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CANO N.

Take any two Unequal Numbers, Multiply feverally the double of the Product of their Multiplication, and the Difference of their Squares by the fide of the given Square, then divide thofe Products feverally by the Sum of the Squares of the two Numbers first taken, and the Quotients fhall be the fides of the two Squares fought.

Queft. 2. To divide a Number given which is compos'd of two known Squares; Suppofe dd the Greater, and bb= the Leffer, into two other Squares.

* For if sr:: d+b.db then the Squares that wou'd be found by the Canon,

wou'd be the fame with

them given.

Square fought, put

1. Take two unequal Numbers, the Greater, and r the Leffer, with this *Caution, viz. That s be not in fuch Proportion to ras d+b to db.

2. For the fide of the firft of the two Squares fought, put ra+b.

3. And for the fide of the second sa―d, or d − s a.

4. Then the firft Square fought, is

ra2+2rba + bb.

5. And the fecond Square fought is s2 a2 — 2 s dat d d. Confequently the Sum of thofe Squares isra's2a3 +2rba-2 sd a + b2 + ď2. da

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7. Which Sum must be

dd + b b.

Hence the following Equation arifeth, Viz.

32 +2 × a2 + 2⋅ b − 2 s d× a + b2 + d2 = b2 + d2, 8. Which Equation after due Reduction, gives

2sd-2rb

s3+rr

Therefore from the eighth and fecond Steps the fide of the firft Square fought is now known and found

2 rsd + ssb-rrb

sstor

10. And from the eighth and third Steps the fide of the feccnd Square fought is likewife known and found

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That is, the former of those two Quantities fhall be the fide of the fecond Square, when sd-rd is Greater than 2rs b. But the latter of thofe two. Quantities fhall be the Side when s'd-r'd is Lefs than 2 rsb: Whence the fide of the fecond Square may be exprefs'd thus,

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Take any two Unequal Numbers, with this Caution, that the Greater may not have the fame Proportion to the Leffer, as the Sum of the fides of the two Squares given hath to the Difference of the Jame Sides. Multiply the double Product of the Multiplication of thofe two Numbers first taken by each of the faid two Sides given, and referve the Products: Multiply alfo the Difference of the Squares of the faid two Numbers first taken, by each of the faid two Sides given, and referve thefe Products. Then add the greater of the two firft referv'd Products to the Leffer of the two Latter, and reServe the Sum for a Dividend. Take alfo the Difference between the Leffer of the two firft Products and the greater of the two Latter for a fecond Dividend. Lastly, Divide severally the faid Dividends by the Sums of the Squares of the two Numbers first taken: So fhall the Quotients be the fides of the two Squares fought.

Quest. 3. To find two Square Numbers whofe Difference fhall be equal to a given Number: Supposed.

Solution.

1. Ler fome Number whofe Square is less than the given Difference be represented by b.

2. For the fide of the Leffer Square fought put 4.

3. For the fide of the Greater Square fought put a +b. 4. Then the Leffer Square is a a.

5. And the Greater Square is a +2ab+bb.

6. And the Difference of thofe Squares is 2ab+b2. 7. But the faid Difference must be equal to the given Difference; therefore

2ab+bb = d

8. Which after due Reduction makes known the Value of

d

bb

the fide of the leffer Square, Viz. 4= =

2 b

9. And

9. And from the 8th and 3d Steps the Value of the fide of the Greater Square is also discovered, Viz. a+b=d+bb

CANON 1.

26

Take any Square Number lefs than the given Difference, and Subtract it from the faid Difference; then divide the Remainder by the double of the fide of the Square first taken, and the Quotient fhall be the fide of the leffer of the two Squares fought. Laftly, This fide added to the fide of the Square first taken, gives the fide of the other Square fought.

Again, Since by the two laft Steps of the preceding Refolution, the Values of the fides of the two Squares fought are,

d-bb d + b b
and

; Therefore, if we fuppofe bed;

26

26

then thofe Sides will be converted to these, viz.

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bc + b b
26

Which laft mention'd Sides after the common

Factor bis caft away, will be reduc'd to cb, and cb. Whence,

CANON 2.

Take two fuch Unequal Numbers that the Product of their Multiplication may be equal to the given Difference, then half the Sum, and half the Difference of those two Numbers shall be the fides of the two Squares fought.

Queft. 4. To find two fuch Square Numbers, that if to the Product of their Multiplication a given Number d be added, the Sum may be a Square.

Solution.

1.. For one of the Squares fought take any known Square Number, which may be reprefented by bb.

2. And for the other Square fought put a a.

3. Then the Product of their Multiplication is ba". 4. To which Product the given Number & being added, the Sum is, ba2 + d.

5. Which

5. Which Sum muft be equal to a Square, the fide where of may be fuppos'd to be ba- any known Number greater than d; fuppofe bac, then the Square of bac, that is, be a 2b ca+c being equated to be a+d, this Equation arifeth, viz.

b2 a2 +d= b2 a2-2 bea+cc.

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For one of the Squares fought take any Square Number, then from any Square Number fubtract the given Number, and divide the Remainder by the double of the Product made by the Multiplication of the fides of thofe two Squares; fo the Quotient fall be the fide of the other Square fought.

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320

PART XVIII.

Of the Alternations and Combinations of QUANTITIES.

CHA P. I.

Of the Alternations of Quantities.

DEFINITION.

Alternation, is a Word ufed by Mathematicians for the different Changes or Alternations of Order in any Number of Things propofed taken one by one, two by two, or three by three, &c.

LEMMA.

The Number of Alternations of m Things at bac, &c. taken n by n, is equal to the number of Alternations of the m-1 things ap

- I 1

1 b 9 cr &c.

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Demonftration.

It is evident, that by placing the Thing a in any deter mined Place; as, fuppose, in the first place of every Alternation which can be made of the m-1 things abc", &c.

9

taken

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