Monday, 11 November 2013

Table of Contents

Babbage's Lectures On Astronomy
Given at The Royal Institution, 1815

Lecture 12: Beyond the Solar System

Lecture 12

Beyond the Solar System*

[script of lecture missing]

* Note: conjectured title of lecture.

Page from the Nautical Almanac and Astronomical Ephemeris for the Year 1815.

Lecture 11: On Comets

Lecture 11

 German engraving of the great comet of 1680

On Comets


Besides the planets which perform their revolutions in orbits nearly circular and which almost always remain within our view  or at least within the reach of our telescopes there is another species of bodies which only present themselves to our view at short and distant intervals, which shine with splendour for a time and then retire into the depths of the heavens. These have been usually called comets and are distinguished from either stars by a long train of light which usually accompanies them and which is always situated in a direction opposite to the Sun and which diminishes in lustre as it recedes from the body of the comet.

Of all parts of philosophy that which was the latest to receive solid improvement was undoubtedly the theory of comets. The stars were considered as meteors little different from the exhalations and luminous appearances which we sometimes behold in the atmosphere. Some philosophers as Apollonius, Seneca and many of the Pythagoreans had more correct notions on this subject but these seeds of truth were extinguished by a weight of prejudice and by the authority of the Aristotelian school.

From this opinion it unfortunately happened that the ancients were very careless in making and transmitting to us observations on these phenomena and we have now only to regret they were so little enlightened on this subject since from the want of materials with which they might have supplied us the decision of some of the most interesting questions of physical astronomy will probably be postponed some centuries.

Until the time of Tycho Brahe we find concerning comets but few reasonable conjectures. This celebrated observer began to open the eyes of astronomers to the real nature of these bodies  by an important discovery. He demonstrated from the smallness of the parallax of these bodies that they are much more distant from our Earth than the Moon is. He even endeavoured to represent their course by supposing them to move in an orbit round the Sun. This however we should observe was an hypothesis purely astronomical and he by no means supposed that they were planets of a peculiar nature revolving round the Sun. This discovery of Tycho [Brahe] was confirmed by the observations of various astronomers of his time. And at the commencement of the 17th century it received new illustrations from Galileo and Kepler.

It was natural as soon as Men were undeceived respecting the situations of these bodies for Men to endeavour to submit their motions to calculations. Tycho [Brahe] had set the example and Kepler soon followed it; this celebrated astronomer imagined he could represent their motions by supposing that they moved in a straight line. He was obliged however to acknowledge that they did not move uniformly in this line. This circumstance ought naturally to have led him to consider their path as curved. But being unwilling to give up the straight line he was obliged to admit an acceleration and retardment in different parts of it. It is singular that Kepler who was in other matters so clear sighted, who possessed a genius peculiarly calculated to penetrate into those causes which contribute to the order, the harmony and the magnificence of the universe, should have been little better acquainted with the nature of these stars than the herd of mankind. He confined himself to supposing that they were new productions of Nature similar to the meteors which sometimes appear in our atmosphere.

The supposition that comets move in a straight line was for a long time the favourite hypothesis among astronomers. The positions of the comet which appeared in 1665 were calculated by this method and it had caused some surprise that the results were not far distant from the truth. We shall however presently see the reasons of this coincidence. It was however from Cassini that this hypothesis derived its greatest celebrity. He applied it to the comet which appeared in 1652 and also to several others and his results were sufficiently near the truth to convince many that he had arrived at the true explanation. It must however be observed that his hypothesis would not satisfy distant observations on the same comet and that to a great number of them it was utterly inapplicable. It will naturally be enquired how it could happen that from a false hypothesis so many observations should be satisfied as to produce for it a considerable reputation for several years. The answer to this question is not difficult. Comets according to more modern observations are found to move in flattened ellipses. In some cases these ovals are so much elongated that they approach very nearly in some parts to the nature of a parabola. A parabola is a curve composed of two branches which at a short distance from the summit approach very much to straight lines. From this circumstance a comet if seen in one part of its orbit will seem to be moving in a straight line. A comet when it is approaching the Sun gradually disappears in its rays and after being hid during some time is seen moving from the Sun. This can not be reconciled with the hypothesis of Cassini and in fact he made a singular mistake respecting the comet of 1680-81. All those who supposed the comet to move in a straight line imagined that there were two different comets which moved in straight lines passing very near the body of the Sun, whereas in fact it was one and the same comet which only disappeared from being lost in the Sun's rays and again became visible in its recess from that body.

It is remarkable that the ellipse which this comet really described is very much elongated so that its two sides approach nearly to straight lines. And it was principally from the accuracy of his predictions relating to this comet that Cassini astonished the world and extended the credit of this theory. But the triumph of this hypothesis was only caused by a fortunate coincidence of circumstances. It was therefore transient and soon gave place to another incomparably more accurate. In fact notwithstanding the theory of Cassini it was soon discovered that the paths of comets are not straight lines, but that they are curved and that the concave part is directed towards the Sun.

            Helvetius recognised this fact and Dr. Hooke demonstrated it. He says that we must positively reject the testimony of observation. It is in contradiction to this. Helvetius imagined comets to be eruptions projected from the body of the Sun or even from the planets. "If" said he, "we project a body from the surface of the Earth it will describe a parabola. Therefore" said he, "these bodies which are projected from the Sun or planets will also describe parabolas." As to the physical construction of these bodies he imagined them to be nothing more than a collection of vapours collected in the atmospheres of the planets which gradually rose higher and higher till at last they were projected from them and moved in different curves according to the velocities they had acquired.

Such was the state of the theory of comets when the celebrated one of 1680 made its appearance. It was first accurately observed in Saxony on the 4th November. It was then moving with increasing velocity towards the Sun. About the 30th it moved at the rate of 5 degrees in a day and shortly after it disappeared. About 22nd December it reappeared moving very swiftly from the Sun and its velocity gradually diminished until the middle of March 1681 when it was no longer visible. On its return from the Sun it had a tail or train of light extending 70 degrees, that is it reached much more than one third of the heavens. It was proved that these two [appearances] were the same comet from the resemblance of the solid nucleus which presented the same appearance before and after its passage near the Sun, and also from the direction of its course which was the same. But the strongest proof was that the calculations which Newton made respecting this comet and which were founded on this supposition agreed accurately with observation.

It was a fortunate circumstance for the progress of Astronomy that the Earth was in a favourable situation to see both the access of this comet to the Sun and also its recess from that body. Without this accidental circumstance the true system of the cometary motions might not perhaps have appeared for a long time. But this singular coincidence hastened its discovery.

            The first outline of the true cometary theory came from Germany. A clergyman named Doerfell, the minister of a small village, had observed the comet with much care. He is an astronomer very little known and has not received that credit which was due to [him for] the manner in which he treated a subject which was at that time both new and difficult. Doerfell proved that the comet which receded from the Sun was the same as the one which had approached it a short time before. He showed that it moved in a parabola having the Sun in its focus. He ascertained the distance at which it passed from the Sun. All these circumstances were published in 1681, but the language in which it was written and probably the little reputation of the author caused it to be neglected, and it was not noticed until long after Newton had established the same truths by other methods.

The anticipation of one of the discoveries of our great countryman does not however in the least denigrate from his glory. It was with Doerfell an astronomical hypothesis, but with Newton it was a physical truth, a branch of his general system. In fact it was impossible for our astronomer after having established the gravitation of the planets towards the Sun and recognising as he did with the astronomers of his time that comets are not the transient meteor of a moment, not to suppose them governed by the same laws which regulate the other bodies of the System. It was therefore necessary to suppose revolving in very eccentric ellipses to account for their not being constantly visible. But Newton still further demonstrated the truth of his method by applying them to the determination of the path of the comet of 1680 and it is remarkable by what accuracy his calculations of the position of the comet agreed with the observations of Flamsteed. The greatest difference only amounted to two minutes of a degree.

The comet of 1680 was remarkable for the long period of its revolution, which is 575 years, and also for its near approach to the Sun. According to Newton it approached so near as to be distant from that body only the ?th part of the distance of the Earth from the Sun. It must therefore have experienced a heat 26,000 times greater than we ever receive from the Sun's rays and if to obtain a more elevated point of comparison it is compared to that of red hot iron it will be found that this comet must have been 2,000 times hotter. From this it appears that the comet must have been composed of very solid matter not to be dissipated by such an intense heat and this affords a new proof of the permanence of these bodies.

            Newton conjectured that this comet as well as others which like it revolve round our Sun approximate continually to this body at each revolution and that the[y] ultimate[ly] fall into it, for the purpose of supplying the loss to which it is continually subject by the emission of particles of light. But this is purely a matter of conjecture and must not be ranked with the astronomical discoveries of Newton, but which are not the less solidly established whatever may be the fate of these conjectures.

With respect to the tails of comets there have been various opinions. It is almost needless to refute that entertained by most of the ancients and by some few of the moderns. They conceived that the tails of comets arose from the refraction of the rays of light through the nucleus of the comet but this is contradicted by the fact that the nuclei of comets are evidently opaque. Kepler once maintained this opinion but in his subsequent writing he gave it up. He then attributed their tails to their atmospheres and to the evaporation of the more volatile parts caused by the heat of the Sun. This is nearly the opinion which Newton embraced and he compared the tails of comets to the smoke which follows a burning body in rapid motion. Such was the most probable explanation of the cause of the tails of comets before De Mairan. This eminent philosopher to whom we are indebted for an explanation of the aurora borealis conjectured with some degree of probability that the tails of comets are formed by the matter of the solar atmosphere which these bodies attract to themselves on their approach to their nearest distance from the Sun, and to account for their tails always appearing on the opposite side to the Sun he supposes that this is the effect of the impulsion of the rays of light. There are some circumstances which render this explanation at least probable. It may be remarked that comets do not begin to exhibit a tail until they have approached nearer the Sun than the semidiameter of the earth's orbit and this is supposed to be about half the extent of the Sun's atmosphere. Those comets on the contrary which have not approached so near to the Sun such as those which appeared in the years 1585, 1718, 1729 etc. have been seen without any tail. The ingenious work of De Mairan in which he establishes this opinion contains several other proofs by which this opinion is  establish

ed. These appendages to cometary bodies present various appearances according to the positions in which they are perceived. If a comet is moving in a direction nearly at right angles to the path of our Earth it will appear to have a tail in the direction opposite to the Sun, but if the comet is moving almost directly towards us or directly from us it will appear to be surrounded with a nebulosity, and in particular cases is said to be bearded.

            After Newton no one contributed more to the improvements of this branch of Astronomy than Dr. Halley. This learned astronomer presented to the Royal Society in 1705 a treatise on Comets in which he applies the principles taught by Newton to the determination of the orbits of comets and he formed tables of their motions similar to those of the planets. Towards the conclusion however he gave other methods of his own on the more accurate supposition of their revolving in ellipses. This was the most valuable and interesting part of the curious communication of the author.

He calculated the orbits of the comets and formed them into a table in order to compare them. By this means he had the satisfaction of verifying the opinion of those who supposed these stars [were] subject to periodical return. In fact from the inspection of the tables he found that the comets which appeared in 1531, 1607  and 1682  had very nearly the same orbit and the intervals between their appearances were nearly 75 years. From this he concluded with a very high degree of probability that it was one and the same comet whose period of revolution is about 75 years. He found that the inclination of all the three orbits was about 18 degrees and that, if the mean distance of the Earth be supposed to be 100, then the least distance of the comet of 1531 was 57, that of 1607 was 58 and that of 1682 was 58. This difference is very small when we consider the imperfection of practical astronomy at the time the observations on which these calculations were made depended.

These were strong reasons for presuming on the identity of the three comets but further circumstances rendered it still more probable. In counting back from 1531 75 or 76 years we find other comets. Thus in the years 1546, 1380 and 1305 [also in 1230, 1155, 1080 and 1006] there appeared other comets. In fact no astronomer has transmitted to us observations by which we may determine decisively their orbits, but by comparing their appearance and motions as transmitted to us by historians with those of the comet we are considering and allowing for the different positions of the Earth Dr. Halley found they agreed very well. Thus assured of its revolution in 75 years he ventured to predict its return in the year 1758 or 59. This is the first prediction that was ever made of the appearance of a comet and it is well known that it was justified by the event. Dr. Halley remarked that the comet observed in 1661 by Helvetius and that of 1532 seen by Appianus were the same. Had this been the case it ought to have returned in 1780 or 81. This however was not the case. By comparing his tables of the orbits of comets Dr. Halley conjectured that the brilliant comet of 1680 had reappeared several time at the distance of 575 years.

He founded this opinion on the following circumstances that in the year 1106 there occurred a beautiful comet whose description much resembled that of 1680. In the year 531 a similar one appeared, and in the year 46 before the Christian era appeared that prodigious comet so celebrated by historians and which followed so nearly the death of Julius Caesar.

            But Dr. Halley went still further in continuing to retrograde 575 years at each step: he found that this same comet must have appeared very nearly at the time of the universal deluge, and he formed the bold conjecture that this was the secondary cause made use of to produce that horrible catastrophe. This body was accompanied by a tail of prodigious extent which according to Newton consisted of vapours raised by the solar heat. Halley supposed that the Earth might have passed through this and that by the effect of gravity these vapours would have fallen on its surface and thus produce the immense body of water by which our globe was inundated. The celebrated Whiston has supported this explanation of the deluge with much ingenuity and seems by his zeal to have acquired the title of its author although it was undoubtedly Halley's. It may be observed that it is scarcely probable that such an effect would result from our globe passing through the tail of a comet. Vapours rarefied to such a degree as these must be even if they exceeded our globe many times in volume would form but an inconsiderable quantity of water insufficient for the ravages of which the traces still remain. It would be easy to prove this from considering what has been demonstrated by Newton that a cubic inch of air would if carried to the distance of the Earth's semidiameter from its surface be rarefied to such a degree as to fill the whole space from the Sun to the orbit of Saturn.

This same comet has been employed by another celebrated writer to explain another point of history. He conjectured that this same comet appeared about the time of Ogyges and that it gave rise to the singular phenomenon which has been mentioned by historians with astonishment. They relate that 40 years before the deluge of Ogyges, the planet Venus was seen to quit its ordinary course and to be accompanied by a long train of light. Upon which the learned writer observes that in the infancy of Astronomy men might easily mistake a comet just disengaging itself from the sun's rays for the planet Venus quitting her usual course and accompanied by a long tail. But so many other comets might have given rise to this mistake that we can unfortunately determine nothing certain as to the date of this deluge from such a circumstance.

            Since the period of Dr. Halley much more extensive tables have been formed. It appears from them that there are nearly as many whose motion is retrograde as there as ones whose motion is direct, and it also appears that their orbits are inclined to the ecliptic at every possible angle. This is another and powerful argument if any further one were wanted against the theory of vortices.

It may also be remarked that the greater number of comets descend towards the Sun within the Earth's orbit. Of the 35 whose orbits were calculated by La Caille there were only six whose least distance from the Sun exceeded the mean distance of the Earth from that body.

            The comets appear to have no fixed zodiac. On the contrary there is scarcely any constellation in the heavens in which some comet or other has not been seen. These different positions and the different inclinations of their orbits do not appear to be the effect of chance but rather affords cause for admiration. If they had moved in the plane of the Earth's orbit or very near to it everytime a comet descended towards the Sun or ascended from it we should be exposed to the danger of a contact, if unfortunately our globe should at that time be situated at the intersection. But by means of this inclination it happens that not one of the orbits of comets yet known cuts that of the Earth. It would indeed be a very curious spectacle to see a comet pass at the distance of one or two diameters from our globe. Possibly these might result [in] some physical changes in our little system which might not be disadvantageous to us. A celebrated naturalist has supposed we might acquire a new moon and has attributed a similar origin to the one we possess. It might however be more happy for us to be deprived of this advantage then to incur the risk of so near a neighbour. Of all the comets which have yet been observed that of 1680  can approach nearest to the Earth. Dr. Halley found from calculation that on the 11th November 1680 at one o'clock it was not further distant from the Earth's orbit than the Moon is from us. But there would have arisen to us no danger from this circumstance. It would only have afforded us an opportunity for curious observations if the Earth had been in a convenient part of her orbit.

It is true we may not always be so fortunate. According to Whiston this comet [has] already been the instrument of divine vengeance [and] may at it return involve us in the burning vapour of its tail, and thus produce a universal conflagration. We must however always separate these bold and sometimes fanciful conjectures from the physical theory by which the motions of the bodies are calculated.

            In consequence of the predictions of Dr. Halley much more attention was paid to the discovery of comets. In the 120 years which elapsed between 1637 to 1757 on 20 had been seen, but in the next 50 years this number was more than tripled. It was observed that the comet whose return was predicted by Dr. Halley had alternately long and short periods and it was therefore supposed that it would return about the year 1759, and the astronomers of the time prepared very diligently to search for it. It was necessary to calculate the part of the heavens in which it would reappear but this and the precise time of its return were alike unknown,  and this much increased the difficulty of the search.

The astronomer Lalande was very ardent in the search and thus assigns his reasons for it. "It has already appeared" said he "6 times with very evident marks of identity particularly the last two or three times. There can therefore be no doubt of its return. Even though astronomers should not observe it they would nevertheless be convinced of it. It well known to the them that its little light and immense distance may possibly hide it from our view. But the public will hardly believe us and they will rank among the number of vague conjectures this discovery which does so much credit to modern astronomy. Disputes will again arise; the terrors of the ignorant will continue and 70 years must elapse before we again have an opportunity of removing these doubts."

            The astronomer Messier was actively employed during 18 months in searching for this comet. This labour was not, however, unrewarded for in consequence of it he discovered a new comet which he observed during several months. It was on the 21 June 1759 that he first had the good fortune of observing the object he was in search of. This day happened to be very severe and as soon as the stars became visible at sunset he profited by this circumstance and directed his telescope towards that part of the heavens in which it was expected to appear. After some time he recognised at about 7 in the evening a feeble light very similar to the small comet he had discovered in 1758. He had frequently imagined he had perceived this much wished for visitor, but he had been deceived by the numerous nebulae which are scattered in that part of the heavens. After a few observations he found from its motion that it was the object he was in search of.

Dr. Halley had observed that the periods of the return of this comet were unequal. They varied from 75 to upwards of 76 years. He attributed this to the attraction of the planets. He knew that the motion of Saturn is very sensibly altered by the attraction of Jupiter and considering that the comet must pass near Jupiter he imagined this planet might cause the change in its periodical time. He was not however able to calculate this effect and offered it merely as a conjecture.

            A little before the reappearance of this comet Clairaut, who concurred in the opinion of Dr. Halley, undertook the calculation of the disturbance it must suffer from the action of Jupiter. This immense undertaking was however beyond the powers of one individual. Clairaut undertook the discovery of the plan to be pursued in these calculations and Lalande and Madame Lepante executed the calculations themselves. After 12 months of fatiguing calculations it was found that the action of Saturn was so considerable that it could not be neglected. The labour was therefore renewed. In November 1758 these immense calculations were completed and Clairaut was able to announce that the period of the comet which was just about to terminate would exceed the preceding one by 618 days. 500 were caused by the attraction of Jupiter and 100 were the result of the action of Saturn. He also announced that the comet would be at it nearest distance from the Sun on the 13 March 1759. It in fact only exceed the specified time by 22 days and arrived there on the 4th of April. Thus did the most celebrated of all the comets confirm the theory of Newton and afford a proof that they revolve round the Sun like planets. It must, however, be observed that this is the only one whose  periodic time is well ascertained. The periods of the revolution of many other comets have been conjectured and their returns expected but with respect to them there is much uncertainty.

The comet which appeared in 1264 and 1556 is expected in 1848. It may perhaps be the same as those recorded to have appeared in 975 and 395. But the observations of 1264 are too imperfect for us to depend much on its reappearance. The comet which appeared in 1770 occupied considerable attention. It was observed for along time, nor could any parabolical orbit be found which suited its motions. After immense calculations it was found by Lexell that it moved in an ellipse and that its period of revolution was 5 years.

            The terror which was excited by the comet of 1773 is one of the most singular circumstances in their history. It was occasioned by a paper of Lalande presented to l'Academie des Sciences, Paris. This memoir was not read at the time, but a report soon spread that Lalande had announced the termination of the world. It was supposed that it would take place in less than a year. As the report spread the time became shortened to a month and then it soon came to a week. The populace were alarmed and the police applied to Lalande to contradict the report. His explanation appeared in the gazette in a few days, but as this was not sufficient to justify him from the numerous absurdities which had been imputed to him he resolved to publish whole paper. It consists of an investigation into the probability of the contact of one of these bodies with the Earth. From a consideration of the orbits of all the comets which had at that time appeared he plainly showed the great improbability of such an occurrence. And since the more recent observation the improbability of the injuring [of] our globe is most materially increased.

If a comet equal in magnitude to our Earth were situated at the distance of about 40,000 miles it would cause a tide of 12,000 [ft.] above the level of the sea, which would be quite sufficient to overflow in succession the whole globe, but, if we consider that the comet is in motion and the earth [is also] in motion, both very rapidly, it will be found that the effect of the comet, during the very short time it would be at the distance, will be wonderfully diminished, and, if to this it be added that as yet we have had no instance of any comet approaching to this magnitude, we shall find that the effect of a comet, even at this short distance, would be comparatively trivial.

            Of all the comets which have been observed that of 1770 approached nearest to the earth. Laplace found that by the attraction of the Earth its time of revolution was diminished two days, and that supposing it to have been of the same magnitude as our globe its reaction would have shortened the length of our sidereal year 2 hours 40 minutes. But according to the most accurate observations the length of the year was certainly not shortened 3 seconds by this comet, from which we may conclude that its mass was not equal to the  ?th of our Earth. These bodies appear in general to exert very little action on the planets they approach. Their wandering courses do not disturb the harmony of the system and though frequently announced by the ignorant as the presages of misfortune they do not appear to have received the power of doing mischief.

There has been much question as to the solidity and planetary nature of these bodies. It has been doubted whether they possess a real nucleus of solid material or whether their centre may not be the densest and most compact part of their nebulosity. According to observations on the comets of 1799 and 1807 by Schroeter and Dr. Herschel it appeared that they possessed solid nuclei of a round form distinct from the nebulosities which accompany them. This nucleus was not subject to the same variations as the vapours which surrounded them. It did not always occupy the centre but generally appeared to incline to wards the Sun. Dr. Herschel is of opinion that the body of the comet shines by its own light, for he observed that when from its situation with respect to the Sun we could not see the whole of its illuminated side, yet the light of the comet appeared by no means diminished but the whole surface shone with one uniform light much more resembling by its vivacity irradiance of the stars than the reflected light of planets and their satellites. This observation however would be satisfied by supposing the centre to consist of a dense mass of vapours through which the Sun's rays are refracted.

In the comet of 1811 the central body was remarkably distinct from the surrounding vapour. In its appearance this was one of the most splendid which have been visible for many years. Its tail or the luminous train which it carried with it in one part of its orbit was 100 million of miles in length and about 15 million in breadth, yet notwithstanding this enormous atmosphere the solid nucleus of the comet was according to the observations of Dr. Herschel not more than 428 miles in diameter. This body would according to the observations made during it appearance describe an ellipse round the Sun in about 2620 years at the mean distance of 190 that of the Earth [is] from the Sun. This period might however be very much altered from the attraction of the planets and if at a very great distance it should be influenced but in a small degree by any unknown body it might suffer a total change in its orbit and perhaps never return to our system.

            Of the numerous comets which have been discovered during the last half century there is scarcely one which on whose certain return we can rely. Many have been found to move in curves called parabolas and hyperbolas, and it is impossible for those again to revisit our system unless some great change take place in their orbits. Laplace has calculated that supposing a comet to move in such an orbit it is 57 to against its being visible to our Earth. On the ground its seems probable that in the last 50 years between 2 and 3 thousand have approached the Sun though the larger part have been entirely unnoticed by us.

It may probably be asked what becomes of those comets which retire from the power of our Sun and never return. Are these bodies lost in the immense deserts which separate our primary from the nearest of the fixed stars. It rather appears from mechanical principles that when a comet has by the action of any extraneous force acquired such a velocity as to cause it to quit the sphere of our Sun that it would pursue its unimpeded course until some other sun should exercise its influence on it and attract it towards a new centre. It might then descend with immense velocity towards this new primary and thus continue to visit system after system making as it were the tour of the universe. If this be really the arrangement of Nature there is doubtless some final cause from which these wanderings are directed. On this we can only form conjecture. It has been supposed that in the transit of these bodies through the immense regions which separate our system from the nearest fixed stars the comets pass through regions of nebulous matter which they convey to distant systems to supply the waste occasioned by the emission of light.

Lecture 10: On the Theory of Universal Gravitation

Lecture 10


 On the Theory of Universal Gravitation

            By observation and experiment we gain the first materials of our knowledge. These are the foundations of all philosophy. But facts however certain or numerous would by themselves be of comparatively small importance if the inquiring spirit of Man did not seize him to generalise his observations, and to form some theory to connect and account for the various appearances presented to him by Nature. The very number of these facts would soon overpower his memory unless he made use of method and arrangement. Thus then the accumulation of his observations would necessarily produce speculative knowledge and give rise to an attempt at theory.

            It is to a legitimate use of theory that the science of Astronomy is particularly indebted. She presents us with some of its most happy illustrations. The indiscriminant zeal against hypothesis so generally avowed by the followers of Bacon has been much encouraged by the strong and decided forms in which they have been reprobated by Newton. But the language of this great man must be qualified and limited by the exemplification he has availed himself given of his general rules. And it should be remembered that they were particularly directed against the vortices of Descartes which were purely fictitious and were the prevailing doctrine of the time. A very learned and acute writer has observed that "the votaries of hypothesis have been challenged to shew one useful discovery in the work of Nature that was ever made in that way." In reply to this challenge it will be sufficient on the present occasion to maintain the Theory of Gravitation and the Copernican System. Of the former I shall presently endeavour to prove from a sketch of its history that it took its rise entirely from a conjecture, or hypothesis suggested by analogy. Nor indeed could it be considered in any other light until that period of Newton's life when, by a calculation founded in an accurate measurement of the Earth by Picard, he evinced the evidence to be true, even the law which regulates the fall of heavy bodies, that power which retains the Moon in her orbit.

The Copernican System offers however a still stronger case, inasmuch as the only evidence which the author was able to offer was the advantage it possessed over every other hypothesis in explaining with beauty and simplicity all the phenomena of the heavens. In the mind of Copernicus therefore this system was nothing more than a hypothesis, but it was an hypothesis conformable to the universal law of nature, always accomplishing her ends by the simplest means.

            Nor is the use of hypothesis confined to these cases in which they have subsequently received confirmation. It may be equally great where they have completely disappointed the explanations of their authors. Indeed any hypothesis which possesses a sufficient degree of plausibility to account for a number of facts will help us to arrange those facts in proper order and will suggest to us proper experiments either to confirm, or refute it. Nor is it solely by the erroneous results of his own hypothesis that the philosopher is assisted in his enquiry after truth.

Similar lengths may often be collected by the errors of his predecessors: it was from a review of the endless and hapless wanderings of preceding enquirers that Bacon inferred the necessity of avoiding every beaten track, and it was this which encouraged him with a confidence in his own powers, amply justified by the event, to explore and to open a new path to the mysteries of Nature.

            In this respect the maturity of reason in the species is analogous to that in the individual. It is not the consequence of any sudden or accidental cause but the fruit of re-iterated disappointment connecting the mistakes of youth and inexperience. "There is no subject," says [Bernard le Bouvier de] Fontenelle, "on which men ever come to a reasonable opinion till they have once exhausted all absurd views which it is possible to take of it. What follies" he adds, "should we not be repeating this day if we had not been anticipated in so many of them by the ancient philosophers."

Those systems which are false are therefore by no means to be regarded as altogether useless. That of Ptolemy, for example, as has well been observed by the elegant historian of Astronomy, is founded on a prejudice so natural and so unavoidable that it may be considered as a necessary step in the progress of astronomical science, and, if it had not been proposed in ancient times, it would infallibly have preceded among the moderns the system of Copernicus and have retarded the period of its discovery.

            Among the numerous discoveries which have rendered illustrious the name of Kepler there are none more important than those with which he enriched Astronomy at the commencement of the 17th century. Galileo had begun the investigation and cleared away some of the difficulties but it was Kepler who transported geometry to the heavens, who discovered the laws of their movement. Let us for a moment follow Kepler in the ideas and conjectures which led him to these memorable results.

By comparing the different velocities of the Sun at different seasons of the year with variation of its apparent diameter, he found that it was impossible to account for the phenomenon by supposing the Earth to revolve [around the Sun] in a circle. He therefore supposed that it might move in an oval, but of this figure there are various kinds, and the first one he hit upon did not answer his purpose. He was however more successful in the next trial and found that the common ellipse would satisfy all the conditions. From this period we may date the knowledge of the elliptic motion of the planets and the destruction of the ancient prejudice which attributed to the heavenly bodies a uniform circular motion which by its simplicity had seduced the ancient philosophers of Greece.

            After discovering that the planets made an ellipse Kepler wished to find some law which might regulate their motions. He knew that when they were nearest the Sun they moved fastest, but was not acquainted with the law of the change of velocity. After numerous trials he found out the following analogy. If we conceive a line drawn from the Sun to the centre of any planet this line will always pass over equal areas in equal time.

Kepler observed that the more distant a planet was from the Sun the longer time it required to perform its revolution round that body, and this led him to the discovery of another law which prevails throughout the planetary system. He found that the square of the time of any planet's revolution always bore a certain proportion to the cube of its distance and that this ratio is constant.

            Mercury           [= 0.13050]         39 x 39 x 39

            Venus               [= 0.1356]           72 x 72 x 72

            Earth                [= 0.1332]         100 x 100 x 100

            Mars                [= 0.1344]         152 x 152 x 152

            Jupiter              [= 0.1335]         520 x 520 x 520

[Table of the square of the number of days required by a planet to revolve round the Sun divided by the cube of its mean distance from that body in millions of miles]

The laws which I have just mentioned were discovered by Kepler after many trials and numerous failures. They rested on no other foundations than experiment and were in precisely a similar situation to the laws relating the planetary distances which I endeavoured to explain in a former lecture. That is to say Mankind were astonished at their coincidence with Nature, but were unable to divine the cause which produced it.

            Kepler is particularly distinguished from the philosophers of his time by the great boldness and frequently by the great correctness of his views in enquiring into the cause which produces the phenomena of Nature. He considered the Sun as the supreme moderator of the celestial bodies. "This star" says he,  "is possessed with a power of moving bodies, which it spreads with immense rapidity throughout all space and hence arise the motions of the planets."

            At one time he compares the weight of heavy bodies on Earth to the gravity of the planets towards the Sun. In another place he suspects that the combined action of the Sun and the Earth produces the irregularities of the [motion of] the Moon. And he imagines that the tides may possibly arise from the attractions of this body. One of his fundamental doctrines is the motion of the Sun on its axis, an hypothesis which was completely justified a few years after by the discovery of the spots which cover its surface. These ideas and conjectures bear the evident stamp of genius, the daring flight of a powerful and comprehensive mind, and they opened to Newton that glorious path which led him to the most sublime discoveries.

Before however we take the steps which led this great philosopher to his theory of gravitation, it will be proper to pass in review the theory of Descartes, which at the time universally prevailed. Philosophers of the highest antiquity had recognised the existence of the heavenly bodies. They had each calculated their distances and appreciated with some accuracy the motions by which they are animated. But no one before the 17th century, had endeavoured to snatch from Nature the secret mechanism by which they hold together the planetary system. The honour of this bold enterprise was reserved for Descartes. Determined to produce a system entirely novel in which everything should be reconciled with his ideas of the harmony of Nature, he conceived himself at the formation of the Universe, and thus presented to himself the spectacle of Creation -an infinity of molecules of matter repose in the immensity of space. All possess an extreme hardness, and their infinitely varied form victoriously opposes the existence of a vacuum. "Creative power" says Descartes, "imposes on them a force, which at the same time carries them forward in space, and causes them to revolve on their axis. And certain laws are prescribed to them by which their actions are to be regulated." Such is the chaos from which Descartes conceives a universe like ours might arise, whose spectacle, although habitual, daily excites fresh reason for surprise and admiration. It is needless to pursue the speculations of this philosopher through their varied course to the existence of vortices of extremely subtle matter, through whose assistance the planets and satellites were conceived to revolve.

We have already seen that they are totally devoid of proof, and are, in fact, physically impossible. Yet were not the speculations of Descartes without their use in the progress of philosophy? They were errors, but it must be confessed that they were splendid ones, and that the mind which could frame such a theory might under better guidance have arrived at more accurate results.

            Kepler, Galileo and Descartes contributed to dissipate the darkness which had for a long time enveloped mankind. They became the benefactors of the human race, but for the shame of the age which produced them, they received as the reward of their labours nothing but injustice, persecution and disgrace. Galileo, whose brilliant discoveries had merited a better fate, was dragged to an unworthy prison. Kepler surrounded by the glory he had acquired by the his sublime views of Nature experienced in his old age all the misery of want and indignance. And it was not until 100 years after the death of Descartes that his grateful country raised even a monument to his memory.

Doubtless nothing would be more satisfactory to the mind than the physical system of Descartes if it could sustain the process of examination and observation. Those vortices, that is to say, those torrents of ethereal matter, which according to this philosopher carry with them the planets round the Sun, present to the mind an intelligible mechanism which enchants by its simplicity. But this theory, which at first glance is so seducing, is subject to many difficulties. It is unfortunately found to agree so little either with the phenomena or with the laws of mechanics that notwithstanding the efforts of many ingenious writers, it is universally allowed that the system of Descartes is not that of Nature.

            Newton pursued a different course, and on the ruins of this system he has erected a new, more solid one, which presents every appearance of the greatest durability. In fact his system exhibits a perpetual coincidence between theory and observation. Whether we regard the grander laws which regulate the Universe, or whether we examine these minute and almost insensible ramifications of observation and geometry [it] has nothing to fear from the vicissitudes of time, or from the more changeable opinions of men. The system of Newton is founded on the principles of Universal Gravitation.

Every particle of matter, whatever may be the mechanism or cause which produces this effect, tends, according to this philosopher, to every other particle with a force which decreases inversely as the square of the distance. This gravity causes on the surface of the Earth the weight of a body, and among the celestial bodies it is the source of the most complicated motions. I shall endeavour to explain the proofs of this principle and the reasoning which lead Newton to it after shortly stating what was known on the subject before the time of his writings.
            We find among the writings of the ancients a glimpse of several of the most brilliant truths. This is particularly the case with the principles of Universal Gravitation, of which we discover some decided marks. Anaxagoras as we have already seen attributed to all the heavenly bodies a tendency towards the Earth, and other traces of the same opinion may be found among the writings of Democritus and Epicurus. It was from this principle that Lucretius drew the bold conclusion that the world is without bounds.

As soon the true system of the world revived by Copernicus arose from its ashes that of Universal Gravitation threw some rays of light. This celebrated astronomer attributed the round figure of bodies to the attraction of their parts. He did not extend this attraction from planet to planet. But Kepler more bold and more systematic made this step. He attributed to the Moon a tendency towards the Earth and said that they could meet in their common centre of gravity if they were not prevented by their rotation. Nobody however before the time of Newton so clearly perceived the principle of that attraction, or more nearly approached in making a proper application of the system of the Universe than Dr. Hooke. The philosophers whom we have mentioned had some of them seized one branch, some another, but Hooke embraced it in all its generality.

            His anticipation of that theory of planetary motions which was soon after to present itself with increased and at length demonstrative evidence to a still more powerful mind furnishes a remarkable instance of this philosophical sagacity. This conjecture I shall state in his own words, and it affords a decisive reply to the undistinguishing censures which have so often been bestowed on the presumptuous vanity of attempting by means of hypothesis to penetrate into the secrets of Nature. "I will explain" says Dr. Hooke, in a communication made to the P[resident of the Royal Society?] in 1666, "a system of the world very different from any yet received. It is founded on the three following positions."

"1st that all the heavenly bodies have not only a gravitation of their parts to their own centre, but that they mutually attract each other within their spheres of action."
            "2ndly that all bodies having a simple motion will continue to move in a straight line unless continually moved out of it by some extraneous force."
            "3rdly that as this attraction is so much greater as the bodies are nearer, as to the proportion in which those forces diminish by an increase of distance."
Dr. Hooke adds "I [on my] own have not discovered it although I have made some experiments for this purpose. I leave this for others who have time and knowledge sufficient for the purpose."

            It should be observed that there is a wide difference between the conjecture of Hooke and the proofs and sublime demonstrations by which Newton supported this law of the Universe. Such however was the state of the question when this profound philosopher appeared. It was in 1666 that he first [began] to suspect the existence of this principle and to endeavour to apply it to the motions of the heavenly bodies. He had retired into the country to avoid the plague, which at that time prevailed in London. His meditations were one day accidentally directed toward the weight of bodies.

His first reflection was that the cause which produced the fall of heavy bodies always acts upon them to whatever height we convey them. It may then be extended much further than we think, possibly as far as the Moon or even beyond. From this he conjectured that it might possibly be this same force which retains the Moon in her orbit. At the same time he considered that though Gravity does not sensibly alter at different heights to which we can attain, yet at greater distances it may vary and these altitudes are too small to conclude that it is the same at all distances. It now remained to discover the law by which it varied,. For this purpose he argued that if gravity retained our Moon in her orbit round the Earth, it must be a similar cause which retains the sattelites of Jupiter in their orbits round that body, and by comparing the periods of these bodies with their distances, he found that gravity must decrease inversely as the square of the distance.

Newton did not however rest here. He continued to examine an account of the Moon's distance from the Earth: the force by which she is attracted will be 3,600 times less than that by which a body falls on the surface of the Earth. If we can compare the space through which the Moon falls towards the Earth in a given time with that through which a body on the Earth's surface falls, we shall have a criterion by which to judge of the truth of our theory. But here arose a difficulty, how shall we find how much the Moon falls towards the Earth in a given time? This difficulty Newton overcame, and these are the means he made use of. This had nearly overthrown his whole edifice. He supposed the terrestrial degrees to contain 60 miles and in consequence of this the two quantities (on whose relation the truth of his theory was to be tried) did not afford a result favourable to it.

            Many philosophers would have been but little troubled by this disagreement and would have continued to construct their theoretical edifice. But this incomparable man, whose object was the discovery of truth and not the formation  of a system, when he found that a single fact overthrew all his conjectures which had hitherto been so well founded, immediately relinquished them.

It was not till 10 years after that he resumed the train of his ideas. In this interval the opinion of Dr. Hooke had been published and a very important step had been made by Ricard in ascertaining the magnitude of the Earth. From this Newton learned that the terrestrial degree was nearly 70 miles in length, and not 60 as he had considered it in his calculations. He now therefore again returned to his theory and having calculated the magnitude of the lunar orbit, he found to his great satisfaction that the space fallen through by the Moon precisely agreed with what it ought from his theory. After this demonstration Newton no longer hesitated to consider the force by which heavy bodies fall at the surface of the Earth and that by which the Moon is retained in her orbit as one and the same.

            He assumed gravity as a well ascertained fact and proceeded to reason upon it. He showed that it followed necessarily from his theory that the planets move in ellipses and he demonstrated the laws which Kepler had only found by induction. By a skillful application of mathematical calculation to the phenomena of Nature aided by the theory of gravity, he unravelled numberless irregularities to which the heavenly bodies are subject. Yet such was the excessive modesty of this great man that it was with the greatest difficulty that he was persuaded to publish his profound discoveries.

At the urgent request of his friend, Dr. Halley, and at the entreaty of the Royal Society, he was persuaded to collect together his discoveries, which he did in his Principia, a work which also is alone sufficient to immortalise its author. This work however in which  this great man has built a new system of natural philosophy upon the most sublime geometry did not at first meet with all that applause it deserved and was one day to receive.

            Two reasons concurred to produce this effect. Descartes' system had at that time got full possession of the world. His philosophy was indeed the creature of a fine imagination; he had given her some of Nature's features and had painted the rest to a seeming resemblance to her. Newton, on the otherhand, had with unparalleled penetration and force of genius pursued Nature up to her most secret abode and was intent to demonstrate her residence to others rather than anxious to describe particularly the way by which he arrived at it himself. But at last that approbation which had been so slowly gained became universal, and nothing was heard from all quarters but one universal burst of applause [and] admiration: "Does Mr. Newton eat, drink or sleep like other men?" said the Marquis de l'Hospital, one of the most enlightened foreigners of the age, to his English visitors, "I represent him to myself as a celestial genius entirely disengaged from matter."

Yet in the midst of these profound enquiries Newton had leisure for other pursuits. When the privileges of the University [of Cambridge] were attacked by James II he appeared as one of the most strenuous defenders. And he made a very successful defence before the high commission court. He was also a member of the Convocation Parliament in which he sat till it was dissolved.

            In his private life Newton was modest and unassuming in the highest degree. His temper was so mild and equal that no accident could disturb it. He would have rather chosen to remain in obscurity than to have the calm of life ruffled by the storms and disputes which genius and learning so frequently draw on those who are eminent for them. From this love of peace arose that unusual horror which he felt for all disputes, and that steady unbroken attention which was his peculiar felicity; he knew and well esteemed its value.

When some objections hostily made to his discoveries concerning light and colours induced him to lay aside his design of publishing his optical lectures, we find him reflecting on that dispute into which he was unavoidably drawn in these terms: "I blamed my own imprudence" said Newton, "for parting with so real a blessing as quiet, to run after a shadow." Yet this shadow was one of the most splendid and most original of these discoveries which have contributed to make his name so illustrious.

            In contemplating the genius of Newton the penetration, the strength and the originality of his mind, in his moral capacity the pre-eminent trait is his modesty and love of quiet. In his intellectual character the most predominant feature is the astonishing power which he possessed of concentrating his attention to the object on which he was employed. It was to this almost supernatural power that he himself attributed his profound discoveries. When he declared that if he had done the world any service it was due to nothing but industry and patient thought; that he kept the subject of consideration constantly before him and waited till the first dawning opened gradually, by little and little into a clear and full light. Such was Newton as a man, and as a philosopher both characters were tinged with a similar colouring. As a man he did not possess that warmth and enthusiasm which we should admire in a friend, but he displayed that calm unruffled serenity, which we should reverence in superior beings. As a philosopher he was not led away by the fire of genius whose too daring grasp by sometimes fostering error, reminding us of his mortal origin. But he was the patient, the accurate investigator of Nature, the deep, the profound philosopher.

To trace the consequences of the law of Universal Gravitation through its numerous and almost endless ramifications would lead us in succession through every branch of astronomical science. Each seeming objection which has successively been brought against its truths has furnished new arguments in its favour and afforded new ground of triumph to the followers of Newton. It has frequently anticipated observation and has predicted to future astronomers irregularities that are yet to be recognised. To such a point of perfection has this science been carried, that there does not now remain one single irregularity of the heavenly bodies of any magnitude which does not follow as a consequence of this law, and whose quantity cannot be calculated by it. It is obvious that an enumeration of these varied irregularities would be of little improvement, but there are some important questions which present themselves and which relate to the law of gravity.

            There are many of the planetary irregularities, which increase and decrease alternately in longer and shorter periods. Some however, since we have observations of them recorded, have been found uniformly to decrease. Such for instance is the obliquity of the ecliptic. If this obliquity were to decrease continually, the ecliptic would at last coincide with the equator. This would not take place until after the lapse of millions of years, but when it did the days and nights all over the world would become equal; the Earth would possess a perpetual spring. Whether this change would be for the advantage of the human race if it then existed, or whether it would not render nearly half the globe uninhabitable, are doubtful questions; yet remote as this change may be it is interesting to enquire whether it is possible, because there are other irregularities which increase much more rapidly and which might if they continued to increase totally derange the system. The question then in the most general sense is to investigate whether the irregularities of the planetary system will continually increase or whether after attaining a certain point they will not decrease and return again in the same order.

This question which is a very important one has occupied some of the greatest philosophers of the age. The progress was very gradual but the solution is complete. It is to the united labours of Lagrange and Laplace that we are indebted for the solution of this interesting question. It has been demonstrated that every irregularity to which the planetary [system] is subject must from its nature be periodical: that is, it will increase to a certain point and then decrease to another point, between which two it will constantly oscillate, never exceeding either of them, just in the same manner as a pendulum which constantly moves to and fro, but never exceeds a certain fixed distance from its point of rest. Almost everything in the system will therefore be in motion, but admidst this universal change some few elements will remain constant: thus the mean distance of each planet from the Sun will be unaltered. Thus then it appears that unless some foreign force disturb the harmony of our System, it will forever continue in its present arrangement, that it does not contain within itself the seeds of destruction, but on the contrary that it is destined to an eternal duration, unless the mechanical laws which govern matter be subverted or some influence foreign to the System be exerted on it.

There were, however, in its original constitution some conditions necessary, that the inclination of the orbit of the planets to the Sun's equator be small, that the ellipses which they describe should be nearly circular and that they should all move round the Sun in the same direction. And if these conditions had not been fulfilled it is possible that the beautiful system from its own action have produced its destruction. If we enquire from the doctrine of chances whether it is probable that the conditions should have been accidentally accomplished we find that the contrary is indicated with the highest possible degree of probability. In fact Laplace, when speaking on this subject, says that we have stronger grounds for believing that the planetary motions and inclinations were all influenced by the same primitive cause, than we have for giving credit to any of the most authentic accounts in history. This wonderful contrivance of an intelligent mind by which the permanence of our System is secured is highly calculated to excite our admiration, yet it has been perverted to the worst, the most unphilosophical purposes. It has been urged as the supporter of fate and of necessity, and has been inconsiderately advanced as an argument against the superintendence and existence of a first cause. It is a singular circumstance that a fact which had a tendency so directly contrary should have been so misunderstood, and it would perhaps be needless to refute it, but that it has been said, though I believe falsely, to have received the sanction of one of the most eminent of the continental philosophers. The difficulty is easily removed. We have only to ask ourselves this question: which is the most skilful artist? he who makes a clock which requires winding up every day and cleaning every year, or he who contrives one which winds itself up and never requires cleaning, to which

it may be further added that among the infinite number of laws by which gravity might act, all equally possible, this of Nature alone aided by the conditions already specified will ensure the stability and permanence of the System. Had there been any other originally established this Universe the beneficent result of creative power would have long since have returned to its primitive chaos.

Lecture 9: The Outer Planets

Lecture 9

William Herschel - Discover of Neptune

The Outer Planets*

[script of lecture missing]

* Note: probable title of lecture.

Lecture 8: On the Minor Planets (Asteroids) and also the History and Development of the Reflecting Telescope

Lecture 8

On the Minor Planets (Asteroids) and also the History and Development of the Reflecting Telescope

William Herschel's 40 foot Reflecting Telescope at Slough

            We have already considered the motions, phases and appearances presented by the planets which are included by the orbit of the Earth. We have also extended our view to the planet Mars, which is next in order to our globe. Beyond this body are situated four small bodies whose diminutive size would have ever hid themselves from our sight without the assistance of the telescope. These are on many accounts remarkably worthy of our attention. The recency of their discovery, the smallness of their magnitudes and the nearly equal periods of their revolution round the Sun, these and the numerous other points in which they differ from the rest of the planetary bodies with which we acquainted combine to give them a singular interest.

            There is, however, another point of view in which these bodies appear in no less striking light. A law has been discovered to which all the planetary bodies are submitted. This law was incomplete until the discovery of these bodies, but it is now found to prevail throughout the system. Some astronomers have contended that it is a law of Nature, whilst others have attributed the coincidence entirely to chance. I now propose to trace the history of this singular question and the consequences to which it leads. In explaining the law itself I fear I shall necessarily appear abstruse, for this the nature of the subject will, I hope, be a sufficient apology.

It is needless to collect the vague notions of a few of the ancients respecting the number of the planets. They were for the most part conjectures without the slightest foundation. Kepler was the first who had some notions real or imaginary respecting the number and distances of the planets. He even pointed out two vacancies in the system in which he supposed new ones ought to be discovered.

            Kepler imagined every thing in Nature must be harmonious. He conceived certain mystical properties to be attached to numbers, and imagined that there must exist some law which should connect together these wandering stars. These were the objects of his constant enquiries of his ardent pursuit and the result was the discovery of those laws which have received his name. These have subsequently been confirmed by the investigations of the mechanical philosophy but at the time of their discovery they were merely the results of trials and were only judged to be true from their coincidence with fact. This is the only kind of evidence which can be offered for the law we are about to consider. Kepler spent a considerable time in endeavouring to find by trial whether there did not exist some relation among the distances of the planets from the Sun, but after a long and unsuccessful labour he gave up the task in despair. All his calculations were overturned from the want of a planet situated between Mars and Jupiter, and that  such an one did exist he strongly suspected but could not discover any law nor assign the distance at which it should be placed from these two bodies.

Titius, a professor of Astronomy at Wittenburg, was the next who applied himself to these researches. After much labour he found out a law to which all the planets then known accorded , and from this he concluded that there must be a planet situated between Mars and Jupiter, and he even determined the distance at which it ought to be placed from the Sun. the law which Titius discovered was this: the distance of the planets Mercury, Venus,  the Earth, Mars, Jupiter and Saturn may be represented by the numbers: 4, 7, 10, 16, 28, 52, 100 [and] 196. If now from each of these quantities we subtract the number 4 and if we divide the remainder by 3 the result will always be a power of two. This is very nearly true for all the planets that were known at the time. Titius lived, but there was a vacancy between Mars and Jupiter at the distance of 28. According therefore to his theory he concluded that some one would be found to be situated between them.

These observations were made before the discovery of Uranus by Dr. Herschel in the year 1781. The proportional distance of that planet is 196, and it is somewhat remarkable that this number corresponds with the law of Titius, for if we subtract four from it there remains 192. This divided by 3 gives 64 which is the 6th power of two. Dr. Herschel's planet then is in a certain sense a proof of the law discovered by Titius. It was found out after the law was known and is situated at the precise distance which that law indicated. We shall find however that there is another perhaps a stronger proof.

            The law of Titius does not seem to have excited that astonishment which so singular such a subject might have been expected to create. It was, however, much considered in Germany, and met with many warm advocates. It excited a strong belief in the assertion of Kepler that another planet must exist between Mars and Jupiter. This was so much increased during the latter years of the last century, that Bode who was quite a convert to the opinion wrote to appoint a meeting to consider of the best means of discovering the supposed planet. Those who found the journey inconvenient sent word that they would undertake a share in any of the labour which might be resolved on for this purpose. Gothe in Saxony was the place appointed for the meeting and here were assembled Bode, Lalande, Schroeter, Harding, Olbers and many others of the most respectable observers in Europe. The result of their consultation was that they would divide the heavens in zones of a few degrees each and that each astronomer should take one of these zones and examine scrupulously every star it contained above a certain magnitude. This was the plan adopted. So each observer was appropriated a zone and to those who were absent an account of their task was sent.

Piazzi, an astronomer at Palermo in Sicily, had one of these zones assigned to him. He was at that time occupied in a description of the starry heavens and consequently had occasion to examine other parts besides that which was appointed to him. In this pursuit he was occupied when he observed one evening the 87th star in the Zodiacal Catalogue of La Caille situated between the Ram and the Bull. Near this he perceived a small star of the 8th magnitude which he thought an unknown one, and it appeared to possess a proper motion of its own. This happened on 1st January 1801 and according to his usual custom he wished to observe it on several of the following days for the purpose of determining its position with better success.

            He made several other observations and perceived a motion in the star and suspected that it might possibly be a new planet. To verify this conjecture he resolved on following the motions of this body very assiduously, but a dangerous illness he was attacked occasioned by excessive fatigue had nearly at once deprived the world of the astronomer and his discovery. When he was sufficiently recovered to pursue his observations the star was no longer visible to the Earth; it had disappeared in the rays of the Sun. Piazzi now reconsidered his former observations. These were the only guides he had to conduct him in his search after this new body. He found that they accorded very well with the supposition of its moving in an ellipse. These conclusions were similar to those of Burckhardt, an astronomer of acknowledged skill, and confirmed him in the idea of its being a planet. He therefore gave it the name Ceres to inform posterity that Sicily which was formerly consecrated to this goddess was the place from which she was first discovered.

The new planet, however, was not easily rediscovered from her extreme minuteness. She escaped all observation. The greater part of the year 1801 was employed in searching for her. After many fruitless attempts she was again found by Zach on the 31st December and by Dr. Olbers on the 1st January 1802, that is just twelve months after her first discovery. These gentlemen, however, were not the first to whom the planet became visible. On the 7th of December it had been observed by Gauss, and as it affords a remarkable instance of the powers of [Mathematical] Analysis I shall mention the circumstances attending it. "All hope" observed this excellent mathematician, "of again gaining a sight of this planetary atom depended entirely on our being able to find its orbit with a sufficient degree of approximation from the few observations that were made on it when visible. I could not" continued he, "desire a better opportunity of trying whether my ideas on this subject were of any practical utility than by using these observations for the determination of the orbit of Ceres." This planet had during 41 days only described an arc of 3 degrees and now after the lapse of a year it was to be sought for in a far, distant part of the heavens. The first applications of Gauss' method was made in the month of October of 1801. And on the first fine night which occurred he directed his telescope to that precise spot in the heavens which it ought to occupy from his calculations and the planet was immediately visible. This is a striking instance of the perfection which theory has attained. With only 3 observations on a new planet we may find its distance from the Sun in a rough manner, and with a very few more we may approach tolerably near the truth.

From the new observations which were now made on this planet it was found that she completes her circuit round the Sun in about 1,618.5 days or in 3 years 7 months 10  degrees. Having thus discovered the nature of her orbit and the principal irregularities to which she is subject there is no longer any danger of her eluding the enquiry of astronomers.

            The discovery of this minute planet has suddenly changed many of the received opinions concerning the Solar System. The extent of the Zodiac in which the motion of the planets was confined was 16 degrees. This was the Zodiac of the ancients, but Ceres has extended these limits and requires a zodiac of 37 degrees, which is more than double the extent of the former. The apparent inclination of her orbit varies from 11 degrees to 18  degrees. She has also disarranged our ideas respecting the rank established among the bodies which constitute the planetary system. Nature appeared to have placed the largest under the immediate dominion of the Sun and around these smaller bodies or satellites revolved, but this arrangement is destroyed. Ceres is one of the smallest bodies of the planetary system. Her apparent diameter does not amount to 1 minute of arc according to Dr. Herschel and from this it would follow that her real diameter is 17 times less than that of the Earth or that our Moon is five times as large as the planet Ceres, and yet this diminutive body does not describe a narrow circle round some primary planet but pursues her lengthened course through the heavens beyond the orbits of the Earth and Mars.

The discovery of Ceres has by some been regarded as the effect of accident, but it should not be considered in such a point of view. It is the honourable fruit of an immense labour. It is the well deserved reward of the care and attention bestowed by its author on the formation of his catalogue of fixed stars. This skilful observer never placed any star in his catalogue until he had viewed it on several successive nights, and it is owing to these repeated observations that the discovery of Ceres must be attributed. It was difficult from its extreme smallness and it has become more glorious to its author from the important consequences which have followed.

            It was in examining the path which this body describes in the heavens that the other planets have been discovered which were before equally unknown. The most singular circumstance attending this new planet is that it occupied the interval between Mars and Jupiter which was predicted by Kepler, and that its distance corresponds very nearly with the law discovered by Titius. In fact 28 = 4 + 3.2. This remarkable law occupied much of the attention of the German philosophers. It must however be confessed that it is not completely accurate, yet it so far agrees with the truth as to excite considerable surprise and to make us almost doubt whether it could be the effect of accident. Whatever it may be planets to whose discovery it contributed will always remain to us and the law itself if found to be fallacious will furnish an example of the happy effect which have sometimes casually resulted from systems entirely erroneous.

On the subject of this law I cannot lay before you the calculations of Professor Schweigger of Nurnberg which would I have no doubt throw some light on this curious question. They are contained in a paper read before the Philosophical Society of Munich on the 6th August 1813. It is entitled a dissertation on a general law which subsists between the distance of the planets and their satellites. From this title I should imagine that its author had discovered some law between the distances of the secondary as well as the primary planets. The volume, however, which contains this paper if it is printed has not yet arrived in this country.

            The third year of the 19th century produced another new planet for whose discovery we are indebted to Dr. Olbers, a physician of Bremen, who was known to the astronomical world as the author of a treatise on comets. On the 28th of March 1802 he was observing with the design of determining the position of Ceres all the stars which form the constellation of the Virgin. At a short distance from that marked 20 near which he had observed the planet about two months before he saw a star of the 7th magnitude which he had not perceived in his former observations. He had some suspicions about this star and examined her more attentively. In the interval of two hours he found that she had altered her situation and on the following two nights he ascertained that she was in motion at the rate of 10 minutes of arc in 24 hours.

If the astronomer considers the accidental discovery of a comet as a piece of the greatest good fortune how much more highly must he estimate the advantage of enriching the system with another planet. Dr. Olbers enjoyed the satisfaction  almost at the moment of his discovery. He had no doubt respecting the nature of the body he was viewing. Its disc was better defined than that of Ceres and it had not the least resemblance to a comet. He had besides learnt from the discovery of Dr. Herschel and Piazzi that the ancient planets were not the only ones belonging to our System. His satisfaction was not therefore interrupted by any of those doubts which had alarmed the former observers. Thus after the first few days of the discovery Dr. Olbers announced the new planet to the astronomical world. The astronomer Burckhardt and Gauss as soon as they were informed of its existence commenced their observations on it. They soon found that it revolved in an ellipse but were much astonished to discover that the inclination of its orbit was greater even than that of Ceres.

A star which embraces in its course from north to south a zone of about 70 degrees wanders too far from the ordinary course of the planets not to leave at first some hesitation as to the rank which ought to be assigned to it. But since this body as well as Ceres is placed between Mars and Jupiter, and since it is not like comets subject to disappear by its recess from the Sun it has been placed among the number of the planets and it received from its discoverer the name of Pallas.

            The effect of this great inclination of its orbit combined with its eccentricity which is larger even than that of Mercury causes the greatest inequalities and perturbations in its motions and at the same time renders their disturbances more difficult to calculate. Burckhardt undertook some of the most laborious calculations with a view to ascertain its elements. Pallas performs it revolution round the Sun in 1,681.7 days. This is about 2  hours longer than Ceres occupies for the same course, so that the two planets are situated almost precisely at the same distance from the Sun. The eccentricity of Pallas is very considerable. She is in one part of her course almost twice as far distant as she is at the opposite part. From the united effect of these considerations it may happen that Pallas whose mean distance from the Sun is greatest may pass between Ceres and the Sun and thus to an observer situated on the planet Ceres there would be a transit or rather from the nearness of the two planets it should be called an eclipse since the Sun would probably be completely hid. If the inclination of their orbits were equal this eclipse might last months or upwards and it would perhaps not again recur for about 33,000 years, but from the difference in the inclination of their orbits it will last but a short time and will occur still less frequently. After the lapse of ages it may happen from the differences of their eccentricities that that which was the inferior shall become the superior planet and that the inhabitants of Pallas shall observe Ceres pass between themselves and the Sun.

To the discovery of the planets of Piazzi and Olbers shortly succeeded that of another new planet by Professor Harding. This astronomer, the worthy colleague of Schroeter, undertook the task of forming a map of that zone of the heavens which contains the paths of Ceres and Pallas. He executed this zodiac of Ceres on twelve large sheets and not only marked down all the stars contained in the different catalogues all of which he found in Lalande's list of the 50,000 he observed, but added a great many from his own observations which had hitherto escaped the attention of astronomers. On the 1st of September 1804 in comparing these maps with the heavens he discovered between two stars whose places were known a new star which had not before been seen in that place.

            On the 4th September he no longer perceived it but at a short distance he saw another which he had not seen 3 days before. He immediately suspected that this might be the same as the first but that its motion had made it appear in two different places. This suspicion was soon changed into certainty: on the next day he plainly observed its movement and as the body presented [had] neither nebulosity nor the appearance of a tail he immediately concluded that it was a planet. This was soon confirmed by the other observations of other astronomers and by the calculations which resulted from them. The new planet received from Professor Harding its discoverer the name of Juno. It performs its revolution round the Sun in 1,590.998 days so that the length of its year is about 90 days shorter than those of Ceres and Pallas. Its distance from the Sun is nearly 26.5 if we consider that of the Earth as represented by the number 10. The eccentricity of Juno is very considerable; it is rather greater even than that of Pallas, but the inclination of its orbit is not so considerable being only 13 degrees.

Since the commencement of this century a fourth planet has been discovered in almost every respect similar to those already described. Although a plan was as we have seen formed for the discovery of a planet between Jupiter and Mars, yet it was not strictly owing to this laborious undertaking that the three first were found. For the discovery of Ceres we are indebted to the formation of a catalogue of stars by Piazzi; that of Pallas arose from the examination of the heavens which [were] undertaken to re-discover Ceres and Juno was found from the investigations undertaken by Harding to form a chart of all the small stars in the path of the two former bodies. The fourth is the only one discovered from pursuing a plan with the express view of finding it.

            The hypothesis on which it was founded is certainly very extraordinary and may perhaps be controverted, but it has been too fortunate in its result to incur the disapprobation of astronomers. The idea itself and the consequences which resulted from it are equally the property of Dr. Olbers. This skilful observer to explain the phenomena presented by the smallness of the new planets and their nearly equal distance from the Sun framed this hypothesis.

That possibly these small bodies might be the fragments of a much more considerable planet which some extraordinary cause had burst in pieces and that these parts continued to circulate round the Sun at the same distance and with equal velocities. This theory does credit to the ingenuity of its author and is not opposed by an argument which has frequently overturned such speculations. It is not repugnant to the principles of mechanics. It is not impossible that such an occurrence should have taken place and if such had been the case it might have happened that several fragments would revolve in nearly an equal time and the orbits of all would cut each other in two points. If however any of these parts should pass within the sphere of attraction of any large body its orbit might be considerably altered. This has perhaps happened in the present case. It is not probable to suppose that the convulsion which thus destroyed a planet should have divided it into precisely the parts which have been discovered. It is more likely that an immense number of pieces of different magnitude should have been formed, the larger parts would revolve regularly in certain orbits but possessing a considerable mass they would only be disturbed by the action of the other planets and would perform their course subject to these irregularities. The smaller fragments would be much more considerably affected by the attractions of the larger, and as they passed within the reach of each new body their orbit would be altered. Thus it might happen that some of these small fragments coming within the sphere of attraction of the Earth may be precipitated on its and thus produce those meteoric stones which are frequently discovered. It is not impossible that at the original disruption one part of the planet might be projected nearly in a right line towards the Sun. This would revolve in a very eccentric ellipse and would consequently become a comet.

This hypothesis of Dr. Olbers will answer another purpose. It was observed that the law proposed by Titius was deficient before the discovery of a planet between Mars and Jupiter. The knowledge that four planets exist there would be equally fatal to this law but, according to Dr. Olbers, they are the remains of an original one and, if we suppose as is most probable that this was situated at the mean distance of all its parts, it will coincide very well with the law alluded to. From the knowledge we possess of the elements of the three planets already noticed it appears that they may at some future period come into contact with each other and that in preceding ages they might have done so before. If we were more completely acquainted with their motions it might be possible to assign with some considerable degree of probability the epoch of the original catastrophe.

            In pursuing his hypothesis Dr. Olbers considered that the orbits of these fragments possessed of different inclinations ought to cut each other in two opposite points of the heavens which would be the common intersection of them all. He thought that if we wished to discover the other scattered fragments of the planet that we should direct our attention to these two points. According to observations on the course of Ceres and Pallas and from calculations on the inclinations of their orbits it was found that one of these points is situated in the constellation of the Virgin and the other towards that of the Ram.

The discovery of Pallas in the first of these points and that of Juno in the second seemed to confirm this ingenious hypothesis and determined Dr. Olbers in his resolution of seeking for some new planet. He resolved therefore three times annually to pass in review all the small stars which compose the opposite constellations of the Virgin and the Ram. Fortune favoured this project and on the 29th March 1807 he discovered in the northern wing of the Virgin a small unknown star whose motion from day to day was very perceptible and it was immediately placed in the rank of planets. She appeared to shine with a pure white light and to be surrounded by a thinner atmosphere than those of her colder sisters Ceres, Pallas and Juno. Vesta is the name assigned to this new planet which soon occupied the attention of the principal observers in Europe. It completes its revolution round the Sun in 1,335.2 days and its mean distance from the Sun is nearly 24 if that of the Earth be considered as represented by ten. The eccentricity of its orbit is considerably less than that of the other recently discovered planets and the plane of its orbit is inclined to the ecliptic only at an angle of 7 degrees. It results from the values which have been assigned to its elements that Vesta is about 36,000,000 miles nearer the Sun than Ceres, Pallas and Juno; that the inclination of her orbit is not much greater than that of Mercury and that its eccentricity is nearly equal to that of Mars.

From these causes it appears that she must be much less exposed to perturbations from the action of Jupiter and that they will be more easy to calculate. In fact Gauss, having compared his calculations with 22 observations of the astronomer Bouvard, found that they differed only 17 minutes of arc. This is certainly a wonderful degree of precision considering the shortness of the time the planet had been discovered.

            The examination by Dr. Olbers of the stars in the constellation of the Virgin and the Ram were crowned by a fortunate result. It must, however, be observed that the intersections of the orbit of Ceres and Pallas and the new planet have not that precise coincidence which was expected. They are separated by an angular distance of about 20 degrees. This however is too small a quantity to afford an argument against the hypothesis of Dr. Olbers.

            The four planets we have just considered present a singular spectacle in the system of the world, differing from all the other. They have among themselves many points of resemblance and appear associated by nature to the destinies. Collectively they fill up the vacancy which was thought to exist between Mars and Jupiter. Placed at a mean distance between these two planets they describe orbits of nearly equal magnitude and move with nearly an equal pace.

Several philosophers at first refused to bestow the name of planets on these stars whose discovery signalised the commencement of the 19th century. The principal reason was their extreme smallness which might make these bodies be regarded as of an inferior order. Dr. Herschel proposed to distinguish them by the name asteroids. But these bodies revolve round [the] Sun as well as the others; like them their elliptical orbit is but little elongated. They are scarcely smaller when compared to Mercury than that body is in respect to Jupiter. The magnitudes of the planets are subject to no law: they have no relation to their distance from the Sun. Mars is further distant than the Earth yet it is smaller. Jupiter is much greater than Saturn though this latter body is most remote.

            It has been objected that these planets are without the limits of the ancient Zodiac, but the bounds of this Zodiac were fixed principally on account of Venus. They would have been much less if this planet had been unknown. It may therefore be extended at will from one pole to the other, that is to say they are artificial limits of no real utility and may be abandoned altogether. In fact there is no reason why there may not exist in the heavens planets whose orbits cut the ecliptic at right angles as the equator of Venus nearly does and as the satellites of Uranus actually do. We should beware of establishing from partial observations arbitrary laws which future discoveries may oblige us to abrogate.

The discussion is however merely verbal and could not long engage the attention of astronomers. The denomination of planet is now universally applied to designate these newly discovered bodies and also all of a similar nature which may be hereafter found.

            The hypothesis of Dr. Olbers respecting the formation of these planets by the explosion of a former one has been considered with some attention by Lagrange whose numerous investigations have contributed much to the splendid progress of Physical Astronomy. he calculated the force necessary to project a body from our Earth so that it should revolve in a very eccentric orbit and become a comet, and found that if a body could be projected with a velocity about a 120 times as great as that of a cannon ball it would quit this globe and revolve in an elongated ellipse round the Sun. Applying similar principles to the case of other planets he found that if a large planet had existed between Jupiter and Mars and if by some internal cause it should be torn asunder, its parts might form small planets and circulate round the Sun in nearly circular orbits provided that at their first projection they moved with a velocity only about 20 times greater than that of a cannon ball. From these calculations it results not only that the hypothesis is a possible one but also since the power required to produce the effect is not exhorbitant it receives from them a certain degree of probability. It appears that the only method of increasing the evidence on which it rests would be by discovering other similar bodies whose orbits intersect those of the small planets already known nearly in the two points before alluded to. Should this ever be the case it will indeed afford us satisfactory evidence and in fact the only kind which the subject admits.

It has already been observed that these new planets are by far the smallest of any we are acquainted with and which revolve as primaries round the Sun. Of their magnitude different opinions have been entertained. Schroeter of Liebenthal whose observations have acquired deserved reputation estimated their apparent diameter at from 2 to 5 seconds of arc.

            This however differs widely from the opinion of Dr. Herschel who undertook a series of experiments of a curious nature purposely with a view to ascertain the diameter of these objects. He found that their extreme smallness rendered the common methods inapplicable and therefore resorted to others of his own invention. Having heated some sealing wax and drawn it out into small threads he passed the ends of them through the flame of a candle. They consequently had at the end of each thread a small round globule of wax. It was now necessary to measure the diameter of these balls and this was accomplished by means of a solar microscope which projected their images on a sheet of paper and their size was thus ascertained with great accuracy. A row of these waxen balls thus arranged was placed on a card at the distance of 7 or 8 hundred feet and viewed with a telescope. By knowing the distance at which they were placed and their real diameters it was easy to calculate the angles under which they would be seen.

Dr. Herschel examined them attentively with different magnifying powers. For instance with a telescope magnifying 150 [times] he could perceive a globule subtending only an angle of  part of a second in diameter. (It results from this that Ceres is about 161 miles in diameter and Pallas 147 according to greatest extent or 40 times smaller than the Moon.)

            For the discovery of the planetary atoms and of numerous other bodies with which we are acquainted the unassisted eye would be utterly incompetent. It is only by the aid of instruments of a most powerful kind that the observations which I have had occasion to notice can be repeated. They are generally carried on by means of the reflecting telescope. Some account therefore of an instrument which has contributed so much to extend our acquaintance with the magnitude of the Creation may not be uninteresting.

            The first idea of the reflecting telescope was undoubtedly entertained by Mersennus and communicated by him to Descartes. But this philosopher rejected the idea and endeavoured to convince his friend of the impossibility of effecting it. Some years after Gregory a young man of uncommon genius was led to the invention by seeking to correct two imperfections of the common telescope. The first was its too great length which made it unmanageable; the other was the incorrectness of the image it produced. These inconveniences he imagined might be obviated by substituting for the object glass a metallic speculum of a parabolic figure to receive the image and afterwards reflect it to a small speculum of the same metal. This was again to return the image to an eye glass placed behind the great mirror which for this purpose [was] to be perforated in the centre.

This construction was published in a work entitled Optica Promota in 1660, a work which in every respect does credit to the talents of its author. But Gregory was he himself declares possessed of no mechanical dexterity, nor could he find any workman capable of realising his invention. And after some fruitless attempts he was obliged to give up the pursuit and probably had not some new discoveries in light and colours been made a reflecting telescope would never more have been thought of, particularly if we consider the difficulty of execution and the little advantage that would accrue from it according to the principles of optics at that time known.

            But Newton whose happy genius for experimental knowledge was equal to that for geometry and to these talents, in a supreme degree joined patience and mechanical abilities, fortunately interposed and saved this noble invention from perishing in its infant state. He also had employed himself at an early period in his life in endeavouring to improve the telescope but imagining that neither Gregory's specula were neither very necessary nor yet easily to be executed he turned his attention towards improving the common telescope. While he was thus employed about three years after the publication of Gregory's book he made that celebrated discovery of the refrangibility of different rays of light. This convinced him of the great errors of refracting telescopes and forced him as it were to turn his thoughts towards reflectors. In a letter to Mr. Oldenburg he observes, "I understood that by their mediation optical instruments might be brought to any degree of perfection imaginable, provided a reflecting substance could be found which would polish as finely as glass transmits and provided also the art of communicating to it a parabolic figure could be obtained. "Amidst these thoughts" he adds, "I was forced from Cambridge by the intervening plague and it was more than two years before I proceeded further." It was not until the end of 1668 or the beginning of the next year that Newton returned to his studies and not relying on any artificer for making his specula he began the work himself. Early in 1672 he completed two small telescopes. One of these he sent to the Royal Society and in the letter which accompanied it he writes that though he then despaired at attaining the parabolic figure by geometrical rules he doubted not that it might in some measure be accomplished by mechanical devices.

But, though the invention was admirable and the theory perfect, the discoveries, even of Newton, were not exempted from the general fatality which so frequently attends great and useful inventions, that of making a slow and vexatious progress to their authors. The fact is that, excepting an unsuccessful attempt made by the Royal Society by employing an artificer to imitate the Newtonian construction, no reflector was heard of for nearly half a century. But when that period had elapsed, a reflecting telescope was at last produced to the world of the Newtonian construction, which the venerable author, ere yet he had finished his much distinguished course,  had the satisfaction to find executed in such a manner as left no room to fear that the invention would continue longer in obscurity. This memorable event was owing to the genius, dexterity and application of Mr. Hadley, the inventor of the reflecting quadrant.

The two telescopes which Newton had made were 6 inches long and were held in the hand for viewing objects. hey were equal in power to about a 6 foot refractor, whereas Hadley's was above 5 foot long, was provided with a well-contrived apparatus for managing it and was equal in power to the celebrated aerial telescope of Huygens of 123 feet in length. Mr. Hadley very liberally communicated to others the results of his experience in the construction of these instruments. About 1734 Mr. James Short of Edinburgh signalised himself by the construction of some excellent reflecting telescopes. Since that time there have been several improvements in the modes of giving them the parabolic figure and also in the composition of the metal made use of for the speculum, but until Dr. Herschel turned his attention to this subject the highest magnifying power which was usually given to telescopes did not exceed a few hundred times. The reason of this deficiency arose from the difficulty of giving to the mirror the requisite parabolic figure.

In viewing heavenly bodies mere magnifying power is not always most favourable for observation. Dr. Herschel found that with different telescopes of the same magnifying power am object did not appear equally distinct. On enquiring into the cause he found that there was another property possessed by telescopes which is totally different from and sometimes interferes with the magnifying power. To this it is necessary to attend in the construction of telescopes and he designates it the power of penetrating into space. It depends on the magnitude of the polished surface.

            A striking instance of the difference of these two powers occurred on the erection of a 20 foot telescope image. One of its effects was that when towards evening on account of the darkness the natural eye could not penetrate far into space, the telescope possessed that power sufficiently to show, by the dial of a distant church steeple what o'clock it was, notwithstanding the naked eye could no longer see the steeple itself. Here it is clear there was a greater penetrating power; for though it might require magnifying power to see the figure of the dial it could require none to see the steeple.

It is this power which is so necessary in resolving nebula into stars and it is on this account principally that Dr. Herschel's large telescope of 40 ft. in length is so powerful an instrument. The smaller telescopes will bear a magnifying power of several thousand. Indeed some of his 7 ft. reflectors will magnify 5 or 6,000 times, which is nearly a big a power as has been used in the larger ones. Their powers of penetrating into space are however very different. If that of a 7 ft. telescope be described by 20 the power of the 40 ft. will be represented by 190. These powers are both susceptible of increase but not to an unlimited extent. With respect to magnifying power Dr. Herschel thinks a telescope of 25 ft. in length may possibly admit of high a power as the nature of our atmosphere will admit. Many of his observations particularly on double and triple stars were made with powers of 4, 5 or even 6,000. The highest amplificative power which he has yet made use of is 7,200, but it is only at very favourable opportunities that such powers can be used.

The penetrating power of telescopes the same skilful observer thinks can be increased. Yet this has its limits which are perhaps more easily found. The greatest penetrating power which it would perhaps be possible to give it would be about 500. This would be about 3 times that of the 40 foot telescope and the diameter of the polished surface required for this purpose would be 10 ft 6 ins..

            Besides the two already mentioned there are several extraneous causes which modify the action of telescopes. These arise chiefly from the various states of the atmosphere and present some unexpected phenomena. A moist state of the atmosphere is generally favourable to observers even if it is so excessive that the vapour condenses and runs from the tube of the telescope. Wind produces a curious effect. It increases the diameter of stars. They all appear like planets. It is otherwise unfavourable. Clouds produce a contrary effect. If they gradually intervene the stars diminish in size and at last become invisible. It is rather singular that a fog which prevents objects being visible at the distance of 40 feet should yet permit excellent observations to the telescope. Frost is not a hindrance. In some cases Dr. Herschel found his feet frozen to the ground while he was making some very favourable observations. And in January 1783 we find in an extract in his journal, "I made a number of very delicate observations yet at four in the morning my ink was frozen and at 5 the frost was so intense that a speculum in the tube of my 20 ft. telescope we off with a crack and broke into two pieces".

Generally speaking a calm state of the atmosphere undisturbed by any motion in the air is most favourable for telescopes of large power and aperture. These occur but rarely and it is calculated that there are not perhaps more than 100 hours occur in the course of a twelve month which are favourable for observations with Dr. Herschel's 40 ft. reflector.