The Arabs contributed nothing new to the theory, but al-Khowârizmî (c. — Negative number

"Abundant quantities are those which are greater than nothing (majores nihilo,) and carry the idea of increase along with them. These have either no symbol prefixed, or this one +, which is the copulative (copula) of increase. Thus, if you are not in debt, and your wealth be estimated at 100 crowns, these may either be noted 100 crowns, or + 100 crowns; and are read a hundred crowns of increase; always signifying wealth and gain. ...Defective quantities are those which are less than nothing (minores nihilo,) and carry the idea of diminution along with them. These are always preceded by this symbol which is the copulative of diminution. Thus, in the estimation of his wealth whose debts exceed his goods by 100 crowns, justly his funds are thus prenoted,—100 crowns, and are read, a hundred crowns of decrease; signifying always loss and defect. I have already shown that defective quantities have their origin in subtracting the greater from the less."

"The next work in which we... find arguments against integrating the negative numbers into the overall conception of real numbers was, in contrast to Fontenelles, a textbook explicitly addressing laymen. ...Clairaut wrote two elementary textbooks ...for an elegant public intent on dabbling in leisurely in mathematics without having to shoulder any real effort. The two textbooks on geometry (1741) and algebra (1746) attempt to realize a methodical approach... which... evolves in a seemingly "natural way" from simple inquiries or from useful problems. ...a particular stumbling block in his eyes was multiplication. ...he adapted part of Fontenelles arguments in favor of separating positive from negative quantities. ...guiding beginners gradually toward an understanding of the necessity of operating with negative numbers, and of the appropriate rules—in particular the rule of signs. ...He ...developed a method of interpreting negative solutions away ...liberating operating with negative quantities from everything "shocking," permitting the reader to recognize the nature of negative problem solutions. One should assume the unknown to be of opposite direction... Clairaut... admitted genuine negative values of unknowns, a fact relieving him from changing the equations... He even extended this conception by permitting negative values as well for the coefficients—in contrast to... quantities preceded by a minus sign, as was then current practice in France. ...He was the first to directly tackle, in a textbook of modern times, the question of how to interpret negative solutions in equations with concrete quantities."

"The "Appendix to the Principles of Algebra by , Esq. F.R.S. Cursitor Baron of his Majestys Court of ," is written as a supplement to Mr. [William] Frends treatise on that science, which we introduced to our readers in our Register for the year 1796. ...what will particularly engage the attention of the mathematical world, it contains an unequivocal and perfect approbation of Mr. Frends doctrine respecting negative numbers. The assent of a person of such eminence in algebraic science to the new opinion, shows, at least, that it has not been adopted without weighty and forcible reasons for its truth; and may, perhaps, encourage other mathematicians to throw off all dread of innovation, all implicit scientific faith, or habit of taking for granted that which has not been previously proved, and to question some other long received dogmas, which certainly wear the appearance of contradiction and absurdity. Is not this the case with the doctrines of infinity and imaginary quantities, as explained by algebraists, and that of the asymptotè in conic sections?"

"— 2 and 4. If negative multiplies with positive their creation is incomplete a kind of un-being negative numbers have empty hearts, must reach out to others alike else we are bound to fall into shadow quadrants that measure loss like childhood fears, doubts trapped in closets"

"While the mathematicians were still looking askance at the Greek gift of the irrational number, the Hindus of India were preparing another brain-teaser, the negative number, which they introduced about A.D. 700. The Hindus saw that when the usual, positive numbers were used to represent assets, it was helpful to have other number represent debts."

"In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers."