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 make 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 augmented, 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

mysteries, even of its obscurities; that novelties can no longer be advanced without peril, and that the text of an author suffices to destroy the strongest reasons...

Not that it is my intention to correct one error by another, and not to esteem the ancients at all because others have esteemed them too much.

I do not pretend to banish their authority in order to exalt reasoning alone, although others have sought to establish their authority alone to the prejudice of reasoning....

To make this important distinction with care, it is necessary to consider that the former depend solely on memory and are purely historical, having nothing for their object except to know what the authors have written; the latter depend solely on reasoning and are entirely dogmatic, having for their object to seek and discover concealed truths.

Those of the former kind are limited, inasmuch as the books in which they are contained......

It is according to this distinction that we must regulate differently the extent of this respect. The respect that we should have for ....

In matters in which we only seek to know what the authors have written, as in history, geography, jurisprudence, languages, and especially in theology; and in fine in all those which have for their principle either simple facts or divine or human institutions, we must necessarily have recourse to their books, since all that we can know of them is therein contained; hence it is evident that we can have full knowledge of them, and that it is not possible to add any thing thereto.

If it is in question to know who was the first king of the French; in what spot geographers place the first meridian; what words are used in a dead language, and all things of this nature; what other means than books can guide us to them? And who can add any thing new to what they teach us, since we wish only to know what they contain?

Authority alone can enlighten us on these. But the subject in which authority has the principal weight is theology, because there she is inseparable from truth, and we know it only through her: so that to give full certainty to matters incomprehensible to reason, it suffices to show them in the

sacred books; as to show the uncertainty of the most probable things, it is only necessary to show that they are not included therein; since its principles are superior to nature and reason, and since, the mind of man being too weak to attain them by its own efforts, he cannot reach these lofty conceptions if he be not carried thither by an omnipotent and superhuman power.

It is not the same with subjects that fall under the senses and under reasoning; authority here is useless; it belongs to reason alone to know them. They have their separate rights: there the one has all the advantage, here the other reigns in turn. But as subjects of this kind are proportioned to the grasp of the mind, it finds full liberty to extend them; its inexhaustible fertility produces continually, and its inventions may be multiplied altogether without limit and without interruption...

It is thus that geometry, arithmetic, music, physics, medicine, architecture, and all the sciences that are subject to experiment and reasoning, should be augmented in order to become perfect. The ancients found them merely outlined by those who preceded them; and we shall leave them to those who will come after us in a more finished state than we received them.

As their perfection depends on time and pains, it is evident that although our pains and time may have acquired less than their labors separate from ours, both joined together must nevertheless have more effect than each one alone.

The clearing up of this difference should make us pity the blindness of those who bring authority alone as proof in physical matters, instead of reasoning or experiments; and inspire us with horror for the wickedness of others who make use of reasoning alone in theology, instead of the authority of the Scripture and the Fathers. We must raise the courage of those timid people who dare invent nothing in physics, and confound the insolence of those rash persons who produce novelties in theology. Nevertheless the misfortune of the age is such, that we see many new opinions in theology, unknown to all antiquity, maintained with obstinacy and received with applause; whilst those that are produced in physics, though small in number, should, it seems,