Two examples of "practical geometry"

Measurement of an inaccessible height

Galileo describes how to find the height of a tree or a building in the same way that practical geometry texts had described it for centuries. The method relies on similar triangles. Recall that any two triangles which agree in all their ANGLES are similar, that is they are just scaled versions of each other. Thus the RATIOS of corresponding sides are equal. The measurement of the distance L and the angle A in the picture below is enough to determine the inaccessible height H using this idea of similarity. The triangle is a right triangle (it has a right angle) and it contains the angle A. Therefore the ratio H/L is the same as in any other such triangle, and depends only on the angle A. This ratio was available in tables to Galileo, and it is available to us in tables and using calculators. It is called the TANGENT of A or tan(A) for short. Since H/L=tan(A), we have H=L.tan(A). When we do this, we have to measure the distance L and the angle A. Then we have to find the tangent of A and do the multiplication. This is not quite what Galileo asks you to do, though. He says you should take L to be 100 of any unit you like. Then you just read H off the Geometric Compass (in that same unit). How is this possible? The answer is that the scale where you would expect to read off the angle does not tell you the angle A but rather 100.tan(A). This is exactly H, just one example of Galileo's cleverness in designing the compass.

A more interesting measurement of an inaccessible height is shown below. In this case you can't move to the foot of building or tree or whatever it is, but must maintain a distance. Galileo says you should take two sightings, finding angles A and B, from locations separated by 100 units (what we call L below). Then he says the height H

is the product of the two readings over the difference. We see that this is true -- since readings on Galileo's compass were actually the TANGENTS of the angles, not the angles themselves. Since we don't actually have Galileo's compass, we will have to measure the angles, take their tangents, and do the arithmetic.

Perspective painting

One of the great scientific achievements of the Renaissance was the invention, possibly the reinvention, of perspective painting, probably around 1420. There are paintings of Giotto in the 1300's that seem to suggest perspective, and there was a general secrecy among artists on this subject until the mid 1400's which makes it hard to be sure how widespread this knowledge was. The painters of the Renaissance believed they were just recovering methods known to the Greeks, and it is certainly true that there are classical paintings that suggest perspective. Some enthusiasts suggested perspective would have to join the subjects of the quadrivium as a fifth "science."

The basic theory of how to construct a painting in perspective is indicated in the figure below:

The idea is that you imagine a cone of lines with vertex at the observing eye, drawn to every point in the scene to be represented. Where each line intersects the plane of the painting, you paint the corresponding point in the scene. Thus the eye, looking at the painting, sees just what it would see if it were looking at the scene itself.

This seems rather obvious, on the one hand, and not very helpful on the other. It doesn't tell you how to paint, unless you take the construction very literally, and really lay out those lines somehow. (Apparently some artists really did devise ways to do this.)

There is a non-obvious consequence of this construction that we will see experimentally. Suppose that the scene contains some straight lines, like sides of buildings. These straight lines in the scene project to straight lines in the painting. This by itself is, of course, not surprising. Now suppose that there is a family of PARALLEL lines in the scene, like the eaves of buildings of various heights, all built methodically along a straight avenue. Since these lines are parallel, they don't intersect each other, even if prolonged, but they are all pointing the same direction. If they were not quite parallel, you could imagine them actually meeting at a very great distance, and the more nearly parallel they became, the farther away would be their intersection. In this sense, you could imagine that parallel lines meet "at infinity." A point at infinity is really just a DIRECTION, the direction of the parallel lines that all meet there. The remarkable thing about perspective painting is that in the projection, the parallel lines really DO meet at the point at infinity! That point is represented by the point in the painting that you determine by drawing a line from the eye in the direction of the parallels. It is sometimes called the "vanishing point," because the space between the parallel lines seems to vanish in the distance.

To see this experimentally, find a window or glass door through which you can see some parallel lines in a (three-dimensional) scene. The lines should be pointing "away" from you, roughly speaking. Arrange to keep your eye steadily in one place to see the scene always in the same way. Then have a friend put masking tape on the glass so that it seems, from your perspective, to lie along various ones of the straight, parallel lines in the scene. When you have represented 4 or 5 of the lines with tape, look at the pattern they make: you will find that these lines, if prolonged, all meet in one point.