TRACT 26 . MEAN DENSITY OF THE EARTH.

SO

small spaces, and f its centre, or the foot of the axis of thepillar whose base is bcde ; the figure awnea being a hori-zontal or level section, through the point A. Join A, f, andwith the centre a describe the arc of the mid circle gfh. Leta denote the length of the axis on the point f, or the meanheight of the pillar on the base bd ; and s the sine of the an-gle of elevation of that pillar, as observed at A, to the radius], or s = - - -. Then will the magnitude of that co-

VV+Af) °

lumn, or its quantity of matter, be expressed by (bc-Pde)x-J-be x a, which is supposed to be all collected into the axis :consequently, if the attraction of each particle of matter bein the reciprocal duplicate ratio of its distance, the attractionof the matter in the pillar so placed, on the plummet at a,in the direction of the meridian an, will be

ic + De2af

X BE X fit X — X C

BC + DE SH

—- XBEXSf = —X BE X SC

2af af

nearly, supposing f to be equally distant from bc and de,and c the cosine of the angle fan, to the radius 1.

But ^ x cis nearly equal to d, the difference of the sinesof the angles ban, can, as is thus demonstrated. Draw gk,fl, hm, perpendicular to aw, and gp parallel to the samejalso draw the chord gh. Then ak, am are the sines of theangles gan, han, to the radius af, their difference being km= gp; also fl is the cosine of fan, to the same radius: con-sequently gp : fl :: d : c. But the triangles lfa, pgh areequiangular, and therefore gp : fl : : gh : af. Conse-quently gh : AF :: d : c ; or ^ x c = tZ. This equation isaccurately true, when gh is the chord of the arc; and, as thesmall arc differs insensibly from its chord, the same equationis sufficiently near the truth when gh is the arc itself. Sub-stituting now d instead of the quantity ^ x c, in the theo-rem above, it will become be x ds for the measure of theattraction of the pillar whose base is bd, in the direction an.Which is as easy and simple an expression for the attractionof a single pillar, as can well be desired or expected.

But to make the application of this theorem still more easy,