8

Resolution of Questions

Boo

given concerning the number d) is to find out a number /, such, that if it multiplysaid + 9, the Product may exceed a Multiple of 28 which is prefixt to d, by the saidto which end the seventh Equation is assumed, to wit, 2 8« + 15 — as. p

5. The eighth and ninth Equations are formed out of the seventh, in like m* nne

the second and third out of the first. di

6 . Because 1 which follows -J- in the ninth Equation , is an Aliquot part of r vV ^Jstands next after —■ in the eighth, the ninth is multiplied by 2 the Denominator of il* e 1

(according to the Rule in Case z. ^Hest. 3.) whence the tenth Equation is p r0 ^

part;

to wit, 5 6 = 54+ r.

7. The eleventh Equation, to wit, 9 9

99 is the summ of the eighth and. tenthsince the said eleventh is free from the signs + and —, a Regressive work nowto fwd out the whole numbers/, & and 4 3 in this manner, viz.- M

8. By dividing either part of the eleventh Equation, to wit, 99, by 9 which is r

to f in the seventh, there ariseth r 1 =/, as in the twelfth Equation. Jf,

9. Then multiplying the numbe^r /, to wit, 11, by 93 , that is, either fart of + ^Equation, and to the Product adding 153, that is, either part of the fifth Eqiuti+’jnltsumm makes 1176 , (as you fee in the thirteenth Equation,) which 1176 is a

of 28, to wit, that which is represented by 28*! in the fourth Equation j Therefore, ^

10. By dividing the said 1176 by 28, the Quotient 42 is the number d } a 5,nfourteenth Equation.

11. Then multiplying the number d, to wit, 42, by 121, that is, either part <+third Equation, and to the product adding 126, that is, either part of the second Eq+l'the summ makes 5208, as you seerin the fifteenth Equation, which 5208 is a M+of 93, to wit, that which is represented by 9; 6 in the first Equation - Therefore, j

12. By dividing either part of the fifteenth Equation, to wit, 5208, by 93, the Qir

5 6 is the number b sought. ■ . /

13. Then from the said 5:208 subtracting 5, to wit, + 5 in the first Equationydividing the Remainder 520; by 121 which is prefixt to a in the first Equation, thegives 43 for the number a sought, as in the seventeenth and last Equation. Th

if 43 be taken for a , and 56 for b, then 1214 + 5 — 93^, which is the Equalsposed in 6. and all the values of 4 and b in whole numbers that are capable o+jj,

stituting thac Equation are the Terms of these two following Arithmetical Prog r4 ‘™whose Construction hath been shewn before in the third step of Sett. j.

Values of 4; 43 , 136 , 229 , 322 , 415 , 508 , &c.

Values of b; 56 , 177 , 298,419 , 540 , 661 , &c.

$

14. After the numbers f and d in the foregoing Resolution of Jgueft. 6. are kaotvn, ( (

numbers e and c in the seventh and fourth Equations may easily be discovered- but ^is no need of their help in the finding out of the desired numbers a and b. * {3 (,

15. But me-thinks I hear the Reader make this Objection, viz. How doth it

thac from every three whole numbers given in such sort as before is declaredthere may infallibly be found out two whole numbers a and b to solve the said Propo* ,[by the Operation before explained in the four Cases before mentioned: For Anh v +,this Objection, I ssiall here /hew how far the Process need be continued at the (i,to find our an Equation having -\- 1 in its latter part; for when such Equation t’tis manifest by the Operation in the third Case explaind in Jguest. 4, and 5. that rvvo ^jjnumbers a and b will infallibly be discovered to satisfie the Proposition, and conse^r,^innumerable other pairs of whole numbers to produce the fame effect. First thenforegoing guest. 6. the given number 121 which is prefixt to 4, being divided by th enumber 93 which is prefixt to b , after the Division isfinilh'd there remains 28, '

-j- 28 in the latter part of the third Equation : Secondly, the said Divisor 93 being ^ Mby the said Remainder 28, after the Division is ended there remains 9, to wit, + + >,iilatter part of the sixth Equation Again, the last Divisor 28 being divided by cRemainder 9, aster this Division is ended there remains 1, that is, + 1 in the U ttet

of the ninth Equation, which Remainder 1 you will alwayes infallibly come unto by a , 3 fftmued Division in that manner, because the two given numbers prefixt to 4( asi the Proposition requires) Prime between themselves; and that continued D lVI +j; jnothing else but the Method of finding out the greatest common Divisor unto two nu^ (i