I am searching for books and articles about applications of Geometry, in particular to the problem of computing distances and lengths which are apparently beyond human reach. As an example, consider the computations of the size of the Earth by Eratosthenes and by Al-Biruni. I am particularly interested in those cases in which (as in the examples that I mentioned) it is known that those methods were actually used in practice.
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asked Nov 22, 2018 at 9:39
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1$\begingroup$ Robert Hahn, The Metaphysics of the Pythagorean Theorem. Thales, Pythagoras, Engineering, Diagrams... (NY: 2018), a book with some examples (height of pyramids, distance at sea, tunnels). $\endgroup$– sand1Nov 22, 2018 at 10:32
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$\begingroup$ There is a very nice book: M. Berger, Geometry, which describes a lot of applications. $\endgroup$– Alexandre EremenkoNov 22, 2018 at 16:24
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1$\begingroup$ You may want to look at Tao's slides on the cosmic distance ladder. $\endgroup$– Andrés E. CaicedoNov 26, 2018 at 13:01
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$\begingroup$ Rather recently, see "The Great Escape," in which POWs used triangulation survey techniques to determine necessary tunnel length. Sadly, they lacked accurate levels; the tunnels sloped slightly up, and became the hypotenuse rather than the leg (hence short of desired destination) $\endgroup$– Carl WitthoftNov 28, 2018 at 13:01
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For the down to Earth geometry see Surveying Instruments of Greece and Rome by Lewis. For the astronomical measurements, which aside from the names mentioned would include Aristarchus, Ptolemy, see Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley by van Helden. The Mathematics of the Heavens and the Earth by van Brummelen focuses specifically on applications of trigonometry. The History of Stellar Measurement website lists some more recent applications to the parallax measurements.