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Ancient Greek Geometry

Ancient Greek scholars were very interested in mathematics. One of the most famous is Euclid, who was born in 330 BC. Euclid wrote a textbook called the Elements which outlined many ideas about shapes in two and three dimensions, as well as the theory of numbers.

Many of Euclid's ideas are still used today. He studied points, lines, planes, triangles, squares, circles, and spheres. He used axioms and definitions to write careful proofs. But Euclid proved ideas very differently from a modern mathematician. He used arguments based on length, area, and volume where modern mathematicians just do algebra. For instance, when he wrote about fractions, he considered the proportional lengths of different line segments. Here is a more complicated example:

One of Euclid's propositions says,

    If there be two straight lines, and one of them be cut into any number of segments whatsoever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

Sam wants to know what this proposition means, so he draws a rectangle. The rectangle is "contained" by the two line segments on the top and along one side.

Sam and a rectangle

Sam cuts the top line segment into pieces.

a rectangle

Then he cuts up the whole rectangle.

three smaller rectangles

But the cut-up rectangles have the same area as the original rectangle!

Sam compares areas

Let's label the length of each line segment.

x (a1 + a2 + a3) = x a1 + x a2 + x a3

Now we can use algebra to record Sam's example as

    x (a1 + a2 + a3) = xa1 + xa2 + xa3

The original proposition let us cut the original line segment into "any number of segments whatsoever". So we could use algebra to write the original proposition as

    x (a1 + a2 + a3 + . . . + an) = xa1 + xa2 + xa3+ . . . + xan

Our algebra looks much simpler than Euclid's proposition, but it does not help Sam draw a picture.

Most Greek mathematicians after Euclid wrote in his style. They made careful proofs based on axioms and definitions, and they based their arguments on length, area, and other geometrical ideas.

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