Thus you have a fhort and general Account of the proper Subjects of those three noble and useful Sciences, Arithmetick, Geometry, and Álgebra. I fhall 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.

Kithmetick, 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 fome Species or thing proposed.

The fecond is that which fuppofes an Unit (and confequently the Quantity or thing represented 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 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 feveral Operations performed in Decimals, differ but little from thofe in Whole Numbers: and therefore it is now become of general Ufe, 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 fome are called Digits.)

viz. {One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Cypher.

2 3 4 5 6 7 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, uide Dr Wallis's Algebra, Page 12.

B 2

The

4

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

Vix. {

That 1+1=2, I+I+I=3.
!+!+1+1+1=5. And fo on to 9.

Nine of thefe Figures were thus compofed of Units, and differently formed to reprefent fo many Units put together into one Sum, as was intended each fhould denote: Nine being the greatest Number of Units that was then thought convenient to be expreffed by one fingle Character; the laft of the Ten is only a Cypher, or (as fome phrafe it) a Nothing, because of it felf it fignifies nothing; for if never 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 feveral Varieties that happen in Computations; I do advife the young Learner to acquaint him felf with the Signification of the following Algebraick Signs or Characters, which he will find of excellent Ufe, as being a much shorter, better, and 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 unis Plus orderftood when feveral 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

÷ } { Bj.

= } { Equal.

: : } { Sa is.

The Sign of Subtraction; 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 in to 6, and fignifies that 9 is to be Multiplied into or with 6.

The Sign of Divifion; as 8-2, 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 z.

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 9-6-3, &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 28: 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.

CHAP. II.

Concerning the Principal Rules in Arithmetick, and how they are performed in Whole Numbers.

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. Numeration (or Potas

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

Sect. 1. Of Rumeration or Notation.

Kumeration or Notation, teacheth to Read or Express the true Value of any Number when writ down; and confequently to write down any propofed 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.

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 fignify it's own fimple Value, viz. fo many Units as that Figure represents.

The fecond Place is that of Tens, and any Figure standing in that Place fignifieth fo many Tens as that Figure reprefents Units.

The

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

For Inftance, fuppofe 759 were propofed to be read or pronounced 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 fignifies but it's own fimple Value, to wit, 9 Units, or 9. The fecond Figure 5 ftands in the Place of Tens, and therefore fignifies Five Tens or Fifty. The Figure 7 ftands 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 ftands 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 outmoft Figure towards the Left-hand, and fo many Figures as are placed together without any Point, Comma, Line, or other Note of Diftinction between them, are all but one Sum, and must be read as fuch.

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 obferved in reading or expreffing the true Value of any Sum or Rank of Numbers confifting 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 arife by Cyphers, as by other Figures; for inftance, 6 ftanding by itfelf, reprefents but Six Units: But if a Cypher be annext to it thus, 60, then it becomes Sixty; for the Cypher poffeffing 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 fignify 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,