On Algebra (1750-1799)

"[negative numbers] darken the very whole doctrines of the equations and to make dark of the things which are in their nature excessively obvious and simple. It would have been desirable in consequence that the negative roots were never allowed in algebra or that they were discarded." (Francis Meseres, 1759)

"First, everything will be said to be a magnitude, which is capable of increase or diminution, or to which something may be added or subtracted […] mathematics is nothing more than the science of magnitudes, which finds methods by which they can be measured." (Leonhard Euler, "Algebra" , 1770)

"We think only through the medium of words. Languages are true analytical methods. Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method. The art of reasoning is nothing more than a language well arranged." (Abbé de Condillac, "System of Logic", cca. 1781)

“[…] 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 algebraic analysis soon makes us forget the main object [of our researches] by focusing our attention on abstract combinations and it is only at the end that we return to the original objective. But in abandoning oneself to the operations of analysis, one is led to the generality of this method and the inestimable advantage of transforming the reasoning by mechanical procedures to results often inaccessible by geometry. Such is the fecundity of the analysis that it suffices to translate into this universal language particular truths In order to see emerge from their very expression a multitude of new and unexpected truths. No other language has the capacity for the elegance that arises from a long sequence of expressions linked one to the other and all stemming from one fundamental idea. Therefore the geometers [mathematicians] of this century convinced of its superiority have applied themselves primarily to extending Its and pushing back its bounds." (Pierre-Simon Laplace, "Exposition du system du monde" ["Explanation on the solar system"], 1796)

"So long as algebra and geometry proceeded separately their progress was slow and their application limited, but when these two sciences were united, they mutually strengthened each other, and marched together at a rapid pace toward perfection." (Joseph-Louis de Lagrange, "Leçons élémentaires sur les mathématiques", 1795)