Negative Numbers: Minus Times Minus

“The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.” (Brahmagupta, “Brahmasphuṭasiddhanta”, cca. 628)
 
"The square of a positive, as also of a negative number, is positive; that the square root of a positive number is twofold, positive and negative. There is no square root of a negative number, for it is not a square.” (Bhaskara, “Lilavati”, 1150)

“And therefore lies open the error commonly asserted that minus times minus produces plus, lest indeed it be more correct that minus times minus produces plus than plus times plus would produce minus” (Cardano, “De Aliza Regulae”, 1570)

 “I see no other answer to this [concerning the proportion argument] than to say that the multiplication of minus by minus is carried out by means of subtraction, whereas all the others are carried out by addition: it is not strange that the notion of ordinary multiplications does not conform to this sort of multiplication, which is of a different kind from the others.” (Antoine Arnauld, “Nouveaux Elémens de Géométrie”, 1683)

“It is not necessary to search for any mystery here: it is not that minus is able to produce a plus as the rule appears to say, but that it is natural that, when too much has been taken away, one puts back the too much that has been taken away.” (Bernard Lamy, 1692)

„Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary. Hence they talk of two roots to every equation of the second order, and the learner is to try which will succeed in a given equation: they talk of solving an equation which requires two impossible roots to make it solvable: they can find out some impossible numbers, which, being multiplied together, produce unity. This is all jargon, at which common sense recoils; but, from its having been once adopted, like many other figments, it finds the most strenuous supporters among those who love to take things upon trust, and hate the labour of a serious thought.“ (William Frend, “The Principles of Algebra”, 1796)

“I thought that mathematics ruled out all hypocrisy, and, in my youthful ingenuousness, I believed that the same must be true of all sciences which, I was told, used it. Imagine how I felt when I realized that no one could explain to me why minus times minus yields plus. […] That this difficulty was not explained to me was bad enough (it leads to truth and so must, undoubtedly, be explainable). What was worse was that it was explained to me by means of reasons that were obviously unclear to those who employed them.” (Stendhal, ”The Life of Henry Brulard”, 1835)

”There are elements of freedom in mathematics. We can decide in favor of one thing or another. Reference to the permanence principle (or another principle) is not a logical argument. We are free to opt for one or another. But we are not free when it comes to the consequences. We achieve harmony if we opt for a certain one (that minus times minus is plus). By making this choice we make the same choice as others in the past and present.” (Ernst Schuberth, “Minus mal Minus”, Forum Pädagogik, Vol. 2, 1988)