
MA232A  Euclidean and nonEuclidean 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 . c ≤ n . 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 threedimensional 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 ThreeDimensional Space.
 Thursday, November 26
 The lecture began by revisiting the discussion of
differentials of smooth functions in subsection 6.8 of
Smooth Surfaces in ThreeDimensional 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 realvalued functions defined on
open sets in threedimensional space.
Back to Module MA232A
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.