astronomers-from-india

"The sun moves towards south or north along the ecliptic every moment. Therefore, the direction [determined by the Indian circle method] appears to be incorrect. The corrected direction will be [obtained by applying a correction] further by [using] the R.Sine of the declination.The difference of the R. Sine of the sun’s declination at the time of the shadow’s entry and exit [to and from the level circle] is multiplied by the hypotenuse [of the shadow] and divided by the R.cosine of the terrestrial latitude. The result is [the correction in terms of ] angulas etc. One should shift the western mark to the opposite direction to the sun’s course (ayana). Otherwise, one should shift the eastern mark to the same direction of the sun’s course. [Thus] the correct east west line is [obtained]."

"Why is it that the actual value is left out and this very near value stated? Let me say. It is impossible to state the actual value. Why? That unit which leaves no remainder when the diameter is measured will leave a remainder if used again for measuring circumference. Likewise, the unit which leaves no remainder in the measure of the circumference will leave a remainder in the diameter if measured by the same unit. Hence if both (the diameter and circumference) are measured by the same unit, a remainderless state is never attained. Even if this is carried out farther to a great extent only diminution of the remainder can be obtained but absence of remainder can never be obtained— this is the meaning."

"One has to accept that [each of ] the five siddhantas had been authoritative at one time [though they might not be so now]. Therefore one has to look for a system which tallies with observation. The said tallying has to be verified by contemporary experimenters at the time of eclipses etc."

"Square the diameter and multiply the product by 12, and extract the root of this product; the root obtained will be the modulus of odd quotes, which if you divide by 3, the quotient will be the modulus of even quotes. Divide each modulus continually by 9, and the quotient thus obtained from the former, divide by double the numbers 1, 3, 5, 7, 9, etc. minus 1 respectively, and the quotient obtained by the latter, by double the number 2, 4, 6, 8, 10, etc. minus 1 respectively, add up the new obtained quotes, and subtract the sum of those gotten from the even from the sum of those gotten from the odd modulus, the remainder is the circumference of the circle. Square the diameter and multiply the product by 12, and extract the root of this product; this root divide continually by 3, and the quotients thus obtained by 1, 3, 5, 7, 9, 11, etc., and subtract the sum of the second, fourth, sixth, eighth of the last obtained quotes from the sum of the first, third, fifth, seventh, ninth, etc. If you do thus, and measure the diameter of a great circle by 100000000000000000 equal parts, the circumference will be equal to 314159265358979324 of such parts."

"The radius into the sine divided by the cosine is the first quote: this multiplied by the square of the sine, and divided by the square of the cosine, is the second quote; this second, and those obtained continually in the same way, multiply and divide by the square of the sine and the square of the cosine respectively: divide the quotes in order by 1, 3, 5, 7,11, etc. respectively, and the difference of the sum of the first, third, fifth, etc. and of the second, fourth, sixth, etc., will be the arc whose sine was taken."

"[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."

"One has to realize that the five siddhantas [i.e. astronomical systems] had been correct at a particular time. Therefore, one should search for a siddhanta that does not show discord with actual observations (at the present time). Such accordance with observation has to be ascertained by (astronomical) observers during times of eclipses etc. When siddhantas show discord, that is, when an earlier siddhanta is in discord, observations should be made of revolutions etc. (which would give results in accord with actual observations) and a new siddhanta enunciated."

"The diameter multiplied by four and divided by unity (is found and saved). Again the products of the diameter and four are divided by the odd numbers like three, five, etc., and the results are subtracted and added in order."

"On any day calculate the longitude of the Sun and the Moon for sunrise or sunset without applying the longitude correction, and therefrom find the time (since sunrise or sunset), in ghatis, of rising or setting of the Moon; and having done this, note the corresponding time in ghatis from the water clock. From the difference, knowledgeable astronomers can calculate the local longitude in time.*?"

"Bhaskara […] does not pretend himself to be the inventor, he assumes no character but that of a compiler."

"Arjuna became furious in the war and in order to kill Karna, picked up some arrows. With half the arrows, he destroyed all of Karna’s arrows. He killed all of Karna’s horses with four times the square root of the arrows. He destroyed the spear with six arrows. He used one arrow each to destroy the top of the chariot, the flag, and the bow of Karna. Finally he cut off Karna’s head with another arrow. How many arrows did Arjuna discharge?"

"Reuben Burrow […] says, he was told by a pundit, that some time ago there were other treatises of algebra."

"The science of calculations with unknowns is the source of the science of calculations with knowns."

"The line which passes through Lanka, Vatsyapura, Avanti, Sthánesvara, and the "abode of the gods" [= Mount Meru = north pole] is the prime meridian [desantara vidhàyani, lit. the prime meridian for longitude differences.]"

"Almost any trouble and expense would be compensated by the possession of the three copious treatises on algebra from which Bhaskara declares he extracted his Bijaganita, and which in this part of India are supposed to be entirely lost."

"(The) Rsine is positive or negative in the upper or lower halves (of the quadrants); and its Rcosine is positive, negative, negative, and positive (respectively) according to the successive quadrants."

"There is no change in infinite (khahara) figure if something is added to or subtracted from it. It is like: there is no change in the infinite Lord Vishnu due to the dissolution or creation of abounding living beings."

"As the one-hundredth part of the circumference of a circle is (scarcely different from) a plane, and as the Earth is an excessively large body, and a man exceedingly small (in comparison), the whole visible portion of the Earth consequently appears to a man on its surface to be perfectly plane."

"On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically: as is particularly noticed by Bhaskara himself, towards the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities."

"A part of a circle is of the form of a bow, so it is called the ‘bow’ (dhanu). The straight line joining its two extremities is the ‘bow-string’ (jiva). It is really the ‘full-chord’ (samasta-jya). Half of it is here (called) the ‘half-chord’ (ardha-jya), and half that arc is called the ‘bow’ of that half-chord. In fact the Rsine (jya) and Rcosine (kotijya) of that bow are always half chords. [24]"

"In a triangle or a polygon, it is impossible for one side to be greater than the sum of the other sides. It is daring for anyone to say that such a thing is possible. If an idiot says that there is a quadrilateral of sides 2, 6, 3, 12 or a triangle with sides 3, 6, 9, explain to him that they don’t exist."

"As the moon moving eastward enters the dark shadow of the earth: therefore its eastern side is first of all involved in obscurity, and its western is the last portion of its disc which emerges from darkness as it advances in its course. (4) As the sun is a body of vast size, and the earth insignificantly small in comparison: the shadow made by the sun from the earth is therefore of a conical form terminating in a sharp point. It extends to a distance considerably beyond that of the moon’s orbit. (5)"

"The Rsines of any two arcs of a circle are reciprocally multiplied by their Rcosines; the products are then divided by the radius; the sum of the quotients is equal to the Rsine of the sum of the two arcs, and their difference is the Rsine of the difference of the arcs."

"The length of the earth’s shadow, and its breadth at the part traversed by the moon, may be easily found by proportion. In the lunar eclipse the earth’s shadow is northwards or southwards of the moon when its latitude is south or north. Hence the latitude of the moon is here to be supposed inverse (i.e. it is to be marked reversely in the projection to find the center of the earth’s shadow from the moon.) (6)"

"The earth attracts inert bodies in space towards itself. The attracted body appears to fall down on the earth. Since the space is homogeneous, where will the earth fall?"

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

"Let ‘so much as’ (yavattavat) be put for the value of the unknown quantity, and doing with that precisely what is proposed in the instance [i.e. what is proposed in the following], let two equal sides be carefully completed, adding or subtracting, multiplying or dividing, [as the case may require]. Subtract the unknown quantity of one side from that of the other; and the known number of the one from that of the other side. Then divide the remainder of the known quantity by the [coefficient of the] remaining unknown: the quotient is the distinct value of the unknown quantity."

"Analysis (bija) is the intellect assisted by the various symbols that has now taken the name algebra (bijaganita)."

"What the learned calculators (sankhyah) describe as the originator of intelligence, being directed by a wise being (satpurusha) and which alone is the primal cause (bija) of all knowns (vyakta), I venerate that invisible Supreme Being as well as that science of calculation with unknowns […] Since questions can scarcely be solved without the reasoning of algebra—not at all by those of dull perceptions—I shall speak, therefore, of the operations of analysis."

"I bow to Lord Shiva whose Lotus feet are rubbed by the foreheads of the Lotus-born (Brahma) and Krisha, to Him, a mere recollection of whose name is a source of fortune and end of misfortune for gods, demons, and men. (Ganitapada, Aryabhatiyabhashya) I bow to Shambhu, who is the cause of creation and destruction of the universe, (the different parts of ) whose body are the Sun etc., and who is as effulgent as tens of thousands of suns. (Ganitapada, Aryabhatiyabhashya). I bow to God Shiva, the support of the three worlds, who though inconceivable by nature, being mercy incarnate, assumedthe eight forms, viz. The Sun, the Moon, the Ether, Fire, Air, Water, Earth, and the Sacrificer, for the good of the entire world. (Kalakriyapada, Aryabhatiyabhashya)I bow to God Shambhu who bears on His forehead a digit of the moon illumining all directions by its rays, to Him whose feet are adored by the gods and who is a source of all knowledge. (Mahabhaskariya, opening verse)"

"A person has 300 rupees and 6 horses; and another person has 10 horses and 100 rupees debt; and the property of the two is equal; and the price of the horses is the same; what then is the value of each?"

"On a plane surface describe a circle of any specified radius with a pair of compasses. Mark on its circumference 360 degrees. Draw the east-to-west and north-to-south lines through its center. These lines will divide the circle into quadrants, which should be taken into consideration in the leftwise manner (i.e. anti-clockwise)."

"Without the knowledge of upapattis, by merely mastering the calculations (ganita) described here, from the madhyamadhikara (the first chapter of Siddhantashiromani) onwards, of the [motion of the] heavenly bodies, a mathematician will not be respected in the scholarly assemblies; without the upapattis he himself will not be free of doubt…."

"The Moon, moving like a cloud in a lower sphere, overtakes the Sun [and obscures its shining disk by its own dark body] hence it arises that the western side of the Sun’s disk is first obscured, and that the eastern side is the last part relieved from the Moon’s dark body: and to some places the Sun is eclipsed and to others is not eclipsed (although he is above the horizon) on account of their different orbits. (1)At the change of the Moon it often so happens that an observer placed at the center of the earth, would find the sun when far from the zenith, obscured by the intervening body of the moon, whilst another observer on the surface of the earth will not at the same time find him to be so obscured, as the moon will appear to him [on the higher elevation] to be depressed from the line of vision extending from his eye to the sun. Hence arises the necessity for the correction of parallax in celestial longitude and parallax in latitude in solar eclipses in consequence of the difference of the distances of the sun and the moon. (2) When the sun and moon are in opposition, the earth’s shadow envelopes the moon in darkness. As the moon is actually enveloped in darkness, and as the earth’s shadow and the moon which enters it, are at the same distance from the earth, there is therefore no call for the correction of the parallax in a lunar eclipse. (3)"

"If the earth were supported by any material substance or living creature, then that would require a second supporter, and for that second a third would be required. Here we have the absurdity of an interminable series. If the last of the series be supposed to remain firm by its own inherent power, then why may not the same power be supposed to exist in the first, that is in the Earth? For is not the Earth one of the forms of the eight-fold divinity i.e. of Shiva? (Goladhyaya, III.4)"

"The property of attraction is inherent in the Earth. By this property the Earth attracts any unsupported heavy thing towards it: The thing appears to be falling [but it is in a state of being drawn to the Earth]. The ethereal expanse being equally outspread all around, where can the Earth fall? (Goladhyaya III.6)"

"From a bunch of lotuses, one third is offered to Lord Shiva, one fifth to Lord Vishnu, one sixth to the sun, one fourth to the goddess. The remaining six are offered to the Guru. Find quickly the number of lotuses in the bunch."

"From the bright half of the month of Magha is the beginning of the northern course of the Sun. The Sun and the Moon begin their rise from Shravishta."

"When the return of the Sun took place from the middle of Aslesha, the solstice was then right. It now takes place from Punarvasu."

"Varāhamihira .... in his Brihat Samhitā.... states that the height of a tall man is 108 angulas, that of a medium man ninety-six angulas, and that of a short man eighty-four angulas (the same heights apply to statues of various deities); using our Harappan angula of 1.76 cm, we get 1.90, 1.69 and 1.48 m respectively, quite consistent with ‘tall’, ‘medium’ and ‘short’."

"There was indeed a time when the Sun’s southerly course began from the middle of the star Ashlesha and the northerly one from the commencement of the star Dhanishtha, as it has been so stated in ancient works. At present the southerly course of the Sun starts from the beginning of Cancer and the other from the initial point of sign Capricorn. The actual state of affairs which goes against the old statement can be verified by direct observation. The Sun’s change of course can be detected by marking everyday the position of a distant object either at sunrise or sunset, or by watching and marking the entry and exit of the shadow of the gnomon fixed at the centre of a big circle drawn on the ground."

"The [Maharashtra Bhushan] award was in the waiting for a long time and he truly deserves it. Having been associated with him for over 40 years, I have seen him work calmly towards achieving all his targets and also encouraging all his colleagues to give work their very best."

"His idea may not solve the mystery of faster-than-light neutrinos, and it may not even shed much light on it – the details of the analysis might be wrong, or be filled with old-fashioned ideas. But I thought it was fascinating that a 50-year-old New Scientist story touched on a topic that is all over the headlines today."

"During another examination the question paper read, ‘do any six questions, all questions carry equal marks’. But I attempted more than six questions and wrote for the examiner, “examine any six questions, all questions carry equal marks."

"In western society, there is a problem similar to India that the income of a research scientist is less than that of a management person. But perhaps the facilities and the infrastructure being much advanced in the western universities, they are able to attract and retain the people. In our case this can happen but not to the extent we would like. A corrective measure to some extent has been taken by the establishment of the Indian Institutes of Science Education and Research (IISERs). They are doing for pure sciences what IIT’s do for applied sciences. Since they are attracting good talent it may happen that you will have more input in research in the next ten years."

"He, a Cambridge physicist who, along with Fred Hoyle, Thomas Gold and Hermann Bondi, believed there was no start to the universe – no "big bang", as Hoyle so memorably (and mockingly) put it."

"He is a cosmologist of international repute and best known for his work on the conformal gravity theory, together with his mentor Fred Hoyle. He also set up the Inter-University Centre for Astronomy and Astrophysics (IUCAA) in Pune. More recently, he was involved in an effort to sample air from the atmosphere at heights of 41km for microorganisms. This study reported at least two strains of bacteria and one fungus that were cultured in the lab. If these findings hold up to further enquiry, they provide a new perspective on life on earth and its beginnings."

"Science popularization involves information about science explaining the things and their implications in detail, or any beliefs the reader might have and also tell them about the newer discoveries so as to enlarge the persons horizons."

"In the Cambridge mathematical tripods, they said, ’do six questions. Complete questions carry proportionately more marks than an equal number of fragments. Till you attempt half of the questions you won’t get full credit’. Upon declaration of result I found that I had scored 110 out of hundred in one paper, 140 in another and likewise in all the rest. I was confused. So my tutor explained to me that although they write ‘do six questions’, you can attempt as many as you want. They award you marks for whatever questions you answer correctly and the ranks are on basis of the score."

"in our society at all levels not only uneducated but educated as well are dominated by superstitions of various kinds. The superstitions of the educated are only more sophisticated than those of the uneducated. So one needs to introduce real knowledge in place of superstitious beliefs. You need to explain to people believing in superstitious ideas that the ideas are not correct and this can be verified experimentally. When doing science popularization you have to express some facts of the real universe or the real world to the person who is not willing to believe in it and if you express it in a very pedantic way or say to the person that you are a fool, you should not believe this then it is not good because then he will believe in his superstitions even more. So you have to persuade him to think for himself, you don’t tell him what to believe but you ask him to try this experiment and see what is the reality. So that way you may be able to convince the person."