Descartes and the Terror of the Negative

When René Descartes first described the system we’ve turned into Cartesian coordinates he didn’t put it forth in quite the way we build them these days. This shouldn’t be too surprising; he lived about four centuries ago, and we have experience with the idea of matching every point on the plane to some ordered pair of numbers that he couldn’t have. The idea has been expanded on, and improved, and logical rigor I only pretend to understand laid underneath the concept. But the core remains: we put somewhere on our surface an origin point — usually this gets labelled O, mnemonic for “origin” and also suggesting the zeroes which fill its coordinates — and we pick some direction to be the x-coordinate and some direction to be the y-coordinate, and the ordered pair for a point are how far in the x-direction and how far in the y-direction one must go from the origin to get there.

The most obvious difference between Cartesian coordinates as Descartes set them up and Cartesian coordinates as we use them is that Descartes would fill a plane with four chips, one quadrant each in the plane. The first quadrant is the points to the right of and above the origin. The second quadrant is to the left of and still above the origin. The third quadrant is to the left of and below the origin, and the fourth is to the right of the origin but below it. This division of the plane into quadrants, and even their identification as quadrants I, II, III, and IV respectively, still exists, one of those minor points on which prealgebra and algebra students briefly trip on their way to tripping over the trigonometric identities.

Descartes had, from his perspective, excellent reason to divide the plane up this way. It’s a reason difficult to imagine today. By separating the plane like this he avoided dealing with something mathematicians of the day were still uncomfortable with. It’s easy enough to describe a point in the first quadrant as being so far to the right and so far above the origin. But a point in the second quadrant is … not any distance to the right. It’s to the left. How far to the right is something that’s to the left?

We have to test our empathy to quite understand the problem. Obviously a point to the left is minus some distance to the right; it should be represented with a negative number. We live in a time that’s grown comfortable with negative numbers, that’s familiar enough with them that they hold no particular terrors. Descartes, genius though he was, was not so fortunate. Negative numbers were still a bit mysterious. Negative numbers were understood to … let’s call it exist, particularly since they corresponded so nicely to a deficit of something and so proved themselves useful in accounting. But they were still somewhat alien, and mathematicians — never mind lay people — had not yet quite worked up the intuitive understanding that since 3 is less than 5, then minus 3 must be greater than minus 5. That a negative number times a negative number is positive could produce lively, angry arguments.

So to the extent they could, people avoided dealing with negative numbers. This would be done by rewriting equations so that only positive numbers have to be dealt with. An equation one was supposed to work on would be broken up into many cases, and terms moved to the other side of the equals sign, in order that negative numbers never needed to be seen. Algorithms would split into positive and negative branches. In J E D Williams’s From Sails To Satellites: The Origin And Development Of Navigational Science, Williams mentions how a set of three navigational equations would be written by Age of Exploration sailors as six, expressing some wonder at this over-complication of their task. I can’t tell whether Williams recognized the conceptual breakthrough which lets us simplify things.

But by this breaking up of the plane into quadrants, Descartes was able to work entirely in comfortable positive numbers: in the second and third quadrants, measuring how far to the left of origin points were, and in the third and fourth quadrants, measuring how far below the origin they were.

Today, we just accept negative numbers as a matter of course, at least past about the point where we learn the rules of multiplying two negative numbers together and somehow getting a positive out of it. Most either accept that idea, or come to accept it after getting some practice with the geometric interpretation of “negative two” as something as far to the left as “positive two” is to the right of the origin, or post the occasional and barely understandable screed to math-oriented newsgroups and web forums, where the can be safely laughed at except by the group’s newest young-enthusiast regular, who feels the need to patiently instruct the cranks until they see the error of their ways, a tradition upheld by the regulars until they see the error of this way.