# Descartes on imaginary unit.

I heard once that Descartes defining the imaginary unit had to talk about the imagining of rise of the spirit over the real numbers because definition based on square root of a negative number could end up for him with ostracism from the scientific community. I will be grateful for pointing me to this fragment in his works (actually I don't know French so exact English translation with reference will also work perfect for me :)).

I've browsed some secondary literature, without finding the reference you are searching for.

I'm quoting from Henk Bos, Redefining Geometrical Exactness : Descartes' Transformation of the Early Modern Concept of Construction (2001), page 235 :

In the notes that Beeckman entered in his journal on occasion of Descartes' visit in October 1628, there occurs a short passage from which it appears that by that time Descartes had come to use the term "imaginary" for numbers:

"Irrationales numeros, qui aliter explicari non possunt, explicat [sc. Descartes] per parabolam; nominat autem quasdam radices veras, quasdam implicatas, id est minores quam nihil, quasdam imaginarias, id est omnino inexplicabiles; ac videt ex tabula vulgari, quot aliqua aequatio radices habere possit quarum una sit quaesita." ([Descartes 1964-1974] vol. 10 pp. 335.)

[Transl. : By a parabola he [Descartes] explains irrational numbers which cannot be explained otherwise. On the other hand, he names some roots 'true', some 'implicit' (that is, less than nothing), and some 'imaginary' (that is, not explicable at all); and, moreover, from a common table he looks at how many roots a certain equation can have, one of which is required. ]

In the Geometry Descartes used the term "imagined" for the non-real roots of equations, stating that an $$n$$-th degree equation had $$n$$ real or imaginary roots.

See into The Geometry of René Descartes (Dover reprint) fo an occurrence of imaginary [imaginaires, page 174 and 175 for translation] regarding non-real roots of an equation.

Usually [see page 186], an imaginary root is called fausse.

• I'm currently waiting for response from my professor who said this anecdote about rising of the spirit - maybe it is just a gap in my memory and its author is not Descartes. However thank you very much for this beautiful answer! Commented Jun 26, 2014 at 9:14