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## CHAPTER X.

MISCELLANEOUS OPTICAL RESEARCHES OF NEWTON — HIS EXPERIMENTS ON THE ABSOLUTE REFRACTIVE POWERS OF BODIES — MORE RECENT EXPERIMENTS — HIS CONJECTURE RESPECTING THE INFLAMMABILITY OF THE DIAMOND, CONFIRMED BY MORE DIRECT EXPERIMENTS — HIS ERRONEOUS LAW OF DOUBLE REFRACTION — HIS OBSERVATIONS ON THE POLARITY OF DOUBLY REFRACTED IMAGES — DISCOVERIES ON DOUBLE REFRACTION IN THE PRESENT CENTURY — HIS EXPERIMENTS ON THE EYE OF A SHEEP — RESULTS OF THEM — HIS THREE LETTERS ON BRIGGS'S NEW THEORY OF VISION — HIS THEORY OF THE SEMI-DECUSSATION OF THE OPTIC NERVES — PARTLY ANTICIPATED BY ROHAULT — OPINIONS OF LATER WRITERS ON VISION, OF REID, BROWN, WOLLASTON, TWINING, AND ALISON, DISCUSSED — THE TRUE LAWS OF SENSATION AND VISION — NEWTON'S OBSERVATIONS ON THE IMPRESSION OF STRONG LIGHT UPON THE RETINA — MORE RECENT OBSERVATIONS — HIS REFLECTING SEXTANT — HIS REFLECTING MICROSCOPE — HIS REFLECTING PRISM FOR REFLECTING TELESCOPES — HIS METHOD OF VARYING THE MAGNIFYING POWER OF NEWTONIAN TELESCOPES — NEWTON'S TREATISE ON OPTICS — HIS LECTIONES OPTICÆ.

Although the discoveries described in the preceding chapters are those on which Newton's reputation in optics chiefly rests, yet it is necessary to notice some of his less elaborate researches, which, though of inferior importance in the science of light, have either exercised an influence over the progress of discovery, or have been associated with the history of other branches of knowledge.

In the second book of his Optics,[1] Newton proves, with much fulness of detail, that "the cause of reflexion is not the impinging of light on the solid or impervious <212> parts of bodies, as is commonly believed;" and that "bodies reflect and refract light by one and the same power variously exercised in various circumstances." He then proceeds to show, that "if light be swifter in bodies than in vacuo, in the proportion of the sines which measure the refraction of the bodies, the forces of the bodies to reflect and refract light are very nearly proportional to the densities of the same bodies, excepting that unctuous and sulphureous bodies refract more than others of the same density." This remarkable exception led our author to point out the connexion between the refractive powers and the chemical composition of bodies. Having obtained measures of the refractive powers and densities, or specific gravities of twenty-two substances varying in density between air and diamond, and having computed their refracting forces, and compared them with their densities, he calculated their refractive powers in respect of their density. From this comparison he found that topaz, selenite, rock-crystal, Iceland spar, common glass, glass of antimony, and air, have their refractive powers almost in the same proportion as their densities, "excepting that the refraction of that strange substate, Iceland spar, is a little bigger than the rest." — " Again," he adds, "the refraction of camphor, olive oil, lintseed oil, spirit of turpentine, and amber, which are fat sulphureous unctuous bodies, and diamond, which probably is an unctuous substance coagulated, have their refractive powers in proportion to one another as their densities, without any considerable variation. But the refractive powers of these unctuous substances are two or three times greater in respect of their densities than the refractive powers of the former substances are in respect of theirs. Water has a refractive power in a middle degree between these two <213> sorts of substances . . . . salts of vitriol between those of earthy substances and water, and spirit of wine between water and oily substances." The following are a few of the numbers on Newton's Table: —

 Refractive Power. Refractive Power. Pseudo topaz,[2] 3,979 Rain water, 7,845 Air, 5,208 Spirit of wine, 10,121 Rock crystal, 5,450 Oil of olives, 12,607 Iceland crystal, 6,536 Amber, 13,654 Rock salt, 6,477 Diamond, 14,556

To the results in this table we have added the following, computed chiefly from observations of our own, and interesting as being, with the exception of three in italics, below the lowest and above the highest in Newton's Table: —

 Refractive Power. Refractive Power. Tabasheer, 976 Realgar artificial, 16,666 Cryolite, 2,742 Ambergris, 17,000 Fluor spar, 3,426 Sulphur, 22,000 Sulphate of Barytes, 3,829 Phosphorus, 28,857 Greenockite, 12,861 Hydrogen, 29,964 Octohedrite, 13,816 Hydrogen, 31,862 Diamond, 13,964

The enormous refractive powers possessed by the last six bodies in the preceding table, when taken in connexion with those given by Newton, exhibit in a striking degree the connexion between a high degree of inflammability and a great refracting force. The conjecture of Newton that the diamond "is an unctuous substance coagulated," has been generally regarded as a proof of singular sagacity, and as an anticipation of the results of chemical analysis; but it is certainly not entitled to such praise. Its solitary position among the oils and inflammable bodies led to the conjecture; but had he known the refractive index and specific gravities of greenockite and octohedrite, he would <214> have drawn the same conclusion respecting them, and been mistaken. The real inference respecting the composition of the diamond, which Newton's Table authorizes, is not that it should consist of carbon, but of sulphur. "So then," says he, "by the foregoing table, all bodies seem to have their refractive powers proportional to their densities, (or very nearly,) excepting so far as they partake more or less of sulphureous oily particles, and thereby have their refractive power made greater or less. Whence it seems rational to attribute the refractive power of all bodies chiefly, if not wholly, to the sulphureous particles with which they abound. For it is probable that all bodies abound more or less with sulphurs. And as light congregated by a burning glass acts most upon sulphureous bodies, to turn them into fire and flame; so since all action is mutual, sulphurs ought to act most upon light"[3]

That diamond is a soft substance coagulated, has been rendered probable by experiments of a more direct nature. We have shown by the examination of a great number of diamonds in polarized light, that the little cavities which many of them contain, have been pressed outward by an elastic force emanating from some gas or fluid with which they had been filled. Several such cavities we found in the Koh-i-noor diamond, and in the two smaller ones which accompanied it; and in a specimen in the British Museum, we found a yellow crystal of diamond that had crystallized upon the cleavage surface of another which was colourless, having been expelled from an adjacent cavity, in which it had existed in a fluid state.[4]

Among the more interesting optical researches of New <215> ton, we rank his observations on the double refraction and polarisation of light. On the 12th of June 1689, when Huygens was in England, during the presidency of Sir Robert Southwell, he attended a meeting of the Royal Society, at which Newton was present. Huygens informed the Society that he was about to publish a treatise concerning the cause of gravity, and another about refraction, giving, among other things, the reasons of the doubly refracting Iceland crystal. ' Mr. Newton, considering a piece of the Iceland crystal, did observe that of the two species wherewith things do appear through that body, the one suffered no refraction when the visual ray came parallel to the oblique sides of the parallelopiped; the other, as is usual in all other transparent bodies, suffered more when the beam came perpendicular to the planes through which the object appeared."[5] It is remarkable that this observation of Newton, which had been made long before by Bartholinus, as Huygens knew at the time, and as the Royal Society ought to have known,[6] should not have been claimed for that author.

In the admirable Treatise on Light, to which Huygens referred at the Royal Society, and which was published in 1690, he has shewn that the observation of Bartholinus, adopted by Newton, is erroneous,[7] and has explained the law of unusual refraction, as exhibited in one of the two pencils formed by the double refraction of Iceland or calcareous spar. This law he deduced from the principles of the undulatory theory, and he confirmed it by direct experiment. Viewing it probably as a theoretical result, Newton seems to have regarded it as incorrect, and though <216> he has given Huygens the credit of describing the phenomena more exactly than Bartholinus, who first discovered and described the remarkable property of this spar, yet without assigning any reason, or even referring to the law of Huygens, he substitutes another in its place. The observations of Newton were first published in his Optics in 1704,[8] fourteen years after the appearance of Huygens's work. The law of unusual refraction, adopted by Newton, is not given as the result of theory. It is stated as an undoubted truth, and no experiments whatever are referred to as having been made either by himself or others. "One of these refractions," he says, "is performed by the usual rule of optics, the sine of incidence out of air into this crystal being to the sine of refraction as five to three. The other refraction, which may be called the unusual[9] refraction, is performed by the following rule." This rule was first shown to be erroneous by the Abbé Hauy,[10] and it has been rejected by all succeeding philosophers.[11]

In his observations on the successive disappearance and re-appearance of two of the four images which are formed when a luminous object is viewed through two rhombs of Iceland spar, one of which is made to revolve upon the other, Newton has been more successful, though he has omitted to give to Huygens the credit of having discovered these curious phenomena. He considers "every ray of light as having four sides or quarters, two of which are originally endued with the property on which the <217> unusual refraction depends, and the other two opposite sides not endued with that property;" and he adds, that "it remains to be inquired whether there are not more properties of light by which the sides of the rays differ, and are distinguished from one another."

In animadverting on Huygens's theory of two vibrating media within the Iceland crystal, he asserts that the unusual refraction depends "not on new modifications, but on the original and unchangeable dispositions of the rays," which, he says, "had Huygens known, he would have found it difficult to explain how these dispositions, which he supposed to be impressed on the rays by the first crystal, could be in them before their incidence on that crystal; and in general, how all rays emitted by shining bodies can have these dispositions in them from the beginning. To me, at least," he adds, "this seems inexplicable, if light be nothing else than pression or motion propagated through ether."

After Newton wrote these imperfect observations, more than a century elapsed before the double refraction and polarisation of light in Iceland spar and other bodies were reduced to regular laws. In 1810, Malus announced to the Institute of France, the remarkable discovery that a ray of light reflected at a particular angle was polarised like one of the pencils formed by Iceland spar, that is, exhibited the same properties in its four sides or quarters which are exhibited in one of the pencils of Iceland spar; and the result of this fine discovery has been the establishment of a new branch of Physical Optics, which possesses the highest interest, not only from the beauty of its laws and the splendour of its phenomena, but from the new power with which it arms the philosopher in detecting organic or inorganic <218> structures, which defy the scrutiny of the eye and the microscope.

Although Sir Isaac Newton has not published any of his opinions or experiments on Vision, or on the structure and functions of the eye, yet we fortunately possess some fragments of his researches, which are both valuable and interesting. Among these is a manuscript in his own handwriting, which we found among the family papers, containing some accurate observations and experiments on the form and dimensions of the eye of a sheep, and accompanied with an outline drawing, on a large scale, of a section of the eye.[12] The following are the most interesting results contained in this manuscript.

In the first part of it, which is written in Latin, he makes the outer surface of the cornea part of a prolate spheroid, the major axis coinciding with the optical axis, or that of the eye, and having to the transverse axis the ratio of 1350 to 972.

He places the focus for parallel rays of the first surface of the cornea at a point behind the eye, and as far beyond the sclerotic coat as one-seventh of the diameter of the eye-ball, which he makes an oblate spheroid, having its vertical axis 1025, and its horizontal one 975, the anterior portion of the spheroid coinciding nearly with the front of the iris.

He represents the crystalline lens as having a great degree of convexity, differing not much from a sphere, and he remarks that the anterior superficies of the crystalline is more full than the posterior surface, which is certainly not the case, and is not so represented in the diagram.

The second part of the manuscript, which contains minute <219> measurements of every part of the eye, is written in English, and concludes with an expression of regret, that "he was prevented by an accident from taking the distance of the crystalline humour from the horny tunic, (the sclerotic coat,) which I would gladly have done to have had the conformity of all the parts one to another, in one and the same eye." The elliptical form of the cornea was detected not many years ago by M. Chossat of Geneva, who, of course, could not know that he had been anticipated by Newton.

We have not been able to ascertain at what time these observations were made, but it appears from the correspondence of Newton with Dr. W. Briggs, published by Mr. Edleston,[13] that in 1682 his attention was called to the subject of binocular vision, in consequence of Dr. Briggs having communicated to the Royal Society on the 15th March, a paper entitled, "A New Theory of Vision."[14] Briggs, who was a contemporary of Newton's at Cambridge, and a Fellow of Corpus Christi College, seems to have sent him a copy of his paper, and to have solicited his opinion of it. The theory which he proposes evinces neither sagacity nor genius. Setting out on the erroneous principle which has so long disfigured the physiology of the senses, that sensation is performed only in the brain, and which has not yet been exploded, he seeks for an explanation of single vision with two eyes, and of other visual phenomena in "the rise of the optic nerve, the position of its fibres, and the manner of their insertion into the eye." He describes the optic nerves as arising "from two gibbous protuberances,"[15] in such a manner <220> that those fibres that are in the zenith or apex of the thalami have the greatest tension, while those in the nadir, or opposite part, have the least tension by reason of a less flexure. Every fibre that passes into the upper part of the right eye from the upper part of one thalamus, has a corresponding one passing from the upper part of the other thalamus into the upper part of the left eye, and the same thing takes place with the lower fibres. The fibres which thus correspond in site correspond also in tension, "so that when any impression from an object without moves both fibres, it causes not a double sensation any more than unisons in two viols struck together cause a double sound." This theory may be called the theory of corresponding fibres, and is doubtless the parent of one more modern though equally inadmissible — the theory of corresponding points.

In his first letter to Briggs, Newton tells him that he has "perused his very ingenious theory of vision, in which (to be free with you as a friend should be) there seems to be some things more solid and satisfactory, others more disputable, but yet plausibly suggested, and well deserving the consideration of the ingenious. The more satisfactory I take to be your asserting that we see with both eyes at once, — your speculation about the musculus obliquus inferior,[16] your assigning every fibre in the optic nerve of one eye to have its correspondent in that of the other, both which make all things appear to both eyes, in one and the same place, and your solving hereby the duplicity of the object in distorted eyes, and confuting the childish opinion about the splitting the optic cone. The more disputable seems your notion about every pair <221> of fellow fibres being unisons to one another, discords to the rest, and this consonance making the object seen with two eyes appear but one, for the same reason that unison sounds seem but one sound." Newton here terminates his letter to "his honoured friend, Dr. Briggs," with the observation that he had intended to state his objections "against this notion," but that he thought it better "to reserve it for discourse at their next meeting."

Briggs, probably anxious for an earlier discussion than one living at Cambridge could concede, seems to have requested him to make his objections in writing. Newton accordingly addressed to his honoured friend a long letter of nearly seven printed pages,[17] a letter of very great interest, and utterly subversive of the theory of his correspondent. In the commencement and conclusion of this letter, which is of a slightly personal nature, we see finely displayed the modesty and peculiar character of its author. "Though I am of all men," he begins, "grown the most shy of setting pen to paper about any thing that may lead into disputes, yet your friendship overcomes me so far, that I shall set down my suspicions about your theory, yet on this condition, that if I can write but plain enough to make you understand me, I may leave all to your use without pressing it further on. For I design not to confute or convince you, but only to present and submit my thoughts to your consideration and judgment."

After shewing that the bending of the nerves in the thalami is no proof of a difference of tension, he states, that when the ear hears two sounds in unison, it does not hear them as one sound, unless they come from nearly the same spot; and for the same reason a similar tension <222> of the optic fibres will make the object appear one to two eyes.

He then proceeds to show that the singleness of the picture arises from the coincidence of the two pictures, and therefore that the cause of single vision must be sought for in the cause that produces the coincidence. "But you will say," he adds, "how is this coincidence made ? I answer, what if I know not ? Perhaps in the sensorium after some such way as the Cartesians would have believed,[18] or by some other way. Perhaps by the mixing of the marrow of the nerves in their juncture before they enter the brain, the fibres on the right side of each eye going to the right side of the head, those on the left side to the left."[19]

In support of his theory, Briggs maintained that "it was not to be imagined that the nerves decussate one another, or are blended together," at the place where they approach each other before they set off to the right and left eye; and he adduces the case of many fishes, where the nerves are joined only by simple contact, "and in the chameleon not at all, (as is said)," admitting, at the same time, that in whitings, and perhaps some other fishes, they do decussate.

To this Sir Isaac replies: "If you say that in the chameleon and fishes the nerves only touch one another without mixture, and sometimes do not so much as touch; 'tis true, but makes altogether against you. Fishes look <223> one way with one eye, the other way with the other; to the right hand with this, to the left hand with that, twisting their eyes severally this way or that as they please. And in those animals which do not look the same way with both eyes, what wonder if the nerves do not join? To make them join would have been to no purpose; and nature does nothing in vain. But then, whilst in these animals, where 'tis not necessary, they are not joined, in all others which look the same way with both eyes, so far as I can yet learn, they are joined. Consider, therefore, for what reason they are joined in the one and not ill the other. For God, in the frame of animals, hath done nothing without reason."

The last objection of Sir Isaac to the new theory is unanswerable. Admitting that consonance unites objects seen with the fibres of two eyes, "much more," says he, "will it unite those seen with those (consonant fibres) of the same eye, and yet we find it much otherwise."

"You have now seen," he says in conclusion, "the sum of what I think of worth objecting, set down in a tumultuary way, as I could get time from my Stourbridge Fair friends. If I have anywhere expressed myself in a more peremptory way than becomes the weakness of the argument, — pray, look on that as done not in earnestness, but for the mode of discoursing. Whether anything be so material as that it may prove any way useful to you, I cannot tell; but pray, accept of it as written for that end. For having laid philosophical speculations aside, nothing but the gratification of a friend would easily invite me to so large a scribble about things of this nature."[20]

<224>

Notwithstanding the force of these objections, Dr. Briggs continued to press his theory on public notice, and in May 1683, he published in Hooke's Philosophical Collections additional explanations of it, and a reply to seven different objections that had been sent him "by Mr. Newton, our worthy Professor of Mathematics at Cambridge, and other friends." It would be out of place to make any observations on this defence of his theory. We hear no more of it for two years; but it appears that Newton had requested Briggs to print a Latin version of it, and we accordingly find that it is published in London in 1685, with a curious letter of Newton's prefixed. This letter[21] must have been solicited by Briggs, in order to call the attention of philosophers to his book; and we confess that we feel great difficulty in appreciating the motives that could have induced its author to express the opinions which it contains.

In this letter,[22] written in Latin, Sir Isaac speaks of Briggs's two treatises[23] as advancing at once two sciences of great name, Anatomy and Optics. He compliments him on having diligently inquired into the mysteries of an organ so skilfully constructed, and he expresses the great delight which he had formerly received from the skill and dexterity with which he had dissected it. He tells him that he had so elegantly developed the muscles of the eye-ball, and expounded the other parts, that we could not only understand, but see the uses and functions <225> of each, and that this shewed that nothing inaccurate could be expected from his scalpel. He then speaks of his excellent anatomical tract, in which he shews the value of accurate observation by "a most ingenious theory." After describing Briggs's theory in a few lines, and mentioning the analogy between unisons in music and in optics, he says that nature is simple — that a great variety of effects may be produced by the same mode of operation, and that this was probable in the causes of the cognate senses. But notwithstanding all this general praise, which is certainly not merited, Newton does not adopt the theory. For though he may suspect that there is another analogy between these senses than that contained in the theory, he must willingly confess that that of Briggs is very ingeniously excogitated. He then remarks that he does not think the second dissertation useless in which he dilutes the objections made against the theory. "Go on, then," he adds, "illustrious sir, as you are doing, and advance these sciences by your very great inventions, and teach the world that those difficulties in investigating physical causes which usually yield with difficulty to vulgar attempts, may be so easily overcome by talent."

While Newton was writing this letter, there is reason to believe that he had himself conceived another theory of single vision with two eyes, proceeding on the supposition that Briggs was wrong in his Anatomy as well as in his Optics. This, we think, is indicated by the "other analogy" of the senses of sight and hearing which he then suspected, and to which he was no doubt led by his correspondence with Briggs. It is evident, that in September 1682, the date of his second letter, he had laid aside philosophical speculations, and that he unwillingly wrote his opinion "about things of that nature;" and it is equally obvious, from his supposition about the mixing of <226> the marrow of the nerves in their juncture before they enter the brain, that if the idea of the semi decussation of the fibres had been then in his view, he had not at that time given it any serious consideration.

That he had studied this subject with peculiar care, is manifest from the 15th Query of his Optics,[24] where he has given a brief abstract of his theory of corresponding points, or of the semi-decussation of the optic nerves, but particularly from an elaborate paper on the subject which was never published in his lifetime, but which was found in MS. among the papers of William Jones, Esq., which form the celebrated Macclesfield Collection of scientific correspondence. A copy of this paper was given to Joseph Harris, who inserted it in his Treatise of Optics,[25] but from the manner in which he has garbled it, we cannot discover whether or not he has published the whole of the manuscript.[26]

The theory of Newton, as published in his Optics, and <227> as more fully developed in the MS. in question, will be understood from the annexed diagram given by himself. Let P, Q represent the two eyes, TVEG, YXEH the optic nerves, crossing at what has been called the sella turcica, GH, and passing between IL or MK towards the brain. Newton observes, that if the nerve be cut crosswise anywhere between TG or YH, the section will "appear full of spots or pimples, which are a little prominent, especially if the nerve be pressed or warmed at a candle; that these shoot into the very eye, and may be seen withinside where the retina grows to the nerve; and that they continue to the very juncture EFGH. But at the juncture they end on a sudden into a more tender white pap, like the anterior part of the brain, and so the nerve continues after the juncture into the brain, filled with a white tender pap, in which can be seen no distinction of parts as betwixt the said juncture and the eye."

"Now I conceive," says he, "that every point in the <228> retina of one eye hath its correspondent point in the other, from which two very slender pipes, filled with a most limpid liquor, do without any interruption, or any other unevenness or irregularity in their process, go along the optic nerves to the juncture EFGH, where they meet either betwixt GF or FH, and there unite into one pipe as big as both of them; and so continue in one passing either betwixt IL or MK into the brain, where they are terminated perhaps at the next meeting of the nerves between the cerebrum and cerebellum, in the same order that their extremities were situated in the retinas. And so there are a vast multitude of these slender pipes which flow from the brain, the one-half through the right-side nerve IL, till they come at the juncture GF, where they are each divided into two branches, the one passing by G and T to the right side of the right eye AB, the other half shooting through the space EF, and so passing by X to the right side of the left eye αβ. And, in like manner, the other half shooting through the left-side nerve MK, divide themselves at FH, and their branches passing by EV to the right eye, and by HY to the left, compose that half of the retina in both eyes which is towards the left side CD and γδ."

From this theory of the semi-decussation of the optic nerves, Newton draws the following conclusions: —

"Hence it appears," says he, —

"1. Why the two images of both eyes make but one image, abcd, in the brain.

"2. Why, when one eye is distorted, objects appear double, for if the image of any object be made upon A in the one eye, and β in the other, that object shall have two images in the brain at a and b . Therefore, the pictures of any objects ought to be made upon the corre <229> sponding points of the two retinas; if upon A in the right eye, then upon α in the left; if upon B, then also upon β. And so shall the motions concur after they have passed the juncture GH, and make one image at a or b more vivid than one eye alone could do.

3. Why, though one thing may appear in two places by distorting the eyes, yet two things cannot appear in one place. If the picture of one thing fall upon A, and of another upon a , they may both proceed to p , but no farther. They cannot both be carried on the same pipe pa into the brain; that which is strongest, or most helped by phantasy, will there prevail, and blot out the other.

"4. Why, if one of the branches of the nerve beyond the juncture, as at G F or F H should be cut, that half of both eyes towards the wounded nerve would be blind, the other half remaining."[27]

This ingenious theory, decidedly superior to that of Briggs, was to a considerable extent anticipated by M. Rohault, in his Traité de Physique, published in 1671, more than ten years before Newton's attention was called to the subject. Rohault gives the very same figure as the preceding one, with this difference, that the nerves neither cross nor split into two at G H. He supposes that the two optic nerves have their corresponding or sympathetic fibres, which unite in one point in the brain; and he thus explains single vision with two eyes, their duplicity by distortion, and the impossibility of two things appearing in one place.[28]

<230>

During the 120 years that have elapsed since the publication of Newton's Optics, we hear nothing more of the Theory of Vision in the 15th Query, and in the manuscript above referred to, till the year 1824, when Dr. Wollaston published in the Philosophical Transactions of that year, a paper On Semi-decussation of the Optic nerves, in which he reproduces the very theory of Newton, in order to account for the curious disease of hemiopsy, or amaurosis dimidiata,[29] in which the patient sees with each eye only half of an object, being blind to the other half. This sympathy between the two eyes may certainly arise from structure, and depend upon "connexion of nervous fibres," and if it does, is very well explained either by the hypothesis of Rohault or of Newton; but we cannot attach any value to the invention of structural hypotheses when the phenomena may be explained by that habitual sympathy of double organs with which we are so well acquainted. This observation is still more applicable to the remark of Wollaston, that by his theory "we clearly grain a step in the solution, if not a full explanation of the long agitated question of single vision with two eyes," because this great fact in vision can be perfectly explained, as we shall presently see, without any hypothesis whatever.

But not only is this theory of semi-decussation uncalled for, it is contradicted by numerous facts. It has been examined with great ability by Mr. Twining, of the Indian Medical Service,[30] who concludes "from anatomical observations respecting the structure of the optic nerves and <231> thalami, and the effects of disease on those parts, that no decussation or semi-decussation of the optic nerves exists in the human subject. No anatomist, indeed, has pretended to say that there is any trace of semi-decussation; and it has been proved that the decussation or crossing at G H, Fig. 16, is only partial, the inner bundles decussating, while the outer bundles remain on the side on which they previously lay."[31]

There is no branch of physical science upon which such unsound views have prevailed as in that which relates to the optical functions of the eye; and in studying the speculations of modern metaphysicians and physiologists, we feel as if we were grappling with the chimeras of Aristotle or Descartes. While Dr. Reid maintains that objects appear single when their images are formed upon corresponding points of the retina, and double in all other circumstances, he gives no explanation whatever of single vision: he merely attaches the name of corresponding points to those upon which the images falls when it is seen single! And when Dr. Brown tells us that it is from association alone we see objects single and erect, by means of double and inverted pictures, he merely asserts his ignorance of the cause; and his assertion is contrary to the most notorious facts and to all experience, as Dr. Reid has shewn.[32] Nor does Dr. Alison, one of the latest writers <232> on the subject, bring us a single step nearer the truth. After controverting the views of Brown and Reid, he apprehends that he has established the following two facts, the one explaining single, and the other erect vision :[33]

1. That images formed on corresponding points of the retinæ of the human eyes, and on those only, naturally affect our minds in the same manner as a single image formed on the retina of one eye; and,

2. That impressions made on different points of the retina of the eye, are naturally followed by inferences as to the relative position of the objects producing these impressions, exactly opposite to those which follow impressions made on different points of the surface of the body.[34]

We are unable to controvert these two palpable facts. They are truisms which explain nothing; and if Nature had been so perverse as to produce three pictures in place of one from two eyes, and had turned round an erect picture 90° in place of 180°, which it does in inverting it, that is, had represented a man upon the retina lying horizontally in place of vertically and inverted, the explanation <233> of Dr. Alison would have been, that in the first case it was natural, and that in the other it was naturally, and exactly half opposite to other impressions on the surface of the body.[35]

From these speculations we venture to solicit the attention of the reader to the true explanation of single and erect vision, and of all the other normal visual phenomena with which we are acquainted, an explanation which has been overlooked by our most distinguished optical writers.

1. The retina[36] is the seat of visual sensation and of vision; and there is a law of visual sensation as well as a law of vision, which can be determined only by experiment.

2. In order to determine the law of visual sensation, or the mental information given by the action of a physical point of light upon the retina, let us make a hole of the smallest size, that of the minimum visible, for example, on a sheet of black paper, and let a ray of the sun's light pass through it and fall upon the eye. This cone of rays, with the pupil for its base, will be refracted by the humours of the eye into a smaller cone, the apex of which falls upon the retina. This apex, or point, is the image of the hole in the paper, and is formed by a cone of rays whose angle we may suppose to be 12°, so that <234> the impression is made by rays falling at all angles on the retina from 0° to 6° on each side of the perpendicular or axis of the cone. If, while looking at the hole in the paper, we stop all the different rays in succession from 0° to 6°, we shall find that the hole is seen by them all in one direction, and that this direction is the axis of the cone, and, as nearly as can be ascertained, the real direction of the hole, or the axis of the incident cone of rays. Hence it follows, that the impression of a ray of light upon the retina, whatever be the angle of its incidence, gives the sensation of having proceeded in a direction perpendicular to the retina, a direction as will be afterwards seen coinciding nearly with the real direction of the hole from which it issues. This is the law of visible direction.

3. In order to determine the law of vision, look at the hole in the paper with both eyes, and it will be found, by opening and shutting each eye alternately, that a single image of the hole is seen, and always in the same place, namely, at a point where the optical axes of the two eyes meet, and consequently at the distance from the eye where these axes meet. The single image seen by both eyes is formed by the superposition or coincidence of the two images. This is the law of visible distance, and the law of single vision; but the law of single vision is true only for visible points. If we had the hundred eyes of Argus in place of two, the hundred images of a point would coincide in one at the point where the hundred axes converge.

4. The law of vision for visual objects is entirely different from that for points. A visual object cannot be seen single at once. Let the object, for example, be a line $\frac{1}{10}$th of an inch long. The two images of it cannot be seen coincident by both eyes. When the right hand extremities of the images are coincident or single, the left hand <235> extremities are not, and vice versa. When the object is a lineal space or superficies, only one point of it is seen single and distinct, the two eyes converging their optic axes on every point of it in succession, and thus obtaining the idea of space. When the object is a solid, such as a cube, only one point of it is seen single and distinct, the two eyes converging their optical axes to the near and remote parts of it in succession, and thus obtaining an idea of the different distances of its parts by the varying angle of the optic axes. This law of vision for solids, includes the theory of the stereoscope.[37]

We have stated that the law of sensation gives a visible direction, which is nearly coincident with the real direction of objects. The celebrated D'Alembert maintained that the action of light upon the retina is conformable to the laws of Mechanics,[38] and therefore that the visible direction of an object should be a line perpendicular to the curvature of the retina at the excited point; but he rejected this law as contrary to observation. By using, however, more correct refractive powers for the humours of the eye, and more accurate measures of its parts, we have shewn that the visible and true direction of points nearly coincide.[39]

By means of these laws all the phenomena of erect vision from an inverted image, — of the single vision of points, — of the vision of plane surfaces and solids, — and of the conversion of two plane pictures into solids or objects in relief, may be calculated with as much accu <236> racy as we can compute the positions of the heavenly bodies.

Among the minor optical labours of Sir Isaac Newton, we must rank some curious observations on the action of strong light upon his own eyes, which have been only recently published by Lord King in his Life of Locke. In his work on Colours, Mr. Boyle has described a curious case, in which a gentleman "eminent for his profound skill in almost all kinds of philological learning, had injured his eyes by looking too fixedly upon the sun through a telescope, without any coloured glass to take off from the dazzling splendour of the object. The excess of light did so strongly affect his eye, that ever since when he turns it towards a window or any white object, he fancies he sees a globe of light of about the bigness the sun then appeared to him, to pass before his eyes; and having inquired of him how long he had been troubled with this indisposition, he replied, that it was already nine or ten years since the accident that occasioned it first befell him."[40] This remarkable case having attracted the attention of Locke, he requested Sir Isaac to give him his opinion on the subject. In his reply, dated Cambridge, June 30, 1691, Sir Isaac sent him the following very interesting observations, made by himself.[41]

"The observation you mention in Mr. Boyle's book of colours, I once made upon myself with the hazard of my eyes. The manner was this : I looked a very little while upon the sun in the looking-glass with my right eye, and then turned my eyes into a dark corner of my chamber, and winked, to observe the impression made, and the <237> circles of colours which encompassed it, and how they decayed by degrees, and at last vanished. This I repeated a second and a third time. At the third time, when the phantasm of light and colours about it were almost vanished, intending my fancy upon them to see their last appearance, I found, to my amazement, that they began to return, and by little and little to become as lively and vivid as when I had newly looked upon the sun. But when I ceased to intend my fancy upon them, they vanished again. After this, I found, that, as often as I went into the dark, and intended my mind upon them, as when a man looks earnestly to see anything which is difficult to be seen, I could make the phantasm return without looking any more upon the sun; and the oftener I made it return, the more easily I could make it return again. And at length, by repeating this without looking any more upon the sun, I made such an impression on my eye, that, if I looked upon the clouds, or a book, or any bright object, I saw upon it a round bright spot of light like the sun, and, which is still stranger, though I looked upon the sun with my right eye only, and not with my left, yet my fancy began to make an impression upon my left eye, as well as upon my right. For if I shut my right eye, or looked upon a book or the clouds with my left eye, I could see the spectrum of the sun almost as plain as with my right eye, if I did but intend my fancy a little while upon it; for at first, if I shut my right eye, and looked with my left, the spectrum of the sun did not appear till I intended my fancy upon it; but by repeating, this appeared every time more easily. And now, in a few hours' time, I had brought my eyes to such a pass, that I could look upon no bright object with either eye, but I saw the sun before me, so that I durst neither write nor read; but to recover the <238> use of my eyes, shut myself up in my chamber made dark, for three days together, and used all means to divert my imagination from the sun. For if I thought upon him, I presently saw his picture, though I was in the dark. But by keeping in the dark, and employing my mind about other things, I began in three or four days to have some use of my eyes again; and, by forbearing to look upon bright objects, recovered them pretty well, though not so well, but that, for some months after, the spectrum of the sun began to return as often as I began to meditate upon the phenomena, even though I lay in bed at midnight with my curtains drawn. But now I have been very well for many years, though I am apt to think, if I durst venture my eyes, I could still make the phantasm return by the power of my fancy. This story I tell you, to let you understand, that in the observation related by Mr. Boyle, the man's fancy probably concurred with the impression made by the sun's light, to produce that phantasm of the sun which he constantly saw in bright objects. And so your question about the cause of this phantasm involves another about the power of fancy, which, I must confess is too hard a knot for me to untie. To place this effect in a constant motion is hard, because the sun ought then to appear perpetually. It seems rather to consist in a disposition of the sensorium to move the imagination strongly, and to be easily moved, both by the imagination and by the light, as often as bright objects are looked upon."

These observations possess in many respects a high degree of interest. The fact of the transmission of the impression from the retina of the one eye to that of the other, or of its production in that eye merely by fancy, is particularly important; and it deserves to be remarked as a singular coincidence, that we had occasion to observe, <239> and to describe the same phenomena above forty years ago,[42] and long before the observations of Sir Isaac were communicated to the scientific world. Æpinus of St. Petersburg observed the circles of colours described by Newton, when produced by looking at the setting sun for fifteen seconds. In the experiments alluded to, we looked at the brilliant image of the sun formed by a concave speculum of 30 inches focus with the right eye tied up, and upon turning our left eye to a white ground, we observed six successions of different colours with their complementary tints when the left eye was shut. Upon uncovering our right eye, and turning it to a white ground, we were surprised to observe the reverse spectra, as if the impression had been conveyed from the left to the right eye, as in Sir Isaac's case. A spectrum of a darkish hue floated before the left eye for many hours, and this was succeeded by severe pains shooting through every part of the head. A slight inflammation, affecting both eyes, continued for several days, and it was not till several years had elapsed that our eyes had recovered their former power.

Among the inventions of Sir Isaac Newton, we may enumerate his reflecting sextant for observing the moon's distance from the fixed stars at sea. The description of this instrument was communicated to Dr. Halley in the year 1700; but, either from having mislaid the manuscript, or from attaching no value to the invention, he never communicated it to the Royal Society, and it remained among his papers till after his death in 1742, when it was read to the Society on the 28th October. The following is Sir Isaac's own description of it, as copied from the original manuscript:[43]

<240>

"In the annexed figure PQRS denotes a plate of brass, accurately divided in the limb DQ, into $\frac{1}{2}$ degrees, $\frac{1}{2}$ minutes, and $\frac{1}{12}$ minutes, by a diagonal scale ; and the $\frac{1}{2}$ degrees, and $\frac{1}{2}$ minutes, and $\frac{1}{12}$ minutes, counted for degrees, minutes, and $\frac{1}{6}$ minutes. A B is a telescope three or four feet long, fixed on the edge of that brass-plate. G is a speculum fixed on the brass-plate perpendicularly as near as may be to the object-glass of the telescope, so as to be inclined forty-five degrees to the axis of the telescope, and intercept half the light which would otherwise come through the telescope to the eye. C D is a moveable index turning about the centre C, and, with its fiducial edge, showing the degrees, minutes, and $\frac{1}{6}$ minutes on the limb of the brass-plate P Q; the centre C must be over against the middle of the speculum G. H is another speculum, parallel to the former, when the fiducial edge of index falls on 0° 0′ 0″; so that the same star may then appear through the telescope in one and the same place, both by the direct rays and by the reflexed ones; but if the index be turned, the star shall appear in two places, whose distance is showed on the brass limb by the index.

<241>

"By this instrument the distance of the moon from any fixed star is thus observed; view the star through the perspicil by the direct light, and the moon by the reflexed, (or on the contrary;) and turn the index till the star touch the limb of the moon, and the index shall show on the brass limb of the instrument the distance of the star from the moon's limb; and though the instrument shake by the motion of the ship at sea, yet the moon and star will move together as if they did really touch one another in the heavens; so that an observation may be made as exactly at sea as at land.

"And by the same instrument, may be observed exactly the altitudes of the moon and stars, by bringing them to the horizon; and thereby the latitude and times of observation may be determined more exactly than by the ways now in use.

"In the time of the observation, if the instrument move angularly about the axis of the telescope, the star will move in a tangent of the moon's limb, or of the horizon; but the observation may notwithstanding be made exactly, by noting when the line, described by the star, is a tangent to the moon's limb, or to the horizon.

"To make the instrument useful, the telescope ought to take in a large angle; and, to make the observation true, let the star touch the moon's limb, not on the outside, but on the inside."

This ingenious contrivance is obviously the very same invention as that which Mr. Hadley produced in 1731;[44] and which, under the name of Hadley's Quadrant, has been of so great service in navigation. But though the merit of this invention is thus transferred to Newton, we must not omit to state, that the germ of it, and something <242> more, had been previously published by Hooke. In giving an account of the inventions of members of the Royal Society, Sprot mentions "a new instrument for taking angles by reflexion, by which means the eye at the same time sees the two objects both as touching on the same point, though distant almost to a semicircle, which is of great use in promoting exact observations at sea."[45] Hooke was the member who made this invention, and there is a drawing and description of it in his Posthumous Works.[46] About the end of the year 1730, Thomas Godfrey of Philadelphia invented an instrument similar to Hadley's, and the Royal Society having found that Hadley's invention could be traced to the summer of 1730, decided that Hadley and Godfrey were independent inventors. The enlargement of this valuable instrument, so as to measure an angle of 120°, was first proposed by Captain Campbell in 1757.[47]

On the 6th February 1672, Sir Isaac communicated to Mr. Oldenburg his "design of a microscope by reflexion, which should have, instead of an object-glass, a reflecting piece of metal, and which seemed as capable of improvement as telescopes, and perhaps more so, because but one reflective piece of metal is requisite in them." This microscope is shown in the annexed diagram, copied from the original, where A B is the object-metal, C D the eye-glass, F their common focus, and O the other focus of the metal in which the object is placed. This ingenious idea has been greatly improved in modern times by Professor <243> Amici, Professor Potter, and Dr. Goring,[48] who make A B a portion of an ellipsoid, whose foci are O and F, and who fix a small plain speculum between O and A B, in order to reflect into the speculum the object which is placed on one side A P, for the purpose of being illuminated.[49]

In another letter to Mr. Oldenburg, dated July 11th in the same year, he suggests an improvement of microscopes by refraction, "which I do," he says, "more willingly, because Mr. Hooke hath made such excellent use of that instrument; and I shall be glad to contribute any thing to your promotion of these his ingenious endeavours, or add to his inventions of that kind. The way is, by illuminating the object in a darkened room with light of any convenient colour not too much compounded; for by that means the microscope will, with distinctness, bear a deeper charge and larger aperture, especially if its construction be such as I may hereafter describe."[50] This happy idea we have some years ago succeeded in realizing, by illuminating microscopic objects with the light of a monochromatic lamp, which discharges a copious flame of pure yellow light of definite refrangibility.[51] Since the time of Newton, the microscope has undergone the greatest improvement, — the single microscopes made of diamond <244> and the other precious stones, — the microscopic doublets, and the magnificent compound microscopes of Ross, Powell, and Nachot fitted up as polarizing microscopes.

In order to remedy the evil of want of light in his reflecting telescope, arising from the weak reflecting power of speculum metal, and from its tarnishing by exposure to the air, Sir Isaac proposed to substitute for the small oval speculum a triangular prism of glass or crystal A B C, Fig. 19. Its side A B b a he supposes to perform the office of that metal, by reflecting towards the eye-glass the light which comes from the concave speculum D F, Fig. 20, the light reflected from which he supposes to enter into this prism at its side C B b c, and lest any colours should be produced by the refraction of these planes, it is requisite that the angles of the prism at A a and B b be precisely equal. This may be done most conveniently, by making them half right angles, and consequently the third angle at C c a right one. The plane A B b a will reflect all the light incident upon it, "especially if the prism be made of crystal;" but in order to exclude unnecessary light, it <245> will be proper to cover it all over with some black substance, excepting two circular spaces of the planes A c and B c, through which the useful light may pass. The length of the prism should be such that its sides A c and B c may be "four square," and so much of the angles B and b as are superfluous ought to be ground off, to give passage for as much light as is possible from the object to the speculum.

One great advantage of this prism, which cannot be obtained from the oval metal, is that without using two glasses, the object may be erected, and the magnifying power of the telescope varied at pleasure, by merely varying the distances of the speculum, the prism, and the eye-glass. This will be understood from Fig. 21, where A I represents the great concave speculum, E F the eye-glass, and B C D the prism of glass, whose sides B C and B D are not flat, but spherically convex. The rays which come from G, the focus of the great speculum A I, will, by the refraction of the first side B D, be reduced to parallelism, and after reflexion from the base C D, will be made by the refraction of the next side B C to converge to the focus H of the eye-glass E F. If we now bring the prism B C D nearer the image at G, the point H will recede from B D, and the image formed there will be greater than that <246> at G; and if we remove the prism B C D from G, the point H will approach to B C, and the image at H will be less than that at G. The prism B C D performs the same part as a convex lens, G and H being its conjugate foci, and the relative size of the images formed at these points being proportional to their distance from the lens. These different contrivances were suggested by some criticisms upon his reflecting telescope by M. Auzout; and Newton does not seem to have executed them, as he recommends "that the first trials be made with prisms whose sides are all of them plane."[52] As more than one-half of the light is lost by reflexion from the small mirror, we have proposed to substitute for it an achromatic prism to refract the rays to the side of the tube.[53] An advantage would be gained by the use of a plane speculum of silver, which reflects much more light than speculum metal. The objection to reflecting prisms arises from the imperfection of the glass, and the difficulty of obtaining three perfectly flat surfaces, and two angles perfectly equal. This construction would be a good one for varying optically the angular distance of a pair of wires placed in the focus of the eye-glass E F; and by bisecting the lenticular prism B C D, and giving the halves a slight inclination, we should be able to separate and to close the two images or discs which the two halves would produce, and thus form a double image micrometer.

In concluding our account of Newton's optical discoveries, some notice of the principal work which contains them will suitably terminate the present chapter. This work, entitled Opticks, or a Treatise on the Reflexions, Refractions, Inflexions, and Colours of Light, was pub <247> lished in London, without a date, on the 16th February, 1704. Newton, from the President's chair, presented it to the Royal Society. Dr. Halley was desired to peruse it and make an abstract of it, and the thanks of the Society were given to the author "for the book, and for being pleased to publish it." In the second edition, with the date of July 16, 1717, the date of April 1, 1704, is added to the advertisement of the first edition, a step of which, as Mr. Edleston observes, "the dispute with Leibnitz had probably taught our philosopher the importance."[54]

In the advertisement to the first edition, we are informed by the author, that "a part of the ensuing discourse about light was written at the desire of some gentlemen of the Royal Society in the year 1675, and then sent to their Secretary and read at their meetings, and the rest was added about twelve years after, to complete the Theory, except the third book, and the last proposition of the second, which were since put together out of scattered papers. To avoid being engaged in disputes about these matters, I have hitherto delayed the printing, and should still have delayed it, had not the importunity of friends prevailed upon me. If any other papers writ on this subject are got out of my hands, they are imperfect, and were perhaps written before I had tried all the experiments here set down, and fully satisfied myself about the laws of refractions and composition of colours. I have here published what I think proper to come abroad, wishing that it may not be translated into another language without my consent." In the advertisement to the second edition, which appeared in 1717, he mentions that he could have added at the end of the third book some <248> questions, (namely, the thirty-one celebrated queries;) "and," he adds, "to shew that I do not take gravity for an essential property of bodies, I have added one question concerning its cause, choosing to propose it by way of a question, because I am not yet satisfied about it for want of experiments."

At the request of Newton, Dr. Samuel Clark prepared a Latin edition of his Optics, which appeared in 1706, and he was generously presented by Sir Isaac with £500, or £100 for each of his five children, as a token of the approbation and gratitude of the author. Demoivre is said to have secured and taken charge of this translation, and to have spared neither time nor trouble in the task. Newton met him every evening at a coffee-house,[55] and when they had finished their work, he took Demoivre home with him to spend the evening in philosophical conversation.[56] Both the English and the Latin editions have been frequently reprinted, both in England and on the Continent, and perhaps there never was a work of profound science more widely circulated.[57]

The only other optical work by Newton was his Lectiones Opticæ, a course of lectures on optics, which he read as Lucasian Professor in the public schools of the University of Cambridge in the years 1669, 1670, and 1671. It was not published till after his death; — an English edition in 1728, in octavo,[58] and the Latin original in 1729 in quarto.

This valuable work is divided into two parts, and contains many beautiful propositions, and interesting and in <249> structive experiments, which are not to be met with in any modern treatise on optics.

In the first part, which is entitled, On the Refraction of the Rays of Light, he treats in four sections: — 1. Of the different refrangibility of the rays of light; 2. Of the measure of refractions; 3. Of the refractions of plane surfaces; and 4. Of the refractions of curved surfaces.

In the second part, which is entitled, On the Origin of Colours, he treats in five sections : — 1. On the doctrine of colours, and its proof by experiments with the prism; 2. On the various phenomena of colours, and on the phenomena of light thrown upon a wall by the prism; 3. On the phenomena of light received in the eye from a prism; 4. On the phenomena of light transmitted through a refracting medium terminated by parallel planes; and, 5. On the phenomena of light transmitted through media terminated spherically, and on the rainbow.[59]

The manuscript from which the Latin edition was printed, was that which had been given by Newton himself to David Gregory, Savilian Professor of Astronomy at Oxford; but after the edition had been printed, the editor learned that a more perfect manuscript, containing several corrections and emendations in Newton's own handwriting, had been preserved in the archives of the University of Cambridge. These emendations, occupying five quarto pages, were therefore printed at the end of the work, and we observe that Bishop Horsley has introduced them into the text in the third volume of his edition of Newton's works.

[1] Part iii., Prop. viii. ix., &c.

[2] Probably Sulphate of Barytes.

[3] Optics, Book ii., Part iii., Prop. x.

[4] See Transactions of the Geological Society, 2d Series, vol. iii. p. 455; and North British Review, vol. xviii. p. 227.

[5] Journal Book of the Royal Society.

[6] It was published in the Phil. Trans. 1671, p. 2039.

[7] Traité de Lumière, chap. v. p. 57 ; and Maseres' Scriptores Opticæ, p. 234.

[8] Query 25th and 26th at the end of the work.

[9] The term unusual, and the ratio of the sines, viz. 5 to 3, were given by Bartholinus in the abstract of his Paper in the Phil. Trans., No. 67, Jan. 1670-1, pp. 20, 39.

[10] Traité de Mineralogie, tom. i. p. 159, Note.

[11] Hauy's Elements of Nat. Phil., by Gregory, vol. ii p. 337.

[12] See Appendix, No. III.

[13] Correspondence, &c. pp. 264-273. From the originals in the British Museum, Add. MSS., 4237, fol. 32 and 34.

[14] Hooke's Collections, March 1682, No. 6, p. 167.

[15] The Thalami Nervorum Opticorum.

[16] Briggs considers this muscle necessary to prevent squinting, by "keeping the eye even and in sight." — Hooke's Coll., March 1682, p. 170.

[17] Dated Trin. Coll. Cambridge, September 12, 1682. Appendix, No. IV.

[18] Descartes himself distinctly states that we see objects single with two eyes in exactly the same way as we feel objects single with two hands, forgetting that we see them double by the displacement of the coincident images, and never feel them double by the two hands. See Descartes' Dioptrice, cap. 6, De Visione, Art. X. The experiment of feeling a pea double between two fingers, is not hostile to this observation.

[19] This is precisely the theory of Rohault, see p. 229.

[20] This letter contains, as will be seen in the Appendix, No. IV., a paragraph respecting the opinions of a Mr. Sheldrake, who, as Mr. Edleston informs us, was a Fellow of Corpus Christi College, and seven years senior to Newton. Mr. Sheldrake <224> states that vision is more distinct when the eye is directed to the object, than when the object is above or below the optic axes. I do not recollect that this curious fact has been stated by any previous writer on vision.

[21] See Appendix, No. V.

[22] Dated Cambridge, May 1685.

[23] The one the Theory of Vision, and the other his Ophthalmographia. Cantab. 1676, and Lond. 1687.

[24] See Appendix, No. VI.

[25] See Appendix, NO. VII.

[26] Although it is evident, from a careful perusal of the 15th Query, that it contains the same doctrine of the semi-decussation of the optic nerves which is given in the MS., yet it has been misunderstood by Dr. Reid, who obviously had not seen the copy of it in Harris's Optics. "Sir Isaac Newton," says Dr. Reid, (Inquiry, cap. vi. sect. 13), "who was too judicious a philosopher and too accurate an observer to have offered even a conjecture which did not tally with the facts which had fallen under his observation, proposes a query with respect to the cause of it, (namely, the relation and sympathy between corresponding points of the two retinæ.)" — Optics, Query 15. Dr. Reid seems not to have detected the doctrine of semi-decussation in the Query, and to have believed that individual nerves, not half-nerves, from the two sides of both eyes, united before they reached the brain, and there produced a joint and single impression; and Dr. Alison has either taken up Dr. Reid's opinion, or misunderstood the Query, and also the theory of semi-decussation. "It is well-known," he says, "that an explanation (of single vision by means of double images) was proposed by Newton, fully considered by Reid, and since supported by Wollaston, (often called the theory of Wollaston, but quite incorrectly,) proceeding on the supposition of a semi-decussation of the human optic nerves at their commissure, whereby the fibres from the right half of the retina go to the right optic lobe in the brain, and vice versa." This is the theory of Rohault, and not of Newton and Wollaston, in which the half-fibres, from the right half of the retina of each eye, unite into one fibre at their commissure GH in Fig. 12, and then go to the right optic lobe.

[27] Sir Isaac draws other four conclusions from his theory, but they will find a fitter place in the Appendix, No. VIII.

[28] A Latin translation of Rohault's work was published in 1708, by Dr. Clarke, "with annotations chiefly from the philosophy of Newton, and yet no notice is taken of Newton's Theory, as contained in his l5th Query, although Dr. Clarke had translated the Optics into Latin. He adds a note stating, that the conjecture <230> respecting the fibres of the optic nerve had not yet been confirmed by dissection. Part I. cap. 31, p. 225, note.

[29] The suffusio dimidians of other authors

[30] See Transactions of the Medical and Physical Society of Calcutta, vol. ii. p. 151; or Edinburgh Journal of Science, July 1828, vol. ix. p. 143.

[31] Wagner's Handwörterbuch der Physiologie, vol. iii. part ii. p. 297.

[32]

If by the sense of touch we could make the two images appear one, then we should also see an object single when it is doubled by looking either at a nearer or a more distant object, or when it is made 100 by a multiplying glass; but if a man were to live a 1000 years, he would still see the two or the hundred images, though he knew there was only one object. In order to illustrate his opinion, Dr. Brown says that the two English words he conquered, excite the same idea as the one Latin world vicit. In reply to this Dr. Whewell says, "that to make this pretended illustration of any value, it ought to be true that when a person has thoroughly learned the Latin language, he can no longer distinguish any separate meaning in he and in conquered." With this assertion we cannot concur. The two words he conquered un <232> doubtedly convey the same meaning as vicit. If we unite the two words thus, heconquered or conqueredhe, we cannot doubt that the word he is as truly included in the termination it of vicit, as he is in the single word heconquered, unless it is alleged that vicit may also mean she conquered.

Dr. Brown's real mistake consists in not taking two exactly similar words, as vicit, vicit, like what he considers as the two exactly similar images. The two words pronounced in succession convey certainly only one idea, but the mind recognised the same in succession or its duplicity, just as it would do the two similar and united images, if one of them were slipped from its superposition on the other by pressing aside one of the eye-balls.

Dr. Brown's views are affected with another error, namely, in the assumption that the pictures in each eye are exactly similar.

[33] Edinburgh Transactions, vol. xiii. p. 479.

[34] There is no opposition between the impressions on the concave retina and on a concave surface of the body. If we hold up the hand vertically, and bend it into a concavity, an impression made on the upper part of the concavity, will be felt as coming from below, and an impression on the lower part of the concavity will be felt as coming from above, exactly as in the case of the concave retina.

[35] We have not noticed the additional explanation adopted by Dr. Alison, "that impressions on the upper part of the retina are impressions on the lower part of the optic lobes, i.e., of the sensorium;" because he has not told us what requires as much explanation as inverted vision, namely, why the lower part of the sensorium makes the object seem lower! Is the sensorium a plane, or a convexity, or a concavity ? If it is a concavity, a physical impression on the lower part will correspond to the top of the object, and an impression on the upper part with the bottom of it.

[36] I omit all consideration of the question, whether the choroid coat or retina is the seat of vision, or whether the foramen centrale is or is not an opening in the retina.

[37] See Edinburgh Transactions, vol. xv. p. 360.

[38] When a ray falls obliquely upon the retina (or any other surface of sensation) its action may be decomposed into two, the one lying in the surface of the membrane, and acting laterally upon the papillæ, and the other perpendicular, and acting in the direction of the axis of the papillæ, and therefore passing to the brain.

[39] See Edinburgh Transactions, vol. xv. pp. 350-353, and North British Review, vol. xvii. p. 165.

[40] Experiments and Considerations touching Colours, chap. ii. § 9, p. 19. Lond. 1664.

[41] King's Life of Locke, vol. i. pp. 404-408. Edit. 1830.

[42] Art. Accidental Colours, in the Edinburgh, Encyclopædia, vol. i. pp. 91, 92.

[43] See Phil. Trans., 1742-43, vol. xlii. p. 155.

[44] Phil. Trans. 1731, p. 147.

[45] Sprot's Hist. of the Royal Society, p. 246. Lond. 1667.

[46] The Posthumous Works of Robert Hooke, M.D., p. 503, tab. xi. fig. 2. Lond. 1705. In the description given of it by Waller, his biographer, the invention is mentioned as "an instrument for taking angles at one prospect, which he found described on a loose paper."

[47] Grant's Hist. of Physical Astronomy, p. 487; and Nautical Mag. vol. i. p. 351.

[48] See Edinburgh Journal of Science, vol. vi. p. 61; Encyclopædia Brit., Art. Microscope, vol. xv. p. 41.

[49] Newtoni Opera, tom. iv. p. 300.

[50] Sir Isaac does not seem to have afterwards described this construction.

[51] See Edinburgh Transactions, vol. ix. p. 433; and the Edinburgh Journal of Science, July 1829, No. I. new series, p. 108.

[52] See Newtoni Opera, tom. iv. p. 276.

[53] Treatise on Optics, edit. of 1853, p. 404.

[54] It is a curious fact, that "there is the same peculiarity about the preface to the Principia{.}" — Edleston's Correspondence &c. &c., pp lviii and lxxi.

[55] " Probably Slaughters' Coffee-house in St. Martin's Lane." — Edleston's Correspondence, p. lxxiv.

[56] Eloge, by Fontenelle. — Mém. Acad. Par. 1727. Hist. p. 151.

[57] The English edition was reprinted at London in 1717, 1721, and 1730, and the Latin one at London in 1719, 1721, 1728, at Lausanne in 1740, and at Padua in 1773.

[58] Biographia Brit. Art. Newton, vol. vii. p. 779.

[59] An analysis of the Lectiones Opticæ has been given by the author of the Life of Newton in the General Dictionary, vol. vii. p. 779, note; but it is by some mistake confined to the first Part, as if there were no second Part. The same mistake is committed in the Biographia Britannica, vol. v. p. 3215, note, where it is obvious that the author knew nothing of the second Part, as he calls the last portion of the first Part the "Last Section of these Lectures."


text:
# SSD...WOW

#### spec

##### Well-Known Member
Last week at night, it rained after few months of winter. Next morning I fell down twice from motorbike. First time I almost got squeezed my balls from leg-guard and second time I got a hole in my pant near the knee. There was a truck sleeping upside down near the road too.
Hope you didn't get hurt too bad and are not speaking with a high voice after the ball squeeze. Also hope they were not your best pants. "truck sleeping upside down" --wonderful term

Motor bikes: great things, I love 'em, but bloody dangerous- just two bits of rubber the size of footprints keeping you from disaster. I had some incidents on bikes too:

It was January 1964, and the weather was quite mild after a cold snap. I was on a 150 mile trip doing around 40 MPH around a bend in the country, near Cambridge UK. Ahead was a long row of people waiting for a bus. Right away I saw the glint of the sun reflected from a long sheet of ice on the road in front of them. There was nothing I could do, so i just hung on. I can remember noticing the peoples legs passing by as the bike slid down the road on its tank and my foot. Finally the bike stopped at the last chap in the queue who said, "I enjoyed that. Can you do it again?"

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#### atferrari

##### Well-Known Member
The message is to regularly back them up (just like i haven't yet).
spec
After loosing all my data reports, laboral and administrative, equivalent to 6 months, I became serious about that.

I recently read that no matter the disk, failures occur always the day before the planned back up.

#### spec

##### Well-Known Member
After loosing all my data reports, laboral and administrative, equivalent to 6 months, I became serious about that.

I recently read that no matter the disk, failures occur always the day before the planned back up.
Yes, too true. You can get very blase about backing your stuff. I have just spent a week on and off recovering my Outlook stuff when Google Gmail IMAP threw a wobbly.

The worst time for me though was in the early IBM PC days. We had been working on a proposal for the MOD for around a month and it had not gone well; often we worked 18hrs straight. The day before the deadline for submission we found another problem and I got the job of doing a fix. At about 2am I had finally finished and went to get a coffee from the vending machine down the hallway. While I was at the vendor there was a power cut which lasted half an hour. About four hours worth of data was lost when my PC was rebooted. That taught me a big lesson and since then I have been very strict about saving and backing up...when I can be bothered that is

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All , Thanks for the backup reminder..... not that my stuff is worth "thousands of $" or even pennies ! I have a small freecom (1x250G) drive, but 3 PC's..... will do it tomorrow ... #### strantor ##### Active Member Just one word of caution though: although SSDs have no moving parts and are reasonably immune to shock and vibration, they are inherently error prone things at the silicon level and should not be used as the only store for any valuable data. It is only by some very sophisticated error correcting algorithms that they are usable. Wow, I had no idea. I learn something new every day! Actually, I think my subconscious demands that I learn something every day. That would explain my insomnia. The message is to regularly back them up (just like i haven't yet). spec Thanks for the reminder! I have that laptop set to automatically back up to my NAS every time it connects to my home network. But my NAS isn't connected right now; hasn't been in months, since I moved into this new house. Still in a box somewhere. I really need to exhume it and use it. #### spec ##### Well-Known Member Most Helpful Member Wow, I had no idea. I learn something new every day! Actually, I think my subconscious demands that I learn something every day. That would explain my insomnia. Thanks for the reminder! I have that laptop set to automatically back up to my NAS every time it connects to my home network. But my NAS isn't connected right now; hasn't been in months, since I moved into this new house. Still in a box somewhere. I really need to exhume it and use it. Yes, it came as a surprise to me too. The adage is always legislate for the worst. So with a NAS for example you have each HDD shadowed and you should also store you data not just in a different location but in a different country on the cloud. I used to think that all this was on the extreme side, but when our house was destroyed by a gas explosion, the shock wave, or heat wrecked most electronic equipment- the items looked ok but malfunctioned in odd ways. Now I am dead careful and think about backing up properly, but somehow never get around to doing it. I don't know how much data you have, but my laptop takes around 24Hrs to back up, which is a real pain. CORRECTION (2016_02_12) 12 Hrs not 24 Hrs Last edited: #### strantor ##### Active Member I don't know how much data you have, but my laptop takes around 24Hrs to back up, which is a real pain. Wow, That's just silly! I don't know exactly how much time it takes because: A. It's been months since I did it. B. Back when I was doing it regularly, I was doing incremental backups; only modified files were synced, so it didn't take long. But IIRC (I probably don't), I thin my first bacup took 4-6hrs. That was about 100GB, over my home hardwired gigabit lan, and it might have been with my old HDD, not sure. If I did it over 10/100Mbit lan or wifi, it might take 24hrs though. Across what type of network are you doing yours? #### spec ##### Well-Known Member Most Helpful Member Wow, That's just silly! I don't know exactly how much time it takes because: A. It's been months since I did it. B. Back when I was doing it regularly, I was doing incremental backups; only modified files were synced, so it didn't take long. But IIRC (I probably don't), I thin my first bacup took 4-6hrs. That was about 100GB, over my home hardwired gigabit lan, and it might have been with my old HDD, not sure. If I did it over 10/100Mbit lan or wifi, it might take 24hrs though. Across what type of network are you doing yours? Silly is a word that worries me My Lenovo T520 laptop has: 500 GB SSD as C: ITB HDD in the Ultra Bay as E: It currently has the following large file groups, in addition to the other usual stuff: 72 GB of images 198 GB of edocs 362 GB of music I also have less immediate data on USB HDDs Why not have this on a NAS or on the cloud? Well I have tried that but it didn't work. Often when visiting in the country or on site somewhere there is no internet coverage so you would be stuck. Our house NAS averages 10MB sec and a system image back up using MS Backup takes around 12 Hrs- I previously said 24 hours but that was an error. I have used Acronis back up in the past but didn't get on with it. Incremental back ups are attractive at first sight but, in my very limited experience, they can go wrong, and when they do, they go wrong in a big way. In reality, backing up is a chore and I have not learned about it sufficiently or configured it properly, mainly due to lack of discipline and lack of interest. My son, runs and hires out servers and is horrified by my caviler attitude and it is only because of his constant nagging that I have a NAS and do backups now and then. One approach that I have found useful is to put all your user files: images, edocs, music, Word, Outlook .ost, etc etc in one directory and back that from time to time as straight files rather than an image. That approach has got me out of a pickle on a number of occasions. Note that you must ensure that the file paths, including file name, do not exceed 255 characters or the file will not be saved and you will get an error message. I think there is a way to fix this shortcoming, but I am not sure. It was great at work: all the housekeeping stuff was done by system admin and the whole thing was backed up every night- trouble is that made me lazy. Last edited: #### spec ##### Well-Known Member Most Helpful Member All , Thanks for the backup reminder..... not that my stuff is worth "thousands of$ " or even pennies ! I have a small freecom (1x250G) drive, but 3 PC's..... will do it tomorrow ...
Yes, very wise- the unexpected, once in a thousand year disaster happens regularly, in my experience. Your data may not be worth much in absolute terms, but I bet it is worth a fortune to you.

One worrying scenario would be if your laptop/PC was stolen. I had a big scare a few months ago. I was in town with my laptop and left it in a cafe by accident. I didn't realise for about 4 hours. Amazingly, when I went back to the cafe the laptop was still under the table where I left it. I did a back up that night.

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#### spec

##### Well-Known Member
My aging battery wouldn't let the laptop run more than ~45min before; now with the SSD it's got better battery life than it did new.
That is a good point that replacing a HDD with an SSD extends your battery life. The other big power gobbler is your display. If you turn the brightness down you can extend battery endurance quite a bit.

Incidentally, when a laptop battery loses capacity, or fails outright, it is often caused by one or two individual battery cells in the pack. It is possible to open up battery packs and either replace the faulty cells (18650), or salvage the good cells

Charging cells in series is never a good idea and sometimes you can discharge and charge a seemingly dud cell and it will recover. Why they do not charge the cells individually in battery packs, I just can't understand

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#### atferrari

##### Well-Known Member
One worrying scenario would be if your laptop/PC was stolen.
Maybe two years ago, I think it was in Facebook (??), the story of a student who lost ALL his data for the final tesis colected along several years. He was literally begging for it not for the stolen laptop. Awful.

EDIT
http://www.freerepublic.com/focus/news/3013002/posts

EDIT/

#### spec

##### Well-Known Member
Maybe two years ago, I think it was in Facebook (??), the story of a student who lost ALL his data for the final tesis colected along several years. He was literally begging for it not for the stolen laptop. Awful.

EDIT
http://www.freerepublic.com/focus/news/3013002/posts
Hola atferrari,

Hell, poor fellow. I suppose it is all too easy to say that he only has himself to blame.

There have been some big stinks in the UK when government officials have lost/had stolen laptops with highly confidential data, which amazingly, was not encrypted. With a few exceptions, I think most individuals, including me, are a wide open to loss of data.

By the way, I understand that Veracrypt is one of the best free encryption packages available at the moment.

My feeling is that back up facilities lag well behind what is technically possible. I don't know if this is feasible but I imagine an environment where backing up is continuous and automatic and part of the operating system, say with a processor solely dedicated to the task, so that the main processing functions are not hobbled.

spec

PS: I just noticed your tag line: In theory, there is no difference between theory and practice. In practice, however, there is.

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#### spec

##### Well-Known Member
I have that laptop set to automatically back up to my NAS every time it connects to my home network.
Hi strantor,
Intresting- feel like explaining how that works. Does the auto backup slow your machine noticeably. Do you back up an image or files. As I said before I don't know much about backing up.

spec

##### Active Member
Re Backup stories, (1) In the 198o's ( if my is memory 1/2 right ) we had a customer,(several employees) a sound lab using a PC with DOS and 2 x 5 1/4" floppy drives. ( that's all) mainly because it had a 16 port I/O board , one black day , it failed to boot, also backup failed to boot, drive had screwed both disks. mainly because both floppies were years old. I had a FE collect disks we and managed to recover some data, seems the whole business was surviving on a few microns of iron oxide......
(2) Large UK retail business , in-store office / stock system . back up done after store closing to removable HDs, system and backup disks then put in the safe ( with money ) , office goes down below zero degrees during the night. Next morning disks removed from safe , placed on drive....... power applied .... very nasty pinging sound ... disks crashed ... happened in several stores, not good , on inspection , water had condensed on HD aluminium platters. (this was only discovered after several such happenings ) not sure who got the blame..

#### spec

##### Well-Known Member
Re Backup stories, (1) In the 198o's ( if my is memory 1/2 right ) we had a customer,(several employees) a sound lab using a PC with DOS and 2 x 5 1/4" floppy drives. ( that's all) mainly because it had a 16 port I/O board , one black day , it failed to boot, also backup failed to boot, drive had screwed both disks. mainly because both floppies were years old. I had a FE collect disks we and managed to recover some data, seems the whole business was surviving on a few microns of iron oxide......
(2) Large UK retail business , in-store office / stock system . back up done after store closing to removable HDs, system and backup disks then put in the safe ( with money ) , office goes down below zero degrees during the night. Next morning disks removed from safe , placed on drive....... power applied .... very nasty pinging sound ... disks crashed ... happened in several stores, not good , on inspection , water had condensed on HD aluminium platters. (this was only discovered after several such happenings ) not sure who got the blame..
Araarg- the number of times that caution has led to disaster. What is the saying: The road to hell is paved with good intentions.

Floppy disks with their massive 360 Kbytes, followed by the incredible 1.4 M bytes: seems hard to believe now.

We had some IBM HDDs in a system at work which were massive and weighed a ton. Each platter was an immense 1M Byte. They regularly crashed and the software boys used the scrap disks as office decoration.

The original consumer HDD were troublesome things too; nothing like the fast, ultra reliable, high capacity types we have now. I read in a science mag that when the research boys sort out molecular memory that capacity will be practically infinite, access will be fast, and retention permanent.

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#### strantor

##### Active Member
I have used Acronis back up in the past but didn't get on with it.
My feeling is that back up facilities lag well behind what is technically possible. I don't know if this is feasible but I imagine an environment where backing up is continuous and automatic and part of the operating system, say with a processor solely dedicated to the task, so that the main processing functions are not hobbled.
You quit Acronis too soon.
Charging cells in series is never a good idea and sometimes you can discharge and charge a seemingly dud cell and it will recover. Why they do not charge the cells individually in battery packs, I just can't understand
I thought they do charge the cells individually. Or at least I assumed that. The battery tab on all the laptops I've had, had like 20 little tabs, I assumed which were part of a per-cell charging circuit; or at least a per-cell monitoring circuit.

Hi strantor,
Intresting- feel like explaining how that works. Does the auto backup slow your machine noticeably. Do you back up an image or files. As I said before I don't know much about backing up.
I use Acronis. (see how here, pg 44). The auto backup does not slow me down that I can remember, but I'm usually not using it at the time when it's backing up. I do an incremental image.

Through the years I have noticed some characteristics of engineering types, not just electrical/electronic, cropping up time and time again:
Here, I'll self-identify, and re-weight the list as it applies to me:
(14) Inquisitive
(17) Independent / (18) Self motivated
(16) Inventive
(xx) Practical (function over fashion)
(12) Horder
(xx) Small spectrum of emotion
(4) Migraine
(xx) Quiet, and Odd apparently, to others
(1) Some sleep problems, ALWAYS
(xx) strong willed
(15) Industrious
(3) Digestive problems.
(19) Good sense of humor
(10) Pedantic or perhaps particular is a better word
(11) Think that females (or other people in general) are also logical
(2) Worrier

Not applicable:
(13) Love of animals, especially cats
(3) Hay-fever and general allergies. Alopecia.
(5) Left handed
(6) Dyslexic
(7) Difficult spelling
(8) Uneasy punctuation and grammar
(9) Difficult second languages

#### spec

##### Well-Known Member
You quit Acronis too soon.

I use Acronis. (see how here, pg 44). The auto backup does not slow me down that I can remember, but I'm usually not using it at the time when it's backing up. I do an incremental image.
Thanks for the info strantor. I obviously have some reading to do.

I thought they do charge the cells individually. Or at least I assumed that. The battery tab on all the laptops I've had, had like 20 little tabs, I assumed which were part of a per-cell charging circuit; or at least a per-cell monitoring circuit.
Hmm, I will have to look into that. I must admit that the last time I messed with battery packs from laptops, power tools, vacuum cleaners etc was about 10 years ago and the equipment was even older. Those batteries were charged in series, either 1 stack of four battery cells or two stacks of four in parallel. The latter is more common on laptops now and is known as 4S:2P, if I remember correctly.

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# Transferring force through a string, including gravity

I couldn't think of a better title to describe this confusion that I have.

Say you pick up a string and pull on both ends so that it is rigid. The string is a collection of an infinite number of objects. For simplicity, assume there are $n$ objects for some large integer $n$, each of which has mass $m$, and number the objects $A_1$, $A_2$, ..., $A_n$ from left to right. Maybe we can take the limit as $n$ goes to infinity; I don't know how to rigorize this.

Your left hand exerts some force $\vec{F}$ onto $A_1$. The only other two forces on $A_1$ are gravity and the force $\vec{F}_{21}$ from $A_2$, since $A_1$ is only in contact with $A_2$. Gravity is $m\vec{g}$. Since the string is not moving, the sum of the forces on $A_1$ is $\vec{0}$, so
$$\vec{F}_{21}+\vec{F}+m\vec{g}=\vec{0}$$ $$\vec{F}_{21}=-\vec{F}-m\vec{g}$$

By Newton's Third Law, the force $\vec{F}_{12}$ from $A_1$ onto $A_2$ equals $-\vec{F}_{21}$, which is $\vec{F}+m\vec{g}$. The only other two forces on $A_2$ are $m\vec{g}$ and the force $\vec{F}_{32}$ from $A_3$ onto $A_2$, so
$$\vec{F}+m\vec{g}+m\vec{g}+\vec{F}_{32}=0$$ $$\vec{F}_{32}=-\vec{F}-2m\vec{g}$$

If we keep going, we add an $m\vec{g}$ each time, until we get that the force from our right hand onto $A_n$ is $-\vec{F}-nm\vec{g}$, which is a lot more than $\vec{F}$, not to mention that $n$ is supposed to go to infinity.

What is wrong with my logic? Do my hands somehow exert $-m\vec{g}$ on each object?

• Why do you think there is a problem? The force on one end equals the force on the other end plus the total weight of the string (which I presume is vertical, so that applied force and weight acts in the same line). – sammy gerbil Aug 22 '16 at 0:11
• @Lelouch: Why are the $m\vec{g}$ vectors different? – user127886 Aug 22 '16 at 0:25
• @sammygerbil: The vector sum of the forces on each end is the force on the string due to gravity, right? – user127886 Aug 22 '16 at 0:32
• If the string is not accelerating, the vector sum of forces at each point on the string is zero. – sammy gerbil Aug 22 '16 at 0:38
• Keep in mind that the sting has a fixed mass, so if you divide it into $N$ identical pieces each piece has mass $m/N$. Then make sure to treat the forces as vectors and you can cover all the bases. – dmckee --- ex-moderator kitten Aug 22 '16 at 4:35

Since a picture sometimes says more than a thousand words:

The first example in the upper left is just a mass hanging on a fixed point(/your hand). Gravitation pulls downwards with $m*\vec{g}$ and therefore the fixed point needs to exert a force $-m*\vec{g}$ on the mass.

The second example in the upper right are multiple masses hanging on each other. Here it is easiest to start from the lowest one (1). Gravitation pulls downwards, so something needs to pull with the same force upwards. And the only possible force here is $F_{21}$ from the connected mass (2). Looking at mass (2) we have gravitation ($m*\vec{g}$), the force from (1) ($m*\vec{g}$) and the force from (3), which therefore needs to be $-2*m*\vec{g}$.
The same principle applies to mass (3), so overall the fixed point/hand needs to exert a force of ($3*m*\vec{g}$) on the rope/chain of masses. Which is pretty clear, because thats the weight of the chain.

In the middle of the picture you see an example of a mass hanging on two fixed points (with massless ropes). We have gravitation pulling downwards with $m*\vec{g}$, so each of the ropes needs to exert a vertical component of $-m*\vec{g}/2$. They might have additional horizontal components $F_{a,h}$ and $F_{b,h}$. In order to have a static system of course it needs to be $F_{a,h} = -F_{b,h}$.
So with each of your hands you would have to exert an overall force of $F_h + m*\vec{g}/2$.
Note that the vertical components again add up to the weight of the 'chain'.

The last example is the trickiest one and the one you were looking for. Leaving aside the horizontal forces which are all equals (besides a sign), it is again the easiest to start in the middle. The gravitational force on mass (2) is 'split' between the two neighbouring masses (1) and (3). Each of them kind of 'adds' its own mass to the overall force needed to hold the system static, until we are finally left with two times $\frac{3}{2}*m*\vec{g}$ which is again the weight of the mass chain.

So basically yes, your hands exert a force $-n*m*\vec{g}$. If they would not, the rope would just fall down. Or better, the rope falls downwards until they actually do, which is the reason why no rope can ever be totally straight.
This vertical force is propagated through the chain of masses, but for every mass one portion $m*\vec{g}$ is 'lost' in order to keep this small mass from falling.

If you divide the string into $n$ parts then each part has mass $m/n$, where $m$ is mass of string. So beginning with one end of the string and proceeding with your argument, by the time you reach the other end the force will be $-\textbf{F}-n\times (m/n)\textbf{g}=-\textbf{F}-m\textbf{g}$, which is as it should because when you consider the string in its entirety, given that it is stationary, vector sum of forces on it must be zero: $\textbf{F}_{one~end}+\textbf{F}_{other~end}+m\textbf{g}=\textbf{0}$.


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# Re: Embellished operators

From: Neil Soiffer <NeilS@dessci.com>
Date: Thu, 23 Feb 2012 22:47:05 -0800
Message-ID: <CAESRWkCOENHgD9kdyB43fQWk5YMGt=_jkx7qBAYb9_+poXO2TA@mail.gmail.com>
To: Frédéric WANG <fred.wang@free.fr>

I hope I'm stating this correctly:  the embellished operator rules are
clear with the exception of the case of an mrow that contains an
embellished operator and space-like elements (the last rule in
http://www.w3.org/TR/MathML3/chapter3.html#id.3.2.5.7.3).

So for your two less trivial vertical stretching rules:

1.  the first should have the arrow stretch to 200px because it is part of
the inner munder. and the other part of the outer mover is smaller.

2.   the second is slightly trickier.  The inner munder's initial width is
100px, but because the arrow is at the "core" and is hence an embellished
operator, laying out the outer mover means that it needs to stretch to
cover the 200px.  Hence this too should be 200px.  This is based on the
following from the spec for *horizontal* stretching:  "...the mo element at
its core, should stretch to cover the width of the other direct
sub-expressions in the given element..."

For the vertical case, it is clear that if there was only one mrow, then
the maximum size would be used (200px).  But that's not the case.  As your
analysis in a subsequent email demonstrated,  the inner mrow should cause
that mrow to be 200px.  But since it should act as an embellished operator,
it is subject to further stretching.

There are two possibilities:
1.  We ignore the stretching caused by the inner mrow and consider the
"normal" height to be be that of the unstretched "|".  In this case, the
answer should be that the result is 100px because it should stretch to the
height f the space in the outer mrow.

2.  We consider the stretched height of "|" caused by the inner mrow's
stretching to be the normal height of the operator.  In this case, the
outer mrow doesn't stretch it further, so the answer is 200px.  If the
outer mrow had a larger stretch than the inner one, than the largest value
should be used.

The spec is silent on what to do here (AFAIK) -- the rule mentioned for
horizontal stretching is not listed as part of the vertical stretching
rules.  I'm leaning towards '1' as being the correct interpretation, so the
result should be 100px.

I haven't looked at the code, but I suspect that MathPlayer chooses the
inner one (even if the outer one is larger) because it doesn't implement
the "mrow whose arguments consist (in any order) of one embellished
operator and zero or more space-like elements" rule.

I tried Mathematica and it behaves like MathPlayer (I didn't implement this
part of the code in MathPlayer, but did in Mathematica).  So both are doing
it wrong (probably because they didn't implement this rule).

Based on what you reported, it sounds like MathJax is doing it wrong under
option '1' above, but correct under '2'.  Firefox sounds is doing the
opposite:  correct under option '1', but wrong under '2'.

If there is a clear consensus on what should happen, we can add an errata
to the spec to clear up this case.

Neil Soiffer
Senior Scientist
Design Science, Inc.
www.dessci.com
~ Makers of MathType, MathFlow, MathPlayer, MathDaisy, Equation Editor ~

On Thu, Feb 23, 2012 at 1:02 AM, Frédéric WANG <fred.wang@free.fr> wrote:

>  Hi all,
>
> I'm thinking again about the rules for embellished operators and it seems
> to me that some elements are particular. For example if we ask how to
> determine the stretching of something like:
>
> $> <mover> > <mo>→</mo> > <mtext>over</mtext> > </mover> >$
>
> The obvious answer is that the arrow should stretch to cover the over
> script. OK. However one can also say that the <mover> is an embellished
> element as a whole. Since is has no siblings, the arrow should have its
> default size.
>
> To give slightly less trivial examples, what should be the size of the
> arrows (100px or 200px?) in these examples:
>
> $> <mover> > <mspace width="100px"/> > <munder> > <mo>→</mo> > <mspace width="200px"/> > </munder> > </mover> >$
>
> and
>
> $> <mover> > <mspace width="200px"/> > <munder> > <mo>→</mo> > <mspace width="100px"/> > </munder> > </mover> >$
>
> An example with vertical stretching rules:
>
> $> <mrow> > <mspace height="50px" depth="50px"/> > <mrow> > <mo>|</mo> > <mspace height="100px" depth="100px"/> > </mrow> > </mrow> >$
>
> (I wonder if an attribute like embellishedop = "false" could help to
> prevent this kind of ambiguity?)
>
> I noticed this because implementing the complete embellished op rules
> caused a regression in Mozilla with MathML code generated by MathJax:
> https://bugzilla.mozilla.org/show_bug.cgi?id=687807
>
>

Received on Friday, 24 February 2012 06:47:44 UTC

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