_Untersuchungen über die unendlich oft oszillierenden und unstetigen Functionen; Ostwald’s Klassiker der exacten Wissenschaften, No. 153, pp. 44-45._ =721.= Without doubt one of the most characteristic features of mathematics in the last century is the systematic and universal use of the complex variable. Most of its great theories received invaluable aid from it, and many owe their very existence to it. --PIERPONT, J. _History of Mathematics in the Nineteenth Century; Congress of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. 474._ =722.= The notion, which is really the fundamental one (and I cannot too strongly emphasise the assertion), underlying and pervading the whole of modern analysis and geometry, is that of imaginary magnitude in analysis and of imaginary space in geometry.--CAYLEY, ARTHUR. _Presidential Address; Collected Works, Vol. 11, p. 434._ =723.= The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast.--RUSSELL, BERTRAND. _The Study of Mathematics; Philosophical Essays (London, 1910), p. 77._ =724.= Induction and analogy are the special characteristics of modern mathematics, in which theorems have given place to theories and no truth is regarded otherwise than as a link in an infinite chain. “_Omne exit in infinitum_” is their favorite motto and accepted axiom.--SYLVESTER, J. J. _A Plea for the Mathematician; Nature, Vol. 1, p. 261._ =725.= The conception of correspondence plays a great part in modern mathematics. It is the fundamental notion in the science of order as distinguished from the science of magnitude. If the older mathematics were mostly dominated by the needs of mensuration, modern mathematics are dominated by the conception of order and arrangement. It may be that this tendency of thought or direction of reasoning goes hand in hand with the modern discovery in physics, that the changes in nature depend not only or not so much on the quantity of mass and energy as on their distribution or arrangement.--MERZ, J. T. _History of European Thought in the Nineteenth Century (Edinburgh and London, 1903), p. 736._ =726.= Now this establishment of correspondence between two aggregates and investigation of the propositions that are carried over by the correspondence may be called the central idea of modern mathematics.--CLIFFORD, W. K.