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the Angle P B G being given, the Side B G is found to be 34018 Perches (which Snellius makes only 33930, for he abated 88 Perches on Account of the Stations where the Elevation of the Pole was observed). But the Arch B G, as was said before, is 71»- min. therefore as 7 ii is to 1 degr. or 60 min. so is 339.30 (or 34018) to 28473 Perches j or the round Number 28500 for 1 degr. equal to 19 Dutch Miles. Or by spherical Trigonometry; having AB, AP, and the Angle ABP given, find the Arch BP 1 degr. 14 min. which equals 34710 Perches; therefore 1 degr. will be 23300 Perches, or 18t Miles. The Reason why this Account differs from that of Snellius is; 1. He did not observe the Elevation of the Pole from the very Tops of the Towers B and P themselves, from whence the Angle GBP was taken, but from some Eminence or rising Ground a little remote from them: yet without Doubt the Altitudes of the Pole were the fame on the Tops of the Towers. 2. Another Reason is, he took B G, B P, P G for right Lines, which are indeed circular; tho', in so small an Arch, the Difference is of little or no Moment. Therefore, granting Snellius'?, Measure of a Degree to be 2.8500 Perches, equal to 18^ Miles, (and mine 28300 making 18*) the Perimeter of the Earth will be, according to Snellius, 10.260.000 Perches, or 123.120.000 Feet, that is 6.840 Holland Miles (a).
[a] The Measure of the Acuities, the curious have not Earth which Snellius with great thought it safe to confide in aIndustry discovered, hath been ny one, tho' the most skillful, deservedly embraced by the Mathematician; which we see Learned; as being much more confirmed by CnJJini the Son accurate than any of the for- of the famous Astronomer of mer. Nevertheless, in a MaN that Name. For he having calter of such Moment, and which culated the Numbers, arising it involved with so many Dif- from Snellius"* Observation, assigned The fifth, but first Terrestrial, Method.
JI Q W to perform the Work, without Celestial Observations, or a Meridian Line, is explained in
signed a much greater Measure to the Earth than Snellius; and also discovered some Errors in his Calculation, which spoiled the whole Process pf his Work. See Hist. Acad. Scien. 1702. Add to this, that the Latitude and Angle ofPofition of Places can now be taken more accurately by Telescopes, which are begun, some Years ago, to be fitted to Astronomical and Surveying Instruments, instead of bare little Pins, which Snellius used. Tho' several others had set about this Work; yet some French Mathematicians, Fellows of the Royal Academy of Sciences, did most successfully perform it: whose Mensuration far exceeds all others, both \n the Number and Accuracy of their Observations, and also in the Furniture of most exquisite Instruments. Wherefore we esteem it well worth the while, to give the whole Method of Operation in short.
The Points in the Figure Which are marked with Roman Letters, fliew the Places chosen for Observation; whose Rearing, or Situation, in respect of the Rpyal Observatory at Paris, is seen in a Geographical Map. (Sie Fig. 8 )
By the same Method of Mensuiation whjch Snellius used,
they proposed to find the Distance between the Parallels of the Places and E, or the Line N * in Fathoms; so that this Distance being known, and the Latitude of each Place N and E, or the Difference of Latitude; that is an Arch of the Meridian intercepted between the two Parallels, being found, it will appear how many Fathoms make any determined Arch of a great Circle of the Earth, such as the Meridian is: from whence it will be easily sound how many Fathoms equal a Degree, or the whole Periphery of the Earth. Afterwards it was thought fit to measure the Line N |3, the Distance between the Parallels of the Places N and Q^_; so that the Latitude at Q, being also observed, there might be had an Arch of the Meridian equal to the whole Distance a,. For by this Means, they could more accurately determine the Measure of the Earth's Periphery, when they had found it the fame by two Operations. These Lines they measured by a continued Series of Triangles drawn from the Line AB; for it being directly plane and straight, they had the Advantage of measuring it with Iron Rods as accurately as could E 2 be, be, and found it to be 5663 Fathoms.
The Latitudes of the Places were taken by an Instrument, whose Radius was 10 Parit Feet; and the Angles of each "Triangle by a Quadrant of a Circle whose Semidiameter was 3! Feet; both which Instruments were accurately divided by Diagonal Lines.
In the first Triangle ABC. There are known by Observation
The SCAB 54°- °4'- 35". JZ< ABC 95. 06. 55. AnS- I ACB 30. 48. 30. Found by Fatb. Fe.
measuring - - A B-5663. 00.
Ia the second Triangle ADC.
In the third Triangle DEC.
Ia the fourth Triangle DCF.
In the fifth Triangle D F G-
In the sixth Triangle GDE-
When they had found the Line GE, by another Series of Triangles, to be 31893 Fatb. 3 Feet, they divided the Difference which made up the lesser Measure 31895 Fatb.
In the seventhTriangle HFG.
In the eighth Triangle HGI.
In the ninth Triangle HIK.
In the tenth Triangle IK L.
Fatk. Fatb. Fe. IK 11683. 00. Hence KL 11188. 02. And IL 11186. 04.
In the eleventh Triangle KLM.
LKM 28. 52. 30. KM L 63. 31. 00. Fatb. Fe. KL in88. 02. Hence LM 6036. 02,
If the Sum of the three Angles ILK, KLM, MLN. be taken from 360 Degrees, there will remain the Angle ILN i\ydegr. 32 min. 40sec.
In the 12th Triangle LMN.
In the 13'* Triangle ILN.
ILN 32. 40.
Here are found three Parts of the Space intercepted between the two Places E and N, viz. EG, G I, IN, not exactly in the Meridian Line it self N *; but so as the Meridional Distances may be found by the following Operations. Also after they had found the Length of G I and IN by another Series of Triangles, as they had done before in the Line G E, they proposed to measure a new straight Line R S (and found it to be 3902 Fatb.) by which the Measures of the a—ES
of which the Line 3*ls compounded, shewing the Distance between Q_(3 and is, Parallels of Latitude erf the Ptacies'Q.and E. For this being found, and an Arch of the Meridian intercepted between the fame PaTallets being known, they had in Effect obtained their Desire, viz. the Measure agreeing to a known Part of the Periphery of the Earth.
Let therefore pN'y J* a, 19, G e be Parts of the Meridian Circle, passing thro' the Places N, I, G; also Qj, ly, G/, and ct E e Parallels of Latitude passing perpendicularly thro' those Meridians in the Places QJ G E.
Then in the Triangle Q 8 N rightangled at (J, the Inclination of the Line Q_N to the Meridian Line N /J is observed, viz.
The Angle QJs £ 18°. 55'. '1 •"; Fath. Fe.
And the Line Nqjs 11161.4. Hence NB 10559. 3.
fn the Triangle N y I fectangled at y ,
vNI 2". 9' 10".
•'. V. Fath. Fe. IN 18967. o. Hence 18893. 3.
In the TriangteG I 9, rect'angled at 9» GIfl i. 9. o.
Fath. Ft. I G 17564. b. Hence I9t oxyS- 17560. 3.
In the Triangle G E e, rectarigled at $,
EG i 66. 26. 00.
GE 31895. d. Hence G5, or ^« 31894. o.
Hence the Distance between
the Parallels of the Places N and E, viz. the Sum of the three Lines, N)/, y <T, / at, is 68348 Fathoms; to which if the Line N 3 be added, it will make up the Distance between the Parallels of the PlacesQjmd E 78967 Fath. % Feet.
Then it remained to observe the Difference of Latitude of the Places E, N» "and Q_; or the Arches of the Meridian intercepted between their Parallels. To Which end there were taken three Stations, a liftle distance from the Places themselves; for the fake of better Observation.
The first Station was distant from the Place E 18 Fathoms Southward; the second from the Place N 65 Fathoms Northward; the third from the Place Q_75 Fathoms Eastward.
The Arch of the Meridian intercepted between the first and second Station was found to be i°. 11'. 57". between the second and third Was izz. 35.
But jf'83 Fath. ('the Sum of 18 arid 65, by which the first and second Station were further than the Piaffe N and E) be added, to'68.348 (that is tothe Line N 4. tlie Distance between the Parallels of the two Places N and E) the Sum will be 68.431