l6l PHILOSOPHICAL TKANSACTIONS. [aNNO ] 726. 



changed, and a cypher is always to be understood where there is no significant 

 figure. The sign of the cypher is neglected ; but where there is occasion to 

 consider it, it is always supposed the same as the sign of the following figure. 



Thus the negativo-affirmative number 729586455982001730 is immediately re- 

 duced to 710585545977998330; and so of all others. 



But on the contrary, common numbers may be reduced to negativo-affirma 

 live numbers a great variety of ways, by substituting instead of the figures J, 



•2, 3, 4, 5, 6, 7, 8, 9, their respective values 19, 18, 17, j6, 15, 14, 13, 12, 11, 

 in any places at pleasure. But the most useful reduction of this kind is what 

 Mr, C. calls a reduction to small figures, which consists in throwing out all the 

 large figures, Q, 8, 7, 6, out of any given number, and introducing in their stead 



the equivalent small figures 11, 12, 13, 14, respectively. Thus I82937462 may 



be reduced to 223143542, consisting only of small figures. But this reduction 

 may be performed more readily by these rules following. 



I. A small figure before a large figure must be increased by a unit. 2. A 

 large figure before a large figure must be changed into its negative complement 

 to 9. 3. A large, figure before a small figure must be changed into its negative 

 complement to 10. Other figures are not to be changed; and 5 will be 

 ambiguous, being to be esteemed either large or small, according as the figure 

 following is either large or small. Some examples of this reduction are as here 

 follow, both in whole numbers and decimal fractions. 



37068259764 = 43132340244 



729528960739957 = 1331531041340043 



9260872395,87294 = 113411 32404, 13314 



Or (9) 926O87239587294 = (10)1134113240413314 



(m) 387916407953 &C. = (m) 412124412153 &c. 



It is to be observed, that in this last example the numbers are what Mr. C. 

 calls interminate, or approximations only ; that is, the first and most valuable 

 figures are expressed, and all the rest (whether finite or infinite in number, 

 whether known or unknown) are omitted as inconsiderable, and insinuated by 

 the mark, &c. Also the index m before the number stands for some integer, 

 expressing the distance of the first figure 3 or 4 from the place of units ; which 

 integer is either affirmative or negative, according as the said first figure stands 



