To the R E A D E R. 3 | Think it needless (and almost endless) to run over all the Usefulness, and Advantages of Mathematicks in General; and all therefore only touch upon those two admirable Sciences, Arithmetick and Geometry; which are indeed the two grand Pillars (or rather the Foundacions) upon which all other parts of Matbematical Learning depend. As to the Usefulness of Arithmetick, it is well known that no Bufiness, Commerce, Trade, or Employment whatsoever, even from the Merchant to the Shop-keeper, &c. can be managed and carried on, without the Ajifance of Numbers. And as to the Usefulness of Geometry, it is as certain, that no curious Art, or Mechanick-Work, can either be invented, improved, or performed, without it's affifling Principles; tho' perhaps the Artist, or Workman, bas but little (nay scarce aoy) Knowledge in Geometry. Then, as to the Advantages that arise from both these Noble Sciences, wben duly joined together, to allijit each other, and then apply'd to Practice, (according as Occafion requires) they will readily be granted by all who conßder the vall Advantages that accrue to Mankind from the Bufiness 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 these, and a great many more very useful Arts, (too many to be enumerated here). wholly depend upon the aforesaid Sciences. And therefore it is no Wonder, That in all Ages so many Ingenioks and Learned Perfons 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 ebay attempted to porase shofe Books, which are generally large Volumes, written in such abftrufe Terms, that young Learners were really afraid of looking into tbose Studies. com These Confiderations 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 firft Rudiments. Therefore I began with their first Ekmenis 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 Çearnór 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 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 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 80. 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 not enlarge upon, but leave the Book to speak for itself; and if it be not able to give Satisfa&tion to the Reader, I am sure all I can say here in it's Bebalf will never ree commend 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, 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 bude 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 Ibado pero formed That, must be left to proper Judges. Dit is on To be brief; as I am not 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 that it suas composed for his Use; by his real Well-wisher, '.' coin .... J. WARD. London, October 10th, 1706. January 20th, 1722. . ii. . . . THE THE CONTENTS · arithmetick. Part I. PRecognita, Concerning the proper Subjects, or Business of Ma. Chap. I. Concerning the several Parts of Arithmetick, and of fuch 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 · 57 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 Chap. IX. of Alligation or Mixing of Things, with all it's DIO Chap. X. Concerning the Specifick Gravities of Metals, &c. 117 Chap. XI. Evolution or Extracting the Roots of all Single Powers, haw high foever they are, by one General Method., 123 ap. I. 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 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- · Of Proportional Quantities, both Arithmetical and Geo- S Chap. VII. Of Proportional Quantities Disjunct, both Simple, Duplicate, and Triplicato; and how turn Equations. Chap. VIII. Of Substitution; and resolving Quadratick Equations. Chap. IX. Of Analysis, or the Method of Resolving Problems, Exemplified by Forty Numerical Questions. 202 Chap. X. The Solution of all kinds of Adfected Equations in Chap. XI. Of Simple Interest, and Annuities in all their various (II. Of Compound Interest, and Annuities both for Years and Lives; and of Purchasing Freehold Estates. 253 Chap. I. Of Geometrical Definitions and Axioms, &c. 283 Chap. II. The First Rudiments or Leading Problenis. in Geo- Chap. III. A Colle&tion of the most useful Theoremş in Plain Geometry, Analytically demonflrated. .. 300 Chap. IV. The Algebraical Solution of Twenty easy Problems in Plain Geometry; which does in part jew the Use of 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 asigned Exactness; by the Solution of one Equation only. Also a Nezu Way of making Natural Sines and Tangents à priore.: 347 . : Tonick Seitions. 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 i arithmetick of Infinites. Part V. Application to Geometry; in demonftrating the Super. ii. ficial and Solid Contents of Circular and Elliptical Fi- : : An Appendix of Pladical Oauging. Wherein all the chief Rules and Problems useful in Gauging, are i 1 | INTRODUCTION TO THE Mathematicks, . PART I. I PRÆCOGNIT A. HE Business of Mathematicks, in all it's Parts, bot Theory and Practice, is only to search out and determine according as Occasion requires. By Quantity of Space is meant the Distance of one thing from 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, are the Subjects of the Mathematicks, but chiefly that of Macter. 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 tbe Branches of these three Sciences, or rather their Application to particular Cafeso Geometri |