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Perimeter of the Earth to be 20.160 Miles, or 20. 340, according to that Measure.
FROM that Time to this none were solicitous about solving the Problem. The Arabs commonly using the Dimensions they had received from their Mathematicians; and the Italians, when they began to study Astronomy, made use of Ptolemy's Measure, viz, 180.000 Furlongs (which make 21.600 Italian Miles, or 5.400 German; fo that 60 of the former, and 15 of the later was thought to make a Degree: but they ought to have reckoned 15 § of the latter, becaufe 32 Furlongs nearly equal a German Mile ; thus the Periphery would be 5625. Germ. Miles). But about 80 Years ago Sneilises, a famous Mathematician, and Professor at Leyden, observing that the Perimeter of the Earth, commonly made use of by Mathematicians (or the length of a Degree, vulgarly supposed 15 Dutch Miles), was queftionable, and founded upon no certain Demonstration; he thereupon applied himself with great Industry to it's Mensuration, and happily finished it; demonstrating the Magnitude of one Degree of the Earth's Periphery to be 28.500 Perches (each containing 12 Rbinland Feet) or 19 Holland Miles; and the whole Periphery to equal 6.840 Miles (reckoning 1.500 Perches, or 18.000 Rbinland Feet, to a Mile).
WE thought fit to premise this short History of the Earth's Mensuration, that the Reader may perceive by what Industry it hath been managed, and with what Difficulty effected. Now we shall treat of the different Methods of Mensuration, all founded upon the Discovery ofthe Earth's spherical Figure, which we have proved in the preceding Chapter. Therefore, considering it globular, if it be cut by a Plane passing thro' the Center, the Section will be a great Circle of the Earth: if not thro'the Center, then the Section will be one of the lesser Circles. Also the Periphery of a great Circle upon the Surface of the
Earth, Earth, is it's Circuit or Measure round. Note, This Periphery is divided (as all others are) into 360 Degr, and because the Extent of the whole cannot be measured at once, we solve the Problem by finding the Length of a Part (viz. of 1 Degr. i Degr. &c.) in known Measures; which Necessity often occurs in other Problems. We also frequently take the Periphery of the Earth to be a Meridian passing thro’the Place of Observation, and the North or Pole-Star; which is more eafy, and less subject to Error. The first Method; used by the Arabians and others
for measuring the Earth.
LE Tour Horizon be bHRSs; then the Perimeter of the Terrestrial Meridian (which lies under, and is concentrical to, that in the Heavens abcd) will be ABCD, (Fig. 6.) and R will be the Center of the Earth. Suppose our Place of Observation at B, whose Zenith is b, and the Terrestrial Pole A lying under that in the Heavens a; then the Elevation of the Pole above our Horizon will be AH, or ab. Let us take another Place in the fame Meridian ABCD under abcd, as G, whose Zenith is 8, and Horizon FFR Tt. Now suppose the Elevation of the Pole to be accurately observed in the Place B, viz. ab or AH; and also in the Place G, viz. fa or FA. Take FA from HA and the Remainder is HF, equal to BG, the Arch intercepted between the two Places. Lastly the Distance BG equal to the Arch bg, is to be accurately measured by some known Measure, as a Perch or a Mile. Then by the Golden Rule say, as BG isto ABGCD, 360 Degr. so is the known Interval in Miles or Perches, to the Miles or Perches contained in the Periphery ABGCD: or as the Arch BG is to I Degr. so are the Miles in the Distance BG, to the Miles or Perches in Degree,
NOTE, If you take the vulgar Computation of the Distance BG, without measuring it, then the Quantity of the Degree will be determined accordingly; as i Degr, will equal 15 such Miles, as B G equals 10, &c.
Example, Let B be Amsterdam, where the Eleva-. tion of the Pole A H or ab is 52 degr. 23 min, and let G be Schoonboven, lying under the fame Meridian with Amsterdam, where the Elevation of the Pole AF or af is 51 degr. 54 min. therefore FH or BG will be 29 min. but the Distance between Amsterdam and Schoonhoven is 91 Dutch Miles, or 13875 Rbinland Perches, 12 Foot each; therefore, as 29 min. is to 60 min. or i degr. so is 9 Miles to 19 Dutch Miles: therefore 19 Dutch Miles equal i degr. and 6,840 make 360, or the whole Periphery.
OR if the Distance B G be supposed 7 German Miles (each equal to 1900 Rbinland Perches) it will be as 29 min. is to 60 min. fo is 71 to 15 of the same German Miles, for a Degr. of which 5.400 make the whole Circumference. Thus the Elevation of the Pole at Prague is 50 degr. 6 min. and at Lincium 48 degr. 16 min. the Difference B G is i degr. 50 min. and the Distance is computed to be 26 German Miles; from whence the Periphery will be 5.105 Miles.
AGAIN, let there be two Places under the fame Meridian ; the one B, Alexandria in Egypt, where Eratosthenes, Keeper of the King's Library, lived; the other G, (Fig. 6.) the Town of Syene, a City in Egypt, under the Tropic of Cancer, and, for that Reason, chosen by Eratosthenes, whose Distance from Alexandria was computed 5000 Furlongs. Let the Distance of the Sun, at Noon, from the Zeniths, & and b, of both Places be
observed by an Instrument on the same Solstitial Day, viz, the 21st of June ; when, at Alexandria, gb or GB equals só Part of the Periphery by Observation (or 7 degr. 12 min.) but at Syene the Sun hath no Distance from the Zenith at Noon, it being exactly vertical that Day. So that the Arch of the Distance B G, intercepted between the two Places is 7 degr. 12 min. but the Distance itself is accounted 5.000 Furlongs (8 of which make an Italian Mile). Therefore by the Golden Rule, as 7 degr. 12 min. is to i degr. (or as já to 360, or as 36 to 5) so is 5000 to 694Furlongs in i degr. Or as jó is to i, (or as I to 50) so is 5000 to 25000 Furlongs, the whole Periphery, according to this Measure. There are divers ways of taking the Meridian Altitude of the Sun, or it's Distance from the Vertex; as by a Quadrant, &c. Eratosthenes found it by a hollow hemispherical Dial ; where the Style BX (Fig. 7.) points to the Zenith, and OXZ is a Ray of the Sun terminating the Shadow of the Style, and shews the Arch B Z equal to O B 7 degr. 12 min, the Distance of the Sun from the Zenith. But at Syene the Style hath no Shadow ; nor hath the Sun any Distance from the Zenith ; being perpendicular to the Plane of the Place. Therefore fince B X Z (Fig. 6, 7.) is equal to the Angle b X O, (whose Measure is B G or bO) B G is equal to BZ 7 degr. 12 min, or so Part of the Periphery, as before.
The third Method, that of Posidonius.
POSIDONIUS took two Places under the same Meridian ; viz. B, Rhodes, the Place where he lived, and G, Alexandria in Egypt; and observed the Altitude of the Star S (Fig. 6.) (a bright Star in the Ship Argo called Canopus) when it came to the Meridian of both Places, on the same,
or which is all one, on different Days. This Star did not rise above the Horizon b Hs at Rhodes, but only glanced upon it at S: tho it was elevated above the Horizon of Alexandria FRT, the Arch tsas Part of the Periphery or 7 degr. 30 min. He tells us the Distance betwixt Alexandria and Rhodes is 5.000 Furlongs. Therefore, as 7 degr. 30 min. is to i degr: (or as to vão, i, e, as 360 to 48) so is 5.000 to 666} Furlongs in i degr, or as 1 : 48 :: 5.000 : 24.000 Furlongs, for the whole Periphery of the Earth, according to Posidonius.
The fourth Method, that of Snellius.
IN the Methods above delivered we have constantly supposed the two Places to lie under the same Meridian; but because Places may lie plainer, and more commodious for this Purpose under different Meridians, we shall propose an Example in this case which is that of Snellius. .
LET therefore ABCD (Fig. 6.) be the Me. ridian of Alcmair, and B, Alcmair itself; where the Elevation of the Pole ba is 52 degr. 40î min. and the Polar Distance BA 37 degr. 19ź min. 30
LET the other Place P be Bergen-op-zoom, whose Meridian is APC, and its Distance from the Pole, or Complement of Latitude (viz. to 51 degr. 29 min.) is AP 38 degr. 31 min. therefore, having drawn PG perpendicular to ABG, the Difference of their Distances from the Pole is BG I degr. 11 min. 30 sec.
AFTER Snellius had taken these Observations, he accurately measured the Distance BP, between Alcmair and Bergen, and found it to be 34710 Rbinland Perches; and the Angle of Position PBG 11 degr. 26 min. 2 sec. therefore in the rightangled Triangle PBG, the Hypotenuse PB and ..