History of trigonometry

"The 14th Book of the Elements, or the book of on 'the Regular Solids', consists of seven propositions... The... treatise on 'Risings'... contains only six propositions, of which the first three, deal... with s... The only interesting proposition is the 4th... Divide the 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."

"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 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 ... 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 ... 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 ), 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 by : 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 es, more correct than Ptolemy's. He invented the method of describing the positions of places by reference to and . 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."

"[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 ', 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 ... ... [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 '."

"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."

"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 , and longitude and for the formation of a table of s 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 ."

"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."

"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 '... 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."

"'... 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."

"Ptolemy... produced his great work on astronomy, which will preserve his name as long as the history of science endures. This... is... the '...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."

"Isaac Argyrus... wrote three astronomical tracts... one on ... 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 , 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 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."

"... 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 ... 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 (...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."

"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 ’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 ... 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 division of angles is derived from the Babylonians through the Greeks. The Babylonian unit angle was the angle of an ; 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 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 , 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."