“They are most certainly one—­a single concept of the only possible equilateral triangle measuring one metre on each face.”

“You told us a moment ago that a universal could not exist wholly and exclusively in two individuals at once.  Does not the universal by definition—­the equilateral triangle measuring one metre on each face—­does it not exist wholly, in its integrity of essence, in each of the two triangles we have conceived?”

“It does—­as a conception.”

“I thank you!  Now, although I fear wearying you, perhaps you will consent to let me add matter to mind.  I have here on my desk an object not uncommon in nature, which I will ask you to describe.”

“It appears to be a crystal.”

“May I ask its shape?”

“I should call it a regular octahedron.”

“That is, two pyramids, set base to base? making eight plane surfaces, each a perfect equilateral triangle?”

“Concedo triangula (I grant the triangles).”

“Do you know, perchance, what is this material which seems to give substantial existence to these eight triangles?”

“I do not.”

“Nor I! nor does it matter, unless you conceive it to be the work of man?”

“I do not claim it as man’s work.”

“Whose, then?”

“We believe all actual creation of matter, united with form, to be the work of God.”

“Surely not the substance of God himself?  Perhaps you mean that this form—­this octahedron—­is a divine concept.”

“I understand such to be the doctrine of the Church.”

“Then it seems that God uses this concept habitually to create this very common crystal.  One question more, and only one, if you will permit me to come to the point.  Does the matter—­the material—­of which this crystal is made affect in any way the form—­the nature, the soul—­of the universal equilateral triangle as you see it bounding these eight plane surfaces?”

“That I do not know, and do not think essential to decide.  As far as these triangles are individual, they are made so by the will of God, and not by the substance you call triangle.  The universal—­the abstract right angle, or any other abstract form—­is only an idea, a concept, to which reality, individuality, or what we might call energy is wanting.  The only true energy, except man’s free will, is God.”

“Very good, Mr. Abelard! we can now reach our issue.  You affirm that, just as the line does not exist in space, although the eye sees little else in space, so the triangle does not exist in this crystal, although the crystal shows eight of them, each perfect.  You are aware that on this line which does not exist, and its combination in this triangle which does not exist, rests the whole fabric of mathematics with all its necessary truths.  In other words, you know that in this line, though it does not exist, is bound up the truth of the only branch of human knowledge