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are joined to any, and to their Sum, or Difference, adjoin the irrational or Surd.

Examples in addition. jil 5V bol 6b Vacl bv aa tcc |

217bol 4 braci vaatio it21 3112 V bc 10b Vac14b vaatio

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Case 2. When the Surd Quantities are Heterogeneal, (viz. their Indices are unlike) they are only to be added, or subtracted by their Signs, viz. t or And from thence will arise Surds cither Binomial, or Residual.

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Sect. 2. Multiplication of Surd Quantities. Celes. W HEN the Quantasies are pure Surds of the same Kind;

W multiply them together, and to their Product prefix their radical Sign.

EXAMPLE S. 111Vb 1vbatda vaa+bb 2 Valvca

vaa-bb 1x231v balv bia atacaalvaana-bbbb

Cafe 2. If Surd Quantities of the fame Kind (as before) are joined to rational Quantities, then múltiply the rational into the rational; and the Surd into the Surd, and join their Products together.

EX A M P L E S. Til dy bc 15cdvbatda 115 vob 12/3bra Izanca

| 5vd Ix2313db v bcal 15 cd av bcaa todca a 75 v abd

Sect. 6. Division of Surd Quantilies, Case 1. W HEN the Quantities are pure Surds of the fame Kind,

W and can be divided off, (viz. without leaving a Remainder) divide them, and to their Quotient prefix their radical

EX A MPLES.
11V bald bcaa tdca avaa aa -bbbb
21 V b. Ivca i

vanbb 1*2131valvbatda vaatio

Sign.

Case 2. If Surd Quantities, of the same Kind, are joined to rational Quantities; then divide the rational by the rational, if it can be, and to their Quotient join the Quotient of the Surd divided by the Surd with it's first radical Sign.

EX A M P L E S.
il 3db v bca 15 cdavocaat dca a 25 vabd
1213bwa zanca

5vd. !-2131 d V bc 50dv bat da 115 val.

Notes

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Sect. 4. Jnvolution of Surd Quantities. Case 1. W HEN the Surds are nut joined to rational Quantities;

w they are involved to the same Height as their Index denotes, by only taking away their radical Sign,

E X AMPLES : Tilvalvocal V aa-bb15a-da I ?21 al bcal aa-bbl 50-da

Case 2. When the Surds are joined to rational Quantities ; involve the rational Quantities to the fame Height as the Index of the Surd denotes; then multiply those involved Quantities into the Surd Quantities, after their radical Sign is taken away, as before.

EX A M PLES.

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The Reason of only taking away the radical Sign, as in Cafe 1. is easily conceived, if you consider that any Root being involved into itself, produces a Square, &C. And from thence the Reason of those Operations performed by the second Case may be thus stated.

Suppose bv a=x. Then Ve= per Axiom 4. and both Sides of the Equation being equally involved, it will be a = Home Then multiplying both Sides of the Equation into bb, it becomes b ba=xx per Axiom 3. Which was to be proved.

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Le {llabybe} Example 1. Cafe 2. 110 3 v bora

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Divifion being the Converse to Multiplication, needs no other

Proof.

CH A P. V.

Concerning the Nature of Equations and bow to prepare

them for a sziučioni.

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When the Number of the Quantities fought exceed the Number of the given Equations, the Question is capable of innumerable Answers.

EX A MPLE. Suppose a Question were proposed thus; there are three such Numbers, that if the first be added to the fecond, their Sum will be 22. And if the second be added to the third, their Sum will be 46. What are those Numbers ?

Let the three Numbers be represented by three Letters, thus, call the first a, the second e, and the third y.

Then Sate=22 Zone
Then I =46 Saccording to the Question.

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