The series here given is 1 - _x_ + _x_² ⁄ 2 + _x_³ ⁄ (2.3) + _x_⁴ ⁄ (2.3.4) - _x_⁵ ⁄ (2.3.4.5), &c. = the number whose hyp. log. is -_x_, and it is required to find the sum of every _nᵗʰ_ term thereof, beginning at the first. Here the quantity sought will (according to the general rule) be truly defined by the _nᵗʰ_ part of the sum of all the numbers whose respective logarithms are -_px_, -_qx_, -_rx_, &c.; which numbers, if _N_ be taken to denote the number whose hyp. log. = 1, will be truly expressed by _N⁻ᵖˣ_, _N⁻⒬ˣ_, _N⁻ʳˣ_, &c. From whence, by writing for _p_, _q_, _r_, &c. their equals 1, α + √(αα - 1), α - √(αα - 1), β + √(ββ - 1), β - √(ββ - 1), &c. and putting α´ = √(1 - αα), β´ = √(1 - ββ), &c. we shall have 1 ⁄ _n_ × (_N⁻ᵖˣ_ + _N⁻⒬ˣ_ + _N⁻ʳˣ_), &c. = 1 ⁄ _n_ into _N⁻ˣ_ + _N⁻ᵃˣ_ × (_N⁻ᵃ‘ˣ_√⁻¹) + (_Nᵃ‘ˣ_√⁻¹) + _N⁻ᵝˣ_ × (_N⁻ᵝ‘ˣ√⁻¹_) + _Nᵝ‘ˣ√⁻¹_) + &c. But _N⁻ᵃ‘ˣ√⁻¹_ + _Nᵃ‘ˣ√⁻¹_ is known to express the double of the co-sine of the arch whose measure (to the radius 1) is α´_x_. Therefore we have 1 ⁄ _n_ into _N⁻ˣ_ + _N⁻ᵃˣ_ × 2 co-s. α´_x_ + _N⁻ᵝˣ_ × 2 co-s. β´_x_, &c. for the true sum, or value proposed to be determined. The solution of this case, in a manner a little different, I have given some time since, in another place; where the principles of the general method, here extended and illustrated, are pointed out. I shall put an end to this paper with observing, that if, in the series given, the even powers of _x_, or any other terms whatever, be wanting, their places must be supplied with cyphers; which, in the order of numbering off, must be reckoned as real terms. CIV. _Observatio Eclipsis Lunæ Die_ 30 Julii 1757. _habita Olissipone à_ Joanne Chevalier, _Congregationis Oratorii Presbytero, é Regia_ Londinensi _Societate. Communicated by_ Jacob de Castro Sarmiento, _M.D. F.R.S._ Tubo optico 8 pedum. [Read Nov. 16, 1758.]