oblique position, terminating in a very short point with a hemispherical shoulder P _a_ M. It is evident that, in this case, the top will not spin upon _a_ the end of the true axis X _a_, but upon P, a point in the circle P M, to which the floor I F is a tangent. Instead, therefore, of revolving upon a fixed and stationary point, the top will roll round upon the small circle P M on its blunt point, with very considerable friction, the force of which may be represented by a line O P at right angles to the floor I F, and to the spherical end of the peg of the top: now it is the action of this force, by its pressure on one side of the blunt point of the top, which causes it to rise in a vertical direction. Produce the line O P till it meets the axis C; from the point C draw the line C T perpendicular to the axis _a_ X, and T O parallel to it; and then, by a resolution of forces, the line T C will represent that part of the friction which presses at right angles to the axis, so as gradually to raise it in a vertical position; in which operation the circle P M gradually diminishes by the approach of the point P to _a_, as the axis becomes more perpendicular, and vanishes when the point P coincides with the point _a_, that is to say, when the top has arrived at its vertical position, where it will continue to _sleep_, without much friction, or any other disturbing force, until its rotatory motion fails, and its side is brought to the earth by the force of gravity.” [Illustration: A top diagonal to a surface, with a diagram describing its rotation.] “I _think_ I understand it,” said Tom, “although I have some doubt about it; but if you would be so kind as to give me the demonstration in writing, I will diligently study it.” “Most readily,” said Mr. Seymour. “Indeed I cannot expect that you should comprehend so difficult a subject, without the most patient investigation; and, in the present state of your knowledge, I am compelled to omit the relation of several very important circumstances, to which I may, hereafter, direct your attention. When, for instance, you have become acquainted with the elements of astronomy, I shall be able to show you that the gyration of the top depends upon the same principles as the precession of the equinoxes.”(22) [Illustration: Two boys spin a top with whips.] ----- Footnote 18: See page 119. Footnote 19: See page 54. Footnote 20: See page 126. Footnote 21: “The wooden engine flies and whirls about,  Admired, with clamours, of the beardless rout:  They lash aloud; each other they provoke,  And lend their little souls at every stroke.”--DRYDEN. Footnote 22: Harl. lib. i. marked 6391. Footnote 23: Page 126. Footnote 24: Short on ‘Serson’s Horizontal Top.’ Phil. Trans. xlvii. 352. ------------------------------------------------------------------------ [Illustration: A boy playing on a swing between two trees.] CHAPTER IX.