tude of the Mountain. In the Triangle BRF there are three Things known. 1. RF the Semidiameter of the Earth. 2. The Right-angle BFR. And 3. Because the Arch FA is 4 Degr. the Angle BRF is alfo 4 Degr. Therefore fay, As the Radius (100000000) is to the Secant of the Angle BRF 4 Degr. (10024419) fo is RF (3440 Italian Miles or 860 German Mifes) 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 affumed as Truths which are actually falfe. 1. It is fuppofed 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 paffing thro' a Part of the Earth, and therefore the Top B cannot be feen 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 caufes the Mountain to be discovered fooner by 1 Degr. (or 15 German Miles) than if there had been no Refraction at all; fo that fuppofing AF but 3 Degr. the Height of the Mountain will be found but 40 Furlongs, or 5 Italian Miles. 2. It is to be confidered, that Sailors allow themfelves a Liberty of fpeaking largely, especially about their Distances; if therefore, in Confideration of this, we deduct half a Degr. more, and suppose the Top first seen at 2 Degr. or 38 German Miles equal to F A; then will the Altitude of the Mountain A B be

found

found by the former Calculation to be a Mile, or thereabouts.

IF a Mountain be first seen at 2 Degr. distance, (fetting afide the Refraction) it will be found 2 Italian Miles high; but if at 1 Degr. or 15 German Miles, it will be half an Italian Mile, or 5 Furlongs high.

Then it will be feen

at the Dillance of | 144 |151|17|18|21|24|29|4

411

BUT thefe are all to be understood without Refraction, whereby the apparent Height and Distance is generally increafed, as may be feen by the Figure; where the refracted Ray T F being produced to N, gives the apparent Altitude N A.

PROPOSITION V.

Having the Altitude of a Mountain given, to find Geographically it's Distance from the Place, whence may be first feen.

it

THIS is but the converfe of the laft 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 fuppofe it to be first seen at F, to find the Distance AF. (Fig. 14.) In the right angled Triangle B FR, 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 fame added to AB, which fuppofe half a German Mile; fo that RF or RA being

860

860 Miles, BR will be 8601. Therefore fay, as RB 860 is to FR 860: fo is the Radius 10000000 to the Sine of the Angle RBF 9994186. 88 degr. 2 min. 40 fec. Wherefore BRF or the Arch AF will be 1. degr. 57 min. 20. fec. which being turned into German Miles make 291, the Distance. from whence a Mountain whofe Altitude is half a Mile, may be first seen without any Refraction, upon which Account we may add 8 Miles, fo that it may be actually feen 37 Miles off. But the Refraction varies according to the different Altitude of the Sun, or the different Denfity of the Air, when the Sun, is below the Horizon; as we fhall fhew more at large, when we come to treat of the Atmosphere; and in the third Part of this Book, where we fhall Difcourfe of the vifible Horizon.

PROPOSITION VI.

The Length of the Shadow of a Mountain, and the Altitude of the Sun at the fame 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 fo high as that it overshadoweth the Ifle of Lemnos, [now called Stalimene] as far as the Market-place of the City of Myrrhina [or Lemnos], when the Sun is in the Summer Solstice; where the ancient Inhabitants for the Curiofity of the Appearance erected a Brazen Calf, at the termination of the Shadow, as is teftified by the old Greek Monoftich, which may be thus Englished.

2

Mount

Mount Atho's Shadow covers half
The Bulk of Lemno's molten Calf. (c)

PLINY writes, that the Distance between Athos and the Ifle of Lemnos, was accounted $7000 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 Myrrhina when Mount Athos, a little before Sun-fet, began to intercept their View of the Sun-Beams; the Sun being then in the fame vertical Circle, which paffeth over Athos and Myrrhina (because Athos is fituated weftward of Myrrhina). We may fuppose the Sun to have been almoft in the very Horizon of Myrrhina F O, and fo the Ray O F, paffing the Top of the Mountain, to have projected the Shadow AF. (Fig. 15). Here OF is a Tangent to the Periphery, and from having the Angle FBR given, and alfo FR, (or FA in the Triangle, BAF taken as a right Line) B A will be found to be 8 Furlongs, or 1 Italian Mile for the Height of the Mountain. But because in this Pofition of the Sun, the Shadow would be infinitely continued, and therefore it's Extent could not be obferved; and as the Interpofition of the Houses in the Town, would alfo intercept the neighbouring Rays, to thofe that bounded the Shadow; therefore, we must allow the Sun to have been elevated at least 2 Degr. above the Horizon of Myrrhina;

(ε) Αθως καλύψει πλευρά faw the Shadow of the Pike of

λημνίας βοός.

Mr Salmon looks upon this to be a very ridiculous Affertion, and tells us that there never was a Shadow difcernable at 10'Miles Distance from the Hill that made it. But in Oppofition to this, Mr Edens fays, that he actually

Teneriff upon the Sea reaching over the Ifland Gomera, and the Shadow of the upper Part, viz. of the Sugarloaf to be imprinted like another Pike in the Sky it felf. See Salmon's Prefent State of all Nat. Vol. 5. Pag. 396. and Philof. Tranf.N° 345. Pag. 317.

For

For Example, to S; fo that SFO may be 2 Degr. and SF a Ray of the Sun paffing 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 1 degr. 6 min. (i. e. the Distance F A 87 Italian Miles, turned into Degr.) hence FTR 86 degr. 54 min. and alfo 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. fo is FR 860, to RT 861 German Miles. So that AT, the Altitude of Mount Athos, is 1 German Mile, or 32 Furlongs, which is too much; for the Grecians account it not above 11 Furlongs

IF we affume the Altitude of the Sun to be but one Degr. the Altitude of the Mountain will be found but 20 Furlongs.

BUT Pliny, I fuppofe, 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 FA feems to be but about 55 Italian Miles; wherefore the Angle FRT will be but about 55 min. So that fuppofing the Sun's Altitude to be 1 degr. 30 min. the Angle TFR will be 91 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 1 degr. 30 min. and FA, fuppofed 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, 1 degr. 30 min. fo is FA 55 Miles to AT, the Altitude of the Mountain.

VOL. I.

K

ΤΟ