AS PRACTISED BY THE ARABS. 55 



For 10 contains 2, or 1 -f 1 figures and all less numbers contain only 

 one, again 100 contains 3 or 2 -f- 1 figures, and all less numbers contain 

 only one or two, again 1000 contains 4 or 3 -f 1 figures, and all less numbers 

 contain only one, two, or three, and 10 = 10\ 1 00 = 10*, 1,000 = 10 3 , &c. 



Lem. 3. Hence the n ttl power of a Digit, as defined in paragraph % 

 cannot contain more than n figures. 



. For let a be any Digit then a n Z. 10 n , but 10 n is the least number 

 which contains n -f- 1 figures, hence a n must contain less than n -f- 1 

 figures, that is not more than n. 



Lem. 4. The greatest number which contains only n figures is 10" — 1. 



For the greatest number with 2 figures is 99 = 100 — 1 zz 10 s — 1. 

 The greatest number with 3 figures is 999 — 1,000—1 = 10 3 — 1. The 

 greatest number with 4 figures is 9,999 = 10,000—1 = 10 4 — 1, &c. 



Lem. 5. Let a be the number of figures in the integer A. Then the 

 number of figures in A n is not greater than na, nor less than n (a — 1) •+- 1. 



For by Lem. 4. since there are a figures in A, so the maximum 



of A is 10 a — 1, and maximum of A n is 10 a — 1 | which is evidently less 



an 



than 10] or 10 an . But 10 an is by Lem. 2. the least number which can 

 contain an -J- 1 figures. And hence (10 a — l) n or A" must contain less than 

 an -|- 1 figures, that is not more than an. 



Again, since there are a figures in A, so by Lemma 2 the minimum 



of A is lO*- 1 and minimum of A° is 10 a -i = 10 <a - ,,n and by Lem. 2 10 (a - ,)0 

 contains (a — 1) n -\- 1 figures. 



Q 



