Combinatory Analysis e-bog

Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. The object of this work is, in the main, to present to mathematicians an account of theorems in combinatory analysis which are of a perfectly general character, and to shew the connexion between them by as far as possible bringing them together as parts of a general doctrine. It may appeal also to others whose reading has not been very extensive. They may not improbably find here some new points of view and suggestions which may prompt them to original investigation in a fascinating subject.<br><br>Little attempt has been hitherto made either to make a general attack upon the territory to be won or to coordinate and arrange the ground that has been already gained. The combinatory analysis as considered in this work occupies the ground between algebra, properly so called, and the higher arithmetic. The methods employed are distinctly algebraical and not arithmetical The essential connecting link between algebra and arithmetic is found in the circumstance that a particular case of algebraical multiplication involves arithmetical addition. Thus the multiplication of a and a, where a, x and y are numerical magnitudes, involves the addition of the magnitudes x and y When these are integers we have the addition which is effective in combinatory analysis. This link was forged by Euler for use in the theory of the partitions of numbers. It is used here for the most general theory of combinations of which the partition of numbers is a particular case. The theory of the partition of numbers belongs partly to algebra and partly to the higher arithmetic. The former aspect is treated here. It is remarkable that in the international organization of the subject-matter of mathematics Partitions is considered to be a part of the Theory of Numbers, which is an alternative name for the Higher Arithmetic, whereas it is essentially a subdivision of Combinatory Analysis which is not considered to be within the purview of the Theory of Numbers. The fact is that up