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 upon the caprice of the men who invented them. For it is clear that to facilitate discourse the name of army has been given to twenty thousand men, that of town to several houses, that of ten to ten units; and that from this liberty spring the names of unity, binary, quaternary, ten, hundred, different through our caprices, although these things may be in fact of the same kind by their unchangeable nature, and are all proportionate to each other and differ only in being greater or less, and although, as a result of these names, binary may not be a quaternary, nor the house a town, any more than the town is a house. But again, although a house is not a town, it is not however a negation of a town; there is a great difference between not being a thing, and being a negation of it.

For, in order to understand the thing to the bottom, it is necessary to know that the only reason why unity is not in the ranks of numbers, is that Euclid and the earliest authors who treated of arithmetic, having several properties to give that were applicable to all the numbers except unity, in order to avoid often repeating that in all numbers except unity this condition is found, have excluded unity from the signification of the word number, by the liberty which we have already said can be taken at will with definitions. Thus, if they had wished, they could in the same manner have excluded the binary and ternary, and all else that it pleased them; for we are master of these terms, provided we give notice of it; as on the contrary we may place unity when we like in the rank of numbers, and fractions in the same manner. And, in fact, we are obliged to do it in general propositions, to avoid saying constantly, that in all numbers, as well as in unity and in fractions, such a property is found; and it is in this indefinite sense that I have taken it in all that I have written on it.

But the same Euclid who has taken away from unity the name of number, which it was permissible for him to do, in order to make it understood nevertheless that it is not a negation, but is on the contrary of the same species, thus defines homogeneous magnitudes: Magnitudes are said to be of the same kind, when one being multiplied several times may exceed the other; and consequently, since unity can,