ON BUILDINGS IN INDIA. 39 L 



Fortunately, we have no rational use for flat roofs. Our cities 

 are roomy, and our habits and their population will for many 

 centuries keep them so. Our houses are low and our yards 

 airy. I cannot conceive a single argument in favour either of 

 the beauty or utility of terrace roofs in our country. Those 

 that have them scarcely ever use them. The cold in winter 

 and the heat in summer drive us from them. A beautiful 

 prospect may justify the partial use of them, in particular situ- 

 ations, but neither architectural beauty, nor the general wants 

 of our wintry climate call for their introduction. — To the south- 

 ward beyond the reach of frost, however, the information con- 

 tained in this paper may be highly useful, 



No. LVII. 



A general method of finding the roots of numeral equations to any 

 degree of exactness ; xiith the application of Logarithms to shorten 

 the operation : by John Garnet t of New Bi'unswkk N. Jersey, 



Read January 20th, 1809. 

 Suppose an equation, ax-}-bx x +cx3-f-dx*-f ex5 &c. = v, to find x. 



RULE. 



Find, by trial, any near root as s.' 



Then, by substitution, »x'-f-bx'2-f-cx'3-f-dx''*-f-ex'5 &c. = v' 



Multiply each term by the index of the power of x', and divide by x". 



Let the products, a+2bx'+3cx'»+4dxJ'-(-5ex4, &c.= A. 



Multiply each term by the power of %', and divide by 2x'. 



Let the products, b-f-3cx'-|-6dx'3_|-10ex'3, & c . = B. 



Multiply each term by the power of x', and divide by 3x'. 



Let the products, c+4dx'-(-10ex'2, &c. = C 

 Multiply each term again by its power of x', and divide by 4x'. 



Let the products, d-f-5ex', &C.—D : and so on, continually, until all the powers of s.' 

 are destroyed ; so that e, &c.=E. 



Then will Ax" -j-Bx"* -fCx"3 -)-Dx"4 -fEx"5 &c. = v-v', be a New Equation 

 whose roots will all be less by x', and the value, v — v', less by v' than the roots 

 of the original equation. And if the roots and value of this new equation be diminished 

 in the same manner, by another near root, as x"', and so on, continually, the root 

 and value may become less than any assignable quantity, and the sum of all the near 

 roots will be equal to x, the root of the original equation. 



F f 



