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iii

THE

PREFACE

I

N the Course of my Studies on this Branch of the Mathematics, I obferv'd much Obfcurity, Irregularity and Deficiency in the Writings of moft Authors on this Subject. I have therefore in the following Treatife endeavour'd to redrefs this Grievance, and to render this Science more plain, eafy, and comprehenfive; to which End I have laid down a large Number of Definitions, Theorems, and Problems, which are abfolutely neceffary to be learned. The Definitions inform us of the Nature, Names, and Differences of the Jeveral Lines and Figures used therein and also other Terms of the Art. The Theorems contain all the Myftery of the Science, on a Demonftration and due underfanding of which depends the Rationale of Trigonometry; that is, the Reason of all the various Operations of the Art; Thirty-one of the beft Theorems I bave collated for this Purpose, which ferve more immediately for understanding the Reafon of the Methods for making the Canon of Natural Sines, Tangents, and Secants.

A

The Theorems ought to be well ftudied. In demonfrating them I have taken a new Method, after the Manner of Mr. Le Clerc, in bis Practical Geometry, which I think is the most clear, perfpicuous and eafy of all others. The Problems are adapted to the Capacity of all, enabling them to make a Geometrical ConftrucA 2

tion

tion of any Procefs or Operation in Trigonometry; in which the Learner ought to be very perfect. In the next Place follow the feveral Methods of conftructing a Canon of Natural Sines, Tangents and Secants, invented by the Learned, and here illuftrated and exemplified in Numbers. This Matter however neglected in Books of Trigonometry, is of the last Importance to finish an Artist in this Part of Learning. Having learn'd this, the Reafon of the Logarithmetic Canon is eafy, as will appear in due Place. After having thus laid the Foundation Principles, I proceed to the Method of falving all the Six Cafes of a Right-angled Triangle, in all the three Varieties, and that by ten feve ral Methods; fome by Natural Numbers, fome by Artificial Numbers, and others by Inftruments.

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I have fhewn how the Method by Logarithms depends on thefe by Natural Numbers; and how the Method by Inftruments depends on both; I let the young Learner at once into the Mystery, or Theory and Prac tice of each ethid, and have taken care to teach him nothing, for the which he may not be able to render a Reafon. Therefore have contriv'd a new Method to convince him of the Rationale of every Analogy, by Logarithms; or why we fay Sine, or Tangent, or Secant; and why the Parts of Analogy are placed as they are, and the Reafon of the large Indices to thofe Sort of Numbers; and have endeavour'd to make the whole clear and intelligible to the Learner by new Schemes for that Purpose.

I have alfo defcrib'd the Nature and Ufe of the feveral Lines on the Inflruments; as the Sector, Plain Scale, Sliding Rule, &c. as I have gone along; and have illuftrated their Ufes in the Doctrine of Triangles. You have alfo an Ample Collection of Anomalous Cafes of a Right Triangle, with their Solution by Algebraic The

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Then follows the Method of folving all the Cafes of Oblique Triangles, by the forefaid new Method, and Schemes; fhewing the Rationale of every Process.

And in the laft Place you have a Variety of Theorems, new and curious, for finding the Area of any Plain Triangle, by having given any of the Sides, or Sides and Angles together.

In the fecond Part of this Work, I have applied the Doctrine of Plain Trigonometry to the ten Mathematical Arts and Sciences mention'd in the Title-Page. In each of which I have given a General Account of the Art, its Principles and Maxims, the Nature and Pro-· perties of its feveral Parts, and the Use of Trigonometry in the whole: By which means the Reader will meet with a Kind of Epitome of the Mathematics, fo far as it depends on Plain Trigonometry: Here are inferted a great Number of very rare and curious Propofitions in Cofmography, Gecgraphy, Aftronomy, Gunnery, Mechanics, Surveying, Optics and Perfpective, which are not to be found in any one Auther (that I know of) on this Subject, and yet fuch as properly appertain thereunto, the Reader may fee the Particulars in the Table of Contents: One Thing I would acquaint the Reader with, that I have been concife, yet I think fufficiently prolix, in thofe Parts which are in every Book that treats of this Art, viz. Navigation, Altimetry, Fortification, and in fome other Parts have perform'd the Operations by the Inftrumental Methods, as by the Gunter, the Sector, &c. and therefore in fuch Cafes, the Reader is not to expect such Mathematical Exactness as by Logarithms, because the Matter does not require it. Upon the whole, I have omitted nothing that I could think of which might render this Work a compleat Guide to the young Trigonometer; and I hope in the Ufe it will be found to be fuch.

As

As for what belongs to Spherical Trigonometry

and its Application, that is contained in the fecond

Volume.

To conclude, having given a general Account of this

Piece, its Occafion, Subject and Method; I can not but
expect the candid Readers will excufe fmall Faults that
may have escaped my Notice; and wish them all the Pro-
fit in the perufal of it, which was intended in its Com-
pofure by him, who will never think much of any Pains
whereby any Benefit may accrue to any of his Fellow
Creatures.

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