On Algebra (-1699)

"Having gotten, with God’s help, to the very desired place, i.e. the mother of all cases called by the people 'the rule of the thing' or the 'Greater Art', i.e. speculative practice; otherwise called Algebra and Almucabala in the Arab language or Chaldean according to some, which in our [language] amounts to saying “restaurationis et oppositionis,” Algebra id est Restau­ ratio. Almucabala id est Oppositio vel contemptio et Solutio, because by this path one solves infinite questions. And one picks out those which cannot yet be solved." (Luca Pacioli, "Summa de arithmetica geometria proportioni et proportionalita", 1494) 

"There remains to be added to the top, as a crown, that type of reasoning which is called popularly by the Arabic name of Algebra [qui vulgo et àrabica voce dicitur Algebra]. I prefer to call it quadratura. In fact this is a rare and subtle practice which the Logista takes from the Geometer as a help." (Jean Borrel, "Logistica", 1559)

"The name of algebra is thought to be Syriac, signifying the ‘art and doctrine of an excellent man’. Now Geber in Syriac signifies ‘man’; it is often a title of honor, as ‘master’ or ‘doctor’ with us. For, there is said to have been some unknown mathematician who sent his algebra, written in the Syriac tongue, to Alexander the Great, and he named it almucabala, that is the 'Book of Occult Things'. Others preferred to call his doctrine algebra. This book is still today very precious among the erudite nations of the East, and it is called by the Indians, who are very studied in these arts, aliabra or alboret, since they are ignorant of the origin of the proper name. Algebra has been called by some Latin A rei et census, as in Regiomontanus. By the Italians it is called ars de la cosa, by others cossa. Many schools today neglect to note how many names, or perhaps even more, algebra has had, in what high regard learned men of all nations have held it and what the loss of the doctrine would mean." (Peter Ramus, "Arithmeticae libri duo et totidem Algebrae", 1560)

"Indeed, many geometric things can be discovered or elucidated by algebraic principles, and yet it does not follow that algebra is geometrical, or even that it is based on geometric principles (as some would seem to think). This close affinity of arithmetic and geometry comes about, rather, because geometry is, as it were, subordinate to arithmetic, and applies universal principles of arithmetic to its special objects." (John Wallis, "Mathesis Universalis", 1657)

"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude." (Gottfried Leibniz, [letter to Christiaan Huygens] 1670)

"After all the progress I have made in these matters, I am still not happy with Algebra, because it provides neither the shortest ways nor the most beautiful constructions of Geometry. This is why when it comes to that, I think that we need another analysis which is properly geometric or linear, which expresses to us directly situm, in the same way as algebra expresses magnitudinem. And I think that I have the tools for that, and that we might represent figures and even engines and motion in character, in the same way as algebra represents numbers in magnitude." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"I found the elements of a new characteristic, completely different from Algebra and which will have great advantages for the exact and natural mental representation, although without figures, of everything that depends on the imagination. Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)

“They that are ignorant of Algebra cannot imagine the wonders in this kind are to be done by it: and what further improvements and helps advantageous to other parts of knowledge the sagacious mind of man may yet find out, it is not easy to determine. This at least I believe, that the ideas of quantity are not those alone that are capable of demonstration and knowledge; and that other, and perhaps more useful, parts of contemplation, would afford us certainty, if vices, passions, and domineering interest did not oppose and menace such endeavors.” (John Locke, “An Essay Concerning Human Understanding”, 1689)

"[…] it is algebraic notation that incarnates, so to speak, the ideal of the characteristic and which is to serve as a model. It is also the example of algebra that Leibniz cites consistently to show how a system of properly chosen symbols is useful and indeed indispensible for deductive thought." (Louis Couturat, [letter to L'Hospital] 1693)

"The Excellence of Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all possible Cases in one view, and in one general Theorem many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases." (Edmund Halley, "An Instance of the Excellence of Modern Algebra in the resolution of the problem of finding the foci of optic glasses universally", Philosophical Transactions, 1694)