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Ancient Mathematics
Note on tranlations: the words ‘arithmos’ (plural: arithmoi) and ‘plêthos’ (plêthê) are left merely transliterated. Both have something of the sense of ‘number’, although ‘arithmos’ is more common outside Euclid's Elements for an ordinary cardinal number (not relevant in the Elements). However, in the Elements, a ‘plêthos’ is any collection that can be put into 1-1 correspondance with another collection, while an ‘arithmos’ is specifically a ‘plêthos’ of units (see def. 2). And so, we find plêthê being put into 1-1 correspondance with the units in an arithmos. I think it better to conceive the two notions within the practice of the Elements.
Prop. 1: When unequal arithmoi are displayed, and the smaller is repeatedly taken away in turn from the larger, if the remaining arithmos never measures out the one before it, until a unit is left from it, the initial arithmoi will be prime in relation to one another.
Corollary: From this, in fact, it is obvious that if an arithmos measures two arithmoi, it will measure the largest common measure of them, which it was required to prove.
Prop. 2: Given arithmoi that are not prime in relation to one another, to find the largest common measure of them.
Prop. 3: Given three arithmoi not prime in relation to one another, to find the largest common measure of them.
Prop 4: Every arithmos smaller than any arithmos is either a part or parts.
Prop 5: If an arithmos is a part of an arithmos, and another is the same part of another, then together it will be the same part of it together that the one is of the one.
Prop 6: If an arithmos is parts of an arithmos, and another the same parts of another, then together both will be the same part of together both that one is of one of them.
Prop 7: If an arithmos is a part of an arithmos, just what an arithmos taken away is of an arithmos taken away, then the remainder will also be the same part of the remainder what the whole is of the whole.
Prop 8: If an arithmos is parts of an arithmos, just what (an arithmos) taken away is of (an arithmos) taken away, then the remainder will also be the same parts of the remainder what the whole is of the whole.
Prop 9: If an arithmos is a part of an arithmos, and another is the same part of another, then alternately (alternando) the part or parts that the first is of the third is the same part or same parts as the second is of the fourth.
Prop 10: If an arithmos is parts or an arithmos, and another is the same parts of another, then alternately (alternando) the parts which he first is of the third will be the same parts, or the same part, as the second of the fourth.
Prop 11: If as a whole is to a whole, so is what is taken away to what is taken away, the remainder will be to the remainder as whole to whole.
Prop 12: If any arithmoi are proportional, as one of the leaders is to one of the followers, so will all the leaders be to all the followers.
Prop 13: If four arithmoi are proportional, they will also be alternando proportional.
Prop 14: If however-many arithmoi and others equal to them in plêthos, by being taken in pairs and in the same ratio, then they will be ex aequali in the same ratio.
Prop 15: If a unit measures an arithmos equal-times as another arithmos measures some other arithmos, then, alternando, the unit will measure the third arithmos equal-times as the second arithmos the fourth.
Prop 16: If two arithmoi by multiplying one another make certain arithmoi, those from them will be equal to one another.
Prop 17: If an arithmos by multiplying two arithmoi makes some arithmoi, they will have the same ratio as those multiplied.
Prop 18: If two arithmoi by multiplying some arithmos make certain arithmoi, those that come about from them will have the same ratio as those multiplied.
Prop. 19: If four arithmoi are proportional, the arithmos that comes about from the first and fourth will be equal to the arithmos that comes about from the second and third; if the arithmos from the first and fourth is equal to that from the second and third, the four arithmoi will be proportional.
Prop. 20: If three arithmoi are proportional, that under the extremes is equal to that from the middle, and if that under the extremes is equal to that from the middle, the three arithmoi are proportional.
Prop. 20 (vulgo): The smallest arithmoi of those having the same ratio measure equal-times those having the same ratio, the larger the larger and the smaller the smaller.
Prop. 21: Primes in relation to one another are the smallest of those having the same ratio as they.
Prop. 22: The least arithmoi of those having the same ratio as they are prime in relation to one another.
Prop. 22 (vulgo) If there are three arithmoi and others equal to them in plêthos, taken two by two and in the same ratio, and the proportion of them is arranged, then they will be in the same ratio from an equal (ex aequali).
Prop. 23: Arithmoi prime in relation to one another are the smallest of those having the same ratio as they.
Prop. 24: The smallest arithmoi of those having the same ratio as they are prime in relation to one another.
Prop. 25: If two arithmoi are prime in relation to one another, the arithmos that comes about from one of them will be prime in relation to the remaining one.
Prop. 26: If two arithmoi are both prime in relation to two arithmoi, to each, the arithmoi that come about from them will also be prime in relation to one another.
Prop. 27: If two arithmoi are prime in relation to one another and each by multiplying itself makes some arithmos, the arithmoi that come about from them will be prime in relation to one another, and if the initial arithmoi by multiplying those that came about make some arithmoi, those will be prime to one another [and this keeps occurring with the last arithmoi].
Prop. 28: If two arithmoi are prime in relation to one another, together the arithmos of them will also be prime in relation to each of them, and if together the arithmos of them is prime in relation to one of them, then the initial arithmoi will be prime in relation to one another.
Prop. 29: Every prime arithmos is prime in relation to every arithmos which it does not measure.
Prop. 30: If two arithmoi, by multiplying one another make some arithmos, and some prime arithmos measures the arithmos that comes about from them, then it will measure one of the initial arithmoi.
Prop. 31: Every composite arithmos is measured by some prime arithmos.
Prop. 32: Every arithmos either is prime or is measured by some prime.
Prop. 33: Given however many arithmoi, to find the smallest of those having the same ratio as they.
Prop. 34: Given two arithmoi, to find the smallest arithmos that they measure.
Prop. 35: If two arithmoi measure some arithmos, the smallest measured by them will also measure it.
Prop. 36: Given three arithmoi, to find the smallest arithmos that they measure.
Prop. 37: If an arithmos is measured by some arithmos, the measured arithmos will have a part homonymous to the measure.
Prop. 38: If an arithmos has any part, it will be measured by an arithmos homonymous with the part.
Prop. 39: To find an arithmos which will be the least arithmos having given parts.