14 A 'Treatise of Algebra.

of different Classes that have different Letters or Quan-tities for their last Factors.

Demonstration of Lemma i.

Cafe r. If the Number of Factors , or Quantities to bemultiply’d together be only 2, as a and b j the Productsab and ba will be equal, by Euc. 1 6. 7.

Cafe 2. If the Number of Factors be z, as a, b and c.

It is plain that of all the different Products or Varia-tions, abc, bac, acb, cab, bca, cba , that can be formed inthis Cafe, those {abc, bac) of the fame Class must be equal,because produced of equal Quantities {ab, ba) multipjy’dby the fame Quantity : Moreover the Products {bac, cab)of any two different Classes will also appear to be equal jfor ba : ca :: b : c {Eu. 17. 7.) whence bay. c = ca x b(Eu. 19. 7.) that is bac —cab.

Cafe 3. If the Number of Factors be 4, as a, b, c and d.

All the Products {abed, aebd, chad, Sic.) of the fameClass will be equal, by the Precedent, being produced ofequal Quantities multiply’d by the fame Quantity : Andthose {abed, abdc) of different Classes are equal - becauseabc : abd :: c : d {Eu. 18. 7.) whence (by Eu. 19. 7) abed— abdc.

Universally. If all the Products, when the Numberof Factors is n, be equal, all the Products when the Num-ber of Factors is «+ 1 will be also equal: For those ofthe fame Class are manifestly equal, because produced ofequal Quantities multiply’d by the fame Quantity : And,to shew that those of different Classes are equal, we needonly take any two Products which differ in their two lastFactors, and have all the preceding ones according tothe fame Order, and prove them to be equal. These twoFactors we will suppose to be represented by r and s,and the Product of all the preceding ones by p\ then thetwo Products themselves will be represented by prs andpsr j which are equal by Case 2.

Demonstration of Lemma 2.

We are to prove that the Product of abed multiply’d byqrst, &c. is equal to the continual Product oiaxbxcxd