Multiplication via similar triangles.
- Reinhard Schmidt
It is well known that the 17th Century mathematician, Rene Descartes, modified Euclid's 12th proposition from Book VI of Elements by assigning a unit to be one of four proportional lines. Before this, the unit was not considered a number as Euclid had defined number as a multitude of units. (Euc. VII Def. 1) Yet what is almost unknown, is how the 13th Century mathematician, Campanus of Novara, presented a difference construction and proof for (Euc. VI Prop. 12). Had Descartes seen the construction of Campanus, he might have produced a diagram to explain the multiplication of two line segments (alongside unity) that produce a line segment as the fourth proportional.
Task 1) Drag the multiplier upwards to increase it and you will see the product change. Task 2) Drag the multiplicand leftwards to increase it and you will see the product change. Task 3) Drag the multiplier downwards to decrease it and you will see the product change. If you drag the multiplier below where an x axis would be, you can see how a negative multiplier multiplied by a negative multiplicand produces a 'positive' product.