The Fourth Dimension

Having held before ourselves this outline of a conjecture of the world as four-dimensional, having roughly thrown together those facts of movement which we can see apply to our actual experience, let us pass to another branch of our subject.

The engineer uses drawings, graphical constructions, in a variety of manners. He has, for instance, diagrams which represent the expansion of steam, the efficiency of his valves. These exist alongside the actual plans of his machines. They are not the pictures of anything really existing, but enable him to think about the relations which exist in his mechanisms.

And so, besides showing us the actual existence of that world which lies beneath the one of visible movements, four-dimensional space enables us to make ideal constructions which serve to represent the relations of things, and throw what would otherwise be obscure into a definite and suggestive form.

From amidst the great variety of instances which lies before me I will select two, one dealing with a subject of slight intrinsic interest, which however gives within a limited field a striking example of the method of drawing conclusions and the use of higher space figures.[1]

The other instance is chosen on account of the bearing it has on our fundamental conceptions. In it I try to discover the real meaning of Kant’s theory of experience.

The investigation of the properties of numbers is much facilitated by the fact that relations between numbers are themselves able to be represented as numbers—e.g., 12, and 3 are both numbers, and the relation between them is 4, another number. The way is thus opened for a process of constructive theory, without there being any necessity for a recourse to another class of concepts besides that which is given in the phenomena to be studied.

The discipline of number thus created is of great and varied applicability, but it is not solely as quantitative that we learn to understand the phenomena of nature. It is not possible to explain the properties of matter by number simply, but all the activities of matter are energies in space. They are numerically definite and also, we may say, directedly definite, i.e. definite in direction.

Is there, then, a body of doctrine about space which, like that of number, is available in science? It is needless to answer: Yes; geometry. But there is a method lying alongside the ordinary methods of geometry, which tacitly used and presenting an analogy to the method of numerical thought deserves to be brought into greater prominence than it usually occupies.

The relation of numbers is a number.

Can we say in the same way that the relation of shapes is a shape?

We can.

To take an instance chosen on account of its ready availability. Let us take two right-angled triangles of a given hypothenuse, but having sides of different lengths (fig. 46). These triangles are shapes which have a certain relation to each other. Let us exhibit their relation as a figure.

Draw two straight lines at right angles to each other, the one HL a horizontal level, the other VL a vertical level (fig. 47). By means of these two co-ordinating lines we can represent a double set of magnitudes; one set as distances to the right of the vertical level, the other as distances above the horizontal level, a suitable unit being chosen.

Thus the line marked 7 will pick out the assemblage of points whose distance from the vertical level is 7, and the line marked 1 will pick out the points whose distance above the horizontal level is 1. The meeting point of these two lines, 7 and 1, will define a point which with regard to the one set of magnitudes is 7, with regard to the other is 1. Let us take the sides of our triangles as the two sets of magnitudes in question.

Then the point 7, 1, will represent the triangle whose sides are 7 and 1. Similarly the point 5, 5—5, that is, to the right of the vertical level and 5 above the horizontal level—will represent the triangle whose sides are 5 and 5 (fig. 48).

Thus we have obtained a figure consisting of the two points 7, 1, and 5, 5, representative of our two triangles. But we can go further, and, drawing an arc of a circle about O, the meeting point of the horizontal and vertical levels, which passes through 7, 1, and 5, 5, assert that all the triangles which are right-angled and have a hypothenuse whose square is 50 are represented by the points on this arc.

Thus, each individual of a class being represented by a point, the whole class is represented by an assemblage of points forming a figure. Accepting this representation we can attach a definite and calculable significance to the expression, resemblance, or similarity between two individuals of the class represented, the difference being measured by the length of the line between two representative points. It is needless to multiply examples, or to show how, corresponding to different classes of triangles, we obtain different curves.

A representation of this kind in which an object, a thing in space, is represented as a point, and all its properties are left out, their effect remaining only in the relative position which the representative point bears to the representative points of the other objects, may be called, after the analogy of Sir William R. Hamilton’s hodograph, a “Poiograph.”

Representations thus made have the character of natural objects; they have a determinate and definite character of their own. Any lack of completeness in them is probably due to a failure in point of completeness of those observations which form the ground of their construction.

Every system of classification is a poiograph. In Mendeléeff’s scheme of the elements, for instance, each element is represented by a point, and the relations between the elements are represented by the relations between the points.

So far I have simply brought into prominence processes and considerations with which we are all familiar. But it is worth while to bring into the full light of our attention our habitual assumptions and processes. It often happens that we find there are two of them which have a bearing on each other, which, without this dragging into the light, we should have allowed to remain without mutual influence.

There is a fact which it concerns us to take into account in discussing the theory of the poiograph.

With respect to our knowledge of the world we are far from that condition which Laplace imagined when he asserted that an all-knowing mind could determine the future condition of every object, if he knew the co-ordinates of its particles in space, and their velocity at any particular moment.

On the contrary, in the presence of any natural object, we have a great complexity of conditions before us, which we cannot reduce to position in space and date in time.

There is mass, attraction apparently spontaneous, electrical and magnetic properties which must be superadded to spatial configuration. To cut the list short we must say that practically the phenomena of the world present us problems involving many variables, which we must take as independent.

From this it follows that in making poiographs we must be prepared to use space of more than three dimensions. If the symmetry and completeness of our representation is to be of use to us we must be prepared to appreciate and criticise figures of a complexity greater than of those in three dimensions. It is impossible to give an example of such a poiograph which will not be merely trivial, without going into details of some kind irrelevant to our subject. I prefer to introduce the irrelevant details rather than treat this part of the subject perfunctorily.

To take an instance of a poiograph which does not lead us into the complexities incident on its application in classificatory science, let us follow Mrs. Alicia Boole Stott in her representation of the syllogism by its means. She will be interested to find that the curious gap she detected has a significance.

A syllogism consists of two statements, the major and the minor premiss, with the conclusion that can be drawn from them. Thus, to take an instance, fig. 49. It is evident, from looking at the successive figures that, if we know that the region M lies altogether within the region P, and also know that the region S lies altogether within the region M, we can conclude that the region S lies altogether within the region P. M is P, major premiss; S is M, minor premiss; S is P, conclusion. Given the first two data we must conclude that S lies in P. The conclusion S is P involves two terms, S and P, which are respectively called the subject and the predicate, the letters S and P being chosen with reference to the parts the notions they designate play in the conclusion. S is the subject of the conclusion, P is the predicate of the conclusion. The major premiss we take to be, that which does not involve S, and here we always write it first.

There are several varieties of statement possessing different degrees of universality and manners of assertiveness. These different forms of statement are called the moods.

We will take the major premiss as one variable, as a thing capable of different modifications of the same kind, the minor premiss as another, and the different moods we will consider as defining the variations which these variables undergo.

There are four moods:—

1. The universal affirmative; all M is P, called mood A.

2. The universal negative; no M is P, mood E.

3. The particular affirmative; some M is P, mood I.

4. The particular negative; some M is not P, mood O.

The dotted lines in 3 and 4, fig. 50, denote that it is not known whether or no any objects exist, corresponding to the space of which the dotted line forms one delimiting boundary; thus, in mood I we do not know if there are any M’s which are not P, we only know some M’s are P.

Representing the first premiss in its various moods by regions marked by vertical lines to the right of PQ, we have in fig. 51, running up from the four letters AEIO, four columns, each of which indicates that the major premiss is in the mood denoted by the respective letter. In the first column to the right of PQ is the mood A. Now above the line RS let there be marked off four regions corresponding to the four moods of the minor premiss. Thus, in the first row above RS all the region between RS and the first horizontal line above it denotes that the minor premiss is in the mood A. The letters E, I, O, in the same way show the mood characterising the minor premiss in the rows opposite these letters.

We have still to exhibit the conclusion. To do this we must consider the conclusion as a third variable, characterised in its different varieties by four moods—this being the syllogistic classification. The introduction of a third variable involves a change in our system of representation.

Before we started with the regions to the right of a certain line as representing successively the major premiss in its moods; now we must start with the regions to the right of a certain plane. Let LMNR be the plane face of a cube, fig. 52, and let the cube be divided into four parts by vertical sections parallel to LMNR. The variable, the major premiss, is represented by the successive regions which occur to the right of the plane LMNR—that region to which A stands opposite, that slice of the cube, is significative of the mood A. This whole quarter-part of the cube represents that for every part of it the major premiss is in the mood A.

In a similar manner the next section, the second with the letter E opposite it, represents that for every one of the sixteen small cubic spaces in it, the major premiss is in the mood E. The third and fourth compartments made by the vertical sections denote the major premiss in the moods I and O. But the cube can be divided in other ways by other planes. Let the divisions, of which four stretch from the front face, correspond to the minor premiss. The first wall of sixteen cubes, facing the observer, has as its characteristic that in each of the small cubes, whatever else may be the case, the minor premiss is in the mood A. The variable—the minor premiss—varies through the phases A, E, I, O, away from the front face of the cube, or the front plane of which the front face is a part.

And now we can represent the third variable in a precisely similar way. We can take the conclusion as the third variable, going through its four phases from the ground plane upwards. Each of the small cubes at the base of the whole cube has this true about it, whatever else may be the case, that the conclusion is, in it, in the mood A. Thus, to recapitulate, the first wall of sixteen small cubes, the first of the four walls which, proceeding from left to right, build up the whole cube, is characterised in each part of it by this, that the major premiss is in the mood A.

The next wall denotes that the major premiss is in the mood E, and so on. Proceeding from the front to the back the first wall presents a region in every part of which the minor premiss is in the mood A. The second wall is a region throughout which the minor premiss is in the mood E, and so on. In the layers, from the bottom upwards, the conclusion goes through its various moods beginning with A in the lowest, E in the second, I in the third, O in the fourth.

In the general case, in which the variables represented in the poiograph pass through a wide range of values, the planes from which we measure their degrees of variation in our representation are taken to be indefinitely extended. In this case, however, all we are concerned with is the finite region.

We have now to represent, by some limitation of the complex we have obtained, the fact that not every combination of premisses justifies any kind of conclusion. This can be simply effected by marking the regions in which the premisses, being such as are defined by the positions, a conclusion which is valid is found.

Taking the conjunction of the major premiss, all M is P, and the minor, all S is M, we conclude that all S is P. Hence, that region must be marked in which we have the conjunction of major premiss in mood A; minor premiss, mood A; conclusion, mood A. This is the cube occupying the lowest left-hand corner of the large cube.

Proceeding in this way, we find that the regions which must be marked are those shown in fig. 53. To discuss the case shown in the marked cube which appears at the top of fig. 53. Here the major premiss is in the second wall to the right—it is in the mood E and is of the type no M is P. The minor premiss is in the mood characterised by the third wall from the front. It is of the type some S is M. From these premisses we draw the conclusion that some S is not P, a conclusion in the mood O. Now the mood O of the conclusion is represented in the top layer. Hence we see that the marking is correct in this respect.

It would, of course, be possible to represent the cube on a plane by means of four squares, as in fig. 54, if we consider each square to represent merely the beginning of the region it stands for. Thus the whole cube can be represented by four vertical squares, each standing for a kind of vertical tray, and the markings would be as shown. In No. 1 the major premiss is in mood A for the whole of the region indicated by the vertical square of sixteen divisions; in No. 2 it is in the mood E, and so on.

A creature confined to a plane would have to adopt some such disjunctive way of representing the whole cube. He would be obliged to represent that which we see as a whole in separate parts, and each part would merely represent, would not be, that solid content which we see.

The view of these four squares which the plane creature would have would not be such as ours. He would not see the interior of the four squares represented above, but each would be entirely contained within its outline, the internal boundaries of the separate small squares he could not see except by removing the outer squares.

We are now ready to introduce the fourth variable involved in the syllogism.

In assigning letters to denote the terms of the syllogism we have taken S and P to represent the subject and predicate in the conclusion, and thus in the conclusion their order is invariable. But in the premisses we have taken arbitrarily the order all M is P, and all S is M. There is no reason why M instead of P should not be the predicate of the major premiss, and so on.

Accordingly we take the order of the terms in the premisses as the fourth variable. Of this order there are four varieties, and these varieties are called figures.

Using the order in which the letters are written to denote that the letter first written is subject, the one written second is predicate, we have the following possibilities:—

There are therefore four possibilities with regard to this fourth variable as with regard to the premisses.

We have used up our dimensions of space in representing the phases of the premisses and the conclusion in respect of mood, and to represent in an analogous manner the variations in figure we require a fourth dimension.

Now in bringing in this fourth dimension we must make a change in our origins of measurement analogous to that which we made in passing from the plane to the solid.

This fourth dimension is supposed to run at right angles to any of the three space dimensions, as the third space dimension runs at right angles to the two dimensions of a plane, and thus it gives us the opportunity of generating a new kind of volume. If the whole cube moves in this dimension, the solid itself traces out a path, each section of which, made at right angles to the direction in which it moves, is a solid, an exact repetition of the cube itself.

The cube as we see it is the beginning of a solid of such a kind. It represents a kind of tray, as the square face of the cube is a kind of tray against which the cube rests.

Suppose the cube to move in this fourth dimension in four stages, and let the hyper-solid region traced out in the first stage of its progress be characterised by this, that the terms of the syllogism are in the first figure, then we can represent in each of the three subsequent stages the remaining three figures. Thus the whole cube forms the basis from which we measure the variation in figure. The first figure holds good for the cube as we see it, and for that hyper-solid which lies within the first stage; the second figure holds good in the second stage, and so on.

Thus we measure from the whole cube as far as figures are concerned.

But we saw that when we measured in the cube itself having three variables, namely, the two premisses and the conclusion, we measured from three planes. The base from which we measured was in every case the same.

Hence, in measuring in this higher space we should have bases of the same kind to measure from, we should have solid bases.

The first solid base is easily seen, it is the cube itself. The other can be found from this consideration.

That solid from which we measure figure is that in which the remaining variables run through their full range of varieties.

Now, if we want to measure in respect of the moods of the major premiss, we must let the minor premiss, the conclusion, run through their range, and also the order of the terms. That is we must take as basis of measurement in respect to the moods of the major that which represents the variation of the moods of the minor, the conclusion and the variation of the figures.

Now the variation of the moods of the minor and of the conclusion are represented in the square face on the left of the cube. Here are all varieties of the minor premiss and the conclusion. The varieties of the figures are represented by stages in a motion proceeding at right angles to all space directions, at right angles consequently to the face in question, the left-hand face of the cube.

Consequently letting the left-hand face move in this direction we get a cube, and in this cube all the varieties of the minor premiss, the conclusion, and the figure are represented.

Thus another cubic base of measurement is given to the cube, generated by movement of the left-hand square in the fourth dimension.

We find the other bases in a similar manner, one is the cube generated by the front square moved in the fourth dimension so as to generate a cube. From this cube variations in the mood of the minor are measured. The fourth base is that found by moving the bottom square of the cube in the fourth dimension. In this cube the variations of the major, the minor, and the figure are given. Considering this as a basis in the four stages proceeding from it, the variation in the moods of the conclusion are given.

Any one of these cubic bases can be represented in space, and then the higher solid generated from them lies out of our space. It can only be represented by a device analogous to that by which the plane being represents a cube.

He represents the cube shown above, by taking four square sections and placing them arbitrarily at convenient distances the one from the other.

So we must represent this higher solid by four cubes: each cube represents only the beginning of the corresponding higher volume.

It is sufficient for us, then, if we draw four cubes, the first representing that region in which the figure is of the first kind, the second that region in which the figure is of the second kind, and so on. These cubes are the beginnings merely of the respective regions—they are the trays, as it were, against which the real solids must be conceived as resting, from which they start. The first one, as it is the beginning of the region of the first figure, is characterised by the order of the terms in the premisses being that of the first figure. The second similarly has the terms of the premisses in the order of the second figure, and so on.

These cubes are shown below.

For the sake of showing the properties of the method of representation, not for the logical problem, I will make a digression. I will represent in space the moods of the minor and of the conclusion and the different figures, keeping the major always in mood A. Here we have three variables in different stages, the minor, the conclusion, and the figure. Let the square of the left-hand side of the original cube be imagined to be standing by itself, without the solid part of the cube, represented by (2) fig. 55. The A, E, I, O, which run away represent the moods of the minor, the A, E, I, O, which run up represent the moods of the conclusion. The whole square, since it is the beginning of the region in the major premiss, mood A, is to be considered as in major premiss, mood A.

From this square, let it be supposed that that direction in which the figures are represented runs to the left hand. Thus we have a cube (1) running from the square above, in which the square itself is hidden, but the letters A, E, I, O, of the conclusion are seen. In this cube we have the minor premiss and the conclusion in all their moods, and all the figures represented. With regard to the major premiss, since the face (2) belongs to the first wall from the left in the original arrangement, and in this arrangement was characterised by the major premiss in the mood A, we may say that the whole of the cube we now have put up represents the mood A of the major premiss.

Hence the small cube at the bottom to the right in 1, nearest to the spectator, is major premiss, mood A; minor premiss, mood A; conclusion, mood A; and figure the first. The cube next to it, running to the left, is major premiss, mood A; minor premiss, mood A; conclusion, mood A; figure 2.

So in this cube we have the representations of all the combinations which can occur when the major premiss, remaining in the mood A, the minor premiss, the conclusion, and the figures pass through their varieties.

In this case there is no room in space for a natural representation of the moods of the major premiss. To represent them we must suppose as before that there is a fourth dimension, and starting from this cube as base in the fourth direction in four equal stages, all the first volume corresponds to major premiss A, the second to major premiss, mood E, the next to the mood I, and the last to mood O.

The cube we see is as it were merely a tray against which the four-dimensional figure rests. Its section at any stage is a cube. But a transition in this direction being transverse to the whole of our space is represented by no space motion. We can exhibit successive stages of the result of transference of the cube in that direction, but cannot exhibit the product of a transference, however small, in that direction.

To return to the original method of representing our variables, consider fig. 56. These four cubes represent four sections of the figure derived from the first of them by moving it in the fourth dimension. The first portion of the motion, which begins with 1, traces out a more than solid body, which is all in the first figure. The beginning of this body is shown in 1. The next portion of the motion traces out a more than solid body, all of which is in the second figure; the beginning of this body is shown in 2; 3 and 4 follow on in like manner. Here, then, in one four-dimensional figure we have all the combinations of the four variables, major premiss, minor premiss, figure, conclusion, represented, each variable going through its four varieties. The disconnected cubes drawn are our representation in space by means of disconnected sections of this higher body.

Now it is only a limited number of conclusions which are true—their truth depends on the particular combinations of the premisses and figures which they accompany. The total figure thus represented may be called the universe of thought in respect to these four constituents, and out of the universe of possibly existing combinations it is the province of logic to select those which correspond to the results of our reasoning faculties.

We can go over each of the premisses in each of the moods, and find out what conclusion logically follows. But this is done in the works on logic; most simply and clearly I believe in “Jevon’s Logic.” As we are only concerned with a formal presentation of the results we will make use of the mnemonic lines printed below, in which the words enclosed in brackets refer to the figures, and are not significative:—

• Barbara celarent Darii ferioque [prioris].
• Caesare Camestris Festino Baroko [secundae].
• [Tertia] darapti disamis datisi felapton.
• Bokardo ferisson habet [Quarta insuper addit].
• Bramantip camenes dimaris ferapton fresison.

In these lines each significative word has three vowels, the first vowel refers to the major premiss, and gives the mood of that premiss, “a” signifying, for instance, that the major mood is in mood a. The second vowel refers to the minor premiss, and gives its mood. The third vowel refers to the conclusion, and gives its mood. Thus (prioris)—of the first figure—the first mnemonic word is “barbara,” and this gives major premiss, mood A; minor premiss, mood A; conclusion, mood A. Accordingly in the first of our four cubes we mark the lowest left-hand front cube. To take another instance in the third figure “Tertia,” the word “ferisson” gives us major premiss mood E—e.g., no M is P, minor premiss mood I; some M is S, conclusion, mood O; some S is not P. The region to be marked then in the third representative cube is the one in the second wall to the right for the major premiss, the third wall from the front for the minor premiss, and the top layer for the conclusion.

It is easily seen that in the diagram this cube is marked, and so with all the valid conclusions. The regions marked in the total region show which combinations of the four variables, major premiss, minor premiss, figure, and conclusion exist.

That is to say, we objectify all possible conclusions, and build up an ideal manifold, containing all possible combinations of them with the premisses, and then out of this we eliminate all that do not satisfy the laws of logic. The residue is the syllogism, considered as a canon of reasoning.

Looking at the shape which represents the totality of the valid conclusions, it does not present any obvious symmetry, or easily characterisable nature. A striking configuration, however, is obtained, if we project the four-dimensional figure obtained into a three-dimensional one; that is, if we take in the base cube all those cubes which have a marked space anywhere in the series of four regions which start from that cube.

This corresponds to making abstraction of the figures, giving all the conclusions which are valid whatever the figure may be.

Proceeding in this way we obtain the arrangement of marked cubes shown in fig. 57. We see that the valid conclusions are arranged almost symmetrically round one cube—the one on the top of the column starting from AAA. There is one breach of continuity however in this scheme. One cube is unmarked, which if marked would give symmetry. It is the one which would be denoted by the letters I, E, O, in the third wall to the right, the second wall away, the topmost layer. Now this combination of premisses in the mood IE, with a conclusion in the mood O, is not noticed in any book on logic with which I am familiar. Let us look at it for ourselves, as it seems that there must be something curious in connection with this break of continuity in the poiograph.

The propositions I, E, in the various figures are the following, as shown in the accompanying scheme, fig. 58:—First figure: some M is P; no S is M. Second figure: some P is M; no S is M. Third figure: some M is P; no M is S. Fourth figure: some P is M; no M is S.

Examining these figures, we see, taking the first, that if some M is P and no S is M, we have no conclusion of the form S is P in the various moods. It is quite indeterminate how the circle representing S lies with regard to the circle representing P. It may lie inside, outside, or partly inside P. The same is true in the other figures 2 and 3. But when we come to the fourth figure, since M and S lie completely outside each other, there cannot lie inside S that part of P which lies inside M. Now we know by the major premiss that some of P does lie in M. Hence S cannot contain the whole of P. In words, some P is M, no M is S, therefore S does not contain the whole of P. If we take P as the subject, this gives us a conclusion in the mood O about P. Some P is not S. But it does not give us conclusion about S in any one of the four forms recognised in the syllogism and called its moods. Hence the breach of the continuity in the poiograph has enabled us to detect a lack of completeness in the relations which are considered in the syllogism.

To take an instance:—Some Americans (P) are of African stock (M); No Aryans (S) are of African stock (M); Aryans (S) do not include all of Americans (P).

In order to draw a conclusion about S we have to admit the statement, “S does not contain the whole of P,” as a valid logical form—it is a statement about S which can be made. The logic which gives us the form, “some P is not S,” and which does not allow us to give the exactly equivalent and equally primary form, “S does not contain the whole of P,” is artificial.

And I wish to point out that this artificiality leads to an error.

If one trusted to the mnemonic lines given above, one would conclude that no logical conclusion about S can be drawn from the statement, “some P are M, no M are S.”

But a conclusion can be drawn: S does not contain the whole of P.

It is not that the result is given expressed in another form. The mnemonic lines deny that any conclusion can be drawn from premisses in the moods I, E, respectively.

Thus a simple four-dimensional poiograph has enabled us to detect a mistake in the mnemonic lines which have been handed down unchallenged from mediæval times. To discuss the subject of these lines more fully a logician defending them would probably say that a particular statement cannot be a major premiss; and so deny the existence of the fourth figure in the combination of moods.

To take our instance: some Americans are of African stock; no Aryans are of African stock. He would say that the conclusion is some Americans are not Aryans; and that the second statement is the major. He would refuse to say anything about Aryans, condemning us to an eternal silence about them, as far as these premisses are concerned! But, if there is a statement involving the relation of two classes, it must be expressible as a statement about either of them.

To bar the conclusion, “Aryans do not include the whole of Americans,” is purely a makeshift in favour of a false classification.

And the argument drawn from the universality of the major premiss cannot be consistently maintained. It would preclude such combinations as major O, minor A, conclusion O—i.e., such as some mountains (M) are not permanent (P); all mountains (M) are scenery (S); some scenery (S) is not permanent (P).

This is allowed in “Jevon’s Logic,” and his omission to discuss I, E, O, in the fourth figure, is inexplicable. A satisfactory poiograph of the logical scheme can be made by admitting the use of the words some, none, or all, about the predicate as well as about the subject. Then we can express the statement, “Aryans do not include the whole of Americans,” clumsily, but, when its obscurity is fathomed, correctly, as “Some Aryans are not all Americans.” And this method is what is called the “quantification of the predicate.”

The laws of formal logic are coincident with the conclusions which can be drawn about regions of space, which overlap one another in the various possible ways. It is not difficult so to state the relations or to obtain a symmetrical poiograph. But to enter into this branch of geometry is beside our present purpose, which is to show the application of the poiograph in a finite and limited region, without any of those complexities which attend its use in regard to natural objects.

If we take the latter—plants, for instance—and, without assuming fixed directions in space as representative of definite variations, arrange the representative points in such a manner as to correspond to the similarities of the objects, we obtain configuration of singular interest; and perhaps in this way, in the making of shapes of shapes, bodies with bodies omitted, some insight into the structure of the species and genera might be obtained.