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1% definitions.tex --- Book I definitions (the 23). Each is a `scope`
2% block: definitions delimit terms used by later claims and edges, but
3% they're not themselves assertions about the world.
4
5\section*{Book I: Definitions}
6\label{sec:def-bookI}
7
8\begin{scope}[Definition I.1: point]
9\label{def:I.1}
10A point is that which has no part.
11\end{scope}
12
13\begin{scope}[Definition I.2: line]
14\label{def:I.2}
15A line is breadthless length.
16\end{scope}
17
18\begin{scope}[Definition I.3: extremities of a line]
19\label{def:I.3}
20The extremities of a line are points.
21\end{scope}
22
23\begin{scope}[Definition I.4: straight line]
24\label{def:I.4}
25A straight line is a line which lies evenly with the points on itself.
26\end{scope}
27
28\begin{scope}[Definition I.5: surface]
29\label{def:I.5}
30A surface is that which has length and breadth only.
31\end{scope}
32
33\begin{scope}[Definition I.6: extremities of a surface]
34\label{def:I.6}
35The extremities of a surface are lines.
36\end{scope}
37
38\begin{scope}[Definition I.7: plane surface]
39\label{def:I.7}
40A plane surface is a surface which lies evenly with the straight lines
41on itself.
42\end{scope}
43
44\begin{scope}[Definition I.8: plane angle]
45\label{def:I.8}
46A plane angle is the inclination to one another of two lines in a
47plane which meet one another and do not lie in a straight line.
48\end{scope}
49
50\begin{scope}[Definition I.9: rectilineal angle]
51\label{def:I.9}
52And when the lines containing the angle are straight, the angle is
53called rectilineal.
54\end{scope}
55
56\begin{scope}[Definition I.10: right angle]
57\label{def:I.10}
58When a straight line set up on a straight line makes the adjacent
59angles equal to one another, each of the equal angles is right, and
60the straight line standing on the other is called a perpendicular to
61that on which it stands.
62\end{scope}
63
64\begin{scope}[Definition I.11: obtuse angle]
65\label{def:I.11}
66An obtuse angle is an angle greater than a right angle.
67\end{scope}
68
69\begin{scope}[Definition I.12: acute angle]
70\label{def:I.12}
71An acute angle is an angle less than a right angle.
72\end{scope}
73
74\begin{scope}[Definition I.13: boundary]
75\label{def:I.13}
76A boundary is that which is an extremity of anything.
77\end{scope}
78
79\begin{scope}[Definition I.14: figure]
80\label{def:I.14}
81A figure is that which is contained by any boundary or boundaries.
82\end{scope}
83
84\begin{scope}[Definition I.15: circle]
85\label{def:I.15}
86A circle is a plane figure contained by one line such that all the
87straight lines falling upon it from one point among those lying within
88the figure are equal to one another.
89\end{scope}
90
91\begin{scope}[Definition I.16: centre]
92\label{def:I.16}
93And the point is called the centre of the circle.
94\end{scope}
95
96\begin{scope}[Definition I.17: diameter]
97\label{def:I.17}
98A diameter of the circle is any straight line drawn through the centre
99and terminated in both directions by the circumference of the circle,
100and such a straight line also bisects the circle.
101\end{scope}
102
103\begin{scope}[Definition I.18: semicircle]
104\label{def:I.18}
105A semicircle is the figure contained by the diameter and the
106circumference cut off by it. And the centre of the semicircle is the
107same as that of the circle.
108\end{scope}
109
110\begin{scope}[Definition I.19: rectilineal figures]
111\label{def:I.19}
112Rectilineal figures are those which are contained by straight lines,
113trilateral figures being those contained by three, quadrilateral those
114contained by four, and multilateral those contained by more than four
115straight lines.
116\end{scope}
117
118\begin{scope}[Definition I.20: kinds of trilateral]
119\label{def:I.20}
120Of trilateral figures, an equilateral triangle is that which has its
121three sides equal, an isosceles triangle that which has two of its
122sides alone equal, and a scalene triangle that which has its three
123sides unequal.
124\end{scope}
125
126\begin{scope}[Definition I.21: kinds of trilateral by angle]
127\label{def:I.21}
128Further, of trilateral figures, a right-angled triangle is that which
129has a right angle, an obtuse-angled triangle that which has an obtuse
130angle, and an acute-angled triangle that which has its three angles
131acute.
132\end{scope}
133
134\begin{scope}[Definition I.22: kinds of quadrilateral]
135\label{def:I.22}
136Of quadrilateral figures, a square is that which is both equilateral
137and right-angled; an oblong that which is right-angled but not
138equilateral; a rhombus that which is equilateral but not right-angled;
139and a rhomboid that which has its opposite sides and angles equal to
140one another but is neither equilateral nor right-angled. And let
141quadrilaterals other than these be called trapezia.
142\end{scope}
143
144\begin{scope}[Definition I.23: parallel straight lines]
145\label{def:I.23}
146Parallel straight lines are straight lines which, being in the same
147plane and being produced indefinitely in both directions, do not meet
148one another in either direction.
149\end{scope}
150
151% ===== Book II definitions =====
152
153\begin{scope}[Definition II.1: rectangle]
154\label{def:II.1}
155Any rectangular parallelogram is said to be contained by the two
156straight lines containing the right angle.
157\end{scope}
158
159\begin{scope}[Definition II.2: gnomon]
160\label{def:II.2}
161And in any parallelogrammic area let any one whatever of the
162parallelograms about its diameter, with the two complements, be
163called a gnomon.
164\end{scope}
165
166% ===== Book III definitions =====
167
168\begin{scope}[Definition III.1: equal circles]
169\label{def:III.1}
170Equal circles are those whose diameters are equal, or whose radii are
171equal.
172\end{scope}
173
174\begin{scope}[Definition III.2: tangent line]
175\label{def:III.2}
176A straight line is said to touch a circle which, meeting the circle
177and being produced, does not cut the circle.
178\end{scope}
179
180\begin{scope}[Definition III.3: tangent circles]
181\label{def:III.3}
182Circles are said to touch one another which, meeting one another, do
183not cut one another.
184\end{scope}
185
186\begin{scope}[Definition III.4: equidistant chords]
187\label{def:III.4}
188In a circle, straight lines are said to be equally distant from the
189centre when the perpendiculars drawn to them from the centre are
190equal.
191\end{scope}
192
193\begin{scope}[Definition III.5: more distant chord]
194\label{def:III.5}
195And that straight line is said to be at a greater distance on which
196the greater perpendicular falls.
197\end{scope}
198
199\begin{scope}[Definition III.6: segment of a circle]
200\label{def:III.6}
201A segment of a circle is the figure contained by a straight line and
202a circumference of a circle.
203\end{scope}
204
205\begin{scope}[Definition III.7: angle of a segment]
206\label{def:III.7}
207An angle of a segment is that contained by a straight line and a
208circumference of a circle.
209\end{scope}
210
211\begin{scope}[Definition III.8: angle in a segment]
212\label{def:III.8}
213An angle in a segment is the angle which, when a point is taken on
214the circumference of the segment and straight lines are joined from
215it to the extremities of the straight line which is the base of the
216segment, is contained by the straight lines so joined.
217\end{scope}
218
219\begin{scope}[Definition III.9: standing on an arc]
220\label{def:III.9}
221And, when the straight lines containing the angle cut off an arc, the
222angle is said to stand upon that arc.
223\end{scope}
224
225\begin{scope}[Definition III.10: sector]
226\label{def:III.10}
227A sector of a circle is the figure which, when an angle is constructed
228at the centre of the circle, is contained by the straight lines
229containing the angle and the arc cut off by them.
230\end{scope}
231
232\begin{scope}[Definition III.11: similar segments]
233\label{def:III.11}
234Similar segments of circles are those which admit equal angles, or in
235which the angles are equal to one another.
236\end{scope}
237
238% ===== Book V definitions (Eudoxean proportion) =====
239
240\begin{scope}[Definition V.1: part]
241\label{def:V.1}
242A magnitude is a part of a magnitude, the less of the greater, when
243it measures the greater.
244\end{scope}
245
246\begin{scope}[Definition V.2: multiple]
247\label{def:V.2}
248The greater is a multiple of the less when it is measured by the less.
249\end{scope}
250
251\begin{scope}[Definition V.3: ratio]
252\label{def:V.3}
253A ratio is a sort of relation in respect of size between two
254magnitudes of the same kind.
255\end{scope}
256
257\begin{scope}[Definition V.4: having a ratio]
258\label{def:V.4}
259Magnitudes are said to have a ratio to one another which are capable,
260when multiplied, of exceeding one another (the Archimedean property).
261\end{scope}
262
263\begin{scope}[Definition V.5: same ratio (Eudoxean equality of ratios)]
264\label{def:V.5}
265Magnitudes are said to be in the same ratio, the first to the second
266and the third to the fourth, when, if any equimultiples whatever be
267taken of the first and third, and any equimultiples whatever of the
268second and fourth, the former equimultiples alike exceed, are alike
269equal to, or alike fall short of, the latter equimultiples respectively
270taken in corresponding order.
271\end{scope}
272
273\begin{scope}[Definition V.6: proportional]
274\label{def:V.6}
275Let magnitudes which have the same ratio be called proportional.
276\end{scope}
277
278\begin{scope}[Definition V.7: greater ratio]
279\label{def:V.7}
280When, of the equimultiples, the multiple of the first magnitude
281exceeds the multiple of the second, but the multiple of the third
282does not exceed the multiple of the fourth, then the first is said
283to have a greater ratio to the second than the third has to the fourth.
284\end{scope}
285
286\begin{scope}[Definition V.8: proportion (three terms)]
287\label{def:V.8}
288A proportion in three terms is the least possible.
289\end{scope}
290
291\begin{scope}[Definition V.9: duplicate ratio]
292\label{def:V.9}
293When three magnitudes are proportional, the first is said to have to
294the third the duplicate ratio of that which it has to the second.
295\end{scope}
296
297\begin{scope}[Definition V.10: triplicate ratio]
298\label{def:V.10}
299When four magnitudes are continuously proportional, the first is said
300to have to the fourth the triplicate ratio of that which it has to
301the second, and so on, in continual proportion of any number of
302magnitudes.
303\end{scope}
304
305\begin{scope}[Definition V.11: corresponding magnitudes]
306\label{def:V.11}
307Antecedents are said to correspond to antecedents, and consequents to
308consequents.
309\end{scope}
310
311\begin{scope}[Definition V.12: alternate ratio]
312\label{def:V.12}
313Alternate ratio means taking the antecedent in relation to the
314antecedent and the consequent in relation to the consequent.
315\end{scope}
316
317\begin{scope}[Definition V.13: inverse ratio]
318\label{def:V.13}
319Inverse ratio means taking the consequent as antecedent in relation
320to the antecedent as consequent.
321\end{scope}
322
323\begin{scope}[Definition V.14: composition of a ratio]
324\label{def:V.14}
325Composition of a ratio means taking the antecedent together with the
326consequent as one in relation to the consequent by itself.
327\end{scope}
328
329\begin{scope}[Definition V.15: separation of a ratio]
330\label{def:V.15}
331Separation of a ratio means taking the excess by which the antecedent
332exceeds the consequent in relation to the consequent by itself.
333\end{scope}
334
335\begin{scope}[Definition V.16: conversion of a ratio]
336\label{def:V.16}
337Conversion of a ratio means taking the antecedent in relation to the
338excess by which the antecedent exceeds the consequent.
339\end{scope}
340
341\begin{scope}[Definition V.17: ratio ex aequali]
342\label{def:V.17}
343A ratio ex aequali arises when, there being several magnitudes and
344another set equal to them in multitude which taken two and two are
345in the same proportion, as the first is to the last of the first
346magnitudes, so is the first to the last of the second magnitudes.
347\end{scope}
348
349\begin{scope}[Definition V.18: perturbed proportion]
350\label{def:V.18}
351A perturbed proportion arises when, there being three magnitudes and
352another set equal to them in multitude, as antecedent is to consequent
353among the first magnitudes, so is antecedent to consequent among the
354second magnitudes, while as the consequent is to a third among the
355first magnitudes, so is a third to the antecedent among the second
356magnitudes.
357\end{scope}
358
359% ===== Book VII definitions (Number theory) =====
360
361\begin{scope}[Definition VII.1: unit]
362\label{def:VII.1}
363A unit is that by virtue of which each of the things that exist is
364called one.
365\end{scope}
366
367\begin{scope}[Definition VII.2: number]
368\label{def:VII.2}
369A number is a multitude composed of units.
370\end{scope}
371
372\begin{scope}[Definition VII.3: part of a number]
373\label{def:VII.3}
374A number is a part of a number, the less of the greater, when it
375measures the greater.
376\end{scope}
377
378\begin{scope}[Definition VII.4: parts]
379\label{def:VII.4}
380But parts when it does not measure it.
381\end{scope}
382
383\begin{scope}[Definition VII.5: multiple]
384\label{def:VII.5}
385The greater number is a multiple of the less when it is measured by
386the less.
387\end{scope}
388
389\begin{scope}[Definition VII.6: even number]
390\label{def:VII.6}
391An even number is that which is divisible into two equal parts.
392\end{scope}
393
394\begin{scope}[Definition VII.7: odd number]
395\label{def:VII.7}
396An odd number is that which is not divisible into two equal parts, or
397that which differs by a unit from an even number.
398\end{scope}
399
400\begin{scope}[Definition VII.8: even-times even]
401\label{def:VII.8}
402An even-times even number is that which is measured by an even number
403according to an even number.
404\end{scope}
405
406\begin{scope}[Definition VII.9: even-times odd]
407\label{def:VII.9}
408An even-times odd number is that which is measured by an even number
409according to an odd number.
410\end{scope}
411
412\begin{scope}[Definition VII.10: odd-times odd]
413\label{def:VII.10}
414An odd-times odd number is that which is measured by an odd number
415according to an odd number.
416\end{scope}
417
418\begin{scope}[Definition VII.11: prime number]
419\label{def:VII.11}
420A prime number is that which is measured by a unit alone.
421\end{scope}
422
423\begin{scope}[Definition VII.12: relatively prime]
424\label{def:VII.12}
425Numbers prime to one another are those which are measured by a unit
426alone as a common measure.
427\end{scope}
428
429\begin{scope}[Definition VII.13: composite number]
430\label{def:VII.13}
431A composite number is that which is measured by some number.
432\end{scope}
433
434\begin{scope}[Definition VII.14: numbers composite to one another]
435\label{def:VII.14}
436Numbers composite to one another are those which are measured by some
437number as a common measure.
438\end{scope}
439
440\begin{scope}[Definition VII.15: multiply]
441\label{def:VII.15}
442A number is said to multiply a number when that which is multiplied
443is added to itself as many times as there are units in the other, and
444thus some number is produced.
445\end{scope}
446
447\begin{scope}[Definition VII.16: plane number]
448\label{def:VII.16}
449When two numbers having multiplied one another make some number, the
450number so produced is called plane, and its sides are the numbers
451which have multiplied one another.
452\end{scope}
453
454\begin{scope}[Definition VII.17: solid number]
455\label{def:VII.17}
456When three numbers having multiplied one another make some number,
457the number so produced is solid, and its sides are the numbers which
458have multiplied one another.
459\end{scope}
460
461\begin{scope}[Definition VII.18: square number]
462\label{def:VII.18}
463A square number is equal multiplied by equal, or a number which is
464contained by two equal numbers.
465\end{scope}
466
467\begin{scope}[Definition VII.19: cube number]
468\label{def:VII.19}
469A cube number is equal multiplied by equal and again by equal, or a
470number which is contained by three equal numbers.
471\end{scope}
472
473\begin{scope}[Definition VII.20: proportional numbers]
474\label{def:VII.20}
475Numbers are proportional when the first is the same multiple, or the
476same part, or the same parts, of the second that the third is of the
477fourth.
478\end{scope}
479
480\begin{scope}[Definition VII.21: similar plane and solid numbers]
481\label{def:VII.21}
482Similar plane and solid numbers are those which have their sides
483proportional.
484\end{scope}
485
486\begin{scope}[Definition VII.22: perfect number]
487\label{def:VII.22}
488A perfect number is that which is equal to the sum of its own parts
489(its proper divisors).
490\end{scope}
491
492% ===== Book X definitions (group 1, before X.1) =====
493
494\begin{scope}[Definition X.1: commensurable magnitudes]
495\label{def:X.1}
496Those magnitudes are said to be commensurable which are measured by
497the same measure, and those incommensurable which cannot have any
498common measure.
499\end{scope}
500
501\begin{scope}[Definition X.2: commensurable in square]
502\label{def:X.2}
503Straight lines are commensurable in square when the squares on them
504are measured by the same area, and incommensurable in square when the
505squares on them cannot possibly have any area as a common measure.
506\end{scope}
507
508\begin{scope}[Definition X.3: rational and irrational straight lines]
509\label{def:X.3}
510With these hypotheses, it is proved that there exist straight lines
511infinite in multitude which are commensurable and incommensurable
512respectively, some in length only, and others in square also, with
513an assigned straight line. Let the assigned straight line be called
514rational, and those straight lines which are commensurable with it,
515whether in length and in square or in square only, rational, but
516those which are incommensurable with it irrational.
517\end{scope}
518
519\begin{scope}[Definition X.4: rational and irrational areas]
520\label{def:X.4}
521And let the square on the assigned straight line be called rational
522and those areas which are commensurable with it rational, but those
523which are incommensurable with it irrational, and the straight lines
524which produce them irrational --- that is, in case the areas are
525squares, the sides themselves; in other cases, the straight lines on
526which the rectangles equal to the areas would be applied.
527\end{scope}
528
529% ===== Book X definitions (group 2, before X.48) =====
530
531\begin{scope}[Definition X(2).1: binomial straight line]
532\label{def:X.II.1}
533Given a rational straight line and a binomial, divided into its terms,
534let the square of the greater term be greater than the square of the
535lesser by the square of a straight line commensurable in length with
536the greater. Then if the greater term is commensurable in length with
537the assigned rational straight line, the whole is called a first binomial.
538\end{scope}
539
540\begin{scope}[Definition X(2).2: second binomial]
541\label{def:X.II.2}
542If the lesser term is commensurable in length with the assigned
543rational straight line, the whole is called a second binomial.
544\end{scope}
545
546\begin{scope}[Definition X(2).3: third binomial]
547\label{def:X.II.3}
548If neither term is commensurable in length with the assigned rational
549straight line, the whole is called a third binomial.
550\end{scope}
551
552\begin{scope}[Definition X(2).4: fourth binomial]
553\label{def:X.II.4}
554If the square of the greater term exceeds the square of the lesser by
555the square of a line incommensurable in length with the greater, and
556the greater term is commensurable in length with the assigned
557rational straight line, the whole is called a fourth binomial.
558\end{scope}
559
560\begin{scope}[Definition X(2).5: fifth binomial]
561\label{def:X.II.5}
562If, in the same case, the lesser term is commensurable in length with
563the assigned rational straight line, the whole is called a fifth
564binomial.
565\end{scope}
566
567\begin{scope}[Definition X(2).6: sixth binomial]
568\label{def:X.II.6}
569If neither term is commensurable in length with the assigned rational
570straight line, the whole is called a sixth binomial.
571\end{scope}
572
573% ===== Book X definitions (group 3, before X.85) =====
574
575\begin{scope}[Definition X(3).1: first apotome]
576\label{def:X.III.1}
577Given a rational straight line and an apotome (i.e. a difference of
578two rationals commensurable in square only), if the square of the
579whole is greater than the square of the annex by the square of a
580straight line commensurable in length with the whole, and the whole
581is commensurable in length with the assigned rational straight line,
582the apotome is called a first apotome.
583\end{scope}
584
585\begin{scope}[Definition X(3).2: second apotome]
586\label{def:X.III.2}
587If the annex is commensurable in length with the assigned rational
588straight line, the apotome is called a second apotome.
589\end{scope}
590
591\begin{scope}[Definition X(3).3: third apotome]
592\label{def:X.III.3}
593If neither the whole nor the annex is commensurable in length with
594the assigned rational straight line, the apotome is called a third
595apotome.
596\end{scope}
597
598\begin{scope}[Definition X(3).4: fourth apotome]
599\label{def:X.III.4}
600If the square of the whole exceeds the square of the annex by the
601square of a straight line incommensurable in length with the whole,
602and the whole is commensurable in length with the assigned rational
603straight line, the apotome is called a fourth apotome.
604\end{scope}
605
606\begin{scope}[Definition X(3).5: fifth apotome]
607\label{def:X.III.5}
608If, in the same case, the annex is commensurable in length with the
609assigned rational straight line, the apotome is called a fifth apotome.
610\end{scope}
611
612\begin{scope}[Definition X(3).6: sixth apotome]
613\label{def:X.III.6}
614If neither the whole nor the annex is commensurable in length with
615the assigned rational straight line, the apotome is called a sixth
616apotome.
617\end{scope}
618
619% ===== Book XI definitions (Solid geometry) =====
620
621\begin{scope}[Definition XI.1: solid]
622\label{def:XI.1}
623A solid is that which has length, breadth, and depth.
624\end{scope}
625
626\begin{scope}[Definition XI.2: extremity of a solid]
627\label{def:XI.2}
628An extremity of a solid is a surface.
629\end{scope}
630
631\begin{scope}[Definition XI.3: line at right angles to a plane]
632\label{def:XI.3}
633A straight line is at right angles to a plane when it makes right
634angles with all the straight lines which meet it and are in the plane.
635\end{scope}
636
637\begin{scope}[Definition XI.4: plane at right angles to a plane]
638\label{def:XI.4}
639A plane is at right angles to a plane when the straight lines drawn
640in one of the planes at right angles to the common section of the
641planes are at right angles to the remaining plane.
642\end{scope}
643
644\begin{scope}[Definition XI.5: inclination of a line to a plane]
645\label{def:XI.5}
646The inclination of a straight line to a plane is, assuming a
647perpendicular drawn from the extremity of the straight line which is
648elevated above the plane to the plane and a straight line joined from
649the foot of the perpendicular to the extremity of the straight line
650which is in the plane, the angle contained by the straight line so
651drawn and the straight line standing up.
652\end{scope}
653
654\begin{scope}[Definition XI.6: inclination of plane to plane]
655\label{def:XI.6}
656The inclination of a plane to a plane is the acute angle contained
657by the straight lines drawn at right angles to the common section at
658the same point, one in each of the planes.
659\end{scope}
660
661\begin{scope}[Definition XI.7: similarly inclined planes]
662\label{def:XI.7}
663A plane is said to be similarly inclined to a plane as another to
664another when the said angles of the inclinations are equal to one
665another.
666\end{scope}
667
668\begin{scope}[Definition XI.8: parallel planes]
669\label{def:XI.8}
670Parallel planes are those which do not meet.
671\end{scope}
672
673\begin{scope}[Definition XI.9: similar solid figures]
674\label{def:XI.9}
675Similar solid figures are those contained by similar planes equal in
676multitude.
677\end{scope}
678
679\begin{scope}[Definition XI.10: equal and similar solid figures]
680\label{def:XI.10}
681Equal and similar solid figures are those contained by similar planes
682equal in multitude and in magnitude.
683\end{scope}
684
685\begin{scope}[Definition XI.11: solid angle]
686\label{def:XI.11}
687A solid angle is the inclination constituted by more than two lines
688which meet one another and are not in the same surface, towards all
689the lines. Otherwise: a solid angle is that which is contained by
690more than two plane angles which are not in the same plane and are
691constructed to one point.
692\end{scope}
693
694\begin{scope}[Definition XI.12: pyramid]
695\label{def:XI.12}
696A pyramid is a solid figure contained by planes which is constructed
697from one plane to one point.
698\end{scope}
699
700\begin{scope}[Definition XI.13: prism]
701\label{def:XI.13}
702A prism is a solid figure contained by planes two of which, namely
703those which are opposite, are equal, similar, and parallel, while the
704rest are parallelograms.
705\end{scope}
706
707\begin{scope}[Definition XI.14: sphere]
708\label{def:XI.14}
709When a semicircle with fixed diameter is carried round and restored
710again to the same position from which it began to be moved, the
711figure so comprehended is a sphere.
712\end{scope}
713
714\begin{scope}[Definition XI.15: axis of a sphere]
715\label{def:XI.15}
716The axis of the sphere is the straight line which remains fixed and
717about which the semicircle is turned.
718\end{scope}
719
720\begin{scope}[Definition XI.16: centre of a sphere]
721\label{def:XI.16}
722The centre of the sphere is the same as that of the semicircle.
723\end{scope}
724
725\begin{scope}[Definition XI.17: diameter of a sphere]
726\label{def:XI.17}
727A diameter of the sphere is any straight line drawn through the
728centre and terminated in both directions by the surface of the sphere.
729\end{scope}
730
731\begin{scope}[Definition XI.18: cone]
732\label{def:XI.18}
733When, one side of those about the right angle in a right-angled
734triangle remaining fixed, the triangle is carried round and restored
735again to the same position from which it began to be moved, the
736figure so comprehended is a cone. And if the straight line which
737remains fixed is equal to the remaining side about the right angle
738which is carried round, the cone will be right-angled; if less,
739obtuse-angled; and if greater, acute-angled.
740\end{scope}
741
742\begin{scope}[Definition XI.19: axis of a cone]
743\label{def:XI.19}
744The axis of the cone is the straight line which remains fixed and
745about which the triangle is turned.
746\end{scope}
747
748\begin{scope}[Definition XI.20: base of a cone]
749\label{def:XI.20}
750And the base is the circle described by the straight line which is
751carried round.
752\end{scope}
753
754\begin{scope}[Definition XI.21: cylinder]
755\label{def:XI.21}
756When, one side of those about the right angle in a rectangular
757parallelogram remaining fixed, the parallelogram is carried round and
758restored again to the same position from which it began to be moved,
759the figure so comprehended is a cylinder.
760\end{scope}
761
762\begin{scope}[Definition XI.22: axis of a cylinder]
763\label{def:XI.22}
764The axis of the cylinder is the straight line which remains fixed and
765about which the parallelogram is turned.
766\end{scope}
767
768\begin{scope}[Definition XI.23: bases of a cylinder]
769\label{def:XI.23}
770The bases are the circles described by the two sides opposite to one
771another which are carried round.
772\end{scope}
773
774\begin{scope}[Definition XI.24: similar cones and cylinders]
775\label{def:XI.24}
776Similar cones and cylinders are those in which the axes and the
777diameters of the bases are proportional.
778\end{scope}
779
780\begin{scope}[Definition XI.25: cube]
781\label{def:XI.25}
782A cube is a solid figure contained by six equal squares.
783\end{scope}
784
785\begin{scope}[Definition XI.26: octahedron]
786\label{def:XI.26}
787An octahedron is a solid figure contained by eight equal and
788equilateral triangles.
789\end{scope}
790
791\begin{scope}[Definition XI.27: icosahedron]
792\label{def:XI.27}
793An icosahedron is a solid figure contained by twenty equal and
794equilateral triangles.
795\end{scope}
796
797\begin{scope}[Definition XI.28: dodecahedron]
798\label{def:XI.28}
799A dodecahedron is a solid figure contained by twelve equal,
800equilateral, and equiangular pentagons.
801\end{scope}
802
803% ===== Book XIII definitions (5 supplementary defs) =====
804
805\begin{scope}[Definition XIII.1: extreme and mean ratio]
806\label{def:XIII.1}
807A straight line is said to have been cut in extreme and mean ratio
808when, as the whole line is to the greater segment, so is the greater
809to the lesser.
810\end{scope}
811
812\begin{scope}[Definition XIII.2: height of a figure]
813\label{def:XIII.2}
814The height of any figure is the perpendicular drawn from the vertex
815to the base.
816\end{scope}
817
818\begin{scope}[Definition XIII.3: medial straight line]
819\label{def:XIII.3}
820A medial straight line is the mean proportional between two rational
821straight lines commensurable in square only.
822\end{scope}
823
824\begin{scope}[Definition XIII.4: minor straight line]
825\label{def:XIII.4}
826A minor straight line is the difference of two straight lines
827incommensurable in square such that the sum of the squares on them
828is rational, but the rectangle contained by them is medial.
829\end{scope}
830
831\begin{scope}[Definition XIII.5: composite irrational]
832\label{def:XIII.5}
833A straight line which produces with a rational area a medial whole
834is the irrational straight line such that the square on it added to
835a rational area makes the whole medial.
836\end{scope}
837