Читать книгу Writings of Charles S. Peirce: A Chronological Edition, Volume 8 - Charles S. Peirce - Страница 37

22

The Architecture of Theories

[Initial Version]

July-August 1890

Houghton Library

Of the fifty or hundred systems of philosophy that have been advanced at different times of the world’s history, perhaps the larger number have been, not so much results of historical evolution, as happy thoughts which have accidentally occurred to their authors. An idea which has been found interesting and fruitful has been adopted, developed, and forced to yield explanations of all sorts of phenomena. The English have been particularly given to this way of philosophizing; witness, Hobbes, Hartley, Berkeley, James Mill. Nor has it been by any means useless labour; it shows us what the true nature and value of the ideas developed are, and in that way affords serviceable materials for philosophy. Just as if a man, being seized with the conviction that paper was a good material to make things of, were to go to work to build a papier mâché house, with roof of roofing-paper, foundations of pasteboard, windows of paraffined paper, chimneys, bath tubs, locks, etc., all of different forms of paper, his experiment would probably afford valuable lessons to builders, while it would certainly make a detestable house, so those one-idea’d philosophies are exceedingly interesting and instructive, and yet are quite unsound.

The remaining systems of philosophy have been of the nature of reforms, sometimes amounting to radical revolutions, suggested by certain difficulties which have been found to beset systems previously in vogue; and such ought certainly to be in large part the motive of any new theory. This is like partially rebuilding a house. The faults that have been committed are, first, that the dilapidations have generally not been sufficiently thorough going, and second, that not sufficient pains have been taken to bring the additions into deep harmony with the really sound parts of the old structure.

When a man is about to build a house, what a power of thinking he has to do, before he can safely break ground! With what pains he has to excogitate the precise wants that are to be supplied! What a study to ascertain the most available and suitable materials, to determine the mode of construction to which those materials are best adapted, and to answer a hundred such questions! Now without riding the metaphor too far, I think we may safely say that the studies preliminary to the construction of a great theory should be at least as deliberate and thorough as those that are preliminary to the building of a dwelling-house.

That systems ought to be constructed architectonically has been preached since Kant; but I do not think the full import of the maxim has by any means been apprehended. What I would recommend is that every person who wishes to form an opinion concerning fundamental problems, should first of all make a complete survey of human knowledge, should take note of all the valuable ideas in each branch of science, should observe in just what respect each has been successful and where it has failed, in order that in the light of the thorough acquaintance so attained of the available materials for a philosophical theory and of the nature and strength of each, he may proceed to the study of what the problem of philosophy consists in, and of the proper way of solving it. I must not be understood as endeavoring to state fully all that these preparatory studies should embrace; on the contrary, I purposely slur over some points in order to give emphasis to my special recommendation of today, namely, to make a systematic study of the conceptions out of which a philosophical theory may be built, in order to ascertain what place each conception may fitly occupy in such a theory and to what uses it is adapted.

I would begin, for example, with inquiring what are the governing conceptions of modern logic,—I do not mean the aimless gabble of Bradley and his opponent Bosanquet,—but the logic of those who keep real scientific, and especially mathematical, reasoning steadily in view, at close range. It is no new thing to attach high philosophical importance to the conceptions of logic. The categories of Aristotle, as well as those of Kant, were derived from the logical analysis of propositions. But what would be new would be to use the conceptions so obtained to form the framework of the philosophical theory.

Three peculiar concepts of great generality, I may well call them category-concepts, run through logic from beginning to end. This appears most clearly in the best development of logic; but were I to undertake to make it clear in that way, it might be thought that my judgment as to the doctrine of logic had been warped by my prepossession in regard to those three conceptions. I will therefore consider logic as it is presented in Aldrich’s Rudiments (1690), a book taught for a century and a half at Oxford, which is as guiltless of any fanciful thinking,—or, for the matter of that, of any independent thinking at all,—as academical treatise ever could be. I will begin at the beginning of this work, and will mention every logical division it contains until I dare task the reader’s patience no further. The conceptions I expect to find are: 1^st, the idea of existing or acting independently, without reference to anything else; 2^nd, the idea of action and reaction between two things; 3^rd, the idea of mediation, or of one thing bringing others into connection. But the conceptions are here stated in a rather too concrete and special form, so as to be readily intelligible. We must not expect precisely these ideas, but ideas resembling them; for ideas which resemble one another contain a common idea.

§1. Aldrich begins by dividing the logical operations of the mind into (1) simple apprehension, (2) judgment, and (3) reasoning. Simple apprehension is conceiving an idea independently of others. Judgment is recognizing two ideas to be bound together in a fact. Reasoning is thought which brings two ideas into connection.

Simple apprehension is incomplex or complex. Here we have the first two concepts, of independent position, and of connectedness.

Judgment is affirmative or negative. An affirmative judgment is one which is made independently of any other; a negative judgment is one which revolts against another possible judgment. Again the first two concepts.

§2. A proposition consists of (1) subject, (2) predicate, and (3) copula. The subject is the term which is conceived as existing independently, the predicate is connected with it, the copula is that which brings subject and predicate into connection.

Parallel to the three operations of the mind, we have the three products, (1) terms, (2) propositions, and (3) syllogisms.

§3. Words are either (1) categorematic, or (2) syncategorems. Categorematic words are those which can be subjects or predicates. Syncategorems are such words as all, some, not. That is, there are words having independent meaning, and words whose meaning depends on their connections with other words.

A name is either (1) singular, or (2) common. That is, it names something having an independent existence or a class existing only in the connection of resemblance between singulars.

A name is either (1) finite, or (2) infinite, that is, formed by the negative particle prefixed to another name, as non-man. This follows the division of propositions into affirmative and negative.

A name is either (1) positive, (2) privative, or (3) negative. A positive name is an independent one, a negative name is one which is connected with a quality not possessed; a privative name is one in which this connection is suggested by a really analogous case where the quality denied is possessed. Thus, when a stone is said to be non-seeing, this is a negative predicate; when a man is said to be blind, this is a privative predicate.

A name is either (1) univocal, (2) equivocal, or (3) analogous. A univocal name has but one meaning, unconnected with any other. An equivocal name has two or more meanings which it reunites in a blind, irrational way. An analogous name has two or more meanings which are connected by a mediating reason.

Anybody who chooses to do so can easily go through the whole treatise in this way, and convince himself that these concepts,—or independency, blind connection, mediation,—enter into every distinction of logic. Independency or arbitrariness is the idea of something being a first, and no second to anything. Blind connection is the idea of something being second to something else, without any third to bring the connection about. Mediation is the idea of thirdness to a first and second that are thus brought into relation.

Of course, many maxims of logic require attention in the prolegomena of philosophy. I will illustrate one of these. The sole justification that any hypothesis can have is that it explains facts and renders them intelligible. If then a philosopher like Herbert Spencer tells us that the principle of the conservation of energy or anything else of the sort is a primitive law of nature, or if another tells me that feeling is an ultimate property of protoplasm, I ask them how they know this. To say that a law is primitive or a fact ultimate is to say that this law or this fact is absolutely inexplicable and unintelligible. But this is a hypothesis. And what does this hypothesis serve to explain? Nothing, except that it is difficult to show from what the law or fact in question is derived; a fact sufficiently explicable by the vera causa of our ignorance. There are, I grant, some facts of which it is absurd to ask for an explanation; as, for example, that things are many and not all alike, for it is of the nature of a hypothesis to explain likenesses or definite relations of unlikeness. The mere absence of definite relations is not a thing requiring any explanation at all. To ask what determines anything to indeterminacy is a senseless and meaningless question. But when a fact or principle is determinate and peculiar, to silence inquiry as to its derivation by the theory that no theory can explain it, this is an affront to reason, and a plain inconsistency. A philosopher who confesses his inability to explain why space has three dimensions, no more and no less, or anything else of like difficulty, may be pardoned, so long as he offers no other excuse than his ignorance. But if he sets up a theory of space which represents its three-dimensional character to be something absolutely inexplicable, now and forever, that theory is to be condemned as involving the hypothesis that no hypothesis is possible. It is the old sophism of the Cretan.

The science which, next after logic, may be expected to throw the most light upon philosophy, is mathematics. It is historical fact, I believe, that it was the mathematicians Thales, Pythagoras, and Plato who created metaphysics, and that metaphysics has always been the ape of mathematics. Seeing how the propositions of geometry flowed demonstratively from a few postulates, men got the notion that the same must be true in philosophy. But of late mathematicians have fully agreed that the axioms of geometry (as they are wrongly called) are not by any means evidently true. Euclid, be it observed, never pretended they were evident; he does not reckon them among his κοιναὶ ἔννοιαι, or things everybody knows,¹ but among the αἴτηματα, postulates, or things the author must beg you to admit, because he is unable to prove them. At any rate, it is now agreed that there is no reason whatever to think the sum of the three angles of a triangle precisely equal to 180°. It is generally admitted that the evidence is that the departure from 180° (if there is any) will be greater the larger the triangle, and in the case of a triangle having for its base the diameter of the earth’s orbit and for its apex the furthest star, the sum hardly can differ, according to observation, so much as 0″.1. It is probable the discrepancy is far less. Nevertheless, there is an infinite number of different possible values, of which precisely 180° is only one; so that the probability is as 1 to ∞, or 0 to 1, that the value is just 180°. In other words, it seems for the present impossible to suppose the postulates of geometry precisely true. The matter is reduced to one of evidence; and as absolute precision is beyond the reach of direct observation, so it can never be rendered probable by evidence, which is indirect observation.

Thus, the postulates of geometry must go into the number of things approximately true. It may be thousands of years before men find out whether the sum of the three angles of a triangle is greater or less than 180°; but the presumption is it is one or the other.

Now what is metaphysics, which has always formed itself after the model of mathematics, to say to this state of things? The mathematical axioms being discredited, are the metaphysical ones to remain unquestioned? I trow not. There is one proposition, now held to be very certain, though denied throughout antiquity, namely that every event is precisely determined by general laws, which evidently never can be rendered probable by observation, and which, if admitted, must, therefore, stand as self-evident. This is a metaphysical postulate closely analogous to the postulates of geometry. Its fate is sealed. The geometrical axioms being exploded, this is for the future untenable. Whenever we attempt to verify a physical law, we find discrepancies between observation and theory, which we rightly set down as errors of observation. But now it appears we have no reason to deny that there are similar, though no doubt far smaller, discrepancies between the law and the real facts. As Lucretius says, the atoms swerve from the paths to which the laws of mechanics would confine them. I do not now inquire whether there is or not any positive evidence that this is so. What I am at present urging is that this arbitrariness is a conception occurring in logic, encouraged by mathematics, and ought to be regarded as a possible material to be used in the construction of a philosophical theory, should we find that it would suit the facts. We observe that phenomena approach very closely to satisfying general laws; but we have not the smallest reason for supposing that they satisfy them precisely.

Philosophy can draw from mathematics many other valuable ideas, at which I can here only hint.

Certain other conceptions of modern mathematics are indispensible to a philosophy which is to be upon the intellectual level of our age. In the first place, there is the conception of a space of more than three dimensions, ordinarily regarded as highly mysterious, but really easy enough. We can have no visual image of a space of three dimensions; we can only see its projections upon surfaces. A perspective view, or picture on a plane surface, represents all we can see at any one time. It would, therefore, be unreasonable to ask how a space of four or five dimensions would look, in any other than a projective sense. Take one of those glass paper-weights cut into the form of a polyhedron. Photographs of such a body of three dimensions taken from one point of view before and after giving it one turn are sufficient to determine how it must look however it be turned. Now, if it had four dimensions instead of three, the difference would be that by a certain peculiar effort we could turn it so as to give a different perspective form from any that the first two pictures would account for. Now, the reader does not know all about the geometry of three or even that of two dimensions; therefore, he cannot ask to have a complete idea of space of four dimensions; but the property that I have just mentioned, that in such a space by a peculiar effort a body could be turned so as to look in a way that ordinary perspective would not account for, this gives a sufficient idea of space of four dimensions. From this, all the other properties of that space could be deduced.

The principal use that philosophy has to make of the conception of n-dimensional space is in explaining why the dimensions of real space are three in number. We see from this study that it is the restriction in the number of dimensions which constitutes the fact to be explained. No explanation of why space has more than two dimensions is called for, because that is a mere indeterminacy. But why bodies should be restricted to move in three dimensions is one of the problems which philosophy has to solve.

Another mathematical conception to be studied is that of imaginary quantities. Several illusory accounts of this conception have been given; yet I believe the true account is the most usual. If one man can lift a barrel of flour, how many men can just lift a bushel? The answer one fourth of a man is absurd, because we are dealing with a kind of quantity which does not admit of fractions. But a similar solution in continuous quantity would be correct. So, there is a kind of quantity which admits of no negative values. Now, as the scheme of quantity with negatives is an extension of that of positive quantity, and as the scheme of positive quantity is an extension of that of discrete quantity, so the scheme of imaginary quantity is an extension of that of real quantity. To determine the position of a point upon a plane requires two numbers (like latitude and longitude) and if we choose to use a single letter to denote a position on a plane and choose to call what that letter signifies a quantity, then that quantity is one which can only be expressed by two numbers. Any point on the plane taken arbitrarily is called zero, and any other is called 1. Then, the point which is just as far on the other side of zero is, of course, −1; for the mean of 1 and −1 is 0. Take any three points ABC forming a triangle, and find another point, D, such that the triangle ACD is similar to the triangle ABC. Then, we naturally write (B − A):(C − A) = (C − A):(D − A). Apply this to the case where A is the zero point, B the point, 1, and D the point −1. Then C will be the point at unit distance from A, but at right angles to AB and this point will represent a quantity i such that i² = −1.

Imaginary quantities are put to two very different uses in mathematics. In some cases, as in the theory of functions, by considering imaginary quantity, and not limiting ourselves to real quantity (which is but a special case of imaginary quantity) we are able to form important generalizations and bind together different doctrines in a manner which leads us to great advances of the most practical kind. In other cases, as in geometry, we use imaginary quantity, because the problems to be solved are too difficult in the case of real quantity. It is very easy to say how many inflexions a curve of a certain description will have, if imaginary inflexions are included, but very difficult if we are restricted to real inflexions. Here imaginaries serve only a temporary purpose, and will one of these days give place to a more perfect doctrine.

To give a single illustration of the generalizing power of imaginaries, take this problem. Two circles have their centres at the distance D, their radii being R₁ and R₂. Let a straight line be drawn between their two points of intersection, at what distances will the two centres be from this line? Let these distances be x₁ and x₂, so that x₁ + x₂ = D. Then the square of the distance from one of the intersections of the circle to the line through their centres will be, by the Pythagorean proposition

These two equations give

Now the two circles may not really intersect at all, and yet x₁ and x₂ continue to be real. In other words, there is a real line between two imaginary intersections whose distance from the line of centres is

I do not know whether the theory of imaginaries will find any direct application in philosophy or not. But, at any rate, it is needed for the full comprehension of the mathematical doctrine of the absolute. For this purpose we must first explain the mathematical extension of the theory of perspective.² In the figure, let O be the eye, or centre of projection, let the line afeDc represent the plane³ of projection seen edgewise, and let the line ABCDE represent a natural plane seen edgewise. Any straight line, as OE, being drawn from the eye to any point on the natural plane, will cut the plane of the picture, or plane of projection, in e, the point which represents the point E. The mathematician (sometimes, the artist, too) extends this to the case where C, the natural point, is nearer the eye than the corresponding point of the picture. He also extends the same rule to the case where A, the natural point, and a, the point of the picture, are on opposite sides of the eye. Here is the whole principle of geometrical projection. Suppose now that three points in nature, say P, Q, R, really lie in one straight line. Then the three lines OP, OQ, OR, from these points to the eye, will lie in one plane. This plane will be cut by the plane of the picture in a straight line (because the intersection of any two planes is a straight line). Hence, the points p, q, r, which are the representations in the picture of P, Q, R, also lie in a straight line, and in general every straight line in nature is represented by a straight line in the picture and every straight line in the picture representing a line in a plane not containing the eye represents a straight line. Now according to the doctrine of Euclid, that the sum of the angles of a triangle is 180°, the parts of a natural plane at an infinite distance are also represented by a straight line in the picture, called the vanishing line of that plane. In the figure f is the vanishing line (seen endwise) of the plane ABCDE. Note how the passage from e through f to a corresponds to a passage from E off to infinity and back from infinity on the other side to A. Euclid or no Euclid, the geometer is forced by the principles of perspective to conceive the plane as joined on to itself through infinity. Geometers do not mean that there is any continuity through infinity; such an idea would be absurd. They mean that if a cannon-ball were to move at a continually accelerated rate toward the north so that its perspective representation should move continuously, it would have to pass through infinity and reappear at the south. There would really be a saltus at infinity and not motion proper. Persons absorbed in the study of projective geometry almost come to think there really is in every plane a line at infinity. But those who study the theory of functions regard the parts at infinity as a point. Both views are fictions which severally answer the purposes of the two branches of mathematics in which they are employed.

As I was saying, if the Euclidean geometry be true and the sum of the angles of a triangle equal 180°, it follows that the parts of any plane at infinity are represented by a right line in perspective. If that proposition be not true, still the perspective representation of everything remains exactly the same. Could some power suddenly change the properties of space so that the Euclidean doctrine should cease to be true,—and all things would look exactly as they did before. Only when you came to measure the real differences between objects you would find those distances, especially the long distances, essentially altered. There would be two possible cases. Either, according to Helmholtz’s supposition, you would find you could measure right through what used to be infinity, nothing being infinitely distant; so that a man might walk round space, somewhat as he might walk round our globe, only he would come round walking on the under side of the floor (if he had anything to hold him on); or a man could part his back hair by looking round space, only he would see himself upside down. Or, on the other hand, according to Lobatchewsky’s hypothesis, you would find that all the points of a certain (geometrical, not physical) sphere or ellipsoid about you were at any infinite distance. On the outside of this sphere there would be another world, which would be in singular geometrical antithesis to this. To every plane in this world (of course, extending through to the Jenseits) would correspond a point in that world. To every point in this world would correspond a plane lying wholly in the other world; and to every line partly in this world would correspond a line wholly in the other world. Any two points in the same world are at real finite distances from one another. Any two points in different worlds are at imaginary finite distances from one another. Any two planes both wholly in the other world or both partly in this are at real angles with one another; any two planes one wholly in the other world, the other partly in this are at imaginary angles with one another. About any axis partly in this world angles can never exceed 360°, about any axis wholly in the other world angles may be infinite and imaginary. Along any line partly in this world distances can be infinite and imaginary. Along any line wholly in the other world distances are all finite.

The manner in which distances are compared in such a space is this.

[…]

After pure mathematics, pass we to dynamics, field in our day of perhaps the grandest discovery science has ever made, I mean the Conservation of Energy. Besides what is to be learned from this, another lesson dynamics has for philosophy, long ago signalized by Whewell, but obscured by Stuart Mill’s skillful-shallow objections. This last mentioned lesson lies in the historical fact that dynamics, most perfect of physical sciences, has been in large part pumped up from the well of truth within us. No doubt observation afforded indispensible correctives and kept us from going quite wrong; but there were no observations at all adequate to determining the precise laws of motion. Only read Galileo, I pray you, and see how much he relies on il lume naturale, and how amazingly slender is his experimentation.

So, in dynamics, at least, simple theories have a tolerable chance of being true. Observe, that the character of a theory as to being simple or complicated depends entirely on the constitution of the intellect that apprehends it. Bodies left to themselves move in straight lines, and to us straight lines appear as the simplest of all curves. This is because when we turn an object about and scrutinize it, the line of sight is a straight line; and our minds have been formed under that influence. But abstractly considered, a system of like parabolas similarly placed, or any one of an infinity of systems of curves, is as simple as the system of straight lines. Again, motions and forces are combined according to the principle of the parallelogram, and a parallelogram appears to us a very simple figure. Yet the whole system of parallelograms is no more simple than any relief-perspective of them, or than any one of an infinity of other systems. As Sir Isaac Newton well said, geometry is but a branch of mechanics. No definition of the straight line is possible except that it is the path of a particle undisturbed by any force; and no definitions of parallels, etc. are possible which do not depend upon the definition of equal distances as measured by a rigid body, or other mechanical means.

Thus, in dynamics, the natural ideas of the human mind tend to approximate to the truth of nature, because the mind has been formed under the influence of dynamical laws. Now, logical considerations show that if there is no tendency for natural ideas to be true, there can be no hope of ever reaching true inductions and hypotheses. So that philosophy is committed to the postulate,—without which it has no chance of success,—[…]

Psychology has only lately become a positive science, and in my humble opinion the new views are now carried too far. I cannot see, for example, why psychologists should make such a bugbear of “faculties.” If in dynamics it has proved safe to rely upon our natural ideas, checked, controlled, and corrected by experience, why should not our natural ideas about mind, formed as they certainly have been under the influence of the true laws of mental action, be likely to approximate to the truth as much as natural ideas of space, force, and the like have been found to do? Upon this point, I must confess to entertaining somewhat heterodox opinions. The Herbartian philosophy, with the mode of reasoning which leads to it, seems to me thoroughly unsound and illusory,—though I fully admit the value and profundity of some of the suggestions of that philosophy. But to trust to such reasoning in the slightest degree seems to me ever so much less safe than trusting to one’s native or natural notions about mind, though these no doubt need to be modified by observation and experiment.

For my part it seems to me that the elementary phenomena of mind fall into three categories. First, we have feelings, comprising all that is immediately present, such as pleasure and pain, blue, cheerfulness, and the feeling which arises upon the contemplation of a complete theory. It is hard to define what I mean by feeling. If I say it is what is present, I shall be asked what I mean by present, and must confess I mean nothing but feeling again. The only way is to state how any state of consciousness is to be modified so as to render it a feeling, although feeling does not essentially involve consciousness proper. But imagine a state of consciousness reduced to perfect simplicity, so that its object is entirely unanalyzed, then that consciousness reduced to that rudimentary condition, unattainable by us, would be a pure feeling, and not properly consciousness at all. Let the quality of blue, for example, override all other ideas, of form, of contrast, of commencement or cessation, and there would be pure feeling. When I say that such impossible states exist as elements of all consciousness, I mean that there are ideas which might conceivably thus exist alone and monopolize the whole mind.

Besides feelings, we have in our minds sensations of reaction, as when a person blindfold suddenly runs against a post, when we make a muscular effort, or when any feeling gives way to another feeling. Suppose I had nothing in my mind but a feeling of blue, which were suddenly to give place to a feeling of red; then, at the instant of transition there would be a sense of reaction, my blue life being transmuted to red life. If I were now also endowed with a memory, that sense would continue for some time. This state of mind would be more than pure feeling, since in addition to the feeling of red a feeling analogous to blue would be present, and not only that but a sense of reaction between the two. This sense of reaction would itself carry along with it a peculiar feeling which might conceivably monopolize the mind to the exclusion of the feelings of blue and red. But were this to happen, though the feeling associated with a sense of reaction would be there, the sense of reaction as such would be quite gone; for a sense of reaction cannot conceivably exist independent of at least two feelings between which the reaction takes place. A feeling, then, is a state of mind having its own living quality, independent of any other. A sense of reaction, or say for short a sensation, is a state of mind containing two states of mind between which we are aware of a connection, even if that connection is no more than a contrast. No analysis can reduce such sensations to feelings. Looking at the matter from a physiological point of view, a feeling only calls for an excited nerve-cell,—or indeed a mere mass of excited nerve-matter without any cell, or shut up in any number of cells. But sensation supposes the discharge or excitation of a nerve-cell, or a transfer of excitement from one part of a mass of nerve-matter to another, or the spontaneous production or cessation of an excited condition.

Besides feelings and sensations, we have general conceptions; that is, we are conscious that a connection between feelings is determined by a general rule; or, looking at the matter from another point of view, a general conception is the being aware of being governed by a habit. Intellectual power is simply facility in taking habits and in following them in cases essentially analogous to, but in non-essentials widely remote from, the normal cases of sensation, or connection of feelings, under which those habits were formed.

The one primordial law of mental action is a tendency to generalization; that is, every connection between feelings tends to spread to neighboring feelings. If you ask what are neighboring feelings, it is like the question that was answered by the parable of the Good Samaritan. A neighboring feeling is simply a connected feeling. These connections are of two kinds, internal or manifest, and external or occult. A feeling is manifestly connected with feelings which it resembles or contrasts with; the connection is merely an identity of feeling. A feeling is occultly connected with feelings bound to it by some external power, as the roll of thunder with the flash of lightning.

The mental law belongs to a widely different category of law from physical laws. A physical law determines that a certain component motion must take place, otherwise the law is violated. But such absolute conformity is not required by the mental law. It does not call for any definite amount of assimilation in any case. Indeed such a precise regulation would be in downright conflict with the law. For it would instantly crystallize thought and prevent all further formation of habit.

The law of mind makes something the more likely to happen. It thus resembles the “non-conservative” forces of physics, such as viscosity and the like, which are due to chance encounters of molecules.

1. Except the proposition that two lines cannot enclose a space, though only one of the three best MSS places even this in the list. But what Euclid meant was that two straight lines can have but one intersection, which is evident.

2. In the main given in Brooke Taylor’s Perspective, 1715.

3. The reader need not be informed that a plane is not a plain. It is flat but need not be level. Thus the vertical wall of a room is a plane.