Mr. James Smith will, I have no doubt, be the most uneclipsed circle-squarerof our day. He will not owe this distinction to his being an influential andrespected member of the commercial world of Liverpool, even though the powerof publishing which his means give him should induce him to issue a wholelibrary upon one paradox. Neither will he owe it to the pains taken with himby a mathematician, who corresponded with him until the joint letters filledan octavo volume. Neither will he owe it to the notice taken of him by SirWilliam Hamilton, of Dublin, who refuted him in a manner intelligible to anordinary student of Euclid, which refutation he calls a remarkable paradoxeasily explainable, but without explaining it. What he will owe it to Iproceed to show.