V. Def. 3. Simson considers that the definitions 3 and 8 are "not Euclid's, but added by some unskilful editor." Other commentators also have rejected these definitions as useless. The last word of the third definition should be quantuplicity, not quantity; so that the definition indicates that ratio refers to the number of times which one magnitude contains another. See De Morgan's Differential and Integral Calculus, page 18.
V. Def. 4. This definition amounts to saying that the quantities must be of the same kind.
V. Def. 5 The fifth definition is the foundation of Euclid's doctrine of proportion. The student will find in works on Algebra a comparison of Euclid's definition of proportion with the simpler definitions which are employed in Arithmetic and Algebra. Euclid's definition is applicable to incommensurable quantities, as -well as to commensurable quantities.
We should recommend the student to read the first proposition of the sixth Book immediately after the fifth definition of the fifth Book; he will there see how Euclid applies his definition, and will thus obtain a better notion of its meaning and importance.
Compound Ratio. The definition of compound ratio was supplied by Simson. The Greek text does not give any definition of compound ratio here, but gives one as the fifth definition of the sixth Book, which Simson rejects as absurd and useless.
V. Defs. 18, 19, 10. The definitions 18, 19, 20 are not presented by Simson precisely as they stand in the original. The last sentence in definition 18 was supplied by Simson. Euclid does not connect definitions 19 and 10 with definition 18. In 19 he defines ordinate proportion, and in 20 he defines perturbate proportion. Nothing would be lost if Euclid's definition 18 were entirely omitted, and the term ex æquali never employed. Euclid employs such a term in the enunciations of V. 20, 21, 11, 23; but it seems quite useless, and is accordingly neglected by Simson and others in their translations.
The axioms given after the definitions of the fifth Book are not in Euclid; they were supplied by Simson.
The propositions of the fifth Book might be divided into four sections. Propositions 1 to 6 relate to the properties of equimultiples. Propositions 7 to 10 and 13 and 14 connect the notion of the ratio of magnitudes with the ordinary notions of
