Euclid inspired Gothic architecture and taught Renaissance painters how to create depth and perspective. More generally, the success of mathematics went to its head, according to some, and created dogmatic individuals dismissive of other branches of learning. Some thought the uncompromising rigour of Euclid went hand in hand with totalitarianism in political and spiritual domains, while others thought creative mathematics was inherently free and liberal.

Transcript

Gothic architecture is known for its pointed arches. Unlike round arches like a classical Roman aqueduct for example. Those are semi-circular, but Gothic arches are steeper, pointier. Gothic buildings, cathedrals, have these arches everywhere: windows, doorways, and so on.

Gothic arches consist of two circular arcs. You can make it like this. First make a rectangular shape. Like a plain window or door. A boring old rectangle. Now let’s spice it up. Take out your compass, and put it along the top side of the rectangle. Draw two circular arcs going up above the rectangle. Use the top side of the rectangle as the radius, and its two endpoints as the two midpoints of the two arcs you are drawing. The two arcs make a pointed extension of the rectangle. Now you have your Gothic window.

If you have your Euclid in fresh memory you will recognize at once that this is precisely the type of construction involved in Proposition 1 of the Elements. Coincidence? No, I don’t think so. The Gothic style of architecture arose in Europe in the early 12th century, within a decade or two of the first Latin translation of Euclid’s Elements. If that’s not cause and effect, it‘s an incredible coincidence.

There is little direct documentation about this, but, I am quoting now from Otto von Simson’s book The Gothic Cathedral, “at least one literary document survives that explains the use of geometry in Gothic architecture: the minutes of architectural conferences held in 1391 in Milan. The question debated at Milan is not whether the cathedral is to be built according to a geometrical formula, but merely whether the figure to be used is to be the square or the equilateral triangle. The minutes of one particularly stormy session relate an angry dispute between the French expert, Jean Mignot, and the Italians. Overruled by them on a technical issue, Mignot remarks bitterly that his opponents have set aside the rules of geometry by alleging science to be one thing and art another. Art, however, he concludes, is nothing without science, ars sine scientia nihil est. This argument was considered unassailable even by Mignot’s opponents. They hasten to affirm that they are in complete agreement as regards this theoretical point and have nothing but contempt for an architect who presumes to ignore the dictates of geometry.”

So the geometrical ethos was very strong indeed. This hardline view probably softened a bit over time. Renaissance art is more expressive, emotive, more alive, one might say, than this rigid late medieval stuff. That’s if we fast-forward two hundred years from these Gothic conferences about how art is nothing without geometry.

Then you have people like Michelangelo who said: “the painter should have compasses in his eyes, not in his hands.” I suppose it means that art should go a little more by feeling and intuition, and not be completely dictated by mathematics. But you still have “compasses in your eyes,” so there’s still a very significant role for geometry, it seems.

It is also revealing, perhaps, that Michelangelo thought it was important to point this out at all. I guess there were a lot of artists with compasses in their hands running around back then. Why else would Michelangelo feel the need to criticise that practice?

In fact, geometry proved useful to art again, in new ways, in the Renaissance. At this time artists discovered (or perhaps rediscovered) the geometrical principles of perspective.

Accurate representation of depth in a painting follows simple geometrical principles. The key construction is that of a tiled floor. Like a chessboard type of pattern of floor tiles, but seen in perspective, so tiles that are further away appear smaller in the picture. There are a lot of tiled floors in Renaissance art, because they are great for conveying a sense of depth.

You draw it like this. Draw two horizontal lines: one is where the floor starts, and one is the horizon. Divide the floor line into many equal pieces, representing the size of the tiles. Connect all these points to a fixed point on the horizon. This is because all the parallel rows of the floor will appear to converge in one point, just as whenever you are looking at parallel lines that go off into the distance, such as railroad tracks for instance, they appear to meet at the horizon.

Next, draw a second horizontal line representing the other edge of the front row of floor tiles. Now here comes the magic step. Draw the diagonal of the first tile, and extend it. Where this line cuts the other lines you have already drawn, those are all corners of other tiles.

This is because of the geometrical principle that a straight line, no matter how you look at it, from whatever angle, will always still be straight. A lot of stuff looks funny in perspective: big things can look small, parallel lines appear to meet, perfectly round things appear to be oval, and so on. Perspective distorts shapes in all kind of ways. But straight lines remain straight lines. That is an invariant in all of this. This is why the diagonal of the first floor tile is also the diagonal of successive tiles. Since that makes a straight line in the real world, it must make a straight line in the picture as well.

Once you have this diagonal it is easy to complete the rest of the floor. It looks great. It creates a photorealistic sense of depth. No wonder so many artists chose to set their scenes in locales that just happened to have lots of tiled floors.

But the insight is much deeper than this. It’s not really about tiled floors; it’s about the correct perspective representation of depth and size generally. Even if you don’t want a tiled floor in your finished picture, it’s still very useful to draw one in pencil on your canvas as a reference grid. You can use the tiles as a guide to transfer and compare sizes at different depths and distances. Then later you can paint it over with some trees or whatever, so nobody can see the grid anymore; you just used it behind the scenes to get the proportions right.

The Greeks probably knew about this stuff, but basically no paintings from antiquity have been preserved. But we know they were skilled artists. One guy is said to have painted grapes so realistically that birds came and pecked at it.

It seems the Greeks knew the geometrical principles of perspective and used it to creates scenes for the theatre for example. As Vitruvius says, “by this deception a faithful representation of the appearance of buildings might be given in painted scenery, so that, though all is drawn on a vertical flat facade, some parts may seem to be withdrawing into the background, and others to be standing out in front.”

This wasn’t just a party trick to the Greeks. It also had philosophical implications. To Plato they raised profound epistemological conundrums. He was concerned that optical illusion painting has “powers that are little short of magical,” “because they exploit this weakness in our nature,” bypassing “the rational part of the soul.” The solution to this problem, as Plato saw it, was a solid mathematical education. Since “sense perception seems to produce no sound result” with these illusory paintings, “it makes all the difference whether someone is a geometer or not.”

“The power of appearance often makes us wander all over the place in confusion, often changing our minds about the same thing and regretting our actions and choices with respect to things large and small.” “The art of measurement,” by contrast, “would make the appearances lose their power” and “give us peace of mind firmly rooted in the truth.”

Those are all Plato’s words. A rousing case for mathematics! But Plato perhaps drew his conclusions a step too far, rejecting categorically the role of observational data in science: “there’s no knowledge of sensible things, whether by gaping upward or squinting downward.” Science must be based on “the naturally intelligent part of the soul,” not observation. For example, “let’s study astronomy by means of problems, as we do geometry, and leave the things in the sky alone.” With such attitudes, perhaps it is no wonder that the Greeks excelled more in mathematics than in the sciences. But indeed the threat of optical illusions is a legitimate argument in Plato’s defense.

When the principles of perspective were rediscovered in Renaissance Italy, they were again at the heart of scientific developments, but this time on the side of empirical science. Galileo looked at the moon through a telescope and concluded that it had mountains and craters. Of course the image one sees through a telescope is flat. But mountains and craters are revealed by the shadows they cast.

This is not necessarily very easy to see, or not necessarily a very evident conclusion. Some scholars have argued that the artistic tradition, and its extensive study of perspective and shadows, was a necessary training for the eye to be able to correctly interpret the telescope data.

Is it a coincidence that Galileo the telescopic astronomer came from the same land as the great Italian Renaissance painters? Galileo was born and raised in Tuscany, right where so many of these masters had worked. Maybe only someone immersed in this artistic culture had the right eyes to interpret the heavens. A far-fetched theory, in my opinion, but it’...

Cultural reception of geometry in early modern Europe