CHAPTER XV 

 MODERN TENDENCIES IN MATHEMATICAL SCIENCE 



Mathematics is the queen of the sciences and arithmetic the 

 queen of mathematics. She often condescends to render service to 

 astronomy and other natural sciences, but in all relations she is en- 

 titled to the first rank. Gauss. 



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 

 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 marshals a whole troop of completed operations 

 in the execution of a new one. Mack. 



The iron labor of conscious logical reasoning demands great per- 

 severance and great caution ; it moves on but slowly, and is rarely 

 illuminated by brilliant flashes of genius. It knows little of that 

 facility with which the most varied instances come thronging into 

 the memory of the philologist or historian. Rather is it an essential 

 condition of the methodical progress of mathematical reasoning that 

 the mind should remain concentrated on a single point, undisturbed 

 alike by collateral ideas on the one hand, and by wishes and hopes on 



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