( SS; )

The fäme Rule also will ferve to sind the Course and bistance betwcen twoPlaces aiiigned, as far as Practice (hall require it; and that without any Ta*ble of hUridional Parts , and yet wirh as much Ease and Exactness.

And fceng all these things do depend on the Solution of this Qiie’tion,

Miether the Artificial Tangent Line he the True Meridian Line} It is tbcrefore,that I undertakc, by God’s Aslistance, to resolve the läid Question. And tolet the World know the Readinefs and Confidence I have to make good thisUnderraking, I am willing to lay a Wäger against any one or more Personsthat have a Mind to engage, for so much as another Inventio» of mine( which is of no less Subtlety, but of a far greater Benefit to the Publick )may be worth to the Inventor.

As for the great Advantage, that all Merchants, Marinen, and consoquentlythe Common-PVca!tb, may reeeive from this other Invention, it is, in my Judg-ment, highly valuable, feeing it will osten times make a Sbip Sail, thoughaccording to the common way of Sailing , the Wind be quite Contrarr, andyet as near to the Place intended, as if the Plind had been favourable : Or ifyou will, it will cnable one to gain fomething in the Intended way, whetherthe Pl'ind be good or no (except only when you go dircctly South or North )but the Advantage will be moil where there is most need of it, that is, whenthe PI 7 ind is contrary: So that one may very osten gain a fiftli, fourth, thirdPart, or more of the Intended Voyage ; according as it is longeror fliorter;vj'~ always more in a longer Voyage, where the Gain is more Confiderable,and more Wclcome, not only by säving Time, but also Vilhtah, iVatcr, Fuel,

Mens Health , and so much Boom in the Ship.

XXXVIII. It was firtt Difcovered by Chance, and as far as I can Learn, The ^inaiogy 0/first Published by Mr. Henry Bond, as an Addition to Norwood's Epitome of Na- ^°S an,hmtc K.wgation , about yo xears lince, that the Meridian Line was Analogous to a MeridianScale of Logarithmick. Tangents of half the Complementi of the Latitudes. Line, Demon- |

For the Demonstration of that Proposition it is Requisite to premiso theso st.'Edm. Hilley.Lemmata. n. 219. p. 202.

Lemma. 1. In the Stereographick. Projeclion of the Sphäre vpon the Plain ofthe Aiquinoclial, the Distances from the Center, which in this Cale is the Pole,are laid dovon by the Tangents of half thosc Distances , that is, of half the Com >plements of the Latitudes. This is evident from Euch 3. xo.

Lemma. X. In the Stereographick. ProjcRion , the Angles under which the di-elet Intcrfifl each other, are in all Cafes equal to the Spherical Angles they repre-fent ; which is a very valuable Property of this Projcftion.

Demonst. Let EPBL be any Great Circle of the Sphere, E the Eye placed 2lJ>in its Circumference; C its Center, P any Point thereof; and let FCO besopposed a Plain erected at Right Angles to the Circle EPBL, on whichFCO we design the Sphere to be Projected. Draw EP Crossing the PlainFCO in p, and p (hall be the Point P projected. To the Point P draw theTangent APG, and on any point thereof, as A, Erect a Perpendicular AD,at right Angles to the Plain EPBL, and draw the Lines PD, AC, DC; andthe Angle APD shall be equal to the Spherical Angle contained betwcen theVol. L Q. q q q Plain» .