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by making more accurate observations, beginning by refining the elements of the orbit of the Earth, reducing the eccentricity from Kepler's value of 0.018 to 0.0173,70 which he derived by combining Tycho's solar theory (in which Tycho had assumed a solar parallax of 3') with Kepler's value of 1' for the solar parallax. This change, which reduced the errors in Tycho's theory by more than

Horrocks' observational procedures were quite sophisticated and often involved the design and construction of his own instruments (Chapman (1990)). Still about 3 per cent too large, Wilson (1980), p. 67.

50 per cent, then had a knock-on effect for all the planetary theories. Horrocks directed most of his efforts on Venus and set about trying to determine its orbital parameters with great accuracy. He concluded from his observations and calculations that the mean solar distance (in terms of the radius of the orbit of the Earth) was 0.7233, exactly as predicted by Kepler's third law, ratherthan Kepler's value of 0.7241. He wrote in his Venus Visible on the Sun (1662), published over 20 years after the author's untimely death aged only 22:

... the proportion that obtains between the periods of the motions of the planets and the semi-diameters of their orbits is most exact, as Kepler, its discoverer, rightly states, and as I by repeated and most certain observation have found; indeed there is not an error of even a minute ...

So Horrocks was a believer in Kepler's first and third laws but, as with many other astronomers, he had problems in applying the second. As we have seen, Kepler had reduced the application of the second law to the solution of the equation t = 0 + e sin 0 for the eccentric anomaly 0. Inthe Rudolphine Tables, Kepler tabulated t (in fact the logarithm of t) for equally spaced values of 0, from which the user of the tables had to interpolate between the non-uniformly spaced values of t. Horrocks instead derived a geometrical approximation, which is illustrated in Figure 7.10, and clearly has its origins in the use of epicycles. Given the mean anomaly t, we locate a point C on the unit circle and then construct a circle of radius e, where e is the eccentricity of the planetary orbit. If we denote the angle AO B by 0 ,an application of the sine rule (Horrocks actually used the law of tangents) to the triangle OBC, BC being parallel to OA, gives

71 Horrocks Venus in sole visa. Quoted from Wilson (1978).

Fig. 7.11. Determining the size of Venus as viewed from the Sun.

Fig. 7.11. Determining the size of Venus as viewed from the Sun.

from which e sin 0 = sin(t — 0) « t — 0 if t and 0 are close together. Horrocks knew that he was only approximating the area law, but he underestimated the magnitude of the error.

With his new orbital parameters for Venus, Horrocks predicted that there would be a transit in 1639 (a supreme test of the accuracy of his theory) and he wrote to his friend William Crabtree asking him to attempt to observe the predicted transit, so as to reduce the chance that bad weather might interfere with the observation. As far as we know, Horrocks and Crabtree were the only two people to observe the event. 3 Horrocks waited patiently through the predicted day of the transit and probably was resigning himself to failure when, not long before sunset,

... the clouds, as if by Divine Interposition, were entirely dispersed ... and I then beheld a most agreeable sight, a spot, which had been the object of my most sanguine wishes, of an unusual size, and of a perfectly circular shape, just wholly entered upon the sun's disc ... I was immediately sensible that this round spot was the planet Venus, and applied myself with the utmost care to prosecute my observations.

Horrocks measured the apparent diameter of Venus at 1' 16" (about 10 times smaller than contemporary wisdom suggested) and used this, together with some Keplerian style speculation, to estimate the Earth-Sun distance. He first calculated the apparent diameter of Venus as it would be observed from the Sun using the method illustrated in Figure 7.11. If we denote the radius of Venus by r, we then have

1 a ~ tan1 a = r/d2, 2P ~ tan 2P = r/d1, so P can be determined in terms of a provided the ratio d2 /dx is known.

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