CHAPTER XII GEOMETRICAL REPRESENTATION OF THE STATES OF THE BEING

In the three-dimensional representation just given, each modality of any state of the being is indicated by a point alone; however, such a modality is itself also capable of developing in the course of a cycle of manifestation involving an indefinite of secondary modifications. Thus, in the corporeal modality of the human individuality, for example, these modifications will be all the moments of its existence (naturally regarded under the aspect of temporal succession, which is one of the conditions to which this modality is subjected), or, which amounts to the same thing, all the acts and motions whatsoever that it will perform in the course of this existence.^[59] If all these modifications are to be included in our representation, then the modality considered will have to be depicted, not merely by a point, but by a whole straight line, each point in which will now be one of the secondary modifications in question; and here it should be carefully noted that this straight line, although indefinite, is none the less limited; in fact, everything indefinite is limited, and so is (if the expression is permissible) every power of the indefinite.^[60] Simple indefinite being represented by a straight line, double indefinite or the indefinite to the power of two will be represented by a plane, and triple indefinite or the indefinite to the power of three by a three-dimensional expanse. If therefore each modality, envisaged as a simple indefinite, is depicted by a straight line, a state of the being, involving an indefinite of such modalities, in other words a double indefinite, will be depicted in its entirety by a horizontal plane, and a being in its totality, with the indefinite of its states, will be represented by a three-dimensional expanse. This new representation is thus more complete than the former one, but it is clear that unless three-dimensional space is departed from, we can here consider only a single being, and not, as previously, the whole of the beings in the Universe, for the consideration of the totality of beings would make it necessary to introduce a further indefinite, which would be of the fourth order, and could not be geometrically depicted except by imagining a fourth dimension superadded to space.^[59]

In this new representation, we see first of all that through each point in the expanse under consideration there pass three straight lines, respectively parallel to the three dimensions of this expanse; each point can therefore be taken as the apex of a trihedral right-angle, constituting a system of co-ordinates to which the whole expanse may be referred, and the three axes of which will form the three-dimensional cross. If the vertical axis of this system be taken as given, it will meet each horizontal plane in a point, which will be the origin of the rectangular co-ordinates to which that plane will be referred, and the two axes of which will form a two-dimensional cross. It can be said that this point is the centre of the plane, and the vertical axis is the locus of the centres of all the horizontal planes; every vertical, in other words every line parallel to this axis, also contains points which correspond to one another in those planes. If in addition to the vertical axis a particular horizontal plane is taken as the basis of the system of co-ordinates, then the trihedral right-angle just mentioned will also be wholly determined thereby. There will be a two-dimensional cross, traced by two of the three axes, in each of the three planes of co-ordinates, one of which is the horizontal plane in question, while the others are the two orthogonal planes each passing through the vertical axis and through one of the horizontal axes; and these three crosses will have as their common centre the apex of the trihedral angle, which is the centre of the three-dimensional cross and may thus be also regarded as the centre of the whole expanse. Every point could be the centre, and, one may say, potentially is so; but in fact it is necessary for one particular point to be given in order to be actually able to draw the cross, in other words to measure the whole expanse, or, analogically, to realize the total scope of the being's possibilities.

Footnotes

[59]This is not the place to deal with the question of the “fourth dimension” of space, which has given rise to many erroneous or fantastic notions, and which would find a more natural place in a study of the conditions of corporeal existence. (See *Le Règne de la Quantité*, ch. XVIII, where these errors are examined.—*Translator*).

[60]The indefinite, which proceeds from the finite, is always reducible thereto, since it is only the development of the possibilities included or implied in the finite. It is an elementary truth, though one too often overlooked, that the alleged “mathematical infinite” (a quantitative indefinite either numerical or geometrical) is not infinite at all, for it is limited by the determination inherent in its own nature. (See also Les principes du Calcul infinitésimal, passim but especially ch. I, II, VII and VIII.—Translator).