Cocker's decimal arithmetick wherein is shewed the nature and use of decimal fractions ... together with tables of interest and rebate ... : whereunto is added, his Artificial arithmetick, shewing the genesis ... of the logarithmes ... : also, his Algebraical arithmetick, containing the doctrine of composing and resolving an equation, with all other rules requisite for the understanding of that mysterious art according to the method used by Mr. John Kerley in his incomparable treatise of algebra / composed by Edward Cocker ... ; perused, corrected, and published by John Hawkins ...

CHAP. XI. Reduction of Equations. (Book 11)

I. AN Equation is an equality between two quantities of different names, whether

the comparison of Equality be between Simple, or Compound Quantities, or both; between which two Quantities there is always this Cha∣ractor viz. =.

So 〈◊〉〈◊〉 •…•…his following Equation, viz. a=3c, a is said to be the first part, and 3c the second part of the Equation, and signifieth that some Number or Quantity represented by a is equal to three times another Number or Quantity repre∣sented by c.

So a=b+c signifieth that some Quantity re∣presented by a is equal to the sum of two other Numbers or quantities represented by b and c.

The manner of composing an Equation will be understood by solving of the several Questions contained in this and other following Chap. But when known, are mingled with unknown Quan∣tities, in an equation they must be so separated or reduced that the unknown Quantity or Quan∣tities may remain intire on the one side, or part, and the known or given Quantities on the other side or part of the Equation, which to performe is the work of Reduction, and which is contained in the several following Rules of this Chap.

Here note that the Quantity unknown or sought in every Equation is represented by the Letter a, or some other Vowel, and the Quan∣tity or quantities known or given are repre∣sented by Consonants, as b, c, d, f, &c.

Reduction by Addition.

II. If equal numbers or quantities be added to equal numbers or quantities, the sums or totals will be equal, and therefore

If it be granted that	〈 math 〉〈 math 〉
Then by adding +8 to each part of the Equation there ariseth	〈 math 〉〈 math 〉
Then because in the first part of the Equation there is +8 and −8, they destroy each other by the Third Rule of the Second Chap. and it followeth that	〈 math 〉〈 math 〉
Again let this Equation be proposed to be reduced, viz.	〈 math 〉〈 math 〉
Then by adding b to each part of the Equation, there ariseth	〈 math 〉〈 math 〉
And because −b and +b are in the first part of the equa∣tion, they destroy each o∣ther, and the Equation is	〈 math 〉〈 math 〉
Likewise if	〈 math 〉〈 math 〉
Then by adding b+c to each part of the equation there ariseth	〈 math 〉〈 math 〉

Now from a due consideration of the pre∣mises it followeth, that if in an Equation there be any Number or Quantity proposed with the sign − before it, then if it be transferred to the other side of the equation, and cancelled on the side or part where it now standeth, the effect will be the same as the adding of that Quan∣tity to each part of the Equation, and

this by Artists is called Transposition. As in the first of the foregoing Examples, where it is gran∣ted

That	〈 math 〉〈 math 〉
And by transposing −8 on the other side of the Equati∣on, making it there +8 it giveth	〈 math 〉〈 math 〉
And in the second Example where	〈 math 〉〈 math 〉
By transposing −b, cancelling it on the first side of the equation, and making it +b on the other, it is	〈 math 〉〈 math 〉
And let it be granted that	〈 math 〉〈 math 〉
Then by transposing of −bb and −d there ariseth	〈 math 〉〈 math 〉

Reduction by Subtraction.

III. If in any Equation there be any number or quantity signed with + (on which side of the equation soever) if it be cancelled on that side, and placed on the other side with the sign − pre∣fixed to it, the work of Reduction is truly per∣formed, and this is also called Transposition, and is only the converse of the foregoing Rule. Ex∣amples.

Let it be granted that	〈 math 〉〈 math 〉
Then if +8 be cancelled, and placed on the other part of the equation with the sign − it will give	〈 math 〉〈 math 〉
Which equation being contracted is	〈 math 〉〈 math 〉

Again let be given	〈 math 〉〈 math 〉
By the Transposition of +b on the first side the Equa∣tion it is	〈 math 〉〈 math 〉
And by Transposition of aa on the second side of the equation it is	〈 math 〉〈 math 〉
Also if	〈 math 〉〈 math 〉
By Transposition of b+c to the second side of the equa∣tion it is	〈 math 〉〈 math 〉
And by the Transposition of ba to the first side of the equation it is	〈 math 〉〈 math 〉

Which method (in reducing of the premised Equation) is deduced from this general Axiom, viz.

If from equal Numbers or Quantities, equal Numbers or Quantities are subtracted, the re∣mainder shall be equal.

So in the second Example there is given this equation, viz.	〈 math 〉〈 math 〉
First by subtracting b from each part of the Equation, there is	〈 math 〉〈 math 〉
Then I subtract aa from each part, and there remaineth	〈 math 〉〈 math 〉

Reduction by Multiplication.

IV. When in an Equation one or both parts are Fractions, then let them be reduced to a com∣mon denominator by the, 2d, 4th, and 5th Rules

of the sixth Chapter, and then casting away the Denominator, use only the Numerators, so shall Equations exprest by Algebraical Fractions be re∣duced to other Equations, consisting altogether of Integers. As in the following Examples.

If	〈 math 〉〈 math 〉
Then by reducing 9 in the se∣cond part of the equation to a Fraction, having 8 for its denominator, it is	〈 math 〉〈 math 〉
And by casting away the de∣nominator which is com∣mon to both it is	〈 math 〉〈 math 〉
Again if	〈 math 〉〈 math 〉
Then by reducing a, on the first side of the equation to to a Fraction haveing a+b for its denominator it is	〈 math 〉〈 math 〉
And by casting away the common denominator a+b the equation is	〈 math 〉〈 math 〉
Likewise if	〈 math 〉〈 math 〉
The quantities being reduced to a common denominator are	〈 math 〉〈 math 〉
And the common denomina∣tor cb being cast away the equation is	〈 math 〉〈 math 〉

V. When either part of an Equation is Com∣posed of a mixed Quantity or Quantities, let the Integral part, or parts be reduced to a Fraction or Fractions, and then proceed as in the last Example.

It is granted that	〈 math 〉〈 math 〉
First it is reduced to	〈 math 〉〈 math 〉
Which Fractional equation being reduced according to the foregoing Rule is	〈 math 〉〈 math 〉

VI. When some power or degree of the num∣ber or quantity sought, is multiplyed into each part, and each member of an Equation, then let that degree or power be cancelled in each part and member so will it quite vanish, and the Equation will be reduced to more Simple Terms. As for Example.

Let it be granted that	〈 math 〉〈 math 〉
Forasmuch as a is a Factor in each part, and member of the equation, therefore it being expunged in each, there ariseth this equation.	〈 math 〉〈 math 〉

VII. When (according to the second, third, fourth, and fifth Rules,) an Equation is reduced, and that some known Number or Quantity is multiplyed into the quantity sought; then divide each part of the Equation by that known Quan∣tity, to the end that the Quantity sought may

have no quantity multiplyed into it but 1 (or unity.) As in Example.

If it be granted that	〈 math 〉〈 math 〉
Then because the Quantity sought is (a) multiplyed by b, I divide each part of the equation by b, and there ari∣seth	〈 math 〉〈 math 〉

VIII. When any one part of an Equation is composed of a surd Quantity, (viz. such as hath the radical sign √ prefixed to it,) and the other part is a rational Quantity; then let that ratio∣nal quantity be raised to the power signified by the Radical sign, and then cast away the said Radical sign, so shall both parts of the Equation be a ra∣tional quantity. As,

If it be proposed that	〈 math 〉〈 math 〉
Square 8, and place its Square in the room of it self, cast∣ing away the radical sign from the first part of the equation, and then it will be	〈 math 〉〈 math 〉
Likewise if	〈 math 〉〈 math 〉
Then by raising the second part of the equation, to its Square, and casting away the radical sign from the first part there ariseth this equa∣tion, viz.	〈 math 〉〈 math 〉

Again, if	〈 math 〉〈 math 〉
The second part of the equa∣tion being squared, and the radical sign cancelled in the first, there ariseth	〈 math 〉〈 math 〉

Reduction by Division.

IX. If equal Quantities be divided by equal Quantities, the Quotients thence arising will be equal. For,

If	〈 math 〉〈 math 〉
Then by dividing each part of the equation by a, there ariseth this equation.	〈 math 〉〈 math 〉
And if	〈 math 〉〈 math 〉
Then by dividing each part of the equation by a there ari∣seth	〈 math 〉〈 math 〉
And da in the second part of the equation being transpo∣sed by the third Rule of this Chapter there ari seth this equation, viz.	〈 math 〉〈 math 〉
And if	〈 math 〉〈 math 〉
Then by dividing each part of the equation by b−c, it is	〈 math 〉〈 math 〉