VOL. XX.] PHILOSOPHICAL TRANSACTIONS. 275 



which, from the time of the Nicene council, till now, amounts to about 

 11 days. So that the equinox which then happened March 21, is now come 

 back to our March 11, or rather March 10. Which, upon Pope Gregory's 

 reforming the Roman calendar, above 100 years since, causes the difference of 

 10 days, between what we call the New Style and the Old. Which, 1 years 

 hence, in the year 1700, and thenceforth for 100 years, will be 11 days. 

 2. It was then supposed, that in 19 years, which is the compass of the golden 

 number, the lunations of new and full moon returned to the same day and hour, 

 as they were 19 years before. This is pretty near the truth, but comes short 

 by about an hour and a half Which hour and a half, in every IQ years since 

 that time, amount to about 4 or 5 days. Whence it happens, that the reputed 

 full moon is later, by 4 or 5 days, than that of the heavens. But our Easter is 

 reckoned according to the reputed full moons, derived from the golden number, 

 and not according to those of the heavens. 



A Method of extracting the Root of an Infinite Equation. By A. De Moivre, 



F.R.S. N°240, p. 190. 



If az + hzz + cz' + dz' + ez' &c. = gy + hjy + iy^ + ktf + ly' &c. 

 then will zbe = -y -\- ^* -(- y^ -j. 



k — Jbb — 2iAC — 3CAAB — (/a" ^ / _ 2Abc — 2Aad — 3cabb - 3caac - 4c?A3r. — cM . . 



a i/ + 2 ^^^• 



For the understanding of this series, and in order to continue it as far as we 

 please, it is to be observed, 1. That every capital letter is equal to the co- 

 efficient of each preceding term ; thus the letter b is equal to the coefficient 



— - — . 2. That the denominator of each coefficient is always a. 3. That 

 the first member of each numerator is always a coefficient of the series °-y + 

 hyy + iy^ &c. viz. the first numerator begins with the first coefficient "■, the 

 second numerator with the second coefficient h, and so on. 4. That in every 

 member after the first, the sum of the exponents of the capital letters, is 

 always equal to the index of the power to which this member belongs : thus 

 considering the coefficient ^ " ^"^ - ^''--- ^^^^b - d.^ ^ ^^.^^ ^^^^^^^ ^^ ^^^ 



power y\ we shall see that in every member Zjbb, 2^ac, 3caab, dA\ the sum 

 of the exponents of the capital letters is 4 ; where I must take notice, that by 

 the exponent of a letter, I mean the number which expresses what place it has 

 in the alphabet ; thus 4 is the exponent of the letter d. Hence I derive this 

 rule for finding the capital letters of all the members that belong to any power ; 



N N 2 



