THE STARTING-POINT 5 



thermometer and note the degrees registered by it that 

 is, we must count. We see plainly enough that some 

 things are bigger than others, but if we would be 

 correct we must measure them by some standard, and 

 this again implies counting. It is the same with 

 respect to weight and motion, with respect to our own 

 past history and the past history of mankind. In matters 

 of antiquarian knowledge, bygone periods of time have 

 to be carefully computed, and sometimes the duration 

 of nations and of dynasties. The velocity of winds and 

 waves, the rapidity of the lightning's flash, as well as 

 the seemingly slow revolutions and displacements of 

 the heavenly bodies, have all to be also estimated by 

 counting that is, by number. 



Thus the one thing which alike pertains to everything 

 we know, terrestrial or celestial, material or mental, 

 is " number." It is a certain numerical relation, or 

 rather various numerical relations, since, for example, a 

 nation is one when compared with other nations, but 

 multitudinous when considered with respect to the 

 individuals that compose it. This truth doubtless under- 

 lay the system of Pythagoras, who, five hundred years 

 and more before our era, taught that number was the 

 principle of all things. 



But the study of that which is thus common to all 

 things, is the study of mathematics ; and therefore 

 mathematics, or the science of number, is and must be 

 the most fundamental of all sciences, since it pertains 

 to every other, and no other can be pursued without it. 

 An introduction to the elements of science must there- 

 fore begin with an introduction to the elements and 

 principles of mathematics, 



