_Philosophy of the Pure Sciences; Lectures and Essays (London, 1901), Vol. 1, p. 402._ =727.= In our century the conceptions substitution and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant and differential parameter, appear more and more clearly as the most important conceptions of mathematics.--LIE, SOPHUS. _Leipziger Berichte, No. 47 (1895), p. 261._ =728.= Generality of points of view and of methods, precision and elegance in presentation, have become, since Lagrange, the common property of all who would lay claim to the rank of scientific mathematicians. And, even if this generality leads at times to abstruseness at the expense of intuition and applicability, so that general theorems are formulated which fail to apply to a single special case, if furthermore precision at times degenerates into a studied brevity which makes it more difficult to read an article than it was to write it; if, finally, elegance of form has well-nigh become in our day the criterion of the worth or worthlessness of a proposition,--yet are these conditions of the highest importance to a wholesome development, in that they keep the scientific material within the limits which are necessary both intrinsically and extrinsically if mathematics is not to spend itself in trivialities or smother in profusion. --HANKEL, HERMANN. _Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), pp. 14-15._ =729.= The development of abstract methods during the past few years has given mathematics a new and vital principle which furnishes the most powerful instrument for exhibiting the essential unity of all its branches.--YOUNG, J. W. _Fundamental Concepts of Algebra and Geometry (New York, 1911), p. 225._ =730.= Everybody praises the incomparable power of the mathematical method, but so is everybody aware of its incomparable unpopularity.--ROSANES, J. _Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13, p. 17._ =731.= Indeed the modern developments of mathematics constitute not only one of the most impressive, but one of the most characteristic, phenomena of our age. It is a phenomenon, however, of which the boasted intelligence of a “universalized” daily press seems strangely unaware; and there is no other great human interest, whether of science or of art, regarding which the mind of the educated public is permitted to hold so many fallacious opinions and inferior estimates.--KEYSER, C. J.