Part IV, Chapter 5: That the Geometers do not appear always to have rightly understood the Difference between the Definition of Words and the Definition of Things
ALTHOUGH there are no authors who have turned the definition of words to better account than the geometers, I feel myself, nevertheless, obliged to remark here, that they have not always regarded the difference which ought to be observed between the definitions of things and the definitions of words, to wit, that the first are open to dispute, and that the others cannot be disputed; for there are some who dispute about the definition of words as earnestly as though they were the things themselves.
Thus we may see, in the Commentaries of Clavius on Euclid, a long and very angry dispute between Pelletier and himself, touching the space between the tangent and the circumference, which Pelletier affirmed was not an angle, when Clavius maintained that it was. Who does not see that all this might have been settled in a word by demanding from each what he understood by the term angle?
We see, again, that Simon Stevin, a very celebrated mathematician of the Prince of Orange, having defined number thus:—Number is that by which the quantity of every thing is explained,—gets immediately into a great rage against those who do not allow unity to be a number, breaking into rhetorical exclamations as though it were a most important discussion. It is true that he mingles with that discourse a question of some importance, which is, Whether the unit is to number what a point is to a line? But it is necessary to distinguish this, in order that we may not confuse two things very different. And thus, to treat separately these two questions—the one, whether the unit is a number; the other, whether the unit is to number what a point is to a line—it must be said about the first that it is only a dispute touching words, and that the unit may be a number, or may not be, according to the definition which we choose to give of number; for, defining it as Euclid does,—number is a multitude of units together—it is plain that the unit is not a number; but that, as this definition of Euclid was arbitrary, and we may thus give another to the word number, we may give to it one such as that which Stevin proposes, according to which unity is a number. Hence the first question is void; and we cannot say any thing against those who choose to call unity a number without a manifest begging of the question, as we may see by examining the pretended demonstrations of Stevin. The first is:—
The part is of the same nature as the whole;
Unity is part of a multitude of units;
Therefore unity is of the same nature as a multitude of units, and, consequently, a number.
This argument is worth nothing at all; for though the part be always of the same nature as the whole, it will not follow that it must always have the same name as the whole; and, on the contrary, it very often happens that it has not the same name. A soldier is one part of an army, and not an army; a room is one part of a house, and not a house; a semicircle is not a circle; a part of a square is not a square. This argument proves, therefore, rather that unity, being part of a multitude of unities, has something in common with the whole multitude of unities, in relation to which we may say that it is of the same nature; but this does not prove that we are obliged to give the same name, number, to a unit and a multitude of units, since we may, if we choose, keep the term, number, for a multitude of units, and give to the unit only the name of unity, or of a part of a number.
The second reason of Stevin is no better: —
If, from a given number we take away no number, the number remains the same;
Therefore, if unity were no number, in taking one from three, the given number would remain the same, which is absurd. But the major here is ridiculous, and supposes the very thing in dispute; for Euclid will deny that the given number remains when we have taken away no number from it, since it is enough for its not continuing what it was, that we take away from it either a number, or a part of a number, such as the unit is. And if this argument were good, we might prove, in the same way, that in taking away a semicircle from a given circle, the given circle must remain, since we have taken away from it no circle.
Thus all the arguments of Stevin prove rather that we may define the word number in such a way that it may apply to unity, inasmuch as unity, and the multitude of unities, have sufficient in common to enable them to be signified by the same name; but they do not prove at all that we may not also define number by restricting this word to a multitude of units, in order that we may not be obliged to except unity whenever we explain the properties which belong to all numbers but unity.
But the second question—that, to wit, whether the unit is to other numbers as the point is to the line—is not of the same nature as the first, and is not a dispute of a word, but of a thing. For it is absolutely false that the unit may be to number as the point is to the line, since unity added to a number makes it greater, whereas, when a point is added to a line, it does not. Unity is part of number, and the point is no part of a line. When unity is taken away from a number, the given number does not remain; and when the point is taken away from the line, the given line does remain.
The same Stevin is full of such disputes on the definition of words, as when he labours zealously to prove that number is not a discrete quantity—that the proportion of numbers is always arithmetical, and not geometrical,—that every root, of any number whatever it may be, is a number;—which proves that he did properly understand what the definition of a word was, and that he has taken the definitions of words which cannot be contested, for the definitions of things which may be very often justly contested.