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That is, I began with an Unit in Arithmetick, and a Point in Geometry; and from these Foundations proceeded gradually on, leading the young Learner Step by Step with all the Plainness I could, &c.

And for that Reafon 1 published this Treatise (Anno 1707) by the Title of the Young Mathematician's Guide; which has answered the Title so well, that I believe I may truly say (without Vanity) this Treatise bath proved a very helpful Guide to near five thousand Perfons; and perhaps most of them such as would never have looked into the Mathematicks at all but for it.

And not only fo, but it hath been very well received amongst the Learned, and (I have been often told) so well approved on at the Universities, in England, Scotland, and Ireland, that it is ordered. to be publickly read to their Pupils, &c.

The Title Page gives a short Account of the several Parts treated of, with the Corrections and Additions that are made to this Fifth Edition, which I fall not enlarge upon, but leave the Book to speak for itself; and if it be not able to give Satisfaction to the Reader, I am sure all I can say here in it's Behalf will never recommend it : But this may be truly said, That whoever reads it over, will find more in it than the Title doth promise, or perhaps be expects: it is true indeed, the Dress is but Plain and Homely, it being wholly intended to instruct, and not to amuse or puzzle the young Learner with hard Words, and obscure Terms: However, in this I fall always have the Satisfaction ; That I have sincerely aimed at what is useful, tho' in one of the meanest Ways; it is Honour enough for me to be accounted as one of the Under-Labourers in clearing the Ground a little, and removing some of the Rubbish that lay in the way to this sort of Knowledge. How well I have performed That, must be left to proper Judges.

To be brief; as I am not sensible of any Fundamental Error in this Treatise, so I will not preiend to say it is without Imperfections, (Humanum eft errare) which I hope the Reader will excuse, and pass over with the like Candour and Good-Will that it was composed for bis Ufe; by his real Well-wisher,

J. WAR D.

London, O&tober toth, 1706.

Corrected, &c. at Chester,

January 20th, 1722.

THE

THE

CONTENTS

Arithmetick. Part I.

P Recognita, Concerning the proper Subjects, or Bufiness of Ma-
thematicks, &c.

Page 1

Chap.

I. Concerning the several Parts of Arithmetick, and of

such Characters as are used in this Treatise. 3

Chap. II. Concerning the Principal Rules in Arithmetick, and

how they are performed in whole Numbers.

5

Chap. III. Concerning Addition, Subtraction, and Reduction of

Numbers that are of different Denominations. 31

Chap. IV. Of Vulgar Fractions, with all their various Rules. 48

Chap. V. Of Decimal Fractions or Parts, with all the useful
Rules, and Contractions, &c.

57
Chap. VI. Of continued Proportion, both Arithmetical and Geo-

matrical; and how to vary the Order of Things. 7?

Chap. VII. Of Disjunct Proportion, or the Golden Rule, both

Direct, Reciprocal or Inverse, and Çompound. 85

Chap. VIII. The Rules of Fellowship, Bartering, and Exchanging

of Coins.

99

Chap. IX. Of Alligation or Mixing of Things, with all it's

Varieties or Cases.

Chap

X. Concerning the Specifick Gravities of Metals, &c. 117

Chap. XI. Evolution or Extracting the Roots of all Single Powers,

bow high foever they are, by one General Merbod. 123

Algebja. Part II.

Chap 1. The Method of noting down Quantities, and tracing

of tbe Steps used in bringing them to an Equation. 143

Chap. II. The Six Principal Rules of Algebraick Aritbmetick, in

whole Quantities.

147

Chap. III. Of Algebraick Fractions, or Broken Quantities. 163

Chap. IV. Of Surds, or Irrational Quantities.

171

Chap. V. Concerning the Nature of Equations, and how to pre-

pare them for a Solution, &c.

175

Chap. VI. Of Proportional Quantities, both Arithmetical and Geo-

metrical continued; alfo of Musical Proportion, 184

Chap

IIQ

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Chap. VII. Of Proportional Quantities Disjunct, both Simple,

Duplicate, and Triplicate; and how turn Equations.

into Analogies, &c.

Page 190

Chap. VIII. Of Subtitution ; and resolving Quadratick Equations.

194
Chap. IX. Of Analysis, or the Method of Resolving Problems,

Exemplified by Forty Numerical Questions.

Chap. X.' The Solution of all Kinds of Adfected Equations in
Numbers:

234

Chap. XI. Of Simple Interest, and Annuities in all their various

Cafes.

245.

Chap. XII. Of Compound Interest, and Annuities both for Years and

Lives; and of Purchasing Freehold Estates. 253

Geometry. Part III.

Chap. I. Of Geometrical Definitions and Axioms, &c. 283

Chap. II. The First Rudiments or Leading Problems in Geo-

metry.

292

Chap. III. A Collection of the most useful Theorems'in Plain

Geometry, Analytically demonstrated.

300 :

Chap. IV. The Algebraical Solution of Twenty časy Problems in

Plain Geometry; which does in part fnew the Use of
the laft Theorems.

320

Chap. V. Practical Problems and Rules, for finding the Area's

of Right lined Superficies, demonstrated.

338

Chap. VI. A New and easy Method of finding the Circle's Pe-

riphery, and Area, to any aligned Exactness; by the

Solution of one Equation only. Also a Neru Way of

making Natural Sines and Tangenis à priore. 347

Conick Sections. Part IV.

Chap. I. Definition of a Cone, and all it's Sections, &c. 36r

Chap II. Goncerning the chief Properties of the Ellipfis, &c.

367

Chap. III. Concerning the chief Properties of the Parabola. 380

Chap. IV. Concerning the chief Properties of the Hyperbala. 386

Srithinetick of Infinites. Part V.

The Arithmetick of Infinites explained, and rendered easy; with it's

Application to Geometry, in demonftrating the Super-
ficial and Solid Contents of Circular and Elliptical Fi-

397
An Appendix of Practical Gauging.

Therein all the chief Rules and Problems useful in Gauging, ara
applied to Practice, &c.

433

AN

gures, &c.

Α Ν

INTRODUCTION

THE
Mathematicks,

TO THE

PART I.

PR Æ COGN IT A.

T

HE Business of Mathematicks, in all it's Parts, both

Theory and Practice, is only to search out and determind the true Quantity; either of Matter, Space, or Motion,

according as Occasion requires. By Quantity of Matter is here meant the Magnitude, or Biga ness of any visible thing, whose Length, Breadth, and Thickness, may either be measured, or estimated.

By Quantity of Space is meant the Distance of one thing from another.

And by Quantity of Motion is meant the Swiftness of any thing moving from one place to another.

The Confideration of these, according as they may be proposed, arx the Subjects of the Mathematicks, but chiefly that of Matter.

Now the Confideration of Matter, with respeel to it's Quantity, Form, and Polition, which may either be Naturals Accidental, or Designed, will admit of infinite Varieties : But all the Varieties that are yet known, or indeed possible to be conceived, are wholly comprized under the due Confideration of these Two, Magnitude and Number, which are the proper Subjects of Geometry; Arithmerick, and Algebra. All other parts of the Mathematicks being only the Branches of these three Sciences, or rather their Application te particular Cafesi

B

Geometry

Geometry is a Science by which we search out, and come to know, either the whole Magnitude, or some part of any proposed Quantity; and is to be obtained by comparing it with another known Quantity of the same Kind, which will always be one of these, viz. A Line, (or Length only) A Surface, (that is, Length and Breadth) or a Solid, (which hath Length, Breadth, and Depth, or Thickness) Nature admitting of no other Dimensions but these Three.

Arithmetick is a Science by which we come to know what Number of Quantities there are (either real or imaginary) of any Kind, contained in another Quantity of the fame Kind: Now this Consideration is very different from that of Geometry, which is only to find out true and proper Answers to all fuck Questions as demand, how Long, how Broad, how Big, &c. But when we consider either more Quantities than one, or how often one Quantity is contained in another, then we have recourse to Arithmetick, which is to find out true and proper Answers to all such Questions as demand, how Many, what Number, or Multitude of Quantities there are. To be brief, the Subje&t of Geometry is that of Quantity, with respect to it's Magnitude only; and the Subject of Arithmetick is Quantities with respect to their Number only.

algeb?a is a Science by which the most abftrufe or difficult Problems, either in Arithmetick or Geometry, are Resolved and Demonstrated; that is, it equally interferes with them both; and therefore it is promiscuously named, being fometimes called Specious Arithmetick, as by Harriot, Vieta, and Dr Wallis, &c. And sometimes it is called Modern Geometry, particularly the ingenious and great Mathematician Dr Edmund Halley, Savilian Professor of Geometry in the University of Oxford, and Royal Afronomer at Greenwich, giving this following Instance of the Excellence of our Modern Algebra, writes thus :

The Excellence of the Modern Geometry (faith he) is in nothing more evident, than in those full and Adequate Solutions & it gives to Problems; representing all the possible Cafes at one View, and in one generał Theorem many Times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the Manner of the Ancients, might well become the Subjects of large Treatises: For whatsoever Theorem folves the most complicated Problem of the Kind, does with a due Reduction reach all the subordinate Cases.' Of which he gives a notable Instance in the Doctrine of Dioptricks for finding the Foci of Optick Glaffes universally. (Vide Philosophical Trani. actions, Numb. 205).

Thus

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