Page:Chance, love, and logic - philosophical essays (IA chancelovelogicp00peir 0).pdf/254

 $$\mathrm{A} \left [ {1 + \frac{p}{q+1}} \right ] {x^p}{y^{q+1}}+ \mathrm{A}\frac{p}{q+ 1} \left [ {1+ \frac{p-1}{q+ 2}} \right ] x^{p-1}y^{q+2}$$ $$= \mathrm{A}\frac{p+q+1}{q+1}x^{p}y{q+1} + \mathrm{A}\frac{p+q+1}{q+1}\frac{p}{q+2}x^{p-1}y^{q+2}$$

It is, thus, seen that the succession of terms follows the rule. Thus if any integral power follows the rule, so also does the next higher power. But the first power obviously follows the rule. Hence, all powers do so.

Such reasoning holds good of any collection of objects capable of being ranged in a series which though it may be endless, can be numbered so that each member of it receives a definite integral number. For instance, all the whole numbers constitute such a numerable collection. Again, all numbers resulting from operating according to any definite rule with any finite number of whole numbers form such a collection. For they may be arranged in a series thus. Let F be the symbol of operation. First operate on 1, giving F(1). Then, operate on a second 1, giving F(1,1). Next, introduce 2, giving 3rd, F(2); 4th F(2,1); 5th, F(1,2); 6th, F(2,2). Next use a third variable giving 7th, F(1,1,1); 8th, F(2,1,1); 9th, F(1,2,1); 10th, F(2,2,1); 11th, F(1,1,2); 12th, F(2,1,2); 13th, F(1,2,2); 14th, F(2,2,2). Next introduce 3, and so on, alternately introducing new variables and new figures; and in this way it is plain that every arrangement of integral values of the variables will receive a numbered place in the series.

The class of endless but numerable collections (so called because they can be so ranged that to each one corresponds