0->

Diophantus\s Algebra explain A.

Book lll

7 7 4

But this latter limit for the chusing of e is useless, for if e be greater than 6 rivvjas appears by the twelfth step, it is evidently greater than i & c .

14. Lastly, from the eleventh, twelfth, third and first steps the following Canon s rl1which will find innumerable Answers to the Question proposed.

1 seA

CANON 1.

Take any number greater than yS 4 H~/+ : 4~ , (w. greater than 6 -,

11

and call the number taken e. Then

ie\b

shall be equal to the number a sought.

1 c. But if it were desired to find a number a, that might be less than 2 ~, and greats

nothing, and make aa~ j— 4^5 —2. to be a square number, then the same Posit' 01 ! 5 JProcess being made as bef ore, saving that —3 is to be used instead of cr~ from the 1°step to the twelfth inclusive, at length there would arise this following

CANON 2.

Take any number (e) greater than y/, but less than V: db -j-/-j- dd: -j -d '■ ^uny number between &c. and 6 r ~~, &c.) Then "-rj-g will give the ^

ber a sought.

An Example of the first Canon.'

For the number e take 8 which exceeds 6 T m , &c. as the first Canon doth ^ ^

Then ee ~zL gives z~i for the number a sought; for 'tis greater than 2 f (o s '2e'\~b _ _ x

and aa -s- 4 a, 2 makes a Square, to wit, -f $£ , whose side is , as was reqoiy ft

Note, That a-\-u might be feigned to be the side of the Square mentioned '' $second step, and thence limits would be discovered to chuse the number u , by whynumber a would consequently be made known j but 1 leave the search of theselimits as an exercise for the Learner. y

%VEST. 13.

To find out a number, call it a ; that (hall be greater than xmake 1 21 -j- 45^ —-■ 9an to be a square number.

RESOLVE 10 N.

but less than 4'

e

b =2

n=

1. First put Consonants to represent the numbers given in the Que- V f — i*

stion y as j • • • • •••••■■••**, ,• .* y ^ — 12 J

/ g = ^ ,

th- 9 j

2. Then the Question requires that ff~\~g a — 1 haa may make a square number >, rside must be so feigned that the value of a may be greater than b , but less thanwhich purpose the said side may be feigned to be f^ea, or f — ua\ (where

do represent numbers unknown : ) First then let the said side be feignedand let its .Square ff -J- zfea -J- eeaa be equated to ff-\~ga — haa above- me”so this following Equation ariseth, viz,.

ff -j- 2 sea -|- eea^ — ff-\- gn — haa.

3. Which Equation, after due Reduction*!*) find out? _ g — 2 fe

the value of a, gives . . . . . . $ a h-

4. And because the Question requires . . . . . J> a cr~ b

5. It follows from the third and fourth steps, that ^ iuNilli. y

6. And 'the

• ee

nd by multiplying each part in the fifth step , by?

: Denominator h -1- ee , it follows, that . -S g

h -j— ee

7. And by subtracting bh from each part in the sixth 7 r u , _

step,..5 g-7fe-bhc._

2 fe c“ bh -I- I stbee