第1章

THE science which has to do with nature clearly concerns itself for the most part with bodies and magnitudes and their properties and movements, but also with the principles of this sort of substance, as many as they may be.For of things constituted by nature some are bodies and magnitudes, some possess body and magnitude, and some are principles of things which possess these.Now a continuum is that which is divisible into parts always capable of subdivision, and a body is that which is every way divisible.A magnitude if divisible one way is a line, if two ways a surface, and if three a body.

Beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all.For, as the Pythagoreans say, the world and all that is in it is determined by the number three, since beginning and middle and end give the number of an 'all', and the number they give is the triad.And so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three in the worship of the Gods.Further, we use the terms in practice in this way.Of two things, or men, we say 'both', but not 'all': three is the first number to which the term 'all' has been appropriated.And in this, as we have said, we do but follow the lead which nature gives.Therefore, since 'every' and 'all' and 'complete' do not differ from one another in respect of form, but only, if at all, in their matter and in that to which they are applied, body alone among magnitudes can be complete.For it alone is determined by the three dimensions, that is, is an 'all'.But if it is divisible in three dimensions it is every way divisible, while the other magnitudes are divisible in one dimension or in two alone:

for the divisibility and continuity of magnitudes depend upon the number of the dimensions, one sort being continuous in one direction, another in two, another in all.All magnitudes, then, which are divisible are also continuous.Whether we can also say that whatever is continuous is divisible does not yet, on our present grounds, appear.One thing, however, is clear.We cannot pass beyond body to a further kind, as we passed from length to surface, and from surface to body.For if we could, it would cease to be true that body is complete magnitude.We could pass beyond it only in virtue of a defect in it; and that which is complete cannot be defective, since it has being in every respect.Now bodies which are classed as parts of the whole are each complete according to our formula, since each possesses every dimension.But each is determined relatively to that part which is next to it by contact, for which reason each of them is in a sense many bodies.But the whole of which they are parts must necessarily be complete, and thus, in accordance with the meaning of the word, have being, not in some respect only, but in every respect.