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

And for that Reafon I published this Treatife (Anno 1707) by the Title of the Young Mathematician's Guide; which has answered the Title fo well, that I believe I may truly fay (without Vanity) this Treatife bath proved a very helpful Guide to near five thousand Perfons; and perhaps most of them fuch 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) fo well approved on at the Univerfities, in England, Scotland, and Ireland, that it is ordered to be publickly read to their Pupils, &c.

The Title Page gives a fhort Account of the feveral Parts treated of, with the Corrections and Additions that are made to this Fifth Edition, which I shall not enlarge upon, but leave the Book to fpeak for itself; and if it be not able to give Satisfaction to the Reader, I am fure all I can fay here in it's Behalf will never re commend it: But this may be truly faid, That whoever reads it over, will find more in it than the Title doth promife, or perhaps he expects: it is true indeed, the Dress is but Plain and Homely, it being wholly intended to inftruct, and not to amufe or puzzle the young Learner with hard Words, and obfcure Terms: However, in this I shall always have the Satisfaction; That I have fincerely 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 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 fenfible of any Fundamental Error in this Treatife, fo I will not pretend to fay it is without Imperfections, (Humanum eft errare) which I hope the Reader will excufe, and pass over with the like Candour and Good-Will that it was compofed for his Ufe; by his real Well-wisher,


London, October 10th, 1706.

Corrected, &c. at Chester,
January 20th, 1722.




Arithmetick. Part I.

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

Page 1


I. Concerning the several Parts of Aritbmetick, and of

such Characters as are used in ihis 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.

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

matrical; 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 Exiratling the Roots of all Single Powers,

bow bigbo foruer they are, by one General Metbed. 123

Algebja. Part II.

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

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

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

whole Quantities.


Chap. lll. Of Algebraick Fraktions, or Broken Quantities. 163

Chap. IV. Of Surds, er Irrational Quantities.


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

pare them for a Solution, &c.


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

metrical continued; also of Musical Proportion,



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

Duplicate, and Triplicate; and how turn Equations.

into Analogies, &c.

Page 190
Chap. VIII. Of Subtitution ; and resolving Quadratick Equations.

Chap. IX. Of Analysis, or the Method of Resolving Problemsy

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.


Geometry. Part III.

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

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



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

Geometry, Analytically demonstrated.


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

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


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

of Right lined Superficies, demonstrated, 338

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

riphery, and Area, to any assigned Exa&tness; by the

Solution of one Equation only

. Alfe a New Way of
making Natural Sines and Tangents à priore. 347

Conick Sections. Part IV.

Chap. I. 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

Hrithinetick of Infinites. Parr V.

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

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


An Appendix of Practical Gauging.

Wherein all the chief Rules and Problems useful in Ganging, art
applied to Practice, &c.



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HE Business of Mathematicks, in all it's Parts, both Theory and Practice, is only to fearch out and determinė the true Quantity; either of Matter, Space, or Motion, according as Occafion requires.


By Quantity of Matter is here meant the Magnitude, or Bignefs of any visible thing, whofe Length, Breadth, and Thickness, may either be measured, or effimated.

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 thefe, according as they may be propofed, are the Subjects of the Mathematicks, but chiefly that of Matter.

Now the Confideration of Matter, with respect to it's Quantity, Form, and Polition, which may either be Natural, Accidental, or Defigned, will admit of infinite Varieties: But all the Varieties that are yet known, or indeed poffible to be conceived, are wholly comprized under the due Confideration of thefe 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.




Geometry is a Science by which we fearch out, and come to know, either the whole Magnitude, or fome Part of any propofed Quantity; and is to be obtained by comparing it with another known Quantity of the fame Kind, which will always be one of thefe, 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 thefe 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 Confideration is very different from that of Geometry, which is only to find out true and proper Answers to all fuch Questions as demand, how Long, how Broad, how Big, &c. But when we confider 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 Anfwers to all fuch Questions as demand, bow Many, what Number, or Multitude of Quantities To be brief, the Subject 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.

there are.

Algebra is a Science by which the most abftrufe or difficult Problems, either in Arithmetick or Geometry, are Refolved and Demonftrated; 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 fometimes it is called Modern Geometry, particularly the ingenious and great Mathematician Dr Edmund Halley, Savilian Profeffor of Geometry in the University of Oxford, and Royal Aftronomer at Greenwich, giving this following Inftance of the Excellence of our Modern Algebra, writes thus:

"The Excellence of the Modern Geometry (faith he) is in • nothing more evident, than in thofe full and Adequate Solutions • it gives to Problems; reprefenting all the poffible Cafes at one • View, and in one general Theorem many Times comprehending • whole Sciences; which deduced at length into Propofitions, and



demonftrated after the Manner of the Ancients, might well become the Subjects of large Treatifes: For whatfoever Theorem • folves the most complicated Problem of the Kind, does with a • due Reduction reach all the fubordinate Cases. Of which he gives a notable Inftance in the Doctrine of Dioptricks for finding the Foci of Optick Glaffes univerfally. (Vide Philofophical Tranf actions, Numb. 205).·

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