5: Geometry
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Those who fear to experiment with their hands will never know anything. -George Sarton (1884-1956)
Mathematical truth is not determined arbitrarily by the rules of some ‘man-made’ formal system, but has an absolute nature and lies beyond any such system of specifiable rules. - Roger Penrose (1930– )
Geometry is in many ways the most natural branch of elementary mathematics through which to convey “the essence” of the discipline.
- The underlying subject matter is rooted in seeing, moving, doing, drawing, making, etc., and so is accessible to everyone.
- At secondary level this practical experience leads fairly naturally to a semi-formal treatment of “geometry as a mental universe”
- — a universe that is bursting with surprising facts, whose statements can be easily understood; and
- — which has a clear logical structure, in terms of which the proofs of these facts are accessible, if sometimes tantalisingly elusive.
This combination of elusive problems to be solved and the steady accumulation of proven results has provided generations of students with their first glimpse of serious mathematics. All readers can imagine the kind of experiences which lie behind the first bullet point above: many of the problems we have already met (such as Problems 4, 19, 20, 26, 27, 28, 29, 30, 31, 37, 38, 39) do not depend on the “semi-formal treatment” referred to in the second bullet point, so can be tackled by anyone who is interested - provided they accept the importance of learning to construct their own diagrams (in the spirit of the George Sarton quotation).
The hand is the cutting edge of the mind.
Jacob Bronowski (1908-1974)
But there is a catch - which explains why the present chapter appears so late in the collection. For many problems to successfully convey “the essence of mathematics” there has to be some shared understanding of what constitutes a solution. And in geometry, many solutions require the construction of a proof. Yet many readers will never have experienced a coherent “semi-formal treatment” of elementary geometry in the spirit of the second bullet point. Hence in Problems 3(c), 18, 21, 32, 34, 36 we committed the cardinal sin of leading the reader by the nose - breaking each problem into steps in order to impose a logical structure. This may have been excusable in Chapter 1; but in a chapter explicitly devoted to geometry, the underlying challenge has to be faced head on: that is, the raw experience of the hand has to be refined to provide a deductive structure for the mind.
As in Chapter 1, some of the problems listed from Section 5.3 onwards can be tackled without worrying too much about the logical structure of elementary geometry. But in many instances, the “essence” that is captured by a problem requires that the problem be seen within an agreed logical hierarchy - a sequencing of properties, results, and methods, which establishes what is a consequence of what - and hence, what can be used as part of a solution. In particular, we need to construct proofs that avoid circular reasoning.
If B is a consequence of A, or if B is equivalent to A, then a ‘proof’ of A which makes use of B is at best dubious, and may well be a delusion.
The need to avoid such circular reasoning arose already in Problem 21 (the converse of Pythagoras’ Theorem), where we felt the need to state explicitly that it would be inappropriate to use the Cosine Rule: (see Problem 192 below).
Such concerns may explain why this chapter on geometry is the last of the chapters relating to elementary ‘school mathematics’, and why we begin the chapter with
- an apparent digression (Section 5.1), and
- an outline of elementary Euclidean geometry (Section 5.2).
Those with a strong background in geometry may choose to skip these sections on a first reading, and move straight on to the problems which start in Section 5.3. But they may then fail to see how the cumulative architecture of Section 5.2 conveys a rather different aspect of the “essence of mathematics”, deriving not just from the individual problems, but from the way a carefully crafted, systematic arrangement of simple “bricks” can create a much more significant mathematical structure.