solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed. That is true, Socrates; but so little seems to be known as yet about these subjects. Why, yes, I said, and for two reasons:—in the first place, no government patronises them; this leads to a want of energy in the pursuit of them, and they are difficult; in the second place, students cannot learn them unless they have a director. But then a director can hardly be found, and even if he could, as matters now stand, the students, who are very conceited, would not attend to him. That, however, would be otherwise if the whole State became the director of these studies and gave honour to them; then disciples would want to come, and there would be continuous and earnest search, and discoveries would be made; since even now, disregarded as they are by the world, and maimed of their fair proportions, and although none of their votaries can tell the use of them, still these studies force their way by their natural charm, and very likely, if they had the help of the State, they would some day emerge into light. Yes, he said, there is a remarkable charm in them. But I do not clearly understand the change in the order. First you began with a geometry of plane surfaces? Yes, I said. And you placed astronomy next, and then you made a step backward? Yes, and I have delayed you by my hurry; the ludicrous state of solid geometry, which, in natural order, should have followed, made me pass over this branch and go on to
Timaeus: of each of these Kinds which I must endeavor to explain to you in an exposition of an unusual type; yet, inasmuch as you have some...
(53) Timaeus: of each of these Kinds which I must endeavor to explain to you in an exposition of an unusual type; yet, inasmuch as you have some acquaintance with the technical method which I must necessarily employ in my exposition, you will follow me. In the first place, then, it is plain I presume to everyone that fire and earth and water and air are solid bodies; and the form of a body, in every case, possesses depth also. Further, it is absolutely necessary that depth should be bounded by a plane surface; and the rectilinear plane is composed of triangles.
Timaeus: meet in a point, they form one solid angle, which comes next in order to the most obtuse of the plane angles. And when four such angles are...
(55) Timaeus: meet in a point, they form one solid angle, which comes next in order to the most obtuse of the plane angles. And when four such angles are produced, the first solid figure is constructed, which divides the whole of the circumscribed sphere into equal and similar parts. And the second solid is formed from the same triangles, but constructed out of eight equilateral triangles, which produce one solid angle out of four planes; and when six such solid angles have been formed, the second body in turn is completed.
Timaeus: had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its...
(32) Timaeus: had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms; but now it is otherwise: for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two. Thus it was that in the midst between fire and earth God set water and air, and having bestowed upon them so far as possible a like ratio one towards another—air being to water as fire to air, and water being to earth as air to water, —he joined together and constructed a Heaven visible and tangible. For these reasons
ANSWER: whether these last should be subdivided, as by the geometers, into those contained by triangular and quadrilateral planes: and whether a further divis...
(13) It has been remarked that the continuous is effectually distinguished from the discrete by their possessing the one a common, the other a separate, limit.
The same principle gives rise to the numerical distinction between odd and even; and it holds good that if there are differentiae found in both contraries, they are either to be abandoned to the objects numbered, or else to be considered as differentiae of the abstract numbers, and not of the numbers manifested in the sensible objects. If the numbers are logically separable from the objects, that is no reason why we should not think of them as sharing the same differentiae.
But how are we to differentiate the continuous, comprising as it does line, surface and solid? The line may be rated as of one dimension, the surface as of two dimensions, the solid as of three, if we are only making a calculation and do not suppose that we are dividing the continuous into its species; for it is an invariable rule that numbers, thus grouped as prior and posterior, cannot be brought into a common genus; there is no common basis in first, second and third dimensions. Yet there is a sense in which they would appear to be equal- namely, as pure measures of Quantity: of higher and lower dimensions, they are not however more or less quantitative.
Numbers have similarly a common property in their being numbers all; and the truth may well be, not that One creates two, and two creates three, but that all have a common source.
Suppose, however, that they are not derived from any source whatever, but merely exist; we at any rate conceive them as being derived, and so may be assumed to regard the smaller as taking priority over the greater: yet, even so, by the mere fact of their being numbers they are reducible to a single type.
What applies to numbers is equally true of magnitudes; though here we have to distinguish between line, surface and solid- the last also referred to as "body"- in the ground that, while all are magnitudes, they differ specifically.
It remains to enquire whether these species are themselves to be divided: the line into straight, circular, spiral; the surface into rectilinear and circular figures; the solid into the various solid figures- sphere and polyhedANSWER: whether these last should be subdivided, as by the geometers, into those contained by triangular and quadrilateral planes: and whether a further division of the latter should be performed.
(14) How are we to classify the straight line? Shall we deny that it is a magnitude?
The suggestion may be made that it is a qualified magnitude. May we not, then, consider straightness as a differentia of "line"? We at any rate draw on Quality for differentiae of Substance.
The straight line is, thus, a quantity plus a differentia; but it is not on that account a composite made up of straightness and line: if it be a composite, the composite possesses a differentiae of its own.
But why is not the product of three lines included in Quantity? The answer is that a triangle consists not merely of three lines but of three lines in a particular disposition, a quadrilateral of four lines in a particular disposition: even the straight line involves disposition as well as quantity.
Holding that the straight line is not mere quantity, we should naturally proceed to assert that the line as limited is not mere quantity, but for the fact that the limit of a line is a point, which is in the same category, Quantity. Similarly, the limited surface will be a quantity, since lines, which have a far better right than itself to this category, constitute its limits. With the introduction of the limited surface- rectangle, hexagon, polygon- into the category of Quantity, this category will be brought to include every figure whatsoever.
If however by classing the triangle and the rectangle as qualia we propose to bring figures under Quality, we are not thereby precluded from assigning the same object to more categories than one: in so far as it is a magnitude- a magnitude of such and such a size- it will belong to Quantity; in so far as it presents a particular shape, to Quality.
It may be urged that the triangle is essentially a particular shape. Then what prevents our ranking the sphere also as a quality?
To proceed on these lines would lead us to the conclusion that geometry is concerned not with magnitudes but with Quality. But this conclusion is untenable; geometry is the study of magnitudes. The differences of magnitudes do not eliminate the existence of magnitudes as such, any more than the differences of substances annihilate the substances themselves.
Moreover, every surface is limited; it is impossible for any surface to be infinite in extent.
Again, when I find Quality bound up with Substance, I regard it as substantial quality: I am not less, but far more, disposed to see in figures or shapes varieties of Quantity. Besides, if we are not to regard them as varieties of magnitude, to what genus are we to assign them?
Suppose, then, that we allow differences of magnitude; we commit ourselves to a specific classification of the magnitudes so differentiated.
Timaeus: when joined together, formed eight solid angles, each composed of three plane right angles; and the shape of the body thus constructed was...
(55) Timaeus: when joined together, formed eight solid angles, each composed of three plane right angles; and the shape of the body thus constructed was cubic, having six plane equilateral quadrangular bases. And seeing that there still remained one other compound figure, the fifth, God used it up for the Universe in his decoration thereof. Now in reasoning about all these things, a man might question whether he ought to affirm the existence of an infinite diversity of Universes or a limited number; and if he questioned aright he would conclude that the doctrine of an infinite diversity is that of a man unversed
Timaeus: of quadrangular bases, being very firmly based, it is a most inelastic form; and so too is everything which is of very dense composition and...
(62) Timaeus: of quadrangular bases, being very firmly based, it is a most inelastic form; and so too is everything which is of very dense composition and most rigid. The nature of “heavy” and “light” would be shown most clearly if, along with them, we examined also the nature of “above” and “below,” as they are called. That there really exist two distinct and totally opposite regions, each of which occupies one-half of the Universe—the one termed “below,” towards which move all things possessing any bodily mass, and the other “above,” towards which everything goes against its will,—