IV. VERIFICATION AND COMPLETION OF THE NEWTONIAN THEORY…

IV. VERIFICATION AND COMPLETION OF THE NEWTONIAN THEORY

Tables of the Moon and Planets.

THE Newtonian discovery of Universal Gravitation, so remarkable in other respects, is also remarkable as exemplifying the immense extent to which the verification of a great truth may be carried, the amount of human labor which may be requisite to do it justice, and the striking extension of human knowledge to which it may lead. I have said that it is remarked as a beauty in the first fixation of a theory that its measures or elements are established by means of a few {551} data; but that its excellence when established is in the number of observations which it explains. The multiplicity of observations which are explained by astronomy, and which are made because astronomy explains them, is immense, as I have noted in the text. And the multitude of observations thus made is employed for the purpose of correcting the first adopted elements of the theory. I have mentioned some of the examples of this process: I might mention many others in order to continue the history of this part of Astronomy up to the present time. But I will notice only those which seem to me the most remarkable.

In 1812, Burckhardt's Tables de la Lune were published by the French Bureau des Longitudes. A comparison of these and Burg's with a considerable number of observations, gave 9-100ths of a second as the mean error of the former in the Moon's longitude, while the mean error of Burg's was 18-100ths. The preference was therefore accorded to Burckhardt's.

Yet the Lunar Tables were still as much as thirty seconds wrong in single observations. This circumstance, and Laplace's expressed wish, induced the French Academy to offer a prize for a complete and purely theoretical determination of the Lunar path, instead of determinations resting, as hitherto, partly upon theory and partly upon observations. In 1820, two prize essays appeared, the one by Damoiseau, the other by Plana and Carlini. And some years afterwards (in 1824, and again in 1828), Damoiseau published Tables de la Lune formées sur la seule Théorie d'Attraction. These agree very closely with observation. That we may form some notion of the complexity of the problem, I may state that the longitude of the Moon is in these Tables affected by no fewer than forty-seven equations; and the other quantities which determine her place are subject to inequalities not much less in number.

Still I had to state in the second Edition, published in 1847, that there remained an unexplained discordance between theory and observation in the motions of the Moon; an inequality of long period as it seemed, which the theory did not give.

A careful examination of a long series of the best observations of the Moon, compared throughout with the theory in its most perfect form, would afford the means both of correcting the numerical elements of the theory, and of detecting the nature, and perhaps the law, of any still remaining discrepancies. Such a work, however, required vast labor, as well as great skill and profound mathematical knowledge. {552} Mr. Airy undertook the task; employing for that purpose, the Observations of the Moon made at Greenwich from 1750 to 1830. Above 8000 observed places of the Moon were compared with theory by the computation of the same number of places, each separately and independently calculated from Plana's Formulæ. A body of calculators (sometimes sixteen), at the expense of the British Government, was employed for about eight years in this work. When we take this in conjunction with the labor which the observations themselves imply, it may serve to show on what a scale the verification of the Newtonian theory has been conducted. The first results of this labor were published in two quarto volumes; the final deductions as to correction of elements, &c., were given in the Memoirs of the Astronomical Society in 1848.[48\A]

[Note 48\A: The total expense of computers, to the end of reading the proof-sheets, was 4300_l._

Mr. Airy's estimate of days' works [made before beginning], for the heavy part of calculations only, was thirty-six years of one computer. This was somewhat exceeded, but not very greatly, in that part.]

Even while the calculations were going on, it became apparent that there were some differences between the observed places of the Moon, and the theory so far as it had then been developed. M. Hansen, an eminent German mathematician who had devised new and powerful methods for the mathematical determination of the results of the law of gravitation, was thus led to explore still further the motions of the Moon in pursuance of this law. The result was that he found there must exist two lunar inequalities, hitherto not known; the one of 273, and the other of 239 years, the coefficients of which are respectively 27 and 23 seconds. Both these originate in the attraction of Venus; one of them being connected with the long inequality in the Solar Tables, of which Mr. Airy had already proved the existence, as stated in Chap. vi. Sect. 6 of this Book.

These inequalities fell in with the discrepancies between the actual observations and the previously calculated Tables, which Mr. Airy had discovered. And again, shortly afterwards, M. Hansen found that there resulted from the theory two other new equations of the Moon; one in latitude and one in longitude, agreeing with two which were found by Mr. Airy in deducing from the observations the correction of the elements of the Lunar Tables. And again, a little later, there was detected by these mathematicians a theoretical correction for the {553} motion of the Node of the Moon's orbit, coinciding exactly with one which had been found to appear in the observations.

Nothing can more strikingly exhibit the confirmation which increased scrutiny brings to light between the Newtonian theory on the one hand, and the celestial motions on the other. We have here a very large mass of the best observations which have ever been made, systematically examined, with immense labor, and with the set purpose of correcting at once all the elements of the Lunar Tables. The corrections of the elements thus deduced imply of course some error in the theory as previously developed. But at the same time, and with the like determination thoroughly to explore the subject, the theory is again pressed to yield its most complete results, by the invention of new and powerful mathematical methods; and the event is, that residual errors of the old Tables, several in number, following the most diverse laws, occurring in several detached parts, agree with the residual results of the Theory thus newly extracted from it. And thus every additional exactness of scrutiny into the celestial motions on the one hand and the Newtonian theory on the other, has ended, sooner or later, in showing the exactness of their coincidence.

The comparison of the theory with observation in the case of the motions of the Planets, the motion of each being disturbed by the attraction of all the others, is a subject in some respects still more complicated and laborious. This work also was undertaken by the same indefatigable astronomer; and here also his materials belonged to the same period as before; being the admirable observations made at Greenwich from 1750 to 1830, during the time that Bradley, Maskelyne, and Pond were the Astronomers Royal.[49\A] These Planetary observations were deduced, and the observed places were compared with the tabular places: with Lindenau's Tables of Mercury, Venus, and Mars; and with Bouvard's Tables of Jupiter, Saturn, and Uranus; and thus, while the received theory and its elements were confirmed, the means of testing any improvement which may hereafter be proposed, either in the form of the theoretical results or in the constant elements which they involved, was placed within the reach of the {554} astronomers of all future time. The work appeared in 1845; the expense of the compilations and the publication being defrayed by the British Government.

[Note 49\A: The observations of stars made by Bradley, who preceded Maskelyne at Greenwich, had already been discussed by Bessel, a great German astronomer; and the results published in 1818, with a title that well showed the estimation in which he held those materials: Fundamenta Astronomiæ pro anno 1775, deducta ex Ohservationibus viri incomparabilis James Bradley in specula Astronomica Grenovicensi per annos 1750-1762 institutis.]

The Discovery of Neptune.

The theory of gravitation was destined to receive a confirmation more striking than any which could arise from any explanation, however perfect, given by the motions of a known planet; namely, in revealing the existence of an unknown planet, disclosed to astronomers by the attraction which it exerted upon a known one. The story of the discovery of Neptune by the calculations of Mr. Adams and M. Le Verrier was partly told in the former edition of this History. I had there stated (vol. ii. p. 306) that "a deviation of observation from the theory occurs at the very extremity of the solar system, and that its existence appears to be beyond doubt. Uranus does not conform to the Tables calculated for him on the theory of gravitation. In 1821, Bouvard said in the Preface to the Tables of this Planet, "the formation of these Tables offers to us this alternative, that we cannot satisfy modern observations to the requisite degree of precision without making our Tables deviate from the ancient observations." But when we have done this, there is still a discordance between the Tables and the more modern observations, and this discordance goes on increasing. At present the Tables make the Planet come upon the meridian about eight seconds later than he really does. This discrepancy has turned the thoughts of astronomers to the effects which would result from a planet external to Uranus. It appears that the observed motion would be explained by applying a planet at twice the distance of Uranus from the Sun to exercise a disturbing force, and it is found that the present longitude of this disturbing body must be about 325 degrees.

I added, "M. Le Verrier (Comptes Rendus, Jan. 1, 1846) and, as I am informed by the Astronomer Royal, Mr. Adams, of St. John's College, Cambridge, have both arrived independently at this result."

To this Edition I added a Postscript, dated, Nov. 7, 1846, in which I said:

"The planet exterior to Uranus, of which the existence was inferred by M. Le Verrier and Mr. Adams from the motions of Uranus (vol. ii. Note (L.)), has since been discovered. This confirmation of calculations founded upon the doctrine of universal gravitation, may be looked upon as the most remarkable event of the kind since the return of Halley's comet in 1757 and in some respects, as a more striking event {555} even than that; inasmuch as the new planet had never been seen at all, and was discovered by mathematicians entirely by their feeling of its influence, which they perceived through the organ of mathematical calculation.

"There can be no doubt that to M. Le Verrier belongs the glory of having first published a prediction of the place and appearance of the new planet, and of having thus occasioned its discovery by astronomical observers. M. Le Verrier's first prediction was published in the Comptes Rendus de l'Acad. des Sciences, for June 1, 1846 (not Jan. 1, as erroneously printed in my Note). A subsequent paper on the subject was read Aug. 31. The planet was seen by M. Galle, at the Observatory of Berlin, on September 23, on which day he had received an express application from M. Le Verrier, recommending him to endeavor to recognize the stranger by its having a visible disk. Professor Challis, at the Observatory of Cambridge, was looking out for the new planet from July 29, and saw it on August 4, and again on August 12, but without recognizing it, in consequence of his plan of not comparing his observations till he had accumulated a greater number of them. On Sept. 29, having read for the first time M. Le Verrier's second paper, he altered his plan, and paid attention to the physical appearance rather than the position of the star. On that very evening, not having then heard of M. Galle's discovery, he singled out the star by its seeming to have a disk.

"M. Le Verrier's mode of discussing the circumstances of Uranus's motion, and inferring the new planet from these circumstances, is in the highest degree sagacious and masterly. Justice to him cannot require that the contemporaneous, though unpublished, labors of Mr. Adams, of St John's College, Cambridge, should not also be recorded. Mr. Adams made his first calculations to account for the anomalies in the motion of Uranus, on the hypothesis of a more distant planet, in 1843. At first he had not taken into account the earlier Greenwich observations; but these were supplied to him by the Astronomer Royal, in 1844. In September, 1845, Mr. Adams communicated to Professor Challis values of the elements of the supposed disturbing body; namely, its mean distance, mean longitude at a given epoch, longitude of perihelion, eccentricity of orbit, and mass. In the next month, he communicated to the Astronomer Royal values of the same elements, somewhat corrected. The note (L.), vol. ii., of the present work (2d Ed.), in which the names of MM. Le Verrier and Adams are mentioned in conjunction, was in the press in August, 1846, a {556} month before the planet was seen. As I have stated in the text, Mr. Adams and M. Le Verrier assigned to the unseen planet nearly the same position; they also assigned to it nearly the same mass; namely, 2½ times the mass of Uranus. And hence, supposing the density to be not greater than that of Uranus, it followed that the visible diameter would be about 3", an apparent magnitude not much smaller than Uranus himself.

"M. Le Verrier has mentioned for the new planet the name Neptunus; and probably, deference to his authority as its discoverer, will obtain general currency for this name."

Mr. Airy has given a very complete history of the circumstances attending the discovery of Neptune, in the Memoirs of the Astronomical Society (read November 13, 1846). In this he shows that the probability of some disturbing body beyond Uranus had suggested itself to M. A. Bouvard and Mr. Hussey as early as 1834. Mr. Airy himself then thought that the time was not ripe for making out the nature of any external action on the planets. But Mr. Adams soon afterwards proceeded to work at the problem. As early as 1841 (as he himself informs me) he conjectured the existence of a planet exterior to Uranus, and recorded in a memorandum his design of examining its effect; but deferred the calculations till he had completed his preparations for the University examination which he was to undergo in January, 1843, in order to receive the Degree of Bachelor of Arts. He was the Senior Wrangler of that occasion, and soon afterwards proceeded to carry his design into effect; applying to the Astronomer Royal for recorded observations which might aid him in his task. On one of the last days of October, 1845, Mr. Adams went to the Observatory at Greenwich; and finding the Astronomer Royal abroad, he left there a paper containing the elements of the extra-Uranian Planet: the longitude was in this paper stated as 323½ degrees. It was, as we have seen, in June, 1846, that M. Le Verrier's Memoir appeared, in which he assigned to the disturbing body a longitude of 325 degrees. The coincidence was striking. "I cannot sufficiently express," says Mr. Airy, "the feeling of delight and satisfaction which I received from the Memoir of M. Le Verrier." This feeling communicated itself to others. Sir John Herschel said in September, 1846, at a meeting of the British Association at Southampton, "We see it (the probable new planet) as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration." {557}

In truth, at the moment when this was uttered, the new Planet had already been seen by Professor Challis; for, as we have said, he had seen it in the early part of August. He had included it in the net which he had cast among the stars for this very purpose; but employing a slow and cautious process, he had deferred for a time that examination of his capture which would have enabled him to detect the object sought. As soon as he received M. Le Verrier's paper of August 31 on September 29, he was so much impressed with the sagacity and clearness of the limitations of the field of observation there laid down, that he instantly changed his plan of observation, and noted the planet, as an object having a visible disk, on the evening of the same day.

In this manner the theory of gravitation predicted and produced the discovery. Thus to predict unknown facts found afterwards to be true, is, as I have said, a confirmation of a theory which in impressiveness and value goes beyond any explanation of known facts. It is a confirmation which has only occurred a few times in the history of science; and in the case only of the most refined and complete theories, such as those of Astronomy and Optics. The mathematical skill which was requisite in order to arrive at such a discovery, may in some measure be judged of by the account which we have had to give of the previous mathematical progress of the theory of gravitation. It there appeared that the lives of many of the most acute, clear-sighted, and laborious of mankind, had been employed for generations in solving the problem. Given the planetary bodies, to find their mutual perturbations: but here we have the inverse problem--Given the perturbations, to find the planets.[50\A]

[Note 50\A: This may be called the inverse problem with reference to the older and more familiar problem; but we may remark that the usual phraseology of the Problem of Central Forces differs from this analogy. In Newton's Principia, the earlier Sections, in which the motion is given to find the force, are spoken of as containing the Direct Problem of Central Forces: the Eighth Section of the First Book, where the Force is given to find the orbit, is spoken of as containing the Inverse Problem of Central Forces.]

The Minor Planets.

The discovery of the Minor Planets which revolve between the orbits of Mars and Jupiter was not a consequence or confirmation of the Newtonian theory. That theory gives no reason for the distance of {558} the Planets from the Sun; nor does any theory yet devised give such reason. But an empirical formula proposed by the Astronomer Bode of Berlin, gives a law of these distances (Bode's Law), which, to make it coherent, requires a planet between Mars and Jupiter. With such an addition, the distance of Mercury, Venus, Earth, Mars, the Missing Planet, Jupiter, Saturn, and Uranus, are nearly as the numbers 4, 7, 10, 16, 28, 52, 100, 196, in which the excesses of each number above the preceding are the series 3, 3, 6, 12, 24, 48, 96. On the strength of this law the Germans wrote on the long-expected Planet, and formed themselves into associations for the discovery of it.

Not only did this law stimulate the inquiries for the Missing Planet, and thus lead to the discovery of the Minor Planets, but it had also a share in the discovery of Neptune. According to the law, a planet beyond Uranus may be expected to be at the distance represented by 388. Mr. Adams and M. Le Verrier both of them began by assuming a distance of nearly this magnitude for the Planet which they sought; that is, a distance more than 38 times the earth's distance. It was found afterwards that the distance of Neptune is only 30 times that of the earth; yet the assumption was of essential use in obtaining the result and Mr. Airy remarks that the history of the discovery shows the importance of using any received theory as far as it will go, even if the theory can claim no higher merit than that of being plausible.[51\A]

[Note 51\A: Account of the Discovery of Neptune, &c., Mem. Ast. Soc., vol. xvi. p. 414.]

The discovery of Minor Planets in a certain region of the interval between Mars and Jupiter has gone on to such an extent, that their number makes them assume in a peculiar manner the character of representatives of a Missing Planet. At first, as I have said in the text, it was supposed that all these portions must pass through or near a common node; this opinion being founded on the very bold doctrine, that the portions must at one time have been united in one Planet, and must then have separated. At this node, as I have stated, Olbers lay in wait for them, as for a hostile army at a defile. Ceres, Pallas, and Juno had been discovered in this way in the period from 1801 to 1804; and Vesta was caught in 1807. For a time the chase for new planets in this region seemed to have exhausted the stock. But after thirty-eight years, to the astonishment of astronomers, they began to be again detected in extraordinary numbers. In 1845, M. Hencke of {559} Driessen discovered a fifth of these planets, which was termed Astræa. In various quarters the chase was resumed with great ardor. In 1847 were found Hebe, Iris, and Flora; in 1848, Metis; in 1849, Hygæa; in 1850, Parthenope, Victoria, and Egeria; in 1861, Irene and Eunomia; in 1852, Psyche, Thetis, Melpomene, Fortuna, Massilia, Lutetia, Calliope. To these we have now (at the close of 1856) to add nineteen others; making up the whole number of these Minor Planets at present known to forty-two.

As their enumeration will show, the ancient practice has been continued of giving to the Planets mythological names. And for a time, till the numbers became too great, each of the Minor Planets was designated in astronomical books by some symbol appropriate to the character of the mythological person; as from ancient times Mars has been denoted by a mark indicating a spear, and Venus by one representing a looking-glass. Thus, when a Minor Planet was discovered at London in 1851, the year in which the peace of the world was, in a manner, celebrated by the Great Exhibition of the Products of All Nations, held at that metropolis, the name Irene was given to the new star, as a memorial of the auspicious time of its discovery. And it was agreed, for awhile, that its symbol should be a dove with an olive-branch. But the vast multitude of the Minor Planets, as discovery went on, made any mode of designation, except a numerical one, practically inconvenient. They are now denoted by a small circle inclosing a figure in the order of their discovery. Thus, Ceres is (1), Irene is (14), and Isis is (42).

The rapidity with which these discoveries were made was owing in part to the formation of star-maps, in which all known fixed stars being represented, the existence of a new and movable star might be recognized by comparison of the sky with the map. These maps were first constructed by astronomers of different countries at the suggestion of the Academy of Berlin; but they have since been greatly extended, and now include much smaller stars than were originally laid down.

I will mention the number of planets discovered in each year. After the start was once made, by Hencke's discovery of Astræa in 1845, the same astronomer discovered Hebe in 1847; and in the same year Mr. Hind, of London, discovered two others, Iris and Flora. The years 1848 and 1849 each supplied one; the year 1850, three; 1851, two; 1852 was marked by the extraordinary discovery of eight new members of the planetary system. The year 1853 supplied four; 1854, six; 1855, four; and 1856 has already given us five. {560}

These discoveries have been distributed among the observatories of Europe. The bright sky of Naples has revealed seven new planets to the telescope of Signer Gasparis. Marseilles has given us one; Germany, four, discovered by M. Luther at Bilk; Paris has furnished seven; and Mr. Hind, in Mr. Bishop's private observatory in London, notwithstanding our turbid skies, has discovered no less than ten planets; and there also Mr. Marth discovered (29) Amphitrite. Mr. Graham, at the private observatory of Mr. Cooper, in Ireland, discovered (9) Metis.

America has supplied its planet, namely (31) Euphrosyne, discovered by Mr. Ferguson at Washington and the most recent of these discoveries is that by Mr. Pogson, of Oxford, who has found the forty-second of these Minor Planets, which has been named Isis.[52\A]

[Note 52\A: I take this list from a Memoir of M. Bruhns, Berlin, 1856.]

I may add that it appears to follow from the best calculations that the total mass of all these bodies is very small. Herschel reckoned the diameters of Ceres at 35, and of Pallas at 26 miles. It has since been calculated[53\A] that some of them are smaller still; Victoria having a diameter of 9 miles, Lutetia of 8, and Atalanta of little more than 4. It follows from this that the whole mass would probably be less than the sixth part of our moon. Hence their perturbing effects on each other or on other planets are null; but they are not the less disturbed by the action of the other planets, and especially of Jupiter.

[Note 53\A: Bruhns, as above.]

Anomalies in the Action of Gravitation.

The complete and exact manner in which the doctrine of gravitation explains the motions of the Comets as well as of the Planets, has made astronomers very bold in proposing hypotheses to account for any deviations from the motion which the theory requires. Thus Encke's Comet is found to have its motion accelerated by about one-eighth of a day in every revolution. This result was conceived to be established by former observations, and is confirmed by the facts of the appearance of 1852.[54\A] The hypothesis which is proposed in order to explain this result is, that the Comet moves in a resisting medium, which makes it fall inwards from its path, towards the Sun, and thus, by narrowing its orbit, diminishes its periodic time. On the other hand, M. Le Verrier has found that Mercury's mean motion has gone on diminishing; {561} as if the planet were, in the progress of his revolutions, receding further from the Sun. This is explained, if we suppose that there is, in the region of Mercury, a resisting medium which moves round the Sun in the same direction as the Planets move. Evidence of a kind of nebulous disk surrounding the Sun, and extending beyond the orbits of Mercury and Venus, appears to be afforded us by the phenomenon called the Zodiacal Light; and as the Sun itself rotates on its axis, it is most probable that this kind of atmosphere rotates also.[55\A] On the other hand, M. Le Verrier conceives that the Comets which now revolve within the ordinary planetary limits have not always done so, but have been caught and detained by the Planets among which they move. In this way the action of Jupiter has brought the Comets of Faye and Vico into their present limited orbits, as it drew the Comet of Lexell out of its known orbit, when the Comet passed over the Planet in 1779, since which time it has not been seen.

[Note 54\A: Berlin Memoirs, 1854.]

[Note 55\A: M. Le Verrier, Annales de l'Obs. de Paris, vol. i. p. 89.]

Among the examples of the boldness with which astronomers assume the doctrine of gravitation even beyond the limits of the solar system to be so entirely established, that hypotheses may and must be assumed to explain any apparent irregularity of motion, we may reckon the mode of accounting for certain supposed irregularities in the proper motion of Sirius, which has been proposed by Bessel, and which M. Peters thinks is proved to be true by his recent researches (Astr. Nach. xxxi. p. 219, and xxxii. p. 1). The hypothesis is, that Sirius has a companion star, dark, and therefore invisible to us; and that the two, revolving round their common centre as the system moves on, the motion of Sirius is seen to be sometimes quicker and sometimes slower.

The Earth's Density.

"Cavendish's experiment," as it is commonly called--the measure of the attractions of manageable masses by the torsion balance, in order to determine the density of the Earth--has been repeated recently by Professor Reich at Freiberg, and by Mr. Baily in England, with great attention to the means of attaining accuracy. Professor Reich's result for the density of the Earth is 5·44; Mr. Baily's is 5·92. Cavendish's result was 5·48; according to recent revisions[56\A] it is 5·52. {562}

[Note 56\A: The calculation has been revised by M. Edward Schmidt. Humboldt's Kosmos, ii. p. 425.]

But the statical effect of the attraction of manageable masses, or even of mountains, is very small. The effect of a small change in gravity may be accumulated by being constantly repeated in the oscillations of a pendulum, and thus may become perceptible. Mr. Airy attempted to determine the density of the Earth by a method depending on this view. A pendulum oscillating at the surface was to be compared with an equal pendulum at a great depth below the surface. The difference of their rates would disclose the different force of gravity at the two positions; and hence, the density of the Earth. In 1826 and 1828, Mr. Airy attempted this experiment at the copper mine of Dolcoath in Cornwall, but failed from various causes. But in 1854, he resumed it at the Harton coal mine in Durham, the depth of which is 1260 feet; having in this new trial, the advantage of transmitting the time from one station to the other by the instantaneous effect of galvanism, instead of by portable watches. The result was a density of 6·56; which is much larger than the preceding results, but, as Mr. Airy holds, is entitled to compete with the others on at least equal terms.

Tides.

I should be wanting in the expression of gratitude to those who have practically assisted me in Researches on the Tides, if I did not mention the grand series of Tide Observations made on the coast of Europe and America in June, 1835, through the authority of the Board of Admiralty, and the interposition of the late Duke of Wellington, at that time Foreign Secretary. Tide observations were made for a fortnight at all the Coast-guard stations of Great Britain and Ireland in June, 1834; and these were repeated in June, 1835, with corresponding observations on all the coasts of Europe, from the North Cape of Norway to the Straits of Gibraltar; and from the mouth of the St. Lawrence to the mouth of the Mississippi. The results of these observations, which were very complete so far as the coast tides were concerned, were given in the Philosophical Transactions for 1836.

Additional accuracy respecting the Tides of the North American coast may be expected from the survey now going on under the direction of Superintendent A. Bache. The Tides of the English Channel have been further investigated, and the phenomena presented under a new point of view by Admiral Beechey. {563}

The Tides of the Coast of Ireland have been examined with great care by Mr. Airy. Numerous and careful observations were made with a view, in the first instance, of determining what was to be regarded as "the Level of the Sea;" but the results were discussed so as to bring into view the laws and progress, on the Irish coast, of the various inequalities of the Tides mentioned in Chap. iv. Sect. 9 of this Book.

I may notice as one of the curious results of the Tide Observations of 1836, that it appeared to me, from a comparison of the Observations, that there must be a point in the German Ocean, about midway between Lowestoft on the English coast, and the Brill on the Dutch coast, where the tide would vanish: and this was ascertained to be the case by observation; the observations being made by Captain Hewett, then employed in a survey of that sea.

Cotidal Lines supply, as I conceive, a good and simple method of representing the progress and connection of littoral tides. But to draw cotidal lines across oceans, is a very precarious mode of representing the facts, except we had much more knowledge on the subject than we at present possess. In the Phil. Trans. for 1848, I have resumed the subject of the Tides of the Pacific; and I have there expressed my opinion, that while the littoral tides are produced by progressive waves, the oceanic tides are more of the nature of stationary undulations.

But many points of this kind might be decided, and our knowledge on this subject might be brought to a condition of completeness, if a ship or ships were sent expressly to follow the phenomena of the Tides from point to point, as the observations themselves might suggest a course. Till this is done, our knowledge cannot be completed. Detached and casual observations, made aliud agendo, can never carry us much beyond the point where we at present are.

Double Stars.

Sir John Herschel's work, referred to in the History (2d Ed.) as then about to appear, was published in 1847.[57\A] In this work, besides a vast amount of valuable observations and reasonings on other subjects {564} (as Nebulæ, the Magnitude of Stars, and the like), the orbits of several double stars are computed by the aid of the new observations. But Sir John Herschel's conviction on the point in question, the operation of the Newtonian law of gravitation in the region of the stars, is expressed perhaps more clearly in another work which he published in 1849.[58\A] He there speaks of Double Stars, and especially of gamma Virginis, the one which has been most assiduously watched, and has offered phenomena of the greatest interest.[59\A] He then finds that the two components of this star revolve round each other in a period of 182 years; and says that the elements of the calculated orbit represent the whole series of recorded observations, comprising an angular movement of nearly nine-tenths of a complete circuit, both in angle and distance, with a degree of exactness fully equal to that of observation itself. "No doubt can therefore," he adds, "remain as to the **prevalence in this remote system of the Newtonian Law of Gravitation."

[Note 57\A: Results of Astronomical Observations made during the years 1834, 5, 6, 7, 8, at the Cape of Good Hope, being the completion of a Telescopic Survey of the whole Surface of the visible Heavens commenced in 1825.]

[Note 58\A: Outlines of Astronomy.]

[Note 59\A: Out. 844.]

Yet M. Yvon de Villarceau has endeavored to show[60\A] that this conclusion, however probable, is not yet proved. He holds, even for the Double Stars, which have been most observed, the observations are only equivalent to seven or eight really distinct data, and that seven data are not sufficient to determine that an ellipse is described according to the Newtonian law. Without going into the details of this reasoning, I may remark, that the more rapid relative angular motion of the components of a Double Star when they are more near each other, proves, as is allowed on all hands, that they revolve under the influence of a mutual attractive force, obeying the Keplerian Law of Areas. But that, whether this force follows the law of the inverse square or some other law, can hardly have been rigorously proved as yet, we may easily conceive, when we recollect the manner in which that law was proved for the Solar System. It was by means of an error of eight minutes, observed by Tycho, that Kepler was enabled, as he justly boasted, to reform the scheme of the Solar System,--to show, that is, that the planetary orbits are ellipses with the sun in the focus. Now, the observations of Double Stars cannot pretend to such accuracy as this; and therefore the Keplerian theorem cannot, as yet, have been fully demonstrated from those observations. But when we know {565} that Double Stars are held together by a central force, to prove that this force follows a different law from the only law which has hitherto been found to obtain in the universe, and which obtains between all the known masses of the universe, would require very clear and distinct evidence, of which astronomers have as yet seen no trace.

[Note 60\A: Connaissance des Temps, for 1852; published in 1849.]

Chapter VI

Sect. 1. Instruments.--2. Clocks.

IN page 473, I have described the manner in which astronomers are able to observe the transit of a star, and other astronomical phenomena, to the exactness of a tenth of a second of time. The mode of observation there described implies that the observer at the moment of observation compares the impressions of the eye and of the ear. Now it is found that the habit which the observer must form of doing this operates differently in different observers, so that one observer notes the same fact as happening a fraction of a second earlier or later than another observer does; and this in every case. Thus, using the term equation, as we use it in Astronomy, to express a correction by which we get regularity from irregularity, there is a personal equation belonging to this mode of observation, showing that it is liable to error. Can this error be got rid of?

It is at any rate much diminished by a method of observation recently introduced into observatories, and first practised in America. The essential feature of this mode of observation consists in combining the impression of sight with that of touch, instead of with that of hearing. The observer at the moment of observation presses with his finger so as to make a mark on a machine which by its motion measures time with great accuracy and on a large scale; and thus small intervals of time are made visible.

A universal, though not a necessary, part of this machinery, as hitherto adopted, is, that a galvanic circuit has been employed in conveying the impression from the finger to the part where time is measured and marked. The facility with which galvanic wires can {566} thus lead the impression by any path to any distance, and increase its force in any degree, has led to this combination, and almost identification, of observation by touch with its record by galvanism.

The method having been first used by Mr. Bond at Cambridge, in North America, has been adopted elsewhere, and especially at Greenwich, where it is used for all the instruments; and consequently a collection of galvanic batteries is thus as necessary a part of the apparatus of the establishment as its graduated circles and arcs.

END OF VOL. I.

HISTORY OF THE INDUCTIVE SCIENCES.

VOLUME II.

HISTORY OF THE INDUCTIVE SCIENCES, FROM THE EARLIEST TO THE PRESENT TIME. BY WILLIAM WHEWELL, D. D., MASTER OF TRINITY COLLEGE, CAMBRIDGE. THE THIRD EDITION, WITH ADDITIONS. IN TWO VOLUMES.

VOLUME II.

NEW YORK: D. APPLETON AND COMPANY, 549 & 551 BROADWAY. 1875.

CONTENTS

OF THE SECOND VOLUME.

THE SECONDARY MECHANICAL SCIENCES.

VIII. HISTORY OF ACOUSTICS

PAGE Introduction. 23

IX. HISTORY OF OPTICS, FORMAL AND PHYSICAL

Introduction. 51 {8}

FORMAL OPTICS.

IX. DISCOVERY OF THE LAWS OF PHENOMENA OF DIPOLARIZED LIGHT. 80

PHYSICAL OPTICS.

XI. EPOCH OF YOUNG AND FRESNEL

Sect. 1. Introduction. 92 Sect. 2. Explanation of the Periodical Colors of Thin Plates and Shadows by the Undulatory Theory. 93 Sect. 3. Explanation of Double Refraction by the Undulatory Theory. 98 Sect. 4. Explanation of Polarization by the Undulatory Theory. 100 Sect. 5. Explanation of Dipolarization by the Undulatory Theory. 105 {9}

XIII. CONFIRMATION AND EXTENSION OF THE UNDULATORY THEORY. 118

1. Double Refraction of Compressed Glass. 119 2. Circular Polarization. 119 3. Elliptical Polarization in Quartz. 122 4. Differential Equations of Elliptical Polarization. 122 5. Elliptical Polarization of Metals. 123 6. Newton's Rings by Polarized Light. 124 7. Conical Refraction. 124 8. Fringes of Shadows. 126 9. Objections to the Theory. 126 10. Dispersion, on the Undulatory Theory. 128 11. Conclusion. 128

X. HISTORY OF THERMOTICS AND ATMOLOGY

Introduction. 137

THERMOTICS PROPER.

I. THE DOCTRINES OF CONDUCTION AND RADIATION

Sect. 1. Introduction of the Doctrine of Conduction. 139 Sect. 2. " " " Radiation. 142 Sect. 3. Verification of the Doctrines of Conduction and Radiation. 143 Sect. 4. The Geological and Cosmological Application of Thermotics. 144 1. Effect of Solar Heat on the Earth. 145 2. Climate. 146 3. Temperature of the Interior of the Earth. 147 4. Heat of the Planetary Spaces. 148 Sect. 5. Correction of Newton's Law of Cooling. 149 Sect. 6. Other Laws of Phenomena with respect to Radiation. 151 Sect. 7. Fourier's Theory of Radiant Heat. 152 Sect. 8. Discovery of the Polarization of Heat. 153 {10}

II. THE LAWS OF CHANGES OCCASIONED BY HEAT

Sect. 1. Expansion by Heat.--The Law of Dalton and Gay-Lussac for Gases. 157 Sect. 2. Specific Heat.--Change of Consistence. 159 Sect. 3. The Doctrine of Latent Heat. 160

ATMOLOGY.

III. THE RELATION OF VAPOR AND AIR

Sect. 1. The Boylean Law of the Air's Elasticity. 163 Sect. 2. Prelude to Dalton's Doctrine of Evaporation. 165 Sect. 3. Dalton's Doctrine of Evaporation. 170 Sect. 4. Determination of the Laws of the Elastic Force of Steam. 172 Sect. 5. Consequences of the Doctrine of Evaporation.--Explanation of Rain, Dew, and Clouds. 176

IV. PHYSICAL THEORIES OF HEAT

Thermotical Theories. 181 Atmological Theories. 184 Conclusion. 187

THE MECHANICO-CHEMICAL SCIENCES.

XI. HISTORY OF ELECTRICITY

Introduction. 191

II. THE PROGRESS OF ELECTRICAL THEORY. 201

Question of One or Two Fluids. 210 Question of the Material Reality of the Electric Fluid. 212 {11}

II. PROGRESS OF MAGNETIC THEORY

Theory of Magnetic Action. 220 Theory of Terrestrial Magnetism. 224 Conclusion. 232

II. RECEPTION AND CONFIRMATION OF THE DISCOVERY OF VOLTAIC ELECTRICITY. 240

CHAPTER III.--DISCOVERY OF THE LAWS OF THE MUTUAL ATTRACTION AND REPULSION OF VOLTAIC CURRENTS.--AMPÈRE. 242

VI. THEORY OF ELECTRODYNAMICAL ACTION

Ampère's Theory. 246 Reception of Ampère's Theory. 249

VII. CONSEQUENCES OF THE ELECTRODYNAMIC THEORY. 250

Discovery of Diamagnetism. 252 {12}

IX. TRANSITION TO CHEMICAL SCIENCE. 256

THE ANALYTICAL SCIENCE.

XIV. HISTORY OF CHEMISTRY

CHAPTER I.--IMPROVEMENT OF THE NOTION OF CHEMICAL ANALYSIS, AND RECOGNITION OF IT AS THE SPAGIRIC ART. 261

IV. DOCTRINE OF ACIDIFICATION AND COMBUSTION.--PHLOGISTIC THEORY

Publication of the Theory by Beccher and Stahl. 267 Reception and Application of the Theory. 271

VI. EPOCH OF THE THEORY OF OXYGEN.--LAVOISIER

Sect. 1. Prelude to the Theory.--Its Publication. 275 Sect. 2. Reception and Confirmation of the Theory of Oxygen. 278 Sect. 3. Nomenclature of the Oxygen Theory. 281

VIII. THEORY OF DEFINITE, RECIPROCAL, AND MULTIPLE PROPORTIONS

Sect. 1. Prelude to the Atomic Theory, and its Publication by Dalton. 285 Sect. 2. Reception and Confirmation of the Atomic Theory. 288 Sect. 3. The Theory of Volumes.--Gay-Lussac. 290

IX. EPOCH OF DAVY AND FARADAY

Sect. 1. Promulgation of the Electro-chemical Theory by Davy. 291 Sect. 2. Establishment of the Electro-chemical Theory by Faraday. 296 Sect. 3. Consequences of Faraday's Discoveries. 302 Sect. 4. Reception of the Electro-chemical Theory. 303

X. TRANSITION FROM THE CHEMICAL TO THE CLASSIFICATORY SCIENCES. 305

THE ANALYTICO-CLASSIFICATORY SCIENCE.

Introduction

Sect. 1. Of the Classificatory Sciences. 313 Sect. 2. Of Mineralogy as the Analytico-classificatory Science. 314

CRYSTALLOGRAPHY.

I. PRELUDE TO THE EPOCH OF DE LISLE AND HAÜY. 316

CHAPTER II.--EPOCH OF ROMÉ DE LISLE AND HAÜY.--ESTABLISHMENT OF THE FIXITY OF CRYSTALLINE ANGLES, AND THE SIMPLICITY OF THE LAWS OF DERIVATION. 320

V. RECEPTION AND CONFIRMATION OF THE DISTINCTION OF SYSTEMS OF CRYSTALLIZATION

Diffusion of the Distinction of Systems. 330 Confirmation of the Distinction of Systems by the Optical Properties of Minerals.--Brewster. 331

VI. CORRECTION OF THE LAW OF THE SAME ANGLE FOR THE SAME SUBSTANCE

Discovery of Isomorphism.--Mitscherlich. 334 Dimorphism. 336

VII. ATTEMPTS TO ESTABLISH THE FIXITY OF OTHER PHYSICAL PROPERTIES.--WERNER. 336

SYSTEMATIC MINERALOGY.

VIII. ATTEMPTS AT THE CLASSIFICATION OF MINERALS

Sect. 1. Proper Object of Classification. 339 Sect. 2. Mixed Systems of Classification. 340

CHAPTER IX.--ATTEMPTS AT THE REFORM OF MINERALOGICAL SYSTEMS.--SEPARATION OF THE CHEMICAL AND NATURAL HISTORY METHODS.

Sect. 1. Natural History System of Mohs. 344 Sect. 2. Chemical System of Berzelius and others. 347 Sect. 3. Failure of the Attempts at Systematic Reform. 349 Sect. 4. Return to Mixed Systems with Improvements. 351

CLASSIFICATORY SCIENCES.

XVI. HISTORY OF SYSTEMATIC BOTANY AND ZOOLOGY

Introduction. 357 {15}

III. FORMATION OF A SYSTEM OF ARRANGEMENT OF PLANTS

Sect. 1. Prelude to the Epoch of Cæsalpinus. 369 Sect. 2. Epoch of Cæsalpinus.--Formation of a System of Arrangement. 373 Sect. 3. Stationary Interval. 378 Sect. 4. Sequel to the Epoch of Cæsalpinus.--Further Formation and Adoption of Systematic Arrangement. 382

IV. THE REFORM OF LINNÆUS

Sect. 1. Introduction of the Reform. 387 Sect. 2. Linnæan Reform of Botanical Terminology. 389 Sect. 3. " " " Nomenclature. 391 Sect. 4. Linnæus's Artificial System, 395 Sect. 5. Linnæus's Views on a Natural Method. 396 Sect. 6. Reception and Diffusion of the Linnæan Reform. 400

VII. THE PROGRESS OF ICHTHYOLOGY. 419

Period of Unsystematic Knowledge. 420 Period of Erudition. 421 Period of Accumulation of Materials.--Exotic Collections. 422 Epoch of the Fixation of Characters.--Ray and Willoughby. 422 Improvement of the System.--Artedi. 423 Separation of the Artificial and Natural Methods in Ichthyology. 426

ORGANICAL SCIENCES.

XVII. HISTORY OF PHYSIOLOGY AND COMPARATIVE ANATOMY

Introduction. 435 {16}

I. DISCOVERY OF THE ORGANS OF VOLUNTARY MOTION

Sect. 1. Knowledge of Galen and his Predecessors. 438 Sect. 2. Recognition of Final Causes in Physiology.--Galen. 442

II. DISCOVERY OF THE CIRCULATION OF THE BLOOD

Sect. 1. Prelude to the Discovery. 444 Sect. 2. The Discovery of the Circulation made by Harvey. 447 Sect. 3. Reception of the Discovery. 448 Sect. 4. Bearing of the Discovery on the Progress of Physiology. 449

III. DISCOVERY OF THE MOTION OF THE CHYLE, AND CONSEQUENT SPECULATIONS

Sect. 1. The Discovery of the Motion of the Chyle. 452 Sect. 2. The Consequent Speculations. Hypotheses of Digestion. 453

IV. EXAMINATION OF THE PROCESS OF REPRODUCTION IN ANIMALS AND PLANTS, AND CONSEQUENT SPECULATIONS

Sect. 1. The Examination of the Process of Reproduction in Animals. 455 Sect. 2. " " " " in Vegetables. 457 Sect. 3. The Consequent Speculations.--Hypotheses of Generation. 459

V. EXAMINATION OF THE NERVOUS SYSTEM, AND CONSEQUENT SPECULATIONS

Sect. 1. The Examination of the Nervous System. 461 Sect. 2. The Consequent Speculations. Hypotheses respecting Life, Sensation, and Volition. 464

VI. INTRODUCTION OF THE PRINCIPLE OF DEVELOPED AND METAMORPHOSED SYMMETRY

Sect. 1. Vegetable Morphology.--Göthe. De Candolle. 468 Sect. 2. Application of Vegetable Morphology. 474

VII. PROGRESS OF ANIMAL MORPHOLOGY

Sect. 1. Rise of Comparative Anatomy. 475 Sect. 2. Distinction of the General Types of the Forms of Animals.--Cuvier. 478 Sect. 3. Attempts to establish the Identity of the Types of Animal Forms. 480 {17}

VIII. THE DOCTRINE OF FINAL CAUSES IN PHYSIOLOGY

Sect. 1. Assertion of the Principle of Unity of Plan. 482 Sect. 2. Estimate of the Doctrine of Unity of Plan. 487 Sect. 3. Establishment and Application of the Principle of the Conditions of Existence of Animals.--Cuvier. 492

THE PALÆTIOLOGICAL SCIENCES.

XVIII. HISTORY OF GEOLOGY

Introduction. 499

DESCRIPTIVE GEOLOGY.

I. PRELUDE TO SYSTEMATIC DESCRIPTIVE GEOLOGY

Sect. 1. Ancient Notices of Geological Facts. 505 Sect. 2. Early Descriptions and Collections of Fossils. 506 Sect. 3. First Construction of Geological Maps. 509

II. FORMATION OF SYSTEMATIC DESCRIPTIVE GEOLOGY

Sect. 1. Discovery of the Order and Stratification of the Materials of the Earth. 511 Sect. 2. Systematic Form given to Descriptive Geology.--Werner. 513 Sect. 3. Application of Organic Remains as a Geological Character.--Smith. 515 Sect. 4. Advances in Palæontology.--Cuvier. 517 Sect. 5. Intellectual Characters of the Founders of Systematic Descriptive Geology. 520

III. SEQUEL TO THE FORMATION OF SYSTEMATIC DESCRIPTIVE GEOLOGY

Sect. 1. Reception and Diffusion of Systematic Geology. 523 Sect. 2. Application of Systematic Geology.--Geological Surveys and Maps. 526 Sect. 3. Geological Nomenclature. 527 Sect. 4. Geological Synonymy, or Determination of Geological Equivalents. 531 {18}

IV. ATTEMPTS TO DISCOVER GENERAL LAWS IN GEOLOGY

Sect. 1. General Geological Phenomena. 537 Sect. 2. Transition to Geological Dynamics. 541

GEOLOGICAL DYNAMICS.

V. INORGANIC GEOLOGICAL DYNAMICS

Sect. 1. Necessity and Object of a Science of Geological Dynamics. 542 Sect. 2. Aqueous Causes of Change. 545 Sect. 3. Igneous Causes of Change.--Motions of the Earth's Surface. 549 Sect. 4. The Doctrine of Central Heat. 554 Sect. 5. Problems respecting Elevations and Crystalline Forces. 556 Sect. 6. Theories of Changes of Climate. 559

VI. PROGRESS OF THE GEOLOGICAL DYNAMICS OF ORGANIZED BEINGS

Sect. 1. Objects of this Science. 561 Sect. 2. Geography of Plants and Animals. 562 Sect. 3. Questions of the Transmutation of Species. 563 Sect. 4. Hypothesis of Progressive Tendencies. 565 Sect. 5. Question of Creation as related to Science. 568 Sect. 6. The Hypothesis of the Regular Creation and Extinction of Species. 573 1. Creation of Species. 573 2. Extinction of Species. 576 Sect. 7. The Imbedding of Organic Remains. 577

PHYSICAL GEOLOGY.

VII. PROGRESS OF PHYSICAL GEOLOGY

Sect. 1. Object and Distinctions of Physical Geology. 579 Sect. 2. Of Fanciful Geological Opinions. 580 Sect. 3. Of Premature Geological Theories. 584

VIII. THE TWO ANTAGONIST DOCTRINES OF GEOLOGY

Sect. 1. Of the Doctrine of Geological Catastrophes. 586 Sect. 2. " " " Uniformity. 588 {19}

ADDITIONS TO THE THIRD EDITION.

VIII. ACOUSTICS

SOUND.

The Velocity of Sound in Water. 599

IX. OPTICS

Photography. 601 Fluorescence. 601

UNDULATORY THEORY.

Direction of the Transverse Vibrations in Polarization. 603 Final Disproof of the Emission Theory. 604

X. THERMOTICS.--ATMOLOGY

THE RELATION OF VAPOR AND AIR.

Force of Steam. 606 Temperature of the Atmosphere. 607

THEORIES OF HEAT.

The Dynamical Theory of Heat. 608

XI. ELECTRICITY

General Remarks. 610 Dr. Faraday's Views of Statical Electrical Induction. 611

XII. MAGNETISM

Recent Progress of Terrestrial Magnetism. 613 Correction of Ships' Compasses. 616 {20}

XIII. VOLTAIC ELECTRICITY

MAGNETO-ELECTRIC INDUCTION.

Diamagnetlc Polarity. 620 Magneto-optic Effects and Magnecrystallic Polarity. 621 Magneto-electric Machines. 623 Applications of Electrodynamic Discoveries. 623

XIV. CHEMISTRY

THE ELECTRO-CHEMICAL THEORY.

The Number of Elementary Substances. 625

XV. MINERALOGY

Crystallography. 627 Optical Properties of Minerals. 629 Classification of Minerals. 630

XVI. CLASSIFICATORY SCIENCES

Recent Views of Botany. 631 " " Zoology. 634

XVII. PHYSIOLOGICAL AND COMPARATIVE ANATOMY

VEGETABLE MORPHOLOGY. 636 ANIMAL MORPHOLOGY. 638 Final Causes. 642

Book XVIII

GEOLOGY. 646

{{21}} BOOK VIII.

THE SECONDARY MECHANICAL SCIENCES.

HISTORY OF ACOUSTICS.

. . . . . . Go, demand Of mighty Nature, if 'twas ever meant That we should pry far off and be unraised, That we should pore, and dwindle as we pore, Viewing all objects unremittingly In disconnexion dead and spiritless; And still dividing, and dividing still, Break down all grandeur, still unsatisfied With the perverse attempt, while littleness May yet become more little; waging thus An impious warfare 'gainst the very life Of our own souls. WORDSWORTH, Excursion.

. . . . . . Ἐσσυμένη δὲ Ἠερίην ἀψῖδα διεῤῥοίζησε πεδίλῳ Εἰς δόμον ἉΡΜΟΝIΗΣ παμμητόρος, ὁππόθι νύμφη Ἴκελον οἶκον ἐναίε τύπῳ τετράζυγι κόσμου Αὐτοπαγῆ NONNUS. Dionysiac. xli. 275.

Along the skiey arch the goddess trode, And sought Harmonia's august abode; The universal plan, the mystic Four, Defines the figure of the palace-floor. Solid and square the ancient fabric stands, Raised by the labors of unnumbered hands.

{{23}} BOOK VIII.

Introduction

The Secondary Mechanical Sciences.

IN the sciences of Mechanics and Physical Astronomy, Motion and Force are the direct and primary objects of our attention. But there is another class of sciences in which we endeavor to reduce phenomena, not evidently mechanical, to a known dependence upon mechanical properties and laws. In the cases to which I refer, the facts do not present themselves to the senses as modifications of position and motion, but as secondary qualities, which are found to be in some way derived from those primary attributes. Also, in these cases the phenomena are reduced to their mechanical laws and causes in a secondary manner; namely, by treating them as the operation of a medium interposed between the object and the organ of sense. These, then, we may call Secondary Mechanical Sciences. The sciences of this kind which require our notice are those which treat of the sensible qualities, Sound, Light, and Heat; that is. Acoustics, Optics, and Thermotics.

It will be recollected that our object is not by any means to give a full statement of all the additions which have been successively made to our knowledge on the subjects under review, or a complete list of the persons by whom such additions have been made; but to present a view of the progress of each of those branches of knowledge as a theoretical science;--to point out the Epochs of the discovery of those general principles which reduce many facts to one theory; and to note all that is most characteristic and instructive in the circumstances and persons which bear upon such Epochs. A history of any science, written with such objects, will not need to be long; but it will fail in its purpose altogether, if it do not distinctly exhibit some well-marked and prominent features. {24}

We begin our account of the Secondary Mechanical Sciences with Acoustics, because the progress towards right theoretical views, was, in fact, made much earlier in the science of Sound, than in those of Light and of Heat; and also, because a clear comprehension of the theory to which we are led in this case, is the best preparation for the difficulties (by no means inconsiderable) of the reasonings of theorists on the other subjects.

I. PRELUDE TO THE SOLUTION OF PROBLEMS IN ACOUSTICS

IN some measure the true theory of sound was guessed by very early speculators on the subject; though undoubtedly conceived in a very vague and wavering manner. That sound is caused by some motion of the sounding body, and conveyed by some motion of the air to the ear, is an opinion which we trace to the earliest times of physical philosophy. We may take Aristotle as the best expounder of this stage of opinion. In his Treatise On Sound and Hearing, he says, "Sound takes place when bodies strike the air, not by the air having a form impressed upon it (σχηματίζομενον), as some think, but by its being moved in a corresponding manner; (probably he means in a manner corresponding to the impulse;) the air being contracted, and expanded, and overtaken, and again struck by the impulses of the breath and of the strings. For when the breath falls upon and strikes the air which is next it, the air is carried forwards with an impetus, and that which is contiguous to the first is carried onwards; so that the same voice spreads every way as far as the motion of the air takes place."

As is the case with all such specimens of ancient physics, different persons would find in such a statement very different measures of truth and distinctness. The admirers of antiquity might easily, by pressing the language closely, and using the light of modern discovery, detect in this passage an exact account of the production and propagation of sound: while others might maintain that in Aristotle's own mind, there were only vague notions, and verbal generalizations. This {25} latter opinion is very emphatically expressed by Bacon.[1\8] "The collision or thrusting of air, which they will have to be the cause of sound, neither denotes the form nor the latent process of sound; but is a term of ignorance and of superficial contemplation." Nor can it be justly denied, that an exact and distinct apprehension of the kind of motion of the air by which sound is diffused, was beyond the reach of the ancient philosophers, and made its way into the world long afterwards. It was by no means easy to reconcile the nature of such motion with obvious phenomena. For the process is not evident as motion; since, as Bacon also observes,[2\8] it does not visibly agitate the flame of a candle, or a feather, or any light floating substance, by which the slightest motions of the air are betrayed. Still, the persuasion that sound is some motion of the air, continued to keep hold of men's minds, and acquired additional distinctness. The illustration employed by Vitruvius, in the following passage, is even now one of the best we can offer.[3\8] "Voice is breath, flowing, and made sensible to the hearing by striking the air. It moves in infinite circumferences of circles, as when, by throwing a stone into still water, you produce innumerable circles of waves, increasing from the centre and spreading outwards, till the boundary of the space, or some obstacle, prevents their outlines from going further. In the same manner the voice makes its motion in circles. But in water the circle moves breadthways upon a level plain; the voice proceeds in breadth, and also successively ascends in height."

[Note 1\8: Hist. Son. et Aud. vol. ix. p. 68.]

[Note 2\8: Ibid.]

[Note 3\8: De Arch. v. 3.]

Both the comparison, and the notice of the difference of the two cases, prove the architect to have had very clear notions on the subject; which he further shows by comparing the resonance of the walls of a building to the disturbance of the outline of the waves of water when they meet with a boundary, and are thrown back. "Therefore, as in the outlines of waves in water, so in the voice, if no obstacle interrupt the foremost, it does not disturb the second and the following ones, so that all come to the ears of persons, whether high up or low down, without resonance. But when they strike against obstacles, the foremost, being thrown back, disturb the lines of those which follow." Similar analogies were employed by the ancients in order to explain the occurrence of Echoes. Aristotle says,[4\8] "An Echo takes place, when the air, being as one body in consequence of the vessel which bounds it, and being prevented from being thrust forwards, is reflected {26} back like a ball." Nothing material was added to such views till modern times.

[Note 4\8: De Animâ, ii. 8.]

Thus the first conjectures of those who philosophized concerning sound, led them to an opinion concerning its causes and laws, which only required to be distinctly understood, and traced to mechanical principles, in order to form a genuine science of Acoustics. It was, no doubt, a work which required a long time and sagacious reasoners, to supply what was thus wanting; but still, in consequence of this peculiar circumstance in the early condition of the prevalent doctrine concerning sound, the history of Acoustics assumes a peculiar form. Instead of containing, like the history of Astronomy or of Optics, a series of generalizations, each including and rising above preceding generalizations; in this case, the highest generalization is in view from the first; and the object of the philosopher is to determine its precise meaning and circumstances in each example. Instead of having a series of inductive Truths, successively dawning on men's minds, we have a series of Explanations, in which certain experimental facts and laws are reconciled, as to their mechanical principles and their measures, with the general doctrine already in our possession. Instead of having to travel gradually towards a great discovery, like Universal Gravitation, or Luminiferous Undulations, we take our stand upon acknowledged truths, the production and propagation of sound by the motion of bodies and of air; and we connect these with other truths, the laws of motion and the known properties of bodies, as, for instance, their elasticity. Instead of Epochs of Discovery, we have Solutions of Problems; and to these we must now proceed.

We must, however, in the first place, notice that these Problems include other subjects than the mere production and propagation of sound generally. For such questions as these obviously occur:--what are the laws and cause of the differences of sounds;--of acute and grave, loud and low, continued and instantaneous;--and, again, of the differences of articulate sounds, and of the quality of different voices and different instruments? The first of these questions, in particular, the real nature of the difference of acute and grave sounds, could not help attracting attention; since the difference of notes in this respect was the foundation of one of the most remarkable mathematical sciences of antiquity. Accordingly, we find attempts to explain this difference in the ancient writers on music. In Ptolemy's Harmonics, the third Chapter of the first Book is entitled, "How the {27} acuteness and graveness of notes is produced;" and in this, after noting generally the difference of sounds, and the causes of difference (which he states to be the force of the striking body, the physical constitution of the body struck, and other causes), he comes to the conclusion, that "the things which produce acuteness in sounds, are a greater density and a smaller size; the things which produce graveness, are a greater rarity and a bulkier form." He afterwards explains this so as to include a considerable portion of truth. Thus he says, "That in strings, and in pipes, other things remaining the same, those which are stopped at the smaller distance from the bridge give the most acute note; and in pipes, those notes which come through holes nearest to the mouth-hole are most acute." He even attempts a further generalization, and says that the greater acuteness arises, in fact, from the body being more tense; and that thus "hardness may counteract the effect of greater density, as we see that brass produces a more acute sound than lead." But this author's notions of tension, since they were applied so generally as to include both the tension of a string, and the tension of a piece of solid brass, must necessarily have been very vague. And he seems to have been destitute of any knowledge of the precise nature of the motion or impulse by which sound is produced; and, of course, still more ignorant of the mechanical principles by which these motions are explained. The notion of vibrations of the parts of sounding bodies, does not appear to have been dwelt upon as an essential circumstance; though in some cases, as in sounding strings, the fact is very obvious. And the notion of vibrations of the air does not at all appear in ancient writers, except so far as it may be conceived to be implied in the comparison of aërial and watery waves, which we have quoted from Vitruvius. It is however, very unlikely that, even in the case of water, the motions of the particles were distinctly conceived, for such conception is far from obvious.

The attempts to apprehend distinctly, and to explain mechanically, the phenomena of sound, gave rise to a series of Problems, of which we most now give a brief history. The questions which more peculiarly constitute the Science of Acoustics, are the questions concerning those motions or affections of the air by which it is the medium of hearing. But the motions of sounding bodies have both so much connexion with those of the medium, and so much resemblance to them, that we shall include in our survey researches on that subject also. {28}

II. PROBLEM OF THE VIBRATIONS OF STRINGS

THAT the continuation of sound depends on a continued minute and rapid motion, a shaking or trembling, of the parts of the sounding body, was soon seen. Thus Bacon says,[5\8] "The duration of the sound of a bell or a string when struck, which appears to be prolonged and gradually extinguished, does not proceed from the first percussion; but the trepidation of the body struck perpetually generates a new sound. For if that trepidation be prevented, and the bell or string be stopped, the sound soon dies: as in spinets, as soon as the spine is let fall so as to touch the string, the sound ceases." In the case of a stretched string, it is not difficult to perceive that the motion is a motion back and forwards across the straight line which the string occupies when at rest. The further examination of the quantitative circumstances of this oscillatory motion was an obvious problem; and especially after oscillations, though of another kind (those of a pendulous body), had attracted attention, as they had done in the school of Galileo. Mersenne, one of the promulgators of Galileo's philosophy in France, is the first author in whom I find an examination of the details of this case (Harmonicorum Liber, Paris, 1636). He asserts,[6\8] that the differences and concords of acute and grave sounds depend on the rapidity of vibrations, and their ratio; and he proves this doctrine by a series of experimental comparisons. Thus he finds[7\8] that the note of a string is as its length, by taking a string first twice, and then four times as long as the original string, other things remaining the same. This, indeed, was known to the ancients, and was the basis of that numerical indication of the notes which the proposition expresses. Mersenne further proceeds to show the effect of thickness and tension. He finds (Prop. 7) that a string must be four times as thick as another, to give the octave below; he finds, also (Prop. 8), that the tension must be about four times as great in order to produce the octave above. From these proportions various others are deduced, and the law of the {29} phenomena of this kind may be considered as determined. Mersenne also undertook to measure the phenomena numerically, that is to determine the number of vibrations of the string in each of such cases; which at first might appear difficult, since it is obviously impossible to count with the eye the passages of a sounding string backwards and forwards. But Mersenne rightly assumed, that the number of vibrations is the same so long as the tone is the same, and that the ratios of the numbers of vibrations of different strings may be determined from the numerical relations of their notes. He had, therefore, only to determine the number of vibrations of one certain string, or one known note, to know those of all others. He took a musical string of three-quarters of a foot long, stretched with a weight of six pounds and five eighths, which he found gave him by its vibrations a certain standard note in his organ: he found that a string of the same material and tension, fifteen feet, that is, twenty times as long, made ten recurrences in a second; and he inferred that the number of vibrations of the shorter string must also be twenty times as great; and thus such a string must make in one second of time two hundred vibrations.

[Note 5\8: Hist. Son. et Aud. vol. ix. p. 71.]

[Note 6\8: L. i. Prop. 15.]

[Note 7\8: L. ii. Prop. 6.]

This determination of Mersenne does not appear to have attracted due notice; but some time afterwards attempts were made to ascertain the connexion between the sound and its elementary pulsations in a more direct manner. Hooke, in 1681, produced sounds by the striking of the teeth of brass wheels,[8\8] and Stancari, in 1706, by whirling round a large wheel in air, showed, before the Academy of Bologna, how the number of vibrations in a given note might be known. Sauveur, who, though deaf for the first seven years of his life, was one of the greatest promoters of the science of sound, and gave it its name of Acoustics, endeavored also, about the same time, to determine the number of vibrations of a standard note, or, as he called it, Fixed Sound. He employed two methods, both ingenious and both indirect. The first was the method of beats. Two organ-pipes, which form a discord, are often heard to produce a kind of howl, or wavy noise, the sound swelling and declining at small intervals of time. This was readily and rightly ascribed to the coincidences of the pulsations of sound of the two notes after certain cycles. Thus, if the number of vibrations of the notes were as fifteen to sixteen in the same time, every fifteenth vibration of the one would coincide with every {30} sixteenth vibration of the other, while all the intermediate vibrations of the two tones would, in various degrees, disagree with each other; and thus every such cycle, of fifteen and sixteen vibrations, might be heard as a separate beat of sound. Now, Sauveur wished to take a case in which these beats were so slow as to be counted,[9\8] and in which the ratio of the vibrations of the notes was known from a knowledge of their musical relations. Thus if the two notes form an interval of a semitone, their ratio will be that above supposed, fifteen to sixteen; and if the beats be found to be six in a second, we know that, in that time, the graver note makes ninety and the acuter ninety-six vibrations. In this manner Sauveur found that an open organ-pipe, five feet long, gave one hundred vibrations in a second.

[Note 8\8: Life, p. xxiii.]

[Note 9\8: Ac. Sc. Hist. 1700, p. 131.]

Sauveur's other method is more recondite, and approaches to a mechanical view of the question.[10\8] He proceeded on this basis; a string, horizontally stretched, cannot be drawn into a mathematical straight line, but always hangs in a very flat curve, or festoon. Hence Sauveur assumed that its transverse vibrations may be conceived to be identical with the lateral swingings of such a festoon. Observing that the string C, in the middle of a harpsichord, hangs in such a festoon to the amount of 1⁄323rd of an inch, he calculates, by the laws of pendulums, the time of oscillation, and finds it 1⁄122nd of a second. Thus this C, his fixed note, makes one hundred and twenty-two vibrations in a second. It is curious that this process, seemingly so arbitrary, is capable of being justified on mechanical principles; though we can hardly give the author credit for the views which this justification implies. It is, therefore, easy to understand that it agreed with other experiments, in the laws which it gave for the dependence of the tone on the length and tension.

[Note 10\8: Ac. Sc. Hist. 1713.]

The problem of satisfactorily explaining this dependence, on mechanical principles, naturally pressed upon the attention of mathematicians when the law of the phenomena was thus completely determined by Mersenne and Sauveur. It was desirable to show that both the circumstances and the measure of the phenomena were such as known mechanical causes and laws would explain. But this problem, as might be expected, was not attacked till mechanical principles, and the modes of applying them, had become tolerably familiar.

As the vibrations of a string are produced by its tension, it appeared to be necessary, in the first place, to determine the law of the tension {31} which is called into action by the motion of the string; for it is manifest that, when the string is drawn aside from the straight line into which it is stretched, there arises an additional tension, which aids in drawing it back to the straight line as soon as it is let go. Hooke (On Spring, 1678) determined the law of this additional tension, which he expressed in his noted formula, "Ut tensio sic vis," the Force is as the Tension; or rather, to express his meaning more clearly, the Force of tension is as the Extension, or, in a string, as the increase of length. But, in reality, this principle, which is important in many acoustical problems, is, in the one now before us, unimportant; the force which urges the string towards the straight line, depends, with such small extensions as we have now to consider, not on the extension, but on the curvature; and the power of treating the mathematical difficulty of curvature, and its mechanical consequences, was what was requisite for the solution of this problem.

The problem, in its proper aspect, was first attacked and mastered by Brook Taylor, an English mathematician of the school of Newton, by whom the solution was published in 1715, in his Methodus Incrementorum. Taylor's solution was indeed imperfect, for it only pointed out a form and a mode of vibration, with which the string might move consistently with the laws of mechanics; not the mode in which it must move, supposing its form to be any whatever. It showed that the curve might be of the nature of that which is called the companion to the cycloid; and, on the supposition of the curve of the string being of this form, the calculation confirmed the previously established laws by which the tone, or the time of vibration, had been discovered to depend on the length, tension, and bulk of the string. The mathematical incompleteness of Taylor's reasoning must not prevent us from looking upon his solution of the problem as the most important step in the progress of this part of the subject: for the difficulty of applying mechanical principles to the question being once overcome, the extension and correction of the application was sure to be undertaken by succeeding mathematicians; and, accordingly, this soon happened. We may add, moreover, that the subsequent and more general solutions require to be considered with reference to Taylor's, in order to apprehend distinctly their import; and further, that it was almost evident to a mathematician, even before the general solution had appeared, that the dependence of the time of vibration on the length and tension, would be the same in the general case as in the {32} Taylorian curve; so that, for the ends of physical philosophy, the solution was not very incomplete.

John Bernoulli, a few years afterwards,[11\8] solved the problem of vibrating chords on nearly the same principles and suppositions as Taylor; but a little later (in 1747), the next generation of great mathematicians, D'Alembert, Euler, and Daniel Bernoulli, applied the increased powers of analysis to give generality to the mode of treating this question; and especially the calculus of partial differentials, invented for this purpose. But at this epoch, the discussion, so far as it bore on physics, belonged rather to the history of another problem, which comes under our notice hereafter, that of the composition of vibrations; we shall, therefore, defer the further history of the problem of vibrating strings, till we have to consider it in connexion with new experimental facts.

[Note 11\8: Op. iii. p. 207.]

III. PROBLEM OF THE PROPAGATION OF SOUND

WE have seen that the ancient philosophers, for the most part, held that sound was transmitted, as well as produced, by some motion of the air, without defining what kind of motion this was; that some writers, however, applied to it a very happy similitude, the expansive motion of the circular waves produced by throwing a stone into still water; but that notwithstanding, some rejected this mode of conception, as, for instance, Bacon, who ascribed the transmission of sound to certain "spiritual species."

Though it was an obvious thought to ascribe the motion of sound to some motion of air; to conceive what kind of motion could and did produce this effect, must have been a matter of grave perplexity at the time of which we are speaking; and is far from easy to most persons even now. We may judge of the difficulty of forming this conception, when we recollect that John Bernoulli the younger[12\8] declared, that he could not understand Newton's proposition on this subject. The difficulty consists in this; that the movement of the parts of air, in which sound consists, travels along, but that the parts {33} of air themselves do not so travel. Accordingly Otto Guericke,[13\8] the inventor of the air-pump, asks, "How can sound be conveyed by the motion of the air? when we find that it is better conveyed through air that is still, than when there is a wind." We may observe, however, that he was partly misled by finding, as he thought, that a bell could be heard in the vacuum of his air-pump; a result which arose, probably, from some imperfection in his apparatus.

[Note 12\8: Prize Dis. on Light, 1736.]

[Note 13\8: De Vac. Spat. p. 138.]

Attempts were made to determine, by experiment, the circumstances of the motion of sound; and especially its velocity. Gassendi[14\8] was one of the first who did this. He employed fire-arms for the purpose, and thus found the velocity to be 1473 Paris feet in a second. Roberval found a velocity so small (560 feet) that it threw uncertainty upon the rest, and affected Newton's reasonings subsequently.[15\8] Cassini, Huyghens, Picard, Römer, found a velocity of 1172 Paris feet, which is more accurate than the former. Gassendi had been surprised to find that the velocity with which sounds travel, is the same whether they are loud or gentle.

[Note 14\8: Fischer, Gesch. d. Physik. vol. i. 171.]

[Note 15\8: Newt. Prin. B. ii. P. 50, Schol.]

The explanation of this constant velocity of sound, and of its amount, was one of the problems of which a solution was given in the Great Charter of modern science, Newton's Principia (1687). There, for the first time, were explained the real nature of the motions and mutual action of the parts of the air through which sound is transmitted. It was shown[16\8] that a body vibrating in an elastic medium, will propagate pulses through the medium; that is, the parts of the medium will move forwards and backwards, and this motion will affect successively those parts which are at a greater and greater distance from the origin of motion. The parts, in going forwards, produce condensation; in returning to their first places, they allow extension; and the play of the elasticities developed by these expansions and contractions, supplies the forces which continue to propagate the motion.

[Note 16\8: Newt. Prin. B. ii. P. 43.]

The idea of such a motion as this, is, as we have said, far from easy to apprehend distinctly: but a distinct apprehension of it is a step essential to the physical part of the sciences now under notice; for it is by means of such pulses, or undulations, that not only sound, but light, and probably heat, are propagated. We constantly meet with evidence of the difficulty which men have in conceiving this undulatory motion, and in separating it from a local motion of the medium as a {34} mass. For instance, it is not easy at first to conceive the waters of a great river flowing constantly down towards the sea, while waves are rolling up the very same part of the stream; and while the great elevation, which makes the tide, is travelling from the sea perhaps with a velocity of fifty miles an hour. The motion of such a wave, or elevation, is distinct from any stream, and is of the nature of undulations in general. The parts of the fluid stir for a short time and for a small distance, so as to accumulate themselves on a neighboring part, and then retire to their former place; and this movement affects the parts in the order of their places. Perhaps if the reader looks at a field of standing corn when gusts of wind are sweeping over it in visible waves, he will have his conception of this matter aided; for he will see that here, where each ear of grain is anchored by its stalk, there can be no permanent local motion of the substance, but only a successive stooping and rising of the separate straws, producing hollows and waves, closer and laxer strips of the crowded ears.

Newton had, moreover, to consider the mechanical consequences which such condensations and rarefactions of the elastic medium, air, would produce in the parts of the fluid itself. Employing known laws of the elasticity of air, he showed, in a very remarkable proposition,[17\8] the law according to which the particles of air might vibrate. We may observe, that in this solution, as in that of the vibrating string already mentioned, a rule was exhibited according to which the particles might oscillate, but not the law to which they must conform. It was proved that, by taking the motion of each particle to be perfectly similar to that of a pendulum, the forces, developed by contraction and expansion, were precisely such as the motion required; but it was not shown that no other type of oscillation would give rise to the same accordance of force and motion. Newton's reasoning also gave a determination of the speed of propagation of the pulses: it appeared that sound ought to travel with the velocity which a body would acquire by falling freely through half the height of a homogeneous atmosphere; "the height of a homogeneous atmosphere" being the height which the air must have, in order to produce, at the earth's surface, the actual atmospheric pressure, supposing no diminution of density to take place in ascending. This height is about 29,000 feet; and hence it followed that the velocity was 968 feet. This velocity is really considerably less than that of sound; but at the time of which {35} we speak, no accurate measure had been established; and Newton persuaded himself, by experiments made in the cloister of Trinity College, his residence, that his calculation was not far from the fact. When, afterwards, more exact experiments showed the velocity to be 1142 English feet, Newton attempted to explain the difference by various considerations, none of which were adequate to the purpose;--as, the dimensions of the solid particles of which the fluid air consists;--or the vapors which are mixed with it. Other writers offered other suggestions; but the true solution of the difficulty was reserved for a period considerably subsequent.

[Note 17\8: Princ. B. ii. P. 48.]

Newton's calculation of the motion of sound, though logically incomplete, was the great step in the solution of the problem; for mathematicians could not but presume that his result was not restricted to the hypothesis on which he had obtained it; and the extension of the solution required only mere ordinary talents. The logical defect of his solution was assailed, as might have been expected. Cranmer (professor at Geneva), in 1741, conceived that he was destroying the conclusiveness of Newton's reasoning, by showing that it applied equally to other modes of oscillation. This, indeed, contradicted the enunciation of the 48th Prop. of the Second Book of the Principia; but it confirmed and extended all the general results of the demonstration; for it left even the velocity of sound unaltered, and thus showed that the velocity did not depend mechanically on the type of the oscillation. But the satisfactory establishment of this physical generalization was to be supplied from the vast generalizations of analysis, which mathematicians were now becoming able to deal with. Accordingly this task was performed by the great master of analytical generalization, Lagrange, in 1759, when, at the age of twenty-three, he and two friends published the first volume of the Turin Memoirs. Euler, as his manner was, at once perceived the merit of the new solution, and pursued the subject on the views thus suggested. Various analytical improvements and extensions were introduced into the solution by the two great mathematicians; but none of these at all altered the formula by which the velocity of sound was expressed; and the discrepancy between calculation and observation, about one-sixth of the whole, which had perplexed Newton, remained still unaccounted for.

The merit of satisfactorily explaining this discrepancy belongs to Laplace. He was the first to remark[18\8] that the common law of the {36} changes of elasticity in the air, as dependent on its compression, cannot be applied to those rapid vibrations in which sound consists, since the sudden compression produces a degree of heat which additionally increases the elasticity. The ratio of this increase depended on the experiments by which the relation of heat and air is established. Laplace, in 1816, published[19\8] the theorem on which the correction depends. On applying it, the calculated velocity of sound agreed very closely with the best antecedent experiments, and was confirmed by more exact ones instituted for that purpose.

[Note 18\8: Méc. Cél. t. v. l. xii. p. 96.]

[Note 19\8: Ann. Phys. et Chim. t. iii. p. 288.]

This step completes the solution of the problem of the propagation of sound, as a mathematical induction, obtained from, and verified by, facts. Most of the discussions concerning points of analysis to which the investigations on this subject gave rise, as, for instance, the admissibility of discontinuous functions into the solutions of partial differential equations, belong to the history of pure mathematics. Those which really concern the physical theory of sound may be referred to the problem of the motion of air in tubes, to which we shall soon have to proceed; but we must first speak of another form which the problem of vibrating strings assumed.

It deserves to be noticed that the ultimate result of the study of the undulations of fluids seems to show that the comparison of the motion of air in the diffusion of sound with the motion of circular waves from a centre in water, which is mentioned at the beginning of this chapter, though pertinent in a certain way, is not exact. It appears by Mr. Scott's recent investigations concerning waves,[20\8] that the circular waves are oscillating waves of the Second order, and are gregarious. The sound-wave seems rather to resemble the great solitary Wave of Translation of the First order, of which we have already spoken in Book vi. chapter vi.

[Note 20\8: Brit. Ass. Reports for 1844, p. 361.]

IV. PROBLEM OF DIFFERENT SOUNDS OF THE SAME STRING

IT had been observed at an early period of acoustical knowledge, that one string might give several sounds. Mersenne and others {37} had noticed[21\8] that when a string vibrates, one which is in unison with it vibrates without being touched. He was also aware that this was true if the second string was an octave or a twelfth below the first. This was observed as a new fact in England in 1674, and communicated to the Royal Society by Wallis.[22\8] But the later observers ascertained further, that the longer string divides itself into two, or into three equal parts, separated by nodes, or points of rest; this they proved by hanging bits of paper on different parts of the string. The discovery so modified was again made by Sauveur[23\8] about 1700. The sounds thus produced in one string by the vibration of another, have been termed Sympathetic Sounds. Similar sounds are often produced by performers on stringed instruments, by touching the string at one of its aliquot divisions, and are then called the Acute harmonics. Such facts were not difficult to explain on Taylor's view of the mechanical condition of the string; but the difficulty was increased when it was noticed that a sounding body could produce these different notes at the same time. Mersenne had remarked this, and the fact was more distinctly observed and pursued by Sauveur. The notes thus produced in addition to the genuine note of the string, have been called Secondary Notes; those usually heard are, the Octave, the Twelfth, and the Seventeenth above the note itself. To supply a mode of conceiving distinctly, and explaining mechanically, vibrations which should allow of such an effect, was therefore a requisite step in acoustics.

[Note 21\8: Harm. lib. iv. Prop. 28 (1636).]

[Note 22\8: Ph. Tr. 1677, April.]

[Note 23\8: A. P. 1701.]

This task was performed by Daniel Bernoulli in a memoir published in 1755.[24\8] He there stated and proved the Principle of the coexistence of small vibrations. It was already established, that a string might vibrate either in a single swelling (if we use this word to express the curve between two nodes which Bernoulli calls a ventre), or in two or three or any number of equal swellings with immoveable nodes between. Daniel Bernoulli showed further, that these nodes might be combined, each taking place as if it were the only one. This appears sufficient to explain the coexistence of the harmonic sounds just noticed. D'Alembert, indeed, in the article Fundamental in the French Encyclopédie, and Lagrange in his Dissertation on Sound in the Turin Memoirs,[25\8] offer several objections to this explanation; and it cannot be denied that the subject has its difficulties; but {38} still these do not deprive Bernoulli of the merit of having pointed out the principle of Coexistent Vibrations, or divest that principle of its value in physical science.

[Note 24\8: Berlin Mem. 1753, p. 147.]

[Note 25\8: T. i. pp. 64, 103.]

Daniel Bernoulli's Memoir, of which we speak, was published at a period when the clouds which involve the general analytical treatment of the problem of vibrating strings, were thickening about Euler and D'Alembert, and darkening into a controversial hue; and as Bernoulli ventured to interpose his view, as a solution of these difficulties, which, in a mathematical sense, it is not, we can hardly be surprised that he met with a rebuff. The further prosecution of the different modes of vibration of the same body need not be here considered.

The sounds which are called Grave Harmonics, have no analogy with the Acute Harmonics above-mentioned; nor do they belong to this section; for in the case of Grave Harmonics, we have one sound from the co-operation of two strings, instead of several sounds from one string. These harmonics are, in fact, connected with beats, of which we have already spoken; the beats becoming so close as to produce a note of definite musical quality. The discovery of the Grave Harmonics is usually ascribed to Tartini, who mentions them in 1754; but they are first noticed[26\8] in the work of Sorge On tuning Organs, 1744. He there expresses this discovery in a query. "Whence comes it, that if we tune a fifth (2 : 3), a third sound is faintly heard, the octave below the lower of the two notes? Nature shows that with 2 : 3, she still requires the unity, to perfect the order 1, 2, 3." The truth is, that these numbers express the frequency of the vibrations, and thus there will be coincidences of the notes 2 and 3, which are of the frequency 1, and consequently give the octave below the sound 2. This is the explanation given by Lagrange,[27\8] and is indeed obvious.

[Note 26\8: Chladni. Acoust. p. 254.]

[Note 27\8: Mem. Tur. i. p. 104.]

V. PROBLEM OF THE SOUNDS OF PIPES

IT was taken for granted by those who reasoned on sounds, that the sounds of flutes, organ-pipes, and wind-instruments in general, {39} consisted in vibrations of some kind; but to determine the nature and laws of these vibrations, and to reconcile them with mechanical principles, was far from easy. The leading facts which had been noticed were, that the note of a pipe was proportional to its length, and that a flute and similar instruments might be made to produce some of the acute harmonics, as well as the genuine note. It had further been noticed,[28\8] that pipes closed at the end, instead of giving the series of harmonics 1, ½, ⅓, ¼, &c., would give only those notes which answer to the odd numbers 1, ⅓, ⅕, &c. In this problem also, Newton[29\8] made the first step to the solution. At the end of the propositions respecting the velocity of sound, of which we have spoken, he noticed that it appeared by taking Mersenne's or Sauveur's determination of the number of vibrations corresponding to a given note, that the pulse of air runs over twice the length of the pipe in the time of each vibration. He does not follow out this observation, but it obviously points to the theory, that the sound of a pipe consists of pulses which travel back and forwards along its length, and are kept in motion by the breath of the player. This supposition would account for the observed dependence of the note on the length of the pipe. The subject does not appear to have been again taken up in a theoretical way till about 1760; when Lagrange in the second volume of the Turin Memoirs, and D. Bernoulli in the Memoirs of the French Academy for 1762, published important essays, in which some of the leading facts were satisfactorily explained, and which may therefore be considered as the principal solutions of the problem.

[Note 28\8: D. Bernoulli, Berlin. Mem. 1753, p. 150.]

[Note 29\8: Princip. Schol. Prop. 50.]

In these solutions there was necessarily something hypothetical. In the case of vibrating strings, as we have seen, the Form of the vibrating curve was guessed at only, but the existence and position of the Nodes could be rendered visible to the eye. In the vibrations of air, we cannot see either the places of nodes, or the mode of vibration; but several of the results are independent of these circumstances. Thus both of the solutions explain the fact, that a tube closed at one end is in unison with an open tube of double the length; and, by supposing nodes to occur, they account for the existence of the odd series of harmonics alone, 1, 3, 5, in closed tubes, while the whole series, 1, 2, 3, 4, 5, &c., occurs in open ones. Both views of the nature of the vibration appear to be nearly the same; though Lagrange's is expressed with an analytical generality which renders it obscure, and Bernoulli has perhaps {40} laid down an hypothesis more special than was necessary. Lagrange[30\8] considers the vibration of open flutes as "the oscillations of a fibre of air," under the condition that its elasticity at the two ends is, during the whole oscillation, the same as that of the surrounding atmosphere. Bernoulli supposes[31\8] the whole inertia of the air in the flute to be collected into one particle, and this to be moved by the whole elasticity arising from this displacement. It may be observed that both these modes of treating the matter come very near to what we have stated as Newton's theory; for though Bernoulli supposes all the air in the flute to be moved at once, and not successively, as by Newton's pulse, in either case the whole elasticity moves the whole air in the tube, and requires more time to do this according to its quantity. Since that time, the subject has received further mathematical developement from Euler,[32\8] Lambert,[33\8] and Poisson;[34\8] but no new explanation of facts has arisen. Attempts have however been made to ascertain experimentally the places of the nodes. Bernoulli himself had shown that this place was affected by the amount of the opening, and Lambert[35\8] had examined other cases with the same view. Savart traced the node in various musical pipes under different conditions; and very recently Mr. Hopkins, of Cambridge, has pursued the same experimental inquiry.[36\8] It appears from these researches, that the early assumptions of mathematicians with regard to the position of the nodes, are not exactly verified by the facts. When the air in a pipe is made to vibrate so as to have several nodes which divide it into equal parts, it had been supposed by acoustical writers that the part adjacent to the open end was half of the other parts; the outermost node, however, is found experimentally to be displaced from the position thus assigned to it, by a quantity depending on several collateral circumstances.

[Note 30\8: Mém. Turin, vol. ii. p. 154.]

[Note 31\8: Mém. Berlin, 1753, p. 446.]

[Note 32\8: Nov. Act. Petrop. tom. xvi.]

[Note 33\8: Acad. Berlin, 1775.]

[Note 34\5: Journ. Ec. Polyt. cap. 14.]

[Note 35\8: Acad. Berlin, 1775.]

[Note 36\8: Camb. Trans. vol. v. p. 234.]

Since our purpose was to consider this problem only so far as it has tended towards its mathematical solution, we have avoided saying anything of the dependence of the mode of vibration on the cause by which the sound is produced; and consequently, the researches on the effects of reeds, embouchures, and the like, by Chladni, Savart, Willis, and others, do not belong to our subject. It is easily seen that the complex effect of the elasticity and other properties of the reed and of the air together, is a problem of which we can hardly {41} hope to give a complete solution till our knowledge has advanced much beyond its present condition.

Indeed, in the science of Acoustics there is a vast body of facts to which we might apply what has just been said; but for the sake of pointing out some of them, we shall consider them as the subjects of one extensive and yet unsolved problem.

Chapter VI

PROBLEM OF DIFFERENT MODES OF VIBRATION OF BODIES IN GENERAL.

NOT only the objects of which we have spoken hitherto, strings and pipes, but almost all bodies are capable of vibration. Bells, gongs, tuning-forks, are examples of solid bodies; drums and tambourines, of membranes; if we run a wet finger along the edge of a glass goblet, we throw the fluid which it contains into a regular vibration; and the various character which sounds possess according to the room in which they are uttered, shows that large masses of air have peculiar modes of vibration. Vibrations are generally accompanied by sound, and they may, therefore, be considered as acoustical phenomena, especially as the sound is one of the most decisive facts in indicating the mode of vibration. Moreover, every body of this kind can vibrate in many different ways, the vibrating segments being divided by Nodal Lines and Surfaces of various form and number. The mode of vibration, selected by the body in each case, is determined by the way in which it is held, the way in which it is set in vibration, and the like circumstances.

The general problem of such vibrations includes the discovery and classification of the phenomena; the detection of their formal laws; and, finally, the explanation of these on mechanical principles. We must speak very briefly of what has been done in these ways. The facts which indicate Nodal Lines had been remarked by Galileo, on the sounding board of a musical instrument; and Hooke had proposed to observe the vibrations of a bell by strewing flour upon it. But it was Chladni, a German philosopher, who enriched acoustics with the discovery of the vast variety of symmetrical figures of Nodal Lines, which are exhibited on plates of regular forms, when {42} made to sound. His first investigations on this subject, Entdeckungen über die Theorie des Klangs, were published 1787; and in 1802 and 1817 he added other discoveries. In these works he not only related a vast number of new and curious facts, but in some measure reduced some of them to order and law. For instance, he has traced all the vibrations of square plates to a resemblance with those forms of vibration in which Nodal Lines are parallel to one side of the square, and those in which they are parallel to another side; and he has established a notation for the modes of vibration founded on this classification. Thus, 5-2 denotes a form in which there are five nodal lines parallel to one side, and two to another; or a form which can be traced to a disfigurement of such a standard type. Savart pursued this subject still further; and traced, by actual observation, the forms of the Nodal Surfaces which divide solid bodies, and masses of air, when in a state of vibration.

The dependence of such vibrations upon their physical cause, namely, the elasticity of the substance, we can conceive in a general way; but the mathematical theory of such cases is, as might be supposed, very difficult, even if we confine ourselves to the obvious question of the mechanical possibility of these different modes of vibration, and leave out of consideration their dependence upon the mode of excitation. The transverse vibrations of elastic rods, plates, and rings, had been considered by Euler in 1779; but his calculation concerning plates had foretold only a small part of the curious phenomena observed by Chladni;[37\8] and the several notes which, according to his calculation, the same ring ought to give, were not in agreement with experiment.[38\8] Indeed, researches of this kind, as conducted by Euler, and other authors,[39\8] rather were, and were intended for, examples of analytical skill, than explanations of physical facts. James Bernoulli, after the publication of Chladni's experiments in 1787, attempted to solve the problem for plates, by treating a plate as a collection of fibres; but, as Chladni observes, the justice of this mode of conception is disproved, by the disagreement of the results with experiment.

[Note 37\8: Fischer, vi. 587.]

[Note 38\8: Ib. vi. 596.]

[Note 39\8: See Chladni, p. 474.]

The Institute of France, which had approved of Chladni's labours, proposed, in 1809, the problem now before us as a prize-question:[40\8]--"To give the mathematical theory of the vibrations of elastic {43} surfaces, and to compare it with experiment." Only one memoir was sent in as a candidate for the prize; and this was not crowned, though honorable mention was made of it.[41\8] The formulæ of James Bernoulli were, according to M. Poisson's statement, defective, in consequence of his not taking into account the normal force which acts at the exterior boundary of the plate.[42\8] The author of the anonymous memoir corrected this error, and calculated the note corresponding to various figures of the nodal lines; and he found an agreement with experiment sufficient to justify his theory. He had not, however, proved his fundamental equation, which M. Poisson demonstrated in a Memoir, read in 1814.[43\8] At a more recent period also, MM. Poisson and Cauchy (as well as a lady, Mlle. Sophie Germain) have applied to this problem the artifices of the most improved analysis. M. Poisson[44\8] determined the relation of the notes given by the longitudinal and the transverse vibrations of a rod; and solved the problem of vibrating circular plates when the nodal lines are concentric circles. In both these cases, the numerical agreement of his results with experience, seemed to confirm the justice of his fundamental views.[45\8] He proceeds upon the hypothesis, that elastic bodies are composed of separate particles held together by the attractive forces which they exert upon each other, and distended by the repulsive force of heat. M. Cauchy[46\8] has also calculated the transverse, longitudinal, and rotatory vibrations of elastic rods, and has obtained results agreeing closely with experiment through a considerable list of comparisons. The combined authority of two profound analysts, as MM. Poisson and Cauchy are, leads us to believe that, for the simpler cases of the vibrations of elastic bodies, Mathematics has executed her task; but most of the more complex cases remain as yet unsubdued.

[Note 40\8: See Chladni, p. 357.]

[Note 41\8: Poisson's Mém. in Ac. Sc. 1812, p. 169.]

[Note 42\8: Ib. p. 220.]

[Note 43\8: Ib. 1812, p. 2.]

[Note 44\8: Ib. t. viii. 1829.]

[Note 45\8: An. Chim. tom. xxxvi. 1827, p. 90.]

[Note 46\8: Exercices de Mathématique, iii. and iv.]

The two brothers, Ernest and William Weber, made many curious observations on undulations, which are contained in their Wellenlehre, (Doctrine of Waves,) published at Leipsig in 1825. They were led to suppose, (as Young had suggested at an earlier period,) that Chladni's figures of nodal lines in plates were to be accounted for by the superposition of undulations.[47\8] Mr. Wheatstone[48\8] has undertaken to account for Chladni's figures of vibrating square plates by this {44} superposition of two or more simple and obviously allowable modes of nodal division, which have the same time of vibration. He assumes, for this purpose, certain "primary figures," containing only parallel nodal lines; and by combining these, first in twos, and then in fours, he obtains most of Chladni's observed figures, and accounts for their transitions and deviations from regularity.

[Note 47\8: Wellenlehre, p. 474.]

[Note 48\8: Phil. Trans. 1833, p. 593.]

The principle of the superposition of vibrations is so solidly established as a mechanical truth, that we may consider an acoustical problem as satisfactorily disposed of when it is reduced to that principle, as well as when it is solved by analytical mechanics: but at the same time we may recollect, that the right application and limitation of this law involves no small difficulty; and in this case, as in all advances in physical science, we cannot but wish to have the new ground which has been gained, gone over by some other person in some other manner; and thus secured to us as a permanent possession.

Savart's Laws.--In what has preceded, the vibrations of bodies have been referred to certain general classes, the separation of which was suggested by observation; for example, the transverse, longitudinal, and rotatory,[49\8] vibrations of rods. The transverse vibrations, in which the rod goes backwards and forwards across the line of its length, were the only ones noticed by the earlier acousticians: the others were principally brought into notice by Chladni. As we have already seen in the preceding pages, this classification serves to express important laws; as, for instance, a law obtained by M. Poisson which gives the relation of the notes produced by the transverse and longitudinal vibrations of a rod. But this distinction was employed by M. Felix Savart to express laws of a more general kind; and then, as often happens in the progress of science, by pursuing these laws to a higher point of generality, the distinction again seemed to vanish. A very few words will explain these steps.

[Note 49\8: Vibrations tournantes.]

It was long ago known that vibrations may be communicated by contact. The distinction of transverse and longitudinal vibrations being established, Savart found that if one rod touched another perpendicularly, the longitudinal vibrations of the first occasion transverse vibrations in the second, and vice versâ. This is the more remarkable, since the two sets of vibrations are not equal in rapidity, and therefore cannot sympathize in any obvious manner.[50\8] Savart found himself {45} able to generalize this proposition, and to assert that in any combination of rods, strings, and laminæ, at right angles to each other, the longitudinal and transverse vibrations affect respectively the rods in the one and other direction,[51\8] so that when the horizontal rods, for example, vibrate in the one way, the vertical rods vibrate in the other.

[Note 50\8: An. Chim. 1819, tom. xiv. p. 138.]

[Note 51\8: An. Chim. p. 152.]

This law was thus expressed in terms of that classification of vibrations of which we have spoken. Yet we easily see that we may express it in a more general manner, without referring to that classification, by saying, that vibrations are communicated so as always to be parallel to their original direction. And by following it out in this shape by means of experiment, M. Savart was led, a short time afterwards, to deny that there is any essential distinction in these different kinds of vibration. "We are thus led," he says[52\8] in 1822, "to consider normal [transverse] vibrations as only one circumstance in a more general motion common to all bodies, analogous to tangential [longitudinal and rotatory] vibrations; that is, as produced by small molecular oscillations, and differently modified according to the direction which it affects, relatively to the dimensions of the vibrating body."

[Note 52\8: Ib. t. xxv. p. 33.]

These "inductions," as he properly calls them, are supported by a great mass of ingenious experiments; and may be considered as well established, when they are limited to molecular oscillations, employing this phrase in the sense in which it is understood in the above statement; and also when they are confined to bodies in which the play of elasticity is not interrupted by parts more rigid than the rest, as the sound-post of a violin.[53\8] And before I quit the subject, I may notice a consequence which M. Savart has deduced from his views, and which, at first sight, appears to overturn most of the earlier doctrines respecting vibrating bodies. It was formerly held that tense strings and elastic rods could vibrate only in a determinate series of modes of division, with no intermediate steps. But M. Savart maintains,[54\8] on the contrary, that they produce sounds which are gradually transformed into one another, by indefinite intermediate degrees. The reader may naturally ask, what is the solution of this apparent {46} contradiction between the earliest and the latest discoveries in acoustics. And the answer must be, that these intermediate modes of vibration are complex in their nature, and difficult to produce; and that those which were formerly believed to be the only possible vibrating conditions, are so eminent above all the rest by their features, their simplicity, and their facility, that we may still, for common purposes, consider them as a class apart; although for the sake of reaching a general theorem, we may associate them with the general mass of cases of molecular vibrations. And thus we have no exception here, as we can have none in any case, to our maxim, that what formed part of the early discoveries of science, forms part of its latest systems.

[Note 53\8: For the suggestion of the necessity of this limitation I am indebted to Mr. Willis.]

[Note 54\8: An. Chim. 1826, t. xxxii. p. 384.]

We have thus surveyed the progress of the science of sound up to recent times, with respect both to the discovery of laws of phenomena, and the reduction of these to their mechanical causes. The former branch of the science has necessarily been inductively pursued; and therefore has been more peculiarly the subject of our attention. And this consideration will explain why we have not dwelt more upon the deductive labors of the great analysts who have treated of this problem.

To those who are acquainted with the high and deserved fame which the labors of D'Alembert, Euler, Lagrange, and others, upon this subject, enjoy among mathematicians, it may seem as if we had not given them their due prominence in our sketch. But it is to be recollected here, as we have already observed in the case of hydrodynamics, that even when the general principles are uncontested, mere mathematical deductions from them do not belong to the history of physical science, except when they point out laws which are intermediate between the general principle and the individual facts, and which observation may confirm.

The business of constructing any science may be figured as the task of forming a road on which our reason can travel through a certain province of the external world. We have to throw a bridge which may lead from the chambers of our own thoughts, from our speculative principles, to the distant shore of material facts. But in all cases the abyss is too wide to be crossed, except we can find some intermediate points on which the piers of our structure may rest. Mere facts, without connexion or law, are only the rude stones hewn from the opposite bank, of which our arches may, at some time, be built. But mere hypothetical mathematical calculations are only plans of projected structures; and those plans which exhibit only one vast {47} and single arch, or which suppose no support but that which our own position supplies, will assuredly never become realities. We must have a firm basis of intermediate generalizations in order to frame a continuous and stable edifice.

In the subject before us, we have no want of such points of intermediate support, although they are in many instances irregularly distributed and obscurely seen. The number of observed laws and relations of the phenomena of sound, is already very great; and though the time may be distant, there seems to be no reason to despair of one day uniting them by clear ideas of mechanical causation, and thus of making acoustics a perfect secondary mechanical science.

The historical sketch just given includes only such parts of acoustics as have been in some degree reduced to general laws and physical causes; and thus excludes much that is usually treated of under that head. Moreover, many of the numerical calculations connected with sound belong to its agreeable effect upon the ear; as the properties of the various systems of Temperament. These are parts of Theoretical Music, not of Acoustics; of the Philosophy of the Fine Arts, not of Physical Science; and may be referred to in a future portion of this work, so far as they bear upon our object.

The science of Acoustics may, however, properly consider other differences of sound than those of acute and grave,--for instance, the articulate differences, or those by which the various letters are formed. Some progress has been made in reducing this part of the subject to general rules; for though Kempelen's "talking machine" was only a work of art, Mr. Willis's machine,[55\8] which exhibits the relation among the vowels, gives us a law such as forms a step in science. We may, however, consider this instrument as a phthongometer, or measure of vowel quality; and in that point of view we shall have to refer to it again when we come to speak of such measures.

[Note 55\8: On the Vowel Sounds, and on Reed Organ-pipes. Camb. Trans. iii. 237.]

{{49}} BOOK IX.

SECONDARY MECHANICAL SCIENCES.

(CONTINUED)

HISTORY OF OPTICS,

FORMAL AND PHYSICAL.

Ω Διὸς ὑψιμέλαθρον ἔχων κράτος αἰὲν ἀτειρὲς Ἄστρων, Ἠελίου τε, Σεληναίης τε μέρισμα Πανδαμάτωρ, πυρίπνου, πᾶσιν ζωοῖσιν ἔναυσμα **Ὑψιφάνης ἌIϴΗΡ, κόσμου στοιχεῖον, **ἄριστον· Ἀγλαὸν ὦ βλάστημα, σελασφόρον, ἀστεροφεγγὲς Κικλήσκων λίτομαι σε, κεκραμένον **εὔδιον εἶναι. ORPHEUS. HYMN.

O thou who fillest the palaces of Jove; Who flowest round moon, and sun, and stars above; Pervading, bright, life-giving element, Supernal ETHER, fair and excellent; Fountain of hope and joy, of light and day, We own at length thy tranquil, steady sway.

{{51}} INTRODUCTION.

Formal and Physical Optics.

THE history of the science of Optics, written at length, would be very voluminous; but we shall not need to make our history so; since our main object is to illustrate the nature of science and the conditions of its progress. In this way Optics is peculiarly instructive; the more so, as its history has followed a course in some respects different from both the sciences previously reviewed. Astronomy, as we have seen, advanced with a steady and continuous movement from one generation to another, from the earliest time, till her career was crowned by the great unforeseen discovery of Newton; Acoustics had her extreme generalization in view from the first, and her history consists in the correct application of it to successive problems; Optics advanced through a scale of generalizations as remarkable as those of Astronomy; but for a long period she was almost stationary; and, at last, was rapidly impelled through all those stages by the energy of two or three discoverers. The highest point of generality which Optics has reached is little different from that which Acoustics occupied at once; but in the older and earlier science we still want that palpable and pointed confirmation of the general principle, which the undulatory theory receives from optical phenomena. Astronomy has amassed her vast fortune by long-continued industry and labor; Optics has obtained hers in a few years by sagacious and happy speculations; Acoustics, having early acquired a competence, has since been employed rather in improving and adorning than in extending her estate.

The successive inductions by which Optics made her advances, might, of course, be treated in the same manner as those of Astronomy, each having its prelude and its sequel. But most of the discoveries in Optics are of a smaller character, and have less employed the minds of men, than those of Astronomy; and it will not be necessary to exhibit them in this detailed manner, till we come to the great generalization by which the theory was established. I shall, therefore, now pass rapidly in review the earlier optical discoveries, without any such division of the series. {52}

Optics, like Astronomy, has for its object of inquiry, first, the laws of phenomena, and next, their causes; and we may hence divide this science, like the other, into Formal Optics and Physical Optics. The distinction is clear and substantive, but it is not easy to adhere to it in our narrative; for, after the theory had begun to make its rapid advance, many of the laws of phenomena were studied and discovered in immediate reference to the theoretical cause, and do not occupy a separate place in the history of science, as in Astronomy they do. We may add, that the reason why Formal Astronomy was almost complete before Physical Astronomy began to exist, was, that it was necessary to construct the science of Mechanics in the mean time, in order to be able to go on; whereas, in Optics, mathematicians were able to calculate the results of the undulatory theory as soon as it had suggested itself from the earlier facts, and while the great mass of facts were only becoming known.

We shall, then, in the first nine chapters of the History of Optics treat of the Formal Science, that is, the discovery of the laws of phenomena. The classes of phenomena which will thus pass under oar notice are numerous; namely, reflection, refraction, chromatic dispersion, achromatization, double refraction, polarization, dipolarization, the colors of thin plates, the colors of thick plates, and the fringes and bands which accompany shadows. All these cases had been studied, and, in most of them, the laws had been in a great measure discovered, before the physical theory of the subject gave to our knowledge a simpler and more solid form.

{{53}} FORMAL OPTICS.

Chapter I

PRIMARY INDUCTION OF OPTICS.--RAYS OF LIGHT AND LAWS OF REFLECTION.

IN speaking of the Ancient History of Physics, we have already noticed that the optical philosophers of antiquity had satisfied themselves that vision is performed in straight lines;--that they had fixed their attention upon those straight lines, or rays, as the proper object of the science;--they had ascertained that rays reflected from a bright surface make the angle of reflection equal to the angle of incidence;--and they had drawn several consequences from these principles.

We may add to the consequences already mentioned, the art of perspective, which is merely a corollary from the doctrine of rectilinear visual rays; for if we suppose objects to be referred by such rays to a plane interposed between them and the eye, all the rules of perspective follow directly. The ancients practised this art, as we see in the pictures which remain to us and we learn from Vitruvius,[1\9] that they also wrote upon it. Agatharchus, who had been instructed by Eschylus in the art of making decorations for the theatre, was the first author on this subject, and Anaxagoras, who was a pupil of Agatharchus, also wrote an Actinographia, or doctrine of drawing by rays: but none of these treatises are come down to us. The moderns re-invented the art in the flourishing times of the art of painting, that is, about the end of the fifteenth century; and, belonging to that period also, we have treatises[2\9] upon it.

[Note 1\9: De Arch. ix. Mont. i. 707.]

[Note 2\9: Gauricus, 1504.]

But these are only deductive applications of the most elementary optical doctrines; we must proceed to the inductions by which further discoveries were made. {54}

II. DISCOVERY OF THE LAW OF REFRACTION

WE have seen in the former part of this history that the Greeks had formed a tolerably clear conception of the refraction as well as the reflection of the rays of light; and that Ptolemy had measured the amount of refraction of glass and water at various angles. If we give the names of the angle of incidence and the angle of refraction respectively to the angles which a ray of light makes with the line perpendicular to surface of glass or water (or any other medium) within and without the medium, Ptolemy had observed that the angle of refraction is always less than the angle of incidence. He had supposed it to be less in a given proportion, but this opinion is false; and was afterwards rightly denied by the Arabian mathematician Alhazen. The optical views which occur in the work of Alhazen are far sounder than those of his predecessors; and the book may be regarded as the most considerable monument which we have of the scientific genius of the Arabians; for it appears, for the most part, not to be borrowed from Greek authorities. The author not only asserts (lib. vii.), that refraction takes place towards the perpendicular, and refers to experiment for the truth of this: and that the quantities of the refraction differ according to the magnitudes of the angles which the directions of the incidental rays (primæ lineæ) make with the perpendiculars to the surface; but he also says distinctly and decidedly that the angles of refraction do not follow the proportion of the angles of incidence.

[2nd Ed.] [There appears to be good ground to assent to the assertion of Alhazen's originality, made by his editor Risner, who says, "Euclideum hic vel Ptolemaicum nihil fere est." Besides the doctrine of reflection and refraction of light, the Arabian author gives a description of the eye. He distinguishes three fluids, humor aqueus, crystallinus, vitreus, and four coats of the eye, tunica adherens, cornea, uvea, tunica reti similis. He distinguishes also three kinds of vision: "Visibile percipitur aut solo visu, aut visu et syllogismo, aut visu et anticipatâ notione." He has several propositions relating to what we sometimes call the Philosophy of Vision: for instance this: "E visibili sæpius viso remanet in anima generalis notio," &c.] {55}

The assertion, that the angles of refraction are not proportional to the angles of incidence, was an important remark; and if it had been steadily kept in mind, the next thing to be done with regard to refraction was to go on experimenting and conjecturing till the true law of refraction was discovered; and in the mean time to apply the principle as far as it was known. Alhazen, though he gives directions for making experimental measures of refraction, does not give any Table of the results of such experiments, as Ptolemy had done. Vitello, a Pole, who in the 13th century published an extensive work upon Optics, does give such a table; and asserts it to be deduced from experiment, as I have already said (vol. i.). But this assertion is still liable to doubt in consequence of the table containing impossible observations.

[2nd Ed.] [As I have already stated, Vitello asserts that his Tables were derived from his own observations. Their near agreement with those of Ptolemy does not make this improbable: for where the observations were only made to half a degree, there was not much room for observers to differ. It is not unlikely that the observations of refraction out of air into water and glass, and out of water into glass, were actually made; while the impossible values which accompany them, of the refraction out of water and glass into air, and out of glass into water, were calculated, and calculated from an erroneous rule.]

The principle that a ray refracted in glass or water is turned towards the perpendicular, without knowing the exact law of refraction, enabled mathematicians to trace the effects of transparent bodies in various cases. Thus in Roger Bacon's works we find a tolerably distinct explanation of the effect of a convex glass; and in the work of Vitello the effect of refraction at the two surfaces of a glass globe is clearly traceable.

Notwithstanding Alhazen's assertion of the contrary, the opinion was still current among mathematicians that the angle of refraction was proportional to the angle of incidence. But when Kepler's attention was drawn to the subject, he saw that this was plainly inconsistent with the observations of Vitello for large angles; and he convinced himself by his own experiments that the true law was something different from the one commonly supposed. The discovery of this true law excited in him an eager curiosity; and this point had the more interest for him in consequence of the introduction of a correction for atmospheric refraction into astronomical calculations, which had been made by Tycho, and of the invention of the telescope. In {56} his Supplement to Vitello, published in 1604, Kepler attempts to reduce to a rule the measured quantities of refraction. The reader who recollects what we have already narrated, the manner in which Kepler attempted to reduce to law the astronomical observations of Tycho,--devising an almost endless variety of possible formulæ, tracing their consequences with undaunted industry, and relating, with a vivacious garrulity, his disappointments and his hopes,--will not be surprised to find that he proceeded in the same manner with regard to the Tables of Observed Refractions. He tried a variety of constructions by triangles, conic sections, &c., without being able to satisfy himself; and he at last[3\9] is obliged to content himself with an approximate rule, which makes the refraction partly proportional to the angle of incidence, and partly, to the secant of that angle. In this way he satisfies the observed refractions within a difference of less than half a degree each way. When we consider how simple the law of refraction is, (that the ratio of the sines of the angles of incidence and refraction is constant for the same medium,) it appears strange that a person attempting to discover it, and drawing triangles for the purpose, should fail; but this lot of missing what afterwards seems to have been obvious, is a common one in the pursuit of truth.

[Note 3\9: L. U. K. Life of Kepler, p. 115.]

The person who did discover the Law of the Sines, was Willebrord Snell, about 1621; but the law was first published by Descartes, who had seen Snell's papers.[4\9] Descartes does not acknowledge this law to have been first detected by another; and after his manner, instead of establishing its reality by reference to experiment, he pretends to prove à priori that it must be true,[5\9] comparing, for this purpose, the particles of light to balls striking a substance which accelerates them.

[Note 4\9: Huyghens, Dioptrica, p. 2.]

[Note 5\9: Diopt. p. 53.]

[2nd Ed.] [Huyghens says of Snell's papers, "Quæ et nos vidimus aliquando, et Cartesium quoque vidisse accepimus, et hinc fortasse mensuram illam quæ in sinibus consistit elicuerit." Isaac Vossius, De Lucis Naturâ et Proprietate, says that he also had seen this law in Snell's unpublished optical Treatise. The same writer says, "Quod itaque (Cartesius) habet, refractionum momenta non exigenda esse ad angulos sed ad lineas, id tuo Snellio, acceptum ferre debuisset, cujus nomen more solito dissimulavit." "Cartesius got his law from Snell, and in his usual way, concealed it." {57}

Huyghens' assertion, that Snell did not attend to the proportion of the sines, is very captious; and becomes absurdly so, when it is made to mean that Snell did not know the law of the sines. It is not denied that Snell knew the true law, or that the true law is the law of the sines. Snell does not use the trigonometrical term sine, but he expresses the law in a geometrical form more simply. Even if he had attended to the law of the sines, he might reasonably have preferred his own way of stating it.

James Gregory also independently discovered the true law of refraction; and, in publishing it, states that he had learnt that it had already been published by Descartes.]

But though Descartes does not, in this instance, produce any good claims to the character of an inductive philosopher, he showed considerable skill in tracing the consequences of the principle when once adopted. In particular we must consider him as the genuine author of the explanation of the rainbow. It is true that Fleischer[6\9] and Kepler had previously ascribed this phenomenon to the rays of sunlight which, falling on drops of rain, are refracted into each drop, reflected at its inner surface, and refracted out again: Antonio de Dominis had found that a glass globe of water, when placed in a particular position with respect to the eye, exhibited bright colors; and had hence explained the circular form of the bow, which, indeed, Aristotle had done before.[7\9] But none of these writers had shown why there was a narrow bright circle of a definite diameter; for the drops which send rays to the eye after two refractions and a reflection, occupy a much wider space in the heavens. Descartes assigned the reason for this in the most satisfactory manner,[8\9] by showing that the rays which, after two refractions and a reflection, come to the eye at an angle of about forty-one degrees with their original direction, are far more dense than those in any other position. He showed, in the same manner, that the existence and position of the secondary bow resulted from the same laws. This is the complete and adequate account of the state of things, so far as the brightness of the bows only is concerned; the explanation of the colors belongs to the next article of our survey.

[Note 6\9: Mont. i. 701.]

[Note 7\9: Meteorol. iii. 3.]

[Note 8\9: Meteorum, cap. viii. p. 196.]

The explanation of the rainbow and of its magnitude, afforded by Snell's law of sines, was perhaps one of the leading points in the verification of the law. The principle, being once established, was applied, by the aid of mathematical reasoning, to atmospheric refractions, {58} optical instruments, diacaustic curves, (that is, the curves of intense light produced by refraction,) and to various other cases; and was, of course, tested and confirmed by such applications. It was, however, impossible to pursue these applications far, without a due knowledge of the laws by which, in such cases, colors are produced. To these we now proceed.

[2nd Ed.] [I have omitted many interesting parts of the history of Optics about this period, because I was concerned with the inductive discovery of laws, rather than with mathematical deductions from such laws when established, or applications of them in the form of instruments. I might otherwise have noticed the discovery of Spectacle Glasses, of the Telescope, of the Microscope, of the Camera Obscura, and the mathematical explanation of these and other phenomena, as given by Kepler and others. I might also have noticed the progress of knowledge respecting the Eye and Vision. We have seen that Alhazen described the structure of the eye. The operation of the parts was gradually made out. Baptista Porta compares the eye to his Camera Obscura (Magia Naturalis, 1579). Scheiner, in his Oculus, published 1652, completed the Theory of the Eye. And Kepler discussed some of the questions even now often agitated; as the causes and conditions of our seeing objects single with two eyes, and erect with inverted images.]

III. DISCOVERY OF THE LAW OF DISPERSION BY REFRACTION

EARLY attempts were made to account for the colors of the rainbow, and various other phenomena in which colors are seen to arise from transient and unsubstantial combinations of media. Thus Aristotle explains the colors of the rainbow by supposing[9\9] that it is light seen through a dark medium: "Now," says he, "the bright seen through the dark appears red, as, for instance, the fire of green wood seen through the smoke, and the sun through mist. Also[10\9] the weaker is the light, or the visual power, and the nearer the color approaches to the black; becoming first red, then green, then purple. But[11\9] the {59} vision is strongest in the outer circle, because the periphery is greater;--thus we shall have a gradation from red, through green, to purple, in passing from the outer to the inner circle." This account would hardly have deserved much notice, if it had not been for a strange attempt to revive it, or something very like it, in modern times. The same doctrine is found in the work of De Dominis.[12\9] According to him, light is white: but if we mix with the light something dark, the colors arise,--first red, then green, then blue or violet. He applies this to explain the colors of the rainbow,[13\9] by means of the consideration that, of the rays which come to the eye from the globes of water, some go through a larger thickness of the globe than others, whence he obtains the gradation of colors just described.

[Note 9\9: Meteor. iii. 3, p. 373.]

[Note 10\9: Ib. p. 374.]

[Note 11\9: Ib. p. 375.]

[Note 12\9: Cap. iii. p. 9. See also Göthe, Farbenl. vol. ii. p. 251.]

[Note 13\9: Göthe, p. 263.]

Descartes came far nearer the true philosophy of the iridal colors. He found that a similar series of colors was produced by refraction of light bounded by shade, through a prism;[14\9] and he rightly inferred that neither the curvature of the surface of the drops of water, nor the reflection, nor the repetition of refraction, were necessary to the generation of such colors. In further examining the course of the rays, he approaches very near to the true conception of the case; and we are led to believe that he might have anticipated Newton in his discovery of the unequal refrangibility of different colors, if it had been possible for him to reason any otherwise than in the terms and notions of his preconceived hypotheses. The conclusion which he draws is,[15\9] that "the particles of the subtile matter which transmit the action of light, endeavor to rotate with so great a force and impetus, that they cannot move in a straight line (whence comes refraction): and that those particles which endeavor to revolve much more strongly produce a red color, those which endeavor to move only a little more strongly produce yellow." Here we have a clear perception that colors and unequal refraction are connected, though the cause of refraction is expressed by a gratuitous hypothesis. And we may add, that he applies this notion rightly, so far as he explains himself,[16\9] to account for the colors of the rainbow.

[Note 14\9: Meteor. Sect. viii. p. 190.]

[Note 15\9: Sect. vii. p. 192.]

[Note 16\9: Meteor. Sect. ix.]

It appears to me that Newton and others have done Descartes injustice, in ascribing to De Dominis the true theory of the rainbow. There are two main points of this theory, namely, the showing that a bright circular band, of a certain definite diameter, arises from the {60} great intensity of the light returned at a certain angle; and the referring the different colors to the different quantity of the refraction; and both these steps appear indubitably to be the discoveries of Descartes. And he informs us that these discoveries were not made without some exertion of thought. "At first," he says,[17\9] "I doubted whether the iridal colors were produced in the same way as those in the prism; but, at last, taking my pen, and carefully calculating the course of the rays which fell on each part of the drop, I found that many more come at an angle of forty-one degrees, than either at a greater or a less angle. So that there is a bright bow terminated by a shade; and hence the colors are the same as those produced through a prism."

[Note 17\9: Sect. ix. p. 193.]

The subject was left nearly in the same state, in the work of Grimaldi, Physico-Mathesis, de Lumine, Coloribus et Iride, published at Bologna in 1665. There is in this work a constant reference to numerous experiments, and a systematic exposition of the science in an improved state. The author's calculations concerning the rainbow are put in the same form as those of Descartes; but he is further from seizing the true principle on which its coloration depends. He rightly groups together a number of experiments in which colors arise from refraction;[18\9] and explains them by saying that the color is brighter where the light is denser: and the light is denser on the side from which the refraction turns the ray, because the increments of refraction are greater in the rays that are more inclined.[19\9] This way of treating the question might be made to give a sort of explanation of most of the facts, but is much more erroneous than a developement of Descartes's view would have been.

[Note 18\9: Prop. 35, p. 254.]

[Note 19\9: Ib. p. 256.]

At length, in 1672, Newton gave[20\9] the true explanation of the facts; namely, that light consists of rays of different colors and different refrangibility. This now appears to us so obvious a mode of interpreting the phenomena, that we can hardly understand how they can be conceived in any other manner; but yet the impression which this discovery made, both upon Newton and upon his contemporaries, shows how remote it was from the then accepted opinions. There appears to have been a general persuasion that the coloration was produced, not by any peculiarity in the law of refraction itself but by some collateral circumstance,--some dispersion or variation of density of the light, in addition to the refraction. Newton's discovery consisted in {61} teaching distinctly that the law of refraction was to be applied, not to the beam of light in general, but to the colors in particular.

[Note 20\9: Phil. Trans. t. vii. p. 3075.]

When Newton produced a bright spot on the wall of his chamber, by admitting the sun's light through a small hole in his window-shutter, and making it pass through a prism, he expected the image to be round; which, of course, it would have been, if the colors had been produced by an equal dispersion in all directions; but to his surprise he saw the image, or spectrum, five times as long as it was broad. He found that no consideration of the different thickness of the glass, the possible unevenness of its surface, or the different angles of rays proceeding from the two sides of the sun, could be the cause of this shape. He found, also, that the rays did not go from the prism to the image in curves; he was then convinced that the different colors were refracted separately, and at different angles; and he confirmed this opinion by transmitting and refracting the rays of each color separately.

The experiments are so easy and common, and Newton's interpretation of them so simple and evident, that we might have expected it to receive general assent; indeed, as we have shown, Descartes had already been led very near the same point. In fact, Newton's opinions were not long in obtaining general acceptance; but they met with enough of cavil and misapprehension to annoy extremely the discoverer, whose clear views and quiet temper made him impatient alike of stupidity and of contentiousness.

We need not dwell long on the early objections which were made to Newton's doctrine. A Jesuit, of the name of Ignatius Pardies, professor at Clermont, at first attempted to account for the elongation of the image by the difference of the angles made by the rays from the two edges of the sun, which would produce a difference in the amount of refraction of the two borders; but when Newton pointed out the calculations which showed the insufficiency of this explanation, he withdrew his opposition. Another more pertinacious opponent appeared in Francis Linus, a physician of Liege; who maintained, that having tried the experiment, he found the sun's image, when the sky was clear, to be round and not oblong; and he ascribed the elongation noticed by Newton, to the effect of clouds. Newton for some time refused to reply to this contradiction of his assertions, though obstinately persisted in; and his answer was at last sent, just about the time of Linus's death, in 1675. But Gascoigne, a friend of Linus, still maintained that he and others had seen what the Dutch physician had described; and Newton, who was pleased with the candor of {62} Gascoigne's letter, suggested that the Dutch experimenters might have taken one of the images reflected from the surfaces of the prism, of which there are several, instead of the proper refracted one. By the aid of this hint, Lucas of Liege repeated Newton's experiments, and obtained Newton's result, except that he never could obtain a spectrum whose length was more than three and a half times its breadth. Newton, on his side, persisted in asserting that the image would be five times as long as it was broad, if the experiment were properly made. It is curious that he should have been so confident of this, as to conceive himself certain that such would be the result in all cases. We now know that the dispersion, and consequently the length, of the spectrum, is very different for different kinds of glass, and it is very probable that the Dutch prism was really less dispersive than the English one.[21\9] The erroneous assumption which Newton made in this instance, he held by to the last; and was thus prevented from making the discovery of which we have next to speak.

[Note 21\9: Brewster's Newton, p. 50.]

Newton was attacked by persons of more importance than those we have yet mentioned; namely, Hooke and Huyghens. These philosophers, however, did not object so much to the laws of refraction of different colors, as to some expressions used by Newton, which, they conceived, conveyed false notions respecting the composition and nature of light. Newton had asserted that all the different colors are of distinct kinds, and that, by their composition, they form white light. This is true of colors as far as their analysis and composition by refraction are concerned; but Hooke maintained that all natural colors are produced by various combinations of two primary ones, red and violet;[22\9] and Huyghens held a similar doctrine, taking, however, yellow and blue for his basis. Newton answers, that such compositions as they speak of are not compositions of simple colors in his sense of the expressions. These writers also had both of them adopted an opinion that light consisted in vibrations; and objected to Newton that his language was erroneous, as involving the hypothesis that light was a body. Newton appears to have had a horror of the word hypothesis, and protests against its being supposed that his "theory" rests on such a foundation.

[Note 22\9: Brewster's Newton, p. 54. Phil. Trans. viii. 5084, 6086.]

The doctrine of the unequal refrangibility of different rays is clearly exemplified in the effects of lenses, which produce images more or {63} less bordered with color, in consequence of this property. The improvement of telescopes was, in Newton's time, the great practical motive for aiming at the improvement of theoretical optics. Newton's theory showed why telescopes were imperfect, namely, in consequence of the different refraction of different colors, which produces a chromatic aberration: and the theory was confirmed by the circumstances of such imperfections. The false opinion of which we have already spoken, that the dispersion must be the same when the refraction is the same, led him to believe that the imperfection was insurmountable,--that achromatic refraction could not be obtained: and this view made him turn his attention to the construction of reflecting instead of refracting telescopes. But the rectification of Newton's error was a further confirmation of the general truth of his principles in other respects; and since that time, the soundness of the Newtonian law of refraction has hardly been questioned among physical philosophers.

It has, however, in modern times, been very vehemently controverted in a quarter from which we might not readily have expected a detailed discussion on such a subject. The celebrated Göthe has written a work on The Doctrine of Colors, (Farbenlehre; Tübingen, 1810,) one main purpose of which is, to represent Newton's opinions, and the work in which they are formally published, (his Opticks,) as utterly false and mistaken, and capable of being assented to only by the most blind and obstinate prejudice. Those who are acquainted with the extent to which such an opinion, promulgated by Göthe, was likely to be widely adopted in Germany, will not be surprised that similar language is used by other writers of that nation. Thus Schelling[23\9] says: "Newton's Opticks is the greatest proof of the possibility of a whole structure of fallacies, which, in all its parts, is founded upon observation and experiment." Göthe, however, does not concede even so much to Newton's work. He goes over a large portion of it, page by page, quarrelling with the experiments, diagrams, reasoning, and language, without intermission; and holds that it is not reconcileable with the most simple facts. He declares,[24\9] that the first time he looked through a prism, he saw the white walls of the room still look white, "and though alone, I pronounced, as by an instinct, that the Newtonian doctrine is false." We need not here point out how inconsistent with the Newtonian doctrine it was, to expect, as Göthe expected, that the wall should be all over colored various colors. {64}

[Note 23\9: Vorlesungen, p. 270.]

[Note 24\9: Farbenlehre, vol. ii. p. 678.]

Göthe not only adopted and strenuously maintained the opinion that the Newtonian theory was false, but he framed a system of his own to explain the phenomena of color. As a matter of curiosity, it may be worth our while to state the nature of this system; although undoubtedly it forms no part of the progress of physical science. Göthe's views are, in fact, little different from those of Aristotle and Antonio de Dominis, though more completely and systematically developed. According to him, colors arise when we see through a dim medium ("ein trübes mittel"). Light in itself is colorless; but if it be seen through a somewhat dim medium, it appears yellow; if the dimness of the medium increases, or if its depth be augmented, we see the light gradually assume a yellow-red color, which finally is heightened to a ruby-red. On the other hand, if darkness is seen through a dim medium which is illuminated by a light falling on it, a blue color is seen, which becomes clearer and paler, the more the dimness of the medium increases, and darker and fuller, as the medium becomes more transparent; and when we come to "the smallest degree of the purest dimness," we see the most perfect violet.[25\9] In addition to this "doctrine of the dim medium," we have a second principle asserted concerning refraction. In a vast variety of cases, images are accompanied by "accessory images," as when we see bright objects in a looking-glass.[26\9] Now, when an image is displaced by refraction, the displacement is not complete, clear and sharp, but incomplete, so that there is an accessory image along with the principal one.[27\9] From these principles, the colors produced by refraction in the image of a bright object on a dark ground, are at once derivable. The accessory image is semitransparent;[28\9] and hence that border of it which is pushed forwards, is drawn from the dark over the bright, and there the yellow appears; on the other hand, where the clear border laps over the dark ground, the blue is seen;[29\9] and hence we easily see that the image must appear red and yellow at one end, and blue and violet at the other.

[Note 25\9: Farbenlehre, § 150, p. 151.]

[Note 26\9: Ib. § 223.]

[Note 27\9: Ib. § 227.]

[Note 28\9: Ib. § 238.]

[Note 29\9: Ib. § 239.]

We need not explain this system further, or attempt to show how vague and loose, as well as baseless, are the notions and modes of conception which it introduces. Perhaps it is not difficult to point out the peculiarities in Göthe's intellectual character which led to his singularly unphilosophical views on this subject. One important {65} circumstance is, that he appears, like many persons in whom the poetical imagination is very active, to have been destitute of the talent and the habit of geometrical thought. In all probability, he never apprehended clearly and steadily those relations of position on which the Newtonian doctrine depends. Another cause of his inability to accept the doctrine probably was, that he had conceived the "composition" of colors in some way altogether different from that which Newton understands by composition. What Göthe expected to see, we cannot clearly collect; but we know, from his own statement, that his intention of experimenting with a prism arose from his speculations on the roles of coloring in pictures; and we can easily see that any notion of the composition of colors which such researches would suggest, would require to be laid aside, before he could understand Newton's theory of the composition of light.

Other objections to Newton's theory, of a kind very different, have been recently made by that eminent master of optical science, Sir David Brewster. He contests Newton's opinion, that the colored rays into which light is separated by refraction are altogether simple and homogeneous, and incapable of being further analysed and modified. For he finds that by passing such rays through colored media (as blue glass for instance), they are not only absorbed and transmitted in very various degrees, but that some of them have their color altered; which effect he conceives as a further analysis of the rays, one component color being absorbed and the other transmitted.[30\9] And on this subject we can only say, as we have before said, that Newton has incontestably and completely established his doctrine, so far as analysis and decomposition by refraction are concerned; but that with regard to any other analysis, which absorbing media or other agents may produce, we have no right from his experiments to assert, that the colors of the spectrum are incapable of such decomposition. The whole subject of the colors of objects, both opake and transparent, is still in obscurity. Newton's conjectures concerning the causes of the colors of natural bodies, appear to help us little; and his opinions on that subject are to be separated altogether from the important step which he made in optical science, by the establishment of the true doctrine of refractive dispersion.

[Note 30\9: This latter fact has, however, been denied by other experimenters.]

[2nd Ed.] [After a careful re-consideration of Sir D. Brewster's asserted analysis of the solar light into three colors by means of {66} absorbing media, I cannot consider that he has established his point as an exception to Newton's doctrine. In the first place, the analysis of light into three colors appears to be quite arbitrary, granting all his experimental facts. I do not see why, using other media, he might not just as well have obtained other elementary colors. In the next place, this cannot be called an analysis in the same sense as Newton's analysis, except the relation between the two is shown. Is it meant that Newton's experiments prove nothing? Or is Newton's conclusion allowed to be true of light which has not been analysed by absorption? And where are we to find such light, since the atmosphere absorbs? But, I must add, in the third place, that with a very sincere admiration of Sir D. Brewster's skill as an experimenter, I think his experiment requires, not only limitation, but confirmation by other experimenters. Mr. Airy repeated the experiments with about thirty different absorbing substances, and could not satisfy himself that in any case they changed the color of a ray of given refractive power. These experiments were described by him at a meeting of the Cambridge Philosophical Society.]

We now proceed to the corrections which the next generation introduced into the details of this doctrine.

IV. DISCOVERY OF ACHROMATISM

THE discovery that the laws of refractive dispersion of different substances were such as to allow of combinations which neutralised the dispersion without neutralizing the refraction, is one which has hitherto been of more value to art than to science. The property has no definite bearing, which has yet been satisfactorily explained, upon the theory of light; but it is of the greatest importance in its application to the construction of telescopes; and it excited the more notice, in consequence of the prejudices and difficulties which for a time retarded the discovery.

Newton conceived that he had proved by experiment,[31\9] that light {67} is white after refraction, when the emergent rays are parallel to the incident, and in no other case. If this were so, the production of colorless images by refracting media would be impossible; and such, in deference to Newton's great authority, was for some time the general persuasion. Euler[32\9] observed, that a combination of lenses which does not color the image must be possible, since we have an example of such a combination in the human eye; and he investigated mathematically the conditions requisite for such a result. Klingenstierna,[33\9] a Swedish mathematician, also showed that Newton's rule could not be universally true. Finally, John Dollond,[34\9] in 1757, repeated Newton's experiment, and obtained an opposite result. He found that when an object was seen through two prisms, one of glass and one of water, of such angles that it did not appear displaced by refraction, it was colored. Hence it followed that, without being colored, the rays might be made to undergo refraction; and that thus, substituting lenses for prisms, a combination might be formed, which should produce an image without coloring it, and make the construction of an achromatic telescope possible.

[Note 31\9: Opticks, B. i. p. ii. Prop. 3.]

[Note 32\9: Ac. Berlin. 1747.]

[Note 33\9: Swedish Trans. 1754.]

[Note 34\9: Phil. Trans. 1758.]

Euler at first hesitated to confide in Dollond's experiments; but he was assured of their correctness by Clairaut, who had throughout paid great attention to the subject; and those two great mathematicians, as well as D'Alembert, proceeded to investigate mathematical formulæ which might be useful in the application of the discovery. The remainder of the deductions, which were founded upon the laws of dispersion of various refractive substances, belongs rather to the history of art than of science. Dollond used at first, for his achromatic object-glass, a lens of crown-glass, and one of flint-glass. He afterwards employed two lenses of the former substance, including between them one of the latter, adjusting the curvatures of his lenses in such a way as to correct the imperfections arising from the spherical form of the glasses, as well as the fault of color. Afterwards, Blair used fluid media along with glass lenses, in order to produce improved object-glasses. This has more recently been done in another form by Mr. Barlow. The inductive laws of refraction being established, their results have been deduced by various mathematicians, as Sir J. Herschel and Professor Airy among ourselves, who have simplified and extended the investigation of the formulæ which determine the best combination of lenses in the object-glasses and eye-glasses of {68} telescopes, both with reference to spherical and to chromatic aberrations.

According to Dollond's discovery, the colored spectra produced by prisms of two substances, as flint-glass and crown-glass, would be of the same length when the refraction was different. But a question then occurred: When the whole distance from the red to the violet in one spectrum was the same as the whole distance in the other, were the intermediate colors, yellow, green, &c., in corresponding places in the two? This point also could not be determined any otherwise than by experiment. It appeared that such a correspondence did not exist; and, therefore, when the extreme colors were corrected by combinations of the different media, there still remained an uncorrected residue of color arising from the rest of the spectrum. This defect was a consequence of the property, that the spectra belonging to different media were not divided in the same ratio by the same colors, and was hence termed the irrationality of the spectrum. By using three prisms, or three lenses, three colors may be made to coincide instead of two, and the effects of this irrationality greatly diminished.

For the reasons already mentioned, we do not pursue this subject further,[35\9] but turn to those optical facts which finally led to a great and comprehensive theory.

[Note 35\9: The discovery of the fixed lines in the spectrum, by Wollaston and Fraunhofer, has more recently supplied the means of determining, with extreme accuracy, the corresponding portions of the spectrum in different refracting substances.]

[2nd Ed.] [Mr. Chester More Hall, of More Hall, in Essex, is said to have been led by the study of the human eye, which he conceived to be achromatic, to construct achromatic telescopes as early as 1729. Mr. Hall, however, kept his invention a secret. David Gregory, in his Catoptrics (1713), had suggested that it would perhaps be an improvement of telescopes, if, in imitation of the human eye, the object-glass were composed of different media. Encyc. Brit. art. Optics.

It is said that Clairaut first discovered the irrationality of the colored spaces in the spectrum. In consequence of this irrationality, it follows that when two refracting media are so combined as to correct each other's extreme dispersion, (the separation of the red and violet rays,) this first step of correction still leaves a residue of {69} coloration arising from the unequal dispersion of the intermediate rays (the green, &c.). These outstanding colors, as they were termed by Professor Robison, form the residual, or secondary spectrum.

Dr. Blair, by very ingenious devices, succeeded in producing an object-glass, corrected by a fluid lens, in which this aberration of color was completely corrected, and which performed wonderfully well.

The dispersion produced by a prism may be corrected by another prism of the same substance and of a different angle. In this case also there is an irrationality in the colored spaces, which prevents the correction of color from being complete; and hence, a new residuary spectrum, which has been called the tertiary spectrum, by Sir David Brewster, who first noticed it.

I have omitted, in the notice of discoveries respecting the spectrum, many remarkable trains of experimental research, and especially the investigations respecting the power of various media to absorb the light of different parts of the spectrum, prosecuted by Sir David Brewster with extraordinary skill and sagacity. The observations are referred to in chapter iii. Sir John Herschel, Prof. Miller, Mr. Daniel, Dr. Faraday, and Mr. Talbot, have also contributed to this part of our knowledge.]

V. DISCOVERY OF THE LAWS OF DOUBLE REFRACTION

THE laws of refraction which we have hitherto described, were simple and uniform, and had a symmetrical reference to the surface of the refracting medium. It appeared strange to men, when their attention was drawn to a class of phenomena in which this symmetry was wanting, and in which a refraction took place which was not even in the plane of incidence. The subject was not unworthy the notice and admiration it attracted; for the prosecution of it ended in the discovery of the general laws of light. The phenomena of which I now speak, are those exhibited by various kinds of crystalline bodies; but observed for a long time in one kind only, namely, the rhombohedral calc-spar; or, as it was usually termed, from the country which supplied the largest and clearest crystals, Iceland spar. These {70} rhombohedral crystals are usually very smooth and transparent, and often of considerable size; and it was observed, on looking through them, that all objects appeared double. The phenomena, even as early as 1669, had been considered so curious, that Erasmus Bartholin published a work upon them at Copenhagen,[36\9] (Experimenta Crystalli Islandici, Hafniæ, 1669.) He analysed the phenomena into their laws, so far as to discover that one of the two images was produced by refraction after the usual rule, and the other by an unusual refraction. This latter refraction Bartholin found to vary in different positions; to be regulated by a line parallel to the sides of the rhombohedron; and to be greatest in the direction of a line bisecting two of the angles of the rhombic face of the crystal.

[Note 36\9: Priestley's Optics, p. 550.]

These rules were exact as far as they went; and when we consider how geometrically complex the law is, which really regulates the unusual or extraordinary refraction;--that Newton altogether mistook it, and that it was not verified till the experiments of Haüy and Wollaston in our own time;--we might expect that it would not be soon or easily detected. But Huyghens possessed a key to the secret, in the theory, which he had devised, of the propagation of light by undulations, and which he conceived with perfect distinctness and correctness, so far as its application to these phenomena is concerned. Hence he was enabled to lay down the law of the phenomena (the only part of his discovery which we have here to consider), with a precision and success which excited deserved admiration, when the subject, at a much later period, regained its due share of attention. His Treatise was written[37\9] in 1678, but not published till 1690.

[Note 37\9: See his Traité de la Lumière. Preface.]

The laws of the ordinary and the extraordinary refraction in Iceland spar are related to each other; they are, in fact, similar constructions, made, in the one case, by means of an imaginary sphere, in the other, by means of a spheroid; the spheroid being of such oblateness as to suit the rhombohedral form of the crystal, and the axis of the spheroid being the axis of symmetry of the crystal. Huyghens followed this general conception into particular positions and conditions; and thus obtained rules, which he compared with observation, for cutting the crystal and transmitting the rays in various manners. "I have examined in detail," says he,[38] "the properties of the {71} extraordinary refraction of this crystal, to see if each phenomenon which is deduced from theory, would agree with what is really observed. And this being so, it is no slight proof of the truth of our suppositions and principles; but what I am going to add here confirms them still more wonderfully; that is, the different modes of cutting this crystal, in which the surfaces produced give rise to refractions exactly such as they ought to be, and as I had foreseen them, according to the preceding theory."

[Note 38\9: See Maseres's Tracts on Optics, p. 250; or Huyghens, Tr. sur la Lum. ch. v. Art. 43.]

Statements of this kind, coming from a philosopher like Huyghens, were entitled to great confidence; Newton, however, appears not to have noticed, or to have disregarded them. In his Opticks, he gives a rule for the extraordinary refraction of Iceland spar which is altogether erroneous, without assigning any reason for rejecting the law published by Huyghens; and, so far as appears, without having made any experiments of his own. The Huyghenian doctrine of double refraction fell, along with his theory of undulations, into temporary neglect, of which we shall have hereafter to speak. But in 1788, Haüy showed that Huyghens's rule agreed much better than Newton's with the phenomena: and in 1802, Wollaston, applying a method of his own for measuring refraction, came to the same result. "He made," says Young,[39\9] "a number of accurate experiments with an apparatus singularly well calculated to examine the phenomena, but could find no general principle to connect them, until the work of Huyghens was pointed out to him." In 1808, the subject of double refraction was proposed as a prize-question by the French Institute; and Malus, whose Memoir obtained the prize, says, "I began by observing and measuring a long series of phenomena on natural and artificial faces of Iceland spar. Then, testing by means of these observations the different laws proposed up to the present time by physical writers, I was struck with the admirable agreement of the law of Huyghens with the phenomena, and I was soon convinced that it is really the law of nature." Pursuing the consequences of the law, he found that it satisfied phenomena which Huyghens himself had not observed. From this time, then, the truth of the Huyghenian law was universally allowed, and soon afterwards, the theory by which it had been suggested was generally received.

[Note 39\9: Quart. Rev. 1809, Nov. p. 338.]

The property of double refraction had been first studied only in Iceland spar, in which it is very obvious. The same property belongs, {72} though less conspicuously, to many other kinds of crystals. Huyghens had noticed the same fact in rock-crystal;[40\9] and Malus found it to belong to a large list of bodies besides; for instance, arragonite, sulphate of lime, of baryta, of strontia, of iron; carbonate of lead; zircon, corundum, cymophane, emerald, euclase, felspar, mesotype, peridote, sulphur, and mellite. Attempts were made, with imperfect success, to reduce all these to the law which had been established for Iceland spar. In the first instance, Malus took for granted that the extraordinary refraction depended always upon an oblate spheroid; but M. Biot[41\9] pointed out a distinction between two classes of crystals in which this spheroid was oblong and oblate respectively, and these he called attractive and repulsive crystals. With this correction, the law could be extended to a considerable number of cases; but it was afterwards proved by Sir D. Brewster's discoveries, that even in this form, it belonged only to substances of which the crystallization has relation to a single axis of symmetry, as the rhombohedron, or the square pyramid. In other cases, as the rhombic prism, in which the form, considered with reference to its crystalline symmetry, is biaxal, the law is much more complicated. In that case, the sphere and the spheroid, which are used in the construction for uniaxal crystals, transform themselves into the two successful convolutions of a single continuous curve surface; neither of the two rays follows the law of ordinary refraction; and the formula which determines their position is very complex. It is, however, capable of being tested by measures of the refractions of crystals cut in a peculiar manner for the purpose, and this was done by MM. Fresnel and Arago. But this complex law of double refraction was only discovered through the aid of the theory of a luminiferous ether, and therefore we must now return to the other facts which led to such a theory.

[Note 40\9: Traité de la Lumière, ch. v. Art. 20]

[Note 41\9: Biot, Traité de Phys. iii. 330.]

VI. DISCOVERY OF THE LAWS OF POLARIZATION

IF the Extraordinary Refraction of Iceland spar had appeared strange, another phenomenon was soon noticed in the same {73} substance, which appeared stranger still, and which in the sequel was found to be no less important. I speak of the facts which were afterwards described under the term Polarization. Huyghens was the discoverer of this class of facts. At the end of the treatise which we have already quoted, he says,[42\9] "Before I quit the subject of this crystal, I will add one other marvellous phenomenon, which I have discovered since writing the above; for though hitherto I have not been able to find out its cause, I will not, on that account, omit pointing it out, that I may give occasion to others to examine it." He then states the phenomena; which are, that when two rhombohedrons of Iceland spar are in parallel positions, a ray doubly refracted by the first, is not further divided when it falls on the second: the ordinarily refracted ray is ordinarily refracted only, and the extraordinary ray is only extraordinarily refracted by the second crystal, neither ray being doubly refracted. The same is still the case, if the two crystals have their principal planes parallel, though they themselves are not parallel. But if the principal plane of the second crystal be perpendicular to that of the first, the reverse of what has been described takes place; the ordinarily refracted ray of the first crystal suffers, at the second, extraordinary refraction only, and the extraordinary ray of the first suffers ordinary refraction only at the second. Thus, in each of these positions, the double refraction of each ray at the second crystal is reduced to single refraction, though in a different manner in the two cases. But in any other position of the crystals, each ray, produced by the first, is doubly refracted by the second, so as to produce four rays.

[Note 42\9: Tr. Opt. p. 252.]

A step in the right conception of these phenomena was made by Newton, in the second edition of his Opticks (1717). He represented them as resulting from this;--that the rays of light have "sides," and that they undergo the ordinary or extraordinary refraction, according as these sides are parallel to the principal plane of the crystal, or at right angles to it (Query 26). In this way, it is clear, that those rays which, in the first crystal, had been selected for extraordinary refraction, because their sides were perpendicular to the principal plane, would all suffer extraordinary refraction at the second crystal for the same reason, if its principal plane were parallel to that of the first; and would all suffer ordinary refraction, if the principal plane of the second crystal were perpendicular to that of the first, and {74} consequently parallel to the sides of the refracted ray. This view of the subject includes some of the leading features of the case, but still leaves several considerable difficulties.

No material advance was made in the subject till it was taken up by Malus,[43\9] along with the other circumstances of double refraction, about a hundred years afterwards. He verified what had been observed by Huyghens and Newton, on the subject of the variations which light thus exhibits; but he discovered that this modification, in virtue of which light undergoes the ordinary, or the extraordinary, refraction, according to the position of the plane of the crystal, may be impressed upon it many other ways. One part of this discovery was made accidentally.[44\9] In 1808, Malus happened to be observing the light of the setting sun, reflected from the windows of the Luxembourg, through a rhombohedron of Iceland spar; and he observed that in turning round the crystal, the two images varied in their intensity. Neither of the images completely vanished, because the light from the windows was not properly modified, or, to use the term which Malus soon adopted, was not completely polarized. The complete polarization of light by reflection from glass, or any other transparent substance, was found to take place at a certain definite angle, different for each substance. It was found also that in all crystals in which double refraction occurred, the separation of the refracted rays was accompanied by polarization; the two rays, the ordinary and the extraordinary, being always polarized oppositely, that is, in planes at right angles to each other. The term poles, used by Malus, conveyed nearly the same notion as the term sides which had been employed by Newton, with the additional conception of a property which appeared or disappeared according as the poles of the particles were or were not in a certain direction; a property thus resembling the polarity of magnetic bodies. When a spot of polarized light is looked at through a transparent crystal of Iceland spar, each of the two images produced by the double refraction varies in brightness as the crystal is turned round. If, for the sake of example, we suppose the crystal to be turned round in the direction of the points of the compass, N, E, S, W, and if one image be brightest when the crystal marks N and S, it will disappear when the crystal marks E and W: and on the contrary, the second image will vanish when the crystal marks N and S, {75} and will be brightest when the crystal marks E and W. The first of these images is polarized in the plane NS passing through the ray, and the second in the plane EW, perpendicular to the other. And these rays are oppositely polarized. It was further found that whether the ray were polarized by reflection from glass, or from water, or by double refraction, the modification of light so produced, or the nature of the polarization, was identical in all these cases;--that the alternatives of ordinary and extraordinary refraction and non-refraction, were the same, by whatever crystal they were tested, or in whatever manner the polarization had been impressed upon the light; in short, that the property, when once acquired, was independent of everything except the sides or poles of the ray; and thus, in 1811, the term "polarization" was introduced.[45\9]

[Note 43\9: Malus, Th. de la Doub. Réf. p. 296.]

[Note 44\9: Arago, art. Polarization, Supp. Enc. Brit.]

[Note 45\9: Mém. Inst. 1810.]

This being the state of the subject, it became an obvious question, by what other means, and according to what laws, this property was communicated. It was found that some crystals, instead of giving, by double refraction, two images oppositely polarized, give a single polarized image. This property was discovered in the agate by Sir D. Brewster, and in tourmaline by M. Biot and Dr. Seebeck. The latter mineral became, in consequence, a very convenient part of the apparatus used in such observations. Various peculiarities bearing upon this subject, were detected by different experimenters. It was in a short time discovered, that light might be polarized by refraction, as well as by reflection, at the surface of uncrystallized bodies, as glass; the plane of polarization being perpendicular to the plane of refraction; further, that when a portion of a ray of light was polarized by reflection, a corresponding portion was polarized by transmission, the planes of the two polarizations being at right angles to each other. It was found also that the polarization which was incomplete with a single plate, either by reflection or refraction, might be made more and more complete by increasing the number of plates.

Among an accumulation of phenomena like this, it is our business to inquire what general laws were discovered. To make such discoveries without possessing the general theory of the facts, required no ordinary sagacity and good fortune. Yet several laws were detected at this stage of the subject. Malus, in 1811, obtained the important generalization that, whenever we obtain, by any means, a polarized ray of light, we produce also another ray, polarized in a contrary {76} direction; thus when reflection gives a polarized ray, the companion-ray is refracted polarized oppositely, along with a quantity of unpolarized light. And we must particularly notice Sir D. Brewster's rule for the polarizing angle of different bodies.

Malus[46\9] had said that the angle of reflection from transparent bodies which most completely polarizes the reflected ray, does not follow any discoverable rule with regard to the order of refractive or dispersive powers of the substances. Yet the rule was in reality very simple. In 1815, Sir D. Brewster stated[47\9] as the law, which in all cases determines this angle, that "the index of refraction is the tangent of the angle of polarization." It follows from this, that the polarization takes place when the reflected and refracted rays are at right angles to each other. This simple and elegant rule has been fully confirmed by all subsequent observations, as by those of MM. Biot and Seebeck; and must be considered one of the happiest and most important discoveries of the laws of phenomena in Optics.

[Note 46\9: Mém. Inst. 1810.]

[Note 47\9: Phil. Trans. 1815.]

The rule for polarization by one reflection being thus discovered, tentative formulæ were proposed by Sir D. Brewster and M. Biot, for the cases in which several reflections or refractions take place. Fresnel also in 1817 and 1818, traced the effect of reflection in modifying the direction of polarization, which Malus had done inaccurately in 1810. But the complexity of the subject made all such attempts extremely precarious, till the theory of the phenomena was understood, a period which now comes under notice. The laws which we have spoken of were important materials for the establishment of the theory; but in the mean time, its progress at first had been more forwarded by some other classes of facts, of a different kind, and of a longer standing notoriety, to which we must now turn our attention.

VII. DISCOVERY OF THE LAWS OF THE COLOURS OF THIN PLATES

THE facts which we have now to consider are remarkable, inasmuch as the colours are produced merely by the smallness of dimensions of the bodies employed. The light is not analysed by any peculiar {77} property of the substances, but dissected by the minuteness of their parts. On this account, these phenomena give very important indications of the real structure of light; and at an early period, suggested views which are, in a great measure, just.

Hooke appears to be the first person who made any progress in discovering the laws of the colors of thin plates. In his Micrographia, printed by the Royal Society in 1664, he describes, in a detailed and systematic manner, several phenomena of this kind, which he calls "fantastical colors." He examined them in Muscovy glass or mica, a transparent mineral which is capable of being split into the exceedingly thin films which are requisite for such colors; he noticed them also in the fissures of the same substance, in bubbles blown of water, rosin, gum, glass; in the films on the surface of tempered steel; between two plane pieces of glass; and in other cases. He perceived also,[48\9] that the production of each color required a plate of determinate thickness, and he employed this circumstance as one of the grounds of his theory of light.

[Note 48\9: Micrographia, p. 53.]

Newton took up the subject where Hooke had left it; and followed it out with his accustomed skill and clearness, in his Discourse on Light and Colors, communicated to the Royal Society in 1675. He determined, what Hooke had not ascertained, the thickness of the film which was requisite for the production of each color; and in this way explained, in a complete and admirable manner, the colored rings which occur when two lenses are pressed together, and the scale of color which the rings follow; a step of the more consequence, as the same scale occurs in many other optical phenomena.

It is not our business here to state the hypothesis with regard to the properties of light which Newton founded on these facts;--the "fits of easy transmission and reflection." We shall see hereafter that his attempted induction was imperfect; and his endeavor to account, by means of the laws of thin plates, for the colors of natural bodies, is altogether unsatisfactory. But notwithstanding these failures in the speculations on this subject, he did make in it some very important steps; for he clearly ascertained that when the thickness of the plate was about 1⁄178000th of an inch, or three times, five times, seven times that magnitude, there was a bright color produced; but blackness, when the thickness was exactly intermediate between those magnitudes. He found, also, that the thicknesses which gave red and {78} violet[49\9] were as fourteen to nine; and the intermediate colors of course corresponded to intermediate thicknesses, and therefore, in his apparatus, consisting of two lenses pressed together, appeared as rings of intermediate sizes. His mode of confirming the rule, by throwing upon this apparatus differently colored homogeneous light, is striking and elegant. "It was very pleasant," he says, "to see the rings gradually swell and contract as the color of the light was changed."

[Note 49\9: Opticks, p. 184.]

It is not necessary to enter further into the detail of these phenomena, or to notice the rings seen by transmission, and other circumstances. The important step made by Newton in this matter was, the showing that the rays of light, in these experiments, as they pass onwards go periodically through certain cycles of modification, each period occupying nearly the small fraction of an inch mentioned above; and this interval being different for different colors. Although Newton did not correctly disentangle the conditions under which this periodical character is manifestly disclosed, the discovery that, under some circumstances, such a periodical character does exist, was likely to influence, and did influence, materially and beneficially, the subsequent progress of Optics towards a connected theory.

We must now trace this progress; but before we proceed to this task, we will briefly notice a number of optical phenomena which had been collected, and which waited for the touch of sound theory to introduce among them that rule and order which mere observation had sought for in vain.

VIII. ATTEMPTS TO DISCOVER THE LAWS OF OTHER PHENOMENA

THE phenomena which result from optical combinations, even of a comparatively simple nature, are extremely complex. The theory which is now known accounts for these results with the most curious exactness, and points out the laws which pervade the apparent confusion; but without this key to the appearances, it was scarcely possible that any rule or order should be detected. The undertaking was of {79} the same kind as it would have been, to discover all the inequalities of the moon's motion without the aid of the doctrine of gravity. We will enumerate some of the phenomena which thus employed and perplexed the cultivators of optics.

The fringes of shadows were one of the most curious and noted of such classes of facts. These were first remarked by Grimaldi[50\9] (1665), and referred by him to a property of light which he called Diffraction. When shadows are made in a dark room, by light admitted through a very small hole, these appearances are very conspicuous and beautiful. Hooke, in 1672, communicated similar observations to the Royal Society, as "a new property of light not mentioned by any optical writer before;" by which we see that he had not heard of Grimaldi's experiments. Newton, in his Opticks, treats of the same phenomena, which he ascribes to the inflexion of the rays of light. He asks (Qu. 3), "Are not the rays of light, in passing by the edges and sides of bodies, bent several times backward and forward with a motion like that of an eel? And do not the three fringes of colored light in shadows arise from three such bendings?" It is remarkable that Newton should not have noticed, that it is impossible, in this way, to account for the facts, or even to express their laws; since the light which produces the fringes must, on this theory, be propagated, even after it leaves the neighborhood of the opake body, in curves, and not in straight lines. Accordingly, all who have taken up Newton's notion of inflexion, have inevitably failed in giving anything like an intelligible and coherent character to these phenomena. This is, for example, the case with Mr. (now Lord) Brougham's attempts in the Philosophical Transactions for 1796. The same may be said of other experimenters, as Mairan[51\9] and Du Four,[52\9] who attempted to explain the facts by supposing an atmosphere about the opake body. Several authors, as Maraldi,[53\9] and Comparetti,[54\9] repeated or varied these experiments in different ways.

[Note 50\9: Physico-Mathesis, de Lumine, Coloribus et Iride. Bologna, 1665.]

[Note 51\9: Ac. Par. 1738.]

[Note 52\9: Mémoires Présentés, vol. v.]

[Note 53\9: Ac. Par. 1723.]

[Note 54\9: Observationes Opticæ de Luce Inflexâ et Coloribus. Padua, 1787.]

Newton had noticed certain rings of color produced by a glass speculum, which he called "colors of thick plates," and which he attempted to connect with the colors of thin plates. His reasoning is by no means satisfactory; but it was of use, by pointing out this as a case in which his "fits" (the small periods, or cycles in the rays of light, of {80} which we have spoken) continued to occur for a considerable length of the ray. But other persons, attempting to repeat his experiments, confounded with them extraneous phenomena of other kinds; as the Duc de Chaulnes, who spread muslin before his mirror,[55\9] and Dr. Herschel, who scattered hair-powder before his.[56\9] The colors produced by the muslin were those belonging to shadows of gratings, afterwards examined more successfully by Fraunhofer, when in possession of the theory. We may mention here also the colors which appear on finely-striated surfaces, and on mother-of-pearl, feathers, and similar substances. These had been examined by various persons (as Boyle, Mazeas, Lord Brougham), but could still, at this period, be only looked upon as insulated and lawless facts.

[Note 55\9: Ac. Par. 1755.]

[Note 56\9: Phil. Trans. 1807.]

IX. DISCOVERY OF THE LAWS OF PHENOMENA OF DIPOLARIZED LIGHT

BESIDES the above-mentioned perplexing cases of colors produced by common light, cases of periodical colors produced by polarized light began to be discovered, and soon became numerous. In August, 1811, M. Arago communicated to the Institute of France an account of colors seen by passing polarized light through mica, and analysing[57\9] it with a prism of Iceland spar. It is remarkable that the light which produced the colors in this case was the light polarized by the sky, a cause of polarization not previously known. The effect which the mica thus produced was termed depolarization;--not a very happy term, since the effect is not the destruction of the polarization, but the combination of a new polarizing influence with the former. The word dipolarization, which has since been proposed, is a much more appropriate expression. Several other curious phenomena of the same kind were observed in quartz, and in flint-glass. M. Arago was not able to reduce these phenomena to laws, but he had a full conviction of their value, and ventures to class them with the great steps in {81} this part of optics. "To Bartholin we owe the knowledge of double refraction; to Huyghens, that of the accompanying polarization; to Malus, polarization by reflection; to Arago, depolarization." Sir D. Brewster was at the same time engaged in a similar train of research; and made discoveries of the same nature, which, though not published till some time after those of Arago, were obtained without a knowledge of what had been done by him. Sir D. Brewster's Treatise on New Philosophical Instruments, published in 1813, contains many curious experiments on the "depolarizing" properties of minerals. Both these observers noticed the changes of color which are produced by changes in the position of the ray, and the alternations of color in the two oppositely polarized images; and Sir D. Brewster discovered that, in topaz, the phenomena had a certain reference to lines which he called the neutral and depolarizing axes. M. Biot had endeavored to reduce the phenomena to a law; and had succeeded so far, that he found that in the plates of sulphate of lime, the place of the tint, estimated in Newton's scale (see ante, chap. vii.), was as the square of the sine of the inclination. But the laws of these phenomena became much more obvious when they were observed by Sir D. Brewster with a larger field of view.[58\9] He found that the colors of topaz, under the circumstances now described, exhibited themselves in the form of elliptical rings, crossed by a black bar, "the most brilliant class of phenomena," as he justly says, "in the whole range of optics." In 1814, also, Wollaston observed the circular rings with a black cross, produced by similar means in calc-spar; and M. Biot, in 1815, made the same observation. The rings in several of these cases were carefully measured by M. Biot and Sir D. Brewster, and a great mass of similar phenomena was discovered. These were added to by various persons, as M. Seebeck, and Sir John Herschel.

[Note 57\9: The prism of Iceland spar produces the colors by separating the transmitted rays according to the laws of double refraction. Hence it is said to analyse the light.]

[Note 58\9: Phil. Trans. 1814.]

Sir D. Brewster, in 1818, discovered a general relation between the crystalline form and the optical properties, which gave an incalculable impulse and a new clearness to these researches. He found that there was a correspondence between the degree of symmetry of the optical phenomena and the crystalline form; those crystals which are uniaxal in the crystallographical sense, are also uniaxal in their optical properties, and give circular rings; those which are of other forms are, generally speaking, biaxal; they give oval and knotted isochromatic lines, with two poles. He also discovered a rule for the tint at each point {82} in such cases; and thus explained, so far as an empirical law of phenomena went, the curious and various forms of the colored curves. This law, when simplified by M. Biot,[59\9] made the tint proportional to the product of the distances of the point from the two poles. In the following year, Sir J. Herschel confirmed this law by showing, from actual measurement, that the curve of the isochromatic lines in these cases was the curve termed the lemniscata, which has, for each point, the product of the distances from two fixed poles equal to a constant quantity.[60\9] He also reduced to rule some other apparent anomalies in phenomena of the same class.

[Note 59\9: Mém. Inst. 1818, p. 192.]

[Note 60\9: Phil. Trans. 1819.]

M. Biot, too, gave a rule for the directions of the planes of polarization of the two rays produced by double refraction in biaxal crystals, a circumstance which has a close bearing upon the phenomena of dipolarization. His rule was, that the one plane of polarization bisects the dihedral angle formed by the two planes which pass through the optic axes, and that the other is perpendicular to such a plane. When, however, Fresnel had discovered from the theory the true laws of double refraction, it appeared that the above rule is inaccurate, although in a degree which observation could hardly detect without the aid of theory.[61\9]

[Note 61\9: Fresnel, Mém. Inst. 1827, p. 162.]

There were still other classes of optical phenomena which attracted notice; especially those which are exhibited by plates of quartz cut perpendicular to the axis. M. Arago had observed, in 1811, that this substance produced a twist of the plane of polarization to the right or left hand, the amount of this twist being different for different colors; a result which was afterwards traced to a modification of light different both from common and from polarized light, and subsequently known as circular polarization. Sir J. Herschel had the good fortune and sagacity to discover that this peculiar kind of polarization in quartz was connected with an equally peculiar modification of crystallization, the plagihedral faces which are seen, on some crystals, obliquely disposed, and, as it were, following each other round the crystal from left to right, or from right to left. Sir J. Herschel found that the right-handed or left-handed character of the circular polarization corresponded, in all cases, to that of the crystal.

In 1815, M. Biot, in his researches on the subject of circular polarization, was led to the unexpected and curious discovery, that this {83} property which seemed to require for its very conception a crystalline structure in the body, belonged nevertheless to several fluids, and in different directions for different fluids. Oil of turpentine, and an essential oil of laurel, gave the plane of polarization a rotation to the left hand; oil of citron, syrup of sugar, and a solution of camphor, gave a rotation to the right hand. Soon after, the like discovery was made independently by Dr. Seebeck, of Berlin.

It will easily be supposed that all those brilliant phenomena could not be observed, and the laws of many of the phenomena discovered, without attempts on the part of philosophers to combine them all under the dominion of some wide and profound theory. Endeavors to ascend from such knowledge as we have spoken of, to the general theory of light, were, in fact, made at every stage of the subject, and with a success which at last won almost all suffrages. We are now arrived at the point at which we are called upon to trace the history of this theory; to pass from the laws of phenomena to their causes;--from Formal to Physical Optics. The undulatory theory of light, the only discovery which can stand by the side of the theory of universal gravitation, as a doctrine belonging to the same order, for its generality, its fertility, and its certainty, may properly be treated of with that ceremony which we have hitherto bestowed only on the great advances of astronomy; and I shall therefore now proceed to speak of the Prelude to this epoch, the Epoch itself, and its Sequel, according to the form of the preceding Book which treats of astronomy.

[2nd Ed.] [I ought to have stated, in the beginning of this chapter, that Malus discovered the depolarization of white light in 1811. He found that a pencil of light which, being polarized, refused to be reflected by a surface properly placed, recovered its power of being reflected after being transmitted through certain crystals and other transparent bodies. Malus intended to pursue this subject, when his researches were terminated by his death, Feb. 7, 1812. M. Arago, about the same time, announced his important discovery of the depolarization of colors by crystals.

I may add, to what is above said of M. Biot's discoveries respecting the circular polarizing power of fluids, that he pursued his researches so as to bring into view some most curious relations among the elements of bodies. It appeared that certain substances, as sugar of canes, had a right-handed effect, and certain other substances, as gum, a left-handed effect; and that the molecular value of this effect was not altered by dilution. It appeared also that a certain element of the {84} substance of fruits, which had been supposed to be gum, and which is changed into sugar by the operation of acids, is not gum, and has a very energetic right-handed effect. This substance M. Biot called dextrine, and he has since traced its effects into many highly curious and important results.**]

{{85}} PHYSICAL OPTICS.

X. PRELUDE TO THE EPOCH OF YOUNG AND FRESNEL

BY Physical Optics we mean, as has already been stated, the theories which explain optical phenomena on mechanical principles. No such explanation could be given till true mechanical principles had been obtained; and, accordingly, we must date the commencement of the essays towards physical optics from Descartes, the founder of the modern mechanical philosophy. His hypothesis concerning light is, that it consists of small particles emitted by the luminous body. He compares these particles to balls, and endeavors to explain, by means of this comparison, the laws of reflection and refraction.[62\9] In order to account for the production of colors by refraction, he ascribes to these balls an alternating rotatory motion.[63\9] This form of the emission theory, was, like most of the physical speculations of its author, hasty and gratuitous; but was extensively accepted, like the rest of the Cartesian doctrines, in consequence of the love which men have for sweeping and simple dogmas, and deductive reasonings from them. In a short time, however, the rival optical theory of undulations made its appearance. Hooke in his Micrographia (1664) propounds it, upon occasion of his observations, already noticed, (chap. **vii.,) on the colors of thin plates. He there asserts[64\9] light to consist in a "quick, short, vibrating motion," and that it is propagated in a homogeneous medium, in such a way that "every pulse or vibration of the luminous body will generate a sphere, which will continually increase and grow bigger, just after the same manner (though indefinitely swifter) as the waves or rings on the surface of water do swell into bigger and bigger circles about a point in it."[65\9] He applies this to the explanation of refraction, {86} by supposing that the rays in a denser medium move more easily, and hence that the pulses become oblique; a far less satisfactory and consistent hypothesis than that of Huyghens, of which we shall next have to speak. But Hooke has the merit of having also combined with his theory, though somewhat obscurely, the Principle of Interferences, in the application which he makes of it to the colors of thin plates. Thus[66\9] he supposes the light to be reflected at the first surface of such plates; and he adds, "after two refractions and one reflection (from the second surface) there is propagated a kind of fainter ray," which comes behind the other reflected pulse; "so that hereby (the surfaces AB and EF being so near together that the eye cannot discriminate them from one), this compound or duplicated pulse does produce on the retina the sensation of a yellow." The reason for the production of this particular color, in the case of which he here speaks, depends on his views concerning the kind of pulses appropriate to each color; and, for the same reason, when the thickness is different, he finds that the result will be a red or a green. This is a very remarkable anticipation of the explanation ultimately given of these colors; and we may observe that if Hooke could have measured the thickness of his thin plates, he could hardly have avoided making considerable progress in the doctrine of interferences.

[Note 62\9: Diopt. c. ii. 4.]

[Note 63\9: Meteor. c. viii. 6.]

[Note 64\9: Micrographia, p. 56.]

[Note 65\9: Micrographia, p. 57.]

[Note 66\9: Micrographia, p. 66.]

But the person who is generally, and with justice, looked upon as the great author of the undulatory theory, at the period now under notice, is Huyghens, whose Traité de la Lumière, containing a developement of his theory, was written in 1678, though not published till 1690. In this work he maintained, as Hooke had done, that light consists in undulations, and expands itself spherically, nearly in the same manner as sound does; and he referred to the observations of Römer on Jupiter's satellites, both to prove that this difference takes place successively, and to show its exceeding swiftness. In order to trace the effect of an undulation, Huyghens considers that every point of a wave diffuses its motion in all directions; and hence he draws the conclusion, so long looked upon as the turning-point of the combat between the rival theories, that the light will not be diffused beyond the rectilinear space, when it passes through an aperture; "for," says he,[67\9] "although the partial waves, produced by the particles comprised in the aperture, do diffuse themselves beyond the rectilinear space, these waves do not concur anywhere except in front of the {87} aperture." He rightly considers this observation as of the most essential value. "This," he says, "was not known by those who began to consider the waves of light, among whom are Mr. Hooke in his Micrography, and Father Pardies; who, in a treatise of which he showed me a part, and which he did not live to finish, had undertaken to prove, by these waves, the effects of reflection and refraction. But the principal foundation, which consists in the remark I have just made, was wanting in his demonstrations."

[Note 67\9: Tracts on Optics, p. 209.]

By the help of this view, Huyghens gave a perfectly satisfactory and correct explanation of the laws of reflection and refraction; and he also applied the same theory, as we have seen, to the double refraction of Iceland spar with great sagacity and success. He conceived that in this crystal, besides the spherical waves, there might be others of a spheroidal form, the axis of the spheroid being symmetrically disposed with regard to the faces of the rhombohedron, for to these faces the optical phenomena are symmetrically related. He found[68\9] that the position of the refracted ray, determined by such spheroidal undulations, would give an oblique refraction, which would coincide in its laws with the refraction observed in Iceland spar; and, as we have stated, this coincidence was long after fully confirmed by other observers.

[Note 68\9: Tracts on Optics, 237.]

Since Huyghens, at this early period, expounded the undulatory theory with so much distinctness, and applied it with so much skill, it may be asked why we do not hold him up as the great Author of the induction of undulations of light;--the person who marks the epoch of the theory? To this we reply, that though Huyghens discovered strong presumptions in favor of the undulatory theory, it was not established till a later era, when the fringes of shadows, rightly understood, made the waves visible, and when the hypothesis which had been assumed to account for double refraction, was found to contain also an explanation of polarization. It is then that this theory of light assumes its commanding form; and the persons who gave it this form, we must make the great names of our narrative; without, however, denying the genius and merit of Huyghens, who is, undoubtedly, the leading character in the prelude to the discovery.

The undulatory theory, from this time to our own, was unfortunate in its career. It was by no means destitute of defenders, but these were not experimenters; and none of them thought of applying it to {88} Grimaldi's experiments on fringes, of which we have spoken a little while ago. And the great authority of the period, Newton, adopted the opposite hypothesis, that of emission, and gave it a currency among his followers which kept down the sounder theory for above a century.

Newton's first disposition appears to have been by no means averse to the assumption of an ether as the vehicle of luminiferous undulations. When Hooke brought against his prismatic analysis of light some objections, founded on his own hypothetical notions, Newton, in his reply, said,[69\9] "The hypothesis has a much greater affinity with his own hypothesis than he seems to be aware of; the vibrations of the ether being as useful and necessary in this as in his." This was in 1672; and we might produce, from Newton's writing, passages of the same kind, of a much later date. Indeed it would seem that, to the last, Newton considered the assumption of an ether as highly probable, and its vibrations important parts of the phenomena of light; but he also introduced into his system the hypothesis of emission, and having followed this hypothesis into mathematical detail, while he has left all that concerns the ether in the form of queries and conjectures, the emission theory has naturally been treated as the leading part of his optical doctrines.

[Note 69\9: Phil. Trans. vii. 5087.]

The principal propositions of the Principia which bear upon the question of optical theory are those of the fourteenth Section of the first Book,[70\9] in which the law of the sines in refraction is proved on the hypothesis that the particles of bodies act on light only at very small distances; and the proposition of the eighth Section of the second Book;[71\9] in which it is pretended to be demonstrated that the motion propagated in a fluid must diverge when it has passed through an aperture. The former proposition shows that the law of refraction, an optical truth which mainly affected the choice of a theory, (for about reflection there is no difficulty on any mechanical hypothesis,) follows from the theory of emission: the latter proposition was intended to prove the inadmissibility of the rival hypothesis, that of undulations. As to the former point,--the hypothetical explanation of refraction, on the assumptions there made,--the conclusion is quite satisfactory; but the reasoning in the latter case, (respecting the propagation of undulations,) is certainly inconclusive and vague; and something better might the more reasonably have been expected, since Huyghens had at least {89} endeavored to prove the opposite proposition. But supposing we leave these properties, the rectilinear course, the reflection, and the refraction of light, as problems in which neither theory has a decided advantage, what is the next material point? The colors of thin plates. Now, how does Newton's theory explain these? By a new and special supposition;--that of fits of easy transmission and reflection: a supposition which, though it truly expresses these facts, is not borne out by any other phenomena. But, passing over this, when we come to the peculiar laws of polarization in Iceland spar, how does Newton's meet this? Again by a special and new supposition;--that the rays of light have sides. Thus we find no fresh evidence in favor of the emission hypothesis springing out of the fresh demands made upon it. It may be urged, in reply, that the same is true of the undulatory theory; and it must be allowed that, at the time of which we now speak, its superiority in this respect was not manifested; though Hooke, as we have seen, had caught a glimpse of the explanation, which this theory supplies, of the colors of thin plates.

[Note 70\9: Principia, Prop. 94, et seq.]

[Note 71\9: Ib. Prop. 42.]

At a later period, Newton certainly seems to have been strongly disinclined to believe light to consist in undulations merely. "Are not," he says, in Question twenty-eight of the Opticks, "all hypotheses erroneous, in which light is supposed to consist in pression or motion propagated through a fluid medium?" The arguments which most weighed with him to produce this conviction, appear to have been the one already mentioned,--that, on the undulatory hypothesis, undulations passing through an aperture would be diffused; and again,--his conviction, that the properties of light, developed in various optical phenomena, "depend not upon new modifications, but upon the original and unchangeable properties of the rays." (Question twenty-seven.)

But yet, even in this state of his views, he was very far from abandoning the machinery of vibrations altogether. He is disposed to use such machinery to produce his "fits of easy transmission." In his seventeenth Query, he says,[72\9] "when a ray of light falls upon the surface of any pellucid body, and is there refracted or reflected; may not waves of vibrations or tremors be thereby excited in the refracting or reflecting medium at the point of incidence? . . . . and do not these vibrations overtake the rays of light, and by overtaking them successively, do they not put them into the fits of easy reflection and easy {90} transmission described above?" Several of the other queries imply the same persuasion, of the necessity for the assumption of an ether and its vibrations. And it might have been asked, whether any good reason could be given for the hypothesis of an ether as a part of the mechanism of light, which would not be equally valid in favor of this being the whole of the mechanism, especially if it could be shown that nothing more was wanted to produce the results.

[Note 72\9: Opticks, p. 322.]

The emission theory was, however, embraced in the most strenuous manner by the disciples of Newton. That propositions existed in the Principia which proceeded on this hypothesis, was, with many of these persons, ground enough for adopting the doctrine; and it had also the advantage of being more ready of conception, for though the propagation of a wave is not very difficult to conceive, at least by a mathematician, the motion of a particle is still easier.

On the other hand, the undulation theory was maintained by no less a person than Euler; and the war between the two opinions was carried on with great earnestness. The arguments on one side and on the other soon became trite and familiar, for no person explained any new class of facts by either theory. Thus it was urged by Euler against the system of emission,[73\9]--that the perpetual emanation of light from the sun must have diminished the mass;--that the stream of matter thus constantly flowing must affect the motions of the planets and comets; that the rays must disturb each other;--that the passage of light through transparent bodies is, on this system, inconceivable: all such arguments were answered by representations of the exceeding minuteness and velocity of the matter of light. On the other hand, there was urged against the theory of waves, the favorite Newtonian argument, that on this theory the light passing through an aperture ought to be diffused, as sound is. It is curious that Euler does not make to this argument the reply which Huyghens had made before. The fact really was, that he was not aware of the true ground of the difference of the result in the cases of sound and light; namely, that any ordinary aperture bears an immense ratio to the length of an undulation of light, but does not bear a very great ratio to the length of an undulation of sound. The demonstrable consequence of this difference is, that light darts through such an orifice in straight rays, while sound is diffused in all directions. Euler, not perceiving this difference, rested his answer mainly upon a circumstance by no means {91} unimportant, that the partitions usually employed are not impermeable to sound, as opake bodies are to light. He observes that the sound does not all come through the aperture; for we hear, though the aperture be stopped. These were the main original points of attack and defence, and they continued nearly the same for the whole of the last century; the same difficulties were over and over again proposed, and the same solutions given, much in the manner of the disputations of the schoolmen of the middle ages.

[Note 73\9: Fischer, iv. 449.]

The struggle being thus apparently balanced, the scale was naturally turned by the general ascendancy of the Newtonian doctrines: and the emission theory was the one most generally adopted. It was still more firmly established, in consequence of the turn generally taken by the scientific activity of the latter half of the eighteenth century: for while nothing was added to our knowledge of optical laws, the chemical effects of light were studied to a considerable extent by various inquirers;[74\9] and the opinions at which these persons arrived, they found that they could express most readily, in consistency with the reigning chemical views, by assuming the materiality of light. It is, however, clear, that no reasonings of the inevitably vague and doubtful character which belong to these portions of chemistry, ought to be allowed to interfere with the steady and regular progress of induction and generalization, founded on relations of space and number, by which procedure the mechanical sciences are formed. We reject, therefore, all these chemical speculations, as belonging to other subjects; and consider the history of optical theory as a blank, till we arrive at some very different events, of which we have now to speak. {92}

[Note 74\9: As Scheele, Selle, Lavoisier, De Luc, Richter, Leonhardi, Gren, Girtanner, Link, Hagen, Voigt, De la Metherie, Scherer, Dizé, Brugnatelli. See Fischer, vii. p. 20.]