732 SCIENTIFIC THOUGHT. 



string together in mathematical reasoning are derived 



from various and heterogeneous sources. We begin 



65. with counting, then we introduce measuring ; in both 



Counting . 



and cases we have definite elements or units which may 



measuring. 



serve to express order or quantity or both, and we 

 have definite conventional operations ; then we have 

 symbols which may denote order or quantity or oper- 

 ation. With these devices we perform on paper 

 certain changes, and we get accustomed to use in- 

 discriminately these heterogeneous conceptions, arith- 

 metical, geometrical, algebraical nay, even dynamical, as 

 when Newton introduced the conception of a flow or 

 fluxion. As mathematics is an instrument for the 

 purpose of solving practical problems, skill in al- 

 ternately and promiscuously using these incongruous 

 methods goes a very long way. Geometrical, mechan- 

 ical evidence helps frequently where pure logic comes 

 to a standstill, and pure logic must help and correct 

 where apparent evidence might deceive us. Mathe- 

 matics and science generally have always progressed 

 by this alternate use of heterogeneous devices, and 

 will probably always do so. The straight line of pure 

 logic has but very meagre resources, and resourcefulness 

 is the soul of all progress. But though this may be 

 so in practice, there are two other interests which govern 

 scientific reasoning. There is the love of consistency and 

 accuracy, and of clean and transparent, as distinguished 

 from muddled and scamped, work. The latter leads 

 inevitably into serious errors and paradoxes, as the 

 great mathematicians, Gauss, Cauchy, Abel, pointed out 

 early in the century. Mathematics then frequently 



