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PHILOSOPHICAL
TRANSACTIONS,
OF THE
ROYAL SOCIETY
OF
LONDON.
FOR THE YEAR MDCCCU.
PART I.
LONDON,
PRINTED BY W. BULMER AND CO. CLEVELAND-ROW, ST. JAMESES ;
AND SOLD BY G. AND W. NICOL, PALL-MALL, BOOKSELLERS TO HIS MAJESTY AND PRINTERS TO THE ROYAL SOCIETY.
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V. ?2_ '
ADVERTISEMENT.
T„e Committee appointed by the Royal Society to direct the pub- lication of the Philosophical Transactions , take this opportunity to acquaint the Public, that it fully appears, as well from the council- books and journals of the Society, as from repeated declarations which have been made in several former Transactions , that the printing of them was always, from time to time, the single act of the respective Secretaries, till the Forty-seventh Volume : the Society, as a Body, never interesting themselves any further in their publication, than by occasionally recommending the revival of them to some of their Se- cretaries, when, from the particular circumstances of their affairs, the Transactions had happened for any length of time to be intermitted. And this seems principally to have been done with a view to satisfy the Public, that their usual meetings were then continued, for the im- provement of knowledge, and benefit of mankind, the great ends of their first institution by the Royal Charters, and which they have ever since steadily pursued.
But the Society being of late years greatly enlarged, and their com- munications more numerous, it was thought advisable, that a Com- mittee of their members should be appointed, to reconsider the papers read before them, and select out of them such as they should judge most proper for publication in the future Transactions ; which was accordingly done upon the s6th of March, 1752* And the grounds
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of their choice are, and will continue to be, the importance and sin* gularity of the subjects, or the advantageous manner of treating them ; without pretending to answer for the certainty of the facts, or pro- priety of the reasonings, contained in the several papers so published, which must still rest on the credit or judgment of their respective authors.
It is likewise necessary on this occasion to remark, that it is an esta- blished rule of the Society, to which they will always adhere, never to give their opinion, as a Body, upon any subject, either of Nature or Art, that comes before them. And therefore the thanks which are frequently proposed from the Chair, to be given to the authors of such papers as are read at their accustomed meetings, or to the persons through whose hands they receive them, are to be considered in no other light than as a matter of civility, in return for the respect shewn to the So- ciety by those communications. The like also is to be said with re- gard to the several projects, inventions, and curiosities of various kinds, which are often exhibited to the Society ; the authors whereof, or those who exhibit them, frequently take the liberty to report, and even to certify in the public news-papers, that they have met with the highest applause and approbation. And therefore it is hoped, that no regard will hereafter be paid to such reports and public notices ; which in some instances have been too lightly credited, to the dishonour of the Society.
*3 5 3 37
CONTENTS.
I. The Croonian Lecture . On the Power of the Eye to adjust
itself to different Distances , when deprived of the Crystalline Lefts. By Everard Home, Esq. F. R. S. page 1
II. The Bakerian Lecture. On the Theory of Light and Colours.
By Thomas Young, M. D. F. R. S. Professor of Natural Phi- losophy in the Royal Institution. p. 12
III. An Analysis of a mineral Substance from North America,
containing a Metal hitherto unknown. By Charles Hatchett, Esq. F. R. S. p. 49
IV. A Description of the Anatomy of the Ornithorhynchus
paradoxus. By Everard Home, Esq. F. R. S. p. 6*7
V. On the Independence of the analytical and geometrical Methods
of Investigation ; and on the Advantages to be derived from their Separation. By Robert Woodhouse, A. M. Fellow of Caius College , Cambridge. Communicated by Joseph Planta, Esq. Sec. R. S. p. 85
VI. Observations and Experiments upon oxygenized and hyper-
oxygenized muriatic Acid; and upon some Combinations of the muriatic Acid in its three States. By Richard Chenevix, Esq. F. R. S. and M. R. I. A. p. 12 6
VII. Experiments and Observations on certain stony and metalline
Substances , which at different Times are said to have fallen on the Earth ; also on various Kinds of native Iron. By Edward Howard, Esq. F. R. S. p. 168
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APPENDIX.
Meteorological Journal kept at the Apartments of the Royal Society , by Order of the President and Council .
THE President and Council of the Royal Society adjudged* for the year 1801, the Medal on Sir Godfrey Copley’s Donation, to Mr. Astley Cooper, for his Papers On the Effects which take place from the Destruction of the Membrana Tympani of the Ear; with an Account of an Operation for the removal of a particular species of Deafness.
ERRATA.
Page 133, line % and 3, for 38*3, read 383.
— — ® 134, — penult, for hyperoxvgenized, read oxygenized.
PHILOSOPHICAL
TRANSACTIONS.
I. The Croonian Lecture. On the Power of the Eye to adjust itself to different Distances, when deprived of the Crystalline Lens . By Everard Horae, Esq. F.R. S.
Read November 5, 1801,
It is intended, on the present occasion, to state some facts and observations, in support of an opinion advanced in a former lecture, that the adjustment of the eye to see objects at different distances, does not depend upon any internal changes in the crystalline lens.
The first of the experiments which will be stated, was made with the assistance of the late Mr. Ramsden; and, had not the death of that valuable member of this Society deprived me of his further aid, the following observations would undoubtedly have been more deserving the attention of my learned audience.
It is impossible for me to mention Mr. Ramsden, from whom I have received so much assistance in every pursuit connected with optics and mathematics, in which I have been engaged,
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Mr. Home’s Lecture on the Power of the Eye ,
without availing myself of this opportunity of paying that tribute of gratitude to his memory, which feelings of delicacy prevented me from offering to him while alive. It is unnecessary here to mention his genius, his merits, or his exertions for the promo- tion of science ; these are equally well known to every member present, as to myself. It is only my individual obligations, in the prosecution of inquiries connected with the objects of this learned Society, that are meant to be taken notice of.
To his friendly and zealous assistance I am indebted for the information which was necessary to enable me to prosecute investigations upon the subject of vision ; and, without such assistance, I should have shrunk from the inquiry. It is also to his early friendship, and his readiness to communicate to me his knowledge, that I look back, as among the sources of my early exertions, and love of philosophical pursuits.
In the year 17 94, I laid before this learned Society some experiments, suggested and made by Mr. Ramsden, upon the comparative powers of adjustment of the eye, when in a perfect state, and when deprived of the crystalline lens. From the result of these experiments it appeared, that the removal of the lens did not deprive the eye of the power of seeing distinctly at different distances. As the person upon whom the experiments were tried did not see very distinctly, without a substitute for the lens, in making them, a double convex glass, of 2^ inches focus, was placed before his eye ; and, to render the image dis- tinct, by correcting the spherical aberrations, the aperture was diminished to -3-ths of an inch ; a less degree of diminution not answering that purpose.
The subject of these experiments was Benjamin Clerk, twenty-one years of age; one of his eyes was in a very perfect
3
when deprived of the Crystalline Lens.
state, and the other without defect, except what arose from the removal of the lens : and the results appeared to be satisfactory in deciding, that the eye, when deprived of the crystalline lens, retains a power of adjustment.
Opportunities of instituting experiments of this kind very rarely occur; the patients who have had their lenses extracted, either not seeing sufficiently well, or being too much advanced in life to be fit subjects for that purpose; but, in the year 1798, the following case came under my care, which enabled me to make some further observations, in confirmation of the former experiments.
Henry Miles, a carpenter, at Westborough Green in Sussex, fifty years of age, applied, in the month of August, 1798, at St. George’s Hospital, to be admitted as a patient, on account of blindness, from having a cataract in each eye ; and was received under my care. Both the cataracts were extracted ; and the eyes recovered from the effects of the operation, without suffer- ing from inflammation. The right eye had the power of seeing objects with unusual distinctness ; but the- left was less perfect, the iris having been slightly torn, by the lens being too big to pass through the aperture, without injuring the membrane.
As soon as this man’s eyes had recovered, I requested Mr. Ramsden to repeat some of the former experiments, on his right -eye; which he readily agreed to do. Before the experiments were made, upon trying what was his power of vision with the naked eye, we were agreeably surprised to find that he saw so distinctly, as to admit of our ascertaining, without the aid of glasses, what were the ranges of his eye’s adjustment.
A piece of pasteboard, with a letter of a moderate size, as an object upon it, was put into his hands ; as he could not read, the
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4 Mr. Home’s Lecture on the Pozver of the Eye,
page of a book might have confused him : he was directed to vary the distance of the pasteboard from his eye, till he had ascertained the nearest and most distant situations, in which the object appeared distinct; these distances, by measurement, were 7 inches, and 18 inches. In repeating this experiment several different times, he brought the object very correctly to the same situations.
This result convinced Mr. Ramsden, that the eye possessed the power of varying its adjustment; and he did not think any more complex experiments would be nearly so satisfactory; consequently, no others were made, and the man was allowed
X
to go into the country.
It was intended to make him a present of a pair of spectacles, allowing him to choose those best adapted to his eye ; but his sight was so very good, that we entirely forgot it, till some time after he was gone.
These experiments confirmed the former ones so very strong- ly, and from their simplicity were so much less liable to error, that Mr. Ramsden and myself considered the object of our inquiry completely attained ; the reason for not, at the time, laying them before this learned Society was, that they estab- lished no new fact, and the former ones did not appear to require their support.
This inquiry, always regarded as highly important by phy- siologists, has continued to engage their attention ; and, in the Bakerian Lecture for last year, Dr. Young has advanced some experiments to prove, that the adjustment of the eye to different distances, depends upon the crystalline lens : he considers the results of the experiments made by Mr. Ramsden, upon Ben- jamin Clerk’s eyes, as inconclusive ; and the phenomena met
5
when deprived of the Crystalline Lens .
with, as arising from the smallness of the aperture, and not from any power of adjustment in the dye. Dr. Young, therefore, with a view to obviate all possibility of deception in future, constructed an optometer, upon the principle of that of Dr. Porterfield. In this instrument, when applied to presbyopic eyes, the eye, by looking along a line through a small convex . lens, before which is placed a card with two narrow slits in it, near enough to each other to be within the limits of the pupil, will see the line as two lines, crossing each other at the point of perfect vision ; and every eye that has the power of adjust- ment, will make the lines cross in different places, when adjusted to different distances.
With this instrument, Dr. Young made experiments upon several eyes which had been deprived of the crystalline lens; and with all of them found, that the crossing of the lines was seen only at one point ; he therefore concludes, that the power of adjustment was lost.
These experiments of Dr. Young led me to reconsider the subject; and it was matter of regret that Benjamin Clerk was not in this country, as making a trial with the optometer on his eye, would have determined, in the most satisfactory manner, whether there had been a fallacy in the former expe- riments.
This not being in my power, I made inquiry after Henry Miles, upon whom the second experiments were tried ; and I had the pleasure to hear, that he was in good health, and that his eyes continued to have very distinct vision, so much so, that he never had occasion to make use of any glasses, from the time the operation had been performed.
With the view of making some experiments on this man’s eyes, with Dr. Young’s optometer, I procured that instrument from
6 Mr. Home’s Lecture on the Rower of the Eye ,
Mr. Cary, the optician, made exactly in 'the same manner as that which had been executed under Dr. Young’s direction. I first, however, tried the experiments upon my own eye ; but had the mortification to find myself unable to make the lines cross in two different situations. This led me to try the eyes of several of my friends ; who were equally unable to make the lines cross any where, except at one point. Young people, indeed all those under thirty years of age, were capable of vary- ing the place of intersection; but none who were above forty, could produce any change in it.
As I could not doubt of my own eye having the power of varying its adjustment, I was led to believe that the instrument required some address in the management, which I had not acquired; and therefore despaired of making Henry Miles Sufficiently master of it, to do justice to my views.
To obviate these difficulties, I adapted the optometer, without the lens, to presbyopic eyes, by making a line 4 feet long, upon strong paper, divided into inches, and having the same slits to look through as in the other. This instrument, and Dr. Young’s, I put into the hands of my friend Sir Henry Englefield, with a request that he would examine them, and, when he had become perfectly master of them, and of the best mode of using them, that he would assist me in making expe- riments with them; for, as he was more in the habit of chang- ing the focus of his eye, in using optical instruments, he would more readily detect the circumstance which prevented me from succeeding in the experiment.
After several trials with this optometer, and seeing its de- fects, Sir Henry Englefield improved it, by having the paper pasted upon a strong board, 4 feet long, which rendered the surface free from the slightest inequalities ; and, instead of
7
when deprived of the Crystalline Lens.
a line marked with ink, a thread of black silk was stretched along the middle of the board. With this instrument, he found that his eye could make the lines cross at two different points, at several inches distance from each other. The readiest mode of making the experiment succeed, was first fixing his eye upon some near object, held above and a little on one side of the silk thread, and, when the focus of his eye was adapted to that distance, then to look at the thread ; afterwards to look at some distant object, and, when that had become very dis- tinct, again to look at the thread. Upon trying the instru- ment with my own eye, in this way, I found the crossing of the lines changed its situation, with every change of adjust- ment ; and, after being accustomed to make this experiment, I was enabled to produce a similar change in the optometer with the lens, but by no means in so satisfactory a manner, nor did it last more than an instant ; my eye probably not being so well fitted as many others, for experiments of this kind.
The optometer without the lens was hence admitted to be the most easily managed, by the eye of a person unaccustomed to such experiments, and therefore it was determined to make use of it in the trials upon Henry Miles's eye ; which we were enabled to do, as his vision was sufficiently distinct without the aid of glasses, and as, from never having used them, he saw much better with his naked eye.
The following experiments were made with the optometer without the lens, on the 27th of August, 1801.
The first trials were upon Sir Henry Englefield’s eye; which, being most familiar with the use of the instrument, be- came a standard with which the others might be compared.
Sir Henry Englefield's eye made the lines to intersect
S Mr. Home’s Lecture on the Power of the Eye ,
each other at 1 <i-\ inches, as the near distance; and at 281- inches, as the furthest distance. The experiment was repeated several different times, and the results were very nearly the same/
My own eye made the lines intersect at i2f- inches, as the near distance ; and at 2 gj inches, as the furthest distance.
A man servant of Sir Henry Englefield's, twenty-five years of age, made the lines intersect at 12 inches, and at 3 if inches.
Henry Miles, fifty years of age, whose eye had been de- prived of the crystalline lens for three years, made the lines intersect at 8y3- inches, as the near distance; and at 13^, as the furthest distance.
This experiment was repeated two different times in the fore- noon, with the same result, and again in the afternoon, without there being any considerable variation; but, upon trying it again, after the eye had been fatigued, he was unable to make the lines cross nearer than 1 if inches, although he could make them cross at 13^ inches; so that adjusting the eye to a near distance, was more difficult after it had been much used, than before.
Henry Miles was unable, in the optometer with the lens, to produce any change in the crossing of the lines, nor did he see them cross with sufficient distinctness to make us consider it a fair experiment.
The following experiment was made upon Miles’s eye, at the suggestion of Sir Henry Englefield, with a view to ascer- tain in another though less decisive way, whether any change took place in it, when directed from a near object to a more distant one.
A piece of pasteboard, in which a black circle, about f of an
9
when deprived of the Crystalline Lens.
inch in diameter, with a dot in the centre, had been described near to its edge, was placed perpendicularly to the horizon, at 5 inches distance from the eye ; another piece of pasteboard, with a circle and dot in it, was placed at the distance of 18 inches ; the farthest circle was made a little larger than the other, that it might appear equally distinct at the greater dis- tance. When the eye was directed towards these two objects, they appeared upon the same level; and the circumference of the circles, had they been projected on the same perpendicular plane, would have been nearly in contact.
Miles was placed opposite these objects, with his head made steady, and prevented from moving : he was then told to look at one, till it became very distinct ; and, when he had done so, this was removed, and he was directed to look at the other, which did not immediately appear to him with the same dis- tinctness. This was equally the case, whether he looked from the near one to the distant one, or the reverse : the eye did not see the object to which it was so suddenly directed, with the same defined outline as that from which it had been with- drawn.
This man sees best in a strong light ; and it was in that light all the experiments were made : he can see very well in any degree of daylight; but his eyes are much fatigued by candle- light. Upon examining the eye attentively, the pupil was rather larger than in perfect eyes; the iris was in a very perfect state ; and the cicatrix of the wound, in the inferior part of the cornea, was scarcely visible.
The sight being so good, without the aid of glasses, is not common; and, had not the lenses been extracted in a public
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Mr. Home’s Lecture on the Power of the Eyei
hospital, before a number of spectators, some doubts might be entertained whether they had been removed.
From the experiments which have been stated, it appeared to Sir Henry Englefield, that Miles’s eye was not deprived of its power of adjustment ; and, by whatever circumstances my own judgment might be deceived, or rendered partial, there was nothing by which his could be biassed, as he could have no object in view, but the promotion of science. His knowledge of optics, and his habit of making experiments, are the best pledges of these having been as accurately performed as the nature of the subject admits of ; for, certainly, the sources of fallacy, in optical experiments, are numerous. Those that have been related, to be made with perfect accuracy, should be tried upon the eye of a person skilled in optics, and accustomed to such experiments ; and whose eye had been deprived of the crystalline lens, without having received the slightest degree of injury in any of its other parts.
The experiments were instituted in the Isle of Wight, which prevented me from requesting several of my friends to be pre- sent at them, whose knowledge of the subject would have made me desirous of their assistance.
Haller mentions the case of a nobleman, from whose eye the crystalline lens had been extracted, who used glasses, and could see with them objects at different distances. As this was an observation made upon a particular friend of his own, and as he refers to Pemberton, who mentions a case of depressed crystalline lens, in which no such effect took place, it is natural to suppose, that he had given considerable attention to the subject; and that, although the experiments he instituted are
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when deprived of the Crystalline Lens .
not mentioned, the opinion was not advanced, without what appeared to him sufficient authority.*
* Et lente ob cataractam extracta vel deposita, oculum tamen ad varias distantias videre, ut coram in nobili viro video, absque ullo experimento, quo earn facultatem recuperaverit. Et si enim tunc, ob diminutas vires, quse radios uniunt, asger lente vitrea opus habet, eadem lens in omnia distantia sufficit.
Haller. Elementa Physiologic. Tom. V. Lib. xvi. §. 25. p. 514.
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II. The Bakerian Lecture . On the Theory of Light and Colours . By Thomas Young, M. D. F. R. S. Professor of Natural Phi- losophy in the Royal Institution.
Read November 12, 1801.
Although the invention of plausible hypotheses, independent of any connection with experimental observations, can be of very little use in the promotion of natural knowledge ; yet the discovery of simple and uniform principles, by which a great number of apparently heterogeneous phenomena are reduced to coherent and universal laws, must ever be allowed to be of considerable importance towards the improvement of the human intellect.
The object of the present dissertation is not so much to pro- pose any opinions which are absolutely new, as to refer some theories, which have been already advanced, to their original inventors, to support them by additional evidence, and to apply them to a great number of diversified facts, which have hitherto been buried in obscurity. Nor is it absolutely necessary in this instance to produce a single new experiment; for of experi- ments there is already an ample store, which are so much the more unexceptionable, as they must have been conducted with- out the least partiality for the system by which they will be explained ; yet some facts, hitherto unobserved, will be brought forwards, in order to show the perfect agreement of that system with the multifarious phenomena of nature.
Dr. Young's Lecture , &c.
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The optical observations of Newton are yet unrivalled ; and, excepting some casual inaccuracies, they only rise in our esti- mation, as we compare them with later attempts to improve on them. A further consideration of the colours of thin plates, as they are described in the second book of Newton’s optics, has converted that prepossession which I before entertained for the undulatory system of light, into a very strong conviction of its truth and sufficiency; a conviction which has been since most strikingly confirmed, by an analysis of the colours of striated substances. The phenomena of thin plates are indeed so sin- gular, that their general complexion is not without great diffi- culty reconcileable to any theory, however complicated, that has hitherto been applied to them ; and some of the principal circumstances have never been explained by the most gratuitous assumptions ; but it will appear, that the minutest particulars of these phenomena, are not only perfectly consistent with the theory which will now be detailed, but that they are all the necessary consequences of that theory, without any auxiliary suppositions ; and this by inferences so simple, that they be- come particular corollaries, which scarcely require a distinct enumeration.
A more extensive examination of Newton's various writings has shown me, that he was in reality the first that suggested such a theory as I shall endeavour to maintain ; that his own opinions varied less from this theory than is now almost uni- versally supposed ; and that a variety of arguments have been advanced, as if to confute him, which may be found nearly in a similar form in his own works ; and this by no less a mathe- matician than Leonard Euler, whose system of light, as far as it is worthy of notice, either was, or might have been,
14 Dr. Young’s Lecture on
wholly borrowed from Newton, Hooke, Huygens, and Male-
BRANCHE.
Those who are attached, as they may be with the greatest justice, to every doctrine which is stamped with the Newtonian approbation, will probably be disposed to bestow on these con- siderations so much the more of their attention, as they appear to coincide more nearly with Newton’s own opinions. For this reason, after having briefly stated each particular position of my theory, I shall collect, from Newton’s various writings, such passages as seem to be the most favourable to its admis- sion ; and, although I shall quote some papers which may be thought to have been partly retracted at the publication of the optics, yet I shall borrow nothing from them that can be sup- posed to militate against his maturer judgment.
HYPOTHESIS i.
A luminiferous Ether pervades the Universe , rare and elastic in a
high degree.
Passages from Newton.
“ The hypothesis certainly has a much greater affinity with “ his own,” that is, Dr. Hooke’s, “ hypothesis, than he seems “ to be aware of ; the vibrations of the ether being as useful and “ necessary in this, as in his.” (Phil. Trans. Vol. VII. p. 5087. Abr. Vol. I. p. 145. Nov. 1672.)
“ To proceed to the hypothesis: first, it is to be supposed “ therein, that there is an ethereal medium, much of the same il constitution with air, but far rarer, subtler, and more strongly c< elastic. — -It is not to be supposed, that this medium is one “ uniform matter, but compounded, partly of the main phleg- “ matic body of ether, partly of other various ethereal spirits.
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the Theory of Light and Colours.
« much after the manner that air is compounded of the phleg- « matic body of air, intermixed with various vapours and « exhalations : for the electric and magnetic effluvia, and gravi- “ tating principle, seem to argue such variety/' (Birch: Hist, ol R. S. Vol. III. p. 249. Dec. 1675.)
tc Is not the heat (of the warm room) conveyed through the “ vacuum by the vibrations of a much subtiler medium than air r « — And is not this medium the same with that medium by which “ light is refracted and reflected, and by whose vibrations light “ communicates heat to bodies, and is put into fits of easy re- “ flection, and easy transmission ? And do not the vibrations of ££ this medium in hot bodies, contribute to the intenseness and <£ duration of their heat ? And do not hot bodies communicate “ their heat to contiguous cold ones, by the vibrations of this me- a dium propagated from them into the cold ones ? And is not this “ medium exceedingly more rare and subtile than the air, and “ exceedingly more elastic and active ? And doth it not readily “ pervade all bodies ? And is it not, by its elastic force, expanded “ through all the heavens ? — May not planets and comets, and “ all gross bodies, perform their motions in this ethereal me- ££ dium ? — And may not its resistance be so small, as to be £C inconsiderable? For instance, if this ether (for so I will call ££ it) should be supposed 700,000 times more elastic than our ££ air, and above 700,000 times more rare, its resistance would ££ be about 600,000000 times less than that of water. And ££ so small a resistance would scarce make any sensible altera- “ tion in the motions of the planets, in ten thousand years. <£ If any one would ask how a medium can be so rare, let him ££ tell me — how an electric body can by friction emit an exha- ££ lation so rare and subtile, and yet so potent ? — And how the
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Dr. Young's Lecture on
“ effluvia of a magnet can pass through a plate of glass, with- “ out resistance, and yet turn a magnetic needle beyond the <r glass?" (Optics, Qu. 18, 22.)
HYPOTHESIS II.
Undulatiofis are excited in this Ether whenever a Body becomes
luminous.
Scholium. I use the word undulation, in preference to vibra- tion, because vibration is generally understood as implying a motion which is continued alternately backwards and forwards, by a combination of the momentum of the body with an acce- lerating force, and which is naturally more or less permanent ; but an undulation is supposed to consist in a vibratory motion, transmitted successively through different parts of a medium, without any tendency in each particle to continue its motion, except in consequence of the transmission of succeeding undu- lations, from a distinct vibrating body ; as, in the air, the vibra- tions of a chord produce the undulations constituting sound.
Passages from Newton.
“ Were I to assume an hypothesis, it should be this, if pro- " pounded more- generally, so as not to determine what light is, “ further than that it is something or other capable of exciting “ vibrations in the ether ; for thus it will become so general and “ comprehensive of other hypotheses, as to leave little room for “ new ones to be invented." (Birch. Vol. III. p. 249. Dec. 1 675. )
(e In the second place, it is to be supposed, that the ether is a “ vibrating medium like air, only the vibrations far more swift <s and minute ; those of air, made by a man’s ordinary voice, succeeding one another at more than half a foot, or a foot
the Theory of Light and Colours. 17
“ distance ; but those of ether at a less distance than the hun- “ dred thousandth part of an inch. And, as in air the vibra- “ tions are some larger than others, but yet all equally swift, “ (for in a ring of bells the sound of every tone is heard at two “ or three miles distance, in the same order that the bells are “ struck,) so, I suppose, the ethereal vibrations differ in big- 4C ness, but not in swiftness. Now, these vibrations, beside their “ use in reflection and refraction, may be supposed the chief “ means by which the parts of fermenting or putrifying sub- “ stances, fluid liquors, or melted, burning, or other hot bodies, “ continue in motion/' (Birch Vol. III. p. 251. Dec. 1675.)
<£ 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 ? — And are not these vibrations propagated “ from the point of incidence to great distances ? And do they <c not overtake the rays of light, and by overtaking them sue- “ cessively, do not they put them into the fits of easy reflection £< and easy transmission described above ?” (Optics. Qu. 17.)
“ Light is in fits of easy reflection and easy transmission, “ before its incidence on transparent bodies. And probably it is “ put into such fits at its first emission from luminous bodies, “ and continues in them during all its progress/’ (Optics. Second Book. Part III. Prop. 13.)
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HYPOTHESIS III.
The Sensation of different Colours depends on the different fre- quency of Vibrations , excited by Light in the Retina.
Passages from Newton.
“ The objector’s hypothesis, as to the fundamental part of it, “ is not against me. That fundamental supposition is, that the “ parts of bodies, when briskly agitated, do excite vibrations in “ the ether, which are propagated every way from those bodies tc in straight lines, and cause a sensation of light by beating “ and dashing against the bottom of the eye, something after “ the manner that vibrations in the air cause a sensation of “ sound by beating against the organs of hearing. Now, the “ most free and natural application of this hypothesis to the “ solution of phenomena, I take to be this : that the agitated ** parts of bodies, according to their several sizes, figures, and “ motions, do excite vibrations in the ether of various depths “ or bignesses, which, being promiscuously propagated through u that medium to our eyes, effect in us a sensation of light of a “ white colour ; but if by any means those of unequal bignesses “ be separated from one another, the largest beget a sensation “ of a red colour, the least or shortest of a deep violet, and “ the intermediate ones of intermediate colours ; much after “ the manner that bodies, according to their several sizes, « shapes, and motions, excite vibrations in the air of various “ bignesses, which, according to those bignesses, make several “ tones in sound : that the largest vibrations are best able to “ overcome the resistance of a refracting superficies, and so “ break through it with least refraction ; whence the vibrations
the Theory of Light and Colours. 19
e< of several bignesses, that is, the rays of several colours, which “ are blended together in light, must be parted from one an- “ other by refraction, and so cause the phenomena ol prisms, ec and other refracting substances ; and that it depends on the “ thickness of a thin transparent plate or bubble, whether a (C vibration shall be reflected at its further superficies, or trans- “ mitted ; so that, according to the number of vibrations, inter- “ ceding the two superficies, they may be reflected or transmitted cc for many successive thicknesses. And, since the vibrations “ which make blue and violet, are supposed shorter than those “ which make red and yellow, they must be reflected at a less “ thickness of the plate : which is sufficient to explicate all the “ ordinary phenomena of those plates or bubbles, and also of “ all natural bodies, whose parts are like so many fragments of “ such plates. These seem to be the most plain, genuine, and “ necessary conditions of this hypothesis. And they agree so fc justly with my theory, that if the animadversor think fit to “ apply them, he need not, on that account, apprehend a divorce “ from it. But yet, how he will defend it from other difficulties, “ I know not.” (Phil. Trans. Vol. VII. p. 5088. Abr. Vol. I. p. 145. Nov. 1672.)
“ To explain colours, I suppose, that as bodies of various “ sizes, densities, or sensations, do by percussion or other “ action excite sounds of various tones, and consequently vi- tc brations in the air of different bigness ; so the rays of light, “ by impinging on the stiff refracting superficies, excite vibra- “ tions in the ether,— of various bigness ; the' biggest, strongest, “ or most potent rays, the largest vibrations ; and others shorter, “ according to their bigness, strength, or power: and therefore “ the ends of the capillamenta of the optic nerve, which pave
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Dr. Young’s Lecture on
“ or face the retina, being such refracting superficies, when the f< rays impinge upon them, they must there excite these vibra- ec tions, which vibrations (like those of sound in a trunk or “ trumpet) will run along the aqueous pores or crystalline pith <c of the capillamenta, through the optic nerves, info the senso- “ rium ; — and there, I suppose, affect the sense with various “ colours, according to their bigness and mixture ; the biggest “ with the strongest colours, reds and yellows ; the least with <£ the weakest, blues and violets ; the middle with green ; and a “ confusion of all with white, much after the manner that, in “ the sense of hearing, nature makes use of aerial vibrations of “ several bignesses, to generate sounds of divers tones ; for the “ analogy of nature is to be observed.” (Birch Vol, III. p. 262. Dec. 1675.) ,
“ Considering the lastingness of the motions excited in the “ bottom of the eye by light, are they not of a vibrating nature ? “ — Do not the most refrangible rays excite the shortest vibra- “ tions, — the least refrangible the largest ? May not the, harmony “ and discord of colours arise from the proportions of the vibra- “ tions propagated through the fibres of the optic nerve into " the brain, as the harmony and discord of sounds arise from “ the proportions of the vibrations of the air ?” (Optics, Qu. 16, 13, 14.)
Scholium. Since, for the reason here assigned by Newton, it is probable that the motion of the retina is rather of a vibra- tory than of an undulatory nature, the frequency of the vibrations must be dependent on the constitution of this substance. Now, as it is almost impossible to conceive each sensitive point of the retina to contain an infinite number of particles, each Capable of vibrating in perfect unison with every possible undulation, it
I
3 3/3 7
the Theory of Light and Colours. 21
becomes necessary to suppose the number limited, for instance, to tiie three principal colours, red, yellow, and blue, of which the undulations are related in magnitude nearly as the numbers 8, 7, and b ; and that each of the particles is capable of being put in motion less or more forcibly, by undulations differing less or more from a perfect unison ; for instance, the undula- tions of green light being nearly in the ratio of 6\, will affect equally the particles in unison with yellow and blue, and pro- duce the same effect as a light composed of those two species : and each sensitive filament of the nerve may consist of three portions, one for each principal colour. Allowing this statement, it appears that any attempt to produce? a musical effect from colours, must be unsuccessful, or at least that nothing more than a very simple melody could be imitated by them ; for the period, which in fact constitutes the harmony of any concord, being a multiple of the periods of the single undulations, would in this case be wholly without the limits of sympathy of the retina, and would lose its effect; in the same manner as the harmony of a third or a fourth is destroyed, by depressing it to the lowest notes of the audible scale. In hearing, there seems to be no permanent vibration of any part of the organ.
■N
HYPOTHESIS IV.
All material Bodies have an Attraction for the ethereal Medium , by means of which it is accumulated within their Substance, and for a small Distance around them, in a State of greater Density, but not of greater Elasticity.
It has been shewn, that the three former hypotheses, which may be called essential, are literally parts of the more compli- cated Newtonian system. This fourth hypothesis differs perhaps
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in some degree from any that have been proposed by former authors, and is diametrically opposite to that of Newton ; but, both being in themselves equally probable, the opposition is merely accidental; and it is only to be inquired which is the best capable of explaining the phenomena. Other suppositions might perhaps be substituted for this, and therefore I do not consider it as fundamental, yet it appears to be the simplest and best of any that have occurred to me.
PROPOSITION i.
All Impulses are propagated in a homogeneous elastic Medium
with an equable Velocity.
i Every experiment relative to sound coincides with the obser- vation already quoted from Newton, that all undulations are propagated through the air with equal velocity; and this is further confirmed by calculations. (Lagrange. Misc. Taur. Vol. I. p. 91. Also, much more concisely, in my Syllabus of a course of Lectures on Natural and Experimental Philosophy, about to be published. Article 289. ) If the impulse be so great as materially to disturb the density of the medium, it will be no longer homogeneous ; but, as far as concerns our senses, the quantity of motion may be considered as infinitely small. It is surprising that Euler, although aware of the matter of fact, should still have maintained, that the more frequent undulations are more rapidly propagated. (Theor. muL and Conject. phys.) It is possible, that the actual velocity of the particles of the luminiferous ether may bear a much less proportion to the veIo= city of the undulations than in sound ; for light may be excited by the motion of a body moving at the rate of only one mile in the time that light moves a hundred millions.
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the Theory of Tight and Colours .
Scholium 1. It has been demonstrated, that in different mediums the velocity varies in the subduplicate ratio of the force directly, and of the density inversely. (Misc.Taur. Vol. I. p. 91. Young’s Syllabus. Art. 294.)
Scholium 2. It is obvious, from the phenomena of elastic bodies and of sounds, that the undulations, may cross each other without interruption. But there is no necessity that the various colours of white light should intermix their undulations *, for, supposing the vibrations of the retina to continue but a five hun- dredth of a second after their excitement, a million undulations of each of a million colours may arrive in distinct succession within this interval of time, and produce the same sensible effect, as if all the colours arrived precisely at the same instant.
PROPOSITION II.
An Undulation conceived to originate from the Vibration of a single Particle , must expand through a homogeneous Medium in a spherical Form, but with different quantities of Motion in different Parts.
For, since every impulse, considered as positive or negative, is propagated with a constant velocity, each part of the undu- lation must in equal times have past through equal distances from the vibrating point. And, supposing the vibrating particle, in the course of its motion, to proceed forwards to a small dis- tance in a given direction, the principal strength of the undula- tion will naturally be straight before it ; behind it, the motion will be equal, in a contrary direction ; and, at right angles to the line of vibration, the undulation will be evanescent.
Now, in order that such an undulation may continue its pro- gress to any considerable distance, there must be in each part of it, a tendency to preserve its own motion in a right line from
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the centre ; for, if the excess of force at any part were commu- nicated to the neighbouring particles, there can be no reason why it should not very soon be equalised throughout, or, in other words, become wholly extinct, since the motions in con- trary directions would naturally destroy each other. The origin of sound from the vibration of a chord is evidently of this nature ; on the contrary, in a circular wave of water, every part is at the same instant either elevated or depressed. It may be difficult to show mathematically, the mode in which this inequality of force is preserved ; but the inference from the matter of fact, appears to be unavoidable ; and, while the science of hydrodynamics is so imperfect that we cannot even solve the simple problem of the time required to empty a vessel by a given aperture, it cannot be expected that we should be able to account perfectly for so complicated a series of phenomena, as those of elastic fluids. The theory of Huygens indeed explains the circumstance in a manner tolerably satisfactory : he sup- poses every particle of the medium to propagate a distinct un- dulation in all directions ; and that the general effect is only perceptible where a portion of each undulation conspires in direction at the same instant ; and it is easy to show that such a general undulation would in all cases proceed rectilinearly, with proportionate force ; but, upon this supposition, it seems to follow, that a greater quantity of force must be lost by the divergence of the partial undulations, than appears to be con- sistent with the propagation of the effect to any considerable distance. Yet it is obvious, that some such limitation of the motion must naturally be expected to take place ; for, if the intensity of the motion of any particular part, instead of conti- nuing to be propagated straight forwards, were supposed to affect the intensity of a neighbouring part of the undulation, an
25
the Theory of Light and Colours,
impulse must then have travelled from an internal to an exter- nal circle in an oblique direction, in the same time as in the direction of the radius, and consequently with a greater velo- city; against the first proposition. In the case of water,, the velocity is by no means so rigidly limited as in that of an elastic medium. Yet it is not necessary to suppose, nor is it indeed probable, that there is absolutely not the least lateral communication of the force of the undulation, but that, in highly elastic mediums, this communication is almost insensible. In the air, if a chord be perfectly insulated, so as to propagate exactly such vibrations as have been described, they will in fact be much less forcible than if the chord be placed in the neighbourhood of a sounding board, and probably in some measure because of this lateral communication of motions of an opposite tendency. And the different intensity of different parts of the same circular undulation may be observed, by holding a common tuning fork at arm's length, while sounding, and turning it, from a plane directed to the ear, into a position per- pendicular to that plane.
PROPOSITION IIIv
A Portion of a spherical Undulation , admitted through an Aper- ture into a quiescent Medium, will proceed to be further propa- gated rectilinearly in concentric Superficies, terminated laterally by weak and irregular Portions of newly diverging Undula- tions.
At the instant of admission, the circumference of each of the undulations may be supposed to generate a partial undulation, filling up the nascent angle between the radii and the surface terminating the medium ; but no sensible addition will be made.
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to its strength by a divergence of motion from any other parts of the undulation, for want of a coincidence in time, as has already been explained with respect to the various force of a spherical undulation. If indeed the aperture bear but a small proportion to the breadth of an undulation, the newly generated undulation may nearly absorb the whole force of the portion admitted ; and this is the case considered by Newton in the Principia. But no experiment can be made under these circum- stances with light, on account of the minuteness of its undula- tions, and the interference of inflection; and yet some faint radiations do actually diverge beyond any probable limits of inflection, rendering the margin of the aperture distinctly visible in all directions ; these are attributed by Newton to some un- known cause, distinct from inflection; (Optics, Third Book, Obs. 5.) and they fully answer the description of this propo- sition.
Let the concentric lines in Fig. 1. (Plate I.) represent the con- temporaneous situation of similar parts of a number of suc- cessive undulations diverging from the point A ; they will also represent the successive situations of each individual undulation: let the force of each undulation be represented by the breadth of the line, and let the cone of light ABC be admitted through the aperture BC ; then the principal undulations will proceed in a rectilinear direction towards GH, and the faint radiations on each side will diverge from B and C as centres, without receiving any additional force from any intermediate point D of the undulation, on account of the inequality of the lines DE and DF. But, if we allow some little lateral divergence from the extremities of the undulations, it must diminish their force, without adding materially to that of the dissipated light; and their
27
the Theory of Light and Colours.
termination, instead of the right line BG, will assume the form CH; since the loss of force must be more considerable near to C than at greater distances. This line corresponds with the boun- dary of the shadow in Newton's first observation, Fig. 1; and it is much more probable that such a dissipation of light was the cause of the increase of the shadow in that observation, than that it was owing to the action of the inflecting atmo- sphere, which must have extended a thirtieth of an inch each way in order to produce it ; especially when it is considered that the shadow was not diminished by surrounding the hair with a denser medium than air, which must in all probability have weakened and contracted its inflecting atmosphere. In other circumstances, the lateral divergence might appear to in- crease, instead of diminishing, the breadth of the beam.
As the subject of this proposition has always been esteemed the most difficult part of the undulatory system, it will be proper to examine here the objections which Newton has grounded upon it.
“ To me, the fundamental supposition itself seems impossible ; “ namely, that the waves or vibrations of any fluid can, like the “ rays of light, be propagated in straight lines, without a con- “ tinual and very extravagant spreading and bending every “ way into the quiescent medium, where they are terminated “ by it. I mistake, if there be not both experiment and demon- “ stration to the contrary." (Phil. Trans. VII. 5089, Abr. I. 146. Nov. 1672.)
“ Motus omnis per fluidum propagatus divergit a recto tra- “ mite in spatia immota."
“ Quoniam medium ibi," in the middle of an undulation
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Dr. Young’s Lecture on
28
admitted, “ densius est, quam in spatiis hinc inde, dilatabit sese <c tam versus spatia utrinque sita, quam versus pulsuum rariora <c intervalla; eoque pacto — pulsus eadem fere celeritate sese in “medii partes quiescentes hinc inde relaxare debent; — ideoque “ spatium totum occupabunt. — Hoc experimur in sonis.” (Prin- cip. Lib. II. Prop. 42.
“ Are not all hypotheses erroneous, in which light is supposed “ to consist in pression or motion, propagated through a fluid “ medium ? — If it consisted in pression or motion, propagated “ either in an instant, or in time, it would bend into the shadow. “ For pression or motion cannot be propagated in a fluid in “ right lines beyond an obstacle which stops part of the motion, <c but will bend and spread every way into the quiescent medium “ which lies beyond the obstacle. — The waves on the surface of “ stagnating water, passing by the sides of a broad obstacle “ which stops part of them, bend afterwards, and dilate them- “ selves gradually into the quiet water behind the obstacle. “ The waves, pulses, or vibrations of the air, wherein sounds t£ consist, bend manifestly, though not so much as the waves <f of water. For a bell or a cannon may be heard beyond a “ hill, which intercepts the sight of the sounding body; and « sounds are propagated as readily through crooked pipes as « straight ones. But light is never known to follow crooked « passages, nor to bend into the shadow. For the fixed stars, “ by the interposition of any of the planets, cease to be seen. “ And so do the parts of the sun, by the interposition of the « moon, Mercury, or Venus. • The rays which pass very near « to the edges of any body, are bent a little by the action of the « body ;— but this bending is not towards but from the shadow.
Dr. Young’s Lecture on
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rectilinear propagation of undulations, Newton has made no reply ; perhaps because of his own misconception of the nature of the motions of elastic mediums, as dependent on a peculiar law of vibration, which has been corrected by later mathematicians. (Phil. Trans, for 1800, p. 11 6.) On the whole, it is presumed, that this proposition may be safely admitted, as perfectly con- sistent with analogy and with experiment.
PROPOSITION IV.
When an Vndulation arrives at a Surface which is the Limit of Mediums of different Densities , a partial Reflection takes place , proportionate in Force to the Difference of the Densities.
This may be illustrated, if not demonstrated, by the analogy . of elastic bodies of different sizes. “ If a smaller elastic body ** strikes against a larger one, it is well known that the smaller “ is reflected more or less powerfully, according to the diffe- “ rence of their magnitudes : thus, there is always a reflection “ when the rays of light pass from a rarer to a denser stratum « of ether ; and frequently an echo when a sound strikes against a cloud. A greater body striking a smaller one, pro- « pels it, without losing all its motion : thus, the particles of a “ denser stratum of ether, do not impart the whole of their “ motion to a rarer, but, in their effort to proceed, they are “ recalled by the attraction of the refracting substance with 44 equal force ; and thus a reflection is always secondarily pro- “ duced, when the rays of light pass from a denser to a rarer 44 stratum/’ (Phil. Trans, for 1800. p. 127.J But it is not ab- solutely necessary to suppose an attraction in the latter case, since the effort to proceed would be propagated backwards without it, and the undulation would be reversed, a rarefaction
the Theory of Light and Colours . 29
« and is performed only in the passage of the ray by the body, « and at a very small distance from it. So soon as the ray is “ past the body, it goes right on.” (Optics, Qu, 28.)
Now the proposition quoted from the Principia does not di- rectly contradict this proposition ; for it does not assert that such a motion must diverge equally in all directions; neither can it with truth be maintained, that the parts of an elastic me- dium communicating any motion, must propagate that motion equally in all directions. (Phil. Trans, for 1800. p. 109 112,)
All that can be inferred by reasoning is, that the marginal parts of the undulation must be somewhat weakened, and that there must be a faint divergence in every direction ; but whe- ther either of these effects might be of sufficient magnitude to be sensible, could not have been inferred from argument, if the affirmative had not been rendered probable by experiment.
As to the analogy with other fluids, the most natural inference from it is this : “ The waves of the air, wherein sounds consist, « bend manifestly, though not so much as the waves of water water being an inelastic, and air a moderately elastic medium ; but ether being most highly elastic, its waves bend very far less than those of the air, and therefore almost imperceptibly. Sounds are propagated through crooked passages, because their sides are capable of reflecting sound, just as light would be pro- pagated through a bent tube, if perfectly polished within.
The light of a star is by far too weak to produce, by its faint
«/
divergence, any visible illumination of the margin of a planet eclipsing it ; and the interception of the sun's light by the moon, is as foreign to the question, as the statement of inflection is inaccurate.
To the argument adduced by Huygens, in favour of the
the Theory of Light and Colours. 31
returning in place of a condensation ; and this will perhaps be found most consistent with the phenomena.
proposition v.
When an Undulation is transmitted through a Surface terminating different Mediums , it proceeds in such a Direction, that the Shies of the Angles of Incidence and Refraction are in the constant Ratio of the Velocity of Propagation in the two Mediums.
(Barrow, Lecc. Opt. II. p. 4. Huygens, de la Lum. cap. 3. Euler, Conj. Phys. Phil. Trans, for 1800, p. 128. Young's Syllabus. Art. 382.)
Corollary 1. The same demonstrations prove the equality of the angles of reflection and incidence.
Corollary 2. It appears from experiments on the refraction of condensed air, that the ratio of the difference of the sines varies simply as the density. Hence it follows, by Schol. I. Prop. I. that the excess of the density of the ethereal medium is in the duplicate ratio of the density of the air ; each particle cooperating with its neighbours in attracting a greater portion of it.
proposition vi.
When an Undulation falls on the Surface of a rarer Medium, so obliquely that it cannot be regularly refracted, it is totally re- flected, at an Angle equal to that of its Incidence.
(Phil. Trans, for 1800, p. 128.)
Corollary. This phenomenon tends to prove the gradual in- crease and diminution of density at the surface terminating two mediums, as supposed in hypothesis iv ; although Huygens has attempted to explain it somewhat differently.
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Dr. Young's Lecture on
PROPOSITION VII.
If equidistant Undulations be supposed to pass through a Medium , of which the Parts are susceptible of permanent Vibrations some- what slower than the Undulations, their Velocity will be some- what lessened by this vibratory Tendency ; and, in the same Medium , the more, a$ the Undulations are more frequent.
For, as often as the state of the undulation requires a change in the actual motion of the particle which transmits it, that change will be retarded by the propensity of the particle to continue its motion somewhat longer : and this retardation will be more frequent, and more considerable, as the difference be- tween the periods of the undulation and of the natural vibration is greater.
Corollary . It was long an established opinion, that heat con- sists in’ vibrations of the particles of bodies, and is capable of being transmitted by undulations through an apparent va- cuum. (Newt. Opt. Qu. 18.) This opinion has been of late very much abandoned. Count Rumford, Professor Pictet, and Mr. Davy, are almost the only authors who have appeared to favour it ; but it seems to have been rejected without any good grounds, and will probably very soon recover its popularity.
Let us suppose that these vibrations are less frequent than those of light; all bodies therefore are liable to permanent vibrations slower than those of light; and indeed almost all are liable to luminous vibrations, either when in a state of ignition, or in the circumstances of solar phosphori ; but much less easily, and in a much less degree, than to the vibrations of heat. It will follow from these suppositions, that the more frequent luminous undulations will be more retarded than the less frequent ; and
33
the Tfjeory of Light and Colours.
consequently, that blue light will be more refrangible than red, and radiant heat least of all ; a consequence which coincides exactly with the highly interesting experiments of Dr. Her- schel. (Phil. Trans, for 1800. p. 284.) It may also be easily conceived, that the actual existence of a state of slower vibra- tion may tend still more to retard the more frequent undulations, and that the refractive power of solid bodies may be sensibly increased by an increase of temperature, as it actually appears to have been in Euler’s experiments. (Acad, de Berlin. 1762. p. 328.)
Scholium. If, notwithstanding, this proposition should appear to be insufficiently demonstrated, it must be allowed to be at least equally explanatory of the phenomena with any thing that can be advanced on the other side, from the doctrine of projec- tiles ; since a supposed accelerating force must act in some other proportion than that of the bulk of the particles ; and, if we call this an elective attraction, it is only veiling under a chemical term, our incapacity of assigning a mechanical cause. Mr. Short, when he found by observation the equality of the velo- city of light of all colours, felt the objection so forcibly, that he immediately drew an inference from it in favour of the undula- tory system. It is assumed in the proposition, that when light is dispersed by refraction, the corpuscles of the refracting sub- stance are in a state of actual alternate motion, and contribute to its transmission ; but it must be confessed, that we cannot at present form a very decided and accurate conception of the forces concerned in maintaining these corpuscular vibrations.
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F
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Dr. Young’s Lecture on
PROPOSITION VIII.
When two Undulations , from different Origins , coincide either perfectly or very nearly in Direction , their joint effect is a Com- bination of the Motions belonging to each.
Since every particle of the medium is affected by each undu- lation, wherever the directions coincide, the undulations can proceed no otherwise than by uniting their motions, so that the joint motion may be the sum or difference of the separate motions, accordingly as similar or dissimilar parts of the undu- lations are coincident.
I have, on a former occasion, insisted at large on the appli- cation of this principle to harmonics; (Phil. Trans, for 1800. p. 130.) and it will appear to be of still more extensive utility in explaining the phenomena of colours. The undulations which are now to be compared are those of equal frequency. When the two series coincide exactly in point of time, it is obvious that the united velocity of the particular motions must be greatest, and, in effect at least, double the separate velocities ; and also, that it must be smallest, and if the undulations are of equal strength, totally destroyed, when the time of the greatest direct motion belonging to one undulation coincides with that of the greatest retrograde motion of the other. In intermediate states, the joint undulation will be of intermediate strength ; but by what laws this intermediate strength must vary, cannot be determined without further data. It is well known that a similar cause produces in sound, that effect which is called a beat ; two series of undulations of nearly equal magnitude co- operating and destroying each other alternately, as they coincide
the Theory of Light and Colours. 35
more or less perfectly in the times of performing their respective motions.
Corollary i. Of the Colours of striated Surfaces.
Boyle appears to have been the first that observed the colours of scratches on polished surfaces. Newton has not noticed them. Mazeas and Mr. Brougham have made some experiments on the subject, yet without deriving any satisfactory conclusion. But all the varieties of these colours are very easily deduced from this proposition.
Let there be in a given plane two reflecting points very near each other, and let the plane be so situated that the reflected image of a luminous object seen in it may appear to coincide with the points ; then it is obvious that the length of the inci- dent and reflected ray, taken together, is equal with respect to both points, considering them as capable of reflecting in all directions. Let one of the points be now depressed below the given plane; then the whole path of the light reflected from it, will be lengthened by a line which is to the depression of the point as twice the cosine of incidence to the radius. Fig. 2.
If, therefore, equal undulations of given dimensions be reflected from two points, situated near enough to appear to the eye but as one, wherever this line is equal to half the breadth of a whole undulation, the reflection from the depressed point will so in- terfere with the reflection from the fixed point, that the pro- gressive motion of the one will coincide with the retrograde motion of the other, and they will both be destroyed ; but, when this line is equal to the whole breadth of an undulation, the effect will be doubled ; and when to a breadth and a half, again destroyed ; and thus for a considerable number of alternations ; and, if the reflected undulations be of different kinds, they will
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Dr, Young’s Lecture on
S6
be variously affected, according to their proportions to the vari- ous length of the line which is the difference between the lengths of their two paths, and which may be denominated the interval of retardation.
In order that the effect may be the more perceptible, a num- ber of pairs of points must be united into two parallel lines ; and, if several such pairs of lines be placed near each other, they will facilitate the observation. If one of the lines be made to revolve round the other as an axis, the depression below the given plane will be as the sine of the inclination ; and, while the eye and luminous object remain fixed, the difference of the length of the paths will vary as this sine.
The best subjects for the experiment are Mr. Coventry’s exquisite micrometers ; such of them as consist of parallel lines drawn on glass, at the distance of one five hundredth of an inch, are the most convenient. Each of these lines appears under a microscope to consist of two or more finer lines, exactly parallel, and at the distance of somewhat more than a twentieth of that of the adjacent lines. I placed one of these so as to reflect the sun’s light at an angle of 450, and fixed it in such a manner, that while it revolved round one of the lines as an axis, I could measure its angular motion ; and I found, that the brightest red colour occurred at the inclinations lof, 2of°, 320, and 450; of which the sines are as the numbers 1, 2, 3, and 4. At all other angles also, when the sun’s light was reflected from the sur- face, the colour vanished with the inclination, and was equal at equal inclinations on either side.
This experiment affords a very strong confirmation of the theory. It is impossible to deduce any explanation of it from any hypothesis hitherto advanced ; and I believe it would be
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the Theory of Light and Colours.
difficult to invent any other that would account for it. There is a striking analogy between this separation of colours, and the production of a musical note by successive echoes from equi- distant iron palisades ; which I have found to correspond pretty accurately with the known velocity of sound, and the distances of the surfaces.
It is not improbable that the colours of the integuments of some insects, and of some other natural bodies, exhibiting in different lights the most beautiful versatility, may be found to be of this description, and not to be derived from thin plates. In some cases, a single scratch or furrow may produce similar effects, by the reflection of its opposite edges.
Corollary if. Of the Colours of thin Plates.
'When a beam of light falls on two parallel refracting sur- faces, the partial reflections coincide perfectly in direction ; and, in this case, the interval of retardation, taken between the sur- faces, is to their distance as twice the cosine of the angle of refraction to the radius. For, in Fig. 3, drawing AB and CD perpendicular to the rays, the times of passing through BC and AD will be equal, and DE will be half the interval of retarda- tion; but DE is to CE as the sine of DCE to the radius. Hence, that DE may be constant, or that the same colour may be re- flected, the thickness CE must vary as the secant of the angle of refraction CED : which agrees exactly with Newton’s expe- riments ; for the correction is perfectly inconsiderable.
Let the. medium between, the surfaces be rarer than the sur- rounding mediums ; then the impulse reflected at the second surface, meeting a subsequent undulation at the first, will render the particles of the rarer medium capable of wholly stopping
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Dr. Young’s Lecture on
the motion of the denser, and destroying the reflection, (prop, iv.) while they themselves will be more strongly propelled than if they had been at rest ; and the transmitted light will be increased. So that the colours by reflection will be destroyed, and those by transmission rendered more vivid, when the double thicknesses, or intervals of retardation, are any multiples of the whole breadths of the undulations ; and, at intermediate thick- nesses the effects will be reversed; according to the Newtonian ^observations.
If the same proportions be found to hold good with respect to thin plates of a denser medium, which is indeed not impro- bable, it will be necessary to adopt the corrected demonstration of prop. iv. but, at any rate, if a thin plate be interposed be- tween a rarer and a denser medium, the colours by reflection and transmission may be expected to change places.
From Newton’s measures of the thicknesses reflecting the different colours, the breadth and duration of their respective undulations may be very accurately determined ; although it is not improbable, that when the glasses approach very near, the atmosphere of ether may produce some little irregularity. The whole visible spectrum appears to be comprised within the ratio of three to five, or a major sixth in music ; and the undulations of red, yellow, and blue, to be related in magnitude as the numbers 8, 7, and 6 ; so that the interval from red to blue is a fourth. The absolute frequency expressed in numbers is too great to be distinctly conceived, but it may be better ima- gined by a comparison with sound. If a chord sounding the tenor c, could be continually bisected 40 times, and should then vibrate, it would afford a yellow^green light : this being
41 40 41
denoted by c, the extreme red would be a, and the blue d.
3$
the Theory of Light and Colours.
The absolute length and frequency of each vibration is ex- pressed in the table ; supposing light to travel in 8|- minutes 500,000,000000 feet.
|
Colours. |
Length of an Undulation in parts of an Inch, in Air. |
Nufnber of Undulations in an Inch. |
Number of Undulations in a Second. |
|
Extreme |
.0000266 |
3764° |
463 millions of millions |
|
Red |
.OOOO256 |
3918° |
482 |
|
Intermediate |
.OOOO246 |
40720 |
501 |
|
Orange |
.OOOO24O |
4l6lO |
512 |
|
Intermediate |
.OOO0235 |
42510 |
523 |
|
Y ellow |
.0000227 |
44OOO |
542 |
|
Intermediate |
.0000219 |
45600 |
561 (= 248 nearly) |
|
Green - |
.0000211 |
4746° |
5H |
|
Intermediate |
.0000203 |
49320 |
607 |
|
Blue - |
.OOOOI96 |
51 up |
629 |
|
Intermediate |
.OOOO189 |
529IO |
652 |
|
Indigo |
.OOOOI85 |
54°7° |
665 |
|
Intermediate |
.OOOOlBl |
35240 |
680 |
|
Violet - |
.OOOOI74 |
57490 |
7°7 |
|
Extreme - |
.OOOOI67 |
59750 |
735 |
Scholium. It was not till I had satisfied myself respecting all these phenomena,' that I found in Hooke’s Micrographia, a pas- sage which might have led me earlier to a similar conclusion. <c It is most evident that the reflection from the under or fur- “ ther side of the body, is the principal cause of the production “ of these colours. — Let the ray fall obliquely on the thin “ plate, part therefore is reflected back by the first superficies, “ - — part refracted to the second surface, — whence it is reflected “ and refracted again. — So that, after two refractions and one
40
Dr. Young's Lecture on
il reflection, there is propagated a kind of fainter ray — ,” and, M by reason of the time spent in passing and repassing, —this u fainter pulse comes behind the” former reflected “ pulse ; so “ that hereby, (the surfaces being so near together that the eye tc cannot discriminate them from one,) this confused or duplicated “ pulse, whose strongest part precedes, and whose weakest fol- “ lows, does produce on the retina, the sensation of a yellow. “ If these surfaces are further removed asunder, the weaker “ pulse may become coincident with the” reflection of the sc second,” or next following pulse, from the first surface, “ and “ lagg behind that also, and be coincident with the third, “ fourth, fifth, sixth, seventh, or eighth — ; so that, if there be <e a thin transparent body, that from the greatest thinness requi- “ site to produce colours, does by degrees grow to the greatest te thickness,— the colours shall be so often repeated, as the ££ weaker pulse does lose paces with its primary or first pulse, 6£ and is coincident with a” subsequent “ pulse. And this, as <£ it is coincident, or follows from the first hypothesis I took of “ colours, so upon experiment have I found it in multitudes of “ instances that seem to prove it.” (P. 65 — 67.) This was printed about seven years before any of Newton's experiments were made. We are informed by Newton, that Hooke was afterwards disposed to adopt his “ suggestion” of the nature of colours ; and yet it does not appear that Hooke ever applied that improvement to his explanation of these phenomena, or inquired into the necessary consequence of a change of obliquity, upon his original supposition, otherwise he could not but have dis- covered a striking coincidence with the measures laid down by Newton from experiment. All former attempts to explain the colours of thin plates, have either proceeded on suppositions
the Theory of Light and Colours. 44
which, like Newton’s, would lead us to expect the greatest irre- gularities in the direction of the refracted rays ; or, like Mr. Michell’s, would require such effects from the change of the angle of incidence, as are contrary to the effects observed; or they are equally deficient with respect to both these circum- stances, and are inconsistent with the most moderate attention to the principal phenomena.
Corollary in. Of the Colours of thick Plates.
1
When a beam of light passes through a refracting surface, especially if imperfectly polished, a portion of it is irregularly scattered, and makes the surface visible in all directions, but most conspicuously in directions not far distant from that of the light itself : and, if a reflecting surface be placed parallel to the refracting surface, this scattered light, as well as the prin- cipal beam, will be reflected, and there will also be a new dis- sipation of light, at the return of fhe beam through the refracting surface. These two portions of scattered light will coincide in direction ; and, if the surfaces be of such a form as to collect the similar effects, will exhibit rings of colours. The interval of retardation is here, the difference between the paths of the principal beam and of the scattered light between the two sur- faces ; of course, wherever the inclination of the scattered light is equal to that of the beam, although in different planes, the interval will* vanish, and all the undulations will conspire. At other inclinations, the interval will be the difference of the secants from the secant of the inclination or angle of refraction of the principal beam. From these causes, all the colours of concave mirrors observed by Newton and others are necessary consequences : and it appears that their production, though mdcccii. G
42
Dr. Young* s Lecture on
somewhat similar, is by no means, as Newton imagined, iden- tical with the production of those of thin plates.
Corollary iv. Of Blackness.
In the three preceding corollaries, we have considered the Refracting and reflecting substances as limited by a mathema- tical surface; but this is perhaps never physically true. The ethereal atmospheres may extend on each side the surface as far as the breadth of one or more undulations ; and, if they be supposed to vary equally in density at every part, the partial reflections from each of the infinite number of surfaces, where the density changes, will very much interfere with each other, and destroy a considerable portion of the reflected light, so that the substance may become positively black; and this effect may take place in a greater or less degree, as the density of the ethereal atmosphere varies more or less equably; and, in some cases, particular undulations being more affected than others, a tinge of colour may be produced. Accordingly, M. Bouguer has observed a considerable loss of light, and in some instances a tinge of colour, in total reflections at the surface of a rarer medium.
Corollary v. Of Colours by Inflection.
Whatever may be the cause of the inflection of light passing through a small aperture, the light nearest its centre must be the least diverted, and the nearest to its sides the most : ano- ther portion of light falling very obliquely on the margin of the aperture, will be copiously reflected in various directions; some of which will either perfectly or very nearly coincide in direc- tion with the unreflected light, and, having taken a circuitous
43
the Theory of Light and Colours .
route, will so interfere with it, as to cause an appearance of colours. The length of the two tracks will differ the less, as the direction of the reflected light has been less changed by its reflection, that is, in the light passing nearest to the margin ; so that the blues will appear in the light nearest the shadow. The effect will be increased and modified, when the reflected light falls within the influence of the opposite edge, so as to interfere with the light simply inflected by that also.
But, in order to examine the consequences more minutely, it will be convenient to suppose the inflection caused by an ethereal atmosphere, of a density varying as a given power of the dis- tance from a centre, as in the eighth proposition of the last Bakerian Lecture. (Phil. Trans, for 1801, p. 83.) Putting r = 3, and x =■§-, I have constructed a diagram, (Fig. 4,) which shows, by the two pairs of curves, the relative position of the re- flected and unreflected portions of any one undulation at two successive times, and also, by shaded lines drawn across, the parts where the intervals of retardation are in arithmetical progression, and where similar colours will be exhibited at different distances from the inflecting substance. The result fully agrees with the observations of Newton’s third book, and with those of later writers. But I do not consider it as quite certain, until further experiments have been made on the inflecting power of dif- ferent substances, that Dr. Hooke’s explanation of inflection, by the tendency of light to diverge, may not have some preten- sions to truth . I am sorry to be obliged to recall here the assent which, at first sight, I was induced to give to a supposed im- provement of a late author. (Phil. Trans, for 1800, p. 128.)
Scholium. In the construction of the diagram, it becomes ne- cessary to find the time spent by each ray in its passage,
G 2 '
Dr. Young's Lecture on
Since the velocity was denoted by a; r , on the supposition of a
X
projectile, it will be as x 7 on the contrary supposition, (Phil. Trans, for 1801, p. 27. Schol. 2. Prop. I.) and the fluxion of the
1
■ ■» • distance described being r7==, that of the time will be -7===*
or-^— . - , of which the fluent is — f-. — . v^t — yv.
*-r '-r s JJ
1
Therefore, with the radius .r1"" r , describe a circle concentric with the surfaces of the inflecting atmosphere, then the angle described by the ray during its passage through the atmosphere, will always be to the angle subtended by the line cut off by this circle from the incident ray produced, in the ratio of r to r — 1; and the time spent in this passage, will be in the same ratio to the time that would have been spent in describing this intercepted portion with the initial velocity. For y, being equal
to is the sine of the inclination of the incident ray to the
radius, where it meets this circle ; therefore, by the proposition quoted, the angle described is in a given ratio to the angle at the centre, which is the difference of the inclinations. Making ^-■fory radius, the sine, instead ofjy, becomes s , and the co- sine v/ ~ — ss, or -1 v/ 1 — yy, and, when y = ss, v/ 1 — ss ; y y y
therefore the line intercepted is to the difference of the fluents as r to r — 1. (See also Young’s Syllabus, Art. 372.)
PROPOSITION IX.
Radiant Light consists in Undulations of the luminiferous Ether.
This proposition is the general conclusion from all the pre- ceding ; and it is conceived that they conspire to prove it in as satisfactory a manner as can possibly be expected from the
45
the Theory of Light and Colours .
nature of the subject. It is clearly granted by Newton, that there are undulations, yet he denies that they constitute light; but it is shown in the three first Corollaries of the last Proposi- tion, that all cases of the increase or diminution of light are referable to an increase or diminution of such undulations, and that all the affections to which the undulations would be liable, are distinctly visible in the phenomena of light ; it may there- fore be very logically inferred, that the undulations are light.
A few detached remarks will serve to obviate some objections which may be raised against this theory.
1. Newton has advanced the singular refraction of the Ice- land crystal, as an argument that the particles of light must be projected corpuscles ; since he thinks it probable that the dif- ferent sides of these particles must be differently attracted by the crystal, and since Huygens has confessed his inability to account in a satisfactory manner for all the phenomena. But, contrarily to what might have been expected from Newton's usual accuracy and candour, he has laid down a new law for the refraction, without giving a reason for rejecting that of
Huygens, which Mr. Hauy has found to be more accurate than
»
Newton's ; and, without attempting to deduce from his own system any explanation of the more universal and striking effects of doubling spars, he has omitted to observe that Huygens's most elegant and ingenious theory perfectly accords with these general effects, in all particulars, and of course derives from them additional pretensions to truth : this he omits, in order to point out a difficulty, for which only a verbal solution can be found in his own theory, and which will probably long remain unexplained by any other.
2. Mr. Michell has made some experiments, which appear* to show that the rays of light have an actual momentum, by
46 Dr. Young's Lecture on
means of which a motion is produced when they fall on a thin plate of copper delicately suspended. (Priestley's Optics.) But, taking for granted the exact perpendicularity of the plate, and the absence of any ascending current of air, yet since, in every such experiment, a greater quantity of heat must be com- municated to the air at the surface on which the light falls than at the opposite surface, the excess of expansion must necessarily produce an excess of pressure on the first surface, and a very perceptible recession of the plate in the direction of the light. Mr. Bennet has repeated the experiment, with a much more sensible apparatus, and also in the absence of air ; and very justly infers from its total failure, an argument in favour of the undu- latory system of light. (Phil. Trans, for 1792, p. 87.) For, granting the utmost imaginable subtility of the corpuscles of light, their effects might naturally be expected to bear some proportion to the effects of the much less rapid motions of the electrical fluid, which are so very easily perceptible, even in their weakest states.
3. There are some phenomena of the light of solar phosphori, which at first sight might seem to favour the corpuscular sys- tem ; for instance, its remaining many months as if in a latent state, and its subsequent re-emission by the action of heat. But, on further consideration, there is no difficulty in supposing the particles of the phosphori which have been made to vibrate by the action of light, to have this action abruptly suspended by the intervention of cold, whether as contracting the bulk of the substance or otherwise; and again, after the restraint is removed, to proceed in their motion, as a spring would do which had been held fast for a time in an intermediate stage of its vibra- tion ; nor is it impossible that heat itself may, in some circum- stances,, become in a similar manner latent. (Nicholson's
47
the Theory of Light and Colours .
Journal. Vol. II. p. 399. ) But the affections of heat may perhaps hereafter be rendered more intelligible to us ; at present, it seems highly probable that light differs from heat only in the frequency of its undulations or vibrations ; those undulations which are within certain limits, with respect to frequency, being capable of affecting the optic nerve, and constituting light; and those which are slower, and probably stronger, constituting heat only ; that light and heat occur to us, each in two predicaments, the vibratory or permanent, and the undulatory or transient state; vibratory light being the minute motion of ignited bodies, or of solar phos- phori, and undulatory or radiant light the motion of the ethereal medium excited by these vibrations; vibratory heat being a motion to which all material substances are liable, and which is more or less permanent ; and undulatory heat that motion of the same ethereal medium, which has been shown by Mr. King, (Mor- sels of Criticism. 1786. p. 99,) and M. Pictet, (Essais de Phy- sique. 1790,) to be as capable of reflection as light, and by Dr. Herschel to be capable of separate refraction. (Phil Trans, for 1800. p. 284.) How much more readily heat is communicated by the free access of colder substances, than either by radiation or by transmission through a quiescent medium, has been shown by the valuable experiments of Count Rumford. It is easy to conceive that some substances, permeable to light, may be unfit for the transmission of heat, in the same manner as particular substances may transmit some kinds of light, while they are opaque with respect to others.
On the whole it appears, that the few optical phenomena which admit of explanation by the corpuscular system, are equally consistent with this theory ; that many others, which have long been known, but never understood, become by these means perfectly intelligible; and that several new facts are
4$
Dr. Young’s Lecture, &c.
found to be thus only reducible to a perfect analogy with other facts, and to the simple principles of the undulatory system. It is presumed, that henceforth the second and third books of New- ton’s Optics will be considered as more fully understood than the first has hitherto been ; but, if it should appear to impartial judges, that additional evidence is wanting for the establishment of the theory, it will be easy to enter more minutely into the details of various experiments, and to show the insuperable dif- ficulties attending the Newtonian doctrines, which, without necessity, it would be tedious and invidious to enumerate. The merits of their author in natural philosophy, are great beyond all contest or comparison ; his optical discovery of the composition of white light, would alone have immortalised his name; and the very arguments which tend to overthrow his system, give the strongest proofs of the admirable accuracy of his experiments.
Sufficient and decisive as these arguments appear, it cannot be superfluous to seek for further confirmation; which may with considerable confidence be expected,' from an experiment very in- geniously suggested by Professor Rori son, on the refraction of the light returning to us from the opposite margins of Saturn’s ring; for, on the corpuscular theory, the ring must be considerably distorted when viewed through an achromatic prism : a similar distortion ought also to be observed in the disc of Jupiter; but, if it be found that an equal deviation is produced in the whole light reflected from these planets, there can scarcely be any re- maining hope to explain the affections of light, by a comparison with the motions of projectiles.
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C 49 3
III. An Analysis of a mineral Substance from North America, containing a Metal hitherto unknown. By Charles Hatchett, Esq. F. R.S.
Read November 2 6, 1801.
In the course of the last summer, when I was examining and arranging some minerals in the British Museum, I observed a small specimen of a dark-coloured heavy substance, which attracted my attention, on account of some resemblance which it had with the Siberian chromate of iron, on which at that time I was making experiments.
Upon referring to Sir Hans Sloane’s catalogue, I found that this specimen was only described as “ a very heavy black stone, “ with golden streaks,” which proved to be yellow mica ; and it appeared, that it had been sent, with various specimens of iron ores, to Sir Hans Sloane, by Mr. Winthrop, of Massachu- sets. The name of the mine, or place where it was found, is also noted in the catalogue ; the writing however is scarcely legible : it appears to be an Indian name, (Natitneauge ;) but I am in- formed by several American gentlemen, that many of the Indian names (by which certain small districts, hills, &c. were forty or fifty years ago distinguished,) are now totally forgotten, and European names have been adopted in the room of them. This may have been the case in the present instance; but, as the other specimens sent by Mr. Winthrop were from the mines ol Massachusets, there is every reason to believe that the mdcccii. H
50
Mr. Hatchett's Analysis of
mineral substance in question came from one of them, although it may not now be easy to identify the particular mine.
§ I. DESCRIPTION OF THE ORE.
The external colour is dark brownish gray.
The internal colour is the same, inclining to iron gray.
The longitudinal fracture is imperfectly lamellated ; and the cross fracture shews a fine grain.
The lustre is vitreous, slightly inclining in some parts to metallic lustre.
It is moderately hard, and is very brittle.
The colour of the streak or powder is dark chocolate brown.
The particles are not attracted by the magnet.
The specific gravity, at temp. 65°, is 5918.*
Experiment 1.
Some of the ore, reduced to fine powder, was digested in boiling muriatic acid for about one hour.
The acid appeared to have acted but slightly upon the powder; as the former remained colourless, and the latter did not seem to be diminished. A portion, however, chiefly of iron, waR found to be dissolved ; for ammonia formed a yellow flocculent pre- cipitate; prussiate of potash produced one which was blue;
* The following results of some experiments which I have purposely made, will shew how much the specific gravity of this ore is different from that of Wolfram, and Siberian chromate of iron.
Pure Wolfram, free from extraneous substances, at temp. 65° - - 6955.
Siberian chromate of iron, containing some of the green oxide - 3728.
Pure Siberian chromate of iron - 4355*
The Siberian chromate of iron, like all other mineral substances which are not crystallized, and which consequently are not always homogeneous, must evidently be liable to considerable variations in specific gravity.
51
a mineral Substance from North America.
and tincture of galls, when the excess of acid had been pre- viously saturated by an alkali, formed a precipitate of a rich purplish brown colour.
Experiment n.
Another portion of the powder was, in like manner, digested with nitric acid; but, excepting some slight traces of iron, this acid afforded nothing worthy of notice ; the action of it upon the ore, was indeed scarcely perceptible.
Experiment hi.
Some of the pulverized ore was digested with concentrated sulphuric acid, in a strongly-heated sand-bath, until nearly the whole of the acid was evaporated ; the edges of the mass then appeared bluish, and became white, when boiling distilled water was added.
This acid certainly acted much more powerfully than those which have been mentioned ; but still only a small part of the ore was dissolved. It must however be observed, that a very copious blue precipitate was obtained by prussiate of potash ; a plentiful purplish brown precipitate was also produced by tinc- ture of galls, after the excess of acid had been saturated by an . alkali; and, lastly, when the yellow ferruginous precipitate formed by ammonia was dissolved in diluted nitric acid, some white flocculi remained, which were completely insoluble in the acid, even when it was added so as to be in considerable excess.
From these experiments it was evident, that the ore could not readily be decomposed by the direct application of the mineral acids; and I therefore had recourse to the following
H s
52 Mr. Hatchett's Analysis of
method, which has frequently been employed with success in similar cases.
ANALYSIS.
A.
A mixture of 200 grains of the powdered ore with five times the weight of carbonate of potash, was exposed to a strong red heat, in a silver crucible. As soon as the matter bewail to flow', a very perceptible effervescence took place ; and, when this had subsided, the whole was poured into a proper vessel.
The mass, when cold, was grayish- brown.
Boiling distilled water was poured upon it ; and the brown residuum, which was considerable, was well edulcorated upon a filter.
The filtrated liquor had a slight yellowish tinge, and, being supersaturated with nitric acid, afforded a copious white floccu- lent precipitate, which speedily subsided ; but, although a very considerable additional quantity of nitric acid was poured upon the precipitate, it was not re-dissolved.
The residuum of the ore was dark brown, and was again melted with potash, and treated as before; but scarcely any effect was thus produced ; the alkali was therefore washed offj and the powder was digested with muriatic acid, which soon assumed the deep yellow colour usually communicated to it by iron. After half an hour, the acid was decanted, and the resi- duum was washed with distilled water.
This powder was now of a much paler colour; and, being mixed with potash, it was melted and treated as before. A considerable precipitate was again obtained by the addition of nitric acid ; and the residuum, after being digested with mu- riatic acid, was again fused with potash, by which means the
53
a mineral Substance from North America.
whole was completely decomposed, after about five repetitions of each operation.
B.
The muriatic solution was diluted, and, being saturated with ammonia, afforded a plentiful ochraceous precipitate; which again was dissolved in cold dilute nitric acid, and afforded a small quantity of a white insoluble substance, similar to that which was obtained from the alkaline solution. From this nitric solution, I then obtained, by means of ammonia, a pre- cipitate of oxide of iron, which, being properly dried, weighed 40 grains.
C.
The different alkaline solutions which had been made subse- quent to that which has been first mentioned, were mixed together, and, being supersaturated with nitric acid, afforded the same white insoluble precipitate; the total quantity of which, obtained from 200 grains of the ore, amounted to about 155 grains.
The liquor from which this precipitate had been separated by nitric acid, was then saturated with ammonia, and, being boiled, afforded about 2 grains of oxide of iron.
I obtained, therefore, from 200 grains of the ore.
Grains.
Oxide of iron “ - - 42 ] Grains.
And of the white precipitated substance 1 55 / ~ W*
But, as I could not repeat the analysis without destroying the remaining part of the only specimen at present known of this ore, I do not wish the above stated proportions to be regarded as rigidly exact ; it will be sufficient, therefore, to say at present, that the ore is composed of about three parts of the white matter, and rather less than one of iron.
Mr. Hatchett’s Analysis of
§ II. PROPERTIES OF THE WHITE PRECIPITATE.
A.
It Is of a pure white, and is not extremely heavy.
It has scarcely any perceptible flavour, nor does it appear to be soluble in boiling water; when, however, some of the powder is placed upon litmus paper moistened with distilled water, the paper in a few minutes evidently becomes red.
B.
1. When examined by the blow-pipe, it is not fusible per se in a spoon of platina, nor upon charcoal, but only becomes of a less brilliant white.
2. Borax does not appear to act upon it; for the white par- ticles are only dispersed throughout the globule.
3. It produces an effervescence when fused with carbonate of soda, and forms a colourless salt ; but, if too much of it be added, then the mass, when cold, appears like a white opaque enamel.
4. When carbonate of potash is employed, the effects are similar in every respect to those of soda ; and it may here be remarked, that the saline combinations thus formed with soda, or potash, are soluble in water ; and that these solutions have the same properties as that which was formed when the ore was decomposed by an alkali. The portion of the white preci- pitate which may be in excess, subsides unaltered, when the globules are dissolved in water.
5. Phosphate of ammonia produces a very marked effect; for, when melted in a platina spoon, if some of the white sub- stance be added, a considerable effervescence takes place, and the two substances rapidly unite. The globule, when cold, is
a mineral Substance from North America, 55
deep blue, with a tinge of purple, but, when held between the eye and the light, it appears of a greenish gray colour.
V
C.
It is perfectly insoluble, and remains unchanged in colour, and in every other respect, when digested in boiling concen- trated nitric acid.
D.
It is dissolved by boiling sulphuric acid, and forms a tran- sparent. colourless solution, which is however only permanent while the acid remains in a concentrated state ; for, if a large quantity of water be added to the solution, or if the latter be poured into a vessel of distilled water, the whole in a few minutes assumes a milay appearance, and a white precipitate is gradually deposited, which cracks as it becomes dry upon the filter, and, fiom wnite, changes to a lavender-blue colour, and again, when completely dry, to a brownish gray. It is then insoluble in water, has not any flavour, is semi-transparent, and breaks with a glossy vitreous fracture.
This substance is much heavier than the original white pre- cipitate ; and in a very slight degree may be dissolved by boiling muriatic acid, or by boiling lixivium of potash.
Upon examining these solutions, I found that both contained the original white substance, together with some sulphuric acid; so that the precipitate obtained from the sulphuric solution by the addition of water, is a sulphate of the white matter.*
The whole is not however precipitated by water; for a part
* This sulphate is also precipitated when the sulphuric solution has been long ex- posed in an open vessel to the air ; and, according as this may be moist or dry, the effect is produced sooner or later.
58 Mr. Hatchett’s Analysis of
remains in solution, which may be separated from the sulphuric acid by either of the fixed alkalis, or by ammonia.
The sulphuric solution is not rendered turbid by the addition of water, until some minutes at least have elapsed ; when, there- fore, some prussiate of potash was added immediately after the water, the colour of the liquor became olive green, and a copious precipitate, of a beautiful olive colour, was gradually deposited.
Tincture of galls, after a few minutes, caused the liquor to become turbid, and a very high orange- coloured precipitate was obtained.
A few drops of phosphoric acid were added to a part of the concentrated sulphuric solution; and, after about 12 hours, the whole became a white opaque stiff jelly, which was insoluble in water.
Potash, soda, and ammonia, whether pure or in the state of carbonates, separate the substance in question from the sul- phuric solution, in the form of a white flocculent precipitate ; and, when these alkalis are added to a considerable excess, they do not redissolve the precipitate, unless they are heated ; then, indeed, the fixed alkalis act upon it, and form combinations which have already been mentioned, but which we shall soon have occasion more particularly to notice,
E.
1 . The white precipitate, when recently separated from pot- ash, is soluble in boiling muriatic acid; and this solution may be considerably diluted with water, without any change being produced.
2. A part was evaporated to dryness, and left a pale yellow substance, which was not soluble in water, and was dissolved
a mineral Substance from North America. 57
with great difficulty, when it was again digested with muriatic acid.
3. Prussiate of potash changed the colour of the muriatic solution to an olive-green ; the liquor then gradually became turbid, and an olive-coloured precipitate was obtained, similar to that which has been lately mentioned. But,
4. If some nitric acid was previously added to the muriatic solution, then the prussiate changed the liquor to a grass-green, but did not produce any precipitate.
5. Tincture of galls, in a few minutes, formed an orange- coloured precipitate, like that which has been mentioned ; but, if the acid was in too great an excess, it wa^ necessary to add a small quantity of lixivium of potash or soda, before the preci- pitate could be obtained.
6. A small quantity of phosphoric acid, being added to the muriatic solution, in a few hours formed a white flocculent precipitate.
7. Potash, soda, and ammonia, also produced white floccu- lent precipitates, which were not redissolved by an excess of the alkalis, unless the liquors were heated ; and, in that case, part was dissolved by the fixed alkalis, but not by ammonia.
8. The muriatic solution did not yield any precipitate, when the muriates of lime, magnesia, and strontian, were added ; but muriate of barytes formed a slight cloud.
9. When a piece of zinc was immersed in the muriatic so- lution, a white flocculent precipitate was obtained.*
* This appears to indicate the obstinacy with which this substance retains a certain portion of oxygen ; tor we here see that zinc does not precipitate it in the metallic state, but only reduces it to an insoluble oxide.
mdccc.il I
58 Mr. Hatchett’s Analysis of
F.
The acetous acid has not any apparent effect on the white precipitate, when long digested with it.
G.
The fixed alkalis readily combine with this substance, both in the dry and in the humid way.
We have already seen, that the former method was employed with success in the analysis of the ore; and the experiments made with the blow-pipe may be regarded as an additional con- firmation. In each of these cases, the white precipitate com- bined with the alkali, as soon as the heat was sufficient to cause the latter to flow'; and, when a carbonate was employed, a portion of carbonic acid was expelled.
The carbonic acid was in like manner disengaged, when the white precipitate was boiled with lixivium of carbonate of pot- ash, or of soda ; and the solutions thus prepared, resembled in every respect those which were formed by dissolving in water the salts which had been produced in the dry way.
It will be proper here to give a more particular account of
these combinations.
i. Some of the white precipitate was digested, during nearly one hour, with boiling lixivium of pure or caustic potash : about one-fourth of the powder was dissolved ; and the remainder, which appeared little if at all altered, subsided to the bottom of the vessel.
The clear solution, which contained a great excess of alkali, was decanted; and, by gentle evaporation, yielded a white glit- tering salt, in scales, very much resembling the concrete boracic
acid.
a mineral Substance from North America. 59
The salt was placed upon a filter, so that the lixivium might be separated. It was then washed with a small quantity of cold distilled water; and, being dried, remained as above described, although constantly exposed to the open air.
This salt had an acrid disagreeable flavour, and contained a small excess of alkali. It did not dissolve very readily in cold water ; but, when dissolved, the solution was perfect and per- manent.
Some nitric acid was added to part of the solution, and im- mediately rendered it white and turbid. In a short time, a white precipitate was collected, similar to that which had been em- ployed to neutralise the potash ; and the clear supernatant liquor, being evaporated, only afforded nitre.
Prussiate of potash was added to another portion ; but did not produce any effect, until some muriatic acid was dropped into the liquor, which then immediately assumed a tinge of olive green, and slowly deposited a precipitate of the same colour.
Tincture of galls did not affect the solution at first; but, when a few drops of muriatic acid had been added, it gradually lost its transparency, and yielded an orange-coloured precipitate.
2. As so large a part of the white precipitate had remained undissolved in the foregoing experiment, it was digested again with another portion of the same lixivium, but without any effect. I therefore washed off the alkali, and boiled some nitric acid with the powder, until the acid was completely evaporated. After this, the powder was exposed to a strong heat in a sand- bath. It was then again digested with the lixivium, and a part was dissolved as before ; but still the residuum required to be treated with nitric acid, before the alkaline liquor could again act upon it ; so that it was necessary to repeat these alternate
I 2
6o "
Mr. Hatchett’s Analysis of
operations several times, before the whole of the powder could be united with the alkali.
3. When the white precipitate was digested with solution of carbonate of potash, or of soda, it was dissolved, much in the same manner as above related ; and the properties of the solu- tions, when examined by reagents, were also similar, excepting that the orange- coloured precipitates produced by tincture of galls were of a paler colour.
Tungstate of potash, molybdate of potash, and cobaltate of ammonia, being severally added to the solution of the white substance in potash, produced white flocculent precipitates.
Hydro-sulphuret of ammonia produced a reddish chocolate- coloured precipitate.
4. As the ore was decomposed by being fused with potash, the following experiment affords a curious instance (among the many already known) of the change in the order of affinities produced by a difference of temperature.
Some of the solution of the white precipitate in potash, was poured into the alkaline solution of iron, which was formerly known by the name of Stahl’s Tinctura Alkalina Martis. Pot- ash was in excess in both of these solutions; but nevertheless a cloud was immediately produced, and a brown ferruginous pre- cipitate was deposited.
Part of this precipitate w7as dissolved in muriatic acid ; and the solution, being examined in the usual way, yielded a blue precipitate when prussiate of potash was added, and a purplish brown precipitate with tincture of galls.
The other part of the precipitate was digested with dilute nitric acid ; which dissolved the ferruginous part, but left un- touched a white flocculent matter, perfectly resembling the
a mineral Substance from North America. 61
substance which has been so often mentioned. The precipitate therefore produced by the mixture of the two alkaline solutions, was a combination of the white matter with oxide of iron, very similar to the original ore.
H.
The white precipitate, when distilled with four parts of sul- phur, remained pulverulent, and, from white, was only changed to a pale ash colour.
Nitric acid was digested on the powder, and, being heated, afforded some nitrous gas ; after this, the powder became white, and in every respect recovered its original properties.
L
Before I conclude this section, I must observe, that when the olive-green precipitates, obtained by prussiate of potash, were digested in an alkaline lixivium, they were decomposed; for the alkali combined with the prussic acid, and with a small part of the white matter ; but the greater part of the- latter remained undissolved, in the same white flocculent state which was noticed when the alkaline combinations were mentioned.
The orange-coloured precipitates, formed by tincture of galls, were also decomposed when digested in boiling nitric acid ; and the white matter was recovered in its original state.
§ III. REMARKS.
The preceding experiments shew, that the ore which has been analysed, consists of iron combined with an unknown sub- stance, and that the latter constitutes more than three-fourths of the whole. This substance is proved to be of a metallic nature, by the coloured precipitates which it forms with prussiate of potash, and with tincture of galls; by the effects which zinc
6<2
Mr. Hatchett's Analysis of
produces, when immersed in the acid solutions ; and by the colour which it communicates to phosphate of ammonia, or rather to concrete phosphoric acid, when melted with it.
Moreover, from the experiments made with the blow-pipe, it seems to be one of those metallic substances which retain oxy- gen with great obstinacy, and are therefore of difficult reduction.
It is an acidifiable metal ; for the oxide reddens litmus paper, expels carbonic acid, and forms combinations with the fixed alkalis. But it is very different from the acidifiable metals which have of late been discovered ; for,
1. It remains white when digested with nitric acid.
2. It is soluble in the sulphuric and muriatic acids, and forms colourless solutions, from which it may be precipitated, in the state of a white flocculent oxide, by zinc, by the fixed alkalis, and by ammonia. Water also precipitates it from the sulphuric solution, in the state of a sulphate.
g. Prussiate of potash produces a copious and beautiful olive- green precipitate.
4. Tincture of galls forms orange or deep yellow precipitates.
5. Unlike the other metallic acids, it refuses to unite with ammonia.
6. When mixed and distilled with sulphur, it does not com- bine with it so as to form a metallic sulphuret.
7. It does not tinge any of the fluxes, except phosphoric acid, with which, even in the humid way, it appears to have a very great affinity.
8. When combined with potash and dissolved in water, it forms precipitates, upon being added to solutions of tungstate of potash, molybdate of potash, cobaltate of ammonia, and the alkaline solution of iron.
These properties completely distinguish it from the other
a mineral Substance from North America. 63
acidifiable metals, viz. arsenic, tungsten, molybdena, and chro- mium; as to the other metals lately discovered, such as ura- nium, titanium, and tellurium, they are still farther removed from it.
The colours of the precipitates produced by prussiate of pot- ash and tincture of galls, approach the nearest to those afforded by titanium. But the prussiate of the latter is much browner; and the gallate is not of an orange colour, but of a brownish red, inclining to the colour of blood. Besides, even if these pre- cipitates were more like each other, still the obstinacy with which titanium refuses to unite with the fixed alkalis, and the insolubility of it in acids when heated, sufficiently denote the different nature of these two substances.
The iron in the ore which has been examined, is apparently in the same state as it is in wolfram, viz. brown oxide; and this oxide is mineralised by the metallic acid which has been described, in the same manner as the oxides of iron and man- ganese are mineralised by the tungstic acid or rather oxide. For, from several experiments made upon a large scale, I have reason to believe that in wolfram, the tungsten has not attained the maximum of oxidation. Several facts in the course of the experiments lately described, seem to prove, that this new metal differs from tungsten and the other acidifiable metals, by a more limited extent of oxidation; for, unlike these, it seems to be incapable of retaining oxygen sufficient to enable the total quantity to combine with the fixed alkalis. In § II. G. 2, this is very evident; for, from the experiment there described it appears, that when the metallic acid or oxide was digested with lixivium of potash, only a part was dissolved; and that the re- mainder was insoluble in the same lixivium, till it had received
6y Mr. Hatchett’s Analysis of
an additional portion of oxygen, by being treated with nitric acid ; also that several of these alternate operations wrere required, before any given quantity of the metallic oxide could be com- pletely combined with the alkali. Now there is much reason to believe, that in this case, wrhen the metallic oxide or acid was digested with potash, the portion which was dissolved, received an accession of oxygen at the expense of the other part, which of course was thus reduced to the state of an insoluble oxide, and therefore required to be again oxidated by nitric acid, before it could combine with the alkaline solution ; but still it appeared, that an adequate proportion of oxygen could never be superinduced, so as to render the oxide totally and imme- diately soluble in the alkalis by one operation, or even by two.
We may, therefore, regard this as an instance of the effects resulting from disposing affinity, and as very similar to those observed in respect to copper, which have been noticed by my ingenious friend Mr. Chenevix, in his valuable analysis of the arseniates of copper and of iron;*
My researches into the properties of this metal, have of course been much limited by the smallness of the quantity which I had to operate upon ; but I flatter myself that more of the ore may soon be procured from the Massachuset mines, particularly as a gentleman now in England, (Mr. Smith, Secretary to the American Philosophical Societ}^) has obligingly offered his as- sistance on this occasion. We shall then be able more fully to investigate the nature of this substance; and shall be more capable of judging how far it may be applicable to useful pur- poses. At present, all that can be said is, that the olive green prussiate and the orange-coloured gallate are fine colours;
* Phil. Trans, for 1801, p. 233,
a mineral Substance from North America. 6*5
and, as they do not appear to fade when exposed to light and air, they might probably be employed with advantage as pigments.
I am much inclined to believe, that the time is perhaps not
«
very distant, when some of the newly-discovered metals, and other substances, which are now considered as simple, primi- tive, and distinct bodies, will be found to be compounds* Yet I only entertain and state this opinion as a probability ; for, until an advanced state of chemical knowledge shall enable us to compose, or at least to decompose, these bodies, each must be classed and denominated as a substance sui generis. Consi- dering, therefore, that the metal which has been examined is so very different from those hitherto discovered, it appeared proper that it should be distinguished by a peculiar name ; and, having consulted with several of the eminent and ingenious chemists of this country, I have been induced to give it the name of Columbium.
POSTSCRIPT.
It appears proper to mention some unsuccessful attempts, which I have lately made to reduce the white oxide.
Fifty grains were put into a crucible coated with charcoal ; and, being covered with the same, the crucible was closely luted, and was exposed to a strong heat, in a small wind-furnace, during about one hour and an half. When the crucible was broken, the oxide was found in a pulverulent state ; and, from white, was become perfectly black.
In order to form a phosphuret, some phosphoric acid was poured upon a portion of the white oxide ; and, being evaporated mdcccii. K
66
Mr.- Hatchett’s Analysis, & c.
to dryness, the whole was put into a crucible coated with char* coal, as above described. The crucible was then placed in a forge belonging to Mr. Chenevix ; and a strong heat was kept up for half an hour.
The inclosed matter was spongy, and of a dark brown ; it in some measure resembled phosphuret of titanium.
After this, we wished to try the effect of a still greater heat ; but in this experiment the crucible was melted.
The above experiments shew, that the white oxide, like several other metallic substances, may be deoxidated to a certain degree, without much difficulty, but that the complete reduction of it is still far from being easily effected.
C 67 D
IV. A Description of the Anatomy of the Ornithorhynchus paradoxus. By Everard Home, Esq. F. R. S.
Read December 17, 1801.
The subjects from which the following description is taken, were sent from New South Wales, to Sir Joseph Banks, who very obligingly submitted them to my examination.
These were two specimens preserved in spirit ; one male, the other female. The male was rather larger than the female, and in every respect a much stronger animal ; they had both arrived at their full growth, or nearly so, as the epiphyses were com- pletely united to the bodies of the bones, which is not the case in growing animals.
The natural history of this animal is at present very little known. Governor Hunter, who has lately returned from New South Wales, where he had opportunities of seeing them alive, has favoured me with the following particulars respecting them.
The Ornithorhynchus is only found in the fresh-water lakes, of which there are many in the interior parts of the country, some three quarters of a mile long, and several hundred yards broad. This animal does not swim upon the surface of the water, but comes up occasionally to breathe, which it does in the same manner as the turtle. The natives sit upon the banks, with small wooden spears, and watch them every time they come to the surface, till they get a proper opportunity of striking
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68
Mr. Home’s Description of the Anatomy
them. This they do with much dexterity; and frequently suc- ceed in catching them in this way.
Governor Hunter saw a native wratch one for above an hour before he attempted to spear it, which he did through the neck and fore leg : when on shore, it used its claws with so much force, that they were obliged to confine it between two pieces of board, while they were cutting off the barbs of the spear, to disengage it. When let loose, it ran upon the ground with as much activity as a land tortoise ; which is faster than the struc- ture of its fore feet would have led us to believe. It inhabits the banks of the lakes, and is supposed to feed in the muddy places which surround them ; but the particular kind of food on which it subsists, is not known.
Description of the external Appearances.
The male is 17I- inches in length, from the point of the bill to the extremity of the tail. The bill is 2^ inches long ; and the tail, measuring from the anus, 4^ inches.
The body of the animal is compressed, and nearly of the same general thickness throughout, except at the shoulders, where it is rather smaller. The circumference of the body is 1 1 inches. There is no fat deposited between the skin and the muscles.
The female measures in length 16^ inches, and' in circumfe- rence 11 inches. The size of the body is rendered proportionally larger than that of the male, by a quantity of fat lying every where under the skin.
The male is of a very dark brown colour, on the back, legs, bill, and tail ; the under surface of the neck and belly is of a silver gray. In the female, the colour of the belly is lighter.
I
of the Ornithorhynchus paradoxus. e 6 g
The hair is made up of two kinds ; a very fine thick fur, \ of an inch long, and a very uncommon kind of hair, -J of an inch long ; the portion next the root has the common appear- ance of hair, but, for ~ of an inch towards the point, it be- comes flat, giving it some faint resemblance to very fine feathers : this portion has a gloss upon it ; and, when the hair is dry, the different reflections from the edges and surfaces of these longer hairs, give the whole a very uncommon appear- ance. The fur and hair upon the belly, is longer than that upon the back.
Externally there is no appearance of the organs of generation, in either sex ; the orifice of the anus being a common opening to the rectum and prepuce in the male, and to the rectum and Vagina in the female.
There is no appearance, that could be detected, of nipples ; although the skin on the belly of the female was examined with the utmost accuracy for that purpose.
The head is rather compressed. The bill, which projects be- yond the mouth, in its appearance resembles that of the duck ; but is in its structure more like that of the spoonbill, the middle part being composed of bone, as in that bird ; it has a very strong cuticular covering.
In the upper portion of the bill, the lip extends for half an inch anteriorly, and laterally, beyond the bony part, and is thick and fleshy. The upper surface of the bill is uniformly smooth, and does not terminate where 'the hair begins, but is continued on for \ of an inch, forming a cuticular flap, which lies loose upon the hair. In the dried specimens that have been brought to Europe, the flap has been contracted in drying, and stands
70
Mr. Home's Description of the Anatomy
up perpendicularly ; this, however, is now ascertained not to be its natural situation.
The under surface of the upper half of the bill is also smooth ; but has two hard ridges of a horny nature, an inch long and to of an inch broad, situated longitudinally, one on each side of the middle line of the bill. :
The lower portion of the bill is much smaller than the upper; and, when opposed to it, the lip of the upper extends beyond it for the whole of its breadth. The edges of the lip of this lower portion have deep seme, in a transverse direction, like those in the duck’s bill, but they are entirely confined to the fleshy lip ; and, immediately within these serrated edges are grooves, lined with a horny substance, which receive, in the closed state of the bill, the ridges of the upper portion above described. There is also a cuticular flap extended upon the hair, as in the upper portion of the bill.
The nostrils are two orifices, very close to each other, near the end of the bill ; the upper lip projecting of an inch beyond them.
The eyes are very small ; they are situated more upon the upper part of the head than is usual, and are directly behind the loose edge of the cuticular flap belonging to the bill. The eyelids are circular orifices, concealed in the hair; and in the male are with difficulty discovered, but in the female there is a tuft of lighter hair, which marks their situation.
The external ears are two oval slits, directly behind the eyes, and much larger than the orifices of the eyelids.
The teeth, if they can be so called, are all grinders; they are four in number, situated in the posterior part of the mouth,
of the Ornithorhyncus paradoxus. 71
one on each side of the upper and under jaw, and have broad flattened crowns. In the smaller specimens before examined, each of these large teeth appeared to be made up of two smaller ones, distinct from each other. The animal, therefore, most probably sheds its teeth as it increases in size. They differ from common teeth very materially, having neither enamel nor bone, but being composed of a horny substance only embedded in the gum, to which they are connected by an irregular surface, in the place of fangs. When cut through, which is readily done by a knife, the internal structure is fibrous, like nail ; the di- rection of the fibres is from the crown downwards.
This structure is similar to that of the horny crust which lines the gizzard in birds.
Between the cheek and the jaw, on each side of the mouth, there is a pouch, as in the monkey tribe, lined with a cuticle. When laid open, it is lj- inch long, and the same in breadth. In the female, it contained a concreted substance, the size of a very small nut, one in each pouch : this, when examined in the microscope, was made up of very small portions of broken crystals.
Besides these grinding teeth, there are two small pointed horny teeth upon the projecting part of the posterior portion of the tongue, the points of which are directed forwards, seemingly to prevent the food from being pushed into the fauces during the process of mastication. This circumstance, of small teeth on the tongue, is, I believe, peculiar to this animal, not being met with in other quadrupeds. In the tongue of the flamingo there is a row of short teeth on each side, but in no other bird that I have seen. The teeth are represented in the annexed drawing.
The fore legs are short, and the feet webbed ; the length of
72 Mr. Home’s Description of the Anatomy
the leg and foot, to the end of the web, is about three inches. On each foot there are five toes, united together by the web, which is very broad, and is continued beyond the points of the toes, for nearly an inch. On each toe there is a rounded straight nail, which lies loose upon the membrane forming the web. The palms of the feet are covered with a strong cuticle; and there is a small prominence at the heel.
The hind legs are nearly of the same length as the fore legs, but stronger. Each leg has five toes, with curved claws ; these are webbed, but the web does not extend beyond the points of the toes. The four outer toes are at equal distances from each other; but the inner one is at a much greater distance from the one next it. The under surface of the foot is defended by a strong cuticular covering.
In the male, just at the setting on of the heel, there is a strong crooked spur, \ an inch long, with a sharp point, which has a joint between it and the foot, and is capable of, motion in two directions. When the point of it is brought close to the leg, the spur is almost completely concealed among the hair ; when di- rected outwards, it projects considerably, and is very conspicu- ous. It is probably by means of these spurs or hooks, that the female is kept from withdrawing herself in the act of copulation ; since they are very conveniently placed for laying hold of her body on that particular occasion. The female has no spur of this kind.
The tail, in its general shape, is very similar to that of the beaver. The hair upon its upper surface is long and strong; it has a coarse appearance. The under surface, if superficially examined, appears to have no hair; but, when more closely inspected, is found to be covered with short straggling hairs.
of the Ornithorhynchus paradoxus.
75
Description of the internal Parts .
The panniculus carnosus, which lies immediately under the skin, and extends over the greatest part of the body, is exceed- ingly strong.
The tongue is two inches long; it lies in the hollow between the two jaws, but does not project any way into the bill, being confined to its situation, except a very small portion at the tip. It is smallest at the point, and becomes larger towards the root ; the posterior portion becomes very large, and rises considerably higher than the rest, forming a projection, on the anterior part of which are the two small teeth already mentioned. The tongue is covered with short cuticular papillae, the points of which are directed backwards.
The velum pendulum of the palate is very broad. The glottis is uncommonly narrow; and the epiglottis proportionally small. The rings of the trachea are broad for their size ; they do not meet behind, but nearly so. The tongue and epiglottis are re- presented in Plate II. Fig. 2.
In the structure of the bones of the chest, there are some peculiarities which deserve notice.
The ribs are sixteen in number : the six superior are united to the sternum, which is narrow and very moveable ; the other ten terminate anteriorly in broad, flattened, oval, bony plates, which overlap each other in the contracted state of the chest, and are united together by a very elastic ligamentous substance, which admits of their being pulled to some distance ; so that the capa- city of the chest can undergo a very unusual degree of change.
The ribs are not connected to the sternum by their cartilages, as in other quadrupeds, but by bone ; the cartilaginous portion
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74 Mr. Home's 'Description of the Anatomy
being only about an inch long, and situated at some distance from the sternum, between two portions of rib, forming a kind of joint at that part. There is no ensiform cartilage.
On the upper end of the sternum is a bone an inch lohg, which at its upper part has two processes that answer the pur- pose of clavicles, and unite with the upper part of the scapulae, keeping them at a proper distance. The scapulae have a very unusual shape: the posterior part is more like the imperfect scapula in the bird ; and the flat part is situated witli one edge under the bone, immediately above the sternum. The other edge forms the glenoid cavity, for the articulation of the os humeri ; so that the fore legs have their connection with the trunk more forward than in other quadrupeds ; and the scapula itself is much more firmly confined to its situation.
This bone above the sternum, with the anterior part of the two scapulas, forms a bony covering of some strength, under which pass the great blood-vessels of the neck, secured from compression.
The appearance of the ribs, sternum, and other bones, is represented in Plate III.
The heart is situated in the middle line of the chest, its apex pointing to the sternum, and is inclosed in a strong pericardium : it is made up of two auricles and two ventricles. The foremen ovale between the auricles was closed, nor was there any com- munication between the ventricles. The right auricle is very large, and has two ascending venae cavae; that to the left winding round the basts of the heart, and forming the subcla- vian and jugular vein of that side, after giving off the vena azygos. This is similar to the kangaroo, beaver, otter, and many other animals. The aorta and other arteries are small.
of the Ornithorhynchus paradoxus, 75
The lungs are large in size, corresponding to the capacity of the chest. On the right side there are two lobes ; there is a small azygos lobe under the heart ; and in the left side only one. Instead of a portion of the lungs being above the heart, as in other animals, the heart may be said to be above the lungs ; for they only embrace its sides, and do not surround its upper sur- face, but extend downwards, into the more moveable part of the cavity of the chest.
The diaphragm is very broad, and every where towards the circumference is muscular, having only a small central portion, which is tendinous, immediately under the heart.
The oesophagus is extremely small, more particularly at its origin behind the larynx, where the fauces terminate in it.
The stomach is a membranous bag, of an oval form, into which the oesophagus can hardly be said to enter, being rather continued along one end of the oval, till it forms the duodenum ; so that the stomach appears to be a lateral dilatation of a canal, which is the oesophagus where the dilatation is formed, and becomes the duodenum immediately afterwards, at which part the. coats are thickened, forming the valve of the pylorus.
The stomach is smaller than in most other animals ; in this respect it is like the true stomach of birds. In the collapsed state it is only if inch long, and ~ of an inch broad. This is exactly double the size of one of the pouches in the cheek.
The duodenum makes a turn in the right side of the abdomen ; then crosses the spine, and becomes a loose intestine. The small intestines are strung upon a loose, broad, transparent mesentery. The origin of the colon is only to be distinguished by a small lateral appendage, inch long, and of an inch in diameter, going off from the side of the intestine, which is not
L 2
7 6 Mr. Home's Description of the Anatomy
altered in its size at this part. This process corresponds to the caecum : it is unlike the caecum in quadrupeds, but resembles that in birds, only is much smaller, and in general they have two; but the bittern and heron have only one. From this part, the colon passes up the left side, fixed to its situation by being attached to the omentum ; then goes across the body, and be- comes rectum, which gradually increases in size, and is very capacious before it terminates at the anus.
The small intestines are four feet four inches long. The colon and rectum are one foot four inches long.
The rectum opens externally at the root of the tail, i\ inch below the pelvis. On each side of the anus is a large solid body, about the size of the testicle, which;proves to be a gland, whose
ducts open by several orifices into thq rectum. In the female,
• * ’ ' - -
the same glands are met with, but of a much smaller size.
The mesentery is free from fat ; nor are there any fatty ap- pendages, or longitudinal bands, on the colon. The mesenteric glands are of the size of millet-seeds;; they are numerous, and scattered over the mesentery. The iacteals are small.
The internal surface of the stomach is uniformly smooth. The duodenum has valvulae conniventes, which are transverse : these are not met with in the jejunum and ilium ; but in them the internal membrane is studded over with glands. There is no appearance whatever of valve at the beginning of the colon ; but there are ten dotted lines, which run in a longitudinal direction, at equal distances from one another, and have their origin at the orifice of the caecum : these dots, upon a close inspection, prove to be the projecting orifices of ducts belonging to the glands of the intestine. The cavity of the small caecum is very cellular, as is shown in Plate II. Fig. 3.
of the Ornithorhynchus paradoxus. 77
The omentum is a thin transparent membrane, without any fat in it, originating from the side of the stomach next the duodenum, and also from that intestine anteriorly : on the left side it hangs loose, and the spleen is connected to it ; but, on the right, after it reaches the gg'lon, it surrounds that gut, and re- turns to the spine ; so thcfealthough the colon is confined by the omentum, there is no ; p$rt of that membranous bag pro- jecting beyond it.
The liver is composed of^pur lobes, besides the small lobe or lobulus Spigelii. The gall-bladder is in the usual situation, and of the common size, i The cystic and hepatic ducts unite into one, and are joined byAe pancreatic duct before their ter- mination in the duoden ui^pvhich is about an inch from the; pylorus.
The pancreas is spread upon the great and little omentum, as in the sea-otter, and is made up of small parts, in a very similar manner.
The spleen consists of tfifevery long slender bodies, united together at one end for thelElgth of half an inch : one of these portions is six inches, the- other four inches long.
The kidnies are conglobate, and lie in the usual situation. The capsulae renales are rather small. The ureters are pellucid and small.
The urinary bladder is not situated in the pelvis, but just above it, in the cavity of the abdomen, and is attached to the peritonaeum lining the abdominal muscles.
The skull is rather flattened upon the upper surface : its cavity is capacious ; and there is a bony process projecting from the cranium, in the place of the falx of the dura mater. This, 1 believe, is not the case in any other quadruped : it is met with.
78 Mr. Home's Description of the Anatomy
in some birds in a less degree, as in the parrot and the spoon- bill; which last bird, in the structure of its beak, bears some analogy to this animal. The tentorium is entirely membranous.
The brain was not in a state to admit of its structure being accurately examined ; but it appears to be made up of the same parts as those of quadrupeds in general.
The olfactory nerves are small, and so are the optic nerves ; but the fifth pair, wrhich supplies the muscles of the face, are uncommonly large. We should be led, from this circumstance, to believe that the sensibility of the different parts of the bill is very great, and therefore that it answers the purpose of a hand, and is capable of nice discrimination in its feeling.*
The eye is very small, and is nearly spherical : the globe is about £ of an inch in diameter ; the cornea Ag- of an inch in diameter. There is a membrana nictitans; and the eyelid is very loose upon the eyeball ; it is probably capable of great dilata- tion and contraction.
The organ of smell, in its construction resembles that of other quadrupeds, and may be said to consist of two turbinated bones in each nostril ; that next the bill is the largest, and has the Ion 2: axis in the direction of the nostril ; its external surface is very irregular. The posterior one is shorter, projects further into the nostril, and is situated transversely, with respect to the nostril. As the external openings of the nose are at the end of the bill, there is a canal of an unusual length for the air to pass through, before it is applied to the immediate organ, unless there is an extension of the branches of the olfactory nerve upon the linings of the cavity, so as to make it a part of it. The external
* The same observations were made by Professor Blum en bach, of Gottingen, who first dissected these nerves.
I
of the Ornithorhynchus paradoxus. yg
opening of the ear is at a great distance from the organ; and there is a cartilaginous canal, the size of a crow-quill, winding round the side of the head, upon the outside of the temporal muscle, leading to the orifice in the temporal bone.
The membrana tympani is larger than in other quadrupeds of the same size : it is of an oval form ; and the central part is drawn in, making its external surface concave. It has only two bones ; one passing directly from the membrane towards the foramen ovale, upon which there is a second bone, imperfectly resem- bling the stapes, having a flat surface of a circular form upon the orifice, and a small neck, by which it is united to the other bone.
This structure of the bones is less perfect or complex than in other quadrupeds ; so that the organ altogether bears a greater resemblance to that of the bird.
The organs of generation in this animal have several pecu- liarities of a very extraordinary nature.
The male organs do not appear externally ; so that the dis- tinguishing mark of the sex is the spur on the hind leg.
. The testicles are situated in the cavity of the abdomen, imme- diately below the kidneys : they are large for the size of the animal. The epididymis is connected to the body of the testicle by a broad membrane, which admits of its lying very loose.
The penis in this animal does not, as in other quadrupeds, give passage to the urine. It is entirely appropriated to the pur- pose of conveying the semen ; and a distinct canal conducts the urine into the rectum, by an opening about an inch from the external orifice of the intestine. The gut, at this part, is de- fended from the acrimony of the urine, by the mucus secreted by two glands already described, which probably for this reason
8o Mr. Home's 'Description of the Anatomy
are very large in the male, but small in the female. The open- ing of the meatus urinarius, and the orifices of the glands, are represented in Plate IV.
The penis is short and small in its relaxed state ; and its body does not appear capable of being very much enlarged when erected. The prepuce is a fold of the internal membrane of the verge of the anus, as in the bird ; and the penis, when retracted, is entirely concealed.
The glans penis is double; one glans having its extremity directed to the right, the other to the left ; and, as they supply two distinct cavities with semen, they may be considered as two penises. This is an approach to the bird, many of which have two. Each glans has, at its extremity, pointed conical papillae, surrounding a central depression. In one glans, the papillae are five in number, in the other four. When the urethra is laid open from the bladder into the rectum, about half an inch from its termination it communicates with the proper urethra of the penis, which afterwards divides into two, one going to each glans, in the centre of which is a cavity communicating di- rectly with the papilke, the points of which are pervious, forming the orifices by which the semen is evacuated.
The vasa deferentia open into the membranous part of the urethra, before it comes to the root of the penis.
Not being aware of so extraordinary a structure, and the parts not being in a perfect state of preservation, they were too much injured by dissection before it was discovered, to admit of their being prepared by injection. The appearance of these parts is .delineated in Plate IV.
There was no appearance of vesiculae seminales.
The female organs open into the rectum, as in the bird. Just
of the Ornithorhynchus paradoxus. 81
within the anus there is a valvular projection, between the rec- tum and vagina, which appears to be the proper termination of the rectum. This also is similar to the bird.
There was no appearance of clitoris, that could be observed.
The vagina is 1^ inch long: its internal membrane is rugous ; the rugae being in a longitudinal direction. At the end of the vagina, instead of an os tineas, as in other quadrupeds, is the meatus urinarius ; on each side of which is an opening leading into a cavity, resembling the horn of the uterus in the quadru- ped, only thinner in its coats. Each of these cavities terminates in a fallopian tube, whiph opens into the capsule of an ovarium.
The ovaria are very small : they were hot in a very perfect state of preservation, but bore a general resemblance to those of other quadrupeds.
This structure of the female organs is unlike any thing hitherto met with in quadrupeds ; since, in all of them that I have examined, there is the body of the uterus, from which the horns go off, as appendages. The opossum differs from all other animals in the structure of these parts, but has a perfectly formed uterus ; nor can I suppose it wanting in any of the class Mammalia.
This animal having no nipples, and no regularly formed uterus, led me to examine the female organs in birds, to see if there was any analogy between the oviducts in any of that class, and the two membranous uteri of this animal ; but none could be observed ; nor would it be easy to explain how an egg could lie in the vagina, to receive its shell, as the urine from the bladder must pass directly over it. Finding they had no resemblance to the oviducts in birds, I was led to compare them with the uteri of those lizards which form an egg, that is afterwards deposited in
MDcecn. M
1
82 Mr. Home's Description of the Anatomy
a cavity corresponding to the uterus of other animals, where it is hatched; which lizards may therefore be called ovi-viviparous; and I find a very close resemblance between them. In these lizards there are two uteri, that open into one common canal or vagina, which is extremely short ; and the meatus urinarius is situated between these openings. The coats of these uteri are thinner than those of the uteri of quadrupeds of the same size.
In the ovi-viviparous dog-fish, the internal organs of the fe- male have a very similar structure. There is therefore every reason to believe, that this animal also is ovi-viviparous in its mode of generation.
EXPLANATION OF THE DRAWINGS.
See Plates II. III. and IV.
Plate II.
Fig. i. Represents the hind leg of the male, in order to shew the situation and appearance of the spur.
Fig. 2. Represents the tongue, in its natural situation; and shows its relative position to the grinding teeth, and the lower portion of the bill ; also the two pointed teeth upon the tongue itself.
On the outside of the jaw, on each side, are the pouches for the food.
The glottis, epiglottis, and oesophagus, are represented of the natural size.
Fig. 3. The loculated caecum, with a portion of the ilium and colon.
Plate III.
Represents the bones of the chest and pelvis, in their relative
of the Ornithorhynchus paradoxus, 83
situation, to show the uncommon shape of the scapulae, which are not connected with the chest, but with a bone placed above the sternum, the upper part of which answers the purpose of clavicles ; the anterior part of each scapula passes under this bone laterally, forming with it a bony case for this part of the neck.
Another peculiarity is, the cartilages of the ribs not being placed next the sternum, but between two portions of the rib. The false ribs have their cartilages terminated by thin bony scales, which slide on one another in the motions of the chest.
The pelvis is unusually small, and has the two moveable bones, attached to the os pubis, which are met with in the kangaroo.
ci cl a. The bone which corresponds to the clavicles in other animals.
hbh . The left scapula.
ccc. The bony scales along the margin of the chest, ddd. The cartilages of the true ribs. ee. The moveable bones of the pelvis.
Plate IV.
Fig. i. Represents the penis in a relaxed state, but drawn out to its full extent, with its relative situation to the rectum and testicles, which are contained in the cavity of the abdomen, a a. The bodies of the testicles. bb. The epididymis. c. The urinary bladder. dd. The rectum.
ee. Two glands, whose ducts enter the rectum by a number of small orifices.
M 2
841 Mr. Home’s Description of the Anatomy , See.
f. The body of the penis, whose external Covering is a con<~ tinuation of the lining of the lower part of the rectum.
gg- The double glans : at the point of the right one are five conical papillae, and at the point of the left only four, which are open at their extremities ; through these orifices the semen passes.
h. The opening of the urethra into the rectum.
Fig. 2. A view of the uteri and vagina.
a a. The vestibulum, common to the rectum and vagina.
bb. The cut edges of the rectum ; the gut being dissected off to expose the vagina.
c. The vagina.
d. The meatus urinarius.
e. The bladder.
ff. The orifices leading to the uteri.
gg. The two uteri.
hh. The fallopian tubes.
ii. The ovaria, enclosed in the capsules.
i
/
«
N
I',hiloi\'£-gm .3kCD C C CIT. Plate IT. p. s,j
I'/ulos. /runs. Ml) CC C.T\Ma/,'. III. /a
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- 5’Av- ? ~ ' .' *
'- _ ,v- - : ’ ^ ,,:';’^'
f*<t ;‘ H ':- '■ V •
■ •" .'.. . . 4 -■ -
-
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. . ;«*•••• ■
-
-
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-
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C &5 3
V. On the Independence of the analytical and geometrical Methods of Investigation ; and on the Advantages to be derived from their Separation. By Robert Woodhouse, A. M. Fellow of Cains College , Cambridge. Communicated by Joseph Planta, Esq. Sec. R. S.
Read January 14, 1802.
One of the objects of the paper which last year I had the honour of presenting to the Royal Society, was to shew the in- sufficiency in mathematical reasoning, of a principle of analogy, by which the properties demonstrated for one figure were to be transferred to another, to which the former was supposed to bear a resemblance ; and the argument for the insufficiency of the principle was this, that the analogy between the two figures was neither antecedent to calculation, nor independent of it, and consequently could not regulate it ; that analogy was the object of investigation, not the guide ; the result of demonstra- tion, riot its directing principle.
Having shewn that analogy could not establish the truth of certain mathematical conclusions, I next endeavoured to shew why such conclusions had been rightly inferred ; not by pro- posing any new excogitated principle, nor by pointing out an hitherto unobserved intellectual process ; but I conceived they might be obtained by operations conducted in a manner similar to that by which all reasoning with general terms is conducted.
8 6 Mr. WoodhOUSE on the Independence of the
and that the relations between the symbols or general terms were to be established by giving the true meaning to the con- necting signs, which indicate not so much the arithmetical computation of quantities, as certain algebraical operations. It was further observed, that, from certain established formulas, abridged symbols or general terms might be formed, which consequently must have their signification dependent on such formulas ; and that, although the parts of certain abridged ex- pressions could not separately be arithmetically computed, yet the expressions themselves might be legitimately employed in all algebraic operations.
The chief object of my paper was to shew, that operations with imaginary quantities, as they are called, were strictly and logically conducted, that is, conducted after the same manner as operations with quantities that can be arithmetically com- puted : the question, whether calculation with imaginary sym- bols is commodious or not, was then slightly discussed. I have since attentively considered it, and, what usually happens in such cases, my inquiries have been extended beyond their origi- nal object ; for, actual research has convinced me of what there were antecedent reasons for suspecting, that not only in the theory of angular functions, demonstration is most easy and direct by giving to quantities their true and natural* represen- tation ; but, that the introduction of expressions and formulas not analytical, into analytical investigation, has caused much
ambiguity, confused notion, and paradox; that it has made
/
. + , -*✓“}, (2V”) ,-w~}
Sec. I call the natural representations of the cosines, sines. Sec. of an arc x ; because, admitting the algebraical notation, they, by strict inference, adequately, unambigu- ously, and solely, represent the cosines, sines. Sec.
analytical and geometrical Methods of Investigation, 87
demonstration prolix, by rendering it less direct, and has made it deficient in precision and exactness, by diverting the mind from the true source and derivation of analytical expression.
The expressions and formulas alluded to are geometrical, that is, taken from the language of geometry, and established by its rules: they are to be found mixed with analytical* ex- pressions and reasonings, in all works on abstract science ; and, as they are certainly foreign and circumlocutory, if it can be shewn that they are not essentially necessary, there will exist an argument for their exclusion, especially if it appears that in analytical investigation they are productive of the evils above mentioned.
That, in algebraical calculation, geometrical expressions and formulas are not essentially necessary, perhaps this short and easy consideration may convince us ; since algebra is an uni- versal language, it ought surely to be competent to express the conditions belonging to any subject of inquiry ; and, if adequate expressions be obtained, then there is no doubt that with such, reasoning or deduction may be carried on.
All expressions and formulas, such as, sin. x , cos. x, hyp, log. x, sin. n x = 2 cos. .r. sin. (n — 1 ) a:-— sin. ( n — 2) x.
* The terms analysis, analytical, algebra, algebraical, have been so often distin- guished, and so often confounded, that I shall not take the trouble again to distinguish them. I use the words analytical, -algebraical, indifferently, in contradistinction to geometrical. The first relates to an arbitrary system of characters ; the latter to a system of signs, that are supposed to bear a resemblance to the things signified, and in which system, lines and diagrams are used as the representatives of quantity : and I am prin- cipally induced to use the words indifferently, because, if analytical were properly defined, another word with a sufficient extent of meaning could not be found ; for, by an improper limitation, the word algebraical has not an extensive signification, being frequently used in contradistinction to transcendental, exponential, &c.
88
Mr. WooDHotrsE on the Independence of the
i
Jx* (i — = circular ar c,fx‘ y/^ = elliptical arc,&c,
are geometrical, or involve geometrical language : they suppose the existence of a particular system of signs, and method of de- duction ; and relate to certain theorems, established conformably to such system and method.
I. Sin. x , cos. x , tang, x, &c. These expressions are borrowed
from geometry ; but, analytically, denote certain functions of x. Typographically considered, these expressions are more commo- dious than (2\/ — l)"1 1 j, (s)-1 | -j-
B—XV~ Sic. but this is the sole advantage; for, all analytical operations with these latter signs are much easier, and more expeditious, than with the former; since they are carried on after a manner analogous to that by which operations with similar expressions are. But, if the geometrical expressions be retained, then, in order to calculate with them, recourse must be had to the geometrical method, proceeding 1 by the similarity of triangles, the doctrines of proportions, and of prime and ulti- mate ratios ; so that, in the same investigation, two methods of deduction, between which there is no similarity, must be em- ployed.
II. The value of/ (i-f- x )~*, is said to be a portion of the
area of .an hyperbola intercepted between two ordinates to its assymptotes ; but this is a foreign and circumlocutory mode of expression; since, to find the value of the area, x\ (1 -j- x)~z must be expanded, and the integrals of the several terms taken; and this same operation must have taken place, in order to ap- proximate to the value of J x- (i if no such curve as the
hyperbola had ever been invented.
III. />•( i — x* | is said to equal the arc of the circle
analytical and geometrical Methods of Investigation. %
rads, j , sin. x ; but nothing is gained by this ; since, in order to find the arc of a circle, x% ( 1 —a:2)— I is expanded, and the inte- grals of the several parts taken and added together. To shew (if it is necessary to add any thing more on so clear a point) Xh&tfx° 1 1 — xzJ—£ =arc circle, is merely a mode of expres- sion borrowed from geometry ; suppose the investigation of the properties of motion to have been prior to the investigation of the properties of extension, for, that the science of geometry was first invented is properly an accidental circumstance, then, such an expression as fx' 1 1 — might have occurred, and its value must have been exhibited as it really is now, that is, by expanding it, and integrating the several terms.
IV. It is an objection certainly against these modes of ex- pression, that they are foreign, and tend to produce confused and erroneous notions ; for the student may be led by them to believe, that the determination of the values of certain analytical expressions, essentially require the existence of certain curves, and the investigation of their properties. But there is a more valid objection against them, which is, that they divert the mind from the true derivation of such expressions as x • ( 1— ^a)~ f. See. and consequently tend to produce ambiguity and indirect methods ; for although, in order to obtain approximately the numerical value off ~,fx- (1— xf~i, &c. it is convenient to expand the expressions, and to take the integrals of the result- ing terms, yet, if the symbol / denotes a reverse operation, f—> Jx ' ( 1 are not properly and by strict inference equal
to (x—i) — i {x-i)' + ^.{x -I)1-, &c. and x+~ +
<2 o ^
+> &c- But> order to explain clearly what I mean, it is
MDCGCII. ]SJ
c)0 Mr. Woodhouse on the Independence of the
necessary to state what I understand by the integral or fluent of an expression.
V. Let <px denote a function of x; if x be increased by o , then $x becomes <p (£ + o), and <p (x + o)} developed according
u R
to the powers of o, becomes <px + + ~T7°* + 7X3 0 ^c‘
where P is derived from <px> Q from P, R from Q, &c. by the same law ; so that the manner of deriving P being known, Q, R, &c. are known. The entire difference or increment of <px is <p (x + 0) — q>x; the differential or fluxion of <px is only a part of the difference or P .0. If, instead of 0, dx , or x*, be put, it is P. dx or Px*; the integral or fluent 01 Px° is that function from which Px* is derived ; and, in order to re- mount to it, we must observe the manner or the operation by which it was deduced ; and, by reversing such operation, the integral or fluent is obtained. Now, in taking the fluxion of certain functions of x, it appears there are conditions to which the indices of x without and under the vinculum are subject : hence, whether or not a proposed fluxion can have its fluent assigned, we must see if the fluxion has the necessary
conditions. Expressions such as ~, ■ 7^7-’ yi- *r» &c' ^lave n0t
\ , ■' . . r r
these conditions; and consequently there is 110 function <px of x, such that the second term of the developement of cp (x + x *) is
or, &c. There are, how-
X* X*
x y or 1+*’ °r VTTX
equal either to
ever, integral equations from which such expiessions may be de-
rived. Thus, let x= 6*, then
— Z', let 1 x = ez . * .
I-f X
z’y let x
— zV-
. X 3/ — I
X'
V i.
X*
.X'
X’
Now, from these equations, the differential equations x
r* z=z ',-?£= =, &c. may, by expunging the exponential V 1— x1 '
analytical and geometrical Methods of Investigation . g i
quantities, be derived ; consequently, if the symbol / is to de- signate a reverse operation, I can only know what that reverse operation is, by attending to the manner by which the expres- sions affected with the symbol / were derived. Hence,
VI. / = z when x = e85.
X
when 1 + x
/*=? =z whcn X = (Sv/~ 1)— }.
In like manner,
fx- {i-\-xl)~i=z,x-\-i/i -t-x‘=e” or X—
Jx- (sx+x')-i = z, l+X+/2I-(.7=f!.
/2X" 1+X
—c- — z,— = ? or x =
£*+1 *
I
2X‘
XV I +X1
^ Vl+x%— I
—
V I J
= e* or \/ 1 + a:1
or x =
£ a— £: Again, suppose
i — f *'»
-£=| 2 \/ — 1 1 !| ^ 1 ~~ e ^ j?
but \/ 1 — ^=2“', j. consequently x'=.z's/ \ ^
or r * = : hence, reversely.
/ v=- = z> * be*ng = (2 v/-i)
In like manner,
/~^r — *, * = 2— . { }.
/ Vzj— j* ^ s> x — (i — ®~‘- 1 r~v-’ + 1— y~ < |j.
* I take no notice, at present, of the arbitrary quantities which may be introduced in the integration of these equations.
9*
Mr. Wood house on the Independence of the
; £1^ — 1 — — Z-^ — I glzV—l 1
J 1ZT“ == ^ ^ = 4- fW-»' I’ 0rvri(^v-1 4- 1 ) ‘
f X' 2
/ == s, .r = — - — i -=■•
And a variety of forms may be obtained, by substituting for x different functions of x , in the expression ; Hence, if
V I — x-
the symbol / is made to denote a reverse operation, the integral equations of the preceding differential equations have been rightly assigned. All other methods of assigning the integrals, by the properties of logarithms, by circular arcs, by logarithmic and hyperbolic curves,* are indirect, foreign, and ambiguous.
VII. An instance or two will shew the advantage of adhering to the true and natural derivation of analytical expressions. Let x and y be the co-ordinates of a circle; then, i = x44- / a, and y — ^ ( l — x* ) , now (arc ) • or z- = x/(x *+ y *a) =, in this instance, x' (l — cc") — I: but it has appeared, that if x ass [2 \/~i }-1^£zV-I^£-zV-i j, z' = x- (i—xa)-i; consequently, in a circle, the co-ordinate x , or, in the language of trigonometry, the sine x = developement of
(2v/~l) {^-1 — pzV'-x
andy or cosine = 2-1. j + £~zl/— 1 } = 1 “
4- — — -
1 i-2.3-4-5
. z4
1.2 ' 1.2. 3.4
&C.
-&C.
1. This method of determining the series for the sine in terms of the arc, is, I think, simple, direct, and exact; it requires no assumption of a series with indeterminate coefficients, nor
• By the strange way of determining the meaning and value of analytical expres- sions from geometrical considerations, it should seem, as if certain curves were believed to have an existence independent of arbitrary appointment.
analytical and geometrical Methods of Investigation. 93
any preparatory process to shew that the value of the first co- efficient must = 1 .*
VIII. Euler demonstrated this formula to be true, viz.
Arc
:sin. arc sin. 2 arc 3- sin. 3 arc — f sin. 4 arc -f &a The following is its analytical deduction,
*-==;s‘{ e-v-' + 1 )!==*'{7^4*i )+%’( ^-'+1 1
ez‘
f £zV~. i ]
:z' { >-> + 1 ] +
v~
+ I
J £zV—i — -j_ — gcc. 1
1 + £— zV— 1 ^lzV— 1 -J- (r—3zl/-T7 &c. )
£32^-
£zV“-‘ — — -j
l ■ —
+
2
-22 V-
— &c.
32
y-x
1
-{- &C. j
2”. cos. COS. ~ . cos. -f- ... cos.——- • sin. —
A 4 O. 2” 2”
and y — (2 — 1) T.| — £~zV~- 1 J — (2%/ — i)“J.
| pzV—i £—2zV~i j. + -H2v/— 1. )—I| £3rV-T7 J — &Cd
which is the analytical translation of Euler's formula.
IX. Euler likewise shewed that sin. x
Which may be thus demonstrated, sin. x= (2s/—i)—ifSxv-;__r-xV-1J;
but (2 v/— 1 )~ 1 j £xV~* — £-xV~ | = 2 . 2~x j j
(2v/“i }~vv:=:7 }
^2.2 — 1 ~f f“~^v —I J . 2 . 2—1 sV— I -(- ) .
( 2 v/ — 1 ) 1 . { £ 1 — £ -T J.
* See Lagrange, Fonctions Analytiques. p. 2 6. Lacroix, Traite du Calcul. djfU ferentiel, &c. p. 56. Le Seur, Sur le Calcul. diff. p. 105. Euler, Anal. Inf. Art- l33’ 134*
94 Mv Woodhouse on the Independence of the
.2.2— {/rV'— + J | 2V/~I }— (f^— - r'^).
or, generally,
= 2”. 2 — 1 \ £ I''— -j- “l'/— }. S— *{ f *V— + £~fv'— } . 2—
i j.
Which is the analytical translation of sin. x— 2”. cos. — . cos.^- &c> Euler, and after him other authors, have demonstrated these formulas by the aid of logarithms, and of theorems drawn from geometry.
X. Euler and Lagrange have treated certain differential equations, which are said to admit for their complete integration an algebraic form, although the integration of each particular term depends on the quadrature of the circle and hyperbola. I purpose to integrate these differential equations, by the method
adopted in Articles V. VI.
Let/r,j/y, denote functions of xandy.
Suppose the differential equation to be £. Z = 0 ; then fx +/y = a when x = £fx,y= zfy* Hence, xy s=s £ fx+jy = gfl = A, a constant quantity, sdly. Let + L=- = o
... fx + fy =V* being = {W~
and y — t )“* • ); or v/(i— -z1) = 2—.
tfW—' £— fW—t), artS v/ 1 — y’= 2—‘. (£-^l/— 1 + £— /jV— . ).
Hence, x. v/(i — -y‘) + jy _ (2 { £(/»+*)✓— — s-(fi+/y)V— }
_ (2v/ — 1)— . j £“v'-1— f— V-q == A, a constant quantity.
analytical and geometrical Methods of Investigation
95
gdly. Let
X’
v' d + bx + CX7-
+
X ■
Let & *-{-
* , bx a
xz+ -f —
c c
+
v' a -f- by -{- cy7-
T
vV c»*+^+-
c c
0.
2 C
v,y +
2 C
V'
+
VcVvz-\-rz 5 VcVvz-b-rz
taking the integrals
c-i fV + V'j =«,» =
v' and r3
= o.
a
C
bz
fV-r*i-V ,_fV-^£-V'
„ , 4/ ■ — — •
/TI A j > /7~r j 7 fV-fV — 4£ — V+\ ) r4 p—x^/c
\ v vr 4-zr-Lir v (r 4-z> = S — .. - r * L~
X 1 2 2
= A, and restoring the values of x and y3
2 cx + b
>/[*+f>y+cy%)+ v/(^ + ^+^a) = A'.
. By the above operation it appears, that certain algebraical ex- pressions, as x \/i --/H-y s/i—x\ s/afhy-fcy'1 &c. may be deduced, which answer the equations f + / — — — &c.
v" i-
But, strictly speaking, such algebraical expressions are not the' integrals : they are rather expressions deduced from the true integral equations, from which other algebraical expressions* besides those put down, might be deduced.*
* For the integration of this sort of differential equations, see Mem. de Turin. Vol. IV. p. 98. “ Sur PIntegration de quelques Equations differentielles, dont les indetermi- “ nees sont separees, mais dont chaque Membre en particulier n’est point integrable.” In this Memoir are given three different methods of integrating .r- (i~xz)~^ y * (1 — yz)~* ; by circular arcs and certain trigonometrical theorems, by impossible logarithms, and by partial integrations. Strictly speaking, all these methods are indi- rect; and the two first are only different but circuitous modes of expressing the method given in Art. X. See likewise Euler, Calc, integral Vol. II. Novi Comm. Petrop. Tom-. VI. p. 37. Tom. VII. p. 1. It is to be observed, that in the present state of analytic science, there is no certain and direct method of integrating differential equa-
96 Mr. Woodhouse on the Independence of the
XI. In the irreducible case of cubic equations, the root, it is said, may be exhibited by means of certain lines drawn in a circle. There is, however, independently of all geometrical con- siderations, a method of analytically expressing the root ; and, from the analytical expression, although it is not the formula which from the time of Cardan mathematicians have been seeking, the value of the root may in all cases be arithmetically computed ; but, previously, it is necessary to shew what are the different symbols that may be substituted for z in the equations, x z=z fis/ — 1 )*-i and v/ (l— .£*) = 2“*
4. r—zV~ | . Let x = 1, and 7 r be the value of % that
answers the equations 1 = (av/ — 1 ) ~ 1 1 jand 0 = -j- which value of tt may be numerically
^•3 a jj»S
computed from the expression . . ?r = % = x + + -jx +
— &c* ix — 1)*
Hence, eW~ = — i~W~' = £mV~ = r2*'’'” = - r
AttV — I . SttV— I __ — 87?^— X __ .
== £ 1 • •} £ £ * • •
x6»V— — i6»V— 1 __ j ffor since 1 ___ , ... 1 ■«.-
£ v £mw\-* 1
and = ~.-,s2mW~ = 1).
£m'7rv — x
Again, since = 1 and = 1, -l = 1; and
tions such as .ar ^ + j* 2+^* ii+^y+<:r3'2,+£b|3 + £3'‘1’ ^ ■<**
because no analytical expression or equation of a finite form has hitherto been in- vented, from which, according to the processes of the differential Calculus, such diffe- rential equations may be deduced. To find the algebraical expressions which answer to these equations, recourse must be had to what are properly to be denominated artifices. For such, see Mem. de Turin. Vol. IV. Comm. Petr. Tom. VI. VII. Lagrange, Fonct, Analyt. p. 80. Lacroix, Calc. di£F. p. 427, &c.
analytical and geometrical Methods of Investigation.
generally 1 =£ 1 — 1} n any number of the pro-
gression o, i, 2, 3, 4, &c.
And, since ^ __ __ t . ^2«V— i x f4w’rV/-'1 —
-2*VZTi x ^~4«wVZT ^ or£(2«+i) arVIT=£-.(a«+i)a*VZ7=_ lf w any number of the progression 0,1,2, 3, 4, 5, &c.
Hence it appears, that if x=(2v/— i)-* j^zr, — 1
— { s } " 1 { e*v~ - ~zV~' } x £V‘”v~' = (since £^~'= f— 4«*v_ I-J J2v/3i)“I| £ (4tt7T+5:)V— I (4«w + a;)V— i J#
Again, since f(2»+0 WZT==f~(2n+i) zb-VUT 2
a: x — i = (2\/ . — 1) ] j(2«-{-i)tV'_i
= (2\/IIl)“'I|-_ ^((2« + i)27r-.ar)VZ7___^-((2«+i)2ff^)^/Z7 consequently,
X= (2\/ — 1 )’~I { £((.2n+l)2V—z)d—l £— ((2«4-l)29T_*)^Z7 I
or the equation * = ( 2v/- 1 )->{ PV=l - ^3 | is ^ wh’en
instead of z is put (477 +«) or (87 7+%), or generally (47277- + #) ; and is moreover true, when instead of z is put
(27T— z), (677— z), or generally (222+1) 27 r—z.
In like manner, the equation y/T^? =2“1(fZv'31+r-s 1/- 1 is true, when instead of z is put 1
4^+2, 87 r-j-z, or 1277+2:, or generally 4 7277-+^; and is moreover true, when instead of z is put
477 z, ,877 — 2, or 1277 — -Zy or generally 422^ %.
Let now x qx~r3 then, by Cardanos solution,
put a —^5— ~ = — b , thenAr=3v/(^+6\/ — i) + \/tf( — h\Z~ 7],
Let 0 + 6 v7—! =772^37 0 — 6 s- mp-~iVzrx
MDCCCII.
O
98
Mr. Woodhouse on the Independence of the
y_, + > 6„m|
Zy/—1
or 2‘
£—zV—l -j.=
}.
'.{^-■+£-^}=7=^>and (2v/-ir‘{^-. -
; but, from what has been premised, these
Vaa+62
equations are true, when instead of %. is put d or 2d -fcz, or 4fd-\-z, or generally ni- \-z, (4^=#).
Hence, — r) }j
f v — 0+2V— 7 ] t f n®+zV —1
or mi\e 5 ~ 1 [, or generally ma] c 3 +
— («0+^)V— 1
)■
there are, however, only 3 different values of x,
0 , / — 30+*V— x
» 30 + ^V I -» ■». — .......
for the index of k in the fourth value is - , and ^
O
«*, T V— 1 £ X £ =1 X-£
£
-jV=7
.-.the fourth value is the same as
the first. Again, the index of e in the fifth value is
49+2
V_i :
(4S+2) / —
; V — x
but, ‘ 3 3 The Ath
value is the same as 2d, and so on~; and, consequently, the
indices of e in the 3 different values of x are .-=±= — V — 1, =f=
L+z v/ITT -±££- v/~.
3 3
If, instead of the index of s in the 3d value,
put, the value of the root remains the same ; for, since eVZ7
f 20 + 2 , —20 + 2
& 1/ i __ 1 - ^ ^ — mi x|_£
f 7 -jZ-VH
m* $ + £
+* , —
T“ V — x
6 +2
V“
- — - — 1 be
1
X £
}•
' V_ I — 0\/— X ,
X£ + £
This mode of representing the roots is not, as has been
analytical and geometrical Methods of Investigation . qq
already stated, according to the conditions* of the formula de- manded by mathematicians. It enables us, however, imme- diately to ascertain that the roots are possible, and to calculate ■their approximate value; for, when \/.i
=
J J l _ v2-
•x% or y = 2
— 1
a
i
when %
f? + -&- + fr + & + &c- },
o y = s"1"-1 j £° -j- s~5 j =s
ec
1 + IT" + TF + •775—+ &c. ]= 7T.
3.2 1 5.8 1 7.16
Hence, we may numerically approximate to the value of % from the expression * = *■ — { 7 +-£- + + &c. ) when y is given, and < 1. Now, in the case of the cubic equation.
y
v 4=; and, since T
<
3rl 3
is < 1, conse-
V a* fb* 5 ' 4 27 '
quently the value of 2; may be obtained ; suppose it t, then the roots are to be approximated to, by means of the series that result from the developements of the forms by which they are repre- sented ; to wit,
B
yj[‘—iSr
— (9+i)"
+ 7
I
2
1-2.3 (2 9-M)’ I-2-31
+ T +
2.3.4
I
2.3.4
J
I.2.3.4
34
3+
(zS + i)4
} }
— &C.'|
- &c. &c.
Now these series converge ; for, since t is finite, we must at length arrive at a term An, in which [n— 1) n is > (
since (w-j-i)th term
p_va
3
and.
A
nq-i
K-f I (W-f 2) * * W-J-I
IS
* The conditions of the formula are, that it should be finite in regard to the num- ber of terms, free from imaginary quantities, and containing only the coefficients q and r. See Mem, de PAcad, 1738.
O 2
ioo Mr. Woodhouse on the Independence of the
< a fortiori, ^+1 is < An+l, and so on; the terms after the n — ith term constantly diminishing.* **
The above method is purely analytical : it has no tacit reference to other methods ; it does not virtually suppose the existence either of an hyperbola or circle. If practical commodi- ousness, however, be aimed at, it is convenient to give a different expression to the values of the roots, or to translate them into geometrical language : and this, because tables have been calcu- lated, exhibiting the numerical values of the cosines, &c. of circular arcs. Now, since it has already appeared that the cosine of an arc z=q~1 | £zV~ _|- e—zV ~ the 3 roots of the equation x3 — qx = r may be said to equal
2 -y/^ -d— . COS. — , 2 \/ COS. 2V/CZI COS. klLt
V 3 3 3 3 V 3 3
XII. In the fifth volume of his Opuscules , -f D’Alembert
* In the Phil. Trans, for 1801. p. 116, I mentioned M. Nicole as the first ma- thematician who shewed the expression of the root in the irreducible case, when expanded, to be real. But the subjoined passage, in Leibnitz’s Letter to Wallis,
causes me to retract my assertion. “ Diu est quod ipse quoque judicavi \/3a-{-bV — 1 “ ~\-V3a-{-b\/ 1 — z esse quantitatem realem, etsi speciem habeat imaginarias ; “ ob virtualem nimirum imaginariae destructionem, perinde ac in destructione actuali “ a-\-b V' — i — 1 —2a. Hinc, si ex \/3a±.b*S — i extrahamus radicem
“ ope seriei infinite, ad inveniendum valorem ipsius z serie tali expressum, efficere possumus, ut reapse evanescat imaginaria quantitas. Atque ita etiam, in casu ima- ginario, regulis Cardanicis cum fructu utimur,” & c. Vol. III. p. 126. See also p. 54.
f “ Elle etoit neanmoins d’autant plus essentielle, que Pexpressiori de l’arc par ie
dx ^
,f sinus, fondee sur la serie connue, qui est l’integrale de — , , — —> ne peut etre regardee
V 1 — xz
u comme exacte, e’est a dire, comme representant a la tois tous les arcs qui ont le raeme sinus ; puisque cette serie ne represente evidemment qu’un seul des arcs qui
** repondent au sinus dont il s’agit, savoir, le plus petit de ces arcs, celui qui est infe- ,e rieur, ou tout au plus egal, a 90 degres. Cependant, e’est d’un autre cote une sorte «« de paradoxe remarquable, que ^expression de l’arc par le sinus ne representant qu’un
analytical and geometrical Methods of Investigation. 101
mentions it as a remarkable paradox, that the series for the arc in terms of the sine represents only one arc, viz. the arc less than go degrees ; whereas the series for the sine, produced by reversion from the former series, exhibits all possible arcs that have the same sine. I shall endeavour to solve this paradox, which, I think, originated partly from the introduction of geo- metrical considerations into an analytical investigation, by which the true derivation of certain expressions was concealed.
It has appeared that the equation j &V—x _ e— ZV~ j,
is true, when instead of z is put, 0-fs, or 20 +2, .... or n9+z,
0 36 2tt-fl f,
or- z, or — % .... or — — 0 —
2 7 2 2
Now, if the fluxions of these equations are taken, and the equa- tions cleared of exponential quantities, there results from each the same equation, to wit, z- = • Hence, if the symbol
/ denotes the operation by which we are to ascend to the ori-
ginal equations from which z’ — strict consequence from fzm — J ■
X'
V'l— .
is derived, the only
X‘
V I — X2
is that x — (^/-s-i) f— zV 1 },
or = (2s/— i)-
- i
or generally
{f(G+*)^-i _ e-(0 + 2;)k_i j (2 __ ~(nQ+z)Viri j,
2H+1 ../ — (2K-fl)
—f—QV-l 1
or
= ( 2\/ — 1)'
— 6
seul arc de go degres au plus* I’expression du sinus par l’arc, qu’on pent deduire (par !a merhode du retour de suites) de i’expression de l’arc par le sinus, represente exactement, etant poussee a i’infini, le sinus de tous les arcs possibles, plus petits ** ou plus grands que go0, et meme que la circonference ou demi circonf’erence, prise “ tant de fois qu’on voudra. Je laisse a d’autres geometres, le sola d’eclaircir ce 44 mystere, ainsique plusieurs autres,” &c. p. 183.
tos
Mr. Woodhouse on the Independence of the
Hence* to answer the equation % • —
x may
or %'
f«3 «v5
_t i . ~
1.2.3 ' x. 2. 3.4.5
t's
1 ,
~ 1.2 ?.4.£
or z" —
.4.2.3
z"3
1.2. 3
3-4-S • 1.2.3 4 5
Vi— xz 5
&C.
Szc.
&c.
| z" , s'", &c. representing 0+tr, 2 0-}-£, 30-j-#, &c. jt
Suppose now it is necessary to deduce z; z1, z", &c. in terms of x and its powers* by reversion of series. What does the reversion of series mean? Merely this; a certain method or operation, according to which, one quantity being expressed in terms of another, the second may be expressed in terms of the first. Hence, in all similar series, the operation must be the same ; consequently, the result, which is merely the exhibition of a formula, must be the same ; so that, whatever is the series ;in terms of x, produced b}^ reversion in
<%■ = % — — V --- —V— &c. the same must be produced
hy reversion in x == % & c.
j 1.2.3 * 1. 2. 3.4.5
in x = -z" — {- &c.
1,2.3 •
&e.
The series produced by reversion in these cases is, x -{ — f - -f-
S
+ .&c. Hence it appears, that we know, a priori, that must happen which D'Alembert considers as a paradox to have happened. Why this paradox found reception in the mind of this acute mathematician, I have stated, as my opinion, one cause to have been, an inattention, from geometrical considera- tions, to the real origin and derivation of certain expressions that appeared in the course of the calculation. Another cause I ap- prehend was, the want of precise notions on the force and
\
I
analytical and geometrical Methods of Investigation. 103
signification of the symbol =. It is true that its signification entirely depends on definition ; but, if the definition given of it in elementary treatises be adhered to, I believe it will be impos- sible to shew the justness and legitimacy of most mathematical processes. It scarcely ever denotes numerical equality. In its general and extended meaning,* it denotes the result of certain operations. Thus, when from
x.
z
1.2.3
z or %' is inferred
+ 7
2.34,5
X
_L
x
X = % - 3-*5
&C.
1.2.3
&c. nothing is affirmed
3.2 * 5.8
concerning a numerical equality; and all that is to be under-
stood is, that x -f — |- + &c. is the result of a certain
3.2 « 5. 8
operation performed on x
z —
1.2.3
JL. - *
• I.2.34.5
&C.
XIII. It appears then, that according to the reversion of series, z, z ', z", See. must all be represented by the same series, proceeding according to the powers of x ; but, if a form for % be required, which shall in all cases afford us a means of numeri- cally computing its value, such a form must involve certain arbitrary quantities. These arbitrary quantities are to be deter- mined by conditions which depend either on the original form of the equation between x and sr, or on the nature of the object to which the calculus is applied.
Let now J
X'
V 1 — ;
mean'f x ~f-
3*2
+
3X
5,8
-fi & C.
* This is consistent with what I advanced in the Phil. Trans, for 3801. p. 99, con- cerning the meaning of the symbols x 4, Sec. It is beside my present purpose, to insist farther on the necessity of attaching precise notions to the symbols employed in calculation ; and the subject deserves a separate and ample discussion.
f It is not so easy to prove as it; may be imagined, that f
X‘
Vt —
x“
~ x 4
3.-*
■p
3xs
4 &c.
104
Mr. Woodhouse on the Independence of the
then, if z represent the arc of a circle, and x the sine, this -equa- lity* z = x + -j~ + ~ — h &c* *s subject to restrictions, for x cannot exceed 1 ; consequently, the greatest value of z that can be determined from the equation, must be so determined
by putting x = i . Let nr — 1 + py + “pr
Now, from the definition of sine and the nature of the circle, the
arcs Qnr—Zf 67 r — Z .... (2ft-{-l) 27 r—% .... ^nr\z .... ^>nr -\~Z ....
have the same sine ; let these arcs be z, z', z", z'", &c.
and let ,r + ““ H — p§ — ^c* == ^ then z' = stt—X, 2;"= for— X, &c. or generally z,,m' **w= 27? — X, or = 4<?i7r -j-X,
n any number of the progression o, 1 , 2, 3, 4, &c.
Or thus, from the conditions contained in the form of the equa-
tion between z and x,
since V 1 — xz~ ezS^~~l + }== 1 + &c*
there is no possible value of % that answers the equation when
a’ is ~7 1,
Let /
X'
= X +
a
V I — X
0 and % = X
But the equation
X'
V x —x2
a
= £•
when z and * begin together,
may be derived from x =s ( 2 V — 1 )~*%
{ },
when instead of z is put s®-— z, Qh—z .... .(a»+i) *«■—*,
* In the expression « =.t+ ~ + -|A + &<=• considered abstractedly from its ori-
gin -and application, there is nothing that limits the value of .r. ^ Again, by applying
the operation of reversion, # is represented by this form, x ygy- + Ii2>3 45 &c*
But there is no method, I believe, of proving (I purposely exclude that unproved pro- position that every equation has as many roots as dimensions) that instead of 2 in
__ &c. = 0, other quantities, as z‘, z", &c. may be substituted.
x —z 4-
1.2.3
analytical and geometrical Methods of Investigation . 105
or 47 r+£ .... 4ft7r-(-2.
Hence, %' or 2tt — %= — X-{- a. Let z=o X=o qtt=u. Again, %" or for— • #== — X+ Let z—o .*. X=o .*. 6tt= a. Hence, the arbitrary quantity a may generally be represented by (2w+i)2tt, or by .-. z"“"m=(Qn-!ri)27r—X,
or = 4«tt+X.
XIV. I shall now shew, by a purely analytical process, what are the divisors of x”+an. It seems a very strange and absurd method, to refer to the properties of geometrical -figures, for the knowledge of the composition of analytical expressions.
1 z .
V— 1 *V— 1
Let x—mn Bn .'. an—m e .\ m—
£ZV 1
r, and m will
c"' — 1
be always positive, if s ~l= 1. But (Art. XI.) the values of %
that answer the equation s J==:i, are o0,=i=0,=i=20, —3^, or ge- nerally =*= s 9, s, any number of the progression o, 1 , 2, 3, &c.
Hence, #=0 s
e
generally,
7V-1
or values of x are a, ae n \ a a n
z9 ,
-vVi — V'-i , tfe n
i- _ Zll —
.*. xn—an—^x— -a) (a:2— <2 ] s n ^ -(- e n
20 —20
— 29
,at n \ &a
+a’) (x* — a
+ £ »
j fa1), See. 71 being odd; when n is even, (and of the form 2 p, p odd,) there must be a number (s) in the progression (o, 1, 2, 3, &c.) that =
£?. f — ■
consequently, there must be a value of 1, 1
= — a, since (Art. XI.) g2^-1 , or s~v“1 == — 1.
Hence, a quadratic divisor of will be (.r— 0). (x-pe), or x a ; when n is even, and of the form 4 p, p even or odd,
P
MDCCCH.
I
106 Mr. Woodhouse on the Independence of the there must be a number (s) in the progression (o, 1, 2, 3 ....)
±s0
= ; consequently, there must be a value of x, a e n
at 4
V-
■V-t
±0
, — — I —
=^x± v — 1, since (Art. XL) e ,ore 4
= ±V — -i.
Hence, one quadratic divisor of xn — a” will be of the form xz-\-a
= (a’-| ~aV — 1). (x — aV — 1); another, as it has been al- ready shewn, will be of the form x*—az.
There are only n different divisors, for ( n odd) the (n*~-i
±n — 1
and ?zth divisors are comprised under the form x—ae~T't the succeeding divisors would be comprised under the form
W — 1
x=as
:n- {- 1 2 n
= ae
i0V-
x e
q Cn — 1 2 n
¥=1
■ 0V“ ± QV~
= as 2n , (since e =1 ) the same as preceding form.
If xn-\-av=o, then m = —v/— -, to have m always positive.
Let e = — i,then (Art. XI.) the values of ^are=t:27r,=t67r...&c. Let 27r = p, then generally /2s+l)fV— 1 __ — j ; consequently,
±(2S+X)
x —as
n
pV-
\s any number of the progression o, 1, 2, See.
&-V=i ^V=7
\asn See.
+ 6 ” V 1 f +«*). (v*-a\ sn
3+V~
or the values of x are a e n
f -Lvn
or xn-j- an= (xa— < <2 ^ e n
— 1
+ e 71 J +<0 &c-
When w is odd, there must be a number (25+1) in the progres- sion (1, 3, 5, 7, &c.) = w; consequently, one value of # must
analytical and geometrical Methods of Investigation. 107
as
p7-
— a, or x -|- a must be a divisor of xn -f- an. XV. Resolution of — 2 la” xn-\-a™ into its quadratic factors
l A 1.
Now, from the equation x”=an j lz±=.\f I — 1 1= A =2= B s/ — 1.
— ~ ~ V— I 27__r /— ZV* T
Let x=zmne 71 me = A -f B V — 1, mi —
2-7—1 , —27 2 ^ £
— 1 f *7— i ~z-d~ | B
Va^+b1
but (Art: XI.) these equations are true, when instead of £ are put 20 -f 45# -J- 2 generally 50 -f s.
±5 0+2
■ ■■■■ ■ ■■■ -*J
Hence, the general value of x is at 71 , and the values
±2 ±0 + 2 . ±20 + 2 .
v — i — ~ — 7 — x — - — 7 — 1
of a? are as 71
»2«
or x2n — 2 lan xn-\ -a
0+2 — 0+2
e
a
* t/-i+£
, as *
<V2 ^ £
as
[ 4-V-,
n
n
+ S " V-,}+a*)xU-
-\-a ) / x &c.
XVI. Such are the analytical processes according to which the resolutions of xn=\ -an, x2n^Man x*+a2n are effected; and