Squaring the Oblique Triangle; Measuring the Sphere of Water
A Euclidean proof for the triangle's area and a plan to gauge the earth by sighting the calm sea.
The upper diagram of two triangles on a single base (lettered a b c and d n e) supports a rule for squaring an oblique triangle: multiply the whole base by half the height dropped from the apex onto the base extended in a straight line, a result Leonardo justifies by a proposition of Euclid equating triangle a b c with b c d. Below, he turns to the size of the sphere of water, stating that it has a smaller circumference than the earth left uncovered by water, and proposes to measure it by taking a known stretch of the sea in calm. A long sighting line drawn across the lower sheet represents the arc of the terrestrial circumference set out to be measured.
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Squaring the oblique triangle b c d
To square an oblique triangle, multiply the whole base b c by half the height d c (that is, by d n), the height being taken from the apex onto the base extended in a straight line to e. Leonardo proves it by a proposition of Euclid showing triangle a b c equal to triangle b c d.
The sphere of water is smaller than the exposed earth
The sphere of water has a smaller circumference than the earth uncovered by the water. This compares the body of the waters with the body of the land standing above it.
Sighting a calm stretch of sea to measure the earth
To measure the sphere of water, take a known stretch of the sea when it is calm. A long sighting line drawn across the lower part of the sheet marks the segment of the terrestrial circumference set out to be gauged.