Rules for the Direction of the Mind

Translator's preface

Rules for the Direction of the Mind

Rule Onerule1

Rule Tworule2

Rule Three

Rule Four

Rule Five

Rule Six

Rule Seven

Rule Eight

Rule Nine

Rule Ten

Rule Eleven

Rule Twelve

Rule Thirteen

Rule Fourteen

Rule Fifteen

Rule Sixteen

Rule Seventeen

Rule Eighteen

Rule Nineteen

Rule Twenty

Rule Twenty-one

Appendix

Rules for the Direction of the Mind

Translator's preface

Descartes' Rules for the Direction of the Mind (Regulae ad Directionem Ingenii) was written in Latin, probably in 1628 or a few years earlier, but was not published during the author's lifetime. A Dutch translation of the work appeared in Holland in 1684, and the first Latin edition was published in Amsterdam by P. and J. Blaeu in 1701.^1

In the inventory of Descartes' papers made at Stockholm shortly after his death in 1650 the work is listed as 'Nine notebooks bound together, containing part of a Treatise on clear and useful Rules for the Direction of the Mind in the Search for Truth'. The original manuscript, which is lost, passed to Claude Clerselier, one of Descartes' staunchest supporters, who showed the work to several scholars, including Antoine Arnauld. The manuscript was seen also by Adrien Baillet, Descartes' biographer, who gave a summary of its contents in his La Vie de Monsieur Des-Cartes (1691). Leibniz bought a copy of the original manuscript in Amsterdam in 1670, and this copy has survived among the Leibniz papers in the Royal Public Library at Hanover.

The Rules was originally intended to contain three parts, each comprising twelve rules. The second set of twelve rules is incomplete, ending at Rule Twenty-one, and only the headings of Rules Nineteen to Twenty-one are given. The final set of twelve Rules is entirely missing; it appears that Descartes left this project unfinished. The first twelve Rules are concerned with simple propositions and the two cognitive operations by means of which they are known, intuition and deduction. The second set deal with what Descartes calls 'perfectly understood problems', i.e. problems in which the object of inquiry is a unique function of the data and which can be expressed in the forms of equations. Problems of this sort are confined largely to the sphere of mathematics. The projected third set of Rules would have dealt with 'imperfectly understood problems', i.e. problems which, owing to the multiplicity of the data involved, resist expression in the form of an equation; problems of this sort are prominent in the empirical sciences. Descartes had intended to show how imperfectly understood problems can be reduced to perfectly understood ones.

The present translation is based primarily on the text in Volume X of Adam and Tannery.^1 There are differences of detail between the Amsterdam edition of 1701 and the Hanover manuscript; they were probably based on different copies of the original manuscript. Where the two texts differ, the 1701 edition in most cases provides the better reading, and Adam and Tannery generally follow this text. In several instances, however, readings other than those adopted by Adam and Tannery have been preferred in the present translation; these are described in footnotes when the variants are not given in Adam and Tannery, or when neither of the alternative variants yields an obviously preferable reading. The critical edition of Giovanni Crapulli ^2 has been a useful supplement to Adam and Tannery, and several of Crapulli's readings have been adopted.

In the footnotes the Amsterdam edition of 1701 is referred to as A, and the Hanover manuscript as H.

D.M.

RULES FOR THE DIRECTION OF THE MIND

Rule One

The aim of our studies should be to direct the mind with a view to forming true and sound judgements about whatever comes before it.

Whenever people notice some similarity between two things, they are in the habit of ascribing to the one what they find true of the other, even when the two are not in that respect similar. Thus they wrongly compare the sciences, which consist wholly in knowledge acquired by the mind, with the arts, which require some bodily aptitude and practice. They recognize that one man cannot master all the arts at once and that it is easier to excel as a craftsman if one practises only one skill; for one man cannot turn his hand to both farming and harp-playing, or to several different tasks of this kind, as easily as he can to just one of them. This has made people come to think that the same must be true of the sciences as well. Distinguishing the sciences by the differences in their objects, they think that each science should be studied separately, without regard to any of the others. But here they are surely mistaken. For the sciences as a whole are nothing other than human wisdom, which always remains one and the same, however different the subjects to which it is applied, it being no more altered by them than sunlight is by the variety of the things it shines on. Hence there is no need to impose any restrictions on our mental powers; for the knowledge of one truth does not, like skill in one art, hinder us from discovering another; on the contrary it helps us. Indeed, it seems strange to me that so many people should investigate with such diligence the virtues of plants,^1 the motions of the stars, the transmutations of metals, and the objects of similar disciplines, while hardly anyone gives a thought to good sense - to universal wisdom. For every other science is to be valued not so much for its own sake as for its contribution to universal wisdom. Hence, we have reason to propose this as our very first rule, since what makes us stray from the correct way of seeking the truth is chiefly our ignoring the general end of universal wisdom and directing our studies towards some particular ends. I do not mean vile and despicable ends such as empty glory or base gain: specious arguments and tricks suited to vulgar minds clearly provide a much more direct route to these ends than a sound knowledge of the truth could provide. I have in mind, rather, respectable and commendable ends, for these are often more subtly misleading - ends such as the pursuit of sciences conducive to the comforts of life or to the pleasure to be gained from contemplating the truth, which is practically the only happiness in this life that is complete and untroubled by any pain. We can indeed look forward to these legitimate fruits of the sciences; but if we think of them during our studies, they frequently cause us to overlook many items which are required for a knowledge of other things, because at first glance they seem of little use or of little interest. It must be acknowledged that all the sciences are so closely interconnected that it is much easier to learn them all together than to separate one from the other. If, therefore, someone seriously wishes to investigate the truth of things, he ought not to select one science in particular, for they are all interconnected and interdependent. He should, rather, consider simply how to increase the natural light of his reason, not with a view to solving this or that scholastic problem, but in order that his intellect should show his will what decision it ought to make in each of life's contingencies. He will soon be surprised to find that he has made far greater progress than those who devote themselves to particular studies, and that he has achieved not only everything that the specialists aim at but also goals far beyond any they can hope to reach.

Rule Two

We should attend only to those objects of which our minds seem capable of having certain and indubitable cognition.

All knowledge ^1 is certain and evident cognition. Someone who has doubts about many things is no wiser than one who has never given them a thought; indeed, he appears less wise if he has formed a false opinion about any of them. Hence it is better never to study at all than to occupy ourselves with objects which are so difficult that we are unable to distinguish what is true from what is false, and are forced to take the doubtful as certain; for in such matters the risk of diminishing our knowledge is greater than our hope of increasing it. So, in accordance with this Rule, we reject all such merely probable cognition and resolve to believe only what is perfectly known and incapable of being doubted. Men of learning are perhaps convinced that there is very little indubitable knowledge, since, owing to a common human failing, they have disdained to reflect upon such indubitable truths, taking them to be too easy and obvious to everyone. But there are, I insist, a lot more of these truths than such people think - truths which suffice for the sure demonstration of countless propositions which so far they have managed to treat as no more than probable. Because they have thought it unbecoming for a man of learning to admit to being ignorant on any matter, they have got so used to elaborating their contrived doctrines that they have gradually come to believe them and to pass them off as true.

Nevertheless, if we adhere strictly to this Rule, there will be very few things which we can get down to studying. For there is hardly any question in the sciences about which clever men have not frequently disagreed. But whenever two persons make opposite judgements about the same thing, it is certain that at least one of them is mistaken, and neither, it seems, has knowledge. For if the reasoning of one of them were certain and evident, he would be able to lay it before the other in such a way as eventually to convince his intellect as well. Therefore, concerning all such matters of probable opinion we can, I think, acquire no perfect knowledge, for it would be presumptuous to hope that we could gain more knowledge than others have managed to achieve. Accordingly, if my reckoning is correct, out of all the sciences so far devised, we are restricted to just arithmetic and geometry if we stick to this Rule.

Yet I do not wish on that account to condemn that method of philosophizing which others have hitherto devised, nor those weapons of the schoolmen, probable syllogisms,^1 which are just made for controversies. For these exercise the minds of the young, stimulating them with a certain rivalry; and it is much better that their minds should be informed with opinions of that sort - even though they are evidently uncertain, being controversial among the learned - than that they should be left entirely to their own devices. Perhaps without guidance they might head towards a precipice, but so long as they follow in their masters' footsteps (though straying at times from the truth), they will surely hold to a course that is more secure, at least in the sense that it has already been tested by wiser heads. For our part, we are very glad that we had a scholastic education of this sort. But we are now freed from the oath which bound us to our master's words and are old enough to be no longer subject to the rod. So if we seriously wish to propose rules for ourselves which will help us scale the heights of human knowledge, we must include, as one of our primary rules, that we should take care not to waste our time by neglecting easy tasks and occupying ourselves only with difficult matters. That is just what many people do: they ingeniously construct the most subtle conjectures and plausible arguments on difficult questions, but after all their efforts they come to realize, too late, that rather than acquiring any knowledge, they have merely increased the number of their doubts.

Of all the sciences so far discovered, arithmetic and geometry alone are, as we said above, free from any taint of falsity or uncertainty. If we are to give a careful estimate of the reason why this should be so, we should bear in mind that there are two ways of arriving at a knowledge of things - through experience and through deduction. Moreover, we must note that while our experiences of things are often deceptive, the deduction or pure inference of one thing from another can never be performed wrongly by an intellect which is in the least degree rational, though we may fail to make the inference if we do not see it. Furthermore, those chains with which dialecticians ^1 suppose they regulate human reason seem to me to be of little use here, though I do not deny that they are very useful for other purposes. In fact none of the errors to which men - men, I say, not the brutes - are liable is ever due to faulty inference; they are due only to the fact that men take for granted certain poorly understood observations,^2 or lay down rash and groundless judgements.

These considerations make it obvious why arithmetic and geometry prove to be much more certain than other disciplines: they alone are concerned with an object so pure and simple that they make no assumptions that experience might render uncertain; they consist entirely in deducing conclusions by means of rational arguments. They are therefore the easiest and clearest of all the sciences and have just the sort of object we are looking for. Where these sciences are concerned it scarcely seems humanly possible to err, except through inadvertence. Yet we should not be surprised if many prefer of their own accord to apply their minds to other arts, or to philosophy. The reason for this is that everyone feels free to make more confident guesses about matters which are obscure than about matters which are clear. It is much easier to hazard some conjecture on this or that question than to arrive at the exact truth about one particular question, however straightforward it may be.

Now the conclusion we should draw from these considerations is not that arithmetic and geometry are the only sciences worth studying, but rather that in seeking the right path of truth we ought to concern ourselves only with objects which admit of as much certainty as the demonstrations of arithmetic and geometry.

Rule Three

Concerning objects proposed for study, we ought to investigate what we can clearly and evidently intuit ^1 or deduce with certainty, and not what other people have thought or what we ourselves conjecture. For knowledge ^2 can be attained in no other way.

We ought to read the writings of the ancients, for it is of great advantage to be able to make use of the labours of so many men. We should do so both in order to learn what truths have already been discovered and also to be informed about the points which remain to be worked out in the various disciplines. But at the same time there is a considerable danger that if we study these works too closely traces of their errors will infect us and cling to us against our will and despite our precautions. For, once writers have credulously and heedlessly taken up a position on some controversial question, they are generally inclined to employ the most subtle arguments in an attempt to get us to adopt their point of view. On the other hand, whenever they have the luck to discover something certain and evident, they always present it wrapped up in various obscurities, either because they fear that the simplicity of their argument may depreciate the importance of their finding, or because they begrudge us the plain truth.

But even if all writers were sincere and open, and never tried to palm off doubtful matters as true, but instead put forward everything in good faith, we would always be uncertain which of them to believe, for hardly anything is said by one writer the contrary of which is not asserted by some other. It would be of no use to count heads, so as to follow the view which many authorities hold. For if the question at issue is a difficult one, it is more likely that few, rather than many, should have been able to discover the truth about it. But even if they all agreed among themselves, their teaching would still not be all we need. For example, even though we know other people's demonstrations by heart, we shall never become mathematicians if we lack the intellectual aptitude to solve any given problem. And even though we have read all the arguments of Plato and Aristotle, we shall never become philosophers if we are unable to make a sound judgement on matters which come up for discussion; in this case what we would seem to have learnt would not be science but history.

Furthermore, we would be well-advised not to mix any conjectures into the judgements we make about the truth of things. It is most important to bear this point in mind. The main reason why we can find nothing in ordinary philosophy which is so evident and certain as to be beyond dispute is that students of the subject first of all are not content to acknowledge what is clear and certain, but on the basis of merely probable conjectures venture also to make assertions on obscure matters about which nothing is known; they then gradually come to have complete faith in these assertions, indiscriminately mixing them up with others that are true and evident. The result is that the only conclusions they can draw are ones which apparently rest on some such obscure proposition, and which are accordingly uncertain.

But in case we in turn should slip into the same error, let us now review all the actions of the intellect by means of which we are able to arrive at a knowledge of things with no fear of being mistaken. We recognize only two: intuition and deduction.^1

By 'intuition' I do not mean the fluctuating testimony of the senses or the deceptive judgement of the imagination as it botches things together, but the conception of a clear and attentive mind, which is so easy and distinct that there can be no room for doubt about what we are understanding. Alternatively, and this comes to the same thing, intuition is the indubitable conception of a clear and attentive mind which proceeds solely from the light of reason. Because it is simpler, it is more certain than deduction, though deduction, as we noted above, is not something a man can perform wrongly. Thus everyone can mentally intuit that he exists, that he is thinking, that a triangle is bounded by just three lines, and a sphere by a single surface, and the like. Perceptions such as these are more numerous than most people realize, disdaining as they do to turn their minds to such simple matters.

In case anyone should be troubled by my novel use of the term 'intuition' and of other terms to which I shall be forced to give a different meaning from their ordinary one, I wish to point out here that I am paying no attention to the way these terms have lately been used in the Schools. For it would be very difficult for me to employ the same terminology, when my own views are profoundly different. I shall take account only of the meanings in Latin of individual words and, when appropriate words are lacking, I shall use what seem the most suitable words, adapting them to my own meaning.

The self-evidence and certainty of intuition is required not only for apprehending single propositions, but also for any train of reasoning whatever. Take for example, the inference that 2 plus 2 equals 3 plus 1: not only must we intuitively perceive that 2 plus 2 make 4, and that 3 plus 1 make 4, but also that the original proposition follows necessarily from the other two.

There may be some doubt here about our reason for suggesting another mode of knowing in addition to intuition, viz. deduction, by which we mean the inference of something as following necessarily from some other propositions which are known with certainty. But this distinction had to be made, since very many facts which are not self-evident are known with certainty, provided they are inferred from true and known principles through a continuous and uninterrupted movement of thought in which each individual proposition is clearly intuited. This is similar to the way in which we know that the last link in a long chain is connected to the first: even if we cannot take in at one glance all the intermediate links on which the connection depends, we can have knowledge of the connection provided we survey the links one after the other, and keep in mind that each link from first to last is attached to its neighbour. Hence we are distinguishing mental intuition from certain deduction on the grounds that we are aware of a movement or a sort of sequence in the latter but not in the former, and also because immediate self-evidence is not required for deduction, as it is for intuition; deduction in a sense gets its certainty from memory. It follows that those propositions which are immediately inferred from first principles can be said to be known in one respect through intuition, and in another respect through deduction. But the first principles themselves are known only through intuition, and the remote conclusions only through deduction.

These two ways are the most certain routes to knowledge that we have. So far as our powers of understanding are concerned, we should admit no more than these and should reject all others as suspect and liable to error. This does not preclude our believing that what has been revealed by God is more certain than any knowledge, since faith in these matters, as in anything obscure, is an act of the will rather than an act of the understanding. And if our faith has a basis in our intellect, revealed truths above all can and should be discovered by one or other of the two ways we have just described, as we may show at greater length below.

Rule Four

We need a method if we are to investigate the truth of things.

So blind is the curiosity with which mortals are possessed that they often direct their minds down untrodden paths, in the groundless hope that they will chance upon what they are seeking, rather like someone who is consumed with such a senseless desire to discover treasure that he continually roams the streets to see if he can find any that a passer-by might have dropped. This is how almost every chemist, most geometers, and many philosophers pursue their research. I am not denying that they sometimes are lucky enough in their wanderings to hit upon some truth, though on that account I rate them more fortunate than diligent. But it is far better never to contemplate investigating the truth about any matter than to do so without a method. For it is quite certain that such haphazard studies and obscure reflections blur the natural light and blind our intelligence. Those who are accustomed to walking in the dark weaken their eye-sight, the result being that they can no longer bear to be in broad daylight. Experience confirms this, for we very often find that people who have never devoted their time to learned studies make sounder and clearer judgements on matters which arise than those who have spent all their time in the Schools. By 'a method' I mean reliable rules which are easy to apply, and such that if one follows them exactly, one will never take what is false to be true or fruitlessly expend one's mental efforts, but will gradually and constantly increase one's knowledge ^1 till one arrives at a true understanding of everything within one's capacity.

There are two points here which we should keep in mind: we should never assume to be true anything which is false; and our goal should be to attain knowledge of all things. For, if we do not know something we are capable of knowing, this is simply because we have never discovered a way that might lead us to such knowledge, or because we have fallen into the opposite error.^2 But if our method properly explains how we should use our mental intuition to avoid falling into the opposite error and how we should go about finding the deductive inferences that will help us attain this all-embracing knowledge, then I do not see that anything more is needed to make it complete; for as I have already said, we can have no knowledge, without mental intuition or deduction. The method cannot go so far as to teach us how to perform the actual operations of intuition and deduction, since these are the simplest of all and quite basic. If our intellect were not already able to perform them, it would not comprehend any of the rules of the method, however easy they might be. As for other mental operations which dialectic ^3 claims to direct with the help of those already mentioned, they are of no use here, or rather should be reckoned a positive hindrance, for nothing can be added to the clear light of reason which does not in some way dim it.

So useful is this method that without it the pursuit of learning would, I think, be more harmful than profitable. Hence I can readily believe that the great minds of the past were to some extent aware of it, guided to it even by nature alone. For the human mind has within it a sort of spark of the divine, in which the first seeds of useful ways of thinking are sown, seeds which, however neglected and stifled by studies which impede them, often bear fruit of their own accord. This is our experience in the simplest of sciences, arithmetic and geometry: we are well aware that the geometers of antiquity employed a sort of analysis which they went on to apply to the solution of every problem, though they begrudged revealing it to posterity. At the present time a sort of arithmetic called 'algebra' is flourishing, and this is achieving for numbers what the ancients did for figures. These two disciplines are simply the spontaneous fruits which have sprung from the innate principles of this method. I am not surprised that, where the simplest objects of these disciplines are concerned, there has been a richer harvest of such fruits than in other disciplines in which greater obstacles tend to stifle progress. But no doubt these too could achieve a perfect maturity if only they were cultivated with extreme care.

That is in fact what I have principally aimed at achieving in this treatise. I would not value these Rules so highly if they were good only for solving those pointless problems with which arithmeticians and geometers are inclined to while away their time, for in that case all I could credit myself with achieving would be to dabble in trifles with greater subtlety than they. I shall have much to say below about figures and numbers, for no other disciplines can yield illustrations as evident and certain as these. But if one attends closely to my meaning, one will readily see that ordinary mathematics is far from my mind here, that it is quite another discipline I am expounding, and that these illustrations are more its outer garments than its inner parts. This discipline should contain the primary rudiments of human reason and extend to the discovery of truths in any field whatever. Frankly speaking, I am convinced that it is a more powerful instrument of knowledge than any other with which human beings are endowed, as it is the source of all the rest. I have spoken of its 'outer garment', not because I wish to conceal this science and shroud it from the gaze of the public; I wish rather to clothe and adorn it so as to make it easier to present to the human mind.

When I first applied my mind to the mathematical disciplines, I at once read most of the customary lore which mathematical writers pass on to us. I paid special attention to arithmetic and geometry, for these were said to be the simplest and, as it were, to lead into the rest. But in neither subject did I come across writers who fully satisfied me. I read much about numbers which I found to be true once I had gone over the calculations for myself. The writers displayed many geometrical truths before my very eyes, as it were, and derived them by means of logical arguments. But they did not seem to make it sufficiently clear to my mind why these things should be so and how they were discovered. So I was not surprised to find that even many clever and learned men, after dipping into these arts, either quickly lay them aside as childish and pointless or else take them to be so very difficult and complicated that they are put off at the outset from learning them. For there is really nothing more futile than so busying ourselves with bare numbers and imaginary figures that we seem to rest content in the knowledge of such trifles. And there is nothing more futile than devoting our energies to those superficial proofs which are discovered more through chance than method and which have more to do with our eyes and imagination than our intellect; for the outcome of this is that, in a way, we get out of the habit of using our reason. At the same time there is nothing more complicated than using such a method of proof to resolve new problems which are beset with numerical disorder. Later on I wondered why the founders of philosophy would admit no one to the pursuit of wisdom who was unversed in mathematics ^1 - as if they thought that this discipline was the easiest and most indispensable of all for cultivating and preparing the mind to grasp other more important sciences. I came to suspect that they were familiar with a kind of mathematics quite different from the one which prevails today; not that I thought they had a perfect knowledge of it, for their wild exultations and thanksgivings for trivial discoveries clearly show how rudimentary their knowledge must have been. I am not shaken in this opinion by those machines ^2 of theirs which are so much praised by historians. These mechanical devices may well have been quite simple, even though the ignorant and wonder-loving masses may have raised them to the level of marvels. But I am convinced that certain primary seeds of truth naturally implanted in human minds thrived vigorously in that unsophisticated and innocent age - seeds which have been stifled in us through our constantly reading and hearing all sorts of errors. So the same light of the mind which enabled them to see (albeit without knowing why) that virtue is preferable to pleasure, the good preferable to the useful, also enabled them to grasp true ideas in philosophy and mathematics, although they were not yet able fully to master such sciences. Indeed, one can even see some traces of this true mathematics, I think, in Pappus and Diophantus ^3 who, though not of that earliest antiquity, lived many centuries before our time. But I have come to think that these writers themselves, with a kind of pernicious cunning, later suppressed this mathematics as, notoriously, many inventors are known to have done where their own discoveries were concerned. They may have feared that their method, just because it was so easy and simple, would be depreciated if it were divulged; so to gain our admiration, they may have shown us, as the fruits of their method, some barren truths proved by clever arguments, instead of teaching us the method itself, which might have dispelled our admiration. In the present age some very gifted men have tried to revive this method, for the method seems to me to be none other than the art which goes by the outlandish name of 'algebra' - or at least it would be if algebra were divested of the multiplicity of numbers and incomprehensible figures which overwhelm it and instead possessed that abundance of clarity and simplicity which I believe the true mathematics ought to have. It was these thoughts which made me turn from the particular studies of arithmetic and geometry to a general investigation of mathematics. I began my investigation by inquiring what exactly is generally meant by the term 'mathematics'^1 and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics, among others, are called branches of mathematics. To answer this it is not enough just to look at the etymology of the word, for, since the word 'mathematics' has the same meaning as 'discipline',^2 these subjects have as much right to be called 'mathematics' as geometry has. Yet it is evident that almost anyone with the slightest education can easily tell the difference in any context between what relates to mathematics and what to the other disciplines. When I considered the matter more closely, I came to see that the exclusive concern of mathematics is with questions of order or measure and that it is irrelevant whether the measure in question involves numbers, shapes, stars, sounds, or any other object whatever. This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis ^3 - a venerable term with a well-established meaning - for it covers everything that entitles these other sciences to be called branches of mathematics. How superior it is to these subordinate sciences both in utility and simplicity is clear from the fact that it covers all they deal with, and more besides; and any difficulties it involves apply to these as well, whereas their particular subject-matter involves difficulties which it lacks. Now everyone knows the name of this subject and without even studying it understands what its subject-matter is. So why is it that most people painstakingly pursue the other disciplines which depend on it, and no one bothers to learn this one? No doubt I would find that very surprising if I did not know that everyone thinks the subject too easy, and if I had not long since observed that the human intellect always bypasses subjects which it thinks it can easily master and directly hurries on to new and grander things.

Aware how slender my powers are, I have resolved in my search for knowledge of things to adhere unswervingly to a definite order, always starting with the simplest and easiest things and never going beyond them till there seems to be nothing further which is worth achieving where they are concerned. Up to now, therefore, I have devoted all my energies to this universal mathematics, so that I think I shall be able in due course to tackle the somewhat more advanced sciences, without my efforts being premature. But before I embark on this task I shall try to bring together and arrange in an orderly manner whatever I thought noteworthy in my previous studies, so that when old age dims my memory I can readily recall it hereafter, if I need to, by consulting this book, and so that, having disburdened my memory, I can henceforth devote my mind more freely to what remains.

Rule Five

The whole method consists entirely in the ordering and arranging of the objects on which we must concentrate our mind's eye if we are to discover some truth. We shall be following this method exactly if we first reduce complicated and obscure propositions step by step to simpler ones, and then, starting with the intuition of the simplest ones of all, try to ascend through the same steps to a knowledge of all the rest.

This one Rule covers the most essential points in the whole of human endeavour. Anyone who sets out in quest of knowledge of things must follow this Rule as closely as he would the thread of Theseus if he were to enter the Labyrinth. But many people either do not reflect upon what the Rule prescribes, or ignore it altogether, or presume that they have no need of it. They frequently examine difficult problems in a very disorderly manner, behaving in my view as if they were trying to get from the bottom to the top of a building at one bound, spurning or failing to notice the stairs designed for that purpose. Astrologers all do likewise: they do not know the nature of the heavens and do not even make any accurate observations of celestial motions, yet they expect to be able to delineate the effects of these motions. So too do most of those who study mechanics apart from physics and, without any proper plan, construct new instruments for producing motion. This applies also to those philosophers who take no account of experience and think that truth will spring from their brains like Minerva from the head of Jupiter.

All those just mentioned are plainly violating this Rule. But the order that is required here is often so obscure and complicated that not everyone can make out what it is; hence it is virtually impossible to guard against going astray unless one carefully observes the message of the following Rule.

Rule Six

In order to distinguish the simplest things from those that are complicated and to set them out in an orderly manner, we should attend to what is most simple in each series of things in which we have directly deduced some truths from others, and should observe how all the rest are more, or less, or equally removed from the simplest.

Although the message of this Rule may not seem very novel, it contains nevertheless the main secret of my method; and there is no more useful Rule in this whole treatise. For it instructs us that all things can be arranged serially in various groups, not in so far as they can be referred to some ontological genus (such as the categories into which philosophers divide things ^1), but in so far as some things can be known on the basis of others. Thus when a difficulty arises, we can see at once whether it will be worth looking at any others first, and if so which ones and in what order.

In order to be able to do this correctly, we should note first that everything, with regard to its possible usefulness to our project, may be termed either 'absolute' or 'relative' - our project being, not to inspect the isolated natures of things, but to compare them with each other so that some may be known on the basis of others.

I call 'absolute' whatever has within it the pure and simple nature in question; that is, whatever is viewed as being independent, a cause, simple, universal, single, equal, similar, straight, and other qualities of that sort. I call this the simplest and the easiest thing when we can make use of it in solving problems.

The 'relative', on the other hand, is what shares the same nature, or at least something of the same nature, in virtue of which we can relate it to the absolute and deduce it from the absolute in a definite series of steps. The concept of the 'relative' involves other terms besides, which I call 'relations': these include whatever is said to be dependent, an effect, composite, particular, many, unequal, dissimilar, oblique, etc. The further removed from the absolute such relative attributes are, the more mutually dependent relations of this sort they contain. This Rule points out that all these relations should be distinguished, and the interconnections between them, and their natural order, should be noted, so that given the last term we should be able to reach the one that is absolute in the highest degree, by passing through all the intermediate ones.

The secret of this technique consists entirely in our attentively noting in all things that which is absolute in the highest degree. For some things are more absolute than others from one point of view, yet more relative from a different point of view. For example, the universal is more absolute than the particular, in virtue of its having a simpler nature, but it can also be said to be more relative than the particular in that it depends upon particulars for its existence, etc. Again, certain things sometimes are really more absolute than others, yet not the most absolute of all. Thus a species is something absolute with respect to particulars, but with respect to the genus it is relative; and where measurable items are concerned, extension is something absolute, but among the varieties of extension length is something absolute, etc. Furthermore, in order to make it clear that what we are contemplating here is the series of things to be discovered, and not the nature of each of them, we have deliberately listed 'cause' and 'equal' among the absolutes, although their nature really is relative. Philosophers, of course, recognize that cause and effect are correlatives; but in the present case, if we want to know what the effect is, we must know the cause first, and not vice versa. Again, equals are correlative with one another, but we can know what things are unequal only by comparison with equals, and not vice versa, etc.

Secondly, we should note that there are very few pure and simple natures which we can intuit straight off and per se (independently of any others) either in our sensory experience or by means of a light innate within us. We should, as I said, attend carefully to the simple natures which can be intuited in this way, for these are the ones which in each series we term simple in the highest degree. As for all the other natures, we can apprehend them only by deducing them from those which are simple in the highest degree, either immediately and directly, or by means of two or three or more separate inferences. In the latter case we should also note the number of these inferences so that we may know whether the separation between the conclusion and the primary and supremely simple proposition is by way of a greater or fewer number of steps. And the chain of inferences - which gives rise to those series of objects of investigation to which every problem must be reduced - is such throughout that the problem can be investigated by a reliable method. But since it is not easy to review all the connections together, and moreover, since our task is not so much to retain them in our memory as to distinguish them with, as it were, the sharp edge of our mind, we must seek a means of developing our intelligence in such a way that we can discern these connections immediately whenever the need arises. In my experience there is no better way of doing this than by accustoming ourselves to reflecting with some discernment on the minute details of the things we have already perceived.

The third and last point is that we should not begin our studies by investigating difficult matters. Before tackling any specific problems we ought first to make a random selection of truths which happen to be at hand, and ought then to see whether we can deduce some other truths from them step by step, and from these still others, and so on in logical sequence. This done, we should reflect attentively on the truths we have discovered and carefully consider why it was we were able to discover some of these truths sooner and more easily than others, and what these truths are. This will enable us to judge, when tackling a specific problem, what points we may usefully concentrate on discovering first. For example, say the thought occurs to me that the number 6 is twice 3: I may then ask what twice 6 is, viz. 12; I may, if I like, go on to ask what twice 12 is, viz. 24, and what twice 24 is, viz. 48, etc. It would then be easy for me to deduce that there is the same ratio between 3 and 6 as between 6 and 12, and again the same ratio between 12 and 24, etc., and hence that the numbers 3, 6, 12, 24, 48, etc. are continued proportionals. All of this is so clear as to seem almost childish; nevertheless when I think carefully about it, I can see what sort of complications are involved in all the questions one can ask about the proportions or relations between things, and in what order the questions should be investigated. This one point encompasses the essential core of the entire science of pure mathematics.

For I notice first that it was no more difficult to discover what twice 6 is than twice 3, and that whenever we find a ratio between any two magnitudes we can always find, just as easily, innumerable others which have the same ratio between them. The nature of the problem is no different when we are trying to find three, four, or more magnitudes of this sort, since each one has to be found separately and without regard to the others. I next observe that given the magnitudes 3 and 6, I easily found ^1 a third magnitude which is in continued proportion, viz. 12, yet, when the extreme terms 3 and 12 were given, I could not find just as easily the mean proportional, 6. If we look into the reason for this, it is obvious that we have here a quite different type of problem from the preceding one. For, if we are to find the mean proportional, we must attend at the same time to the two extreme terms and the ratio between them, in order to obtain a new ratio by dividing this one.^1 This is a very different task from that of finding a third magnitude, given two magnitudes in continued proportion.^2 I can go even further and ask whether, given the numbers 3 and 24, it would be just as easy to find one of the two mean proportionals, viz. 6 and 12. Here we have another sort of problem again, an even more complicated one than either of the preceding ones. We have to attend not just to one thing or to two but to three different things at the same time, if we are to find a fourth.^3 We can go even further and see whether, given just 3 and 48, it would be still more difficult to find one of the three mean proportionals, viz. 6, 12 and 24. At first sight it does indeed seem to be more difficult. But then the thought immediately strikes us that this problem can be split up and made easier: first we look for the single mean proportional between 3 and 48, viz. 12; then we look for a further mean proportional between 3 and 12, viz. 6; then another between 12 and 48, viz. 24. In that way we reduce the problem to one of the second kind described above.

Moreover, from these examples I realize how in our pursuit of knowledge of a given thing we can follow different paths, one of which is much more difficult and obscure than the other. If, for example, we are asked to find the four proportionals, 3, 6, 12, 24, given any two consecutive members of the series, such as 3 and 6, or 6 and 12, or 12 and 24, it will be a very easy task to find the others. In this case we shall say that the proposition we are seeking is investigated in a direct way. But if two alternate numbers are given, such as 3 and 12, or 6 and 24, and we are to work out the others from these, in that case we shall say that the problem is investigated indirectly by the first method. Likewise, if we are to find the intermediate numbers, 6 and 12, given the two extremes, 3 and 24, then the problem will be investigated indirectly by the second method. I could thus go on even further and draw many other conclusions from this one example. But these points will suffice to enable the reader to see what I mean when I say that some proposition is deduced 'directly' or 'indirectly', and will suffice to make him bear in mind that on the basis of our knowledge of the most simple and primary things we can make many discoveries, even in other disciplines, through careful reflection and discriminating inquiry.

Rule Seven

In order to make our knowledge ^1 complete, every single thing relating to our undertaking must be surveyed in a continuous and wholly uninterrupted sweep of thought, and be included in a sufficient and well-ordered enumeration.

It is necessary to observe the points proposed in this Rule if we are to admit as certain those truths which, we said above, are not deduced immediately from first and self-evident principles. For this deduction sometimes requires such a long chain of inferences that when we arrive at such a truth it is not easy to recall the entire route which led us to it. That is why we say that a continuous movement of thought is needed to make good any weakness of memory. If, for example, by way of separate operations, I have come to know first what the relation between the magnitudes A and B is, and then between B and C, and between C and D, and finally between D and E, that does not entail my seeing what the relation is between A and E; and I cannot grasp what the relation is just from those I already know, unless I recall all of them. So I shall run through them several times in a continuous movement of the imagination, simultaneously intuiting one relation and passing on to the next, until I have learnt to pass from the first to the last so swiftly that memory is left with practically no role to play, and I seem to intuit the whole thing at once. In this way our memory is relieved, the sluggishness of our intelligence redressed, and its capacity in some way enlarged.

In addition, this movement must nowhere be interrupted. Frequently those who attempt to deduce something too swiftly and from remote initial premisses do not go over the entire chain of intermediate conclusions very carefully, but pass over many of the steps without due consideration. But, whenever even the smallest link is overlooked the chain is immediately broken, and the certainty of the conclusion entirely collapses.

We maintain furthermore that enumeration is required for the completion of our knowledge.^1 The other Rules do indeed help us resolve most questions, but it is only with the aid of enumeration that we are able to make a true and certain judgement about whatever we apply our minds to. By means of enumeration nothing will wholly escape us and we shall be seen to have some knowledge on every question.

In this context enumeration,^2 or induction, consists in a thorough investigation of all the points relating to the problem at hand, an investigation which is so careful and accurate that we may conclude with manifest certainty that we have not inadvertently overlooked anything. So even though the object of our inquiry eludes us, provided we have made an enumeration we shall be wiser at least to the extent that we shall perceive with certainty that it could not possibly be discovered by any method known to us. If we have managed to examine all the humanly accessible paths towards the object of our inquiry (which we often do), we shall be entitled confidently to assert that knowledge of it lies wholly beyond the reach of the human mind.

We should note, moreover, that by 'sufficient enumeration' or 'induction' we just mean the kind of enumeration which renders the truth of our conclusions more certain than any other kind of proof (simple intuition excepted) allows. But when our knowledge of something is not reducible to simple intuition and we have cast off our syllogistic fetters, we are left with this one path, which we should stick to with complete confidence. For if we have deduced one fact from another immediately, then provided the inference is evident, it already comes under the heading of true intuition. If on the other hand we infer a proposition from many disconnected propositions, our intellectual capacity is often insufficient to enable us to encompass all of them in a single intuition; in which case we must be content with the level of certainty which the above operation allows. In the same way, our eyes cannot distinguish at one glance all the links in a very long chain; but, if we have seen the connections between each link and its neighbour, this enables us to say that we have seen how the last link is connected with the first.

I said that this operation should be 'sufficient', because it can often be deficient and hence liable to error. For sometimes, even though we survey many points in our enumeration which are quite evident, yet if we make even the slightest omission, the chain is broken and the certainty of the conclusion is entirely lost. Again, sometimes we do cover everything in our enumeration, yet fail to distinguish one thing from another, so that our knowledge of them all is simply confused.

The enumeration should sometimes be complete, and sometimes distinct, though there are times when it need be neither. That is why I said only that the enumeration must be sufficient. For if I wish to determine by enumeration how many kinds of corporeal entity there are or how many are in some way perceivable by the senses, I shall not assert that there are just so many and no more, unless I have previously made sure I have included them all in my enumeration and have distinguished one from another. But if I wish to show in the same way that the rational soul is not corporeal, there is no need for the enumeration to be complete; it will be sufficient if I group all bodies together into several classes so as to demonstrate that the rational soul cannot be assigned to any of these. To give one last example, say I wish to show by enumeration that the area of a circle is greater than the area of any other geometrical figure whose perimeter is the same length as the circle's. I need not review every geometrical figure. If I can demonstrate that this fact holds for some particular figures, I shall be entitled to conclude by induction ^1 that the same holds true in all the other cases as well.

I said also that the enumeration must be well-ordered, partly because there is no more effective remedy for the defects I have just listed than a well-ordered scrutiny of all the relevant items, and partly because, if every single thing relevant to the question in hand were to be separately scrutinized, one lifetime would generally be insufficient for the task, for either there would be too many such things or the same things would keep cropping up. But if we arrange all of the relevant items in the best order, so that for the most part they fall under definite classes, it will be sufficient if we look closely at one class, or at a member of each particular class, or at some classes rather than others. If we do that, we shall at any rate never pointlessly go over the same ground twice, and thanks to our well-devised order, we shall often manage to review quickly and effortlessly a large number of items which at first sight seemed formidably large.

In such cases the order in which things are enumerated can usually be varied; it is a matter for individual choice. For that reason, if our choice is to be intelligently thought out we should bear in mind what was said in Rule Five. In the more frivolous of man's skills there are many things whose method of invention consists entirely in arranging things in this orderly way. Thus if you want to construct a perfect anagram by transposing the letters of a name, there is no need to pass from the very easy to the more difficult, nor to distinguish what is absolute from what is relative, for these operations have no place here. All you need do is to decide on an order for examining permutations of letters so that you never go over the same permutations twice. The number of these permutations should, for example, be arranged into definite classes, so that it becomes immediately obvious which ones present the greater prospect of finding what you are looking for. If this is done, the task will seldom be tedious; it will be mere child's play.

Now, these last three Rules should not be separated. We should generally think of them together, since they all contribute equally to the perfection of the method. It was immaterial which of them we expounded first. We are giving only a brief account of them here, for our task in the remainder of the treatise will be confined almost entirely to explicating in detail what we have so far covered in general terms.

Rule Eight

If in the series of things to be examined we come across something which our intellect is unable to intuit sufficiently well, we must stop at that point, and refrain from the superfluous task of examining the remaining items.

The three preceding Rules prescribe and explain the order to be followed; the present Rule shows when order is absolutely necessary, and when it is merely useful. It is necessary that we examine whatever constitutes an integral step in the series through which we must pass when we proceed from relative terms to something absolute or vice versa, before considering all that follows in the series. Of course if many things belong to a given step, as is often the case, it is always useful to survey all of them in due order. But we are not forced to follow the order strictly and rigidly; generally we may proceed further, even although we do not have clear knowledge of all the terms of the series, but only of a few or just one of them.

This Rule is a necessary consequence of the reasons I gave in support of Rule Two. But it should not be thought that this Rule contributes nothing new to the advancement of learning, even though it seems merely to deter us from discussing certain things and to bring no truth to light. Indeed, all it teaches beginners is that they should not waste their efforts, and it does so in practically the same manner as Rule Two. But it shows those who have perfectly mastered the preceding seven Rules how they can achieve for themselves, in any science whatever, results so satisfactory that there is nothing further they will desire to achieve. If anyone observes the above Rules exactly when trying to solve some problem or other, but is instructed by the present Rule to stop at a certain point, he will know for sure that no amount of application will enable him to find the knowledge ^1 he is seeking; and that not because of any defect of his intelligence, but because of the obstacle which the nature of the problem itself or the human condition presents. His recognition of this point is just as much knowledge ^1 as that which reveals the nature of the thing itself; and it would, I think, be quite irrational if he were to stretch his curiosity any further.

Let us illustrate these points with one or two examples. If, say, someone whose studies are confined to mathematics tries to find the line called the 'anaclastic' in optics ^2 - the line from which parallel rays are so refracted that they intersect at a single point - he will easily see, by following Rules Five and Six, that the determination of this line depends on the ratio of the angles of refraction to the angles of incidence. But he will not be able to find out what this ratio is, since it has to do with physics rather than with mathematics. So he will be compelled to stop short right at the outset. If he proposes to learn it from the philosophers or derive it from experience, he will achieve nothing, for that would be to violate Rule Three. Besides, the problem before him is composite and relative; and it is possible to have experiential knowledge which is certain only of things which are entirely simple and absolute, as I shall show in the appropriate place. Again, it is no use his assuming some particular ratio between the angles in question, one he conjectures to be most likely the real one; for in that case what he was seeking to determine would no longer be the anaclastic - it would merely be the line which was the logical consequence of his supposition.

Now take someone whose studies are not confined to mathematics and who, following Rule One, eagerly seeks the truth on any question that arises: if he is faced with the same problem, he will discover when he goes into it that the ratio between the angles of incidence and the angles of refraction depends upon the changes in these angles brought about by differences in the media. He will see that these changes depend on the manner in which a ray passes through the entire transparent body,^1 and that knowledge of this process presupposes also a knowledge of the nature of the action of light. Lastly, he will see that to understand the latter process he must know what a natural power in general is - this last being the most absolute term in this whole series. Once he has clearly ascertained this through mental intuition, he will, in accordance with Rule Five, retrace his course through the same steps. If, at the second step, he is unable to discern at once what the nature of light's action is, in accordance with Rule Seven he will make an enumeration of all the other natural powers, in the hope that a knowledge of some other natural power will help him understand this one, if only by way of analogy - but more of this later.^2 Having done that, he will investigate the way in which the ray passes through the whole transparent body. Thus he will follow up the remaining points in due order, until he arrives at the anaclastic itself. Even though the anaclastic has been the object of much fruitless research in the past, I can see nothing to prevent anyone who uses our method exactly from gaining a clear knowledge of it.

But let us take the finest example of all. If someone sets himself the problem of investigating every truth for the knowledge of which human reason is adequate - and this, I think, is something everyone who earnestly strives after good sense should do once in his life - he will indeed discover by means of the Rules we have proposed that nothing can be known prior to the intellect, since knowledge of everything else depends on the intellect, and not vice versa. Once he has surveyed everything that follows immediately upon knowledge of the pure intellect, among what remains he will enumerate whatever instruments of knowledge we possess in addition to the intellect; and there are only two of these, namely imagination and sense-perception. He will therefore devote all his energies to distinguishing and examining these three modes of knowing. He will see that there can be no truth or falsity in the strict sense except in the intellect alone, although truth and falsity often originate from the other two modes of knowing; and he will pay careful heed to everything that might deceive him, in order to guard against it. He will make a precise enumeration of all the paths to truth which are open to men, so that he may follow one which is reliable. There are not so many of these that he cannot easily discover them all by means of a sufficient enumeration;^1 this will seem surprising and incredible to the inexperienced. And as soon as he has distinguished, with respect to each individual object, between those items of knowledge which merely fill and adorn the memory and those which really entitle one to be called more learned - an easy task to accomplish . . .^2 he will take the view that any lack of further knowledge on his part is not at all due to any lack of intelligence or method, and that whatever anyone else can know, he too is capable of knowing, if only he properly applies his mind to it. He may often be faced with many questions which this Rule prohibits him from taking up; yet, because he sees clearly that these questions are wholly beyond the reach of the human mind, he will not regard himself as being more ignorant on that account. On the contrary, his very knowing that the matter in question is beyond the bounds of human knowledge will, if he is reasonable, abundantly satisfy his curiosity.

Now, to prevent our being in a state of permanent uncertainty about the powers of the mind, and to prevent our mental labours being misguided and haphazard, we ought once in our life carefully to inquire as to what sort of knowledge human reason is capable of attaining, before we set about acquiring knowledge of things in particular. In order to do this the better, we should, where the objects of inquiry are equally simple, always begin our investigation with those which are more useful.

Our method in fact resembles the procedures in the mechanical crafts, which have no need of methods other than their own, and which supply their own instructions for making their own tools. If, for example, someone wanted to practise one of these crafts - to become a blacksmith, say - but did not possess any of the tools, he would be forced at first to use a hard stone (or a rough lump of iron) as an anvil, to make a rock do as a hammer, to make a pair of tongs out of wood, and to put together other such tools as the need arose. Thus prepared, he would not immediately attempt to forge swords, helmets, or other iron implements for others to use; rather he would first of all make hammers, an anvil, tongs and other tools for his own use. What this example shows is that, since in these preliminary inquiries we have managed to discover only some rough precepts which appear to be innate in our minds rather than the product of any skill, we should not immediately try to use these precepts to settle philosophical disputes or to solve mathematical problems. Rather, we should use these precepts in the first instance to seek out with extreme care everything else which is more essential in the investigation of truth, especially since there is no reason why such things should be thought more difficult to discover than any of the solutions to the problems commonly set in geometry, in physics, or in other disciplines.

But the most useful inquiry we can make at this stage is to ask: What is human knowledge and what is its scope? We are at present treating this as one single question, which in our view is the first question of all that should be examined by means of the Rules described above. This is a task which everyone with the slightest love of truth ought to undertake at least once in his life, since the true instruments of knowledge and the entire method are involved in the investigation of the problem. There is, I think, nothing more foolish than presuming, as many do, to argue about the secrets of nature, the influence of the heavens on these lower regions, the prediction of future events, and so on, without ever inquiring whether human reason is adequate for discovering matters such as these. It should not be regarded as an arduous or even difficult task to define the limits of the mental powers we are conscious of possessing, since we often have no hesitation in making judgements about things which are outside us and quite foreign to us. Nor is it an immeasurable task to seek to encompass in thought everything in the universe, with a view to learning in what way particular things may be susceptible of investigation by the human mind. For nothing can be so many-sided or diffuse that it cannot be encompassed within definite limits or arranged under a few headings by means of the method of enumeration we have been discussing. But in order to see how the above points apply to the problem before us, we shall first divide into two parts whatever is relevant to the question; for the question ought to relate either to us, who have the capacity for knowledge, or to the actual things it is possible to know. We shall discuss these two parts separately.

Within ourselves we are aware that, while it is the intellect alone that is capable of knowledge,^1 it can be helped or hindered by three other faculties, viz. imagination, sense-perception, and memory. We must therefore look at these faculties in turn, to see in what respect each of them could be a hindrance, so that we may be on our guard, and in what respect an asset, so that we may make full use of their resources. We shall discuss this part of the question by way of a sufficient enumeration, as the following Rule will make clear.

We should then turn to the things themselves; and we should deal with these only in so far as they are within the reach of the intellect. In that respect we divide them into absolutely simple natures and complex or composite natures. Simple natures must all be either spiritual or corporeal, or belong to each of these categories. As for composite natures, there are some which the intellect experiences as composite before it decides to determine anything about them: but there are others which are put together by the intellect itself. All these points will be explained at greater length in Rule Twelve, where it will be demonstrated that there can be no falsity save in composite natures which are put together by the intellect. In view of this, we divide natures of the latter sort into two further classes, viz. those that are deduced from natures which are the most simple and self-evident (which we shall deal with throughout the next book), and those that presuppose others which experience shows us to be composite in reality. We shall reserve the whole of the third book for an account of the latter.^2

Throughout this treatise we shall try to pursue every humanly accessible path which leads to knowledge of the truth. We shall do this very carefully, and show the paths to be very easy, so that anyone who has mastered the whole method, however mediocre his intelligence, may see that there are no paths closed to him that are open to others, and that his lack of further knowledge is not due to any want of intelligence or method. As often as he applies his mind to acquire knowledge of something, either he will be entirely successful, or at least he will realize that success depends upon some observation which it is not within his power to make - so he will not blame his intelligence, even though he is forced to come to a halt; or, finally, he will be able to demonstrate that the thing he wants to know wholly exceeds the grasp of the human mind - in which case he will not regard himself as more ignorant on that account, for this discovery amounts to knowledge ^1 no less than any other.

Rule Nine

We must concentrate our mind's eye totally upon the most insignificant and easiest of matters, and dwell on them long enough to acquire the habit of intuiting the truth distinctly and clearly.

We have given an account of the two operations of our intellect, intuition and deduction, on which we must, as we said, exclusively rely in our acquisition of knowledge. In this and the following Rule we shall proceed to explain how we can make our employment of intuition and deduction more skilful and at the same time how to cultivate two special mental faculties, viz. perspicacity in the distinct intuition of particular things and discernment in the methodical deduction of one thing from another.

We can best learn how mental intuition is to be employed by comparing it with ordinary vision. If one tries to look at many objects at one glance, one sees none of them distinctly. Likewise, if one is inclined to attend to many things at the same time in a single act of thought, one does so with a confused mind. Yet craftsmen who engage in delicate operations, and are used to fixing their eyes on a single point, acquire through practice the ability to make perfect distinctions between things, however minute and delicate. The same is true of those who never let their thinking be distracted by many different objects at the same time, but always devote their whole attention to the simplest and easiest of matters: they become perspicacious.

It is, however, a common failing of mortals to regard what is more difficult as more attractive. Most people consider that they know nothing, even when they see a very clear and simple cause of something; yet at the same time they get carried away with certain sublime and far-fetched arguments of the philosophers, even though these are for the most part based on foundations which no one has ever thoroughly inspected. It is surely madness to think that there is more clarity in darkness than in light. But let us note, those who really do possess knowledge, can discern the truth with equal facility whether they have derived it from a simple subject or from an obscure one. For once they have hit upon it, they grasp each truth by means of a single and distinct act which is similar in every case. The difference lies entirely in the route followed, which must surely be longer if it leads to a truth which is more remote from completely absolute first principles.

Everyone ought therefore to acquire the habit of encompassing in his thought at one time facts which are very simple and very few in number - so much so that he never thinks he knows something unless he intuits it just as distinctly as any of the things he knows most distinctly of all. Some people of course are born with a much greater aptitude for this sort of insight than others; but our minds can become much better equipped for it through method and practice. There is, I think, one point above all others which I must stress here, which is that everyone should be firmly convinced that the sciences, however abstruse, are to be deduced only from matters which are easy and highly accessible, and not from those which are grand and obscure.

If, for example, I wish to inquire whether a natural power can travel instantaneously to a distant place, passing through the whole intervening space, I shall not immediately turn my attention to the magnetic force, or the influence of the stars, or even the speed of light, to see whether actions such as these might occur instantaneously; for I would find it more difficult to settle that sort of question than the one at issue. I shall, rather, reflect upon the local motions of bodies, since there can be nothing in this whole area that is more readily perceivable by the senses. And I shall realize that, while a stone cannot pass instantaneously from one place to another, since it is a body, a power similar to the one which moves the stone must be transmitted instantaneously if it is to pass, in its bare state, from one object to another. For instance, if I move one end of a stick, however long it may be, I can easily conceive that the power which moves that part of the stick necessarily moves every other part of it instantaneously, because it is the bare power which is transmitted at that moment, and not the power as it exists in some body, such as a stone which carries it along.^1

In the same way, if I want to know how one and the same simple cause can give rise simultaneously to opposite effects, I shall not have recourse to the remedies of the physicians, which drive out some humours and keep others in; nor shall I prattle on about the moon's warming things by its light and cooling them by means of some occult quality. Rather, I shall observe a pair of scales, where a single weight raises one scale and lowers the other instantaneously, and similar examples.

Rule Ten

In order to acquire discernment we should exercise our intelligence by investigating what others have already discovered, and methodically survey even the most insignificant products of human skill, especially those which display or presuppose order.

The natural bent of my mind, I confess, is such that the greatest pleasure I have taken in my studies has always come not from accepting the arguments of others but from discovering arguments by my own efforts. It was just this that attracted me to the study of the sciences while I was still in my youth. Whenever the title of a book gave promise of a new discovery, before I read any further I would try and see whether perhaps I could achieve a similar result by means of a certain innate discernment. And I took great care not to deprive myself of this innocent pleasure through a hasty reading of the book. So frequently was I successful in this that eventually I came to realize that I was no longer making my way to the truth of things as others do by way of aimless and blind inquiries, with the aid of luck rather than skill; rather, after many trials I had hit upon some reliable rules of great assistance in finding the truth, and I then used these to devise many more. In this way I carefully elaborated my whole method, and became convinced that the method of study I had pursued from the outset was the most useful of all.

Still, since not all minds have such a natural disposition to puzzle things out by their own exertions, the message of this Rule is that we must not take up the more difficult and arduous issues immediately, but must first tackle the simplest and least exalted arts, and especially those in which order prevails - such as weaving and carpet-making, or the more feminine arts of embroidery, in which threads are interwoven in an infinitely varied pattern. Number-games and any games involving arithmetic, and the like, belong here. It is surprising how much all these activities exercise our minds, provided of course we discover them for ourselves and not from others. For, since nothing in these activities remains hidden and they are totally adapted to human cognitive capacities, they present us in the most distinct way with innumerable instances of order, each one different from the other, yet all regular. Human discernment consists almost entirely in the proper observance of such order.

It was for this reason that we insisted that our inquiries must proceed methodically. In these somewhat trivial subjects the method usually consists simply in constantly following an order, whether it is actually present in the matter in question or is ingeniously read into it. For example, say we want to read something written in an unfamiliar cypher which lacks any apparent order: what we shall do is to invent an order, so as to test every conjecture we can make about individual letters, words, or sentences, and to arrange the characters in such a way that by an enumeration we may discover what can be deduced from them. Above all, we must guard against wasting our time by making random and unmethodical guesses about similarities. Even though problems such as these can often be solved without a method and can sometimes perhaps be solved more quickly through good luck than through method, nevertheless they might dim the light of the mind and make it become so habituated to childish and futile pursuits that thereafter it would always stick to the surface of things and would be unable to penetrate more deeply. But for all that we must not fall into the error of those who occupy their minds exclusively with serious and lofty issues, only to find that after much toil they gain, not the profound science they desired, but mere confusion. We must therefore practise these easier tasks first, and above all methodically, so that by following accessible and familiar paths we may grow accustomed, just as if we were playing a game, to penetrating always to the deeper truth of things. In this way we shall gradually find - much sooner than we might expect - that it is just as easy to deduce, on the basis of evident principles, many propositions which appear very difficult and complicated.

Some will perhaps be surprised that in this context, where we are searching for ways of making ourselves more skilful at deducing some truths on the basis of others, we make no mention of any of the precepts with which dialecticians ^1 suppose they govern human reason. They prescribe certain forms of reasoning in which the conclusions follow with such irresistible necessity that if our reason relies on them, even though it takes, as it were, a rest from considering a particular inference clearly and attentively, it can nevertheless draw a conclusion which is certain simply in virtue of the form. But, as we have noticed, truth often slips through these fetters, while those who employ them are left entrapped in them. Others are not so frequently entrapped and, as experience shows, the cleverest sophisms hardly ever deceive anyone who makes use of his untrammelled reason; rather, it is usually the sophists themselves who are led astray.

Our principal concern here is thus to guard against our reason's taking a holiday while we are investigating the truth about some issue; so we reject the forms of reasoning just described as being inimical to our project. Instead we search carefully for everything which may help our mind to stay alert, as we shall show below. But to make it even clearer that the aforementioned art of reasoning contributes nothing whatever to knowledge of the truth, we should realize that, on the basis of their method, dialecticians are unable to formulate a syllogism with a true conclusion unless they are already in possession of the substance of the conclusion, i.e. unless they have previous knowledge of the very truth deduced in the syllogism. It is obvious therefore that they themselves can learn nothing new from such forms of reasoning, and hence that ordinary dialectic is of no use whatever to those who wish to investigate the truth of things. Its sole advantage is that it sometimes enables us to explain to others arguments which are already known. It should therefore be transferred from philosophy to rhetoric.

Rule Eleven

If, after intuiting a number of simple propositions, we deduce something else from them, it is useful to run through them in a continuous and completely uninterrupted train of thought, to reflect on their relations to one another, and to form a distinct and, as far as possible, simultaneous conception of several of them. For in this way our knowledge becomes much more certain, and our mental capacity is enormously increased.

This is a good time to explain more clearly what was said about mental intuition in Rules Three and Seven. In one passage we contrasted it with deduction,^1 and in another only with enumeration,^2 which we defined as an inference drawn from many disconnected facts. But in the same passage we said that a simple deduction of one fact from another is performed by means of intuition.

It was necessary to proceed in that way, because two things are required for mental intuition: first, the proposition intuited must be clear and distinct; second, the whole proposition must be understood all at once, and not bit by bit. But when we think of the process of deduction as we did in Rule Three, it does not seem to take place all at once: inferring one thing from another involves a kind of movement of our mind. In that passage, then, we were justified in distinguishing intuition from deduction. But if we look on deduction as a completed process, as we did in Rule Seven, then it no longer signifies a movement but rather the completion of a movement. That is why we are supposing that the deduction is made through intuition when it is simple and transparent, but not when it is complex and involved. When the latter is the case, we call it 'enumeration' or 'induction', since the intellect cannot simultaneously grasp it as a whole, and its certainty in a sense depends on memory, which must retain the judgements we have made on the individual parts of the enumeration if we are to derive a single conclusion from them taken as a whole.

All these distinctions had to be made in order to make clear the meaning of this Rule. Rule Nine dealt only with mental intuition; Rule Ten only with enumeration. The present Rule explains the way in which these two operations aid and complement each other; they do this so thoroughly that they seem to coalesce into a single operation, through a movement of thought, as it were, which involves carefully intuiting one thing and passing on at once to the others.

There is, we should point out, a twofold advantage in this fact: it facilitates a more certain knowledge of the conclusion in question, and it makes the mind better able to discover other truths. As we have said, conclusions which embrace more than we can grasp in a single intuition depend for their certainty on memory, and since memory is weak and unstable, it must be refreshed and strengthened through this continuous and repeated movement of thought. Say, for instance, in virtue of several operations, I have discovered the relation between the first and the second magnitude of a series, then the relation between the second and the third and the third and fourth, and lastly the fourth and fifth: that does not necessarily enable me to see what the relation is between the first and the fifth, and I cannot deduce it from the relations I already know unless I remember all of them. That is why it is necessary that I run over them again and again in my mind until I can pass from the first to the last so quickly that memory is left with practically no role to play, and I seem to be intuiting the whole thing at once.

One cannot fail to see that in this way the sluggishness of the mind is redressed and its capacity even enlarged. But in addition we must note that the greatest advantage of this Rule lies in the fact that by reflecting on the mutual dependence of simple propositions we acquire the habit of distinguishing at a glance what is more, and what is less, relative, and by what steps the relative may be reduced to the absolute. For example, if I run through a number of magnitudes which are continued proportionals, I shall be struck by the following points. It is just as easy for me to recognize the relation between the first and the second magnitude, as between the second and the third, the third and fourth, etc., and the act of conceiving is exactly similar in each case. But it is more difficult for me to form a simultaneous conception of the relation of the second magnitude to the first and the third; and it is much more difficult still to conceive the way in which it depends on the first and fourth magnitudes, etc. These considerations enable me to understand why it is that, given only the first and second magnitudes, I can easily find the third and fourth, etc.: the reason is that the discovery is made by means of particular and distinct acts of conceiving. But if only the first and the third are given, it will not be so easy for me to discern the intermediate magnitude, for this can be done only by means of an act of conceiving which simultaneously involves two of the acts just mentioned. If only the first and the fourth magnitudes are given, it is even more difficult to intuit the two intermediate ones, for in this case three acts of conceiving are simultaneously involved. So, as a logical consequence, it might seem even more difficult to find the three intermediate magnitudes given the first and fifth. Yet this is not the case, owing to a further reason, which is that, although four acts of conceiving are joined together in the present example, they can nevertheless be separated, since four is divisible by another number. So I can obtain the third magnitude alone on the basis of the first and the fifth, then the second on the basis of the first and the third, etc. If one is used to reflecting on these and similar matters, whenever one investigates a new problem one will immediately recognize the source of the difficulty and the simplest method for dealing with it. And that is the greatest aid to knowledge of the truth.

Rule Twelve

Finally we must make use of all the aids which intellect, imagination, sense-perception, and memory afford in order, firstly, to intuit simple propositions distinctly; secondly, to combine ^1 correctly the matters under investigation with what we already know, so that they too may be known; and thirdly, to find out what things should be compared with each other so that we make the most thorough use of all our human powers.

This Rule sums up everything that has been said above, and sets out a general lesson the details of which remain to be explained as follows.

Where knowledge of things is concerned, only two factors need to be considered: ourselves, the knowing subjects, and the things which are the objects of knowledge. As for ourselves, there are only four faculties which we can use for this purpose, viz. intellect, imagination, sense-perception and memory. It is of course only the intellect that is capable of perceiving the truth, but it has to be assisted by imagination, sense-perception and memory if we are not to omit anything which lies within our power. As for the objects of knowledge, it is enough if we examine the following three questions: What presents itself to us spontaneously? How can one thing be known on the basis of something else? What conclusions can be drawn from each of these? This seems to me to be a complete enumeration and to omit nothing which is within the range of human endeavour.

Turning now to the first factor, I should like to explain at this point what the human mind is, what the body is and how it is informed ^1 by the mind, what faculties within the composite whole promote knowledge of things, and what each particular faculty does; but I lack the space, I think, to include all the points which have to be set out before the truth about these matters can be made clear to everyone. For my aim is always to write in such a way that I make no assertions on matters which are apt to give rise to controversy, without first setting out the reasons which led me to make them and which I think others may find convincing too.

But since I cannot do that here, it will be sufficient if I explain as briefly as possible what, for my purposes, is the most useful way of conceiving everything within us which contributes to our knowledge of things. Of course you are not obliged to believe that things are as I suggest. But what is to prevent you from following these suppositions if it is obvious that they detract not a jot from the truth of things, but simply make everything much clearer? This is just what you do in geometry when you make certain assumptions about quantity, which in no way weaken the force of the demonstrations, even though in physics you often take a different view of the nature of quantity.

Let us then conceive of the matter in the following way. First, in so far as our external senses are all parts of the body, sense-perception, strictly speaking, is merely passive, even though our application of the senses to objects involves action, viz. local motion; sense-perception occurs in the same way in which wax takes on an impression from a seal. It should not be thought that I have a mere analogy in mind here: we must think of the external shape of the sentient body as being really changed by the object in exactly the same way as the shape of the surface of the wax is altered by the seal. This is the case, we must admit, not only when we feel some body as having a shape, as being hard or rough to the touch etc., but also when we have a tactile perception of heat or cold and the like. The same is tr