History Of Mathematics

"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 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..."

"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..."

"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."

"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."

"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."

"[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."

"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."

"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."

"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."

"[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."

"[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."

"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."

"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."

"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] Cunns 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."

"[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?"

"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]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."

"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."

"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."

"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."

"Such weak foundations Spherical Trigonometry has had to this day!"

"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 ... 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."

"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."

"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 s 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, thenge : 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."

"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 ' ή μεγίστη [maximum]... In it... he reproduces the theorem of the six segments... and founds upon it a system of trigonometry, plane and spherical."

"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.""

"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."

"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 ..."

"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 '."

"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."

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

"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 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 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."

"[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."