Surveying: measuring mountain heights and leveling land
A graduated altimeter, triangulating a mountain across a plain, and leveling a route of many miles with lamp-levels.
The sheet lays out a practical surveying method for finding the true height of any mountain. A graduated instrument p S, with rules divided into degrees, points and minutes, is set on a plain at least two miles off; the line 3 9 is aimed at the summit and fixed by a thread in a waxed notch, then the height is read from how many times the segment a b enters b c, refined by stepping back until the count changes from ten to eleven. A companion procedure levels a route of many miles using a ten-braccia level fitted with two small oil lamps whose reservoirs drip to keep the flame, and so the sight-line, at a constant height while staffs are leapfrogged forward in 100- to 400-braccia stages. Rows of poles, a mountain profile and lamp details illustrate the operations.
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The graduated altimeter p-S
The instrument p S is built with the rule p b and the rules 3 1 f and 3 9, each divided by degrees, points, minutes and smallest parts. From f 3 the length is two braccia and from r S two braccia. Letters marked include p, wax [cera], h 9, m, a, c b, 3 g n r f and S.
Finding the true height of a mountain
Set the instrument p S on a plain at least two miles away and align its level r S with its counter-marks; raise the line 3 9 to the summit and fix it with the thread h in the waxed notch p. Then note how many times a b enters b c, for that many times the mountain rises to the angle 3; if it enters ten, step back until it enters eleven, and from ten to eleven is the mountain's true height. A second instrument f n with the level a t, using staffs planted every 100 braccia (m b c d e), levels the ground when the span is great.
Height read by proportion of a b to b c
The measurement rests on similar triangles: the height is judged by how many times the short segment a b is contained in b c, refined by moving back until the ratio steps from ten to eleven. The interval between those counts fixes the true altitude.
Leveling a plain of many miles
To level a plain of many miles, set a level a b about ten braccia long with the upper level K, place a lamp in line at c 200 braccia off, then d at a further 400, remove b c and set e at another 400 in line with d a, then leapfrog d e forward 400 braccia to f, continuing bit by bit until the whole route is leveled.
Water-level with two oil lamps
The leveling rig carries the level a b with lamps at its ends, fixed in the ground with the upper level K over a span of ten braccia, lettered b A a and g f e d c. Each lamp's little basin h drips drop by drop into the lamp i so the oil stays at height m, keeping the flame, and thus the sight-line, from falsifying the level as the oil falls.