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. XII. To Convert Analogies into Equations, and Equations into Analogies. (Book 12)

I. THis is deduced from this universal The∣orem, viz. That if four quantities are proportionals, the product of the two means is equal to the product of the two Extreams, and if three numbers are proportio∣nals, the product of the two extreams is equal to the Square of the means.

1. Let there be proposed these four proportionals.	〈 math 〉〈 math 〉
Then by the said Theorem this equation will follow, viz.	〈 math 〉〈 math 〉
2 Let there be proposed these 3 continual proportionals, viz.	〈 math 〉〈 math 〉
That is to say	〈 math 〉〈 math 〉
whence there followeth this equation, viz.	〈 math 〉〈 math 〉

II. From a due consideration of the premises it is evident that Equations may oftentimes be re∣solved into proportionals, viz. when the Pro∣duct of two Quantities is found equal to the pro∣duct

of two other quantities: Then as any one of the Factors in the first side of the equation is to any one of the Factors in the second part of the Equation, so is the remaining Factor of the second part, to the remaining Factor in the first part: And the Converse,

Suppose that	〈 math 〉〈 math 〉
From thence may be drawn this Analogie.	〈 math 〉〈 math 〉

The truth of which may be proved by the first Rule of this Chapter, for thereby the said Ana∣logy may be reduced to the given equation, viz. bc=ad.

Again if	〈 math 〉〈 math 〉
Then from thence may be de∣duced this Analogy, viz.	〈 math 〉〈 math 〉
or	〈 math 〉〈 math 〉
or	〈 math 〉〈 math 〉
Likewise if	〈 math 〉〈 math 〉
Then may that equation be resolved into these propor∣tionals.	〈 math 〉〈 math 〉
And if	〈 math 〉〈 math 〉
Then it will be found that	〈 math 〉〈 math 〉

III. When it happens that there is an Equati∣on between an Algebraical Fraction, and an In∣teger, if the Numerator of the said Fraction can be resolved into two such Quantities, as be∣ing multiplyed the one by the other, will pro∣duce the said Numerator, then will the said equa∣tion

produce this proportion, viz.

As the Denominator of the Fraction is to one of the Factors of which the Numerator is produ∣ced, so is the other Factor to the Integer, unto which the said Fraction is equal. Examples.

If it be granted that	〈 math 〉〈 math 〉
Then may that equation be resolved into this Analogy.	〈 math 〉〈 math 〉
For,	〈 math 〉〈 math 〉
Again if	〈 math 〉〈 math 〉
Then may that equation be resolved into this Analogy.	〈 math 〉〈 math 〉
And also if.	〈 math 〉〈 math 〉
Then may the said equation be resolved into this Analo∣gy, viz.	〈 math 〉〈 math 〉

The Practice of the two last Rules will be plainly discovered in the next Chapter (in the resolution of Questions producing simple equati∣ons) to be of most excellent use in discovering or laying down of Theorems for the ready so∣lution of the Question proposed, or any other of the same nature, which Theorems are to be reserved in store for the finding out of new, and the confirmation of old Truths.