CASE III. Given as in Cafe II. to find the third Side BC.

In the whole Triangle BDA, you have given the Angle ABD 45, and the Hypotenufe BA 560; alfo by Confequence, the Angle BAD, which is alfo 45, to find the whole Side BD; but in this Cafe the acute Angle being equal, viz, 45 Degrees, the Leg BD is equal to AD 396; then having found CD 106, by the fecond Operation in Cafe II. fubtract it from the whole Side BD 396, the Remainder 290, is the Side BC required.

Given

CASE. IV.

The Side AC

4107

The Side AB- 560 Required the Angle at B
The Angle at A-30.0

In the Triangle AEC, is given the Angle at A 30, and the Hypoten ufe AC 410, to find CE, which by the firft Cafe hereof is to be found to be 205, and Fig. 50. therefore I need not repeat the Operation. Then,

in the fame Triangle ACE, there is given the Sides A C -410, and CE 205, to find the Side AE, by Rule III.

The Leg AE 355 fubtracted from the whole Side AB 560, reft E B 205; then in the Triangle BEC, you have given BE 205, and EC 205, to find CB by

Rule

Rule II. and the Angle B by Rule IV. but in this Cafe, BC and EB being equal, the Angle at B is proved 45 Degrees, without Calculation.

CASE V.

Given as in Gâfe IV. to find the third Side BC.

Although this is the fifth Cafe in the Trigonometrical Operation, yet the Side B C is neceffarily found th Cafe IV. before the Angle at B can be found; and, therel fore, although the Operation in Cafe IV. be fomewhat tedious, yet both the Fourth and Fifth Cafes are included

in it.

Find AE by the Rule laid down in Axim IV. of Plane Triangles. As the Bafe AB 560, to the Fig. 51. Sum of the other two Sides, 700 So the Difference of the faid Sides 120, to the Difference of the Segments of the Bafe AD 150, as by the Operation below.

120

700

84000

56.0)8400.0(150
280

To the half Diff. 75 add the half Bafe-280, the Sum 355 is the greater Base AE; but fubtracted the Difference is the leffer Bafe EB 205.

Then in the Triangle AEC, there is given AC 410, and A E 355, to find CE by Rule III. and the Angle at A by Rule IV.

The

Then, by Rule IV. find the Angles at A.

As 587.5, to 86: So CE 205, to the Angle oppofite at

578.5)17630.2(30 The Angle at A.

A 30.

205
86

5

required.

17630

Although this Method be not altogether fo expeditious for Oblique Triangles, as the Calculation by Logarithms, because you are obliged to divide every Oblique Triangle into two Right-angled ones, which fometimes requires two Operations; yet I thought fit to infert it to make the Method compleat, it being of great Ufe when Tables are wanting, and of fufficient Exactnefs for moft Ufes in Navigation: but the Right-angled Cafes, as performed hereby, I fhall recommend to the Reader, as a Thing very useful, fufficiently exact, and as expeditious as any Method commonly in Ufc.

SECT.

SECT. IV.

How to find the Difference of Longitude, and keep a Reckoning both in Latitude and Longitude, by this New Method of Trigonometry, (as applied to Navigation) without the Help of any Tables or Inftruments whatsoever, according to Middle Latitude, which is of fufficient Exactness for the Working fo fhort a Distance as a Day's Run, and confequently of great Ufe in Navigation.

You may remember, that in Middle Latitude Sailing Trigonometrical, there is a Proportion for finding the Difference of Longitude, which is, as Sine Complement of Middle Latitude, is to the Departure; So is Radius, to the Difference of Longitude. And therefore, in Middie Latitude Sailing Geometrical, one Way which I have proposed for projecting Middle Latitude Sailing, is, by conftituting a right angled plane Triangle, whofe Angle at the Bafe is equal to the Complement of Middle Latitude, and the Perpendicular is equal to the Departure: And then by that known Proportion of oppofite Sides, oppofite Angles, it will neceffarily follow that the Hypotenuse muft needs reprefent the Difference of Longitude; which being granted, there is no more to do for finding the Difference of Longitude, but only the Solution of the faid Right-angled Triangle: Of the feveral Varieties of which you have had fufficient Inftances in the fix Cafes of Plane Sailing before going, where any two Parts being given, the other two are eafily found. Neverthelefs, that nothing may be wanting for the Reader's Inftruction, I fhall inftance in one Queftion for Example's Sake, which I fhall firft work by this new Method, and then

by

by the Method propofed in Middle Latitude Sailing Trigonometrical; and laftly, fhall work the fame by Mercator, to let the Reader fee the Sufficiency and Exactness of this new and useful Invention.

Question. A Ship in Latitude 58d. oom. North, fails South 25 Degrees Eafterly 96 Miles: I demand the Latitude come to, and also her Departure and Difference of Longitude.

The Course and Distance is the fame as in the Question in Cafe I. of Plane Sailing, as performed by this New Method, and therefore I fhall refer you to the Operation there, for finding the Difference of Latitude and Departure. The Difference of Latitude being there found to be 87, and the Departure 40 or 40.5. and therefore (the Course being Southerly) the Latitude come to is 56d. 33m. and confequently the Middle Latitude, found by the Direction and Caution laid down in Cafe I. of Middle Latitude Sailing Trigonometrical is 57d. 17m. and the Complement of Middle Latitude is 32d. 43m. From hence by the foregoing Directions, is conftituted the Fig. 52. Triangle ABC, wherein the Angle at A is equal

to the Complement of Middle Latitude 32d. 43m. and the Side oppofite BC, is equal to the Departure 40.5, both which are given to find the Hypotenufe AC, equal to the Difference of Longitude required; and here the Side oppofite to leffer Angle being given, I fhall find Natural Radius by Method II. And here obferve, that although in Queftions of Plane Sailing, you need not regard Minutes in the Angle of the Course, because whole Degrees are exact enough to keep Account of a Ship's Way; yet in this Cafe you must not omit the odd Minutes in the Angle; and therefore reduce the odd Minutes to Tenths of a Degree, accounting 6 Minutes for one Tenth of a Degree, and 12 for two Tenths, &c. And then 42 Minutes is 7 Tenths; and this Angle being 32d. 43m. I fhall call it 32.7, viz. 32 and 7 Tenths, it being but one Minute more, which cannot cause any great Error in the Operation.

For