Equation, This equation is the defining equation for all the major orbits of the Solar System. The equation is first used to calculate the value of 'F'. 'F' is the offset constant, calculated to ten decimals .... F = 0.50077209786..... Structure and use of main equation. A circle of circumference three (arbitrary units) has a radius of (3/2p) and an area of (9/4p). These two related nonrandom values serve as exponents in the production of an equation and a graph of the orbits of the Solar System. Inner mean planetary orbits AU­(9/4p) are to be related to other mean planetary orbits AU­(3/2p). (Note..Orbital periods in Years can be substituted for mean distance, the exponents need to be changed to (3/2p) or (1/p), respectively.) The upper limit for the inner system is declared to be at the value of 'p' astronomical units, and so the inner range is terminated at this value .... ....p­(9/4p) Outer system is declared to limit at 20p AU, so outer range is terminated at .... (20p)­(3/2p). The equation is :- [( p­(9/4p).Ln30) - F]­(2p/3) = 20p The equation is illustrated in graphical form in fig 1

Fig 1, graph of ... [( p­(9/4p).Ln30) - F]­(2p/3) = 20p

Application of Equation, Fig 2 The first 'p' symbol, on the left of the equation, represents the input number, it also represents the upper limit for that number. Any number equal to or less than 'p' may be put into the equation in this place. Any input number less than 'p' will give an output result that is less than 20p. All the inner orbits are less than 'p' in astronomical units, and all the outer orbits are less than 20p. This means that the orbits of Mercury, Venus, Earth, Mars and the Asteroids can all be used as inputs to the equation, but only when expressed in astronomical units, a point which is to be discussed later. When a mean value for one of the inner planets is input into the equation, the result is the mean orbit of another planet. Mercury gives Mars, Venus gives Jupiter, Earth gives Saturn and Mars gives Uranus. The result accuracy is better than 0.2% in all cases except for Saturn, which is 2.5%. These input and output conditions for the equation are represented in graphical form on fig 2 It is noted that not all the planets are represented on graph fig 2.

Fig 2 [( p­(9/4p).Ln30) - F]­(2p/3) = 20p

[( AU(9/4p).Ln30) - F]­(2p/3) = Outer AU

The Orbits of Venus and Mars, Fig 3 The main equation relates inner orbits to outer orbits. The distribution of the planets is also a matter that demonstrates mathematical order. The spaces between the inner planets on graph fig 3 are marked with the letters a,b,c. These letters represent the arithmetical differences between the values for the adjacent planets.

Fig 3 [( p­(9/4p).Ln30) - F]­(2p/3) = 20p

[( AU(9/4p).Ln30) - F]­(2p/3) = Outer AU

For example 'a' is equal to ... Venus mean AU ­(9/4p) minus Mercury mean AU ­(9/4p), which, if Nortons data is regarded as absolute, works out as :- Value of 'a' is 0.792972486 - 0.506756171 = 0.286216314... Earth is unity, and unity raised to any power remains unity, so we subtract the figure for Venus from 1.0000 to obtain 'b'. 1.0000 - 0.792972486 = 0.207027514 So 'b' = 0.207027514 and 'a' = 0.286216314 Notice that the ratio b/a is a pertinent number... 0.723325342 Nortons figure for Venus mean is ... 0.7233322 The figures are the same to within 0.001% Calculating Venus If we accept that V = b/a we can calculate 'V' from a knowledge of Mercury, and Earth, by successive approximation. (See section Venus Equation) When this is done, a slightly different result is obtained. By successive approximation V= 0.723331002... Which differs from Nortons figure by 0.000166%

Pythagoras Calculations imply that the inner orbits are related in some way by the rule of Pythagoras, but only after the exponent has been applied. It is true that the rule of Pythagoras can be used to calculate orbits to a high degree of accuracy. Calculate Mars from Mercury, Venus, and Earth. Apply the rule of Pythagoras. Calculate 'c' from the rule c2 = a2 + b2, and having found 'c' add it to unity and then raise the result to the inverted power (4p/9) to convert back to AU. We know 'a' and 'b'. By Pythagoras 'c' = 0.353242367.. Add this to Unity (Earth) and raise to the power (4p/9). Result is 1.525579777 AU which differs from Nortons figure for Mars by 0.124%. The important thing to note is that the values on the axes appear to be related by the rule of Pythagoras, and of course this must apply to both the axes and the coordinate points on the line, since the one is obtained from the other. We cannot prove the point we are making from any clever mathematics, such that Pythagoras himself used, but rather it is an empirical observation. . That is to say that if we assume Pythagoras applies, and use it to calculate orbits, the results are accurate to within 0.2%, so there is a pragmatic case for the use of Pythagoras, but not a formal proof. The rule of Pythagoras is applied.

Repeating Patterns, Fig 4 We might well wonder what would happen if we repeated the Pythagorean pattern in the empty regions of the axes ? We cannot do this at random, we must replicate the exact values, such that (see fig 4) a = a, b = b , c = c. When we do this we obtain three new points on the vertical axis which are labeled as P1, P2, and P3.

Fig 4 [( p­(9/4p).Ln30) - F]­(2p/3) = 20p

[( AU(9/4p).Ln30) - F]­(2p/3) = Outer AU

These points are easily calculated by addition of the three values a,b,c in summing succession to the Pythagorean point for Mars, which is 1 + 'c'. Remember that the results are figures already on the graph axis, which means they are seen as orbits 'already raised' to the exponent. Having obtained the axis values for these new points, we then calculate them out, to see what values they correspond to in Astronomical Units in the inner solar system, and we also put them through the main equation, to see what values they yield in the outer solar system. For point P1 P1 = 1 + c + a = 1.639458683 This figure is processed through the equation to yield an output value of 30.02787608 AU. This is the same as Nortons figure for Neptune mean to within 0.1%. P1 is raised to (4p/9) to give an inner system value of :- 1.994 AU, which is a good figure for the nominal inner edge of the Asteroid belt, often quoted as 2AU. For Point P2 To obtain the axis value for P2 we simply add 'b' to P1 to give P2 = 1.846486197.. We process this through the equation to find an outer orbit of 39.41830763 AU which just happens to differ from Nortons figure for the mean orbit of Pluto by 0.0525% When P2 is converted direct to inner system units, it turns out to be a rather meaningless figure lost in the asteroid belt. Though it is meaningless it does not detract from the overall scheme of things, so it could be seen as a 'null' or 'neutral' point. Approx = (3p/4)AU For Point P3 P3 Calculates to 3AU in the inner system, (actual 3.00633) and thus can be seen as the nominal outer edge of the Asteroid Belt, which is often quoted as 3AU. When processed through the main equation P3 produces a figure of 58.545AU which is beyond the orbit of Pluto and does not correspond to any known planet. The designation 'Planet X' has been added on the figures. The mathematical scheme under discussion predicts the possible existence of a planet of mean orbit 58.5 AU. It may be worth while for astronomers to look for such a planet, though it should be noted that if no such planet is found it would not have much effect on the relationships discussed in these pages, which remain true with regard to those planets that are currently known. Calculations continue in next section