# How could ancient Greeks study solid figures?

(1) Analytic geometry was invented by Descartes. Ancient Greeks studied geometry by drawing with straightedge and compass.

(2) Perspective was developed by Renaissance painters and mathematized by Desargues and others in the XVII century. Before that, people couldn't properly represent 3 dimensional figures.

But, for example, last three books in Euclid's Elements are about solid figures. In modern versions of the book, I see figures that I don't think were possible in the time of Euclid.

So, how did ancient Greeks study solid figures?

• The first thought I had was "Egyptian pyramids." Jul 29, 2021 at 12:09
• you mean Descartes invented analytic geometry? Jul 29, 2021 at 12:11
• You don't necessarily need accurate drawings to do sinthetic geometry. Also, there is nothing especially accurate about perspective: it's just a systematic way to draw a realistic trompe-l'œil.
– user562983
Jul 29, 2021 at 12:11
• The <a href="smarthistory.org/niobid-krater/">Niobid Krater</a> is one example of Ancient Greek use of perspective. In addition, the columns of many Greek temples, such as the Parthenon, were built tapering towards the top so they would appear taller. Jul 29, 2021 at 12:15
• @John Gowers, you can add links in comments by using [text to be displayed](link-adress) Jul 29, 2021 at 12:19

I think you might have made some implicit assumptions that aren't necessarily accurate.

For (1), you can actually do a lot of 3D geometry using straightedge and compass in appropriately chosen construction planes. You mention the last three books of Euclid, did you see how he constructed the 3D shapes in the text? The diagrams are mostly just guides anyway, and I'm not certain the original even had many. You were expected to go through the constructions and make your own.

For (2), that seems a little strong. Perspective in art was figured out later, but why is that the only 'proper' way to depict 3D images on a 2D medium? The Greeks understood orthogonal projection, and that is a very good way to display a 3D object, it's still used today in technical drawings. https://www.britannica.com/technology/orthographic-projection-engineering

And why are you assuming that they must have drawn their 3D shapes. People have been making 3D models for a very long time. https://www.georgehart.com/virtual-polyhedra/neolithic.html

My favorite Greek proof in 3D is Archytas' doubling of the cube (https://lsusmath.rickmabry.org/rmabry/live3d/archytas1.htm), which constructs $$\sqrt[3]{2}$$ by intersecting a cone, a cylinder, and a torus. It shows a skill at visualizing and understanding 3D shapes and their interactions that I don't think most mathematicians today could match, myself included, given the constraints (no algebra, no analytic geometry, no computer graphing...).

• (a) the orthographic projection as it is used in engineering was initiated by Gaspard Monge around 1780 (b) you say I assume they must have drawn their 3D shapes, when I am guessing rather the opposite (c) you downplay my assumptions saying "you can actually do a lot of 3D using straightedge and compass" but you finalize with "I don't think most mathematicians today could match given the constraints". Jul 29, 2021 at 15:21
• @FCardelle Monge described one technique using projections, but they were understood well before his time, e.g. the orthographic map projection. For the second part, I think you misunderstood my point - I was emphasizing that the Greeks DID do a lot of 3D geometry using the techniques described, and that their brilliance using the tools available to them is a tribute to how talented some of the ancient mathematicians were. There's a tendency to think that the ancients weren't as smart as we are today, but take a look at Archytas' proof and you'll see evidence of a first-rate geometric mind. Jul 29, 2021 at 18:24

You are asking about illustrations. A good sketch is very useful in geometry, and I usually insist upon it. It is not always necessary though. Greek geometers of Euclid's era were probably quite adept at visualizing these figures with no illustration at all.

The Archimedes Codex is a recent book documenting efforts to retrieve text and images from a copy of some works of Archimedes. The copy was probably produced in the 10th century AD or thereabouts. The years have not been kind to this book, and it was scribed more than a thousand years after Archimedes anyway. It might have been a faithful representation of the original though. I have tried to reproduce one of the images here.

The figure on the left is similar to one found on the ancient parchment. It is a diagram for Proposition 30 of On the Sphere and Cylinder, Book I. It represents a regular dodecagon inscribed in a circle, with another circle inscribed in the dodecagon. The sides of the dodecagon are drawn as arcs, which of course is quite a wild distortion. If that looks bad, consider the figure on the right, a geometrically precise representation of the same thing. There is so little space between the circles that it is difficult to see what is going on.

My point here is that it is unlikely that Archimedes or his contemporaries were much concerned with precise drawings. Conceptual representations, warts and all, would have served their needs.

• Yes sure they had a good mental representation of the problem. But a mental representation is not a mathematical proof. How can they prove this kind of theorems without Cartesian coordinates or a precise 3D drawing? Aug 16, 2021 at 16:05
• To see how proofs were effected, I recommend a look at the very book I cited. It has many propositions and proofs, and modern translations are easy to find. Also, let me say that the precision of a drawing may aid the visualization, but adds nothing at all to the validity of the proof. I am now recalling a certain physics professor who told his students to go ahead and use 3 as an approximation for $\pi$. I would not carry things so far, but his point was valid. He was interested in the procedure, not the bottom line.
– Pope
Aug 17, 2021 at 6:10