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To the READER.

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Think it needlefs (and almof endless) to run over all the
Usefulness, and Advantages of Mathematicks in General; and

ball therefore only touch upon those two admirable Sciences,
Arithmetick and Geometry; which are indeed the two grand:
Pillars (or rather the Foundations) upon which all other parts of
Mathematical Learning depend.

As to the Usefulness of Arithmetick, it is well known that no Business, Commerce, Trade, or Employment whatsoever, even from the Merchant to the Shop-keeper, &c. can be managed and carried en, without the Afiftance of Numbers.

And as to the Usefulness of Geometry, it is as certain, that no curicus Art, or Mechanick-Work, can either be invented, improved, er performed, without it's alifting Principles; tho' perhaps the Arif, or Workman, bas but little (nay scarce aoy) Knowledge in Geometry.

Ther, as to the Advantages that arise from both these Noble Sciences, wben duly joined together, to all each other, and then apply'd to Practice, (according as Occafion requires) they will Teadily be granted by all who conßder the vast Advantages that accrue to Mankind from the Business of Navigation only. As also from that of Surveying and Dividing of Lands betwixt Party and Party. Besides the great pleasure and Use there is from Timekeepers, as Dials, Clocks, Watches, &c. All thefe, and a great many more very useful Arts, (too many to be enumerated here) wbelly depend upon the aforesaid Sciences.

And therefore it is no Wonder, That in all Ages so many Ingenious and Learned Persons bave employed themselves in writing upon the Subject of Mathematicks; but then most of those Authors seem to presuppose that their Readers had made fome Progress in that sort of Learning before they attempted to porufe those Books, which are generally large Volumes, written in such abftrufe Terms, that young Learners were really afraid of looking into abole Studies.

These Confideration's first put me (many Years ago) upon the Thoughts of endeavouring to compose such a plain and familiar Introduction to the Mathematicks, as might encourage those that were willing (to spend some Time that Way) to venture and proceed on with Chearfulness; tho' perhaps they were wholly ignorant of it's forft Rudiments. Therefore I began with their first Elemenes or Principles.

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 Reason 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 fay (without Vanity) this Treatise hath proved a very helpful Guide to near five thousand Persons"; and perhaps most of them such as would never bave looked into the Mathematicks at all but for it.

And not only fo, but it bath been very well received among At 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 fort. Account of the several Parts treated of, with the Corrections and Additions that are made to ibis Fifth Edition, which I fall nat 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 faid, Thut whoever reads it over, will find more in it than the Title doth promise, or perhaps he expects: it is irue indeed, the Dress is but Plain and Homely, is being wholly intended to instruct, and not to amuse: or puzzle: the young Learner with hard Words, and obscure Terms: However, in this I shall 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 fome of the Rubbishi that lay in the Way to this Sort of Knowledge. How well libave pero formed That, must be left to proper Judges.

To be brief; as I am noi sensible of any Fundamental: Error in this Treatife, so I will not pretend to say it is without Imperfections, (Humanum eft errare) which I hope the Reader will excufe, and pass over with the like Candour and Good-Will thar it was composed for his Use; by his real Well-wisher,

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Arithmetick. Part I.

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

Page 1

Chap. I. Concerning the several Parts of Arithmetick, and of

Juch Characters as are used in this Treatise

. 3

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

how they are performed in whole Numbers.


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.

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

metrical; and how to vary the Order of Things. 72

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

Direct, Reciprocal or Inverse, and Compound. 85

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

of Coins.


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

Varieties or Cafes.

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

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

baw high foever they are, by one General Method., 123

Algebja. Part II.

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

of the Steps ufed in bringing them to an Equation. 143

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

whole Quantities.


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

Chap. IV. Of Surds, or Irrational Quantities,


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

pare them for a Solution, &c.


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

metrical continued; alfo of Musical Proportion, 184



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

Duplicate, and Triplicate; and how turn Equations.

into Analogies, &c.

Page 190

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


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



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



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 Problenis. in. Geo-



Chap. III. A Collection of the most useful Theoremş in Plain

Geometry, Analytically demonflrated.


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

Plain Geometry; which does in part few the Use of

the laft Theorems.


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

of Right lined Superficies, demonstrated. 338

Chap. VI. A New and caly Method of finding the Circle's Deo

riphery, and Area, to any assigned Exaltness; by the
Solution of one Equation only

. Alfe a Neiu Way of

making Natural Sines and Tangents à priore.


Tonick Setions. Part IV.

Chap. 1. Definition of a Cone, and all it's Sections, &c. 361

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


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

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

Arithmetick of Infinites. Part V.

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

Application to Geometry; in demonstrating the Super:
ficial and Solid Contents of Circular and Elliptical Fi-

An Appendix of Practical Gauging.

1 herein all the chief Rules and Problems useful in Gauging, are

Applied to Practice, &c.










HE Business of Mathematicks, in all it's Parts, both
Theory and Practice, is only to search out and determine
the true Quantity; either of Matter, Space, or Motion,

according as Occasion requires.
By Quantity of Matter is here meant the Magnitude, or Bigf
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 Swiftnefs of any thing moving from one place to another.

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

Now the Confideration of Matter, with respect to it's Quantity, Form, and Position, which may either be Natural, 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, Arithmetick, and Algebra. All other Parts of the Mathematicks being only the Branches of these three Sciences, or rather their Application to particular Cafes,



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