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Euclid, Elements, Book V, Definitions©
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Propositions

Definitions Brief comments

1 A magnitude is a part of a magnitude, the smaller of the larger, whenever it measures the larger,

2. and the larger is a multiple of the smaller whenever it is measured by the smaller.

3. Ratio is a sort of condition of two magnitudes of the same kind according to their size.

4. Magnitudes which are able by being multiplied to exceed one another are said to have a ratio to one another. Some have read this as stating a basic principle about continuity and what it is for two magnitudes to be of the same kind. Knorr pointed out that this actually sets up necessary conditions for the next definiiton. Is this a definition?

5. Magnitudes are said to be in the same ratio, first to second and third to fourth, whenever equal multiples of the first and third are either together greater than or together equal to or together less than the equal multiples of the second and fourth, according to any respective multiplications, Controversy about this definition seems to begin in the 16th cent.

6. and let magnitudes having the same ratio be called 'proportional', The Greek word 'analogon' is formed from a prepositional phrase and is adverbial. 'Proportional' is the standard translation, but 'in-ratio' would be better. The difficulty is that one needs to be able to form the noun from the preposition, for 'analogia' (cf. def. 9), while 'in-rationality' or something like that is too artificial.

7. and whenever of equal multiples the multiple of the first exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first to the second is said to have a greater ratio than the third to the fourth,

8. and a proportion in three terms is least, That is, one cannot have a proportion involving two terms. Note that this is not a definition in any ordinary sense.

9. and whenever three magnitudes are proportional, the first to the third is said to have duplicate the ratio it has to the second.

10 and whenever four magnitudes are proportional, the first to the fourth is said to have triplicate the ratio it has to the second, and in each case in succession it holds similarly, however the proportion holds.

11. Corresponding magnitudes (homologous) are said to be the leading terms to the the leading terms and the following terms to the following terms.

12. Alternate ratio is taking the leading term to the leading term and the following term to the following term.

13. Inverse ratio is taking the following term as leading to the leading term as following.

14. Composition of ratio is taking the leading term with the following term as one to the following term itself.

15. Division of ratio is taking the excess by which the leading term exceeds the following term to the following term itself.

16. Conversion of ratio is taking the leading term to the excess by which the leading term exceeds the following term.

17. A ratio through an equal (ex aequali ratio) is, given that there are several magnitudes and others equal to them in number are taken two by two and in the same ratio, whenever in the first group of magnitudes the first is to the last, so in the second group of magnitudes the first is to the last. Or, alternatively, taking the extreme terms by removing the middle terms.

18. A perturbed proportion is when given three magnitudes and others equal to them in number it happens that (1) as, in the first group of magnitudes, a leading term is to a following term, so, in the second group of magnitudes, a leading term is to a following term, while (2) as, in the first group of magnitudes, a following term is to some other term, so, in the second group of magnitudes, some other term is to a leading term.

Propositions:

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	Definitions	Brief comments
1	A magnitude is a part of a magnitude, the smaller of the larger, whenever it measures the larger,
2.	and the larger is a multiple of the smaller whenever it is measured by the smaller.
3.	Ratio is a sort of condition of two magnitudes of the same kind according to their size.
4.	Magnitudes which are able by being multiplied to exceed one another are said to have a ratio to one another.	Some have read this as stating a basic principle about continuity and what it is for two magnitudes to be of the same kind. Knorr pointed out that this actually sets up necessary conditions for the next definiiton. Is this a definition?
5.	Magnitudes are said to be in the same ratio, first to second and third to fourth, whenever equal multiples of the first and third are either together greater than or together equal to or together less than the equal multiples of the second and fourth, according to any respective multiplications,	Controversy about this definition seems to begin in the 16th cent.
6.	and let magnitudes having the same ratio be called 'proportional',	The Greek word 'analogon' is formed from a prepositional phrase and is adverbial. 'Proportional' is the standard translation, but 'in-ratio' would be better. The difficulty is that one needs to be able to form the noun from the preposition, for 'analogia' (cf. def. 9), while 'in-rationality' or something like that is too artificial.
7.	and whenever of equal multiples the multiple of the first exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first to the second is said to have a greater ratio than the third to the fourth,
8.	and a proportion in three terms is least,	That is, one cannot have a proportion involving two terms. Note that this is not a definition in any ordinary sense.
9.	and whenever three magnitudes are proportional, the first to the third is said to have duplicate the ratio it has to the second.
10	and whenever four magnitudes are proportional, the first to the fourth is said to have triplicate the ratio it has to the second, and in each case in succession it holds similarly, however the proportion holds.
11.	Corresponding magnitudes (homologous) are said to be the leading terms to the the leading terms and the following terms to the following terms.
12.	Alternate ratio is taking the leading term to the leading term and the following term to the following term.
13.	Inverse ratio is taking the following term as leading to the leading term as following.
14.	Composition of ratio is taking the leading term with the following term as one to the following term itself.
15.	Division of ratio is taking the excess by which the leading term exceeds the following term to the following term itself.
16.	Conversion of ratio is taking the leading term to the excess by which the leading term exceeds the following term.
17.	A ratio through an equal (ex aequali ratio) is, given that there are several magnitudes and others equal to them in number are taken two by two and in the same ratio, whenever in the first group of magnitudes the first is to the last, so in the second group of magnitudes the first is to the last. Or, alternatively, taking the extreme terms by removing the middle terms.
18.	A perturbed proportion is when given three magnitudes and others equal to them in number it happens that (1) as, in the first group of magnitudes, a leading term is to a following term, so, in the second group of magnitudes, a leading term is to a following term, while (2) as, in the first group of magnitudes, a following term is to some other term, so, in the second group of magnitudes, some other term is to a leading term.