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6 Equal to co
4.b.c.da As a is to b, ro is b to c, fo is c rod,
A'.. 6:: 6 d As a is to b, fo is c to d.
4.b.c.d. a, b, c, d, have equal Differences.

The Difference of a and b, is equal ta 4.

that of c and d. by cor bro 6 Is Greater than f. b hac or bac ! bIsLess than c.

The Difference between b and c, when it is not known which of them is the Greater. b Is to be involved, or raised to lome Power.

b Is to be Evolved, or some Root to be buna

Extracted out of it. Nb, (or jb, b, ,&c. Signify the Square Roi of ! the Cubę. Root of b, the Biquadrar-Roor of b, co respeto

bas G

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Besides the foregoing Signs, which are commonly us’d) I make use of others, whicb are not common, and are as follows. be

Some Quantity indefinitely Less than b; is to

be added to be bo

s Some Quantity indefinitely Less than b, iş

to be Subftracted from b. o

Either 8 ore, when not material which of

them it is. There are other Signs which are to be usd in the Geometrical Part of Algebra, and which I will explain in Book II.


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All Quantities concerned in any Question or Problem may stand in any Order at pleasure, viz. The most convenient for Operatión; as a tb-d, may ftand thus, b-dta, or thus, a dtb d+b, or thus, - d tatb, &c; these still being the samç;

, tho'differently plac'd.

Thar Quantity which hath no Sign before it (as generally the leading Quantity hath not) is always understood to bave the sign + before it, as a is -ta, or bad is + b-d, &c. For the Sign + is an affirmative Sign, and therefore all leading, or Positive Quantities are understood to have it, as well as they chac are to be added.

But the Sign - being a negative Sign, or Sign of Defect, there is a necefsity of prefixing it to that Quantity to which it belongs, whereever the Quantity stands.

When any Quantity is taken more than once, you must prefix its Number to it, as za stands for three times a, and 76 ftands for seven times b, c.

All Numbers thus prefixt to any Quantities, are calld Coefficients, or Fellow-Factors; because they Multiply the Quantity i and if any Quantity be without a Coefficient, it is always suppofed or understood to have an Unit prefixt to it; as a is i a, or bc, is ibè, &c.

All Quantities that are express'd in Numbers only, (as in Vulgar, Arithmetick) are called absolute Numbers.

Those Quantities that are represented by single Letters, as a, b, c, d, &c. Or by several Letters that are immediately join'd together, as ab, cd, or 7bd, &c. are call’d Simple, är single whole Quantities.

But when different Quantities represented by different or unlike Letters are connected together by the Signs + or -, as a



+b, or a-b, or a b to do, or a + aa. They are call'd Compound whole Quantities.

And when Quantities are express'd, or set down like Valgar Fractions, thus a +d abt de

&c. They are calb

C~P led Fractional or broken Quantities.

Like Quantities are those which are express'd by the same Letters under the same Power; as b and b, a 4 and a a, cd b and 6cdb,&c.

Unlike Quantities are such as are expressd by different Letçers, or by the same Letters under different Powers, as a and b, cdf and cd, b3 and bs &c.

Sect. 2. Of Tracing the Steps used in bringing

Quantities to an Æquation. The Method of tracing the Steps used in bringing the Quanricies concern'd in any Queftion to an Equation, is best perform'd by Regiftring the several Operations with Figures and Signs pla. ced in the Margent of the Work, according as the several Operations require ; being very useful in long and tedious Operations.

For Instance, if it be required to set down and Register the Sum of the two Quantities a and b, the Work will ftand Thus, o

First set down the propos d Quantities a and b b, over against the Figures 1 and 2, in the 1

small Column (which are called Steps) and +2lzla + b

against 3, (the third Step) set down the Sum, viz. a +b; then againft the third Step, ser down it 2, in the Margin ; which denotes that the Quantities against the first and second Steps are added together, and that thole in the third Step are their Şum.

To illuftrate this in Numbers, suppose a=9, and b=6, then it will be Thus, |!|a=





1 +21314-.6=9+6=15

Agzip, if i wes required to ser down the Difference of the same two Qu'ntities, Then it will be Thus, 112:39

26 - 6

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Or if it were required to ser down their product, then it will be Thus,

9 1:1

b6 xelilaxb or ad

9 x6 = 54

ಲೇ. Note, Letters Set, or joind immediately together, like a Word) Gignifie the Rectangle, or Product of those Quantities they Represent; as in the last Example, wherein a b = 54, is the Product of a 5%, and b = 6.

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d =

Į. If equal Quantities be added to equal Quantities, the
Suins of those Quantities will be equal,

As if a be=b, and o=d, then a te will be =b+d, or 4+d=b+c, by the first Axion.

2. If egual Quantities be taken from equal Quantities, the Quantities remaining will be equal; fo if a be=b, and cd, then f, will be

d, or a cb

c, by the second Axiom.

3. If equal Quantities be Multiplied by equal Quantities, the Products will be equal. So if a be=b, and c=d, then 4 e will be bd, or a d=bc, by the third Axiom.

4. If equal Quantities be Divided by equal Quantities, their Quotients

will be equal. Example, if a be=b, and o =d, then (or a + c) will be b

b 6 5. Those Quantities that are equal to one and the same thing, are equal to one another. As for Instance, if a be = g, and b =9, then a will be =b, by the fifth Axiom.

a cor bu dz) or a = by the fourth Axiomi

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Of whole Quantities.

CH A P. 11.

Addition of whole quantities. ADdition in Algebra may be eafily Learn'd

, by observing the Rule 1. When Simple and like Quantities having like Sigas are to be added,

Add the Coefficients, or prefixt-Numbers together, and to their Sum adjoin the Letters common to each, or in either of the Said Quantities. Lastly, to this Sum prefix the common Siga, and you'll have the Sum required.

N.B. Ex. fignifies Example.

Ex. 3. Ex. 4. b

2 ob cald b

3.4bca- 29 bc 1 +213126


Ex. 1.

Ex. 2.

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Ex. S.

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Ex. 6. 654abcd

- 9233 382 abcd

- 2427 + 2.1 abcd

4 +134 9 abcd - 5829 iti +3 +4151 240 6 abcdl

240 6 abedli8479 The Reason of these Additions is evident from the Work of Common Arithmetick; for suppose b to represent i Crown, to which if I add i Crown, the Sum wilt be Crowns, or ab, as in Ex. I.

Or if we suppose - a to represent the Want, or Debt of one Crown, to which if another Want, or Debt ef i Crown be added, the Sum must needs be the Want or Debt of z Crowns, as in Ex. 2. And so for all the reft.

Rule 2. When Simple and like Omantities having unlike Signs are co be added,

Add all the Affirmative ones into one Sum, and all the Negative ones into another, (by the firft Rule) then prefix the Difference of



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