Introduction to Euclid:
Philosophy: Euclid does geometry without using real numbers. He uses finite line segments instead of numbers, so he wants to be able to compare them, i.e. to say when they are equal, or whether one is shorter than the other, without assigning a numerical length to them. To do this he uses the concept of a straight line, and the principle of betweenness for points on a line. These concepts are not made quite precise in Euclid, but we can see some of their properties in his language.
About Euclid’s definitions:
Euclid attempts to define all concepts, but without complete success it seems. Indeed some of these ideas were not made clear for centuries after him, but he does make important progress. In particular he tries to distinguish objects of different dimensions, and gives some hint of the modern way of doing this. In definition 1, he calls a “point” something with “no part”, which is an attempt to define a zero dimensional object. We prefer now simply to say we are given certain fundamental objects called points of which all other objects of study will be composed. We don’t define the points, we just say they are given and we give some of their properties.
In definition 2, Euclid defines a “line”, [we would call it a “curve”, allowing it to possibly be straight], as something with only length but no breadth, an attempt to say it has only one dimension. This is not a precise definition, but in definition 3 he says that the extremities of a line are points. This gives a clue to the modern inductive description of dimension. Namely we have some way to recognize the border of an object, and an object should be one dimensional if its border is zero dimensional, i.e. if the border consists of a finite number of points.
The same pattern occurs in definitions 5 and 6, where in 5 a surface is something two dimensional, and in 6, we see that the border of something two dimensional should be one dimensional. This is a general pattern, that a border should have dimension one less than the thing it borders.
Today we focus more on the relationship between our objects than on the nature of those objects. So in different situations, what are called points or lines or surfaces could be different things, but in all situations the points will be related to the lines in the same way. I.e. whatever the points are, they should border the curves, and whatever the curves are they should border the surfaces, etc…
So today mathematicians tend to ignore Euclid’s definition 1, and to consider definitions 2 and 5 to be clarified by definitions 3 and 6. Unfortunately definition 4 of what it means for a curve to be straight, is not clarified by any additional property, and we will need one in Prop. I.4. The usual one taken nowadays as basic for straight lines is that two different lines which are both straight, can only meet in one point. This is related to Euclid’s 1st Postulate, that one can draw a straight line between any two points, but only if that means one and only one straight line, so this is the usual modern postulate. So to guarantee that two different lines can only meet once, we need more or less the converse of Euclid’s 1st postulate. I don’t know the original Greek, so I do not know if the words “a straight line” used in that postulate mean “exactly one straight line”.
Terminology that Euclid used differently from mathematicians today
Euclid seems to mean by “straight line” only a finite portion of an infinite straight line. Today we call such finite pieces of lines, line segments, or finite line segments. When Euclid wants to speak of an infinite straight line, he speaks of a (finite) straight line being extended indefinitely or calls it explicitly an infinite straight line. So what he calls a line today we call a curve, what he calls a straight line today we call a line segment, and what he calls a line segment extended infinitely in both directions, or an infinite straight line, we just call a line.
Definition 8 describes an angle as the “inclination” made by two straight line segments which meet but are not in the same straight line. It is not clear to me whether they meet at an extremity, but apparently in that case he considers only the convex angle they make together. E.g. the outside of a 90 degree angle is not considered by him as a 270 degree angle. (Since he does not consider 270 degree angles, it is harder for him to “add” two 135 degree angles.) He defines a right angle as one of the angles formed by two lines that form equal angles. Presumably in this case the lines do not meet only at extremities, since they form more than one convex angle.
Definition 15 describes a circle, but again not quite completely. He says a circle consists of a point called the center, together with a collection of line segments all having that center as an extremity, and all having the same length. But he does not say whether all segments of that length are included, as presumably they should be. E.g. a semicircle seems to satisfy the description given, since all line segments from the center to any point of the semi circle are equal to one another.
We assume he meant that a circle is the figure formed by a center and a segment with that center as extremity, plus all other segments having the same center as an extremity, and which are equal to the first segment. Thus he includes the entire region on and within the circle, whereas today we mean by “circle” only what he calls the circumference or boundary of his circle. I.e. we take a center point A and a segment XY, and we consider the circle to consist of all those points B such that the segment connecting B to the given center A, is equal to the segment XY. It follows that two circles with the same center have either the same circumference, i.e. are the same circle, or else their circumferences are disjoint, i.e. have no common points at all. He is not quite consistent since later he says a circle cannot cut another circle at more than 2 points, apparently referring to their circumferences.
Euclid’s five postulates:
Here are the postulates Euclid explicitly stated (slightly paraphrased):
1. Given any two points, one can draw a straight line (segment) joining them.
2. Given a finite line segment, one can extend it continuously in a straight line, (presumably infinitely in both directions).
3. Given any point as center, and any other point (distance), one can describe a circle centered at the first point and with the other point on the circumference.
4. All right angles are equal.
5. If two lines cross a third line so as to make interior angles on one side total less than a straight angle (two right angles), then the two lines meet on that same side of the third line.
Note Euclid clearly assumes in postulate 5 that a line has two sides. Also there is nothing here asserting that parallel lines exist - rather this has the opposite flavor, guaranteeing that certain lines are not parallel. So this is not the usual parallel postulate I learned in high school. (Through a point off a line, there passes one and only one line parallel to the given line.)
This postulate will imply there is not more than one line parallel to a given line and containing a given point off that line. In the other direction, Euclid will actually prove there is at least one such parallel, using his “exterior angle” theorem.
The properties that Euclid used most without stating them concern how lines and circles meet each other. In modern mathematics we discuss these in terms of connectivity or separation properties. A set is convex if for every pair of points in the set, the straight line segment joining them is also in the set. E.g. a straight line segment is convex. Then Euclid seems to assume basic facts like the following: removal of a point other than an extremity separates a segment into two convex pieces. Removal of an infinite line from the plane separates the plane into two convex “sides”. Removal of the circumference of a circle from the plane separates the plane into two pieces, one of which: the inside, is convex, and the other: the outside is at least “connected” [in what sense?].
What do we mean by “separates”? We mean the segment joining a point inside to a point outside should meet the border which was removed. So if two points of the plane are on opposite sides of a line, then the segment joining them should meet the line. Thus the line separates the two ides of the plane, and forms the border of both sides. If one point is inside and another point is outside a circle, the segment joining them should meet the circle. We can say something about the shape of a circle if we agree that any (infinite) line containing a point inside a circle should meet the circle exactly twice. And we might be wise to agree that a circle that contains a point inside and a point outside another circle also meets that circle exactly twice.
Some of these facts about how circles and lines meet can be proved, and Euclid does so, but others cannot be proved. In general one can prove that circles and lines cannot meet more than expected, but I do not know how to prove that they do meet as often as expected, without more assumptions than Euclid has made. Today many people assume that lines correspond to real numbers, which does guarantee that lines and circles meet as often as expected, since the axioms for real numbers guarantee this. However most geometry books which make these assumptions about lines do not bother to explain the relevant axioms for real numbers, so to me not much clarity is gained, and perhaps some is lost.
Euclid has one postulate (#5) guaranteeing that two lines do meet under certain conditions, but he was criticized for centuries for including this postulate. It turns out he was right, as this postulate cannot be omitted without broadening the possible geometric worlds he was trying to describe. People were unable to imagine any other geometry than Euclid’s however for a long time where this postulate could fail. A Jesuit priest, Girolamo Saccheri, showed that if we deny Euclid’s 5th postulate then there would not exist any rectangles. This and other consequences seemed so impossible to Saccheri that he concluded Euclid’s axiom must be automatically true, and thus did not need to be stated explicitly. However, there is another plane geometry in which there are no rectangles, called hyperbolic geometry, and unless we assume Euclid’s 5th postulate we cannot be sure we are not in that world instead. Today the results Saccheri correctly deduced , but considered impossible, are regarded as theorems in hyperbolic geometry due to him.
So we regard Euclid’s stated definitions and postulates, plus the ones he used but did not state, as rules for the game we are going to play. They tell us what we can do, and we want to deduce as many consequences from them as possible, without violating the rules.
The problem of congruence
If two triangles have vertices A,B,C and X,Y,Z, a correspondence between their vertices, e.g. A→Y, B→X, C→Z, induces correspondences between the sides: AB→YX, AC→YZ, BC→XZ, and the angles: <ABC→<YXZ, <ACB→<YZX, <BAC→<XYZ.
If a correspondence between the vertices induces correspondences of sides and angles such that every side and every angle equals the one it corresponds to, we call the correspondence a “congruence”.
Notice a congruence must be given by a specific correspondence. It is not sufficient just to say two triangles are congruent, one must say what correspondence induces the congruence. E.g. the triangles ABC and XYZ may be congruent under the correspondence A→Y, B→X, C→Z, but not under the congruence A→X, B→Y, C→Z. Other very symmetrical triangles may be congruent under more than one correspondence, but we should always say what correspondence we mean.
Exercise: Given an example of two triangles that are congruent by more than one correspondence.
The first question we ask in plane geometry is when two triangles are congruent, given only a smaller amount of information. The basic criteria are sometimes called SAS, SSS, ASA, and AAS. E.g. “SSS” is shorthand for the fact that if a correspondence of vertices induces a correspondence of sides such that all corresponding pairs of sides are all equal, then all corresponding pairs of angles are also equal, and hence the triangles are congruent. “SAS” refers to the fact that if two corresponding pairs of sides are equal, as well as the pairs of included angles, then the triangles are congruent. Etc…
Once one knows these basic criteria, most geometry courses proceed in the same way, at least for while. Getting started thus means establishing these basic congruence facts. Some books assume them all, while some assume only a few of them and deduce the others. Euclid proves them all, but only by making some implicit assumptions that he has not included among his axioms. See if you can spot some of those assumptions.