Astronomical Observations on the Rotation of the Planets Round Their Axes, Made with a View to Determine Whether the Earth's Diurnal Motion is Perfectly Equable. In a Letter from Mr. William Herschel of Bath to William Watson, M. D. F. R. S.

X. Astronomical Observations on the Rotation of the Planets round their Axes, made with a View to determine whether the Earth's diurnal Motion is perfectly equable. In a Letter from Mr. William Herschel of Bath to William Watson, M. D. F. R. S. Read January 11, 1781. SIR, Bath, October 18, 1780. The various motions of the planet we inhabit; the annual revolution in its orbit; the diurnal rotation round its axis; the menstrual motion round the common center of gravity of the moon and earth; the precession of the equinoctial points; the diminution of the obliquity of the ecliptic; the nutation of the earth's axis: in short, every one of the motions that arise from the actions of the sun, moon, and planets, combined with the spheroidal figure of the earth, and the projectile and rotatory motions first impressed upon it, have all been considered by astronomers, and their real and apparent inequalities investigated. And to the great honour of modern astronomers it must be confessed, that no science has ever made such considerable strides towards perfection in so short a time as astronomy has done since the invention of the telescope. There is one of the motions of the earth however which, it seems, has hitherto escaped the scrutiny of observers; I mean the diurnal rotation round its axis. The principal reason why why this has not been looked into, is probably the difficulty of finding a proper standard to measure it by; since it is itself used as the standard by which we measure all the other motions. We have, indeed, no cause to suspect any very material periodical irregularity, either diurnal, menstrual, or annual; for the great perfection of our present time-pieces would have discovered any considerable deviation from that equability which we have hitherto ascribed to the diurnal motion of the earth. And yet, it is not perhaps altogether impossible but that inequalities may exist in this motion which, in an age where observations are carried to such a degree of refinement, may be of some consequence. To shew how far time-keepers, though ever so perfect, are from being a proper, or at least a sufficient, standard to examine the diurnal motion of the earth by, I may ask, whether it is probable, that any clock would have discovered to us the aberration of the fixed stars? And yet that aberration produces a change in longitude, and of consequence in right ascension, which causes an annual irregularity in a star's coming to the meridian, which a time-piece, were it a sufficient standard, would soon have discovered, and which we might have attributed to an inequality of the earth's diurnal motion, had we not been acquainted with its real cause. And if we were to find out any apparent irregularity, acceleration, or retardation, should we not much rather suspect the clock than the diurnal motion? I may therefore venture to say, that the aberration of the fixed stars, though attended with the above mentioned consequence, would for ever have remained a secret to us, if it had not been found out by other methods than time-keepers. Now, if time-pieces do fail us in this critical case, where we stand in the greatest need of their assistance, it is almost in vain vain to expect any help from another quarter; for what mechanical movement on earth, or motion of the heavens, is there that can measure out such equal portions of time as we require to compare the diurnal motion of the earth to? However, to proceed, since we have already great proofs that the diurnal motion of the earth is, if not perfectly equable, at least more so than any other motion we are acquainted with, it will not appear absurd to suppose the diurnal rotation of the other planets to be so likewise. This suggested to me the thoughts of estimating the diurnal motion of one planet very exactly by that of another, making each the standard of the other. In this manner we may obtain a comparative view, by which future astronomers, if they shall hereafter be inclined to pursue the subject, may be enabled to make some estimate of the general equability of the rotatory motions of the planets. For if in length of time they should perceive some small retardation in the diurnal motion of a planet occasioned by some resistance of a very subtle medium in which the heavenly bodies perhaps move; or, on the other hand, if there should be found an acceleration from some cause or other, they might then ascribe the alteration either to the diurnal motion of the earth, or to the gyration of the other planet, according as circumstances, or observed phenomena, should make one or the other of these opinions most probable. Now, this method of comparing together different rotations of several planets, simple as it may appear, was not without some difficulties. In the first place it was evident, that the common account of their diurnal motions*, which makes that * Venus spatio 23 horarum gyrationem circa axem ab occidente in orientem perficit. Mars similem rotationem, horis 24 min. 40 absoluit. Macularum revolutionibus saepius observatis, Ds. Cassini comperuit periodum Jovis circa proprium Axem esse horarum 9, minutorum 56, KEILL, Astr. Lect. V. of Jupiter 9 h. 56', of Mars 24 h. 40', how true soever it may be in a general way, was much too inaccurate for this critical purpose. The gyration of Venus was still less to be depended upon, being only noted to the hour without the minutes: it became, therefore, necessary to proceed to observations of a more determinate kind. From what I had already seen of the rotation of the planets, I concluded, that Mars on several accounts would be the most eligible planet for my purpose: for the spots on Jupiter change so often that it is not easy, if at all possible, to ascertain the identity of the same appearance, for any considerable length of time. Nor do the dark spots only change their place, which may be supposed to be large black conglomerations of vapours and clouds swimming in the atmosphere of Jupiter; but also the bright spots, though they may adhere firmly to the body of Jupiter, may undergo some apparent change of situation by being differently covered or uncovered on one side or the other, by alterations in the belts. It will be seen hereafter, that I have observed the revolution of a very bright spot, not suspected of any change of situation, to be first, by one set of observations, at the rate of 9 h. 51' 45",6; and afterwards, by another set immediately following at the rate of 9 h. 50' 48". As the principal belts on Jupiter are equatorial, and as we have certain constant winds upon our planet, especially near the equator, that regularly, for certain periods, blow the same way *, it is easily supposed, that they may form equatorial belts by gathering together the vapours which swim in our atmosphere, and carrying them about in the same direction. This will, by analogy, account for all the irregularities of Ju- * See Acta Eruditorum, 1687. Dr. Halley's Account of periodical Winds. piter's revolutions, deduced from spots on his disk that may have changed their situation; for if we suppose the rotation of Jupiter, according to Cassini, to be 9 h. 56', then some spots that I have observed must have been carried through about 60° of Jupiter's equator in 22 of his revolutions or days. This would certainly be a very great velocity in the clouds, which is, however, not unparalleled by what has happened in our own atmosphere. But to return to my purpose: on the planet Mars we see spots of a different nature; their constant and determined shape, as well as remarkable colour, shew them to be permanent and fastened the body of the planet. These will give the revolution of his equator to a great certainty, and by a great number of revolutions, to a very great exactness also. Supposing then, that, by a method I shall hereafter describe, we can determine whether a spot on the disk of Mars is, or is not, in the line which joins the center of the earth and the center of that planet, to half an hour's time with certainty (I believe ten or twelve minutes will be found sufficient for that purpose), in this case we shall in 30 days have the revolution true to a minute; and, by continuing these observations for three months, we shall have it to 20''. When we are so far certain, we can easily arrive to a much greater degree of exactness; for as we now can no longer mistake a whole revolution, if we take the time of any particular spot's being in the line which joins the centers of the planets during one opposition of Mars, and take the same again at or near the next opposition, we shall have an interval of about 780 days, which will give the diurnal motion of that planet true to about 2''. The next opposition will give it to one, and so forth; by which means, and by taking a proper number of such periods, we may determine the rotation of Mars. Mars to as great an exactness as we shall think necessary for the purpose of our comparative view. Had such observations as these been made two thousand, or perhaps only so many hundred years ago, we might now, by repeating them, most probably become acquainted with some curious minute changes of the solar system that have hitherto passed unnoticed. There is a certain circumstance which would almost create a suspicion that there has been some retardation in the diurnal motion of the earth. The difference between the equatorial and polar diameters of the earth, by actual measurement has been found to be about 36 English miles and 9 tenths; but by a calculation wherein the present rotation is made use of it will only amount to about 33 miles and 8 tenths: from which it should seem probable, that when the earth assumed the present form, the diurnal rotation was somewhat quicker than it is at present, by which means the centrifugal force bore a greater proportion to the force of gravity to which it is contrary, and thus occasioned a higher elevation of the equatorial parts. But I would not lay much stress upon this argument; for, in the calculation, it has been supposed, that the earth is nearly of an equal density at the surface and towards the center, which it seems is not agreeable to some late curious experiments and calculations that have been made under the conduct of the Astronomer Royal upon the attraction of a mountain*, the result of which ought now to be taken into consideration, and the calculation repeated. If all the data could be exactly depended upon, it would be practicable enough from the laws of * See Mr. Hutton's Account of the Calculations made from the Survey and Measures taken at Schehallien, in order to ascertain the mean Density of the Earth. Phil. Trans. 1778. gravity, and the present rotation and given form of the earth, to find the centrifugal force required to produce that form, and thence to shew what must have been its diurnal motion when it assumed the same. However, these are researches that in my present situation I neither have opportunity nor perhaps ability enough to investigate properly; and which, therefore, I hope some of our excellent mathematicians will think worth while to look into. I shall now relate my observations on Jupiter and Mars. The telescopes I used are of my own construction; and are, a twenty-feet Newtonian reflector, a ten-feet reflector of the same form, and a seven-feet reflector already mentioned in my paper on the mountains of the moon. My time I gained by equal altitudes taken with a brass quadrant of two-feet radius, carrying a telescope which magnifies about 40 times; for the correction of altitudes taken of the sun I used De La Lande's tables. I kept my time by two very good pieces; one having a deal pendulum-rod, the other a compounded one of brass and iron, both having a proper contrivance not to stop when winding up. The rate of going of my clocks I determined by the transit of stars. Observations on Jupiter in the year 1778. February 24. Clock 1' 10'' too soon. About 9 o'clock I saw a bright belt on one part of the disk of Jupiter, see tab. V. fig. 1. About 10 o'clock it was advanced as far as the center, fig. 2. 11 h. The white belt still more advanced, fig. 3. 11 h. 25'. It approached towards the edge of the disk; and at 12 h. was extended all over, as in fig. 4. February 25. 8 h. The same bright belt I observed yesterday extends all over. 8 h. 45'. It is divided by a darkish spot, situated at some distance from the center, as in fig. 5. 9 h. 5'. The small dark division is advanced a little farther than the center, as in fig. 6. 9 h. 23'. The spot is visibly advanced a considerable deal farther. March 2. 8 h. 2'. The darkish spot, with some alteration in its shape, is now in the middle of the disk, see fig. 7. March 3. 10 h. 34'. The bright belt on the south of the equator is now in the middle; that is to say, if a line be drawn perpendicular to the equatorial belt, and through the center, the end of the equatorial belt now touches it, fig. 8. 13 h. 49'. The darkish spot, in which there has been some alteration since yesterday, seems now to be in the center, fig. 9. March 14. The clock altered to true equated time; but the rate of going not changed, being well regulated. 7 h. 35'. The spot is now in the center, but does not seem quite to fill the white belt; nor is it so large and distinct as it was before, fig. 10. April 7. 9 h. 31'. There are three dark spots in the equatorial belt nearly in the center, see fig. 11. April 12. 7 h. 50'. The three dark spots are in the center. The fourthermost of the three is nearly quite vanished; the other two are also much fainter. They are, however, distinct enough to be known, fig. 12. Observations on Jupiter in 1779. April 14. Clock 52'' too late. 8 h. 48' A remarkable bright spot in the equatorial belt towards the north is in the center, see fig. 18. 8 h. 58'. The spot is a little past the center. April 19. Clock true mean time. 7 h. 10'. There is a bright spot just now in the center, which, from its shape, I take to be the same that was there April 14th. 7 h. 20'. The spot is visibly past the center. April 23. Clock shews true time. 9 h. 38'. The same bright spot is in the center. 9 h. 43'. It is past the center. Memorandum, my time-piece may be depended upon to a few seconds. It will not be amiss to observe, that the spots, as well as a great many other phenomena, were watched as they came on, passed over the center, and went off the disk of Jupiter; but I have only selected those observations that were necessary to my present purpose. Comparing together the observations that were made in the year 1778, February 24th and March 3d, we obtain an interval of 7 days 34 minutes, which being divided by 17 revolutions made by Jupiter on his axis, we have the time of one synodical revolution equal to 9 h. 54' 56'',4. The dark spot on February 25 was observed some time before, and also just after it was past the center; therefore I have supposed it to be in the center about 8 h. 58': and we have, Mr. Herschel's Astronomical Observations | D. | H. | M. | |----|----|----| | February 25 | 8 | 58 | | March | 2 | 8 | 2 | Divided by 12 rev. | 4 | 23 | 4 | 1 revolution = 9 h. 55' 20"' | D. | H. | M. | |----|----|----| | February 25 | 8 | 58 | | March | 3 | 13 | 49 | 15 revol. | 6 | 4 | 51 | 1 revolution = 9 h. 55' 24"' | D. | H. | M. | |----|----|----| | February 25 | 8 | 58 | | March | 14 | 7 | 36 | 10 | Allowing 1' 10" for the alt. of the clock. 41 revol. | 16 | 22 | 38 | 10 | 1 revolution = 9 h. 55' 4", 6. | D. | H. | M. | |----|----|----| | March | 2 | 8 | 2 | 3 | 13 | 49 | 1 revol. | 1 | 5 | 47 | 1 revolution = 9 h. 55' 40"' | D. | H. | M. | |----|----|----| | March | 2 | 8 | 2 | 14 | 7 | 36 | 10 | 29 revol. | 11 | 23 | 34 | 10 | 1 revolution = 9 h. 54' 58", 2. March on the Rotation of the Planets, &c. D. H. M. S. March 3 13 49 0 14 7 36 10 Divided by 26 rev. 10 17 47 10 1 revolution = 9 h. 54' 53'',4. April 7 9 31 12 7 50 12 revol. 4 22 29 1 revolution = 9 h. 51' 35''. Again, comparing together the observations of 1779, which were made with the utmost attention to time, we have, April 14 8 48 52 April 19 7 10 0 12 revol. 4 22 21 8 1 revolution = 9 h. 51' 45''6. April 19 7 10 April 23 9 38 10 revol. 4 2 28 1 revolution = 9 h. 50' 48''. And taking both together, April 14 8 48 52 April 23 9 38 0 22 revol. 9 0 49 8 1 revolution = 9 h. 51' 19'',4. These several results are so exceedingly various, that it is evident Jupiter is not a proper planet for the critical purpose of a comparative view of the diurnal motions; nor can this great variety proceed from any inaccuracy in the observations: for, in my opinion, it is not well possible to make a mistake in the situation of a spot that shall amount to so much as five minutes of time. The observation of April 23, 1779, was made with a view to ascertain this point, when it was found that five minutes of time made a sensible difference in the situation of a spot when near the center. If we reduce the synodical revolutions to sydereal ones, the result will be so little different from the above, that I have not thought it worth while to do it in this place. By a comparison of the different periods it appears, that a spot which is carried about in the atmosphere of Jupiter generally suffers an acceleration, or, which is the same thing, performs its revolutions by degrees in less time than it did at first; for the spot observed in 1778 moved at the following rates. From February 25. to March 2. in 9 h. 55' 20''; to March 3. nearly the same; to March 14. in 9 h. 55' 4''; from March 2. to March 3. in 9 h. 55' 40''; to March 14. in 9 h. 54' 58''; from March 3. to March 14. in 9 h. 54' 53''. In 1779 a spot moved from April 14. to April 19. at the rate of 9 h. 51' 45''; to April 23. in 9 h. 51' 24''; and from April 19. to April 24. in 9 h. 50' 48''; all which is agreeable enough to the theory of equatorial winds, since it may probably take up sometime before a spot can acquire a sufficient velocity to go as fast as those winds may blow. And, by the by, if Jupiter's spots should be observed in different parts of his year, and be found in some to be accelerated, in others to be retarded, it would almost amount to a demonstration of his monsoons and their periodical changes; but if his axis axis should not be inclined enough to his orbit, to occasion such a change, they may probably always blow in the same direction. Observations on Mars in the year 1777. Twenty-feet Newtonian reflector; power 300. April 8. 7 h. 30'. I observed two spots upon Mars, with a bright belt or partition between them. The belt was not very well defined, see tab. VI. fig. 14. 9 h. 30'. The spots are advanced, and more spotted parts are visible, fig. 15. 10 h. The revolution of Mars on his axis is now very evident, fig. 16. April 17. Ten-feet Newtonian reflector; power about 211. 7 h. 50'. Mars appeared as in fig. 17. At \(a\) and \(b\) there were two bright spots, so luminous that they seemed to project beyond the disk. At \(c\) and \(d\) there were two very dark spots, joined by a lesser black line in the middle, which however was crossed at \(e\) and \(f\) by a very faint whitish partition. April 26. Ten-feet reflector; power 211. 9 h. 5'. The spots on the planet are very faint, and much about as in fig. 18. April 27. Ten-feet reflector; power 324. 8 h. 40'. The evening very fine: my telescope in compleat order. The spots as in fig. 19. Observations on Mars in the year 1779. May 9th. Clock 15'' too fast; by equal altitudes on the 14th of April, and by the transit of a star, is found to lose 1', 45 per day. 11 h. 1' by the clock, I found the situation of the spots on Mars as in figure 20; there is a very remarkable dark spot not far from the center. 11 h. 30'. The figures are gone from the center. May 11. Clock 12'' too fast. 10 h. 18'. The same spot that was visible May 9. is on the disk, the darkest place being entirely south-east of the center, see fig. 21. 11 h. 43'. The darkest part is almost arrived at the center, fig. 22. 12 h. 17'. The dark spot is with its edge just near the center, as in fig. 23. May 13. Seven-feet reflector; power 222. Clock 9'' too fast. 11 h. 26'. Mars seems now to be in the same situation he was the 11th, at 10 h. 8'. May 22. Clock 4'' too slow. 12 h. 5'. The figure of May 11th is not on the disk; but some other fainter spots are visible. The air is full of vapours. June 6. The clock set by ten equal altitudes taken to-day, and by the transit of δ Scorpii loses 1'', 9 per day. What I have perhaps improperly called a transit is the occultation of a star passing behind the perpendicular edge of a high building at about 40 yards distance, observed with a fixed telescope directed to the place where it vanishes. 10 h. 10'. The same figure is upon the disk of Mars which was there April 8, 1777, at 7 h. 30'. June June 15. Clock 17'' too slow. 9 h. 45'. The same figure is upon Mars that was there May 9. at 11 h. 1'; but it is more advanced. I suppose it to be the same, and in the same situation, as April 17, 1777, at 7 h. 50'. June 17. Clock 20'' slow. 9 h. 12'. The dark spot on Mars is rather more advanced than it was May 11th, at 10 h. 18'. 10 h. The spot is visibly advanced: I suppose it will take near an hour to come to the center. 10 h. 15'. A very thick fog obscures the sky. 11 h. 15'. The same darkness. June 19. Clock 22'' too slow by the transit of δ Scorpii observed this evening. 8 h. 40'. The figure on the disk of Mars appears now to be as it was April 26, 1777, at 9 h. 5', see fig. 18. 11 h. 30'. The figure of May 11. which I have been hitherto watching, is not come to the position it was then at 11 h. 43', but cannot be far from it. I fear, as Mars approaches the horizon, I shall not be able to follow him till the figure comes to the center. 11 h. 47'. The state of the air near the horizon is very unfavourable. With much difficulty I can but just see that the figure is not quite so far advanced as it was May 11th, at 11 h. 43', but can certainly not be above two or three minutes from it. 11 h. 51'. The undulation of the air prevents all further observation. Let us now examine the result of the above mentioned observations: comparing together the two following short intervals of the year 1779, we have, Mr. Herschel's Astronomical Observations D. H. M. S. From May 9 11 0 45 to May 11 12 16 48 Divided by 2 revol. 2 1 16 3 Gives 1 revolution = 24 h. 38' 1'', 5'. A second small interval. May 11 10 17 48 13 11 25 51 2 revol. 2 1 8 3 1 revolution = 24 h. 34' 1'', 5'. Here we have two very short intervals that agree to 4', which is more than we could have expected in such short periods of time. Comparing together observations that were made at a greater distance, we find, First monthly period, May 11 10 17 48 June 17 9 9 20 { allowing 3' because the obs. says the spot was rather more advanced. 36 revol. 36 22 51 32: 1 revolution = 24 h. 38' 5'', 9. Second monthly period, May 11 11 42 48; June 19 11 50 22 { allowing 3' for the time the spot would have taken to come to the place mentioned. 38 revol. 39 0 7 34 1 revolution = 24 h. 38' 5'', 4. Third Third monthly period, D. H. M. s. May 13 11 25 51 June 17 9 9 20 34 revol. 34 21 43 29 1 revolution = 24 h. 38' 20'' 3 This last is, perhaps, as likely to be near the truth as any, since the same spot was here observed for the third time, and therefore its motion become more familiar. Here we have three longer periods that agree to fifteen seconds, which is quite sufficient for extending the interval of time to those observations that were made in the year 1777. But as these are the synodical revolutions, it will be necessary first to reduce them to sydereal rotations. In figure 24. let us suppose the orbit of Mars, \( \text{MABC} \), to be in the same plane with the orbit of the earth, \( \text{EDFG} \); and the axis of Mars to be perpendicular to his orbit. Let \( m, e, m, e \) be the situations of Mars and the earth on the 13th of May and 17th of June; then will the line \( EM \), that connects the centers of Mars and the earth, point out the geocentric place of Mars on the 13th of May; and the line \( em \), the geocentric place of the same planet on the 17th of June. Draw \( er \) and \( ms \) parallel to \( ER \); then will \( er \) point out the geocentric place of Mars on the 13th of May; and the angle \( sme \) is equal to the angle \( mer \). Now, by an ephemeris * the geocentric place of Mars, May 13. at 11 h. 26' was 7 f. 20 d. 59' 21''; * The Nautical Almanac gives the geocentric place of Mars only to every sixth day; for which reason I used White's Ephemeris, where it is given for every day, though perhaps not with so much exactness as I could wish. and on the 17th of June, at 9 h. 9', it was 7 f. 12 d. 27' 22'', by which we obtain the difference or angle \( \text{rem} = \text{ems} = 8 \text{d. } 31' 59'' \). Now a spot on Mars, situated in the direction \( \text{ME} \), will have made a sydereal revolution when it returns to the same, or a parallel direction \( \text{MS} \). From which we gather, that the spot on the 17th of June, after coming to the line \( \text{ME} \), where it finishes the synodical revolution, will have to go through an arch of 8 d. 31' 59'', in order to arrive into the direction of the line \( \text{MS} \), where it finishes the sydereal rotation. The time it will take to go through this arch, at the sydereal rate of 24 h. 39' 20'' to 360 degrees, or 4'',109 per minute of a degree, will be 35' 3'',8; this being divided by the numbers of revolutions 34, gives 1' 1'',8; which, added to 24 h. 38' 20'',3, gives us 24 h. 39' 22'',1 for the sydereal revolution of Mars, as found by the third of the monthly periods. This quantity will help us to find a proper divisor for the three following long biennial periods. It is to be observed, that Mars has been retrograde in the above example, for which reason the measure of the angle \( \text{EMS} \) was to be added to the synodical revolution when we wanted to find the sydereal rotation; but if he had been direct, or if his place had been more advanced in the ecliptic than that to which we compared it, as at \( \mu \), then the line \( \mu\sigma \) parallel to \( \text{EM} \) would be the direction to which the spot should return, in order to accomplish a sydereal revolution, and therefore the quantity of the angle \( \sigma\mu\epsilon = \mu\epsilon r \), or difference of the geocentric places ought to be subtracted from the synodical revolution to obtain the sydereal one. First biennial period, 1777, April 8 7 3° 1779, June 6 10 10 The geocentric places of Mars at those times were, | S. D. | |-------| | 6 6 31 26 | | 7 13 48 30 | turned into time at 4", 109 per minute of a degree and subtracted, because Mars is more advanced in the ecliptic, is. \[ \frac{789}{2} = 33 \text{ 11},8 \] Divided by 768 rev. \[ \frac{789}{768} = 0 \text{ 6 48},2 \] 1 revolution = 24 h. 39' 23",03. Second biennial period, 1777, April 17 7 5° 0 1779, June 15 9 45 17 | S. D. | |-------| | 6 3 31 27 | | 7 12 40 23 | Turned into time and subtracted \[ \frac{789}{2} = 39 \text{ 18},94 \] Third biennial period, 1777, April 26 9 5 0 1779, June 19 8 40 22 783 23 35 22 Geocentric places 6 1 24 36 7 12 31 48 1 11 7 12 Turned into time 783 23 35 22 and subtracted — 2 45 15,6 763 revol. 783 20 50 6,4 1 revolution = 24 h. 39' 23'',64. As these three periods are supported by observations of equal validity, I shall take a mean of them all for the nearest approximation to the true sydereal revolution of Mars on his axis, which therefore is 24 h. 39' 21'',67. It remains now only to see how far we may depend upon this determination of Mars's diurnal rotation as coming near the truth; and looking over those causes which may possibly produce any errors, we find, first of all, that in the long biennial periods a mistake in the number of revolutions would produce a considerable deviation from truth. Secondly, in the observations of a spot which moves so slow, we are also liable to some considerable mistake in estimating the time when it comes to a certain place; and the more so, if that place is not the center. Lastly, the time itself is liable to inaccuracy. As to the first, it appears from the three monthly periods observed in the year 1779, when the proper allowances for the geocentric places are made, that the sydereal revolution of Mars cannot well be less than $24 \text{ h. } 39' 5''$, nor more than $24 \text{ h. } 39' 22''$; but if we should divide any one of the three biennial periods by a supposed number of revolutions, only one more or one less than we have done, the difference would be so considerable, that nothing but a mistake in every one of the three monthly periods, of at least one whole hour, could justify such a supposition; and that such a mistake in the situation of a spot on Mars cannot have been made in those observations, I think, is evident enough from the exactness with which they were made, and from their agreement with each other. The second cause of error, which is the uncertainty in assigning the exact time when a spot comes to the center, is of some force. But it seems to me highly probable, from the manner in which I have seen the spots on Mars pass over the disk of that planet, that there can hardly be so great an error as $10'$ in an observation of any remarkable spot's coming to the center. However, not being willing to trust more to the eye than I ought to do, I had recourse to the following experiment. I drew several circles of one inch radius, taking care to make no visible impression of a center; and placed in each a fine point at the several distances of $.0424$, $.0636$, $.0848$, in ten thousands of an inch from the real center; some to the right, others to the left. These measures are the fines to radius one, of $2^\circ 26'$, $3^\circ 39'$, and $4^\circ 52'$, which are the arches a spot on Mars passes over in $10$, $15$, $20'$ minutes respectively. I exposed them to several persons unacquainted with my designs, and found, that not one of them made a single mistake in saying whether the point was, or was not, in the center of the circle. circle, and which way it deviated from it. As the direction of the motion of a spot on Mars is known, I thought the persons who were to judge of the place of the points were intitled to be acquainted with the line in which they were placed, which for that reason was always to the right and left only. The points that answer to the eccentricity of 15 and 20' are indeed visibly out of the center, that I believe we may safely say, that any mistake, in estimating the time of a spot on Mars coming to the center, cannot well exceed a quarter of an hour at the outside. As for the third and last occasion of error, the time itself, I believe my manner of obtaining and keeping it in the year 1779 will appear satisfactory, and may, I think, be depended upon to a few seconds; but the observations of the year 1777, indeed, are far from having the same advantage. I was not then provided with an altitude instrument, therefore set my clock by a good sun-dial, with the equation of time contained in the Nautical Almanac, and found it to agree generally to a minute or two with the time calculated for the eclipses of Jupiter's first satellite, as I deduced it for Bath from the Nautical Almanac. However, it was certainly liable to an error of several minutes; therefore, allowing no less than 10' for the clock in 1777, and 20' for an error in estimating the situation of a spot in 1779, it will both amount to half an hour: then, if we take a mean of the three numbers, whereby we have divided the three biennial periods, we have $766\frac{1}{3}$; and half an hour, divided by $766\frac{1}{3}$, will therefore give us the quantity to which, it seems, can amount, all the uncertainty in the sydereal diurnal rotation of Mars, which is $2''$,34. A nearer approximation to truth I hope to obtain at the next opposition, which will happen about the middle of July 1781. I have I have ventured to calculate the times for that opposition, when the edge of the remarkable dark spot will be seen near the center, as it is in figure 23, or, which is the same thing, as it was the 11th of May 1779, at 12 h. 17'. The spot not being visible at the time of the opposition, I have taken the nearest period, before and after, in which it will pass over the disk. There is, however, a circumstance which may make the appearance of the spot not quite similar to the figure I have drawn, even though the rotations should perfectly answer as to the times; for the position of the axis of Mars being still in some measure unknown, I could make no allowance for a change, which a difference in the situation of no less than two signs may occasion, though in all probability it will not be very considerable. Those who are provided with proper telescopes will have an opportunity to see how far the calculated times agree with the spot's appearance; and it is by this means I also hope to correct and improve the tables I have drawn up for this purpose, and further to approximate to a true theory of the gyration of this planet. Not knowing the exact difference of meridians between Greenwich and this place, I have calculated the spot's appearance for the meridian of Bath. From an eclipse or two of Jupiter's satellites, of which, by the favour of the Rev. Mr. Hornsby, I have seen correspondent observations, I suppose the difference cannot be much less than 9' west of Greenwich; and at the same time I join an account of the solar eclipse of the 24th of June 1778, which may be depended upon as a very compleat observation, and may serve to ascertain the longitude of this place. Eclipse of the sun observed at Bath. June 24, 1778. Beginning by equated or mean time, 3 30 10.7 End, - - - 5 18 7.7 Calculations, or (as the principles on which they are founded are still established upon a few observations only, and require some time for mature confirmation) I would rather, if I might be allowed the expression, call them calculated conjectures of the times when the remarkable dark spot will be seen near the center of the disk of Mars. For June, July, and August, of the year 1781. | D. | H. M. S. | D. | H. M. S. | |----|----------|----|----------| | June 28 | 9 48 39 | July 8 | 16 13 23 | | 29 | 10 27 10 | August 3 | 8 11 6 | | 30 | 11 5 42 | 4 | 8 49 57 | | July 1 | 11 44 13 | 5 | 9 28 48 | | 2 | 12 22 32 | 6 | 10 7 39 | | 3 | 13 0 51 | 7 | 10 46 31 | | 4 | 13 39 10 | 8 | 11 25 44 | | 5 | 14 17 29 | 9 | 12 4 56 | | 6 | 14 55 46 | 10 | 12 44 8 | | 7 | 15 34 4 | I have the honour to be, &c.