Of the four classical mathematical arts, geometry is arguably the most significant to the development of modern mathematics. Simply put, geometry is the art of measuring lines and shapes against each other to reach conclusions about things like length, perimeter, area, or volume. Most mathematicians until about the seventeenth century depended entirely on geometry for their certainty in mathematical ideas. With that kind of philosophical weight, you can bet geometry has a fascinating history.
The earliest geometrical manuscripts we have from the West are, like those of arithmetic, from the ancient Egyptian and Babylonian civilizations. Both pursued geometry for its practical value (like they did for arithmetic), and even though they weren’t interested in abstract math like we are, they still ran into fundamental geometrical concepts that we still use today. For example, they knew about the Pythagorean Theorem: if I take a triangle with a square corner, the two short sides (when they’re squared) add up to be the area of the longest side (when it’s squared). Another example is the Egyptians’ and Babylonians’ knowledge of pi (π, about 3.14). They estimated pi’s value, but they did recognize that it takes about three times as much length to measure around a circle as it does to measure across it. You can see how these kinds of ideas would be natural for people to discover when working with real-life structures and materials.
Geometry gets very interesting once you step forward in time to Greek mathematical thought (c. 600 B.C.). Although practical application was sometimes the thrust of pursuing new mathematical ideas, the main concern of Greek mathematicians was for the ideas themselves. This concern was largely due to the thought of Plato, a philosopher who taught that ideas, not physical objects, are the ultimate reality. If you were a Greek geometer, imagine the excitement in knowing that you’re discovering reality with nothing more than your mind (and maybe a piece of papyrus)! For the Greeks, ideas like triangles, circles, lines, and points were the stuff of mathematical truth. Now, we Christians, who believe in the Incarnation, certainly believe that the physical world, not just the world of ideas, is real. Nonetheless, the Greeks were still onto something correct about abstract mathematical ideas. After all, although you could never draw a “perfect” triangle with a pencil, a “perfect” triangle can, in a way, still exist in your mind!
By the time the city of Alexandria was flourishing (c. 300 B.C.), Greek geometry was at its peak. It was also believed to be the most important math to study, because unlike today, the only kinds of rigorous proofs that existed at the time were geometric ones. That meant that the only way you could prove mathematical ideas was by constructing them with a compass and a straightedge, very much the same way you did if you took high school geometry.
Thanks to the Greek mathematician Euclid, we have a huge written summary of all Greek mathematical thought up through his time: Elements. Since geometry was the name of the game for the Greeks, Euclid shows all the mathematical ideas in Elements through geometric proof. Elements, being such an all-encompassing summary, became the standard book for math education (our high schoolers even studied some of Elements last year). It’s also estimated to be the second-most printed book of all time, next to the Bible. As you can imagine, that much popularity and circulation made Elements incredibly influential to mathematics. In fact, largely thanks to Euclid, the Greek idea that geometry is fundamental to math continued to be the norm in the Western world for over a thousand years.
But, after that millennium, some mathematicians started to question strongly the idea that geometry is fundamental—a shift that changed Western math forever…
I’ll leave that as a cliffhanger for next time. For today, I want to point out that geometry (although its fundamentality has been questioned since the time of the Greeks) is still fundamental to a solid math education: first, because it takes realities we observe in creation (triangles, circles, etc.) and turns them into tools we can use to learn more complex ideas about the real world; second, because it gives us access to the Greek mathematical thinking that still influences mathematics to this day.
In Christ,
Mr. Hahn
