# How the heck did people do geometry before the Descartes?

So as I understand it, Rene Descartes invented the coordinate system of an x- and y-axis in the 1600s. Obviously, pretty complicated math had been done way before that. Trigonometry is necessary for the most basic engineering and architecture. But thinking about trigonometry without having a set of axes to work with is tough.

Euclidean geometry is difficult too. How do you calculate stuff without using x and y? It seems like it makes algebra impossible - and wikipedia hints at this, saying that the Cartesian plane was a way of uniting algebra and geometry. I can’t even separate the two in my mind - how does geometry work without algebra?

And while we’re on the topic of old-timey maths, how did people do math before the Arabic numerals and decimal system became standard? Trigonometry using I, V, X, C, etc and no decimals just sounds close to impossible.
Edit: Um… not “the Descartes.” Just “before Descartes.” If someone could change that, I’d be a satisfied customer.

Guess and check?

By thinking. You start with a text problem. You formulate the way through to the solution as a series of steps. And voila.

Here’s how you do geometry:

And here are is an example of Babylonian quadratic problems and solutions:

http://it.stlawu.edu/~dmelvill/mesomath/Rectangular.html

Pick up a copy of Euclid’s Geometry and see – you can get the translation from Dover Books. It’svery straightforward. Geometry doesn’t need a coordinate system at all, although that is an immensely powerful tool.

Have you ever taken a high school geometry course or the equivalent? It seems to me that most simple geometric problems (for an appropriate notion of “most”) are solved without casting points into (x, y) coordinates and then chasing through algebra. What sort of examples are you thinking of which you cannot see how the ancients would have dealt with?

That was certainly the case in my high school geometry class. Almost all of it consisted of: given these shapes and angles, prove these facts. All of this relied on the various Euclidean axioms and rules derived from them.

Here is an example of the kind of problems we did in high school geometry. No coordinates needed.

To his credit, Descartes’ coordinate system provides a way of formally uniting Euclidean geometry with algebra, which hadn’t been possible in classical Greece and was perhaps the single most pivitol advance of modern mathematics.

To do multiplication in Roman numerals, you break down the parts of the numbers, multiply each by each and then total them up. So CXXVI multiplied by LIX would be C(L) +C(IX) +X(L) +X(IX) +X(L) +X(IX) +V(L) +V(IX) +I(L) +I(IX). Note that that’s still what you do to multiply any two large numbers but the decimal system streamlines things considerably.

The Ancient Romans did their sums using abacuses, not with Roman numerals.

HS geometry, AFAIK, doesn’t usually include the details on how we get the areas and volumes of things, beyond just saying “They imagined slicing it up into very very small bits…”. That’s reasonably intuitive, but doesn’t answer the question of how they actually proceeded to to do it. Nor can it, since a great many students in those classes have not mastered sufficient algebra to understand such details, not least how you can end up with an infinite number of slices, each zero in value, and compute an actual number. The details are quite beautiful when you do learn them; if they could do this for more kids of early HS age there might be less math-phobia and innumeracy around.

Can I admit something? I’ve never actually taken a high school geometry course or the equivalent.

There are others that agree with you. There’s an assertion that kids are not taught the beauty of math. They’re actually just taught the mechanical rules and terminology of it – which of course, most kids will detest.

Paul Lockhart wrote a beautiful essay about this problem. Click on the A Mathematician’s Lament pdf link at: http://www.maa.org/devlin/devlin_03_08.html

I wish every teacher and parent could read and absorb the wisdom from that short article.

Lucky you. I’ve found it difficult to unlearn.

Yeah - the simple ones are doable. I want to know how to build an aqueduct. How do you figure out the engineering behind the dome of the Pantheon? And if you only want basic problems, is there any way to do even simple vector problems with Roman numerals and no coordinate system? How do you do the trigonometry behind basic siege weapons to figure out where that rock you’re chucking is going to end up?

Naita, those answers are fascinating. Are there examples of more complicated applied math?

Before Descartes’ time math mostly was geometry, in the style of Euclid.* Algebra was far less developed, and, indeed, only arrived at something like its modern form in Descartes’ work. People like Archimedes and Galileo would solve problems that we would now do with pure algebra by using geometrical arguments. Actually, I understand that Newton still did a lot of the math in the Principia this way, even though he was familiar with Descartes’ work.

When I was at high school we spent several years on the Euclidean style geometry before ever getting to the Cartesian stuff. Actually, it was my favorite area of math (not that that is saying much!). It was all about things such as congruent and similar triangles, parallel lines, etc., and proving that different angles or line segments were equal or unequal to one another. For the most part, we did not deal with actual values for lengths or angles at all (except for special cases like right angles), but just with equality or inequality.

WoodenTaco seems to have a whole field of basic math missing from his (her?) education. I doubt whether it would be practical to teach him about it here.

*Not to be confused with Euclidean geometry in the modern sense, i.e., flat-space geometry where parallel lines meet at infinity (i.e., Euclid’s fifth axiom holds). Cartesian geometry is (or was, before Bolyai, Lobachevsky, and Gauss got into the act) still Euclidean in this sense, but it is not geometry done in the Euclidean, non-algebraic and non-numeric, style.

I’m not sure why you make that assumption. I took 9th grade Euclidean geometry too, and did proofs of similar triangles. I guess I mistitled my post - I want to know how you do engineering. Proofs are clear. I’m not sure how to do applied math if all you have is proofs.

Heh, actually, it perhaps wasn’t the best thing to say. Actually, much of the condescension towards high school geometry in Lockhart’s Lament rings true with my own experience from observing others (students developing this weird conflation of Geometry and Proof, as though the practice of reasoning to establish a fact was only something mathematicians did in this one narrow context [and in this one bureaucratic style at that], the rest of math still being just about carrying out set calculation procedures…). Yeah, it’s a head-noddingly good read.

I should add that my high-school Euclid-style geometry did not get into curves apart from circles and circular arcs. However, it must be possible to deal with at least conic section curves in this style. Ancient mathematicians such as Appolonius worked on conic sections, and Galileo’s work on parabolic trajectories, and Kepler’s on elliptical planetary orbits, must all have been done without the aid of Cartesian style algebraic geometry. I never learned how to do any of this, though.

A well-known example of applied math using just 9th grade geometry is Erastosthenes method for determining size of the Earth:

The only extra thing he did that’s not emphasized in high-school geometry was assign a number to the distance between the cities of Alexandria and Syene. Most proofs (constructions) in school geometry work without measuring a specific length to any segment. So the only leap to make geometry “applied” in the real world was the distance measurement – not exactly a quantum leap in human reasoning. No x,y Cartesian coordinates needed.

Following up on my previous post, the other misconflation is possible, I suppose: thinking that “directly geometric” reasoning is just good for proofs and not calculations, as such. You can reason directly geometrically to calculate lengths and angles in just the same way as you might reason directly geometrically to establish a conjecture, without having to switch over to some imposed system of Cartesian coordinates and work purely numerically. It’s just the same sort of thing.

What’s a specific engineering-inspired math problem which perplexes you?

As I recall in my youth, the order in which basic math was taught was arithmetic, algebra, geometry, then calculus (in which the Cartesian coordinate system was introduced). You don’t need the Cartesian system to do geometry. Euclid was doing geometry about two thousand years before Descartes. And you can certainly measure things just fine without the Cartesian system of coordinates.