# MA232A     Michaelmas Term 2015

## Syllabus for MA232A at the Annual Examination 2016, and Sample Paper

MA232A - Syllabus for Examination 2016
MA232A - Sample Paper 2016
Note that the final two pages contain a description of Selected Propositions from Euclid. This is not ideally suited to the questions that appear on the sample paper, but is a verbatim reproduction of the corresponding pages appended to the actual examination paper. The Selected Propositions also gives names to some of the propositions (SAS Congruence Rule, Alternate Angles Theorem) that match or resemble names attached to these results in some Project Maths textbooks. Some propositions whose statements are included in the Selected Propositions are nevertheless included in the formal list of examinable material, as candidates are advised to be familiar with those propositions.

### Recently noted Errata

• Proof of Lemma 5.11, second paragraph: “the angle between the vectors nV and nW is also a right angle” changed to “the angle between the vectors. w and nW is also a right angle” (May 15, 2016)
• Proof of Proposition 5.14: corrections changing r' to r'', and changing L'' to L', at appropriate places (April 13, 2016)
• Statement of Corollary 8.3 corrected (May 16, 2016)

## Note on Labelling of Congruence Rules and Other Theorems

The following table may assist in matching up labelled theorems and congruence rules to propositions in Euclid.
RuleProposition
SAS Congruence RuleEuclid, Book I, Proposition 4
SSS Congruence RuleEuclid, Book I, Proposition 8
ASA Congruence RuleEuclid, Book I, Proposition 26
SAA Congruence RuleEuclid, Book I, Proposition 26
Isosceles Triangle TheoremEuclid, Book I, Proposition 5
Vertically Opposite Angles TheoremEuclid, Book I, Proposition 15
Alternate Angles TheoremEuclid, Book I, Proposition 27 and 29
Corresponding Angles TheoremEuclid, Book I, Proposition 29
Pythagoras's TheoremEuclid, Book I, Proposition 47

Note that not all of the above are included in the Selected Propositions for 2016.

To interpret congruence rule labellings: “SAS” stands for “side-angle-side” and encodes the fact that a first side is incident on the angle which in turn is incident on the second side.

Note also that there is no valid “ASS” congruence rule!

## Summary of Lectures

For an indication of what was been covered in lectures prior to Study Week, consult the Lecture Log.

## Resource Pages

Resources related to Euclid's Elements of Geometry

Resources related to Gauss's General Investigations of Curved Surfaces

## Curriculum

Euclid's Elements of Geometry (6 weeks)
We explored selected topics from the first six books of Euclid's Elements of Geometry, including in particular the following topics:-
• in Book I, the Definitions, Postulates and Common Notions
• in Book I, the early propositions, including in particular the Pons Asinorum taking note in particular of places where proofs make unspecified assumptions regarding existence of points of intersection and of the homogeneity of the Euclidean plane;
• in Book I, rigidity of triangles, showing how triangles are uniquely determined once appropriate lengths and angles are given;
• in Book I, the theory of parallels (Proposition 27 to 33);
• in Book I, propositions (from Proposition 35 onwards) concerning the areas of triangles and parallelograms, and concern transformation of areas;
• in Book I, the proof of Pythagoras's Theorem (Proposition 47)
• the geometry of circles in Book III, up to Proposition 22;
• Eudoxus's Theory of Proportion, in Book V, covering the basic definitions and Propositions 1, 2 and 8;
• Applications of the Theory of Proportion in Propositions 1 and 2 of Book VI.
Gauss's General Investigations of Curved Surfaces (4 weeks)
This work by Gauss, originally published in Latin, gave a considerable boost to the theory of curved surfaces, as not much about them had been discovered before Gauss. Included is the Theorema Egregium that ensures that Gaussian Curvature (one of the measures of the curvature of a surface at a point) is invariant under isometries between surfaces. We aim to concentrate on sections 1 to 20.
Geometry of the Hyperbolic Plane(1 week)
We investigate 2-dimensional hyperbolic geometry, making use of the Poincaré Disk model of the hyperbolic plane

## Texts and other Resources on the Internet

Extensive use was made of online texts of mathematical works that are freely available over the internet. Ideally participants should be prepared to download selected texts themselves for online viewing on desktop and laptop computers etc., maybe printing off selected portions for further study. Note that Euclid's Elements of Geometry has a length of several hundred pages, so it may make sense to restrict printing of texts to selected portions. Photocopied portions of significant extracts may be distributed from time to time.

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