History of geometry
The word geometry comes from Greek words meaning earth measuring. People needed it to mark farm boundaries again after river floods, to build houses and to plan temples and fire altars.
- Egypt and Mesopotamia: rules for areas of fields and volumes of grain stores, mostly as recipes without reasons.
- India: the Harappan towns (about 2500 BCE) had well-planned streets and bricks in fixed ratios. Later the Sulbasutras (about 800–500 BCE) gave exact rope rules for building altars.
- Greece: thinkers like Thales and Pythagoras began to ask why a rule is true. Around 300 BCE, Euclid in Alexandria collected geometry into a book series called the Elements, where each result is proved from simple starting facts.
Euclid's big idea is the method: accept a few obvious facts, then prove everything else step by step. This is how all of mathematics works today.
Squares in Baudhayana's Sulbasutras
The Sulbasutras are Indian texts about using a sulba (cord or rope) to build altars of exact shapes and sizes. The one by Baudhayana is among the oldest.
Making a square with a rope
Fix two pegs for one side. Using a rope, mark points at equal distance to get a right angle at each corner (for example with a knotted rope in the ratio 3 : 4 : 5), then complete the four equal sides.
Doubling a square
The rope stretched along the diagonal of a square makes a new square with twice the area. If the side is 1, the diagonal square has area 2, so the diagonal is √2. Baudhayana even gave a very close value of √2 (about 1.4142).
The diagonal rule
For a rectangle, the square on the diagonal equals the square on the length plus the square on the breadth. This is the same result we now call the Pythagoras theorem: d² = l² + b².
Definitions, axioms and the five postulates
Definitions
Euclid began by describing basic words: a point is that which has no part; a line is breadthless length; the ends of a line are points; a surface has length and breadth only. These descriptions use other words that are not defined, so today we treat point, line and plane as undefined terms that we understand from pictures.
Axioms (true in all of maths)
- Things equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide (fit exactly) are equal.
- The whole is greater than the part.
- Things which are double of the same thing are equal.
- Things which are halves of the same thing are equal.
Postulates (special to geometry)
- A straight line can be drawn from any point to any other point (and it is unique).
- A terminated line (segment) can be extended without end.
- A circle can be drawn with any centre and any radius.
- All right angles are equal to one another.
- If a line falling on two lines makes the interior angles on one side together less than two right angles (180°), the two lines, if extended, meet on that side.
An axiom is a common-sense fact used everywhere; a postulate is an assumption about geometry. A theorem is a statement proved from these. Example theorem: two distinct lines cannot have more than one point in common.
Equivalent form of the fifth postulate
Playfair's version: through a point not on a line, exactly one line can be drawn parallel to it. For 2000 years people tried to prove the fifth postulate from the other four and failed; this led to new, non-Euclidean geometries.
Try it yourself (practical)
Double a square: Cut a paper square of side 10 cm. Fold along the diagonal and measure it (about 14.1 cm). Draw a square of side 14.1 cm. Now cut two more 10 cm squares along one diagonal each. The 4 half-squares fit exactly inside the big square, long sides on its edges. So its area is double.
Rope circle: Tie a string to a pencil, hold the other end fixed and draw. That is postulate 3.
In the 3D: go to the last step. Predict: with angles 90° and 90°, do the lines meet? Then check.
Key formulas and definitions
- Axiom 1: if a = c and b = c, then a = b
- Axiom 5: the whole > any of its parts
- Postulate 1: two points → exactly one line
- Postulate 5: inside angles < 180° → lines meet on that side
- Playfair: one parallel through a point not on the line
- Baudhayana: diagonal² = length² + breadth²; diagonal square = 2 × original square
Worked examples
1. AB = PQ and PQ = XY. What can you say about AB and XY? Which axiom?
AB = XY. Things equal to the same thing are equal to one another (axiom 1).
2. Point C lies between A and B on a line, and AC = BC. Show that AC = ½ AB.
AC + BC = AB (C lies between A and B, so the parts make the whole). Since AC = BC, AC + AC = AB, so 2AC = AB and AC = ½ AB. (Things double of the same thing are equal.)
3. How many lines can pass through (a) one point, (b) two distinct points?
(a) Infinitely many lines. (b) Exactly one line (postulate 1).
4. A square altar has side 3 m. What is the area of the square built on its diagonal?
Area of the altar = 9 m². The square on the diagonal has twice the area: 18 m². (Diagonal = 3√2 m, and (3√2)² = 18.)
5. Prove that every line segment has one and only one midpoint.
Suppose AB has two midpoints C and D. Then AC = ½AB and AD = ½AB, so AC = AD (things equal to the same thing). This is only possible if C and D are the same point. So the midpoint is unique.
6. A transversal makes inside angles 85° and 90° on the right with two lines. Where do the lines meet?
85° + 90° = 175°, less than 180°. By postulate 5 the lines meet on the right side.
Common mistakes
- Mixing up axioms and postulates: axioms are common facts for all maths; postulates are Euclid's assumptions for geometry.
- Thinking a postulate is proved: postulates and axioms are accepted without proof; theorems are proved.
- Reading the fifth postulate backwards: the lines meet on the side where the inside angles add to LESS than 180°.
- Thinking a line has a thickness or an end: a line (in geometry) has no width and goes on forever both ways.