All the definitions which I have given have a common property; they are entirely or chiefly descriptive, they do not go down to the essentials of the matter. In criticising them in this way I do not wish to be misunderstood; the question is not one of mere criticism, but of the relative importance of the fundamental propositions upon which a scientific study is built up, for the whole matter begins with the definition. It may be objected to this that the definition can in this case only be determined after the object defined has been thoroughly investigated,—its position at the beginning is therefore artificial. The statement is perfectly true: it is, and must be, true however for every definition. The beginning must presuppose the end. A scientific study is not a chronological account of its own investigations. It is not necessary, however, that the learner should know at the very commencement the full meaning of the definitions. The explanation and development of the subject, as they are carried on, refer him back always to these first propositions, which he finds, as he proceeds, to be a mirror in which he can see a reflection of each later one,—until at the end of his work he discovers the full and complete justification of the definitions with which he started. It has seemed to grow more and more full of meaning as he advanced, until finally it has shown itself to be a true representation of the whole matter in a form of the greatest possible conciseness.
An incomplete or merely descriptive definition forming the key-stone of any branch of science reflects the condition in which that study must be. We shall have an opportunity of showing in the course of this work how the machine has gradually shaken itself free from the problems of Pure Mechanics. The student only of this science, in whose work the machine is merely an accident, does not feel the drawbacks of its incorrect definition. Nor do these present themselves even to the engineering student, so long as his science has not yet been put in the form of rigidly logical propositions. I cannot do better than quote here an excellent sentence from Mill (Logic I., bk. i., chap, viii., § 4) bearing upon the subject. He says:—"What is true of the definition of any term of Science, is of course true of the definition of a science itself: and accordingly the definition of a science must necessarily be progressive and provisional. Any extension of knowledge or alteration in the current opinions respecting the subject matter, may lead to a change more or less extensive in the particulars included in the science; and its composition being thus altered, it may easily happen that a different set of characteristics will be found better adapted as differentia fpr defining its name."
8 (P. 47.) The immense importance of this proposition will appear greater to the reader presently than it can do at present. Its principle seems to have been hitherto entirely unrecognised. I have found but a single trace of it. This is in Chasles (Aperçu historique sur l'Origine ... des Méthodes en Geometrie, 1837, note xxxiv., p. 408, et seq., and later on), where he speaks of the elliptic chuck of Leonardo da Vinci. The consequence in that case of fixing another link of the kinematic chain Chasles has taken for an indication of the great law of duality, upon which he enlarges at great length. His reasoning, however, is not well grounded, and goes seriously wrong; we have here not duality, but a most characteristic plurality, which contains naturally
