Kepler's Third Law

In 1619, Kepler discovered, I know not how, his third law, which relates the period, P, and the mean distance, R, of the six known planets:
P2 = C*R3
where C is a constant. This is the equation for a semi-cubical parabola.

If P is measure in days, and R in AU (astronomical units, in which R = 1 for the Earth), then the constant C is about 134,000. In fact, here is modern data for the six planets, again from (Moore/Hunt, 1983, p. 404). In the last column we have computed, for comparison, the value of the constant for each planet. The average value is 133,428.2.

Planet R, AU P, days P2 / R3
Mercury 0.387 87.959 133,483.5
Venus 0.723 224.701 133,703.6
Earth 1.000 365,256 133,411.9
Mars 1.524 686.980 133,331.7
Jupiter 5.203 4332.590 133,270.4
Saturn 9.539 10759.200 133,367.9
And here is a graph of the modern data, P vs R.
This curve is actually the graph of the equation above, with the six data points taken from the table above. The question is: How did Kepler see the equation in the table? Did he use the graph? And if so, how did he recognize the semi-cubical parabola?
Rev'd 10 jan 2002 by ralph abraham