Kriyakramakari
Encyclopedia
Kriyakramakari is an elaborate commentary in Sanskrit
written by Sankara Variar
and Narayana, two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics, on Bhaskara II's well-known textbook on mathematics Lilavati
. Kriyakramakari ('Operational Techniques'), along with Yuktibhasa
of Jyeshthadeva, is one of the main sources of information about the work and contributions of Sangamagrama Madhava, the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a ninth century CE astronomer from Kerala.
Sankara Variar
(circa. 1500 - 1560 CE), the first author of Kriyakramakari, was a pupil of Nilakantha Somayaji
and a temple-assistant by profession. He was a prominent member of the Kerala school of astronomy and mathematics. His works include Yukti-dipika an extensive commentary on Tantrasangraha by Nilakantha Somayaji. Narayana (circa. 1540-1610 CE), the second author, was a Namputiri Brahmin
belonging to the Mahishamangalam family in Puruvanagrama (Peruvanam in modern-day Thrissur District
in Kerala
).
Sankara Variar wrote his commentary of Lilavati
up to stanza 199. Variar completed this by about 1540 CE when he stopped writing due to other preoccupations. Sometimes after his death, Narayana completed the commentary on the remaining stanzas in Lilavati.
based on Kriyakramakari, stanza 199 of Lilavati reads as follows (Harvard-Kyoto convention
is used for the transcription of the Indian characters):
This could be translated as follows;
Taking this verse as a starting point and commenting on it, Sanakara Variar in his Kriyakrakari explicated the full details of the contributions of Sangamagrama Madhava towards obtaining accurate values of π. Sankara Variar commented like this:
Sankara Variar then cites a set of four verses by Madhava that prescribe a geometric method for computing the value of the circumference
of a circle
. This technique involves calculating the perimeter
s of successive regular circumscribed polygon
s, beginning with a square
.
To translate these verses into modern mathematical notations, let C be the circumference
and D the diameter
of a circle
. Then Madhava's easier method to find C reduces to the following expression for C:
This is essentially the series known as the Gregory-Leibniz series for π. After stating this series, Sankara Variar follows it up with a description of an elaborate geometrical rationale for the derivation of the series.
of a circle. This yields the infinite series expansion of the arctangent function. This result is also ascribed to Madhava.
The above formulas state that if for an arbitrary arc
θ of a circle
of radius
R the sine
and cosine are known and if we assume that sinθ < cos θ, then we have:
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
written by Sankara Variar
Sankara Variar
Sankara Variar was an astronomer-mathematician of the Kerala school of astronomy and mathematics who lived during the sixteenth century CE...
and Narayana, two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics, on Bhaskara II's well-known textbook on mathematics Lilavati
Lilavati
Lilavati was Indian mathematician Bhāskara II's treatise on mathematics. It is the first volume of his main work Siddhānta Shiromani, Sanskrit for "Crown of treatises," alongside Bijaganita, Grahaganita and Golādhyāya.- Name :The name comes from his daughter Līlāvatī...
. Kriyakramakari ('Operational Techniques'), along with Yuktibhasa
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...
of Jyeshthadeva, is one of the main sources of information about the work and contributions of Sangamagrama Madhava, the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a ninth century CE astronomer from Kerala.
Sankara Variar
Sankara Variar
Sankara Variar was an astronomer-mathematician of the Kerala school of astronomy and mathematics who lived during the sixteenth century CE...
(circa. 1500 - 1560 CE), the first author of Kriyakramakari, was a pupil of Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
and a temple-assistant by profession. He was a prominent member of the Kerala school of astronomy and mathematics. His works include Yukti-dipika an extensive commentary on Tantrasangraha by Nilakantha Somayaji. Narayana (circa. 1540-1610 CE), the second author, was a Namputiri Brahmin
Brahmin
Brahmin Brahman, Brahma and Brahmin.Brahman, Brahmin and Brahma have different meanings. Brahman refers to the Supreme Self...
belonging to the Mahishamangalam family in Puruvanagrama (Peruvanam in modern-day Thrissur District
Thrissur district
Thrissur is a revenue district of Kerala situated in the central part of that state. Spanning an area of about 3,032 km2, Thrissur district is home to over 10% of Kerala’s population. Thrissur district was formed on July 1, 1949, with the headquarters at Thrissur City. Thrissur is known as...
in Kerala
Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
).
Sankara Variar wrote his commentary of Lilavati
Lilavati
Lilavati was Indian mathematician Bhāskara II's treatise on mathematics. It is the first volume of his main work Siddhānta Shiromani, Sanskrit for "Crown of treatises," alongside Bijaganita, Grahaganita and Golādhyāya.- Name :The name comes from his daughter Līlāvatī...
up to stanza 199. Variar completed this by about 1540 CE when he stopped writing due to other preoccupations. Sometimes after his death, Narayana completed the commentary on the remaining stanzas in Lilavati.
On the computation of π
As per K.V. Sarma's critical edition of LilavatiLilavati
Lilavati was Indian mathematician Bhāskara II's treatise on mathematics. It is the first volume of his main work Siddhānta Shiromani, Sanskrit for "Crown of treatises," alongside Bijaganita, Grahaganita and Golādhyāya.- Name :The name comes from his daughter Līlāvatī...
based on Kriyakramakari, stanza 199 of Lilavati reads as follows (Harvard-Kyoto convention
Harvard-Kyoto
The Harvard-Kyoto Convention is a system for transliterating in ASCII the Sanskrit language and other languages that use the Devanāgarī script...
is used for the transcription of the Indian characters):
- vyAse bha-nanda-agni-hate vibhakte kha-bANa-sUryais paridhis sas sUkSmas/
- dvAviMzati-ghne vihRte atha zailais sthUlas atha-vA syAt vyavahAra-yogyas//
This could be translated as follows;
- "Multiply the diameter by 3927 and divide the product by 1250; this gives the more precise circumference. Or, multiply the diameter by 22 and divide the product by 7; this gives the approximate circumference which answers for common operations."
Taking this verse as a starting point and commenting on it, Sanakara Variar in his Kriyakrakari explicated the full details of the contributions of Sangamagrama Madhava towards obtaining accurate values of π. Sankara Variar commented like this:
- "The teacher Madhava also mentioned a value of the circumference closer [to the true value] than that: "Gods [thirty-three], eyes [two], elephants [eight], serpents [eight], fires [three], three, qualities [three], Vedas [four], naksatras [twentyseven], elephants [eight], arms [two] (2,827,433,388,233)—the wise said that this is the measure of the circumference when the diameter of a circle is nine nikharva [10^11]." Sankara Variar says here that Madhava’s value 2,827,433,388,233 / 900,000,000,000 is more accurate than "that", that is, more accurate than the traditional value for π."
Sankara Variar then cites a set of four verses by Madhava that prescribe a geometric method for computing the value of the circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. This technique involves calculating the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
s of successive regular circumscribed polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s, beginning with a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
.
An infinite series for π
Sankara Variar then describes an easier method due to Madhava to compute the value of π.- "An easier way to get the circumference is mentioned by him (Madhava). That is to say:
-
- Add or subtract alternately the diameter multiplied by four and divided in order by the odd numbers like three, five, etc., to or from the diameter multiplied by four and divided by one.
-
- Assuming that division is completed by dividing by an odd number, whatever is the even number above [next to] that [odd number], half of that is the multiplier of the last [term].
-
- The square of that [even number] increased by 1 is the divisor of the diameter multiplied by 4 as before. The result from these two (the multiplier and the divisor) is added when [the previous term is] negative, when positive subtracted.
-
- The result is an accurate circumference. If division is repeated many times, it will become very accurate."
To translate these verses into modern mathematical notations, let C be the circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
and D the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. Then Madhava's easier method to find C reduces to the following expression for C:
- C = 4D/1 - 4D/3 + 4D/5 - 4D/7 + ...
This is essentially the series known as the Gregory-Leibniz series for π. After stating this series, Sankara Variar follows it up with a description of an elaborate geometrical rationale for the derivation of the series.
An infinite series for arctangent
The theory is further developed in Kriyakramakari. It takes up the problem of deriving a similar series for the computation of an arbitrary arcArc
Arc may refer to:-Mathematics:*Arc , a segment of a differentiable curve*Arc , a particular type of set of points of a projective plane*Arcminute, a measure used for angles, equal to 1/60th of a degree...
of a circle. This yields the infinite series expansion of the arctangent function. This result is also ascribed to Madhava.
- "Now, by just the same argument, the determination of the arc of a desired Sine can be [made]. That is as [follows]:
-
- The first result is the product of the desired Sine and the radius divided by the Cosine. When one has made the square of the Sine the multiplier and the square of the Cosine the divisor,
-
- now a group of results is to be determined from the [previous] results beginning with the first. When these are divided in order by the odd numbers 1, 3, and so forth,
-
- and when one has subtracted the sum of the even[-numbered results] from the sum of the odd ones], [that] should be the arc. Here, the smaller of the Sine and Cosine is required to be considered as the desired [Sine].
-
- Otherwise there would be no termination of the results even if repeatedly [computed]."
The above formulas state that if for an arbitrary arc
Arc
Arc may refer to:-Mathematics:*Arc , a segment of a differentiable curve*Arc , a particular type of set of points of a projective plane*Arcminute, a measure used for angles, equal to 1/60th of a degree...
θ of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
of radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
R the sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
and cosine are known and if we assume that sinθ < cos θ, then we have:
- θ = (R sin θ)/(1 cos θ) − (R sin3 θ)/(3 cos3 θ) + (R sin5 θ)/(5 cos5 θ) − (R sin7 θ)/(7 cos7 θ)+ . . .