THEORY OF APPROXIMATION. 103 



differ only by an imperceptible quantity. To the car- 

 penter anything less than the hundredth part of an inch 

 is non-existent ; there are few arts or artists to which the 

 hundred-thousandth of an inch is of any account. Since 

 all coincidence between physical magnitudes is judged by 

 one or other sense, we must be restricted to a knowledge 

 of apparent equality. 



In reality even apparent equality is rarely to be ex- 

 pected. More commonly experiments will give only 

 probable equality, that is results will come so near to 

 each other that the difference may be ascribed to un- 

 important disturbing causes. Thus physicists often assume 

 quantities to be equal provided that they fall within the 

 limits of probable error of the processes employed. We 

 cannot expect observations to agree with theory more 

 closely than they agree with each other, as Newton re- 

 marked of his investigations concerning Halley's Comet. 



Arithmetic of Approximate Quantities. 



Considering that almost all the quantities which we 

 treat in physical and social science are approximate only, 

 it seems desirable that some attention should be paid in 

 the teaching of arithmetic to the correct interpretation 

 and treatment of approximate numerical statements. We 

 ought carefully to distinguish between 2*5 when it means 

 exactly two and a half, and when it means, as it usually 

 does, anything between 2*45 and 2*55 It would be better 

 in the latter case to write the number as 2*5 .... and we 

 might then distinguish 2*50 .... as meaning anything 

 between 2*495 . . . . and 2*505. When approximate 

 numbers are added, subtracted, multiplied, or divided, 

 it becomes a matter of some complexity to determine 

 the degree of accuracy of the result. There are few 

 persons, for instance, who could assert straightway that 



