The murmur broke out again, and Hoskins van Huysman blew his nose somewhat aggressively. His scientific bile was beginning to rise. He disapproved very strongly of the tone which his rival had begun. Its quiet confidence was somewhat ominous. The lecturer continued without this time noticing the interruption, and proceeded to give a lengthy and learned but singularly lucid _resume_ of the more recent progress in the higher mathematics and the deeply interesting speculations to which it had given rise. This, with certain demonstrations which he made on the great black-board beside him, occupied nearly an hour. When he had finished there was another murmur, which this time was wholly of applause, for this part of the lecture had not only been masterly but entirely orthodox. Then silence fell again, the silence of expectant waiting, for every one felt that the "Examination" was coming now. He began again in a slightly altered voice. "What I have just been saying was necessary to my subject as far as it went, but for all that it was chiefly introductory to what I am now going to bring to your notice. But this is a matter rather for illustration and discussion than for mere disquisition. Therefore, to save your time as much as possible, I will proceed at once to the illustration, and then we will have the discussion." Professor van Huysman snorted again, even as a war-horse that snuffs the fray. This time Franklin Marmion seemed to recognise the implied challenge, for he looked round the crowded theatre with a curious smile, which seemed to say: "Yes, gentlemen, I see that some of you are getting ready for a tussle. I am in hopes of being able to oblige you." "Now," he continued, "it is generally conceded that an ounce of practice is worth a good many pounds of precept, so I will get to the practice. I need hardly remind you that ever since mathematics became an exact science, three problems have been recognised as impossible of solution--trisecting the triangle, squaring the circle, and doubling the cube. I have now the pleasure of announcing that I have had the great good fortune to discover certain formulae which, so far, at least, as I can see, make the solution of those problems not only possible, but comparatively easy--to those who know how to use them."