_Elements of the Philosophy of the Human Mind, Part 3, chap. 1, sect. 3._ =238.= No process of sound reasoning can establish a result not contained in the premises.--MELLOR, J. W. _Higher Mathematics for Students of Chemistry and Physics (New York, 1902), p. 2._ =239.= ... we cannot get more out of the mathematical mill than we put into it, though we may get it in a form infinitely more useful for our purpose.--HOPKINSON, JOHN. _James Forrest Lecture, 1894._ =240.= The iron labor of conscious logical reasoning demands great perseverance and great caution; it moves on but slowly, and is rarely illuminated by brilliant flashes of genius. It knows little of that facility with which the most varied instances come thronging into the memory of the philologist or historian. Rather is it an essential condition of the methodical progress of mathematical reasoning that the mind should remain concentrated on a single point, undisturbed alike by collateral ideas on the one hand, and by wishes and hopes on the other, and moving on steadily in the direction it has deliberately chosen. --HELMHOLTZ, H. _Ueber das Verhältniss der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden, Bd. 1 (1896), p. 178._ =241.= If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called _lemmas_, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry.--LEIBNITZ, G. W. _New Essay on Human Understanding [Langley], Bk. 4, chaps. 2, 8._ =242.= In Pure Mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those _hypotheses_ which are the principles of the science), an arrangement is beautiful in proportion as the principles are few; and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises.