12$ SECURITY AGAINST FIRE. 



remainder ever occur, it must necessarily do so before the di- 

 vidend exceeds D digits. 



Let therefore 111111 &c. to D digits, when divided by D, 

 give q for a quotient and r for a remainder, now sir.ee this 

 remainder will recur again, whether 111111 &c. be or be 

 not divisible by D, if the --umber of digits D in thedividend 

 be increased, let therefore 111111 &c. to D + d digits, when 

 divided by D, giveQ for a quotient and r for a remainder. — 

 In the first case we have 1 1 1 1 1 1 &c. to D digits rz D q -f r. 

 and in the second case we have 111111 &c. to L> + d digits 

 zDQ + r, the difference of these equations gives 111111 

 &c. 000000, &c. — DQ ~-D q, where it is evident the units 

 will consist of d digits and the ciphers of D places, whence 



_ 111111 &c. ..to d digits 000000 &c. D places 

 Q_ q _ __ 



z=a whole number, where the divisor D is any number even or 

 odd. Now the 111111 &c->to<i digits must, without the 

 ciphers, be divisible by every odd number not greater than 

 the divisor D (5 and its multiples excepted) for if it were not, 

 whatever were the remainder, suppose K for instance, we 

 should have R 0000 &c- »to D ciphers divisible by an odd 

 number not a multiple of 5, which is impossible; moreover 

 llllll &c. • -to d digits, will not divide by 2, or by 5, or by 

 any multiple of these numbers, since no multiple of 2 or 5 

 can terminate with 1 : hence llllll &c. to d digits is divi- 

 sible by any odd number, except 5 and its multiples, where 

 d the number of digits in the dividend can never be greater 

 than D the divisor, since the recurrence of any remainder 

 must take place in D digits of the dividend. 



Q. E. D. 



VII. 



Qn the Use of Iron for Stairs, and instead of the Timbers of 

 Houses, as a Security against Fire. In a Letter from Mr. 

 Benjamin Cook. 



To Mr. NICHOLSON. 

 SIR, 



Ironrecom- Jj[N a former paper I threw out some loose hints on the ad- 

 s'airs ° °* vantage of employing iron in various articles of furniture, 



as 



