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It is a pleasant surprise to him (the pure mathematician) and an added problem if he finds that the arts can use his calculations, or that the senses can verify them, much as if a composer found that sailors could heave better when singing his songs.
In J. R. Newman (ed.), The World of Mathematics, New York: Simon and Schuster, 1956.
The Rule of False Position and Geometric Problems
Construction of the Fourth Proportional
Let a, b, and c be three line segments.
On ray Or trace segments OA = a and AC = c. On ray Os trace segment OB = b. From point C trace CX parallel to AB. We can verify that x is the fourth proportional with respect to the segments a, b, and c. That is, a:b = c:x.
Indeed, triangles OAB and OCX are similar. Therefore a:b = (a + c):(b + x). So a:b = [(a + c) - a]:[(b + x) - b] = c:x, as desired.
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