On Trigonometry (1800-1849

"By use of the symbol √-1 and of the forms proved to obtain in the combination of real quantities, a mode of notation is obtained, by which we may express sines and cosines, relatively to their arc." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"I am convinced that the motion of the vibrating string is as exactly represented in all possible cases by trigonometric developments as by the integration which contains the arbitrary functions [...]" (Jean-Baptiste-Joseph Fourier, 1807)

"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept." (Hermann GGrassmann, "Ausdehnungslehre", 1844)

"Trigonometry contains the science of continually undulating magnitude: meaning magnitude which becomes alternately greater and less, without any termination to succession of increase and decrease." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)