History Of Mathematics

"[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity which is the kernel of modern classical analysis."

"Let us define the job of the astronomer in the classical phrase as "saving the appearances" of the celestial movements. ...[A]n astronomical theory must "save" in the sense of "preserve"– ...[i.e.,] it must not deny any of the apparent celestial movements as appearances, and in this bare sense, it might merely comprise a record of observed positions... [I]n order to take into account all the apparent movements, it must... predict apparent movements in the future from those observed in the past. ...[T]o be able to look backwards and forwards beyond recorded positions of the planets, it must arrange the celestial movements in a pattern of orderly recurrence. ...[B]y setting up this pattern of order, it saves... in a second sense... [I]t gives them salvation... by making them intelligible and... explicating them in terms of a permanent order."

"[I]t does not follow that because heaven moves in a circle that the earth or something else rests at its center... because circular movement... does not require... any body at rest at the center... [I]t is possible to imagine that the earth moves with heaven in its daily movement... [A]ssuming that the earth moves with or contrariwise to heaven, it does not follow... that celestial movement would stop; so... this circular movement of heaven does not require that the earth should remain motionless at the center of the world. ...[I]t is not impossible that the whole earth moves, with a different movement or in another way... For otherwise the parts near the center would never reach the place where they are destroyed and would be perpetual... Against this objection and against the principal argument is the manifest evidence of heaven itself, for to save appearances and from our observations of celestial movements... there are spherical bodies called epicycles in heaven, and that each epicycle has its own proper circular movement about its center... different from the... heavenly sphere... [I]t is impossible... that any body should be at rest in the center of this epicycle."

"[T]he principal reason why the heliocentric idea fell perfectly flat, was the rapid rise of practical astronomy, which had commenced from the time when the Alexandrian Museum became a centre of learning in the Hellenistic world. Aristarchus had no other phenomena to "save" except the stationary points and retrograde motions of the planets as well as their change of brilliancy; he may even have neglected the inequality of the sun's apparent motion originally discovered by Euktemon and recognized by Kalippus. But when similar and much more marked inequalities began to be perceived in the motions of the other planets, the hopelessness of trying to account for them by the beautifully simple idea of Aristarchus must have given the deathblow to his system, which thereby even among mathematicians lost its only claim to acceptance, that of being able to "save the phenomena." Most likely, as we have already said, these new inequalities had already more or less dimly commenced to make themselves felt in the days of Apollonius... and in that case we can understand why he did not feel disposed to simplify the system of movable excentrics by gathering the reins of all the unruly planetary steeds into one mighty hand, that of ."

"Here Pharnaces... broke in... you are not going to draw me on... to answer your charges against the Stoics, unless we first get an account of your conduct in turning the universe upside." Lucius smiled : "Yes, my friend," he said, "only do not threaten us with... heresy, such as used to think that the Greeks should have had served upon Aristarchus of Samos, for shifting the hearth of the Universe, because that great man attempted 'to save phenomena' with his hypothesis that the heavens are stationary, while our earth moves round in an oblique orbit, at the same time whirling about her own axis. ...[W]hy are those who assume that the moon is an earth turning things upside down, any more than you who fix the earth where she is, suspended in mid air, a body considerably larger than the moon? At least mathematicians tell us so, calculating the magnitude of the obscuring body from... eclipses, and from the passages of the moon through the shadow. For the shadow of the earth is less as it extends, because the illuminating body is greater, and its upper extremity is fine and narrow, as even Homer... did not fail to notice. He called night 'pointed' because of the sharpness of the shadow. Such... is the body by which the moon is caught in her eclipses, and yet she barely gets clear by a passage equal to three of her own diameters. Just consider how many moons go to make an earth, if the earth cast a shadow as broad at its shortest as three moons. Yet you have fears for the moon lest she should tumble, while as for our earth, Aeschylus has perhaps satisfied you that Atlas'Stands, and the pillar which parts Heaven and Earth His shoulders prop, no load for arms t' embrace!'Then you think that under the moon there runs light air, quite inadequate to support a solid mass, while the earth, in Pindar's words, 'is compassed by pillars set on adamant.' And this is why Pharnaces has no fear... of the earth's falling, but pities those who lie under the orbit of the moon... Yet the moon has that which helps her against falling, in her very speed and the swing of her passage round, as objects placed in slings are hindered from falling by the whirl of the rotation. For everything is borne on in its own natural direction unless this is changed by some other force. Therefore the moon is not drawn down by her weight, since that tendency is counteracted by her circular movement. ...[B]ut the earth, being destitute of any other movement, might naturally be moved by its own weight; being heavier than the moon not merely in proportion to its greater bulk, but because the moon has been rendered lighter by heat and conflagration. It would actually seem that the moon, if she is a fire, needs earth all the more, a solid substance whereon she moves and to which she clings, so feeding and keeping up the force of her flame. For it is impossible to conceive fire as maintained without fuel. But you Stoics say that our earth stands firm without foundation or root." "Of course," said Pharnaces, "it keeps its proper and natural place, as being the essential middle point, that place around which all weights press and bear, converging towards it from all sides. But all the upper region, even if it receive any earth-like body thrown up with force, immediately thrusts it out hitherward, or rather lets it go, to be borne down by its own momentum.""

"When Copernicus, instead of leaving the earth at rest in the center of the world, gave it not only two rotations on its own center, but... an annual revolution around the sun, astronomers were able to maintain that these hypotheses are not... realities, that it suffices for them to be fictions by which the phenomena are saved in a simpler... more exact manner than... Ptolemy's devices. But physicists did not willingly use this loophole; they not only saw in the system of Copernicus a model enabling them to construct new tables of celestial movements, they also imagined something... that claims to reveal a truth. They imagined that the earth is a planet of the same nature as Venus, Mars, or Jupiter. The problem... can each of the... wandering stars be a world similar to the world in which we are living, having at its center an earth covered by water, surrounded by air?"

"The system of Anaxagoras, like that of Empedokles, aimed at reconciling the Eleatic doctrine that corporeal substance is unchangeable with... a world which... presents the appearance of coming into being and passing away. The conclusions of Parmenides are... accepted and restated. Nothing can be added to all things; for there cannot be more than all, and all is always equal... Nor can anything pass away. What men commonly call coming into being and passing away is... mixture and separation... This... reads almost like a prose paraphrase of Empedokles (fr. 9); and it is... probable... Anaxagoras derived his theory... from his younger contemporary, whose poem was most likely published before his own treatise. ...Empedokles sought to save the world of appearance by maintaining that the opposites—hot and cold, moist and dry—were things, each...real in the Parmenidean sense. Anaxagoras regarded this as inadequate. ...[T]hings of which the world is made are not "cut off with a hatchet" ...the true formula must be: There is a portion of everything in everything."

"In the 1590s... Kepler adopted the ideas of Copernicus. In the heliocentric model... the simultaneous motion of the earth around the sun and about its own axis explained the observed motion of the planets and stars. Kepler set out to prove that this... hypothesis... an attempt to "save the appearances", did... correspond with reality. In doing so, however, he noticed that the circular orbits... proposed by Copernicus were not in keeping with his... observations. ...Kepler wanted... to glorify God, who... was responsible for the harmonious arrangement of the universe... This aim is... in the... first lines of the preface to The Secret of the Cosmos: "It is my intention... to show... that the most great and good Creator, in the creation of this moving universe and the arrangement of the heavens, looked to these five regular solids... so celebrated from the time of Pythagoras and Plato... and that he fitted to the nature of those solids the number of the heavens, their proportions and the law of their motions.""

"In his commentary to the Physics of Aristotle, Simplicius gives us an interesting quotation from a commentary to the Meteorology of Posidonius, written by ... Dealing with the difference between physics and astronomy, Geminus says... to the former... belongs the examination of the nature, power, quality, birth, and decay of the heavens and the stars, but astronomy does not attempt... this, it makes known the arrangement of the heavenly bodies, it investigates the figure and size and distance of earth and sun and moon, the eclipses and conjunctions of stars and the quality and quantity of their motions... with help from arithmetic and geometry. But although the astronomer and the physicist often prosecute the same research... they do not proceed in the same manner, the latter seeking for causes and moving forces, while the astronomer finds certain methods, adopting which the observed phenomena can be accounted for. "For why do sun, moon, and planets appear to move unequally? Because, when we assume their circles to be excentric or the stars to move on an epicycle, the appearing anomaly can be accounted for.., and it is necessary to investigate in how many ways the phenomena can be represented, so that the theory of the wandering stars may be made to agree with the ... Therefore also... Herakleides of Pontus... said that also when the earth moved... and the sun stood still.., could the irregularity observed relatively to the sun be accounted for. ...[I]t is not the astronomer's business to see what by its nature is immovable and of what kind the moved things are, but framing hypotheses as to some things being in motion and others being fixed, he considers which hypotheses are in conformity with the phenomena in the heavens. He must accept as his principles from the physicist, that the motions of the stars are simple uniform, and regular, of which he shows that the revolutions are circular, some along parallels, some along oblique circles." This... distinguishes clearly between the physically true causes of observed phenomena and a mere mathematical hypothesis which (whether true or not) is able to "save the phenomena." This expression is ... a favourite... with Simplicius, who doubtless had it from the authors long anterior to himself, from whose works he derived his knowledge. It means that a certain hypothesis is able to account for the apparently irregular phenomena revealed by observation, which at first sight are puzzling and seem to defy all attempts to make them agree with the assumed regularity of all motions, both as to velocity and direction. In this passage Geminus points out that an astronomer's chief duty is to frame a theory which can represent the observed motions and make them subject to calculation, while it is for this purpose quite immaterial whether the theory is physically true or not."

"[I]n Plutarch's book On the face in the disc of the Moon...[o]ne of the persons in the dialogue, being called to account for turning the world upside down, says that he is quite content so long as he is not accused of impiety, "like as Kleanthes held that Aristarchus of Samos ought to be accused of impiety for moving the hearth of the world.., as the man in order to save the phenomena supposed... that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis.""

"The language... as to the Moon's movements and the Epicyclic Theory... settled later on by Ptolemy... deserve careful examination... Astronomy had... become... technical and mathematical, sharply distinguished from general physical enquiry. Even Hipparchus... "though he loved truth above everything," yet was not versed in "natural science," and was content to explain the motions of the heavenly bodies by an hypothesis mathematically consistent, without care for its physical truth... Take the case of the Moon. Ptolemy was content to "save the phenomena"... by a system which admirably accounted for her very complex movements, but which involved the consequence that her distance from us at the nearest must he half that at the farthest, and her angular diameter therefore double!"

"Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle... Reason is employed in another way, not as furnishing a sufficient proof... but... confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained [saved] (possunt salvari apparentia sensibilia); not, however, as if this proof were sufficient, forasmuch as some other theory might explain them."

"The first book contains the general description of the universe and the foundations by which he undertakes to save the appearances and the observations of all ages. He adds as much of the doctrine of sines and plane and spherical triangles as he deemed necessary to the work."

"For it is now clear to me that there are no solid spheres in the heavens... But there really are not any spheres in the heavens.... and those which have been devised by the authors to save the appearances exist only in the imagination, for the purpose of permitting the mind to conceive the motion which the heavenly bodies trace in their course and, by the aid of geometry, to determine the motion numerically through the use of arithmetic... Of course, almost the whole of antiquity and also very many recent philosophers consider as certain and unquestionable the view that the heavens are made of a hard and impenetrable substance, that it is divided into various spheres, and that the heavenly bodies, attached to some of these spheres, revolve on account of the motion of these spheres. But this opinion does not correspond to the truth of the matter..."

"The statement of Diogenes, that Herakleides attended the Pythagorean schools is of... importance... as it is... likely... their influence (which is also perceptible in his ideas about atoms, which he calls masses...), tended to convince him of the truth of the... simple explanation of the daily motion of the stars proposed by Hiketas and Ekphantus. ... He first alludes to Herakleides when discussing the chapter in which Aristotle considers the motion of the starry vault. Aristotle... remarks that, taking for granted that the earth is at rest, the starry sphere... and the planets might either both be at rest, or both be in motion, or one be at rest and the other in motion. And these cases he considers (says Simplicius) "on account of there being some, among whom were Herakleides of Pontus and Aristarchus, who believed they could save the phenomena (account for the observed facts) by making the heavens and the stars be immovable, but making the earth move round the poles of the equator... from the west, each day one revolution as near as possible; but 'as near as possible' is added on account of the [daily] motion of the sun of one part (degree); so that, if then the earth does not move, which presently he (Aristotle) is going to show, the hypothesis of both being at rest cannot possibly save the phenomena.""

"Now, so far as appearances go, it... the same thing whether the heavens, that is, all space with its contents, revolve round a spectator at rest in the earth's centre, or whether that spectator... turn round in the opposite direction in his place, and view them in succession. The aspect of the heavens, at every instant, as referred to his horizon (which must be supposed to turn with him), will be the same in both suppositions. And since... appearances are also, so far as the stars are concerned, the same to a spectator on the surface as to one at the centre, it follows that, whether we suppose the heavens to revolve without the earth, or the earth within the heavens, in the opposite direction, the diurnal phenomena, to all its inhabitants, will be no way different. The Copernican astronomy adopts the latter as the true explanation of these phenomena, avoiding... the necessity of otherwise resorting to the cumbrous mechanism of a solid but invisible sphere, to which the stars must be supposed attached, in order that they may be carried round the earth without derangement of their relative situations inter se [among themselves]. Such a contrivance would..., suffice to explain the diurnal revolution of the stars, so as to "save appearances;" but the movements of the sun and moon, as well as those of the planets, are incompatible with such a supposition... On the other hand, that a spherical mass of moderate dimensions (or, rather, when compared with the surrounding and visible universe, of evanescent magnitude), held by no tie, and free to move and to revolve, should do so, in conformity with those general laws which, so far as we know, regulate the motions of all material bodies, is so far from being a postulate difficult to be conceded, that the wonder would rather be should the fact prove otherwise. As a postulate, therefore, we shall henceforth regard it... The earth's rotation on its axis so admitted, explaining, as it evidently does, the apparent motion of the stars in a completely satisfactory manner, prepares us for... its motion, bodily, in space... to explain... the apparently complex and enigmatical motions of the sun, moon, and planets. The Copernican astronomy adopts this idea in its full extent, ascribing to the earth, in addition to its motion of rotation about an axis, also one of translation or transference through space, in such a course or orbit, and so regulated in direction and celerity, as, taken in conjunction with the motions of the other bodies of the universe, shall render a rational account of the appearances they successively present... [i.e.,] an account of which the several parts, postulates, propositions, deductions, intelligibly cohere, without contradicting... experience. In this view of the Copernican doctrine it is rather a geometrical conception than a physical theory, inasmuch it simply assumes the requisite motions, without attempting to explain their mechanical origin, or assign them any dependence on physical causes. The Newtonian theory of gravitation supplies this deficiency, and, by showing that all the motions required by the Copernican conception must, and that no others can, result from a single, intelligible, and very simple dynamical law, has given a degree of certainty to this conception, as a matter of fact, which attaches to no other creation of the human mind."

"Rosen quotes various passages from De Revolutionibus in which Copernicus uses without distinction, the terms: principle, assumption and hypothesis, for fundamental s: "Furthermore astronomy, that divine rather than human science, which inquires into the loftiest things, is not free from difficulties. Especially with regard to its principles (principia) and assumptions (assumptiones), which the Greek call 'hypotheses' (hypotheses)..." These axioms, in order to be recognized as true, must satisfy two conditions: 1) apparentias salvare (save the appearances): "the results deduced from them must agree with the observed phenomena within satisfactory limits of error."..: 2) aequalitatem tueri [to protect equality]: "They must be consistent with certain preconceptions, called 'axioms of physics,' such as that every celestial motion is circular, every celestial motion is uniform, and so forth.""

"Greek astronomers observed intricate motions of the sun, moon, and planets on the two-dimensional sky. They explained them—saved the appearances—by positing simple regular motions... in three dimensions. The success... [was] brought to a triumphant conclusion by Kepler..."

"When Newton wrote his Mathematical Principles of Natural Philosophy and System of the World, he distinguished the phenomena to be saved from the reality he postulated. He distinguished the "absolute magnitudes" that appear in his axioms from their "sensible measures" which are determined experimentally. He discussed carefully the ways in which, "the true motions of particular bodies [may be determined] from the apparent," via the assertion that "the apparent motions... are the differences of true motions.""

"He gives the Greek text of the Placita Philosophorum... about Philolaus, Herakleides and Ekphantus, and continues: " Occasioned by this I also began to think of a motion of the earth, and although the idea seemed absurd, still, as others before me had been permitted to assume certain circles in order to explain the motions of the stars, I believed it would readily be permitted me to try whether on the assumption of some motion of the earth better explanations of the revolutions of the heavenly spheres might not be found. And thus I have, assuming the motions which I in the following work attribute to the earth, after long and careful investigation, finally found that when the motions of the other planets are referred to the circulation of the earth and are computed for the revolution of each star, not only do the phenomena necessarily follow therefrom, but the order and magnitude of the stars and all their orbs and the heaven itself are so connected that in no part can anything be transposed without confusion to the rest and to the whole universe." According to this statement, Copernicus first noticed how great was the difference of opinion among learned men as to the planetary motions; next he noticed that some had even attributed some motion to the earth, and finally he considered whether any assumption of that kind would help matters. ...It must then have struck him as a strange coincidence that the revolution of the sun round the and the revolution of the epicycle-centres of Mercury and Venus round the zodiac should take place in the same period, a year, while the period of the three outer planets in their epicycles was the synodic period, i.e. the time between two successive oppositions to the sun. This curious relationship between the sun and the planets must have struck scores of philosophers, but at last the problem was taken up by a man of a thoroughly unprejudiced mind and with a clear mathematical head. Probably it suddenly flashed on him that perhaps each of the deferents of the two inner planets and the epicycles of the three outer ones simply represented an orbit passed over by the earth in a year, and not by the sun! His emotion on finding that this assumption would really "save the phenomena," as the ancients had called it, that it would explain why Mercury and Venus always kept near the sun and why all the planets annually showed such strange irregularities in their motions, his emotion on finding this clear and beautifully simple solution of the ancient mystery must have been as great as that which long after overcame Newton when he discovered the law of universal gravitation. But Copernicus is silent on this point. This may have been the way followed by Copernicus, but we cannot be sure..."

"[T]he heliocentric idea of Aristarchus might just as well have sprung out of the epicyclic theory as from that of movable excentrics... But with regard to the curious dependence of each planet on the sun in the Ptolemaic system.., the zodiacal inequality of the planets showed that in any case a simple circular motion would not "save the phenomena"; while the discovery of a strongly marked inequality of the moon, depending on its position with regard to the sun, confirmed the notion that the sun was mixed up in the theories of all the celestial bodies alike. ...For more than fourteen hundred years it remained the Alpha and Omega of theoretical astronomy, and whatever views were held as to the constitution of the world, Ptolemy's system was almost universally accepted as the foundation of astronomical science."

"While knowledge of the dimensions of the universe had... advanced, philosophers found it... difficult to agree with regard to the physical constitution of... heavenly bodies, though all acknowledged that they were of a fiery nature, the Stoics in... supposing them... of... pure fire or ether, which pervaded... upper regions of space. ...[T]he peculiar appearance of the "face of the moon" pointed to its being... different... and... Anaxagoras and Demokritus... recognized... it was a solid mass having mountains and plains, while Plato held it to be chiefly... earthlike matter. ...[In] Plutarch "On the face in the disc of the moon"... opinion of the Stoics [that the moon is a mixture of air and gentle fire] is refuted, since the moon ought not... be invisible at new moon if it did not borrow all its light from the sun; and this... proves... it is not... a substance like glass or crystal, since s would... be impossible. The manner in which the sunlight is reflected... and... absence of a bright, reflected image of the sun and... earth, prove... the substance of the moon is not polished but is like... earth. ...Plutarch ...to combat the idea that the moon cannot be like the earth since it is not in the lowest place ...asserts ...it is not proved ...earth is in the centre of the universe, as space is infinite and therefore has no centre; ...if everything heavy and earthy were crowded together ...we should expect all ...fiery bodies ...likewise brought together."

"[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."

"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."

"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."

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

"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."

"[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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

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

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

"[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. 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."

"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."

"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!"

"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."

"[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?"