ponendo

Ponendo is a Latin mathematical term used in the study of proportions. The word itself means “placing” or “putting,” and in the tradition of classical algebra it refers to a family of techniques for manipulating proportions, often alongside composendo, dividendo, and the combined method known as componendo et dividendo.

The essential idea of ponendo is to derive new equivalent relations by placing the same quantities into the terms of a known proportion. In a typical formulation, if a/b = c/d, then a new ratio formed by adding the corresponding terms, (a+c)/(b+d), preserves the same value. This holds because, if a/b = c/d, then a and c can be expressed as multiples of b and d (a = kb, c = kd for some k), giving (a+c)/(b+d) = k = a/b = c/d. Thus ponendo allows the construction of an additional equivalent ratio from an existing one.

Historically, ponendo appears in early modern European algebra and in various mathematical traditions that treated proportions in a mnemonic or rule-based way. It is commonly taught together with other proportional techniques, such as componendo and dividendo, as part of a toolkit for solving proportion problems without explicit arithmetic or roots.

In contemporary mathematics, ponendo is mainly of historical interest. Modern treatments prefer direct algebraic manipulation, cross-multiplication, and the broader framework of ratio and similarity, while recognizing ponendo as one of several classical methods for reasoning about proportions.