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)

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)

On Trigonometry (-1799)

"When the arc and its chord are bisected, we call that half-chord the right sine of the half-arc." (Joannes Regiomontanus, "De triangulis omnimodis" ["On triangles of every kind"], 1464)

"Thus the analysis of angular sections involves geometric and arithmetic secrets which hitherto have been penetrated by no one." (François Viète, cca 1615)

"In these examples s stands for sine: t for tangent: sc for sine complement [i.e., cosine]: tc for tangent complement: sec for secant." (Richard Norwood, 1631) [first known use of abbreviations] 

"And having thus passed the principles of arithmetic, geometry, astronomy, and geography, with a general compact of physics, they may descend in mathematics to the instrumental science of trigonometry, and from thence to fortification, architecture, engineering, or navigation. And in natural philosophy they may proceed leisurely from the history of meteors, minerals, plants, and living creatures, as far as anatomy. Then also in course might be read to them out of some not tedious writer the institution of physic. […] To set forward all these proceedings in nature and mathematics, what hinders but that they may procure, as oft as shall be needful, the helpful experiences of hunters, fowlers, fishermen, shepherds, gardeners, apothecaries; and in other sciences, architects, engineers, mariners, anatomists." (John Milton, "On Education", 1644)

"I have finally discovered the true solution: in the same way that to one sine there correspond an infinite number of different angles I have found that it is the same with logarithms, and each number has an infinity of different logarithms, all of them imaginary unless the number is real and positive; there is only one logarithm which is real, and we regard it as its unique logarithm." (Leonhard Euler, [letter to Cramer] 1746)

“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential."  (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748)

On Trigonometry (1850-1899)

"The employment of the uninterpretable symbol √-1 the intermediate processes of trigonometry furnishes an illustration of what has been said. I apprehend that there is no mode of explaining that application which does not covertly assume the very principle in question." (George Boole, "Laws of Thought", 1854)

"A Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space, by a very easy calculation. It is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions."  (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skillful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures."(Johann F Herbart, 1890)

"There is perhaps nothing which so occupies, as it were, the middle position of mathematics, as trigonometry." (Johann F Herbart, "Pestalozzi's Idee eines ABC der Anschauung", 1890)

"He [General Nathan Bedford Forrest] possessed a remarkable genius for mathematics, a subject in which he had absolutely no training. He could with surprising facility solve the most difficult problems in algebra, geometry, and trigonometry, only requiring that the theorem or rule be carefully read aloud to him." (John A Wyeth, "Life of General Nathan Bedford Forrest", 1899) 

On Trigonometry (2000-)

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"For centuries, mathematicians had been listening to the primes and hearing only disorganised noise. These numbers were like random notes wildly dotted on a mathematical stave with no discernible tune. Now Riemann had found new ears with which to listen to these mysterious tones. The sine-like waves that Riemann had created from the zeros in his zeta landscape revealed some hidden harmonic structure." (Marcus du Sautoy, "The Music of the Primes", 2003)

"What a wealth of insight Euler’s formula reveals and what delicacy and precision of reasoning it exhibits. It provides a definition of complex exponentiation: It is a definition of complex exponentiation, but the definition proceeds in the most natural way, like a trained singer’s breath. It closes the complex circle once again by guaranteeing that in taking complex numbers to complex powers the mathematician always returns with complex numbers. It justifies the method of infinite series and sums. And it exposes that profound and unsuspected connection between exponential and trigonometric functions; with Euler’s formula the very distinction between trigonometric and exponential functions acquires the shimmer of a desert illusion." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"[…] all roads in mathematics lead to infinity. At any rate, most of the attempts to do the impossible have called upon infinity in one way or another: not necessarily the infinitely large, not necessarily the infinitely small, but certainly the infinitely many." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"Either a logarithmic or a square-root transformation of the data would produce a new series more amenable to fit a simple trigonometric model. It is often the case that periodic time series have rounded minima and sharp-peaked maxima. In these cases, the square root or logarithmic transformation seems to work well most of the time." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"Euler’s general formula, e^iθ = cos θ + i sin θ, also played a role in bringing about the happy ending of the imaginaries’ ugly duckling story. [...] Euler showed that e raised to an imaginary-number power can be turned into the sines and cosines of trigonometry." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The trig functions’ input consists of the sizes of angles inside right triangles. Their output consists of the ratios of the lengths of the triangles’ sides. Thus, they act as if they contained phone-directory-like groups of paired entries, one of which is an angle, and the other is a ratio of triangle-side lengths associated with the angle. That makes them very useful for figuring out the dimensions of triangles based on limited information." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Basis real and imaginary numbers have eternal and necessary reality. Complex numbers do not. They are temporal and contingent in the sense that for complex numbers to exist, we first have to carry out an operation: adding basis real and imaginary numbers together. Complex numbers therefore do not exist in their own right. They are constructed. They are derived. Symmetry breaking is exactly where constructed numbers come into existence. The very act of adding a sine wave to a cosine wave is the sufficient condition to create a broken symmetry: a complex number. The 'Big Bang', mathematically, is simply where a perfect array of basis sine and cosine waves start entering into linear combinations, creating a chain reaction, an 'explosion', of complex numbers - which corresponds to the 'physical' universe." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"It is in fact mathematics itself that is simplest in hypothesis and also richest in phenomena (i.e. the simple source of all complexity). In ontological mathematics, all of existence comprises sinusoidal waves arranged into autonomous units called monads, and these are all that are required to explain everything." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"[…] the derivative of a sine wave is another sine wave, shifted by a quarter cycle. That’s a remarkable property. It’s not true of other kinds of waves. Typically, when we take the derivative of a curve of any kind, that curve will become distorted by being differentiated. It won’t have the same shape before and after. Being differentiated is a traumatic experience for most curves. But not for a sine wave. After its derivative is taken, it dusts itself of f and appears unfazed, as sinusoidal as ever. The only injury it suffers - and it isn’t even an injury, really - is that the sine wave shifts in time. It peaks a quarter of a cycle earlier than it used to." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

♾️On Complex Numbers XIX (Euler's Formula II)

"The equation e^πi+1 = 0 is true only by virtue of a large number of profound connections across many fields. It is true because of what it means! And it means what it means because of all those metaphors and blends in the conceptual system of a mathematician who understands what it means. To show why such an equation is true for conceptual reasons is to give what we have called an idea analysis of the equation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The significance of e^πi+1 = 0 is thus a conceptual significance. What is important is not just the numerical values of e, π, i, 1, and 0 but their conceptual meaning. After all, e, π, i, 1, and 0 are not just numbers like any other numbers. Unlike, say, 192,563,947.9853294867, these numbers have conceptual meanings in a system of common, important nonmathematical concepts, like change, acceleration, recurrence, and self-regulation. [...] They are not mere numbers; they are the arithmetizations of concepts. When they are placed in a formula, the formula incorporates the ideas the function expresses as well as the set of pairs of complex numbers it mathematically determines by virtue of those ideas." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"We will now turn to e^πi+1 = 0. Our approach will be there as it was here. e^πi+1 = 0 uses the conceptual structure of all the cases we have discussed so far - trigonometry, the exponentials, and the complex numbers. Moreover, it puts together all that conceptual structure. In other words, all those metaphors and blends are simultaneously activated and jointly give rise to inferences that they would not give rise to separately. Our job is to see how e^πi+1 = 0 is a precise consequence that arises when the conceptual structure of these three domains is combined to form a single conceptual blend." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"[…] the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Thus, while feelings may be the essence of subjectivity, they are by no means part of our weaker nature - the valences they automatically generate are integral to our thought processes and without them we’d simply be lost. In particular, we’d have no sense of beauty at all, much less be able to feel (there’s that word again) that we’re in the presence of beauty when contemplating a work such as Euler’s formula. After all, e^iπ + 1 = 0 can give people limbic-triggered goosebumps when they first peer with understanding into its depths." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Today, Euler’s formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It’s arguable that the formula’s ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Euler’s formula - although deceptively simple - is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

♾️On Complex Numbers VII

“The remark which you make concerning roots that can not be extracted, and containing imaginary quantities which when added together give none the less a real quantity, is surprising and entirely new. One would never have believed that √(1 + √-3) + √(1 - √- 3) would make √6, and there is something hidden in this which is incomprehensible.” (Christaan Huygens, [letter to Gottfried W Leibniz] cca. 1670)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)

“Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality.” (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)

“For it ought to be considered that both –b   and –c  , as they stand alone, are, in some Sense, as much impossible Quantities as √(-b)  and √(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination.” (Thomas Simpson, “A Treatise of Algebra”, 1745) 

“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential."  (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748)

“Moreover, the whole method has the essential disadvantage that it occupies the mind with the distinction of a great number of cases that can be recognized only by inner intuition, and thus neutralizes an important part of that which algebra is supposed to accomplish, which is relieving the power of inner intuition. Finally, in such a treatment algebra loses a great part of the generality that it can obtain by the mutual connection of different problems, which becomes evident so easily when one uses isolated negative quantities. [...] Since imaginary quantities have to occur, science would certainly not win that much by avoiding negative quantities than it would lose in terms of clarity and generality.” (Johann P W Stein,  “Die Elemente der Algebra: Erster Cursus”, 1828) 

"Originally assuming the concept of the absolute integers, it extended its domain step by step; integers were supplemented by fractions, rational numbers by irrational numbers, positive numbers by negative numbers, and real numbers by imaginary numbers. This advance, however, occurred initially with a fearfully hesitant step. The first algebraists preferred to call negative roots of equations false roots, and it is precisely these where the problem to which they refer was always termed in such a way as to ensure that the nature of the quantity sought did not admit any opposite.” (Carl F Gauss, “Theoria residuorum biquadraticum. Commentatio secunda. [Selbstanzeige]”, Göttingische gelehrte Anzeigen 23 (4), 1831)

“[T]he notion of a negative magnitude has become quite a familiar one […] But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry. […] considering the prominent position which the notion occupies-say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it.” (Arthur Cayley, [address before the meeting of the British Association at Southport] 1870) 

“A satisfactory theory of the imaginary quantities of ordinary algebra, which is essentially a simple case of multiple algebra, with difficulty obtained recognition in the first third of this century. We must observe that this double algebra, as it has been called, was not sought for or invented; - it forced itself, unbidden, upon the attention of mathematicians, and with its rules already formed. [...] But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.” (Josiah W Gibbs, “On multiple Algebra”, Proceedings of the American Association for the Advancement of Science Vol. 35, 1886)