A N INTRODUCTION TO THE Mathematicks. PART I. PRECOGNIT A. T HE Bufinefs of Mathematicks, both in Theory and By Quantity of Matter is here meant the Magnitude, or Bignefs of Bodies (of the fame Denfity) whofe Length, Breadth, and Thickness, may either be measured, or effimated. By Quantity of Space is meant the Distance which a Body moves over in a given Time, with a certain Velocity. And by Quantity of Motion is meant the Swiftness of any moving Body drawn into its Quantity of Matter. The Confideration of thefe, 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 Pofition, which may either be Natural, Accidental, or Defigned, may paffibly admit of infinite Varieties: But all the Varieties that are yet known, or indeed can 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 enly the Branches of these three Sciences, or rather their Application to particular Cafes. B. Geometry 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 obtained by comparing it with another known Quantity of the fame Kind, which will always be one of these, 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 Anfwers 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 recourfe to Arithmetick, which is to find out true and proper Answers 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 respect to it's Magnitude only; and the Subject of Arithmetick is Quantities with respect to their Number only. Algebra is a Science by which the most abftrufe or difficult Problems, in Arithmetick or Geometry, are Refolved and Demonftrated; that is, it equally interferes with them both; and therefore it is promifcuously named, being fometimes called Specious Arithmetick, as by Harriot, Vieta, and Dr Wallis, &c. And fometimes it is called Modern Geometry, particularly by the late ingenious and great Mathematician Dr Edmund Halley, Savilian Profeffor of Geometry in the University of Oxford, and Royal Aftronomer at Greenwich; who, in giving the 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; representing 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 be-, · come the Subjects of large Treatifes: For whatsoever Theorem folves the most complicated Problem of the Kind, does with a due Reduction reach all the fubordinate Cafes.' Of which he gives a notable Inftance in the Doctrine of Dioptricks for finding the Foci of Optic Glaffes univerfally. (Vide Philofophical Tranfactions, Numb. 205). ، Thus Thus you have a fhort and general Account of the proper Subjects of thofe noble and ufeful 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 reft are meddled withal. CHAP. I. Concerning the feveral Parts of Arithmetick, with the Definition of fuch Characters as are used in this Treatife. Rithmetick, or the Art of Numbering, is fitly divided into three diftinct Parts, two of which are properly called Natural, and the third Artificial. The first, being the moft plain and eafy, is commonly called Vulgar Arithmetick in whole Numbers; because every Unit or Integer concerned in it, represents one whole Quantity of fome Species or thing proposed. The fecond is that which fuppofes an Unit (and confequently the Quantity or thing reprefented by that Unit) to be Broken or Divided into any Number of equal Parts, and confiders of them either as pure Parts, viz. each lefs than an Unit, or elfe of Parts and Integers intermixt. And is usually 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 Ufe, efpecially in Geometrical Computations. Arithmetick (in all it's Parts) is performed by the various ordering and difpofing of Ten Arabick Characters or Numeral Figures (which by some are called Digits.) One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Cypher. 2 3 4 5 6 7 8 9 0 viz. { The Ufe of thefe Characters is faid to be firft introduced into England more than fix hundred Years ago, viz. about the Year 1130, vide Dr Wallis's Algebra, Page 12. The first of thefe Characters is called Unity, and reprefents one, of any Kind of Species or Quantity. As one World, one Star, one Man, &c. Viz. Unity is that by which every thing that is, is called one, (Euclid. 7. Def. 1.) and is the beginning of all Numbers. That is to fay, Number is a Multitude of Units. Euclid. 7. Def. 2. For, one more one, make Two; and one, more one, more one, make Three, &c. Which is the firft and chief Poftulate, or rather an Axiom in Arithmetick. Viz. { That 1+1=2. 1+1+1=3. 1+1+1+1=4. 1+1+1+1+1=5. And fo on to 9. Nine of thefe Figures were thus composed of Units, and differently formed to reprefent fo many Units put together into one Sum, as was intended each should denote: Nine being the greatest Number of Units that was then thought convenient to be expreffed by one fingle Character; the last of the Ten is only a Cypher, or (as fome phrase it) a Nothing, because of itself it fignifies nothing; for if ever fo many Cyphers be Added to, or Subftracted from, any Number, they can neither increase nor diminish that Number; but yet, as a Cypher (or Cyphers) may be placed, the other Figures, will become of different Values from what they were before, as will appear further on. For the more convenient ordering of the aforefaid Numeral Figures according to the several Varieties that happen in Computations; I do advise the young Learner to acquaint himself with the Signification of the following Algebraick Signs or Characters, which he will find of excellent Ufe, as being a fhorter, better, and much more fignificant Way of denoting what is to be done, (in moft Operations) than can otherwife be expreffed in Words at length. Signs Names. +}{ SIGNIFICATION S. The Sign of Addition; as 8+7 is 8 more 7, and fignifies that the Numbers 8 and 7 are to be added into one Sum. The like is to be unS Plus orderstood when several Numbers are connected together with the Sign +. more. As 34+22+9+45, &c. denotes these are all to be added into one Sum. The Into less. x} {with. -} { By. or = }{ Equal. :: } { So is. { The Sign of Subftraction; as 9-6 is 9 lefs 6, and fignifies that 6 is to be taken from 9, that fo their Difference may be found. The Sign of Multiplication; as 9x6, is 9 into 6, and fignifies that 9 is to be Multiplied into or with 6. The Sign of Divifion; as.82, is 8 by 2, and fignifies that 8 is to be Divided by 2, alfo thus 2) 8 (4 or thus; each fignifying the fame thing, to wit, 8 Divided by 2. The Sign of Equality or Equation, viz. when this Sign is placed betwixt Numbers (or Quantities) it denotes them to be equal; as 9=9, or 9+6= 15, or 9—6—3, &c. That is, 9 is Equal to 9, or 9 more 6 is Equal to 15, and 9 lefs 6 is Equal to 3, &c. The Sign of Proportion, or that commonly called the Golden Rule, or Rule of Three, and :: is always placed betwixt the Two middle Terms or Numbers in Proportion. Thus 2:8:: 6:24. To be read thus; as 2, is to 8; fo is 6, to 24. Thefe Signs and their Significations, being perfectly learnt, will help to fhorten the Work. CHA P. II. Concerning the Principal Rules in Arithmetick, and bow they are performed in Whole Numbers. T HE Rules by which Numerical Operations are performed in all the Parts of Arithmetick, are many and various, feveral of them being formed and raifed as Occafion requires, when applied to Practice; yet they are all comprehended within the due Confideration of these Six, viz. Pumeration (or 20 tation) |