history of mathematics - National STEM Centre

products will be in arithmetical progression, as above;
but the first two products are 12 and 0; therefore the
third will be -12; therefore -4 multiplied into +3,
produces -12.
Case 3d. To prove that + 4 multiplied into -3
produces -12; multiply + 4 into +3, 0 and -3
successively, and the products will be in arithmetical
progression; but the first two are 12 and 0, therefore the
third will be -12 ; therefore +4 multiplied into -3,
produces -12.
Case 4th. Lastly, to demonstrate, that -4 multiplied
into -3 produces +12, multiply -4 into 3, 0 and -3
successively, and the products will be in arithmetical
progression; but the two first products are -12 and 0, by
the second case; therefore the third product will be +12;
therefore -4 multiplied into -3. produces +12.
10 'Two minuses make a plus'
Cas. 2nd + 4, 0, -4 Cas. 3d + 4, + 4, +4
+3, +3, +3
+3, 0, -3
+12, 0, -12. +12, 0, -12.
Cas. 4th - 4, - 4, - 4
3, 0, -3
-12, 0, +12.
These 4 cases may be also more briefly demonstrated
thus: +4 multiplied into +3 produces +12 ; therefore -4
into +3, or +4 into -3 ought to produce something
contrary to +12, that is, -12: but if-4 multiplied into +3
produces -12, then -4 multiplied into -3 ought to
produce something contrary to -12, that is, +12; so that
this last case, so very formidable to young beginners,
appears at last to amount to no more than a common
principle in Grammar, to wit, that two negatives make an
affirmative; which is undoubtedly true in Grammar, though
perhaps it may not always be observed in languages.
If possible, work on this activity in groups of three to six. One or two students
should designate themselves as representing Euler, another student Saunderson,
and another student Arnauld, as introduced above. Then work through the activities
below. If you are working on your own, alternative activities are suggested.
1 a Everyone should read through the extracts from Euler's and Saunderson's
books. Then, together, draw up a list of assumptions made by each author to enable
him to derive the rules relating to the multiplication of negative numbers.
b If you are representing either Euler or Saunderson, imagine yourself in his place
and prepare to defend his position against the objections raised by Arnauld. If you
are representing Arnauld, prepare to argue your case against either Euler or
Saunderson. Stage a debate in front of your fellow students and ask them to judge
who makes the most convincing case.
If you are working on your own, but within a group of other mathematics students,
prepare an oral presentation summarising Arnauld's views and either Euler's or
Saunderson's. Make the presentation to the other students. As part of your
preparation, make clear which 'side' you would be on and why.
Otherwise, write an essay summarising Arnauld's views and either Euler's or
Saunderson's. Give your own opinions on the validity of the respective arguments.
If you were working in groups on Activity 10.11, you may have reached the position
at the end of the debate that your fellow students were unable to choose which pair
made the most convincing argument. As Euler or Saunderson, you might have found
it difficult to convince Arnauld that his objections to negative numbers were
incorrect. Likewise, as Arnauld, you might have found it hard to establish that your
objections really do correspond to a flaw in Euler's or Saunderson's arguments.
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