What is geometry? One young lady, when asked this question, answered without hesitation, “Oh, that is the subject in which we proved things.” When pressed to give an example of one of the “things” proved, she was unable to do so. Why was it a good idea to prove things also eluded her. This girl’s reactions are typical of those of a large number of people who think they have studied geometry. They forget all the subject matter, and they do not realize why the course was taught.
Forgetting the theorems is no tragedy. We forget much of the factual material we learn — or should I say encounter? — during our so-called education. Nevertheless, it is a pity if a whole course is so dull that it fails to impress any of its content on the memories of the students. It must be admitted that traditional geometry was (and still is) guilty of this fault. They, why was it taught? Because it was supposed to present to young people a unified logical system on a level intelligible to them. Presumably, some students got the point, but others became so involved in the details of the proof that they lost sight of the main objective.
The “new math” (some years ago in the USA) that had been introduced in most forward-looking schools had done much toward remedying these defects. Less time is being spent on the complicated details of Euclidean (especially solid geometry) geometry, and more on the idea of a geometric system. Other logical systems, much less elaborate, are being presented in order to give the student some notion of what a small forest looks like, while reducing the chance of his getting lost in the trees.
We are not going to tackle these educational problems. This and the following articles are intended for people who liked geometry (and perhaps, even some who did not), but sensed a lack of intellectual stimulus and wondered what was missing, or felt that the play was ending just when the plot was at last beginning to become interesting.
The theorems of classical elementary geometry are nearly all too obvious to be worthy of study for their own sake. Their importance lies in the role they play in the chain of reasoning. It is regrettable that so few non-trivial theorems can be proved within the framework of the traditional geometry course when so many startlingly good ones lie just around the corner, hidden from the view of the young student. It is my purpose to present some of these to you, to recapture or reawaken your interest, with the hope that you may find that geometry is not so dull as you may have thought.
The material we are going to present is not at all new Most of it, though unknown to the Greeks to be sure, has been in existence for a number of years. Why, then, was it not made available to you? If there wasn’t time in school, why not in college? Because you were born too later — or too soon: too late to participate in the wave of enthusiasm for geometry that swept through mathematics in the nineteenth century, when many of these things were being discovered; too soon for the “revival in geometry” now taking in many colleges and universities. What might be called advanced elementary geometry somehow fell from favour during the first half of the twentieth century, crowded out, probably, by the multitude of other subjects that demanded a place in the curriculum.
The question, “What is geometry?” has many answers today. There are different kinds of geometries: foundational, topological, non-Euclidean, n-dimensional, many others. Our aim will be much more modest: to look into some of the readily accessible topics that require no formidable array of new definitions and abstractions. We shall deal mostly in the kind of geometry you already know about: the lines and points involved will be, with a few exceptions, the “ordinary” lines and points of “ordinary” geometry. We shall draw only from the kind of material that is either self-evident in the classical sense or very easy to prove. Our postulates and axioms will be those of Euclid (school geometry) unless otherwise stated, and our tools the straightedge ruler, the compass, and a little thought!
This approach does not please the professional mathematician. He must needs the start from the beginning, with a set of assumptions, and derive everything from these. No real mathematics can proceed along any other path. But that may have been just the trouble with your high school course: it was too formal, too cold, too bald — and hence, uninspired and uninspiring. To avoid this catastrophe we will beg the forgiveness of the mathematicians, skip the formalities, and take our chances with the rest.
Of course, we shall prove things. There is not much point (or fun) in being told that something is so without finding out how we know it is so. Our proofs will make liberal use of diagrams. The future blogs on this topic will be full of diagrams, and we should come to understanding at the outset about the role that they are to play.
Have you seen a geometric point or a point? Perhaps, you will agree that you have not. A point has no size, no length, no mass, no breadth, no height and “there is nothing there to see.” But what about a circle? You may not be so willing to admit that you have never seen a circle, but it is very certain that you have not. A circle is defined as a set of all points in a plane equidistant from a fixed point. Already we have guaranteed its invisibility: if a point has no thickness, neither has a “row” of them, and there is still nothing to see. What you do see when you draw a circle with a compass is only an attempt to picture a circle, and a poor attempt at that. It is not a circle because: (1) It is not made up of points strung out along a line: the alleged “line” has width. (2) Even if the line (or rather its picture) were to be made microscopically thin, accurate measurement would detect unequal distances from the centre — assuming that the centre could be located in a meaningful way, which it could not. (3) The alleged circle does not lie in a planc; a piece of paper is very far from a true geometric plane (4) Even if the paper were a plane, the ink has a thickness building up away from the paper. And, so on!
Have you ever seen any geometric figures? Certainly not. They are defined in such a way that they can never have physical or tangible existence. It is a tribute to out powers of mental abstraction that even though none of us have ever seen points or lines, we can talk about them with confidence. When I say “straight line”, you have no trouble visualizing exactly what I mean. It is in this inner realm of mental visions that all geometry really takes place — not on the paper. One must guard against thinking, “The diagram proves it.” Appearances are often misleading; diagrams are useful only as an aid for picturing things that can (at least theoretically) be stated and proved without them. Yet they are useful in clarifying our thinking that only the most abstract purists attempt to dispense with them entirely.
The abstract nature of geometry was at least partially understood and appreciated by the Greeks. That is why the “permissible tools” of classical geometry are straightedge ruler and compass only. Consider the problem of trisecting an angle, which has no solution with these tools alone. Then, why not use a protractor? Just measure the angle, divide the number of degrees by three, and there you are. But where are you? This superficial solution to the problem disturbs our feeling of what is acceptable and proper in geometric society, so to speak. It is the very purity of the ruler and
compass that suits them so well to the purity (abstractness) of the subject. If your sensibilities are outraged by the idea of measuring the number of degrees in an angle, you have already taken a long stride into geometry.
Are you ready for a mathematical ramble? Then, let’s go…until the next blog then,