Chapter VIII: Letter XIX: gives a definition of the ellipsis, which would be a (2)

The length of an undulation, _d_, depends on two things: first, on the promptitude with which the motion is propagated in the fluid; and secondly, the duration of the complete oscillation of the vibrating plane; for the longer this duration, and the more rapid the propagation of the motion, the greater will be the distance to which the first agitation has been extended at the instant of the return of the solid plane to its initial situation. If the oscillations are all performed in the same medium, the velocity of propagation remaining the same, the length of the undulations will be simply proportional to the duration of the oscillations of the vibrating particles from which they originate. As long as the vibrating particles continue to be subjected to the same forces, it follows from the principles of mechanics that each of their minute oscillations will occupy the same time, whatever their extent may be; so that the corresponding undulations of the fluid will continue to be of the same length; they will only differ from each other in the greater or less extent of the elementary vibrations of the particles, which will be proportional to the extent of the luminous particles; for it appears from what has already been stated, that each stratum of the fluid repeats exactly all the motions of the vibrating particle. The greater or less amplitude of the oscillations of the strata of the fluid determines the degree of absolute velocity with which they move, and consequently the energy, but not the nature of the sensation which they excite, which must depend, according to every analogy, upon the duration of the oscillations. It is thus that the nature of the sounds, transmitted by the air to our ears, depends entirely on the duration of each of the oscillations executed by the air, or by the sonorous [p121] body which puts it in motion; and that the greater or less amplitude or energy of the oscillations only augments or diminishes the intensity of the sound, without changing its nature, that is, its tone, or pitch.

The intensity of the light must depend then on the intensity of the vibrations of the ether; and its nature, that is to say, the sensation of colour that it produces, will depend on the duration of each oscillation, or on the length of the undulation, the one of these being proportional to the other. [We find, however, nothing in light of the same colour that is at all analogous to the different register, quality, or _timbre_ of a sound; by which, for instance, the sound of a violin differs from that of a flute in unison with it: the subordinate, or harmonic tones of the sound having nothing in light to correspond with them. TR.]

The duration of the elementary oscillation remaining the same, the absolute velocity of the ethereal particles, at the corresponding periods of the oscillatory motions, is, as we have seen, proportional to its extent. It is the square of this velocity, multiplied by the density of the fluid, that represents what is called the living force in mechanics, or otherwise the energy or impetus of the particles, which is to be taken as the measure of the sensation produced, or of the intensity of the light: thus, for example, if in the same medium, the amplitude of the oscillation is doubled, the absolute velocities will also be doubled, and the living force, or the intensity of the light, will be quadrupled.

We must, however, take care not to confound this absolute velocity of the particles of the fluid with the velocity of the propagation of the agitation. The first varies according to the amplitude of the oscillations; the second, which is nothing but the promptitude with which the motion is communicated from one stratum to the other, is independent of the intensity of the vibrations. It is for this reason, that a weak sound is transmitted by the air with the same velocity as a stronger one; and that the least intense light is propagated with the same rapidity as the brightest. When we speak of the velocity of light, we always speak of the velocity of its propagation. Thus, when we say that light passes through 200 thousand [p122] miles in a second, we do not mean, according to the undulatory system, that such is the absolute velocity of the ethereal particles; but that the motion communicated to the ether employs only a second to pass to a stratum at the distance of 200 thousand miles from its origin.

In proportion as the undulation becomes more distant from the centre of agitation, the motion, spreading over a greater distance, must be weakened in every part of the wave. It is shown by calculation, that the amplitude of the oscillatory motion, or the absolute velocity of the particles concerned in it, is inversely proportional to the distance from the centre of agitation. Consequently, the square of this velocity is inversely proportional to the square of the distance, and the intensity of the light must be inversely as the square of the distance from the luminous point. It must be remarked, that, for the same reasons, the sum of the living forces of the whole undulation remains unaltered; for, on one side the length of the undulation _d_, which may also be called its thickness, is invariable, and its extent of surface augmenting in proportion to the square of the distance from the centre, the quantity, or mass of the fluid agitated, is proportional to the same square: and since the squares of the absolute velocities are diminished in the same proportion as the masses have augmented, it follows that the sum of the products of the masses by the squares of the velocities, that is to say, the sum of the living forces, remains unaltered. It is a general principle of the motion of elastic fluids, that however the motion may be extended or subdivided, the total sum of the living forces remains constant; and this is the principal reason why the living force must be considered as the measure of light, of which the total quantity always remains very nearly the same, at least as long as it continues to pass through perfectly transparent mediums.

It may be remarked, that black substances, and even the most brilliant metallic surfaces, by no means reflect the whole of the light which falls on them; bodies which are imperfectly transparent, and even the most transparent, when of great thickness, absorb also, to use a common expression, a considerable portion of the light that is passing through [p123] them: but it must not be inferred that the principle of living forces is inapplicable to these phenomena; it follows, on the contrary, from the most probable idea that can be formed of the mechanical constitution of bodies, that the sum of the living force must remain always the same, as long as the accelerating forces tending to bring the particles to their natural positions remain unchanged, and that the quantity of living force which disappears in the state of light, instead of being annihilated, is reproduced in the form of heat.

In order to obtain a correct idea of the manner in which the oscillation of a small solid body occasions undulations in an elastic fluid, it has been only necessary to consider a complete oscillation of the solid plane, which produces an entire undulation. If we suppose the oscillations of the plane to be continually repeated, we shall have a series of undulations instead of a single one: and they will follow each other without intermission, provided that the vibrations of the particle first agitated have been regular. Such a series of regular and uninterrupted luminous motions I call a _system of undulations_.

It is natural to suppose, on account of the prodigious rapidity of the vibrations of light, that the luminous particles may perform a great number of regular oscillations in each of the different mechanical situations in which they are placed during the combustion or the incandescence of the luminous body, although these circumstances may still succeed each other in extremely short periods; for the millionth part of a second is sufficient to exhibit, for example, 545 millions of undulations of yellow light; so that the mechanical disturbances, which derange the regular succession of the vibrations of the luminous particles, or which even change their nature, might be repeated a million times in a second without preventing the regular succession of more than 500 millions of consecutive undulations in each state of the particle. We shall soon have occasion to apply this observation to the determination of the circumstances in which the interference of luminous waves is capable of producing sensible effects.

We have seen that each undulation produced by an oscillatory motion was composed of two semiundulations, which [p124] occasioned in the particles of the fluids velocities exactly equal in their intensity, though opposite in the direction of the motions. Let us at first suppose that two whole undulations, moving in the same line and in the same direction, differ half an undulation in their progress: they will then be superinduced on each other through one half of their length, or of their breadth, as we should say in speaking of the waves of a liquid: but I here use in preference the term length as applied to the interval between the two points which are similarly affected by the motions of two consecutive undulations. In the supposed case of the coincidence of one half of each of the undulations, the interference will only take place with respect to the parts so coinciding: that is, to the latter half of the first undulation, and the preceding half of the second: and if these two semiundulations are of equal intensity, since they tend to give, to the same points of the ether, impulses directly opposite, they will wholly neutralise each other, and the motion will be destroyed in this part of the fluid, while it will subsist without alteration in the two other halves of the undulations. In such a case, therefore, half of the motion only would be destroyed.

If now we suppose that each of these undulations, differing in their progress by half the whole length of each, is preceded and followed by a great number of other similar undulations; then, instead of the interference of two detached undulations, we must consider the interference of two systems of waves, which may be supposed equal in their number and their intensity. Since, by the hypothesis, they differ half an undulation in their progress, the semiundulations of the one, which tend to cause in the particles of ether a motion in one direction, coincide with the semiundulations of the other, which urge them in the opposite direction, and these two forces hold each other in equilibrium, so that the motion is wholly destroyed in the whole extent of these two systems of waves, except the two extreme semiundulations, which escape from the interference. But these semiundulations will always constitute a very small part of the whole series to be considered.

This reasoning is obviously applicable to such systems only [p125] as are composed of undulations of the same length; for if the waves were longer one than the other, however small their difference might be, it would happen at last that their relative position would not be the same throughout the extent of the groups; and while the first destroyed each other almost completely, the following ones would be less in opposition, and would ultimately agree completely with each other: hence there would arise a succession of weak and strong vibrations analogous to the beatings which are produced by the coincidence of two sounds differing but little from each other in their tone; but these alternations of weaker and stronger light, succeeding each other with prodigious rapidity, would produce in the eye a continuous sensation only.

It is very probable that the impulse of a single luminous semiundulation, or even of an entire undulation, would be too weak to agitate the particles of the optic nerve, as we find that a single undulation of sound is incapable of causing motion in a body susceptible of a sympathetic vibration. It is the succession of the impulse, which, by the accumulation of the single effects, at last causes the sonorous body to oscillate in a sensible manner; in the same manner as the regular succession of the single efforts of a ringer is at last capable of raising the heaviest church bell into full swing. Applying this mechanical idea to vision, supported as it is by so many analogies, we may easily conceive that it is impossible for the two remaining semiundulations, which have been mentioned, to produce any sensible effect on the retina; and that the result of such a combination of the two systems must be the production of total darkness.

If again we suppose the second system of undulations to be again retarded half an undulation more, so as to make the difference of the progress an entire undulation, the coincidence in the motions of the two groups will be again restored, and the velocities of oscillation will conspire and be augmented in the points of superposition; the intensity of the light being then at its maximum.

Adding another semiundulation to the difference in the progress of the two systems, so as to make it an interval and [p126] a half, it is obvious that the semiundulations, superinduced on each other, will now possess opposite qualities, as in the case of the half interval first supposed: and that all the undulations must in this manner be neutralised, except the extreme three semiundulations on each side, which will be free from interference. Thus almost the whole of the motion will again be destroyed, and the combination of the two pencils of light must produce darkness, as in the case first considered.

Continuing to increase the supposed difference by the length of a semiundulation at each step, we shall have alternately complete darkness and a maximum of light, accordingly as the difference amounts to an odd or an even number of semiundulations: that is, supposing always that the systems of undulations are of equal intensity: for if the one series were less vivid than the other, they would be incapable of destroying them altogether: the velocities of the one series would be subtracted from those of the other, since they would tend to move the particles of the ether in contrary directions, but the remainders would still constitute light, though feebler than that of the strongest single pencil. Thus the second pencil would still occasion a diminution of the light: but the diminution would be the less sensible as the pencil is supposed to be weaker.

Such are the consequences of the principle of the interference of undulations, which agree perfectly, as we have seen, with the law of the mutual influence of the luminous rays which is deduced from experiment: for the results are expressed precisely in the same words, if we give the name of _length of undulation_ to the difference of routes which had been represented by the symbol _d_. Admitting, therefore, as there is every reason to believe, that light consists in the undulations of a subtile fluid, the period _d_, after which the same effects of interference are repeated, must be the length of an undulation.

It appears from the table already given for the seven principal kinds of coloured rays, that this period _d_, or the length of the undulation, varies greatly, according to the [p127] colour of the light, and that for the extreme red rays, for example, it is [more than] half as great again as for the violet rays situated at the other extremity of the spectrum.

It may easily be imagined that the number of different undulations is not limited to the seven principal ones which are indicated in the table, and that there must be a multitude of intermediate magnitudes, and others beyond the red and the violet rays: for the ponderable particles, of which the oscillations give rise to them, must be subjected to forces that are infinitely varied, in the combustion or the incandescence of the bodies which excite the motions of the ether: and it is on the energy of these forces that the duration of each oscillation depends, and consequently the length of the undulation produced by it. It is found that all the undulations comprehended [in the air] between the lengths .0000167 E.I. and .0000244, are visible; that is, are capable of exciting vibrations in the optic nerve: the rest are only sensible by their heat, or by the chemical effects which they produce.

It has been remarked, that when two systems of waves differ half an undulation in their progress, two of the semiundulations must escape from interference; that six must be exempt when the difference amounts to three semiundulations; and that, in general, the number of undulations exempt from interference is equal to the number of lengths of a semiundulation separating the corresponding points of the two systems. While this number is very small in proportion to that of the waves contained in each system, the motion must be nearly destroyed, as in the case of the exemption of a single undulation. But it may be imagined that, as we increase the difference of the progress of the two pencils, the undulations exempted from interference may become a material portion of each group, and that it may finally become so great as to separate the groups entirely from each other; and in this case the phenomena of interference would no longer be observable. If, for example, the groups of undulations consisted but of a thousand each, a difference of one-twentieth of an inch in their routes would be much more than sufficient to prevent the interference of the rays of all kinds. [p128]

But there is another much more powerful reason which prevents our perceiving the effects of the mutual influence of the systems of waves when the difference of their routes is considerable; which is the impossibility of rendering the light sufficiently homogeneous: for the most simple light that we can obtain consists still of an infinity of heterogeneous rays, which have not exactly the same length of undulation; and however slight the difference may be, when it is repeated a great number of times, it produces of necessity, as we have already seen, an opposition between the modes of interference of the various rays, which then compensates for the weakening of some by the strengthening of others; [while the shades of colour are not sufficiently distinct to allow the eye to remark the difference.] This is without doubt the principal reason why the effects of the mutual interference of the rays of light become insensible when the difference of the routes is very considerable, so as to amount to 50 or 60 times the length of an undulation.

It has already been laid down as one of the conditions necessary for the appearance of the phenomena of interference, that the rays which are combined should have issued at first from a common source: and it is easy to account for the necessity of this condition by the theory which has now been explained.

Every system of waves, which meets another, always exercises on it the same influence when their relative positions are the same, whether it originates from the same source or from different sources; for it is clear that the reasons, by which their mutual influence has been explained, would be equally applicable to either case. But it is not sufficient that this influence should exist, in order that it may become sensible to our eyes: and for this purpose the effect must have a certain degree of permanence. Now this cannot happen when the two systems of waves which interfere are derived from separate sources. For it is obvious that the particles of luminous bodies, of which the vibrations agitate the ether, and produce light, must be liable to very frequent disturbances in their oscillations, in consequence of the rapid changes which are taking place around them, which may [p129] nevertheless be perfectly reconciled, as we have seen, with the regular continuance of a great number of oscillations in each of the series separated by these perturbations. This being admitted, it is impossible to suppose that these perturbations should take place simultaneously and in the same manner in the vibrations of separate and independent particles; so that it will happen, for example, that the motions of the one will be retarded by an entire semioscillation, while those of the other will be continued without interruption, or will be retarded by a complete oscillation, a change which will completely invert the whole effects of the interference of the two systems of undulations which originate from them; since if they had agreed on the first supposition, they would totally disagree on the second. Now these opposite effects, succeeding each other with extreme rapidity, will produce in the eye a continuous sensation only, which will be a mean between the more or less lively sensations that they excite, and will remain constant, whatever may be the difference of the routes described.

But the case is different when the two luminous pencils originate from a common source: for then the two systems of waves, having originated from the same centre of vibration, undergoing these perturbations in the same manner and at the same instant, undergo no changes in their relative positions: so that if they disagreed in the first instance at any given point, they would continue to disagree at all other times; and if their motions cooperated at first, they would continue to agree as long as the centre of vibration continued to be luminous: so that in this case, the effects must remain constant, and must therefore be sensible to the eye. This is therefore a general principle, applicable to all the effects produced by luminous undulations; that in order to become sensible, they must be permanent.

We have hitherto supposed that the two systems of waves were moving exactly in the same direction, and that consequently their elementary motions, to be combined with each other, were precisely limited to one single line: this is the simplest case of interference, and the only one in which the one motion can be completely destroyed by the other: [p130] for in order that this effect may be produced, not only the two forces must be equal and in contrary directions, but they must also act in the same right line, or be directly opposed to each other.

The phenomenon of coloured rings, and that of the colours developed by polarised light in crystallised plates, present a particular case of interference, in which the undulations are exactly parallel. But in the phenomena of diffraction, or in the experiment with the two mirrors, which has been already described, the rays which interfere always form sensible though very small angles with each other. In these cases the impulses to be combined with each other at the same points, as belonging to the two systems of undulations, will also act in directions forming sensible angles with each other: but on account of the smallness of these angles, the result of the two impulses is almost exactly equal to their sum, when the impulses act in the same direction, and to their difference, when they are in contrary directions. Thus, in the points of agreement or disagreement, the intensity of the light will be the same as if the directions agreed more perfectly; at least the nicest eye will not be able to discover any difference in them. But although, with respect to the intensity of the light, this case of interference resembles that which has already been considered, there are other differences which modify the phenomenon very greatly, both with respect to its general form, and to the circumstances necessary for producing it.

We may take, as a convenient example, the case of diverging rays originating from the same luminous point, and reflected by two mirrors slightly inclined to each other, so as to produce two pencils meeting each other in a sensible angle: the two systems of waves will then meet each other with a slight inclination; and it follows from this obliquity, that if a semiundulation of the first system coincides perfectly in one point with a semiundulation of the second, urging the fluid in the same direction, it must separate from it to the right and left of the point of intersection, and must coincide, a little further off, on one side with the preceding semiundulation which is in a contrary direction, [p131] and on the other side with the following semiundulation, and then be separated from this again, and at a distance twice as great as the first, must coincide with the second semiundulation before and behind it, of which the actions will coincide with its own: whence there will arise, on the surface of this undulation, a series of lines, at equal distances from each other, in which the motion is destroyed and doubled alternately by the action of the second series. Thus if we receive this luminous undulation on a white card, we shall observe on it a series of dark and bright stripes, if the light employed is homogeneous; or coloured fringes of different tints, if we employ white light for the experiment.

This will be more easily understood by the inspection of a figure, which represents a section of the two mirrors and of the reflected undulations, formed by a plane drawn from the luminous point perpendicularly to the mirrors represented by DE and DF. The luminous point is supposed to be S, and A and B are the geometrical positions of its two images, which are determined by the perpendiculars SA and SB falling from S on the mirrors, taking in them PA = SP [p132] and QB = SQ. The points A and B, thus found, are the centres of divergence of the rays reflected from the respective mirrors, according to the well known law of reflection. Thus, in order to have the direction of the ray reflected at any point G of the mirror DF, for example, it is sufficient to draw a right line through B and G, which will be the direction of the reflected ray. Now it must be remarked, that, according to the construction by which the position of B is found, the distances BG and SG will be equal, and thus the whole route of the ray coming from S and arriving at _b_, is the same as if it had come from B. This geometrical truth being equally applicable to all the rays reflected by the same mirror, it is obvious that they will arrive at the same instant at all the points of the circumference _n′bm_, described on the point B as a centre, with a radius equal to B_b_; consequently this surface will represent the surface of the reflected undulation when it arrives at _b_, or, more correctly speaking, its intersection with the plane of the figure: the surface of the undulation being understood as relating to the points which are similarly agitated at the same instant: the points being all, at the commencement of the whole oscillation, for example, or at the middle or the end, completely at rest; and in the middle of each semioscillation, possessed of the maximum of velocity.

In order to represent the two systems of reflected undulations, there are drawn, with the points A and B for their centres, two different series of equidistant arcs, separated from each other by an interval which is supposed equal to the length of a semiundulation. In order to distinguish the motions in opposite directions, the arcs on which the motions of the ethereal particles are supposed to be direct, are represented by full lines, and the maximum of the retrograde motions are indicated by dotted lines. It follows that the intersections of the dotted lines with the full lines are points of complete discordance, and of course show the middle of the dark stripes; and, on the contrary, the intersections of similar arcs show the points of perfect agreement, or the middle of the bright stripes. The intersections of the arcs of the same kind are joined by the dotted lines _b′p′_, _br_, _b′p′_, and those of arcs of [p133] different kinds by the full lines _n′o′_, _no_, _no_, _n′o′_: these latter representing the successive positions or the trajectories of the middle points of the dark stripes, and the former the trajectories of the bright bands.

It has been necessary to magnify very greatly in this figure the real length of the luminous undulations, and to exaggerate the mutual inclination of the two mirrors, so that we must not expect an exact representation of the phenomenon, but merely a mode of illustrating the distribution of the interferences, in undulations which cross each other with a slight inclination.

It is easy to deduce from geometrical considerations, that the length of these fringes is in the inverse ratio of the magnitude of the angle made by the two pencils which interfere, and that the interval, comprehended between the middle points of two consecutive dark or bright bands, is as much greater than the length of the undulation, as the radius is greater than the sine of the angle of intersection.

In fact the triangle _bni_, formed by the right line _bi_, and the two circular arcs _ni_ and _nb_, may be considered as rectilinear and isosceles, on account of the smallness of the arcs; and the sine of the angle _bni_, considered as very small, may be called _ib_/_bn_: so that _bn_ being the radius, _ib_ will represent the sine of the angle _bni_, which has its legs perpendicular to those of the angle A_b_B: consequently, these angles being equal, one of them may be substituted for the other; and representing by _i_ the angle A_b_B, formed by the reflected rays, we have _bn_ = _ib_/sin _i_; consequently _nn_, which is twice _bn_, will be equal to 2_ib_/sin _i_. But _nn_ is the distance between the middle points of two consecutive dark stripes, and is the distance which has been called the breadth of a fringe; and _ib_ being the breadth of a semiundulation, according to the construction of the figure, 2_ib_ will be that of a whole undulation; consequently the breadth of a fringe may be said to be equal to the length of an undulation divided by the [numerical] sine of the angle made by the reflected rays [p134] with each other, which is also the angle under which the interval AB would appear to an eye placed at _b_. We find another equivalent formula, by remarking that the two triangles, _bni_ and A_b_B, are similar, whence we have the proportion _bn_: _bi_ = A_b_ : AB, and _bn_ = (_bi_ × A_b_)/AB, or 2_bn_ = (2_bi_ × A_b_)/AB: which implies that we may find the numerical breadth of a fringe by multiplying the length of an undulation by the distance of the images A and B from the plane on which the fringes are measured, and dividing the product by the distance of the two images.

It is sufficient to inspect the figure, in order to be convinced of the necessity of having the two mirrors nearly in the same plane, if we wish to obtain fringes of tolerably large dimensions; for in the little triangle _bni_, the side _bi_, which represents the length of a semiundulation, being little more than the hundred thousandth of an inch for the yellow rays, for example, the side _bn_, which measures the half breadth of a fringe, can only become sensible when _bn_ is very little inclined to _in_, so that their intersection may be remote from _ib_; and the inclination of _bn_ to _in_ depends on the distance AB, which is the measure of the inclination of the mirrors.

If A and B, instead of being the images of the luminous point, were the projections of two very fine slits cut in a screen RN, through which the rays of light were admitted from a luminous point placed behind the screen in the continuation of the line _b_DC, the two paths described between the point and the slits A and B being equal, it would be sufficient to compute the paths described by the rays, beginning from A and B, in order to have the differences of their lengths; and it is obvious in this case, that the calculations which we have been making of the breadth of the fringes, produced by the two mirrors, would remain equally applicable, at least as long as each slit remained narrow enough to be considered as a single centre of undulation, relatively to the inflected rays which it transmits. It may therefore be said that the breadth of the fringes, produced by two very fine slits, is equal to the length of an undulation supposed [p135] to be multiplied by the interval between the two slits, and divided by the distance of the screen from the wires of the micrometer employed for measuring the fringes.

This formula is also applicable to the dark and bright stripes which are observed in the shadow of a narrow substance, substituting the breadth of this substance for the interval which separates the two slits, as long as the stripes are far enough from the edges of the shadow: for when they approach very near to the edges, it is shown, both by theory and by experiment, that this calculation does not represent the facts with sufficient accuracy; and it is not perfectly correct in all cases, either for the fringes within the shadow, or for those of the two slits, but only for the fringes produced by the mirrors, which exhibit the simplest case of the interference of rays slightly inclined to each other. In order to obtain from the theory, a rigorous determination of the situation of the dark and light stripes in the two former cases, it is not sufficient to calculate the effect of two systems of undulations, but those of an infinite number of similar groups must be combined, according to a principle which will shortly be explained, in treating of the general theory of diffraction.

ii. _Rule for the Correction of a_ LUNAR OBSERVATION. _By_ Mr. WILLIAM WISEMAN, _of Hull_.

RULE.

Add together the reserved logarithm (found as directed, page 111 and 112 of the Appendix to the third edition of the Requisite Tables) the log. sines of half the sum, and half the difference of the apparent distance, and difference of apparent altitudes, and 0.3010300, the log. of 2. Then, to the natural number corresponding to the sum of these four logarithms, add the natural verse sine of the difference of true altitudes, and the sum will be the natural verse sine of the true distance.

Or, having obtained the natural number, as directed above, subtract it from the natural cosine of the difference of the true altitudes, and the remainder will be the natural cosine of the true distance. [p136]

EXAMPLE.

(_From page 112, Appendix to Requisite Tables_.)

Reserved log. from Tables (Req.) 9th and 11th 9.9938860 Log. sin. 43° 23′ 5″ = 1/2 sum of app. dist. and diff. app. altitudes 9.8368895 Log. sin. 6° 45′ 36″ = 1/2 diff. ditto ditto 9.0708157 Log. of 2 0.3010300 --------- Nat. num. to sum of 4 logarithms .1594488 9.2026212 Nat. vers. 37° 13′ 12″ = diff. true altitudes .2036812 ------- Nat. vers. 50° 26′ 28″ = true distance .3631300

Or, Nat. cos. 37° 13′ 12″ = diff. true altitudes .7963188
Nat. number found above .1594488
-------
Nat. cosin. 50° 26′ 28″ = true distance .6368700

DEMONSTRATION OF THE RULE.

Let M′, S′, D′, d′ and M, S, D, d, respectively denote the true and apparent altitudes, distances, and differences of true and apparent altitudes of the moon and sun (or a star); then will the theorem answering to the above rule be expressed by

vers. D′ = ((2 cos M′ cos S′)/(cos M cos S)) sin 1/2(D + d) × sin 1/2(D−d) + vers. d′.

By Bonnycastle’s Trig. p. 175, the cosine of the angle contained by the co-altitudes is

(cos D − sin M sin S)/(cos M cos S) = (cos D′ − sin M′ sin S′)/(cos M′ cos S′);

consequently the verse sine of the same angle

= 1−(cos D − sin M sin S)/(cos M cos S) = 1−(cos D′ − sin M′ sin S)/(cos M′ cos S′); that is,

(cos M cos S + sin M sin S − cos D)/(cos M cos S) = (cos M′ cos S′ + sin M′ sin S′ − cos D′)/(cos M′ cos S′).

Substituting cos d and cos d′ for cos M cos S + sin M sin S and cos M′ cos S′ + sin M′ sin S′. (Bon. Trig. p. 282), we have

(cos d−cos D)/(cos M cos S) = (cos d′−cos D′)/(cos M′ cos S′); whence

cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S)) (cos d−cos D);

or, which is the same,

cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S)) (vers D−vers d);

or, (Bon. Trig. p. 286.) [p137]

cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S))· (2 sin^2 ((1/2)D) − 2 sin^2 ((1/2)d)); that is,

cos D′ = cos d′−((2 cos M′ cos S′)/(cos M cos S)) sin ((1/2)(D + d))· sin 1/2(D−d); whence

also vers D′ = vers d′ + ((2 cos M′ cos S′)/(cos M cos S)) · sin ((1/2)(D + d)) sin ((1/2)(D−d)).

It may be observed, that Requisite Tables 9–11, answer logarithmically to (cos M′ cos S′)/(cos M cos S); and the verse sines, and the cosines can be very readily taken out of the tables in the Appendix. Also no ambiguity can arise from the application of the rule before given: for all the arcs concerned in the operation will always be (each of them) less than a quadrant, except the resulting true distance, which cannot cause any ambiguity; and the verse sines are given in the Appendix, to 126°.

EXAMPLE.

(_Example 2nd, p. 39, Requisite Tables_.)

Reserved log. from Tables 9 and 10 9.995307 Log. sin 62° 45′ 56″ = 1/2 sum app. dis. and diff. app. alts. 9.948971 Log. sin 40 43 31 = 1/2 diff. ditto ditto 9.814536 Log. 2 0.301030 -------- Nat. num. corres. 1.147741 0.059844 Nat. vers. 22° 48′ 16″ = diff. true alts. 0.078167 -------- -------- Nat. vers. 103 3 23 = true distance 1.225908 --------

_De l’Influence des Agens Physiques sur la Vie_. Par W. F. Edwards, D.M., Membre associé de l’Académie royale de Médicine de Paris, Membre de la Société Philomatique, de la Société de Médicine de Dublin, &c.

The researches of science among the phenomena of the physical world have long obtained a high degree of estimation and interest in general society; but it is of late years only that their application to living functions has attracted much of the attention of the literary world.

The laws which govern the action of animal organs (the proper department of Physiology) have usually been investigated by the medical profession, to which they especially [p138] refer. Now we find the public take some pains, and with reason, to inform themselves upon subjects connected with physiological knowledge. A well-educated person, disposed to philosophical inquiries, is not merely contented with the consciousness of living, and the common information he derives of its means by experience, but he seeks also to comprehend the relations subsisting between his own organisation and the matters with which he is surrounded, and which at once furnish him with nutrition, life, and support, and assail him with disease and annihilation. His own instincts and observation, joined to the more learned experience of his medical advisers, help him through the precarious stages of life, and these may perhaps be sufficient for all its purposes; and under this impression many will seek to know no more of the secrets of nature.

But we live in an inquiring and scrutinising age, when the demand for scientific principles is very generally urgent. All, therefore, relating to organisation seems of equal interest with that appertaining to what is termed the physical creation or inert matter.

Under this impression we have perused the book before us with great satisfaction, and propose to present our readers with an analysis of the valuable materials which it contains. We have some knowledge of Dr. Edwards, a countryman domiciliated in France, and long resident in Paris. We have confidence in his reports, and highly estimate his philosophical skill, extensive acquirements, and accuracy of observation, ranking him among the first physiologists of the age.

The work, now under consideration, contains an elaborate account of a long series of experiments, instituted for the purpose of ascertaining the influence of the physical agents upon animal life. These agents comprehend the atmospheric air, water, and temperature; the two first constituting the media in which all animals exist, and the last influencing in common the inhabitants of both media. It is true, this is a subject by no means new, for it has engaged the attention of experimenters from the earliest days of science. But Dr. Edwards has diligently and patiently sought to investigate the subject himself, to correct previous errors, and to embody the facts which he has accumulated into a more complete and regular system than heretofore adopted. In this attempt he has been eminently successful, and has effected more perhaps than all who preceded him, availing himself, nevertheless, of the experience of former inquiries.

The extent of his book, and the number of the experiments [p139] are indeed somewhat appalling, but his clear and distinct method of arrangement greatly facilitates the reader’s endeavours to master the extensive subjects of his pages. As a book of reference it should find a place in the library of every scientific society, and no individual devoted to philosophy should omit the possession of it.

The agency of the air around us, water, and heat and cold, have often been the objects of _chemical_ inquiry, from their known great influence upon the animal economy. The changes effected by the phenomena of animal life upon these agents have been accurately examined, and partly reduced to a mathematical precision of calculation.

Spallanzani and others have viewed the subject as it regards physiology, but with such results as left the field open to subsequent investigation. Dr. Edwards seems to have seized upon the deficiencies of his predecessors, and, by going over their ground, and extending his own inquiries, he has arrived at most interesting and important results. These he has divided into four parts, as they relate to the different orders of the animal creation. The first part includes some of the lower animals, particularly tenacious of life, and of cold blood, such as frogs, toads, and salamanders. The second part is devoted to other animals of cold blood, and of the vertebrated order, as fish, and those reptiles which include lizards, snakes, and turtles. The third part refers to warm-blooded animals; and the fourth part of the work is dedicated to the influence of the physical agents upon the human race and vertebrated animals. To these the author has added the discoveries of modern times, relative to electricity on the animal economy, in an Appendix. A collection of tables is appended to the work, exhibiting the principal series of his experiments, as they regard the relative influence of physical agents on the duration of life, and the phenomena resulting from their mutual action.

The great importance of the four grand divisions of the work forbids our hastily reviewing them, and we will endeavour to condense so much of the information they contain as may forward the objects of our analysis. Dr. Edwards thus announces the arrangement of his work:—

“Ces recherches auront donc rapport à l’air dans les conditions de quantité, de mouvement et de repos, de densité et de raréfaction; à l’eau liquide et à la vapeur aqueuse; à la température, dans ses modifications de degré et de durée; à la lumière et à l’électricité. Ces causes agissent à la fois sur l’économie animale, ordinairement d’une manière sourde et imperceptible; et toujours [p140] l’impression qu’on reçoit est le résultat de toutes ces actions combinées.”

“Lors même que, par l’intensité de l’une d’elles, il nous arrive de distinguer la cause qui nous affecte, l’observation de l’effet se borne le plus souvent à la sensation, et les autres changemens qui l’accompagnent nous échappent. On conçoit par la que l’observation la plus attentive des phénomènes tels que la nature nous les présente, ne saurait démêler dans cette combinaison d’actions l’effet propre à chaque cause, ni reconnaître des effets qui ne seraient pas révélés par la sensation.

“Il est une méthode qui règle les conditions extérieures, qui fait varier celle dont on veut apprécier l’action, et qui fait juger, par la correspondance entre ce changement et celui qui survient dans l’économie, du rapport de cause et d’effet: c’est la méthode expérimentale; c’est celle que j’ai suivie. Pour en tirer parti il fallait, d’une part, déterminer l’intensité de la cause, d’autre part celle de l’effet. La physique nous fournit ordinairement les moyens de remplir la première indication.”

In the true spirit of philosophical investigation, Dr. Edwards, in the first place, proceeds to examine the action of physical agents upon the simplest forms, and least elaborately developed organised beings, extending his inquiries upwards, in the scale of the animal world, to man, the most perfect creature, and the ultimate object of all physiological researches.

The peculiarity of constitution belonging to cold-blooded reptiles, among which there is so little mutual dependance of organs, renders these the best tests of the relative and proportionate influence of the different agents, the intense action of which is liable to destroy the more perfect animals; and the great development of the nervous system in the higher orders gives them a wider and more acute range of sensibility. It is difficult, at all times, and often impossible, to insulate corporeal functions among the warm-blooded classes, so as to ascertain the amount and limits of physical agency. The four classes of vertebrated animals, or such as are furnished with true spines, afford ample means of comparative illustrations; and these departments have engaged the author’s attention, in order to display the result of the action of the same agent exercising a uniform influence upon constitutions very differently constructed. The _air_, for example, exercises its influence uniformly upon he four mentioned classes of vertebratæ, and their different families are similarly exposed to the action of the atmosphere by respiration.

Curious and interesting as is this subject, it is singular [p141] that, while it was among the first to be noticed, it has been the latest in producing satisfactory results. Among the opposing causes of the advancement of knowledge in this department, the ignorance of our ancestors in _chemical_ science seems to be the principal. Without chemical aid it is perfectly useless to attempt the investigation. The composition of the air respired must be well understood; the different gases must be carefully examined, or the physiological inquiry will be darkened and obscured.

Dr. Priestley laid the foundation of our chemical knowledge of gases in their relation to respiration; but some time elapsed before it was understood in what manner the air was connected with animal organisation. Oxygen gas, one of the known constituents of atmospheric air, was Priestley’s discovery, in its effect upon the blood, of converting this fluid from a dark purple to a bright crimson. Lavoisier founded a chemical theory upon this discovery of the agency of air, which was subsequently applied by Goodwin to physiology. The latter author demonstrated, by a series of excellent and correct experiments, that the exclusion of atmospheric air produces death in animals, in consequence of the dark-coloured blood usually circulating in the veins being prevented from becoming crimsoned. The state in which any animal may be thus placed, is known by the term ASPHYXY, and by which is to be understood a deficient or suspended aërification of the blood, from whatever cause it may proceed that the atmospheric air is prevented from access to the blood as it circulates through the lungs.

The great French anatomist, Bichat, pursued this subject still farther, and published a treatise on Asphyxy. He sought, by numerous experiments, to determine the threefold relation of the air to the nervous system, respiration, and the circulation; and he arrived at this great and important conclusion, that the VENOUS OR DARK BLOOD CIRCULATING THROUGH THE BRAIN, CREATES A CESSATION OF THE FUNCTIONS OF THAT ORGAN, AND THAT IN CONSEQUENCE THE HEART LOSES ITS ACTION. This discovery shows us at once the direct cause of asphyxy in all its different degrees, according, in effect, to the vitiated state of the blood from its deficient or suspended aërification.

Le Gallois also investigated the subject of asphyxy; and he found that, when _dark blood circulated through the spinal marrow, the motions of the heart ceased_; and thus he not only determined the relations of the nervous system to atmospheric air, but also those of the respiration and the circulation, [p142] explaining the action of the air upon animals physiologically.

In this inquiry warm-blooded animals were almost exclusively referred to.

Spallanzani certainly investigated the action of the air on animals of cold blood, but less in relation to the three grand objects of Bichat and Le Gallois; and Spallanzani had the misfortune to live in an age when neither chemistry nor physiology had made such advances as the present age has produced.

Messrs. Humboldt and Provençal have, indeed, supplied much of this deficiency, by their researches into the respiratory functions of fishes. Nevertheless, the ground was still open, and our author has justly appreciated the extent of former inquiries, and observed that the phenomena of cold-blooded animals were too extraordinary to be noticed lightly, and required much more extensive observation than was previously bestowed upon them. With this impression, he proceeded to form an estimate of the comparative influence of the air and water upon the nervous and muscular systems of cold-blooded animals, which the singular modifications of life among reptiles in particular afford ample means of ascertaining.

We know that these animals possess the extraordinary property of existing a considerable time after the removal of the heart, with the free exercise of their senses and of voluntary motion, notwithstanding the suppression of the circulation. Dr. Edwards accordingly selected _salamanders_ for his first investigations, and removed the heart, with the bulb of the aorta. Two of these were exposed to the free action of the air, and the other two were submersed in water previously deprived of air by boiling; a similar temperature being maintained in each medium. In four or five hours, those submersed in the non-arëated water ceased to be active, unless irritated, when they still appeared to retain voluntary power. One died in eight, and the other in nine hours. The salamanders in air lived from twenty to twenty-six hours and upwards. These comparisons were frequently repeated, and upon frogs and toads, with the same results, showing the experiments in air to be far more favourable to their existence than with the animals submersed in the water. Eight hours were about the maximum of the duration of life among the animals submersed in the water, and twenty-nine among those exposed to the air; so that, independently of respiration, the air is thus proved to be the most proper [p143] medium for the action of their nervous and muscular systems, in their insulated state, the respiration and the circulation of the blood being both suspended. As a further corroboration of the superior vivifying property of the air over simple water, when the same animals were plunged into unaërated water during a certain time, as soon as they were, removed into the atmosphere, they instantly revived; and their nervous and muscular systems were acted on according as they were placed in either medium. Dr. Edwards also confirmed the observation of Goodwin relative to the effect produced on the colour of the blood. Properly speaking, the asphyxy comes on the instant the air is excluded, the shades of difference in the colour of the blood being referrible to the air left in the lungs after cessation of respiration.

The next point to determine was the influence of the air upon the same animals exercising the respiratory function, and retaining their circulation, compared with those deprived of these functions.

The difference of time in the two cases developes the influence which the general circulation of the blood, free from aërial contact, exercises upon the nervous system.

To ascertain this point, an equal number of frogs, deprived of the power to exercise their respiratory and circulating functions, together with others left entire, were respectively plunged into disaërated water. At times the difference in favour of the untouched animals was twenty-four hours in favour of the duration of life. Similar trials with toads and salamanders produced the same results. In each case _asphyxy_ came on; but the existence of the animals which lived without the respiration and circulation was much shortened. Thus the relative powers of life between the sole and insulated action of the nervous system, and its action combined with the circulation of dark blood, were estimated. The inference to be deduced, therefore, is, that although disaërated blood furnishes but an ephemeral sort of existence, it nevertheless exercises a comparatively favourable influence upon the nervous and muscular systems, since it tends to the prolongation of the action of these animal functions.