Difference of Latitude 140, Departure 81, the Course is found by Cafe VI. to be an Angle of 30 Degrees from the Meridian, and the Distance 162, as you may fee by the Traverie Table.

Then for Difference of Longitude, the Middle Latitude is 57 25, but we must call it in this Cafe 58, which found in the Degrees, there is not fuch a Number as 81 in the Column of Latitude; therefore find its half 40: 5, or the nearest to it, which is 40: 3, and against it, in the Column of Diftance, you find 76, which doubled is 152, the Difference of Longitude required.

Note, Although in particular Questions the Longitude found this Way differs fometimes a Minute or two from that found by Calculation: because we are obliged to use whole Degrees in the Table for Middle Latitude; yet the Error being fometimes a Minute or two too much, and fometimes as much too little, the Difference is not difcern able in a long Reckoning.

CHAP.

CHAP. IV.

Arithmetical Navigation.

SECT. I.

Plane Sailing Arithmetical.

HAVING thus finished all the Kinds and Cafes of Navigation in the foregoing Book, according to the Method commonly taught, and practifed both at Sea and a-fhore, viz. by a Canon of Logarithmical Sines, Tangents and Secants, (together with the Help of the Logarithms and Natural Numbers) my next Work is (according to my Promife) to fhew, how all the aforefaid Kinds of Navigation may be compleatly performed without any Canon, only by the Help of fome given Numbers, which fhall be inferted in their proper Places, which are of Ufe where the Courfe is either given or required: And first in Order, I fhall begin with Plane Sailing, in which the three firft Cafes are wrought only by the Table of given Numbers, as alfo are thofe Questions in the three laft Cafes where the Courfe is required: But the Solution of all Queftions in Plane Sailing, when two Sides are given to find a third (as Distance and Difference of Latitude given to find Departure, &c.) are grounded upon that known Propofition in Euclid, Lib. 1. Prob. 47. viz. That the Square of the Hypotenufe (called there the Base, because it is the longeft Side) of a right-angled Triangle, is equal to the Squares of both Legs added together. Hence then (the Hypotenuse representing the Distance, and the two Legs the Difference of Latitude

and

and Departure (if you fquare each Leg, that is, multiply each Leg by itself) these two Sums added together fhall be equal to the Square of the Hypotenufe; and confequently, if from the Square of the Hypotenufe you fubtract the Square of one Leg, the Remainder is the Square of the other Leg, as may be proved by those three known Numbers, 3, 4, and 5, which three Numbers make a Right-angled Triangle; the two Legs being 3 and 4, and the Hypotenufe 5: Now if you fquare the two Legs, 3 times 3 is 9, and 4 times 4 is 16, which two Squares of 16 and 9 added together, the Sum is 25, the Square of 5, the Hypotenufe. Again, if from 25, the Square of the Hypotenufe, you fubtract 16, the Square of one Leg, there remains 9, the Square of the other Leg, &c. And when the Square of any Side is thus found, the Square Root thereof (found as is fhewn in the firft Part hereof) is the Side required.

Alfo with refpect to the given Numbers, if Course and Distance be given, multiply the Distance by the given Numbers, as the Title of the Table directs for Latitude and Departure, (whofe Ufe follows in feveral Examples) the Sum, abating as many Figures to the Right-hand as you multiply by, is the Difference of Latitude or Departure required: Now if the Hypotenufe multiplied by the given Number produce a Leg, only two Figures to be cut off, the faid Leg being given, and two Figures (or Cyphers) added to it, and divided by the faid given Number, the Quotient muft needs be the Hypotenufe; and if fo, then if the faid Leg, with two or more Figures (or Cyphers) added to it, be divided by the Hypotenuse, the Quotient must needs be the given Number, as is apparently evident to almost every Body that have but learned the firft Rudiments of Arithmetick, and underitand (what every School-Boy is taught) how Multiplication and Divifion will undo and prove each other; and hence any two Parts of a Question of Plane Sailing being given, the reft may be found by the Square Root, and one Table of given Numbers, all the Cafes except the fixth, being wrought immediately by the given Numbers; and even in the fixth Cafe, where the two Legs are gi ven, the Hypotenufe or Diftance may be found by the

Square

3

Square Root, and then the Courfe as in the other Cafes ; as I fhall explain by the following Example.

A TABLE of Given
Numbers.

The Ufe of this Table is for working the fix Cafes of Plane Sailing, in which obferve, that in the first and last Columns you have the

Points Latitude Depart. Points Rumbs or Points of the

3

Compafs, with the half

Points, and quarter

Points, numbered from

the Meridian, or North and South Line, either Eaft or Weft, accord6ing as they stand in Order; thus S. S. W. is two Points, because South is upon the Meridian, then S. by W. is one Point, and S.S. W. two Points, &c. So likewife N. N. E. and N. N. W. is two Points from the North.

2047 : 13 5

85 7951 38 5 183 1555

56 5

80: 3359

55 4

77

2963

45 4

74

1067

4 70

15 4

71 70 71 4

Again, fuppofe a Ship fails S. W. S. that is, three Points and an half, and N. N. E. E. is two Points and a Quarter; which being a Thing fo commonly known, I fhall not need to add any more Examples.

Points Depart. Latitude Points

The Number of Points thus found, obferve on which Side of the Table it is found, for if you find the Course at the Left-hand, then find the Denomination of Latitude and Departure at the Top: But if the Course be

on

on the Right-hand find the Denomination of Latitude and Departure at the Bottom; for the Left-hand Column contains all the Points under four Points, and the Right-hand Columns contains all the reft of the Points to eight. Now if the Courfe be found in the first Columns, (viz, lefs than four Points) then the fecond Column is the given Number for Difference of Latitude, and the third Column is the given Numbers to find Departure: But if the Course be in the laft Column, (viz. about four Points) then the third Column is the given Numbers to find Difference of Latitude, and the fecond is for the Departure, &c. And the Ufe of the Table being thus known for the firft Cafe, the reft is eafily found by Confequence, as we have hinted before, and fhall make it evident and intelligible to the meanest Capacity, by the following Examples.

Note; If you have a given Number, with a Cypher on the Left-hand, (as in 7 and 7 Points) although a Cypher adds nothing to the Product (being on the Left-hand) yet it caufeth a Figure more to be cut off from the Decimal Fraction; and having cut off as many Figures towards the Right-hand as you multiply by, the reft is the Latitude or Departure fought, and thofe cut off are a Numerator of a Decimal Fraction to a Denominator, confifting of as many Cyphers as you cut off Figures, with a 1 at the Left-hand; fo if you cut off 97, it is, or if you cut off 342, it is, and fo of the reft.

Note; When you have found the given Number in the Table, you need not always use all the four Figures, especially, if the third or fourth be a small Digit, as 1 or 2, as fuppole your given Number be 8315, here the third Figure being a 1, you may cut off the two laft Figures, and ufe only 83 to work with. Again, if the third Figure be one of the highest Digits, as 8 or 9, you may cut off the two laft only add one to the fecond Figure, and fo work as before; as fuppofe your Number be 2899, you may add 1 to the fecond Figure, viz. 8, and then cut off the two laft, and then your Number to work with will be 29; the Reason of all which is very evident to any that understand the Nature of Decimals,

and