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Notes.

Page 11, line 28. It is worth while to compare the following passage from Plato's "Republic," Book vii. (Jowett's translation):

After plane geometry, we took solids in revolution instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have been followed.

It is true, Socrates; but these subjects seem to be as yet hardly explored.

Why, yes, I said, and for two reasons; in the first place, no government patronises them, which leads to a want of energy in the study of them, and they are difficult; in the second place, students cannot learn them unless they have a teacher. But then a teacher is hardly to be found, and even if one could be found, as matters now stand the students of these subjects, who are very conceited, would not mind him; that, however, would be otherwise if the whole state patronised and honoured them, then they would listen, and there would be continuous and earnest search, and discoveries would be made; since even now, disregarded as they are by the world, and maimed of their fair proportions, and although none of their votaries can tell the use of them, still these studies force their way by their natural charm, and very likely they may emerge into light.

P. 22, 1. 3. Compare with this the latter part of Plato's "Philebus," on knowledge and the handicraft arts; also Prof. Jowett's Introduction thereto.

P. 25,1. 15. See "Trattato della Pittura," by Leonardo da Vinci; also the Memoir on the MSS. of L. d. V., by Venturi, 1797.

P. 25,1. 24. "The Gentleman and Cabinet Maker's Director," by Thomas Chippendale, London, 1754.

"The Cabinet Maker and Upholsterer's Drawing Book," by Thomas Sheraton, London, 1793.

P. 26,1. 30. See Sorby's Address to the Microscopical Society, 1876.

P. 27, 1. 5. Phil. Trans, of the Royal Society, 1870, p. 333; and 1876, p. 27.

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Page 27, line 10. Phil. Trans. 1877, p. 149.

P. 27, 1. 23. "On Attraction and Repulsion resulting from Radiation," Phil. Trans. 1874, p. 501; 1875, p. 519; 1876, p. 325.

P. 27. 1. 26. Philosophical Magazine, April 1878.

P. 27, 1. 28. Philosophical Magazine, 1875, Vol. ii., pp. 337, 446; 1877, Vol. i., p. 321; 1878, Vol. i., p. 161.

P. 28, 1. 7. Poggendorff's Annalen, Tom. xxxv., p. 337.

P. 28. 1. 9. Royal Society's Proceedings, 1878.

P. 28, 1. 18. The Papers on the Telephone are too numerous to specify.

P. 28, 1. 19. See various Papers in "Nature," and elsewhere, during the last twelve months.

P. 28, 1. 25. Royal Society's Proceedings, May 9, 1878.

P. 28, 1. 32. Phil. Trans., Vol. 169, pp. 55 and 155, and other Papers catalogued in the Appendix to Part II. of the Memoir.

P. 30, 1. 4. See Maxwell "On Heat," chap. xxii.

P. 32, 1. 1. Grunert's Archiv., Vol. vi., p. 337; also separate work, Berlin, 1862.

P. 32, 1. 1. "Linear Associative Algebra," by Benjamin Peirce, Washington City, 1870.

P. 33, 1. 1. Sir W. Thomson, "Cambridge Mathematical Journal," vol. iii., p. 174. Jcvons' "Principles of Science," Vol. ii., p. 438.

But an explanation of the difficulty seems to me to be found in the fact that the problem, as stated, is one of the conduction of heat, and that the "impossibility" which attaches itself to the expression for the "time" merely means, that previous to a certain epoch the conditions which gave rise to the phenomena were not those of conduction, but those of some other action of heat. If, therefore, we desire to comprise the phenomena of the earlier as well as of the later period in one problem, we must find some more general statement; viz., that of physical conditions which at the critical epoch will issue in a case of conduction. I think that Prof. Clifford has somewhere given a similar explanation.

P. 38, 1. 13. S. Newcomb "On Certain Transformations of Surfaces." American Journal of Mathematics," Vol. i., p. i.

P. 38, 1. 14. Tait "On Knots." Transactions of the Royal Society of Edinburgh," Vol. xxviii., p. 145. Klein, "Mathematische Annalen," ix., p. 478.

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Page 48, line 31. Royal Society's Proceedings," February 3rd, 1876, and May 9, 1878.

P. 53, 1. 17. For example, in Herbart's "Psychologie."

P. 53, 1. 19. A specimen will be found in the Moralia of Gregory the Great, Lib. I., c. xiv., of which I quote only the arithmetical part:

"Quid in septenario numero, nisi summa perfectionis accipitur? Ut enim humanæ rationis causas de septenario numero taceamus, quæ afferent, quòd idcirco perfectus sit, quia exprimo pari constat, et primo impari; ex primo, qui dividi potest, et primo, qui dividi non potest; certissimè scimus, quòd septenarium numerum Scriptura Sacra pro perfectione ponere consuevit. . . . A septenario quippe numero in duodenarium surgitur. Nam septenarius suis in se partibus multiplicatus, ad duodenarium tenditur. Sive enim quatuor per tria, sive per quatuor tria ducantur, septem in duodecim vertuntur. . . . . Jam superiùs dictum est, quòd in quinquagenario numero, qui septem hebdomadibus ac monade additâ impletur, requies designatur; denario autem numero summa perfection is exprimetur."