Volume 1

The elements of that mathematical art commonly called algebra, expounded in four books / By John Kersey.

  • John Kersey the elder
Date:
1673
Catalogue details

Licence: Public Domain Mark

Credit: The elements of that mathematical art commonly called algebra, expounded in four books / By John Kersey. Source: Wellcome Collection.

    286/350 (page 266)
    Book ll. %t6 Concerning the Kefolntion Which is yet too liftle j therefore 6. I fuppofe. Thence. . . . . . =3 40 . . aaa~\—z6a ~ 6 5'04O Which 65040 being greater than 401 88, it ftiews me that the true Root or value of 4 is Icfs than 40 j but by the fifth tryal its greater than 30, and confequently the fir ft figure af the Root is 3. Now the fecond Charafter of the Root muft neceffarily be one of thefe, viz,, o, i, 2, 3,' 4> 5>7» 9 5 becaufe it hath been difcovered that the true value of the Root a is greater than 30, the fecond Charafter cannot be o , I therefore make tryal with i, and (lippofe 4 = 31, which proving too little, I make tryal with 32,33,34, feverally, in like manner as before, and at length 1 find 34 to be the true number a fought, by which the Equation propos’d may be expounded • for if 4 = 34, then confequcntly ^44 264 = 40188, II. But if after tryals made ( as before ) the value of 4 the Root fought happens to fall between two whole numbers that differ by Unity; then tryals are to be made with the lefTcr whole number increafed with 7^, 7^, rJ-, &c. until you have found the value of 4 in forae raixt number confifting of a whole number and fome certain tenth parts of an Unit; But if the faid value of a happens not to be expreft exadly by the faid leffer whole number increafed with certain tenth parts, then you are to make tryals with the faid IcRer whole number increafed with a decimal Fraftion having for its Numerator a number greater than 10, but lefs than 100 ; and for its Denominator loo, as with and by proceeding in that manner you may find the exaft value of the Root a when its fractio¬ nal part is exactly equal to fome decimal Fraction, or elfe approach infinitely near to the faid exact value when ’tis irrational or furd, as in this following ’ ^EST, 2. • , 4444-}-504 = 184638.6801 ; (or, 1846387 i) what is -iLioi 10000 9 the number 4 ? RESQLVTION. Firfl:, 1 fuppofe ‘4=1, but this proving too little I put 4 = i o, thij^ alfo proving tooi little, I afl'ume az=ioo , which after tryal 1 find to be greater than the true number 4, and ’ confequently the number 4 falls between 10 and too • then making tryal with 20 1 find it too little, but making tryal with 3*0 I find this too great, and therefore the true Root a falls between 20 and 30; Again, making tryal with 21 I find it too great, but 20 was before found too little • therefore the true Roof 4 is between 20 and 21 j then 1 make tryal with 20.1, (that is, 207^,) 20.2 ; 20.3 , &c. and at length find 20.7 to be the true number 4 fought . for if 4 = 20.7 ( that is, 207^,) it will make 4444-}-* 504 = 18463 8.6 801 the Equation propofed. But if 2 0.7 had proved too little, and 20.8 too great, then tryals muft have been made with 20.71, (that is, 207^75) 20.72 ; 20.73 , '&c. In like manner if 20.7 had been too little, but 20.71, (that is, 20777) too great, then tryals muft have been made with ' 20.701, (that is, 207^17;) 20.702; 20.703, &'£, This will be partly exercis'd in refolving the Equation in this following ^'>EST. 3. If r ^ . ‘444-}^ 2044 =: 1954 j vvhat is the number 4 ? Anfrfi, a = 8.308, &c. found out by tryals, as before. ! 1I I. When the value of ( 4 ) the required Root of an Equation happens to be left than Unity, then tryal is to be made with 77 j but if this prove too gi;eat, then with 77^, &c» Now fuppofe .1 (that is, 7o) be too great, but .01 (that is, 770) too little; then tryal muft be made with .02 j .03 j .04 | &c. until you have found out the greateft figure that muft ftand in the fecond place of the decimal Fraction exprefting the Root fought; fuppofing then fuch figure to be found 8, viz,, that .08 (or 77®) is lefs, but .09 (or 77^) is greater tha n the Root, try al muft be made with .081, (chat is, 7717,) .082 j .083 j &c, as in this following - ^VESr, 4* i 444-]-32404 = what is the number 4 H a =z ,083, &c. that is,' ^