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Long-time listener, first-time caller here. I've got what I guess is a history of math question. I've been teaching a college course on Euclid's Elements (surprisingly fun, btw). I've been using David Joyce's (Joyce is a Professor of Math/CS at Clarke University) online guide to the Elements as a resource, and in his commentary on one theorem (VI.12), he says the following: "Descartes (1591–1661) is well known for his coordinate geometry which he and Fermat developed in the 16th century...[But] Descartes was equally interested is using geometry to solve algebraic problems, though using a method quite distinct from that in the Elements." He then goes on to give some examples (e.g., construct a line whose length is equal to the product of the lengths of two given lines).

On the one hand, it's perfectly clear what is meant. If you can convert a geometrical question into an algebraic one using Cartesian coordinates, then you can presumably convert an algebraic problem into a geometrical one. (Side question: Do we need to place some constraints on the kind of algebraic problem at issue? It's hard to imagine what the geometrical analogue to an algebraic problem involving complex numbers, or one involving more than three variables, would be.) What I've been puzzled about is this: It's perfectly clear why it's useful to convert a geometrical problem into an algebraic one (as anyone who has tried to work through Euclid will be painfully aware). But I struggle to see why it would be useful to be able to do the opposite. Are there examples of problems which are easier to solve in their geometrical form than in their algebraic form? Or does the interest of the topic lie elsewhere? E.g., maybe if we can convert all geometrical questions into algebraic ones and algebraic questions into geometrical ones, that sort of supports the idea that algebra and geometry are different ways of expressing one and the same underling mathematical reality?

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  • $\begingroup$ You may also be interested in sites History of Science and Mathematics Stack Exchange and Mathematics Educators Stack Exchange $\endgroup$ Commented Dec 10 at 23:50
  • $\begingroup$ Not all algebraic problems can be translated within the scope of classical rules. The Delian problem for instance, extracting the cube root of $2$, needs more flexibilitity. $\endgroup$ Commented Dec 10 at 23:55
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    $\begingroup$ Thank you both! I hadn't realized that there was a history of sci/math stack exchange as well, probably my question would have made more sense there. But thanks for the example of an algebraic problem that can't be translated, lulu. $\endgroup$ Commented Dec 11 at 0:12

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None of the existing answers mention hard limitations of geometric constructions. Compass-and-straightedge constructions can only construct lengths that can be obtained from given lengths by using the four basic arithmetic operations (+,−,·,/) and square-root. So there is no method to construct given any length $x$ some other length $y$ such that $(y/x)^3 = 2$. If you allow extra kinds of geometric constructions, there will be similar limitations.

These limitations were only known to mathematicians after Galois, so it is not surprising that mathematicians before that might have believed that every algebraic problem may have a geometric solution.

Nevertheless, there are some interesting examples where synthetic geometry is more conceptually fruitful than Cartesian geometry. For example, Dandelin spheres.

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Arguably in some scenarios the geometric solution provides some extra insight into how the result would apply in physics or engineering. But I suspect that's not the main reason for Descartes' inclination.

I suspect the reason is mostly philosophical. The ancient Greeks of Euclid's time regarded geometry to be the foundation of mathematics. Notice that even in Book VII of the elements, which deals with number theory (i.e. the mathematics strictly of positive integers), the diagrams illustrate positive-integer quantities as line segments of various lengths. This geometry-centric philosophy persisted to a significant degree even into Descartes' time. For example, this article confirms something I recall reading earlier about Isaac Newton, namely "Newton was convinced that only geometrical (as opposed to algebraic) proofs can be considered certain."

As you alluded to, the geometrical approach to algebra has significant limitations, although I don't think it's primarily to do with the number of variables (note that Book II, Prop. 12 describes the relationship between four different line segments). A bigger issue has to do with regarding magnitudes as inherently geometric. For example, the reason we call $x^2$ "x squared" and call $x^3$ "x cubed" is because in the ancient Greek conception, non-integer (in today's terms "real-valued") magnitudes were (as far as I know) thought of strictly in geometric terms. Thus if $x$ was the length of a line segment, then what we would call $x^2$ was for them the area of a square of side-length $x$ and $x^3$ was the volume of a cube of side-length $x$. The problem with this, of course, is that it gives you no way to talk about higher powers of $x$ (powers greater than 3).

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I think geometric interpretations can be quite helpful in solving some inequalities. There's quite a nice geometric proof for the Quadratic Mean - Arithmetic Mean - Geometric Mean - Harmonic Mean inequality. Some other inequalities such as Holder and Minkowski benefit from arguments about geometric convexity.

Several other problems can be solved geometrically. Though, they are often contrived to do so. As an example,

If $a$, $b$, $c$ are positive reals, then $\sqrt{a^2 + ac + c^2} \le \sqrt{a^2-ab+b^2} + \sqrt{b^2-bc+c^2}$

You can view this as a geometric problem. Consider a quadrilateral $ABCD$, where $\angle ADB = 60^{\circ}$, $\angle BDC = 60^{\circ}$, and let $AD = a$, $BD = b$, $CD = c$. An application of the cosine rule turns this into the triangle inequality for $\triangle ABC$.

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An example of a problem easier to solve synthetically:

Consider Euclid's very first proposition: to construct an equilateral triangle on a segment. The synthetic construct by hand takes 10 seconds and goes to the heart of the matter via congruent circles. Algebraically, depending on the segment and its inclination, the calculation by hand could be a few minutes or longer. There are many other examples where the geometric approach is direct and efficient, and is complete by the time you would have written down the relevant equations to just start an algebraic approach.

A different kind of answer to your question can be seen in a frequent request on this forum: "I can solve [this problem], but does someone see a synthetic approach?" Many of us have a desire to find a geometric solution, even if we have other solutions in hand. Descartes may have been like that too.

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Euclid's first proof of the Pythagorean theorem, which shows that the perpendicular from the right angle vertex to the hypotenuse divides the square on the hypotenuse into two rectangles each of which has the same area as one of the squares on the legs, gives a method of constructing a square with the same area as a given rectangle.

However, it requires knowledge of which side of the rectangle is longer.

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