mathematics

"Multiply the arc by the square of the arc, and take the result of repeating that [any number of times]. Divide [each of the above numerators] by the squares of successive even numbers increased by that number [lit. the root] and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jı̄vā, as collected together in the verse beginning with “vidvān” etc."

"This method is supreme above all praise; it is certainly the finest thing accomplished in number theory before Lagrange."

"This is as I have said in my Arithmetic:—The area of a circle is equal to the product of the circumference by one-fourth of the diameter [πr²]. That result multiplied by 4 gives the surface of the sphere [4πr²], which is like the net surrounding a hand ball; the same (surface of a sphere) when multiplied by the diameter and divided by six [4/3πr³] becomes invariably the volume of the sphere."

"The method represents a best approximation algorithm of minimal length that, owing to several minimization properties, with minimal effort and avoiding large numbers automatically produces the best solutions to the equation. The chakravala method anticipated the European methods by more than a thousand years. But no European performances in the whole field of algebra at a time much later than Bhaskara's, nay nearly equal up to our times, equalled the marvellous complexity and ingenuity of chakravala."

"A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed.The axis of a sphere is the fixed straight line about which the semicircle revolves.The centre of a sphere is the same with that of the semicircle.The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere."

"The process of reasoning called "Mathematical Induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus. [...] By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite."

"Having fixed a pole, on a level piece of ground, and having described a circle by a cord attached to the pole, one should mark the points with pegs, where the shadow of the top of the pole touches the circle. The line joining these pegs is the West to East line. Having increased the length of the cord (which is equal to the distance of the two pegs fixed in the East West line) by itself, one should provide two slings at each end of the cord. Fix the slings on the two pegs (already fixed) stretch the cord with its mid point towards the south and fix a peg at the place, where the mid point of the cord touches the ground. He should similarly proceed towards the North. The line joining these pegs will be the South to North line."

"‘…the transition [to the Hindu number system], far from being immediate, extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals [more precisely, Hindu numerals] were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code.’"

"Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him."

"The first writer to attribute this proposition to Pythagoras is Vitruvius, hardly a reliable witness. From then on, this account became more widespread, but always in connection with the famous hecatomb that Pythagoras is said to have offered in celebration of discovering the proposition—an anecdote that severely undermines the credibility of the entire story. This sacrifice is incompatible with the strict prohibition of all bloody sacrifices, which writers of the same period, indeed often the very same ones who elsewhere recount the hecatomb, have handed down to us from the Pythagorean ritual laws. Cicero himself took offense at this anecdote, and in the latest Neopythagorean tradition, the bloody sacrifice is replaced by that of an ox formed from flour. For this reason, the hecatomb is not only incongruous with the Pythagoreans, but also with the Pythagoreans themselves. Proclus, an insightful writer, expresses himself remarkably vaguely: ‘When we listen to those who want to tell old stories, we find that they trace this theorem back to Pythagoras.’ As this shows, he too was unaware of any reliable source."

"However, considering the increasingly clear connections between India and the Babylonian cultural sphere, particularly in recent years, the Indian evidence for the study of Babylonian mathematics certainly comes into play. The difficulties which arise from assuming a direct borrowing by the Greeks from India fall away on the assumption of a common origin in Babylonia.As for Greece itself, the origin of the Pythagorean theorem is, as is well known, shrouded in mystery. However, there is no doubt that Pythagoras and his school were strongly influenced by the Orient. The possibility is therefore obvious that Pythagoras (or his school) had learned the theorem later named after him from oriental sources, which would be very compatible with seeing him as the discoverer of the 'Pythagorean method' for constructing integers that satisfy the relation a² + b² = c². This would only fit with the entire number-speculative tendency of this circle."

"If we consider the results obtained together, we will not be able to doubt the conclusion to be drawn from them. The ancient priestly geometry of the Indians not only knew the Pythagorean theorem, but it even played the main role in their calculations; with its help, they constructed elements that the Greeks found in a completely different way; with its help, they also found the irrational quantities. And it was precisely these two things that Pythagoras introduced into the Greek-Italian world; these two things, according to the Greeks, he invented. Indeed, even more! The way in which Pythagoras proved his theorem was also, in all likelihood, the same as that which we find in the Vedic Shulba Sutras. After examining the Shulba Sutras, we could have said: If Pythagoras really was in India, as we previously suggested, and initiated himself into the priestly wisdom of the Brahmins, then he could have brought precisely these theorems of geometric science to Greece; — and history has been telling us for several millennia now that this was indeed the case!"

"Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)"

"This present art, in which we use those twice five Indian figures, is called algorismus."

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast."

"‘The change did not come about without obstruction from the representatives of custom thought. An edict of A.D. 1259 forbade the bankers of Florence to use the infidel symbols, and the ecclesiastical authorities of the University of Padua in A.D. 1348 ordered that the price list of books should be prepared not in “ciphers”, but in plain letters.’"

"This boke is called the boke of algorym or Augrym after lewder use. And this boke tretys the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor, and he made this craft. And aft his name he called hit algory."

"The diagonal cord of a rectangle makes both (the squares) that the vertical side and the horizontal side make separately."

"‘These operations are all founded on a very distinct conception of what happens in the case of an eclipse, and on the knowledge of this theorem, that, in a right-angled triangle, the square on the hypotenuse is equal to the squares of the other two sides. It is curious to find the theorem of PYTHAGORAS in India, where, for aught we know, it may have been discovered, and from whence that philosopher may have derived some of the solid, as well as the visionary speculations, with which he delighted to instruct or amuse his disciples.’"

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately."

"All the algorithms for fractions now used were invented by the Hindus. The Greek treatment of fractions never advanced beyond the level of the Egyptian Rhind papyrus. […] This inability to treat a fraction as a number on its own merits is the explanation of a practice [which] was as useless as it was ambiguous. […] When we remember that the Greeks and Alexandrians continued this extraordinary performance, there is nothing remarkable about the small progress which they achieved in their arithmetic. What is remarkable is that a few of them like Archimedes should have discovered anything at all about series of numbers involving fractional quantities."

"Moreover, the complete details of the complication of the structure of Mandelbrot's set cannot really be fully comprehended by anyone of us, nor can it be fully revealed by any computer. It would seem that this structure is not just part of our minds, but it has a reality of its own. ... The computer is being used in essentially the same way that the experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there!"

"The dependence of so many results on Riemann's challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word 'hypothesis' has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. 'Conjecture', in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann's riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated."

"The last few decades have provided abundant evidence for physics beyond the two standard models of particle physics and . As is now known, the by far largest part of our universe's matter/energy content lies in the `dark' and consists of and . Despite intensive efforts on the experimental as well as the theoretical side, the origins of both are still completely unknown. Screened scalar fields have been hypothesized as potential candidates for dark energy or dark matter. Among these, some of the most prominent models are the , , and environment-dependent ."

"2. Are there any new elementary scalars not yet discovered with masses below the mass of the -like Higgs boson? For example, do -like particles exist? ... 8. If additional scalars are discovered, how will these discoveries impact the question of the stability of the ? 9. Do neutral (inert) scalars comprise a significant fraction of the dark matter?"

"Let's take some extra time to talk about one: Only the number one can create all numbers with this simple equation, 111111111 × 111111111 = 12345678987654321. One, expressed nine times, multiplied by itself, produces all subsequent numbers progressively and then inversely."

"Sol Robeson: This is insanity, Max. Max Cohen: Or maybe it's genius! I have to get that number. Sol Robeson: Hold on! You have to slow down. You're losing it. You have to take a breath. Listen to yourself. You're connecting a computer bug I had with a computer bug you might have had and some religious hogwash. If you want to find the number 216 in the world, you will be able to find it everywhere: 216 steps from your street corner to your front door, 216 seconds you spend riding in the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere: 320, 450, 22, whatever. You've chosen 216, and you will find it everywhere in nature. But Max, as soon as you discard scientific rigor, you are no longer a mathematician—you're a numerologist."

"Abulafia himself wrote an autobiographical account of this conversion attempt [of Pope Nicholas III] in his Sefer HaEdot (Book of Testimonies). In this account, he calls himself Raziel, an anonym that he frequently used, this being the name of the angel who taught the mysteries to Adam. Raziel () has a numerical value of 248, this being the same as that of Abulafia's first name, Abraham (). As Abulafia himself indicates, this relationship is more than a simple gematria, but it is a mishkal (balance), since both the numerical value and the number of letters in both names are equal."

"When HaShem commanded him [Moshe] to go down to Egypt and liberate the Jewish people from bondage, he asked HaShem, "When I come to the children of Israel and I say to them, 'The God of your forefathers sent me to you,' they will ask me, 'What is His name,' What [] should I say to them?" ... The final letters are the name HaShem-, while the beginning letters equal "The Secret of the Name" (Sod Shem–) [30- + 40- + 300- + 40- = 60- + 6- + 4- + 300- + 30- = 410]. Therefore, the beginning and final letters make up the phrase "The Secret of the Name HaShem–.""

"Lenny Meyer: The ancient Jews used Hebrew as their numerical system. May I? Each letter's a number. Like, the Hebrew A, Aleph [], is 1. B, Bet [], is 2. You understand? But look at this. The numbers are interrelated. Like, take the Hebrew word for "father", av []. Aleph, Bet. 1, 2. Equals 3. All right? Hebrew word for "mother", em []. Aleph, Mem []. 1, 40. Equals 41. The sum of 3 and 41? 44. All right? Now, Hebrew word for "child", all right? Mother, father, child. Yeled []. That's 10, 30 and 4. It's 44. Torah is just a long string of numbers. Some say that it's a code, sent to us from God. Max Cohen: That's kind of interesting."

"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."

"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."

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

"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. 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."

"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."

"[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."

"[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?"

"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."

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

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

"[T]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."

"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."

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

"[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."

"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 ."

"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.""