Chapter XXII: Part IX: , p. 475, 1890), Dr. Johnstone Stoney, F. R. S., published an (12)

“The recognition by one person of another’s personality takes
place by means to some extent identical with the means by which
he is conscious of his own personality. The idea of the second
personality, which is as much as to say that second personality
itself, enters within the field of direct consciousness of
the first person, and is as immediately perceived as his ego,
though less strongly. At the same time, the opposition between
the two persons is perceived, so that the externality of the
second is recognised.” (“The Law of Mind,” p. 558.)

This is the scheme of “otherness” which, in the case of the Neo-Kantians, particularly the French section, represented by M. Pillon, M. Renouvier, and others, has proved such a snare. To these thinkers, (as indeed to the late Prof. T. H. Green, of Oxford, though in a less degree,) the so-called external world lies in “other” thinking subjects—in “foreign centres of representations.” The free-trade doctrine has verily penetrated to the philosophic region—the wholesale admission of foreign wares to the detriment of home products. Why should I place the content of that so-called external world, which, external or internal, is my very own inalienably, in a centre of representation other than my own, thus making my cognition of it rest entirely upon the “ejective” plane? It is only when I discover, as I must sooner or later, that there is nothing in the report of an “outsider” (or in any number of them) beyond what I credit him or her with in my own consciousness; and that the outsider is on the same plane as other objects, it is only then that the mystification is cleared up. I do _not_ cognise, or recognise, the external at second-hand. The “note” of otherness is simply another term more or less in the cosmical series.

It is, however, not only with the familiar “other selves” of ordinary life that we are confronted in Synechism. In the creed of animism

“Millions of spiritual creatures walk the earth,”

and Mr. Peirce speaks of “spiritual influences” (p. 559) as having at least no hindrance presented to them by his doctrine. But he has some other shadowy personalities at command, which, it must be confessed, are well calculated to give us pause. “There should be something like[78] personal consciousness in bodies of men who are in intimate and intensely sympathetic communion.... None of us can fully realise which the minds of corporations are.... But the law of mind clearly points to the existence of such personalities.” It is probably true that the “minds of corporations,” must ever present an insoluble riddle of perversity to the suburban dweller, vexed with the mockery of paving and lighting. But we need not linger over this speculation, for there are other shades behind.

“If such a fact is capable of being made out anywhere it
should be in the Church.... Surely a personality ought to have
developed in that Church, in that ‘bride of Christ,’ as they
call it.” (“Man’s Glassy Essence,” pp. 21-22.)

A PERSONAL CREATOR.

Bearing our ecclesiastical divisions in mind, it is difficult to conceive the unity of a “corporate personality” of this kind, but, to let that pass, it may be remarked that, when any one begins to imagine that there are others in the universe besides himself, he is not, as a rule, content with two or three companions of his solitude. They come in battalions. Thus, behind the other selves, corporate personalities and spiritual influences of Synechism, there looms a transcendent personality. “A genuine evolutionary philosophy,” we are told, “... is so far from being antagonistic to the idea of a personal Creator, that it is really inseparable from that idea.” And a philosophy of pseudo-evolutionism is “hostile to all hopes of personal relations to God.” (“The Law of Mind,” p. 557.)

Mr. Peirce thus assigns to his first cause a place in the _continuum_ of ideas, and says that if there is a personal God we must have a direct perception of that person and “indeed be in personal communication with him.” The difficulty, he admits, is that if this be so, how is it possible that the existence of this being should ever have been doubted by anybody. And the only answer he can at present make is, that “facts that stand before our face and eyes, and stare us in the face, are far from being in all cases the ones most easily discerned. That,” he adds, “has been remarked from time immemorial.” (“The Law of Mind,” pp. 558-559.)

One of the ablest of living philosophical writers, Professor Veitch, of Glasgow University, puts it somewhat similarly, though with his own realistic coloring, when he says:

“God, if at all, must rise above the line of finite regress; He
cannot be a cause in that; He cannot be a cause dependent on
another cause; He must be somewhere, or at some point, in the
line of an otherwise endless scientific regress, there, above
it, yet related to it, and in it; otherwise He is nothing for
us.” (“Knowing and Being,” p. 320.)

The parallelism is worth noting. Those views embody what has been the contention of the present writer throughout this paper, _with this most notable difference_: that no term of a series may thus transcend the series, or be other than on a level with the other terms, being itself only a term, a link, in the series itself. And with this falls forever the idea of a cause uncaused.

Yet am _I_ not _in_ the series? For all that is in the series is mine every percept, every concept; so that, “extravagant” as it may appear, it is _I who am the series_. In other words, the ego is the universe-synthesis, and the universe-synthesis the ego.

Is Mr. Peirce prepared to take the consequences of that which his Synechism leads to?

G. M. MCCRIE.

FOOTNOTES:

[67] From τύχη, chance.

[68] _Tychism_ again comes to the front in the succeeding number of _The Monist_, (Vol. III, No. 1,) in an article by Mr. Peirce, entitled “Man’s Glassy Essence.”

[69] Dr. Carus, in his review of Mr. Peirce’s doctrines, (_The Monist_, Vol. II, No. 4, p. 575,) notes this positivistic-constructionism.

[70] Cf. T. H. Green, _Prolegomena to Ethics_, Ch. II, p. 63.

[71] Nos. 258, 59, 61, August, 1892. _Miss Naden’s World-Scheme._

[72] In a note to this passage was appended a quotation from a pamphlet by Dr. E. Cobham Brewer as a practical instance of the objective being, on the antiquated subject-object plane, actually superseded. Suppose a very remote star to become extinct, the “vibrations” would continue to “travel” towards a spectator situated on our planet for years, it may be for centuries. So that the spectator, ultimately, “sees” that which does not even exist. Dr. Brewer’s comment, which cannot be considered any contribution to a satisfactory _rationale_, is: “the objects, however, must have existed, or no messenger could have been sent from their courts.” Evidently, in this case, that which is sent is, at least, as good as the sender—is, in fact, the self-same thing. Only, in that case, what of the extinct object?

[73] Or to put it in another form, any one idea, and the timing of this idea are really two ideas, although, as we shall see later, they may be inseparable in practice.

[74] Cf., in this connection, the results of experiments by Cheselden, as far back as 1727 on congenitally blind persons, couched for double cataract.

[75] Much more inclusive, also, than the Relational Theory of the Neo-Kantians.

[76] _The Monist_, Vol. III, No. 1.

[77] Mr. Peirce uses the word “instant” to mean a point of time, and “moment” to mean an infinitesimal duration.

[78] The phrase, “something like,” is significant, when we remember, (see _ante_,) that with Mr. Peirce the excitants were “something like” the excited feelings.

THE FOURTH DIMENSION.

MATHEMATICAL AND SPIRITUALISTIC.

INTRODUCTORY.

The tendency to generalise long ago led mathematicians to extend the notion of three-dimensional space, which is the space of sensible representation, and to define aggregates of points, or spaces, of more than three dimensions, with the view of employing these definitions as useful means of investigation. They had no idea of requiring people to imagine four-dimensional things and worlds, and they were even still less remote from requiring of them to believe in the real existence of a four-dimensioned space. In the hands of mathematicians this extension of the notion of space was a mere means devised for the discovery and expression, by shorter and more convenient ways, of truths applicable to common geometry and to algebra operating with more than three unknown quantities. At this stage, however, the spiritualists came in, and coolly took possession of this private property of the mathematicians. They were in great perplexity as to where they should put the spirits of the dead. To give them a place in the world accessible to our senses was not exactly practicable. They were compelled, therefore, to look around after some _terra incognita_, which should oppose to the spirit of research inborn in humanity an insuperable barrier. The residence of the spirits had to be a place inaccessible to our senses and full of mystery to the mind. This property the four-dimensioned space of the mathematicians possessed. With an intellectual perversity which science has no idea of, these spiritualists boldly asserted, first, that the whole world was so situated in a four-dimensioned space as a plane might be situated in the space familiar to us, secondly, that the spirits of the dead lived in such a four-dimensioned space, thirdly, that these spirits could accordingly act upon the world and, consequently, upon the human beings resident in it, exactly as we three-dimensioned creatures can produce effects upon things that are two-dimensional; for example, such effects as that produced when we shatter a lamina of ice, and so influence some possibly existing two-dimensioned _ice_-world.

Since spiritualism, under the leadership of the Leipsic Professor Zöllner, thus proclaimed the existence of a four-dimensioned space, this notion, which the mathematicians are thoroughly master of,—for in all their operations with it, though they have forsaken the path of actual representability, they have never left that of the truth,—this notion has also passed into the heads of lay persons who have used it as a catchword, ordinarily without having any clear idea of what they or any one else mean by it. To clear up such ideas and to correct the wrong impressions of cultured people who have not a technical mathematical training, is the purpose of the following pages. A similar elucidation was aimed at in the tracts which Schlegel (Riemann, Berlin, 1888) and Cranz (Virchow-Holtzendorff’s Sammlung, Nos. 112 and 113) have published on the so-called fourth dimension. Both treatises possess indubitable merits, but their methods of presentation are in many respects too concise to give a lay mind any profound comprehension of the subject. The author, accordingly, has been able to add to the reflections which these excellent treatises offer, a great deal that appears to him necessary for a thorough explanation in the minds of non-mathematicians of the notion of the fourth dimension.

I.

THE CONCEPT OF DIMENSION.

Many text-books of stereometry begin with the words: “Every body has three dimensions, length, breadth, and thickness.” If we should ask the author of a book of this description to tell us the length, breadth, and thickness of an apple, of a sponge, or of a cloud of tobacco smoke, he would be somewhat perplexed and would probably say, that the definition in question referred to something different. A cubical box, or some similar structure, whose angles are all right angles and whose bounding surfaces are consequently all rectangles is the only body of which it can at all be unmistakably asserted that there are three principal directions distinguishable in it, of which any one can be called the length, any other the breadth, and any third the thickness. We thus see that the notions of length, breadth, and thickness are not sufficiently clear and universal to enable us to derive from them any idea of what is meant when it is said that every body possesses three dimensions, or that the space of the world is three-dimensional.

This distinction may be made sharper and more evident by the following considerations: We have, let us suppose, a straight line on which a point is situated, and the problem is proposed to determine the position of the point on the line in an unequivocal manner. The simplest way to solve this is, to state how far the point is removed in the one or the other direction from some given fixed point; just as in a thermometer the position of the surface of the mercury is given by a statement of its distance in the direction of cold or heat from a predetermined fixed point—the point of freezing water. To state, therefore, the position of a point on a straight line, the sole datum necessary is a single number, for beforehand we have fixed upon some standard line, like the centimetre, and some definite point to which we give the value zero, and have also previously decided in what direction from the zero-point, points must be situated whose position is expressed by positive numbers, and also in what direction those must lie whose position is expressed by negative numbers. This last mentioned fact, that a _single_ number is sufficient to determine the place of a point in a straight line, is the real reason why we attribute to the straight line or to any part of it a single dimension.

More generally, we call every totality or system, of infinitely numerous things, _one_-dimensional, in which _one_ number is all that is requisite to determine and distinguish any particular one of these things amidst the entire totality. Thus, time is one-dimensional. We, as inhabitants of the earth, have naturally chosen as our unit of time, the period of the rotation of the earth about its axis, namely, the day, or a definite portion of a day. The zero-point of time is regarded in Christian countries as the year of the birth of Christ, and the positive direction of time is the time _subsequent_ to the birth of Christ. These data fixed, all that is necessary to establish and distinguish any definite point of time amid the infinite totality of all the points of time, _is a single number_. Of course this number need not be a whole number, but may be made up of the sum of a whole number and a fraction in whose numerator and denominator we may have numbers as great as we please. We may, therefore, also say that the totality of all conceivable numerical magnitudes, or of only such as are greater than one definite number and smaller than some other definite number, is one-dimensional.

We shall add here a few additional examples of one-dimensioned magnitudes presented by geometry. First, the circumference of a circle is a one-dimensional magnitude, as is every curved line, whether it returns into itself or not. Further, the totality of all equilateral triangles which stand on the same base is one-dimensional, or the totality of all circles that can be described through two fixed points. Also, the totality of all conceivable cubes will be seen to be one-dimensional, provided they are distinguished, not with respect to position, but with respect to magnitude.

In conformity with the fundamental ideas by which we define the notion of a one-dimensional manifoldness, it will be seen that the attribute _two_-dimensional must be applied to all totalities of things in which _two_ numbers are necessary (and sufficient) to distinguish any determinate individual thing amid the totality. The simplest two-dimensioned complex which we know of is the plane. To determine accurately the position of a point in a plane, the simplest way is to take two axes at right angles to each other, that is, fixed straight lines, and then to specify the distances by which the point in question is removed from each of these axes.

This method of determining the position of a point in a plane suggested to the celebrated philosopher and mathematician Descartes the fundamental idea of analytical geometry, a branch of mathematics in which by the simple artifice of ascribing to every point in a plane two numerical values, determined by its distances from the two axes above referred to, planimetrical considerations are transformed into algebraical. So, too, all kinds of curves that graphically represent the dependence of things on time, make use of the fact that the totality of the points in a plane is two-dimensional. For example, to represent in a graphical form the increase of the population of a city, we take a horizontal axis to represent the time, and a perpendicular one to represent the numbers which are the measures of the population. Any two lines, then, whose lengths practical considerations determine, are taken as the unit of time, which we may say is a year, and as the unit of population, which we will say is one thousand. Some definite year, say 1850, is fixed upon as the zero point. Then, from all the equally distant points on the horizontal axis, which points stand for the years, we proceed in directions parallel to the other axis, that is, in the perpendicular direction, just so much upwards as the numbers which stand for the population of that year require. The terminal points so reached, or the curve which runs through these terminal points, will then present a graphic picture of the rates of increase of the population of the town in the different years. The rectangular axes of Descartes are employed in a similar way for the construction of barometer curves, which specify for the different localities of a country the amount of variation of the atmospheric pressure during any period of time. Immediately next to the plane the surface of the earth will be recognised as a two-dimensional aggregate of points. In this case geographical latitude and longitude supply the two numbers that are requisite accurately to determine the position of a point. Also, the totality of all the possible straight lines that can be drawn through any point in space is two-dimensional, as we shall best understand if we picture to ourselves a plane which is cut in a point by each of these straight lines and then remember that by such a construction every point on the plane will belong to some one line and, _vice versa_, a line to every point, whence it follows that the totality of all the straight lines which pass through the point assigned are of the same dimensions as the totality of the points of the imagined plane.

The question might be asked, In what way and to what extent in this case is the specification of _two_ numbers requisite and sufficient to determine amid all the rays which pass through the specified point a definite individual ray? To get a clear idea of the problem here involved, let us imagine the ray produced far into the heavens, where some quite definite point will correspond to it. Now, the position of a point in the heavens depends, as does the position of a point on all spherical surfaces, on two numbers. In the heavens these two numbers are ordinarily supplied by the two angles called altitude, or the distance above the plane of the horizon, and azimuth, or the angular distance between the circle on which the altitude is measured and the meridian of the observer. It will be seen thus that the totality of all the luminous rays that an eye, conceived as a point, can receive from the outer world is two-dimensional, and also that a luminous point emits a two-dimensional group of luminous rays. It will also be observed, in connection with this example, that the two-dimensional totality of all the rays that can be drawn through a point in space is something different from the totality of the rays that pass through a point but are required to lie in a given plane. Such a group of objects as the last-named one, is a one-dimensional totality.

Now that we have sufficiently discussed the attributes that are characteristic of one and two-dimensional aggregates, we may, without any further investigation of the subject, propose the following definition, that, generally, _an n-dimensional totality of infinitely numerous things is such, with respect to which the specification of n numbers is necessary and sufficient to indicate a definite individual amid the totality of all the infinitely numerous individuals of the group_.

Accordingly, the point-aggregate made up of the world-space which we inhabit, is a three-dimensional totality. To get true bearings in this space and to define any determinate point in it, we have therefore to lay through any point which we take as our zero-point three axes at right angles to each other, one running from right to left, one backwards and forwards, and one upwards and downwards. We then join each two of these axes by a plane and are enabled thus to specify the position of every point in space by the three perpendicular distances by which the point in question is removed in a positive or negative sense from these three planes. It is customary to denote the numbers which are the measures of these three distances by _x_, _y_, and _z_, the positive _x_, positive _y_, and positive _z_ ordinarily being reckoned in the right hand, the forward, and the upward directions from the origin. If now, with direct reference to this fundamental axial system, any particular specification of _x_, _y_, and _z_ be made, there will, by such an operation, be cut out and isolated from the three-dimensional manifoldness of all the points of space a totality of less dimensions. If, for example, _z_ is equal to seven units or measures, this is equivalent to a statement that only the two-dimensional totality of the points is meant, which constitute the plane that can be laid at right angles to the upward-passing _z_-axis at a distance of seven measures from the zero-point. Consequently, every imaginable equation between _x_, _y_, and _z_ isolates and defines a two-dimensional aggregate of points. If two different equations obtain between _x_, _y_, and _z_, two such two-dimensional totalities will be isolated from among all the points of space. But as these last must have some one-dimensional totality in common, we may say that the co-existence of two equations between _x_, _y_, and _z_ defines a one-dimensional totality of points, that is to say a straight line, a line curved in a plane, or even, perhaps, one curved in space. It is evident from this that the introduction of the three axes of reference forms a bridge between the theory of space and the theory of equations involving three variable quantities, _x_, _y_, _z_. The reason that the theory of space cannot thus be brought into connection with algebra in general, that is, with the theory of indefinitely numerous equations, but only with the algebra of three quantities, _x_, _y_, _z_, is simply to be sought in the fact that space, as we picture it, can only have three dimensions.

We have now only to supply a few additional examples of _n_-dimensional totalities. All particles of air are four-dimensional in magnitude when in addition to their position in space we also consider the variable densities which they assume, as they are expressed by the different heights of the barometer in the different parts of the atmosphere. Similarly, all conceivable spheres in space are four-dimensional magnitudes, for their centres form a three-dimensional point-aggregate, and around each centre there may be additionally conceived a one-dimensional totality of spheres, the radii of which can be expressed by every numerical magnitude from zero to infinity. Further, if we imagine a measuring stick of invariable length to assume every conceivable position in space, the positions so obtained will constitute a five-dimensional aggregate. For, in the first place, one of the extremities of the measuring stick may be conceived to assume a position at every point of space, and this determines for one extremity alone of the stick a three-dimensional totality of positions; and secondly, as we have seen above, there proceeds from every such position of this extremity a two-dimensional totality of directions, and by conceiving the measuring-stick to be placed lengthwise in every one of these directions we shall obtain all the conceivable positions which the second extremity can assume, and consequently, the dimensions must be 3 plus 2 or 5. Finally, to find out how many dimensions the totality of all the possible positions of a square, invariable in magnitude, possesses, we first give one of its corners all conceivable positions in space, and we thus obtain three dimensions. One definite point in space now being fixed for the position of one corner of the square, we imagine drawn through this point all possible lines, and on each we lay off the length of the side of the square and thus obtain two additional dimensions. Through the point obtained for the position of the second corner of the square we must now conceive all the possible directions drawn that are perpendicular to the line thus fixed, and we must lay off once more on each of these directions the side of the square. By this last determination the dimensions are only increased by one, for only one one-dimensional totality of perpendicular directions is possible to one straight line in one of its points. Three corners of the square are now fixed and therewith the position of the fourth also is uniquely determined. Accordingly, the totality of all equal squares which only differ from one another by their position in space, constitutes a manifoldness of six dimensions.

II.

THE INTRODUCTION OF THE NOTION OF FOUR-DIMENSIONAL POINT-AGGREGATES, PERMISSIBLE.

In the preceding section it was shown that we can conceive not only of manifoldnesses of one, two, and three dimensions, but also of manifoldnesses of _any_ number of dimensions. But it was at the same time indicated that our world-space, that is, the totality of all conceivable _points_ that differ only in respect of position, cannot in agreement with our notions of things possess more than three dimensions. But the question now arises, whether, if the progress of science tends in such a direction, it is permissible to extend the notion of space by the introduction of point-aggregates of more than three dimensions, and to engage in the study of the properties of such creations, although we know that notwithstanding the fact that we may conceptually establish and explore such aggregates of points, yet we cannot picture to ourselves these creations as we do the spatial magnitudes which surround us, that is, the regular three-dimensional aggregates of points.

To show the reader clearly that this question must be answered in the affirmative, that the extension of our notion of space is permissible, although it leads to things which we cannot perceive by our senses, I may call the reader’s attention to the fact that in arithmetic we are accustomed from our youth upwards to extensions of ideas, which, accurately viewed, as little admit of graphic conception as a four-dimensional space, that is, a point-aggregate of four dimensions. By his senses man first reaches only the idea of whole numbers—the results of counting. The observation of primitive peoples[79] and of children clearly proves that the essential decisive factors of counting are these three: First, we abstract, in the counting of things, completely from the individual and characteristic attributes of these things, that is, we consider them as homogeneous. Second, we associate individually with the things which we count other homogeneous things. These other things are even now, among uncivilised peoples, the ten fingers of the two hands. They may, however, be simple strokes, or, as in the case of dice and dominoes, black points on a white background. Third, we substitute for the result of this association some concise symbol or word; for example, the Romans substituted for three things counted, three strokes placed side by side, namely: III; but for greater numbers of things they employed abbreviated signs. The Aztecs, the original inhabitants of Mexico, had time enough, it seems, to express all the numbers up to nineteen by equal circles placed side by side. They had abbreviated signs only for the numbers 20, 400, 8000, and so forth. In speaking, some one same sound might be associated with the things counted; but this method of counting is nowadays employed only by clocks: the languages of men since prehistoric times have fashioned concise words for the results of the association in question. From the notion of number, thus fixed as the result of counting, man reached the notion of the addition of two numbers, and thence the notion that is the inverse of the last process, the notion of subtraction. But at this point it clearly appears that not every problem which may be propounded is soluble; for there is no number which can express the result of the subtraction of a number from one which is equally large or from one which is smaller than itself. The primary school pupil who says that 8 from 5 “won’t go” is perfectly right from his point of view. For there really does not exist any result of counting which added to eight will give five.

If humanity had abided by this point of view and had rested content with the opinion that the problem “5 minus 8” is not solvable, the science of arithmetic would never have received its full development, and humanity would not have advanced as far in civilisation as it has. Fortunately, men said to themselves at this crisis: “If 5 minus 8 won’t go, we’ll _make it go_; if 5 minus 8 does not possess an intelligible meaning, we will simply give it one.” As a fact, things which have not a meaning always afford men a pleasing opportunity of investing them with one. The question is, then, what significance is the problem “5 minus 8” to be invested with?

The most natural and, therefore, the most advantageous solution undoubtedly is to abide by the original notion of subtraction as the inverse of addition, and to make the significance of 5 minus 8 such, that for 5 minus 8 plus 8 we shall get our original minuend 5. By such a method all the rules of computation which apply to real differences will also hold good for unreal differences, such as 5 minus 8. But it then clearly appears that all forms expressive of differences in which the number that stands before the minus direction is less by an equal amount than that which follows it may be regarded as equal; so that the simplest course seems to be to introduce as the common characteristic of all equal differential forms of this description a common sign, which will indicate at the same time the difference of the two numbers thus associated. Thus it came about, that for 5 minus 8, as well as for every differential form which can be regarded as equal thereto the sign “-3” was introduced. But in calling differential forms of this description numbers, the notion of number was extended and a new domain was opened up, namely, the domain of negative numbers.

In the further development of the science of arithmetic, through the operation of division viewed as the inverse of multiplication, a second extension of the idea of number was reached, namely, the notion of fractional numbers as the outcome of divisions that had led to numbers hitherto undefined. We find, thus, that the science of arithmetic throughout its whole development has strictly adhered to the principle of conformity and consistency and has invested every association of two numbers, which before had no significance, by the introduction of new numbers, with a real significance, such that similar operations in conformity with exactly the same rules could be performed with the new numbers, viewed as the results of this association, as with the numbers which were before known and perfectly defined. Thus the science proceeded further on its way and reached the notions of irrational, imaginary, and complex numbers.

The point in all this, which the reader must carefully note, is, that all the numbers of arithmetic, with the exception of the positive whole numbers, are artificial products of human thought, invented to make the language of arithmetic more flexible, and to accelerate the progress of science. All these numbers lack the attributes of representability.

No man in the world can picture to himself “minus three trees.” It is possible, of course, to know that when three trees of a garden have been cut down and carried away, that three are missing, and by substituting for “missing” the inverse notion of “added,” we may say, perhaps, that “minus three trees” are added. But this is quite different from the feat of imagining a negative number of trees. We can only picture to ourselves a number of trees that results from actual counting, that is, a positive whole number. Yet, notwithstanding all this, people had not the slightest hesitation in extending the notion of number. Exactly so must it be permitted us in geometry to extend the notion of space, even though such an extension can only be mentally defined and can never be brought within the range of human powers of representation.

In mathematics, in fact, the extension of any notion is admissible, provided such extension does not lead to contradictions with itself or with results which are well established. Whether such extensions are necessary, justifiable, or important for the advancement of science is a different question. It must be admitted, therefore, that the mathematician is justified in the extension of the notion of space as a point-aggregate of three dimensions, and in the introduction of space or point-aggregates of more than three dimensions, and in the employment of them as means of research. Other sciences also operate with things which they do not know exist, and which, though they are sufficiently defined, cannot be perceived by our senses. For example, the physicist employs the ether as a means of investigation, though he can have no sensory knowledge of it. The ether is nothing more than a means which enables us to comprehend mechanically the effects known as action at a distance and to bring them within the range of a common point of view. Without the assumption of a material which penetrates everything, and by means of whose undulations impulses are transmitted to the remotest parts of space, the phenomena of light, of heat, of gravitation, and of electricity would be a jumble of isolated and unconnected mysteries. The assumption of an ether, however, comprises in a systematic scheme all these isolated events, facilitates our mental control of the phenomena of nature, and enables us to produce these phenomena at will. But it must not be forgotten in such reflections that the ether itself is even a greater problem for man, and that the ether-hypothesis does not solve the difficulties of phenomena, but only puts them in a unitary conceptual shape. Notwithstanding all this, physicists have never had the least hesitation in employing the ether as a means of investigation. And as little do reasons exist why the mathematicians should hesitate to investigate the properties of a four-dimensioned point-aggregate, with the view of acquiring thus a convenient means of research.

III.

THE INTRODUCTION OF THE IDEA OF FOUR-DIMENSIONED POINT-AGGREGATES OF SERVICE TO RESEARCH.

From the concession that the mathematician has the right to define and investigate the properties of point-aggregates of more than three dimensions, it does not necessarily follow that the introduction of an idea of this description is of value to science. Thus, for example, in arithmetic, the introduction of operations which spring from involution, as involution and its two inverse operations proceed from multiplication, is undoubtedly permitted. Just as for “_a_ times _a_ times _a_” we write the abbreviated symbol “_a_³,” (which we read, _a_ to the third power,) and investigate in detail the operation of involution thus defined, so we might also introduce some shorthand symbol for “_a_ to the _a_ᵗʰ power to the _a_ᵗʰ power” and thus reach an operation of the fourth degree, which would regard _a_ as a passive number and the number 3, or any higher number, as the active number, that is, as the number which indicates how often _a_ is taken as the base of a power whose exponent may be _a_, or “_a_ to the _a_ᵗʰ,” or “_a_ to the _a_ᵗʰ to the _a_ᵗʰ power.”

But the introduction of such an operation of the fourth degree has proved itself to be of no especial value to mathematics. And the reason is that in the operation of involution the law of commutation does not hold good. In addition, the numbers to be added may be interchanged and the introduction of multiplication is therefore of great value. So, also, in multiplication the numbers which are combined, that is, the factors, may be changed about in any way, and thus the introduction of involution is of value. But in involution the base and the exponent cannot be interchanged, and consequently the introduction of any higher operation is almost valueless.

But with the introduction of the idea of point-aggregates of multiple dimensions the case is wholly different. The innovation in question has proved itself to be not only of great importance to research, but the progress of science has irresistibly forced investigators to the introduction of this idea, as we shall now set forth in detail.

In the first place, algebra, especially the algebraical theory of systems of equations, derives much advantage from the notion of multiple dimensioned spaces. If we have only three unknown quantities, _x_, _y_, _z_, the algebraical questions which arise from the possible problems of this class admit, as we have above seen, of geometrical representation to the eye. Owing to this possibility of geometrical representation, some certain simple geometrical ideas like “moving,” “lying in,” “intersecting,” and so forth, may be translated into algebraical events. Now, no reason exists why algebra should stop at three variable quantities; it must in fact take into consideration any number of variable quantities.

For purposes of brevity and greater evidentness, therefore, it is quite natural to employ geometrical forms of speech in the consideration of more than three variables. But when we do this, we assume, perhaps without really intending to do so, the idea of a space of more than three dimensions. If we have four variable quantities, _x_, _y_, _z_, _u_, we arrive, by conceiving attributed to each of these four quantities every possible numerical magnitude, at a four-dimensioned manifoldness of numerical quantities, which we may just as well regard as a four-dimensioned aggregate of points. Two equations which exist on this supposition between _x_, _y_, _z_, and _u_, define two three-dimensioned aggregates of points, which intersect, as we may briefly say, in a two-dimensioned aggregate of points, that is, in a surface; and so on. In a somewhat different manner the determination of the contents of a square or a cube by the involution of a number which stands for the length of its sides, leads to the notion of four-dimensioned structures, and, consequently, to the notion of a four-dimensioned point-space. When we note that _a_² stands for the contents of a square, and _a_³ for the contents of a cube, we naturally inquire after the contents of a structure which is produced from the cube as the cube is produced from the square and which also will have the contents _a_⁴. We cannot, it is true, clearly picture to ourselves a structure of this description, but we can, nevertheless, establish its properties with mathematical exactness.[80] It is bounded by 8 cubes just as the cube is bounded by 6 squares; it has 16 corners, 24 squares, and 32 edges, so that from every corner 4 edges, 6 squares, and 4 cubes proceed, and from every edge 3 squares and 3 cubes.

Yet despite the great service to algebra of this idea of multiple-dimensioned space, it must be conceded that the conception although convenient is yet not indispensable. It is true, algebra is in need of the idea of multiple dimensions, but it is not so absolutely in need of the idea of _point_-aggregates of multiple dimensions.

This notion is, however, necessary and serviceable for a profound comprehension of geometry. The system of geometrical knowledge which Euclid of Alexandria created about three hundred years before Christ, supplied during a period of more than two thousand years a brilliant example of a body of conclusions and truths which were mutually consistent and logical. Up to the present century the idea of elementary geometry was indissolubly bound up with the name of Euclid, so that in England where people adhered longest to the rigid deductive system of the Grecian mathematician, the task of “learning geometry” and “reading Euclid” were until a few years ago identical. Every proposition of this Euclidian system rests on other propositions, as one building-stone in a house rests upon another. Only the very lowest stones, the foundations, were without supports. These are the axioms or fundamental propositions, truths on which all other truths are, directly or indirectly, founded, but which themselves are assumed without demonstration as self-evident.

But the spirit of mathematical research grew in time more and more critical, and finally asked, whether these axioms might not possibly admit of demonstration. Especially was a rigid proof sought for the eleventh axiom of Euclid, which treats of parallels.

After centuries of fruitless attempts to prove Euclid’s eleventh axiom, Gauss, and with him Bolyai and Lobatschewsky, Riemann and Helmholtz, finally stated the decisive reasons why any attempt to prove the axiom of the parallels must necessarily be futile. These reasons consist of the fact that though this axiom holds good enough in the world-space such as we do and can conceive it, yet three-dimensioned spaces are ideally conceivable though not capable of mental representation, where the axiom does not hold good. The axiom was thus shown to be a mere fact of _observation_, and from that time on there could no longer be any thought of a deductive demonstration of it. In view of the intimate connection, which both in an historical and epistemological point of view exists between the extension of the concept of space and the critical examination of the axioms of Euclid, we must enter at somewhat greater length into the discussion of the last mentioned propositions.

Of the axioms which Euclid premises to his geometry, only the following three are really geometrical axioms:

_Eighth axiom_: Magnitudes which coincide with one another are equal to one another.

_Eleventh axiom_: If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.

_Twelfth axiom_: Two straight lines cannot inclose a [finite] space.

The numerous proofs which in the course of time were adduced in demonstration of these axioms, especially of the eleventh, all turn out on close examination to be pseudo-proofs. Legendre drew attention to the fact that either of the following axioms might be substituted for the eleventh:

_a_) Through a point there can be drawn to a straight line, within the plane which joins the point with the line, one and one line only which shall not intersect the first (parallels) however far the two lines may be produced;

_b_) If two parallel lines are cut by a third straight line, the interior alternate angles will be equal.

_c_) The sum of the angles of a triangle is equal to two right angles, that is, to the angle of a straight line or 180°.

By the aid of any one of these three assertions, the eleventh axiom of Euclid may be proved, and, _vice versa_, by the aid of the latter each of the three assertions may be proved, of course with the help of the other two axioms, eight and twelve. The perception that the eleventh axiom does not admit of demonstration without the employment of one of the foregoing substitutes may best be gained from the consideration of congruent figures. Every reader will remember from his first instruction in geometry that the congruence of two triangles is demonstrated by the superposition of one triangle on the other and by then ascertaining whether the two completely coincide, no assumptions being made in the determination except those above mentioned.

In the case of triangles which are congruent as are I and II in the preceding cut, this coincidence may be effected by the simple _displacement_ of one of the triangles; so that even a two-dimensional being, supposed to be endowed with powers of reasoning, but only capable of picturing to itself motions within a plane, also might convince itself that the two triangles I and II could be made to coincide. But a being of this description could not convince itself in like manner of the congruence of triangles I and III. It would discover the equality of the three sides and the three angles, but it could never succeed in so superposing the two triangles on each other as to make them coincide. A three-dimensioned being, however, can do this very easily. It has simply to turn triangle I about one of its sides and to shove the triangle, thus brought into the position of its reflection in a mirror, into the position of triangle III. Similarly, triangles II and III may be made to coincide by moving either out of the plane of the paper around one of its sides as axis and turning it until it again falls in the plane of the paper. The triangle thus turned over can then be brought into the position of the other.

Later on we shall revert to these two kinds of congruence: “congruence by displacement” and “congruence by circumversion.” For the present we will start from the fact that it is always possible within the limits of a plane to take a triangle out of one position and bring it into another without altering its sides and angles. The question is, whether this is only possible in the plane, or whether it can also be done on other surfaces.

We find that there are certain surfaces in which this is possible, and certain others in which it is not. For instance, it is impossible to move the triangle drawn on the surface of an egg into some other position on the egg’s surface without a distension or contraction of some of the triangle’s parts. On the other hand, it is quite possible to move the triangle drawn on the surface of a sphere into any other position on the sphere’s surface without a distension or contraction of its parts. The mathematical reason of this fact is, that the surface of a sphere, like the plane, has everywhere the same curvature, but that the surface of an egg at different places has different curvatures. Of a plane we say that it has everywhere the curvature zero; of the surface of a sphere we say it has everywhere a positive curvature, which is greater in proportion as the radius is smaller. There are surfaces also which have a constant negative curvature; these surfaces exhibit at every point in directions proceeding from the same side a partly concave and a partly convex structure, somewhat like the centre of a saddle. There is no necessity of our entering in any detail into the character and structure of the last-mentioned surfaces.

Intimately related with the plane, however, are all those surfaces, which, like the plane, have the curvature zero; in this category belong especially cylindrical surfaces and conical surfaces. A sheet of paper of the form of the sector of a circle may, for example, be readily bent into the shape of a conical surface. If two congruent triangles, now, be drawn on the sheet of paper, which may by displacement be translated the one into the other, these triangles will, it is plain, also remain congruent on the conical surface; that is, on the conical surface also we may displace the one into the other; for though a bending of the figures will take place, there will be no distension or contraction. Similarly, there are surfaces which, like the sphere, have everywhere a constant positive curvature. On such surfaces also every figure can be transferred into some other position without distension or contraction of its parts. Accordingly, on all surfaces thus related to the plane or sphere, the assumption which underlies the eighth axiom of Euclid, that it is possible to transfer into any new position any figure drawn on such surfaces without distortion, holds good.

The eleventh axiom in its turn also holds good on all surfaces of constant curvature, whether the curvature be zero or positive; only in such instances instead of “straight” line we must say “shortest” line. On the surface of a sphere, namely, two shortest lines, that is, arcs of two great circles, always intersect, no matter whether they are produced in the direction of the side at which the third arc of a great circle makes with them angles less than two right angles, or, in the direction of the other side, where this arc makes with them angles of more than two right angles. On the plane, however, two straight lines intersect only on the side where a third straight line that meets them makes with them interior angles less than two right angles.

The twelfth axiom of Euclid, finally, only holds good on the plane and on the surfaces related to it, but not on the sphere or other surfaces which, like the sphere, have a constant positive curvature. This also accounts for the fact that one of the three postulates which we regarded as substitutes for the eleventh axiom, though valid for the plane, is not true for the surface of a sphere; namely, the postulate that defines the sum of the angles of a triangle. This sum in a plane triangle is two right angles; in a spherical triangle it is more than two right angles, the spherical triangle being greater, the greater the excess the sum of its angles is above two right angles. It will be seen, from these considerations, that in geometries in which curved surfaces and not fixed planes are studied, the axioms of Euclid are either all or partially false.

The axioms of geometry thus having been revealed as facts of experience, the question suggested itself whether in the same way in which it was shown that different two-dimensional geometries were possible, also different three-dimensional systems of geometry might not be developed; and consequently what the relations were in which these might stand to the geometry of the space given by our senses and representable to our mind. As a fact, a three-dimensional geometry can be developed, which like the geometry of the surface of an egg will exclude the axiom that a figure or body can be transferred from any one part of space to any other and yet remain congruent to itself. Of a three-dimensional space in which such a geometry can be developed we say, that it has no constant measure of curvature.

The space which is representable to us, and which we shall henceforth call the _space of experience_, possesses, as our experiences without exception confirm, the especial property that every bodily thing can be transferred from any one part of it to any other without suffering in the transference any distension or any contraction. The space of experience, therefore, has a constant measure of curvature. The question, however, whether this measure of curvature is zero or positive, that is, whether the space of experience possesses the properties which in two-dimensioned structures a plane possesses, or whether it is the three-dimensioned analogon of the surface of a sphere is one which future experience alone can answer. If the space of experience has a constant positive measure of curvature which is different from zero, be the difference ever so slight, a point which should move forever onward in a straight line, or, more accurately expressed, in a shortest line, would sometime, though perhaps after having traversed a distance which to us is inconceivable, ultimately have to arrive from the opposite direction at the place from which it set out, just as a point which moves forever onward in the same direction on the surface of a sphere must ultimately arrive at its starting point, the distance it traverses being longer the greater the radius of the sphere or the smaller its curvature.