## Wednesday, February 3, 2016

### Ancient Mathmatics Part 3

Ancient Indian Mathematics. Part Three.
Numerals and the decimal number system
It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear.
The earliest extant script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system.
The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.
There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."
A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. According to Plofker 2009, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left. The earliest reference employing object numbers is a ca. 269 CE Sanskrit text, Yavanajātaka (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.[53] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.
It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. According to Plofker 2009,
These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."
Bakhshali Manuscript
The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit" in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously dated—as early as the "early centuries of the Christian era" and as late as between the 9th and 12th century CE.The 7th century CE is now considered a plausible date, albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work."
The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi) and algebra (simultaneous linear equations and quadratic equations), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."[55] Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following:
One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant.
The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.
Classical Period (400–1600)
This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā). This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika[61] (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 CE)—of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.
Fifth and sixth centuries
Surya Siddhanta
Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry.[citation needed] Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.
This ancient text uses the following as trigonometric functions for the first time: Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). It also contains the early uses of tangent,.secant.
.
Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.
Chhedi calendar
This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).
Aryabhata I
Aryabhata (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:
Trigonometry
The value of π, correct to 4 decimal places.
Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:
Introduced the trigonometric functions.
Defined the sine (jya) as the modern relationship between half an angle and half a chord.
Defined the cosine (kojya).
Defined the versine (utkrama-jya).
Defined the inverse sine (otkram jya).
Gave methods of calculating their approximate numerical values.
Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.
Spherical trigonometry.
Arithmetic: Continued fractions. Algebra:
Whole number solutions of linear equations by a method equivalent to the modern method.
General solution of the indeterminate linear equation .
Mathematical astronomy:
Accurate calculations for astronomical constants, such as the:
Solar eclipse.
Lunar eclipse.
The formula for the sum of the cubes, which was an important step in the development of integral calculus.
Varahamihira
Varahamihira (505–587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:
\sin^2(x) + \cos^2(x) = 1
\sin(x)=\cos\left(\frac{\pi}{2}-x\right)
\frac{1-\cos(2x)}{2}=\sin^2(x)
Seventh and eighth centuries
Brahmagupta's theorem states that AF = FD.
In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:
Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.
Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).
Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by
A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \,
where s, the semiperimeter, given by s=\frac{a+b+c+d}{2}.
Brahmagupta's Theorem on rational triangles: A triangle with rational sides a, b, c and rational area is of the form:
a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w)
for some rational numbers u, v, and w .
Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers and is considered the first systematic treatment of the subject. The rules (which included a + 0 = \ a and a \times 0 = 0 ) were all correct, with one exception: \frac{0}{0} = 0 . Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:
\ ax^2+bx=c
To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.
This is equivalent to: x = \frac{\sqrt{4ac+b^2}-b}{2a}
Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,
\ x^2-Ny^2=1,
where N is a nonsquare integer. He did this by discovering the following identity:
Brahmagupta's Identity: \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 which was a generalisation of an earlier identity of Diophantus: Brahmagupta used his identity to prove the following lemma:
Lemma (Brahmagupta): If x=x_1,\ \ y=y_1 \ \ is a solution of \ \ x^2 - Ny^2 = k_1, and, x=x_2, \ \ y=y_2 \ \ is a solution of \ \ x^2 - Ny^2 = k_2, , then:
x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \ is a solution of \ x^2-Ny^2=k_1k_2
He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem:
Theorem (Brahmagupta): If the equation \ x^2 - Ny^2 =k has an integer solution for any one of \ k=\pm 4, \pm 2, -1 then Pell's equation:
\ x^2 -Ny^2 = 1
also has an integer solution. Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: Example (Brahmagupta): Find integers \ x,\ y\ such that: \ x^2 - 92y^2=1
In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was:
\ x=1151, \ y=120
Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced:
Solutions of indeterminate equations.
A rational approximation of the sine function.
A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.
Ninth to twelfth centuries.
Virasena
Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which:
Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order.
The same concept for base 3 (trakacheda) and base 4 (caturthacheda).
Virasena also gave:
The derivation of the volume of a frustum by a sort of infinite procedure.
It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.
Mahavira
Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:
Zero
Squares
Cubes
square roots, cube roots, and the series extending beyond these
Plane geometry
Solid geometry
Problems relating to the casting of shadows
Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle.
Mahavira also:
Asserted that the square root of a negative number did not exist
Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.
Solved cubic equations.
Solved quartic equations.
Solved some quintic equations and higher-order polynomials.
Gave the general solutions of the higher order polynomial equations:
\ ax^n = q
a \frac{x^n - 1}{x - 1} = p
Solved indeterminate cubic equations.
Solved indeterminate higher order equations

### Ancient Indian Mathematics. Part Two.

Ancient Indian Mathematics. Part Two.
Pingala
Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala (piṅgalá) (fl. 300–200 BCE),a musical theorist who authored the Chhandas Shastra (chandaḥ-śāstra, also Chhandas Sutra chhandaḥ-sūtra), a Sanskrit treatise on prosody. There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both the Pascal triangle and Binomial coefficients, although he did not have knowledge of the Binomial theorem itself. Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru). Although the Chandah sutra hasn't survived in its entirety, a 10th-century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as Meru-prastāra (literally "the staircase to Mount Meru"), has this to say:
Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ...
The text also indicates that Pingala was aware of the combinatorial identity:
{n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n-1} + {n \choose n} = 2^n
Katyayana
Katyayana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra, which presented much geometry, including the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places.
Jain Mathematics (400 BCE – 200 CE)
Although Jainism as a religion and philosophy predates its most famous exponent, the great Mahavira (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period."
A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, they went on to define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations (beejganita samikaran). Jain mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. More than a millennium later, their appellation became the English word "zero" after a tortuous journey of translations and transliterations from India to Europe.
In addition to Surya Prajnapti, important Jain works on mathematics included the Vaishali Ganit (c. 3rd century BCE); the Sthananga Sutra (fl. 300 BCE – 200 CE); the Anoyogdwar Sutra (fl. 200 BCE – 100 CE); and the Satkhandagama (c. 2nd century CE). Important Jain mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called Tiloyapannati; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics, composed a mathematical work called Tattwarthadhigama-Sutra Bhashya.
Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (paṇḍita "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vyākaraṇa), exegesis (mīmāṃsā) and logic (nyāya)." Memorisation of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia."
Styles of memorization
Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order. The recitation thus proceeded as:
word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...
In another form of recitation, dhvaja-pāṭha (literally "flag recitation") a sequence of N words were recited (and memorised) by pairing the first two and last two words and then proceeding as:
word1word2, wordN − 1wordN; word2word3, wordN − 3wordN − 2; ..; wordN − 1wordN, word1word2;
The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to (Filliozat 2004, p. 139), took the form:
word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ...
That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (ca. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. 500 BCE).
The Sutra genre
Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Vedas, which took the form of works called Vedāṇgas, or, "Ancillaries of the Veda" (7th–4th century BCE). The need to conserve the sound of sacred text by use of śikṣā (phonetics) and chhandas (metrics); to conserve its meaning by use of vyākaraṇa (grammar) and nirukta (etymology); and to correctly perform the rites at the correct time by the use of kalpa (ritual) and jyotiṣa (astrology), gave rise to the six disciplines of the Vedāṇgas. Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedāṇgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in a highly compressed mnemonic form, the sūtra (literally, "thread"):
The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable.
Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language," using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. The sūtras create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called Guru-shishya paramparai, 'uninterrupted succession from teacher (guru) to the student (śisya),' and it was not open to the general public" and perhaps even kept secret. The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE).
The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely. The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana Śulba Sūtra, this procedure is described in the following words:
II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three.
II.65. In another layer one places the [bricks] North-pointing.
According to (Filliozat 2004, p. 144), the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju, f.), two pegs (Sanskrit, śanku, m.), and clay to make the bricks (Sanskrit, iṣṭakā, f.). Concision is achieved in the sūtra, by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the East-West direction, but that too is implied by the explicit mention of "North-pointing" in the second stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory.
With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation.
India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally.
The earliest mathematical prose commentary was that on the work, Āryabhaṭīya (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to (Hayashi 2003, p. 123), "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti). Bhaskara I's commentary on the Āryabhaṭīya, had the following structure:
Rule ('sūtra') in verse by Āryabhaṭa
Commentary by Bhāskara I, consisting of:
Elucidation of rule (derivations were still rare then, but became more common later)
Example (uddeśaka) usually in verse.
Setting (nyāsa/sthāpanā) of the numerical data.
Working (karana) of the solution.
Verification (pratyayakaraṇa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then.
Typically, for any mathematical topic, students in ancient India first memorized the sūtras, which, as explained earlier, were "deliberately inadequate" in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta (fl. 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit: dhulikarman).

### Ancient Indian Mathematics. Part 1

Ancient Indian Mathematics.
Important facts the world should know about Indian Mathematics. This is a long article, it clearly explains the great achievements of Indian mathematicians ,and therefore of the accomplishments on the advancement of science and technology. Part One.
It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by Indian mathematicians are presently culturally attributed to their Western counterparts, as a result of Eurocentrism. According to G. G. Joseph's take on "Ethnomathematics":
[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing"
The historian of mathematics, Florian Cajori, suggested that he and others "suspect that Diophantus got his first glimpse of algebraic knowledge from India." However, he also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".
More recently, as discussed in the above section, the infinite series of calculus for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described (with proofs) in India, by mathematicians of the Kerala school, remarkably some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China and Arabia, and, from around 1500, with Europe. The existence of communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place. According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."
Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. However, they were not able, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware." This is an active area of current research, especially in the manuscripts collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre National de Recherche Scientifique in Paris.
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1600 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.
Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE.
A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.
Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of "practical mathematics". The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.
The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length
Samhitas and Brahmanas
The religious texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurvedasaṃhitā- (1200–900 BCE), numbers as high as 1012 were being included in the texts.[2] For example, the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:
Hail to śata ("hundred," 102), hail to sahasra ("thousand," 103), hail to ayuta ("ten thousand," 104), hail to niyuta ("hundred thousand," 105), hail to prayuta ("million," 106), hail to arbuda ("ten million," 107), hail to nyarbuda ("hundred million," 108), hail to samudra ("billion," 109, literally "ocean"), hail to madhya ("ten billion," 1010, literally "middle"), hail to anta ("hundred billion," 1011,lit., "end"), hail to parārdha ("one trillion," 1012 lit., "beyond parts"), hail to the dawn (uṣas), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to svarga (the heaven), hail to martya (the world), hail to all.
The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4):
With three-fourths Puruṣa went up: one-fourth of him again was here.
The Satapatha Brahmana (ca. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.
Śulba Sūtras
Main article: Śulba Sūtras
The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars.[22] Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"[23] that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.
According to (Hayashi 2005, p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."
The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."
Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations. They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."
Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37),[28] as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."] It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." Baudhayana gives a formula for the square root of two:
\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} = 1.4142156 \ldots
The formula is accurate up to five decimal places, the true value being 1.41421356... This formula is similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900–1600 BCE):
\sqrt{2} \approx 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = 1.41421297 \ldots
which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places (after rounding).
According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE[32] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[33] indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say:
As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily.
In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava (fl. 750–650 BCE) and the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.
Vyakarana
An important landmark of the Vedic period was the work of Sanskrit grammarian, Pāṇini (c. 520–460 BCE). His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages).
Continues...........
Wikipedia The Free Encyclopedia

### Muslim war with Europe in 2016 predictions

• नास्त्रेदमस की परंपरा में एक और भविष्यवक्ता का नाम सामने आया है वांजेलिया पांडेवा डिमित्रोवा का. इस नेत्रहीन बुल्गारियाई भविष्यदृष्टा को बाबा वैंगा के नाम से भी जाना जाता है. इनकी कुछ बड़ी भविष्यवाणियों में एक यह भी है कि 2016 में यूरोप पर विशाल मुस्लिम आक्रमण होगा.
• अपनी मौत के बाद की भी हजारों भविष्यवाणियां करने वाली वैंगा की 1996 में मृत्यु हो चुकी है. लेकिन उनके द्वारा की गई सबसे बड़ी वैश्विक आपदाओं की भविष्यवाणियों में से 2004 की सुनामी और 9/11 हमले को देखे जा चुका है.

अविश्वासी औऱ नास्तिक लोगों का ध्यान भविष्यवाणी जैसी चीजों पर शायद न जाय. अगर बात किसी ज्योतिष की हो तो फिर इसकी संभावना एकदम खत्म हो जाती है. अगर आप तर्क संगत तरीके से सोचते हैं और प्लूटो के चश्में से दुनिया के भविष्य की कल्पना करते हैं तो यह आपको समझ नहीं आएगा.

हां, हम सभी को बचपन से ही बताया गया है कि फ्रांसीसी भविष्यवक्ता नास्त्रेदमस ने हिटलर, नेपोलियन समेत तमाम बड़ी हस्तियों के उदय की सही-सही भविष्यवाणी कर दी थी.

और इस सप्ताह हफिंगटन पोस्ट द्वारा बताया जा रहा है कि नास्त्रेदमस की एक उत्तराधिकारी वांजेलिया पांडेवा डिमित्रोवा नाम की महिला हैं. इन नेत्रहीन बुल्गारियाई भविष्यदृष्टा को बाबा वैंगा के नाम से भी जाना जाता है जिन्होंने 10 भयानक भविष्यवाणियां की हैं. इनमें 2016 में यूरोप में होने वाले विशाल मुस्लिम आक्रमण की भविष्यवाणी भी शामिल है.

#### कौन थीं (स्वर्गीय) बाबा वैंगा?

यह ऐसी कहानी है जिसपर संभवत: बॉलीवुड में लोगों ने काम करना शुरू कर दिया गया होगा.

स्थानीय किंवदंती है कि बाबा वैंगा या बाल्कन की नास्त्रेदमस ने पहली बार भविष्य देखने की क्षमता तब प्राप्त की थी जब 12 साल की उम्र में एक भयानक तूफान में उसकी आंखों की रोशनी चली गई थी.

16 साल की उम्र में ही उसने भविष्यवाणी करनी शुरू कर दी और "30 साल की उम्र पूरी होने पर पहले से ही जानने की उसकी शक्तियां और मजबूत हो गईं."

1952 की शरद ऋतु में यह महिला भविष्यवक्ता काफी मुसीबत में पड़ गई जब उसने कहा कि जोसेफ स्टालिन को पाताल लोक में जाना होगा. इस भविष्यवाणी के परिणामस्वरूप उसे जेल में बंद कर दिया गया, हालांकि वो जल्द ही बाहर आ गईं.

वर्ष 1967 में उन्हें 'सरकारी अधिकारी' के रूप में नियुक्त किया गया बावजूद इसके कि कम्युनिस्ट शासन में तमाम ऐसे लोग थे वेंगा बाबा को चुड़ैल समझते थे. कहानी यह भी है कि एक दिन हिटलर उनसे मिलने पहुंचा था लेकिन बाद में वो बहुत परेशान हो गया था.

अपनी मौत के बाद की भी हजारों भविष्यवाणियां करने वाली वैंगा की अंततः 1996 में मृत्यु हो गई. लेकिन उनके द्वारा की गई सबसे बड़ी वैश्विक आपदाओं की भविष्यवाणियों के संदर्भ में 2004 की सुनामी और 9/11 हमले को देखा जाता है.

#### 2016 में क्या हो सकता है?

अपनी मौत के 20 साल बाद यानी वर्ष 2016 की अपनी भविष्यवाणी के चलते बाबा वैंगा वापस खबरों में आ गई हैं. क्योंकि उन्होंने कहा था कि 2016 वो वर्ष होगा जब, "मुसलमानों द्वारा यूरोप पर आक्रमण किया जाएगा."

यूरोप और पश्चिम एशिया का राजनीतिक माहौल स्वाभाविक रूप से इस चिंता को बल दे रहा है. अप्रवासी संकट के चलते अभूतपूर्व रूप से यूरोप पर काफी दबाव है और विभिन्न देशों द्वारा इस मुद्दे पर अलग-अलग राय रखने के चलते इस संघ में दरारें फैलती जा रही हैं.

तो निकट भविष्य के लिए वैंगा की अन्य प्रमुख भविष्यवाणी क्या हैं? इनमें 2018 में चीन दुनिया की सर्वोच्च 'महाशक्ति' बन जाएगा. उनके मुताबिक इसी साल एक अंतरिक्ष यान वीनस पर 'ऊर्जा के एक नए रूप' की खोज करेगा.

#### क्यों लोग इनसे आकर्षित नहीं हुए?

उन पर विश्वास करने वालों का दावा है कि वैंगा ने पिछले दशकों की कुछ प्रमुख वैश्विक घटनाओं की भविष्यवाणी की थीः

• भारतीय प्रधानमंत्री इंदिरा गांधी और राजीव गांधी की हत्या.
• कुस्क परमाणु पनडुब्बी आपदा
• ग्लोबल वॉर्मिंग
• 9/11: "डरावना, भयावह! अमेरिकी भाइयों पर स्टील पक्षियों के द्वारा हमला किए जाने के बाद वह गिर जाएंगे."
• 2004 की सुनामी: "एक विशाल लहर एक बड़े तट को घेर लेगी. जिसमें लोगों, कस्बों के साथ सबकुछ पानी के नीचे गायब हो जाएगा. सब कुछ सिर्फ बर्फ की तरह पिघल जाएगा."
• एक "विशाल मुस्लिम युद्ध"
• बराक ओबामा का चुनाव: उन्होंने कहा था कि अमेरिका का 44वां राष्ट्रपति एक अफ्रीकी अमेरिकी होगा. उसने यह भी कहा कि वह आखिरी अमेरिकी राष्ट्रपति होगा.
• द्वितीय विश्व युद्ध
• ज़ार बोरिस III की मृत्यु की तारीख
• चेकोस्लोवाकिया का विघटन
• सोवियत संघ का टूटना
• यूगोस्लाविया का टूटना
• पूर्व और पश्चिम जर्मनी का एकीकरण
• बोरिस येल्तसिन का चुनाव
• चेर्नोबिल आपदा
• स्टालिन की मौत की तारीख
• सीरिया में संघर्ष
• क्रीमिया का अलग होना

#### आपको उलझन में क्यों होना चाहिए

विशेषज्ञों' ने गणना की है कि उसकी 68 फीसदी भविष्यवाणी सच साबित हुई हैं.

लेकिन इन भविष्यवाणियों के साथ सबसे बड़ी समस्या यह है कि इनका कोई आधिकारिक स्रोत नहीं है. जहां नास्त्रेदमस ने अपनी भविष्यवाणियों को लिखित रूप में छोड़ा था, बाबा वैंगा ने ऐसा नहीं किया. यही कारण है कि उन्होंने शायद जो कहा था घटनाओं के बाद उसके अर्थ बदलते रहे. शायद यह संभव नहीं लेकिन संभावित है. जैसे चीन का फुसफुसाने वाला खेल हो रहा है.

अगर फिर भी आप जानना चाहते हैं कि भविष्य के बारे में वो क्या सोचती थीं?
• 2016: मुसलमानों का यूरोप पर आक्रमण
• 2018: अमेरिका को पीछे करते हुए चीन विश्व की महाशक्ति बन जाएगा
• 2023: पृथ्वी की कक्षा में बड़े बदलाव होंगे
• 2025-2028: भुखमरी खत्म हो जाएगी
• 2025: युद्ध के परिणामस्वरूप यूरोप की जनसंख्या गायब हो जाएगी
• 2028: ऊर्जा के अन्य स्रोतों को खोजने की उम्मीद के साथ वीनस जैसे अन्य ग्रहों की यात्रा का प्रयास होगा
• 2033: ध्रुवों के पिघलने से जल स्तर में वृद्धि
• 2045: हिमच्छादित चोटियों का अस्तित्व ही नहीं होगा
• 2076: साम्यवाद वापस आ जाएगा
• 2084: प्रकृति का पुनर्जन्म
• 2100: ग्रह के अंधेरे पक्ष को रोशनी देते एक नए सूर्य का जन्म.
• 2130: हम एलियंस के साथ संपर्क कर सकते हैं
• 2170: वैश्विक सूखा
• 2262: मंगल ग्रह को एक धूमकेतु का खतरा
• 2304: मनुष्य समय की यात्रा करने में सक्षम हो जाएगा
• 2480: दो कृत्रिम सूर्यों के टकराने से पृथ्वी पर अंधेरा हो जाएगा
• 3005: मंगल ग्रह पर युद्ध से ग्रह की गति बदल जाएगी
• 3010: चंद्रमा पर एक धूमकेतु के पहुंचने से पृथ्वी पत्थरों और धूल से घिर जाएगी
• 3797: पृथ्वी खत्म हो जाएगी लेकिन मानवजाति इतनी तरक्की कर चुकी होगी कि एक नई सौर प्रणाली के पास चली जाएगी
• 4674: मानव जाति एलियंस के साथ मिल जाएगी
• 5079: ब्रह्मांड खत्म हो जाएगा

तो आप क्या सोचते हैं?

(इन भविष्यवाणियों को सुनकर भयानक भविष्य के लिए खुद को तैयार करने से पहले बस यह याद रखें: बाबा वैंगा ने नवंबर 2010 से अक्टूबर 2014 तक न्यूक्लियर विश्व युद्ध की भविष्यवाणी की थी. इतना ही नहीं उन्होंने संभवत: बराक ओबामा, निकोलस सरकोजी, व्लादिमीर पुतिन और गॉर्डन ब्राउन जैसे चार राष्ट्र प्रमुखों की हत्या के प्रयासों के बारे में भी कहा था. ये सभी ठीक हैं.)

## Tuesday, February 2, 2016

### Subramanian Swamy

Subramanian Swamy, mythical phoenix bird, he reinvents himself from the very flames that others think he burnt down with.

The Man and His Machines!

Teacher, economist, mathematician, politician, rebel, crusader, dog lover, all this and much more, was born into a family of intellectuals....

Not one to let go a fight and not one who forgets easily, survivor and loner, Swamy has bounced back to cast an imprint on contemporary history that few individuals in India can lay claim to.

1. His Father Was a Well Known Mathematician

Swamy was born in Mylapore, Chennai on Sept 15, 1939.

His father, Sitaram Subramanian was at one time director of the Central Statistical Institute.

2. He Graduated in Mathematics from Hindu College (Finished 3rd in DU)

It was in the very stars that he was born under. He was not six months old, when his mathematician father Sitaram Subramanian, in 1940, changed jobs and moved from Chennai (then Madras) to Delhi, the seat of power.

Swamy graduated from the prestigiousHindu College in B.A. (Hon.), finishing 3rd in the Delhi University.

3. Enrolled at Indian Statistical Institute (ISI), Kolkata for Post Graduation

From Delhi, the seat of power, Swamy moved to Kolkata (then Calcutta) for PG studies. It was going to be his first battle ground.

4. Director of the Institute Happened to a be Professional Rival of Swamy’s Father

The institute at that time was headed by PC Mahalanobis who happened to be a professional rival of Swamy’s father. So when Mahalanobis learnt about Swamy, the latter began to get lower grades. Too bad (for Mahalanobis).

Mahalanobis was the brain behind setting up of the Planning Commission, something that Prime Minister Narendra Modi after many decades intends to dismantle.

He was the kind of person that no one (at least not someone studying at his institute) would want to develop animosity with.

5. Swamy Taught the Big Guy a Lesson

Swamy’s ability at crunching numbers and postulating theories, pitched him against PC Mahalanobis.

His paper ‘Notes on Fractile Graphical Analysis’ published in Econometrica, 1963 had questioned a Mahalanobis statistical analysis method as not being original but only a differentiated form of an older equation, was an early expression of the rebel that Swamy is, a trait that has found expression both as an intellectual and as a politician.

6. Got a Recommendation for Harvard

Having demonstrated his ability for research,Hendrik S Houthakker, the American economist who was the referee for the paper published in Econometrica, recommended Swamy’s admission to Harvard.

7. Completed PhD from Harvard at 24

Backed by a full Rockefeller scholarship, in two and a half years, Swamy at 24 completed his PhD.

At Harvard, having cut his teeth in mathematics in the early 1960s and armed with a doctorate at 24 years of age, by 27 he was a teacher at Harvard.

8. Co-Authored a Paper with the 1st American Who Won Nobel Memorial Prize in Economic Sciences

Swamy co-authored a paper on theory of index numbers with Paul Samuelson. The paper was published in 1974

9. Became an Expert on the Chinese Economy

In 1975, Swamy wrote a book titled “Economic Growth in China and India, 1952–70: A Comparative Appraisal”

He learnt Chinese/Mandarin in just 3 months (when someone challenged him to learn this tough-language-to-learn in a year).

Till this day, Swamy is considered an authority on the Chinese economy and especially comparative analysis of Indian and China.

10. Got an Invite from Amartya Sen to Join DSE (Delhi School of Economics)

As an advocate of free markets economy, much before Manmohan Singh’s 1991 budget made it fashionable, Swamy’s market friendly views after moving from Harvard to Delhi School of Economics in 1968 were simply too radical and not palatable with Indira Gandhi’s socialist ‘Garibi Hatao’ India slogans.

Anyhow, Swamy accepted Amartya Sen’s offer.

The position earmarked for a young academician with market friendly views was a full professorial chair on Chinese studies.

But, by the time he traveled from Harvard to DSE, other traveler academics at the famed institute had changed their views on Swamy.

He was just offered a Reader’s rank at DSE. A sharp U-turn.

Students backed Swamy.

11. Moved to IIT in 1969

Swamy taught economics to students at the IIT.

He would often meet students at the hostels and discuss political and international views.

He suggested that India ought to do away with Five Year Plans and stop relying on foreign aid.

According to him, it was possible to achieve 10% growth.

12. Indira Took Note of Swamy in 1970

Indira, one of India’s most powerful prime ministers, in a 1970 budget debate dismissed Swamy as a “Santa Claus with unrealistic ideas.”

This was probably the first that a national leader of her stature had gone to the extent of directly mimicking Swamy’s ideas.

Swamy continued with his work nonetheless.

13. Establishment Goes After Swamy

The hostility cost him his IIT job from where he was unceremoniously sacked in December 1972.

Swamy in 1973 sued the prestigious institute for wrongful dismissal. He won the suit in 1991 and to prove his point, he joined only for a day before resigning.

14. Political Inning Began in 1974

With a young wife, a new born daughter and no job, Swamy was contemplating heading back to America when fate intervened and launched him into politics.

A phone call by Jan Sangh stalwart Nanaji Deshmukh picking Swamy to represent the party in the Rajya Sabha had him elected to parliament in 1974.

15. Dared the Establishment during the Emergency Days

Independence and the gross human tragedy that unfolded after partition, was something that a young Swamy saw up close. He was witness to the partition survivors’ daily struggle taking place just outside the family’s government allotted house at Turkman Gate, Delhi.

The emergency (1975-77) made a political hero out of him. Swamy defied and evaded arrest warrants for the entire 19 month period.

His most daring act during emergency was coming into India from America, breaking through security cordons of parliament, attending a Lok Sabha session on 10th August 1976, managing to slip out of parliament, escaping from the country and returning to America.

16. Founding Member of the Party that Won Elections after Emergency

Swamy was one of the founding members of the Janata Party that swept the Indira Gandhi emergency regime out of power in 1977.

Though the party splintered but Swamy stuck on and was its president since 1990 till the party was merged with BJP in 11 August, 2013. Opposition often joked about him heading Janta Party as being a general without an army. But, he has been that way for a long time.

17. Swamy’s Blueprint – A Guiding Light for Manmohan Singh in Early 1990s

As the country’s commerce and law minister during Chandra Shekhar’s brief term as Prime Minister in 1990-91, Swamy laid the foundations of economic reforms in India by creating a blue print.

Dr. Manmohan Singh, then FM presented interim budget for 1991-92 under Congress PM Narasimha Rao.

The same blue print was later picked up by finance minister Manmohan Singh under Prime Minister PV Narasimha Rao to deliver the country of Nehruvian socialism.

18. Given a Cabinet Rank When in Opposition

Being president of Janta Party and an opposition leader, Swamy has the distinction of being handed out a cabinet rank by the ruling party.

It is said that Swamy stood by Narasimha Rao Even in the Wilderness

Prime Minister PV Narasimha Rao in 1994 appointed Swamy as Chairman with Commission of Labor Standards and International Trade with a cabinet rank.

Contrary to what most Indians believe, Swamy, as pointed out above, is a mathematician by education.

It was the turn of events in his life that turned him to politics and law.

20. Played a Crucial Role in Exposing 2G Scam

After a long hibernation, Swamy writing to Prime Minister Manmohan Singh in 2008 seeking permission to prosecute A Raja over illegal allotment of mobile spectrum bands unraveled the colossal 2G Scam.

21. Made it Possible for Indians to Access the Kailash Mansarovar and protected Ram Sethu from destruction.

Subramanian Swamy played an important role in making it possible for people of Hindu faith in India to access the Kailash Mansarovar religious pilgrimage route. Also petitioned against destruction of Ram Sethu, which has environmental and religious impact.

To make it happen, he had met Deng Xiaoping China’s top guy of the time (April 1981).

22. Took on mighty Jayalalitha and Gandhi family in corruption cases.

Tamil Nadu CM was arrested and Gandhis had to apply for bail based on Swamy's efforts as anticorruption crusader.

23. Filed petition in Delhi high court on Nirbhaya case.

Requested high court against release of juvenile violent rapist irrespective of juvenile criminal bill still pending in Rajya sabha by blockade of opposition.Has pledged to fight in court and give Nirbhaya family justice.

### MAKAR SANKRANTI

WHY WE CELEBRATE MAKAR SANKRANTI ON JANUARY 14-15
(Extract from the article of Dr. Mayank Vahia)

On January 14 every year, we celebrate Makar Sankranti. It is the only Indian festival celebrated on a fixed calendric day of the solar calendar. All other Indian festivals are celebrated as per the lunar calendar, which make their days of celebration on the solar calendar vary every year.
...
The difference is easy to see. In India, we follow a lunar calendar; the moon goes from new moon to new moon or full moon to full moon in 29.5 days. We get 12 full moons in 354 days, making a lunar calendar year 354 days long. However, the Sun returns to the same spot in the sky every 365.25 days. So, there is a difference of 11.25 days between the solar and lunar years. Every 2.5 years, therefore, an intercalary month (the Adhik Maas) is added to the lunar calendar to roughly synchronise the two.
This is crucial because weather patterns follow the solar calendar, not the lunar. On the other hand, accurate ‘mahurat’ (or ‘muhurat’) calculations are better done with the relatively faster moving moon. In fact, to make such calculations more accurate, the path of the moon, which is slightly off from the path of the sun, is divided into 27 ‘nakshatras’ while the path of the sun is divided into 12 ‘rashis’.
All this is simple and good. The exact calculations are a bit more complex since the fractions given above are not exact. Moreover, Indian calculations are done with natural numbers rather than fractions, so numbers have to be magnified accordingly.

But the problem of Makar Sankranti is unique: it goes entirely by the solar calendar. The clue to this mystery lies in the fact that Makar Sankranti is also called Uttarayan, or the day on which the sun begins its northward journey.

The solar calendar itself is fairly rigid (except for the 0.25 at the end of 365 which in reality is 365.256363004 days). So this additional 0.006363004 over 365.25 means that we slightly over compensate when in a leap year we add February 29. To re-correct for it, we don’t have a leap year in the years ending with 00. This gives a reasonably accurate and stable calendar.

In this system, the sun enters different zodiacs on a fixed day with an error of one day on either side depending on how close you are to the leap year. In the Indian system, this correction mechanism is more subtle and complex, and involves use of additional days with the same lunar date etc.

The path of the sun over one year is divided into ‘rashis’, which are identical to the zodiacs.
So around mid-December, the sun rises in the Sagittarius or Dhanush, in January in Capricorn or Makara, and so on.

We celebrate January 14 as the day on which the sun begins to rise in the Makara Rashi, Sankranti meaning entering.

But then there is an additional problem. The Earth’s axis of rotation moves north-south. So if you look to the sky everyday and note where the sun rises, you will notice that in one year it drifts from north of east to south of east. The same will be true in the west if you track sunset. For the northern hemisphere, the further south the sun is, the less it remains in the sky and the colder it is on earth.
So the point at which the sun rises is crucial in deciding the seasons. When the sun starts moving south, the days get colder. In reality, because the Earth can store some heat and its daily motion is relatively small, it takes a few weeks after the sun begins to move south to get colder.

Hence, from time immemorial, the days on which the sun touches its northernmost and southernmost points are noted. These are called solstices – winter or summer. In Sanskrit, the journey southwards is called Dakshinayan, and the one northward is called Uttarayan, ‘dakshin’ and ‘uttar’ being south and north respectively. Now the question is how to map this directional movement of the sun with its movement in the zodiac. In principle it is easy. We know the winter solstice falls on December 21, and hence Uttarayan begins on that day, while the summer solstice falls on June 21, when Dakshinayan begins.

So, why do we celebrate Uttarayan on Makar Sankranti, when, as you must have realised, it should be celebrated on Dhanu Sankranti? This is where history comes in.

While the exact day on which the winter or summer solstice occurs remains steady (within one day error), there is a slight change in the way the Earth’s rotation axis is aligned to the sun. Hence, over a period of a few hundred years, this drift means that even though the sun begins its Uttarayan on December 21, it is not in the Makara rashi as it was about 1,500 years ago. So, 1,500 years ago, during the time of Aryabhata, the Uttarayan and Makar Sankranti coincided.

### Evidence of democracy in Rig Veda

Evidence of democracy in Rig Veda
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Evidence of a Democratic system of government in India is originally found in Rig Veda, which mentions a thriving republican form of Government in India.
These are the slokas from Rig Veda which were to be sung in unison at the beginning of the republican assembly
...
sáM sam íd yuvase vRSann ágne víshvAny aryá Á
iLás padé sám idhyase sá no vásUny Á bhara ||
devÁ bhAgáM yáthA pÚrve saMjAnAnÁ upÁsate ||
samAnó mántraH sámitiH samAnÍ samAnám mánaH sahá cittám eSAm
samAnám mántram abhí mantraye vaH samAnéna vo havíSA juhomi ||
samAnÍ va ÁkUtiH samAnÁ hRdayAni vaH
samAnám astu vo máno yáthA vaH súsahÁsati || [Rig Veda 10.191.1-4]

Translation :THOU, mighty Agni, gatherest up all that is precious for thy friend. Bring us all treasures as thou art enkindled in libation’s place.
Assemble, speak together: let your minds be all of one accord, As ancient Gods unanimous sit down to their appointed share.

The place is common, common the assembly, common the mind, so be their thought united. A common purpose do I lay before you, and worship with your general oblation.
One and the same be your resolve, and be your minds of one accord. United be the thoughts of all that all may happily agree.

However, not much historic evidence is available today to prove implementation of democractic and republic ideas in ancient India.
The terms Sabha,( gathering) , Samiti, ( smaller Gathering or Committee ) Rajan or Raja,( Householder, Leader), exists and are found in Vedic literature.
Sabha is found eight times in the Rig Veda, which is accepted as the oldest Veda.

The term Rajan denoted Householder, Head of the Household. One who was eligible to take part in the assembly or gathering or the sabha.It did not mean a King, but simply meant a ‘Leader’, a leader who was elected.
The term Raja came to mean in time, a feudal king, a monarch. The elected leader or elected king would, as is usual with human nature, wish his offspring to follow in his footsteps, and take the leadership or Kingship position after him.

Rig Veda also says that the position of the King(Leader) was not absolute, and he could be removed by the Sabha or the Assembly.
The term Sabha is still used today, as the Indian Elected Parliament is called the ‘Lok Sabha- Assembly of the People’, Nominated Parliament is called ‘Rajya Sabha‘ and state assemblies are called as ‘Vidhana Sabha’.

Democracy today functions at 3 basic level in India :
The village level- The Panchayat – Council of Five (Panch = Five)
Province or State level – Assembly
National – Lok Sabha and Rajya Sabha in Parliament.

It is known to modern world that Issac Newton discovered universal gravitation in 16th century when he observed an apple fall from a tree .

But Prasnopanishad(6000 BC) described the force that pulls objects down and keeps us grounded on earth without floating(Sage/philosopher Kanad -2nd century BCE, from school of Vaisheshika wrote commentaries based on vedas and upanishads)

In Prashnopanishad c...hapter 3 while describing the panchapranas, the apana vayu is said to be residing in the anus and genitals – paayoopasthepaanam.

It is responsible for throwing out from the body faeces, urine, semen, menstrual blood and foetus. Further, the upanishad says :

prithivyaam yaa devataa saisha purushasya apaanamavashatabhyaantaraa ||

"The devata that is in earth she supports this apana. She helps apana for throwing out from the body. "

Usually Space travelers face difficulty in excretion due to absence of gravitational force there. The link between apana and the earth aiding it is quite clear in this upanishad."

‘tathaa prithivyaam abhimaaninee yaa devataa prasiddhaa saishaa purushasya apaanavrithimavashtabhyaakrishya vasheekrityaadha evapakarshanenanugraham kurvati vartata ityarthaha. anyathaa hi shareeram gurutvat patetsavakashe vaa udagacheta’.||

"This devata blesses by supporting apana by pulling in the downward direction. Or else, the body would have floated up."

### Vashishtha Narayan Singh- From Bhojpur to Berkley

The Great Mathematician, solving 4-6 OUT OF 8 unsolved Maths problems of ARYABHATT and challenged works of Great Scientist Albert Einstein, suffers from Schizophrenia.

Born in 1942 in Village Basantpur, under Sadar Block of Dist. Bhojpur. In the year 1962, he passed his matriculation examination, topping in the entire state of Bihar.

After his school education, he got admission in the prestigiou...s Patna Science College. During that time, Dr. P. Nagendra, a great mathematician, was the principal of Science College. He truly realized the hidden talent in young Vashishtha. Coincidentally, at the same time, the great American scholar Prof. Kelly was visiting Patna to participate in the World Mathematics Conference. Prof. Nagendra arranged an interview of Vashishtha with Prof. Kelly. Prof. Kelly quizzed the young student with various types of questions and Vashishtha answered all his questions correctly. Seeing the immense talent in Vashishtha Babu, Prof. Kelly expressed his desire to teach him in America under his guidance. Dr. Nagendra showed promptness and immediately arranged for a special examination for Vashishtha Babu and he cleared this examination with cent percent marks. Thus, in 1963, he went to California, USA as a research scholar. There, he conducted research on the Cycle Vector Space Theory and his research work catapulted him to great heights in the world of Science.

After completing his research, Vashishtha Babu came back to India, but he was destined to return soon to America. During his second stint in the USA, he was appointed an Associate Professor of Mathematics in Washington. He returned to India in 1971 and was appointed a professor in IIT Kanpur. After spending barely eight months at IIT Kanpur, he joined as a professor in Tata Institute of Fundamental Research. After a year, in 1973, he was appointed as permanent professor in Indian Statistical Institute, Calcutta.

An alumnus of the well-known Netarhat School, Singh worked on space theory at Nasa before returning home. He tied the knot in 1974, but his wife deserted him after he suffered his first attack of schizophrenia in 1976.

Even now, Vashishtha Babu keeps writing something or the other.

Vashishtha Narayan Singh, who had been languishing in penury, has joined the BhupendraNarayanMandal University (BNMU) in Madhepura as a visiting professor.( April 2013)

In the Pic: Vashishtha Narayan Singh, Update on the University of California, BERKELEY and John L Kelley.

### Myth of CASTE SYSTEM in India -It was Varna system

DOCUMENTARY PROOF IT WAS VARNA OR CLASSIFICATION OF PROFESSIONAL WORK IN ANCIENT INDIA!!
In ancient India society was based upon the varna (class) according to their vocation.
Four orders of society were recognized based upon the four duties of human beings and established society accordingly. These four groups were the Brahmins, the priests or teacher class; the Kshatriya, the nobility or administrative class; the Vaishya, the merchants and farmers; and the Shudras or helpers for above all.
These four orders of society were called "varna", which means a "veil". As color it does not refer to the color of the skin of people, but to the qualities or energies of human nature. As a veil it shows the four different ways in which the Divine Self is hidden in human beings.
In ancient India, these divisions were not based on birth but based on qualifications. According to the Bhagavad Gita this Aryan family system started breaking down in India at the time of attacks of Mlechhas .Hence after so many years this system of determining natural aptitude has degenerated into the caste system which resembles it now only in form.
1. The Suta and the Magadha were traditionally the bards and the chroniclers, in fact the preservers of the early Indian historical tradition. They were close to the king not only because of their profession, but we are told that the presence of the Suta was essential to one of the rites in a royal sacrifice.
2. That the members of the samkirna jatis did not necessarily in fact have a low social status is indicated by the sources. The Aitareya Brahmana mentions an Ambastha king.~ Aitereya Brahmana, VIII, 21; The Ambastha tribe is frequently identified by modern scholars with Ambastanoi of Arrian and the Sambastoi of Diodorus.H.C. Raichaudhury, Political History of Ancient India, (Calcutta, 1952), p. 255.
3. The Taittiriya Brahmana refers to the material well-being of the Ugras, one of whom is mentioned as a king's officer.~Taittiriya Brahmana, III, 8, 5.
There is no word like "caste" in sanskrit! Casta is an Iberian word (existing in Spanish, Portuguese and other Iberian languages since the Middle Ages), meaning "lineage", "breed" or "race." It is derived from the older Latin word castus, "chaste," implying that the lineage has been kept pure. Casta gave rise to the English word caste during the Early Modern Period.[3][4].https://en.wikipedia.org/wiki/Casta#Etymology

## Monday, February 1, 2016

### Myth of Vasco de gama discover sea route to India

Vasco da Gama discovered sea route to India . Really ? Read on
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Portugese explorer Vasco da Gama, who is credited with discovering the sea route to India, actually followed a Gujarati trader from Zanzibar, a new book claimed. Suresh Soni, author of ‘India’s Scientific Heritage’, quoting archaeologist Dr Vishnu Shridhar Wakankar, said ”He no doubt came to India but not as a discoverer sea-farer but following an Gujarati trader from Zanzibar.” According to Dr Wakankar, Vasco da Gama had recorded in his diary that upon his arrival at Zanzibar in Africa he saw a docked ship three times bigger than his own. He took an African interpreter to meet the owner of that ship Chandan, a Gujarati trader who used to bring pine wood and teak from India along with spices and take back diamonds to Cochin. Vasco da Gama followed Chandan to reach the shores of India, a fact very few in independent India know about, regrets Soni. ”This should have been told to the new generation but this is not done,” he added.

The author said Venetian trader and explorer Marco Polo, as early as 13th century, had recounted that ships in India had double boards which were joined together with strong nail and crevices, filled with special kind of gum and were so huge that 300 boatmen were needed to row them. These vessels could take a load of 3000 to 4000 gunny bags having small rooms and arrangements for comfort. Additional layers were added to the bottom, when it gets damaged. Some ships had as many as six layers, the book says. In the 15th century another traveller Nicolo Conti found Indian ships were much bigger than their own ships and their bases were made of three boards to weather formidable storms. Some ships were built in a such a manner that if one part was damaged, the rest could substitute for it. Another traveller Berthma had written how wooden boards were joined to prevent even a drop of water seeping into the ship and that it would take eight days to come to Iran from Cape Comorin (Kanyakumari), the book records.