Rhombus and Aryabhata: Difference between pages

Revision as of 04:18, 4 November 2006

Revision as of 13:20, 21 May 2007

Line 1:

[[Image:2064_aryabhata-crp.jpg|thumb|Statue of Aryabhata on the grounds of [[Inter-University Centre for Astronomy and Astrophysics|IUCAA]], [[Pune]].]]

:''For other uses of the word rhombus, see [[Rhombus (disambiguation)]]''

'''Āryabha{{Unicode|ṭ}}a''' ([[Devanāgarī]]: आर्यभट) (AD [[476]] – [[550]]) is the first of the great mathematician-astronomers of the classical age of [[Indian mathematics]]. He was born at Muziris (the modern day Kodungallour village) near [[Thrissur]], [[Kerala]]. Available evidence suggest that he went to [[Patna|Kusumapura]] for higher studies. He lived in Kusumapura, which his commentator [[Bhaskara I|Bhāskara I]] (AD [[629]]) identifies as Pataliputra (modern [[Patna, India|Patna]]).

Aryabhata was the first in the line of brilliant mathematician-astronomers of classical Indian mathematics, whose major work was the '''''[[Aryabhatiya]]''''' and the ''Aryabhatta-siddhanta''. The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple [[Bhaskara I]] (''Bhashya'', ca. 600) and by

[[Image:Rhombus.svg|frame|right|This shape is a rhombus]]

[[Nilakantha Somayaji]] in his ''Aryabhatiya Bhasya,'' (1465).

The number place-value system, first seen in the

3rd century [[Bakhshali Manuscript]] was clearly

in place in his work.<ref>P. Z. Ingerman, 'Panini-Backus form', Communications of the ACM 10

(3)(1967), p.137</ref> He may have been the first mathematician to

use letters of the alphabet to denote unknown quantities.<ref>

History of Hindu Mathematics/Bibhutibhushan Dutta and Avadhesh Narayan Singh, Asia Publishing House, 1962. (reprint ISBN 81-86050-86-8). </ref>

Aryabhata's system of astronomy was called the ''audAyaka system'' (days are reckoned from ''uday'', dawn at ''lanka'', equator). Some of his later writings on astronomy, which apparently

In [[geometry]], a '''rhombus''' (or '''rhomb''') is a [[quadrilateral]] in which all of the sides are of equal length, i.e., it is an [[equilateral]] quadrangle. In colloquial usage the shape is often described as a ''diamond'' or ''lozenge''.

proposed a second model (''ardha-rAtrikA'', midnight), are lost, but can be partly

reconstructed from the discussion in [[Brahmagupta]]'s ''khanDakhAdyaka''. In some texts he seems to ascribe the apparent motions of the heavens to the earth's rotation.

==The Two definitions Aryabhatiya==

{{Mergefrom|Aryabhatiya|date=April 2007}}

If a figure meeting the above description has a right angle, then all its angles are right angles and it is a [[square (geometry)|square]]. There is disagreement as to whether a square is considered a kind of rhombus or not. If it is, the above definition of rhombus is the complete one; if not, the definition additionally requires the angles not to be right angles.

===Pi as Irrational===

The number system we use today known as Hindu-Arabic number system was developed by Indian mathematicians and spread around the world by Arabs. In Aryabhatiya, Aryabhatta stated "''Stanam Stanam Dasa Gunam''" or in English "''Place to Place Ten Times in Value''". As per Tobias Denzig, discovery of the place value notation is a world event. Later zero was added to the Aryabhatta's number system by [[Brahmagupta]].

Aryabhata worked on the approximation for [[Pi]], and may have realized that <math>\pi</math> is irrational. In the second part of the Aryabhatiyam. In other words, <math>\pi \approx 62832/20000 = 3.1416</math>, correct to five digits. The commentator [[Nilakantha Somayaji]], ([[Kerala School]], 15th c.) has argued that the word ''āsanna'' (approaching), appearing just before the last word, here means not only that this is an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 ([[Johann Heinrich Lambert|Lambert]]).

Aryabhata's greatest contribution is signified by 0 (Zero). Notation for placeholders in positional numbers is found on stone tablets from ancient (3,000 B.C.) Sumeria. Yet, the Greeks had no concept of a number like zero. In terms of modern use, zero is sometimes traced to the Indian mathematician Aryabhata who, about 520 A.D., devised a positional decimal number system that contained a word, "kha," for the idea of a placeholder. By 876, based on an existing tablet inscription with that date, the kha had become the symbol "0". Meanwhile, somewhat after Aryabhata, another Indian, Brahmagupta, developed the concept of the zero as an actual independent number, not just a place-holder, and wrote rules for adding and subtracting zero from other numbers. The Indian writings were passed on to al-Khwarizmi (from whose name we derive the term algorithm) and thence to Leonardo Fibonacci and others who continued to develop the concept and the number.

===Mensuration and Trigonometry===

== Properties ==

In any rhombus, opposite sides will be [[parallel (geometry)|parallel]]. Thus, the rhombus is a special case of the [[parallelogram]]. One suggestive analogy is that the rhombus is to the parallelogram as the square is to the [[rectangle]]. A rhombus is also a special case of a [[kite (geometry)|kite]], that is, a quadrilateral with two pairs of equal adjacent sides. The opposite sides of a kite are not parallel unless the kite is also a rhombus.

In Ganitapada 6, Aryabhata gives the '''area of triangle''' as

The rhombus has the same [[symmetry]] as the rectangle (with [[symmetry group]] ''D''₂, the [[Klein four-group]]) and is its [[Duality (mathematics)|dual]]: the vertices of one correspond to the sides of the other.

: ''tribhujasya phalashariram samadalakoti bhujardhasamvargah'' (for a triangle, the result of a perpendicular with the half-side is the area.)

But he gave an incorrect rule for the volume of a pyramid.<ref>{{cite book|first=Roger|last=Cooke|authorlink=Roger Cooke|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Mathematics of the Hindus|pages=205|isbn=0471180823|quote=Aryabhata gave the correct rule for the are of a triangle and an incorrect rule for the volume of a pyramid. (He claimed that the volume was half the height times the area of the base.)}}</ref> Aryabhata was not concerned with demonstrating his formulas.<ref>{{cite book|first=Roger|last=Cooke|authorlink=Roger Cooke|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Mathematics of the Hindus|pages=205|isbn=0471180823|quote= It is clear therefore that Aryabhata was not concerned with demonstration;}}</ref> Aryabhata, in his work ''Aryabhata-Siddhanta'', first defined the sine as the modern relationship between half an angle and half a chord.

A rhombus in the plane has five [[degrees of freedom (physics and chemistry)|degrees of freedom]]: one for the shape, one for the size, one for the [[orientation (rigid body)|orientation]], and two for the position.

He also defined the [[cosine]], [[versine]], and inverse sine. He used the words ''jya'' for sine, ''kojya'' for cosine, ''ukramajya'' for versine, and ''otkram jya'' for inverse sine.

Aryabhata's tables for the sines (from which the rest can be computed), is presented in a single rhyming stanza, with each syllable standing for increments

The diagonals of a rhombus are [[perpendicular]] to each other. Hence, by joining the midpoints of each side, a [[rectangle]] can be produced.

at intervals of 225 minutes of arc or 3 degrees 45'. Using a compact alphabetic code called ''varga/avarga'', he defines the sines for a circle of circumference

21600 (radius <math>\approx</math> 3438). He uses the alphabetic code to define a set of increments :''makhi bhakhi fakhi dhakhi Nakhi N~akhi M~akhi hasjha ...''.

Here "makhi" stands for 25 (ma) + 200 (khi), and

the corresponding sine value (for 225 minutes of arc) is 225 / 3438. The value corresponding to the eighth term (hasjha, 199 (ha=100 + s=90 + jha=9), is the sum of all the increments before it, totalling 1719.

The entire table for 90 degrees is given as follows:

: 225,224,222,219,215,210,205,199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,22,7

So we see that sin(15) (sum of first four terms) = 890/3438 = 0.258871 (correct value = 0.258819, correct to four significant digits). The value of sin(30) (corresponding to ''hasjha'') is 1719/3438 = 0.5; this is of course, exact. His alphabetic code (there are many such codes in Sanskrit) has come to be known as the [[Aryabhata cipher]].

One of the five 2D [[lattice (group)|lattice]] types is the rhombic lattice, also called centered rectangular lattice.

===Motions of the Solar System===

Adjacent angles of a rhombus are supplementary.

Aryabhata described a [[geocentric]] model of the solar system, in which the Sun and Moon are each carried by [[epicycle]]s which in turn revolve around the Earth. In this model, which is also found in the ''Paitāmahasiddhānta'' (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller ''manda'' (slow) epicycle and a larger ''śīghra'' (fast) epicycle.<ref>David Pingree, "Astronomy in India", pp. 127-9.</ref> The positions and periods of the planets were calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn move around the Earth at specific speeds representing each planet's motion through the zodiac. Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic [[Greek astronomy#Hellenistic astronomy|Greek astronomy]]. Another element in Aryabhata's model, the ''śīghrocca'', the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying [[heliocentric]] model. Aryabhata defines the sizes of the planets' orbits in terms of these periods.<ref>Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," ''Scripta Mathematica'', 22(1956): 165-192; reprinted in Otto Neuge bauer, ''Astronomy and History: Selected Essays,'' New York: Springer-Verlag, 1983, pp. 129-156. ISBN 0-387-90844-7</ref><ref>Hugh Thurston, ''Early Astronomy,'' New York: Springer-Verlag, 1996, pp. 178-189. ISBN 0-387-94822-8</ref>

== Proof ==

=== The diagonals are perpendicular. ===

Let A, B, C and D be the [[vertices]] of the rhombus, named in agreement with the figure (higher on this page). Using <math>\overrightarrow{AB}</math> to represent the vector from A to B, one notices that

<br><math>\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}</math>

<br><math>\overrightarrow{BD} = \overrightarrow{BC}+ \overrightarrow{CD}= \overrightarrow{BC}- \overrightarrow{AB}</math>.<br>

The last equality comes from the parallelism of CD and AB.

Taking the [[inner product]],

<br><math>

<\overrightarrow{AC}, \overrightarrow{BD}> = <\overrightarrow{AB} + \overrightarrow{BC}, \overrightarrow{BC} - \overrightarrow{AB}> = <\overrightarrow{AB}, \overrightarrow{BC}> - <\overrightarrow{AB}, \overrightarrow{AB}> + <\overrightarrow{BC}, \overrightarrow{BC}> - <\overrightarrow{BC}, \overrightarrow{AB}> = 0

</math><br>

since the norms of AB and BC are equal and since the inner product is [[bilinear]] and symmetric. The inner product of the diagonals is zero if and only if they are perpendicular. {{Unicode|■}}

He states that the [[Moon]] and planets shine by reflected sunlight. He also correctly explains eclipses of the Sun and the Moon, and presents methods for their calculation and prediction.

== Area ==

The [[area]] of any rhombus is one half the product of the lengths of its [[diagonal]]s:

<br><math>A=\frac{D_1 \times D_2}{2}</math><br>

Because the rhombus is a [[parallelogram]] with four equal sides, the area also equals the length of a side (B) multiplied by the perpendicular distance between two opposite sides(H):<br><math>A=B \times H</math>

In the fourth book of his Aryabhatiya, ''Goladhyaya'' or Golapada, Aryabhata is dealing with the celestial sphere, shape of the earth, cause of day and night etc. In golapAda.6 he says:

== Origin ==

: ''bhugolaH sarvato vr.ttaH'' (The earth is circular everywhere)

The origin of the word ''rhombus'' is from the [[Greek language|Greek]] word for something that spins. [[Euclid]] uses the word ρομβος; and in his translation Heath says it is apparently drawn from the Greek word ρεμβω, to turn round and round. He also points out that [[Archimedes]] used the term solid rhombus for two right circular [[cone (geometry)|cone]]s sharing a common base. For more on the origin of the word, see ''rhombus'' at the [http://www.pballew.net/rhomb.html MathWords web page].

Another statement, referring to ''Lanka '', describes the movement of the stars as a relative motion caused by the rotation of the earth:

== External links ==

:Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (i.e. on the equator) as moving exactly towards the West. [''achalAni bhAni samapashchimagAni'' - golapAda.9]

*[http://www.mathopenref.com/rhombus.html Rhombus definition. Math Open Reference] With interactive applet.

*[http://www.mathopenref.com/rhombusarea.html Rhombus area. Math Open Reference] Shows three different ways to compute the area of a rhombus, with interactive applet.

However, in the next verse he describes the motion of the stars and planets as real: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.

[[Category:Quadrilaterals]]

''Lanka'' here is a reference point to mean the equator, which was known to pass through [[Sri Lanka]]. Aryabhatta make numerous references to Lanka where there is a doubt whether he was originally from Sri Lanka, island nation south of India.

[[vls:Rutte]]

Aryabhata's computation of Earth's [[circumference]] as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation improved on the computation by the Alexandrinan mathematician

[[bg:Ромб]]

[[Erastosthenes]] (c.[[200 BC]]), whose exact computation is not known in modern units.

[[ca:Rombe]]

[[da:Rombe]]

====Sidereal periods====

[[de:Raute]]

[[es:Rombo]]

Considered in modern English units of time, Aryabhata calculated the [[sidereal rotation]] (the rotation of the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the [[sidereal year]] at 365 days 6 hours 12 minutes 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.

[[fr:Losange]]

[[gl:Rombo]]

====Heliocentrism====

[[ko:마름모]]

[[ht:Lozanj]]

Aryabhata's computations are consistent with an [[Ellipse|elliptical]]<ref>J. J. O'Connor and E. F. Robertson, [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Aryabhata_I.html Aryabhata the Elder], ''[[MacTutor History of Mathematics archive]]'':

[[it:Rombo]]

<br>{{quote|"He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses."}}</ref> [[heliocentrism|heliocentric]] motion of the planets orbiting the sun and the earth spinning on its own axis. While he is not the first to say this, his authority was certainly most influential. The earlier Indian astronomical texts ''[[Shatapatha Brahmana]]'' (c. 9th-7th century BC), ''[[Aitareya Brahmana]]'' (c. 9th-7th century BC) and ''[[Vishnu Purana]]'' (c. 1st century BC) contain early concepts of a heliocentric model.<ref>Teresi, Dick (2002). ''Lost Discoveries: The Ancient Roots of Modern Science - from the Babylonians to the Maya''. [[Simon & Schuster, Inc.|Simon & Schuster]], New York. ISBN 0-684-83718-8.</ref><ref>[[Madame Blavatsky|Blavatsky, Helena P.]] (1877). ''[[Isis Unveiled]]'', [http://www.sacred-texts.com/the/iu/iu000.htm Part One, Chapter I]. [[Theosophical Society in America|Theosophical University Press]]. ISBN 0-911500-03-0.</ref><ref>[[Subhash Kak|Kak, Subhash C.]] (2000). 'Birth and Early Development of Indian Astronomy', p. 31-33. In Selin, Helaine (2000). ''Astronomy Across Cultures: The History of Non-Western Astronomy'' (303-340). Kluwer, Boston. ISBN 0-7923-6363-9.</ref> Though [[Aristarchus of Samos]] (3rd century BC) and sometimes [[Heraclides of Pontus]] (4th century BC) are usually credited with knowing the heliocentric theory, the version of [[Greek astronomy]] known in ancient India, ''[[Paulisa Siddhanta]]'' (possibly by a [[Paulus Alexandrinus|Paul]] of [[Alexandria]]) makes no reference to a Heliocentric theory. The [[8th century]] [[Arabic]] edition of the ''Āryabhatīya'' was translated into [[Latin]] in the [[13th century]], well before Copernicus and may have influenced European astronomy, though a direct connection with [[Copernicus]] cannot be established. 10th century Arabic scholar [[Abū al-Rayhān al-Bīrūnī|Al-Biruni]] states that Aryabhata's folowers believe Earth to revolving around the sun. Then he casually adds that this notion does not create any mathematical difficulties. In Indian astronomy, the Sun is always at the center in the "sugrocha" system. The concept of the Earth revolving around the Sun was known to Aryabhatta at least 1,000 years before [[Copernicus]].<ref name=Teresi>Teresi, Dick (2002). ''Lost Discoveries: The Ancient Roots of Modern Science - from the Babylonians to the Maya''. [[Simon & Schuster, Inc.|Simon & Schuster]], New York. ISBN 0-684-83718-8.</ref><ref>Ragep, J. (2002). [http://www.npr.org/templates/story/story.php?storyId=885213 Ancient Roots of Modern Science], ''[[Talk of the Nation]]''.</ref>

[[he:מעוין]]

[[lt:Rombas]]

===Diophantine Equations===

[[li:Roet]]

A problem of great interest to [[Indian mathematicians]] since very ancient times concerned [[diophantine equations]]. These involve integer solutions to equations such as ax + b = cy. Here is an

[[hu:Rombusz]]

example from [[Bhaskara]]'s commentary on Aryabhatiya: :

[[nl:Ruit (meetkunde)]]

: Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9 and 1 as the remainder when divided by 7.

[[nds:Ruut]]

i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general,

[[no:Rombe]]

diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text [[Sulba Sutras]], the more ancient parts of which may date back to [[800 BCE]]. Aryabhata's method of solving such problems, called the ''kuttaka'' (कूटाक) method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm,

[[pl:Romb]]

as elaborated by Bhaskara in AD [[621]], is the standard method for solving first order Diophantine equations,

[[pt:Losango]]

and it is often referred to as the [[Aryabhata algorithm]]. See details of the Kuttaka method in this

[[ru:Ромб]]

[http://www.ias.ac.in/resonance/Oct2002/pdf/Oct2002p6-22.pdf|article].

[[fi:Neljäkäs]]

[[sv:Romb]]

== Continued Relevance ==

[[vi:Hình thoi]]

[[uk:Ромб]]

Aryabhata's astronomical calculation methods have been in continuous use for the practical purposes of fixing the [[Panchanga]]

[[zh:菱形]]

[[Hindu calendar]].

Recently Aryabhata was a theme in the RSA Conference 2006, [[Indocrypt]] 2005, which had a session on Vedic mathematics.

The [[lunar crater]] [[Aryabhata (crater)|Aryabhata]] is named in his honour.

==Notes==

{{reflist}}

==References==

*William Eugene Clark, ''The Aryabhatiya of Aryabhata, An Ancient Indian Work on Mathematics and Astronomy'', University of Chicago Press (1930); reprint: Kessinger Publishing (2006), ISBN 978-1425485993.

* {{Harvard reference

| Surname1 = Dutta

| Given1 = B.

| Surname2 = Singh

| Given2 = A.N.

| Year = 1962

| Title = History of Hindu Mathematics

| Publisher = Asia Publishing House, Bombay

}}

* [[Subhash Kak|Kak, Subhash C.]] (2000). 'Birth and Early Development of Indian Astronomy'. In {{Harvard reference

| Surname1 = Selin

| Given1 = Helaine

| Year = 2000

| Title = Astronomy Across Cultures: The History of Non-Western Astronomy

| Publisher = Kluwer, Boston

| ID = ISBN 0-7923-6363-9

}}

* {{Harvard reference

| last = Pingree

| first = David

| authorlink = David Pingree

| contribution = Astronomy in India

| editor-last = Walker

| editor-first = Christopher

| title = Astronomy before the Telescope

| pages = 123-142

| publisher = British Museum Press

| place = London

| year = 1996

| ID = ISBN 0-7141-1746-3

}}

* {{Harvard reference

| Surname1 = Rao

| Given1 = S. Balachandra

| Year = 1994/1998

| Title = Indian Mathematics and Astronomy: Some Landmarks

| Publisher = Jnana Deep Publications, Bangalore

| ID = ISBN 81-7371-205-0

}}

* Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.

* {{Harvard reference

| Surname1 = Thurston

| Given1 = H.

| Year = 1994

| Title = Early Astronomy

| Publisher = Springer-Verlag, New York

| ID = ISBN 0-387-94107-X

}}

==External links==

* {{MacTutor Biography|id=Aryabhata_I}}

* Amartya K Dutta, [http://www.ias.ac.in/resonance/Oct2002/pdf/Oct2002p6-22.pdf Diophantine equations: The Kuttaka], Resonance, October 2002. Also see earlier overview: [http://www.ias.ac.in/resonance/April2002/pdf/April2002p4-19.pdf ''Mathematics in Ancient India,'']

* [http://2006.rsaconference.com/us/conference/theme.aspx RSA Conference 2006]

* [http://www.cse.iitk.ac.in/~amit/story/19_aryabhata.html ''Aryabhata and Diophantus' son'', [[Hindustan Times]] Storytelling Science column, Nov 2004]

[[Category:Indian mathematics]]

[[Category:476 births]]

[[Category:550 deaths]]

[[Category:5th century mathematicians]]

[[Category:6th century mathematicians]]

[[Category:Indian astronomers]]

[[Category:Ancient Indian mathematicians]]

[[Category:Medieval astronomers]]

[[Category:Hindus]]

[[Category:People with craters of the Moon named after them]]

[[bn:আর্যভট্ট]]

[[de:Aryabhata]]

[[es:Aryabhata]]

[[fr:Âryabhata]]

[[gu:આર્યભટ્ટ]]

[[hi:आर्यभट]]

[[is:Aryabhata]]

[[kn:ಆರ್ಯಭಟ]]

[[ml:ആര്യഭടന്‍]]

[[nl:Aryabhata]]

[[ja:アリヤバータ]]

[[pms:Aryabhata]]

[[pt:Aryabhata]]

[[sa:आर्यभट:]]

[[sl:Aryabhata I.]]

[[fi:Aryabhata]]

[[sv:Aryabhata]]

[[ta:ஆரியபட்டா]]

[[uk:Аріабхата I]]