history of mathematics - National STEM Centre

Activity 10.10 Objections to negative numbers
Interestingly Euler and
Saunderson were both
blind, Euler only for the
last 17 years of his life,
but Saunderson from the
age of 1, when he
suffered from smallpox.
10 'Two minuses make a plus'
In the following extract, Arnauld raises strong objections to the use of negative
numbers.
... je ne comprens pas que le quarre de -5 puisee etre la meme chose que
le quarre de +5, et que I'un et I'autre soit +25, Je ne sgai de plus comment
ajuster cela au fondement de la multiplication, qui est que I'unite doit etre
a I'une des grandeurs que Ton multiplie, comme I'autre est au produit. Ce
qui est egalement vrai dans les entiers et dans les fractions. Car 1 est a 3,
comme 4 est a 12. Et 1 est a \ comme \ est a -fe . Mais je ne puis ajuster
cela aux multiplications de deux moins. Car dira-t-on que +1 est a -4,
comme -5 est a +20? Je ne le vois pas. Car +1 est plus que -4. Et au
contraire -5 est moins que +20. Au lieu que dans toutes les autres
proportions, si le premier terme est plus grand que le second, le troisieme
doit etre plus grand que le quatrieme.
1 a Translate the extract from Arnauld's letter into English. If you believe that
your French is not up to it, then ask the help of a friend - but not before having tried
to get an impression for yourself of what Arnauld claims. (You should only refer to
the hint as a last resort!)
b Which properties of numbers is Arnauld using in his argument?
c From the point of view of modern-day mathematics, what is the fallacy in his
argument? What part of your A-level course can you use to argue against Arnauld?
It took until the middle of the 18th century for further developments to take place;
then they accelerated. Leonard Euler (1707-1783) and Nicholas Saunderson (1682-
1739) were partly responsible for this. They produced some of the first algebra
textbooks, which dwelt at some length on the rules for multiplying negative
numbers. This emphasis was quite natural, because negative numbers and their rules
of combination had not been described in texts until that time. Both texts were
intended for teaching and were considered well designed for this purpose, and many
editions of each of them were published. It has been said that the textbooks were
easily understandable partly because of their authors' blindness; Euler, for example,
had to dictate his text and make it intelligible to a relatively unskilled person.
Activity 10.11 A debate about negative numbers
The following is an extract from Euler's Elements of Algebra, which first appeared
in 1774. Many editions were published centuries later. This one is from the 1828
edition.
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