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Geometry is a Science by which we search out, and come to know, either the whole Magnitude; or some part of any proposed Quantity; and is to be obtained by comparing it with another known

Quantity of the same Kind, which will always be one of these, * viz. A line, for 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 there 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 such Questions as : demand, how Long, how Broad, how Big, &c. But when we

consider 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 Answers to all such questions as.. demand, how Many, what Number, or Multitude of Quantities there are. To be brief, the Subject of Geometry is that of Quan. tity, with respect to it's Magnitude only; and the Subject of Arithmetick is Quantities with rejpect to their Number only.

algebra is a Science by which the most abftrufe or dificult Problems, either in Arithmetick or Geometry, aré Resolved and Demonstrated; that is, it equally interferes witb them both;

and therefore it is promiscuousy named, being sometimes called Specious Arithmetick, as by Harriot, Vieta, and Dr Wailis, &c.

And sometimes it is called Modern Geometry, particularly the in"genious and great Mathematician Dr Edmund Halley, Savilian Professor of Geometry in the University of Oxford, and Royal

Astronomer at Greenwich, giving this following Instance of the Ex:cellence of our Modern Algebra, writes thus :

** The Excellence of the Modern Geometry (faith he) is in

nothing more evident, than in those full and Adequate Solutions ' "it gives to Problems ; representing all the possible Cafes at one "D.View, and in one general Theorem many Times comprehending

o whole Sciences; which deduced at length into Propositions, and se demonstrated after the Manner of the Ancients, might well be

come the Subjects of large Treatises: For whatsoever Theorem • *folves the most complicated Problem of the Kind, does with a

due Reduction reach all the subordinate Cafes. Of which he

gives a notable Infance in the Doctrine of Dioptricks for finding .: the Foci of Optick Glasses universally. (Vide Philosophical Trans...ections, Numb. 205). .

Thus

of Georeenwich.ebra, wrn" Geomena Adequacases at ding

"The Excellevodern Algebrang this following Oxford, and Relian

Tous you have a short and general Account of the proper Subjects of these three noble and useful Sciences, Arithmetick, Geometry,

d' Algebra. I shall now proceed to give a particular Account of seb; and firit of Arithmetick, which is the Balis or Foundation

all Arts, both Mathematick and Mechanick; and therefore it sagot to be well understood before the rest are meddled withal.

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Whitbmetick, or the Art of Numbering, is fitly divided into » three diftinet Parts, two of which are properly called Natural, and the third Artificial.

The first, being the most plain and easy, is commonly called Valgar Arithmetick in whole Numbers; because every Unit or Integer concerned in it, represents one whole Quantity of some Species or thing proposed.

The second is that which supposes an Unit (and consequently the Quantity or thing represented by that Unit) to be Broken or Divided into equal Parts (either even or uneven) and confiders of tbem either as pure Parts, viz. Each less than an Unit, or else 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 easy Way than that of Vulgar Fractions : For the several Operations performed in Decimals, differ but little from those in Whole Numbers: and therefore it is now become of general Use, elpecially in Geometrical Computations.

Arithmctick (in all it's Parts) is performed by the various ordering and disposing of Ten Arabick Characters or Numeral Fi. gares (which by some are called Digits.) viz. vis s One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Cypher. .

9 ? 1 2 3 4 5 6 7 8 9 0 The Use of thefe Characters is said to be first introduced into England near fix hundred Years ago, viz, about the Year 1130, vide Dr Wallis's Algebra, Page 12. B 2 :

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The first of these Characters is called Unity, and represents one, of any kind of Species or Quantity. As one World, one Star, one Man, &c.

Viz. Únity 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 say, Number is a Multitude of Units. Euclid. 7. Def. 2.

For, one more one, makes Two; and one, more one, more one, makes Three, &c. Which is the first and chief Poftulate, or rather Axiom to Arithmetick.

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Nine of these Figures were thus composed of Units, and differently formed to represent so 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 expressed by one single Character; the last of the Ten is only a Cypher, or (as some phrase it) a Nothing, because of itself it signifies nothing; for if never so many Cyphers be Added to, or Substracted 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 aforesaid 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 Use, as being a much shorter, better, and more fignificant Way of denoting what is to be done, (in most Operations) than can otherwise be expressed in Words at length.

SIGNIFICATIONS.

Signs Names. 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 un

or{derstood when several Numbers are connected ts more. · together with the Sign to

As 34 +22+9+45, &c. denotes these are (all to be added into one Sum.

The © The Sign of Subtraction; as 9–6 is 9 less U Minus

36, and signifies that 6 is to be taken from 9, Nor lefs. Ithat fo their Difference may be found.

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C The Sign of Multiplication; as 9x6, is 9 in3 to 6, and signifies that 9 is to be Multiplied (into or with 6.

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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; lso is 6, to 24.

These Signs and their Significations, being perfectly learnt, will help to Thorten the Work.

CHA P. II.

Concerning the Principal Rules in Arithmetick, and how

they are performed in Whole Numbers.

THE Rules by which Numerical Operations are performed

in all the Parts of Arithmetick, are many and various, several of them being formed and raised as Occasion requires, when applied to Practice; yet they are all comprehended within the due Consideration of these Six, viz. Pumeration (or Potas nion) Addition, Subtraction, multiplication, Dibition, and Evolution, or Extraction of Roots.

Sect. 1. Of Fumeration or potation.

Lumeration or Notation, teacheth to Read or Express the true Value of any Number when writ down; and consequently to write down any proposed Number according to it's true Value when it is named: And this confifteth of Two Parts.

1. The due Order of placing down Figures.
2. The true valuing of each Figure in it's Place.
Both which are plainly exhibited in the following Table.

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* By this Numeration Table it is apparent, that the Order of Places is reckoned from the Right-hand towards the Left ; the first Place of any Number being always that which is the outmoft Figure to the Right-hand : and whatever Figure ftands in that Place, doth only signify it's own fimple Value, viz. fo many Units as that Figure represents. . The second place is that of Tens, and any Figure standing in that place signifieth so many Tens as that Figure represents Units.

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