Measuring an unknown distance with similar triangles
A surveying method by right-angled triangles and proportion
This sheet sets out a practical method for finding an unknown distance without traversing it, by pacing a chosen baseline at a right angle and constructing a right-angled triangle. Leonardo argues that any smaller triangle cut parallel to the base is similar to the whole, so measuring the small triangle yields the large unknown distance. Lettered diagrams (a, b, c, d, e) give two versions of the construction, and a profile of a hill carries a superimposed triangle. A closing note advises drawing back far enough that a portion of the figure becomes known.
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Finding an unknown distance by a right-angled triangle
Let a be the object to be measured and b your own position. Walk toward c at a right angle for any convenient distance, then imagine the diagonal from c to a: you have made a triangle. Wherever you cut it at right angles to b c, the proportion of dc to de equals that of cb to ba.
A second construction of the survey triangle
To learn how far it is from a to b without going there, walk from a to c at a right angle for any distance, then from c imagine an orthogonal line at right angles to B, making a right-angled triangle cut at d e. If d c goes twice into d E, then C a goes twice into a B.
The small triangle makes the large one known
A section parallel to the base a B forms a smaller right-angled triangle similar to the large one, d C being cut by the line d E. Once the small triangle and the proportion of its sides are known, the large one is known too, its sides being proportioned like the smaller's.
Draw back until a portion becomes measurable
The rule for an unknown distance in the field: withdraw far enough that a portion of the figure becomes known to you.
Hill profile with a superimposed survey triangle
A drawn profile of a hill carries a triangle laid over it, illustrating how the distance to elevated terrain is taken by the similar-triangle method rather than by direct measurement.