Introduction to Euclid's Geometry

Around 300 BCE, the Greek mathematician Euclid collected all the known geometrical facts of his time into a single set of 13 books called the Elements. Instead of just listing rules, Euclid built geometry like a chain of reasoning: he started from a small list of definitions, then a small list of axioms (common notions) and postulates (assumptions specific to geometry), and used only logical reasoning to prove every other result, called a theorem.

Euclid gave 23 definitions in Book 1 of the Elements. Some of the most important ones used even today are:

• A point is that which has no part.
• A line is breadthless length.
• The ends of a line are points.
• A straight line is a line which lies evenly with the points on itself.
• A surface is that which has length and breadth only.
• The edges of a surface are lines.
• A plane surface is a surface which lies evenly with the straight lines on itself.

Worked example: Explain why "a line is breadthless length" is only a description and not a strict mathematical definition.

Solution: To define "breadthless" we would need to define "breadth" first, and to define "length" we would need still other terms. Euclid's definitions describe these basic terms using other terms that are themselves not formally defined. In modern geometry, point, line and plane are simply taken as undefined terms on which the rest of geometry is logically built.

Euclid called certain self-evident assumptions, common to all branches of mathematics and not just geometry, axioms or common notions. Unlike postulates, which are specific to geometry, axioms are general truths that need no proof. The axioms used in the current textbook are:

1. Things which are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.

Worked example: In a figure, AB = PQ and PQ = XY. Show that AB = XY, and name the axiom used.

Solution: Since AB = PQ and PQ = XY, both AB and XY are equal to the same segment PQ. By Axiom 1 ("things which are equal to the same thing are equal to one another"), AB = XY.

Worked example: If AB = CD, prove that AB + BC = CD + BC.

Solution: AB = CD is given. Adding the same segment BC to both sides (equals added to equals), by Axiom 2, AB + BC = CD + BC, that is, AC = BD.

While axioms are general, Euclid's postulates are assumptions specific to geometry. Euclid stated five postulates:

1. Postulate 1: A straight line may be drawn from any one point to any other point.
2. Postulate 2: A terminated line (line segment) can be produced indefinitely.
3. Postulate 3: A circle can be drawn with any centre and any radius.
4. Postulate 4: All right angles are equal to one another.
5. Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Postulate 1 tells us that a line can be drawn joining two given points; combined with axiom-based reasoning, Euclid also concluded that only one such line can pass through two distinct points, a result known as Theorem 5.1.

Worked example: A line segment AB is given. Using Euclid's postulates, describe how to obtain a full line through A and B that extends beyond both points.

Solution: By Postulate 1, the segment AB can be drawn joining A and B. By Postulate 2, this terminated line AB can be produced indefinitely on both sides, beyond A and beyond B, giving the complete straight line.

Euclid's fifth postulate is more detailed than the other four because it tells us exactly what happens when a straight line crosses two other straight lines.

Suppose a straight line n falls on two straight lines l and m, forming a pair of interior angles on one side of n — call them ∠1 and ∠2. Euclid's fifth postulate says:

• If ∠1 + ∠2 is less than two right angles (180°), then lines l and m, when produced indefinitely, meet on the side where this angle sum is less than 180°.

If the interior angles on both sides of n add up to exactly 180° each, the postulate's condition for meeting is never satisfied on either side, so the two lines never meet, however far they are produced — this is the everyday idea of two lines running parallel to each other.

Worked example: A line n falls on two lines l and m, making interior angles of 85° and 90° on the right side of n. What can you conclude about lines l and m?

Solution: The sum of the interior angles on the right side is 85° + 90° = 175°, which is less than 180°. By Euclid's fifth postulate, lines l and m will meet on the right side of n when produced indefinitely.

Worked example: A line n falls on two lines l and m so that the interior angles on the right side are 90° and 90°, adding to exactly 180°. What can you conclude?

Solution: Since the sum is exactly 180°, not less than 180°, the condition in Euclid's fifth postulate for the lines to meet on the right side is not satisfied. The interior angles on the left side also add up to 180°, so the condition fails there too. Hence lines l and m will never meet, however far they are produced.

In Euclid's system, once definitions, axioms and postulates are laid down, no new geometric fact may simply be assumed from a diagram; it must be logically deduced using only what has already been accepted or previously proved. A statement proved this way is called a theorem, and once proved, it becomes available to prove further theorems.

Worked example (Theorem): If a point C lies between two points A and B such that AC = BC, prove that AC = (1/2)AB.

Solution: Since C lies between A and B, AC + CB = AB (the whole segment AB equals the sum of its two parts). It is given that AC = BC. Adding AC to both sides of AC = BC gives AC + AC = BC + AC, so 2AC = BC + AC. Since BC + AC = AB, we get 2AC = AB, so AC = (1/2)AB. Hence proved.

Worked example (Theorem 5.1): Prove that two distinct lines cannot have more than one point in common.

Solution: Suppose, if possible, two distinct lines l and m have two points, say P and Q, in common. Then both l and m pass through the two distinct points P and Q. But by Euclid's Postulate 1, only one line can pass through two distinct points. This contradicts the assumption that l and m are distinct lines through P and Q. Hence, two distinct lines cannot have more than one point in common.