Dividing a Long Line in the Same Proportion as a Short One
Transferring unequal divisions by a triangle of parallels
Leonardo sets the problem in his own words: 'I have a small line divided into unequal parts and I wish to divide a large one in the same proportion.' A triangle is formed from the short divided line and the long line, its points marked a, d(?), c, b, so that parallels carry the divisions of the small line onto the large one. This is the Euclidean construction for dividing an uncut line like a given divided line. The sketch is faint pencil.
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Proportional division of a line
The note asks how to divide a large line into the same unequal parts as a small one already divided. Setting the two lines as sides of a triangle and drawing parallels from the division points transfers the proportions from the short side to the long one.
Triangle of the divided lines
The accompanying figure is a triangle formed from the said divided lines, its vertices lettered a, d(?), c, b. A diagonal crosses the triangle so that parallel lines can be dropped to the base.