VOL. LXVIII.] PHILOSOPHICAL TKANSACTIONS. 451 



secured wooden building, quite contiguous to this vast heap of fire, was not at 

 all damaged, except some parts of the outer coat of plaster-work. This experi- 

 ment was intended to represent a wooden town on fire, and to show how effectu- 

 ally even a wooden building, if secured according to his new method, would 

 stop the progress of the flames on that side, without any assistance from fire- 

 engines, &c. 



§ 20. The last experiment made that day, was the attempting to burn a wooden 

 stair-case, secured according to the simple method of under-flooring. The under 

 side of the stair-case was extra-lathed. Several very large kiln faggots were laid, 

 and kindled, under the stair-case, round the stairs and on the steps; this wooden 

 stair-case notwithstanding resisted, as if it had been of fire-stone, all the attempts 

 that were made to consume it. His lordship afterwards made 5 other still stronger 

 fires on this same stair-case, without having repaired it, having also filled the 

 small place in which this stair-case is entirely with shavings and large faggots; 

 yet the stair-case is still standing, being but little damaged. 



§ 21. In most houses it is necessary only to secure the floors; and that accord- 

 ing to the method of single under-flooring described above, in '^ 2, 3, 4, and 5. 

 The extra-expence of it, all materials included, is only about Q pence per square 

 yard, unless there should be particular difficulties attending the execution, in 

 which case it will vary a little. When quick lime is used, the expence is a trifle 

 more. The extra-expence of the method of extra lathing, is no more than 6 

 pence per square yard for the timber side-walls and partitions ; but for the cieling 

 about 9 pence per square yard. No extra-lathing is necessary in the generality 

 of houses. 



XLf. A Method of finding, by the Help of Sir Isaac Newton's Binomial The- 

 orem, a Near Value of the very Slowly Converging Infinite Series x -\- iar' 

 -\. ^^ -\- \x'^ -\- Xx^ -\- &c. when x is very nearly equal to 1 . By Francis 

 Maseres, Esq., F.R.S., &c. p. 895. 



If the capital letters a, b, c, d, e, &c. be put for the numeral co-efRcients of 

 the powers of or in the said series, so that A shall be = 1, B = -l, c = 4-, d = a., 

 E = -J-, and so on, we shall have b = ^a, c = ^b, d = ac, e = 4-d, &c., and 

 the proposed series x + -i.r'^ -|- -ijr^ + -^x* -j- iz* -j- &c. will be = o^' + iAx^ -}- 

 ■l-Bar^ + ^cx* -f ^Dx^ -f AE.r® -|- tFx'' -\- ^-Gx^ -\- &c.; in which series the frac- 

 tions i, -f , ^, ^, f, -f , •!-, &c. which generate the co-efficients of the powers of x 

 in the several terms after the first term x, are derived from each other by the 

 continual addition of 1 to both their numerators and their denominators. 



This observation suggests a method of finding a near value of the sum of this 

 series by the help of Sir Isaac Newton's binomial theorem, which may be ex- 

 plained as follows. If m and n represent any two whole numbers, the reciprocal 



3 M 2 



