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Copy of a letter sent by an independent mathematician to an astronomer colleague at University College London. The mathematician wishes to remain anonymous, as does his colleague in London.

Here it is............

Peter, I hope you can help me with this problem: SOME RELATIONSHIPS CONNECTING THE SEMI-MAJOR AXES OF PLANETARY ORBITS. Introduction Although I have attempted to simplify the mathematical treatment of the analysis that follows, this work appears to be entirely attributable to Mr G. Curtis, who first discussed the matter with me a few months ago. He claims to have limited mathematical background but, he has been able to obtain a number of unexpected relationships between the lengths of the semi-major axes of the planets' elliptical orbits. As far as I am able to judge, the relationships do not derive from any fundamental mathematical theory (eg Kepler's laws) and yet, in the main, they so closely reflect the truth that I am reluctant to dismiss the results as a consequence of pure chance. Unfortunately, my own knowledge of astronomy is so sparse that I am not able to judge the merits of these findings, so I am seeking the views of a more authoritative and reliable source. Data (Throughout, AU units are employed.) In the table below, semi-major planetary axes lengths, (the arithmetic mean of aphelion and perihelion distances) are denoted by z; with x and y values being taken as z(3/2p) and z(9/4p) respectively. Planet Z value X value Y value Mercury (Me) 0.3870987   0.5067562 Venus (V) 0.7233322   0.7929725 Earth (E) 1.0000000   1.0000000 Mars (Ma) 1.5236915 1.2227197 1.3520425 Jupiter (J) 5.2028039 2.1977496   Saturn (S) 9.5388437 2.9354505   Uranus (U) 19.1818710 4.0976543   Neptune (N) 30.0579240 5.0777748   Pluto (P) 39.4390000 5.7809400   Table 1 Preliminary x-y graph (not to scale) It is now 'speculated' that the four points shown on the graph may be joined by a straight line of gradient 1/ln30 which passes through the point ((20p)3/2p,(p)9/4p). The equation of this line is (y-(p)9/4p) = (ln30)-1(x-(20p)3/2p), i.e. y=0.2940141 x +0.1472341 (Equation 1) My own determination of the linear regression line for the four points shown gives the equation y=0.2933088 x +0.1464114 (Equation 2) with correlation coefficient r = 0.999900. Clearly the two equations are very similar and comparison of the z values, calculated from the two equations, with the original data, is shown in the following table.. Planet Z (data) Z (equation 1) Z (equation 2) Mercury 0.3870987 0.3870718 0.3852757 Venus 0.7233322 0.7238812 0.7208603 Earth 1.0000000 1.0144079 1.0103543 Mars 1.5236915 1.5236281 1.5177892 Table 2 This shows that Eric's equation and the regression equation produce very accurate results but that, with Eric's equation, three out of the four z estimates are better than those obtained from the regression equation. Graphical development If x values for Neptune and Pluto are now introduced onto the graph then the corresponding y values may be determined from equation (1) as 1.6401715 and 1.84691 respectively. (These correspond to z values of 1.99545533 AU and 2.35520343 AU ) On the developed graph shown below, labels (P1) and (P2) indicate positions on the y axis. Alternatively, the following approach may be employed : Considering the y differences along the y axis, Table 1 gives (V) - (Me) = 0.2862163 = a (say), (E) - (V) = 0.2070275 = b (say) and (Ma) - (E) = 0.3520425 = c (say); then (P1) is taken as (Ma)+a = 1.3520425 + 0.2862163 = 1.6382588 and (P2) is taken as (P1) + b = 1.6382588 + 0.2070275 = 1.8452863 By comparing these y values with those above we see that the differences are both about 0.1%---- a surprising result ! (For these values of y the corresponding z values are 1.99220695 and 2.35231288.) If further extrapolation along the y axes is considered, then another position (P3) is determined such that (P3) = (P2) +c = 1.8452863 + 0.3520425 = 2.1973288 , and the corresponding z value is 3.001757 AU. As a final graphic development, the y value for (P3) can be substituted into equation (1) to find the x value for a hypothetical planet Px (as shown on the graph). This is found to have an x value of 6.9895438 which corresponds to a z value of 58.6961334 AU. It appears that Eric has used (P1) and (P3) (z values of 2 AU and 3 AU respectively) as rough boundaries for the asteroid belt, but I am unable to comment on this. Eric has also marked in x scale differences A,B, and C, which correspond to the a,b,and c differences on the y axis. Further findings a,b,c / Pythagoras: The square root of (a2 + b2) = 0.3532423 and c= 0.3520425, which differs by about 0.3%. Thus, a2 +b2 ~ c2 ! (Consequently A2 + B2 ~ C2 ) b/a and Venus The ratio b/a = 0.2070275 / 0.2862163 = 0.7233253, but the z value for Venus is 0.7233322 --- ---a difference of less than 0.001% ! An amazing result. Comment Some aspects of the above investigation are not completely precise but, even allowing for such deficiencies, I find many of the results so surprising and apparently without any mathematical explanation. If there is some 'well-known' theory which accounts for such strange results, I would be grateful to receive details of it but, in any case, an informed second opinion would be welcomed by both Eric and myself. Many thanks for your help, Regards..

(Letter was sent to University College London on 23/7/01. Further information may be available if and when a reply is received..)