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ference of Latitude. This done, as the ' Latitude of the one is already had, that of the other is lb too.
In Mercatofs Sailing. Place the Compals on the Charts before; and by the given Rhumb, draw the Rhumb-line, a b. Draw a Meridian E F through the given Place a; and with the Interval of the Difference of Longitude FL, draw another, LM, for that the Veffel is arrived ar. Where this interfecls the Rhumb- line, is the Place c that the Veffel is arrived at. Where fore, if through c be drawn K O parallel to A B ; N A will be the Latitude of the Place. The Distance run a c is eafily reduced into Miles by the Scale.
A3y the labia. Under the given Rhumb, leek the Diftance run, and the Difference of Longitude an- fwering to the given Latitude. If the Veffel have Jailed towards the Pole, the Difference of Longitude is to be Added to the given Difference of Longitude ; if to- wards the Equator, 'tis to be fubtrafted from the lame. In the former Cafe, defcend in the Table, and in the latter, afcend j 'till in the fir ft, the Aggregate, in the lat- ter, the Difference be feen in the Column of Longitude. The Latitude anfwering hereto in the firft Column, is that fought. And from the Diftance anfwering to this Latitude in the firft Cafe, the Tabular Diftance is to be fubtracted; or in the latter Cafe, that Diftance to be fubtracled from the Tabular Diftance. What remains, is the Diftance run.
From the Solution of thefe Cafes in Sailing 'tis evi- dent, fome are more eafily performed by the Charts than the Tables ; and that the Mercatcr's Charts are prefera- ble to the Plain ones; fince in the latter, the Diftance is not reduced by the Map, but by a particular Scale for that Purpole.
Do&rine of Circular Sailing.
% The Latitude and Longitude of the Places failed to and front) being given ; to find the Angle M (Fig. 8.) which a ■ Ship's Way MO proceeding in a Circular Courfe, includes, with the Meridian?. M. of the 'Place failed from.
Since in the Triangle PMN, we have PM and PK\ the Complements of the given L at ' tu des H M, and IN, together with the Angle M P N, meafured by the Arch H I, the Difference of the given Longitudes H and I ; the Angle PMN is/found by Spherical Trigonometry. See Triangle.
II. The Latitude H M, and the Longitude H, of the Place failed from, M, with the 'Diftance run-, and the Lati- tude of the 'Place L S the Ship iuaCircular Voyage ar- rives at, being given ; to find theLongitude of the Place L, and the Angle P L M comprehended between the Ship's May M L, and the Meridian P S.
In the Triangle P M L, we have given P M the Com- plement of the Latitude H M, and P L the Complement or the Latitude L S. Wherefore, if the Ship's Way ML be turned into Degrees of the Equator 5 we fhall find the Angle M P L, which is meafured by the Dif- ference of Longitudes H S j and likewife the Angle PLM by Spherical Trigonometry. See Spherical Triangle.
After the like Manner may other Problems be folved j but as 'tis eafier and better Sailing by Rhumbs, than by Circles, and as this latter Way is but very little in Ufc ; we chufe to pals them over. See Globular Sailing- Sailing, in a more confined Senfe, is the Art of con- ducting a Veffel from Place to Place, by the working or handling of her Sails and Rudder; though what is done by Means of this latter, is more properly called Steering. See Sthering.
To brin3 Sailing to certain Rules, a late Author com- putes the Force of the Water, againft the Ship's Rud- der, Stem or Side; and that of the Wind again!! her Sails: in order to this, he confiders all Fluid bodies, as the Air or Water, ?$c. as being compofed of little Par- ticles, which, when rhey a£l upon, or move again!! any Surface, do all move parallel one to another, or ftrike againft the Surface after the fame Manner. He confiders, 2, That the Motion of any Body, with regard to a Sur- face, on which it is to ftrike, rauft be either Perpendicu- lar, Parallel, or Oblique. In the firft Cafe, the Body ftrikes with all its Force, which will be greater or lets, according as the Body moves fwifter or flower. In the fecond Cafe, the Line of Motion a b, {Tab. Navigation Fig. 3.) will not affect the Surface at all, becaufe it is no way oppofed to it ; nor can the moving Body ftrike up- on it, or touch it. In the third, If the Line of Motion, A D, be oblique to the Surface D C, fo that the Angle of Incidence be A D C, then the Motion of the Body in the Line A D may be refblved into Two Directions, viz. into A E, or DB, and into AB. But the Direction or Line
of Motion A E bein£ parallel to the Surface D C, cannot affect it at all ; fo that the whole Motion of the Body A in that oblique Manner or* linking on the Surface, will' be expounded by the Perj-endkti.ar Line, A B. And if D A be made the Radius of a Circle, whole Center is at D, B A will be the Line of the Angle ot Inci- dence, ADC. Hence we deduce, That the Force of a Particle of Air or Water, as A, linking againft the Sur- face DC, whjch may reprefent, either a Sail or the Rud- der of a Ship, in the oblique Direction A D, will be to the Perpendicular Force there, as B A is to D A : that is, as the Line of the Angle of Incidence is to ihe Ra- dius. And fince what is thus true of one Partkle, fing- ry confidered, will be true of all the Particles of any Fluid Body collectively, ir will follow, That the Force of the Air or Water linking perpendicularly upon a Sail or Rudder, to the Force of the lame^n any oblique Im- pingency, will be, as the Square of the Radius, to the Square of rhe Sine or the Angle of Incidence : and confequently, that all oblique Forces of the Wind againft the Sails, or of the Water againft the Rudder, will be to one another, as the Squares of the Lines of the An- gles of Incidence. If the different Degrees of Velocities be confidered, it will be found, that the Forces will then be as the Squares of the Velocities of the moving Air or Water 5 that is, That a Wind that blows thrice as ftrong, or moves thrice as t'wtfi as another, will have nine Times the Force upon the Sail. And it be- ing alio indifferent, whether \ou ennfider the Motion of a Solid in a Fluid, whole Particles are at reft $ or of thole Particles moving all parallel againft a Solid that is at reft, the reciprocal Impreftions being always the fame : If a Solid be moved with different Velocities in the fame Fluid Matter (as fuppofe Water) the different Refiftances which it will receive from that Water, will be in the fame Pro* portion, as the Squares of the Vilocitiesof that Body.
Let HM (Fig. 4) reprefent a Ship, CD the Po- sition of the Sail, and A B the Omrfe of the Wind blowing towards B. Draw B G perpendicular to the Sail, and GK perpendicular to the Line of the Keel produced H M K. By what is laid above, the Sail CD, will be driven by the Wind A B, according to the Direction of the Line B G. So that if Ihe could divide the Water every Way with the fame Facility, as fhe doth with her Head, the Ship would go directly to the Point G, along the Line B G. And if H K reprefent her direct Courfe, fhe would have got forward the Length B K, and Sideways Ihe would have gone the Quantity G K. But as her Length is much greater than her Breadth, fo fhe will divide the Water, or make her Way in it with more Difficulty with her Side, than with her Head or Stern 5 on which Account, fhe will nor run Sideways fo far as K G, but fall fhort of it in Proportion to the /aid Difficulty of dividing the Water with her Side; t':at s, If the Refiftance fhe finds in her palling thro' the Water Sideways, be to that of palling Lengthways, fuppofe, as ten to one, then will not the Ship get Sideways above a tenth Part of the Line GK. Wherefore if K G be found to G L, in the Ratio of the Refiftance of the Side to that of the Stern, and the Line B L, be drawn ; the Ship will go to the Point L, along the Line B L, in the fame Time as it would have gone to G, if ir could have divided the Water every Way equally. This Parr, K L, is called the Drift ot Lee -way of a Ship, and the Angle KBL is her Degrees of L^ee.-ivay ; as the Angle A BK, ex- prefles how near the Wind fhe lies. After this, he proceeds to demonftrate, Thar the belt Pofition or Si- tuation of a Ship, fo as fhe may make the beft Lee- way, but go to Wind-wari as much as is poffible, is this. That, let the Sail have what Situation it will, the Ship be always in a Line biff cling the Complement of the Winds Angle of Incidence upm the Sail; that is, Sup- uofing the Sail in the Pofition BC (Fig. 4.) the Wind blowing from k to B, and confequently, the Angle of the Wind's Incidence on the Sail ABC, and its Comple- ment C B E ; then mull the Ship be put into the Pofition B K, or move in the Line B K, bi fleeting the Angle BGE. He fhews farther, That the Angle which the Sail ought- to make with the Wind, i. e.. the Angle ABC, ought to be but 24 Degrees ; that being the moft advantageous Situation to go to Wind-ward, the moft that is poffible. And in order to bring this to bear in Practice, he directs to put Marks to the Sheets, Braces, and Bow lines of the lower Sails, to know when they are in their beft Situation 5 and then, even in the Night, when the Marks of a Brace or of a Sheet fhall come to the Cleat, one may be pretty well allured, that the Sail trims well. To this may be added, many curious Things from Borelli de V, Percuflionis concerning the different Di- rection given to a Veffel from the Rudder, when Sailing with a .Wind, or. floating. without Sails in a Current ; in
the
