(^h

9 * out of Binomials in Numbers.

'<? J ----— -;-;—

h T<,> an d i>b less than b and consequently the Numerator 2 be

-e-\- b: wherefore

s6z

' n °mi n;

ator

the bb

b bib is less than the

is less than

1.

Aft,

th

er the fame manner it may be proved that c -j- b

Jn 2S,

Nc

cut — ee

»s greater

but this excess also shall be less than x: which was to be shewn.

^ivgr, 0 " 10 apply-the preceding three Propositions to the Demonstration of the Rule before\vh 0 s’J et .‘t be required to extract the cubick Root out of the Binomial 100 -j- V'7^ 0 ? J, Rational part 100 is greater than the ot her part Here we may suppose

%5^ t0 be too , and 3 bl^/d-^dyjd ( or \bb\^d x yd ) to be ^7803 j so thatbd more ibb^dx y'd may design the given Binomial loo-j-y^oz-; and itsJ'hen ro °t b -j- \'d the Root sought, whose greater part may be b , and the lesser yd .*’ ^cording to the Rule

To extraB y(z) out °f • ♦ 100-J-y^ 803.

first

Sub ’/ rom the Square ps too, that is * from . . .

the Square os yy 803, that is, 7

3'i, e *^ e mainder is.2

Cubick root of that Remainder is

100007803•ip 713

( — bb — d. )

^itioJ- C h Root 13 is (by Prop. 1.) equal to the difference of the Squares of the parts of the

C , root sou § ht -

^tsf) y» 6hd outaRattohal number greater than the summ of the parts of the Cubick■y Ui tr * VV,t ^ 1 £ ^ ,s Caution » that the excess may not be above f^, me greater part of the given Binomial, that is, to . .

V78 ^ the nearest value in whole numbers of the other part 7

jj o °3 i that isj . ... . $

giv^ ye summ soews, that the value in whole numbers of.the £

z , Mt.

too

88 of 89188 and 1 89.

, n oinomial falls between -b

so J^nce the Cubick root of the given Binomial is greater than 5t, but less than 6 ;

the excess of 6 above the true Roof sought is less than f.ofy’^iy, having found out ( as above ) 1 3 the true difference of the Squares of the partsof Cubick root sought, and 6 a Rational number which exceeds not the true summil)ij esa me parts above £; We may by the help Of Prep 3, and 1. find out the parts severallybanner, vie.

>hi

r ^ # ; i i

Divide the said . ♦ - • • * * *. ^

By the said ... 3 i

And the Quotient is.. * r * * ' ol

Which added to the said Divisor 6 > makes the summ • • • * 6

r rt °f th ^ Umtn doth (by Prop. 3.) exceed the double of the greater (to wit, the Rational)bd " e Cubick Root fought, but the excess is less than 1 - therefore 7 j is less than the

'I^PPof j ^ Ut ^6 ' s g reat er than the fame : and consequently, because the said greater part«teat n. t0 a ^ at * ona ^ whole numbers the double thereof must necestarily be b‘, (to wit*)Jybich b • w bole number between and 8^, ) and therefore the said part it self is 4 :r Sq u , e,£) §, found out, it is easie to find the other part. For, ( by Prop. 1.) if from 16^ thg v 6 °* the said greaser part 4, there be subtracted 13, the Cubick root of the differencePa areS £ be parts of the given Binomial, there will remain 3 , the Square of thetk ^ the r ' *0 that the Cubick root sound out is 4 —y 3,, which will appear by the Proof' given c'? 6 ^oot sought; for 4 >y. yz being multiplied into it self eubically producetfiBinnrv,;„i ... i /_o_, a„j .u„ . yz is the Cubick root

of

‘Oq

H

.^Or

soomial ioo-py78o3. And for the fame reason 4v'7803.

Or wore briefly , the Proof ma] be made thus.

to Ube 4 the Rational part of the Rooc found out,^

^ u ar e induct 0 f t hrifcc that part multiplied into the ?

AiX(j m e , r d part found out, i>t£. the Product . . C

the fomm

64, that is, bbb

36, that is, 3 bd

1OQ, that is, bbb -j- ^kd.

Which