E/NE 1: Things Euclid Never Told Me
Oct. 2nd, 2005 11:40 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowIn this series of posts, I'm going to talk about "the Euclidean axioms" from time to time, so I'd better clarify what I mean by that. Now, Euclid's Elements was a groundbreaking work; it established a format for mathematical writing and reasoning that still has great influence more than two thousand years later. As a methodologist, Euclid has no real rival until the time of Descartes. But, still, there were flaws in his work, which were not corrected until the late nineteenth century. I'd like to talk about that a bit before going on to non-Euclidean geometry.
Euclid begins with five postulates: that a unique straight line can be drawn between any two points; that any straight line can be extended indefinitely; that a circle can be drawn with any given center and radius; that any two right angles are congruent; and that, given a line l and a point P not on l, there is exactly one line through P parallel to l. (Actually, that last is not Euclid's fifth postulate, but they're equivalent, and this version is simpler.) He then proceeds to build a great edifice of theorems from them. The problem is that he does a few things which can't be justified purely on the basis of these postulates.
The first problem is a tiny one, easy to overlook; there must be at least two points. Euclid never mentions this; indeed, he never assumes that there are any points. I'm not going to say anything about that one.
The second problem is more serious. It's easy to show, from Euclid's postulates, that two lines intersect in exactly one point, unless they're parallel (in which case they don't intersect at all). But in the very first proposition, he refers to the intersection of two circles, and soon after to the intersection of a circle and a line - and nowhere does he say anything to ensure that these intersections exist. This problem can be resolved if we make the assumption that the points on a line correspond, in one-to-one fashion, with the real numbers. (To keep this from being circular reasoning, we have to have some way of describing the real numbers without referring to points on a line; fortunately, two ways of doing this were developed in the nineteenth century - Cauchy sequences and Dedekind cuts.)
Euclid never actually uses the word "congruent" (or its Greek equivalent); he speaks of triangles being "equal" and the like. But he uses the word in more than one sense; sometimes he evidently means "of equal area", but other times he means what we would call "congruent", and he never clearly defines the latter notion. Today, we define it in terms of "isometries" - transformations which preserve distances. There are three principal kinds of isometries: translations, in which every point of a figure is moved a specified distance in a specified direction; rotations, in which every point of a figure is rotated by a specified angle about a specified center; and reflections, in which figures are flipped over a specified line. Today's version of Euclidean geometry includes an axiom which guarantees the existence of these isometries. (This saves a lot of trouble. Proposition I.5 of the Elements states that, if two sides of a triangle are equal, the angles opposite those sides are also equal. The proof Euclid gives is notoriously difficult; comprehension of that proof was long regarded as a critical test of a student's mathematical abilities, and it was called, for this reason, the pons asinorum - the "bridge of asses". The proposition is much easier to prove once the concept of isometry is brought in.)
Finally, Euclid frequently uses the notion of "betweenness", without ever defining it. Such concepts as the interior of a triangle, for example, cannot be discussed without some idea of what it means for one point to lie between two others. Modern Euclidean geometry adds several axioms describing the properties of "betweenness". One important result of this is a theorem which, again, Euclid uses without mentioning or proving it: if a straight line enters a triangle through one side, it must exit through another. In particular, if a straight line enters a triangle through a vertex, it must exit through the opposite side. We'll be using this quite a bit.
In the remaining posts of this series, then, when I refer to the Euclidean axioms, I will mean, in addition to those Euclid himself used, axioms guaranteeing the existence of at least two points, identifying the points on a line with the real numbers, allowing the use of isometries, and clarifying the properties of betweenness.
Euclid begins with five postulates: that a unique straight line can be drawn between any two points; that any straight line can be extended indefinitely; that a circle can be drawn with any given center and radius; that any two right angles are congruent; and that, given a line l and a point P not on l, there is exactly one line through P parallel to l. (Actually, that last is not Euclid's fifth postulate, but they're equivalent, and this version is simpler.) He then proceeds to build a great edifice of theorems from them. The problem is that he does a few things which can't be justified purely on the basis of these postulates.
The first problem is a tiny one, easy to overlook; there must be at least two points. Euclid never mentions this; indeed, he never assumes that there are any points. I'm not going to say anything about that one.
The second problem is more serious. It's easy to show, from Euclid's postulates, that two lines intersect in exactly one point, unless they're parallel (in which case they don't intersect at all). But in the very first proposition, he refers to the intersection of two circles, and soon after to the intersection of a circle and a line - and nowhere does he say anything to ensure that these intersections exist. This problem can be resolved if we make the assumption that the points on a line correspond, in one-to-one fashion, with the real numbers. (To keep this from being circular reasoning, we have to have some way of describing the real numbers without referring to points on a line; fortunately, two ways of doing this were developed in the nineteenth century - Cauchy sequences and Dedekind cuts.)
Euclid never actually uses the word "congruent" (or its Greek equivalent); he speaks of triangles being "equal" and the like. But he uses the word in more than one sense; sometimes he evidently means "of equal area", but other times he means what we would call "congruent", and he never clearly defines the latter notion. Today, we define it in terms of "isometries" - transformations which preserve distances. There are three principal kinds of isometries: translations, in which every point of a figure is moved a specified distance in a specified direction; rotations, in which every point of a figure is rotated by a specified angle about a specified center; and reflections, in which figures are flipped over a specified line. Today's version of Euclidean geometry includes an axiom which guarantees the existence of these isometries. (This saves a lot of trouble. Proposition I.5 of the Elements states that, if two sides of a triangle are equal, the angles opposite those sides are also equal. The proof Euclid gives is notoriously difficult; comprehension of that proof was long regarded as a critical test of a student's mathematical abilities, and it was called, for this reason, the pons asinorum - the "bridge of asses". The proposition is much easier to prove once the concept of isometry is brought in.)
Finally, Euclid frequently uses the notion of "betweenness", without ever defining it. Such concepts as the interior of a triangle, for example, cannot be discussed without some idea of what it means for one point to lie between two others. Modern Euclidean geometry adds several axioms describing the properties of "betweenness". One important result of this is a theorem which, again, Euclid uses without mentioning or proving it: if a straight line enters a triangle through one side, it must exit through another. In particular, if a straight line enters a triangle through a vertex, it must exit through the opposite side. We'll be using this quite a bit.
In the remaining posts of this series, then, when I refer to the Euclidean axioms, I will mean, in addition to those Euclid himself used, axioms guaranteeing the existence of at least two points, identifying the points on a line with the real numbers, allowing the use of isometries, and clarifying the properties of betweenness.
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Date: 2005-10-02 06:06 pm (UTC)Five?
Date: 2005-10-03 02:20 am (UTC)Was that all part of the reformation of the "New Math" ? Or am I just remembering the count of the basics wrong?
Re: Five?
Date: 2005-10-03 08:42 am (UTC)At the beginning of the Elements, Euclid gives a long string of definitions, followed by five "postulates" - the five I named above - and then five "common notions". The common notions are:
"Things which are equal to the same thing are also equal to each other";
"If equals be added to equals, the wholes are equal";
"If equals be subtracted from equals, the remainders are equal";
"Things which coincide with one another are equal to one another";
"The whole is greater than the part".
The content of these five isn't specifically geometric, and so they are usually set aside. (The geometric matters they do refer to are usually handled in a rather different way nowadays, and at a different level.)