PROCEEDINGS OF THE BOARD OP REGENTS. 119 



Beport on an improved system qf numeration, hy W. B. Taylor, Esq. 



Umted States Patent Office, 



Washington, March 22, 18G7. 



I have examined tlie paper referred to, on tlie subject of an improved numera- 

 tion for arithmetical operations, and have, respectfull}', to offer the following 

 remarks: The proposal is simply to interpolate six additional '' digits" (if the 

 term may be allowed) between the nine and the ten of our common arithmetical 

 scale, in every order or place of figures ; in other words, to substitute a senide- 

 nary for the received denary radix of numeration. This suggestion has been 

 made, I believe, more than once before. In 1859, Mr. J. W. Nystrom, of Phila- 

 delphia, published an essay on what he called the tonal system, (ton being 

 the name he assigned to the senidenary ten,) advocating the adoption of the 

 number 16 as the basis of a universal arithmetic and metrology. 



All who have given the subject of weights and measures much consideration 

 will agree in the proposition that a scheme of continual bisections and doublings 

 would prove a great convenience in all the operations of concrete arithmetic, and 

 were it not for the enormous lal)or of a reconstruction, and the gi'cat time required 

 for its general introduction among civilized nations, some such reform might be 

 accepted as advantageous or desirable. 



So early as the beginning of the last century, the illustrious Leibnitz elaborated 

 a scheme of binar}- arithmetic, (whose only characters w'ere 1 and 0,) and pub- 

 lished a treatise in its exposition and support. A paper of his upon the subject 

 will be found in the Memoirs of the Academic Royal des Sciences for the year 

 1703, page 85, in which he says he had himself employed this ratio of computa- 

 tion for many years, and that he regarded it as " la perfection de la science des 

 nombres;" an opinion which, from such an authority, is entitled to very high 

 respect. 



It may well be questioned, however, whether the senidenarj' scale favored by 

 your correspondent would fulfil the true desideratum — a minimum of arithmetical 

 labor. There are considerations tending to show that even our present denary 

 ratio is too high for the most complete and general facility. In balancing the 

 two opposite conditions of conciseness of expression, and simplicity of construc- 

 tion, it must be borne in mind that while the number of places required to express 

 a given value is diminished, simply as the logarithm of the radix increases, 

 the mental labor required in using any scale is increased in a considerably higher 

 ratio than the arithmetical increment of the radix ; probably in a geometrical 

 progression, or as some low power of the base. I am inclined to believe, there- 

 fore, that as between the binary and senidenary systems, the former is decidedly 

 to be preferred ; that the economy of places or of expression in the latter would 

 prove but a trival compensation for its much larger range and variety of symbols 

 and the far greater complexity of all the tables and processes necessary in its 

 employment. 



For all popular uses, either the quarternary or octonary scale would probably 

 be found much more convenient than either of these suggested extremes, and 

 certainly much more available for the distribution of weights and measures. 



In 1719, Swedenl>org published an Octonary Computus, and a project of an 

 octaval system of weights, measures, and coins. It is said that Charles XII, of 

 Sweden, had contemplated the experimental adoption of the scheme not long 

 before his death, in 1718. 



It may not be considered irrelevant to here briefly compare the four different 

 scales above mentioned with our established scale, in point of expressiveness. 



