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TWO:Inherent or cooling strains in castings is much more intricate than shrinkage: it is, in fact, one of the most uncertain and obscure matters that pattern-makers and moulders have to contend with. Inherent strains may weaken castings, or cause them to break while cooling, or sometimes even after they are finished; and in many kinds of works such strains must be carefully guarded against, both in the preparation of designs and the arrangement of patterns, especially for wheels and pulleys with spokes, and for struts or braces with both ends fixed. The main difficulty resulting from cooling strains, however, is that of castings being warped and sprung; this difficulty is continually present in the foundry and machine-shop, and there is perhaps no problem in the whole range of mechanical manipulation of which there exists more diversity of opinion and practice than of means to prevent the springing of castings. This being the case, an apprentice can hardly hope for much information here. There is no doubt of springing and strains in castings being the result of constant causes that might be fully understood if it were not for the ever-changing conditions which exist in casting, both as to the form of pieces, the temperature and quality of metal, mode of cooling, and so on.

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TWO:Such a view was essentially unfavourable to the progress of science, assigning, as it did, a higher dignity to meagre and very questionable abstractions than to the far-reaching combinations by which alone we are enabled to unravel the inmost texture of visible phenomena. Instead of using reason to supplement sense, Aristotle turned it into a more subtle and universal kind of sense; and if this disastrous assimilation was to a certain extent imposed upon him by the traditions of Athenian thought, it harmonised admirably with the descriptive and superficial character of his own intelligence. Much was also due to the method of geometry, which in his time had already assumed the form made familiar to us by Euclid��s Elements. The employment of axioms side by side with definitions, might, indeed, have drawn his attention to the existence and importance of judgments which, in Kantian terminology, are not analytic but synthetic��that is, which add to the content of a notion instead of simply analysing it. But although he mentions axioms, and states that mathematical theorems are deduced from them, no suspicion of their essential difference from definitions, or of the typical significance which they were destined to assume in the theory of reasoning, seems ever to have crossed his mind; otherwise he could hardly have failed to ask how we come by our knowledge of them, and to what they correspond in Nature. On the whole,385 it seems likely that he looked on them as an analysis of our ideas, differing only from definition proper by the generality of its application; for he names the law of contradiction as the most important of all axioms, and that from which the others proceed;277 next to it he places the law of excluded middle, which is also analytical; and his only other example is, that if equals be taken from equals the remainders are equal, a judgment the synthetic character of which is by no means clear, and has occasionally been disputed.278

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