Perhaps some pupil who has read Thoreau's descriptions of outdoor life may be interested in what he says of his crude mathematics. He writes, "I borrowed the plane and square, level and dividers, of a carpenter, and with a shingle contrived a rude sort of a quadrant, with pins for sights and pivots." With this he measured the heights of a cliff on the Massachusetts coast, and with similar home-made or school-made instruments a pupil in geometry can measure most of the heights and distances in which he is interested.
Theorem. If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse:
1. The triangles thus formed are similar to the given triangle, and are similar to each other.
2. The perpendicular is the mean proportional between the segments of the hypotenuse.
3. Each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side.
To this important proposition there is one corollary of particular interest, namely, The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter. By means of this corollary we can easily construct a line whose numerical value is the square root of any number we please.
Thus we may make AD = 2 in., DB = 3 in., and erect DC ⊥ to AB. Then the length of DC will be √6 in., and we may find √6 approximately by measuring DC.
Furthermore, if we introduce negative magnitudes into geometry, and let DB = +3 and DA = -2, then DC will equal √(-6). In other words, we have a justification for representing imaginary quantities by lines perpendicular to the line on which we represent real quantities, as is done in the graphic treatment of imaginaries in algebra.