Estimating some Very Big Things

Suppose you want to know the height of a tall tree, how could you estimate it? (Without climbing to the top with a long piece of string!) The same problem was put, it was said, to the Greek mathematician and philosopher Thales (pronounced thahlees) when he visited Egypt in the sixth century BCE, only in his case it was to find the height of one of the Egyptian pyramids. Here's how he did it - but we'll use a tree.

Measure your own height, h. At the same time of day measure both the length s of your shadow and that S of the tree (on level ground of course). Then, the sun's rays being parallel and so making the same angle with level ground, we have two similar triangles. Hence, if H is the height of the tree


H
S
 = h
s
 ( = tanq)   so that H = S×h
s
 .
Actually, Thales is said to have been even cleverer than this and avoided doing any measuring or calculating at all, by choosing the time of day carefully so that his shadow was as long as he was (q = 450). He only needed to lie down first to mark that length. At the same moment, he got someone else to mark where the pyramid's shadow lay, and then informed his admiring audience that the pyramid was as high as the shadow was long.

Thales was also said to have estimated how far out to sea a ship was. One way of doing this is to take sightings of the ship from the water's edge and also from the top of a high sand-dune or cliff, measuring the angles a, b, and the distance l, as in the diagram.

Problems [Answers on another page]

  1. How would you work out the distance d knowing a, b and l?

  2. Now imagine you are Thales - all you have is a pair of hinged rulers to measure the physical angle by sighting along them, some string and some flat sand to do geometry on. No protractor, no trig. tables, no tape measure... You can take your unit length to be a thumb-joint length (`inch'), a handwidth (`hand'), a handstretch (`span'), a footlength (`foot'), a forearmlength (`cubit'), an armspan (`fathom'), or a pace (`yard'). How would you find the distance of the ship from shore?

  3. Finally, see if you can work out how, like Thales, you can be so clever as to avoid doing any measuring at all!



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On 13 Jan 2001, 12:02.