_A Plea for the Mathematician, Nature, Vol. 1, p. 261._ =203.= Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,--we see in all this but a faint reflection of the intellectual activity of a _Lagrange_ or _Cauchy_, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.--MACH, E. _Populär-wissenschafliche Vorlesungen (1908), pp. 224-225._ =204.= Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols. --DILLMANN, E. _Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 94._ =205.= The essence of mathematics lies in its freedom. --CANTOR, GEORGE. _Mathematische Annalen, Bd. 21, p. 564._ =206.= Mathematics pursues its own course unrestrained, not indeed with an unbridled licence which submits to no laws, but rather with the freedom which is determined by its own nature and in conformity with its own being.--HANKEL, HERMANN. _Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), p. 16._ =207.= Mathematics is perfectly free in its development and is subject only to the obvious consideration, that its concepts must be free from contradictions in themselves, as well as definitely and orderly related by means of definitions to the previously existing and established concepts.--CANTOR, GEORGE. _Grundlagen einer allgemeinen Manigfaltigkeitslehre (Leipzig, 1883), Sect. 8._