b G. H. KNIBBS. 



emotions and will, the cultivation of our understanding and 

 imagination, the education of our thought and of its expression, 

 and indeed all those things that are the concern not only of science,, 

 but of polite literature, philosophy, and religion, are also factors 

 in civilisation, of transcendent importance. If in reviewing our 

 indebtedness to the physical sciences, it may appear that the 

 greater things have been forgotten, it is in appearance only. 



That this is so, is obvious when we recognise that the glory of 

 physical science lies not so much in the fact that it satisfies our 

 material needs with relatively so small an expenditure of physical 

 energy, or that it swiftly places at our disposal the products of 

 industrial effort; but rather, that in doing these things, it becomes 

 an instrument of culture, by intensifying, enlarging and enriching 

 our life, and by giving us a finer appreciation of the possibilities 

 of existence. 



No science is so intimately interwoven in the plexus of human 

 affairs, or may so justly be regarded as constituting the very warp 

 and woof of the fabric of modern civilisation, as mathematics; and 

 yet no equally weighty fact is perhaps more completely suppressed 

 in our everyday consciousness. Touching even the simplest of its- 

 elements, the trivialities of ordinary arithmetic, which one might 

 think almost beneath notice, it would be difficult to compute the 

 magnitude of our debt to that scheme of notation, which assigns 

 a constant ratio of increase in the unit value of each place of 

 figures proceeding to the left, and to the congruity therewith of 

 our system of numeration. 1 We are apt to overlook the value of 



i The popular impression that the advantage of our notational and 

 numerational systems depends upon its decimal character is of course 

 erroneous. In respect to the facility of our ordinary arithmetical opera- 

 tions, the base or radix is a matter of comparative indifference, and a 

 sexagesimal or duodecimal system has merits absent in the decimal system. 

 The laws and processes of ordinary arithmetic are independent of the 

 base of the system, and are dependent only upon its notational scheme, 

 or * law of position/ but that arithmetic may involve a minimum of mental 

 labour, it is essential to have a congruous scheme of numeration. Our 

 cumbrous English systems of weights and measures afford many examples 

 of the difficulty introduced by absence of congruity. 



