ARITHMETICAL INSTRUMENTS. 27 



been instantaneous if the idea of employing a distinct symbol for 

 zero had occurred to those who used the abacus. But it is pre- 

 cisely the introduction of this symbol which forms the central 

 point of the whole decimal notation ; and it may be admitted that 

 in the abacus itself there was nothing to suggest its introduction. 

 The nearest approximations in the modern world to the ancient 

 abacus are the bean-tables, multiplication boards, and other 

 similar 'appliances employed in elementary education ; * and the 

 marking boards in use in certain games. 



The monetary transactions of the ancient world were occa- 

 sionally on a scale approachfhg those of our own times. When 

 Vespasian became emperor he found that, after the profligate 

 expenditure of Nero, and the subsequent civil wars, the indebted- 

 ness and pressing requirements of the imperial and public treasuries 

 amounted to no less a sum than about three hundred and thirty 

 millions sterling. However rudely the accounts of these vast 

 liabilities may have been kept, they must have required an enor- 

 mous amount of calculation ; and all this calculation must have 

 been performed with the abacus ; for it would have been almost 

 impossible with the written characters of the Roman system of 

 numeration. Perhaps no single instance could better serve to 

 show the great saving of human labour which has been effected 

 by the use of a decimal notation. 



The arithmetic of whole numbers, of which we are here speaking, 

 has its theoretical as well as its practical part. This theoretical 

 part is called the Theory of Numbers, and is perhaps the only 

 branch of pure mathematics against which the charge of useless- 

 ness has ever been seriously alleged. Nevertheless, at all periods 

 of the history of mathematical science it has excited a keen 

 interest, and to it, rather than to researches of more obvious 

 utility, we owe the development of the practical branch of arith- 

 metic. As early as the second or third century before Christ, the 



* A series of these appliances have been contributed to the Exhibition by 

 the Committee of the Russian Pedagogical Museum. 



