mathematics

"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."

"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. Simpsons Algebra, Geom. Probl. 15. Dr. Robertsons Navigation, Emerson, and [Benjamin] Martin. The analogy of the prop. in particular, is to be met with in Trigonometria Britannica, [Henry] Sherwins Tables, de la Caille, Dr. Simpson, and Ward."

"[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 ..., 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."

"[C]oncerning ... 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 notation, which the Greeks afterwards adopted for astronomical purposes. Herodotus expressly states that the polos and ' (...sundials) and the twelve parts of the day were made known to the Greeks from Babylon. Much of the trigonometry and of the later Greeks may... have been... derived from Babylonian sources."

"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."

"'... 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..."

"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."

"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."

"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."

"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."

"[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."

"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."

"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."

"[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."

"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... 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..."

"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..."

"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."

"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."

"[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 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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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 (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."

"As in the rest of mathematical sciences, so in trigonometry, were the Arabs pupils of the Hindus…"

"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 ) and its excellent if still imperfect measurements of and , correcting the flagrant errors of Ptolemy."

"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."

"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 , 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."

"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."

"Now, it is worth remarking, that this property of the table of sines, which has been so long known in the East, was not observed by the mathematicians of Europe till about two hundred years ago […] If we were not already acquainted withthe high antiquity of the astronomy of Hindostan, nothing could appear more singular than to find a system of trigonometry, so perfect in its principles, in a book so ancient as the Surya Siddhanta […]’ ‘In the progress of science […] the invention of trigonometry is to be considered as a step of great importance, and of considerable difficulty. It is an application of arithmetic to geometry […] (and) a little reflection will convince us, that he, who first formed the idea of exhibiting, in arithmetical tables, the ratios of the sides and angles of all possible triangles, and contrived the means of constructing such tables, must have been a man of profound thought, and of extensive knowledge. However, ancient, therefore, any book may be, in which we meet with a system of trigonometry, we may be assured, that it was not written in the infancy of science.’ ‘As we cannot, therefore, suppose the art of trigonometrical calculation to have been introduced till after a long preparation of other acquisitions, both geometrical and astronomical, we must reckon far back from the date of the Surya Siddhanta, before we come to the origin of the mathematical sciences in India […] Even among the Greeks […] an interval, of at least 1000 years, elapsed from the first observations in astronomy, to the invention of trigonometry; and we have surely no reason to suppose, that the progress of knowledge has been more rapid in other countries."

"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. , 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 '; 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 ."

"[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?"

"[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?"

"[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 Euclids Elements, Archimedes de Sphaera et Cylindro, and Dr. Hamiltons Conic Sections."

"[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."

"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."

"Similarly to the spread of the Indian place-value system, Indian trigonometry came to Europe via the Arab world, for example through the work on astronomy and trigonometry of Abu Abdallah Mohammad ibn Jabir al-Battani (ca. 850–929), also known as Albategnius, whose Kitab al-Zij was translated into Latin."