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Katz (1998), p. 418 gives details of how Napier actually applied his logarithms to trigonometric problems.

Napier's invention greatly impressed Henry Briggs, who was at the time the Professor of Geometry at Gresham College, London. The two of them agreed that the system should be modified to make it easier to use. Briggs tells the story as follows:

I myself... remarked that it would be more convenient that 0 be kept for the logarithm of the whole sine [as it was in Napier's original work], but that the logarithm of the tenth part of the whole sine, that is to say 5° 44' 21'' [i.e. the angle whose sine is i] should be 1010. And concerning that matter I wrote immediately to the author himself; and ... I journeyed to Edinburgh, where being most hospitably received by him, I lingered for a whole month. But as we talked over the change in the logarithms he said that he had been for some time of the same opinion and had wished to accomplish it; he had, however, never published those he had already prepared, until he could construct more convenient ones if his affairs and his health would permit of it. But he was of the opinion the change should be effected in this manner, that 0 should be the logarithm of unity, and 1010 that of the whole sine, which I could not but admit was by far the most convenient.

These modified logarithms are, apart from the position of the decimal point, just logarithms to the base 10 (log10) or, as they became known, common logarithms. Briggs went on, after Napier's death, to produce tables of common logarithms accurate to fourteen decimal places, and these formed the bases of tables of logarithms right up until the arrival of the electronic calculator. As an aid to computation, common logarithms are simpler than Napier's original logarithms because setting log 1 = 0 ensures that there is no term equivalent to the 107ln107 inEqn(7.1)to subtract, i.e. log10(xix2) = log10(xi) + log10(x2).

Napier's work contained a discussion on the use of decimal fractions and his notation was the same as that which we use today. Because logarithms were welcomed widely throughout the scientific community, the publication of Napier's work and then Briggs' tables had the effect of spreading the use of decimal fractions across Europe.

Kepler carried out a study of Napier's logarithms during the winter of 1621 /2 and wrote a short book on the subject.52 He constructed his own version of logarithms, following Napier's approach rather than Briggs', which are related to our modern natural logarithms through

Keplerian logx = 105ln(105/x).

The Rudolphine Tables contains tables of these logarithms and was the first book to require the use of logarithms in a scientific application. Kepler was thrilled with this new device, though his former teacher Michael Mastlin commented,

51 Quoted from Coolidge (1949). 52 Chilias logarithmorum, eventually published in 1624.

'it is not seemly for a professor of mathematics to be childishly pleased about any shortening of the calculations'.53

The great success of the Rudolphine Tables in predicting planetary positions is well illustrated by the reaction of Peter Crüger, Professor of Mathematics at Danzig, who until their appearance had been unimpressed by Kepler's work. Responding to Philip Müller, Professor of Mathematics at Leipzig, on the subject of improving the planetary tables produced by Longomontanus, he wrote (in 1629):

But I should have thought that it would be a waste of time now that the Rudolphine Tables have been published, since all astronomers will undoubtably use these ... For myself, so far as other less liberal occupations allow, I am wholly occupied with trying to understand the foundations upon which the Rudolphine rules and tables are based, and I am using for this purpose the Epitome of Copernican Astronomy previously published by Kepler as an introduction to the tables. This epitome which previously I had read so many times and so little understood and so many times thrown aside, I now take up again and study with rather more success seeing that it was intended for use with the tables and is itself clarified by them ... I am no longer repelled by the elliptical form of the planetary orbits; Kepler's proofs in his New Astronomy have convinced me.54

We know now that the errors in Kepler's tables were about 30 times smaller than those in previous astronomical tables, though this would not have been apparent when they were first published, and not all astronomers were convinced immediately.55

Whereas Kepler's planetary theory, in the form of the Rudolphine Tables,was a success, the same could not be said for his physics. One of the most significant astronomical treatises published between those of Kepler and Newton was that written by the Parisian librarian Ismael Boulliau in 1645. Boulliau supported strongly the idea of elliptical orbits, but was not prepared to accept Kepler's physical theory. In the introduction he wrote:

After I had long considered Kepler's Commentaries on Mars and his Epitome of Copernican Astronomy, and had seen that his elliptical hypothesis represents the observed celestial motions more exactly than all the others, I did not cease praising and commending his felicitous ability and cleverness. At last I determined to seek the truth of a hypothesis so apt and appropriate, and to confirm the truth thus found

53 Quoted from Caspar (1993), p. 309. 54 Quoted from Russell (1964).

The Polish Jesuit Michael Boym introduced the Rudolphine Tables to the Far East, declaring in

1646 that they were 'of inestimable value in calculating partial and complete solar eclipses,

56 together with celestial movements' (Szczesniak (1949)).

Astronomia Philolaica, the reference to the Pythagorean Philolaus reinforcing the fact that the Earth is not immobile.

by reasons. But I saw that in the explanation of the hypothesis that man had left many roughnesses, had enunciated many things obscurely, and had thrust abstruse physical causes upon us in place of demonstrations; also, he had not demonstrated certain things that required demonstration.

Boulliau took particular exception to Kepler's supposition that the effect of the motive virtue in the Sun decays inversely with distance in the ecliptic plane, and argued instead that, if such a physical force existed, it would have to decay in inverse proportion to the square of the distance, just as the intensity of light does. However, like many of his contemporaries, Boulliau did not believe any such force did exist; instead, his view was that we should look to geometry for the causes of celestial motion, i.e., planets moved as they did because the geometrical form of an ellipse represented the natural motion of such bodies.

Kepler's second law had its roots in his physical theory and, having discarded the physics, Boulliau could discard the troublesome second law, too. In the mid seventeenth century, application of the second law involved the solution of the equation t = 0 + e sin 0 for 0, which involved messy and non-geometrical trial-and-error-type approaches and thus was considered unsatisfactory by those who looked for elegance in planetary motion. Boulliau replaced the second law with one that was, to him, more appropriate. As he was still fixated with the aesthetic beauty of uniform circular motion, he proposed a scheme in which a planet always was moving instantaneously on a circular path at constant angular speed, but in which the circles were of continually changing radius. These circles, taken together, made up a surface in three-dimensional space, and he took this surface to be a cone, and then the planetary path was the intersection of this cone with the plane of the orbit, i.e. an ellipse!

In 1653, Seth Ward, Savilian Professor of Astronomy at Oxford, demonstrated that Boulliau's hypothesis was equivalent to treating the empty focus of the elliptical orbit as an equant point, about which a planet would move with uniform angular speed (something Boulliau had never suspected and that Kepler had examined and discarded in the Epitome of Copernican Astronomy58). Boulliau's theory - which gave meaning to the otherwise purposeless empty focus of the elliptical orbit - was followed widely, but Boulliau realized eventually that his replacement law did not predict planetary positions satisfactorily.59

57 Quoted from Wilson (1970).

In Book V, Part I, 5. In the Rudolphine Tables, Kepler claimed that one of his friends, the Jesuit Albert Curz (Curtius in Latinized form) used the empty focus as an equant. Another attempt to make the empty focus an equant point was made around 1690 by G. D. Cassini. He proposed that planetary orbits took the form of ovals defined by the condition that the product of the distances of a point on the curve from the two foci is constant. Such curves are now referred to as 'Cassini ovals', or 'cassinoids'.

Fig. 7.7. Boulliau's modified version of Kepler's second law.

In 1657, he produced a modified theory that provided answers in much better agreement with the true area law, though still not equivalent to it.

This modified theory is illustrated in Figure 7.7, in which the heavy curve represents the elliptical orbit, and the lighter curve the circumscribing circle. The Sun is at S, and the empty focus of the ellipse is F. In Boulliau's original theory, the planet was at B, where the angle AFB is the mean anomaly, t. In the modified theory, we construct the line BN, perpendicular to the axis of the ellipse which, when extended, cuts the circumscribing circle at C. Finally, we join C to the focus F, and the planet P is situated where this line cuts the ellipse. Based on the Tychonic data of the time, this version of the second law actually is as accurate as Kepler's law.60

If we denote the eccentricity of the orbit by e, then we know from the basic geometry of an ellipse that |BN|/|CN| = V1 - e2. Thus, writing u for the angle subtended by the planet at the empty focus, we have tan t = VI —

As Newton would later demonstrate (see p. 268), this is correct up to second order in e.

Notwithstanding the success of Keplerian astronomy, many influential astronomers were still not prepared to accept the idea of a heliocentric universe. One such was Giambattista Riccioli, a well-respected professor at the Jesuit College in Bologna, who published his New Almagest in 1651. Riccioli was a serious astronomer and knew that Ptolemy's universe could no longerbe upheld,

60 A comparison of the accuracy of the two methods is given in Wilson (1970).

but his religious beliefs forced him to argue against the Copernican hypothesis:

... all Catholics are obliged by prudence and obedience ... not to teach categorically the opposite of what the decree lays down.

In the New Almagest, he produced forty-nine arguments that were in favour of heliocentrism, and seventy-seven that were against, and thus the weight of the argument favoured an Earth-centred cosmology! However, even Riccioli recognized that in terms of predictive power, astronomy owed much to Kepler, particularly with regard to the motion of Mars.

As far as Jupiter and Saturn were concerned, the Rudolphine Tables were not really an improvement on other competing planetary theories. In his Reformed Astronomy of 1665, Riccioli listed seventy-one observations of Saturn, both ancient and modern, and compared them with various planetary tables. He concluded that the tables he himself had computed and those of Boulliau were the best, while those of Kepler and Longomontanus were not far behind.63 John Flamsteed, the first Astronomer Royal, wrote in 1674:

... the places of the planet Jupiter have been, for these last two years, some 13 or 14 minutes forwarder in the heavens, than Kepler's numbers represent; and ... his motions are not much better solved by any others ... .

In other words, all the planetary tables were similarly inaccurate. One problem here was the accuracy of the orbital parameters that Kepler had computed, and Flamsteed did manage to improve on Kepler's theory for Jupiter by recalculating these parameters. The real problem remained hidden, though. In the eighteenth century, Laplace showed that the gravitational effect of these two large planets on each other causes deviations in their orbits from perfect Keplerian ellipses, that are of far greater significance than the errors introduced by using eccentric circles and equants rather than elliptical orbits.

Quoted from Russell (1989). According to Russell, Riccioli asserted that Catholics were under no obligation to believe that the Copernican system was heresy (as the Holy Office decree of 1633 had stated), but that out of respect for the Church they ought not to maintain its truth in public. There were, in fact, no serious attempts by the Holy Office in Rome to discipline those of a Copernican persuasion and outside Italy astronomers were pretty much free to write what they believed.

Almost literally. The frontispiece of Riccioli's New Almagest shows his own world system (described briefly in note 21 on p. 209) being weighed against that of Copernicus while Ptolemy's system lies discarded on the ground.

Riccioli's observations and calculations contain numerous errors (Wilson (1970)). 64 Quoted from Wilson (1970).

Fig. 7.8. Transits of inferior planets.

Transits of Mercury and Venus

One of the reasons that many astronomers were impressed by Kepler's tables was their success at predicting the transit of Mercury, which took place in 1631. Transits of Mercury or Venus take place when the planet lies directly between the Earth and the Sun, and so a necessary condition for a transit is that we have an inferior conjunction. The orbits of the inferior planets do not lie in the ecliptic plane but are instead inclined slightly to it, so in order for a transit to occur the planet must either be at its ascending node (A in Figure 7.8) with the Earth at E1, or at its descending node D with the Earth at E2.

In 807, an observed spot on the Sun was interpreted as a transit of Mercury even though it lasted for 8 days and there were a number of other medieval reports of transits of Mercury or Venus, though none has been authenticated. Kepler claimed also to have observed a transit in 1607 but later, when he realized there could not have been one on the day of his observation, he acknowledged his error. In 1629, he did, however, predict a transit that 2 years later (after his death) became the first to be observed. Transits of Mercury last at most about 4 h and, since pre-Keplerian theories of Mercury were in error by several days in the times of predicted inferior conjunctions, Kepler's successful prediction (even though he was out by about 5 h) was extremely impressive.

Apart from providing strong evidence for the accuracy of the Rudolphine Tables, this transit was important for two reasons. First, it helped greatly in the accurate determination of Mercury's orbit (since the same error in angular measurement results in a much greater uncertainty in the planet's position in its

Goldstein (1969) discusses the transit reports of al-Kindi, ninth century, Ibn Sina (Avicenna) eleventh century, Ibn Rushd (Averroes) and Ibn Bajja (Avempace) twelfth century. The only one that could possibly have been a transit is that of Ibn Sina.

In terms of the predicted longitude, Kepler's error amounts to 14' 24''. By comparison, the error in the set of tables published by the Dutch astronomer Philip Lansberg in 1631-32 was 1° 21', and that in the Prussian Tables based on Copernicus's theory was about 5° (Wilson (1970)).

orbit near elongation than when directly in front of the Sun67) and, second, it provided the first indisputable quantitative measure of the apparent magnitude of a planetary disc. Pierre Gassendi, who was one of the three people who are known to have observed the transit of 1631, was very surprised at Mercury's small size (he measured its angular diameter at about 20'', whereas Tycho Brahe had estimated its apparent diameter at mean distance as 2' 10"). Johannes Hevelius made accurate observations of the transit of 1661, and discovered that Mercury was even smaller than Gassendi had thought. Hevelius also found that only those astronomical tables based on Kepler's theory of elliptical orbits predicted a transit on the correct day.

The British astronomer Vincent Wing used a modified version of the second law rather than Kepler's area law, but clearly he believed that Kepler's successful predictions of Mercury transits represented very strong empirical evidence for his elliptical orbits, since he wrote in the posthumously published Astronomía Britanica of 1669:

But this is proved especially by the planet Mercury, which on 28 Oct 1631, and again on 23 Oct 1651 and 23 April 1661 was interposed between our vision and some part of the body of the sun; on each occasion the Keplerian tables, conforming to the Copernican hypothesis, best agreed with the truth, while the tables of Longomontanus and Argolus, conforming to the Tychonic system, contained errors of many days.

Transits of Venus are much rarer than those of Mercury. Whereas there were fourteen transits of Mercury in the twentieth century, only five transits of Venus have ever been observed - those in 1639, 1761, 1769, 1874 and 1882 (though the sixth is on 7 June 2004). The 8-year gap between the two transits in the eighteenth century and again between the two transits in the nineteenth century is due to the fact that thirteen sidereal periods of Venus is about 2921 days, which is very nearly 8 years. Thus, if the Sun, Venus and the Earth are aligned at a given time, they will be again 8 years later. The orbit of Venus is inclined at a little over 3° to that of the Earth and the discrepancy between the two periods leads to a change in ecliptic latitude of Venus between the two alignments of about 22', which is less than the angular diameter of the solar disc (32'), and so, provided the first transit is not too close to the centre of the solar disc, there will be another after an 8-year gap. However, there cannot be another 8 years after that.

67 Transits of Mercury actually occur about thirteen times per century, though each one is visible only from certain parts of the globe. They became less significant when Halley became able to observe the planet in daylight to within 15° of the Sun. Quoted from Wilson (1973). The dates are Old Style (see note 3 on p. 120).

Fig. 7.9. Transits of Venus between 1396 and 2012.

Figure 7.9 shows the paths of Venus across the Sun for the eleven transits between 1396 and 2012. There was only one transit in the fourteenth century, with Venus passing just above the solar disc in 1388, but since then all transits have occurred in pairs. Venus was near its ascending node during the transits in the fourteenth, seventeenth and nineteenth centuries, but near its descending node during the transits in the sixteenth and eighteenth centuries, as it will be also in the twenty-first century.

Kepler predicted the transit of Venus in 1631, but this turned out to be visible only in America. He failed to predict the transit of 1639 which was visible from Europe, however, since the Rudolphine Tables implied that Venus would pass below the Sun. As a result, astronomers were not ready with their telescopes to observe the spectacle. The transit did not go unobserved though, thanks to the efforts of the young and largely self-taught Englishman, Jeremiah Horrocks.

Horrocks obtained a copy of the Rudolphine Tables in 1637 and comparisons of observations with Kepler's and other contemporary tables soon convinced him of the superiority of Keplerian astronomy (though he did not subscribe to the physical side of this new science). He set about trying to improve the tables

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