1. 1 Presented at Central University of Finance and Economics 中央财经大学 Beijing by 卜若柏 Robert Blohm Chinese Economics and Management Academy 中国经济与管理研究院 http://www.blohm.cnc.net June 8, 15 & 22, 2008 2008 年 6 月 8 日和 15 日和 22 日 Extract from From the Renaissance to the Scientific Revolution

2. 2 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World  Newton Proof that Kepler’s laws are a special case of Newton’s Gravitation law in Andrew T Hyman “A simple Cartesian treatment of planetary motion” European Journal of Physics 14 (1990) 145-147, posted at http://www.blohm.cnc.net/philosophy/newtonEU.pdf  “The task of demonstrating the relationship between the laws of Kepler and Newton was 'the major scientific problem of the [seventeenth] century' (Cohen, I. 1982 Physics, ed P Tipler (New York Worth ).”  The following detailed mathematical demonstration is only mentioned, but not shown, in Hyman’s accordingly short paper.

3. 3 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d)   is eccentricity    a circle    an ellipse   a parabola    a hyperbola The Sun is at the origin and the planet’s directrix is a straight line perpendicular to the x-axis at a distance D/  from the Sun. [D is the planet’s distance from the sun when the planet crosses the y-axis] and is called the ‘semi-latus-rectum’ of the conic section R/  R

4. 4 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) (2) Kepler’s Second Law of Motion: fixed proportionality of angular area to time-elapse _ = (1) Kepler’s First Law of Motion: elliptical motion = rectangular area under ellipse = area of last triangle to subtract to get angular area under ellipse

5. 5 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) (3) Kepler’s Third Law of Motion: fixed proportion of square of revolution period t to cube of average distance R = a to focus for all orbits around that focus. C 2 /D = K = vertical distance, R = Y, to focus = constant Derivation of (3) from characteristics a, b, C, D of ellipse:

6. 6 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of a in terms of D &  used in derivation of (3): semi-major axis average dis- tance of ellipse from x1x1 x2x2 y

7. 7 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) 1st of 2 alternative derivations of b in terms of D &  used in derivation of (3): x b R a a=R x semi-minor axis b :

8. 8 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) 2nd of 2 alternative derivations of b in terms of D &  used in derivation of (3): x b R a a=R x semi-minor axis:

9. 9 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Newton’s law of planetary acceleration/gravitation : = _ Differentiation of Kepler’s First Law (1): (1) Differentiation of Kepler’s Second Law (2): (2) (4) (5)

10. 10 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Newton’s law of planetary acceleration/gravitation (cont.d): from (4): from (5): substituting these in (5): (6)

11. 11 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Newton’s law of planetary acceleration/gravitation (cont.d): from (4): from (5): substituting these in (5): (7)

12. 12 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Newton’s law of planetary acceleration/gravitation (cont.d): Differentiating (5) [which was derived from Kepler’s 2nd law]: Differentiating (6) [which was derived from Kepler’s 1st & 2nd laws] & applying (9) & Kepler’s 3rd law: Calculating partial acceleration ( ) by 2 equations in 2 unknowns: (8) (9) Intermediate derivation: (10)

13. 13 Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Newton’s law of planetary acceleration/gravitation (cont.d): By (8) & (10): Calculating partial acceleration ( ) by 2 equations in 2 unknowns(cont.d): (11) Calculating total acceleration by (10) & (11): x y (12) Proportional to solar mass m 1, & planetary mass m 2 very small

14. 14 (8) is derivative of (5) :  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Kepler’s laws (1), (2), (3) as special cases of Newton’s law of gravitation (12). Proving : (12) decomposes into (11) & (10): Proving : (3) used in derivation of (10): (12) decomposes into (11) & (10): Proving : (8) used in derivation of (11): (5) is derivative of (2) : Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)

15. 15  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Kepler’s laws (1), (2), (3) as special cases of Newton’s law of gravitation (12). Proving (cont.d): Solving for : Substitute (5) for in (9): Proving : Substitute (10) for R 3 in (13), then integrate: Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d) (13) (14)

16. 16  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Kepler’s laws (1), (2), (3) as special cases of Newton’s law of gravitation (12). Proving (cont.): Solving for : Interchange x & y in (9) and substitute (5) into (9) to get: Proving (cont.d): Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d) (15) Substitute (11) for R 3 in (15), then integrate: (16)

17. 17  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Kepler’s laws (1), (2), (3) as special cases of Newton’s law of gravitation (12). Proving (cont.): Plug (14) & (16) into (5) to get (1): Proving (cont.d): Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d) (16’) Substitute (14’) & (16’) into (5): (17) (14’); (17) is a conic section of eccentricitywith directrix

18. 18  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Kepler’s laws (1), (2), (3) as special cases of Newton’s law of gravitation (12). Proving (cont.): Plug (14) & (16) into (5) to get (1) (cont.d): Proving (cont.d): Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d) (1) as a special case of (17): (3) (17’) [(17’) & A=0 ] (1) directrix: distance from origin/focus to directrix:

19. 19  Newton (cont.d) Proof that Kepler’s laws are a special case of Newton’s Gravitation law … (cont.d) Derivation of Kepler’s laws (1), (2), (3) as special cases of Newton’s law of gravitation (12). Proving (cont.): Proof of logical transitivity rule used in proving : Rise of Modern Science: 17th Cen- tury Inaugurates Modern World (cont.d)