History of trigonometry

"Euler wrote... in 1748 his Introductio in Analysin Infinitorum... intended... as an introduction to pure analytical mathematics. The first part... contains... the matter... found in modern text-books on algebra, theory of equations, and trigonometry. In the algebra he paid particular attention to the expansion of various functions in series, and to the summation of given series; and pointed out explicitly that an infinite series cannot be safely employed unless it is convergent. In the trigonometry, much of which is founded on F. C. Mayer’s Arithmetic of Sines... published... 1727, Euler developed the idea of John Bernoulli that the subject was a branch of analysis and not a mere appendage of astronomy or geometry: he also introduced (contemporaneously with Simpson) the current abbreviations for the trigonometrical functions, and shewed that the trigonometrical and exponential functions were connected by the relation \cos\theta + i\sin\theta = e^{i\theta}."

"Claudius Ptolemaeus, a celebrated astronomer, was a native of Egypt. ...The chief of his works are the Syntaxis Mathematica (or the ', as the Arabs call it) and the Geographica, both of which are extant. ...Ptolemy did considerable for mathematics. He created, for astronomical use, a trigonometry remarkably perfect in form. ...The fact that trigonometry was cultivated not for its own sake, but to aid astronomical inquiry, explains the rather startling fact that came to exist in a developed state earlier than plane trigonometry. ...Ptolemy has written other works which have little or no bearing on mathematics, except one on geometry. Extracts from this book made by Proclus indicate that Ptolemy did not regard the parallel-axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it."

"John Bernoulli (1667-1748)... treated trigonometry by the analytical method, studied caustic curves and trajectories."

"De Moivre enjoyed the friendship of Newton and Halley. His power as a mathematician lay in analytic rather than geometric investigation. He revolutionised higher trigonometry by the discovery of the theorem known by his name [(\cos x + i \sin x)^n = \cos nx + i \sin nx,] and by extending the theorems on the multiplication and division of sectors from the circle to the ."

"[Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \textstyle \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi} \textstyle \int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"The tables of the numerical values of the trigonometric functions had now attained a high degree of accuracy, but their real significance and usefulness were first shown by the introduction of logarithms."

"Napier is usually regarded as the inventor of logarithms, although Cantor's review of the evidence leaves no room for doubt that Bürgi was an independent discoverer. His Progress Tabulen, computed between 1603 and 1611 but not published until 1620 is really a table of antilogarithms. Bürgi's more general point of view should also be mentioned. He desired to simplify all calculations by means of logarithms while Napier used only the logarithms of the trigonometric functions."

"The greatest of Hindu astronomers and mathematicians, Aryabhata, discussed in verse such poetic subjects as quadratic equations, sines, and the value of π; he explained eclipses, solstices and equinoxes, announced the sphericity of the earth and its diurnal revolution on its axis, and wrote, in daring anticipation of Renaissance science: "The sphere of the stars is stationary, and the earth, by its revolution, produces the daily rising and setting of planets and stars.""

"The Hindus were not so successful in geometry. In the measurement and construction of altars the priests formulated the ... several hundred years before the birth of Christ. Aryabhata, probably influenced by the Greeks, found the area of a triangle, a trapezium and a circle, and calculated the value of π... at 3.1416—a figure not equaled in accuracy until the days of Purbach... Bhaskara crudely anticipated the differential calculus, Aryabhata drew up a table of sines, and the ' provided a system of trigonometry more advanced than anything known to the Greeks."

"[T]he invention of the infinitesimal calculus... foreshadowed the speedy end of trigonometry as an independent and growing branch of mathematics... By the end of the eighteenth century Leonhard Euler and others had exhibited all the theorems of trigonometry as corollaries of complex function theory."

"Three categories of people confronted or confront situations that are in many ways closely analogous. ...[T]he original inventor of a theorem or techinique... applies to the solution of his problems the mathematical tools he has inherited but slowly or quickly intuits more powerful ways... [T]he historian... seeks, hampered by his... hindsight, to retrace via texts and artifacts the thought processes of the inventor. ...[T]he student, for whom a given problem is as new as ever ...People in all three categories share... an inability to grasp the full implications of their own accomplishments. Both the historian and the student can gain understanding from the example of the inventor."

"[A]t least three thousand years ago man was employing implicity the notion of a function. For each scheme... there is a unique shadow length... Between these primitive landmarks and the road... to trigonometry there lies a great gap... The basic tool... the chord function, still tabulated in engineering handbooks, the precurser of the sine rather than the tangent."

"For serious computation a place-value system for representing numbers was requisite, but since the second millenium B.C. this had been available in the system developed in ..."

"At the end of Book I, Chapter 11, of the ' of... Ptolemy... there is a table of the function \mathrm {crd} \theta ... calculated to three significant sexigesimal places for the domain \theta = 1/2^{\circ}, 1^{\circ}, 1\ 1/2^{\circ}, \ldots , 180^{\circ } and having a column of tabular differences for . In Book I, 9, Ptolemy explains how this table was computed. ...[I]n developing the theory frequent recourse is had to the geometry and geometrical algebra... in the Elements of Euclid. ...Ptolemy is giveing a systematic exposition of... a doctrine well known in his time. This would have included... the work of Hipparchus..."

"[E]xcept for tantelizing hints... from two old cuneiform tablets, there is no way of determining how the trigonometry of chords came into being."

"The ' is a compendium of astronomy made up of cryptic rules in Sanskrit verse, with little explanation and no proofs... [N]umbers were written... in strings of words which conform to the poetic scheme... [A]lmost all...primitive sine functions [are] defined in terms of a circle whose radius, R, is not unity. We distinguish all these with from the modern sine function by the use of an intitial capital... explicitly displaying the parameter R when useful, as a subscript. In general, then, \mathrm{Sin}_R \theta \equiv R \sin \theta... and analogously for the other trigonometric functions..."

"Beginning with the ninth century, the number of people working in trigonometry increased markedly. Astronomers all, they lived in and traveled widely... from India to Spain. ...Of extraordinarily varied ethnic background—Persian, Arabic, Turkish... [etc.]—almost all shared a common faith, Islam, and a common language of science, Arabic."

"The "rule of four quanitites" marked a stage in the transition from a calculus dealing with arcs of a spherical quadrilateral to spherical trigonometry proper, involving the sides and angles of a spherical triangle. This theorem states that in a pair of right spherical triangles having an acute angle (A and A') in common or equal...\sin a/a' = \sin c/\sin c'...[A]lthough it utilizes triangles, angles are not dealt with. ...[A] proof by means of ... [is] straightforward. ...The Menelaus equation... can be stated in terms of sines by use of the identity \sin \theta \equiv \frac{1}{2} \mathrm{Crd} 2 \theta."

"By the ninth century instead of primitive schemes... tables... were common... [and] gave as a function of the sun's altitude the shadow lengths cast by it... Lengths were measured in units of a standard vertical ... These are tables of \mathrm{Cot} \theta \equiv R \cot \thetausually calculated for \theta = 1^{\circ}, 2^{\circ}, 3^{\circ}, \ldots , 90^{\circ} ."

"A few... recognized and corrected the inconvenience of the parameter R \neq 1. Since computations in the sexigesimal system were customary, this inconvenience could be minimized by putting R = 60 = 1,0 , in imitation of the Ptolemaic chord function. The astronomer Habash... tabulated... the function 1,0 \cdot \tan \theta ...[M]ultiplication or division by R becomes a... matter (as we would put it) of shifting the sexigesimal point..."

"... [i]n his unprinted manual of trigonometry... expounded the prosthaphæretic method aimed at simplifying... trigonometric computations. This method first appeared in print in 1588, in Nicolas Reimerus (Ursus)' '. It was of great value... and was going to be a direct competitor to the method of logarithms."

"... had been devised by ... and was likely brought to by... ... in 1580... This method was based on...\sin A \sin B = \frac{1}{2}[ \cos (A-B)- \cos (A+B) ], \cos A \cos B = \frac{1}{2}[ \cos (A-B)+ \cos (A+B) ].With... a table of sines, these... could... replace multiplications by additions and subtractions, something...Wittich found out, but apparently Werner didn’t realize."

"Thanks to Brahe’s manual of trigonometry, the fame of the method of prosthaphæresis spread abroad and it was brought by Wittich to Kassel in 1584... This is probably how Jost Bürgi... then an instrument maker working for the Landgrave of Kassel, learned of it."

"Bürgi not only used this method, but... improved it. He found the second formula, for Brahe and Wittich only knew the first. In addition, he improved the computation of the ...\cos c = \cos a \cos b + \sin a \sin b \cos CUsing the method of prosthaphæresis... cos a cos b and sin a sin b... could be computed but two new multiplications were... left... Bürgi realized that... prosthaphæresis could be used a second time, and... all multiplications could be replaced by additions or subtractions."

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table... Bürgi... seems to have been reluctant at publishing it and in 1592, Brahe wrote that he did not understand why he was keeping the table hidden, after... a look at it."

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table..."

"All [previous] procedures for calculating chords and sines... [were] based in principle on the method which Ptolemy had presented in his '. Totally different... is a procedure which Jost Bürgi...invented... [H]e was able to compute the sine of each angle with any desired accuracy in a... short time... Bürgi explained his procedure in ...'."

"In 1605 Bürgi went to Prague and lived there as a at the Emperor’s Chamber until 1631... [and] was... involved in astronomical observations and interpretations. ...... worked at the court. Among... temporary visitors... ... and Ursus... who had both worked with . Wittich... brought... knowledge of ... by which multiplications and divisions can be replaced by additions and subtractions of trigonometrical values... based on the identity\sin \alpha \cdot \sin \beta = \frac{1}{2}[\sin (90^\circ - \alpha + \beta) - \sin(90^\circ - \alpha -\beta)],this... achieves the same as logarithms, which were invented... decades later."

"[A] manuscript has survived in which Bürgi... explains his method. From the preface... we know that Bürgi presented it to... Rudolf II... The manuscript consists of 95 folios and contains a[n]... extensive work on by Bürgi, entitled '."

"Bürgi... found an arithmetic procedure for computing sine values with arbitrary accuracy. By dividing the right angle into 90 parts, Bürgi is able to calculate the sines of all degrees from sin 1° to sin 90°."