*LEONHARD EULER, Swiss Mathematician in his sixties and also totally blind by this time, is dictating a portion of his Elements of Algebra to a very inexperienced SCRIBE in St Petersburg about 1770.*^{19}

EULER^{20} [*Throughout, where indicated by [+] or [-], Euler gestures in the air to show the confused scribe how to write plus as "+" and minus as "-"* ]

Section thirty one. Hitherto we have considered only positive numbers; and there can be no doubt, but that the products which we have seen arise are positive also: viz. plus [+]*a* by plus [+]*b* must necessarily give plus [+]*ab*. But we must separately examine what the multiplication of plus [+]*a* by minus [-]*b* and of minus [-]*a* by minus [-]*b* will produce.

Section thirty two. Let us begin by multiplying minus [-]*a* by 3 - or *plus* [+]3. Now, since minus [-]*a* may be considered as a debt, it is evident that if we take that debt three times, it must thus become three times greater, and consequently the required product will be negative; and it may be laid down as a rule, that whereas, firstly, plus [+] by plus [+] makes plus [+] or "plus" -

[*Scribe shows consternation; Euler irascibly and with some difficulty communicates that he wants "+ by + makes + , or plus"; this interchange should be ad-libbed* ]

Where was I after all that? Hmm... it may be laid down as a rule that firstly, plus by plus makes plus, or "*plus* ", - and that, on the contrary, plus [+] by minus [-], or minus [-] by plus [+] gives minus [-], or "*minus* "...

[*Euler makes signs furiously in the air again, to the confusion of Scribe* ]

Section thirty three. It remains to resolve the case in which minus [-] is multiplied by minus [-]; or, for example, minus [-]*a* by minus [-]*b*. It is evident, at first sight, with regard to the letters, that the product will be *ab*; but it is doubtful whether the sign plus [+], or the sign minus [-], is to be placed before it; all we know is, that it must be one or the other of these signs. Now, I say that it cannot be the sign minus [-]: for minus [-]*a* by plus [+]*b* gives minus [-]*ab*, and minus [-]*a* by minus [-]*b* *cannot* produce the same result as minus [-]*a* by plus [+]*b*; but it must produce a contrary result, that is to say, plus [+]*ab*; and consequently, we have the following rule: minus [-] multiplied by minus [-] produces plus [+], that is, the same as plus [+] multiplied by plus [+]...

[*Euler nods off, and Scribe retires quietly, looking very relieved* ] [CURTAIN]