Lastly, if they find it surprising that a small space has as many parts as a great one, let them understand also that they are smaller in measure, and let them look at the firmament through a diminishing glass, to familiarize themselves with this knowledge, by seeing every part of the sky in every part of the glass.

But if they cannot comprehend that parts so small that to us they are imperceptible, can be divided as often as the firmament, there is no better remedy than to make them look through glasses that magnify this delicate point to a prodigious mass; whence they will easily conceive that by the aid of another glass still more artistically cut, they could be magnified so as to equal that firmament the extent of which they admire. And thus these objects appearing to them now easily divisible, let them remember that nature can do infinitely more than art.

For, in fine, who has assured them that these glasses change the natural magnitude of these objects, instead of re-establishing, on the contrary, the true magnitude which the shape of our eye may change and contract like glasses that diminish?

It is annoying to dwell upon such trifles; but there are times for trifling.

It suffices to say to minds clear on this matter that two negations of extension cannot make an extension. But as there are some who pretend to elude this light by this marvellous answer, that two negations of extension can as well make an extension as two units, neither of which is a number, can make a number by their combination; it is necessary to reply to them that they might in the same manner deny that twenty thousand men make an army, although no single one of them is an army; that a thousand houses make a town, although no single one is a town; or that the parts make the whole, although no single one is the whole; or, to remain in the comparison of numbers, that two binaries make a quaternary, and ten tens a hundred, although no single one is such.

But it is not to have an accurate mind to confound by such unequal comparisons the immutable nature of things with their arbitrary and voluntary names, names dependent

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, be

ing multiplied several times, exceed any number whatsoever, it is precisely of the same kind with numbers through its essence and its immutable nature, in the meaning of the same Euclid who would not have it called a number.

It is not the same thing with an indivisible in respect to an extension. For it not only differs in name, which is voluntary, but it differs in kind, by the same definition; since an indivisible, multiplied as many times as we like, is so far from being able to exceed an extension, that it can never form any thing else than a single and exclusive indivisible; which is natural and necessary, as has been already shown. And as this last proof is founded upon the definition of these two things, indivisible and extension, we will proceed to finish and perfect the demonstration.

An indivisible is that which has no part, and extension is that which has divers separate parts.

According to these definitions, I affirm that two indivisibles united do not make an extension.

For when they are united, they touch each other in some part; and thus the parts whereby they come in contact are not separate, since otherwise they would not touch each other. Now, by their definition, they have no other parts; therefore they have no separate parts; therefore they are not an extension by the definition of extension which involves the separation of parts.

The same thing will be shown of all the other indivisibles that may be brought into junction, for the same reason. And consequently an indivisible, multiplied as many times as we like, will not make an extension. Therefore it is not of the same kind as extension, by the definition of things of the same kind.

It is in this manner that we demonstrate that indivisibles are not of the same species as numbers. Hence it arises that two units may indeed make a number, because they are of the same kind; and that two indivisibles do not make an extension, because they are not of the same kind.

Hence we see how little reason there is in comparing the relation that exists between unity and numbers with that which exists between indivisibles and extension,

But if we wish to take in numbers a comparison that rep

resents with accuracy what we are considering in extension, this must be the relation of zero to numbers; for zero is not of the same kind as numbers, since, being multiplied, it cannot exceed them: so that it is the true indivisibility of number, as indivisibility is the true zero of extension. And a like one will be found between rest and motion, and between an instant and time; for all these things are heterogeneous in their magnitudes, since being infinitely multiplied, they can never ma any thing else than indivisibles, any more than the indivisibles of extension, and for the same reason. And then we shall find a perfect correspondence between these things; for all these magnitudes are divisible ad infinitum, without ever falling into their indivisibles, so that they all hold a middle place between infinity and nothingness.

Such is the admirable relation that nature has established between these things, and the two marvellous infinities which she has proposed to mankind, not to comprehend, but to admire; and to finish the consideration of this by a last remark, I will add that these two infinites, although infinitely different, are notwithstanding relative to each other, in such a manner that the knowledge of the one leads necessarily to the knowledge of the other,

For in numbers, inasmuch as they can be continually aug mented, it absolutely follows that they can be continually diminished, and this clearly; for if a number can be multiplied to 100,000, for example, 100,000th part can also be taken from it, by dividing it by the same number by which it is multiplied; and thus every term of augmentation will become a term of division, by changing the whole into a fraction. So that infinite augmentation also includes necessarily infinite division.

And in space the same relation is seen between these two contrary infinites; that is, that inasmuch as a space can be infinitely prolonged, it follows that it may be infinitely diminished, as appears in this example: If we look through a glass at a vessel that recedes continually in a straight line, it is evident that any point of the vessel observed will continually advance by a perpetual flow in proportion as the ship recedes. Therefore if the course of the vessel is extended ad infinitum, this point will continually recede; and

yet it will never reach that point in which the horizontal ray carried from the eye to the glass shall fall, so that it will constantly approach it without ever reaching it, unceasingly dividing the space which will remain under this horizontal point without ever arriving at it. From which is seen the necessary conclusion that is drawn from the infinity of the extension of the course of the vessel to the infinite and infinitely minute division of this little space remaining beneath this horizontal point.

Those who will not be satisfied with these reasons, and will persist in the belief that space is not divisible ad infinitum, can make no pretensions to geometrical demonstrations, and although they may be enlightened in other things, they will be very little in this; for one can easily be a very capable man and a bad geometrician.

But those who clearly perceive these truths will be able to admire the grandeur and power of nature in this double infinity that surrounds us on all sides, and to learn by this marvellous consideration to know themselves, in regarding themselves thus placed between infinitude and a negation of extension, between an infinitude and a negation of number, between an infinitude and a negation of movement, between an infinitude and a negation of time. From which we may learn to estimate ourselves at our true value, and to form reflections which will be worth more than all the rest of geometry itself.

I have thought myself obliged to enter into this long discussion for the benefit of those who, not comprehending at first this double infinity, are capable of being persuaded of it. And although there may be many who have sufficient enlightenment to dispense with it, it may nevertheless happen that this discourse which will be necessary to the one will not be entirely useless to the other.

PREFACE TO THE TREATISE ON VACUUM

THE respect that we bear to antiquity is at the present day carried to such a point on subjects in which it ought to have less weight, that oracles are made of all its thoughts and