On Trigonometry (1900-1999)

"The Universe repeats itself, with the possible exception of history.” Of all earthly studies history is the only one that does not repeat itself. [...] Astronomy repeats itself; botany repeats itself; trigonometry repeats itself; mechanics repeats itself; compound long division repeats itself. Every sum if worked out in the same way at any time will bring out the same answer. [...] A great many moderns say that history is a science; if so it occupies a solitary and splendid elevation among the sciences; it is the only science the conclusions of which are always wrong." (G. K. Chesterton, "A Much Repeated Repetition", 1904

"But in the mathematical or pure sciences, - geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves, - we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline." (Joshua Fitch,"Lectures on Teaching", 1906)

"The science of trigonometry was in a sense a precursor of the telescope. It brought faraway objects within the compass of measurement and first made it possible for man to penetrate in a quantitative manner the far reaches of space." (Stanley L Jaki, "The Relevance of Physics", 1966)

"Such a close connection between trigonometric functions, the mathematical constant 'e', and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view." (Keith Devlin, "Mathematics: the Science of Patterns", 1994)

"The simplest musical tone is a pure tone; it is produced by a sine wave, or - to use a term from physics - by simple harmonic mation. A pure tone can be generated by an electronic synthesizer, but all natural musical instruments produce tones whose wave profiles, while periodic, are rather complicated. Nevertheless, these tones can always be broken down into their simple sine components - their partial tones - according to Fourier's theorem. Musical tones, then, are compound tones, whose constituent sine waves are the harmonics of the fundamental (lowest) frequency." (Eli Maor, "Trigonometric Delights", 1998)

"There is a certain ambiguity in the concept of angle, for it describes both the qualitative idea of 'separation' between two intersecting lines, and the numerical value of this separation-the measure of the angle. (Note that this is not so with the analogous 'separation' between two points, where the phrases line segment and length make the distinction clear.) Fortunately we need not worry about this ambiguity, for trigonometry is concerned only with the quantitative aspects of line segments and angles." (Eli Maor, "Trigonometric Delights", 1998)