{evidence} 1546 1547\begin{claim}[Proposition X.115: Medials yield infinitely many irrationals] 1548\label{prop:X.115} 1549From a medial straight line there arise irrational straight lines 1550infinite in number, and none of them is the same with any preceding. 1551\end{claim} 1552\begin{evidence}[Proof of X.115] 1553\label{ev:X.115} 1554By repeated mean-proportional construction (VI.13) on the medial, 1555each new line is irrational with respect to all earlier ones (using 1556the unique-decomposition results X.42--X.47, X.79--X.84). 1557\dependson{X.115}{VI.13} 1558\dependson{X.115}{X.21} 1559\dependson{X.115}{X.114} 1560\end{evidence} 1561

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

Figure