This does not mean that one must have complete command of the entire range of those earlier sciences in order to pursue a later one. Mere human limitations would prevent the fulfilment of such a demand. As a matter of fact, successful work can be done in one of the later sciences even if only the most general features of the earlier ones have been clearly grasped. Nevertheless, the rapidity and certainty of the results are very considerably increased by a more thorough knowledge of the earlier sciences, and the investigator, accordingly, should seek a middle road between the danger of insufficient preparation for his special science and the danger of never getting to it from sheer preparation. In any circumstances he must be prepared always, even though it be in later age, to acquire those fundamental aids so soon as he feels the need of them for carrying out any special work. It is generally acceded that without logic the adequate pursuit of science is impossible. Nevertheless, the opinion is widely current, even among men of science, that everybody has command of the needful logic without having studied it. No more than a man can learn of himself to use the calculus, even if he may have discovered unaided some of its elementary principles, can he acquire certainty and readiness in the use of the logical rules generally necessary, unless he has made the necessary studies. It is true that the scientific works of the great pioneers and leaders in the special sciences furnish practical examples of such logical activity. But complete freedom and security are acquired only on the basis of conscious knowledge. We have now seen how, from the physiological construction of our mental apparatus, the process of concept formation and the experience of concept connections are the basis of the whole of mental life. The laws of the mutual interaction of the most general or elementary concepts operated in the formation of the concepts, _thing_, _group_, _co-ordination_. Here were found the fundamentals of logic or the science of concepts. A special process of abstraction yielded the concept of _number_, and with it the corresponding field of _mathematics_, arithmetic, algebra, and the theory of numbers. By means of the second fundamental fact of physiology, the _threshold_, another elementary fact was explained, that of _continuity_. The co-ordination of individual things under the influence of this concept was expanded into the _co-ordination of continuous phenomena-series_, and yielded the correspondingly more general concept of the _function_. From the application of the number concept to continuous things, the idea of _measurement_ resulted. In mathematics the concept of continuity led to higher _analysis_ and the _theory of functions_. Finally, the concept of continuity proved to be an inexhaustible aid for the extension of scientific knowledge and for the formulation of natural laws in mathematical form. PART III THE PHYSICAL SCIENCES