"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."