1. The great ſucceſs of the experiment, lately made by the Royal Society, on the hill Schehallien, to deter⯑mine the univerſal attraction of matter, and the im⯑portant conſequences that have reſulted from it, may probably give occaſion to other experiments of the ſame kind to be made elſewhere: and as all poſſible means of accuracy and facility are to be deſired in ſo delicate and laborious an undertaking; it has occurred to me that it might not be unuſeful to add, by way of ſup⯑plement to my paper of calculations relative to the a⯑bove-mentioned experiment, an inveſtigation of the height above the bottom of a hill, at which its horizon⯑tal attraction ſhall be the greateſt; ſince that is the height at which commonly the obſervations ought to be made, and ſince this beſt point of obſervation has never been any where determined that I know of, but has been va⯑riouſly ſpoken of or gueſſed at, it being ſometimes ac⯑counted at ⅓, and ſometimes at ½ of the height of the hill; whereas from theſe inveſtigations it is found to be generally at about only ¼ of the altitude from the bottom.

2. Let ABCEDA be part of a cuneus of matter, its ſides or faces being the two ſimilar right-angled trian⯑gles ABC, ADE meeting in the point A, and forming the indefinitely ſmall angle BAD. Then of any ſection [6]bced, perpendicular to the planes ABD and ADE, the attraction on a body at A in the direction AB, is equal to the conſtant quantity ss; where s = ſin. ›BAC and s = ſin. ›BAD, to the ra⯑dius 1.

For, firſt, ſince the magnitude of the flowing ſec⯑tion is every where as Ab2, and the attraction of the particles of matter inverſely as the ſame, or as 1/Ab2; therefore their product or Ab2/Ab2 or 1 (a conſtant quantity) is as the force of attraction of bced.

Then to find what that quantity is. Put AB = a, and BC = x; then BD or CE (the diſtance between the two planes at the diſtance AB) is = as. Now the force of a particle in the line CE is as 1/AC² in the direction AC, and therefore it is as AB / AC³ in the direction AB; conſequently the force of the whole lineola CE in the direction AB is AB.CE / AC³ ; and therefore the fluxion of the force of the ſection BCED or ḟ is [...]; and the fluent gives [...] for the attraction itſelf.

[7]3. To find now the attraction of the whole right-angled cuneus on a body at A in the direction AB.— Since the force of each ſection is ss by the laſt article, therefore the force of all the ſections, the number of them being AB or a, is ass = s . AB. BC / AC the force of the whole cuneus ABCEDA.

4. To find the attraction of the rect⯑angular part ABCD on A in the direction AB; ABCD being one ſide of the cune⯑us, and AD its edge.—Put AD = BC = b, and AB = x. Then, the force of any ſection as BC being always as ss by Art. 2, the fluxion of the force or ḟ will be [...]; and the fluent is [...] the attraction of ABCD.

5. To find the attraction of the right-angled part BCD of a cuneus whoſe edge paſſes through A the place of the body attracted.—Put AB = a, BC = b, BD = c, DA = d = a − c, DC = e, AC = g, and DP = x. Then, the force of any ſection PQ being ſtill as ss, the fluxion of the force of the part DPQ is [...]; and the correct fluent [8]when x = c is [...] the force of a body at A in the direction AB.

6. To find the attraction of the right-angled part BCD on the point A.—Uſing here again the ſame notation as in the laſt article, we have [...]. The correct fluent of which, when x = c, is [...].

7. To apply now theſe premiſes to the finding of the place where the attraction of a hill is greateſt, it will be neceſſary to ſuppoſe the hill to have ſome certain fi⯑gure. That poſition is moſt convenient for obſerving the attraction, in which the hill is moſt extended in the eaſt and weſt direction. Suppoſing then ſuch a poſition of a hill, and that it is alſo of a uniform height and meridional ſection throughout; the point of obſerva⯑tion muſt evidently be equally diſtant from the two ends. But inſtead of being only conſiderably extended, I ſhall ſuppoſe the hill to be indefinitely extended to the eaſt and to the weſt of the point of obſervation, in order that the inveſtigation may be mathematically true, and yet at the ſame time ſufficiently exact for the before-ſaid limited extent alſo. It will alſo come neareſt to the [9]practical experiment, to ſuppoſe the hill to be a long triangular priſm, ſo that all its meridional ſections may be ſimilar triangles. Let therefore the triangle ABC re⯑preſent its ſection by a verti⯑cal plane paſſing through the meridian, or one ſide of an indefinitely thin cuneus whoſe edge is in PG; or rather PBCGP the ſide of one cuneus, and PAG the ſide of another, their common edge being the line PG perpendicular to the baſe AC; P being the required point in the ſide AB where the attrac⯑tion of the ſection ABC, [...] indefinitely thin cuneus, ſhall be greateſt in a direction parallel to the horizon AC. And then from the foregoing ſuppoſitions, it is evident that in whatever point of AB the attraction of ABC is greateſt, there alſo will the attraction of the whole hill be the greateſt.

8. Now draw HPDEF parallel to AC; and AH, PG, BI, CF, perpendicular to the ſame. Then it is evident that at the point P, in the direction PF, the attraction of PBCGP is affirmative, and that of PAG negative. But PBCGP is = PBD + BDE + PFCG − EFC; and PAG = PHAG − PHA. Therefore the attractions of PBD, BDE, PFCG, PHA, are affirmative; and thoſe of EFC, PHAG, negative.

[10]Put now BI = a, AI = b, IC = c, AB = d, BC = e, AC = g = b + c, and PG = x, the altitude of the point P above the bottom. Alſo let s = the ſine of the indefinitely ſmall angle of the cuneus to rad. 1; and [...].

Then by Art. 3, the attraction

• of PBD is s. PD. BD / BP = sb × a−x/d,
• of PHA is s. PH. PG / PA = sb × x/d.

By Art. 4, the attraction

• PFCG is s. PG × h. l. PF+PC / PG = sx × h. l. ag−bx+qq/ax,
• PGAH is s. PG × h. l. PH+PA / PG = sx × h. l. b+d/a.

By Art. 5, the attraction of EFC is [...].

Laſtly by Art. 6, the attraction of BDE is [...].

Theſe quantities being collected together with their proper ſigns, and contracted, we have [...] for the whole attraction in the direction PE.

[11]9. Having now obtained a general formula for the meaſure of the attraction in any ſort of triangle, if the particular values of the letters be ſubſtituted which any practical caſe may require, and the fluxion of this at⯑traction be put = 0, the root of the reſulting equation will be the required height from the bottom of the hill.

10. But for a more particular ſolution in ſimpler terms, let us ſuppoſe the triangle ABC to be iſoſceles, in which caſe we ſhall have d = e, and g = 2b = 2c, and then the above general formula will become [...] for the value of the attraction in the caſe of the iſoſ⯑celes triangle, where q2 is [...]. And the fluxion of this expreſſion being equated to 0, the equation will give the relation between a and x for any values of b and d, by a proceſs not very trouble⯑ſome.

11. Now it is probable that the relation between a and x, when the attraction is greateſt, will vary with the various relations between b and d, or between b and a. Let us therefore find the limits of that relation, between which it may always be taken, by uſing two particular extreme caſes, the one in which the hill is [12]very ſteep, and the other in which it is very flat, or a very ſmall in reſpect of b or d.

12. And firſt let us ſuppoſe the triangular ſection to be equilateral; in which caſe the angle of elevation is 60°, which being a degree of ſteepneſs that can ſcarce⯑ly ever happen, this may be accounted the firſt extreme caſe. Here then we ſhall have d = 2b = ⅔a√3, and the formula in Art. 10, will become s × : 2a−r−x/2 + x × h. l. 2a+2r−x/3x + a−x/4 × h. l. a+2r+x/a−x for the value of the attraction in the caſe of the equilateral triangle, in which r is [...].

13. Or if we take x = na, where n expreſſes what part of a is denoted by x, the laſt formula will become [...] for the caſe of the equila⯑teral triangle.

14. To find the maximum of the expreſſion in the laſt article, put its fluxion = 0, and there will reſult this equation [...]; the root of which is n = .251999. Which ſhews that, in the equilateral triangle, the height from the bottom to the point of greateſt attraction, is [13]only 1/500^th part more than ¼ of the whole altitude of the triangle. And this is the limit for the ſteepeſt kind of hills.

15. Let us find now the particular values of the mea⯑ſure of attraction ariſing by taking certain values of n varying by ſome ſmall difference, in order to diſcover what part of the greateſt attraction is wanting by ob⯑ſerving at different altitudes.

16. And firſt uſing the value of n (.251999) as found in the 14^th article, the general formula in Art. 13, gives sa × 1.0763700 for the meaſure of the greateſt at⯑traction.

17. If n = 3/10, or x = 3/10a; the ſame formula gives sa/20 × : 17 − √79 + 6h. l. 17+2√79/9 + 7/2 h. l. 13+2√79/7 = sa × 1.0702512 for the attraction at 3/10 of the altitude, which is ſomething leſs than the other.

18. If n = 4/10 = ⅖ the formula gives sa/20 × : 16 − √76 + 8h. l. 8+√76/6 + 3h. l. 7+√76/3 = sa × 1.0224232 for the attraction at 4/10 or ⅖ of the altitude; leſs again than the laſt was.

19. If n = 5/10 = ½; the formula gives [...] for the attraction at half way up the hill; ſtill leſs again than the laſt.

[14]20. If n = 6/10 = ⅗; the formula gives sa/20 × : 14 − √76 + 12h. l. 7+√76/9 + 2h. l. 8+√76/2 = sa × .8109843 for the attraction at 6/10 or ⅗ of the altitude from the bottom; being ſtill leſs than the laſt was. And thus the quantity of attraction is continually leſs and leſs the higher we aſcend up the hill above the .251999 part, or in round numbers .252 part of the altitude. Let us now deſcend, by trying the numbers below .252; and firſt,

21. If n = .25 = ¼; the ſame formula in Art. 13, gives ⅛sa × : 7 − √13 + 2 h. l. 7+2√13/3 + 3/2 h. l. 5+2√13/3 = sa × 1.0763589 for the attraction at ¼ of the altitude; and is very little leſs than the maximum.

22. If n = 2/10 = ⅕; the formula gives 1/10 sa × :9 − √21 + 2 h. l. 9+2√21/3 + 2 h.l. 3+√21/2 = 1/10 sa × 9 − √21 + 2 h. l. 23+5√21/2 = sa × 1.0684622 for the attraction at 2/10 or ⅕ of the altitude; and is ſomething leſs than at ¼ of the altitude.

23. If n = 1/10; the formula gives sa/20 × : 19 − √91 + 2 h.l. 19+2√91/3 + 9/2 h. l. 11+2√91/9 = sa × .9986188 for the attraction at 1/10 of the altitude; ſtill leſs than the laſt was. And, laſtly,

24. If n = 0, or the point be at the bottom of the [15]hill; the formula gives [...] for the attraction at the bottom of the hill; which is between ⅔ and ¾ of the greateſt attraction, being ſome⯑thing greater than ⅔ but leſs than ¾ of it.

25. The annexed table exhibits a ſummary of the

calculations made in the preceding articles; where the firſt column ſhews at what part of the altitude of the hill the obſervation is made; the ſecond column contains the correſponding numbers which are proportional to the attraction; and the third column ſhews what part of the greateſt attraction is loſt at each reſpective place of obſervation, or how much each is leſs than the greateſt.

26. Having now ſo fully illuſtrated the caſe of the firſt extreme, or limit, let us ſearch what is the limit for the other extreme, that is, when the hill is very low or flat. In this caſe b is nearly equal to d, and they are both very great in reſpect of a; conſequently the for⯑mula for the attraction in Art. 10, will become barely [...]; the fluxion of which being put = 0, we obtain [...]; hence therefore [...], and [...]. [16]Which ſhews that the other limit is 29/100; that is, when the hill is extremely low, the point of great⯑eſt attraction is at 29/100 of the altitude, like as it is at 25/100 when the hill is very ſteep. And between theſe limits it is always found, it being nearer to the one or the other of them, as the hill is flatter or ſteeper.

27. Thus then we find that at ¼ of the altitude, or very little more, is the beſt place for obſervation, to have the greateſt attraction from a hill in the form of a tri⯑angular priſm of an indefinite length. But when its length is limited, the point of greateſt attraction will deſcend a little lower; and the ſhorter the hill is, the lower will that point deſcend. For the ſame reaſon, all pyramidal hills have their place of greateſt attraction a little below that above determined. But if the hill have a conſiderable ſpace flat at the top, after the manner of a fruſtum, then the ſaid point will be a little higher than as above found. Commonly, however, ¼ of the altitude may be uſed for the beſt place of obſervation, as the point of greateſt attraction will ſeldom differ ſenſibly from that place. And when uncommon circumſtances may produce a difference too great to be intirely neglect⯑ed, the obſerver muſt exerciſe his judgment in gueſſing at the neceſſary change he ought to make in the place of obſervation, ſo as to obtain the beſt effect which the concomitant circumſtances will admit of.