Descartes’ La Geometrie actually proved over my head– something which I address at the end of this post. However, I was able to glean a few scattered notions as I trailed behind the master…
Probably what La Geometrie is most famous for today is its demonstrations showing that some physical lines and curves can be described by mathematical equations, and some mathematical equations can describe physical lines and curves. In case the reader misses the point, Descartes states this fundamental contention straight out on page 323 of the original printing…
“No matter how we conceive a curve to be described, provided it be one of those which I have called ‘geometric,’ it is always possible to find” […] “an equation determining all its points.”
Descartes also shows that complex equations which seem impossible to represent by physical curves can actually sometimes be broken down and restated in simpler equations which CAN be graphed.
I also read in the Second Book, page 316, that Descartes felt that the equations of mathematics could more precisely describe geometrical conditions than human hands could ever construct (try free-handing a series of perfect circles or perfectly parallel lines!).
In the same paragraph, he says that– unlike “the ancients”– we should not admit defeat in the face of complex curves– not if there is hope to come up with a formula or equation for them. For Descartes, this required only that the curve be produced by some variety of continuous or otherwise predictable motions– with every point on the curve capable of being derived from our knowledge of the points coming before or after it. Says Descartes…
“It seems very clear to me that: IF we make the usual assumption that Geometry is precise and exact while Mechanics is not” […] “THEN we have not more right to exclude the more complex curves than the simpler ones, provided they can be conceived of as described by a continuous motion or by several successive motions, each motion being completely determined by those which precede.”
Additionally, and I’m not sure if he was the first to state this, but Descartes said that … “Every equation can have as many distinct roots (values of the unknown quantity) as the number of dimensions of the unknown quantity in the equation.”
[BTW: I think I’ve read before that it was Descartes who CAME UP with the “exponent” notation (a small number to the upper right of the base number) for those values which are multiplied by themselves a certain number of times.]
Descartes explains the connection between exponent-value and number-of-solutions this way… Consider, he says, the equation from the end result and work backwards… If the solutions to the equation are… x=2, x=3, and x=4 (and yes he does use “x” to stand for the unknown– the first person to consistently use small letters from the end of the alphabet to stand for variables/unknowns!)… then if you work backwards and start rebuilding the original equation, you’d have…
x-2 = 0 and x-3 = 0 and x-4 = 0. Multiplying (x-2) (x-3) you get… x^2 – 5x + 6 = 0. Then multiplying THAT by (x-4) you get… x^3 – 9x^2 + 26x – 24 = 0… As you can see… the highest exponent of “x” is 3… and relatedly, the equation will have the same number of possible answers, in this case, x=2, 3, 4. [NOTE: not ALL answers always prove workable as real-world solutions].
One thing that I noticed while reading Descartes– and which has always struck me about the early mathematicians in general– is how fixated they were on the concept of “conic sections.” In their minds– in their fundamental mathematical worldview– they could hardly approach a parabola or ellipse without seeing it with their mind’s eye as a slice taken out of a cone.
This traditional approach to curves was not communicated at all to me when I was taught mathematics. Therefore, I think of curves on a page as just that– a curve on a page. I can’t explain exactly why, but this difference in fundamental conception has always intrigued me, helping to remind me that humans from different times and cultures and languages– as similar as we APPEAR to be– can actually be moving around in quite different worlds; in other words, the world I’m seeing is not the world they are seeing– they are using different lenses and filters from the ones I’m peering through.
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And now for a personal disclosure…
So I guess I was getting a little cocky. I’ve read about 125 books so far in my three year quest to digest 300 books. I’ve tackled Spinoza and Wittgenstein, delved deeply into Mach and Einstein and Schrodinger. Along the way, I’ve become one of the world’s number one internet sources for… 1) the interpretation of Rawl’s “veil of ignorance,” 2) the best short stories of Borges, 3) the comparison of Hunger Games and Animal Farm, and more.
But La Geometrie of Rene Descartes overwhelmed me. I’m a beaten man. My flounce has been trounced.
In my conceit, I did not approach the work through an annotated type of book or one with an accompanying study guide. And I chose not to read a book ABOUT Descartes’ Geometry, one which could explain to me the work in modern English, using the modern symbols and terms of today’s mathematics. Oh no. Not me– not the man who, in his own mind, recently completed some of the freshest, spot-on reviews of the poetical works of Holderlin and Rimbaud and Adrienne Rich. Only the original work, itself would do for this guy– which of course has the additional complication of being written in French. I condescended only so much as to obtain a dual-language version of the work, with the English text facing the facsimile pages of the original French text– I was excited by the prospect of reading the work just the way it appeared when published as a sort of appendix to Descartes’ great work, Discours de la Methode. I had this.
Descartes’ math was over my head from the very first page. A humiliating defeat.
Nevertheless, I must pick myself up out of the dirt and dust myself off and continue on… humbled yes, but not destroyed. You don’t carve mountains into gods without losing a few men (I’m thinking of Mount Rushmore here, and of the workers who plummeted to their deaths while working hard to release from the mountain the faces of our American gods). The trick in life is to walk that wire between confidence and humility– which coincidentally can also be the dividing line between bravery and stupidity.
One day, perhaps, when I am better prepared, I will come back to La Geometrie of Descartes– but most definitely armed with study aids. Or perhaps this will be a work I will never fully absorb in its original form (ah, pride’s wound doth stingeth!). But if I do return to La Geometrie, it will not be during the course of my walk (really a very long sprint) through The 300. I would need a few months of prep work (geometry review, french review, history of mathematics review, etc…) just to hope to understand some majority percentage of the ideas presented in the book.