EDITORIAL PREFACE

THE WORKS presented in this section have suffered unwarranted neglect at the hands of some Spinoza scholars. Wolfson, for example, dismisses them by saying that “If these two works are not to be altogether disregarded by the student of the Ethics, they may be considered only as introductory to it.” (1, 1:32) The first he characterizes as a summary of the first two parts of Descartes’ Principles, together with a fragment of the third; the second, as a “summary of certain philosophic views of scholastic origin.” But this is most misleading. Though there is much in both works that Spinoza would not accept, in neither work is Spinoza merely summarizing anyone’s views.

In Descartes’ Principles, as Meyer remarks in his preface, Spinoza frequently offers proofs different from Descartes’—or offers proofs where Descartes had indulged in mere assertion. He makes use of other Cartesian works—notably the Correspondence, the Dioptrique, and the Meditations (including the Replies to objections)—to help interpret Descartes where Descartes is obscure or too brief. And sometimes he criticizes Descartes, though rarely does he do so openly.

The most important acknowledged points of difference are probably those Meyer notes (at Spinoza’s request) at the end of his preface (I/132): Spinoza does not think the will is distinct from the intellect, or endowed with the liberty Descartes ascribes to it; he does not think that the mind is a substance; and he does not think there is anything that surpasses our understanding, provided that we seek the truth in a way different from Descartes’. (See also the interesting Scholium to IP7.)

But there is a good deal of thinly veiled criticism in Spinoza’s exposition of the Principles. In the Introduction, for example, Spinoza comments that Descartes’ reply (or what he takes to be Descartes’ reply) to the charge of reasoning in a circle “will not satisfy some people.” He does not say who will be dissatisfied, or why, but immediately goes on to offer an answer of his own. In so doing, he subtly invites the reader to put his own critical faculties to work. Similar instances occur at IP8D, IP9S, IP15S, IP21Note, IIP2CS, and in the Metaphysical Thoughts at I/239, 255, 274, 276, and 277-281.

And quite apart from exposition, interpretation, and criticism of Descartes, there is much, particularly in the Metaphysical Thoughts, which is simply independent of Descartes. According to Meyer’s Preface (I/131) it was Spinoza’s intention—both in his exposition of the Principles and in the Appendix—to set out Descartes’ opinions, and their demonstrations, as they would be found in his writings or as they ought to be deduced validly from the fundamental principles of Descartes’ philosophy. But it would take a very generous interpretation of this last clause to justify everything that appears here. Spinoza’s discussions of Zeno’s paradoxes (IIP6S), or of truth (I/246-247), or of good and evil (I/247-248), go well beyond Descartes’ sketchy reflections on those topics. The reader who is familiar with Spinoza’s mature philosophy will find many passages which foreshadow the Ethics or the Theological-Political Treatise (e.g., at I/240-243, 250-252, 264-265, 266). And it is all the more interesting to see these anticipations developed as deductions from Cartesian principles.

The frequency with which Spinoza’s own opinions emerge in the Metaphysical Thoughts gave rise, at the turn of the century, to an interesting debate among German scholars about the relation of that work to Descartes’ Principles. Kuno Fischer (I:285) saw the Metaphysical Thoughts as having been written against Descartes, for the purpose of clarifying and emphasizing the disagreements which Meyer had alluded to in his Preface.1 Freudenthal, however, showed (i) that this work was certainly written before the first part of the Principles, and quite probably before any of the Principles (though, of course, it would have been revised somewhat before being printed), and (ii) that it was directed more against the scholastics than against Descartes.

And indeed, if Spinoza did write these works in the order suggested, there would be a certain logic to the presentation. The Metaphysical Thoughts begin with the definition of being and its division into those beings whose essence does, and those whose essence does not, involve existence, the latter being subdivided into substances and modes. There follows a discussion of various putative beings, which do not in fact qualify as beings (time, truth, etc.). The second part deals principally with the nature of the one infinite substance, God, and his relation to the world, but closes with a brief discussion of the nature of finite thinking substances. And the second part of the Principles completes the discussion by describing the most general features of finite corporeal substances. The work would thus be an introduction to modern philosophy, written from a broadly Cartesian point of view, for someone who already had some familiarity with the scholastic philosophy still dominant in Dutch universities.2 It would then be the first part of the Principles, not the “appendix,” which would be the afterthought.

Freudenthal (in Freudenthal 3) provided a great service to students of Spinoza by identifying some of the medieval and late scholastic authors who formed the background for the Metaphysical Thoughts: Aquinas and Maimonides, of course, were part of that background, but of the better-known scholastics Suarez was probably the most important. Also important were two now obscure Dutch writers, Burgersdijk and Heereboord.

Burgersdijk was a professor of philosophy at the University of Leiden from 1620 to 1635, and the author of a number of manuals which had a considerable influence on the teaching of philosophy in Holland. Dibon (277) comments that while, for Burgersdijk, Aristotle remained the master, “true fidelity to the spirit of Aristotle required each philosopher to adapt the traditional philosophy to the requirements of his own reflection, taking account of what preceding thinkers had contributed.” Burgersdijk’s openness to change clearly inspired his pupil Heereboord to be receptive to the new philosophy.

Heereboord was a professor of logic and ethics at Leiden from 1641 until his death in 1661. Like many Dutch philosophers of his time, he hoped to achieve a synthesis of the new philosophy and the old. By comparison with a Descartes or a Spinoza, he appears a reactionary figure. But he was, in fact, one of the reasons why the University of Leiden became known as a center of Dutch Cartesianism (see Bouillier, I, 270-271). Thijssen-Schoute (2, 96-105) reports that he embarrassed Descartes by his excessive praise of him.

Since there is some reason to think that Spinoza may have studied at the University of Leiden after his excommunication (see Revah 1, 32, 36), and since Heereboord may have been one of his teachers, the man and his work deserve to be less obscure. Spinoza frequently seems to have Heereboord in mind when he criticizes unnamed opponents, and Heereboord shares with Aristotle the distinction of being one of only two opponents to be both named and quoted. Nor does Spinoza refer to Heereboord only to disagree with him. Sometimes he adopts from Heereboord doctrines and distinctions which are of great importance in his mature philosophy (at I/240-241, for example). Whether or not Heereboord was formally his teacher, Spinoza learned from him.

So there is far more in these works than mere summary of Cartesian and scholastic doctrine. They are of the greatest importance for the study of Spinoza’s development. But they also hold much that is of interest to the student of Descartes. No less a scholar than Gilson has commended Spinoza as “an incomparable commentator” (Gilson 2, 68ff.) To say that Spinoza is always faithful to Descartes’ thought would be to claim too much, even if we considered only those passages where Spinoza intends merely to expound Descartes’ view.3 But even the errors of a Spinoza are interesting.

Finally, a word about Balling’s Dutch translation of this work. This appeared in 1664, the year after the first Latin edition. It is more than a translation, though something rather less than the second edition Meyer hoped for (cf. his Preface I/131). A number of new passages have been added, which were not consistently taken into account by any of Spinoza’s editors before Gebhardt. There seems to be no reasonable ground for doubting that these additions were made by Spinoza.

“EP” designates the reading of the first edition, “B” a reading taken from Balling’s translation.

PARTS I AND II OF DESCARTES’ PRINCIPLES OF PHILOSOPHY

Demonstrated in the geometric manner,

By Benedictus de Spinoza, of Amsterdam.

To which are added his

Metaphysical Thoughts

In which are briefly explained the more difficult problems which arise both in the general and in the special part of Metaphysics.

Amsterdam,
Johannes Riewerts,
1663

To the Honest Reader Lodewijk Meyer Presents His Greetings

[I/127] Everyone who wishes to be wiser than is common among men agrees that the best and surest Method of seeking and teaching the truth in the Sciences is that of the Mathematicians, who demonstrate their Conclusions from Definitions, Postulates, and Axioms. Indeed, this [10] opinion is rightly held. For since a certain and firm knowledge of anything unknown can only be derived from things known certainly beforehand, these things must be laid down at the start, as a stable foundation on which to build the whole edifice of human knowledge; otherwise it will soon collapse of its own accord, or be destroyed by the slightest blow.

No one who has even the most cursory acquaintance with the noble [15] discipline of Mathematics will be able to doubt that the things which are there called Definitions, Postulates, and Axioms, are of that kind. For Definitions are nothing but the clearest explanations of the words and terms by which the things to be discussed are designated; and Postulates and Axioms, or common Notions of the mind, are Propositions [20] so clear and evident that no one can deny his assent to them, provided only that he has rightly understood the terms themselves.

But in spite of this you will find hardly any sciences, other than Mathematics, treated by this Method. Instead the whole matter is arranged and executed by another, almost totally different, Method, [25] in which Definitions and Divisions are constantly linked with one another, [I/128] and problems and explanations are mixed in here and there. For almost everyone has been convinced, and many who have applied themselves to founding and writing about the sciences still are convinced, that the Mathematical Method is peculiar to the Mathematical disciplines, and does not apply to any of the rest.

[5] The result is that none of the things they produce are demonstrated by conclusive reasonings, but that they try to construct only probable arguments, foisting on the public a huge heap of huge books, in which you will find nothing that is firm and certain. All of their works are full of strife and disagreement, and whatever is corroborated by some [10] slight, insufficient reasoning is soon rebutted by another, and destroyed and torn apart by the same weapons. So the mind, which has longed for an unshakable truth, and thought to find a quiet harbor, where, after a safe and happy journey, it could at last reach the desired haven of knowledge, finds itself tossed about on a violent sea of opinions, surrounded everywhere by storms of dispute, hurled up and [15] dragged down again endlessly by waves of uncertainty, without any hope of ever emerging from them.

Nevertheless, there have been some who have thought differently, and, taking pity on the wretched plight of Philosophy, have departed from the common way of treating the sciences, and entered on an [20] arduous new path, one beset indeed with many difficulties, that they might leave to posterity the other parts of Philosophy, beyond Mathematics, demonstrated by the mathematical Method and with mathematical certainty. Some of these have put into mathematical order and communicated to the world of letters the Philosophy already received and customarily taught in the school; others have done this with a new philosophy, discovered through their own struggle.

And though many undertook that task for a long time without success, [25] at last there appeared that brightest star of our age, René Descartes. By this new Method he first brought out of darkness and into the light, whatever in Mathematics had been inaccessible to the ancients, and whatever could be desired in addition to that by his own Contemporaries. Then he uncovered firm foundations for Philosophy, foundations on which a great many truths can be built, with Mathematical [30] order and certainty, as he himself really demonstrated, and as manifests itself more clearly than the Noon light to anyone who diligently studies those writings of his, which can never sufficiently be praised.

Although the Philosophical writings of this most Noble and Incomparable Man contain a Mathematical manner and order of demonstration, [I/129] nevertheless they are not written in the style commonly used in Euclid’s Elements and in the works of other Geometricians, the style in which the Definitions, Postulates, and Axioms are set out first, followed by the Propositions and their Demonstrations. Instead they are written in a very different manner, which he calls the true and [5] best way of teaching, the Analytic. For at the end of his Reply to the Second Objections, he recognizes two ways of demonstrating things conclusively, by Analysis, “which shows the true way by which the thing was discovered, methodically, and as it were a priori,”1 and by Synthesis, “which uses a long series of definitions, postulates, axioms, [10] theorems, and problems, so that if a reader denies one of the consequences, the presentation shows him that it is contained immediately in the antecedents, and so forces his assent from him, no matter how stubborn and contrary he may be.”

But though a certainty which is placed beyond any risk of doubt is found in each way of demonstrating, they are not equally useful and [15] convenient for everyone. For since most men are completely unskilled in the Mathematical sciences, and quite ignorant, both of the Synthetic Method, in which they have been written, and of the Analytic, by which they have been discovered, they can neither follow for themselves, nor present to others, the things which are treated, and demonstrated conclusively, in these books. That is why many who have been led, either by a blind impulse, or by the authority of someone [20] else, to enlist as followers of Descartes, have only impressed his opinions and doctrines on their memory; when the subject comes up, they know only how to chatter and babble, but not how to demonstrate anything, as was, and still is, the custom among those who are attached to Aristotle’s philosophy.

To bring these people some assistance, I have often wished that someone who was skilled both in the Analytic and the Synthetic order, [25] and possessed a thorough knowledge of Descartes’ writings and Philosophy, would be willing to take on this work, to render in the Synthetic order what Descartes wrote in the Analytic, and to demonstrate it in the manner familiar to the geometricians. Indeed, I myself, though quite unequal to so great a task, and fully conscious of my weakness, frequently thought of doing this, and even began it. [30] But other occupations distracted me so often that I was prevented from completing it.

Therefore I was very pleased to learn from our Author that he had dictated, to a certain pupil of his, whom he was teaching the Cartesian [I/130] Philosophy, the whole Second Part of the Principles, and part of the Third, demonstrated in that Geometric manner, along with some of the principal and more difficult questions which are disputed in Metaphysics, and had not yet been resolved by Descartes,2 and that in response to the entreaties and demands of his friends, he had agreed [5] that, once he corrected and added to them, these writings might be published. So I too commended this project to him, and at the same time gladly offered my help in publishing, if he should require it.

Moreover, I advised him—indeed entreated him—to render also the first part of the Principles in a like order, and set it before what he had already written, so that by having been arranged in this manner from the beginning, the matter could be better understood and more pleasing. [10] When he saw the soundness of this argument, he did not wish to deny both the requests of a friend and the utility of the reader. And he entrusted to my care the whole business of printing and publishing, since he lives in the country, far from the city, and so could not be present.3

This, then, honest Reader, is what we give you in this little book: [15] the first and second parts of Descartes’ Principles of Philosophy, together with a fragment of the third, to which we have attached, as an Appendix, our Author’s Metaphysical Thoughts. But when we say, and when the title of the book promises, the first part of the Principles, we do not mean that everything Descartes says there is demonstrated here [20] in Geometric order, but only that the main matters which concern Metaphysics, and were treated by Descartes in his Meditations, have been taken from there (leaving aside whatever is a matter of Logic, or is recounted only historically).4

To do this more easily, our Author has carried over, word for word, [25] almost all the things which Descartes put in Geometrical order at the end of his Reply to the Second Objections—beginning with all of Descartes’ Definitions and inserting Descartes’ Propositions among his own, but not annexing the Axioms to the Definitions without interruption. He has placed the Axioms taken from Descartes after the fourth Proposition and altered their order, so they could be demonstrated more [30] easily. He has also omitted certain things which he did not require.

Our Author realizes that these Axioms could be demonstrated as Theorems (as Descartes himself says in the 7th postulate), and that they would be more elegantly treated as Propositions. And though we [I/131] asked him to do this, more important business in which he was involved allowed him only two weeks in which to complete this work. So he was unable to satisfy his desire and ours. Annexing at least a brief explanation, which can take the place of a proof, he has put off [5] a fuller explanation, complete in every respect, till another time. Perhaps, after this printing is exhausted, a new one will be prepared. If so, we shall also try to get him to enrich it by completing the Third Part, On the visible World (we have added here only a fragment of that Part, since our Author ended the instruction of his pupil at that point, and we did not wish to deprive the reader of it, however little [10] it was). For this to be done properly, it will be necessary to introduce certain Propositions concerning the nature and properties of Fluids in the Second Part. I shall do my best to see that our Author accomplishes this at that time.

Our Author quite frequently departs from Descartes, not only in [15] the arrangement and explanation of the Axioms, but also in the demonstration of the Propositions themselves, and the rest of the Conclusions; he often uses a Proof very different from Descartes’. Let no one take this to mean that he wished to correct that most distinguished Man in these matters; it was done only so as to better retain the order he had already taken up, and not to increase unduly the number of [20] Axioms. For the same reason he has also been forced to demonstrate quite a number of things which Descartes asserted without any demonstration, and to add others which he completely omitted.

Nevertheless, I should like it to be particularly noted that in all these writings—not only in the first and second parts of the Principles, and in the fragment of the third part, but also in his Metaphysical [25] Thoughts—our Author has only set out the opinions of Descartes and their demonstrations, insofar as these are found in his writings, or are such as ought to be deduced validly from the foundations he laid. For since he had promised to teach his pupil Descartes’ philosophy, he considered himself obliged not to depart a hair’s breadth from Descartes’ opinion,5 nor to dictate to him anything that either would not [30] correspond to his doctrines or would be contrary to them. So let no one think that he is teaching here either his own opinions, or only those which he approves of. Though he judges that some of the doctrines are true, and admits that he has added some of his own, nevertheless there are many that he rejects as false, and concerning which he holds a quite different opinion.

[I/132] An example of this—to mention only one of many—is what is said concerning the will in the Principles IP15S and in the Appendix, II, 12, although it seems to be proved with sufficient diligence and preparation. For he does not think that the will is distinct from the Intellect, [5] much less endowed with such freedom. Indeed in asserting these things—as is evident from the Discourse on Method, Part IV, the Second Meditation, and other places—Descartes only assumes, but does not prove that the human mind is a substance thinking absolutely. Though our Author admits, of course, that there is a thinking substance in nature, he nevertheless denies that it constitutes the essence of the [10] human Mind; instead he believes that just as Extension is determined by no limits, so also Thought is determined by no limits. Therefore, just as the human Body is not extension absolutely, but only an extension determined in a certain way according to the laws of extended nature by motion and rest, so also the human Mind, or Soul, is not [15] thought absolutely, but only a thought determined in a certain way according to the laws of thinking nature by ideas, a thought which, one infers, must exist when the human body begins to exist. From this definition, he thinks, it is not difficult to demonstrate that the Will is not distinct from the intellect, much less endowed with that liberty which Descartes ascribes to it; that that faculty of affirming [20] and denying is a mere fiction; that affirming and denying are nothing but ideas; and that the rest of the faculties, like Intellect, Desire, etc., must be numbered among the fictions, or at least among those notions which men have formed because they conceive things abstractly, like humanity, stone-hood, and other things of that kind.

[25] Again we must not fail to note that what is found in some places—viz. that this or that surpasses the human understanding—must be taken in the same sense, i.e., as said only on behalf of Descartes. For it must not be thought that our Author offers this as his own opinion. He judges that all those things, and even many others more sublime and [30] subtle, can not only be conceived clearly and distinctly, but also explained very satisfactorily—provided only that the human Intellect is guided in the search for truth and knowledge of things along a different path from that which Descartes opened up and made smooth. The foundations of the sciences brought to light by Descartes, and the [I/133] things he built on them, do not suffice to disentangle and solve all the very difficult problems that occur in Metaphysics. Different foundations are required, if we wish our intellect to rise to that pinnacle of knowledge.

[5] Finally—to put an end to prefacing—we wish our Readers to know that all the things treated here are published with no purpose except that of searching out and propagating the truth, and rousing men to strive for a true and genuine Philosophy; so in order that men may be able to harvest that rich fruit which we sincerely desire each of them to have, we warn them, before they set themselves to read this book, [10] to insert in their place certain things which have been omitted, and to correct accurately the Typographical errors which have crept in. For some of them could be an obstacle to a correct perception of the Author’s intention, and the force of the Demonstration, as anyone who inspects them will easily see.

[I/141] The Principles of Philosophy Demonstrated In the Geometric Manner
[5] Part I

PROLEGOMENON

Before we come to the Propositions themselves and their Demonstrations, it seems desirable to explain concisely why Descartes doubted [10] everything, how he brought to light solid foundations for the sciences, and finally, by what means he freed himself from all doubts. We would have reduced even all these things to Mathematical order, if we had not judged that the prolixity required by such a presentation would [15] prevent them from being understood as they ought to be. For they should all be seen in a single act of contemplation, as in a picture.

Descartes, then, in order to proceed as cautiously as possible in the investigation of things, attempted

(1) to lay aside all prejudices,

[20] (2) to discover the foundations on which all things ought to be built,

(3) to uncover the cause of error,

(4) to understand all things clearly and distinctly.

That he might be able to attain the first, second and third of these, he [25] sought to call all things into doubt, not as a Skeptic would, who has no other end than doubting, but to free his mind from all prejudices, so that in the end he might discover firm and unshakable foundations of the sciences. In this way, if there were any such foundations, they [30] could not escape him. For the true principles of the sciences must be [I/142] so clear and certain that they need no proof, that they are beyond all risk of doubt, and that nothing can be demonstrated without them. These he found, after a long period of doubting. And after he had discovered these principles, it was not difficult for him to distinguish [5] the true from the false, to uncover the cause of error, and so to put himself on guard against assuming something false and doubtful as true and certain.1

To obtain the fourth and last, i.e., that he might understand all things clearly and distinctly, his chief rule was to enumerate and examine [10] separately all the simple ideas of which all the rest of his ideas were compounded. For when he could perceive the simple ideas clearly and distinctly, he would undoubtedly understand, with the same clarity and distinctness, all the rest, which have been constructed from those simple ideas.

[15] With this as preface, we shall explain briefly how he called all things into doubt, discovered the true principles of the Sciences, and extricated himself from the difficulties of his doubts.

Doubt concerning all things

First, then, he considered all those things which he had received from the senses, viz. the heavens, the earth, and the like, and even his [20] own body. All these he had till then thought to exist in nature. And he came to doubt their certainty because he had realized that the senses sometimes deceived him, because in dreams he had often persuaded himself of the existence outside himself of many things, concerning which he afterwards discovered himself to have been deluded, and [25] finally, because he had heard others claim, even while awake, that they felt pain in limbs which they had long lacked.2 So it was not without reason that he was able to doubt the existence of his own body.

From all this he was able to conclude truly that the senses are not that most firm foundation on which every science should be built (for [30] they can be called into doubt), but that certainty depends on other principles, of which we are more certain.

To investigate such principles then, he considered second all universals, such as corporeal nature in general, and its extension, figure, [I/143] quantity, and also all Mathematical truths. And though these seemed more certain to him than all those he had derived from the senses, nevertheless he discovered a reason for doubting them: for others had erred even about these matters, and most important, deeply rooted in [5] his mind was an old opinion, according to which there is a God who can do all things and by whom he was created such as he was. Perhaps this God had made him so that he would be deceived even about those things that seemed clearest to him. And this is the way he called all things in doubt.

The discovery of the foundation of the whole science

To discover the true principles of the sciences, he asked next whether [10] he had called into doubt everything which could fall under his thought. His purpose was to examine whether, perhaps, there was not something remaining which he had not yet doubted. And if he did, by doubting in this way, discover something which could be called into doubt by none of the preceding reasons, nor by any other, he rightly [15] judged that he should set it up as the foundation on which he might build all his knowledge.

And though it seemed that he had already doubted everything—for he had doubted both the things he had derived from the senses and those he had perceived by the intellect alone—nevertheless, there was [20] something remaining which should be examined, viz. he himself who was doubting in this way. Not himself insofar as he consisted of a head, hands, and the other members of the body, since he had doubted these things, but only himself insofar as he was doubting, thinking, etc.

And when he considered it accurately, he discovered that he could not doubt it for any of the previously mentioned reasons. For whether [25] he thinks waking or sleeping, he still thinks and is. And though others, and even he himself had erred concerning other things, since they were erring, they were. Nor could he feign any author of his nature so cunning3 as to deceive him about this. For it will have to be conceded that he exists, so long as it is supposed that he is deceived. [30] Finally, whatever other reason for doubting might be thought up, none could be mentioned that did not at the same time make him most certain of his existence. Indeed, the more reasons for doubting are brought up, the more arguments are brought up that convince him of [I/144] his existence. So in whatever direction he turns in order to doubt, he is forced to break out with these words: I doubt, I think, therefore I am.

Hence, because he had laid bare this truth, he had at the same time also discovered the foundation of all the sciences, and also the measure [5] and rule of all other truths: Whatever is perceived as clearly and distinctly as that is true.4

That there can be no other foundation of the sciences than this, is more than sufficiently evident from the preceding. For we can call all the rest in doubt with no difficulty, but we can not doubt this in any [10] way.

But what we must note here, above all else concerning this foundation, is that this formula, I doubt, I think, therefore I am, is not a syllogism in which the major premise is omitted. For if it were a syllogism, the premises would have to be clearer and better known [15] than the conclusion itself, therefore I am. And so, I am would not be the first foundation of all knowledge. Moreover, it would not be a certain conclusion. For its truth would depend on universal premises which the Author had previously put in doubt. So I think, therefore I am is a single proposition which is equivalent to this, I am thinking.5

[20] Next, to avoid confusion in what follows, we need to know what we are (for this is a matter that ought to be perceived clearly and distinctly). Once we do understand it clearly and distinctly, we shall not confuse our essence with others. To deduce it from the above, our Author proceeded as follows.

[25] He recalled all the thoughts which he had formerly had of himself, e.g., that his soul was something tenuous, like wind, or fire, or air, infused throughout the grosser parts of his body, that the body was better known to him than the soul, and that he perceived it more [30] clearly and distinctly. And he observed that all these thoughts are clearly incompatible with those which up to this point he had understood. For he was able to doubt his own body, but not his own essence, insofar as he was thinking. Moreover, he perceived these thoughts neither clearly nor distinctly, and consequently, according to the rule of his method, he was obliged to reject them as false.

[I/145] Since he could not understand such things to pertain to himself, insofar as he was known to himself up to this point, he proceeded to inquire further what did properly pertain to his essence, which he could not put in doubt, and on account of which he was forced to infer his existence. But these were such things as: that he wished to take [5] care lest he be deceived; that he desired to know many things; that he doubted all things which he could not understand; that so far he affirmed only one thing; that he denied all the rest and rejected them as false; that he imagined many things, even though unwilling to; and finally that he perceived many things as if coming from the senses. Since he could infer his existence from [10] each of these things equally clearly, and could count none of them among those which he had called in doubt, and finally, since they can all be conceived under the same attribute, it followed that all these things were true and pertained to his nature. So when he said, I think, [15] all these modes of thinking were understood, viz. doubting, understanding, affirming, denying, willing, not willing, imagining, and sensing.6

But here the chief things to be noted—because they will be very useful later, when we deal with the distinction between mind and body—are (i) that these modes of thinking are understood clearly and [20] distinctly without the rest, concerning which there is still doubt, and (ii) that the clear and distinct concept we have of them is made obscure and confused, if we wish to ascribe to them any things concerning which we still doubt.

Liberation from all doubts

Finally, in order to become certain of the things he had called in [25] doubt and to remove all doubt, Descartes proceeded to inquire into the nature of the most perfect Being, and whether such a Being existed. For when he discovers that there is a most perfect being, by whose power all things are produced and conserved, and with whose nature being a deceiver is incompatible, then that reason for doubting which he had because he was ignorant of his cause will be removed. [30] He will know that a God who is supremely good and veracious did not give him the faculty of distinguishing the true from the false so that he might be deceived. Hence neither Mathematical truths nor any of those that seem most evident to him can be at all suspected.

Next, to remove the remaining causes of doubt, he went on to ask [I/146] how it happens that we sometimes err. When he discovered that this occurs because we use our free will to assent even to things we have perceived only confusedly, he was able to conclude immediately that he could guard against error in the future, provided he gave his assent [5] only to things perceived clearly and distinctly. Each of us can easily accomplish this by himself, since each has the power of restraining the will, and so of bringing it about that it is contained within the limits of the intellect.

But because we have absorbed at an early age many prejudices from [10] which we are not easily freed, he went on next to enumerate and examine separately all the simple notions and ideas of which all our thoughts are composed, so that we might be freed from our prejudices, and accept nothing but what we perceive clearly and distinctly. For if he could take note of what was clear and what obscure in each, [15] he would easily be able to distinguish the clear from the obscure and to form clear and distinct thoughts. In this way he would discover easily the real distinction between the soul and the body, what was clear and what obscure in the things we have derived from the senses, and finally, how a dream differs from waking states. Once this was [20] done, he could no longer doubt his waking states nor be deceived by the senses. So he freed himself from all the doubts recounted above.

But before we finish, it seems we must satisfy those who make the following objection. Since God’s existence does not become known to [25] us through itself, we seem unable to be ever certain of anything; nor will we ever be able to come to know God’s existence. For we have said that everything is uncertain so long as we are ignorant of our origin, and from uncertain premises, nothing certain can be inferred.

To remove this difficulty, Descartes makes the following reply.7 [30] From the fact that we do not yet know whether the author of our origin has perhaps created us so that we are deceived even in those things that appear most evident to us, we cannot in any way doubt the things that we understand clearly and distinctly either through themselves or through reasoning (so long, at any rate, as we attend to [II/147] that reasoning). We can doubt only those things that we have previously demonstrated to be true, and whose memory can recur when we no longer attend to the reasons from which we deduced them and, indeed, have forgotten the reasons. So although God’s existence cannot [5] come to be known through itself, but only through something else, we will be able to attain a certain knowledge of his existence so long as we attend very accurately to all the premises from which we have inferred it. See Principles I, 13; Reply to Second Objections, 3, and Meditation 5, at the end.

[10] But since this answer does not satisfy some people, I shall give another.8 When we previously discussed the certainty and evidence of our existence, we saw that we inferred it from the fact that, wherever we turned our attention—whether we were considering our own nature, [15] or feigning some cunning deceiver as the author of our nature, or summoning up, outside us, any other reason for doubting whatever—we came upon no reason for doubting that did not by itself convince us of our existence.

So far we have not observed this to happen regarding any other [20] matter. For though, when we attend to the nature of a Triangle, we are compelled to infer that its three angles are equal to two right angles, nevertheless we cannot infer the same thing from [the supposition] that perhaps we are deceived by the author of our nature. But from [this supposition] we did most certainly infer our existence. So [25] here we are not compelled, wherever we direct our attention, to infer that the three angles of a Triangle are equal to two right angles. On the contrary, we discover a ground for doubting, viz. because we have no idea of God which so affects us that it is impossible for us to think [30] that God is a deceiver. For to someone who does not have a true idea of God (which we now suppose ourselves not to have) it is just as easy to think that his author is a deceiver as to think that he is not a deceiver. Similarly for one who has no idea9 of a Triangle, it is just as easy to think that its three angles are equal to two right angles, as to think that they are not.

[I/148] So we concede that we can not be absolutely certain of anything, except our own existence, even though we attend properly to its demonstration, so long as we have no clear and distinct concept of God [5] that makes us affirm that he is supremely veracious, just as the idea we have of a Triangle compels us to infer that its three angles are equal to two right angles. But we deny that we cannot, therefore, arrive at knowledge of anything.

For as is evident from everything we have said just now, the crux [10] of the whole matter is that we can form a concept of God which so disposes us that it is not as easy for us to think that he is a deceiver as to think that he is not, but which now compels us to affirm that he is supremely veracious. When we have formed such an idea, that reason for doubting Mathematical truths will be removed. Wherever we [15] then direct our attention in order to doubt some one of them, we shall come upon nothing from which we must not instead infer that it is most certain—as happened concerning our existence.

E.g., if, after we have discovered the idea of God, we attend to the nature of a Triangle, the idea of this will compel us to affirm that its [20] three angles are equal to two right angles; but if we attend to the idea of God, this too will compel us to affirm that he is supremely veracious, and the author and continual conserver of our nature, and therefore that he does not deceive us concerning that truth. Nor will it be less impossible for us to think that he is a deceiver, when we attend [25] to the idea of God (which we now suppose ourselves to have discovered), than it is for us to think that the three angles of a Triangle do not equal two right angles, when we attend to the idea of a Triangle. And just as we can form such an idea of a Triangle, even though we do not know whether the author of our nature deceives us, so also we [30] can make the idea of God clear to ourselves and put it before our eyes, even though we still doubt whether the author of our nature deceives us in all things. And provided we have it, however we have acquired it, it will suffice to remove all doubt, as has just now been shown.

Therefore, from these premises we reply as follows to the difficulty [I/149] raised. We can be certain of nothing—not, indeed, so long as we are ignorant of God’s existence (for I have not spoken of this)—but as long as we do not have a clear and distinct idea of him.

So if anyone wishes to argue against me, his objection will have to [5] be this: we can be certain of nothing before we have a clear and distinct idea of God; but we cannot have a clear and distinct idea of God so long as we do not know whether the author of our nature deceives us; therefore, we can be certain of nothing so long as we do not know whether the author of our nature deceives us, etc.

[10] To this I reply by conceding the major and denying the minor. For we have a clear and distinct idea of a Triangle, although we do not know whether the author of our nature deceives us; and provided we have such an idea (as I have just shown abundantly), we will be able [15] to doubt neither his existence, nor any Mathematical truth.

With this as preface, let us now come to the matter itself.

DEFINITIONS10

D1: Under the word thought I include everything which is in us and of which we are immediately conscious.

[20] So all operations of the will, the intellect, the imagination and the senses are thoughts. But I have added immediately to exclude those things that follow from thoughts, e.g., voluntary motion does have thought as its principle, but it is still not itself a thought.

D2: By the term idea I understand that form of each thought through [25] the immediate perception of which I am conscious of the thought itself.

So if I understand what I say, I cannot express anything in words, without its being certain from this that there is in me an idea of what is signified by those words.11 And so I do not call only images depicted in the fantasy ideas. [30] Indeed I do not here call them ideas at all, insofar as they are depicted in the corporeal fantasy, i.e., in some part of the brain, but only insofar as they give form to the mind itself which is directed toward that part of the brain.

[I/150] D3: By the objective reality of an idea I understand the being of the thing represented by the idea, insofar as it is in the idea.

In the same way, one can speak of objective perfection, or objective artifice, etc. For whatever we perceive as in the objects of the ideas is in the ideas [5] themselves objectively.

D4: The same things are said to be formally in the objects of the ideas when they are in the objects as we perceive them, and eminently when they are in the objects, not indeed as we perceive them, but to such an extent as to be able to take the place of such things.

[10] Note that when I say the cause contains the perfections of its effect eminently, I mean that the cause contains the perfections of the effect more excellently than the effect itself does. See also A8.

D5: Everything in which there is immediately, as in a subject, or through which there exists, something we perceive, i.e., some property, [15] or quality, or attribute, of which there is a real idea in us, is called Substance.12

For of substance itself, taken precisely, we have no idea, other than that it is a thing in which exists formally or eminently that something which we [20] perceive, or, which is objectively in one of our ideas.13

D6: A substance in which thought is immediately is called a Mind.

I speak here of mind [mens] rather than soul [anima], because the word soul is equivocal and is often taken for a corporeal thing.14

D7: A substance which is the immediate subject of extension15 and of [30] accidents which presuppose extension, like figure, position, local motion, etc., is called a body.

But whether the substance called mind is one and the same as that called body, or whether they are two different substances, will need to be asked later.

D8: The substance which we understand to be through itself16 supremely perfect, and in which we conceive nothing which involves any defect or limitation of perfection, is called God.

D9: When we say that something is contained in the nature or concept [I/151] of something, that is the same as saying that it is true of that thing, i.e., can be truly affirmed of it.17

D10: Two substances are said to be really distinct when each of them can exist without the other.

[5] We have omitted Descartes’ postulates here, because we infer nothing from them in what follows. Still we earnestly ask the reader to read through them, to consider them, and to meditate on them carefully.

AXIOMS18

A1: We do not arrive at knowledge and certainty of an unknown thing [10] except by the knowledge and certainty of another thing which is prior19 to it in certainty and knowledge.

A2: There are reasons which make us doubt the existence of our body.

[15] This has been shown in the Prolegomenon, and so it is made an axiom here.

A3: If we have anything beyond a mind and a body, it is less known to us than the mind and the body.

It should be noted that these axioms make affirmations concerning no things outside us, but only concerning those things which we find in us, insofar as we are thinking things.20

[20] PROPOSITIONS

P1: We cannot be absolutely certain of anything, so long as we do not know that we exist.

Dem.: This proposition is evident through itself. For whoever absolutely [25] does not know that he is, equally does not know that he is affirming or denying, i.e., that he certainly affirms or denies.21

But it should be noted here that although we affirm and deny many things with great certainty without attending to the fact that we exist, nevertheless, unless this is presupposed as indubitable it is possible for everything to be called [30] in doubt.

[I/152] P2: I am must be known through itself.

Dem.: If you deny this, then it will not become known except through [5] something else, the knowledge and certainty of which (by A1) will be prior in us to this proposition, I am. But this is absurd (by P1). Therefore, it must be known through itself, q.e.d.

P3: I, insofar as I am a thing consisting of a body, am, is not the first thing [10] known, nor is it known through itself.

Dem.: There are certain things which make us doubt the existence of our body (by A2); therefore (by A1), we shall not arrive at certainty of [the existence of our body] except through the knowledge and certainty [15] of another thing, which is prior to it in knowledge and certainty. Therefore, the proposition that I, insofar as I am a thing consisting of a body, am, is not the first thing known, nor is it known through itself, q.e.d.

[20] P4: I am cannot be the first thing known except insofar as we think.

Dem.: The proposition I am a corporeal thing or one consisting of a body is not the first thing known (P3). Nor am I certain of my existence [I/153] insofar as I consist of anything else besides a mind and a body. For if we consist of anything else different from the mind and the body, this is less known to us than the body (A3). So I am can not be the first thing known except insofar as we think, q.e.d.

[5] Cor.: Hence it is evident that the mind, or thinking thing, is better known than the body. For a fuller explanation, see Principles I, 11 and 12.

[10] Schol.: Everyone perceives most certainly that he affirms, denies, doubts, understands, imagines, etc., or that he exists doubting, understanding, affirming, etc., or in a word, thinking. Nor can this be [15] called in doubt. So the proposition I think, or I am Thinking is the unique (P1) and most certain foundation of the whole of Philosophy.

Now to be completely certain of matters in the sciences nothing more can be sought or desired than to deduce all things from the firmest principles and to render them as clear and distinct as the principles [20] from which they are deduced. So clearly whatever is equally evident to us, whatever we perceive as clearly and distinctly as the principle we have already discovered, and whatever so agrees with this principle, and so depends on it that if we should wish to doubt it we would have to doubt this principle as well, must be held most true.22

[25] But to proceed as cautiously as possible in examining these matters, I shall admit in the beginning, as equally evident and as perceived by us with equal clarity and distinctness, only those things that each of us observes in himself, insofar as he is thinking. E.g., that he wills [30] this and that, that he has ideas of a certain sort, that one idea contains in itself more reality and perfection than another, that the idea which contains objectively the being and perfection of substance is far more [I/154] perfect than the one which contains only the objective perfection of some accident, and finally that the idea of a supremely perfect being is the most perfect of all. These, I say, we perceive not only with equal evidence and clarity, but even, perhaps, more distinctly. For [5] they affirm not only that we think, but also how we think.

Next we shall also say that those [propositions] agree with this principle which cannot be called into doubt unless at the same time this unshakable foundation of ours should be put in doubt. E.g., if someone should wish to doubt whether something comes from nothing, he [10] will at the same time be able to doubt whether we exist when we think. For if I can affirm something of nothing—viz. that it can be the cause of something—I shall be able at the same time, with the same right, to affirm thought of nothing, and to say that I am nothing when I think. But since I cannot do that, it will also be impossible for me [15] to think that something may come from nothing.

Having considered these matters, I decided to set out here, in order, the things which at present seem necessary to enable us to go further, and to add to the number of Axioms. They are put forward as axioms by Descartes, at the end of the Replies to the Second Objections, and [20] I do not wish to be more accurate than he is.23 Nevertheless, so as not to depart from the order already begun, I shall try to make them somewhat clearer and to show how one depends on the other and how they all depend on this principle—I am thinking—or agree with it in evidence and reason.

[25] AXIOMS TAKEN FROM DESCARTES

A4: There are different degrees of reality, or being: for a substance has more reality than an accident or mode, and the infinite substance more than a finite; accordingly there is more objective reality in the [30] idea of a substance than in that of an accident, and in the idea of the infinite substance than in that of a finite [substance].

This axiom comes to be known just from the contemplation of our ideas, of [I/155] whose existence we are certain, because they are modes of thinking. For we know how much reality or perfection the idea of substance affirms of a substance, and how much the idea of mode affirms of a mode.24 Hence we necessarily find that the idea of substance contains more objective reality than that [5] of some accident. See P4S.

A5: If a thinking thing knows any perfections which it lacks, it will immediately give them to itself, if they are in its power.

Everyone observes this in himself, insofar as he is a thinking thing. Consequently [10] (by P4S) we are most certain of it. And for the same reason we are no less certain of the following, viz.,

A6: Existence—either possible or necessary—is contained in the idea, or concept, of every thing (see Descartes, A10). Necessary existence, in the concept of God, or of a supremely perfect being (for otherwise he would be [15] conceived as imperfect, contrary to what is supposed to be conceived); but contingent, or possible, in the concept of a limited thing.

A7: No actually existing thing and no actually existing perfection of a thing can have nothing, or a thing not existing, as the cause of its existence.

[20] In P4S I have demonstrated that this axiom is as evident to us as I am thinking.

A8: Whatever reality, or perfection, there is in any thing, exists formally or eminently in its first and adequate cause.

[25] I understand that the reality is in the cause eminently when the cause contains the whole reality of the effect more perfectly than the effect itself, but formally when it contains it as perfectly.

This axiom depends on the preceding one. For if it were supposed that there was either nothing in the cause, or less in the cause than in the effect, then the [30] nothing in the cause would be the cause of the effect. But this (by A7) is absurd. So not anything can be cause of an effect, but only that in which there is every perfection which is in the effect either eminently or at least formally.

A9: The objective reality of our ideas requires a cause in which the [I/156] same reality itself is contained, not only objectively, but formally or eminently.

Though many people misapply this axiom, it is acknowledged by everyone. For whenever anyone has conceived something new, there is no one who does [5] not look for the cause of that concept, or idea. When they can assign one which contains formally or eminently as much reality as that concept contains objectively, they are satisfied. This is explained adequately by the example of the machine which Descartes uses in the Principles (I, 17).

Again, if anyone should ask from what source a man has the ideas of his [10] thought and of his body, no one fails to see that he has them from himself, as containing formally all that the ideas contain objectively. So, if a man were to have some idea which contained more objective reality than he contained formally, we would be driven by the natural light to look for another cause, outside the man himself, which contained all that perfection formally or eminently. [15] Nor has anyone ever assigned any other cause, except this one, which he conceived as clearly and distinctly.

As for the truth of this axiom, it depends on the preceding one. For (by A4) there are different degrees of reality or being in ideas, and therefore by (A8) [20] the more perfect they are, the more perfect the cause they require. But the degrees of reality which we perceive in our ideas are not in the ideas insofar as they are considered as modes of thinking, but rather insofar as one represents a substance and another represents only a mode of substance—or, in a word, insofar as they are considered as images of things.a So clearly, there can be no [25] other first cause of ideas except that which (as we have just shown) everyone understands clearly and distinctly by the natural light: viz., one in which there is contained either formally or eminently the same reality which the ideas have objectively.

That this conclusion may be more clearly understood, I shall explain it with one or two examples. Suppose someone sees two books—one the work of a distinguished [30] philosopher, the other that of some trifler, but both written in the same hand. If he attends to the meaning of the words (that is, does not attend to them insofar as they are like images), but only to the handwriting and to the order of the letters, he will recognize no inequality between them which [I/157] compels him to look for different causes. They will seem to him to have proceeded from the same cause in the same way. But if he attends to the meaning of the words and the discourses,25 he will find a great inequality between them. And so he will conclude that the first cause of the one book was very different from the first cause of the other, and really more perfect than it in proportion to the [5] differences he finds between the meaning of the discourses of each book, or between the words considered as images. I speak of the first cause of the books; there must be a first cause, though I concede—indeed I assume—that one book can be copied from another, as is obvious in itself.

The same [axiom] can also be explained clearly by the example of a portrait—say [10] of some Prince. For if we attend only to its materials, we shall find between it and other portraits no inequality which would force us to look for different causes. On the contrary, nothing will prevent us from being able to think that it has been painted from another picture, and that one again from [15] another, and so on to infinity. For we shall discern [clearly] enough that no other cause is required for drawing it. But if we attend to the image insofar as it is an image we shall immediately be forced to look for a first cause which contains, formally or eminently, what the image contains by representation. I do not see what more could be desired for the confirmation and clarification of this axiom.

[20] A10: No less a cause is required for preserving a thing than for first producing it.

From the fact that we are thinking now, it does not necessarily follow that we shall be thinking afterwards. For the concept which we have of our thought does not involve, or contain, the necessary existence of the thought. I can [25] conceive the thought clearly and distinctly even though I suppose that it does not exist.b

But the nature of every cause must contain or involve in itself the perfection of its effect (by A8). From this it follows clearly that there must be something, either in us or outside us, which we have not yet understood, whose concept, [30] or nature, involves existence and which is the cause of our thought’s having begun to exist, and also of its continuing to exist. For though our thought has begun to exist, its nature and essence does not on that account involve necessary existence any more than before it existed. So it needs the same power to persevere [I/158] in existing as it needed to begin existing. And what we say here about thought, must also be said about anything whose essence does not involve necessary existence.

A11: Nothing exists of which it cannot be asked, what is the cause, or reason, why it exists. See Descartes’ A1.

[5] Since existing is something positive, we cannot say that it has nothing as its cause (by A7). Therefore we must assign some positive cause, or reason, why [a thing] exists—either an external one, i.e., one outside the thing itself, or an internal one, i.e., one comprehended in the nature and definition of the existing thing itself.

[10]

The following four propositions are taken from Descartes.

P5: God’s existence is known from the consideration of his nature alone.

[15] Dem.: To say that something is contained in the nature, or concept, of something is the same as saying that it is true of that thing (by D9). But necessary existence is contained in the concept of God (by A6). So, it is true to say of God that necessary existence is in him, or that he exists.

[20] Schol.: Many excellent things follow from this proposition. Indeed, almost all that knowledge of God’s attributes through which we are led to the love of him, or the highest blessedness, depends on this alone: that existence pertains to the nature of God, or that the concept of God involves necessary existence, as the concept of a triangle involves [25] that its three angles are equal to two right angles, or that his existence, no less than his essence, is an eternal truth. So it would be [I/159] very desirable for the human race at last to embrace these things with us.

I confess, of course, that there are certain prejudices that stand in the way of everyone’s understanding this so easily.c But if anyone [5] moved by a good intention and by the simple love of the truth and of his own true advantage, should wish to examine the matter and to weigh carefully the things considered in the Fifth Meditation and at the end of the Replies to the First Objections, as well as what we say about eternity in our Appendix (II, 1), he will doubtless understand [10] it as clearly as possible, and no one will be able to doubt whether he has an idea of God, which, of course, is the first foundation of human blessedness. For he will see that the idea of God is very different from the ideas of other things as soon as he understands that God differs in every way from other things, with respect both to his essence and to [15] his existence. So there is no need to detain the Reader longer about these matters.

P6: God’s existence is demonstrated a posteriori from the mere fact that there is an idea of him in us.

[20] Dem.: The objective reality of any of our ideas requires a cause in which the very same reality is contained, not only objectively, but formally or eminently (by A9). But we have an idea of God (by D2 and D8), [25] and the objective reality of this idea is not contained either formally or eminently in us (by A4); nor can it be contained in any other thing except God himself (by D8). So this idea of God which is in us requires God as its cause, God, therefore, exists (by A7).

[I/160] Schol.: There are some who deny that they have any idea of God, and who nevertheless (so they say) worship and love him. And though you may put before them a definition of God, and God’s attributes, [5] you will still gain nothing by it, no more than if you labored to teach a man blind from birth the differences between the colors, just as we see them. But unless we should wish to regard them as a new kind of animal, between men and the lower animals, we must not bother too [10] much about their words. How, I ask, can we make the idea of any thing known except by propounding its definition and explaining its attributes? Since we offer this concerning the idea of God, there is no reason for us to be delayed by the words of men who deny that they have an idea of God merely because they can form no image of him in their brain.

[15] Next we should note that when Descartes cites A4 to show that the objective reality of our idea of God is not contained in us, either formally or eminently, he supposes that everyone knows that he is not an infinite substance—i.e., supremely intelligent, powerful, etc. He is [20] entitled to suppose this because he who knows that he thinks, knows also that he has doubts about many things and does not understand everything clearly and distinctly.

Finally, we must note that it also follows clearly from D8 that there can not be more than one God, as we clearly demonstrate in P11 and in our Appendix, II, ii.

[25] P7: The existence of God is also demonstrated from the fact that we ourselves who have an idea of him exist.

Schol.: To demonstrate this proposition Descartes assumes these [I/161] two axioms: (1) What can bring about the greater, or more difficult, can also bring about the lesser; (2) It is greater to create, or (by A10) to preserve, a substance than the attributes, or properties, of a substance. But what he means by this I do not know. What does he call easy, and what difficult? [5] Nothing is said to be easy or difficult absolutely, but only in relation to a cause. So one and the same thing can at the same time be called both easy and difficult in relation to different causes.d

But if he calls difficult those things that can be accomplished [by a cause] with great labor, and easy, those that can be accomplished by [10] the same cause with less labor—as a force which can lift 50 pounds will be able to lift [25] pounds twice as easily—then of course, the axiom will not be absolutely true, nor will he be able to demonstrate from it what he wants to. For when he says [AT VII, 168], if I had the power of preserving myself, I would also have the power of giving myself all the perfections I lack (because they do not require such a great power), [15] I would concede this to him. The powers I expend in preserving myself could bring about many other things far more easily, if I did not require them for preserving myself. But so long as I use them for preserving myself, I deny that I can expend them to bring about other things, even though they are easier, as is clear in our example.

[20] It does not remove the difficulty if it is said that since I am a thinking thing I would necessarily have to know whether I spend all my powers in preserving myself, and also whether this is the cause of my [25] not giving myself the remaining perfections. The dispute now does not concern this, but only how the necessity of this proposition follows from this axiom. Moreover, if I knew it, I would be greater, and perhaps would require greater powers to preserve myself in that greater perfection than those I have.26

And then I do not know whether it is a greater work to create (or preserve) a substance than to create (or preserve) attributes. To speak [30] more clearly and Philosophically, I do not know whether a substance does not require its whole power and essence, by which it perhaps preserves itself, for preserving its attributes.27

But let us leave these things to examine further what our most noble Author means here, i.e., what he understands by easy and difficult. I [I/162] do not think, nor can I in any way persuade myself, that by difficult he understands what is impossible (so that it cannot in any way be conceived how it happens), and by easy, what implies no contradiction (so that it can easily be conceived how it happens). It is true that he [5] seems at first glance to mean this, when he says in the Third Meditation [AT VII, 48]: I must not think that perhaps the things I lack are more difficult to acquire than those now in me. On the contrary, it is evident that it was far more difficult for me—i.e., a thing, or substance, which thinks—to emerge from nothing than, etc. But that would not be consistent with [10] the author’s words and would not be worthy of his genius.

For, to pass over the first consideration, there is nothing in common between the possible and the impossible, or between the intelligible and the unintelligible, just as there is nothing in common between something and nothing; and power does not agree with impossibilities [15] any more than creation and generation do with nonexistent things, so they ought not to be compared in any way. Moreover, I can compare things with one another and know the relation between them only if I have a clear and distinct concept of each of them. Hence I deny that it follows that if someone can do the impossible, he should also be [20] able to do what is possible.

What sort of conclusion is this? If someone can make a square circle, he will also be able to make a circle all of whose radii are equal, or, if someone can bring it about that nothing28 is acted on, and can use it [25] as a material from which to produce something, he will also have the power to make something from some [B: other] thing. As I have said, between these and similar things there is neither agreement, nor proportion, nor comparison, nor anything whatsoever in common. Anyone can see this, if he gives the matter any attention at all. I think [30] Descartes was too intelligent to have meant that.

But when I consider the second axiom of the two just cited, it seems that by greater and more difficult he means more perfect, and by less and easier, more imperfect. But this is also very obscure. There is the [I/163] same difficulty here as before. I deny, as before, that he who can do the greater, should be able at the same time and by the same work (as must be supposed in the Proposition) to do the lesser.

Again, when he says: it is greater to create or preserve a substance than to create or preserve its attributes, he can surely not understand by attributes [5] what is contained formally in substance and is distinguished from substance itself only by reason.29 For then creating a substance is the same as creating its attributes. For the same reason he also cannot understand [by attributes] the properties of a substance which follow necessarily from its essence and definition.

[10] Much less can he understand what he nevertheless seems to mean, viz. the properties and attributes of another substance. So, for example, if I say that I have the power of preserving myself, a finite thinking substance, I cannot on that account say that I also have the power [15] of giving myself the perfections of the infinite substance which differs in its whole essence from my essence. For the power, or essence,e by which I preserve myself in my being differs entirely from the power, or essence, by which the absolutely infinite substance preserves itself, from which its powers and properties are only distinguished by reason. [20] Hence, even though I were to suppose that I preserve myself, if I should wish to conceive that I could give myself the perfections of the absolutely infinite substance, I would be supposing nothing but this—that I can reduce my whole essence to nothing and create afresh an infinite substance. This, of course, would be much greater than [25] only supposing that I can preserve myself, a finite substance.

Since, then, he can understand none of these things by attributes or properties, nothing else remains, except the qualities that the substance [30] itself contains eminently (as, this or that thought in the mind, which I clearly perceive to be lacking in me), but not those another substance contains eminently (as, this or that motion in extension; for such perfections are not perfections for me, a thinking thing, and so are not lacking to me). But then Descartes cannot in any way infer from this axiom the conclusion he wants to demonstrate; i.e., that if I [I/164] preserve myself, I also have the power of giving myself all the perfections that I clearly find to pertain to a supremely perfect being.

This is quite evident from what has just been said. But not to leave the matter undemonstrated, and to avoid all confusion, it seemed best [5] to demonstrate the following lemmas first, and afterwards to construct a demonstration of P7 on that basis.

Lemma 1: The more perfect a thing is by its own nature, the greater and more necessary is the existence it involves; conversely, the more necessary the [10] existence it involves by its own nature, the more perfect it is.

Dem.: Existence is contained in the idea, or concept, of everything (by A6). Let A be a thing that has ten degrees of perfection. I say that [15] its concept involves more existence than it would if it were supposed to contain only five degrees of perfection. For since we can affirm no existence of nothing (see P4S), then the more we take away its perfection [20] in thought, and so the more we conceive it as participating in nothing,30 the more possibility of existence we also deny it. Hence, if we should conceive its degrees of perfection to be diminished infinitely to zero, it will contain no existence, or absolutely impossible existence. On the other hand, if we increase its degree [of perfection] infinitely, [25] we shall conceive it as involving existence in the highest degree, and therefore as involving supremely necessary existence. This was the first thing to be proven.

And the second thing proposed for demonstration follows clearly from the fact that these two things [necessary existence and perfection] cannot be separated in any way (as is sufficiently established by A6, and by this whole first part).

[30] Note 1. Although many things are said to exist necessarily from the mere fact that there is a determinate cause to produce them, we are not speaking about these things here, but only about that necessity and possibility which [I/165] follow solely from the consideration of the nature, or essence, of the thing, without regard to any cause.

Note 2. We are not speaking here about beauty and the other ‘perfections’ which men have wished, in their superstition and ignorance, to call perfections. [5] By perfection I understand only reality, or being. E.g., I perceive that more reality is contained in substance than in modes, or accidents. Hence I understand clearly that it contains a more necessary and perfect existence than accidents do, as is plain enough from A4 and A6.

[10] Cor.: Hence it follows that whatever involves necessary existence is a supremely perfect being, or God.

Lemma 2: The nature of him who has the power of conserving himself involves [15] necessary existence.

Dem.: Whoever has the power to preserve himself also has the power to create himself (by A10), i.e. (as everyone will readily concede), he requires no external cause in order to exist; rather, his own nature [20] alone will be a sufficient cause of his existing, either possibly (see A10) or necessarily. But he does not exist possibly. For then (by what we have demonstrated concerning A10), from the fact that he existed now, it would not follow that he would exist afterwards (which is contrary to the hypothesis). So he exists necessarily, i.e., his nature involves [25] necessary existence, q.e.d.

Demonstration of P7: If I had the power to preserve myself, I would [I/166] be of such a nature that I would involve necessary existence (by L2). So (by L1C) my nature would contain all perfections. But I find in myself, insofar as I am a thinking thing, many imperfections—that I [5] doubt, desire, etc.—of which I am certain (P4S). Therefore, I have no power to preserve myself. I cannot say that the reason I now lack those perfections is that I wish to deny them to myself, for that would clearly be incompatible with L1 and with what I clearly find in myself (by A5).

Next, I cannot now exist without being preserved as long as I exist, [10] either by myself, if in fact I have that power, or by another who has it (by A10 and All). But I exist (by P4S) and nevertheless I do not have the power to preserve myself, as was just now proved. Therefore, I am preserved by another. But not by another who does not have the power to preserve himself (by the same reasoning by which [15] I just demonstrated that I cannot preserve myself). So I am preserved by another who has the power of preserving himself, i.e. (by L2), whose nature involves necessary existence, i.e. (by L1C), who contains all the perfections which I understand to pertain clearly to a supremely [20] perfect being. And therefore a supremely perfect being, i.e. (by D8), God, exists, as was to be demonstrated.

Cor.: God can bring about whatever we clearly perceive, as we perceive it.

[25] Dem.: All these things clearly follow from the preceding Proposition. For it is proved there that God exists from the fact that there must exist someone who has all the perfections of which there is some idea in us. But there is in us the idea of a power so great that the [30] heaven, and the earth, and all the other things which I understand to be possible, can be made, unaided, by him in whom the power exists. [I/167] So all these things have been proved about God together with his existence.

P8: Mind and body are really distinct.

[5] Dem.: Whatever we perceive clearly can be made by God as we perceive it (P7C). But (by P3 and P4) we clearly perceive the mind, i.e. (by D6), a thinking substance, without the body, i.e. (by D7), [10] without any extended substance. Conversely, we perceive the body clearly without mind (as everyone will readily concede). So the mind can exist without the body and the body can exist without the mind—at least by divine power.

Now substances which can exist without one another are really distinct [15] (by D10). But the mind and the body are substances (by D5, 6, and 7), which can each exist without the other (as we have just proven). So the mind and the body are really distinct.

See Descartes’ P4 (at the end of the Replies to the Second Objections) [20] and the Principles I, 22-29. For I do not judge it worthwhile to transcribe here the things said there.31

P9: In the highest degree, God understands.

[25] Dem.: If you deny this, then God will understand either nothing, or not everything, or only certain things.

But to understand only certain things and be ignorant of others supposes a limited and imperfect intellect, which it is absurd to ascribe to God (by D8).

[I/168] But that God should understand nothing would indicate either that he lacks any intellection, as men do when they understand nothing, and so would involve imperfection, which cannot be in God (by D8), or that his understanding something would be incompatible with his perfection.

[5] Since intellection would thus be denied to him altogether, he would not be able to create any intellect (by A8). But since we perceive intellect clearly and distinctly, God can be its cause (P7C). Hence it is not at all true that it is incompatible with God’s perfection for him to understand something.

[10] Consequently, he will, in the highest degree, understand, q.e.d.

Schol.: Although it must be conceded that God is incorporeal, as is demonstrated in P16, still this must not be taken to mean that all the [15] perfections of Extension are to be denied him. Extension is to be rejected only insofar as its nature and properties involve some imperfection. The same thing must also be said about God’s intellection, as everyone who wants to be wiser than the ordinary run of Philosophers [20] confesses. This will be explained fully in our Appendix (II, vii).32

P10: Whatever perfection is found in God, is from God.

Dem.: If you deny this, suppose there is some perfection in God [25] which is not from God. It will be in God, either from itself or from something different from God. If it is from itself, then it will have necessary, or not [merely] possible, existence (by P7L2).33 And so (by P7L1C) it will be something supremely perfect, and therefore (by D8) God. Accordingly, if it should be said that there is something in God [I/169] which is from itself, it is said at the same time to be from God, q.e.d. But if it is from something different from God, then God cannot be conceived as supremely perfect through himself, contrary to D8. Therefore, whatever perfection is found in God is from God, q.e.d.

[5] P11: There is not more than one God.34

Dem.: Suppose you deny this. Conceive, if possible, that there is more than one God, e.g., A and B. Then both A and B must, in the [10] highest degree, understand (by P9), i.e., A understands everything, including both himself and B, and B, in turn, will understand himself and A. But since A and B exist necessarily (by P5), then the cause of the truth and necessity of the idea of B which is in A, is B. Conversely, the cause of the truth and necessity of the idea of A which is [15] in B is A. Consequently, there will be a perfection in A which is not from A, and one in B which is not from B. So (by P10) neither A nor B will be Gods. Therefore, there is not more than one God, q.e.d.

It should be noted here that it follows necessarily from the mere fact that some thing involves necessary existence from itself (as God does) that it is unique. [20] Everyone will be able to see this for himself, provided he meditates attentively. I could also have demonstrated it here, but not in a way perceptible by everyone, as has been done in this proposition.

[25] P12: Whatever exists is preserved by the power of God alone.

Dem.: If you deny this, let it be supposed that something preserves [I/170] itself. Then (by P7L2), its nature involves necessary existence, and so it would be God (by P7L1C). There would thus be more than one God, which is absurd (by P11). So nothing exists which is not preserved by the power of God alone, q.e.d.

[5] Cor. 1: God is the creator of all things.

Dem.: God (by P12) preserves all things, i.e. (by A10), he has created whatever exists, and even now continuously creates it.

[10] Cor. 2: Things have no essence from themselves which might be the cause of God’s knowledge; rather, God is the cause of things even with respect to their essences.

[15] Dem.: Since no perfection is found in God which is not from God (by P10), things will have no essence from themselves which could be the cause of God’s knowledge. On the contrary, since God has not generated everything from something else, but has created all things completely (P12 and P12C1), and since that act of creation admits of [20] no cause except the efficient (for so I define creation), which is God, it follows that things were nothing at all before the creation; and so God was also the cause of their essence, q.e.d.

It should be noted that this corollary is also evident from the fact that God is the cause, or creator, of all things (by C1) and that the [25] cause must contain in itself all the perfections of the effect (by A8), as everyone can easily see.

[I/171] Cor. 3: From this it clearly follows that God does not sense or, strictly speaking, perceive. For his intellect is not determined by anything outside itself. Rather, all things proceed from him.

[5] Cor. 4: God is prior in causality to the essence and to the existence of things, as follows clearly from Corollaries 1 and 2 of this Proposition.

P13: God is supremely veracious, and not at all a deceiver.

[10] Dem.: We can attribute nothing to God (by D8) in which we find any imperfection. Now (as is known through itself) every deception, or will to deceive, proceeds only from malice or fear.f Fear implies a [15] lesser power, and malice a privation of goodness. So no deception, or will to deceive, ought to be ascribed to God, i.e., to a being supremely powerful and supremely good. On the contrary, he must be said to be supremely veracious, and not at all a deceiver, q.e.d. See the Replies to the Second Objections, No. 4.

[25] P14: Whatever we perceive clearly and distinctly is true.

Dem.: The faculty of distinguishing the true from the false, which (as everyone discovers in himself and can be seen from all that we [I/172] have now demonstrated) is in us, has been created, and is continuously preserved by God (by P12 with its corollary), i.e. (by P13), by a being supremely veracious and not at all a deceiver. Nor has he given us (as everyone discovers in himself) any faculty of holding back from, or [5] not assenting to, those things we perceive clearly and distinctly. So if we were deceived concerning them, we would be deceived entirely by God, and he would be a deceiver, which (by P13) is absurd. Therefore, whatever we perceive clearly and distinctly is true, q.e.d.

[10] Schol.: Since the things to which we must necessarily assent, when we perceive them clearly and distinctly, must be true, and since we have the faculty of not assenting to those that are obscure and doubtful, or that are not deduced from most certain principles (as everyone [15] discovers in himself), it clearly follows that we can always prevent ourselves from falling into errors and from ever being deceived, provided that we decide resolutely to affirm nothing we do not perceive clearly and distinctly, or which is not deduced from principles clear and certain through themselves. This will be understood even more [20] clearly from what follows.

P15: Error is not something positive.35

Dem.: If error were something positive, it would have God alone [25] as its cause, by whom it would be continuously created (by P12). But this is absurd (by P13). So, error is not something positive, q.e.d.

[I/173] Schol.: Since error is not something positive in man, it can be nothing but a privation of the proper use of liberty (by P14S). So God should not be called the cause of error, except in the sense in which [5] we say that the absence of the Sun is the cause of darkness, or that God is the cause of blindness, because he has made a child like others, except in respect of vision. For he has given us an intellect that extends only to a few things.

To understand this more clearly, and to understand as well how [10] error depends solely on the misuse of our will and, finally, how we can guard against error, we should recall the modes of thinking that we have. They can all be grouped into two kinds: modes of perceiving (like sensing, imagining, and purely understanding) and modes of willing [15] (like desiring, shunning, affirming, denying, and doubting).

It must be noted about these [modes of thinking]:

(1) that the mind can be deceived neither insofar as it understands things clearly and distinctly and assents to them (by P14), nor insofar as it merely perceives things and does not assent to them. For though [20] I may now perceive a winged horse, it is certain that this perception contains no falsity, so long as I do not assent to its being true that there is a winged horse, nor also so long as I doubt whether there is a winged horse. And since assenting is nothing but determining the will, it follows that error depends solely on the use of the will.

[25] To make this still clearer, note:

(2) that we have the power of assenting not only to those things which we perceive clearly and distinctly, but also to those which we perceive in any other way. For our will is not determined by any limits. Anyone can see this clearly, provided he attends to the fact [30] that if God had wished to make our faculty of understanding infinite, he would not have needed to give us a greater faculty of assenting [I/174] than the one we now have in order for us to be able to assent to everything we understand. The same faculty we now have would suffice to our assenting to infinitely many things. And we know from our experience that we assent to many things that we have not deduced from certain principles.

[5] Again, it is clear from these considerations that we should never fall into error (by P14) if either the intellect extended itself as widely as the faculty of willing, or if the faculty of willing could not extend itself more widely than the intellect, or finally, if we could contain the faculty of willing within the limits of the intellect.

[10] But we have no power to bring about the first two. For it involves a contradiction both that the will should not be infinite and that the created intellect should not be finite. There remains, then, the third possibility to be considered, viz., whether we have the power to contain our faculty of willing within the limits of the intellect. Now since [15] the will is free to determine itself, it follows that we do have the power to contain our faculty of assenting within the limits of the intellect, and so can bring it about that we do not fall into error. Hence it is quite evident that it depends entirely on the use of the freedom of the [20] will that we are ever deceived. That our will is free is demonstrated in Principles I, 39 and in the Fourth Meditation.36 We have also shown it fully in the last chapter of our Appendix.

And though we cannot but assent to a thing when we perceive it clearly and distinctly, that necessary assent does not depend on the weakness of our will, but only on its freedom and perfection. For to [25] assent is truly a perfection in us (as is known sufficiently by itself), and the will is never more perfect or more free than when it completely determines itself. Since this can happen when the mind understands something clearly and distinctly, it will necessarily give that perfection to itself immediately (by A5). So it is far from being the [30] case that we understand ourselves to be less free from the fact that we are not at all indifferent in embracing the truth. On the contrary, we have established it as certain that the more we are indifferent, the less we are free.

All that remains to be explained, therefore, is how error is nothing [I/175] but a privation in relation to man, but is only a negation in relation to God. We shall see this easily if we first observe that because we perceive many things in addition to those we understand clearly37 we are more perfect than we would be if we did not perceive them. This [5] is clearly established by the fact that if we could perceive nothing clearly and distinctly, but only confusedly, we would have nothing more perfect than perceiving things confusedly. Nor could anything else be desired for our nature. Next, assenting to things, even to confused things, is a kind of action, and as such, it is a perfection. This [10] will also be plain to anyone who supposes (as above) that perceiving things clearly and distinctly is contrary to man’s nature. For then it will be evident that it is far better for man to assent even to confused things and to exercise his freedom, than to remain always indifferent, [15] i.e. (as we have just shown), in the lowest degree of freedom. And if we also consider what is needed and advantageous in human life, we shall find it absolutely necessary, as daily experience sufficiently teaches everyone.

Since, then, all the modes of thinking we have are perfect, insofar [20] as they are considered in themselves alone, what constitutes the form of error cannot be in them, considered in themselves. But if we consider the modes of willing, as they differ from one another, we shall discover that some are more perfect than others, insofar as some render the will less indifferent, i.e., more free, than others. Next we shall [25] also see that, so long as we assent to confused things, we make the mind less fit to distinguish between the true and the false, and bring it about that we lack the best liberty. So assenting to confused things, insofar as it is something positive, contains neither any imperfection [B: in itself], nor the form of error; [it contains imperfection] only [30] insofar as we thereby deprive ourselves of the best freedom, which belongs to our nature and is in our power. So the whole imperfection of error will consist solely in the privation of the best freedom, which is called error. It is said to be a Privation because we are deprived of a perfection which is suited to our nature; but [it is said to be] Error [I/176] because we lack that perfection through our own fault, insofar as we do not contain the will within the limits of the intellect to the extent that we can.

Since, then, error is nothing, in relation to man, but a privation of [5] the perfect, or right, use of freedom, it follows that it is not placed in any faculty which man has from God, nor in any operation of faculties, insofar as it depends on God. Nor can we say that God has deprived us of a greater intellect than he could have given us, and so has brought it about that we can fall into error. For nothing is such [10] that its nature can require anything of God, nor does anything pertain to anything except what the will of God has willed to bestow on it. For nothing existed prior to the will of God, nor can anything be conceived prior to it. (This is fully explained in our Appendix, II, vii and 8.) So God has no more deprived us of a greater intellect, or a [15] more perfect faculty of understanding, than he has deprived a circle of the properties of a sphere or its circumference of the properties of a spherical surface.

Therefore, since none of our faculties, however it is considered, can show any imperfection in God, it clearly follows that that imperfection [20] in which the form of error consists, is a privation only in relation to man. But considered in relation to God as its cause, it cannot be called a privation, only a negation.

P16: God is incorporeal.

[25] Dem.: Body is the immediate subject of local motion (by D7). So if God were corporeal, he would be divided into parts. Since this clearly involves an imperfection, it is absurd to affirm it of God (by D8).

[I/177] Alternative Dem.: If God were corporeal, he could be divided into parts (by D7). Now each part could either subsist through itself, or it could not. If the latter were the case, then it would be like the other [5] things created by God, and so, like every created thing, would be created continually by God by the same power (by P10 and A11), and would no more pertain to the nature of God than the other created things do. But that is absurd (by P5). On the other hand, if each part exists through itself, then each one must also involve necessary existence [10] (by P7L2), and consequently each one would be a supremely perfect being (by P7L2C). But that is also absurd (by P11). Therefore, God is incorporeal, q.e.d.

P17: God is an entirely simple being.

[15] Dem.: If God were composed of parts, the parts would have to be at least prior in nature to God (as everyone will easily concede). But that is absurd (by P12C4). Therefore, he is an entirely simple being, q.e.d.

[20] Cor.: From this it follows that God’s intellect and his will, or his Decree, and his power, are only distinguished by reason from his essence.38

[I/178] P18: God is immutable.

Dem.: If God were mutable, he could not be changed only in part, [5] but would have to be changed in respect to his whole essence (by P17). However, the essence of God exists necessarily (by P5, 6, and 7). Therefore, God is immutable, q.e.d.

P19: God is eternal.

[10] Dem.: God is a supremely perfect being (by D8), from which it follows (by P5) that he exists necessarily. If we now ascribe a limited existence to him, the limits of his existence must be understood—at least by God himself, if not by us, because he understands in the [15] highest degree (by P9). So beyond those limits God will understand himself (i.e., a supremely perfect being, by D8) as not existing. That is absurd (by P5). So God has, not a limited, but an infinite existence, which we call eternity. See our Appendix, II, 1. Therefore, God is [20] eternal, q.e.d.

P20: God has preordained all things from eternity.

Dem.: Since God is eternal (by P19), his understanding is eternal, [25] because it pertains to his eternal essence (by P17C). But his intellect [I/179] is not really distinct from his will, or decree (by P17C). So when we say that God has understood things from eternity, we are saying at the same time that he has willed, or decreed, them so from eternity, q.e.d.39

[5] Cor.: From this Proposition it follows that God is supremely constant in his works.

P21: Substance extended in length, breadth and depth really exists; and we are [10] united to one part of it.40

Dem.: The extended thing, as we perceive it clearly and distinctly, does not pertain to God’s nature (by P16), but can be created by God [15] (by P7C and P8). Now we perceive clearly and distinctly (as everyone finds in himself, insofar as he thinks) that extended substance is a sufficient cause for producing in us pleasure, pain, and similar ideas, or sensations. These are continually produced in us, even though we [20] are unwilling. But if we wish to feign some other cause of our sensations, beyond extended substance—say God or an Angel—we immediately destroy the clear and distinct concept which we have. Hence, so long as we attend rightly to our perceptions,g so that we admit nothing but what we have perceived clearly and distinctly, we shall [25] be wholly disposed (or not at all indifferent) to assent that extended substance is the only cause of our sensations. Hence [we will be wholly disposed] to affirm that the extended thing created by God exists.41 And in this we can surely not be deceived (by P14 and P14S). So it [I/180] is affirmed truly that substance extended in length, breadth, and depth exists. This was the first thing to be demonstrated.

Next, we observe that among our sensations, which must be produced in us by extended substance (as we have now demonstrated), [5] there is a great difference, viz. when I say that I sense, or see, a tree, and when I say that I am thirsty or in pain. But I clearly see that I cannot perceive the cause of this difference unless I first understand that I am closely united to one part of matter and not to others. Since [10] I understand this clearly and distinctly and cannot perceive it in any other way, it is true that I am united to one part of matter (by P14 and P14S). This was the second thing to be demonstrated.

Note: Unless the Reader here considers himself only as a thinking thing, lacking a body, and puts to one side, as prejudices, all the reasons he previously [15] had for believing that body exists, any effort to understand this proof will be in vain.

[I/181] Principles of Philosophy Demonstrated in the Geometric Manner
[5] Part II

POSTULATE: Here I ask only that everyone attend to his perceptions as accurately as possible, so as to be able to distinguish the clear from the obscure.

[10] DEFINITIONS

D1: Extension is what consists of three dimensions; but by extension we do not understand the act of extending, or anything distinct from quantity.1

D2: By substance we understand what requires only the concurrence of [15] God to exist.2

D3: An atom is a part of nature which is, by its nature, indivisible.3

D4: Indefinite is that whose limits (if it has any) cannot be discovered by the human intellect.4

D5: A vacuum is extension without corporeal substance.5

[20] D6: We make only a distinction of reason between space and extension, or they are not really distinct. Read Principles II, 10.

D7: What we understand to be divided in thought is at least potentially divisible.6

D8: Local motion is the transfer of one part of matter, or one body, [25] from the vicinity of those bodies that touch it immediately, and are considered as resting, to the vicinity of others.

[I/182] Descartes uses this definition to explain local motion. To understand it properly, we must consider:

(1) That he understands by a part of matter whatever is transferred at the same time, even though it, in turn, may consist of many parts.7

[5] (2) That for the sake of avoiding confusion he speaks in this definition only of what is constantly in the mobile thing, viz. the transfer, in order not to confuse this, as others frequently do, with the force or action which moves it. It is commonly thought that this force or action is required only for motion, and not for rest. But those who so think are thoroughly deceived. For as is [10] known through itself, the force which is needed to impart certain degrees of motion to a body at rest is also required to take away those certain degrees of motion from the body so that it is wholly at rest.

Indeed this is also proved by experience. For we use nearly the same force to [15] put in motion a boat resting in still water as we use to check suddenly the same boat when it is moving. The force would surely be exactly the same if we were not aided in checking the motion by the weight and resistance of the water the boat displaces.

(3) That he says the transfer takes place from the vicinity of contiguous bodies into the vicinity of others, and not from one place to another.8 For place, as [20] he has explained in II, 13, is not something real, but depends merely on our thought, so that the same body can be said at the same time both to change and not to change place. But it cannot be said at the same time both to be transferred and not to be transferred from the vicinity of a contiguous body. For only certain bodies can be contiguous to the same mobile [thing] at the same moment of time.

[25] (4) That he does not say absolutely that the transfer takes place from the vicinity of contiguous bodies, but only from the vicinity of those which are regarded as being at rest. If body A is transferred from body B, which is at rest, the same force and action are required on the one part as on the other.9 This is evident from the example [30] of a boat which is stuck in mud or sand at the bottom of the water. To free it, the force exerted on the bottom will have to equal that exerted on the boat. So the force by which bodies must be moved is expended equally on the body which is moved and on the one at rest. The transfer is, in fact, reciprocal; for if the boat is separated from the sand, the sand is also separated from the boat.

[I/183] Therefore, if we should wish to ascribe equal motions absolutely to two bodies which are separated from one another—one in one direction, the other in another—and should wish not to regard one as being at rest, simply because the [5] same action which is in one is in the other, then we would also be compelled to ascribe just as much motion to bodies which everyone takes to be at rest (like the sand from which the boat is separated) as we do to the bodies which move. For as we have shown, the same action is required on the one part as on the other and the transfer is reciprocal. But this would be too inconsistent with the [10] ordinary manner of speaking. Still, even though those bodies from which others are separated are regarded as being at rest, and are so spoken of, nevertheless we shall remember that whatever is in the body in motion, in virtue of which it is said to move, is also in the body at rest.

(5) Finally, it is also clear from the Definition that each body has only one [15] motion proper to it, since it is understood to depart from certain bodies only, viz. those that are contiguous to it and at rest. Nevertheless, if the body in motion is a part of other bodies, having other motions, we understand clearly that it can also participate in countless other motions. But because we cannot [20] easily understand so many at once, or even recognize all of them, it will suffice to consider in each body that one motion which is proper to it. Read Principles II, 31.

D9: By the circle of moved bodies we understand only [25] what occurs when the last body which is moved on account of the impulse of another body immediately touches the first of the bodies in motion, even though the line which is described by all the bodies at once through the impulse of one motion may be very twisted.10

AXIOMS

A1: Nothing has no properties.11

[30] A2: If something can be removed from a thing, while that thing remains intact, it does not constitute the thing’s essence, but if something, on being taken away, takes the thing away, it does constitute the thing’s essence.12

[I/184] A3: The senses do not indicate anything to us in hardness, nor do we understand anything clearly and distinctly about it, except that the parts of hard bodies resist the motion of our hands.13

A4: If two bodies move, either toward one another or away from one [5] another, they will not, on that account, occupy more or less space.

A5: Whether a part of matter moves away or resists, it does not, on that account, lose the nature of body.

A6: Motion, rest, figure and the like cannot be conceived without extension.14

[10] A7: Beyond the sensible qualities there is nothing in body except extension and its affections, enumerated in Part I of the Principles.15

A8: One space, or extension, cannot be larger at one time than at another.16

[15] A9: Every extension can be divided, at least in thought.17

No one who has learned even the elements of Mathematics doubts the truth of this axiom. For the space between a circle and a line tangent to the circle can always be divided by infinitely many other larger circles. The same thing is also evident from the Asymptotes of the Hyperbola.

[20] A10: No one can conceive the limits of any extension, or space, unless at the same time he conceives other spaces beyond them, i.e., immediately following them.18

A11: If there were more than one sort of matter and one did not touch the other immediately, each would be comprehended within limits [25] beyond which there is no matter.19

A12: The smallest bodies yield easily to the motion of our hands.

A13: One space does not penetrate another, nor is it larger at one time than at another.

[30] A14: If a pipe, A, is of the same length as another pipe, C, but C is twice as wide as A, and some fluid matter passes twice as quickly through A as what passes through C, the quantity of matter which passes through [I/185] A in a given interval of time will be the same as that which passes through C. And if the same quantity passes through A as through C, the former will move twice as quickly.20

A15: If two things agree with a third, they agree with one another. And if they are each double the same third thing, they are equal to [5] one another.

A16: Matter that moves in various ways has at least as many parts into which it is actually divided as the different degrees of speed that are observed in it at the same time.

A17: The shortest line between two points is a straight line.

[10] A18: If a body, A, in motion from C toward B, is repelled by an opposite impulse, it will be moved along the same line toward C.

A19: When two bodies which have opposite modes come into contact with one another, either both are constrained to suffer some variation, [15] or else at least one of them is.

A20: A variation in any thing proceeds from a stronger force.

A21: If, when body 1 moves toward body 2 and sets [20] it in motion, body 8 is moved toward body 1 by this impulse, bodies 1, 2, and 3, etc., cannot be in a straight line, but all of them will form a whole circle, up to body 8. See D9.

[25] L1: Where there is Extension, or Space, there is necessarily a Substance.

Dem.: Extension, or space, cannot be a pure nothing (by A1). Therefore, it is an attribute, which must necessarily be attributed to some thing. But not to God (by IP16); therefore to a thing which [30] requires only the concurrence of God to exist (by IP12), i.e. (by D2), to a substance, q.e.d.

[I/186] L2: We conceive Rarefaction and Condensation clearly and distinctly, even though we do not grant that bodies occupy a larger space when rarefied than [5] when condensed.

Dem.: We can clearly and distinctly conceive Rarefaction and Condensation as occurring when the parts of a body merely move away from one another or approach one another. Therefore (by A4) they [10] will not occupy a larger or smaller space; for if the parts of a body—say, a sponge—by, approaching one another expel the bodies which fill the intervals between the parts, this in itself renders that body more dense, and its parts will not on that account occupy a smaller space than before (by A4). And if they should move away from one [5] another again and their path should be filled by other bodies, rarefaction will take place, and yet they will not occupy a larger space. What we perceive clearly in a sponge with the aid of the senses, we can conceive by the intellect alone concerning all bodies, even though the intervals between their parts completely escape our senses [B: on account [20] of their smallness]. So we do conceive Rarefaction and Condensation clearly and distinctly, etc., q.e.d.

It seemed desirable to set these things out first so that the intellect might lay aside its prejudices about Space, Rarefaction, etc., and be made suited to understand the things that follow.

[25] P1: Even though the hardness, weight, and the rest of the sensible qualities are separated from a body, the nature of the body will still remain whole.

Dem.: Sensation does not indicate anything else to us in the hardness [30] of, say, this stone than that the parts of hard bodies resist the [I/187] motion of our hands, nor do we clearly and distinctly understand anything else about it (by A3). So hardness will also be nothing else than that (by IP14). But if that body should be reduced to as fine a [5] powder as possible, its parts will move away easily (by A12), and nevertheless it will not lose the nature of body (by A5), q.e.d.

The proof proceeds in the same way for weight and the rest of the sensible qualities.

[10] P2: The nature of Body, or Matter, consists in extension alone.

Dem.: The nature of the body is not taken away when the sensible qualities are taken away (by P1). Therefore, they do not constitute its essence (by A2). Nothing remains, then, except extension and its affections [15] (byA7), [B: which (by A6) cannot be conceived without extension]. So if extension is taken away, nothing will remain that pertains to the nature of the body, but it will be entirely taken away. Therefore, the nature of Body consists in extension alone (by A2), q.e.d.

[20] Cor.: Space and body do not really differ.

Dem.: Body and extension do not really differ (by P2), and space and extension do not really differ (by D6); therefore (by A15) space [25] and body do not really differ, q.e.d.

[I/188] Schol.: Though we say that God is everywhere, we do not thereby concede that God is extended,a i.e. (by P2), corporeal. For being [5] everywhere is related only to God’s power and concurrence, by which he preserves all things, so that God’s omnipresence is related no more to extension, or body, than to angels or human souls. But it should be noted that when we say that his power is everywhere, we do not exclude his essence. For where his power is, there his essence is also [10] (by IP17C). We exclude only corporeality, i.e., God is everywhere not by some corporeal power, but by the divine power, or essence, which is common to the preservation both of extension and of thinking things (by IP17). For God would not really have been able to preserve [15] the latter if his power, i.e., his essence, were corporeal.21

P3: It involves a contradiction that there should be a vacuum.

Dem.: By a vacuum is understood extension without corporeal substance [20] (by D5), i.e. (by P2), body without body, which is absurd.

For a fuller explanation, and to correct the prejudice about the vacuum, Principles II, 17-18 should be read. The main point there is that bodies [25] between which nothing lies must touch one another, and also that nothing has no properties.

[I/189] P4: One part of a body does not occupy a larger space at one time than at another; conversely, the same space does not contain more body at one time than at another.

[5] Dem.: Space and body do not really differ (P2C). So when we say that a space is not larger at one time than at another (by A13), we are saying thereby that a body cannot be larger, i.e., occupy a larger space, at one time than at another. That was the first thing to be [10] proven. Next, from the fact that space and body do not really differ, it follows that when we say that the body cannot occupy a larger space at one time than at another, we are saying thereby that the same space cannot contain more body at one time than at another.

[15] Cor.: Bodies which occupy equal space—say, gold and air—have just as much matter, or corporeal substance.

Dem.: Corporeal substance does not consist in the hardness of the [20] gold, nor in the softness of the air, nor in any of the sensible qualities (by P1); it consists rather in extension alone (by P2). But since (by Hypothesis) there is as much space, or (by D6) extension, in the one as in the other, there will also be as much corporeal substance, q.e.d.

[I/190] P5: There are no atoms.

Dem.: Atoms are parts of matter which are, by their own nature, [5] indivisible (by D3). But since the nature of matter consists in extension (by P2), which is, by its nature, divisible, however small it may be (by A9 and D7), a part of matter, however small, is, by its nature, divisible. I.e., there are no Atoms, or parts of matter which are, by [10] their nature, indivisible, q.e.d.

Schol.: The dispute about Atoms has always been an important and complicated one. Some maintain that there are Atoms on the ground that one infinite cannot be greater than another. If two quantities, A [15] and its double, should be divisible to infinity, they will also be able to be actually divided into infinitely many parts by the power of God, who understands their infinitely many parts in one intuition. Therefore, since, as has been said, one infinite is not greater than another, quantity A will be equal to its double, which is absurd. Again, they [20] also ask whether half of an infinite number is also infinite, whether it is even or odd, and the like.

To all these questions Descartes replies that we must not reject the things that fall under our intellect and therefore that we conceive clearly and distinctly, because of others that exceed our intellect, or grasp, [25] and that we therefore perceive only quite inadequately.22 But the infinite, and its properties, exceed the human understanding, which is finite by nature. So it would be foolish to reject as false, or to doubt, what we conceive clearly and distinctly about space, because we do [I/191] not comprehend the infinite. For this reason Descartes considers those things in which we do not perceive any limits—like the extension of the world, or the divisibility of parts of matter—as indefinite. Read Principles I, 26.

[5] P6: Matter is indefinitely extended and the matter of the heavens is one and the same as that of the earth.23

Dem.: (1) We cannot imagine any limits to extension, i.e. (by P2), [10] to matter, unless we conceive other spaces which follow immediately beyond them (by A10), i.e. (by D6), extension or matter, and that indefinitely. This was the first thing to be proven.

(2) The essence of matter consists in extension (by P2), and it is [15] indefinite (by part 1), i.e., cannot perceived by the human intellect under any limits (by D4). So (by A11) there is not more than one kind of matter, but it is one and the same everywhere. This was the second thing to be proven.

[20] Schol.: So far we have dealt with the nature, or essence, of extension. That it exists, created by God just as we conceive it, we have demonstrated in IP21. From IP12 it follows that it is now preserved by the same power by which it was created. We have also demonstrated [25] in IP21 that, insofar as we are thinking things, we are united to a part of that matter, with the aid of which we perceive that all those variations actually exist, which we know matter to be capable of from merely contemplating it. E.g., divisibility and local motion, or [I/192] the passage of one part from one place to another.24 We perceive local motion clearly and distinctly, provided that we understand that other parts of matter take the place of those that are moving.

We conceive this division and motion in infinite ways, and so can [5] conceive infinite variations of matter. I say that we conceive them clearly and distinctly so long as we conceive them as modes of extension, but not as things really distinct from extension. This is fully explained in Part I of the Principles.25 And though philosophers have [10] feigned many other motions, we who admit nothing we do not conceive clearly and distinctly, must admit no motion except local motion, since we clearly and distinctly understand that extension is not capable of any motion except local motion, nor can we even imagine any other motion.

[15] Zeno, they say, denied local motion, because of various arguments, which were refuted by Diogenes the Cynic in his fashion—i.e., by walking about the School in which Zeno was teaching these doctrines and thus disturbing those who were listening to Zeno. When he perceived that one of the listeners was holding him back to prevent his [20] walking, he reproached him, saying “How is it that you have thus dared to refute your master’s arguments?”26

Nevertheless, someone who is deceived by Zeno’s argument might think that the senses show us something (viz. motion) which the intellect finds absolutely contradictory, so that the mind would be deceived even about those things that it perceives clearly and distinctly [25] with the aid of the intellect. To prevent any such confusion, I shall set out here Zeno’s main arguments and show that they rest only on false prejudices, because he did not have a true concept of matter.

First, then, Zeno is reported to have said27 that if there were local motion, the motion of a body moving circularly with the greatest speed, [30] would not differ from rest. But the latter is absurd, so the former is also. He proves the consequence as follows. If all of the points of a body remain continuously in the same place, it is at rest. But all the points of a body moving circularly with the greatest speed remain continuously in the same place. Therefore, etc.

They say he explained this by using the example [I/193] of a wheel—say ABC. If it moves around its center with a certain speed, point A will complete the circle through B and C more quickly than it would if it [5] moved more slowly. Suppose that when it begins to move slowly, after an hour has passed it is in the same place as that from which it began. If it should move twice as quickly, it would be in that place after the passage of half an hour. If it should move four [10] times as quickly, then after a quarter of an hour. And if we conceive this speed to be increased to infinity and the time diminished to a moment, then when point A is at that greatest speed it will be at every moment, or continuously, in the place from which it began28 to be moved; so it will always remain in the same place. What we understand [15] to be the case concerning A, must also be understood to apply to all the points of this wheel. So all the points, when at that greatest speed, remain continuously in the same place.

To answer this argument, I must call attention to the fact that it is more an argument against the greatest speed of motion than against [20] motion itself. We shall not examine here whether Zeno argues rightly, but shall rather uncover his prejudices, on which this whole argument—insofar as he thinks that it attacks motion—rests. He supposes, first, that bodies can be conceived to move so quickly that they cannot [25] move more quickly, and second, that time is composed of moments, just as others have conceived that quantity is composed of indivisible points.

Both assumptions are false. For we can never conceive a motion so fast that we do not at the same time conceive a faster one. Our intellect [30] finds a contradiction in conceiving a motion so fast that there cannot be a faster one, no matter how short its course may be. The same is true of slowness. The concept of a motion so slow that there cannot be a slower one also implies a contradiction. We maintain the same thing about time, which is the measure of motion, viz. that our intellect [I/194] clearly finds a contradiction in conceiving a time so short that there cannot be a shorter one.

To prove all these assertions, let us follow in Zeno’s footsteps. Suppose, therefore, as he did, that a Wheel, ABC, moves about its center with such speed that point A is at [5] every moment in the place, A, from which it moves. I say that I clearly conceive a speed indefinitely faster [10] than this, and so moments infinitely less than these. For suppose that while the wheel, ABC, moves around its center, with the aid of a belt it makes another wheel, DEF, move around its center. Let DEF be [15] half the size of ABC. It is plain that in this case, DEF moves twice as fast as ABC, and consequently that at each half moment the point D is again in the place from which it began to move. Again, if we attribute the motion of DEF to ABC, DEF will move four times as fast as [20] [ABC did]29 before. And if we again ascribe this last speed of DEF to ABC, then DEF will move eight times faster, and so on, to infinity.

But this is clearest from the concept of matter alone. For we have proven that the essence of matter consists in extension, or space, which [25] is always divisible; and there is no motion without space. We have also demonstrated that one part of matter cannot occupy two spaces at the same time. For that would be the same as if we were to say that one part of matter is equal to its double, as is evident from what has been demonstrated above. So if a part of matter moves, it moves through [30] some space, which will be divisible, no matter how small it is feigned to be. Consequently the time by which that motion is measured will also be divisible, and the duration of that motion, or time, will be divisible, and this to infinity, q.e.d.

Let us go on now to the other sophism which he is said to have [I/195] used.30 Viz., if a body moves, it either moves in a place in which it is or in one in which it is not. But not in a place in which it is, for if it is somewhere, then it must be at rest. And not in a place in which it is not. Therefore, the body does not move.

[5] But this argument is just like the previous one. For it, too, supposes that there is a time than which no time is shorter. If we reply to the argument by saying that the body does not move in a place, but rather from the place in which it is to the place in which it is not, he will ask whether it has not been in the places in between. We may reply by drawing a distinction: if by has been he understands has rested, then [10] we deny that it has been anywhere while it was moving; but if by has been he means has existed, we say that, while it was moving, it must have existed.

Again, he will ask, where has it existed, while it was moving? We may reply once more: if by where has it existed? he means what place has [15] it stayed in? while it was moving, we say that it did not stay in any. But if he means what place has it changed? we say that it changed whatever places he might wish to assign in the space through which it was moving.

He will continue by asking whether it could, at the same moment of time, both occupy a place and change it. To this we will reply [20] finally, by drawing this distinction: if by a moment of time he understands a time than which none can be shorter, he asks something which has been adequately shown to be unintelligible and hence unworthy of an answer. But if he takes time in the sense I have explained above, i.e., in its true sense, he can never assign a time so short that, even [25] though an indefinitely shorter one might be supposed, a body could not both occupy and change its place. This is sufficiently plain to anyone who pays attention. So what we were saying above is now clearly evident—that he is supposing a time so short that there can not be a shorter one. Hence, he proves nothing.

[30] In addition to these two, still another argument of Zeno’s is commonly mentioned. This can be read, together with its refutation in the next to the last of Descartes’ Letters, Volume one.31

But here I should like my Readers to note that I have opposed my reasonings to Zeno’s reasonings, and therefore that I have refuted him [I/196] by reason, not by the senses, as Diogenes did. For the senses cannot provide anything else to one who is seeking the truth except the Phenomena of Nature, by which he is determined to investigate their causes. They can never show him that something is false that the [5] intellect has clearly and distinctly found to be true. For so we judge. And therefore, this is our Method: to demonstrate the things we put forward by reasons perceived clearly and distinctly by the intellect, and to regard as negligible whatever the senses say that seems contrary [10] to those reasons. As we have said, the senses can only determine the intellect to inquire into this matter rather than that one. They cannot convict it of falsity, when it has perceived something clearly and distinctly.

P7: No body enters the place of another unless at the same time that other body [15] enters the place of some other body.

Dem.: If you deny this, assume (if it can be done) that body Ab enters the place of body B, which, I also assume, is equal to A and does not yield its place. Therefore the space which used to contain [20] only B, now contains (by Hypothesis) both A and B. So it contains twice as much corporeal substance as before, which (by P4) is absurd. Therefore no body enters the place of another unless, etc., q.e.d.32

[25] P8: When any body enters the place of another, at the same moment of time the place left by it is occupied by another body which touches it immediately.

[I/197] Dem.: If body B moves toward D, then either bodies A and C will, at the same moment of time, approach and touch one another or they will not. If they should approach and touch one another, then what we have [5] maintained is conceded. But if they should not approach one another, then the whole space left by B would lie between A and C. Therefore, a body equal to B lies between A and C (by P2C and [10] P4C). But (by Hypothesis) the body is not identical with B. Therefore, it is another body which enters B’s place at the same moment of time. And since it enters at the same moment of time, it can be none other than one which is immediately touching. For in P6S we demonstrated [15] that there is no motion from one place to another which does not require a time such that there is always a shorter one. From this it follows that body B’s space cannot be occupied at the same moment of time by another body which would have to move through some space before it entered B’s place. So only a body which touches B [20] immediately enters its place at the same moment of time, q.e.d.

Schol.: Since the parts of matter are really distinct from one another (by Principles I, 61), one can exist without another (by IP7C), and they [25] do not depend on one another. So all those fictions about Sympathy and Antipathy are to be rejected as false. Moreover, since the cause of an effect must always be positive (by IA8), it should never be said that a body moves in order that there not be a vacuum. A body moves only on account of the impulse of another body.

[I/198] Cor.: In every motion a whole Circle of bodies moves at the same time.33

Dem.: At the same time that body 1 enters the [5] place of body 2, body 2 must enter the place of another body, say 3, and so on (by P7). Next, at the same moment of time at which body 1 enters the place of body 2, the place left behind by body 1 must be occupied by another [10] body (by P8), say 8, or another, which touches 1 immediately. Since this happens only on account of the impulse of another body (by P8S) which is here taken to be 1, not all of these bodies can be in motion in the same straight line (by A21); instead they describe a [15] whole circle (by D9), q.e.d.

P9: If a circular pipe, ABC, is full of water, and is four times as wide at A as at B, when the water (or other fluid body) which is at A begins to move [20] toward B, the water which is at B will move four times as fast.34

Dem.: Since all the water which is at A moves [25] toward B, at the same time just as much water must enter its place from C, which immediately touches A (by P8). And as much water will have to enter C’s place from B (by P8). Therefore (by A14), [30] it will move four times as quickly, q.e.d.

What we say here about a circular pipe must [I/199] also be understood to be true of all the unequal spaces through which bodies which move at the same time are forced to pass. For the demonstration will be the same in the other cases.

[5] Lemma: If two semicircles are described about the same center, as A and B are, the space between their perimeters will be equal everywhere. But if they are described about [10] different centers, as C and D are, the space between their perimeters will be unequal everywhere.

The demonstration is obvious from the definition of the circle alone.

P10: A fluid body moving through a pipe, ABC,c receives indefinite degrees [15] of speed.

Dem.: The space between A and B is unequal everywhere (by the preceding Lemma). So (by P9), the speed with which a fluid body moves through a pipe, ABC, will be unequal everywhere. Now, since [20] we conceive in thought indefinite spaces between A and B, which are always less and less (by P5), we shall also conceive the inequalities, which are everywhere, as indefinite. Hence (by P9), the degrees of speed will be indefinite, q.e.d.

[25] P11: There is a division into indefinite parts in the matter which flows through a pipe, ABC.d

[I/200] Dem.: The matter that flows through the pipe, ABC, acquires at the same time indefinite degrees of speed (by P10). Therefore (by A16), it has indefinite parts which are really divided, q.e.d. Read [5] Principles II, 34 and 35.35

Scholium: So far we have dealt with the nature of motion. Now it is necessary for us inquire into its cause, which is twofold. There is the primary, or general, cause, which is the cause of all the motions [10] that there are in the world, and there is the particular cause, by which it comes about that the individual parts of matter acquire motions that they did not have before. As far as the general cause is concerned, since nothing is to be admitted except what we perceive clearly and distinctly (by IP 14 and P15S), and since we do not clearly and distinctly [15] understand any other cause except God (i.e., the creator of matter), it is evident that no other general cause except God should be admitted.36 And what we say here about motion should also be understood to be true of rest.

[20] P12: God is the principal cause of motion.

Dem.: Examine the immediately preceding Scholium.

P13: God still preserves, by his concurrence, the same quantity of motion and [25] rest which he first imparted to matter.37

[I/201] Dem.: Since God is the cause of motion and of rest (by P12), he still preserves them by the same power by which he created them (by [5] IA10), and indeed, in that same quantity in which he first created them (by IP20C), q.e.d.

Schol.: (1) Although it may be said in Theology that God does many things from his good pleasure and to show his power to men, [10] nevertheless, since those things which depend only on his good pleasure do not become known except by divine revelation, they are not to be admitted in Philosophy, where we inquire only into what reason tells us, lest Philosophy be confused with Theology.38

(2) Although motion is nothing in the matter that moves but a mode [15] of it, nevertheless it has a certain determinate quantity. How this is to be understood will be evident from what follows. Read Principles II, 36.

P14: Each thing, insofar as it is simple, undivided, and considered in itself [20] alone, always perseveres in the same state as far as it can.39

This proposition is like an axiom to many; nevertheless, we shall demonstrate it.

[25] Dem.: Since nothing is in any state except by God’s concurrence alone (IP12) and God is supremely constant in his works (IP20C), if we attend to no external, i.e., particular causes, but consider the thing by itself, we shall have to affirm that insofar as it can it always perseveres [30] in the state in which it is, q.e.d.

[I/202] Cor.: Once a body moves, it always continues to move unless it is impeded by external causes.

[5] Dem.: This is obvious from P14. Nevertheless, to correct the prejudice about motion, read Principles II, 37-38.

P15: Every body in motion tends of itself to continue to move in a straight line, not in one which is curved.40

[10] It would be proper to number this proposition among the axioms, but I shall demonstrate it as follows from what has been shown above.

Dem.: Because the motion has only God as its cause (P12),41 it never [15] has any power to exist of itself (IA10), but is as it were created by God at every moment (by those things which are demonstrated concerning the axiom just cited). Hence, so long as we attend only to the nature of the motion, we shall never be able to attribute to it, as pertaining to its nature, a duration that can be conceived to be greater [20] than another. But if it should be said to pertain to the nature of a body in motion that it describes a curved line by its own motion, a greater duration would be attributed to the nature of the motion than when it is supposed to be of the nature of the moving body to tend to [25] continue to move in a straight line (A17). But since (as we have just demonstrated) we cannot attribute such a duration to the nature of the motion, neither can we assert that it is of the nature of the moving body to continue to move in a curved line, but only in a straight line, q.e.d.

[I/203] Scholium: To many, perhaps, it will seem that this Demonstration shows no more that it does not pertain to the nature of motion to describe a curved line, than that it does pertain to the nature of motion [5] to describe a straight line. For no straight line can be assigned such that there is none shorter, whether straight or curved, nor any curved line such that there is not also another shorter curve. But though I have considered these things, I still judge that the demonstration proceeds correctly. It reaches the conclusion proposed for demonstration [10] from the universal essence, or essential difference, of lines, not from the quantity of anything, or not from an accidental difference.

But to demonstrate a thing already clear enough in itself might make it more obscure. So I shall refer the Readers simply to the definition of motion [D8], which affirms nothing of motion except that it is the [15] transfer of one part of matter from the vicinity, etc., into the vicinity of others, etc. So unless we conceive this transfer most simply, i.e., as occurring in a straight line, we add something to the motion which is not contained in its definition, or essence. So it does not pertain to its nature.

[20] Cor.: From this proposition it follows that every body which moves in a curved line continuously deviates from the line along which it would, of itself, go on moving; and this occurs by the force of some external cause (P14).

[25] P16: Every body which moves in a circle, as for example, a stone in a sling, is continuously determined to go on moving along a tangent.42

[I/204] Dem.: A body that moves in a circle is continuously prevented by an external force from continuing to move in a straight line (by P15C). If this [5] force ceases, the body of itself will continue to move in a straight line (by P15). I say, moreover, that a body moving in a circle is determined by an external [10] cause to continue to move along a tangent. For, if you deny this, suppose there is a stone at B which is determined to move by a sling, not along the tangent BD, but along another line from the same point, [15] a line conceived as either outside the circle or within it—say, along BF, when the sling is assumed to come from L toward B, or along BG, if it is supposed to come from C toward B (I understand BG and BF to form equal angles with the line BH which is drawn from the [20] center of the circle and cuts the circumference at B). But if the stone at B is supposed to be determined by the sling, which is moving in a circle from L toward B, so that the stone continues to move toward F, then necessarily (by A18) when the sling moves in an opposite direction, from C toward B, the stone will be determined to continue [25] to move along the same line, BF, in an opposite direction. Hence, it will tend toward K, not toward G. But this is contrary to the hypothesis. And since, except for the tangent, there can be no line drawn through B which makes equal angles with the line BH (as DBH and [30] ABH do),e there can be none except the tangent which can preserve the hypothesis, whether the sling moves from L toward B or from C toward B. So there can be no line except the tangent along which it tends to move, q.e.d.

[I/205] Alt. Dem.: Conceive, instead of a circle, a Hexagon, ABH, inscribed in a circle, and a body [5] C at rest on one side, AB. Then conceive that a ruler, DBE (with one end fixed in the center [10] at D and the other end movable) moves around the center, continuously cutting the line AB. It is evident that if the ruler, DBE, while it is conceived [15] to move in this way, meets body C at the time when it cuts the line AB at right angles, the ruler will determine C by its impulse, to continue to move [20] toward G along the line FBAG, i.e., along the line AB extended indefinitely.

But because we have assumed a Hexagon at our pleasure, we shall have to say the same of any other figure which we conceive can be inscribed in this circle—i.e., that when a body, C, lying at rest on [25] one side of the figure, is struck by a ruler, DBE, at the same time DBE cuts the side at right angles, it will be determined by that ruler to continue to move along that side produced indefinitely. Conceive, then, instead of a Hexagon, a rectilinear figure of infinitely many sides [30] (i.e., a circle, according to Archimedes’ definition). It is evident that the ruler, DBE, whenever it meets C, always meets it at that time when it cuts some side of the figure at right angles. Hence it will never [I/206] meet C without determining it at the same time to continue to move along that side. And since any side, extended in any direction, must always fall outside the figure, this side, indefinitely extended, will be [5] tangent to a figure of infinitely many sides, i.e., a circle. Therefore, if we conceive, instead of a ruler, a sling moving in a circle, it will continuously determine this stone to go on moving along a tangent, q.e.d.

Here it should be noted that both of these demonstrations can be adapted to [10] any curved figure.

P17: Every body that moves in a circle strives to move away from the center of the circle that it describes.43

[15] Dem.: As long as a body moves in a circle, it is compelled by an external cause. When this ceases, it continues at the same time to move along a tangent (by P16), all of whose points, except that which [20] touches the circle, fall outside the circle (by Euclid’s Elements III P16), and so are further away from its center. So when a stone which moves in [25] a circle in the sling EA is at point A, it strives to continue along a line whose points are all further away from the center, E, than are all of the points of the circumference LAB. Doing this is nothing but striving to move away from the center of the circle [30] which it describes, q.e.d.

[I/207] P18: If a body, say A, moves toward another body at rest, B, and nevertheless B does not lose any of its rest on account of the impetus of A, then A will also [5] not lose any of its motion, but it will retain entirely the same quantity of motion that it had before.44

Dem.: If you deny this, suppose that body A loses some of its motion, and nevertheless does not transfer [10] what it has lost to another, say to B. Then, when this happens, there will be a lesser quantity of motion in nature than there was before, which is absurd (P13). The Demonstration proceeds in the same way with respect to the rest in body B. So if the one transfers nothing to the other, B will retain all its rest, and A will retain all its [15] motion, q.e.d.

P19: Motion, considered in itself, is different from its determination in some definite direction; nor is it necessary that a moving body be at rest for a time, [20] in order to move in an opposite direction, or be repelled.45

Dem.: Assume, as in the preceding proposition, that body A is moving in a straight line toward B, and is prevented by B from continuing further. Therefore (P18) A will retain its whole motion, and [25] will not be at rest for any interval of time, however small. Nevertheless, although it continues to move, it does not move in the same direction in which it was moving before. For it is supposed to be prevented by B. Hence since its motion remains intact, and its former [I/208] determination [in a certain direction] is lost, it will move in the opposite direction, and not in any other (by those things said in chap. 2 of the Dioptric). Hence (by A2) its determination [in a certain direction] does not pertain to the essence of the motion, but differs from it. Nor does a moving body, when it is repelled, remain at rest for some time, [5] q.e.d.

Cor.: From this it follows that motion is not contrary to motion.

P20: If body A meets body B, and takes B with it, A will lose as much motion [10] as B acquires because of its meeting with A.46

Dem.: If you deny this, suppose that B acquires more or less motion from A than A loses. That whole difference will have to be added to or subtracted from [15] the quantity of motion of the whole of nature, which is absurd (by P13). Since, therefore, body B can acquire neither more nor less motion, it will acquire as much as A loses, q.e.d.

[20] P21: If body A is twice as large as Bf and is moving with equal speed, A will also have twice as much motion as B, or twice as much force for retaining a speed equal to B’s.

[25] Dem.: Let twice B be put in place of A, i.e. (by Hypothesis), one A divided in two equal parts; each B has a force for remaining in the state in which it is (P14), and that force is equal in each of them (by [I/209] Hyp.); now if these two Bs are joined, their speed remaining the same, there will result one A, whose force and quantity will be equal to two Bs, or twice one B, q.e.d.

Note that this also follows simply from the definition of motion; for the greater a moving body is, the more matter there is which is separated from [5] other matter; there is, therefore, more separation, i.e. (by D8), more motion. See our fourth note on the definition of motion [I/182ff.].

P22: If body A is equal to body B, and A is moving twice as fast as B, the [10] force, or motion, in A will be twice that of B.g

Dem.: Assume that when B first acquired a certain force for moving, it acquired four degrees of speed. If nothing comes near it, it will [15] continue to move (by P14) and to persevere in its state. Suppose it again acquires, from some new impulse, another new force equal to the first, so that it acquires, in addition to the first four, four more degrees of speed, which it will also maintain (by P14). I.e., it will move twice as quickly as before, i.e., as quickly as A; and it will at [20] the same time have twice the force, i.e., a force equal to A’s. So the motion in A is twice that in B, q.e.d.

Note that here, by force in moving bodies, we understand a quantity of motion, which must be greater, in bodies of equal size, as the speed of motion is greater, insofar as the equal bodies are, by that speed, separated more, in the [25] same time, from bodies immediately touching them, than they would be if they were moving more slowly. Therefore (by D8) they also have more motion. But in bodies at rest we understand by force of resisting a quantity of rest.47 From which it follows that

[30] Cor. 1: The more bodies are moving slowly, the more they participate in [I/210] rest. For they offer more resistance to bodies moving more quickly which meet them and have less force than they. They are also separated less from bodies which touch them immediately.

[5] Cor. 2: If body A moves twice as quickly as body B, and B is twice as large as A, there is just as much motion in the larger B as in the smaller A; and hence there is also an equal force.

Dem.: Let B be twice as large as A, and A move twice as quickly [10] as B; next, let C be half as large as B, and move with half the speed of A. Then B (by P21) will have a motion twice as great as C’s, and A (by P22) will have a motion twice as great as C’s. Therefore (by A15) B and A have equal motion, for the motion of each is twice that [15] of the same third thing, C, q.e.d.

Cor. 3: From these it follows that motion is distinguished from speed. For we conceive that of bodies which have equal speed, one can have [20] more motion than another (P21); and on the other hand, those which have unequal speed can have equal motion (by P22C2). This can also be inferred just from the definition of motion. For it is nothing but the transfer of one body from the vicinity, etc.

But here it should be noted that this third Corollary is not inconsistent with [25] the first. For we conceive of speed in two ways: either insofar as a body is separated more or less in the same time from the bodies immediately touching it (and to that extent it participates more or less in motion or rest) or insofar as it describes a greater or lesser line in the same time (and to that extent it is distinguished from motion).

[I/211] I could have added other propositions here to explain P14 more fully, and to explain the forces of things in each state, as we have done here concerning motion. But it will be enough to read over Principles II, 43, and to annex one proposition which is necessary for understanding those which follow.

P23: When the modes of a body are forced to suffer variation, that variation will always be the least that there can be.

[10] Dem.: This proposition follows clearly enough from P14.

P24, Rule 1: If two bodies, A and B, absolutely equal, and moving in a straight line toward each other with equal speed,h meet, each will be reflected [15] in the opposite direction without losing any part of its speed.48

In this hypothesis it is clearly evident that, to remove the contrariety of these two bodies, either each of them must be reflected in an opposite direction, or one must take the other with it. For they are contrary to one another only with respect to their determination in a [20] certain direction, not with respect to their motion.

Dem.: When A and B meet, they must undergo some variation (by A19). But since motion is not contrary to motion (by P19C), they will [25] not be forced to lose any of their motion (by A19). Hence the change will occur only in the determination [in a certain direction]. But we cannot conceive that the determination of only one—say B—is changed, unless we should suppose that A, by which it would have to be changed, [I/212] has more force (by A20). But this would be contrary to the hypothesis. Therefore, since the change of determination in a certain direction cannot occur only in one, it will occur in each of them, with both A and B moving off in an opposite direction (but not in just any other [5] direction—according to what is said in chapter 2 of the Dioptric and retaining its motion intact, q.e.d.49

P25, Rule 2: If two bodies are unequal in bulk, B being larger than A, and the rest assumed to be as before,i then only A will be reflected, and each will [10] continue to move with the same speed.50

Dem.: Since A is supposed to be smaller than B, it will also have less force than B (P21). But since, in this hypothesis (as in the preceding one), there is contrariety only in the determination [in a certain [15] direction], then, as we demonstrated in P24, the variation must occur only in the determination [in a certain direction]. It will occur in A and not in B (A20). So only A will be reflected in the opposite direction by B, which has more force, with A retaining its speed intact, q.e.d.

[20] P26: If two bodies are unequal both in bulk and speed, B being twice as large as A, but the motion in A being twice as fast as that in B, and the rest assumed to be as before,j then each will be reflected in an opposite direction, [25] and each will retain the speed it had.

Dem.: When A and B move toward each other, according to the hypothesis, there is just as much motion in the one as in the other (P22C2), and51 the motion of the one will not be opposed to that of [I/213] the other (P19C), and the forces in each are equal (P22C2). Hence this hypothesis is just like the hypothesis of P24. Therefore, A and B are [5] reflected in an opposite direction, each retaining its motion intact (by the demonstration of P24), q.e.d.