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tude of the Mountain. In the Triangle BRF there are three Things known. I. RF the Semidiameter of the Earth. 2. The Right-angle BFR. And 3. Because the Arch FA is 4 Degr. the Angle BRF is also 4 Degr. Therefore say, : As the Radius (100000000) is to the Secant of the Angle BRF 4 Degr. (10024419) so is RF (3440 Italian Miles or 860 German Mises) to RB (3448 Italian Miles or 860 German Miles); substract RA (3440 or 860) and there will remain BA (8 Italian Miles, or 2 German Miles, for the Height of the Mountain (which is extraordinary, and even above the Computations of the Antients). Therefore we must know that there are two Things assumed as Truths which are actually falfe. i. It is supposed that the Ray of Light which first strikes the Eye, comes from B in a right Line, when it is known on the contrary to be curved, or refracted, by Reason of the Density of the Atmosphere. For a Right Line cannot be drawn from the Top B to F (FA being 4 Degr.) without passing thro’ a Part of the Earth, and therefore the Top B cannot be seen in a right Line from the Place F, but by the bowed Ray BTF, being the first of the refracted Rays that can touch F. From whence we may reasonably infer, that this Refraction causes the Mountain to be discovered sooner by i Degr. (or 15 German Miles) than if there had been no Refraction at all, so that supposing A F but 3 Degr. the Height of the Mountain will be found but 40 Furlongs, or 5 Italian Miles. 2. It is to be considered, that Sailors allow themselves a Liberty of speaking largely, especially about their Distances ; if therefore, in Consideration of this, we deduct half a Degr. more, and suppose the Top first feen at 2ż Degr. or 3.8 German Miles equal to F A; then will the Altitude of the Mountain A B be
found by the former Calculation to be a Mile, or thereabouts.
I Fa Mountain be first seen at 2 Degr: distance, (setting aside the Refraction) it will be found 2 Italian Miles high; but if at i Degr. or 15 German Miles, it will be half an Italian Mile, or 5 Furlongs high.
Then it will be seen at the Distance of
at the Distance of 144 (15817_183|21|24|29|41|
BUT these are all to be understood without Refraction, whereby the apparent Height and Distance is generally increased, as may be seen by the Figure ; where the refracted Ray TF being produced to N, gives the apparent Altitude N A.
Having the Altitude of a Mountain given, to find
Geographically il's Distance from the Place, whence it may be first seen.
THIS is but the converse of the last Propofition, and may be had from the foregoing Table : but Calculation will give a more accurate Solution.
LET therefore A B be the Height of a Mountain given, and suppose it to be first seen at F, to find the Distance AF. (Fig. 14.) In the right angled Triangle BFR, the Angle F is a right Angle, and the two Sides FR, RB are given, the former being the Semidiameter of the Earth, and the later the same added to AB, which suppofe half a German Mile ; so that RF or RA being
860 860 Miles, BR will be 860ị. Therefore fay, as RB 860ż is to FR 860: so is the Radius 10000000 to the Sine of the Angle RBF 9994186. 88 degr. 2 min. 40 sec. Wherefore BRF or the Arch AF will be 1. degr. 57 min. 20. sec. which being turned into German Miles make 29į, the Distance from whence a Mountain whose Altitude is half a Mile, may be first seen without any Refraction, upon which Account we may add 8 Miles, so that it may be actually seen 373 Miles off. But the Refraction varies according to the different Altitude of the Sun, or the different Density of the Air, when the Sun, is below the Horizon; as we shall shew more at large, when we come to treat of the Atmosphere ; and in the third Part of this Book, where we shall Discourse of the visible Horizon.
The Length of the Shadow of a Mountain, and the
Altitude of the Sun at the same Time, being given, to find the Altitude of the Mountain,
WE propose this Problem more for the Antiquity and Elegancy of it, than for any Accuracy we believe to be in the Method. Plutarch and Pliny have writ, that Mount Athos, on the Macedonian Shore, is so high as that it overshadowech the Ise of Lemnos, (now called Stalimene] as far as the Market-place of the City of Myrrhina for Lemnos), when the Sun is in the Summer Solstice ; where the ancient Inhabitants for the Curiosity of the Appearance erected a Brazen Calf, at the termination of the Shadow, as is testified by the old Greek Monoftich, which may be thus Engliseda
Mount Atho's Shadow covers half
PLINY writes, that the Distance between Athos and the Ife of Lemnos, was accounted 87000 Paces, or 87 Italian Miles, but neither he nor any other Author have determined the Altitude of the Sun, at the Time of this Shadow ; tho' it is probable, it was projected upon the Town of Myrrhing when Mount Aibos, a little before Sun-set, began to intercept their View of the Sun-Beams; the Sun being then in the same vertical Circle, which passeth over Athos and Myrrhina (because Athos is situated westward of Myrrhina). We may suppose the Sun to have been almost in the very Horizon of Myrrhina F O, and so the Ray O F, passing the Top of the Mountain, to have projected the Shadow AF. (Fig. 15). Here O F is a Tangent to the Periphery, and froin having the Angle FBR given, and also FR, (or FA in the Triangle, BAF taken as a right Line) B A will be found to be 8 Furlongs, or i Italian Mile for the Height of the Mountain. But because in this Position of the Sun, the Shadow would be infinitely continued, and therefore it's Extent could not be observed ; and as the Interposition of the Houses in the Town, would also intercept the neighbouring Rays, to those that bounded the Shadow ; therefore, we must allow the Sun to have been elevated at least 2 Degr. above the Horizon of Myrrhina;
(c) "Antws nonefes aneupci faw the Shadow of the Pike of amp.vies Boós.
Teneriff upon the Sea reaching Mr Salmon looks upon this to over the Island Gomera, and the be a very ridiculous Assertion, Shadow of the upper Part, viz. and tells us that there never was of the Sugarloaf to be imprinted a Shadow discernable at 10 Miles like another Pike in the Sky it Distance from the Hill that made self. See Salmon's Present State it. But in Opposition to this, of all Nat. Vol. 5. Pag: 396. and Mr Edens says, that he actually Philos. Trans. N 345. Pag. 317.
For Example, to S; so that SFO may be 2 Degr. and SF a Ray of the Sun passing the Vertex of the Mountain T, and terminating the Shadow in F.
THEREFORE in the oblique angled Triangle RFT, the Angle TFR 92 Degr. and FRT i degr. 6 min. (i. e. the Distance F A 87 Italian Miles, turned into Degr.) hence FTR 86 degr. 54 min. and also the Semidiameter FR, 860 German Miles, being all given; the Side T R may be found by this Proportion. As the Sine of the Angle FTR 86 degr. 54 min. is to the Sine of the Angle TFR 92 degr. so is FR 360, to RT 861 German Miles. So that AT, the Altitude of Mount Athos, is i German Mile, or 32 Furlongs, which is too much; for the Grecians account it not above 11 Furlongs
IF we assume the Altitude of the Sun to be but one Degr. the Altitude of the Mountain will be found but 20 Furlongs.
BUT Pliny, I suppose, has given us too large a Distance betwixt Aibos and Myrrhina, which may perhaps be a Reason, that too great a Height arises from this Calculation : and in most of our modern Maps of Greece, the Distance F A seems to be but about 55 Italian Miles; wherefore the Angle FRT will be but about 55 min. So that supposing the Sun's Altitude to be i degr. 30 min. the Angle TFR will be gi degr. 30 min. and FTR 87 degr. 35 min. Therefore in the Triangle FRT, as the Sine of the Angle FRT 87 degr. 35 min. is to the Sine of the Angle TFR 91 degr. 30 min. fo is FR 860 to RT.
OR in the Triangle TFA right angled at A, TFA will be i degr. 30 min. and FA, supposed a right Line, 55 Miles, from whence the Height TA will be found by this Proportion. As the Radius is to the Tangent of the Angle TFA, i degr. 30 min. fo is FA 55 Miles to AT, the Altitude of the Mountain, VOL. I.