The first Chapter I advise the Learner to ftudy well before
he proceeds, it being preparative to the next, as indeed they are
all depending on one another, as Links in a Chain.

II. Plane Trigonometry is next, in which are many use-

ful Notes and Definitions, with the Axioms, and alfo the Cafes
depending on each Axiom, orderly fet down in 8 Problems,
containing 13 Cafes.

And here I must advertise the Young Student that would

work Trigonometry by the Logarithms, to confult Chapter

1. The Explanation and general Ufe of the Table of

Logarithms, and Tables of Sines, Tangents, and Secants,

towards the latter End of the Book, in Pape 294.

III. Then follows Plane Trigonometry, applied in Pro-

blems of Sailing by the Plane Sea-Chart, commonly called

Plane Sailing.

And that nothing be wanting, I begin with the common

Notes of the Julian Calendar (in this Edition tranfmuted

for the Gregorian or New Calendar, or by the late Act of

Parliament required) fhewing how to find the Prime, Epact,

Dominical-Letter, Eafter-Day, the Moon's-Age, South-

ing, and Time of Full-Sea, or High-Water, In 9 Prob.

Then proceeding to the Description and Ufe of the Plane-
Chart, in 5 Problems, before I come to the Cafes of Plane
Sailing, which I divide into three Parts.

1. In a Right-angled Triangle, relating to a fingle

Curfe, in which are 6 Cafes, commonly called the 6 Cafes

of Plane Sailing.

2. In a Right-angled Triangle, relating to feveral Courfes

called a Traverse.

3. In an Oblique-Triangle, in which are but 4 Cafes,

though there may be a Multitude of various Questions; of
which you have a Taste in Turning to Windward, and
Sailing in Currents, in 21 Problems.

V. In Chapter 4th is Mercator's Sailing; To the right

understanding of which, 'tis neceffary to defcribe Mr.

Wright's Projection, commonly known by the Name of

Mercator's Chart, and fhew the Ules of it, before I treat of

the Problems of Sailing by it; which you will find performed

in 12 Problems: In the first 9 the Table of Meridional

Parts, or the Meridional Line on Gunter's-Scale is used:

And in Cafe that Table or Line be wanting, to fupply their

Room I have added Problems of Sailing by the Middle

Latitude, which will nearly agree with Mercator's Sailing,

a Tbing of good Ufe, In 4 Problems.

V. Spheric Trigonometry, or the Dorine of Spheric

Triangles Rectangular and Oblique, is next in Order; and
it being fo neceffary you should understand how to make a
Spheric Triangle, and also how to measure any of its Parts,
before the framing and working Proportions therein, I have
fully explained that Matter in the beginning of this Chapter,
being in a Manner a New Invention, which I call Spheric
Geometry; This you have in 21 Problems.

And in Spheric Trigonometry properly fo called (the

next in Order) you have all the Axioms and Cafes, both

in Rectangular and Qbliquangular Triangles, explain'd

with neceffary Notes on each Cafe; as to know when a re-

quired Angle is Acute or Obtufe, and when a required Side,

is more or less than a Quadrant, In 12 Problems, containing

28 Cafes.

VI. The Defcription and Ufe of both Globes, is the

next to be confider'd; in which I have plainly and familiarly
explain'd and fhewed the Ufe of the most neceffary Things be-
longing or relating to each of them, In 24 ufeful Problems.

To which is annexed a fhort Defcription and Ufe of the

Hemifpheres, projected on the Plane of the Ecliptic.

VII. Geography is the fubject of this Chapter, which is

the Application of Spheric Trigonometry in finding the
True Diftance of Places in the Variety of their Situation on
the Globe of the Earth, In 4 Problems.

VIII. Great Circle Sailing comes next, which as it's the
most accurate Way of Sailing, fo it's the most difficult, and
bardly poffible for a Ship exactly to fail by; yet it's of great
Advantage to keep conveniently near it; for which Purpose
you'll find all that neceffarily belongs or relates to it, both
as to the Projective Part, and that both Stereographic and
Gnomonic; as alfo in the Calculate Part, (which requires

the Application of both Spheric and Plane Trigonometry,
fully made out; With an Intimation of fhortning the Work,
by fhewing how to defcribe the Arch of a Great Circle on a
Mercator's Chart; the Whole in 4 Problems.

IX. Next you have Spheric Trigonometry, applied in
fundry Aftronomic Problems useful in Navigation, wherein
the Circles of the Sphere are defcribed, and the neceffary
Terms of Art explained to the meaneft Capacity, with Refpect
to the diurnal Motion; and that either,

1. According to the Ptolomaic. Syftem, wherein you
bave in a Right-Angle Spheric Triangle, all the Variety of
Questions and Examples that relate to the Sun, with Re-
Spect to bis Longitude, Declination, Right or Oblique Afcen-
fion, or Difcenfion, Rifing, Setting, Amplitude, Altitude and
Azimuth, at the Hour of Six; Altitude and Hour of the
Day, when Eaft or Weft; Hour, Azimuth, and Altitude,
when he is in the Equinoctial, In 24 Problems. Also,

In an Oblique Spheric Triangle, you have great Variety
both with Refpect to the Sun, or a Star, in many Questions
and Examples relating to the Sun's Altitude, Azimuth, and
Hour of the Day, in any Place, at any Time of the Year:
And relating to a Star, as to its Longitude, Latitude, Decli-
nation, Right Afcenfion, Rifing, Setting, Amplitude, Alti-
tude, Azimuth, Hour of the Night, its Altitude on the Me-
ridian, and Time of its coming to it, In 12 Problems. Or,

2. According to the Pythagorean, or Copernic System,
which is now generally receiv'd, as most agreeable to the ob-
Served experienc'd Motion of the Heavenly Bodies; wherein
Spheric Trigonometry is applied in variety of Questions and
Examples relating to the Earth's Diurnal Motion about its
own Axis, once in 24 Hours, whereby all the visible Ap-
pearances of the Sun and Fixed Stars are folved, with the
Defcription of the Circles of the Sphere, and how they are
drawn Stereographically on the Plane of the Earth's Eclip-
tic, In 9 Problems.

X. Then follow very eafy Rules to find the Variation of
the Compafs, and how to rectify the fame, and to Correct
the Course thereby, both Arithmetically and Inftrumentally,

Alfo here you have the way of Projecting the Sphere Or-
thographically, In 2 Problems, with many Examples.

XI. An Obfervation, either of Sun or Star; what it is,
bow or with what, and when 'tis taken; with Rules to find
the Latitude of the Place of Obfervation, and all the Varieties
therein reduced into one Propofition, containing two General
Cafes, fully explained by many Examples.

XII. You have next the Ufe of all the foregoing Inftruc-
tions fummarily comprehended in a new Form of keeping a
Sea Reckoning or Journal, wherein the Log-Book and
Journal in Words at Length, and Tabular in Figures, are
kept together in one Book; whereby the whole Proceedings,
and every particular Tranfaction of any Voyage at all times
may be feen and known, which will be no fmall Satisfaction to
thofe concerned in Ships and Goods, nor a little Augmenta-
tion to the MARINER's Credit and Reputation.

And that fo ufeful and beneficial a Method may be prac
tifed by all who defire to keep a compleat and exact Reckoning,
and fo be enabled to render a good Account thereof, I have
largely defcribed the Form and Manner of it, with plain Rules
and Directions how to Correct the Reckoning by the ob-
ferved Latitude, and how to find what Latitude and Lon-
gitude the Ship is in every Day; with an Example of a
Seven Days Journal; and how each Days failing is managed,
in taking it from the Log-Board cafting it up, and bring-
ing it into one Courfe and Distance; with framing the Dead
Reckoning, and transferring it into the Journal; fo that
by this the Whole is made more intelligible.

Laftly, In the Tabular Part, you have firft a Traverfe
Table; which tho' it ftands in fo little Room as two Pages,
yet by it the Difference of Latitude and Departure from the
Meridian may be found for any Distance under 10,000, and
for every Quarter Point of the Compafs.

Next to that, A Table of Meridional Parts to every 5
Minutes of Latitude, which together with the Table of Pro-
portional Parts annexed, the Meridional Parts for each fingle
Minute are found.

And

And next adjoining is a Table of 10,000 Logarithms. After which you have a Triangular Cannon Logarithmic, or a Table of Artificial Sines, Tangents and Secants, to every Degree and Minute of the Quadrant, which are corrected with more than ordinary Care, there being no Volume (when this Book was first published in 1686) extant that had Secants befides this: The Defcription and general Uses of thefe Tables are comprehended in 4 Chapters, containing 13 Propofitions, and fet just before the Tables, beginning at Page 294.

The Schemes or Figures are contained in 10 CopperPlates, inferted in their proper Places, being orderly Numbered with proper References for the more eafy turning unto, upon Occafion.

Thus have you a Summary of what's here treated; what my Labour and Pains have been herein, I leave you to judge who are most like to reap the Fruit and Profit (my Share being a very fmall Part thereof) tho' I dare aver, it's the Compleatest and most Portable Pile of Inftructions (for a Young Learner of Navigation) now extant; it's the very Method I have used for now 50 Years, finding it ever fuccefsful, even to the most indifferent Capacity, among the many Hundreds I have Taught: Therefore if my Reader would be a Proficient herein, let him begin chearfully, proceed gradually, and the End will crown bis Endeavours with anfwerable Succefs.

Let not Sloth perfuade to give out at the meeting of any Difficulty, but rather remember that Love, Labour, and Conftancy will overcome the greatest Difficulty; And,

That my Learner may fo read as to understand, and fo understand as to be a Proficient, is the Defire of him, who wifheth the Young Student's Welfare, and the Progress of ARTS.

JAMES ATKINSON.