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
EULER20 [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]