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In Mercator's Sailing. The Reduction is much more commodioufly performed in Mercator's Charts ; whereili the Arch intercepted between the Two Meridijins, is ap- plied to an Arch of the Meridian intercepted between the Two Parallels; and the Diflance it there roeafures, gives the -Departure, ot -Miles of Longitude required. See De- parture and Longitude.
Cafe II. The Longitude and Latitude of Two Tlaces, to and from which, a Ship is to fail, being-given; To find the Rhumb to be failed on, and the Diftance to be run.
In plain Sailiig. i. Find the "Departure by the laft Cafe. 2- From the Departure, and Difference of La- titudes, find the Iftsodromjc Angle or Rhumb-line ; which is done by this Proportion : as the Difference of Latitude is to the Depatture, lb is the whole Sine to the Tangent of the Angle of the Rhumb line. The Diflance then, to be run on this Rhumb, is to the Departure, as the whole Sine to the Sine of the Angle of the Rhumb. See Rhumb. In Mercator's Sailing, i. Apply the Center of the Manners Compafs on the Place Jailed from, on the Mercator's Chart, as a, (Tab. Navigation Fig. 7.) and fo as that the North or South Line thereof be parallel to fbme of the Meridians. 2. Mark the Rhumb of the Compafs, wherein the Place failed to, as b, is placed. For this is the Rhumb to be failed on. 3. 1 he fame Rhumb is likewile found by drawing a right Line from mob; and with a Protrailor, finding the Angle the Rhumb makes with any Meridian it cuts. 4. The Quantity or Diflance a b is found by applying the Part a 1 to 1 K, 1 a to k /, 2 b to / m.
Note, the Rhun.b and Diflance may alfo be found af- ter the fame Manner on a plain Chart.
The fame may likewile be found by Loxodromic Ta- bles ; thus, j. Chufe a Rhumb at Pleafure, and under the fame, in the Tables, find rhe Longitudes correspond- ing to the given Latitudes. The Difference whereof, if it coincide with the Difference of the given Longitudes, the Rhumb is well choien ; orhcrwife another muff be pitched on, either more or lefs oblique, 'till the Tabular Difference agree with the given Difference. 2. The Rhumb thus found, the Dillances anfwering to the given Latitudes, inuit be taken from the Tables, and the lefs fubtrafted from the greater ; the Remainder is the Di- flance fought.
Cafe III. The Rhumb and Diftance failed being given; To find the Longitude and Latitude of the 'Place arrived, at.
hi Tlain Sailing. 1. From the Data, find the Differ- ence of Latitude of the the Two Places: (by the Pro. pottion delivered under the Article RuaML-line.') This Difference added to the Latitude of the Place failed from, or fubrracfed from the fame, the Sum, or the Remainder, leaves the Latitude of the Place failed to. 2. From the fame, find the Departure ; and thence the Latitudes of the Place failed to (as directed under Rhumb&«.)
In Mercator's Sailing. 1. Place the Mariners Compafs on the Chart, with the Center over the Place a ; and the Meridian, or North or South Line, parallel to the Meri- dian thereof. 2- From rhe Point a, draw a right Line, as a b, fur the Ship's Courfe. Take the Diflance by Parts, in Parts of the Meridian IK, KL, i£c. and fet it off upon the right Line ab ; E. gr. from a to c ; Then will c be the Place the Ship is atrived at 5 the Longitude and Latitude whereof are given by the Chart. See Chart-
'By the Loxodromic Tables. 1. Under the given Rhumb, feek the Diflance anfwering to the Latitude of the Place failed from ; and either add it to, or fubtradt it from the given Diflance ; as the Latitude of the Place failed to is greater, or lefs than that failed from. 2. Under the fame Rhumb, afcend or defcend further, till you meet with the Diflance corrected. 3. The Latitude anfwer- ing thereto in the firfl Column, is the Latitude of the Place failed to. 4. From rhe fecond Column of the Table take the Longitude correfponding to the Latitudes of the Places failed to and from. Their Difference is the Difference of Longitudes of the Places failed to and from.
Cafe IV. The Latitudes of the places failed to and from, together with the Rhumb failed in, being given ; To find the Diftance and Difference of Latitudes.
In 'Plain Sailing. From the Difference of Latitude and the Rhumb given, find the Diflance ; and from the fame Data, the Departure. See Rhhm*. This ajnverted in-
to Degrees of a great Circle (See Choree) .exhibits the Difference of Longitudes fought.
In Mercator's Sailing. ,. Place the Compafs on the Chart, as in the preceding Cafe. From the Place failed from, a,, draw the Rhumb.line ab,. failed in; till it cut the Parallel of the given Latitude. 2. The Point of Inter- feron will be the Place arrived in. 3. Hence its Longi- tude is eafily found ; and the Diflances. See Rhumb
Sy the Tables, Take both the Longitudes and the Di- flances anfwering to the Latitude of the given Places out of the Tables ; then, liibtracr both the Longitudes and the Diflances from each other. The firfl Remain- der is the Difference of Longitude, the latter the Di- ftance ot the Places.
Cafe V. The Latitudes of the Tlaces failed from, and to, with the Diftance given ; To find the Rhumb and the Difference of Longitude.
In Tlain Sailing. From the Difference of Latitude, and the Diflance, find the Rhumb, and from the fame Data find the Departure; which may be alfo determined from the Rhumb now found, and the Differences of Lati- tude, or from the Rhumb and the Diflance run. Laflly, from the Departure find the Difference of Longitude. See Rhumb.
In Mercator's Sailing. On the Map draw the Parallel the Ship atrives at, CD. Reduce the Diftance run into Parts proportional ro the Degrees of the Map. The Di- flance reduced, being a 2 ; from a defcribe an Arch cut- ting the Parallel C D in 2 : Then will 2 be the Place in the Map ; whofe Longitude accordingly is eafily found.
"By the Tables. Subtract the given Latitudes from each. other ; and in the Tables leek the Rhumb ; under which, the Diflance run anfwers to the given Difference of Latitude. Subtract the Longitude under the Rhumb s ' anfwering to the Latitude of the Place failed to ; and - that under rhe fame Rhumb againft the Latitude of the Term failed to, from each other, the Remainder is the Difference of Longitude fought.
Cafe VI. The Difference of Longitudes of the 'Places failed to, and from, with the Latitude of one of the. 'Places, and the Diftance run, being given ; To find the Rhumb, and the Latitude of the other.
In Tlain Sailing. Convert the Difference of Longi- tudes into Miles of Longitude or the Departure; from the given Departure and Diflance run, feek the Rhumb: And from rhe fame, and the Rhumb, feek the Differ- ence of Latitude ; which, and the Latitude of one Place being had, the Latitude of the other readily follows. See Rhumb.
In Mercator's Sailing. Through the given Place a in the Map, draw a right Line E F parallel ro the Meridi- an 1H; and make FL equal to the Difference of Longitudes. From F draw L M parallel to E F, which will be the Meridian the Ship is arrived at. Then from a, with the Interval of the Diflance run, a c, defcribe an Arch interfering the Meridian ML; the Place fought will be inc. If then a Compafs be placed on the Map, as before directed, the Rhumb.iine will fall in with a c ; and confequently the RhumbwiU be known. Laflly, If through c be drawn N O, parallel to AB; N A will be the Latitude of the Place required.
'By the Tables. Take a Rhumb at Pleafure, and under the fame in rhe Tables, find the Longitude, and the Diftance anfwering ro the given Latitude. Add the gi- ven Diflance to the Diflance found in the Tables, if the Veffel failed ffom the Equator ; or fubtrail it there- from, if it failed towards the fame. With the fame Sum, or the Difference, enter the Tables ; and the Longitudes found againft it, f'ubtrafl or add from that jufl found. If the Remainder be found the given Dif- ference of Longitudes, the Rhumb is well taken. Other- wife, it muft be changed for a more or lefs oblique one, till the fame Operation being repeated, the Remainder be found the Difference of Longitudes ; then the La- titude in the firfl Column, correfponding to the Diflance, will be the Latitude of the other Place.
Cafe VII. ihe Difference of Longitude, and the Latitude of one of the 'Places, being given, together with the Rhumb ; To find the Diftance run, and the Latitude of the other Tlsce.
In 'Plain Sailing. Reduce the Difference of Longi. tude into Miles of Longitude, or Departure, as under the firfl Cafe. From the Departure and the Rhumb, find the Diflance run. (See Rhumb.) And from thefe, or from the Rhumb, and the Diflance run, find the Dif- ference
