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    Delphi Collected Works of René Descartes (Illustrated)
    Delphi Collected Works of René Descartes (Illustrated)
    Delphi Collected Works of René Descartes (Illustrated)
    Ebook863 pages15 hours

    Delphi Collected Works of René Descartes (Illustrated)

    By René Descartes

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    About this ebook

    The father of modern western philosophy, René Descartes formulated the first modern version of mind-body dualism, promoting the development of a new science grounded in observation and experiment. Applying an original system of methodical doubt, he dismissed apparent knowledge derived from the senses and reason, establishing a new epistemic foundation on the basis of intuition, expressed in the dictum: “I think, therefore I am” (Cogito, ergo sum). This comprehensive eBook presents Descartes’ collected works, with numerous illustrations, rare texts appearing in digital print for the first time, informative introductions and the usual Delphi bonus material. (Version 1)


    * Beautifully illustrated with images relating to Descartes’ life and works
    * Concise introductions to the treatises and other texts
    * All the major treatises, with individual contents tables
    * Features rare treatises appearing for the first time in digital publishing
    * Images of how the books were first published, giving your eReader a taste of the original texts
    * Excellent formatting of the texts
    * Features two biographies - discover Descartes’ literary life
    * Scholarly ordering of texts into chronological order


    Please visit www.delphiclassics.com to browse through our range of exciting titles


    CONTENTS:


    The Books
    RULES FOR THE DIRECTION OF THE MIND
    THE SEARCH FOR TRUTH
    THE WORLD
    DISCOURSE ON THE METHOD
    MEDITATIONS ON FIRST PHILOSOPHY
    SELECTIONS FROM ‘THE PRINCIPLES OF PHILOSOPHY’
    NOTES DIRECTED AGAINST A CERTAIN PROGRAMME
    PASSIONS OF THE SOUL


    The Biographies
    RENÉ DESCARTES by William Wallace
    BRIEF BIOGRAPHY: RENÉ DESCARTES by Clodius Piat


    Please visit www.delphiclassics.com to browse through our range of exciting titles or to purchase this eBook as a Parts Edition of individual eBooks


    LanguageEnglish
    PublisherPublishdrive
    Release dateJan 7, 2017
    ISBN9781786560643
    Delphi Collected Works of René Descartes (Illustrated)
    Author

    René Descartes

    René Descartes (1596–1650) was a French philosopher and mathematician. Considered the father of modern philosophy, Descartes is perhaps most famous for the dictum I think, therefore I am. This notion was a groundbreaking departure from the Aristotelian school of thought that had been dominant up to that point. Descartes is also credited with founding analytic geometry.

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      Book preview

      Delphi Collected Works of René Descartes (Illustrated) - René Descartes

      The Collected Works of

      RENÉ DESCARTES

      (1596-1650)

      Contents

      The Books

      RULES FOR THE DIRECTION OF THE MIND

      THE SEARCH FOR TRUTH

      THE WORLD

      DISCOURSE ON THE METHOD

      MEDITATIONS ON FIRST PHILOSOPHY

      SELECTIONS FROM ‘THE PRINCIPLES OF PHILOSOPHY’

      NOTES DIRECTED AGAINST A CERTAIN PROGRAMME

      PASSIONS OF THE SOUL

      The Biographies

      RENÉ DESCARTES by William Wallace

      BRIEF BIOGRAPHY: RENÉ DESCARTES by Clodius Piat

      The Delphi Classics Catalogue

      © Delphi Classics 2017

      Version 1

      The Collected Works of

      RENÉ DESCARTES

      By Delphi Classics, 2017

      COPYRIGHT

      Collected Works of René Descartes

      First published in the United Kingdom in 2017 by Delphi Classics.

      © Delphi Classics, 2017.

      All rights reserved.  No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of the publisher, nor be otherwise circulated in any form other than that in which it is published.

      With thanks to Jean Mahoney for permission to include the 2000 translation of ‘The World’ by Michael S. Mahoney, late Professor of the History of Science at Princeton University.

      ISBN: 978 1 78656 064 3

      Delphi Classics

      is an imprint of

      Delphi Publishing Ltd

      Hastings, East Sussex

      United Kingdom

      Contact: sales@delphiclassics.com

      www.delphiclassics.com

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      Then you’ll love these eBooks:

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      Explore Philosophy at Delphi Classics

      The Books

      Descartes’ birthplace in La Haye en Touraine

      Descartes was born in La Haye en Touraine (now Descartes, Indre-et-Loire), a commune in the Indre-et-Loire department in central France. It is approximately 13 miles east of Richelieu and about 24 miles east of Loudun, on the banks of the Creuse River.

      RULES FOR THE DIRECTION OF THE MIND

      Translated by Elizabeth Haldane and G. R. T. Ross

      In the late 1620’s Descartes began work on an unfinished treatise regarding the proper method for scientific and philosophical thinking, titled Regulae ad directionem ingenii, which outlined the basis for his later work on complex problems of mathematics, science and philosophy. In total thirty-six rules were planned, although only twenty-one were actually written. The treatise was not published during the author’s lifetime and first appeared in a Dutch translation in 1684, with the first Latin edition being published in 1701.

      The first 12 rules concern Descartes’ proposed scientific methodology in general; scholars now consider them to be early versions of principles that he expanded upon in his later writings.

      Portrait of René Descartes by Jan Baptist Weenix, c. 1649

      CONTENTS

      RULE I

      RULE II

      RULE III

      RULE IV

      RULE V

      RULE VI

      RULE VII

      RULE VIII

      RULE IX

      RULE X

      RULE XI

      RULE XII

      RULE XIII

      RULE XIV

      RULE XV

      RULE XVI

      RULE XVII

      RULE XVIII

      RULE XIX

      RULE XX

      RULE XXI

      RULE I

      The end of study should be to direct the mind towards the enunciation of sound and correct judgement on all matters that come before it.

      Whenever men notice some similarity between two things, they are wont to ascribe to each, even in those respects to which the two differ, what they have found to be true of the other. Thus they erroneously compare the sciences, which entirely exist in the cognitive exercise of the mind, with the arts, which depend upon an exercise and disposition of the body. They see that not all the arts can be acquired by the same man, but that he who restricts himself to one, most readily becomes the best executant, since it is not so easy for the same hand to adapt itself both to agricultural operations and to harp-playing, or to the performance of several such tasks as to one alone. Hence they have held the same to be true of the sciences also, and distinguishing them from one another according to their subject matter, they have imagined that they ought to be studied separately, each in isolation from all the rest. But this is certainly wrong. For since the sciences taken all together are identical with human wisdom, which always remains one and the same, however applied to different subjects, and suffers no more differentiation proceeding from them than the light of the sun experiences from the variety of the things which it illuminates, there is no need for minds to be confined at all within limits; for neither does the knowing of one truth have an effect like that of the acquisition of one art and prevent us from finding out another, it rather aids us to do so. Certainly it appears to me strange that so many people should investigate human customs with such care, the virtues of plants, the motions of the stars, the transmutations of metals, and the objects of similar sciences, while at the same time practically none bethink themselves about good understanding. Wisdom, though nevertheless all other studies are to be esteemed not so much for their own value as because they contribute something to this. Consequently we are justified in bringing forward this as the first rule of all, since there is nothing more prone to turn us aside from the correct way of seeking out truth than this directing of our inquiries, not towards their general end, but towards certain, special investigations. I do not here refer to perverse and censurable pursuits like empty glory or base gain; obviously counterfeit reasonings and quibbles suited to vulgar understanding open up a much more direct route to such a goal than does a sound apprehension of the truth. But I have in view even honourable and laudable pursuits, because these mislead us in a more subtle fashion. For example take our investigations of those sciences conducive to the conveniences of life or which yield that pleasure which is found in the contemplation of truth, practically the only joy in life that is complete and untroubled with any pain. There we may indeed expect to receive the legitimate fruits of scientific inquiry; but if, in the course of our study, we think of them, they frequently cause us to omit many facts which are necessary to the understanding of other matters, because they seem to be either of slight value or of little interest. Hence we must believe that all the sciences are so inter-connected, that it is much easier to study them all together than to isolate one from all the others. If, therefore, anyone wishes to search out the truth of things in serious earnest, he ought not to select one special science; for all the sciences are conjoined with each other and interdependent: he ought rather to think how to increase the natural light of reason, not for the purpose of resolving this or that difficulty of scholastic type, but that his understanding may light his will to its proper choice in all the contingencies of life. In a short time he will see with amazement that he has made much more progress than those who are eager about particular ends, and that he has not only obtained all that they desire, but even higher results than fall within his expectation.

      RULE II

      Only those objects should engage our attention, to the sure and indubitable knowledge of which our mental powers seem to be adequate.

      Science in its entirety is true and evident cognition. He is no more learned who has doubts on many matters than the man who has never thought of them; nay he appears to be less learned if he has formed wrong opinions on any particulars. Hence it were better not to study at all than to occupy one’s self with objects of such difficulty, that, owing to our inability to distinguish true from false, we are forced to regard the doubtful as certain; for in those matters, any hope of augmenting our knowledge is exceeded by the risk of diminishing it. Thus in accordance with the above maxim we reject all such merely probable knowledge and make it a rule to trust only what is completely known and incapable of being doubted. No doubt men of education may persuade themselves that there is but little of such certain knowledge, because, forsooth, a common failing of human nature has made them deem it too easy and open to everyone, and so led them to neglect to think upon such truths; but I nevertheless announce that there are more of these than they think — truths which suffice to give a rigorous demonstration of innumerable propositions, the discussion of which they have hitherto been unable to free from the element of probability. Further, because they have believed that it was unbecoming for a man of education to confess ignorance on any point, they have so accustomed themselves to trick out their fabricated explanations, that they have ended by gradually imposing on themselves and thus have issued them to the public as genuine.

      But if we adhere closely to this rule we shall find left but few objects of legitimate study. For there is scarce any question occurring in the sciences about which talented men have not disagreed. But whenever two men come to opposite decisions about the same matter one of them at least must certainly be in the wrong, and apparently there is not even one of them who knows; for if the reasoning of the second were sound and clear he would be able so to lay it before the other to succeed in convincing his understanding also. Hence apparently we cannot attain to a perfect knowledge in any such case of probable opinion, for it would be rashness to hope for more than others have attained to. Consequently if we reckon correctly, of the sciences already discovered, Arithmetic and Geometry alone are left, to which the observance of this rule reduces us.

      Yet we do not therefore condemn that method of philosophizing which others have already discovered, and those weapons of the schoolmen, probable syllogisms, which are so well suited for polemics. They indeed give practice to the wits of youth and, producing emulation among them, act as a stimulus; and it is much better for their minds to be moulded by opinions of this sort, uncertain though they appear, as being objects of controversy amongst the learned, than to be left entirely to their own devices. For thus through lack of guidance they might stray into some abyss, but as long as they follow in their masters’ footsteps, though they may diverge at times from the truth, they will yet certainly find a path which is at least in this respect safer, that it has been approved by more prudent people. We ourselves rejoice that we in earlier years experienced this scholastic training; but now, being released from that oath of allegiance which bound us to our old masters and since, as become our riper years, we are no longer subject to the ferule, if we wish in earnest to establish for ourselves those rules which shall aid us in scaling the heights of human knowledge, we must admit assuredly among the primary members of our catalogue that maxim which forbids us to abuse our leisure as many do, who neglect all easy quests and take up their time only with difficult matters; for they, though certainly making all sorts of subtle conjectures and elaborating most plausible arguments with great ingenuity, frequently find too late that after all their labours they have only increased the multitude of their doubts, without acquiring any knowledge whatsoever.

      But now let us proceed to explain more carefully our reason for saying , as we did a little while ago, that of all the sciences known as yet, Arithmetic and Geometry alone are free from any taint of falsity or uncertainty. We must note then that there are two ways by which we arrive at the knowledge of facts, viz. by experience and by deduction. We must further observe that while our inferences from experience are frequently fallacious, deduction, or the pure illation of one thing from another, though it may be passed over, if it is not seen through, cannot be erroneous when performed by an understanding that is in the least degree rational. And it seems to me that the operation is profited but little by those constraining bonds by means of which the Dialecticians claim to control human reason, though I do not deny that that discipline may be serviceable for other purposes. My reason for saying so is that none of the mistakes which men can make (men, I say, not beasts) are due to faulty inference; they are caused merely by the fact that we found upon a basis of poorly comprehended experiences, or that propositions are posited which are hasty and groundless.

      This furnishes us with an evident explanation of the great superiority in certitude of arithmetic and Geometry to other sciences. The former alone deal with an object so pure and uncomplicated, that they need make no assumptions at all which experience renders uncertain, but wholly consist in the rational deduction of consequences. They are on that account much the easiest and clearest of all, and possess an object such as we require, for in them it is scarce humanly possible for anyone to err except by inadvertence. And yet we should not be surprised to find that plenty of people of their own accord prefer to apply their intelligence to other studies, or to Philosophy. The reason for this is that every person permits himself the liberty of making guesses in the matter of an obscure subject with more confidence than in one which is clear, and that it is much easier to have some vague notion about any subject, no matter what, than to arrive at the real truth about a single question however simple that may be.

      But one conclusion now emerges out of these considerations, viz. not, indeed, that Arithmetic and Geometry are the sole sciences to be studied, but only that in our search for the direct road towards truth we should busy ourselves with no object about which we cannot attain a certitude equal to that of the demonstrations of Arithmetic and Geometry.

      RULE III

      In the subjects we propose to investigate, our inquiries should be directed, not to what others have thought, not to what we ourselves conjecture, but to what we can clearly and perspicuously behold and with certainty deduce; for knowledge is not won in any other way.

      TO STUDY the writings of ancients is right, because it is a great boon for us to be able to make use of the labours of so many men; and we should do so, both in order to discover what they have correctly made out in previous ages, and also that we may inform ourselves as to what in the various sciences is still left for investigation. But yet there is a great danger lest in a too absorbed study of these works we should become infected with their errors, guard against them as we may. For it is the way of writers, whenever they have allowed themselves rashly and credulously to take up a position in any controverted matter, to try with the subtlest arguments to compel us to go along with them. But when, on the contrary, they have happily come upon something certain and evident, in displaying it they never fail to surround it with ambiguities, fearing, it would seam, lest the simplicity of their explanation should make us respect their discovery less, or because they grudge us an open vision of the truth.

      Further, supposing now that all were wholly open and candid, and never thrust upon us doubtful opinions as true, but expounded every matter in good faith, yet since scarce anything has been asserted by any one man the contrary of which has not been alleged by another, we should be eternally uncertain which of the two to believe. It would be no use to total up the testimonies in favour of each, meaning to follow that opinion which was supported by the greater number of authors: for if it is a question of difficulty that is in dispute, it is more likely that the truth would have been discovered by few than by many. But even though all these men agreed among themselves, what they teach us would not suffice for us. For we shall not e.g. all turn out to be mathematicians though we know by heart all the proofs that other have elaborated, unless we have an intellectual talent that fits us to resolve difficulties of any kind. Neither, though we have mastered all the arguments of Plato and Aristotle, if yet we have not the capacity for passing a solid judgment on these matters, shall we become Philosophers; we should have acquired the knowledge not of science, but of history.

      I lay down the rule also, that we must wholly refrain from ever mixing up conjectures with our pronouncements on the truth of things. This warning is of no little importance. There is no stronger reason for our finding nothing in the current Philosophy which is so evident and certain as not to be capable of being controverted, than the fact that the learned, not content with the recognition of what is clear and certain, in the first instance hazard the assertion of obscure and ill-comprehended theories, at which they have arrived merely by probable conjecture. Then afterwards they gradually attach complete credence to them, and mingling them promiscuously with what is true and evident, they finish by being unable to deduce any conclusion which does not appear to depend upon some proposition of the doubtful source and hence is not uncertain.

      But lest we in turn should slip into the same error, we shall here take note of all those mental operations by which we are able, wholly without fear or illusion, to arrive at the knowledge of things. Now I admit only two, viz. intuition and induction. (Sense here seems to require deduction.) By intuition I understand, not the fluctuating testimony of the senses, not the misleading judgment that proceeds from the blundering constructions of imagination, but the conception which an unclouded and attentive mind gives us so readily and distinctly that we are wholly freed from doubt about that which we understand. Or, what comes to the same thing, intuition is the undoubting conception of an unclouded and attentive mind, and springs from the light of reason alone; it is more certain than deduction itself, in that it is simpler, though deduction, as we have noted above, cannot by us be erroneously conducted. Thus each individual can mentally have intuition of the fact that he exists, and that he thinks; that the triangle is bounded by three lines only, the sphere by a single superficies, and so on. Facts of such kind are far more numerous than many people think, disdaining as they do to direct their attention upon such simple matters.

      But in case anyone may be put out by this new use of the term intuition and of other terms which in the following pages I am similarily compelled to dissever from their current meaning. I here make the general announcement that I pay no attention to the way in which particular terms have of late been employed in the schools, because it would have been difficult to employ the same terminology while my theory was wholly different. All I take note of is the meaning of the Latin of each word, when, in cases where an appropriate term is lacking, I wish to transfer to the vocabulary that expresses my own meaning those that I deem most suitable.

      This evidence and certitude, however, which belongs to intuition, is required not only in the enunciation of propositions, but also in discursive reasoning of whatever sort. For example consider the consequence: 2 and 2 amount to the same 3 and 1. Now we need to see intuitively not only that 2 and 2 make 4, and that likewise 3 and 1 make 4, but further that the third of the above statements is a necessary conclusion from these two. Hence now we are in a position to raise the question as to why we have, besides intuition, given this supplementary method of knowing, viz. knowing by deduction by which we understand all necessary inference from other facts which are known with certainty. This, however, we could not avoid, because many things are known with certainty, though not by themselves evident, but only deduced from true and known principles by the continuous and uninterrupted action of a mind that has a clear vision of each step in the process. It is in a similar way that we know that the last link in a long chain is connected with the first, even though we do not take in by means of one and the same act of vision all the intermediate links on which that connection depends, but only remember that we have taken them successively under review and that each single one is united to its neighbor, from the first even to the last. Hence we distinguish this mental intuition from deduction by the fact that into the conception of the latter there enters a certain movement or succession, into that of the former there does not. Further deduction does not require an immediately presented evidence such as intuition possesses; its certitude is rather conferred upon it in some way by memory. The upshot of the matter is that it is possible to say that those propositions indeed which are immediately deduced from first principles are known now by intuition, now by deduction, i.e. in a way that differs according to our point of view. But the first principles themselves are given by intuition alone, while, on the contrary, the remote conclusions are furnished only by deduction.

      These two methods are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous. But this does not prevent us from believing matters that have been divinely revealed as being more certain than our surest knowledge, since belief in these things, as all faith in obscure matters, is an action, not of our intelligence, but of our will. They should be heeded also since, if they have any basis in our understanding, they can and ought to be, more than all things else, discovered by one of the ways abovementioned, as we hope perhaps to show at greater length on some future opportunity.

      RULE IV

      There is need of a method for finding out the truth.

      So blind is the curiosity by which mortals are possessed, that they often conduct their minds along unexplored routes, having no reason to hope for success, but merely being willing to risk the experiment of finding whether the truth they seek lies there. As well might a man burning with no unintelligent desire to find treasure, continuously roam the streets, seeking to find something that a passer by might have chanced to drop. This is the way in which most Chemists, many Geometricians, and Philosophers not a few prosecute their studies. I do not deny that sometimes in these wanderings they are lucky enough to find something true. But I do not allow that this argues greater industry on their part, but only better luck. But however that may be, it were far better never to think of investigating truth at all, than to do so without a method. For it is very certain that unregulated inquiries and confused reflections of this kind only confound the natural light and blind our mental powers. Those who so become accustomed to walk in darkness weaken their eye-sight so much that afterwards they cannot bear the light of day. This is confirmed by experience; for how often do we not see that those who have never taken to letters, give a sounder and clearer decision about obvious matters than those who have spent all their time in the schools? Moreover by a method I mean certain and simple rules, such that, if a man observe them accurately, he shall never assume what is false is true, and will never spend his mental efforts to no purpose, but will always gradually increase his knowledge and so arrive at a true understanding of all that does not surpass his powers.

      These two points must be carefully noted, viz. never to assume what is false as true, and to arrive at a knowledge which takes in all things. For, if we are without the knowledge of any of the things which we are capable of understanding, that is only because we have never perceived any way to bring us to this knowledge, or because we have fallen into the contrary error. But if our method rightly explains how our mental vision should be used, so as not to fall into the contrary error, and how deduction should be discovered in order that we may arrive at the knowledge of all things, I do not see what else is needed to make it complete; for I have already said that no science is acquired except by mental intuition or deduction. There is besides no question of extending it further in order to show how these said operations ought to be effected, because they are the most simple and primary of all. Consequently, unless our understanding were already able to employ them, it could comprehend none of the precepts of that very method, not even the simplest. Bus as for the other mental operations, which Dialectic does its best to direct by making use of these prior ones, they are quite useless here, rather they are to be accounted impediments, because nothing can be added to the pure light of reason which does not in some way obscure it.

      Since then the usefulness of this method is so great that without it study appears to be harmful than profitable, I am quite ready to believe that the greater minds of former ages had some knowledge of it, nature even conducting them to it. For the human mind has in it something that we may call divine, wherein are scattered the first germs of useful modes of thought. Consequently it often happens that however much neglected and choked by interfering studies they bear fruit of their own accord. Arithmetic and Geometry, the simplest sciences, give us an instance of this; for we have sufficient evidence that the ancient Geometricians made use of a certain analysis which they extended to the resolution of all problems, though they grudged the secret to posterity. At the present day also there flourishes a certain kind of arithmetic, called Algebra, which designs to effect, when dealing with numbers, what the ancients achieved in the matter of figures. These two methods are nothing else than the spontaneous fruit sprung from the inborn principles of the discipline here in question; and I do not wonder that these sciences with their very simple subject matter should have yielded results so much more satisfactory than others in which greater obstructions choke all growth. But even in the latter case, if only we take care to cultivate them assiduously, fruits will certainly be able to come to full maturity.

      This is the chief result which I have had in view in writing this treatise. For I should not think much of these rules, if they had no utility save for the solution of the empty problems with which Logicians and Geometers have been wont to beguile their leisure; my only achievement thus would have seemed to be an ability to argue about trifles more subtly than others. Further, though much mention is here made of numbers and figures, because no other sciences furnish us with illustrations of such self-evidence and certainty, the reader who follows my drift with sufficient attention will easily see that nothing is less in my mind than ordinary Mathematics, and that I am expounding quite another science, of which these illustrations are rather the outer husk than the constituents. Such a science should contain the primary rudiments of human reason, and its province ought to extend to the eliciting of true results in every subject. To speak freely, I am convinced that it is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others. But as for the outer covering I mentioned, I mean not to employ it to cover up and conceal my method for the purpose of warding of the vulgar; rather I hope so to clothe and embellish it that I may make it more suitable for presentation to the human mind.

      When first I applied my mind to Mathematics I read straight away most of what is usually given by the mathematical writers, and I paid special attention to Arithmetic and Geometry, because they were said to be the simplest and so to speak the way to all the rest. But in neither case did I then meet with authors who fully satisfied me. I did indeed learn in their works many propositions about numbers which I found on calculation to be true. As to figures, they in a sense exhibited to my eyes a great number of truths and drew conclusions from certain consequences. But they did not seem to make it sufficiently plain to the mind itself why those things are so, and how they discovered them. Consequently I was not surprised that many people, even of talent and scholarship, should, after glancing at these sciences, have either given them up as being empty and childish or, taking them to be very difficult and intricate, been deterred at the very outset from learning them. For really there is nothing more futile than to busy one’s self with bare numbers and imaginary figures in such a way as to appear to rest content with such trifles, and so to resort to those superficial demonstrations, which are discovered more frequently by chance than by skill, and are a matter more of the eyes and the imagination than of the understanding, that in a sense one ceases to make use of one’s reason. I might add that there is no more intricate task than that of solving by this method of proof new difficulties that arise, involved as they are with numerical confusions. But when I afterwards bethought myself how it could be that the earliest pioneers of Philosophy in bygone ages refused to admit to the study of wisdom any one who was not versed in Mathematics, evidently believing that this was the easiest and most indispensable mental exercise and preparation for laying hold of other more important sciences, I was confirmed in my suspicion that they had knowledge of a species of Mathematics very different from that which passes current in our time. I do not indeed imagine that they had a perfect knowledge of it, for they plainly show how little advanced they were by the insensate rejoicings they display and the pompous thanksgivings they offer for the most trifling discoveries. I am not shaken in my opinion by the fact that historians make a great deal of certain machines of theirs. Possibly these machines were quite simple, and yet the ignorant and wonder-loving multitude might easily have lauded them as miraculous. But I am convinced that certain primary germs of truth implanted by nature in human minds — though in our case the daily reading and hearing of innumerable diverse errors stifle them — had a very great vitality in that rude and unsophisticated age of the ancient world. Thus the same mental illumination which let them see that virtue was to be preferred to pleasure, and honour to utility, although they knew not why this was so, made them recognize true notions in Philosophy and Mathematics, although they were not yet able thoroughly to grasp these sciences. Indeed I seem to recognize certain traces of this true Mathematics in Pappus and Diophantus, who though not belonging to the earliest age, yet lived many centuries before our own times. But my opinion is that these writers then with a sort of low cunning, deplorable indeed, suppressed this knowledge. Possibly they acted just as many inventors are known to have done in the case of their discoveries, i.e. they feared that their method being so easy and simple would become cheapened on being divulged, and they preferred to exhibit in its place certain barren truths, deductively demonstrated with show enough of ingenuity, as the results of their art, in order to win from us our admiration for these achievements, rather than to disclose to us that method itself which would have wholly annulled the admiration accorded. Finally there have been certain men of talent who in the present age have tried to revive this same art. For it seems to be precisely that science known by the barbarous name Algebra if only we could extricate it from that vast array of numbers and inexplicable figures by which it is overwhelmed, so that it might display the clearness and simplicity which, we imagine ought to exist in a genuine Mathematics.

      It was these reflections that recalled me from the particular studies of Arithmetic and Geometry to a general investigation of Mathematics. and thereupon I sought to determine what precisely was universally meant by that term, and why not only the above mentioned sciences, but also Astronomy, Music, Optics, Mechanics and several others are styled parts of Mathematics. Here indeed it is not enough to look at the origin of the word; for since the name Mathematics means exactly the same thing as scientific study, these other branches could, with as much right as Geometry itself, be called Mathematics. Yet we see that almost anyone who has had the slightest schooling, can easily distinguish what relates to Mathematics in any question from that which belongs to the other sciences. But as I considered the matter carefully it gradually came to light that all those matters only were referred to Mathematics in which order and measurement are investigated, and that it makes no difference whether it be in numbers, figures, stars, sounds or any other object that the question of measurement arises. I saw consequently that there must be some general science to explain that element as a whole which gives rise to problems about order and measurement, restricted as these are to no special subject matter. This, I perceived, was called Universal Mathematics, not a far fetched designation, but one of long standing which has passed into current use, because in this science is contained everything on account of which the others are called parts of Mathematics. We can see how much it excels in utility and simplicity the sciences subordinate to it, by the fact that it can deal with all the objects of which they have cognizance and many more besides, and that any difficulties it contains are found in them as well, added to the fact that in them fresh difficulties arise due to their special subject matter which in it do not exist. But now how comes it that though everyone knows the name of this science and understands what is its province even without studying it attentively, so many people laboriously pursue the other dependent sciences, and no one cares to master this one? I should marvel indeed were I not aware that everyone thinks it to be so very easy, and had I not long since observed that the human mind passes over what it thinks it can easily accomplish, and hastens straight away to new and more imposing occupations.

      I, however, conscious as I am of my inadequacy, have resolved that in my investigation into truth I shall follow obstinately such an order as will require me first to start with what is simplest and easiest, and never permit me to proceed farther until in the first sphere there seems to be nothing further to be done. This is why up to the present time to the best of my ability I have made a study of this universal Mathematics; consequently, I believe that when I go on to deal in their turn with more profound sciences, as I hope to do soon, my efforts will not be premature. But before I make this transition I shall try to bring together and arrange in an orderly manner, the facts which in my previous studies I have noted as being more worthy of attention. Thus I hope both that at a future date, when through advancing years my memory is enfeebled, I shall, if need be, conveniently be able to recall them by looking in this little book, and that having now disburdened my memory of them I may be free to concentrate my mind on my future studies.

      RULE V

      Method consists entirely in the order and disposition of the objects towards which our mental vision must be directed if we would find out any truth. We shall comply with it exactly if we reduce involved and obscure propositions step by step to those that are simpler, and then starting with the intuitive apprehension of all those that are absolutely simple, attempt to ascend to the knowledge of all others by precisely similar steps.

      IN THIS alone lies the sum of all human endeavour, and he who would approach the investigation of truth must hold to this rule as closely as he who enters the labyrinth must follow the thread which guided Theseus. But many people either do not reflect on the precept at all, or ignore it altogether, or presume not to need it. Consequently, they often investigate the most difficult questions with so little regard to order, that, to my mind, they act like a man who should attempt to leap with one bound from the base to the summit of a house, either making no account of the ladders provided for his ascent or not noticing them. It is thus that all Astrologers behave, who, though in ignorance of the nature of the heavens, and even without having made proper observations of the movements of the heavenly bodies, expect to be able to indicate their effects. This is also what many do who study Mechanics apart from Physics, and rashly set about devising new instruments for producing motion. Along with them go also those Philosophers who, neglecting experience, imagine that truth will spring from their brain like Pallas from the head of Zeus.

      Now it is obvious that all such people violate the present rule. But since the order here required is often so obscure and intricate that not everyone can make it out, they can scarcely avoid error unless they diligently observe what is laid down in the following proposition.

      RULE VI

      In order to separate out what is quite simple from what is complex, and to arrange these matters methodically, we ought, in the case of every series in which we have deduced certain facts the one from the other, to notice which fact is simple, and to mark the interval, greater, less, or equal, which separates all the others from this.

      ALTHOUGH this proposition seems to teach nothing very new, it contains, nevertheless, the chief secret of method, and none in the whole of this treatise is of greater utility. For it tells us that all facts can be arranged in certain series, not indeed in the sense of being referred to some ontological genus such as the categories employed by Philosophers in their classification, but in so far as certain truths can be known from others; and thus, whenever a difficulty occurs we are able at once to perceive whether it will be profitable to examine certain others first, and which, and in what order.

      Further, in order to do that correctly, we must note first that for the purpose of our procedure, which does not regard things as isolated realities, but compares them with one another in order to discover the dependence in knowledge of one upon the other, all things can be said to be either absolute or relative.

      I call that absolute which contains within itself the pure and simple essence of which we are in quest. Thus the term will be applicable to whatever is considered as being independent, or a cause, or simple, universal, one, equal, like, straight, and so forth; and the absolute I call the simplest and the easiest of all, so that we can make use of it in the solution of questions.

      But the relative is that which, while participating in the same nature, or at least sharing in it to some degree which enables us to relate it to the absolute and to deduce it from that by a chain of operations, involves in addition something else in its concept which I call relativity. Examples of this are found in whatever is said to be dependent, or an effect, composite, particular, many, unequal, unlike, oblique, etc. These relatives are the further removed from the absolute, in proportion as they contain more elements of relativity subordinate the one to the other. We state in this rule that these should all be distinguished and their correlative connection and natural order so observed, that we may be able by traversing all the intermediate steps to proceed from the most remote to that which is in the highest degree absolute.

      Herein lies the secret of this whole method, that in all things we should diligently mark that which is most absolute. For some things are from one point of view more absolute than others, but from a different standpoint are more relative. Thus though the universal is more absolute than the particular because its essence is simpler, yet it can be held to be more relative than the latter, because it depends upon individuals for its existence, and so on. Certain things likewise are truly more absolute than others, but yet are not the most absolute of all. Thus relatively to individuals, species is something absolute, but contrasted with genus it is relative. So too, among things that can be measured, extension is something absolute, but among the various aspects of extension it is length that is absolute, and so on. Finally also, in order to bring out more clearly that we are considering here not the nature of each thing taken in isolation, but the series involved in knowing them, we have purposely enumerated cause and equality among our absolutes, though the nature of these terms is really relative. For though Philosophers make cause and effect correlative, we find that here even, if we ask what the effect is, we must first know the cause and not conversely. Equals too mutually imply one another, but we can know unequals only by comparing them with equals and not per contra.

      Secondly, we must note that there are but few pure and simple essences, which either our experiences or some sort of light innate in us enable us to behold as primary and existing per se, not as depending on any others. These we say should be carefully noticed, for they are just those facts which we have called the simplest in any single series. All the others can only be perceived as deductions from these, either immediate and proximate, or not to be attained save by two or three or more acts of inference. The number of these acts should be noted in order that we may perceive whether the facts are separated from the primary and simplest proposition by a greater or smaller number of steps. And so pronounced is everywhere the inter-connection of ground and consequence, which gives rise, in the objects to be examined, to those series to which every inquiry must be reduced, that it can be investigated by a sure method. But because it is not easy to make a review of them all, and besides, since they have not so much to be kept in the memory as to be detected by a sort of mental penetration, we must seek for something which will so mould our intelligence as to let it perceive these connected sequences immediately whenever it needs to do so. For this purpose I have found nothing so effectual as to accustom ourselves to turn our attention with a sort of penetrative insight on the very minutest of the facts which we have already discovered. Finally, we must in the third place note that our inquiry ought not to start with the investigation of difficult matters. Rather, before setting out to attack any definite problem, it behoves us first, without making any selection, to assemble those truths that are obvious as they present themselves to us, and afterwards, proceeding step by step, to inquire whether any others can be deduced from these, and again any others from these conclusions and so on, in order. This done, we should attentively think over the truths we have discovered and mark with diligence the reasons why we have been able to detect some more easily than others, and which these are. Thus, when we come to attack some definite problem we shall be able to judge what previous questions it were best to settle first. For example, if it comes into my thought that the number 6 is twice 3,1 may then ask what is twice 6, viz. 12; again, perhaps I seek for the double of this, viz. 24, and again of this, viz. 48. Thus I may easily deduce that there is the same proportion between 3 and 6, as between 6 and 12, and likewise 12 and 24, and so on, and hence that the numbers 3, 6, 12, 24, 48, etc. are in continued proportion. But though these facts are all so clear as to seem almost childish, I am now able by attentive reflection to understand what is the form involved by all questions that can be propounded about the proportions or relations of things, and the order in which they should be investigated; and this discovery embraces the sum of the entire science of Pure Mathematics. For first I perceive that it was not more difficult to discover the double of six than that of three; and that equally in all cases, when we have found a proportion between any two magnitudes, we can find innumerable others which have the same proportion between them. So, too, there is no increase of difficulty, if three, or four, or more of such magnitudes are sought for, because each has to be found separately and without any relation to the others. But next I notice that though, when the magnitudes 3 and 6 are given, one can easily find a third in continued proportion, viz. 12, it is yet not equally easy, when the two extremes, 3 and 12, are given, to find the mean proportional, viz. 6. When we look into the reason for this, it is clear that here we have a type of difficulty quite different from the former; for, in order to find the mean proportional, we must at the same time attend to the two extremes and to the proportion which exists between these two in order to discover a new ratio by dividing the previous one; and this is a very different thing from finding a third term in continued proportion with two given numbers. I go forward likewise and examine whether, when the numbers 3 and 24 were given, it would have been equally easy to determine one of the two intermediate proportionals, viz. 6 and 12. But here still another sort of difficulty arises more involved than the previous ones, for on this occasion we have to attend not to one or two things only but to three, in order to discover the fourth. We may go still further and inquire whether if only 3 and 48 had been given it would have been still more difficult to discover one of the three mean proportionals, viz. 6,12, and 24. At the first blush this indeed appears to be so; but immediately afterwards it comes to mind that this difficulty can be split up and lessened, if first of all we ask only for the mean proportional between 3 and 48, viz. 12, and then seek for the other mean proportional between 3 and 12, viz. 6, and the other between 12 and 48, viz. 24. Thus, we have reduced the problem to the difficulty of the second type shown above.

      These illustrations further lead me to note that the quest for knowledge about the same thing can traverse different routes, the one much more difficult and obscure than the other. Thus, to find these four continued proportionals, 3, 6, 12, and 24, if two consecutive numbers be assumed, e.g. 3 and 6, or 6 and 12, or 12 and 24, in order that we may discover the others, our task will be easy. In this case we shall say that the proposition to be discovered is directly examined. But if the two numbers given are alternates, like 3 and 12, or 6 and 24, which are to lead us to the discovery of the others, then we shall call this an indirect investigation of the first mode. Likewise, if we are given two extremes like 3 and 24, in order to find out from these the intermediates 6 and 12, the investigation will be indirect and of the second mode. Thus I should be able to proceed further and deduce many other results from this example ; but these will be sufficient, if the reader follows my meaning when I say that a proposition is directly deduced, or indirectly, and will reflect that from a knowledge of each of these matters that are simplest and primary, much may be discovered in other sciences by those who bring to them attentive thought and a power of sagacious analysis.

      RULE VII

      If we wish our science to be complete, those matters which promote the end we have in view must one and all be scrutinized by a movement of thought which is continuous and nowhere interrupted; they must also be included in an enumeration which is both adequate and methodical.

      IT is necessary to obey the injunctions of this rule if we hope to gain admission among the certain truths for those which, we have declared above, are not immediate deductions from primary and self-evident principles. For this deduction frequently involves such a long series of transitions from ground to consequent that when we come to the conclusion we have difficulty in recalling the whole of the route by which we have arrived at it. This is why I say that there must be a continuous movement of thought to make good this weakness of the memory. Thus, e.g. if I have first found out by separate mental operations what the relation is between the magnitudes A and B, then what between B and C, between C and D, and finally between D and E, that does not entail my seeing what the relation is between A and E, nor can the truths previously learnt give me a precise knowledge of it unless I recall them all. To remedy this I would run them over from time to time, keeping the imagination moving continuously in such a way that while it is intuitively perceiving each fact it simultaneously passes on to the next; and this I would do until I had learned to pass from the first to the last so quickly, that no stage in the process was left to the care of the memory, but I seemed to have the whole in intuition before me at the same time. This method will both relieve the memory, diminish the sluggishness of our thinking, and definitely enlarge our mental capacity.

      But we must add that this movement should nowhere be interrupted. Often people who attempt to deduce a conclusion too quickly and from remote principles do not trace the whole chain of intermediate conclusions with sufficient accuracy to prevent them from passing over many steps without due consideration. But it is certain that wherever the smallest link is left out the chain is broken and the whole of the certainty of the conclusion falls to the ground.

      Here we maintain that an enumeration [of the steps in a proof] is required as well, if we wish to make our science complete. For resolving most problems other precepts are profitable, but enumeration alone will secure our always passing a true and certain judgment on whatsoever engages our attention; by means of it nothing at all will escape us, but we shall evidently have some knowledge of every step. This enumeration or induction is thus a review or inventory of all those matters that have a bearing on the problem raised, which is so thorough and accurate that by its means we can clearly and with confidence conclude that we have omitted nothing by mistake. Consequently as often as we have employed it, if the problem defies us, we shall at least be wiser in this respect, viz. that we are quite certain that we know of no way of resolving it. If it chances, as often it does, that we have been able to scan all the routes leading to it which lie open to the human intelligence, we shall be entitled boldly to assert that the solution of the problem lies outside the reach of human knowledge.

      Furthermore, we must note that by adequate enumeration or induction is only meant that method by which we may attain surer conclusions than by any other type of proof, with the exception of simple intuition. But when the knowledge of some matter cannot be reduced to this, we must cast aside all syllogistic fetters and employ induction, the only method left us, but one in which all confidence should be reposed. For whenever single facts have been immediately deduced the one from the other, they have been already reduced, if the inference was evident, to a true intuition. But if we infer any single thing from various and disconnected facts, often our intellectual capacity is not so great as to be able to embrace them all in a single intuition; in which case our mind should be content with the certitude attaching to this operation. It is in precisely similar fashion that though we cannot with one single gaze distinguish all the links of a lengthy chain, yet if we have seen the connection of each with its neighbour, we shall be entitled to say that we have seen how the first is connected with the last.

      I have declared that this operation ought to be adequate because it is often in danger of Mire defective and consequently exposed to Tor. For sometimes, even though in our enumeration we scrutinize many facts which are highly evident, yet if we omit the smallest step the chain is broken and the whole of the certitude of the conclusion falls to the ground. Sometimes also, even though all the facts are included in an accurate enumeration, the single steps are not distinguished from one another, and our knowledge of them all is thus only confused. Further, while now the enumeration ought be complete, now distinct, there are times when it need have neither of these characters; it was for this reason that I said only that it should be adequate. For if I want to prove by enumeration how many genera there are of corporeal things, or of those that in any way fa11 under the senses, I shall not assert that they are just so many and no more, unless I previously have become aware that I have included them all in my enumeration, and have distinguished them each separately from all the others. But if in the same way I wish to prove that the rational soul is not corporeal, I do not need a complete enumeration; it will be sufficient to include all bodies in certain collections in such a way as to be able to demonstrate that the rational soul has nothing to do with any of these. If, finally, I wish to show by enumeration that the area of a circle is greater than the area of all other figures whose perimeter is equal, there is no need for me to call in review all other figures; it is enough to demonstrate this of certain others in particular, in order to get thence by induction the same conclusion about all the others.

      I added also that the enumeration ought to be methodical. This is both because we have no more serviceable remedy for the defects already instanced, than to scan all things in an orderly manner; and also because it often happens that if each single matter which concerns the quest in hand were to be investigated separately, no man’s life would be long enough for the purpose, whether because they are far too many, or because it would chance that the same things had to be repeated too often. But if all these facts are arranged in the best order, they will for the most part be reduced to determinate classes, out of which it will be sufficient to take one example for exact inspection, or some one feature in a single case, or certain things rather than others, or at least we shall never have to waste our time in traversing the same ground twice. The advantage of this course is so great that often many particulars can, owing to a well devised arrangement, be gone over in a short space of time and with little trouble, though at first view the matter looked immense.

      But this order which we employ in our enumerations can for the most part be varied and depends upon each man’s judgment. For this reason, if we would elaborate it in our thought with greater penetration, we must remember what was said in our fifth proposition. There are also many of the trivial things of man’s devising, in the discovery of which the whole method lies in the disposal of this order. Thus if you wish to construct a perfect anagram by the transposition of the letters of a name, there is no need to pass from the easy to the difficult, nor to distinguish absolute from relative. Here there is no place for these operations; it will be sufficient to adopt an order to be followed in the transpositions of the letters which we are to examine, such that the same arrangements are never handled twice over. The total number of transpositions should, e.g. be split up into definite classes, so that it may immediately appear in which there is the best hope of finding what is sought. In this way the task is often not tedious but merely child’s play.

      However, these three propositions should not be separated, because for the most part we have to think of them together, and all equally

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