CONTENTS. Arithmetick. Part I. Pages pRecognita, Concerning the proper Subjects, or Bufness of Chap. I. Concerning the several Parts of Arithmetick, and of how they are perform'd in Whole Numbers. 5 Chap. III. Concerning Addition, Subftraction, and Redu&ion of Numbers that are of different Denominations. 31 Chap. IV. Of Vulgar Fradions, with all their various Rules. 48 57 Chap. VI. Of Continued Proportion, both Arithmetical and Geo- metrical ; And how to vary the Order of Things. 72 Chap. VII. Of Disjun&t Proportion, or the Golden Rule, boch 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 its Varieties or Cafes. Chap. X. Concerning the Specifick Gravities of Metals,&c. 117 Chap. XI. Evolution or Extrading the Roots of all Single Powers, how High foever they are, by one General. of Noting dovin Quantities, and Tracing Chap. II. The Six Principal Rules of Algrebraick Arithmetick, Chap. III. Of Algebraick Fractions, or Broken Quantities. 163 Chap. IV. Of Surds, or Irrational Quantities, Chap. V. Concerning the Nature of Æquations, and how to prepare them for a Solution, &c. IIO Chap. VII. Of Proportional Quantities Disjunct, both Simple, Chap. IX. Of Analyfis, or the Method of Refolving Problems, Chap. I. Of Geometrical Definitions and Axioms, &c. Chap. IV. The Algebraical Solution of Twenty eafy Problems in Plain Geometry; which does in part fhew the Chap. V. Practical Problems and Rules, for finding the Area's of Right lin❜d Superficies, Demonftrated. 338 Chap. VI. A New and Eafy Method of finding the Circle's Pe- riphery, and Area, to any affigned Exactnefs; by the Solution of Oue Equation only. Alloa New Way of making Natural Sines and Tangents a Priore. 347 Chap. I. Definition of a Cone, and all its 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 Hyperbolą, 386 Arithmetick of finites. Part V. The Arithmetick of Infinites Explain'd, and rendered Eafy; with its Application to Geometry, in Demonftrating An Appendix of Practical Gauging. Wherein all the Chief Rules and Problems useful in Gauging, are Applied to Practice, &c. 433 ΑΝ I AN INTRODUCTION TO THE Mathematicks. T PART I Præcognita. HE Bufinefs of Mathematicks in all its Parts, both By Quantity of Matter is here meant the Magnitude or Big nefs of any visible thing, whofe Length, Breadth, and Thickness may either be measured, or estimated. By Quantity of Space is meant the diftance of one thing from another; other with respect to time or pola cer. And by Quantity of Motion is meant the fwiftness of dug 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 refpect to its Quantity, Form and Pofition, which may either be Natural, Accidental, or Defigned, will admit of infinite Varieties; But all the Varieties that are yet known, or indeed possible to be conceived, dre wholly comprised 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 bring only the Branches of these three Sciences; or taiker ilj:17 Application to particular Cafes B coming Geometry is a Science by which we fearch out and come to know either the whole Magnitude, or fome part of any proposed 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 bath Length, Breadth and Depth, or Thickness) Nature admitting of no other Dimenfions but thefe Three. Arithmetick is a Science by which we come to know what Number of Quantities there are (either real or imaginarg) 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 are to confider either of more Quantities than one, or how often one Quantity is contained in another, then we have recourfe to Arithmetick, which is to find out true and proper Anfwers to all fuch Questions as demand how Many, what Number, or Multitude of Quantities there are. To be brief, the Subject of Geometry is that of Quantity, with refpect to its Magnitude only; and the Subject of Arithmetick is Quantities with respect to their Number only. Algebra is a Science by which the most at ftrufe or difficult Problems either in Arithmetick or Geomerry are Refolved and Demonftrated, that is, it equally interferes with them both; and therefore it's promiscuously named, being fometimes called Specious Arithmetick, as by Harriot, Vieta, and Doctor Wallis, &c. And fometimes its called Modern Geometry, particularly the ingenious and great Mathematician, Mr. Edmund Halley, Savilian Profeffor of Geometry in the University of Oxford, 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 whatsoever I'heorem folves the most complicated Problem of the kind, does with a due Reduction reach all the Subordinate Cafes. Of which he gives a notable Inftance in the Doctrine of Dioptricks for finding the Foci of Optick Glasses universally, (vide Philofophical Trantactions, Numb. 205.) Thus Thus you have a fhort and general Account of the proper Subjects of thofe three Noble and Useful Sciences, Arithmetick, Geometry and Algebra. I shall now proceed to give a particular Account of each; and firft of Arithmetick, which is the Bafis or Foundation of all Arts, both Mathematick and Mechanick; and therefore it ought to be well understood before the reftare medled withal. C. HA P. I. Concerning the feveral Parts of Arithmetick, with the Definition of fuch Characters as are used in this Treatise. A Rithmetick, or the Art of Numbering, is fitly divided into three diftinct Parts, two of which are properly called Natu ral, and the third Artificial The first being the most plain and eafieft, is commonly called Vulgar Arithmetick in whole Numbers; because every Unit or Integer concerned in it, reprefents one whole Quantity of fome Species or thing propofed. f The fecond is that which fuppofes an Unit (and confequently the Quantity or thing reprefented by that Unit) to be Broken or Divided into equal Parts (either even or uneven) and confiders of them either as pure Parts, viz. Each lefs than an Unit, or elfe of Parts and Integers intermixt. And is ufually called the Doctrine of Vulgar Fractions. The third, or Artificial Part, is called Decimal Arithmetick; being an Artificial Invention of managing Fractions or Broken Numbers, by a much more commodious and eafy Way than that of vulgar Fractions: For the feveral Operations performed in Decimals, differ but little from thofe in Whole Numbers; and therefore it is now become of general Use, especially in Geome trical Computations. Arithmetick (in all its Parts) is performed by the various ordering and difpofing of Ten Arabick Characters or Numeral Figures (which by fome are called Digits.) One Two Three Four Five Six Seven Eight Nine Cypher 8 9% The ufe of thefe Characters is faid to be first introduced into England near fix hundred Years ago, viz. about the Year 1130, vide Doctor Wallis's Algebra, Page 12. B 2 The |