Draft:Theory of proportion

Book V of the Elements concerns ratios of magnitudes (intuitively, how much bigger or smaller one shape is relative to another) and the comparison of ratios.^[1] Thomas Heath and other translators have formulated its first six propositions in symbolic algebra, as forms of the distributive law of multiplication over division and the associative law for multiplication. However, Leo Corry argues that this is anachronistic and misleading, because Euclid did not treat magnitudes as numbers, nor taking a ratio as a binary operation from numbers to numbers.^[2]

Much of Book V was probably ascertained from earlier mathematicians, perhaps Eudoxus,^[3] although certain propositions, such as V.16, dealing with "alternation" (if a : b :: c : d, then a : c :: b : d) likely predate Eudoxus.^[4]

Christopher Zeeman has argued that Book V's focus on the behavior of ratios under the addition of magnitudes, and its consequent failure to define ratios of ratios, was a flaw that prevented the Greeks from finding certain important concepts such as the cross ratio (central to projective geometry).^[5]

Book VI utilizes the theory of ratios from Book V in the context of plane geometry,^[6] especially the construction and recognition of similar figures. It is built almost entirely of its first proposition:^[7] "Triangles and parallelograms which are under the same height are to one another as their bases". That is, if two triangles have the same height, the ratio of their areas is the same as the ratio of lengths of their two base segments (and analogously for two parallelograms of the same height). This proposition provides a connection between ratios of lengths and ratios of areas.^[8] Proposition 25 constructs, from any two polygons, a third polygon similar to the first and with the same area as the second. Plutarch attributes this construction to Pythagoras, calling it "more subtle and more scientific" than the Pythagorean theorem. The famous ancient Greek problem of doubling the cube, now known impossible with compass and straightedge, is a special case of the analogous 3d problem of constructing a figure with a specified shape and volume.^[9] The book ends as it begins, by connecting two types of ratios: ratios of angles, and ratios of circular arc lengths, in proposition 33.^[10]

Of the Elements, book X is by far the largest and most complex, dealing with (in modern terms) irrational numbers in the context of magnitudes.^[3][11] For Euclid, rather than treating magnitudes as numbers, this means considering the commensurability of lengths of line segments: whether two line segments can both be measured by an integer number of copies of a common subunit.^[11] Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. Euclid calls a length irrational when the square on that length is commensurable with the unit square: that is, for Euclid, a length such as ${\displaystyle {\sqrt {2}}}$ that is the square root of a rational area is itself rational.^[12] Proposition 9 proves the irrationality (in modern terms) of the square roots of non-square integers such as ${\displaystyle {\sqrt {2}}}$ (again, constructed as lengths of line segments relative to a fixed unit segment).^[13] This book classifies irrational lengths into thirteen disjoint categories.^[14]