It is not certain that the monumental treatise known as the Elements (late 4th century BC) was written entirely by Euclid, a mysterious figure about whom virtually nothing else is known. What is certain, however, is that this thirteen-book work laid the foundations of an axiomatic approach to mathematics, profoundly influencing its development and, more broadly, that of science. The first four books of the Elements establish the foundations of plane geometry and examine figures such as triangles and circles. The next two (Books V and VI) deal with proportions. Books VII to X then focus more on arithmetic, while the final three concern solid geometry. It is mainly in Book VI that proportions are applied to geometry.
The problem of incommensurable magnitudes
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For the Greeks, one magnitude measures another, larger one when the latter consists of a certain number of copies of the former. Thus, in the figure below, the line segment AU measures the line segment AB, whose length is three times as great; the same line segment AU also measures CD. AB and CD have a common measure, AU, and are said to be commensurable.
Seven times AB equals three times CD. Here too, AB is said to measure CD, since the length of CD is 7/3 times that of AB.
