has given to the word Proportion agrees with them. For that there is between the first number and the second compared together according to quantity, a habitude like to that between the third and the fourth.

Now that he might avoid falling into this Inconvenience he should have observed, that there are two ways of comparing two Magnitudes; one by considering how far the one surpasses the other, and the second, by considering after what manner the one is contained in the other. And in regard these two habitudes are different, he ought to have given them different Names, to the first the name of Difference, to the second the name of Reason. Afterwards he ought to have defined Proportion, the Equality of the one or the other of these two sorts of Habitudes, that is, of Difference or Reason. And as this makes two Species, to have distinguished them also by two several names, calling the Equality of Difference, Arithmetical Proportion; and equality of Reason, Proportion Geometrical. And because the latter is much more beneficial than the former, the Readers are to be admonished, that when Proportion or Proportional Magnitudes are barely named, it is to be understood of Geometrical Proportion; but for Arithmetical Proportion, it is never to be understood, but when it is expressed. Which would have unveiled all obscurity, and taken away all Equivocation.

This shews us that we are not to make an ill use of that Maxim, That the Definitions of words are Arbitrary. But that great heed is to be taken to design so clearly and exactly the Idea to which we affix the word, that is to be defined, that we may not be deceived in the Series of the Discourse; by taking the word in another Sense than that which is given it by the Definition; so that we may always substitute the Definition to the thing defined without falling into Absurdity.

CHAP. V.

That the Geometricians seem not to have rightly understood the

difference between the Definitions of words and things.

ALTHOUGH there are not any Writers, who make a better use of the Definitions of Words, than the Geometricians, yet I cannot but observe, that they have not rightly understood the difference between the Definitions of words and things; which is, that the first are disputable, the second not to be controverted: For I find some that raise Disputes about the Definitions of words with the same heat, as if they were disputing about the things themselves,

Thus we find in the Commentaries of Clavius upon Euclid, a long and violent dispute between Pelletier and himself, touching the space between the Tangent and the Circumference, which Clavius denies, Pelletier affirms to be an Angle. Who does not see, that all this might be determined in one word, by demanding of Both, what they meant by the word Angle?

We find also the Famous Simon Stevin, Mathematician to the Prince of Orange, having defined Number to be, That by which is explained the quantity of every Thing, he becomes so highly inflamed against those that will not have the Unit to be a Number, as to exclaim against Rhetoric, as if he were upon some solid Argument. True it is that he intermixes in his Discourses a question of some Importance, that is, whether a Unit be to Number, as a Point is to a Line. But here he should have made a distinction, to avoid the confusing together of two different things. To which end these two questions were to have been treated apart; whether a Unit be Number, and whether a Unit be to Number, as a Point is to a Line; and then to the first he should have said, that it was only a Dispute about a Word, and that an Unit was, or was not a Number, according to the Definition, which a Man would give to Number. That according to Euclid's Definition of Number; Number is a Multitude of Units assembled together it was visible, that a Unit was no Number. But in regard this Definition of Euclid was arbitrary, and that it was lawful to give another Definition of Number, Number might be defined as Stevin defines it, according to which Definition a Unit is a Number; so that by what has been said, the first question is resolved, and there is nothing farther to be alleged against those that denied the Unit to be 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 with the whole,
The Unit is a Part of a Multitude of Units,

Therefore the Unit is of the same Nature with a Multitude of
Units, and consequently of Number.

This Argument is of no validity. For though the part were always of the same nature with the whole, it does not follow that it ought to have always the same name with the whole; nay it often falls out, that it has not the same Name. A Soldier is part. of an Army, and yet is no Army; a Chamber is part of a House, and yet no House; a Half Circle is no Circle; a Part of a Square is no square. This Argument therefore proves no more, than that, -Unit being part of a Multitude of Units, has something common

with a Multitude of Units, and so it may be said to have something common with them; but it does not prove any necessity of giving the same name of Number to Unit, as to a Number of Units: Because if we would we could not reserve the name of Number to a multitude of Units, nor give to Unit more than its name of Unit, or part of Number.

The Second Argument which Stevin produces is of no more force.

If from a Number given we subtract any Number, the Number given remains.

If then the Unit were not a Number, Subtracting one out of three, the Number given would remain, which is absurd.

But here the major is ridiculous, and supposes the Thing in Question. For Euclid will deny that the Number given remains after subtraction of another Number. For to make it another Number than what was given, there needs no more than to subtract a Number from it, or a part of a Number, which is the Unit. Besides, if this Argument were good, we might prove in the same manner, that by taking a half Circle from a Circle given, the Circle given would remain, because no Circle is taken away.

So that all Stevins's Arguments prove no more, than that Number may be defined in such a manner, that the word Number may agree with Unity, because that Unit and multitude of Units accord so well together, as to be signified by the same word, yet they no way prove that number can be no way defined, by restraining the word to the Multitude of Units, that we may not be obliged to except the Unit, every time we explain the properties that belong to all numbers, except the Unit.

But the second Question, Whether an Unit be to Number, as a Point is to a Line, is a dispute concerning the thing? For it is absolutely false, that an Unit is to number as a point is to a Line. Since an Unit added to number makes it bigger, but a Line is not made bigger by the addition of a point. The Unit is a part of Number, but a Point is no part of a Line. An Unit being subtracted from a Number, the Number given does not remain; but a point being taken from a Line, the Line given remains.

Thus doth Stevin frequently wrangle about the Definition of words, as when he perplexes himself to prove that Number is not a quantity discreet, that Proportion of Number is always Arithmetical, and not Geometrical, that the Root of what Number soever, is a Number, which shews us that he did not properly understand the definition of words, and that he mistook the definition of words, which were disputable, for the definition of things that were beyond all Controversy.

CHAP. VI.

Of the Rules in Reference to Axioms.

ALL Men agree, that there are some Propositions so clear and evident of themselves, that they have no need of being demonstrated and that all that are not demonstrated, ought to be such, that they may become the principle of true Demonstration. For if they be sullied with the least uncertainty, it is clear, that they cannot be the ground of a conclusion altogether certain.

But there are some who do not apprehend wherein this clearness and evidence of a Proposition consists. For it is not to be imagined, that a Proposition is then only clear and certain, when nobody contradicts it: Or that it ought to be questioned, or at least that we should be obliged to prove it, when we meet with any one that denies it. For if that were so, there would be nothing clear and certain, in regard there are a sort of Philosophers that question every thing; and others, who assert that there is no proposition more probable than its contrary. And therefore we must not judge of certainty or truth by the contest among men. For there is nothing about which we may not contend, especially in words : But we are to take that for clear and certain, which appears to be so to all those, who will take the pains diligently to consider things, and who are no less sincere and ingenuous to discover what inwardly they think of them. It is, therefore, a great saying of Aristotle, that Demonstration relates more to the inward Eviction of the mind, than to the forcing of an outward belief. For that there is nothing which can be so evidently demonstrated, which may not be denied by a Person truly opinionated; who many times engages himself in disputes about things, of which he is inwardly persuaded to the contrary. Which is a sign of forward Disposition, and an ill contrived genius: Though it be too true, that this humour is frequently predominant in the Schools of Philosophy, wherein custom of brangling has prevailed, and it is thought dishonourable to submit in the least; he being accounted to have most wit, who is most ready at shifts and evasions. It is, however, the Character of an ingenuous Man to yield his Arms to Truth, as soon as she comes to be perceived, and to admire her even in the Mouth of his Adversary.

Secondly, all Philosophers, who affirm that our Ideas proceed from our senses, maintain also, that all certainty and evidence of Propositions proceed either immediately or mediately from the senses. For, say they, this Axiom, than which there can be nothing desired more clear and evident. The whole is greater than

a part, has gained no belief in our understandings, but only because we have particularly observed from our Infancy, that every Man is bigger than his Head, that a House is bigger than a Chamber, a Forest than a Tree, and the whole Heavens than a Star.

This Imagination is as false as that which we have refuted in the first part, That all our Ideas proceed from our Senses. For if we were not assured of this Truth, That the Whole is bigger than a part, but by our observations from our Infancy, in regard Induction is no certain means to know a thing, but when we are assured the Induction is entire. There being nothing more frequent, than to discover the falsity of what we have believed upon the credit of Inductions, which seemed to us so general, that it was thought impossible to make any exceptions against them.

Thus it was formerly thought a thing not to be questioned, that the Water contained in an Arched Vessel, having one side much more capacious than the other, kept always at an even level, not rising higher on the greater side than on the less, because we seemed to be assured of it, by an infinite number of experiments. But afterwards this was discovered to be false, provided that one of the sides of the Vessel be very narrow, for then the water will rise higher on that, than on the other side. This shews us, that Inductions only can give us no solid assurance of any Truth, unless we could be certain they were general, which is impossible. And by consequence we could be but probably assured of the Truth of this Axiom; The whole is bigger than the part, were we no other way assured of it, but because we have seen a Man bigger than his Head, a Forest bigger than a Tree, a House bigger than a Chamber, or the Heavens than a Star. For that we should have always reason to doubt whether there were not some other whole, not so big as its part, that had escaped our knowledge.

It is not then upon the observations we have made from our Infancy, that the certainty of this Axiom depends, there being nothing more likely to precipitate us into error, than to trust to the prejudices of our Infancy: But it solely depends upon that which is contained in the clear and distinct Ideas of the whole, and a part; that is, that the whole is bigger than a part, and a part less than the whole. And as for all our former observations of a Man's being bigger than his Head, a House than a Chamber, they only furnish us with an occasion to consider more diligently the Ideas of the whole and a part. But it is absolutely false, that they are the absolute and undeniable causes of the Truth of this Axiom.