# MA232A - Euclidean and non-Euclidean Geometry Michaelmas Term 2015 Dr. David R. Wilkins Outline of Lectures, Michaelmas Term 2015

## Lectures on Euclid

Wednesday, September 30, 2015
Course Overview
Thursday, October 1
Discussion of the Definitions in Euclid, Book I, based on the text in Heath, pp. 153, 154
Friday, October 2
Discussion of Postulates, Common Notions and Axioms in Euclid, Book I, comparing the presentation in terms of five postulates and five common notions found in more modern editions of Euclid derived from Heiberg's critical edition of the Greek text, such as Heath's translation, pp. 154, 155 with the presentation in terms of three postulates and twelve axioms in older editions of Euclid derived from Commandinus's Latin translation, such as Todhunter' Edition, pp. 5, 6. For a comparison of the labelling and wording of the axioms in the various editions, see The Axiom System of Book I of Euclid's Elements of Geometry
Wednesday, October 7 (noon)
Printed notes for John Casey's Edition of the First Six Books of Euclid were distributed. This was followed by a review of the structure of the axiom system, with particular reference to the statements of postulates and axioms in Casey's edition, and Casey's comments on some of the axioms. The lecture concluded with a discussion of Propositions 1, 2 and 3 of Euclid, Book I (Casey's Edition).
Wednesday, October 7 (evening)
This lecture covered Propositions 4 and 5 of Euclid, Book I (Casey's Edition). In this course of this lecture, the class crossed the Pons Asinorum (‘Bridge of Asses’). This is the name traditionally given to Proposition 5 of Book I of Euclid's Elements of Geometry which asserts that, in an isosceles triangle where the two sides incident on the base are equal, the angles at the endpoints of the base are equal. The proof of this result is based on applications of the preceding proposition, Proposition 4, which asserts that, if ABC and DEF are triangles, with AB = DE, AC = DF and angle BAC = angle EDF, then BC = EF, and moreover the angles of the triangle ABC at B and C are equal to the angles of the triangle DEF at E and F respectively. The proof of this proposition makes assumptions about the homogeneity of Euclidean space that are not explicit in the Postulates and Axioms (or Common Notions) of Book I of Euclid.
Thursday, October 8
This lecture covered Propositions 6 and 7 of Euclid, Book I (Casey's Edition). This was followed by a discussion of the significant differences between the statement and proof of the result in Casey's Edition (written as a textbook for schools and colleges) and the original text of Euclid, as presented in Euclid, Book I, Proposition 7, in T.L. Heath's translation. In particular, the full proof requires consideration of a number of special cases, which are presented in Casey's edition. But, in accordance with the practice of his time, Euclid only presented a proof in one representative case. Casey includes, in addition to the case considered by Euclid, a proof of the result in another special case that is due to Proclus, and is to be found in T.L. Heath's discussion of Book I, Proposition 7.
Wednesday, October 14 (noon)
This lecture covered Propositions 8 to 18 of Euclid, Book I (Casey's Edition).
Wednesday, October 14 (evening)
This lecture covered Propositions 19 to 25 of Euclid, Book I (Casey's Edition).
Thursday, October 15
This lecture covered Propositions 26 to 29 of Euclid, Book I (Casey's Edition). Proposition 28 includes the result that when a line crosses two parallel lines, the sum of the interior angles on one side is equal to two right angles. The converse of this result is the Parallel Postulate, which is Postulate 5 (in modern editions based on Heiberg's text), or Axiom 12 (in older editions based, directly or indirectly, on Commandinus's Latin translation). The Parallel Postulate is used to justify equivalent results which are included in the statement of Proposition 29.
Wednesday, October 21 (noon)
This lecture began with a discussion of the theory of parallels as presented in Euclid's Elements of Geometry, based on the scholarly edition by T.L. Heath. We discussed in particular Playfair's Axiom, which appeared as an alternative to Euclid's Parallel Postulate in John Playfair's edition of Euclid's Elements of Geometry, published in 1795. This postulate formed the basis of Playfair's proof of Proposition 29 of Book I of Euclid. The lecture finished with a discussion of Propositions 30 to 33 of Euclid, Book I (Casey's Edition).
Wednesday, October 21 (evening)
This lecture covered Propositions 34 to 43 of Euclid, Book I (Casey's Edition). However the proof of Proposition 35, which states that “Parallelograms which are on the same base and in the same parallels are equal to one another” was based on the discussion of Euclid, Book I, Proposition 35, in T.L. Heath's translation.
Thursday, October 22
This lecture covered Propositions 44 to 48 of Euclid, Book I (Casey's Edition), completing the discussion of the First Book of Euclid, and including in particular the important Proposition 44 on the “application of areas” and Pythagoras' Theorem (Proposition 47).
Wednesday, October 28 (noon)
This lecture commenced the study of Book III of Euclid's Elements of Geometry, which is concerned with the theory of the circle. The lecture began with a detailed discussion of Propositions 1 and 2 of Book III (Heath's translation, Internet Archive) (transcribed here). Proposition 1 provides a construction for finding the centre of a circle. Proposition 2 asserts that, given two points on the boundary of a circle, the line segment joining them must lie within the circle. Euclid approached the proof of this by showing that an impossibility would result were the line segment to go through some point outside the circle, and claimed that the same would be true were the line segment to pass through some point on the circumference of the circle between its endpoints. Euclid's approach is presented in Euclid Book III, Proposition 2 (Heath's translation, Internet Archive). John Casey's textbook provided a direct proof (see Book II, Proposition 2 (John Casey's edition)). Casey is nevertheless closely related to Euclid's proof. Propositions 3 and 4 (Heath's translation, Internet Archive) were discussed and the statements of Propositions 5 and 6 (Heath's translation, Internet Archive) were noted. (Propositions 5 and 6 together ensure that concentric circles do not intersect: this is an obvious consequence of the definition of a circle.)
Wednesday, October 28 (evening)
This lecture discussed Propositions 7 to 13 of Book III (Heath's translation, Internet Archive). Also mentioned was the more straightforward proof of Proposition 10 in Casey's edition.
Thursday, October 29
This lecture continued the discussion of Book III of Euclid's Elements of Geometry with Proposition 14 to 17 (Heath's translation, Internet Archive). Proposition 16 of Book III (Heath's translation, Internet Archive) (transcribed here) is the only proposition in Euclid to make use of ``horn angles'', such as occur between a circle and its tangent lines. The topic of horn angles led to controversies between mathematicians and philosophers between the 13th and 17th centuries. Isaac Newton, in his Philosophiae Naturalis Principia Mathematica included several lemmas concerned with relationships between horn angles and limits (see Newton's Principia, Book I, Section 1, Lemmas 6, 7 and 8 (Andrew Motte's translation) and the Scholium concluding Book I, Section 1 of Newton's Principia (Andrew Motte's translation).) The lecture concluded with a discssion of Proposition 17 (Heath's translation, Internet Archive). Proposition 16 of Book III (Heath's translation, Internet Archive) (transcribed here).
Wednesday, November 4 (noon)
This lecture was devoted to a discussion of the theory of Proportion attributed to Eudoxus of Cnidus (408—355 B.C.), which is the subject matter of Euclid, Book V (linked here to Heath's translation, Internet Archive). The discussion in this lecture focussed on Definitions 1 to 7 of Book V of Euclid (linked here to Heath's translation, Internet Archive) (transcribed, with comments here). A large part of the lecture was devoted to discussion of Definition 5. Suppose that one is given four magnitudes a, b, c and d, where some multiple of a exceeds b, some multiple of b exceeds a, some multiple of c exceeds d and some multiple of d exceeds c. We say that “a is to b as c is to d” if and only if, given any positive integers m and n, the following conditions are satisfied: m . a exceeds n . b if and only if m . c exceeds n . d; m . a equals n . b if and only if m . c equals n . d; m . a is less than n . b if and only if m . c is less than n . d. (see Definition 5 in Heath's translation, Internet Archive, transcribed, with comments here). Also we say that “a has a greater ratio to b than c has to d” if and only if there exist positive integers m and n such that m . a > n . b but m . cn . d. (see Definition 7 in Heath's translation, Internet Archive, transcribed, with comments here). The ancient Greek mathematicians did not explicitly employ a theory of numbers that includes irrational numbers. But suppose that the ratio of a to b is represented, according to modern mathematical practice, by some real number x, and that the ratio of c to d is represented by some real number y. Then (as explained here) one can prove that a is to b as c is to d, in accordance with Definition 5 of Book V of Euclid's Elements of Geometry, if and only if x = y. And of course a has a greater ratio to b than c has to d if and only if x > y. The proof of this uses the fact that a positive real number x determines and is determined by the set of positive rational numbers q that satisfy q < x. This provides the link between Eudoxus's Theory of Proportion, developed in Book V of Euclid's Elements of Geometry and the theory of Dedekind sections developed by Richard Dedekind in his essay Stetigkeit und irrationale Zahlen published in 1872 (see the authorized translation by Wooster Woodruff Beman published as Continuity and irrational numbers in the volume of translations of works of Dedekind entitled Essays in the Theory of Numbers, Open Court Publishing Company, Chicago, 1901).
Wednesday, November 4 (evening)
This lecturer began with a summary of Eudoxus's theory of Proportion that had been introduced in the previous lecture. Then the lecture continued with discussion of Proposition 1 of Book V (Heath's translation, Internet Archive) (transcribed, with comments here), focussing in particular on De Morgan's comments on Propositions 1 to 6 that these propositions are “simple propositions of concrete arithmetic, covered in language which makes them unintelligible to modern ears“, followed by the observation that Proposition 1 corresponds to the result that ma + mb + mc + ... = m ( a + b + c + ) for any positive integer m and finite list a, b, c... of magnitudes capable of being added to one another. This was followed by a brief dicussion of Proposition 2 of Book V (Heath's translation, Internet Archive) (transcribed, with comments here). The statements of Propositions 3 to 7 were briefly surveyed, and the algebraic identities equivalent to them were noted. Then discussion moved to Proposition 8 of Book V (Heath's translation, Internet Archive) (transcribed, with comments here), focussing not on the details of Euclid's proof (as translated by Euclid), but rather on the algebraic paraphrase of the proof presented in the lecturer's note on Proposition 8 of Book V). Discussion then moved to Book VI of Euclid's Elements of Geometry (Heath's translation, Internet Archive), which is primarily concerned with applications of Eudoxus's theory of proportion (developed in Book V) to problems of plane geometry. There was detailed discussion of Proposition 1 of Book VI (Heath's translation, Internet Archive), which shows that, for triangles with the same apex and collinear bases, the triangles are proportional (in area) to their bases. This result was applied in Euclid's proof of Proposition 2 of Book VI (Heath's translation, Internet Archive), which shows that, where a line cuts a triangle parallel to the base of the triangle, the segments of the legs of the triangle cut off by this line are proportional to the legs themselves.
Thursday, October 29
Discussion moved back to the theory of the circle in Book III of Euclid's elements of geometry, and, in particular Proposition 18 to 22 of Book III (Heath's translation, Internet Archive) (transcribed here [link may need to be updated in future to the correct page]). The lecture concluded with a discussion of De Morgan's method, on pages 71 and 72 of The Connexion of Number and Magnitude: an attempt to explain the Fifth Book of Euclid, by Augustus De Morgan, 1836 (Internet Archive), for proving the result of Proposition 2 of Book VI of Euclid's Elements of Geometry (Heath's translation, Internet Archive). This result corresponds to Theorem 12 of Strand 2 of the current Irish Leaving Certificate Mathematics syllabus (2013). The lecture concluded with a discussion of proofs of Theorem 12 in the Leaving Certificate syllabus and the preceding Theorem 11.
Wednesday, November 18 (noon)
The lecture began with a general discussion of Gauss's General Investigations of Curved Survaces. It was noted that section 2 of that paper was concerned with spherical trigonometry. Though Gauss's proof involved standard trigonometric identities such as the Cosine Rule of Spherical Trigonometry, the algebra of vectors in three-dimensional space provides an alternative means to derive such identities. The lecture continued with a discussion of subsections 5.1 and 5.2 of Notes on Vector Algebra and Spherical Trigonometry.
Wednesday, November 18 (evening)
The lecture covered subsections 5.3 to 5.5 of Notes on Vector Algebra and Spherical Trigonometry, together with the first proof of Proposition 5.6 and the historical discussion of Hamilton's discovery of the vector triple product identity in his own quaternion notation, from subsection 5.6 of those notes.
Thursday, November 19
This lecture discussed the remainder of subsection 5.6, together with subsections 5.7 and 5.8 of Notes on Vector Algebra and Spherical Trigonometry.
Wednesday, November 25 (noon)
This lecture discussed the identities of spherical trigonometry stated and proved in subsection 5.9 and Notes on Vector Algebra and Spherical Trigonometry, and noted the application and proof of some of these identities in section 2 of Gauss's General Investigations of Curved Survaces.
Wednesday, November 25 (evening)
This lecture began by discussing Gauss's use of the term “continuity” and the use of the language of “infinitesmals” in sections 3 and 4 of General Investigations of Curved Survaces. Gauss is initiating here is discussion of the curvature of smooth surface. The lecture continued with subsections of the lecturer's presentation of some of these ideas in Notes on the Gauss Map. The lecture also discussed the representation of differentials of smooth functions on a surface as linear functions on tangent spaces to that surface as discussed in Subsection 6.8 of Smooth Surfaces in Three-Dimensional Space.
Thursday, November 26
The lecture began by revisiting the discussion of differentials of smooth functions in subsection 6.8 of Smooth Surfaces in Three-Dimensional Space. The lecture continued with material from subsection 6.10 of those notes, discussing the statement of the Inverse Function Theorem in three dimensions and a special case of the Implicit Function Theorem concerning the smoothness of zero sets of smooth real-valued functions defined on open sets in three-dimensional space.

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