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Thus you have a short and general Account of the proper Subjects of those three noble and useful Sciences, Arithmetick, Geometry, and Algebra. I fall now proceed to give a particular_Account of cach; and first of Arithmetick, which is the Basis or Foundation of all Arts, both Mathematick and Mechanick; and therefore it ought to be well understood before the rest are meddled withal.

CH A P I.

Concerning the several Parts of arithmetick, with the De

finition of such Characters as are used in this Treatise. AWithmetick, 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 most plain and easy, is commonly called Vulgar 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 fupposes 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 considers of them 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, especially in Geometrical Computations.

Arithmetick (in all it's Parts) is performed by the various ordering and difpofing of Ten Arabick Characters or Numeral Figares (which by some are called Digits.)

viz. { One

, Two
, Three
, Four, Fige

, Six
, Seven, Eight

, Nine

, Cypher.

The Use of these Characters is faid to be first introduced into England' near fix hundred Years ago, viz. about the Year 1130, uide Dr Wallis's Algebra, Page 12.

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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. 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 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 Axioin to Arithmetick.

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That iti=2. ititi=3. ititi+1=4 .

?titititi=5. And so on to 9. 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 expreffed by one single Character; the last of the Ten is only a Cypher, or (as fome 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 significant 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 72 and signifies that the Numbers 8 and 7 are to be added into one Sum. ' The like is to be un

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. order with Numbers are connected

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 -}{ Ş Minus S fo 6, and fignifies that 6 is to be taken from

9 or less.

chat fo-their Difference may be found.

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

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The Sign of Division; as 8+2, is 8 by 2, and fignifies that 8 is to be Divided by 2, alfo

thus 2) 8 (4 or thus t each signifying the same thing, to wit, 8 Divided by 2.

=}{Equal.

The Sign of Equality or Equation, viz. whenever this Sign = is placed betwixt Numbers (or Quantities) it denotes them to be Equal; as 9=9, or 9+6=15, or 9-633, &c. That is, 9 is Equal to 9, or 9 more 6 is Equal to 15, and 9 less 6 is Equal to 3, &c.

The Sign of Proportion, or that commonly

called the Golden Rule, or Rule of Three, and Sa is.

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

CHA P. II.

Concerning the Principal Rules in arithmetick, and bow

they are performed in Whole Numbers.

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HE Rules by which Numerical Operations are performed several of them being formed and raised as Occasion requires, when applied to Practice; yet they are all comprehended within the due Confideration of these șix, viz. Rumeration (or Potas

cjon) tion) Addition, Subtraction, spultiplication, Division, and Ebolution, or Extraction of Roots.

Sect. 1. Of Fumeration or Rotation.

Fumeration 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 consisteth 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 fumeration 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 out1 moft Figure to the Right-hand : and whatever Figure sands in that Place, doth only fignify 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 fignifieth so many Tens as that Figure represents Units.

The

The third Place is Hundreds, the fourth Place Thousands, &c. That is, each Place towards the Left-hand is Ten Times the Value of that next it, towards the Right.

For Instance, suppose 759 were proposed to be read or pru. nounced according to the Value of each Figure as they now ftand. The first Figure in this Sum is 9, because it stands in the Place of Units, and therefore signifies but it's own simple Value, to wit, 9 Units, or 9. The second Figure 5 ftands in the Place of Tens, and therefore signifies Five Tens or Fifty. The Figure 7 stands in the third Place, or Place of Hundreds, and therefore it fignifies Seven Hundred; and the whole Sum is to be read or pronounced thus, Seven Hundred Fifty Nine.

Note, Although the Figure 7 stands in the third Place (according to the Order of Numbering) yet when the whole Sum comes to be read, it is firft pronunced; the reading of Numbers being performed like that of Letters or Words, always beginning with the outmost Figure towards the Left-hand, and so many Figures as are placed together without any Point, Comma, Line, or other Note of Distinction between them, are all but one Sum, and must be read as such.

For Example, 763596 is but one entire Sum or Number, notwithstanding it confifts of fix Places of Figures, and is thus read; Seven Hundred Sixty Three Thousand, Five Hundred Ninety Six.

The like is to be observed in reading or expressing the true Value of any Sum or Rank of Numbers consisting of Seven, Eight, Nine, or more Places of Figures, each Figure being to be valued according to it's Distance from the Place of Unity: As in the foregoing 'Table.

Now fuch Values may as well arise by Cyphers, as by other Figures; for instance, 6 ftanding by itself, represents but Six Units: But if a Cypher be annext to it thus, 60, then it becomes Sixty; for the Cypher possessing the Place of Units, hath hereby removed the 6 into the Place of Tens; and another Cypher more would make it 600, Six Hundred, &c.

Whence it may be noted, that although a Cypher of itself signify nothing (as hath been said before) yet being placed on the Right-hand of any Figure, it augments the Value of that Figure by advancing it into a higher Place than otherwise it would have been, had not the Cypher been there,

Take one Example more in Numeration (if you please, that in the Table) viz. 678987654321, which is, according as is there fignified,

Six

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