T I To the READER. 1 Think it needless (and almost endless) to run over all the Usefulness, and Advantages of Mathematicks in General; and shall 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 on, without the Assistance 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 assisting Principles; the perhaps the Artist, or Workman, has but little (nay scarce any) Knowledge in Geometry. Then, as to the Advantages that arise from both these Noble Sciences, when duly joined together, to affift each other, and then apply'd to Practice, (according as Occafion requires) they will readily be granted by all who confider 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 Timekupers, 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 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 some Progress in that Sort of Learning before they attempted to peruse those Books, which are generally large Volumes, written in fuch abstruse Terms, that young Learners were really afraid of looking into those Studies. the 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 first Rudiments. Therefore I began with their first Elements or Principles. That 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 I 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 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 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 shall 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 faid, Thut whoever reads it over, will find more in it than the Title doth promise, or perhaps he 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 shall always have the Satisfaction; That I have fincerely aimed at what is useful, thơ' 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 pretend 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 his Ufe; by his real Well-wisher, THE CONTENTS. Arithmetick. Part I. Precognita, Concerning the proper Subjects, or Business of Ma- Chap. I. Concerning the several Parts of Arithmetick, and of Juch Characters as are used in this Treatise. 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. 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- 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. X. Concerning the Specifick Gravities of Metals, &c. 117 Chap. XI. Evolution or Extracting the Roots of all Single Powers, Chap. 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 metrical continued; also of Musical Proportion. 184 : Exemplified by Forty Numerical Questions. Chap. IV. The Algebraical Solution of Twenty easy Problems in of Right lined Superficies, demonstrated. Chap. VI. A New and easy Method of finding the Circle's Pe- riphery, and Area, to any affigned Exactness; by the I. Definition of a Cone, and all it's Sections, &c. Chap. II. Concerning the chief Properties of the Ellipsis, &c. Chap. III. Concerning the chief Properties of the Parabola. 380 Chap. IV. Concerning the chief Properties of the Hyperbola. 386 The Arithmetick of Infinites explained, and rendered easy; with it's Application to Geometry; in demonstrating the Super- An Appendix of Practical Gauging. applied to Practice, &c. 397 433 AN T HE Business of Mathematicks, in all it's Parts, both By Quantity of Matter is here meant the Magnitude, or Big 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, 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, B Geometr hrirch |