Part IV, Chapter 4: More particular Exposition of these Rules, and first of those which relate to Definitions
ALTHOUGH we have already spoken in the First Part touching the utility of the definition of terms, it is nevertheless so important, that we cannot have it too much impressed on our minds, since we may by it clear up a number of disputes, which have as their subject often only the ambiguity of terms, which one takes in one sense, and another in another. So that some of the greatest controversies would cease in a moment, if one or other of the disputants took care to mark out precisely, and in a few words, what he understands by the terms which are the subject of dispute.
Cicero has remarked that the greater part of the disputes between the ancient philosophers, and especially between the Stoics and the Academics, were founded only on this ambiguity of words,—the Stoics being delighted, in order to elevate themselves, to take several terms in a different sense from others. This created the belief that their morality was much more severe and perfect, although in reality this pretended perfection was only in words, and not in things. The wise man of the Stoics did not less enjoy all the pleasures of life than the philosophers of other sects, apparently not so strict, and did not avoid with less care its evils and inconveniences, with this single difference, that while other philosophers employed the common terms of good and evil, the Stoics, in enjoying pleasures, did not call them good things, but preferable things (προηγμένα); and in avoiding evils, they did not call them evils, but simply things to be rejected (ἀποπροηγμένα).
It is a caution very useful to cast away from all disputes everything which is founded only on the equivocation of words, by defining them in other terms so clear, that it is impossible for them to be any longer mistaken.
For this, the first of the rules which we have laid down avails: never to leave any term at all obscure or equivocal without defining it.
But in order to derive all the profit which we ought to do from these definitions, it is necessary still to add the second rule,—to employ in the definitions only terms perfectly well known, or already explained.
For when we have not marked out with sufficient precision and distinctness the idea to which we wish to attach a word, it is almost impossible for us, in the course of the argument, to avoid passing to another idea than that which we had marked out,—that is to say, instead of mentally substituting, every time we use the word, the same idea which we had designated, we substitute for it another word with which nature furnishes us; and it is easy to discover this by formally substituting the definition for the thing defined. For this ought not to change the proposition at all if we have always kept to the same idea, whereas it will change it if we have not done so.
All this will be better comprehended by some examples. Euclid defines a plane rectilinear angle the meeting of two right lines which incline towards each other in the same plane. If we consider this definition as the simple definition of a word, so that the word angle be considered as having been deprived of all signification in order to receive that of the meeting of two right lines, there is nothing to censure in it; for Euclid may be permitted to call the word angle the meeting of two lines. But he is bound to remember this, and never to take the word angle in any other sense. Now, in order to do this, it is only necessary to substitute for the word angle, wherever he uses it, the definition of it which he has given; and if, in substituting this definition, there be found any absurdity in what is said of an angle, it will follow that he has not kept to the same idea as he had designated, but that he has unconsciously passed to another, which is that of nature. He tells us, for example, how to divide an angle in two. Substitute his definition: Who does not see that it is not the meeting of two lines which we divide into two,—but it is not the meeting of two lines which has sides and a base or subtendant,—but that all this belongs to the space between the lines, and not to the meeting of the lines?
It is plain that what perplexed Euclid, and withheld him from designating an angle by the words—space comprised within two lines which meet together—was, that he saw that this space might be larger or smaller when the sides of the angle were longer or shorter, without the angle within being greater or less. But he ought not to have concluded from this that the rectilinear angle was not a space, but simply that it was a space contained between two right lines which meet together, indeterminate in relation to the one of the two dimensions, which answers to the length of these lines,—and determinate in relation to the other by the proportional part of a circumference, which has for its centre the point in which these lines meet.
This definition designates so exactly the idea which all men have of an angle, that it is at once the definition of a word and of a thing, except that the word angle comprises also, in common discourse, a solid angle, whereas, by this definition, it is restricted to signify a plane rectilinear angle. And when we have thus defined an angle, it is indubitable that everything which we may afterwards say of a plane rectilinear angle (such as we find in all rectilinear figures) will be true of this angle thus defined, without our ever being obliged to change the idea, and without our meeting with any absurdity in substituting the definition for the thing defined. For it is that spaces thus explained, which we may divide into two, into three, into four; it is that space which has two sides, between which it is contained; it is that space which we may terminate on the side which is itself indeterminate, by a line which is called the base or subtendant; it is that space which is not considered as greater or less for being contained between lines longer or shorter, because, being indeterminate in relation to this dimension, it is not from this that we ought to measure its greatness or smallness. By this definition, too, we obtain the means of judging whether one angle is equal to another angle, or greater or less; for since the size of that space is only determined by the proportional part of a circumference, which has for its centre the point in which the lines which contain the angle meet, when two angles have for their measure equal aliquot parts of its circumference, they are equal, as, for instance, the tenth part; and if one has the tenth, and the other the twelfth, that which has the tenth is greater than that which has the twelfth. Whereas, according to the definition of Euclid, we cannot understand in what the equality of two angles consists, which produces a terrible confusion in his Elements, as Ramus has remarked, though he himself makes scarcely any improvement.
The following are other definitions of Euclid, in which he commits the same fault as in that of the angle. " Ratio” says he, " is the habitude of two magnitudes of the same kind compared together, according to quantity. Proportion is a likeness of ratios”
According to these definitions, the term ratio ought to comprehend the habitude which is between two magnitudes, when we consider how far one exceeds the other; for it cannot be denied that this is a habitude of two magnitudes compared in relation to their quantity; and, consequently, four magnitudes will have a proportion together when the difference of the first to the second is equal to the difference of the third to the fourth. Nothing, therefore, can be said against these definitions of Euclid, provided that he always keeps to the notions which he has designated by these words, and to which he has given the names of ratio and proportion. But he does not always keep to them, since, according to what follows in his book, these four numbers, 3, 5, 8, 10, are not proportional, although the definition which he has given to the word proportion agrees with them, since there is between the first number and the second, compared according to quantity, a like habitude to that which exists between the third and the fourth.
It is necessary, therefore, in order not to be deceived by this disagreement, to remark that we may compare two magnitudes in two ways, one by considering how much one exceeds the other, and the other, in what way one is contained in another; and since these two habitudes are different, it is necessary to give them different names, giving to the first the name of difference, and to the second the name of ratio. It is necessary, accordingly, to define proportion as the equality of one or other of these kind of habitudes, that is to say, of the difference or of the ratio; and since this makes two species, to distinguish them also by two different names, by calling the equality of the differences arithmetical proportion, and the equality of the ratios geometrical proportion. And since this last is of much greater use than the first, we might still further premise, that when we simply speak of proportion, or proportional magnitudes, we mean geometrical proportion, and that we mean arithmetical only when it is so expressed. This would have cleared up all obscurity, and have removed the equivocation.
All this shows us that we ought not to abuse that maxim, that the definition of words is arbitrary, but that great care ought to be taken to designate so accurately and clearly the idea to which we wish to connect the word which we define that we cannot be deceived by it in the subsequent discourse, by changing that idea, that is, by taking the word in another sense from that which we had given to it in the definition, so that we cannot substitute the definition for the thing defined without falling into some absurdity.