William of Champeaux, lecturing on dialectics or logic, comes to the question of universals, which he says, are substances. Starting from the highest substance, God, all being descends through created substances by stages, until it reaches the substance animality, from which it descends to the substance humanity: and humanity being, like other essences or substances, indivisible, passes wholly into each individual, becoming Socrates, Plato, and Aristotle, much as the divine substance exists wholly and undivided in each member of the Trinity. Here Abelard interrupts. The divine substance, he says, operates by laws of its own, and cannot be used for comparison. In treating of human substance, one is bound by human limitations. If the whole of humanity is in Socrates, it is wholly absorbed by Socrates, and cannot be at the same time in Plato, or elsewhere. Following his favourite reductio ad absurdum, Abelard turns the idea round, and infers from it that, since Socrates carries all humanity in him, he carries Plato, too; and both must be in the same place, though Socrates is at Athens and Plato in Rome. The objection is familiar to William, who replies by another commonplace:-- "Mr. Abelard, might I, without offence, ask you a simple matter? Can you give me Euclid's definition of a point?" "If I remember right it is, 'illud cujus nulla pars est'; that which has no parts." "Has it existence?" "Only in our minds." "Not, then, in God?" "All necessary truths exist first in God. If the point is a necessary truth, it exists first there." "Then might I ask you for Euclid's definition of the line?" "The line is that which has only extension; 'Linea vocatur illa quae solam longitudinem habet.'" "Can you conceive an infinite straight line?" "Only as a line which has no end, like the point extended." "Supposing we imagine a straight line, like opposite rays of the sun, proceeding in opposite directions to infinity--is it real?" "It has no reality except in the mind that conceives it." "Supposing we divide that line which has no reality into two parts at its origin in the sun or star, shall we get two infinities?--or shall we say, two halves of the infinite?" "We conceive of each as partaking the quality of infinity." "Now, let us cut out the diameter of the sun; or rather--since this is what our successors in the school will do,--let us take a line of our earth's longitude which is equally unreal, and measure a degree of this thing which does not exist, and then divide it into equal parts which we will use as a measure or metre. This metre, which is still nothing, as I understand you, is infinitely divisible into points? and the point itself is infinitely small? Therefore we have the finite partaking the nature of the infinite?" "Undoubtedly!"