Among Carlyle’s Edinburgh connections in the Kirkcaldy days, one comes to us in a book form. It was in 1817 that Professor Leslie, not yet Sir John Leslie, brought out the third edition of his _Elements of Geometry and Plane Trigonometry_, being an improvement and enlargement of the two previous editions of 1809 and 1811. The geometrical portion of the volume consists of six books, intended to supersede the traditional six books of Euclid, and containing many propositions not to be found there. The seventeenth proposition of the sixth book is the problem “_To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle_.” The solution, with diagrams, occupies a page; and there is an additional page of “scholium,” pointing out in what circumstances the problem is impossible, and calling attention to the value of the proposition in the construction of quadratic equations. So much for the text of the proposition at pp. 176–177; but, when we turn to the “Notes and Illustrations” appended to the volume, we find, at p. 340, this note by Leslie:— “The solution of this important problem now inserted in the text was suggested to me by Mr. Thomas Carlyle, an ingenious young mathematician, formerly my pupil. But I here subjoin likewise the original construction given by Pappus; which, though rather more complex, has yet some peculiar advantages.” Leslie then proceeds to give the solution of Pappus, in about two pages, and to add about three pages of further remarks on the application of the problem to the construction of quadratics. The mention of Carlyle by Leslie in this volume of 1817 is, I believe, the first mention of Carlyle by name in print; and it was no small compliment to prefer, for text purposes, young Carlyle’s solution of an important problem to the old one that had come down from the famous Greek geometrician. Evidently Carlyle’s mathematical reputation was still kept up about the Edinburgh University, and Leslie was anxious to do his favourite pupil a good turn.[16]