6 '4

Resolution of Arithmetical Omftions

Book I*

J.

A..

' /«a

Qyitifr: 2

gJJEST. l°

There are two numbers whose Summ is r6, (or £,) and their difference, (to witjexcess of the greater above the lesser) is 8, (or c ; ) What are the Numbers ?

RESQLVTION: Numeral ,

%6-

24-

-26

24 — 26 = 8

4 = 17

p, that is,

Literal.

4

b — a

24-—Æ

4- k — s

gVEST. 2.

There are two numbers whose Sumro is 40, (or £,) and the greater number hath ^proportion to the lesser as 3 to 2, or, as r to §;) What are the Numbers ?

1. For the greater number sought put . .

2. Then to find the lesser number, savbv theRule of Three,

If 3 . 2 :: 4 ; ™

3. >

Or, r . s n a .

whence the lesser number is , ; :

J

*4

Z

£4

r

=iH-*

r. For the greater number put . . . .

2. Then subtracting that number a from the'given Sumrn, the Remainder will be the lesser^

number, to wit,.*

z. And by subtracting the lesser number from.the greater, the Remainder will be their)difference, to wit,.V

4. Which difference found out in the last step"must be equal to the given difference 8, (or c )!whence this Equation ariseth, .....

5. From which Equation, after it is duly re-'

duced according to Sett. 3. and 5. of Chap. 12.the greater number sought will be discovered,^to wit, ..

6. And consequently from the fifth and second,

steps the lesser number is also discovered,'to wit,.

So the numbers sought are found 17 and tZ, whose summ is 2 6, and their difference is ^as was prescribed.

Moreover, If the two last steps of the literal Resolution be exprest by words, they ^give this

THE O RE M.

Half the difference of any two numbers added to half their Summ, gives the gre Jtrtnumber: But half the difference of any two numbers subtracted from half theirleaves the lesser number.

Therefore the Summ and difference of any two numbers being given severally, the nv^bers themselves are also given by the said Theorem - but it presupposeth that the nutf*®*given for the Difference must not exce ed the number given for the Summ.

Note here once for all, That the numbers given in a Question cannot alwayes be chof*"at pleasure, but sometimes they must be subject to one or more Determinations, v»h ,c ”for the most part (though not alwayes ) are discoverable by the Theorem or Canonresulteth from the Resolution. But how limits or Determinations are discovered, I ^ 1have occasion to shew hereafter in my second, third, and fourth Books of AlgebrElements,