Cartesian Space vs. Euclidean Space

[NoahsArk]

TL;DR Summary

Why was the Invention of Cartesian space such a jump forward?

For a while I've been trying to get a better understanding of how Descartes' invention of Cartesian space revolutionized math. It seems like an invention on par with the invention of agriculture in how it led to human progress. I am still having trouble, though, pinpointing examples of what can be done in Cartesian space that can't be done in Euclidean space, or which can only be done with much greater difficulty in Euclidean space than it can be in Euclidean space. Part of my confusion might be conceptual, and the other part might come from not knowing the history.
I'm reading the book, Infinite Ascent: A Short History of Mathematics, where there is a chapter on analytical geometry which discusses the invention of Cartesian space. Euclidean space had points just like Cartesian space does, but in Cartesian space the points have a pair of x y coordinates. The author talks about how the invention of a "map of the world" preceded the invention of Cartesian space (and maybe this was partly an inspiration to invent Cartesian space?). Here is a quote from the book:

"With the introduction of points having coordinates, and a formula for distance, the first step in a great drama of identification begins: If points correspond to pairs of numbers, there is no reason that geometrical figures more complicated than points- lines and curves - might not correspond to mathematical objects more complicated than pairs of numbers. The geometrical world then becomes coordinated with the world of numbers. A straight line in a Euclidean plane is simple a straight line, of no address, and in no way distinguished from any other straight line. In a Cartesian plane they get the dignity of a fixed identity. Two numbers are again required: slope and intercept."

The above I found very thought provoking. I paused reading the book, though, to try and find concrete examples of what can be done now that couldn't have been in Euclidean space. For example, there were also lines in Euclidean space, and you could find the distance of lines using the Pythagorean theorem. Cartesian space seems to have led to the distance formula for finding the distance between two points, but why couldn't this have been done in Euclidean space? Is it because you'd have to physically measure the horizontal and vertical distances corresponding to a diagonal line (i.e. the A and B sides of the triangle) before you could use the Pythagorean theorem to calculate its distance? In Cartesian space the two points both already have coordinates from which you can get the vertical and horizontal distance between the points, but in order to assign coordinates in the first place you still need to do some form of measurement.

Another example is finding area, say under a line like y = 2x at the point x =4. The height at that point is 8 and the base is four so you can use the triangle formula for area to get an area of 16. The Greeks knew how to do this already, though.

Also, the very concept of having a pair of numbers describing horizontal and vertical distance seems to far pre-date Descartes. The system of latitude and longitude, for example, was invented by Eratosthenes in the third century BC.

So, I'm still trying to get a better understanding of what made the invention of Cartesian space such a leap.

 

[fresh_42]

Here is what Dieudonné has to say about it.

As for "geometry", what was understood by it at the beginning of the seventeenth century was the geometry of the Greeks, as it had been handed down by Euclid, Archimedes, Apollonius and Pappus. In all problems concerning "quantities" (lengths, areas, volumes, angles), it had never been separated from algebra; for example, Apollonius's study of conic sections is carried out primarily by means of their equation in relation to two conjugate diameters.

The discovery of the method of Cartesian coordinates (which was called "analytic geometry" at the end of the eighteenth century) by Descartes and Fermat considerably increased the importance of algebra in geometric questions, since it made it possible, among other things, to conceptualize algebraic curves and algebraic surfaces, which are more general than conic sections or quadrics.

Until about 1680, however, the only general concept that was used was the degree of a plane curve, and the study of such curves did not go beyond a few special cases. At least the methods of solving algebraic equations by intersecting plane curves (such methods were highly valued since the Greeks) got people used to operating with equations of algebraic curves and to "eliminating" unknowns from such curve equations in order to determine intersection points of these curves.

Following this argument, I was looking for examples of something that is hard to solve by pure geometric means and more or less easily doable in coordinates.
https://www.physicsforums.com/threads/math-challenge-march-2021.1000319/ (problem #8)
or
https://math.stackexchange.com/ques...late-area-of-astroid-represented-by-parameter
or
https://www.physicsforums.com/threads/ready-for-a-summer-math-challenge.950689/ (problem #5)

Coordinates opened up the door to problems that classical geometry could hardly approach. The toolset was what was revolutionized. You may not forget the time and the classical education mathematicians then had.

 

[NoahsArk]

Thanks @fresh_42. I will check out Dieudonne.

I was thinking yesterday that there must have been equations representing relationships between quantities long before Descartes. What seems part of the novelty is having equations for shapes/geometrical objects like lines, circles, etc. Without an equation for a parabola, for example, I don't see how you could calculate the area underneith it. To get the limit of a Riemann sum, for example, you have to sum up the area of an infinite number of rectangles underneith the curve. All of these rectangles need a height to be able to calculate the area. Each height is a function of the width at that point. So, without a function, you'd need to take measurements of an infinite number of heights along the curve instead of plugging numbers into the formula for the limit of a Riemann Sum. Seems you can only do this when you have a relationship between the x and y values of the points on the parabola.

I haven't looked into it yet, although based on what I've read in the past I am assuming that the first uses of Cartesian space where for studying the path of projectile motion of objects which takes the shape of a parabola. Finding the speed at a given moment of the object or area under the parabola would be very hard without Cartesian coordinates.

 

[fresh_42]

Well, that's basically true, but don't underestimate the capabilities mathematicians already had in the 17th century. Archimedes calculated $3+\dfrac{10}{71} <\pi <3+\dfrac{10}{70}$ ~250 BC!

Not sure, whether projectile motion was the driving application, but physics and mathematics evolved in parallel back then, far more than today. The crucial point of using a coordinate system was the link between geometry and algebra. What was once restricted to Euclid's Elements became algebraic expressions! That was the revolution, less what could be done by it or not.

 

[NoahsArk]

What was once restricted to Euclid's Elements became algebraic expressions!

If you can give a few examples that would be helpful. Thanks

 

[fresh_42]

If you can give a few examples that would be helpful. Thanks

I thought I did, three of them.

See the second quote of Dieudonné. Coordinates allowed the description of algebraic curves that weren't conic sections or quadrics.

 

[fresh_42]

Here is another example:

Since the beginning of calculus in the seventeenth century, mathematicians have tried to calculate the length of arcs: this is the problem of rectification. Archimedes had already dealt with the case of the circle. The general problem quickly proved difficult, and Descartes considered the rectification of algebraic curves in closed form impossible, but individual results were achieved.

Rectification of elliptic curves in a closed form wasn't even a question without describing them algebraically because "closed form" would be meaningless.

 

[NoahsArk]

@fresh_42 thank you

 

[fresh_42]

Dieudonné further states that there were three major steps in the development of differential geometry:
1. Descartes and Fermat developed the coordinate method.
2. Gauß introduced curved coordinates.
3. Riemann extended Gauß's ideas and introduced infinite-dimensional spaces.

This may not be what you were looking for, but it supports the assessment of a "revolution" in geometry.

(I love that book!)

 

[NoahsArk]

@fresh_42 These ideas are very subtle and I hope one day to have a better handle on them and to be more proficient in understanding the development of math ideas. I believe this is a difficult task given the long history.

 

[fresh_42]

@fresh_42 These ideas are very subtle and I hope one day to have a better handle on them and to be more proficient in understanding the development of math ideas. I believe this is a difficult task given the long history.

If you take a look at physics, no matter where, you will find yourself within coordinates all over the place, from simple vector calculations in mechanics, over differential geometry in relativity, up to groups of invariants in quantum field theory. I would go as far as to say that studying physics is the adoption of coordinate acrobatics (duck and cover).