History of trigonometry

History of trigonometry begins with the early study of triangles, traced to the 2nd millennium BC, in Ancient Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century BC), who discovered the versine, sine and cosine functions.

When during the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa' al-Buzjani. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Latin translations of the 12th century for Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus.

The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).

Quotes

Isaac Newton... went to school at Grantham and in 1661 came up as a subsizar to Trinity. ...He had not read any mathematics before coming into residence but was acquainted with Sanderson's Logic, which was then frequently read as preliminary to mathematics. At the beginning of his first October term he... picked up a book on astrology, but could not understand it on account of the geometry and trigonometry. He therefore bought a Euclid, and was surprised to find how obvious the propositions seemed. He thereupon read Oughtred's Clavis and Descartes's Geometry, the latter of which he managed to master by himself though with some difficulty. The interest he felt in the subject led him to take up mathematics rather than chemistry as a serious study. His subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures. At a later time on reading Euclid more carefully he formed a very high opinion of it as an instrument of education, and he often expressed his regret that he had not applied himself to geometry before proceeding to algebraic analysis. ...He was elected to a scholarship in 1663.
- W. W. Rouse Ball, A History of the Study of Mathematics at Cambridge (1889) pp. 51-52.

[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and Bonaventura Cavalieri... from Italy; several... as Henry Briggs.., Thomas Harriot.., and William Oughtred... were English; two... Simon Stevin... and Albert Girard... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany.
- Carl Benjamin Boyer, A History of Mathematics (1968, 1991)

At its higher levels the golden age of Muslim civilization was both an immense scientific success and a exceptional revival of ancient philosophy. These were not its only triumphs... but they eclipse the rest. ...[T]he Saracens ...made the most original contributions [to science]. These, in brief, were nothing less than trigonometry and algebra... In trigonometry the Muslims invented the sine and the tangent. The Greeks had measured an angle only from the chord of the arc it subtended: the sine was half the chord. The Chosranian (...Mohammed Ibn-Musa) published in 820 an algebraic treatise which went as far as quadratic equations: translated into Latin in the sixteenth century, it became a primer for the West. Later, Muslim mathematicians resolved biquadratic equations.
Equally distinguished were Islam's mathematical geographers, its astronomical observatories and instruments (in particular the astrolabe) and its excellent if still imperfect measurements of latitude and longitude, correcting the flagrant errors of Ptolemy.
- Fernand Braudel, A History of Civilizations (1994) p. 80.

Euler wrote... Introductio in Analysin infinitorum, 1748, which was intended to serve as an introduction to pure analytical mathematics. ...He ...showed that the trigonometrical and exponential functions are connected by the relation $\cos \theta +i\sin \theta =e^{i\theta }$ . Here too we meet the symbol e used to denote the base of the Naperian logarithms, namely the incommensurable number 2.7182818... The use of the single symbol to denote the incommensurable number 2.7182818... seems to be due to Cotes, who denoted it by M. Newton was probably the first to employ the literal exponential notation, and Euler using the form a^z, had taken a as the base of any system of logarithms. It is probable that the choice of e for a particular base was determined by its being a vowel consecutive to a, or, still more probable because e is the initial of the word exponent.
- Benjamin Franklin Finkel, A Mathematical Solution Book (1893) p. 449.

The development of Indian trigonometry, based on sine as against chord of the Greeks was another of Aryabhatiya's achievements which was necessary for astronomical calculations. Because of his own concise notation, he could express the full sine table in just one couplet, which students could easily remember. For preparing the table of sines, he gave two methods, one of which was based on the property that the second order sine differences were proportional to sines themselves.
- Radha Charan Gupta, "Āryabhata" Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures p. 245.

The second part of the book... contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity.
- E. W. Hobson, A Treatise on Plane Trigonometry (1891) Preface.

The idea of the logarithm probably had its source in the use of... trigonometric formulas that transformed multiplication into addition and subtraction. ...[I]f one needed to solve a triangle using the law of sines, a multiplication and division were required. ...[C]alculations were long and errors... made. Astronomers realized... multiplication and division could be replaced by additions and subtractions. To accomplish this... sixteenth century astronomers used formulas... as $2\sin \alpha \sin \beta =cos(\alpha -\beta )-\cos(\alpha +\beta )$ . ...A second source of the... logarithm was probably found in... algebraists as Stifel and Chuquet, who both displayed tables relating the powers of 2 to the exponents and showed that multiplication in one table corresponded to addition in the other. But because these tables had large gaps, they could not be used for necessary calculations. ...[T]wo men... independently, the Scot John Napier... and the Swiss Jobst Bürgi... came up with the idea of producing an extensive table... to multiply any... numbers... (not just powers of 2)... Napier published... first.
- Victor J. Katz, A History of Mathematics: An Introduction (1993, 1998)

We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks.
- Joseph Needham, as quoted in R.C. Majumdar, "Nationalist Historians" Historians of India, Pakistan and Ceylon ed. Philips, from E. Sreedharan, A Textbook of Historiography, 500 B.C. to A.D. 2000.

Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations.
- Martin Andrew Nordgaard, A Historical Survey of Algebraic Methods of Approximating the Roots of Numerical Higher Equations Up to the Year 1819 (1922) p.13.

Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks.
- George Pólya, Mathematical Methods in Science (1977)

The sum and difference formulas are vital to building trigonometric tables finer than the traditional 24 entries per 90°. ...they can also be used to generate many other identities. In particular, formulas for Sin 2θ, Cos 2θ, Sin 3θ, Cos 3θ, and higher multiples may be generated simply by writing nθ = θ + θ +... + θ and applying the sum formulas repeatedly. This was done by... Kamalākara in his Siddhānta-Tattva-Viveka (1658) up to the sine and cosine of 5θ; he quotes Bhāskara II (who clearly knew this could be done) for the addition and subtraction laws. Kamalākara's sine triple-angle formula...was $\mathrm {Sin} 3\theta =\mathrm {Sin} \theta (3-{\frac {(\mathrm {Sin} \theta )^{2}}{(\mathrm {Sin} \,30^{\circ })^{2}}})$ ,equivalent to the modern formula $\sin 3\theta =3\sin \theta -4\sin ^{3}\theta$ ; ...The identity ...has special significance, since it may be used to get an accurate estimate of sin 1° from sin 3°—provided one is able to solve cubic equations.
- Glen Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry (2009) Ch. 3 India, p.107. Also see angle trisection.

After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley.
- Bartel Leendert van der Waerden, Science Awakening (1954); Ontwakende wetenschap (1950)

Elements of Plane and Spherical Trigonometry (1786)

Written originally in Russian, and translated into English by the authors, Basil Nikitin & Prochor Souvoroff. A source @archive.org.

The British mathematicians have been the greatest, nay... the only improvers of Trigonometry within these two centuries. ...[N]ot to mention the extraordinary inventions of Lord Neper... nor the analogies, wherein sines and tangents of half-arches are used, nor the applications of Trigonometry to stereographic projection of the sphere, for all which we are indebted... the very words of Trigonometry, cosine, cotangent, &c. have been first used by the writers of that nation.

The most conspicuous authors of both Trigonometries amongst the British, who have been consulted by us, are Caswell, in Wallis's Works, Keil, Simpson, Robertson, Mr. Thomas Simpson, Emerson, and [Benjamin] Martin; and of those who have treated of them occasionally, Oughtred, in his Clavis Mathematica, and Circles of Proportion, Wallis, Jones, Wingate, [Henry] Sherwin, and Gardiner, in their Tables of Logarithms; Sir Isaac Newton, in his Univ. Arithm. Geometrical Problem II. Harris and Chambers in their dictionaries. Plane Trigonometry alone has been treated by [Philip] Ronayne, Mr. Thomas Simpson, Maseres, and Muller. However, the merits of some foreigners also cannot, without injustice, be suppressed. Such are Copernicus, in his Astronomia Instaurata; Balanus, a modern Greek; Simon Stevin, commented upon by Albert Girard; Clavius; M. de la Caille; M. de la Lande, in his Astronomie, tom. III. de Chales; Ozanam; Segnerus; and the labours of Schottus, Tacquet, and others, are commendable. We need not mention the parents of these sciences, Theodosius, Ptolemy, Menelaus, and Regiomontanus.
- Ref: 1) Philip Ronayne, A Treatise on Algebra (1717) 2) Benjamin Martin, The Young Trigonometer's Compleat Guide (1736)

[N]otwithstanding the labours and exertions of so many eminent men... in this branch of mathematics many things are deficient, many superfluous; some are too general, others too particular; some are too much dwelt upon, others want a great deal of explanation; in many there is hardly any order, or connexion, or demonstration, in some too much unnecessary precision.

[W]hat else... the reason of doubts arising in solutions... of plane and spherical triangles, but the want of accurate determinations and explanations?

From what have proceeded disputes in Spherical Trigonometry, not solved either by Cunn, or Ham, but from the inaccurate notion of a supplemental triangle?

[T]o what can this be owing, but to the want of sufficient principles, the neglect of enumerating and distinguishing: cases of a proposition, and the inattention to rendering the subject as complete as possible?

[T]here is as yet no classical book of Trigonometry in any language... fit to give to learners a solid foundation in them... as are Euclid's Elements, Archimedes de Sphaera et Cylindro, and Dr. Hamilton's Conic Sections.
- Ref: Hugh Hamilton, De Sectionibus Conicis: Tractatus Geometricus (1758)

[T]he reason, why in Algebra and Fluxions, expressions for trigonometrical lines always run out into infinite series... is because the number of arches, to which any one of such lines belongs is always infinite.

Prop. 14. and its corollaries deserve... examination. It is hard to say, by whom they were invented, though... probably by the English; and perhaps corr. 3 & 2. of prop. 29. in spherics, have given rise to them all, as they are to be found in most books of the last age. They are all to be seen in Caswell... Wallace, Newton Univ. Arithm. Geom. Probl. 11. Thos. Simpson's Algebra, Geom. Probl. 15. Dr. 'Robertson's Navigation, Emerson, and [Benjamin] Martin. The analogy of the prop. in particular, is to be met with in Trigonometria Britannica, [Henry] Sherwin's Tables, de la Caille, Dr. Simpson, and Ward.
- Ref: Henry Sherwin, William Gardiner, Sherwin's Mathematical Tables (1717) 3rd Edition Source (1742)

With regard to demonstration, the old and perplexed one (used formerly for the area of a triangle, and accommodated here by Dr. Simpson) is laid aside, and another... easier demonstration is substituted... after the manner of Dr. Robertson's, but greatly improved by the change of a side of the triangle; the same indeed nearly, which has been communicated in Russia some years ago by... Professor Robison of Glasgow.

But in Book II. i.e. in Spherical Trigonometry, the greatest pains were to be taken, and the greatest difficulties to be overcome. For though... Spherical Trigonometry is not so easy as the Plane, as it wants those previous helps and that determination, which the Plane... has; yet, when out of three parts of a proposition one only or two are laid down; when a proposition is demonstrated only in one case out of several; when, on account of bad definitions, several things, wanting demonstration, are passed by, or dignified with the name of axioms; when an argument turns in circulo... when whole steps are omitted; when, instead of a direct way, we go round about; when things are scattered about without order; when a whole set of triangles is neglected, &c. &c. surely, all this is not the fault of Spherical Trigonometry.

The whole doctrine of axes and poles is to this day both incomplete and inaccurate. The authors endeavoured to their utmost, to remedy such extraordinary defects in so important a subject. ...In truth, the subject of poles of circles, as it is laid down here, seems exhausted: several of their properties are exhibited, over and above those in Theodosius's Spherics, and Dr. Barrow's additions thereto; and this is done in a lesser number of lines, than they have pages.

In prop. 5. and cor. the confused and inaccurate ideas of arches being measures of angles, of arches being equal to angles, and of arches being the supplements and complements of angles, and v. v. so much prevailing even among the best geometricians, are attempted to be rectified: for it is manifest enough, that nothing can be a measure of another thing, or equal to it, or a supplement and complement of it, unless it be homogeneous with it. For want of such a plain consideration, and afterwards most probably from habit, people have debased many propositions, both in their enunciations and demonstrations; and often it is not without some trouble that they are corrected.

Prop. 7. claims particular notice on account of its use, of its application, and even on account of the disputes it has occasioned. Its application is so... extensive, that the doctrine of spherics, by means only of it, may be reduced almost to half the number of its propositions. The invention of it may be ascribed perhaps to Philip Langsberg. Vid. Simon Stevin Liv. 3. de la Cosmographie, prop. 31. & Alb. Girard in loc. ...[W]e have been obliged to form for it an enunciation entirely new; and were happy to find afterwards, that Mr. Cotes in his Æstim. Error. iem. 4. gives the first part of it exactly the same.

It is from the vagueness of the proposition... and from misunderstanding the terms supplement and complement, that disputes have arisen in spherics: these may be seen at the end of [Samuel] Cunn's Euclid, in his remarks and the appendix. Whatever be the mistakes of Mr. Heynes... the respectable names of Dr. Keil, Mr. Caswell, and Dr. Harris, whom Mr. Cunn joins in company with Heynes, are treated by him... injuriously; especially as he himself had not examined his subject with sufficient attention. His own rule... is indeed true... but it is more troublesome to the memory. Mr. Ham... awards his own rule, which, notwithstanding, is much more unmanageable... it using subtraction of natural versed sines, to whose difference therefore (and every one knows the thing is not easy) logarithms are to be accommodated. But it were time, long ago, to bury these worthless disputes in oblivion, that learners of spherics should not be discouraged by seeing them printed and reprinted so often.
- Ref: John Keil, Tr. & Ed. Samuel Cunn, Euclid's Elements of Geometry, from the Latin translation of Commandine. To which is added, A Treatise of the Nature and Arithmetick of Logarithms; Likewise another of Plain and Spherical Trigonometry.. (1723)

Of prop. 10. no one gives an accurate demonstration, except Menelaus : Dr. Keil's need not be mentioned, and Dr. Simpson's leaves out a case, and is at the same time very prolix. That which is offered here... seems remarkably short and easy, and is derived from Dr. Simpson's Elements of Euclid, Book XI. prop. A. ...[H]e is... to be praised for his merits in his... Euclid; but... there are still many inaccuracies...
- Ref: Robert Simpson, Euclid's Elements (1756) with Annexed Elements of Plane and Spherical Trigonome. See also: 1838 edition.

Prop. 18 & 19. though... the... foundation of... Spherical Trigonometry... have not been demonstrated... except in one single case... it does not seem... long ago, that they... appear anew in their present form. V. Keil.

Such weak foundations Spherical Trigonometry has had to this day!

A Treatise on Plane and Spherical Trigonometry (1809, 1827)

by Robert Woodhouse 5th edition was published 1827. A source @archive.org

The first considerable extension of Trigonometry, beyond its original object, was made about twenty years after the death of Newton. It was then, on the ground-work laid down by that great man, that... Clairaut, Dalembert and Euler, and Thomas Simpson... began to establish a system of Physical Astronomy more perfect... [T]hey laid aside the Geometrical method which Newton had used... and adopted the Analytical. ...[T]hey perceived the formulæ of Trigonometry to be of continual use and recurrence, and the language, by which the process of demonstration was conducted... in a great degree, of symbols and phrases borrowed from that science. ...[T]he advancement of Trigonometry, the pure and subsidiary science, was contemporaneous with that of Astronomy, the mixed and principal one.

Clairaut and Dalembert in their Lunar Theories... introduce... several, now commonly known, Trigonometrical formulæ. In... Thomas Simpson... the Author evidently intended the one... at p. 76, as preparatory to the... Theory of the Moon; and Euler... states as a reason for cultivating the algorithm of sines, its great utility in the mixed Mathematics.

Spherical [Trigonometry]... has not, like... [plane Trigonometry], been extended beyond its original purpose. It has no collateral and indirect uses; it has not enriched the general language of analysis... But... its propositions are more easily established by the Analytical method than the Geometrical. ...[T]his would be the case, even if there existed no similarity and artificial connexion, between the processes by which the series of formulæ in the two branches of Trigonometry were... established. [T]he corresponding propositions can be deduced by methods so analogous, that to know the one is almost to know the other.

We shall... find similar Algebraical derivations of formulæ from two fundamental expressions for the cosine of an angle. The principle of the derivation... is not new; it originated with Euler, who inserted in the Acta Acad. Petrop. for 1779, a Memoir entitled Trigonometria Spherica Universa, ex primis principiis breviter et dilucide derivata. Gua next, in the Memoirs of the Academy of Sciences for 1783, p. 291, deduced... Spherical Trigonometry "from the Algebraical solution of the simplest of its Problems." In 1786, Cagnoli... derived fundamental expressions for the sine and cosine of the sum of two arcs. And lastly, Lagrange and Legendre, the one in the Journal de L Ecole Polytechnique, the other in his Elemens de Geometrie, have followed and simplified Euler’s method, and instead of three fundamental expressions, have shewn one to be sufficient.

Memoirs of John Napier of Merchiston (1834)

His Lineage, Life and Times, with a History of the Invention of Logarithms by Mark Napier. A source. @archive.org

Kepler's indefatigable inquiries, for nine years... amounts to this, that Byrgius had made some observations upon the adaptation of an arithmetical to a geometrical progression, very naturally occurring to him in trigonometrical calculations.

Longomontanus and Byrgius, are all whom Dr Hutton can find to represent his learned calculators of the sixteenth and seventeenth centuries, who anticipated or coincided with Napier in the discovery. ...But he is contradicted by the history of science... and by every philosopher of greatest name... ...[O]ur philosopher's invention... removed a pressure, long and severely felt, and which might have crushed the temple of science, had that not possessed such a pillar as Kepler. To use the expressions of a distinguished writer, " What all mathematicians were now wishing for, the genius of Neper enabled him to discover; and the invention of Logarithms introduced into the calculations of trigonometry a degree of simplicity and ease, which no man had been so sanguine as to expect." Kepler, Ursine, Speidell, Gunter, Briggs, Vlacq, [Petrus] Cugerus, Cavalieri, Wolff, Wallis, Halley, Keill, and a host of others, all bear witness against Dr Hutton, in the honourable and enthusiastic manner they acknowledge Napier as the only author of that revolution in science.
- Quote Ref: Review of Woodhouse's Trigonometry, Edin. Review (1810) Vol. XVII. p. 124.

An Introduction to the Ancient and Modern Geometry of Conics (1881)

Being a Geometrical Treatise on the Conic Sections with a Collection of Problems and Historical Notes and Prolegemena by Charles Taylor, M.A. Fellow at St. John's College, Cambridge. A source @archive.org.

In continuation of... the most brilliant period of ancient geometry, the century of Euclid, Archimedes and Apollonius, recourse must again be had to the Collectio of the much later writer Pappus, for information about the lost three books of Porisms of Euclid. ...In the Sphœrica of Menelaus, a geometer and astronomer of the first century A.D., is found the theorem (lib. III. lemma 1 p. 83, Oxon. 1758): If the sides $ag,gd,da$ of a plane triangle be met by any transversal in the points $erb$ respectively, then $ge:ea=gr.db:rd.ba$ ,or the product of three non-adjacent segments of the sides of the triangle by any transversal is equal to the product of the remaining three. This was... extended to spherical triangles... as a basis for the spherical trigonometry of the ancients.

[T]he property of the six segments in plano... [led to] great results... long after, especially in the hands of Desargues.

Claudius Ptolemseus was "le plus ceélèbre, sans contredit, mais non le plus véritablement grand astronome de toute l'antiquité." [the most famous, without a doubt, but not the truly greatest astronomer of all antiquity] Thus writes Delambre in the Biographie Universelle (vol. 36... 1823).

In a work on the three dimensions of bodies, Ptolemy introduced the idea of determining the position of a point in space by referring it to three rectangular axes of coordinates... His chief work, which he called a mathematical Σύνταξις [Syntax], was further described by his admirers as ή μεγύλη, and by the Arabs as Almagest ή μεγίστη [maximum]... In it... he reproduces the theorem of the six segments... and founds upon it a system of trigonometry, plane and spherical.
- Ref: 1) (ibid. p. 272) 2) Dictio Prima, cap. 12. fol. 95. Venet. 1515. 3) Delambre, Histoire de L'Astronomie Ancienne (Paris 1817) tome II, comparing the Præfatio to the 3rd volume of Fridericus Hultsch, Pappi Alexandrini collectionis quae supersunt (1876)

A Short History of Greek Mathematics (1884)

by James Gow

Nicomachus turns to the discussion of proportion... which, he says, is very necessary for "natural science, music, spherical trigonometry and planimetry and particularly for the study of the ancient mathematicians."

Of the great Greek mathematicians, Archimedes alone (in his Circuli Dimensio) ventures to introduce actual numbers into a geometrical discussion, and to divide a line by another line. He finds the value of π and some other similar ratios but does not himself pursue such investigations further and is not followed by any other writer. Trigonometry was used only for astronomical purposes and did not form part of geometry at all.
- Footnote, p.106.

[T]he word 'geometry'... means 'land-measurement,' that the Egyptians gave this science to the world and that among the Egyptians... it... was confined almost entirely to the practical requirements of the surveyor. The work ["Directions for obtaining the knowledge of all dark things" in the Rhind collection] of Ahmes..., contains, beside sums in arithmetic, a great many geometrical examples... Ahmes proceeds to calculate the contents of... receptacles... The rectilineal figures of which Ahmes calculates the areas are the square, oblong, isosceles triangle and isosceles parallel-trapezium (...part of an isosceles triangle cut by a line parallel to the base). As to the last two, the areas... are incorrect. ...The errors in these cases are small... The area of a circle is found (in no. 50) by deducting from the diameter 1/9th of its length and squaring the remainder. Here π is taken $=({\frac {16}{9}})^{2}=3.1604...$ , a... fair approximation. ...Lastly, the papyrus contains (nos. 56 to 60) some examples which seem to imply a rudimentary trigonometry. In these... the problem is to find the uchatebt, piremus or seqt of a pyramid or obelisk.
- Ref: August Eisenlohr, Ein Mathematisches Handbuch der alten Aegypter (Papyrus Rhind des British Museum) (1877) pp. 126—129.

[C]oncerning Babylonian mathematics... The Chaldees... almost contemporaneous with Ahmes... had made advances, similar to the Egyptian, in arithmetic and geometry, and were especially busy with astronomical observations. ...[T]hey had divided the circle into 360 degrees, and... obtained a fairly correct... ratio of the circumference of a circle to its diameter. They used... a sexagesimal notation, which the Greeks afterwards adopted for astronomical purposes. Herodotus expressly states that the polos and gnomon (...sundials) and the twelve parts of the day were made known to the Greeks from Babylon. Much of the trigonometry and spherical geometry of the later Greeks may... have been... derived from Babylonian sources.

In 1120, Adelhard of Bath obtained in Spain a copy of Euclid's Elements and translated... into Latin. Translations from the Arabic of other Greek works, especially... Aristotle, soon followed. About 1186 Gherardo of Cremona made another translation of the Elements and... in 1260, Giovanni Campano reproduced Adelhard's translation under his own name and obtained... wide celebrity. The fruit of these translations soon followed. In 1220, Leonardo of Pisa... published... Practica Geometriae which though it deals with the calculation of areas and numerical ratios of spaces, is founded on Euclid and Archimedes and Ptolemy, and contains some trigonometry and conics.

The century which produced Euclid, Archimedes and Apollonius was... the time at which Greek mathematical genius attained its highest development. For many centuries... geometry remained a favourite study, but no substantive work... compared with the Sphere and Cylinder or the Conics... One great invention, trigonometry, remains to be completed, but trigonometry with the Greeks remained always the instrument of astronomy and was not used in any other branch of mathematics, pure or applied. The geometers who succeed to Apollonius are professors who signalised themselves by this or that pretty little discovery or by some commentary on the classical treatises.

Hypsicles'... astronomical work, Aναφορικός, does not use the trigonometry which was certainly introduced by Hipparchus, and would have been absurdly antiquated if written after Hipparchus' time...

The 14th Book of the Elements, or the book of Hypsicles on 'the Regular Solids', consists of seven propositions... The... treatise on 'Risings'... contains only six propositions, of which the first three, deal... with arithmetical progressions... The only interesting proposition is the 4th... Divide the zodiac into 360 local degrees and the time of its revolution into 360 chronic degrees. Then, given the ratio, for any place on the earth, of the longest day to the shortest, we can deduce the number of chronic degrees for each number of local degrees. Here, for the first time in any Greek work, we find a circle divided in the Babylonian manner into 360 degrees.

Hypsicles introduces the division as if it were a novelty. He does not, however, take the next step, to trigonometry. ...This was ...taken by Hipparchus ...upon whose work the whole system of Greek astronomy was founded. ...[T]hough the Almagest of Ptolemy is clearly derived almost entirely from writings of Hipparchus, none of the works of the earlier astronomer have survived, save a commentary in three books on the Phenomena of Aratus... In the Second Book... he claims to have invented a method of solving spherical triangles for the purpose of finding the exact eastern point of the ecliptic. The treatise... is lost. Theon, in his commentary on the Almagest... states that Hipparchus calculated a "table of chords" (i.e. practically of sines) in twelve books. ...[T]herefore ...Hipparchus was the founder of trigonometry, though we are obliged to look elsewhere for ... the progress of the Greeks in this department ...

Hipparchus... the following little summary, taken from Delambre, will shew what manner of man he was. ...[H]e ...determined (...not with absolute accuracy) the precession of the equinoxes, the inequality of the sun, and the place of its apogee, as well as its mean motion: the mean motion of the moon, its nodes and its apogee: the equation of the centre of the moon and the inclination of its orbit. He had discovered a second inequality of the moon (the evection), of which he could not, for want of proper observations, find the period and the law. He had commenced a more regular course of observations for the purpose of supplying his successors with the means of finding the theory of the planets. He had both a spherical and a plane trigonometry. He had traced a planisphere by stereographic projection: he knew how to calculate eclipses of the moon and to use them for the improvement of the tables: he had an approximate knowledge of parallaxes, more correct than Ptolemy's. He invented the method of describing the positions of places by reference to latitude and longitude. What he wanted was only better instruments. Yet in his determination of the equations of the centres of the sun and moon and of the inclination of the moon, he erred only by a few minutes. For 300 years after his time astronomy was stationary. Ptolemy followed him with little originality. Some 800 years later the Arabs added a few more discoveries and more accurate determinations and then the science is stationary again till Copernicus, Tycho and Kepler.
- Ref: Jean Baptiste Joseph Delambre, Biographie Universelle Vol. 36 (1823)

[T]hough Heron's ability is sufficiently indicated by... [his] proofs, as a general rule he confines himself merely to giving directions and formulae. ...[H]e availed himself of the highest mathematics of his time. Thus in the Dioptra, two chapters treat of the mode of drawing a plan of an irregular field and of restoring, from a plan, the boundaries of a field in which only a few landmarks remain. ... The method is closely similar to the use of latitude and longitude introduced by Hipparchus. So...Heron gives, for finding the area of a regular polygon from the square of its side, formulae which imply a knowledge of trigonometry. Suppose $F_{n}$ to be the area of a regular polygon of which ${a_{n}}$ is a side, and let $c_{n}$ be the coefficient by which ${a_{n}}^{2}$ is to be multiplied in order to produce the equation $F_{n}=c_{n}{a_{n}}^{2}$ then it is easy to see that $c_{n}={\frac {n}{4}}\cot {\frac {180^{\circ }}{n}}$ . ...[H]is approximations are generally near enough. We need not be surprised... Hipparchus made a table of chords... [i.e.] the coefficients $k_{n}$ were known, with the aid of which $a_{n}=k_{n}r$ , where $r$ is the radius. Then $c_{n}={\frac {n}{4}}{\sqrt {{\frac {4}{{k_{n}}^{2}}}-1}}$ , and Heron was competent to extract such square roots. But Heron does not use the sexagesimal fractions, and... sexagesimal fractions were always, as... afterwards called, astronomical fractions... [S]ave by Heron, trigonometry was generally conceived to be a chapter of astronomy and was not used for the calculation of terrestrial triangles.

Heron was by no means a geometer of the Euclidean School. He is a practical man who will use any means to attain his end and is... untrammelled by... classical restrictions. He is... a mechanician who, unlike Archimedes, is... proud of his... ingenuity. He adds... almost nothing, to the geometry of his time but he is learned in the... bookwork. On the other hand... he is the first Greek writer who uses a geometrical nomenclature and symbolism, without the geometrical limitations, for algebraical purposes, who adds lines to areas and multiplies squares by squares and finds numerical roots for quadratic equations. Hence, for a similar reason to... de Morgan... it is now commonly believed that Heron was an Egyptian. ...[T]he ...style of his work recalls ...Ahmes ... [A]lgebra was an Egyptian art and ...the symbolism of Diophantus was of Egyptian origin. ...[I]f Heron was not a Greek, he relied almost entirely on Greek learning and did not resort to the ...priestly tradition ...He is a man who writes in Greek upon Greek subjects, but who thinks in Egyptian. [Following is in the footnote.] Let it be remembered that the seqt-calcalation of Ahmes leads to trigonometry: his hau-calculation to algebra. Almost the first sign of both appears in Heron... An algebraic symbolism first appears in Diophantus, but the symbols are probably not Greek and probably are Egyptian. Both Heron and Diophantus were Alexandrians. This is all the evidence that trigonometry and algebra were of Egyptian origin, but does it not raise a shrewd suspicion? Proclus... speaks... as if Heron founded a school.

Practically all that we know of the trigonometry of the Greeks, is derived from two chapters of the famous Μεγαλή Σύνταξις [Great Compilation] of Claudius Ptolemæus. This work contains many astronomical observations by Ptolemy... The common name μεγαλή Σύνταξις [Great Syntax] was altered by... fervent admirers into μεγίστη [maximum] and this word was adopted by the Arabs... The Arabic article was... added and the name corrupted into Almidschisti, whence is derived its common mediaeval title Almagest.
- Note: Ptolemy's title is μαΘηματική Σύνταξις [Mathematical Syntax].

Ptolemy's method of calculating chords seems to be his own. The measures of the sides of regular polygons, as chords of certain arcs, were known in terms of the diameter. He next proves the proposition, now appended to Euclid VI. (D), that "the rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to both the rectangles contained by its opposite sides", and then... how from the chords of two arcs that of their sum and difference [may be found] and how from the chord of any arc that of its half may be found.
- Ref: Michel Chasles, Aperçu Historique sur l'origine et le Développement des Méthodes en Géométrie (1837, 1875) p. 27, note I. says that Carnot in his Géométrie de Position shewed that all rectilineal trigonometry could be deduced from this theorem in quotes above. Note: Ptolemy's proofs follow on pp. 294-296 of Gow.

Chapter X., which follows, is on the obliquity of the ecliptic... The next, XI., XII. contain spherical geometry and trigonometry "enough for the determination of the connexion between the sun's right ascension, declination and longitude and for the formation of a table of declinations to each degree of longitude."

Chap. XI. contains προλαμβανόμενα [preventable], "preliminaries to the spherical demonstrations". These begin with the lemma of Menelaus, the regula sex quantitatum, borrowed without any acknowledgement. After proving this, he gives four propositions. ...This ...contains ...the whole of Greek trigonometry. The further progress... is due mainly to the Indians and after them to the Arabians.

The Indians never used "the chord of twice of the arc", as the Greeks... but half that chord. This they called jyârdha or ardhajyâ, but the name of the whole chord jyâ or jivâ... The Arabs... transliterated it to dschîba... later... altered for...Arabic... dschaib, which... means 'bosom' and was therefore translated 'sinus' by Plato of Tivoli in his Latin version ('De Motu Stellarum') of the astronomy of Albategnius. In this way, sine came to be a technical term of modern trigonometry.

The applications of trigonometry in Book II. of the Almagest and the geometry of eccentric circles and epicycles in Book III. belong... by language and purpose, to the history of astronomy.

To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress.

[N]o Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life.

[D]uring this time practical astronomy had been making rapid strides in the hands of Eudoxus, Aristarchus, Eratosthenes and others down to Hipparchus. Now the needs of the practical astronomer are in many respects similar to those of the surveyor, the engineer and the architect. Each of these is chiefly concerned, not to find the general rules which govern all similar cases, but to find under what general rules a particular case, presented to them, falls. But the question whether an angle is acute, or a triangle isosceles, can be determined only by measurement, and hence about 130 B.C., in the time of Heron and Hipparchus, we find the results of geometry applied to measured figures, for the purpose of finding some other measurement as yet unknown. Trigonometry and an elementary algebraical method are thus introduced.

The learning of the Greeks passed over in the 9th century to the Arabs and with them came round into the West of Europe. But no material advance was made by the Arabs in geometry and it was their arithmetic, trigonometry and algebra which chiefly interested the mediaeval Universities. In the 16th century Greek geometry again became known in the original and was studied with intense zeal for about 100 years, until Descartes and Leibnitz and Newton, the best of its scholars, superseded it.

A Short Account of the History of Mathematics (1893, 1908)

by W. W. Rouse Ball. A source @archive.org.

Ahmes then goes on to find the area of a circular field … and gives the result as (d - 1/9d)², where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416.

Ahmes gives some problems on pyramids. ...Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes's geometrical results were intended only as approximations correct enough for practical purposes.

Arab missionaries who had come to China in the course of the thirteenth century, and while there introduced a knowledge of spherical trigonometry.

The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. ...they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival|the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry.

Archimedes... work... The following is a fair specimen of the questions considered. A solid in the shape of a paraboloid of revolution of height $h$ and latus rectum $4a$ floats in water, with its vertex immersed and its base wholly above the surface. If equilibrium be possible when the axis is not vertical, then the density of the body must be less than $(h-3a)^{2}/h^{2}$ (book II. prop. 4). When it is recollected that Archimedes was unacquainted with trigonometry or analytical geometry, the fact that he could discover and prove a proposition such as that... will serve as an illustration of his powers of analysis.

The third century before Christ, which opens with... Euclid and closes with the death of Apollonius, is the most brilliant era in the history of Greek mathematics. But the great mathematicians of that century were geometricians... It was not till after... nearly 1800 years that the genius of Descartes opened the way to any further progress in geometry... [R]oughly... during the next thousand years Pappus was the sole geometrician of great ability; and... almost the only other pure mathematicians of exceptional genius were Hipparchus and Ptolemy who laid the foundations of trigonometry, and Diophantus who laid those of algebra.

Ptolemy’s great treatise, the Almagest... was founded on the observations and writings of Hipparchus, and from the notes there given we infer that the chief discoveries of Hipparchus, and from the notes there... we infer that the chief discoveries of Hipparchus... [H]is observations and calculations... placed the subject for the first time on a scientific basis. ...[His] theory accounted for all the facts which could be determined with the instruments then in use, and... enabled him to calculate... eclipses with considerable accuracy. ...No further advance in the theory of astronomy was made until the time of Copernicus, though the principles laid down by Hipparchus were extended and worked out in detail by Ptolemy.
Investigations such as these naturally led to trigonometry, and Hipparchus must be credited with the invention of that subject. ...[I]n plane trigonometry he constructed a table of chords of arcs... practically the same as... natural sines; and... in spherical trigonometry he had some method of solving triangles: but his works are lost, and we can give no details.

It is believed... that the elegant theorem, printed as Euc. VI. D... known as Ptolemy’s Theorem, is due to Hipparchus and was copied... by Ptolemy. It contains implicitly the addition formulæ for $sin(A\pm B)$ and $cos(A\pm B)$ ; and Carnot shewed how the whole of elementary plane trigonometry could be deduced from it.

Hero of Alexandria... placed engineering and land surveying on a scientific basis. ...He was ...acquainted with the trigonometry of Hipparchus, but ...nowhere quotes a formula or expressly uses the value of the sine, and it is probable that like the later Greeks he regarded trigonometry as forming an introduction to, and being an integral part of, astronomy.

[T]hroughout the first century after Christ... the only original works of any ability were... by Serenus and... Menelaus. ...Those by Serenus... were on the plane sections of the cone and cylinder... edited by E. Halley... 1710. That by Menelaus... was on spherical trigonometry, investigated in the Euclidean method... translated by E. Halley... 1758. The fundamental theorem... is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them (book III. prop. 1). Menelaus also wrote on the calculation of chords... plane trigonometry; this is lost.
- Ref: 1) Tr. Edmond Halley, Menelai Sphaericorum libri III (1758) 2) Edmond Halley, Conics of Apollonius (1710) with the treatise by Serenus, De sectione cylindri et coni.

Ptolemy... produced his great work on astronomy, which will preserve his name as long as the history of science endures. This... is... the Almagest...founded on the writings of Hipparchus, and, though it did not... advance the theory... it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise.

Ptolemy made observations at Alexandria from the years 125 to 150... .but an indifferent practical astronomer, and the observations of Hipparchus are... more accurate...

The work is divided into thirteen books. ...[T]he first... treats of trigonometry, plane and spherical; gives a table of chords, i.e. of natural sines (... substantially correct and... probably taken from... Hipparchus); and explains the obliquity of the ecliptic... It became... the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be... the centre of the solar system.

The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed... But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly... as a convenient way of expressing known facts, it is not only legitimate but convenient. It was as good a theory as with their instruments and knowledge it was possible to frame, and... it exactly corresponds to the expression of a given function as a sum of sines or cosines, a method... of frequent use in... analysis.

Ptolemy had shewn... geometry could be applied to astronomy, but... indicated how new methods of analysis like trigonometry might be... developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence... Pappus...

Pappus wrote several books, but... only one which has come down to us is his Συναγωγή [Synagoge], a collection of mathematical papers... in eight books of which... part... have been lost... published by F. Hultsch... 1876—8. This collection was intended to be a synopsis of Greek mathematics... with comments and additional propositions... we rely largely on it for... knowledge of... works now lost. ...[T]he sixth [book deals] with astronomy including, as subsidiary subjects, optics and trigonometry ...His work... and... comments shew... he was a geometrician of great power; but it was his misfortune to live at a time when but little interest was taken in geometry, and... the subject, as then treated, had been practically exhausted.
- Ref: Fridericus Hultsch, Pappi Alexandrini collectionis quae supersunt (1876)

Isaac Argyrus... wrote three astronomical tracts... one on geodesy... one on geometry... and one on trigonometry, the manuscript of which is in the Bodleian at Oxford.

The Mathematics of the Middle Ages and the Renaissance... begins about the sixth century, and may be said to end with the invention of analytical geometry and infinitesimal calculus. The characteristic feature of this period is the creation of modern arithmetic, algebra, and trigonometry.

Arya-Bhata... is frequently quoted by Brahmagupta, and... many commentators [write that] he created algebraic analysis though it has been suggested that he may have seen Diophantus’s Arithmetic. ...[H]is Aryabhathiya... consists of the enunciations of... rules and propositions... in verse. There are no proofs, and the language is... obscure and concise... [I]t long defied all efforts to translate it.
The book is divided into four parts: of these three are devoted to astronomy and the elements of spherical trigonometry; the remaining part... enunciations of thirty-three rules in arithmetic, algebra, and plane trigonometry. It is probable that Arya-Bhata, like Brahmagupta and Bhaskara... regarded himself as an astronomer, and studied mathematics only so far as... was useful... in his astronomy. ...In trigonometry he gives a table of natural sines of the angles in the first quadrant, proceeding by multiples of 3 3/4° defining a sine as the semichord of double the angle. ...A large proportion of the geometrical propositions which he gives are wrong.
- Resources 1) ed., H. Kern, Sanskrit text of Aryabhathiya (1874); 2) H. Kern article, Journal of the Asiatic Society, London, 1863, Vol. XX., pp. 371-387. 3) L. Rodet, French translation of algebra & trigonometry portion, Journal Asiatique (1879) Series 7, Vol. XIII., pp. 393-434.

Brahmagupta... wrote a work in verse... Brahma-Sphuta-Siddhanta... system of Brahma in astronomy. ...Chaps. XII. and XVIII ...are devoted to arithmetic, algebra, and geometry... It is impossible to say whether the whole of Brahmagupta’s results... are original. He knew of Arya-Bhata’s work, for he reproduces the table of sines... and it is likely that some progress in mathematics had been made by Arya-Bhata’s... successors, and that Brahmagupta was acquainted with their works; but there seems no reason to doubt that the bulk of Brahmagupta’s algebra and arithmetic is original, although perhaps influenced by Diophantus... the origin of the geometry is more doubtful, probably some... is derived from Hero...

Bhaskara... is said to have been... lineal successor of Brahmagupta as head of an astronomical observatory at Ujein... sometimes written Ujjayini. He wrote an astronomy... Lilavati is on arithmetic... Bija Ganita is on algebra; the third and fourth... on astronomy and the sphere... [I]t is... probable that Bhaskara was acquainted with... Arab works... written in the tenth and eleventh centuries, and with... Greek mathematics... transmitted through Arabian sources. ...[F]rom the ...table of contents ...Arithmetical progressions, and sums of squares and cubes. Geometrical progressions. Problems on triangles and quadrilaterals. Approximate value of π. Some trigonometrical formulae. ..[T]he book ends with a few questions on combinations.
This is the earliest known work which contains a systematic exposition of the decimal system of numeration. ...Chapters on algebra, trigonometry, and geometrical applications exist, and fragments of them have been translated by Colebrooke. Amongst the trigonometrical formulae is one... equivalent to... $d(\sin \theta )=\cos \theta d\theta$ .

Like the Greeks, the Arabs never used trigonometry except... with astronomy; but they introduced the trigonometrical expressions... now current, and worked out the plane trigonometry of a single angle. They were also acquainted with the elements of spherical trigonometry.

The trigonometrical ratios seem to have been the invention of Albategni... who was among the earliest of the many distinguished Arabian astronomers. He wrote the Science of the Stars (published by Regiomontanus... 1537)... [where] he determined his angles by "the semi-chord of twice the angle," i.e. by the sine of the angle (taking the radius vector as unity). Hipparchus and Ptolemy... had [also] used the chord.

Albuzani... also known as Abul-Wafa... introduced all the trigonometrical functions, and constructed tables of tangents and cotangents. He was celebrated not only as an astronomer but as one of the most distinguished geometricians of his time.

The Arabs were at first content to take the works of Euclid and Apollonius for their text-books in geometry without attempting to comment on them, but Alhazen issued in 1036 a collection of problems something like the Data of Euclid, this was translated by Sédillot... in 1836. Besides commentaries on the definitions of Euclid and on the Almagest Alhazen also wrote a work on optics which shews that he was a geometrician of considerable power: this was published at Bale in 1572, and served as the foundation for Kepler’s treatise.

Bhaskara, ... there is every reason to believe ...was familiar, with the works of the Arab school... and... that his writings were... known in Arabia.

The Arab schools continued to flourish until the fifteenth century... [T]he work of the Arabs in arithmetic, algebra, and trigonometry was of a high order of excellence. They appreciated geometry and the applications of geometry to astronomy, but they did not extend the bounds of the science.

The earliest Moorish writer of distinction... is Geber ibn Aphla... His works... chiefly... astronomy and trigonometry, were translated into Latin by Gerard... 1533. He seems to have discovered the theorem that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.

Leonardo Fibonacci... known as Leonardo of Pisa... in 1202 published... Algebra et almuchabala (the title being taken from Alkarismi’s work) but... known as the Liber Abaci. He there explains the Arabic system of numeration, and remarks on its great advantages over the Roman system. He then gives an account of algebra, and points out the convenience of using geometry to get rigid demonstrations of algebraical formulae. He shews how to solve simple equations... All the algebra is rhetorical. ...Roger Bacon ...recommends the algorism (...the arithmetic founded on the Arab notation) ...[B]y the year 1300, or at the latest 1350, these numerals were familiar both to mathematicians and to Italian merchants. ...He ...wrote a geometry termed Practica Geometriae ...1220. This is a good compilation and some trigonometry is introduced; among other propositions and examples he finds the area of a triangle in terms of its sides.

The Mathematics of the Renaissance... Mathematicians had barely assimilated the knowledge obtained from the Arabs, including their translations of Greek writers, when the refugees who escaped from Constantinople after the fall of the eastern empire brought the original works and the traditions of Greek science into Italy. Thus by the middle of the fifteenth century the chief results of Greek and Arabian mathematics were accessible to European students.
The invention of printing about that time rendered the dissemination of discoveries comparatively easy. ...[W]hen a mediaeval writer "published" ... the results were known to only a few of his contemporaries. This had not been the case in classical times for... until the fourth century of our era Alexandria was the... centre for the reception and dissemination of new works and discoveries. In mediaeval Europe... there was no common centre through which men of science could communicate with one another, and to this cause the slow and fitful development of mediaeval mathematics may be partly ascribed.
The last two centuries of this period... described as the renaissance, were distinguished by great mental activity in all branches of learning. The creation of a fresh group of universities... testify to the... desire for knowledge. The discovery of America in 1492 and the discussions that preceded the Reformation flooded Europe with new ideas... ut the advance in mathematics was at least as well marked as that in literature and... politics.
During the first part of this time the attention of mathematicians was to a large extent concentrated on syncopated algebra and trigonometry.

Regiomontanus was among the first to take advantage of the recovery of the original texts of the Greek mathematical works... the earliest notice in modern Europe of the algebra of Diophantus is [his] remark... that he had seen a copy... at the Vatican. He was also well read in the works of the Arab mathematicians.
The fruit of this study... his De Triangulis... 1464... the earliest modern systematic exposition of trigonometry, plane and spherical, though the only trigonometrical functions introduced are... the sine and cosine. It is divided into five books. The first four... plane trigonometry... in particular... determining triangles from three given conditions. The fifth book is... spherical trigonometry. The work was printed in five volumes... 1533, nearly a century after the death of Regiomontanus.

[A]lgebra and trigonometry were still only in the rhetorical stage of development, and when every step of the argument is expressed in words at full length it is by no means easy to realise all that is contained in a formula.

Regiomontanus did not hesitate to apply algebra to the solution of geometrical problems. An... illustration of this is to be found in his discussion of a question... in Brahmagupta’s Siddhanta... to construct a quadrilateral, having its sides of given lengths, which should be inscribable in a circle. The solution given by Regiomontanus was effected by means of algebra and trigonometry: this was published by C. G. von Murr... 1786.

Georg Purbach, first the tutor and then the friend of Regiomontanus... wrote a work on planetary motions... 1460; an arithmetic... 1511; a table of eclipses... 1514; and a table of natural sines... 1541.

There are but few special symbols in trigonometry, I... however add here... all that I have been able to learn... The current sexagesimal division of angles is derived from the Babylonians through the Greeks. The Babylonian unit angle was the angle of an equilateral triangle; following their usual practice... this was divided into sixty equal parts or degrees, a degree was subdivided into sixty equal parts or minutes, and so on. The word sine was used by Regiomontanus and was derived from the Arabs: the terms secant and tangent were introduced by Thomas Finck... in his Geometriae Rotundi, Bâle, 1583: the word cosecant was (I believe) first used by Rheticus in his Opus Palatinum, 1596: the terms cosine and cotangent were first employed by E. Gunter in his Canon Triangulorum, London, 1620. The abbreviations sin, tan, sec were used in 1626 by Albert Girard, and those of cos and cot by Oughtred in 1657; but these contractions did not come into general use till Euler re-introduced them in 1748. The idea of trigonometrical functions originated with John Bernoulli, and this view of the subject was elaborated in 1748 by Euler in his Introductio in Analysin Infinitorum.

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 $\cos \theta +i\sin \theta =e^{i\theta }.$

A History of Mathematics (1893)

by Florian Cajori. A source @Archive.org.

Claudius Ptolemaeus, a celebrated astronomer, was a native of Egypt. ...The chief of his works are the Syntaxis Mathematica (or the Almagest, 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 spherical trigonometry 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 hyperbola.

[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.

A Brief History of Mathematics (1903)

by Karl Fink, Tr. Wooster Woodruff Beman & David Eugene Smith, 2nd revised edn.

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.

Our Oriental Heritage (1935)

Being a history of civilization in Egypt and the Near East to the death of Alexander, and in India, China and Japan from the beginning to our own day; with an introduction on the nature and foundations of civilization in the book series, The Story of Civilization, Part I by Will Durant.

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."
- p. 526.

The Hindus were not so successful in geometry. In the measurement and construction of altars the priests formulated the Pythagorean theorem... 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 Surya Siddhanta provided a system of trigonometry more advanced than anything known to the Greeks.
- p. 528.

The History of Trigonometry (1969)

an overview by Edward S. Kennedy, in Historical Topics for the Mathematics Classroom (1969) A source.

[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 sexigesimal system developed in Mesopotamia...

At the end of Book I, Chapter 11, of the Almagest 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 interpolation. 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 Surya Siddhanta 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 Menelaus's theorem... [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 gnomon... These are tables of $\mathrm {Cot} \theta \equiv R\cot \theta$ usually 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...

Bürgi’s "Progress Tabulen" (1620): logarithmic tables without logarithms (2010, 2013)

by Denis Roegel (2010, last updated 2013) A source @LOCOMAT, The Loria Collection of Mathematical Tables

Tycho Brahe... [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)' Fundamentum astronomicum. It was of great value... and was going to be a direct competitor to the method of logarithms.
- Ref: John Louis Emil Dreyer, "On Tycho Brahe’s manual of trigonometry" The Observatory (1916) Vol. 39, pp. 127–131. Ref: Nicolas Reimarus, Fundamentum astronomicum: id est, nova doctrina sinuum et triangulorum (1588) Strasbourg: Bernhard Jobin.

Prosthaphaeresis... had been devised by Johannes Werner... and was likely brought to Tycho Brahe by... Paul Wittich... 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.
- Ref: Victor E. Thoren, The lord of Uraniborg: A biography of Tycho Brahe (1990) p. 237.

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 spherical law of cosines... $\cos c=\cos a\cos b+\sin a\sin b\cos C$ Using 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...

"Jost Bürgi’s Method for Calculating Sines" (Feb 2, 2016)

by Menso Folkerts, Dieter Launert, Andreas Thom, Version 2, p. 8, arXiv:1510.03180.

All [previous] procedures for calculating chords and sines... [were] based in principle on the method which Ptolemy had presented in his Almagest. 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 ...Fundamentum Astronomiae.

In 1605 Bürgi went to Prague and lived there as a watchmaker at the Emperor’s Chamber until 1631... [and] was... involved in astronomical observations and interpretations. ...Christoph Rothmann... worked at the court. Among... temporary visitors... Paul Wittich... and Nicolaus Reimers Ursus... who had both worked with Tycho Brahe. Wittich... brought... knowledge of prosthaphaeresis... 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 trigonometry by Bürgi, entitled Fundamentum Astronomiae.

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°.

External links

Wikipedia

Wikipedia has an article about:

History of trigonometry

The trigonometric functions under Historical Topics @MacTutor History of Mathematics Archive
Trigonometric Delights by Eli Maor @Princeton University Press