Dividing a line into equal parts by a triangle
A compass-and-triangle method proved by Euclid and a rule of perspective
Leonardo shows how, with a single fixed opening of the compass, a given line can be divided into any stated number of equal parts, whether even or odd. He builds an equilateral triangle on the base a b by intersecting two circles to find the apex c, drops the perpendicular c f, and draws a parallel d e which he divides with the compass. He proves the method by a proposition of Euclid on triangles cut parallel to their base and by a rule of perspective that equal things at equal distances appear equal.
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Dividing a given line into any number of equal parts
With one given opening of the compass a given line is to be divided into a stated number of equal parts, whether that number is even or odd. Leonardo presents this as a single general problem before giving its construction.
Triangle construction on base a b with apex c
On the given line a b the point c is found at equal distance from the ends by intersecting two circles, exactly as an equilateral triangle is built on its base. The perpendicular c f then bisects a b, and across it the parallel d e is drawn equidistant to a b and divided with the compass into the required parts.
Proof by Euclid and by a proposition of perspective
Leonardo grounds the method in a proposition of Euclid: all triangles cut by a line equidistant to their base have their sides proportioned like the whole. He adds a rule of perspective, that equal things set at equal distances appear equal among themselves, so that the divisions of d e transfer to a b.
Line divided by parallels (n m, o p)
A subsidiary figure shows a line divided into equal parts by means of parallels, labelled n m and o p. It illustrates the same principle of equal spacing carried across parallel lines.