Karanapaddhati

Key takeaways

Karanapaddhati belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Karanapaddhati to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Karanapaddhati from memory before moving on to harder problems.

Reference excerpt

Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the attention of European scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit. The sixth chapter contains series expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and inverse tangent functions.

Author and date of Karanapaddhati Nothing definite is known about the author of Karanapaddhati. The last verse of the tenth chapter of Karanapaddhati describes the author as a Brahamin residing in a village named Sivapura. Sivapura is an area surrounding the present day Thrissur in Kerala, India. The period in which Somayaji lived is also uncertain. There are several theories in this regard.

C.M. Whish, the first westerner to write about Karanapaddhati, based on his interpretation that certain words appearing in the final verse of Karanapaddhati denote in katapayadi system the number of days in the Kali Yuga, concluded that the book was completed in 1733 CE. Whish had also claimed that the grandson of the author of the Karanapaddhati was alive and was in his seventieth year at the time of writing his paper. Based on reference to Puthumana Somayaji in a verse in Ganita Sucika Grantha by Govindabhatta, Raja Raja Varma placed the author of Karanapaddhati between 1375 and 1475 CE. An internal study of Karanapaddhati suggests that the work is contemporaneous with or even antedates the Tantrasangraha of Nilakantha Somayaji (1465–1545 CE).

Synopsis of the book A brief account of the contents of the various chapters of the book is presented below.

Chapter 1 : Rotation and revolutions of the planets in one mahayuga; the number of civil days in a mahayuga; the solar months, lunar months, intercalary months; kalpa and the four yugas and their durations, the details of Kali Yuga, calculation of the Kali era from the Malayalam Era, calculation of Kali days; the true and mean position of planets; simple methods for numerical calculations; computation of the true and mean positions of planets; the details of the orbits of planets; constants to be used for the calculation of various parameters of the different planets. Chapter 2 : Parameters connected with Kali era, the positions of the planets, their angular motions, various parameters connected with Moon. Chapter 3 : Mean center of Moon and various parameters of Moon based on its latitude and longitude, the constants connected with Moon. Chapter 4 : Perigee and apogee of the Mars, corrections to be given at different occasions for the Mars, constants for Mars, Mercury, Jupiter, Venus, Saturn in the respective order, the perigee and apogee of all these planets, their conjunction, their conjunctions possibilities. Chapter 5 : Division of the kalpa based on the revolution of the planets, the number of revolutions during the course of this kalpa, the number of civil and solar days of earth since the beginning of this kalpa, the number and other details of the manvantaras for this kalpa, further details on the four yugas. Chapter 6 : Calculation of the circumference of a circle using variety of methods; the division of the circumference and diameters; calculation of various parameters of a circle and their relations; a circle, the arc, the chord, the arrow, the angles, their relations among a variety of parameters; methods to memorize all these factors using the katapayadi system. Chapter 7 : Epicycles of the Moon and the Sun, the apogee and perigee of the planets; sign calculation based on the zodiacal sign in which the planets are present; the chord connected with rising, setting, the apogee and the perigee; the method for determining the end-time of a month; the chords of the epicycles and apogee for all the planets, their hypotenuse. Chapter 8 : Methods for the determination of the latitude and longitude for various places on the earth; the R-sine and R-cosine of the latitude and longitude, their arc, chord and variety of constants. Chapter 9 : Details of the Alpha aeries sign; calculation of the positions of the planets in correct angular values; calculation of the position of the stars, the parallax connected with latitude and longitude for various planets, Sun, Moon and others stars. Chapter 10 : Shadows of the planets and calculation of various parameters connected with the shadows; calculation of the precision of the planetary positions.

Infinite series expressions The sixth chapter of Karanapaddhati is mathematically very interesting. It contains infinite series expressions for the constant π and infinite series expansions for the trigonometric functions. These series also appear in Tantrasangraha and their proofs are found in Yuktibhāṣā.

Series expressions for π Series 1

Series 2

Series 3

Series expansions of trigonometric functions

References

Venketeswara Pai R, K Ramasubramanian, M S Sriram and M D Srinivas, Karanapaddhati of Putumana Somayaji, Translation with detailed Mathematical notes, Jointly Published by HBA (2017) and Springer (2018).

Further references Open Library reference to Karana-paddhati with two commentaries.[1] Bag, Amulya Kumar (1976). "Madhava's sine and cosine series" (PDF). Indian Journal of History of Science. 11 (1). Indian National Academy of Science: 54–57. Archived from the original (PDF) on 14 February 2010. Retrieved 17 December 2009. Bag, Amulya Kumar (1975). "The method of integral solutions of indeterminate equations of the type BY=AX ± C in ancient and medieval India" (PDF). Indian Journal of History of Science. 12 (1). Indian National Academy of Science: 1–16. Retrieved 12 January 2010. P.K. Koru, ed. (1953). Karanapaddhati of Puthumana Somayaji. Cherpu, Kerala, India: Astro Printing and Publishing Company. Indian National Science Academy has started a project in 2007–08 titled "A Critical Study of Karana-paddhati of Putumana Somayaji and Preparation of English Translation with Mathematical Notes" by Dr. K Ramasubramanian, Assistant Professor, Dept. of History, Indian Institute of Technology, Powai, Mumbai 400076.[link removed] (Retrieved on 13 January 2010)

Worked examples

Example 1 — a first encounter with Karanapaddhati

Start with the simplest possible case. Write down what Karanapaddhati claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Karanapaddhati before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Karanapaddhati ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Karanapaddhati

In research: Karanapaddhati appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Karanapaddhati in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Karanapaddhati is common in secondary-school and first-year university syllabi. It links to neighbouring topics Astronomy books, Hindu astronomy, Indian astronomy texts, so understanding it makes those chapters shorter.
In everyday life: Look for Karanapaddhati outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Karanapaddhati in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Karanapaddhati means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Karanapaddhati out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Karanapaddhati in simple terms?

Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain.

Why does Karanapaddhati matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Karanapaddhati?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Karanapaddhati.