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Ancient Mathematics
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[(a_{1}, b_{1}), ..., (a_{n}, b_{n})] ⇒ Equi-mult[(a_{i},b_{i}), (a_{i} + ... + a_{n}, b_{i} + ... + b_{n})]
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[(a_{1}, a_{2}), (a_{3}, a_{4})] & Equi-mult[(a_{5}, a_{2}),(a_{6}, a_{4})] ⇒ Equi-mult[(a_{1} + a_{5}, a_{2}), (a_{3} + a_{6}, a_{4})], 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[(a_{1}, a_{2}), (a_{3}, a_{4})] & Equi-mult[(x, a_{1}),(y, a_{3})] ⇒ Equi-mult[(x, a_{2}), (y, a_{4})]. 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 just what 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 > b ⇒ c : 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 : c ⇒ a = b and c : a ≈ c : b ⇒ a = b (converse of 7)
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 : c ⇒ a > 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. a_{1}: b_{1} ≈ ... ≈ a_{n} : b_{n} ⇒ a_{i}: b_{i} ≈ (a_{1} + ... a_{n}) : (b_{1} + ... b_{n})
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. a_{1} : a_{2} ≈ a_{3} : a_{4} & a_{3} : a_{4} > a_{5} : a_{6} ⇒ a_{1} : a_{2} > a_{5} : a_{6}
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. a_{1} : a_{2} ≈ a_{3} : a_{4} ⇒ a_{1} > a_{3} → a_{2} > a_{4 }& a_{1} = a_{3} → a_{2} = a_{4} & a_{1} < a_{3} → a_{2} < a_{4}
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 : d ⇒ a : b ≈ c : d and inversely. We can also express this as: a : b ≈ c : d ⇒ a+b : b ≈ c+d : d
Prop. 19: If it is: as a whole to a whole so a magnitude taken way to one taken away, the remainder to the remainder will be as whole to whole. a < b & c < d & a : c ≈ b : d ⇒ b : d = b-a : d-c
Corollary: From this, in fact, it is obvious that if added magnitudes are proportional, they will also be proportion in conversion (convertendo), just what it was required to show. a : b ≈ c : d ⇒ a: a-b ≈ c : c-d
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. a_{1} : a_{2} ≈ b_{1} : b_{2} & a_{2} : a_{3} ≈ b_{2} : b_{3} ⇒ a_{1} > a_{3} → b_{1} > b_{3} & a_{1} = a_{3} → b_{1} = b_{3} & a_{1} < a_{3} → b_{1} < b_{3}
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. a_{1} : a_{2} ≈ b_{2} : b_{3} & a_{2} : a_{3} ≈ b_{1} : b_{2} ⇒ a_{1} > a_{3} → b_{1} > b_{3} & a_{1} = a_{3} → b_{1} = b_{3} & a_{1} < a_{3} → b_{1} < b_{3}
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. a_{1} : a_{2} ≈ b_{1} : b_{2} & ... & a_{n-1} : a_{n} ≈ b_{n-1} : b_{n} ⇒ a_{1} : a_{n} ≈ b_{1} : b_{n}
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. a_{1} : a_{2} ≈ b_{2} : b_{3} & a_{2} : a_{3} ≈ b_{1} : b_{2} ⇒ a_{1} : a_{3} ≈ b_{1} : b_{3}
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. a_{1} : a_{2} ≈ a_{3} : a_{4} & a_{5} : a_{2} ≈ a_{6} : a_{4} ⇒ a_{1}+a_{5} : a_{2} ≈ a_{3}+a_{6} : a_{4}
Prop. 25: If four magnitudes are proportional, the largest of them and the smallest are larger than the two remaining.