Q:

When did the term and taught technique 'cross multiplication' enter into common use?

The title says it all, I suppose. I'm interested to know when/where the term/technique cross multiply came into use. Sources would be nice.
In case it's unfamiliar to anyone, or in case the usage of the term varies, I'm referring to the compound arithmetic operation where an equation of two fractions is multiplied on both sides by its two denominators. It's named because it can be illustrated with a big cross:
$$\frac{a}{b} = \frac{c}{d}$$

$$ad = cb$$

A:

The earliest example of the OP's use I can find on Google Books is 1908. It was also used to describe adding fractions about the same time (although there is a use in a professional mathematics article in 1833).  It starts to be used often in the late 1930s and shoots up again in mid 1970s.
Some early references to school mathematics from Google Books:

adding fractions (1903)
adding fractions (1905)
solving fractional equations (1908)

Earlier uses of the term in professional articles.  They seem merely descriptive, although the usage to describe multiplying number systems with place values seems common (ones-tens-hundreds, feet-inches-parts).  The 1833 paper uses the term to describe an operation in simplifying an fairly complicated algebraic fraction.

multiplying lengths in the English system (1803)
adding fractions (1833)
systems of equations (1843)
multiplying digits (1851)

Ngram viewer results:

