This astonishing Pre-Socratic was a brilliant mind. I guess everyone knows his ‘Theorem” about right angle triangles. The saying above might not be quite so well known (or his enjoiner, ‘Eschew beans’! I think I know why!), but we can use some of the proportions of the human body and the a property of an Isosceles Triangle (ie one with two equal sides) to do some pretty handy estimations.
You hold a stick at arm’s length as shown in the drawing on the left so that the top of the branch exactly aligns with the top of the object whose height you want to measure. You can do one of two things: drop the stick over as in the drawing on the right, or pace the distance between you and the object. The height of the tree will be exactly the same as the distance from you to the object (Isosceles triangle, you see) Or the point on the ground where the stick on the right indicates.
You have formed a little isosceles triangle with your eye, your arm and the stick This triangle projects forward to the larger isosceles triangle formed by your feet, the distance to the base of the tree and the height of the tree itself, so the height of the tree is always exactly the distance from you to the tree..
Here is an interesting proportion. The distance between your eyes is almost exactly 1/9th of the length of your arm to the tip of your thumb (as shown below). By alternatively closing one eye and then the other, and estimating how far the object aimed at with your thumb ‘jumps’ sideways, then multiplying that estimated distance by 9 (it might be easier to multiply by ten which is close enough really) you can get a pretty good estimate of the distance to that remote object (ie it will be distant roughly ten times the distance your thumb jumps!) Neat eh?
This can be useful if you are taking a long shot (eg at a deer) with a rest, or eg if the object is your destination and you wish to know how far away it is, or if you need to cross a river and you want to know how wide it is so that you can judge how far upstream you need to start swimming or paddling (on your Thermarest Neoair mat) to safely get across. Always cross at the deepest, slowest straightest spot. You will already have measured the speed of the current by throwing s floating twig in and timing it.
If you dropped the stick to the side as in the right hand drawing in the first illustration, you can use the approximate number of tree lengths to estimate how far your thumb has jumped. If you assume that a similar tree near you which you measured by pacing is the same height as the one on the far bank, you will have a very clear idea of the distance to the remote object in tree lengths. From then on, it s only a matter of simple multiplication.
The featured image is Da Vinci’s famous ‘Vitruvian Man’ where Da Vinci sets out his ideas of the ideal human proportion: https://en.wikipedia.org/wiki/Vitruvian_Man