Roots of Unity

## In Which Omar Khayyam Is Grumpy with Euclid

My math history class is currently studying non-Euclidean geometry, which means we’ve studied quite a few “proofs” of Euclid’s fifth postulate, also known as the parallel postulate. I’ve written about this postulate before. There are many statements that are equivalent to the parallel postulate, including the fact that parallel lines in a plane are equidistant. This [...]

Roots of Unity

## Look Ma, No Zero!

As I told my class on Thursday, the theme of the first week of our math history course was “easy algebra is hard in base 60.” We started the semester in ancient Mesopotamia, trying to understand Babylonian* mathematical notation and decipher Plimpton 322, an enigmatic tablet from about 1800 BCE. The Babylonian number system uses [...]

Roots of Unity

## What’s the Deal with Euclid’s Fourth Postulate?

In February, I wrote about Euclid’s parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry. I included the text of the five postulates, from Thomas Heath’s translation of Euclid’s Elements: “Let the following be postulated: 1) To draw a straight line [...]

Roots of Unity

## Felix Klein on Mathematical Progress

I just finished reading a set of lectures the great mathematician Felix Klein delivered at the 1893 World’s Columbian Exposition in Chicago. The lectures are now in the public domain, and you can download them for free here. (Unfortunately, not all the mathematical notation survived digitization, so a good amount of creative interpretation—also known as [...]

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