As a follow up question with respect to : Who wrote down minus times minus is equal to plus? and to : Historically, how did people define multiplication for negative numbers?, it can be interesting to trace the first "modern" justification for the rule :
"minus multiplied by a minus makes a plus".
If it is quite intuitive that "if you have a debt of $3$ by $3$ people, then you have a debt of $9$" (we can justify it reducing multiplication to repeated sum) it is not so easy to imagine a debt of $3$ by "minus" $3$ people.
For a (negative) reference, see :
- Isaac Newton, Universal Arithmetick : or, A Treatise of Arithmetical Composition and Resolution. To which is Added, Dr.Halley's Method of Finding the Roots of Equations Arithmetically (latin manuscript : Arithmetica Universalis (edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge, in 1707, based on Newton's lecture notes).
See page 3 :
Quantities are either Affirmative, or greater than nothing; or Negative, or less than nothing. Thus in humane Affairs, Possessions or Stock may be calld affirmative Goods, and Debts negative ones. [...] A negative Quantity is denoted by the Sign $-$ ; the Sign $+$ is prefix'd to an affirmative one [...].
In an Aggregate of Quantities the Note $+$ signifies, that the Quantity it is prefix'd to, is to be added, and the Note $-$, that it is to be subtracted. And we usually express these Notes by the Words Plus (or more) and Minus (or less). Thus $2+3$, or $2$ more $3$, denotes the Sum of the Numbers $2$ and $3$, that is $5$. And $5-3$ or $5$ less $3$, denotes the Difference which arises by subducting $3$ from $5$, that is $2$. [Note the clear distinction of the two usages of the signs $+$ and $-$.]
Then see page 16 :
Simple Algebraick Terms are multiply'd by multiplying the Numbers into the Numbers, and the Species into the Species, and by making the Product Affirmative, if both the Factors are Affirmative, or both Negative : and Negative if otherwise. Thus $2a$ into $3b$, or $- 2a$ into $- 3b$ make $6ab$, or $6ba$; for it is no Matter in what Order they are plac'd. Thus also $2a$ by $- 3b$, or $- 2a$ by $3b$ make $- 6ab$.
Useful "post-Newtonian" references :
- William Jacob s'Gravesande, The Elements of Universal Mathematics or Algebra (1728 - Latin original : 1727), page 9 :
if the Signs of Multiplicand and Multiplicator are familiar (or the same) the Product will be Affirmative, but Negative if they are different.
The explanation of the minus times minus case is in terms of "symmetry", followed by an intuitive example :
There is taken away in this Case a negative Quantity, by which the Negation vanishes. So to take away a Debt is to pay it [emphasis added].
It is useful to compare with the fully "algebraic" explanation by one of the most brilliant Newton's followers:
By the definitions, $+a-a=0$; therefore, if we multiply $+a-a$ by $n$, the product must vanish or be $0$ because the factor $a-a$ is $0$. [...] Therefore $-a$ multiplied by $+n$ gives $-na$.
In like manner, if we multiply $+a-a$ by $-n$, the first term of the product being $-na$, the latter term of the product must be $+na$, because the two together must destroy each other, or they amount be $0$, since one of the factors (viz.$a-a$) is $0$. Therefore $-a$ multiplied by $-n$ must give $+na$.