#### Authors:

Prabir Chandra Bhattacharyya,#### DOI NO:

https://doi.org/10.26782/jmcms.2022.06.00004#### Keywords:

Bhattacharyya’s Coordinate System,Cartesian Coordinate system,Equation of the circles,Quadratic equation,Theory of Dynamics of Numbers,#### Abstract

The concept of the circle has been known to human beings since before the beginning of recorded history. With the advent of the wheel, the study of the circle in detail played an important role in the field of science and technology. According to the author, there are three types of circles, 1) Countup circle, 2) Countdown circle, and 3) Point circle instead of two types of circles as defined by René Descartes in real plane coordinate geometry and Euler in the complex plane. The author has been successful to solve the equations of three types of circles in the real plane by using three fundamental recent (2021 – 2022) inventions, 1) Theory of Dynamics of Numbers, 2) Rectangular Bhattacharyya’s Co-ordinate System, 3) The novel Concept of Quadratic Equation where the author becomes successful to solve the quadratic equation of x^{2}+ 1 = 0 in real number instead of an imaginary number. In the present paper, the author solved successfully the problem where radius if g

^{2}+ f

^{2}< c, c the constant term of the general form of the equation of a circle x

^{2}+ y

^{2}+ 2gx + 2fy + c = 0 by using Bhattacharyya’s Coordinate system without any help from the complex plane where Euler solved it by using a complex plane. According to Bhattacharyya’s Co-ordinate System, the equation of the countdown circle is as follows : where, the coordinates of the moving point P are (x, y) with Centre C (a, b) and radius = – r The concept of a countdown circle is very much interesting and it exists really in nature. We may consider that the rotational motion of the Earth around the Sun is a countdown rotational motion.

#### Refference:

I. Arthur Koestler, The Sleepwalkers: A History of Man’s Changing Vision of the Universe (1959)

II. Bibhutibhushan Datta, The Jaina School of Mathematics, Bull. Cal. Math. Soc. 21 (1929), 115 -145.

III. Bibhutibhushan Datta, Mathematics of Nemicandra, The Jaina Antiquary, 1 (1935),

IV. Chronology for 30000 BC to 500 BC Archived 2008-03-22 at the Wayback Machine. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.

V. Circumcircle – from Wolfram MathWorld Archived 2012-01-20 at the Wayback Machine. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

VI. D. Fowler and E. Robson, Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context, Historia Math. 25 (1998) 366 – 378

VII. Dr. S.N. De, ‘Higher Mathematics’, New Edition. March, 2006. P : C4.13

VIII. G. Thibaut, The Sulvasutras, The Journal, Asiatic Society of Bengal, Part I, 1875, Printed by C.B. Lewis, Baptist Mission Press, Calcutta, 1875.

IX. Gamelin, Theodore (1999). Introduction to topology. Mineola, N.Y: Dover Publications. ISBN 0486406806.

X. George Gheverghese Joseph, A passage to infinity, Medieval Indian mathematics from Kerala and its impact, Sage Publications, Los Angeles, CA, USA, 2009.

XI. Incircle – from Wolfram MathWorld Archived 2012-01-21 at the Wayback Machine. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

XII. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007.

XIII. Katz, Victor J. (1998), A History of Mathematics / An Introduction (2nd ed.), Addison Wesley Longman, p. 108, ISBN 978-0-321-01618-8.

XIV. Kim Plofker, Mathematics in India : 500 BCE – 1800 CE, Princeton University Press, NJ, USA, 2008. 17

XV. Kirkus Archived 2013-11-06 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus.

XVI. Meskhishvili, Mamuka (2020). “Cyclic Averages of Regular Polygons and Platonic Solids”. Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340.

XVII. Prabir Chandra Bhattacharyya, “AN INTRODUCTION TO RECTANGULAR BHATTACHARYYA’S CO-ORDINATES: A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, November (2021). pp 76-86.

XVIII. Prabir Chandra Bhattacharyya ,“AN INTRODUCTION TO THEORY OF DYNAMICS OF NUMBERS: A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, January (2022). pp 37-53.

XIX. Prabir Chandra Bhattacharyya, : ‘A NOVEL CONCEPT IN THEORY OF QUADRATIC EQUATION’. J. Mech. Cont. & Math. Sci., Vol.-17, No.-3, March (2022) pp 41-63.

XX. Proclus, The Six Books of Proclus, the Platonic Successor, on the Theology of Plato Archived 2017-01-23 at the Wayback Machine Tr. Thomas Taylor (1816) Vol. 2, Ch. 2, “Of Plato”

XXI. Raghunath P. Kulkarni, Char Sulbasutra (in Hindi), Maharshi Sandipani Rashtriya ´ Vedavidya Pratishthana, Ujjain, 2000.

XXII. R.C. Gupta, Bh¯askara I’s approximation to sine, Indian J. History Sci. 2 (1967), 121 – 136.

XXIII. R.C. Gupta, Madhavacandra’s and other octagonal derivations of the Jaina value π = √ 10, Indian J. Hist. Sci. 21 (1986), no. 2, 131 – 139.

XXIV. R.C. Gupta, New Indian values of π from the Manava Sulbasutra, Centaurus 31 (1988), no. 2, 114 – 125.

XXV. R.C. Gupta, Sulvasutras: earliest studies and a newly published manual, Indian J. ´ Hist. Sci. 41 (2006), 317 – 320.

XXVI. S.G. Dani, Geometry in the Sulvasutras, in ´ Studies in History of Mathematics, Proceedings of Chennai Seminar, Ed. C.S. Seshadri, Hindustan Book Agency, New Delhi, 2010.

XXVII. S.N. Sen and A.K. Bag, The Sulvasutras of Baudhayana, Apastamba, Katyayana, and Manava, Indian National Science Academy, 1983.

XXVIII. Squaring the circle Archived 2008-06-24 at the Wayback Machine. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.

XXIX. The papers of R.C. Gupta cited here are also available in the compilation of Gan. Itananda, edited by K. Ramasubramanian, Published by the Indian Society for History of Mathematics (ISHM), 2015.