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As the Tangent of LI= 32 58 9.8119641
Is to the Tangent of AL = 12 33 9.3475454
So is Radius
To the Co-sine of ALI = 69 56 9.5355813
Thus Jerusalem bears from London ES E, and 20 26'

3. To find how London bears from Jerusalem, or the Angle at 1.


As the Sine of LI= 32 58 Co. Ar. 0.2642805 Is to the Sine of LPI= 37 24 9.7837878 So is the Sine of L P = 38 28

9.7938317 To the Sine of LIP = 44 or


Thus London bears from Jerusalem NW, and one 59' to the North.

This Problem might be diversified and branched into several others, by having other Data ; but that if the Reader pleases he may make his own Exercise.


Given the Latitude of two Places ( one in North, the other in South Latitude, ) and their Distance from each other, to find their Difference of Longitude, and Position from one another.




Cape Horn

Admit the La-

titude of London
be 51° 32' North;
and the Latitude
of Cape

Cape Horn (the farthest South

A Point of America) Æ be 58° 00' South; and the Distance from London to Cape Hornbe 8890 English Miles; I demand the Dif

S ference of Longitude ?

Let L represent London, and C Cape Horn; through both these Places draw proper Circles, and there will be formed the Oblique Triangle PLC, or rather its Supplemental one COS, in which all the sides are given ; for OS = P L= 38° 28' the Co-Latitude of London ; CS= 32° oo', the Co-Latitude of Cape Horn ; and O C= 52° 06', the Complement of LC

8890 Miles = 127° 54' to a Semicircle ; to find the Difference of Longitude, or the Angle O SC, by Case 5, Of Oblique Triangles

. Here - SC= A M = 6° 28'. Then the Sine of OS= 38° 28' Co. Ar. 0.2061683

the Sine of S C= 32° oo'Co. Ar. 0.2757903 Add the Sine of


= 29° 177 9.6894232 OC-AM the Sine of


= 22° 49' 9.5885896



The Sum = 19.7599714 thereof is the Sine of LOSC=490 20 = 9.8799857


The Double of which is 98° 40' = 0 SC, whose Complement AS Q = 810 20', the Difference of Longitude fought.

2. To find how the said Cape bears from London, or the Angle at 0.


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As the Sine of OC= 52 06 Co. Ar. 0.1028767 Is to the Sine of O SC = 98 40

9.9950126 So is the Sine of SC= 32 00

9.7242097 To the Sine of SOC = 41 35 9.8220990

Now the Angle at O is equal to the Angle at L, therefore Cape Horn bears S. W. by S. and almost to the W. of London.

3. Thus also the Angle at C will be found to be about 51 Degrees, by the Gunter ; so London bears from Cape Horn N E, and almost half to the E.

These Problems being well understood, any thing further on this Head will be unnecessary,

I shall now subjoin ( for the Reader's Curiosity, ) a Scheme of the Dimensions of all the chief Kingdoms of the World in length and breadth, the Metropolis of each, with its Distance and Bearing from the Metropolis of the whole World, viz. the City of London.

Note, The Dimensions following are to be understood of English Statute-Miles.




Kingdoms Leng, Bread. Metropolis Dift. Bearing

Europe. France 520

208 S. by E. Spain 620 480 Madrid

760 S. Germany 540 510 Vienna

860 E. by S. Poland 780 600 Cracow


E. Muscovy 1630 1500 Muscow 1530E. N. E. Sweden

Stockholm N. Denmark


700E. N. E. Norway


799/N. N. E. Italy 760 134 Rome

8441 S. E. Turkey 770 660 Constantinople 1640 E. by S.

Asia. Tartary

3000 2250 Chambalu China

13801260 Peckin India 1680 1690 Agra

5290 E. by N. Persia 1440 1260 Tpaban

3200 E. by S. Turkey 2100 1740 Aleppo

2350 E. S. E. Africa. Egypt 650 310|Cairo

2154 E Barbary 2300 380 Fez

12301S. S. W. Biledulgerid 2040

300 Dara

1590S. S.W. Zaara 12340 330 Zuenziga 20851S. S. E. Negroland 2280 600 Tombute

25701S. S. W. Guinea 1320

360 Arda

2919 S. by E. Nubia 840

570 Duncala

|3475 S.E. by E. Ethiopia 3600 2180 Monomotapa 5421S. S. E.

America. Mexico

2520 480 Mexico 5699 W.N.W. Florida 1000

600 Coca

4170W. N.W. Terra Canad. 1500 1920 Boston

3197W. N.W. Terra Firmali 260 480 St. Fee

152 12W. by S. Peru 1440 480 Lima

6463 W. S.W. Brasil 1600 1380 St. Salvador 4865 S. W. Chili

480 St. Jago

17297 W. Paraguay 1560 1500 ljumption 15907 S.W.



Spherical Trigonometry applied to Navi

gation; shewing how to find the Differences of Latitude, and Longitude ; the Course and Distance failed, and the Departure from the first Meridian ; all in the Arches of a Great Circle of the Globe.


REAT Circle Sailing is of all others the most perfect and expeditious ; the Reafon is

evident, because an Arch of a great Circle is the nearest Distance between any two Places on the Spherical Surface of the Earth, and consequently is the foonest failed.

But because the Motion of a Ship is directed by the Wind, 'cwill be somewhat difficult to steer a Ship directly on a Circular Arch ; however 'tis possible to keep pretty near it, and the nearer the better.

As this Method is most perfect and expeditious, so it is the truest and most natural of any ; yea could the Ship be all along conducted on the Arch of a Circle, it would then be perfectly true and natural.

The Difference between Plain, Mercator's, and Middle Latitude Sailing, and this, is obvious in the Form of the Charts, and in the Solutions thereby ; the Charts there being all Rectilineal Plans, but in this the Chart is only a Stereographic Projection of VOL. II.



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