Euler's Investigations on the Roots of Equations - Euler's Notation

The reader who is new to Euler might be surprised to discover how similar Euler's notation is to our own, and how modern his view of mathematics is.

Euler does use the terms "simple factor" and "double factor" where we might say "linear factor" and "quadratic factor", but after one reads it Euler's way, one might wonder why we do it differently. Euler also tends to write xx instead of x², and one gets used to that quickly enough. This was probably done to make the typesetting easier, and was common at the time.

In this translation, I have sometimes written i where Euler has written \( \sqrt{-1} \). We must note, however, that this substitution is not completely unproblematical, and should not be done mechanically. Euler knew very well, and articulates in several places in this article, that \( \sqrt{-1} \) has a dual nature, that every quantity has two square roots, -1 included. So when Euler is referring to that dual nature, it can be odd, or even misleading, to replace \( \sqrt{-1} \) with i. At other times, Euler's use of \( \sqrt{-1} \) is close enough, although not identical to, our notion of i that it seems justified to use i, in order not to distract the contemporary reader. But the reader should note that, in this article, Euler never wrote i, but always wrote \( \sqrt{-1} \).

Editor's note: This article was published in October of 2005.