Negative Numbers: Direction

"[H]ere negation is […] contrariety […] that is to say, in the contrary direction. As the west is contrary of east; and the south the converse of north. Thus, of two countries, east and west, if one be taken as positive, the other is relatively negative. So when motion to the east is assumed to be positive, if a planets motion be westward, then the number of degrees equivalent to the planets motion is negative.” (Bhāskara II, "Bijaganita", 12th century)

“Magnitudes have more or less reality as their being takes them further from zero, and they have less reality when their non-being takes them further from this same zero. It became customary to call positive or true every magnitude which adds to zero, and negative or false every magnitude which takes away from this same zero.” (Jean Prestet, 1675) 

“It is evident that zero, or nothing, is the term between the positive and negative magnitudes that separates them one from the other. The positives are magnitudes added to zero; the negatives are, as it were, below zero or nothing; or to put it a better way, zero or nothing lies between the positive and negative magnitudes; and it is as the term between the positive and negative magnitudes, where they both begin.” (Charles-René Reyneau, 1714)

“From this it follows that the idea of positive or negative is added to those magnitudes which are contrary in some way. […] All contrariness or opposition suffices for the idea of positive or negative. […] Thus every positive or negative magnitude does not have just its numerical being, by which it is a certain number, a certain quantity, but has in addition its specific being, by which it is a certain Thing opposite to another. I say opposite to another, because it is only by this opposition that it attains a specific being (Bernard le Bouyer de Fontenelle, “Éléments de la géométrie de l'Infini“, 1727)

“It should be remarked that negative quantities are magnitudes opposite to positive quantities. […] With this notion of positive and negative quantities, it follows that both are equally real and that, consequently, negatives are not the negation or absence of positives; but they are certain magnitudes opposite to those which are regarded as positive (Dominique-François Rivard, “Élémens de Mathématique”, 1744)

“When two quantities equal in respect of magnitude, but of those opposite kinds, are joined together, and conceived to take place in the same subject, they destroy each other’s effect, and their amount is nothing.” (Colin MacLaurin, “A Treatise of Algebra”, 1748)

“If two quantities are in such a relation to each other that the one decreases just as much as the other one increases, and vice versa, then they are called opposite quantities. […] Such opposite quantities, considered for themselves, are quantities of a different kind, or are to be regarded as having different denominations. However, they are always situated under a common principal concept, and can in so far be considered as quantities of the same kind.” ” (Wenceslaus J G Karste, 1768)

“With respect to magnitude, a negative quantity is not distinct from a positive one at all, but it is distinct with respect to the operation which is to be executed with this quantity.” (Moses Mendelssohn)

“[…] direction is not a subject for algebra except in so far as it can be changed by algebraic operations. But since these cannot change direction (at least, as commonly explained) except to its opposite, that is, from positive to negative, or vice versa, these are the only directions it should be possible to designate. […] It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic. “ (Casper Wessel, „On the Analytical Representation of Direction“, 1787)

"The words positive and negative are general terms, that indicate the different states a quantity can be in, and that in special cases will have interpretations such as capital and debt, east and west, right and left, up and down, ascending and descending, winning and losing, etc. In each particular case it is up to us to choose which of the two states we wish to call positive, and thereby denote with the + sign, but once this is determined, we must consistently call the other state negative, and indicate it by the sign −." (Sylvestre-François Lacroix, "Beginselen der Stelkunst", 1821)