which claims absolute certainty for human processes.  You admit that this line and triangle, which are mere figments of our human imagination, not only exist independent of us in the crystal, but are, as we suppose, habitually and invariably used by God Himself to give form to the matter contained within the planes of the crystal.  Yet to this line and triangle you deny reality.  To mathematical truth, you deny compulsive force.  You hold that an equilateral triangle may, to you and all other human individuals, be a right-angled triangle if you choose to imagine it so.  Allow me to say, without assuming any claim to superior knowledge, that to me your logic results in a different conclusion.  If you are compelled, at one point or another of the chain of being, to deny existence to a substance, surely it should be to the last and feeblest.  I see nothing to hinder you from denying your own existence, which is, in fact, impossible to demonstrate.  Certainly you are free, in logic, to argue that Socrates and Plato are mere names—­that men and matter are phantoms and dreams.  No one ever has proved or ever can prove the contrary, Infallibly, a great philosophical school will some day be founded on that assumption.  I venture even to recommend it to your acute and sceptical mind; but I cannot conceive how, by any process of reasoning, sensual or supersensual, you can reach the conclusion that the single form of truth which instantly and inexorably compels our submission to its laws—­is nothing.”

Thus far, all was familiar ground; certainly at least as familiar as the Pons Asinorum; and neither of the two champions had need to feel ruffled in temper by the discussion.  The real struggle began only at this point; for until this point was reached, both positions were about equally tenable.  Abelard had hitherto rested quietly on the defensive, but William’s last thrust obliged him to strike in his turn, and he drew himself up for what, five hundred years later, was called the “Coup de Jarnac":—­

“I do not deny,” he begins; “on the contrary, I affirm that the universal, whether we call it humanity, or equilateral triangle, has a sort of reality as a concept; that it is something; even a substance, if you insist upon it.  Undoubtedly the sum of all individual men results in the concept of humanity.  What I deny is that the concept results in the individual.  You have correctly stated the essence of the point and the line as sources of our concept of the infinite; what I deny is that they are divisions of the infinite.  Universals cannot be divided; what is capable of division cannot be a universal.  I admit the force of your analogy in the case of the crystal; but I am obliged to point out to you that, if you insist on this analogy, you will bring yourself and me into flagrant contradiction with the fixed foundations of the Church.  If the energy of the triangle gives form to the crystal, and the energy of the line gives reality to the triangle,