Editing Ratio (section)

===Euclid's definitions===
Book V of [[Euclid's Elements]] has 18 definitions, all of which relate to ratios.<ref>Heath, reference for section.{{full citation needed|date=April 2025}}</ref> In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a ''part'' of a quantity is another quantity that "measures" it and conversely, a ''multiple'' of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning [[aliquot part]]) is a part that, when multiplied by an integer greater than one, gives the quantity.

Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity ''measures'' the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.

Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.<ref>"Geometry, Euclidean", ''[[Encyclopædia Britannica Eleventh Edition]]'', p.&nbsp;682.</ref> Euclid defines a ratio as between two quantities ''of the same type'', so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities ''p'' and ''q'', if there exist integers ''m'' and ''n'' such that ''mp''>''q'' and ''nq''>''p''. This condition is known as the [[Archimedes property]].

Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's definition of equality is that given quantities ''p'', ''q'', ''r'' and ''s'', ''p'':''q''∷''r'':''s'' if and only if, for any positive integers ''m'' and ''n'', ''np''&nbsp;<&nbsp;''mq'', ''np''&nbsp;=&nbsp;''mq'', or ''np''&nbsp;>&nbsp;''mq'' according as ''nr''&nbsp;<&nbsp;''ms'', ''nr''&nbsp;=&nbsp;''ms'', or ''nr''&nbsp;>&nbsp;''ms'', respectively.<ref>Heath, p. 114.</ref> This definition has affinities with [[Dedekind cuts]] as, with ''n'' and ''q'' both positive, ''np'' stands to ''mq'' as {{sfrac|''p''|''q''}} stands to the rational number {{sfrac|''m''|''n''}} (dividing both terms by ''nq'').<ref>Heath, p. 125.</ref>

Definition 6 says that quantities that have the same ratio are ''proportional'' or ''in proportion''. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".

Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities ''p'', ''q'', ''r'' and ''s'', ''p'':''q''&nbsp;>&nbsp;''r'':''s'' if there are positive integers ''m'' and ''n'' so that ''np''&nbsp;>&nbsp;''mq'' and ''nr''&nbsp;≤&nbsp;''ms''.

{{anchor|EuclidDef8}}As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms ''p'', ''q'' and ''r'' to be in proportion when ''p'':''q''∷''q'':''r''. This is extended to four terms ''p'', ''q'', ''r'' and ''s'' as ''p'':''q''∷''q'':''r''∷''r'':''s'', and so on. Sequences that have the property that the ratios of consecutive terms are equal are called [[geometric progression]]s. Definitions 9 and 10 apply this, saying that if ''p'', ''q'' and ''r'' are in proportion then ''p'':''r'' is the ''duplicate ratio'' of ''p'':''q'' and if ''p'', ''q'', ''r'' and ''s'' are in proportion then ''p'':''s'' is the ''triplicate ratio'' of ''p'':''q''.