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Definitions

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 definition.  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 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 (all are theorems):

Note on translation: plêthos is not translated. In Euclid's Elements, it is any collection of countable things, as opposed to an arithmos, which is a plêthos of units.

Below, Equi-mult[(a,b),..,(c,d)] will mean that a is equally-times a multiple of b as ... as c of d.

Part[a, b] will mean that a is a part of b. Obviously, if Part[a, b], b will be a multiple of a.

a < b will mean that a is a portion of b or is equal to a portion of b. Hence, b - a will be a portion of b that remains when a portion of b equal to a is removed or a magnitude equal to that remaining portion.

Prop. 1: If however-many magnitudes are equally-times a multiple of so-many magnitudes equal in plêthos, each of each, as-many-a-multiple as one is of one of the magnitudes, so-many-multiples will all-of-them be of all-of-them. Equi-mult[(a1, b1), ..., (an, bn)] ⇒ Equi-mult[(ai,bi), (ai + ... + an, bi + ... + bn)]

Prop. 2: If first is equally-times a multiple of second as third is of fourth, and fifth is also equally-times a multiple of second as a sixth is of fourth, then first and fifth added will be equally-times a multiple of second as third and sixth are of fourth. Equi-mult[(a1, a2), (a3, a4)] & Equi-mult[(a5, a2),(a6, a4)] ⇒ Equi-mult[(ai + a5, a2), (a3 + a6, a4)], which is a form of distribution.

Prop. 3: If first is equally-times a multiple of second as third is of fourth, but equally-times multiples of first and third are taken, then through an equal (ex aequali) each of those taken will be equally-times a multiple of each, one of the second and the other of the fourth. Equi-mult[(a1, a2), (a3, a4)] & Equi-mult[(x, a1),(y, a3)] ⇒ Equi-mult[(x, a2), (y, a4)]. Ex aequali is a form of transitivity for equi-multiples.

Prop. 4: If first to second has the same ratio as third to fourth, the equal-times multiples of the first and third to the equal-times multiples of the second and fourth, according to any multiplication, will have the same ratio taken to one another. a : b  ~ c : d & Equi-mult[(w, a), (x,c)] & Equi-mult[(y, b),(z, d)] w : y ~ x : z. Multiplying corresponding terms preserves proportionality.

Prop. 5: If a magnitude is equally-times a multiple of a magnitude, which something taken away is of something taken away, the remainder will also be equally-times a multiple of the remainder, so-many-multiples-as the whole is of the whole. b < a & d < c & Equi-mult[(a, c), (b, d)] ⇒ Equi-mult[(a, c), (a - b, c - d)]

Prop. 6: If two magnitudes are equal-times multiples of two magnitudes, and some magnitudes taken away from them are equal-times multiples of the same, the remainders will be either equal to them or equal-times multiples of them. Equi-mult[(a, b), (c, d) & w < a & x < b & y < c & z < d & Equi-mult[(w, x),(y, z)] ⇒ a -w = b - x & c - y = d - z or Equi-mult[(a -w, b- x), (c - y, d - z)]

Prop. 7: Equals have the same ratio to the same and the same to equals. a = b  ⟺ a : c ~ b : c and a = b  ⟺ c : a ~ c : b

Corollary: From this, in fact, it is obvious that if some magnitudes are proportional, they will be inversely proportional, just what it was required to show. a : b ~ c : d  ⟺ b : a ~ d : c

Prop. 8: The larger of unequal magnitudes has a larger ratio to the same than the smaller, and the same has a larger ratio to the smaller than to the larger. a > b ⇒ a : c > b : c and a > bc : a < c : b

Prop. 9: Things having the same ratio to the same are equal to one another. And those to which the same has the same ratio, they are equal. a : c ~ b : ca = b and c : a ~ c : b ⇒ a = b

Prop. 10: Of those having a ratio to the same, that having the larger ratio is larger, and that to which the same has the larger ratio is smaller. a : c > b : ca > b and c : a > c : b ⇒ a < b

Prop. 11: Those ratios that are the same as the same ratio are the same as one another. a : b ~ c : d & a : b ~ e : f c : d ~ e : f. Transitivity of proportionality.

Prop. 12: If however-many magnitudes are proportional, as one of the leaders is to one of the followers, so are all the leaders to all the followers. a1: b1 ~ ...  ~ an : bnai: bi ~ (a1 + ... an) : (b1 + ... bn)

Prop. 13: If first has the same ratio to second as third to fourth, but third has a larger ratio to fourth than fifth to sixth, first will also have a larger ratio to second than fifth to sixth. a1 : a2 ~ a3 : a4 & a3 : a4 > a5 : a6 a1 : a2 > a5 : a6

Prop. 14: If first has the same ratio to second as third to fourth, but the first is larger than the third, second will also be larger than fourth, and if equal, equal, and if smaller, smaller. a1 : a2 ~ a3 : a4a1 > a3a2 > a4 & a1 = a3a2 = a4 & a1 < a3a2 < a4

Prop. 15: The parts have the same ratio as the multiples in the same way, taken respectively. Equi-mult[(a, b),(c , d) ⇒ b : d ~ a : c

Prop. 16: If four magnitudes are proportional, they will also be alternately (alternando) proportional. a : b ~ c : d a : c ~ b : d

Prop. 17: If added magnitudes are proportional, divided (separando) they will also be proportional. a+b : b ~ c+d : d a : b ~ c : d and inversely. We can also express this as: a : b ~ c : d ⇒ a-b : b ~ c-d : d

Prop. 18: If divided magnitudes are proportional, added (componendo) they will also be proportional. a-b : b ~ c-d : da : b ~ c : d and inversely. We can also express this as: a : b ~ c : d ⇒ a+b : b ~ c+d : d

Corollary: From this, in fact, it is obvious that if added magnitudes are proportional, they will also be proportion in conversion convertando, just what it was required to show. a : b ~ c : d ⇒ a: a-b ~ c : c-d

Prop. 19: If as a whole is to a whole so is a magnitude taken way to one taken away, the remainder will also be to the remainder as whole to whole. a < b & c < d & a : c = b : db : d = b-a : d-c

Prop. 20: If three magnitudes and others equal to them in plêthos, taken by two's and in the same ratio, and through an equal (ex aequali) the first is larger than the third, the fourth will be larger than the sixth, and if equal, equal, and if smaller, smaller. a1 : a2 ~ b1 : b2 & a2 : a3 ~ b2 : b3a1 > a3b1 > b3 & a1 = a3b1 = b3 & a1 < a3b1 < b3

Prop. 21: If three magnitudes and others equal to them in plêthos, taken by two's, are also in the same ratio, and their proportion is perturbed (perturbando), and through an equal (ex aequali) the first is larger than the third, the fourth will also be larger than the sixth, and if equal, equal, and if smaller, smaller. a1 : a2 ~ b2 : b3 & a2 : a3 ~ b1 : b2a1 > a3b1 > b3 & a1 = a3b1 = b3 & a1 < a3b1 < b3

Prop. 22: If however-many magnitudes and others equal to them in plêthos, taken by two's, are also in the same ratio, they will be through an equal (ex aequali) in the same ratio. a1 : a2 ~ b1 : b2 & ... & an-1 : an ~ bn-1 : bn a1 : an ~ b1 : bn

Prop. 23 (with Gerard of Cremona's translation and a comparison): If three magnitudes and others equal to them in plêthos, taken by two's, are in the same ratio, but their proportion is perturbed, they will also be through an equal (ex aequali) in the same ratio. a1 : a2 ~ b2 : b3 & a2 : a3 ~ b1 : b2a1 : a3 ~ b1 : b3

Prop. 24: If first has the same ratio to second as third to fourth, but fifth also has to second the same ratio as sixth to fourth, added first and fifth will also have to second the same ratio as third and sixth to fourth. a1 : a2 ~ a3 : a4 & a5 : a2 ~ a6 : a4 a1+a5 : a2 ~ a3+a6 : a4

Prop. 25: If four magnitudes are proportional, the largest of them and the smallest are larger than the two remaining.


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