Principles of Cartesian Philosophy

It is admitted by all who have any claims to superior intelligence that the method of mathematics, viz., the method by which conclusions are demonstrated from definitions, postulates, and axioms is the best method of obtaining and imparting truth. And rightly so; for as certain knowledge of an unknown object can only be obtained through facts previously known, there must of necessity be certain premises on which the whole superstructure of human knowledge rests, provided it does not fall of its own weight, or succumb to some slight attack from without. No one who has paid any attention to the noble study of mathematics can doubt its definitions or postulates or axioms. For definitions are but a very open explanation of the terms and names under which the subject is discussed, and the postulates and axioms of mathematics, or the general ideas of the mind, cannot be denied by any one who understands the use of his vocabulary.

Nevertheless mathematicians are almost the only ones committed to such a method. Others employ a method radically different from this, namely, a method where the end is attained through definitions and logical division interspersed with numerous questions and explanations. For almost all believe, and many well informed persons have asserted that this method is peculiar to mathematics and should be abjured in all other branches of study. Therefore they, in their discussions, are unable to offer apodeictic proof, but are compelled to reason by analogy and from probable evidence. They produce a whole medley of ponderous volumes in which nothing is established with certainty, but which are full of contending views; what is in one place asserted is presently, and for a similar reason, denied. So much so that the mind eager for eternal truth, when it had hoped to find the tranquil expanse of its own desire, and crossing this with propitious speed to gain the haven of true cognition, finds itself on a tempestuous sea of thought tossed about and overcome, surrounded by storms of contending belief, and lost amid waves of uncertainty, without hope of rescue.

There are some, however, who, regretting this wretched plight of Philosophy, in order that they may leave to posterity some studies beside mathematics established with absolute certainty, have departed from the ancient method to this new path, arduous though it be. Some of these have put into literary form the philosophy now accepted and accustomed to be taught in the schools; others have set in order new systems elaborated through their own reflection. Although for years the task was undertaken in vain, at length that splendid star of our century Rene Descartes arose, who, after he had made clear the mathematical truth that was inaccessible to the ancients, and everything desired by his contemporaries, also discovered this fundamental principle of all knowledge. By means of this truth he was able to elaborate and establish many things with mathematical certainty. To any one who attends to his writings, which cannot be too highly praised, this will be as evident as the midday sun.

Although the philosophical writings of this incomparable man contain a method of demonstrating mathematics, it is not the method found in Euclid and in other geometries. Descartes’ method, which he called Analysis and maintained was the best way to discover truth, was widely different from this. In the end of his “Response to the Second Objection,” he recognizes two kinds of apodeictic demonstration. The one, Analysis, which he showed to be the true method, by which truth is discovered methodically and as it were a priori; the other by Synthesis, the method in which a long series of definitions, and premises and axioms, and theorems, and problems is used so that if anything is denied in the conclusion, it is immediately shown to have been contained in the premises. By this means assent is extorted from the reader however unwilling or unyielding he may be.

Granted, however, that truth may be established beyond all chance of doubt by these two methods, still they do not have an equal value. For many plainly unlearned in mathematical knowledge and so wholly ignorant of the method by which such truth is discovered (analysis), and by which it is set in order (synthesis), are not only unable to teach this truth to others but are unable to follow it for themselves. Whence it happens that many who have made his opinions and dogmas only a matter of memory, carried away by some thoughtless attack or influenced by the authority of others, have defamed the name of Descartes, and, when a discussion of these things arises, since they cannot demonstrate anything, garrulously repeat what has always been ascribed to the Peripatetic Philosophy. Wherefore, in order that this state of affairs might be improved, I have often desired that some one, alike skilful in Analysis and Synthesis, well versed in the writings of Descartes, and thoroughly master of his philosophy, might give his attention to this work and what he has put in analytic form remold in synthetic order and demonstrate in the more familiar forms of geometry. I myself, although fully conscious of my unfitness for that task, have often been inclined to undertake it. Other occupations, however, have filled my time and prevented me from acting on my desire.

I was very glad to know, therefore, that our author had put into geometrical form, for a pupil whom he was teaching Descartes’ philosophy, the entire second Part of the Principles and a part of the third, together with some important questions and difficulties usually discussed in Metaphysics, and not yet discussed by Descartes; and that he had consented, at the urgent request of his friends, that these, corrected and revised by himself, should be published. I, therefore, approving this purpose, offered my services, if he had any need of them, in helping to publish the work. I asked and even urged him that he should put the first Part into similar form and let it precede the two already done, in order that the work might be complete and therefore more intelligible. Although for good reasons he did not wish to do this, he was unwilling to refuse his friends or to neglect anything which might be done for the benefit of the reader. He entrusted to my care the entire management of the publication since he had departed from the city into the country and was unable to be present.

This is what we offer you, therefore, dear reader, in this little volume: viz., the first and second Parts of Descartes’ Principles of Philosophy, together with a fragment of the third, to which we have added under the name of an appendix the Cogitata Metaphysica of our author. But when we speak of the first Part of the Principles, and the title of the book suggests the same thought, we do not wish it understood that all that Descartes has said in this is here presented demonstrated in geometrical order, but only that preferable terms have been selected and those principles which Descartes treated in his Meditations, which relate more particularly to Metaphysics. Those matters which are only of logical interest, however, or which Descartes expressed for their historical value, he has omitted. That he might the more easily effect this end our author, so far as the order is concerned, has transposed almost all that Descartes had put into geometrical form in the end of his “Response to the Second Objection.” This he did by placing all of Descartes’ definitions first and by inserting propositions of his own; by placing the axioms not together with his definitions, but a part of them after the fourth proposition; finally by omitting those not needed and by changing the order so that they might more easily be understood. It did not escape the notice of our author that these axioms might be demonstrated in the manner of theorems (as Descartes also held, Postulate 7.) and might even properly be classed as propositions. We, indeed, urged him to do this, but the amount of work with which he was employed only left him two short weeks in which to complete the work. For this reason he was unable to satisfy his own and our desire, but merely adding a brief explanation in place of the demonstration, he postponed until another time the complete and perfected volume, if perchance after this imperfect edition a new one should be demanded. To this end we shall urge him to complete the third Part concerning the visible world. (Of this we have added the fragment our author had completed, for we were unwilling that the reader should be deprived of any part of his work, however small.) To better accomplish his purpose, certain propositions concerning the nature of fluid bodies had to be inserted in Part II. For my part, I shall urge him to speedily complete the work.

Not only in regard to the axioms, but also in demonstrating the propositions and in other conclusions our author often differs from Descartes; for example, the term apodeictic is used in a widely different sense. However, let no one think that he wished to correct that most illustrious man in these things, but only that he did this in order to retain the current order and not increase the number of axioms unduly. For this and for many reasons he was compelled to demonstrate many things which Descartes had stated without demonstration and to add much that he had omitted.

Nevertheless, I wish it to be noted first of all that in all of these Parts, viz., in the first and second Parts of the Principles and in the fragment of the third, as well as in the Cogitata Metaphysica, our author is merely expressing the opinions of Descartes with their demonstrations so far as they are found in his writings or as they logically follow from his premises. For when he promised to teach a pupil the philosophy of Descartes it was a matter of principle with him not to depart in the least from his opinions or to teach anything that did not follow from his dogmas, or was contrary to them. Wherefore, let no one think that he is teaching here his own opinions or only what he approves. Although he adjudges certain things to be true, he affirms that others are opposed to his belief. Many things he rejects as false, from which he holds a far different opinion. Of this nature, to mention only one from many, are those conclusions concerning the Will, Schol. Prop. 15, Part I., and Chapter 12, Part II. of the Appendix, although they seem to be proved with painstaking care. For he did not think that the Will was something distinct from Intellect, much less endowed with such freedom. For in these assertions, as is evident from his dissertation concerning method Part 4, Meditation 2, and in other places, Descartes merely affirms and does not prove that the human mind is an absolute thinking substance. Although our author indeed admits that there is a thinking substance in Nature, he denies that this constitutes the essence of the human mind. He believed that in the same way that there are no limits to Extension, so Thought is in no way determined. And as the human body is not absolute, but its extension is determined according to natural laws of motion and of rest, so also the mind or human spirit is not absolute but is determined through ideas by natural laws of thought. These, we ought to conclude are given when the body begins to exist. From this definition it is easy to show that the Will and the Intellect cannot be distinguished, much less, as Descartes affirmed, can we say that the Will is endowed with liberty. To say that it is the faculty of affirming or of denying is wrong, for to affirm or to deny is only a form of idea. Indeed, those faculties, as Intellect, Desire, etc., ought to be placed in the list of fictitious thoughts, or at least of those ideas which men have because of their powers of abstraction, as for example, with humanity, lapidity, etc.

Reference must also be made to another point which was prominent in the mind of Descartes, namely, that this or that surpasses human knowledge. For it must not be thought that our author states this as his own opinion. He believed that all these things, and even many things more subtle and more sublime, could not only be clearly and distinctly conceived by us, but even readily explained if only the human mind were led in the way which Descartes opened up and made possible for investigating truth and for acquiring knowledge. Therefore the principles of knowledge which Descartes set forth and the philosophy based upon these do not suffice for solving all those extremely difficult problems which relate to metaphysics, but others are required if we desire the intellect to sound the depths of cognition.

And finally, to bring our preface to an end, we wish the reader to know that these papers are published for no other purpose than to discover and to impart truth, and to incite in men a desire for a true and a sincere philosophy. So let every one, having been diligently warned, before he undertakes to read this work, determine to correct as far as possible certain typographical errors which have crept in, and to insert the omissions in order that he may receive the full benefit which we earnestly desire for every reader. For these obstacles, as any one can readily see, may easily prevent the force of the demonstration and the thought of the author from being easily seen.

Before giving these propositions and their demonstration it seems best to recall briefly why Descartes came to doubt all things, how he discovered the fundamental principle of all knowledge, and finally how he liberated himself from this universal doubt. All of this we would have put in mathematical order if we had not thought that such prolixity would have impeded our understanding of these things which should be seen as clearly as though presented in a picture.

In order to proceed with his investigation with the utmost caution Descartes was compelled:

1. To lay aside all prejudices.

2. To find the fundamental truth on which all knowledge rests.

3. To discover the cause of error.

4. To understand everything clearly and distinctly.

In order to accomplish the first three points he doubted all things, not, however, as a sceptic who doubts merely for the sake of doubting, but in order to free his mind of all prejudices, so that he might find at length the firm and certain truth on which all knowledge rests. By using this method, such a truth, if any such existed, could not escape him. For this principle must be so clear that it needs no proof, and cannot under any circumstances be doubted; every demonstration must presuppose it. Such a truth he found after a long period of doubt. And after he had once gained this truth it was not difficult to distinguish the true from the false, or to detect the cause of error. And thus he could be on his guard lest he accept anything doubtful and false for what is certain and true.

To accomplish the last point, viz., to understand everything clearly and distinctly, his principal rule was to examine separately all the simple ideas from which all others are composed. For when he clearly and distinctly understood these simple ideas, he was enabled to understand in the same thorough way, all others into which they entered as component parts. Having prefaced our remarks with these few words we shall proceed with our purpose as stated above, namely, to explain why he doubted everything, how he found the fundamental truth of all knowledge, and how he extricated himself from the difficulties of these doubts.

In the first place he calls attention to all of those things perceived through the senses, the heavens, the earth and all external objects. So also, even concerning those things which he thought to be most certain he doubted, because he knew that his senses had sometimes deceived him, and in sleep he had often persuaded himself that many things existed in which he later found he had been deceived. And finally because he had heard competent witnesses affirm that they sometimes felt pain in limbs recently lost. It was not without reason, therefore, that he doubted everything even the existence of his own body. Hence from all these reasons he was able to conclude that sense perception is not a certain foundation for knowledge (all that the senses give may well be called in doubt), but certainty rests upon some more indubitable principle than this. To investigate further he next notices the universal attributes of corporeal matter, as extension, form, quantity, etc., as well as all mathematical truth. Although these seem more certain than the objects of sense perception, nevertheless, he finds a cause for doubting them as well. Some err even in these, and beside there is an old idea that God, who is omnipotent, and has created us with our present faculties, has perhaps so made us that we are deceived even in those things which seem most certain. These are the causes that led him to doubt all things.

In order to find the fundamental truth in knowledge, he afterward inquired whether all things which are subjects of cognition could be doubted, if perchance there was anything which he had not yet called in question. Doubting in this way he believed that if anything was found, which, for none of the reasons given above, should be doubted, this might be considered the foundation on which all knowledge rests. And although, as it now seemed he had doubted everything (for he had called in question all that the senses give, and all that comes from the understanding), there was something left the certainty of which had not been doubted, namely, he himself who was doubting. Not, however, so far as he consisted of head or hands or other bodily members, for he had doubted the existence of these, but even while he was doubting he was thinking, etc. Carefully examining this fact he found that for no reason whatever could it be doubted, for, whether waking or sleeping, if thinking at all he must therefore exist. And even though he and others might fall into error, since they were in error they must exist. Nor could he conceive of a creator so skilful in deceit that he could deceive him about this truth. For if it is supposed that he is deceived, it must also be supposed that he exists. Finally, whatever reason for doubt may be conceived, there is none which does not at the same time make one more certain of his own existence. Indeed the more arguments that can be assigned as cause for doubt the more there are which convince him of his own existence. So true is this that whoever begins to doubt will nevertheless exclaim, “I doubt, I think, therefore I am.”

In this truth he finds the ground of all knowledge as well as the measure of all other truth, viz., Whatever is as clearly and distinctly perceived as this is true.

That nothing but this Cogito ergo sum can be the fundamental truth in all knowledge is evident from what has already been said. Concerning this it should be noted in the first place, that it is not a syllogism in which the major premise is omitted. If it were, the premise cogito ought to be better known than the conclusion, ergo sum. And if this were so the ergo sum would not be the foundation of all knowledge. Beside it would not be a certain conclusion, for its truth depends upon universal premises which the author had called in question. Therefore Cogito ergo sum is one proposition equivalent to the statement ego sum cogitans.

To avoid confusion hereafter (for the matter ought to be thoroughly understood), we must know what we are. For if I clearly understand this our essence will not be confused with other things. To deduce this from what precedes our author thus proceeds:

He now recalled all those opinions formerly held, as for example, that his mind was something very fine in texture, like the wind, or fire, or air, interplaced with the coarser particles of the body; and that his body was better known than his mind and could be more clearly perceived. These opinions he now saw were at variance with what he had discovered. For he could doubt the existence of his body, but not his reality so far as he was a thinking being. Beside this, the body could not be clearly and distinctly known and therefore, according to his own dictum, should be rejected as non-existent. Therefore, since the body cannot be accepted as pertaining to his essence, so far as it is known, he further inquires what there is about his being which compels him to believe in his own existence. Such things were these: that he had determined to be on his guard lest he be deceived; that he had desired to understand so many things; that he had doubted everything he was not able to know; that he had affirmed only one thing at a time; that he had denied all else, and even rejected it as false; that he had conceived many things though reluctantly; and finally that he had considered many things as though derived from the senses. Since his existence was so evidently bound up with each one of these actions, and since none of them belonged to the things which he had doubted, and finally since they all may be considered as forms of thought, it follows that these are all true and pertain to his nature. So when he said cogito these modes of thought were all implied, viz., to doubt, to understand, to affirm, to deny, to wish, to be unwilling, to imagine, and to feel.

Some distinctions must here be noted that will have importance when we come to discuss the distinction between mind and body, (1) That modes of clear and distinct thought may be known even when some things are still in doubt. (2) That we render a clear and a distinct concept obscure and confused when we ascribe to it something concerning which we are still in doubt.

Finally, in order that he might be certain and remove all doubt from those things he had called in question, he further proceeds to inquire into the nature of a perfect Being and whether such a Being exists. For when he has discovered that this Being, by whose power all things are created and preserved and to whose nature it would be repugnant to be a deceiver, exists, then he has removed that reason for doubt which is found in the fact that he was ignorant of the cause of his existence. For he knew that the power of discerning the true from the false would not have been given to him by a God of perfect goodness and truth in order that he might be deceived. Mathematical truth, therefore, and all other of like certainty cannot be doubted. To remove other causes for doubt he inquires next why it is we sometimes fall into error. For when he discovered how error arose, and that we use our free will to assert what we perceive only confusedly, he concluded straightway that we could avoid error by withholding assent from that which is seen only indistinctly.

As every one has the power of inhibiting the will he can easily restrain it to the limits of the understanding. And since in youth we form many prejudices from which we free ourselves only with difficulty, he enumerates and examines separately all of our simple ideas to assist us in casting these prejudices aside. His object was to determine what was clear and what was obscure in each. Thus he was able to distinguish the clear from the obscure and to form clear and distinct ideas. By this means he easily found the real distinction between mind and body; what was clear and what obscure in those ideas derived from the senses; and finally how sleep differs from waking. When this was done he could doubt no longer concerning the waking life, nor could he be deceived by his senses. In this way he was able to free himself from all his recent doubt.

Before I close this part of the discussion it seems that some satisfaction should be given to those who argue, that since it is not known that God exists per se it is impossible for us ever to know that God does exist. For from uncertain premises (and we have said that all things are uncertain so long as we are ignorant of our origin), nothing can be concluded with certainty.

In order to remove this difficulty Descartes responded in this fashion; although we do not know whether the creator of our nature has created us so that we are deceived in those things which seem most certain, nevertheless, we cannot doubt those things we understand clearly and distinctly, so long as we attend merely to them. But we only doubt those things previously demonstrated, and now recalled to memory, when we no longer attend closely to the reasons from which they were deduced, which perchance are even forgotten. Therefore, though we cannot know directly that God exists, but must learn this by deduction, still, we are able to know this certainly, provided we attend very accurately to the premises from which the conclusion is deduced. Vid. Prin. Pt. I. Art. 13, and Response to Second, Obj. No. 3, and end of Med. 5.

But since this reply is not sufficient we will offer another.

We saw above, when speaking of the evidence and certainty of our existence, that this was found in the fact that, consider what we will, we meet no argument for doubt which does not at the same time convince us of the certainty of our existence. This is true whether we consider our own nature, or conceive of God as a skilful deceiver, or adduce some extraneous reason for doubt. For example, considering the nature of a triangle, though we are now compelled to believe that its three angles are equal to two right angles we are not forced to the conclusion that this is really true if perchance we are deceived by our Creator. In the same way we deduce the certainty of our existence. We are not compelled to believe that under any conditions the three angles of a triangle are equal to two right angles. On the contrary we find reason for doubt, for we have no idea of God which compels us to believe that it is impossible for God to deceive us. It is equally easy for one who has no true conception of God to think that he is a deceiver or that he is not. So for those who have no right conception of a triangle it is equally easy for them to think that the sum of the angles is equal to two right angles, or that it is not. Therefore, we grant that we cannot be absolutely certain of anything except of our own existence, however closely we attend to the proof, until we have a clear concept of God which compels us to affirm (in the same way that the concept of a triangle compels us to affirm that the sum of its angles is equal to two right angles) that he is perfectly true in His being. But we deny that we are unable to come to any certain knowledge of the external world. For, as now appears, the whole matter hinges upon this, viz., whether we can form such a concept of God that it is not as easy for us to think of Him as a deceiver as to believe that He is perfectly true in His being. When we obtain such a concept as this, all cause for doubting mathematical truth is removed. For, when we consider how the doubt of this affects our own existence, if we doubt this still we ought not to even affirm our own existence. If now having obtained this concept of God we consider the nature of a triangle we are compelled to affirm that the sum of its three angles is equal to two right angles; or if we consider the nature of God, and this also compels us to affirm that He is perfectly true and the author and continual preserver of our being, we are not deceived. Nor is it less impossible for us to think when we once have obtained this idea of God (which we suppose to be already found), that He is a deceiver, then when we consider the nature of a triangle to think that the sum of its angles is not equal to two right angles. As we can form such an idea of a triangle although we are not certain that God is not deceiving us, so we can form this idea of God, although we do not know whether or not He is deceiving us. And, provided only that we have such an idea of God, however it may have been obtained, it is sufficient to remove all doubt.

This point having been made clear I shall remark upon this difficult proposition: we can be certain of nothing not merely as long as we are ignorant of God’s existence (for I have not yet spoken of this), but as long as we do not have a clear and a distinct idea of His being. Hence if any one should desire to oppose our conclusions, his argument should be as follows: We cannot be certain of anything so long as we have no clear and distinct idea of God. But we cannot have a clear and a distinct idea of God as long as we do not know whether or not he is deceiving us. Therefore we cannot be certain of anything as long as we do not know whether or not our Creator is deceiving us, etc. To this I reply by conceding the major premise but, denying the minor. For we have a clear and a distinct idea of a triangle although we do not know whether or not God is deceiving us. And in the same way we have a clear and a distinct idea of God as I have already shown, and, therefore, cannot doubt His existence, nor any mathematical truth.

Our prefatory remarks being thus completed we proceed now to the main problem.

I. Under the term thought (cogitatio) I comprehend all mental phenomena of which we are immediately conscious.

Thus volition, understanding, imagination and sense perception are all forms of thought. I have added the term immediately to exclude phenomena which directly depend upon and follow from these mental states. Thus voluntary motion arises as the direct result of some form of thought but is not itself a mental state.

II. By the term idea (idea) I understand any form of thought of which we are conscious through immediate perception.

I cannot express anything in words, therefore, without thus making it certain that I have some idea which these words are meant to signify. Therefore I would even call the images depicted in phantasy, ideas, not, however, so far as they are corporeal, i. e., as they affect some portion of the brain, but only so far as they affect the mind in that portion of the brain.

III. By the objective reality of an idea, I understand the object represented by the idea.

In the same manner I may speak of objective perfection, or of an objective art, etc. For whatever we perceive in the objects of our ideas are objective in the ideas themselves.

IV. These characteristics are said to be formally (formaliter) contained in the objects of our ideas when they really are just as we perceive them. They are said to be eminently (eminenter) contained when they are not just as we perceive them but so great that they can easily supply what we perceive.

Note that when I say a cause eminently contains the perfection of its own effect, I mean that the cause contains the perfection of the effect more completely than the effect itself. Vid. Ax. 8.

V. Every object to which belongs as to a subject, some property, or quality, or attribute, or through which some things which we perceive exist, or of which we have some real idea is called substance.

Properly speaking, indeed we have no other idea of substance than that it is an object in which either formally or eminently something else exists which we perceive, or that it is objective in something apart from our ideas.

VI. Substance in which thoughts are immediately present, is called mind.

I use the term mind (mens), rather than spirit (animus), for the latter term is equivocal, often being used to mean a corporeal object.

VII. Substance, which is the immediate subject of extension, and of accidents, which presupposes forms of extension as figure, position, and motion, etc., is called body (corpus).

Whether mind and body are one and the same substance will be inquired into later.

VIII. Substance which we know to be perfect in the highest degree, and in which nothing can be conceived implying a defect or limitation, is called God.

IX. When we say that something is contained in the nature of the thing itself or in its concept, it is the same as to affirm that this is true.

X. Substances are said to be distinct when the one can exist alone and apart from others.

We have here omitted the Postulates of Descartes because we were unable to deduce any conclusions from them in what is to follow. Nevertheless, we earnestly ask the reader that he does not fail to carefully read them over and give them his earnest attention.

I. The knowledge and certainty of an unknown object depends upon the cognition of objects previously known.

II. There are reasons for doubting the existence of our own bodies.

(This was shown in the Prolegomenon, so may be placed here as an axiom.)

III. If our being comprises anything beside mind and body it is not so well known as these.

(These axioms, it should be noted, do not affirm objective existence, but only deal with objects as a part of our mental life.)

We cannot be absolutely certain of anything until we know that we really exist.

This proposition is self-evident. For he who does not know that he exists, cannot know that he is affirming or denying. It should be noted, too, that although we affirm and deny many things that have no reference to our existence, nevertheless unless this fact is accepted as indubitable all things are in doubt.—Q. E. D.

The proposition ego sum is self-evident.

If you deny that it is self-evident, it can be known only through some truth, prior to the proposition ego sum (per. Ax. 1), which is absurd (per Ibid.). Therefore it is self-evident.—Q. E. D.

The primary truth is not that I am a corporeal being, neither is this fact self-evident.

There are some reasons for doubting the existence of our bodies (Vid. Ax. 2). Hence (per Ax. 1) we must derive this truth through something previously and more indubitably known. Therefore, the primary truth is not that I am a corporeal being nor is this fact self-evident.—Q. E. D.

Ego sum is the primary fact in cognition only so far as I am a thinking being.

The assertion that I am a corporeal being is not the primary fact in cognition (per Prop. III.) neither am I certain of my existence except as I am mind and body. For if I comprise in my being any thing beside mind and body, it is not so well known to me as body (per Ax. 3). Therefore ego sum is the primary fact of cognition only so far as I am a thinking being.—Q. E. D.

From the last proposition it is evident that the mind is better known than the body. (For a fuller explanation, see Art. 11 and 12, Part I. of the Principles.)

Every one is certain that he affirms, he denies, he doubts, he understands, he imagines, etc., or that he is a doubting, an understanding, an affirming — in a word — a thinking being. This truth no one can doubt. Therefore the proposition cogito, or sum cogitans, is the fundamental truth of all Philosophy. And, since for certain knowledge nothing more can be demanded or desired than that we deduce all things from certain premises so that all our conclusions are as certain as our premises, it follows that all that we deduce from our principle so that if we doubt the conclusion we must also doubt the premises, must be held to be perfectly true. In order to proceed as cautiously as possible, in the beginning I shall admit to be of equal certainty only those things which we perceive in ourselves so far as we are thinking beings. As, for example, that one desires this or that, that one has certain ideas, and that one thing contains more perfection than another; namely, that which contains objective perfection of substance is far more perfect than that which contains only objective perfection of some accident. Finally, that that is the most perfect substance which contains the highest degree of perfect being. These things, I say, are not only all as clear as our first principle but, perhaps, are even more certain. For they not only affirm that we think but that we think in this particular way. And we shall find, when we come to test them, that they are not only indubitable, but that we cannot doubt their verity without doubting the fundamental truth of all knowledge. For example, if some one should say he is in doubt whether something can arise from nothing, he might also doubt his own existence even when he is thinking. For if I can affirm that something can exist without a cause I can, by the same right, affirm that thought may exist without a cause and that I think although I am nothing. Since this is impossible I cannot believe that something can arise from nothing. Leaving these matters for the present, it seems necessary, in order to proceed, to add to the number of Axioms we have already given. In the end of his “Response to the Second Objection,” Descartes has given certain truths as axioms, and I could not wish to be more accurate than he. Nevertheless, in order to preserve the order now begun, and to render them a little clearer, I shall attempt to show how they depend one upon the other and all upon the principle Ego sum cogitans, or that they are all as certain as this truth.

IV. There are different degrees of reality or being. For substance has more reality than accidents or mode; and infinite substance than finite. So, too, there is more objective reality in the idea of substance than in the idea of accident; and in the idea of infinite substance than in the idea of finite substance.

This axiom is known as true from a consideration of those ideas of which we are certain because they are modes of thought. For we know how much reality or perfection the idea of substance affirms of substance and how much the idea of mode affirms of mode. And since this is true we know that the idea of substance contains more objective reality than the idea of its accidents, etc. (Vid. Schol. Prop. 4).

V. A thinking being, if it were possible, would immediately add to itself any attribute of perfection in which it was lacking.

Every one observes this in himself so far as he is a thinking being; therefore (per Schol. Prop. 4) we know that this is true. And for the same reason we are equally certain of the inference.

VI. In the idea or concept of everything, existence either as possible or necessary is contained (Vid. Axioms of Descartes, No. 10).

In the concept of God or an absolutely perfect being, existence is necessary. For otherwise it would be imperfect which is contrary to the hypothesis.

VII. No object or quality of an object already existing can exist without some existing object as the cause of its existence.

In the Scholium to Prop. 4 I have shown that this axiom is equal in truth to the proposition Ego sum cogitans.

VIII. Whatever reality or perfection an object contains, this exists either formally (formaliter) or eminently (eminenter) in its primary or adequate cause.

By the term eminently I mean that the cause contains the perfection of the effect more fully than the effect itself. By the term formally I mean that the cause and the effect contain the perfection to a like degree.

This axiom depends upon the previous ones. For if it is supposed that there is less perfection in the cause than in the effect we have a result without a cause, and this is absurd (per Ax. 7).

Therefore nothing can be the cause of a given effect except that in which is contained eminently or at least formally, all the perfection found in the effect.

IX. The objective reality of our ideas requires a cause in which this same reality is not only objectively contained, but one in which it is found formally, or eminently.

Although this axiom is evident to all, many misuse it. For when some one forms some new idea everyone wishes to know why he did so. When they can assign some cause that contains formally or eminently all the perfection found in the concept they are content. Descartes has sufficiently explained this in his example of a machine (Vid. Prin. of Phil., Pt. I., Art. 17). So also if one inquires from whence man derives the ideas of his own thought and body, he finds that they are derived from himself. He discovers that he formally or at least constantly contains all that these ideas objectively contain. Therefore if one has some idea which contains more objective reality than he himself contains, impelled by reason he would seek some other cause outside of himself which formally or eminently contains all the perfection he is seeking to understand. Nor would any one ever assign any other reason for doing this than that he had conceived this with equal clearness and distinctness and that he had comprehended the truth of this axiom as it depends directly upon those preceding it. Namely (per Ax. 4), different degrees of reality or being are given in our ideas; and (per Ax. 8) for these degrees of perfection, some cause with equal perfection is required. But since these degrees of reality in our ideas are not merely in thought, but represent something in substance and its modes, in a word, so far as they are considered as images of things, it clearly follows that no other cause for this can be assigned than that all the reality they objectively contain is contained either formally or eminently in reason. This we have shown above and it is evident to all.

In order to make this perfectly clear I will illustrate with one or two examples. If one should see two books (for example, one written by a great philosopher and another by an uncultured man) written in the same hand, and should consider not the meaning of the words, i. e., the mental images they represent, but only the delineation of the characters in which the thoughts are expressed, he would discover no dissimilarity. So he would not be led to look for different authors for the books but would believe they were written by the same person and with a common end in view. But attending not to this but to the meaning of the words and of the discussions he would find great differences, and would conclude that they certainly had a different origin. He would find that the sense of the words being considered, that is, the concepts they represent, the one is far more perfect than the other. I speak here of the first cause of the books. Although as is evident the one might even have been derived from the other.

We may illustrate further by the statue of some leader. Here, if we attend only to the material used we will find no cause for seeking a different sculptor for this, and for some copy. Indeed, nothing hinders us from thinking that the first is a copy of the second, this again of a third and so on ad infinitum. If the material alone is considered we do not need a separate cause for each. But if we consider the statue as a statue we are immediately compelled to seek a first cause which contained either formally or eminently all that is presented to us. I do not see that this axiom requires any further elucidation or confirmation.

X. No lesser cause is required for the conservation of an object than for its first creation.

Because at the present time we are thinking it does not at all follow that we must continue to think. Our concept of thought does not contain nor involve necessary existence. For I can clearly conceive of thought although I suppose that it does not exist. (This every one knows from his own experience so far as he is a thinking being). But since the nature of any cause ought to contain and involve in itself the perfection of its effect (per Ax. 8), it follows that there is something in ourselves or without us (as yet we do not know which) whose nature involves necessary existence. And this something is the primary cause of our thought both of its beginning and its continuance. For, although our thought began to exist, its nature and essence does not imply a necessary existence any more than it did before it began to be. It is therefore preserved in its existence by the same force that determined that it should exist. What we here affirm of thought is true also for every thing whose essence does not involve a necessary existence.

XI. Nothing exists of which we may not ask, what is the cause (or reason) of its existence. (Vid. Ax. I. of Descartes).

If anything positive exists we cannot say that it exists without a cause (per Ax. 7). Therefore we must assign some positive cause for its existence. This may be external, i. e., some cause outside of the object itself, or internal, i. e., something comprehended in the nature and definition of the object.

God’s existence is known merely from the consideration of his nature.

It is equivalent to saying that a thing is true to say that it is contained in its nature or in its concept. (per Def. 9). The concept of God includes necessary existence. Therefore it is true to say that he has a necessary existence in Himself, or that He exists.—Q. E. D.

Many important truths follow from this proposition. Indeed upon this truth alone, namely, that existence belongs to the nature of God, or that the concept of God involves a necessary existence as that of a triangle that the sum of its angles is equal to two right angles, or again that His existence and His essence are eternal truth, depends almost all our knowledge of God’s attributes by which we are led to a love of God (or to the highest blessedness). Therefore it is extremely desirable that the human race should sometime consider this. I confess that there are certain prejudices which make this truth hard to see. But if any one with earnest purpose, impelled by the love of truth and its utility, wishes to examine into this, we recommend that he consider what is given in Meditation V. and in the end of his “Response to the Sec. Obj.,” and also, what we have said of Eternity in Ch. I. Pt. II. of our Appendix. He would then understand very clearly, nor could he doubt that we do have an idea of God which is indeed the foundation of human blessedness; he would see clearly that the idea of God differs greatly from that of other objects; He would see, when he understands the essence and existence of God, that he differs toto genere from all other things. But there is no need to detain the reader longer.

The existence of God may be demonstrated a posteriori from this, viz., that we possess this idea of such a Being.

The objective reality of anything requires a cause apart from our ideas, in which cause this reality is not only objective, but in which it is contained either formally, or eminently (per. Ax. 8). We have the idea of God, and the objective reality of this idea as it is not in our minds either formally or eminently cannot be anywhere but in God himself. Therefore this idea of God as we have it requires God for its cause and He, therefore, exists.

There are certain ones who say they have no idea of God, although, as they affirm, they love and worship him. And although you place before their eyes the definition and attributes of God, you have accomplished nothing. No more, by Hercules, than if you should attempt to teach a man blind from birth the different colors as we see them. Indeed, we ought to give their words very little attention unless we wish to consider them as a new species of animal half way between man and the lower beasts. In what way do I attempt to set forth the idea of anything except by giving a definition and explaining its attributes? Indeed when we are discussing the idea of God, it is not so much that men deny the words as that they are unable to form some image corresponding to these words.

Then it should be noted that Descartes when he cites Axiom 4 to show that the objective reality of the idea of God is not in us either formally or eminently, supposes that every one knows that he is not infinite substance nor perfect in knowledge or power, etc. This he was justified in doing, for whoever thinks at all knows that there are many things he does not understand clearly and distinctly, and that he is even in doubt in regard to much that he sees.

Finally it should be noted that there are not many gods, as clearly follows from Axiom 8, but only one as we have shown in Proposition II. of this part and in Pt. II., Chapter II., of our Appendix.

The existence of God is demonstrated in the fact that we, having the idea of existence, also exist.

To prove this proposition Descartes laid down two axioms, viz., (1) “Whatever is able to do that which is more difficult is able to do that which is less so. (2) It is greater to create or (per Ax. 10) conserve substance than attributes or properties of substance.” What he meant by these I do not know. For these terms are not used absolutely but only in respect to a definite cause.[1]

[1] If you wish an example, consider the spider which easily spins its web, but for man this would be almost impossible. On the other hand, men easily do many things which perhaps are impossible for angels.

So one and the same thing at the same time, in respect to different causes may be easy or difficult. If you call that difficult which requires more exertion, and that easy which requires less in the same case, as for example, the force which sustains fifty pounds could sustain twenty-five with double ease, clearly the axiom would not be true; neither does it demonstrate what he intended it should. For when he said, “If I have the power of preserving myself I have the power also of giving to myself all the works of perfection which I lack” (for that would require only as great power) I would concede that this energy used for self-preservation might be able to do many other things far more easily if I did not need it for conserving myself. But so long as I use this energy for self-preservation, I deny that it is possible to use it for accomplishing other things, though they be never so easy. This is clearly evident from our example. Nor does he take away the difficulty by saying, that as I am a thinking being I shall know this necessarily, for I employ all my strength in preserving myself which is the reason I do not give myself the attributes of perfection which I lack. For (although we are not now discussing this, but only how the necessity of this proposition follows from this axiom) if I know this I would be greater and perhaps would require, for preserving myself in such perfection, greater power than I now possess. And then I do not know that it is any greater task to create (or to conserve) substance than attributes, i. e., to speak clearly and more philosophically, I do not know but that substance requires all the virtue and essence by which it conserves itself, to conserve its attributes. But we leave this and will inquire, as this worthy author intended we should, into what is meant by the terms “easy” and “difficult.” I do not think that by any means I could persuade myself that he understood by the term “difficult” that which is impossible (and so could not be conceived as existing), and by the term “easy” that which implies no contradiction (and so is easily conceivable). Although in the Third Meditation and in the observation he seems to mean that when he says: “Nor ought I to think that those things which are wanting in my nature are more difficult to acquire than the powers which I now possess. For, on the contrary, it is manifestly far more difficult for me as a being or a thinking substance to arise from nothing than, etc.” For this is not in keeping with the words of the author nor consonant with his ability. And, indeed, though for the present I shall overlook it, between the possible and the impossible, or between that which is conceivable and inconceivable there is no relation, just as there is none between something and nothing. Power does not quadrate better with that which is impossible than creation and generation with non-being; such terms are not capable of relationship. Beside this it should be remembered that I can compare and understand only those things of which I have a clear and a distinct concept. I cannot conclude, therefore, that one who is able to do impossible things is able also to do that which is possible. I ask what conclusion is this? If any one can square a circle he can also make a circle whose radii are not equal; or if one can endue nothing with the qualities of matter he can also produce something from nothing. As I have said there is no analogy, or relation, or means of comparison between such terms. Any one who reflects upon this even a little can clearly see that this is true. Therefore I believe that something else was meant by the ingenious Descartes. Considering the second axiom given above, Descartes seemed to mean by the terms greater and more difficult that which was more perfect, and by the opposite terms that which was less perfect. This also certainly seems obscure. It is the same difficulty found above, and I deny here, as there, that he who has power to do the greater thing has power also at the same time to do the lesser. According to the above proposition this must be granted. Then when he says “it is greater to create or conserve substance than its attributes,” we cannot understand by attributes that which is formally contained in substance and only distinguished from it by reason. For then it would be the same thing to create substance as to create attributes. For the same reason we cannot think that he meant the properties of substance. This follows necessarily from its essence and definition, much less can we understand by this, however, as he seemed to wish, the properties and attributes of some other substance; as for example, if I say that because I have the power of conserving myself, a thinking, finite substance, so I have the power of giving to myself all the perfection of infinite substance which differs by its whole essence from me. For the power or essence by which I conserve my being differs toto genere from the power and essence by which absolute or infinite substance conserves itself. The power and properties of infinite substance are not differentiated per se but only by reason;[2] so (while I may concede that I conserve myself), if I wish to think that I have the power to give to myself all the perfection of infinite substance I suppose nothing else than that I have the power to annihilate my being and to create infinite substance anew. Which clearly presupposes more than that I am able to conserve the finite substance of my being. If then none of these interpretations can be given to the terms attributes or properties, nothing remains but the qualities which are eminently contained in the substance (as this or that thought which I clearly see are wanting in me). Not, however, what some other substance eminently contains; for these attributes even though wanting in me are not imperfections so far as I am considered to be a thinking being. This, then, which Descartes wished to infer from his axioms does not logically follow; namely, that if I have the power to conserve myself, I have the power also of giving to myself all the marks of perfection of the Absolute Being. This is evident from what has been said. But to avoid confusion, and to make the matter more certain, it seems best to demonstrate the following Lemmata first and give the demonstration of the seventh proposition afterward.

[2] It may be noted here that the power by which substance conserves itself is nothing but its essence and only differs from that in name. Which we will clearly show when in the Appendix we discuss the nature of God.

An object of a higher degree of perfection, by virtue of this fact involves a fuller existence and a greater necessity of existence. Conversely, that which by nature involves a greater necessity of existence, is more perfect.

Existence is contained in the idea or concept of every object (per Ax. 6). Let us suppose A to be an object with ten degrees of perfection. I say that this object involves more existence than if it is supposed to contain but five. For, as we cannot affirm existence of nothing (Vid. Schol. Prop. 4), as we detract from the perfection of a concept and conceive its content to approach zero as its limit, so much do we detract from its possible existence. If we conceive this degree of perfection to be infinitely diminished, even to zero, it will contain no existence, or but an absolutely impossible one. On the other hand if we increase this degree of perfection to infinity we conceive that it has the highest possible existence and so to be absolutely necessary. This was the first point to be proven. Then, as I am by no means able to separate these two (as appears from Ax. 6 and the whole of Pt. I.) it clearly follows that the other is likewise true.

Note I. Although many things are said to exist necessarily simply because the cause producing them is given we do not now speak of such objects; but only of that necessity and possibility which follows from the mere consideration of the nature and essence of the thing itself, no reason being held as to its cause.

Note II. We do not here speak of beauty and other marks of perfection which men from ignorance and tradition are accustomed to esteem as such. But by perfection I understand only so much reality or being. As for example, I perceive that there is more reality in substance than in modes or qualities. And so far, I know clearly that there is necessity, and a more perfect existence in the first than in the latter two, as is evident from Axioms 4 and 6.

Hence it follows that that which absolutely involves a necessary existence is perfect Being, or God.

He who has the power of conserving himself, involves, by his nature, a necessary existence.

Whoever has the power of conserving himself, has also the power of self-creation (per Ax. 10), that is, (as all will readily concede), he needs no external cause of his existence, but his own nature is sufficient cause that he should exist, either problematically or necessarily. But not problematically; for (according to what I have shown in Ax. 10) from the mere fact of existence it does not follow that an object will continue to exist; this being contrary to the hypothesis. Therefore necessarily: that is, his nature involves existence. Q. E. D.

If I had in myself the power of self-conservation I would by nature have a necessary existence (per Lemma II.), and (per Coroll. Lemm. I.); my nature would contain all the attributes of perfection. But as a thinking being I am certain that there are many imperfections in me (per Schol. Prop. 4) as that I doubt, I desire, etc. Therefore I do not have the power of self-conservation, nor can I say that I choose thus to limit my being for this is clearly opposed to Lemma I. and to what I actually experience in myself. (Per Ax. 5).

Since then it is impossible for me to exist except as I am conserved, as long as I exist, I must exist either by my own power (provided I possess such power), or by the power of another. But I exist (per Schol. Prop. 4) and yet have not the power of self-conservation as is now positively proven. Therefore I am conserved by another; but not by a being who does not possess the power of self-conservation (for the same reason that I myself do not possess this power); therefore by some being who has this power, i. e. (per Coroll. Lemm. I.) by one whose nature involves a necessary existence, and contains all perfection which I recognize as belonging to an absolutely perfect being. Therefore this perfect being, i. e., God, exists. Q. E. D.

God is able to do all that we clearly understand, just as we so understand it.

This all follows from the preceding Proposition. For it was proven that God does exist from this, viz., that it is necessary for some being to exist in whom is found all the perfection we can clearly conceive. Moreover, there is in us the idea of some power so great that by it alone all things exist which are understood by me as possible, the heavens, the earth and all other things. Therefore with God’s existence all of these statements are likewise proven.

Mind and body are essentially different.

Whatever we clearly conceive, can be realized by God just as we so conceive it (per Coroll, of the preceding). But we clearly conceive of mind, a thinking substance (per Def. 6) apart from body, i. e. (per Def. 7), apart from extended substance (per Props. 3 and 4); and vice versa body apart from mind (as all will concede). Therefore, through divine power mind can exist apart from body and body apart from mind.

Substances which can exist the one apart from the other are essentially different (per Def. 10); body and mind are substances (per Defs. 5, 6, 7) which can so exist; therefore they are essentially different.

See Prop. 4 of Descartes in the end of his Response to the Second Objection; and also what is found in Pt. I. of the Principles, Arts. 22-29. For I consider that these things here do not give the value of the work.

God is omniscient.

If you deny it, then God either knows nothing or only a certain limited amount. But to understand some things and be ignorant of others implies a limitation to God’s perfection, which is absurd (per Def. 8). If God understands nothing, it either indicates in God a want of intelligence and involves in Him, as in men who understand nothing, an imperfection, which is impossible with God (Ibid.), or it indicates, which is also repugnant to the idea of His being, that He understands only some things. But if intelligence is so denied to Him it is impossible for Him to create intellect (per Ax. 8). Since intellect is clearly and distinctly conceived by us, God is able to be its cause (per Coroll. Prop. 7). Therefore, it is far from being the case that he understands only some things, this being opposed to God’s perfection. Therefore, God is omniscient. Q. E. D.

Although it must be conceded that God is incorporeal, as will be proven in Proposition 16, this does not mean only that all perfection of extension is wanting in Him, but only that the imperfections of extension must not be attributed to Him. The same is true of God’s intelligence, as all, who wish to be above the rank and file of philosophers, will readily admit. This will be further explained in our Appendix, Pt. II., ch. 7.

Whatever perfection is found in God arises from His own being.

If you deny it, let it be supposed that there is some perfection in God which does not have its source in Himself. Either it would be in God by virtue of itself or by virtue of something apart from God. But if its cause was in itself it would have a necessary, or at least a problematical existence (per Lemma II. Prop. 7), and so far (per Coroll. Lemma I. Ibid.), have some absolute perfection and (per Def. 8) thus be God. If, therefore, we say that there is some perfection in God whose cause is itself, we affirm that it arises from God Q. E. D. But if it has arisen from some other source than God, then He is not an absolutely perfect being, which is contrary to Def. 8. Therefore whatever perfection is found in God arises from His own being. Q. E. D.

There are not many gods.

If you deny this, conceive, if possible, that there are many gods, for example A and B. Then necessarily (per Prop. 9) A as well as B will be omniscient; that is, A will understand all things himself and B, and likewise B will understand himself and A. But since A and B exist necessarily (per Prop. 5), the cause of the truth and the necessity of the idea of B which A has is B himself; and likewise the cause of the truth and the necessity of the idea of A in B is in A himself. Therefore there will be some perfection in A that is not self-caused, and likewise with B. And so far A and B would not be gods. Therefore there is only one God. Q. E. D.

It should be noted here that because there is something which in itself involves a necessary existence as does God’s being, He is the only being of whom this is true, as any one who reflects carefully will clearly see. I might also demonstrate this, but it is evident in all that I have shown in this Proposition.

All existing things are conserved by God’s power alone.

If you deny this, let it be supposed that something conserves itself. Then (per Lemma II. Prop. 7) its nature involves necessary existence. And so (per Coroll. Lemma I. Prop. 7), it would be God, and there would be more than one God, which is absurd (per Prop. supra). Therefore, nothing exists which is not conserved by God’s power alone. Q. E. D.

God is the Creator of all things.

God (per the preceding) conserves all things, i. e. (per Axiom 10), he has created all things and is continually creating them.

Objects have in themselves no essence which is the cause of God’s knowledge of them.

Since God’s perfection is self-derived (per Prop. 10), objects can have no self-caused essence which could be the cause of God’s knowledge of them. On the other hand, since God has created all things, not from other objects, but by the mere fiat of His will, (per Prop. 12 with Coroll.), and since He knows no other power beside His own (for so I define creation), it follows that before creation nothing existed, and that God is the cause of the essence of all things. Q. E. D.

It may be noted, also, that this corollary is evident from the fact that God is the cause or creator of all things (per Coroll. I.), and that the cause must contain in itself all the perfection of the effect (per Axiom 8).

It clearly follows, therefore, that God does not, properly speaking, perceive or form precepts, for His understanding is not determined by any external object, but all things arise from Himself.

God’s causality is prior to the essence and existence of things. This clearly follows from Corollaries I. and II. above.

God is never a deceiver, but in all things is perfectly true.

We can attribute nothing to God in which we find any imperfection (per Ax. 8).[3] All deception (as is evident) or desire of deceiving, arises either from malice or fear. Fear, moreover, presupposes a limited power; malice a privation of some good. No deception, therefore, can be ascribed to God, a being omnipotent and of perfect goodness, but on the contrary, it must be agreed that He is in no way a deceiver. Q. E. D. See “Response to Second Objection,” number 4.

[3] I have not put this down as an Axiom with the others, as I could not see the need of so doing. I do not use it except in demonstrating this proposition, and also, while we have not yet proved God’s existence I did not wish to assume anything as true which I could not deduce from the primary truth Ego sum as I said in Scholium Prop. 4. Further, I have not given among the others the definitions of malice and fear for no one is ignorant of them, and I do not use them except in this place.

Whatever we clearly and distinctly conceive, is true.

The faculty we possess of discerning the true from the false (as every one finds in himself and is evident from all that has been said) has been created by God and is continually conserved by Him (per Prop. 12 and Coroll.), that is (per the above), by a Being of absolute truth and not a deceiver. Neither has He given to us (as every one knows) any power of withholding assent to what we thus clearly conceive. Wherefore if we are deceived in this, we are deceived in everything by God, and He is a deceiver, which, by the above, is absurd. Therefore, whatever we clearly and distinctly conceive is true. Q. E. D.

Since those things to which we are constrained to assent when we clearly and distinctly conceive them are necessarily true; and since we have the power of withholding assent from those things which are obscure and doubtful, and not derived from certain premises (as every one understands from his own experience), it clearly follows that we are able to be on our guard lest we fall into error and are deceived (which will be made clearer as we proceed). We may, in this manner, determine in ourselves to affirm nothing which we do not clearly and distinctly conceive, or which is not deduced from certain premises.

Error is nothing positive.

If error were something positive, God would be its cause, and by Him it would continually be procreated (per Prop. 12). But this is absurd (per Prop. 13). Therefore error is nothing positive. Q. E. D.

Since error is nothing positive in man, its cause will be merely the lack of a correct use of our freedom (per Schol. Prop. 14). We cannot say, therefore, that God is the cause of error in any sense, except as we say the absence of the sun is the cause of darkness, or as we say that God is the cause of blindness in a child having all his faculties except sight. For He has given to us understanding for a few things only. In order that it may be clearly understood how error depends entirely upon the misuse of the will, and how we may be able to avoid all error, we will call to mind the different modes of thought which we have, viz.: All modes of conception (as sensation, imagination, and pure cognition) and of volition (as desire, aversion, affirming, denying, and doubt); for all forms of thought may be referred to these two classes.

Concerning these things it may be noted: 1. That so far as mind knows objects clearly and distinctly and assents to them, it cannot be deceived (per Prop. 14); and also so far as it knows things and does not assent to them. For, although I can conceive of a winged horse, it is certain that I do not fall into error as long as I do not assent to the proposition that such a creature exists, or even while I am in doubt about it. And since to assent is nothing else than to determine the will, it follows that error depends entirely upon the use of the will.

As now more evidently appears, it should be noted: 2. That we not only have the power of assenting to those things which we clearly and distinctly conceive, but also of assenting to things conceived in some other way. For our will is determined by no limits. If one but consider for a moment it will be evident that if God should choose to give us infinite knowledge there would be no necessity for bestowing upon us a more ample power of volition in order that we might approve all that would be known under such conditions. But the power we now possess would be sufficient for assenting to infinite things. From this we learn that we give our assent to many things not deduced from certain principles. And further, it is evident that if knowledge extended as far as the power of volition, or if we could not exercise our power of volition beyond the limits of understanding, or finally, if we could but keep volition within the bounds of knowledge, we should never fall into error (per Prop. 14).

We do not have the power of attaining the first two conditions, however, for that would imply that the will was not created infinite in its nature and the understanding finite. The third condition alone remains, viz., whether we have the power of limiting the action of volition to the limits of the understanding. Since the will is free to determine itself, it follows that we have the power of restricting this faculty of assent within the limits of understanding. So, also, we can prevent ourselves from falling into error. Whence it is perfectly evident that whether or not we fall into error depends entirely upon the use we make of our free will. That our will is free is demonstrated in Art. 39, Pt. I. of the Principles, and in Meditation 4, and in our Appendix, the last chapter, it is also clearly shown. Although it is true that when we clearly and distinctly conceive something we cannot withhold assent, this necessity of assent does not depend upon some defect in the will, but upon its freedom and perfection. For to assent to the truth is a mark of perfection in us, as is sufficiently evident in itself; neither is the will ever more perfect or more free than when it directly determines itself. If it were possible for the mind so to do, it would give to itself this same perfection, viz., to assent necessarily to what is clearly and distinctly conceived. Wherefore it is far from being the case, that because we are not indifferent in comprehending truth, we know we are less free. On the other hand, we know that the more indifferent we are under such conditions the less freedom we possess.

It only remains to show how, in regard to man, error is privation, and in regard to God mere negation. This we will easily see if we consider first, that seeing many things beside those which we understand clearly, we are more perfect than if we did not perceive them. This is evident because, if it be supposed that we are able to conceive nothing clearly and distinctly, but only confusedly, we would have nothing more perfect than these confused concepts, neither would anything further be desired. Under such conditions, to assent to what we perceive only in a confused way, so far as the act is concerned, would be the perfect thing to do. This will be evident to any one, if, as above, he supposes that it is repugnant to human nature to know anything clearly or distinctly. For though he does not attain to clearness in his knowledge, it is far better to assent to what is perceived only confusedly, and thus to exercise his freedom, than to remain indifferent, that is, as will be shown, to remain in a lower degree of freedom. If we wish to appeal to experience and utility, we will find that daily experience teaches this same truth.

Since, therefore, all our modes of thought, considered in themselves, are perfect, the source of error is not in the understanding. But if we consider the different forms of volition as they differ from one another, some are found to be more perfect than others, for there are some that show less indifference of will, that is, are more free. We know, also, that as long as we give our assent to what is not clearly and distinctly known, we are rendering ourselves the more unfit to discern the true from the false. And thus we do not possess the highest liberty. Therefore, to assent to what is only obscurely perceived, so far as it is anything positive, is not in itself an imperfection or error. But it deprives us of the highest freedom for which we are fitted. All imperfection of error, therefore, consists in the privation of the highest form of liberty and is called error. It is called privation because it deprives us of some perfection which is consonant with our nature. It is called error because, from our own fault, we are without that perfection which we might possess, did we but keep, as far as possible, volition within the bounds of knowledge. Since error in men, therefore, is nothing else than a privation of the perfect use of freedom, it follows that this freedom is not connected with any faculty which man has obtained from God, nor even in the operation of a faculty so far as it depends upon God. Nor can we say that He has deprived us of a more perfect knowledge with which, in order that we should not fall into error, He might have endowed us. For no one has a right to demand anything of God, nor has an object any properties except those which God of His own free will has given it. Nothing existed before the will of God, nor, as we will clearly show in chaps. 7 and 8 of our Appendix, can anything be conceived to have existed. God, therefore, has no more deprived us of a fuller understanding, or of the faculty of a more perfect knowledge, than He has deprived the circle of the properties of the globe or its periphery of the properties of the sphere.

Since, then, nothing in our powers, however considered, reveals any imperfection in God, it clearly follows that error in man is nothing but privation; but relative to God as its cause, it is not privation, but negation.

God is incorporeal.

Matter is the immediate subject of motion (per Def. 7); therefore, if God is corporeal, He may be divided into parts. This, however, since it involves an imperfection, it is absurd to affirm.

If God were corporeal, He might be divided into parts (per Def. 7). Now, either each part would be able to subsist per se or it would not. If the former, each part would be similar to other things erected by God and constantly conserved by His power (per Prop. 10 and Ax. 11). These parts would then pertain no more to the nature of God than do other created objects, as is evident from Prop. 5. But if each part exists by its own power, they would each involve a necessary existence (per Lemma II. Prop. 7), and consequently would be a perfect being (per Coroll. Lemma II. Prop. 7). But this also is absurd (per Prop. II.). Therefore God is incorporeal. Q. E. D.

God is simple being (ens simplissimum).

If God were composite in His nature, these parts, as all will readily concede, should be prior, even down to the most insignificant one, to the nature of God, which is absurd (per Coroll. 4, Prop. 12). Therefore God is simple being. Q. E. D.

Hence it follows that God’s understanding, His volition, His decrees and His power are only distinctions of reason.

God is unchangeable.

If God were changeable, He would not change in part, but His whole essence would change (per Prop. 7). But God’s essence is necessarily what it is (per Props. 5, 6 and 7); therefore God is unchangeable. Q. E. D.

God is eternal.

God is a perfect being (Def. 8), and therefore necessarily exists. If we attribute only a limited existence to Him these limits must be known, if not by us, by God Himself (per Prop. 9), who is omniscient. But then God who is omniscient (per Def. 8), would know no existence beyond these limits, which is absurd (per Prop. 5). Therefore God does not have a limited but an infinite existence, which we call eternity. (Vid. Chap. I., Part II., of our Appendix.) God, therefore, is eternal. Q. E. D.

God has preordained everything from eternity.

Since God is eternal (per Prop. 19), His understanding is eternal because it pertains to His eternal essence (per Coroll. Prop. 7). Hence His understanding, and His will, and His decrees are one (Ibid.) Therefore when we say God knows all things from eternity, we say also that He has willed, and decreed them from eternity. Q. E. D.

From this proposition it follows that God is unchangeable in all His works.

Extended substance has three dimensions, length, breadth and depth. We are united with each of these three.

Extended substance, so far as we clearly understand it, does not pertain to the nature of God (per Prop. 10). It can, however, be created by God (per Coroll. Prop. 7, and per Prop. 8). Then we clearly and distinctly perceive (as every thinking person knows) that extended substance produces in us titulations or pain and other similar sensations or ideas, at times even contrary to our desires. If we attempt to find some other cause of our sensations, as for example God or an angel, we immediately destroy the clear concept which we had before. Therefore (Vid. demonstration Prop. 14 and Schol. Prop. 15), as long as we properly attend to our perceptions and do not admit what is not clearly and distinctly known, we lose our indifference and are led to admit that extended substance alone is the cause of our sensations. So, also, we will see and admit that extended things were created and exist by God’s power. In this we are clearly not deceived (per Prop. 14 with Schol.). Therefore it is truly affirmed that extended substance has length, and breadth, and depth, which was the first point.

And further, as I have already proven, we observe great differences between our various sensations, as for example, when I say I perceive or see a tree; or when I say I am thirsty, or suffer, etc. It is evident that I cannot see or understand the reasons for these differences, unless I know that as a being, I am united to certain portions of matter and not to others. When I understand this clearly, and there is no other way to know it, it is evident that I am united to a certain part of matter. This was the second point, and it is now proven. Q. E. D.

Note.—Unless the reader considers himself merely as a thinking being and free from his body, and lays aside as prejudices all the reasons he has heretofore held as proving the existence of the body, he will attempt in vain to understand this demonstration.

A Postulate.

It is only asked here that each one attend as accurately as possible to his concepts in order to be able to distinguish the clear from the obscure.

I. Extension is that which consists of three dimensions. We do not understand by the term the act of extending or anything else distinct from quantity.

II. By substance we understand that which depends only upon the concurrence of God for its existence.

III. An Atom as a part of matter, by nature is indivisible.

IV. That is indefinite, the limits of which, if it has any, cannot be investigated by the human mind.

V. A vacuum is extension without corporeal substance.

VI. Space is distinguished from extension only by the reason; in reality they are one and the same thing. See Art. 10, Pt. II. of the Principles.

VII. That which we understand to be divisible, is divisible, at least potentially.

VIII. Local motion is the transference of a particle of matter or of a body from the vicinity of other contiguous bodies considered as in a state of rest, to the vicinity of others.

This definition Descartes used to explain local motion. In order to understand this rightly it should be noted:

1. That by a particle of matter he understood all that which is transferred at the same time, although it may itself be composed of many parts.

2. That to avoid confusion in this definition he spoke only of that which is always in moving bodies, viz., transference, lest, as has often happened, this be confused with the force or action which transfers them. This force or action, it is generally believed, is required only for motion and not for rest, which belief is plainly wrong. For, as is self-evident, the same force is required to give to a body at rest a certain velocity as is required to bring the same body with that given velocity to rest. This is proved also by experience. Almost the same force is used in starting a ship at rest in quiet water as in suddenly stopping it when in motion. Plainly this force would be the same except that we are assisted in retarding the motion of the ship by the weight and viscosity of the retarding water.

3. That, he says, the transference is made from the vicinity of contiguous bodies to the vicinity of others and not from one place to another. For place (as he himself explained Art. 13, Pt. 2) is not something in the object, but it depends upon our thought, so much so that the same body may be said at the same time to change its place and not to change it; but not at the same time to be transferred from the vicinity of contiguous bodies and not to be transferred. For only one body at the same moment of time can be contiguous to the same moving body.

4. That he did not say absolutely that a transference was made from the vicinity of contiguous bodies, but only so far as they were considered to be at rest. For in order that the body A be transferred from the body B at rest, the same force is required whether in this direction or in that. This is evident from the example of a boat aground or on the sand in shallow water. For in order that the boat may be moved an equal force must be exerted against the boat and against the ground. Therefore the force by which bodies are moved is expended equally on the moving body and on the one at rest. The action and the reaction are equal. If the boat is moved from the sand, the sand is likewise moved from the boat. If, of bodies which are mutually separated, the one to this place, the other to that, we attribute equal motion, then regard one of them as at rest, it is because the same action is in one as in the other. Then also even to bodies which are regarded by all as at rest, e. g., the sand from which the boat is separated, we are compelled to attribute to this a motion equal to the motion of the boat; for, as we have shown, the same action is required in the one part as in the other, and the transposition is reciprocal. But this is too much at variance with the common way of speaking. In truth, although, those bodies from which others are separated are regarded as at rest and are also said to be so, nevertheless we affirm that everything in the moving body on account of which it is said to be moving is also in the body at rest.

5. Finally, from the definition it is evident that every body has for itself its own one proper motion, since only in regard to contiguous quiet bodies is it said to recede. Nevertheless, if a moving body is a part of other bodies having other motions, we clearly see that it is also able to participate in these as well. But because so many things can not easily be understood, nor will all recognize this, it will suffice to consider that alone which is peculiar to each body. See Art. 31, Pt. 2, Principles.

IX. By a circle of moving bodies we understand such an arrangement that when one is impelled by the impulse of another the last immediately touches the first one of the series; although the line described by the motion of these bodies may plainly be contorted.

I. To non-being there are no properties.

II. Whatever can be detracted from an object, without destroying the completeness of that object, does not constitute its essence; that which, when taken away, destroys the object does constitute its essence.

III. As to hardness sense indicates nothing else, nor do we clearly and distinctly know more than that the parts of hard bodies resist the motion of our hands.

IV. Whether two bodies are mutually approaching one another, or whether receding, they occupy the same amount of space.

V. A particle of matter, whether it gives way to or resists another, does not lose its character.

VI. Motion, rest, form, and similar ideas cannot be conceived without the concept of extension.

VII. Beside the sensible qualities of bodies, nothing remains except extension and its affects as given in Part I. of the Principles.

VIII. One space or portion of extension is no greater than another.

IX. All extension can be divided, at least, in thought. Concerning the truth of this axiom no one can doubt who has learned even the elements of mathematics. For the space between a given circle and its tangent can always be divided by an infinite number of greater circles. Which is also true as regards the asymptote of the hyperbole.

X. No ends of extension or space can be conceived except as another space is conceived to immediately follow such limits.

XI. If matter were manifold and one part did not immediately touch the other, each part would necessarily be comprehended under limits beyond which no matter is given.

XII. A very minute body easily recedes before the motion of our hands.

XIII. One space does not penetrate another, nor is the one greater than the other.

XIV. If the tube A is of equal length with C, but C is twice as large as A, then if some liquid flows through A with double the velocity of that which passes through C, in the same time an equal amount will have passed through each. And if, in an equal time, an equal quantity has passed through each, the velocity through A will be double that of C.

XV. Things which agree in a third part agree in the whole. And things which are each double a third part are equal to one another.

XVI. Matter which is moved in different ways has at least as many divided parts as there were degrees of swiftness observed at any given time.

XVII. A straight line is the shortest distance between two points.

XVIII. A body A moving from C toward B, if repelled by a contrary impulse, will move along the same line toward C.

XIX. Bodies having motions in opposite directions, when they come in contact, undergo some change.

XX. Variation in any object proceeds from a stronger force.

XXI. If when body 1 is moved toward body 2 and impels it, and body 8 from this impulse is moved toward 1, bodies 1, 2, 3, etc., cannot be in a straight line. But all of them from 1 to 8 compose a complete circle (Vid. Def. 9).

Where there is extension or space, there from necessity substance also exists.

Extension or space (per Ax. 1) cannot be pure nothing; therefore it is an attribute which must be attributed to something. But not to God (per Prop. 16, Part I.); therefore to some object which needs only God’s concurrence for its existence (per Prop. 12, Part I.), that is (per Def. 2), to substance. Q. E. D.

We clearly and distinctly conceive of rarefaction and condensation. We would not concede, however, that a body occupies more space under rarefaction than under condensation.

We can have a clear and a distinct concept of these because we can conceive that the parts of a body mutually recede, or mutually approach one another. Therefore (per Ax. 4), they will not occupy more or less space. If the parts of some body, for example a sponge, are compressed the bodies between the parts will be occupied. What we can thus clearly perceive occupy less space than before (per Ax. 4). And if, again, the body expands and the pores are filled by some body, there is a rarefaction, but no more space will be occupied. What we can thus clearly perceive by the senses in the case of the sponge, we can conceive by the understanding to be true with all bodies, although the pores of these cannot be perceived by our senses. Therefore, rarefaction and condensation are clearly conceived, etc. Q. E. D.

It seemed best to give this at this place in order to overcome these prejudices concerning space, rarefaction, etc., and in order that the mind may be ready to understand what follows.

Although hardness, weight, and the other sensible qualities of a body be removed, the whole nature of that body will nevertheless remain.

In the hardness of this stone, for example, sense indicates nothing to us, nor can we clearly and distinctly conceive anything, except that its parts resist the movements of our hands (per Ax. 3). Therefore (per Prop. 14, Part I.), hardness is nothing but this. If that body were pulverized into very small particles, these parts would easily give way (per Ax. 12); nevertheless, they do not lose the nature of the body (per Ax. 5). Q. E. D.

In regard to weight and to other sensible qualities, the same demonstration is valid.

The nature of body or matter (corporis sive materiae) consists in extension alone.

The nature of body is not destroyed by the loss of sensible qualities (per Prop. 1 above); therefore these do not constitute its essence (per Ax. 2). Nothing remains except extension and its affects (per Ax. 7). Therefore, if extension is destroyed, nothing will remain which pertains to the nature of body, but it is destroyed; therefore (per Ax. 2), the nature of body consists in extension alone. Q. E. D.

Space and Body are the same.

Body and extension are the same (per the preceding); space and extension are the same thing (per Def. 6); therefore (per Ax. 15), space and body are the same. Q. E. D.

Although we have said [4] that God is omnipresent we do not believe that God is extension, that is, (per the preceding) body. His omnipotence pertains only to his power and his concurrence by which all things are conserved. So far, therefore, His omnipresence refers no more to extension or body than to angels or to the human mind. It should be noted, too, that when we say His power is everywhere we do not exclude His essence: for where His power is, there His essence is also (Coroll. Prop. 17, Part I.). We would exclude corporeality, that is, God is not everywhere in some corporeal power, but in divine essence, which is common in the preservation of extension and in thinking being (Prop. 17, Part I.). These He would not be able to perfectly conserve if His power or essence were corporeal.

[4] Vid. Appendix, Pt. II., Chaps. III. and IX.

It is a contradiction to say that a vacuum exists.

By a vacuum is meant extension without corporeal substance (per Def. 5), that is (per Prop. 2), body without body, which is absurd.

For a fuller explanation of this proposition, and to correct the prejudices men have concerning a vacuum, Articles 17 and 18, Part II. of the “Principles,” should be read. In these it is noted more especially that bodies between which nothing intervenes necessarily touch one another, and, also, that there are no properties to non-being.

One particle of a body occupies at one time no more space than another; conversely, the same space at a given time will contain no more of one body than of another.

Space and body are the same thing (per Coroll. Prop. 2): therefore when we say that one portion of space is no larger than another (per Ax. 13) we affirm that a body cannot occupy more space in one place than in another, which was the first point to be proved.

Further, from the fact that space does not differ from body, it follows, that when we say that a body cannot occupy more space in one place than in another, we likewise affirm that the same space cannot contain more of one body than another. Q. E. D.

Bodies which occupy an equal amount of space, as for example some gold and some brass, have an equal amount of matter or of corporeal substance.

Corporeal substance does not consist in hardness, e. g. of gold, nor in softness, e. g. of brass, nor in any sensible quality (per Prop. 1, above), but in extension alone (per Prop. 2 above). Moreover, since by hypothesis there is an equal amount of space or (per Def. 6) of extension in the one as in the other, there is a like amount of corporeal substance. Q. E. D.

There are no Atoms.

By their nature atoms are indivisible parts of matter (per Def. 3). But since the nature of matter consists in extension (per Prop. 2 above), which by nature is divisible, however small the part (per Ax. 9 and Def. 7), it follows that any part of matter, however small, is divisible. That is, there are no atoms or indivisible parts of matter. Q. E. D.

The question of atoms has always been a great and an intricate one. Some affirm that atoms must exist because one infinity cannot be greater than another and if two bodies, A and one double the size of A, are divisible to infinity by the power of God, who understands their infinite parts in one intuition, they can actually be so divided. Therefore, as it is said, since one infinity is no greater than another, one part of A will be equal to one double its size, which is absurd. Then they ask also, whether a part divided half way to infinity would still be infinite, and whether it would be equal or unequal, and other things of this kind. To this question Descartes replied that we ought not to reject what we properly understand on account of other things which surpass our understanding, and which consequently cannot adequately be conceived. Infinity and its properties are beyond the power of the human intellect, which is by nature finite. It would, then, be improper to reject as false what we clearly and distinctly conceive, or to doubt this because we do not understand the infinite. Hence, Descartes held that those things which have no limits, such as the extension of the world or the divisibility of a part of matter, should be called indefinite. See Art. 26, Part I. Principles.

Matter is indefinitely extended, and is the same throughout the heavens and the earth.

Point I. No limit to extension or matter can be conceived (per Prop. 2 above) except as we conceive of another space, that is (per Def. 6), extension or matter immediately following this (per Ax. 10), and so on indefinitely; which was the first point to be proved.

Point II. The essence of matter consists in extension (per Prop. 2 above), and this is indefinite (per Point I.). That is (per Def. 4), it cannot be conceived to be bounded by any limits; therefore (per Ax. 11), it is not manifold in its nature but everywhere one and the same; which was the second point to be proved.

We have already discussed the nature and essence of matter. In the last Proposition to Part I. we showed that matter created by God’s power exists, as it is conceived by us, and from Proposition 12 of the same part it follows that it is conserved by the same power that created it. Also, in the last Proposition to Part I. we showed that so far as we are thinking beings, we are united to some part of matter. Hence, we are certain that all the changes in matter are real, which we, by the contemplation of matter, perceive as possible. As, for example, that matter is divisible or capable of motion, that there may be a transference of some parts of matter from one place to another, which, indeed, we clearly and distinctly know, provided we understand that other parts take the place of those which are moved. This division and motion is conceived by us in infinite modes, hence an infinite variation of matter is conceived as possible. I say that these things are clearly and distinctly conceived by us (as was clearly explained in Part I. of the Principles) so long as they are regarded as modes of extension, and not as objects apart from extension. And although some philosophers conceive of many forms of motion, we only admit that there is local motion. For it is evident to us, who admit only what is clearly and distinctly perceived, that extension is capable only of local motion, neither can any other form be imagined.

Zeno, indeed, it is said, for various reasons denied that there was motion in space (motum localum), which assertion the cynic Diogenes refuted in a characteristic way, namely, by walking about in the school where Zeno was teaching, and thus disturbing his pupils. When he was asked by a certain listener to stop his walking he began to find fault with him by saying: “Why do you thus dare to refute the teaching of your master?” But lest any one, perchance, deceived by this argument of Zeno, should think that the senses show anything to us, as for example motion, which is at variance with the understanding, so that the mind is deceived about those things, which, by the help of the intellect are perceived clearly and distinctly, I shall give his principal reasons, and show that these are only supported by false prejudices; namely, because he had no true concept of matter.

In the first place it is said that he argued that if there is motion in space, the motion of a body moving in a circle with the greatest possible speed does not differ from a body at rest; and this is absurd, therefore, that also. Consequently, he affirmed this. That body is at rest, all of whose parts constantly remain in the same place; but all the parts of a body moving in a circle with the greatest possible velocity constantly remain in the same place. Therefore, etc., This, it is said, he explained by the example of a wheel, for example A, B, C. Which, if it moves with a certain velocity about its center, the point A will move more rapidly through the points B and C than if it rotated more slowly. Let it be supposed for example, that when moving slowly for an hour, the point A occupies the same point as when it began. If it moves with double this velocity, in a halfhour it will occupy the same point, and if with a velocity four times as great, then in a quarter of an hour. If, now, we conceive that the velocity is increased to infinity, and the time to be diminished even to a moment, then the point A, moving with this infinite velocity every moment, or continually, will be in the place from which it began to move. So far it will always remain in the same place. And this, which is true of the point A, is also true of every other point of the wheel. Therefore, all points of a body moving with the highest velocity remain in the same place.

Indeed, as I would reply, it should be noted that this is more an argument against infinite motion than against motion itself. We shall not, however, inquire whether Zeno argued rightly, but rather would detect those prejudices on which the whole argument rests so far as he thought this to annul the idea of motion. In the first place it is supposed that a body may be conceived to be moving so fast that a greater velocity is impossible. Then, again, it is supposed that time is composed of moments, as some think that quantity is made up of indivisible points. Both suppositions are false. For we are not able to conceive of a motion than which there can be no greater. It is contrary to reason to think there is a motion, however small the line it describes, so rapid that no more rapid one can be given. The same thing holds true in regard to slowness. For it implies that we can conceive of a motion so slow that a slower one can not be given. Concerning time also, which is a measure of motion, we affirm that the same thing is true, and that it is contrary to reason to think of a time so short that no shorter can be. given. All of which, as we will prove, follows from the words of Zeno. Let it be supposed, therefore, as he said, that the wheel A, B, C, rotates with such speed that the point A at all moments is in the point A from which it moves. I say that I can clearly conceive that this swiftness is indefinitely increased and these moments of time to be diminished in inverse ratio. For let it be supposed that while the wheel A, B, C, is rotating, another wheel, D, E, F, (which I suppose half as large as the other), is made to rotate by the chord H. Since the wheel D, E, F is only half the size of A, B, C, it is evident that the first one will rotate twice as fast as the latter. Then if the wheel A, B, C is supposed to have the motion of D, E, F, the movement of D, E, F will be four times the original motion of A, B, C. And if we suppose this motion of D, E, F to be given to A, B, C, the motion of D, E, F will be eight times the motion of our original wheel. And so on to infinity. This is perfectly evident from the very concept of matter. For as we have proven the essence of matter consists in extension or in space always divisible. There is no motion except in space. We showed also that the same part of matter cannot occupy two points of space at the same time. This would be equal to saying that one part of matter is equal to another part twice its size. Therefore, if a particle of matter is moved it is moved through some space. This space, and the time that serves to measure this as well, however small they be conceived to be, will always be divisible. Q. E. D.

We turn now to another argument of the same nature. If a body is moved, does it move in the place in which it is, or in some other? It does not move in the place where it is, for if it is any where it is necessarily at rest. Neither can it move in a place where it is not. Therefore, a body does not move. This argument is plainly similar to the first, for it also supposes that there is a time given than which there can be no smaller. For if you reply that a body does not move in the place it is, but from that place to another, he will ask whether it does not also move through the intervening places. We reply by making a distinction — if through the term was we understand to be at rest then we deny that the body was at any of the places through which it moved: but if by was existence is meant, then we say that it necessarily existed in that point although it was moving. But he would also ask whether it existed any where while it was moving. We reply, if he meant to ask whether the body remained in any one place, that it did not; but if he wished to ask whether it changed its position, we reply that it has, through all the points in the given distance. Then he would inquire whether it could occupy, and move from a point at the same moment of time. To this we reply by making another distinction. If by a moment of time he understands a duration so short that no shorter is conceivable, as was shown above, he asks a question that is not intelligible, and hence unworthy of reply.

But if he take time in the true sense explained above, however small the duration assigned, it will never be so small that a body may not both occupy it and be moving at the same time. This is evident to any one who considers the matter. For it is evident, as we said above, that he supposes a time given than which no smaller is possible. Hence, he proves nothing.

Beside these two there is another argument of Zeno which, together with its refutation, is given in Descartes (Vol. I. “Epis.,” last letter but one).

I wish here to remind the reader that my argument is opposed to the reasonings of Zeno; and that as far as he argued from reason not from sense, he followed the argument of Diogenes. Nor does sense ever give any truth to the inquirer, except the mere phenomena of Nature, whose causes he is impelled to investigate; never does it show anything to be false which the understanding clearly comprehends as true. So we believe, and so far this is our Method:—to demonstrate the things we set forth, by reasons clearly and distinctly perceived by the understanding; holding to these, whatever the senses may give that seems contrary to this; which, as we have said, can only determine the understanding as it inquires about this or that, but cannot prove the falsity of anything which is clearly and distinctly perceived.

No body moves to the place of another, except as that other moves into the position of some other.

If you deny this, let it be supposed, if possible, that a body A take the position of a body B, and is equal to it, and also that B does not recede from its place. Then the space which before only contained B (by hypothesis) will contain A and B, and so twice as much corporeal substance as it before contained. Which (per Prop. 4 of this Part) is absurd. Therefore, no body can take the place of another, etc. Q. E. D.

When some body takes the place of another, at the same moment the place left by the one is occupied by another which is immediately contiguous to it.

If the body B moves toward D, either the bodies A and C mutually approach and touch one another, or they do not. If they mutually approach and are contiguous the question is conceded. If they do not approach one another, but the space left by B lies between them, then (per Coroll. Prop. 2 supra, and Coroll. Prop. 4) some body equal to B lies between. But (per hypothesis) not B: therefore, some other body, which at the same moment takes its place. But since it is at the same moment it is no other than the one immediately contiguous (per Schol., Prop. 6). There it was shown that there can not be motion from one place to another which does not require a duration than which a shorter may always be conceived. Hence, it follows that the space of the body B cannot be occupied at the same moment by a body which must be moved from some other position. Therefore, only the body immediately contiguous to B can occupy this space at the given moment. Q. E. D

Since the parts of matter are really distinct from one another (per Art. 61, Part I. Principles) one part can exist apart from the other (per Coroll. Prop. 7, Part I.) and they are not dependent upon one another. Therefore, all the fancies about sympathy and antipathy should be rejected as false. And further, since the cause of an effect must be positive (per Ax. 8, Part I.) we can not say that the cause of motion is a vacuum, but that it is due to the impulse of some other body.

In all motion the whole circle of bodies is moved.

At the moment when body 1 takes the place of body 2, this one must move into place of body 3, etc. (per Prop. 7). Then at the moment when 1 is occupying the place of 2, the place it formerly held must (per Prop. 8) be filled by some other body, for example by 8, or some other body contiguous to 1. But since this can only come from the impulse of another body (per Schol. sup.) which is here supposed to be 1, the series cannot lie in a straight line (per Ax. 21) but (per Def. 9) describes a complete circle. Q. E. D. Proposition IX.

If a circular canal, A, B, C, is filled with water or with some other fluid, and at A the canal is four times as broad as at B, when the water ( or liquid) at A begins to move toward B, the water at B will move four times as fast as the water at A.

When the water at A moves toward B, the water at C, which is contiguous to A, takes its place (per Prop. 8); then from B an equal quantity must replace that at C. (per eandem). Therefore (per Ax. 14), it will move four times as fast. Q. E. D.

What we have just said concerning circular channels is also true of all unequal spaces through which water is forced. For the proof would be the same in all such cases.

If two semicircles are described from the same center as for example A and B, the distance between them is equal at all points. But if they are described from different centers as are C and D. the distance between them is unequal at all points. This is evident from the definition of the circle. Proposition X.

A fluid body moving through the channel A, B, C, changes its velocity by indefinite degrees.

(Vid. Figure to Prop. IX). The space between A and B is unequal at all points (per Lemm. sup.); therefore (per Prop. 9) the velocity with which a fluid moves through this channel is everywhere unequal. And since we can conceive that the space between A and B to be indefinitely divided (per Prop. 5), the inequalities also will be indefinitely changing and also (per Prop. 9) the motion by indefinite degrees. Q. E. D.

In the matter which passes through the channel A, B, C, there is a division into indefinitely small parts.

(Vid. Fig. Prop. 9). The matter flowing through A, B, C, has a motion changing by indefinite degrees (per Prop. 10); therefore (per Ax. 16) its parts must be indefinitely divided Q. E. D. See also Articles 34 and 35, Part II. “Principles.”

We have already spoken of the nature of motion. It behooves us here to inquire into its cause, which is twofold: the primary or general cause, which is the cause of all the motion in the world, and then more specifically, how does it happen that particular objects which have no motion, acquire it. Regarding the general cause of motion it is clear, since we ought not to admit anything except what is clearly and distinctly perceived (per Prop. 14, Pt. I. and Schol. Prop. 17, Pt. II.), and since we understand no other cause except God (the creator of matter), that no other general cause of motion can be admitted except God. And what we have said of motion is also true of rest.

God is the principal cause of motion.

See the Scholium just given.

God by his power conserves the same quantity of motion and rest which he once gave to matter.

Since God is the cause of motion and of rest (per Prop. 12), he conserves these by the same power by which he also created them (per Ax. 10 Pt. I.), and indeed with the same amount of power (per Coroll. Prop. 20, Pt. I.) Q. E. D.

I. Although, in Theology, it is said that God does many things because He is pleased to do so, and to show His power to man, since these acts are known only through divine revelation, they should not be admitted into the body of philosophical truth where reason is the criterion of truth, lest Theology and Philosophy become confused.

II. Although motion is nothing but a mode of a moving body, nevertheless, it has a certain definite quantity. How this is possible will appear below. See Art. 36, Pt. II. of the “Principles.”

Every object so far as it is simple and individual considered in itself alone, has a certain unchanging quantity.

To many this proposition is, as it were, an axiom; nevertheless we will give a demonstration of its truth.

Since an object can exist in a certain state only by the concurrence of God (per Prop. 12, Pt. I.), and since God is unchanging in all His works (per Coroll. Prop. 20, Pt. I.), if we consider no external causes (i. e., particular ones) but consider the object in itself, it must be admitted that its quantity always remains the same. Q. E. D.

A body when once in motion will continue to move unless hindered by some external forces.

This is evident from the preceding Proposition. Nevertheless for correcting certain prejudices concerning motion read Articles 37 and 38, Part II. “Principles.” Proposition XV.

Every moving body, in itself, tends to move in a straight line not in a curved one. This proposition might be given as an axiom but I will demonstrate it from the preceding ones.

Motion since it has God alone as its cause ( per Prop. 12, Pt. II.) has in itself no power of existence (per Ax. 10, Pt. I.), but is, as it were, procreated every single moment by God (per that which was demonstrated with the axiom just cited). Therefore, as long as we consider the mere nature of motion we cannot attribute to it, as pertaining to its nature, a duration so great that a greater may not be conceived. But if it is said that it pertains to the nature of a moving body to move in a curved line, a longer duration is attributed to the nature of motion than if it is supposed to be the nature of motion to move in a straight line (per Ax. 8). Since (as we have already demonstrated) we cannot assign such a duration to the nature of motion, we cannot suppose that it is in the nature of a moving body to move in a curved line but it must tend to move in a straight line. Q. E. D.

To many, perhaps, this demonstration will not seem to prove that a moving body tends to describe a straight line rather than a curved one, for no straight line can be assigned so small that there may not be a smaller either curved or straight, neither is there any curved line so small that there may not also be another curved one still smaller. Although I have considered these objections I do not consider them to be valid. For we have based our conclusion upon the universal essence of these lines, not upon’ the quantity of each or accidental differences. But that I may not in this demonstration render obscure what is clear, I refer the reader to the definition of motion, which affirms nothing of motion except a transference of one part of matter from one vicinity to the position of others, etc. This is true so far as we conceive of a simple transference, that is, that this is made in a straight line. So far as we go beyond this we assign something to motion which is not in the definition and so far does not pertain to its nature.

From this it follows that every body moving in a curved line is continually deflected from the line which it tends to follow. This is done by some external force (per Prop. 14, Pt. II.).

Every body moving in a circular orbit, as for example a stone in a sling, tends constantly to move off at a tangent.

A body moving in a circumference is continuously restrained by some external force from moving in a straight line (per Coroll., Prop. XV. Pt. II.). When this ceases to act the body at once moves off in a straight line (per Prop. 15). I say also, that a body describing a circle continually tends to move off at a tangent. For, if you deny it let it be supposed that the stone, for example, in the sling at B, does not tend to follow the line BD, but some other line, either within or without the circle, for example, BF. Or the line BG (which I understand intersects the line BH, drawn from the center of the circle at B and makes with it an angle equal to the angle FBH), if it is supposed that the sling is moving from C toward B. But if the stone moving from L to B at B, tends to move in the line BF, then (per Axiom 18) when the sling moves from C toward B, it should tend to move toward K, not toward G, which is contrary to the hypothesis. And since there is no line that can be drawn through the point B except the tangent AD, which keeps the angles DBH, and ABH equal, there is no line except this tangent able to fulfil the hypothesis, when the sling is moving either from L to B or from C to B. Therefore no other line except the tangent can be drawn on which the body tends to move. Q. E. D.

In place of a circle let the hexagon A, B, H, be inscribed in a circle and a body C be at rest on one side AB. Then let the ruler DBE (one end fixed at D and the other end free) be moved about the center D, continually intersecting the line AB. It is evident that if the ruler DBE, while it is thus conceived to move meet some body at C, when the ruler intersects the line AB at a right angle it will tend to move C in the line FBAG toward G, that is, along the line AB produced indefinitely. Indeed, since we can consider the number of sides of the polygon to be increased ad libitum, it can be affirmed of any figure whatever that can be inscribed in a circle that when a body C at rest on one of its sides, is impelled by a ruler fixed at the center, when the angle found by the side of the polygon and the ruler is a right angle, the body will tend to move in the line of that side indefinitely produced.

Let us conceive instead of the hexagon a polygon of an infinite number of sides (that is, the circle according to the definition of Archimedes); then it is clear that wherever the ruler shall come in contact with the body C, it would always meet it at a right angle. Hence it would never come in contact with C, without C at the same time tending to move in the line of that side produced. Any side whatever when produced will lie wholly outside the figure, and this side indefinitely produced is the tangent of one side of the figure of an infinite number of sides, that is, of a circle. Therefore if we think of sling moving in a circle in place of the ruler the stone will constantly tend to move in a tangent to that circle. Q. E. D.

It should be noted that this demonstration can be applied to any curved figure. Proposition XVII.

Every body moving in a circle tends to move from the center of the circle it describes.

As long as a body is moving in a circular path, so long is it held in its course by some external force; this force being removed it at once begins to move off at a tangent (per the above) all of whose points except that which touches the circle fall outside of the circle (per Prop. 16, Lib. 3, Elements), and so are further removed from the center of its path. Therefore, when the stone in the sling EA, is at the point A it will tend to move along a straight line whose points are all further from the center E than those of the circumference LAB. This is to do nothing else than to recede from the circle which it is describing. Q. E. D.

If some body A is moved against another body B at rest, and B acquires no motion from the impact, then A has lost none of its motion but retains all it had before.

If you deny it, let it be supposed that A has lost some of its motion but has not transferred it to another body as, for example, B. If this happens there will be less motion in Nature than before which is absurd (per Prop. 13). The demonstration in respect to the rest in B is the same. Therefore if no motion is transferred B will be in the same state of rest and A will retain the same amount of motion. Q. E. D.

Motion, considered in itself, by its own determination moves in a given direction; nor is there any need for a moment of rest before it can change its direction or be repelled.

Let it be supposed as in the preceding proposition that a body A is moved directly against B and impeded by B but that B is not moved. Therefore (by the above) A will retain all of its motion, nor does it remain at rest even for a moment. Nevertheless when it moves it does not move in the same direction as before, for it is supposed to be impeded by the body B. Therefore its motion remaining entire and its prior determination being lost it will move in some other direction (per what was said in Chap. 2, Diopt.); and so far (per Ax. 2) determination does not pertain to the essence of motion, nor is a moving body when repelled at rest at any time. Q. E. D

Hence it follows that motion is not the opposite of motion. Proposition XX.

If a body A meet a body B and they move on together the gain of motion in B and the loss of motion in A are equal.

If you deny it, let it be supposed that B acquires less motion from A than A loses. Then that quantity of motion must be added or substracted from the total motion in Nature, which is absurd (per Prop. 13 above). Since therefore, B can acquire neither more nor less motion than A loses it must receive just what is lost by A. Q. E. D.

If a body A is twice as great as B and moves with an equal velocity, it will have twice the motion of B or a force for retaining a motion equal to that of B.

Demonstration. Let it be supposed, for example, that in place of A there are two Bs that is (by hypothesis) A is divided into two equal parts. Each one then has the same inertia and (by hypothesis) the force in each is equal. If now these two parts are joined together their velocity remaining the same there will be a body A whose force and quantity will be equal to two Bs or double that of one B. Q. E. D.

This follows also from the definition of motion, for when a larger body is moved there is more matter separated from the surrounding matter. There is, therefore, more of a separation, that is (per Def. 8) more motion. See also Def. VIII.

If a body A is equal to another body B and moves with twice the velocity of B, the force or motion in A is also double that of B. (Vid. Fig. Prop. 20 ).

Let it be supposed that B, when it first received a certain force acquired a velocity of four degrees. If it is not acted upon by some external force it will continue to move with the same velocity (per Prop. 14). Suppose now, that by a new impulse it receives a new force equal to that which it first received. It will thus acquire four other degrees of velocity beside those that it had before, which (per the same Prop.) it will retain. Thus it will be moving with twice its former velocity, or with the velocity of A, and have double its former force or a force equal to that of A. Therefore the motion in A is double that in B. Q. E. D.

It should be noted here, that by force (vis) we understand the quantity of motion. This, in bodies of equal size, will vary according to the velocity, for in a given time the distance by which equal bodies are separated from those tangent to them varies with the velocity. Therefore those moving more swiftly have more motion (per Def. 8). In bodies at rest by the force of resistance we understand the quantity of rest. From which follow: Corollary I.

The more slowly bodies move, the more they partake of rest, for bodies having a greater velocity meeting those which have less force, resist more and are not separated so far from bodies immediately contiguous to them.

Corollary II.

If a body A moves with double the velocity of a body B which is twice as large as A, they contain an equal amount of motion and force.

If B is twice as large as A, but A moves with twice the velocity of B, and C is only half as large as B and moves only half as fast as A (per Prop. 21), B will have twice the force of C and A will have twice the motion of C (per Prop. 22). Therefore (per Ax. 15) B and A have an equal motion, for the motion of each is double that of the third body C. Q. E. D.

Corollary III.

From these corollaries it follows that motion must be distinguished from velocity. For we can conceive of bodies which have an equal velocity, but one of them having more motion than the other (per Prop. 21). And on the other hand, bodies with an unequal velocity may have an equal motion (per Coroll. II. above). This, also, is evident from the definition of motion which is nothing but the transference of one body from the vicinity, etc.

It should be noted that this corollary is not at variance with the first. For velocity is conceived in two ways: Either so far as a body in a given time is further or nearer removed from those bodies immediately contiguous to it, and so far partakes more or less of rest, or so far as in a given time it describes a longer or a shorter line and so far is distinguished from motion.

I might add other propositions to explain more fully Proposition 14, in regard to other points, as we have done in regard to motion. But it is sufficient to read Art. 43, Part II. of the Principles, and to add one Proposition only in order to understand what follows.

When the modes of a body are forced to suffer change, this change is always, under the circumstances, a minimum one.

This proposition follows sufficiently clearly from Prop. 14.

If two bodies, for example A and B (vid. Fig. Prop. 20) are equal, and are moving toward one another with equal velocity, when they meet each will be reflected in an opposite direction without any loss of velocity.

In this hypothesis it is evident that in order that this opposition may be removed, either they must both be reflected in opposite directions or one must take the other on with it. For they are opposed only in regard to their determination, not as to their motion. DEMONSTRATION.

Since A and B are mutually approaching one another they must suffer some change (per Ax. 19 ). But since the motion of the one is not opposed to the motion of the other (per Coroll. Prop. 19) they do not necessarily lose any of their motion (per Ax. 19). Therefore, the change is in their determination alone. But we can not say that the determination of only one, e. g. B, is changed unless A, by which it was changed, is supposed to be stronger (per Ax. 20). This, however, is contrary to the hypothesis. Therefore, since a change in the determination of only one is impossible, it will be in both, A and B being deflected in opposition directions (per what was said in ch. 2, Dioptric.), each retaining its original motion. Q. E. D.

If they are unequal in mass, namely, B being greater than A (vid. Fig. Prop. 20) other things being as before, then A alone will be deflected and each will move with its former velocity.

Since it is supposed that A is less than B, it will have less force than B. And, since in this hypothesis as in the last, the opposition is only in their determination, as we showed above, the variation therefore will be in their determination alone. It will be merely in A and not in B (per Ax. 20). Therefore, A alone will be reflected in an opposite direction by the greater body B, its former velocity being returned. Q. E. D. Proposition XXVI.

If the mass and velocity are unequal, for example B twice as large as A (vid. Fig. Prop. 20), but A moving with double the velocity of B, other things being as before, both will be reflected in opposite directions, and each will retain its former velocity.

Since A and B are moving toward one another, and according to hypothesis the motion in one is equal to that in the other (per Coroll. 2, Prop. 22); therefore, the motion of the one is not opposed to the motion of the other (per Coroll., Prop. 19). This hypothesis, therefore, is so far similar to the hypothesis of Proposition 24, and by the same demonstration A and B will be reflected in opposite directions by retaining each its former motion. Q. E. D.

From the three preceding Propositions it is evident that in order that a body be moved the determination of that body requires a force equal to its motion. Whence, it follows that a body which has lost more than half of its determination and more than half of its motion suffers more change than one which has lost all of its determination.

If A and B are equal in mass, but B moves a little more rapidly than A, not only is A reflected back in an opposite direction, but B gives to it one-half of its excess of motion, and both move in the same direction with an equal velocity.

A (by hypothesis) is opposed to B not only by its determination, but also by its slowness, so far as it partakes of the nature of rest (per Coroll., Prop. 22). Whence, although it is reflected in an opposite direction and its determination is changed, all of its opposition (contrarietas) is not destroyed. Therefore (per Ax. 19), there ought to be a variation both in its determination and in its motion. But since by hypothesis, B moves faster than A, B will have a greater force than A (per Prop. 22). Therefore, the change proceeds from B to A, which will be reflected in an opposite direction, which was the first point to be proved.

Then, so long as A moves more slowly than B, it will be opposed to B (per Coroll. 1, Prop. 22). Consequently there ought to be a variation in A until it moves with a velocity equal to B’s (per Ax. 19). But in this hypothesis it is not impelled by some stronger force so that it should move more rapidly than B. Since it is impelled by B, it can not move slower than that body, nor can it move faster: therefore, it must move with a velocity equal to B’s.

Further, if B gave less than one-half of its excess of velocity to A, A would move slower than B; if more than one-half, then faster, which as we have shown, is absurd. Therefore, the variation will continue until B has given one half of its excess of velocity to A, which B would therefore lose. So, also, they will both move without any opposition with an equal velocity in the same direction. Q. E. D.

It follows, that a body moving with a greater velocity has the more determination, so that it tends the more to move in a straight line; and, on the other hand, a body moving more slowly has less determination.

Lest the reader confuse the force of determination (vim determinationis) with the force of motion (vi motus), it seems best to explain these terms. If now the bodies A and C are equal, and are moving toward one another with an equal velocity, they will be deflected back (per Prop. 24), each retaining its former velocity. If there is a body C at B and it moves obliquely toward A its determination is less than its motion equaling the line B D or the line C A. Therefore, although the motion in the two cases is the same, the force of determination of C moving directly toward A is greater than the force of determination of C moving obliquely from B toward A in the ratio of AB to CA. Since it is here supposed that A and B move with an equal velocity, the time B consumes in moving from B to A or A C, which measures its opposition to A, will be to A’s time as B A to A C. When the body C, moving from B along the line B A, strikes A, it will be deflected along the line of A B,’ B’ being on B C produced so that B’ C is equal to B C. And since the motion of the two bodies is equal the time B consumes in traversing the perpendicular distance A C is greater than the time of A in moving an equal distance or, to such a degree it is opposed to the determination of A, which is the stronger. In order that the determination of C moving from B toward A, may be equal to the determination of C (or from the hypothesis, to A), it is necessary that the motion from B be to the motion from C as B A is to C A. Then when it strikes the body A obliquely, A will be reflected back to A’ and B toward B’, each retaining its former velocity. But if B is as much greater than A as the line B A than C A, then B will repel A toward A’ and give of its motion until the motion of B is to the motion of A as the line B A is to C A, and by losing the motion which it has transferred to A it will proceed in the direction it was first moving. For example, if the line A C is to the line A B as I to 2, and the motion of A to the motion of B as I to 5, then B will give to A one degree of motion, and will repell it in an opposite direction, while B, having lost one-fifth of its motion, will move on in the same direction as before.

If a body A (vid. Fig., Prop. 27), a little larger than B, is at rest, it will not be moved, however great the velocity of B, but B will be deflected at an angle retaining its former motion.

Note.—Of these bodies there is opposition of three kinds: One when the one meets the other and they both move on with an equal velocity; the second when one is reflected in an opposite direction the other remaining at rest; the third when one is deflected from its course and gives some part of its motion to the body at rest. There is no fourth kind as is seen from Prop. 13. Therefore it will be evident (per Prop. 23) that conformable to our hypothesis the least possible change occurs in these bodies.

If B moves A, until both move with the same velocity, it must give from its own motion (per Prop. 20) all that A acquires and consequently would lose more than half of its motion (per Prop. 21) as well as more than half of its determination (per Coroll., Prop. 27). So far (per Coroll., Prop. 26) it undergoes more change than if it lost all of its determination. And if A loses a part of its rest, but not so much that it moves with a velocity equal to B’s, then the opposition of the two bodies is not destroyed. For A, so far as it partakes of rest, will be opposed to the motion of B, and so far B will be deflected from its course and will lose all of its determination and that part of its motion which it has given to A. And this also is a greater change than if it had lost its determination alone. The change, therefore, under our hypothesis, since it is in the determination alone, is the least possible that can come in these bodies, and therefore (per Prop. 23) the only possible one. Q. E. D.

It should be noted that in the demonstration of this Proposition, and it holds true likewise in other places, that we have not cited Prop. 19, where it was demonstrated, that the entire determination of a body may change, the motion remaining the same. Nevertheless this should be remembered in order that the force of this argument may be seen. For in Prop. 23 we do not say that the variation is the least absolutely, but the least possible under the given conditions. That such a change as this is possible, that is, a change in the determination alone, is evident from Props. 18 and 19 with the Corollary.

If there is a body A, less than B, at rest (vid. Fig. Prop. 30.), then, however slowly B moves toward A, B will give such a portion of its motion to A that they will move on together (Read Art. 50, Part II. Principles).

In this rule, as in the preceding, there are three possible cases by which their opposition may be destroyed. We will show that under this hypothesis there is the least possible change in these bodies, and so (per Prop. 23) this is the only variation.

By hypothesis, B transfers to A (per Prop. 21) less than half of its motion, and (per Coroll., Prop. 17) less than half of its determination. But if B did not carry A on with it, but should be reflected in an opposite direction, it would lose all of its determination and there would be a greater variation (per Coroll., Prop. 26). And much greater even, if it should lose all of its determination, and at the same time a part of its motion, as is supposed in the third case. Therefore, the change under our hypothesis is the least possible. Q. E. D.

If a body A, at rest, is exactly equal to B, a body moving toward it, when B strikes A, A will be impelled and B repelled.

Here, as in the preceding, there are three possible cases, and we will show that the resulting change is the least possible.

If the body B should take A with it until both move with an equal velocity, then the motion in each would be the same (per Prop. 22) and (per Coroll., Prop. 27) B would lose half of its determination and half of its motion (per Prop. 20). If B is repelled in an opposite direction it will lose all of its determination but retain all its motion (per Prop. 18). In the latter case the change is equal to that of the former (per Coroll., Prop. 26). But neither of these is possible, for if A remains at rest and still changes the determination of B, it must needs (per Ax. 20) be greater than B, which is contrary to the hypothesis. And if B act on A until both move together with an equal velocity, B is greater than A, which is also contrary to the hypothesis. Since neither of these results is possible, the third case must be the result, namely, that B impels A a little, and is repelled by A. Q. E. D. Read Art. 51, Part II. of the Principles.

If B and A (vid. Fig. above) are moving in the same direction, A a little more slowly than B, which is following, so that it will impinge on A, and if A is greater than B, but the excess in magnitude is not equal to B’s excess of velocity, then B will give a part of its velocity to A, so that they will move on together. But if the excess of motion in B does not equal the excess of magnitude in A, then B will be reflected back, each retaining its former velocity.

See Art. 52, Part of II. of the Principles. Here, as above, there are three possible cases.

Point one: B, which is supposed to be stronger than A, cannot be reflected in an opposite direction (per Props. 21 and 22, and Ax. 20). Therefore, since B is stronger than A it will move A on with it so that the two advance together, for as appears from the previous proposition, there is less change in this way than in any other. Point two: Since here B is not so strong as A it can not move A (per Props. 21 and 22), neither (per Ax. 20) can it give it any of its motion. Therefore (per Coroll., Prop. 14), it will retain its former motion. Not, however, in the same direction, for it is supposed to be impeded by A. Therefore (per what was said in Chap 2, Diopt.), it will be reflected in an opposite direction, each body retaining its former motion (per Prop. 18). Q. E. D.

It should be noted that here, as in the preceding propositions, we have assumed as proven that every body meeting another by which it is absolutely impeded, advances no further in its former direction, but on the contrary, is reflected in an opposite direction. To understand this read Chapter 2 of the Diopts.

For explaining the changes of bodies which are mutually impelled we have so far considered two bodies as though separated from all others, no account being taken of other impinging bodies. For we will consider the state and changes of those in place of the bodies by which they are impinged on all sides.

If a body B is impinged on all sides by moving particles which tend to move it in all directions, as long as there is no other cause it will remain unmoved.

This proposition is self-evident: for if a body is moving from the impulse of corpuscles coming from a certain direction, the corpuscles which move it impel it with greater force than others coming from other parts, and striking it are unable to produce a sensible effect (per Ax. 20). But this is contrary to the hypothesis.

Conditions being as in the last Proposition, a body B can be moved in a certain direction by a force however small.

Since, by hypothesis, B is at rest, and the particles contiguous to it are in motion, these particles (per Prop. 28) in touching B are repelled, each one retaining its former velocity. B, therefore, is being continually left by those particles contiguous to it, and no action is required for separating it from those particles which come in contact with it (according to what was remarked concerning Def. 8). Therefore, however small the external force impinging on B, it is still greater than that required to retain B in its position (for we have shown that there is no force in the bodies immediately tangent to B), and which, added to the impact of the particles moving in the same direction, is not greater than the impact of particles moving in an opposite direction (for we suppose that these particles are acting equally on all sides.) Therefore (per Ax. 20), any external force, however small, will move B in a certain direction. Q. E. D.

A body B, the conditions being as above, can not be moved with a velocity greater than the velocity of the body impelling it, although the particles of that body may be moving much more rapidly.

The corpuscles which, together with the external force, impel B in a certain direction, although they move much more rapidly than the external force is able to move, nevertheless, since (by hypothesis) they have no greater force than those particles which tend to drive B in another direction, all strength of their determination is used in resisting these, and hence (per Prop. 32) they give no acceleration to the body. Therefore, since there are, according to supposition, no other elements or causes except this external force, nor does it receive acceleration from any other bodies except this external force, it can not be moved with a greater velocity than the impulse of this body. Q. E. D.

When a body B is so moved by an external force, it receives the greater part of its motion from those bodies which immediately surround it, and not from an external force.

The body B, however large it is supposed to be, will be moved by the continued impulse of a body however small (per Prop. 33). Let us suppose that B is four times as large as an external body by which it is moved. Since (by the preceding Proposition) both will move with an equal velocity there will be four times as much motion in B as in the body by which it is impelled (per Prop. 21). Therefore (per Ax. 8, Part I.), it does not receive the principal part of its motion from this external force. And since no other factors are present except those bodies by which it is continually being impelled (for B is supposed to be at rest), it receives the greater part of its motion from these bodies continually acting upon it and not from some external force, Q. E. D.

Let it be noted here that we cannot say, as above, that the motion of particles coming from one direction is needed to counterbalance the motion of parts coming from an opposite direction. For bodies moving toward one another (as is here supposed) with an equal motion are opposed to one another in their determination alone and not in their motion [5] (per Coroll. Prop. 19); hence when resisting one another they are opposed only in their determination and not in their motion. Beside B cannot receive from circumjacent bodies any determination and consequently no increase of velocity so far as it is distinguished from motion. It does, however, receive motion; therefore, some adventitious force being present, it must needs be moved by these particles as we have shown in this Proposition, and this is clear also from the method by which we demonstrate Proposition 33.

[5] See Proposition 24 of this book, where it was demonstrated that bodies resisting one another are opposed only in their determination and not in their motion.

If some body, for example my hand, moves with a uniform motion in any direction so that it in no way resists any bodies, nor do any bodies in any way resist it, in the space through which it moves the bodies must necessarily be moving in all directions and with a velocity equal to that of my hand.

A body can move through no space that is not a plenum (per Prop. 3). That is, the space through which my hand is moving is filled with bodies which move under the condition given above. If you deny it let it be supposed that they are at rest or are moving in some other way. If they were at rest they necessarily resist the motion of the hand (per Prop. 14) until its motion is communicated to them so that they move in the same direction and with a velocity equal to that of my hand (per Prop. 20). But it is supposed in the hypothesis that they do not resist. Therefore, they are moving, which is the first point to be proved. These bodies must be moving in all directions. If you deny it let it be supposed that they do not move in one direction, say from A to B. If, therefore, the hand is moving from A to B it will necessarily meet with moving particles (according to what has just been said and according to hypothesis) with a different determination from the determination of the hand. Therefore, they would resist it (per Prop. 14) until they are moved in the same direction as the hand itself (per Prop. 24 and Schol. Prop. 27). And since (by hypothesis) they do not resist the hand, they will be moving in the same direction, which was the second point.

Again, these bodies in whatever direction they move, will all have an equal velocity. For if it be supposed that they do not move with an equal velocity, let it be supposed that those which move from A toward B, do not have as great a velocity as those which move from A toward C. Therefore, if the hand moves from A to B with the same velocity as the bodies moving from A to C (for it is supposed to be able to move with an equal velocity in any direction without resistance), the bodies moving from A toward B will resist the hand until they move with the hand with an equal velocity (per Props. 14 and 31). But this is contrary to hypothesis. Therefore they will move in all directions with an equal velocity, which was the third point.

Finally, if these bodies do not move with an equal velocity to the hand, the hand moves either slower or faster than they. If the former the hand would resist the bodies which are following it in the same direction (per Prop. 31); if these particles move slower than the hand, then they will resist the hand, both cases being contrary to our hypothesis. Therefore, since the hand does not move slower or faster than the particles it must move with a velocity equal to that of the particles amid which it moves. Q. E. D.

If you inquire why with an equal degree of velocity, I would reply that it is not with a velocity absolutely equal. See Schol. Coroll. Prop. 27. If then you would ask whether the hand while, for example, it was moving from A to B would not resist those particles at that time moving from B to A, read Prop. 33, in which you will see that the force of these bodies is equalized by the force of bodies moving from A to B (for by the third part of this demonstration this force is equal to that).

If some body, A for example, can be moved in a certain direction by a force however small, it is necessarily surrounded by bodies which are all moving with equal velocity in all directions.

This body A is surrounded on all sides by bodies (per Prop. 6), which are moving in all directions with an equal velocity. For if they were at rest the body A (as is supposed), could not be moved in a given direction by a force however small, but only by a minimum force which was able to move A together with the bodies which immediately surround it (per Ax. 20). Then if the bodies by which A is surrounded are moved in one direction with a greater force than in another, for example, from B to C than from C to B, since as we have already shown it is surrounded on all sides by moving bodies, it will necessarily be moved along the line from B to C (per Prop. 30). Hence a force however small would not suffice to move A from C to B, but one sufficient to overcome this excess of motion toward B would be required (Ax. 20). Therefore these bodies must be moving in all directions with an equal force.

Since these things are true of fluid bodies, it follows that fluid bodies are those which are divided into minute parts, which are moving in all directions with equal velocity. And, although these particles cannot be seen even by the eye of a lynx, still the truth of what we have thus demonstrated cannot be denied. For, in Propositions 10 and 11 it was shown that the subtlety of nature is so great that it cannot be known (I shall not say by the senses). Beside, as was also shown above, since bodies only by their rest resist other bodies, and as the senses tell us hardness is nothing else than the resistance the parts of a body offer to our hands, we conclude that those bodies are hard whose particles are, with regard to another at rest, and near together.

Read Articles 54, 55, and 56, Part II. Prin.

The universal principles of nature having been presented, we must proceed now to explain those things which follow from them. But since the things which follow from these principles are more than can ever be known, and since we are not determined by them to one thing rather than another, first of all a brief history should be given of the principal phenomena whose causes we are to investigate. This you have in Articles 5 to 15, Part III. of the Principles. And from Articles 20 to 40 the hypothesis is expressed which Descartes deemed best fitted not only to explain the phenomena of the world, but also for investigating their natural causes.

The best way to understand the nature of plants or of men is to see how they arise and develop from their germ cells. Principles perfectly simple and easily known should be found from which we can demonstrate that the heavens, the earth, and all the visible world could have arisen as from cells, although we know full well that they have not actually done so. For in this way we will explain their nature much better than if we only describe them.

I say that we are seeking for principles which are simple and easily known. Unless such are found we do not desire any at all. For we are only adding the first principles to what has been given in order that the nature of these things may more easily be known, and that we may, according to the method of mathematics, advance from what is perfectly clear to that which is more obscure, and from the simple to what is more complex.

Therefore, we said that we are seeking the principles from which we might demonstrate that the heavens, the earth, etc., could have arisen. We are not seeking merely the causes which are sufficient, for explaining the phenomena of the heavens as is now and then done by astronomers. But we are seeking for those principles which will lead us to a knowledge of all those things in the earth (for we believe that all those things which we observe in the earth should be included in the phenomena of nature). In order that these may be found, the following points should be observed in a valid hypothesis:

I. That (considered in itself alone), it implies no contradiction.

II. That it should be as simple as possible.

III. That what follows from it may be easily known.

IV. That all things observed in all nature can be deduced from it.

Finally, we said, that we might assume an hypothesis from which, as from a cause we are able to deduce the phenomena of nature although we know that these phenomena have not thus arisen. In order that this may be understood I will give an example. If a person should find on a paper the curved line which we call a parabola and wished to investigate its nature it would make no difference whether he regarded it as first cut from some cone and then placed upon the paper, or whether he regarded it as generated from the movement of two straight lines, or as derived in some other way. In whatever way he conceives it to have been generated, he wishes to demonstrate all the properties of the parabola from it. Indeed, although he knows that it was made from the cone, he will be free to assign some other cause which seems to him better adapted to explain all its properties. So also in order to explain the forms of nature we may assume any hypothesis at will, provided we deduce from it, through mathematical inference, all the phenomena of nature. And, what is even more worthy of note, we can scarcely assume anything, from which we may not, perhaps with more labor, through the laws of nature given above, deduce the same results. For since in accordance with these laws, matter assumes successively all the forms of which it is capable, if we consider these forms in order we will come finally to the form in which the world exists. We need not fear, therefore, the error of a false hypothesis.

We ask that it be conceded that, in the beginning, all matter of which the visible world is composed was divided by God into particles as nearly as possible equal to one another. These particles, however, from which now the heavens and the stars are composed were not spherical, for a number of spheres joined together do not fill up the space they occupy; but they, small in size, were fashioned in some other way. These particles had in them just as much motion as there is in the world to-day and they were all moving with an equal velocity. Not only did single particles mutually separate from one another move about their own center as if they composed a fluid body, such as we think the heavens to be, but there are many moving together around certain other points, equally remote and like disposed and now the centers of fixed bodies; then also, there were some around other points which equal the number of the planets. And so they compose as many vortices as there are stars in the world. Vid. Fig. Art. 47, Part III. of the Principles.

This hypothesis considered in itself implies no contradiction; for it attributes nothing to matter except divisibility and motion, which we have already demonstrated to really exist in matter. And since we have shown that matter is indefinite, and the earth and the heavens are one and the same, we can suppose without a trace of contradiction, that these modifications are in all matter.

Then this hypothesis is a very simple one, because there is no irregularity or dissimilarity in the particles into which matter was divided at the beginning, nor in their motion. For these reasons this hypothesis is also very easy to understand. This is evident also from the fact that in this hypothesis nothing is assigned to matter except that which is known to any one from the concept of matter alone, namely, divisibility and motion in space.

We shall attempt to show, as far as it may be done and in the following order, that all that we observe in nature can be deduced from this alone. In the first place we will deduce the fluidity of the heavens from this postulate and explain how this is the cause of light. Then we shall proceed to consider the nature of the sun and those things which are observed in the fixed stars. Afterward we shall speak of comets and of the planets and their phenomena.

I. By the Equator (per Eclipticam) we understand that part of a rotating body, which as it turns on its axis, describes the greatest circle.

II. By the Poles we understand those parts of a rotating body most remote from the Equator, or which describe minimum circles.

III. By the Conatus to move (conatum ad motum) we do not understand some form of thought, but only that a part of matter is so placed and impelled to move that, if it is not impeded by some external cause, it will really move somewhere.

IV. By an Angle we understand that part of a body which extends beyond its spherical form.

I. A number of spheres joined together cannot fill the space they occupy.

II. A portion of matter divided into angular parts requires more space, if these parts each move about its own center, than if they are all at rest and the sides of all are immediately and mutually tangent to one another.

III. A smaller part of matter is easier divided by a given force than a larger one.

IV. Parts of matter moving in the same direction which do not in this motion recede from one another, are not really divided.

The parts into which matter was first divided are not spherical but angular.

In the beginning all matter was divided into equal and similar parts (per Postulate). Therefore (per Ax. 1 and Prop. 2, Part II.) they are not spherical but (per Def. 4) thus far angular. Q. E. D.

The force which causes the particles of matter to move about their own centers, also causes the angles of these particles to be worn away.

In the beginning all matter was divided into equal and angular parts, (per Postulate and Prop. 1). If the angles were not worn away when they began to move around their centers, all matter would occupy (per Ax. 2) more space than if they were at rest. But this is absurd (per Prop. IV., Part II). Therefore their angles begin to wear away as soon as they began to move. Q. E. D.