Rule IV
We need a method if we are to investigate the truth of things.
Rule V
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 knowledge of all the rest.
Rule VI
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.
Rule VII
In order to make our knowledge 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.
The observance of these precepts is necessary in order that we may admit to the class of certitudes those truths which, I previously said, are not immediate deductions from the first self-evident principles. For sometimes the succession of inferences is so long that when we arrive at our results we do not readily remember the whole road that has led us so far; and therefore I say that we must aid the weakness of our memory by a continuous movement of thought.
For instance, suppose that by excessive mental acts I have learnt first the relation between the magnitudes A and B, then that between B and C then that between C and D, and finally that between D and E; I do not on this account see the relation between A and E; and I cannot form a precise conception of it from the relations I know already, unless I remember them all. So I will run through these several times over in a continuous movement of the imagination, in which intuition of each relation is simultaneous with transition to the next, until I have learnt to pass from the first to the last so quickly that I leave hardly any parts to the care of memory and seem to have a simultaneous intuition of the whole. In this way memory is aided, and a remedy found for the slowness of the understanding, whose scope is in a way enlarged.
I add that the movement must be uninterrupted because it often happens that people who try to make some deduction in too great haste and from remote principles do not run over the whole chain of intermediate conclusions with sufficient care to avoid making many unconsidered jumps. But assuredly the least oversight immediately breaks the chain and destroys all the certainty of the conclusion. Further, I say that enumeration is required in order to complete our knowledge. For other precepts are helpful in resolving very many questions, but it is only enumeration that enables us to form a true and certain judgment about anything whatever that we apply our mind to, and, by preventing anything from simply escaping our notice, seems to give us some knowledge of everything.
This enumeration, or induction, ranging over everything relevant to some question we have set before us, consists in an inquiry so careful and accurate that it is a certain and evident conclusion that no mistaken omission has been made. When, therefore, we perform this, if the thing we are looking for still eludes us, we are at any rate so much the wiser, that we can see with certainty the impossibility of our finding it by any way known to us; and if we have managed to run over all the ways of attaining it that are humanly practicable (as will often be the case) then we may boldly affirm that knowledge of it has been put quite out of reach of the human mind.
It must further be observed that by adequate enumeration or induction I mean exclusively the sort that makes the truth of conclusions more certain than any other type of proof, apart from simple intuition, makes it. Whenever a piece of knowledge cannot be reduced to simple intuition (if we throw off the fetters of syllogism), this method is the only one left to us that we must entirely rely on. For whenever we have deduced one thing from others, if the inference was an evident one, the case is already reduced to genuine intuition. If on the other hand, we make a single inference from many separate data, our understanding is often not capacious enough to grasp them all in one act of intuition, and in that case we must content ourselves with the certitude of this further operation. In the same way, we cannot visually distinguish all the links of a longish chain in one glance (intuitu); but nevertheless, if we have seen the connexion of each with the next, this will justify us in saying that we have actually seen how the first is connected to the last.
I said this operation must be adequate, because it may often be defective, and consequently liable to error. For sometimes our enumeration includes a number of very obvious points; nevertheless, the least omission breaks the chain and destroys all the certainty of the conclusion. Again, sometimes our enumeration covers everything but the items are not all distinguished, so that we have only a confused knowledge of the whole.
Sometimes, then, this enumeration must be complete, and sometimes it must be distinct; but sometimes neither condition is necessary., This is why I say merely that the enumeration must be adequate. For example, if I want to establish by enumeration how many kinds of things are corporeal, or are in some way the objects of sensation, I shall not assert that there are just so many without first assuring myself that my enumeration comprises all the kinds and distinguishes each from the others. But if I want to show in the same way that the rational soul is not corporeal, a complete enumeration will not be needed; it will be enough to comprise all bodies in a certain number of classes and show that the rational soul cannot be referred to any of these. Again, if I want to show by enumeration that the area of a circle is greater than the areas of all other figures of equal periphery, I need not give a list of all figures; it is enough to prove this in some particular cases, and then we may inductively extend the conclusion to all other figures.
I added further that the enumeration must be orderly for the defects already enumerated cannot be remedied more directly than they are by an orderly scrutiny of all items. Again, it is often the case that nobody could live long enough to go through each several item that concerns the matter in hand; either because there are too many such items, or because we should keep going back to the same items. But if we arrange these items in the ideal order, then as a rule they will be reduced to certain classes; and it may be enough to have an exact view of one class, or of some member of each class, or of some classes rather than others; at any rate, we shall not ever go futilely over and over the same point. This is a great help; a proper arrangement often enables us to deal rapidly and easily with an apparently unmanageable multitude of details.
This order of enumeration is variable, and depends on the free choice of the individual; skill in devising it requires that we bear in mind the terms of Rule V. There are, indeed, a good many ingenious, trivialities where the device wholly consists in effecting this sort of arrangement. For example, suppose you want to make the best anagram you can by transposing the letters of a certain name. Here there is no need to advance from easy to difficult cases, or to distinguish between what is underived and what is dependent; for these problems do not arise here. It will be enough to determine an order for examining transpositions of letters, so that you never go over the same arrangement twice over, and to divide the possible arrangements into certain classes in a way that makes the most likely source of a solution immediately apparent. The task will then often be no long one-child’s play, in fact.
Really, though, these last three Rules are inseparable; in most cases they have all to be taken into account at once, and they all go together towards the completeness of the method. The order of setting them forth did not much matter; I have explained them here briefly because almost all the rest of this treatise will be a detailed exposition of what is here summed up in a general way.
Rule VIII
If in the series of things to be examined we come across something which our inte
llect is unable to intuit sufficiently well, we must stop at that point, and refrain from the superfluous task of examining the remaining items.
Rule IX
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.
Rule X
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.
Rule XI
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.
It is in place here to give a clearer exposition of what I said before about intuition (Rules III and VII). In the one place I contrasted intuition with deduction; in the other, merely with enumeration. (I defined enumeration as an inference made from many separate data put together; the simple deduction of one thing from another is made, I said, by intuition.) This procedure was necessary because intuition must satisfy two conditions: first, our understanding of a proposition must be clear and distinct; secondly, it must be one simultaneous whole without succession. Now if we are thinking of the act of deduction, as in Rule III, it has not the appearance of being a simultaneous whole; rather, it involves a movement of the mind in which we infer one thing from another. Here, then, we were justified in distinguishing it from intuition. If on the other hand we attend to deduction as something already accomplished, as in the notes on Rule VII, then the term does not stand any longer for such a movement, but for the result of the movement. In that sense, then, I assume that a deduction is something intuitively seen, when it is simple and clear, but not when it is complex and involved; for that, I used the term ‘ enumeration’ or ‘induction ‘. For the latter sort of deduction cannot be grasped all at once; its certainty depends in a way on memory, which must retain judgments about the various points enumerated in order that we may put them all together and get some single conclusion.
All these distinctions had to be made in order to bring out the meaning of the present Rule. Rule IX dealt only with intuition, and Rule X only with enumeration; then comes this Rule, explaining how these two activities cooperate-operate and supplement one another-seem, in fact, to merge into a single activity, in which there is a movement of thought such that attentive intuition of each point is simultaneous with transition to the next.
I mention two advantages of this: the greater certainty in our knowledge of the conclusion we have in view, and the greater aptitude of our mind for making further discoveries. As I said, when conclusions are too complex to be held in a single act of intuition, their certainty depends on memory; and since memory is perishable and weak, it must be revived and strengthened by this continuous and repeated movement of thought. For example, suppose I have learnt, in a number of successive mental acts, the relations between magnitudes 1 and 2, magnitudes 2 and 3, magnitudes 3 and 4, and, finally, magnitudes 4 and 5; this does not make me see the relation between magnitudes 1 and 5, nor can I deduce it from the ones I already know, unless I remember them all; accordingly, I must run over them in thought again and again, until I pass from the first to the last so quickly that I have hardly any parts to the care of memory, but seem to have a simultaneous intuition of the whole.
In this way, as no-one can fail to see, the slowness of the mind is remedied, and its capacity enlarged. But it must further be noticed, as the chief advantage of this Rule, that by reflection upon the interdependence of simple propositions we acquire the practice of rapidly discerning their degrees of derivativeness and the steps of their reduction to what is underived. For example, if I run through a series of magnitudes in continued proportion, I shall reflect on all the following points: it is by concepts of the same level that I discern the ratio of term 1 to term 2, of term 2 to term 3, of term 3 to term 4, and so on, and there are no degrees of difficulty in conceiving these ratios; but it is more difficult for me to conceive the way that term 2 depends on terms 1 and 3 together, and still more difficult to conceive how the same term 2 depends on terms 1 and 4, and so on. This shows me the reason why, given merely terms 1 and 2, 1 can easily find terms 3, 4, etc.; for this is done by means of particular and distinct concepts. But given merely terms 1 and 3, I cannot so easily find their (geometric) mean; this can be done only by means of a concept involving two together of the concepts just mentioned. Given only terms 1 and 4, it is still more difficult to get an intuition of the two mean (proportionals), since this involves three simultaneous concepts. Consequently it might seem to be even more difficult to find three mean (proportionals) given terms 1 and 5; but, for a further reason, this is not the case. Although we have here four concepts joined together, they can be separated, because 4 is divisible by another number; so I can begin by trying to find term 3 from terms 1 and 5, and then go on to find term 2 from terms 1 and 3
. He who is accustomed to reflect on such matters recognizes at once, when he examines each new problem, the source of the difficulty and the simplest method ; and this helps very much towards knowledge of the truth.
Rule XII
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 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 all that has been said already, and gives a general account of the various particulars that had to be explained: as follows.
Only two things are relevant to knowledge: ourselves, the subjects of knowledge; and the objects to be known. In ourselves there are just four faculties that can be used for knowledge: understanding, imagination, sense, and memory. Only the understanding is capable of perceiving truth, but it must be aided by imagination, sense, and memory, so that we may not leave anything undone that lies within our endeavor. On the side of the object of knowledge, it is enough to consider three points: first, what is obvious on its own account; secondly, the means of knowing one thing by another; lastly, the inferences that can be made from any given thing. This enumeration seems to me to be complete, and not to leave out anything that can be attained by human endeavor.
Turning therefore to the first point , I should like to expound here the nature of the human mind and body, the way that the soul is the form of I the body, the various cognitive faculties that exist in the whole composed and their several activities; but I think I have not enough space to contain all that would have to be premised before the truth on these matters could be made clear to everybody. For it is my aim always to write in such a way that, before making any assertion on the ordinary controversial points, I give the reasons that have led me to my view and might, in my opinion, convince other people as well.
Since such an exposition is now impossible, I shall content myself with explaining as briefly as possible the way of conceiving our means of knowledge that is most useful for our purpose. You need not, if you like, believe that things are really so; but what is to stop us from following out these suppositions, if it appears that they do not do away with any facts, but only make everything much clearer ? In the same way, geometry makes certain suppositions about quantity; and although in physics we may often hold a different view as to the nature of quantity, the force of geometrical demonstrations is not in any way weaker on that account.
My first supposition, then, is that the external senses qua bodily organs may indeed be actively applied to their objects, by locomotion, but their having sensation is properly something merely passive, just like the shape Wuram) that wax gets from a seal. You must not think this expression is just an analogy; the external shape of the sentient organ must be regarded as really changed by the object, in exactly the same way as the shape of the surface of the wax is changed by the seal. This supposition must be made, not only as regards tactual sensations of shape, hardness, roughness, etc., but also as regards those of heat, cold, and so on. So also for the other senses. The first opaque part of the eye receives an image (figuram) in this way from many-colored illumination; and the first membrane of the ears, nostrils, or tongue that is impervious to the object perceived similarly derives a new shape from the sound, odor, or savor.
It is of great help to regard all these facts in this way; for no object of sense is more easily got than shape, which is both felt and seen. And no error can follow from our making this supposition rather than any other, as may be proved thus: The concept of shape is so common and simple that it is involved in every sensible object. For example, on any view of color it is undeniably extended and therefore has shape. Let us then beware of uselessly assuming, and rashly imagining, a new entity; let us not deny anyone else’s view of color, but let us abstract from all aspects except shape, and conceive the difference between white, red, blue, etc., as being like the difference between such shapes as these: