#HE General Approbation of this Book, since it; of first Impression in 1686, hath obliged me not f only to publish my Grateful Acknowledgment thereof; but also excited my diligent Search o 3 into, and exačt Survey of every Part of its Struślure, to find out the Defetis, to know all its Deftciencies, which I have so far laboured in, That in this Impression, I have now Reşified what was amiss; Altered what was Disorderly; Explained what was Dark or Obscure ; Enlarged where it was Scanty; Added where it was Wanting ; And, These Rećlifications, Alterations, Emendations, and Additions, being in diverse Places, I thought it needless to Enumerate them particularly ; for they are obvious to a diligent Reader, that will but compare the Rules, Precepts, and Examples of this with the former Impressions, in the plain and easy Method th;3 are placed in, but which in general is thus; *. I. 2%u bave ProtoGeometry, explained by Definitions, Problems and Proportions; In this Chapter is taught the making the most useful Geometric Figures, with the measuring all Superficies and Solids; Also the Application thereof in Pratlical Measuring, Board, Glass, Plaistering, Painting, Paving, and Land; Timber and Stone ; Gauging of Casks, and a Ship's Hold: All being illustrated with Rules, Proportions and Examples, easy to the Understanding, and not burthensome to the Memory ; and so fated that they may be performed both by Arithmetic, and by the Line of Numbers on Gunter's-Stale, in 37 Problems, A 3 The The first Chapter I advise the Learner to study well before

he proteeds, 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

useful Notes and Definitions, with the Axioms, and also the
Cases depending on each Axiom, orderly set down in 8 Prob-
blems, containing 13 Cases.
And bere I must advertise the 7'oung Student that would

work Trigonometry by the Logarithms, to consult Chap-

ter 1. The Explanation and general Use of the Table

of Logarithms, and Tables of Sines, Tangents, and Se-

cants, towards the latter End of the Book, in Page 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 transmuted

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

Parliament required) shewing how to find the Prime, Epačt,

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

ing, and Time of Full-Sea, or High-Water, In 9 Prob.
Then proceeding to the Description and Use of the Plane-
Chart, in 5 Problems, before I come to the Cases of Plane
Sailing, which I divide into three Parts.
I.T., a Right-angled Triangle, relating to a single

Course, in which are 6 Case", commonly called the 6 Cases

of Plane Sailing. ~ \- -

2. In a Right-angled Triangle, oto several Courses

called a Traverse.

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

though there may be a Multitude of various Questions ; of
which you have a Taffe 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, 'tir necessary to describe Mr.

Wright's Projećtion, commonly known by the Name of

Mercator's Chart, and shew the Uses of it, before I treat of

the Problems of Sailing by it which you will find performed

$s,

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

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

And in Case that Table or Line be wanting, to supply ther

Room I have added Problems of Sailing by the Middle

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

a Thing of good Use, In 4 Problems.

V. Spheric Trigonometry, or the Doğrine of Spheric

Triangles Rectangular and Oblique, is next in Order; and
it being so necessary you should understand how to make a
Spheric Triangle, and also bow 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 so called (the

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

in Rectangular and Obliquangular Triangles, explain'd

with necessary Notes on each Case; as to know when a re-

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

o: loss than a Quadrant, In 12 Problems, containing
2 ö Cales. -

VI. The Description and Use of both Globes, is the

next to be consider'd ; in which I have plainly and familiary
explain’d and shewed the Use of the most necessary Things be-
longing or relating to each of them, In 24 useful Problems.
To which is annexed a short: Description and Use of the
Hemispheres, proječied on the Plane of the Ecliptic.
WII. Geography...itoulied of this Chapter, which

if the Application of' Spheric Trigonometry in finding the

True Distance 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, so it’s the most difficult, and
hardly possible for a Ship exačily to sail by ; yet it’s of great
Advantage to keep conveniently near it; for which Purpos:
you'll find all that necessarily belongs or relates to it, both
as to the Projećtive Part, and that both Stereographic and
Gnomonic; as also in the Calculate Part, (which requires
the Application of both Spheric and Plane Trigonometry,
fully made out; With an Intimation of shortning the H/ork,
by shewing how to describe the Arch of a Great Circle on
Mercator’s Chart; the Whole in 4 Problems, -
IX. Next you have Spheric Trigonometry, applied in
fundry Astronomic Problems useful in Navigation, wherein
the Circles of the Sphere are described, and the necessary
Terms of Art explained to the means: Capacity, with Respe:
to the diurnal Motion; and that either,
1. According to the Ptolomaic System, wherein you
bave in a Right-Angle Spheric Triangle, all the Variety of
Questions and Examples that relate to the Sun, with Re-
sped to his Longitude, Declination, Right or Oblique Ascen-
Jion, or Descension, Rising, Setting, Amplitude, Altitude and
Azimuth, at the Hour of Six ; Altitude and Hour of the
Day, when East or West ; Hour, Azimuth, and Altitude,
when be is in the Equinoëtial, In 24 Problems. Also,
In an Oblique Spheric Triangle, you have great Pariety
both with Respeš: to the Sun or a Star, in many Qasiions
and Examples relating to the Sun's Altitude, Azimutb, 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 Ascension, Rising, 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 Pytiagorean, or Copernic System,
which is now generally receiv as most agreenble to the ob-
ferved experienc'd Motion of Thoeo, 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 Asp-
pearances of the Sun and Fixed Stars are solved, with the
Description of the Circles of the Sphere, and how they are
drawn Stereographically on the Plane of the Earth's Ecliptic,
In 9 Problems. -> -
X. Then follow very easy Rules to find the Variation of