trees; the lamina, which have the appearance of wood, being always horizontal, according to the situation of the pieces in the strata: or could we suppose a number of fossil trees to be brought together, and ranged in this regular manner. in the several strata, yet by the form and roundness of their trunks, they must be in a great measure encompassed by the soil, in which they are buried; whereas there is not the least mixture of earth, or any other aperture in the Bovey strata of coal, except a few crevices, common to this sort of fossil, which divide the pieces from each other in all directions, and seem to be inconsistent with the nature and fibrous texture of wood. If the basis or matrix of this fossil were wood, it would acquire, by being impregnated with bitumen, a greater degree of inflammability; whereas it neither kindles nor consumes so fast as wood. Dr. M. mentions a number of other places where such fossil substances are found; and then concludes with observing the several particulars in which all the species of the bituminous fossils resemble each other. They seem to be generally found between beds of clay or stone; are of a dark brown, or black colour, of a laminated texture; pliable when moist, and fresh dug, but crisp and brittle when dry; full of cracks, and easily breaking transversely; they all sink in water, and emit the same nauseous and bituminous smell; they differ in being more or less solid, heavy, and inflammable, according to the proportions and principles of which they consist; and if any doubt could remain of their being a mineral substance, it must be removed by the following analysis. One pound of Bovey coal, of the woody kind, powdered, put into a glass retort, and distilled in sand, yielded 44oz. of phlegm, which had the appearance of common water, but somewhat of a bituminous smell and taste; near 4 oz. of a turbid whitish bituminous liquor, of an intolerable fetid smell, and extremely pungent to the tongue; about 2 drs. of a heavy bituminous matter, which would not mix with the liquor above mentioned, but sunk entirely to the bottom, and (which is very remarkable) there was not the least appearance of any light oil floating on the bituminous liquor. There remained in the retort about 7 oz. of a very black powder, which had the same bituminous smell, not very heavy ; some of which being put on a red-hot iron, emitted a little smoke, but no flame, The ashes of this fossil, when burnt, being boiled in water, and the water evaporated, there remained no salt behind. LIV. A New Method of Computing the Sums of Certain Series. By Mr. Communicated by Mr. Thomas Simpson, F. R. S. p. 553. John Landen. 1. Supposing to be the sine of the circular arc z, whose radius is 1, correct fluents, we have hyp. log. Hence, by taking the Hence, writing a for one 4th of the periphery of the circle whose radius is 1, and taking æ equal to the said radius, we find hyp. log. sequently hyp. log. √1= 1 a = √; and con a and hyp. log. 2a - hyp. log. 22 4.39 &c. F' = x+ F= 3. By writing, in the first equation in the preceding article, instead of r, we have + hyp. log. − 1 = ± 26 + x + hyp. log. being put for, and x for the hyp. log. of x. It is evident, therefore, that 1 1. x 2 3 Hyp. log. —— is = − 26 −x+x++, &c. where, of the two signs prefixed to 2b, the upper one takes place, when the hyp. log. of 1 is taken equal to likewise when a is taken equal to -1; and the lower one takes place, when the hyp. log. of 1 is taken equal to 1 taken equal to therefore, if we observe to take the value of hyp. log. of — 1, as last mentioned, and a equal to instead of ✔=1, we need retain 4. For brevity sake, we shall, in what follows, put the series 5. Multiplying the last equation in art. 3, by and taking the correct fluents we have F′ = 2r" + 2bx — X — — x 2 Whence, by multiplying by, and taking the fluents, we get F” = 2p′′x + bx2 − 23 + x2 + + + + 7, 23 39 &c. Again, multiplying the last equation by, and taking the correct fluents, we p" From which equation, by taking + 1, −, &c. = p′′ + b2 = p" — a2; and, by taking æ equal to Therefore 4p" - 3a2 is pa2: hence p" is found = = 2a2 3 1 + 12 22 31 we have &c. r" a', as found above; we, by subtraction, a2 &c. (= 20′′) = a2, and, consequently a" = 0. Hyp. log. of — = 1 − x + (-)+(-), &c. and consequently x = — 1 -X 2 (1 Moreover the fluent of hyp. log. ofis = x + 1 + × -x vanishes when a vanishes; and the fluent of x x is = (1 − x) + 3 1 -x + (1_-_*)", &c. — p′′, being corrected so as to vanish when a vanishes. 32 1 But the fluent of hyp. log. of fluent of × + hyp. log. of ——, which also vanishes when a vanishes. (1-x)1 22 X X is = x X 1 - I Whence, by taking x equal to 4, we find square of hyp. log. of 2 = 2 x (12.21 + 2121 + 3:23, &c.) — p′′: hence, p′′ being before found = 7. Further, ++, &c. the value of F" in art. 2, must be equal to +x++, &c. the value of p" in art. 5, when Whence, by taking x equal to -1, we have 4bp" + 4b3 — 2.3* 2.3 = 0; and, we find Whence the values of p, p, &c. q", q", &c. q', q'i, &c. may tained, in terms of a. 9. Hyp. log. (—-—±—=)1 being G' = fluent of hyp. log. () be easily ob 10. By writing, in the first equation in the preceding article, we have Hyp. log. (――) = x2+=+==,&c. 1+ But the hyp. log. of (—) is = hyp. log. (+1) = hyp. log. (±) hyp. log. ✔=1= ±b+ hyp. log. (1±5)3. It is manifest, therefore, that Hyp. log. (1±1)3 is = 5 where, with respect to the two signs prefixed to b, the same observation may be made as in art. 3. 11. Multiplying the last equation by and taking the correct fluents, we have G' = 2a" + bx X-1 7-7, &c. - ཀླག་ Whence, by multiplying by, and taking the fluents, we get Again, multiplying the last equation by, and taking the correct fluents, we |