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| ISBN-13: | 9780486158198 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 07/18/2012 |
| Series: | Dover Books on Mathematics , #2 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 464 |
| File size: | 29 MB |
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The Thirteen Books of Euclid's Elements Volume II
By Thomas L. Heath
Dover Publications, Inc.
Copyright © 1956 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-15819-8
CHAPTER 1
DEFINITIONS.
1. Equal circles are those the diameters of which are equal, or the radii of which are equal.
2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
3. Circles are said to touch one another which, meeting one another, do not cut one another.
4. In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.
5. And that straight line is said to be at a greater distance on which the greater perpendicular falls.
6. A segment of a circle is the figure contained by a straight line and a circumference of a circle.
7. An angle of a segment is that contained by a straight line and a circumference of a circle.
8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.
9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.
10. A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
11. Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.
Definition 1.
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Many editors have held that this should not have been included among definitions. Some, e.g. Tartaglia, would call it a postulate; others, e.g. Borelli and Playfair, would call it an axiom; others again, as Billingsley and Clavius, while admitting it as a definition, add explanations based on the mode of constructing a circle; Simson and Pfleiderer hold that it is a theorem. I think however that Euclid would have maintained that it is a definition in the proper sense of the term; and certainly it satisfies Aristotle's requirement that a "definitional statement" ([TEXT NOT REPRODUCIBLE IN ASCII]) should not only state the fact [TEXT NOT REPRODUCIBLE IN ASCII]) but should indicate the cause as well (De anima [TEXT NOT REPRODUCIBLE IN ASCII]. 2, 413 a 13). The equality 01 circles with equal radii can of course be proved by superposition, but, as we have seen, Euclid avoided this method wherever he could, and there is nothing technically wrong in saying "By equal circles I mean circles with equal radii." No flaw is thereby introduced into the system of the Elements; for the definition could only be objected to if it could be proved that the equality predicated of the two circles in the definition was not the same thing as the equality predicated of other equal figures in the Elements on the basis of the Congruence-Axiom, and, needless to say, this cannot be proved because it is not true. The existence of equal circles (in the sense of the definition) follows from the existence of equal straight lines and I. Post. 3.
The Greeks had no distinct word for radius, which is with them, as here, the (straight line drawn) from the centre [TEXT NOT REPRODUCIBLE IN ASCII] ([TEXT NOT REPRODUCIBLE IN ASCII]); and so definitely was the expression appropriated to the radius that [TEXT NOT REPRODUCIBLE IN ASCII] was used without the article as a predicate, just as if it were one word. Thus, e.g., in III. I [TEXT NOT REPRODUCIBLE IN ASCII] means "for they are radii": cf. Archimedes, On the Sphere and Cylinder II. 2, [TEXT NOT REPRODUCIBLE IN ASCII], BE is a radius of the circle.
Definition 2.
[TEXT NOT REPRODUCIBLE IN ASCII].
Euclid's phraseology here shows the regular distinction between [TEXT NOT REPRODUCIBLE IN ASCII] and its compound [TEXT NOT REPRODUCIBLE IN ASCII], the former meaning "to meet" and the latter "to touch" The distinction was generally observed, by Greek geometers from Euclid onwards. There are however exceptions so far as [TEXT NOT REPRODUCIBLE IN ASCII] is concerned; thus it means "to touch" in Eucl. IV DEF 5 and sometimes in Archimedes. On the other hand, [TEXT NOT REPRODUCIBLE IN ASCII] is used by Aristotle in certain cases where the orthodox geometrical term would be [TEXT NOT REPRODUCIBLE IN ASCII]. Thus in Meteorologica III. 5 (376 b 9) he says a certain circle will pass through all the angles ([TEXT NOT REPRODUCIBLE IN ASCII]), and (376 a 6) M will lie on a given (circular) circumference ([TEXT NOT REPRODUCIBLE IN ASCII] M). We shall find [TEXT NOT REPRODUCIBLE IN ASCII] used in these senses in Book iv. Deff. 2, 6 and Deff. 1, 3 respectively. The latter of the two expressions quoted from Aristotle means that the locus of M is a given circle, just as in Pappus [TEXT NOT REPRODUCIBLE IN ASCII] means that the locus of the point is a straight line given in position.
Definition 3.
[TEXT NOT REPRODUCIBLE IN ASCII].
Todhunter remarks that different opinions have been held as to what is, or should be, included in this definition, one opinion being that it only means that the circles do not cut in the neighbourhood of the point of contact, and that it must be shown that they do not cut elsewhere, while another opinion is that the definition means that the circles do not cut at all. Todhunter thinks the latter opinion correct. I do not think this is proved; and I prefer to read the definition as meaning simply that the circles meet at a point but do not cut at that point. I think this interpretation preferable for the reason that, although Euclid does practically assume in III. II—13, without stating, the theorem that circles touching at one point do not intersect anywhere else, he has given us, before reaching that point in the Book, means for proving for ourselves the truth of that statement. In particular, he has given us the propositions III. 7, 8 which, taken as a whole, give us more information as to the general nature of a circle than any other propositions that have preceded, and which can be used, as will be seen in the sequel, to solve any doubts arising out of Euclid's unproved assumptions. Now, as a matter of fact, the propositions are not used in any of the genuine proofs of the cheorems in Book III.; III. 8 is required for the second proof of III. 9 which Simson selected in preference to the first proof, but the first proof only is regarded by Heiberg as genuine. Hence it would not be easy to account for the appearance of III. 7, 8 at all unless as affording means of answering possible objections (cf. Proclus' explanation of Euclid's reason for inserting the second part of I. 5).
External and internal contact are not distinguished in Euclid until III. II, 12, though the figure of III. 6 (not the enunciation in the original text) represents the case of internal contact only. But the definition of touching circles here given must be taken to imply so much about internal and external contact respectively as that (a) a circle touching another internally must, immediately before "meeting" it, have passed through points within the circle that it touches, and (b) a circle touching another externally must, immediately before meeting it, have passed through points outside the circle which it touches. These facts must indeed be admitted if internal and external are to have any meaning at all in this connexion, and they constitute a minimum admission necessary to the proof of III. 6.
Definition 4.
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Definition 5.
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Definition 6.
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Definition 7.
[TEXT NOT REPRODUCIBLE IN ASCII].
This definition is only interesting historically. The angle of a segment, being the "angle" formed by a straight line and a "circumference," is of the kind described by Proclus as "mixed." A particular "angle" of this sort is the "angle of a semicircle? which we meet with again in III. 16, along with the so-called "horn-like angle "([TEXT NOT REPRODUCIBLE IN ASCII]), the supposed "angle "between a tangent to a circle and the circle itself. The "angle of a semicircle "occurs once in Pappus (VII. p. 670, 19), but it there means scarcely more than the corner of a semicircle regarded as a point to which a straight line is directed. Heron does not give the definition of the angle of a segment, and we may conclude that the mention of it and of the angle of a semicircle in Euclid is a survival from earlier text-books rather than an indication that Euclid considered either to be of importance in elementary geometry (cf. the note on III. 16 below).
We have however, in the note on 1. 5 above (Vol. I. pp. 252—3), seen evidence that the angle of a segment had played some part in geometrical proofs up to Euclid's time. It would appear from the passage of Aristotle there quoted (Anal, prior, 1. 24, 41 b 13 sqq.) that the theorem of 1. 5 was, in the text-books immediately preceding Euclid, proved by means of the equality of the two "angles of" any one segment. This latter property must therefore have been regarded as more elementary (for whatever reason) than the theorem of 1. 5; indeed the definition as given by Euclid practically implies the same thing, since it speaks of only one "angle of a segment," namely "the angle contained by a straight line and a circumference of a circle." Euclid abandoned the actual use of the "angle "in question, but no doubt thought it unnecessary to break with tradition so far as to strike the definition out also.
Definition 8.
[TEXT NOT REPRODUCIBLE IN ASCII].
Definition 9.
[TEXT NOT REPRODUCIBLE IN ASCII].
Definition 10.
[TEXT NOT REPRODUCIBLE IN ASCII].
A scholiast says that it was the shoemaker's knife, [TEXT NOT REPRODUCIBLE IN ASCII], which suggested the name [TEXT NOT REPRODUCIBLE IN ASCII] for a sector of a circle. The derivation of the name from a resemblance of shape is parallel to the use of [TEXT NOT REPRODUCIBLE IN ASCII] (also a shoemaker's knife) to denote the well known figure of the Book of Lemmas partly attributed to Archimedes.
A wider definition of a sector than that given by Euclid is found in a Greek scholiast (Heiberg's Euclid, Vol. V. p. 260) and in an-Nairizi (ed. Curtze, p. 112). "There are two varieties of sectors; the one kind have the angular vertices at the centres, the other at the circumferences. Those others which have their vertices neither at the circumferences nor at the centres, but at some other points, are for that reason not called sectors but sector-like figures [TEXT NOT REPRODUCIBLE IN ASCII])." The exact agreement between the scholiast and an-Nairizi suggests that Heron was the authority for this explanation.
The sector-like figure bounded by an arc of a circle and two lines drawn from its extremities to meet at any point actually appears in Euclid's book On divisions ([TEXT NOT REPRODUCIBLE IN ASCII]) discovered in an Arabic MS. and edited by Woepcke (cf. Vol. I. pp. 8—10 above). This treatise, alluded to by Proclus, had for its object the division of figures such as triangles, trapezia, quadrilaterals and circles, by means of straight lines, into parts equal or in given ratios. One proposition e.g. is, To divide a triangle into two equal parts by a straight line passing through a given point on one side. The proposition (28) in which the quasi-sector occurs is, To divide such a figure by a straight line into two equal parts. The solution in this case is given by Cantor (Gesch. d. Math. I3, pp. 287—8).
If ABCD be the given figure, E the middle point of BD and EC at right angles to BD, the broken line AEC clearly divides the figure into two equal parts.
Join AC, and draw EF parallel to it meeting AB in F.
Join CF, when it is seen that CF divides the figure into two equal parts.
Definition 11.
[TEXT NOT REPRODUCIBLE IN ASCII].
De Morgan remarks that the use of the word similar in "similar segments" is an anticipation, and that similarity of form is meant. He adds that the definition is a theorem, or would be if "similar" had taken its final meaning.
Proposition 1.
To find the centre of a given circle.
Let ABC be the given circle; thus it is required to find the centre of the circle ABC.
Let a straight line AB be drawn 5 through it at random, and let it be bisected at the point D; from D let DC be drawn at right angles to AB and let it be drawn through to E; let CE be bisected at F; 10 I say that F is the centre of the circle ABC.
For suppose it is not, but, if possible, let G be the centre, and let GA, GD, GB be joined.
15 Then, since AD is equal to DB, and DG is common, the two sides AD, DG are equal to the two sides BD, DG respectively; and the base GA is equal to the base GB, for they are 20 radii; therefore the angle ADG is equal to the angle GDB. [I.8]
But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right; [I. Def. 10]
25 therefore the angle GDB is right.
But the angle FDB is also right; therefore the angle FDB is equal to the angle GDB, the greater to the less: which is impossible.
Therefore G is not the centre of the circle ABC.
30 Similarly we can prove that neither is any other point except F.
Therefore the point F is the centre of the circle ABC.
Porism. From this it is manifest that, if in a circle a straight line cut a straight line into two equal parts and at 35 right angles, the centre of the circle is on the cutting straight line.
Q. E. D.
12. For suppose it is not. This is expressed in the Greek by the two words [TEXT NOT REPRODUCIBLE IN ASCII], but such an elliptical phrase is impossible in English.
17. the two sides AD, DG are equal to the two sides BD, DQ respectively. As before observed, Euclid is not always careful to put the equals in corresponding order. The text here has "GD, DB."
Todhunter observes that, when, in the construction, DC is said to be produced to E, it is assumed that D is within the circle, a fact which Euclid first demonstrates in III. 2. This is no doubt true, although the word [TEXT NOT REPRODUCIBLE IN ASCII], "let it be drawn through" is used instead of [TEXT NOT REPRODUCIBLE IN ASCII], "let it be produced" And, although it is not necessary to assume that D is within the circle, it is necessary for the success of the construction that the straight line drawn through D at right angles to AB shall meet the circle in two points (and no more): an assumption which we are not entitled to make on the basis of what has gone before only.
Hence there is much to be said for the alternative procedure recommended by De Morgan as preferable to that of Euclid. De Morgan would first prove the fundamental theorem that "the line which bisects a chord perpendicularly must contain the centre," and then make III. I, III. 25 and IV. 5 immediate corollaries of it. The fundamental theorem is a direct consequence of the theorem that, if P is any point equidistant from A and B, then P lies on the straight line bisecting AB perpendicularly. We then take any two chords AB, AC of the given circle and draw DO, E O bisecting them perpendicularly. Unless BA, AC are in one straight line, the straight lines DO, E O must meet in some point O (see note on IV. 5 for possible methods of proving this). And, since both DO, EO must contain the centre, O must be the centre.
This method, which seems now to be generally preferred to Euclid's, has the advantage of showing that, in order to find the centre of a circle, it is sufficient to know three points on the circumference. If therefore two circles have three points in common, they must have the same centre and radius, so that two circles cannot have three points in common without coinciding entirely. Also, as indicated by De Morgan, the same construction enables us (I) to draw the complete circle of which a segment or arc only is given (III. 25), and (2) to circumscribe a circle to any triangle (IV. 5).
But, if the Greeks had used this construction for finding the centre of a circle, they would have considered it necessary to add a proof that no other point than that obtained by the construction can be the centre, as is clear both from the similar reductio ad absurdum in III I and also from the fact that Euclid thinks it necessary to prove as a separate theorem (III. 9) that, if a point within a circle be such that three straight lines (at least) drawn from it to the circumference are equal, that point must be the centre. In fact, however, the proof amounts to no more than the remark that the two perpendicular bisectors can have no more than one point common.
And even in De Morgan's method there is a yet unproved assumption. In order that DO, EO may meet, it is necessary that AB, AC should not be in one straight line or, in other words, that BC should not pass through A. This results from III. 2, which therefore, strictly speaking, should precede.
To return to Euclid's own proposition III. I, it will be observed that the demonstration only shows that the centre of the circle cannot lie on either side of CD, so that it must lie on CD or CD produced. It is however taken for granted rather than proved that the centre must be the middle point of CE. The proof of this by reductio ad absurdum is however so obvious as to be scarcely worth giving. The same consideration which would prove it may be used to show that a circle cannot have more than one centre, a proposition which, if thought necessary, may be added to III. I as a corollary.
(Continues...)
Excerpted from The Thirteen Books of Euclid's Elements Volume II by Thomas L. Heath. Copyright © 1956 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
BOOK III.DEFINITIONS
PROPOSITIONS
BOOK IV.
DEFINITIONS
PROPOSITIONS
BOOK V.
INTRODUCTORY NOTE
DEFINITIONS
PROPOSITIONS
BOOK VI.
INTRODUCTORY NOTE
DEFINITIONS
PROPOSITIONS
BOOK VII.
DEFINITIONS
PROPOSITIONS
BOOK VIII.
BOOK IX.
GREEK INDEX TO VOL. II.
ENGLISH INDEX TO VOL. II.
