Chapter IX (2)

There are various other Lighthouses, which, in themselves, are sufficiently deserving of a separate notice, were it not that they have more or less something in common with those already described, which are unquestionably the most remarkable edifices of the kind. The first design for an Iron Lighthouse, is that by my Father for the Bell Rock, in the year 1800. The invention of Mr MITCHELL of Belfast, for applying the principle of the screw to the erection of Lighthouses on soft foundations, deserves a longer notice than is consistent with the nature of these notes. It must therefore be sufficient to say, that the principal Lighthouses on this plan (those of the Maplin, Fleetwood, and Belfast Lough) consist of piles or of hollow pillars of cast-iron, grouped together in the form of a truncated pyramid, and closely resembling, in the general arrangement of their parts, the Beacon shewn in Plate XXX., and that erected on the Carr Rock in 1821. The lower end of each pillar is furnished with a flat screw or worm and a sharp point, which is screwed into the sand, clay, or gravel, or other soft subsoil. Mr ALEXANDER GORDON of London also fitted up a Lighthouse, composed of cast-iron plates, which was erected at Morant, in the West Indies, a style of building in itself by no means eligible, and which seems suitable only where stone cannot be easily obtained, or conveniently applied. Both those plans (except in so far as the screw is concerned, which is indeed the distinguishing feature of Mr MITCHELL’S ingenious plan) are to be found in one of my Father’s designs for the Bell Rock Lighthouse (see his Account, at Plate VII., figs. 2, 3, 4, and 5, and pp. 499, 500). Dr POTTS has also invented a method of driving piles by means of atmospheric pressure, which has been used at the South Galliper Beacon, on the Goodwin Sands.

~Early modes of Illumination.~

Having thus hastily described the most interesting and celebrated Lighthouses, I proceed to the proper object of these Notes, which are chiefly intended to make known the various methods now in use for the illumination of Lighthouses. There can be little doubt, that down to a very late period, the only mode of illumination adopted in the Lighthouses, even of the most civilized nations of Europe, was the combustion of wood or coal in _chauffers_, on the tops of high towers or hills. It consists with the personal knowledge of many persons now living, that the Isle of May Light, in the Frith of Forth, previous to its being assumed by the Commissioners of the Northern Lights in 1786, was of that kind; and, even in England, the art of illumination had made so little progress, that the magnificent Tower of the Eddystone, for about forty years after it came from the hands of SMEATON, could boast of no better Light than that derived from a few miserable tallow candles. Such methods were most imperfect, not only in point of efficiency and power, but also as respects the distinction of one light from another, an object which, on a difficult and rugged coast, may be considered as of almost equal importance with the distance at which the Light can be seen.

~Flame.~

Solid substances which remain so throughout their combustion, are only luminous at their own surface, and exhibit phenomena, such as the dull red heat of iron, or of most kinds of pit-coal, and are therefore more suited for the purpose of producing heat than light. But by using substances which are formed into inflammable vapours, at a temperature below that which is required for the ignition of the substances themselves, gas is obtained and _flame_ is produced. Much light is thus evolved at a comparatively low temperature. The gas necessarily rises _above_ the combustible substance from which it is evolved, owing to its being formed at a temperature considerably higher than that of the surrounding air, than which it is necessarily rarer. Of this description are the flames obtained by the burning of the various oils, which are generally employed in the illumination of lighthouses. In the combustion of oil, wicks of some fibrous substance, such as cotton, are used, into which the oil ascends by capillary action, and being supplied in very thin films, is easily volatilized into vapour or gas by the heat of the burning wick. The gas of pit-coal has been occasionally used in lighthouses; it is conveyed in tubes to the burners, in the same manner as when employed for domestic purposes. There are certain advantages, more especially in dioptric lights, where there is only one large central flame, which would render the use of gas desirable. The form of the flame, which is an object of considerable importance, would thus be rendered less variable, and could be more easily regulated, and the inconvenience of the clock-work of the lamp would be wholly avoided. But it is obvious, that gas is by no means suitable for the majority of lighthouses, their distant situation and generally difficult access rendering the transport of large quantities of coal expensive and uncertain; whilst in many of them there is no means of erecting the apparatus necessary for manufacturing gas. There are other considerations which must induce us to pause before adopting gas as the fuel of lighthouses; for, however much the risk of accident may be diminished in the present day, it still forms a question, which ought not to be hastily decided, how far we should be justified in running even the most remote risk of explosion in establishments such as lighthouses, whose sudden failure might involve consequences of the most fatal description, and whose situation is often such, that their re-establishment must be a work of great expense and time. Gas is, besides, far from being suitable in catoptric lights, to which, in many cases (especially when the frame is moveable, as in revolving lights), it could not be easily applied. The oil most generally employed in the Lighthouses of England is the sperm oil of commerce, which is obtained from the South Sea whale (_Physeter macrocephalus_). In France, the colza oil, which is expressed from the seed of a species of wild cabbage (_Brassica oleracea colza_), and the olive oil are chiefly used; and a species of the former has lately been successfully introduced into the Lighthouses of Great Britain. Of all these oils, the purified sperm oil has hitherto been generally considered the most advantageous for lighthouse purposes; but there is every reason for anticipating that the late adoption of the colza oil in many of the British Lights, on the suggestion of Mr JOSEPH HUME, M.P., while chairman of a select committee of the House of Commons on Lighthouses, will lead to an important saving, as its combustion produces an equal quantity of light at somewhat more than one-half of the expense for spermaceti oil. Careful trials have been made of this oil; and on the 10th of March 1847, I was enabled to report the results to the Commissioners of Northern Lighthouses in the following terms:

“1. The colza oil possesses the advantage of remaining fluid at temperatures which thicken the spermaceti oil so that it requires the application of the frost lamp.

“2. It appears, from pretty careful photometrical measurements of various kinds, that the light derived from the colza oil is, in point of intensity, a little superior to that derived from the spermaceti oil, being in the ratio of 1·056 to 1.

“3. The colza oil burns both in the Fresnel lamp and the single Argand burner with a thick wick during seventeen hours without requiring any coaling of the wick or any adjustment of the damper; and the flame seems to be more steady and free from flickering than that from spermaceti oil.

“4. There seems (most probably owing to the greater steadiness of the flame) to be less breakage of glass chimneys with the colza than with the spermaceti oil.

“5. The consumption of oil, in so far as that can be ascertained during so short a period of trial, seems in the Fresnel lamp to be 121 for colza, and 114 for spermaceti; while in the common Argand, the consumption appears to be 910 for colza, and 902 for spermaceti.

“6. If we assume the means of these numbers, 515 for colza, and 508 for spermaceti, as representing the relative expenditure of these oils, and if the price of colza be 3s. 9d., while that of spermaceti is 6s. 9d. per imperial gallon, we shall have a saving in the ratio of 1 to 1·755, which, at the present rate of supply for the Northern Lights, would give a saving of about L.3266 per annum.

“Of these conclusions, the three last may be considered as more or less conjectural, being founded on data derived from too short a trial; but the striking agreement of the results obtained at the six lights in which the experiments were made, tends in some measure to supply the place of a longer period of trial; and I have no hesitation, therefore, in recommending the Board at once to introduce the use of the colza oil into all the dioptric lights, except that of Skerryvore, where some special reasons induce me to defer the change for another season. In the catoptric lights, the only reason for not making an equally extensive trial is the necessity for renewing all the burners, which require to be so constructed as to receive thick wicks of brown cotton; and it may perhaps be considered prudent to proceed with some caution in changing the apparatus, so as to suit it for burning a patent oil, the circumstances attending the regular and extensive supply of which are not yet fully known. I may remark, that I have burnt the colza oil in the solar lamp alluded to in my last report; but I disapprove of it as tending to elongate the flame vertically, and thus to decrease its horizontal volume. The elongated form of flame increases the divergence vertically where the light is lost, and so far circumscribes its horizontal range where it is most required. I have therefore substituted the thick wick burner for the solar lamp, whereby an equally complete combustion is obtained, and the proper form of the flame is at the same time preserved.”[36]

[36] Since the above report was written, the price of colza oil has
risen; and other circumstances have occurred to justify the caution
as to the universal adoption of that oil.

* * * * *

~Drummond and Voltaic Lights.~

The application of the Drummond and Voltaic lights[37] to lighthouse purposes is, owing to their prodigious intensity, a very desirable consummation; but it is surrounded by so many practical difficulties that, in the present state of our knowledge, it may safely be pronounced unattainable. The uncertainty which attends the exhibition of both these lights, is of itself a sufficient reason for coming to this conclusion. But other reasons unhappily are not wanting. The smallness of the flame renders them wholly inapplicable to dioptric instruments, which require a great body of flame in order to produce a degree of divergence sufficient to render the duration of the flash in revolving lights long enough to answer the purpose of the mariner. M. FRESNEL made some experiments on the application of the Drummond light to dioptric instruments, which completely demonstrate their unfitness for this combination. He found that the light obtained by placing it in the focus of a great annular lens was much more intense than that produced by the great lamp and lens; but the divergence did not exceed 30′; so that, in a revolution like that of the Corduan Light, the flashes would last only 1¹⁄₃ second, and would not, therefore, be seen in such a manner as to suit the practical purposes of a revolving light. The great cylindric refractor used in fixed lights of the first order, was also tried with the Drummond light in its focus; but it gave coloured spectra at the top and bottom, and only a small bar of white light was transmitted from the centre of the instrument. The same deficiency of divergence completely unfits the combination of the Drummond light with the reflector for the purposes of a fixed light, and even if this cause did not operate against its application in revolving lights on the catoptric plan, the supply of the gases, which is attended with almost insurmountable difficulties, would, in any case, render the maintenance of the light precarious and uncertain in the last degree.

[37] The Drummond light is produced by the ignition or combustion of
a ball of lime (³⁄₈ inch diameter) in the united flames of hydrogen
and oxygen gases, and is equal to about 264 flames of an ordinary
Argand Lamp with the best Spermaceti oil. It derives its name from
the late LIEUT. DRUMMOND, R. E., who first applied it in the focus of
a paraboloïd for geodetical purposes, and afterwards proposed it for
Lighthouses. (See his Account of the Light in the Phil. Trans. for
1826, p. 324, and for 1830, p. 383.) The Voltaic light is obtained
by passing a stream of Voltaic electricity from a powerful battery
between two _charcoal points_, the distance between which requires
great nicety of adjustment, and is the chief circumstance which
influences the stability and the permanency of the light. The Voltaic
light greatly exceeds the Drummond light in intensity, as ascertained
by actual comparison of their effects; but the ratio of their power
has not been accurately determined. It was first exhibited in the
focus of a reflector by Mr JAMES GARDNER, formerly engaged in the
Ordnance Survey of Great Britain.

~Mr Gurney’s Lamp.~

In 1835, Mr GURNEY proposed the combination of a current of oxygen with the flame of oil, in order to obtain a powerful light of sufficient size to produce the divergence required for the illumination of lighthouses. The Trinity-House of London entertained the proposal, and made some experiments on this important subject; but the plan was finally rejected as disadvantageous in practice.

~Argand Burners.~

Until the invention by ARGAND (about the year 1784), of the lamp with a double current of air, the art of illumination seems to have received no improvement, and to have occupied very little attention from the time of CARDAN, or at all events of Dr HOOK, who, about the year 1677, in a monograph entitled “Lampas,” made some important observations on the constitution of _flame_, so as to make one wonder that he should have stopped short of the discoveries of later inventors. Before ARGAND’S time, every wick consisted of a solid cord, whose flame was fed only by the current of air on its outside; and the consequence of this arrangement is, that the stream of vapour or smoke, especially from the centre of thick wicks, escapes unburnt, because, before it reaches the height at which the combustion of the central stream can take place, its temperature has become too low to admit of its ignition.[38] The chief improvements which had been made, consisted in varying the level of the oil in the cistern, or in attempts to render that level constant, by mechanical means, and in lessening the thickness of the wick, by spreading its substance into a flat form, thus reducing the stream of gas which escapes from the centre of a thick cylindric wick without being burnt, and thereby causing a more complete combustion, and producing less smoke and a whiter flame. To ARGAND belongs the great merit of having first formed the wick into a hollow cylinder, thus supplying the flame with two currents of air, one of which, as in the case of the solid wick, envelopes the flame, and the other, passing through the centre of the wick, is enveloped by the flame itself. He also added a chimney, which served to defend the flame from irregular draughts of air, and to regulate the proportion between the velocities of the currents of air and the stream of gas. This was indeed a most important step in the art of illumination, and causes the great difference between the incomplete combustion, which, owing chiefly, as we have seen, to a defect in the supply of air, always takes place with a solid wick (from which much unburnt gas escapes in the form of smoke), and that more perfect combustion in which passage is given for a free current of air through the centre of the wick. The invention of ARGAND came nearly perfect from his hands; and but a few slight modifications of his original arrangement have been introduced. The Argand burner consists of two concentric tubes or cylinders, separated by a small annular space, which is shut at the bottom, and communicates by a pipe with the oil fountain, whose level ought to be a little _below_ the level of the upper edge of the cylinders. In this annular space, partly filled with oil from the fountain, stands a cylindric wick of cotton, loosely wove, into which the oil rises freely by capillary action. The wick has its lower edge fixed to a metallic ferule or ring, called a wickholder, which (by means of a peculiar arrangement, to be afterwards described) gives the power of raising or depressing the wick to any convenient level with regard to the burner. A cylinder of glass, of greater diameter than the burner, rests on a gallery or ring which hangs from the burner and surrounds it. This glass cylinder, or chimney as it is generally called, should stand vertically with its axis coincident with that of the burner itself. The effect of this arrangement is obvious, and has already in part been indicated. The flame is thus necessarily bounded on all sides by two conical concentric surfaces, one external and concave, and the other internal and convex, both of which receive a free current of air. The flame is therefore very thin in every direction; and, as a consequence of the mutual radiation of its different parts on each other, it is throughout its entire surface of more equal temperature than can ever be attained in the thick solid wick or the narrow flat one. The glass cylinder also increases the force of the two currents which pass outside and inside of the flame; and the union of so many favourable circumstances produces a greater amount of pure light than has yet been obtained by any other method. The contraction of the glass chimney (known by the technical name of the _shoulder_) at a point a little above the level of the wick, tends to direct the current of air inwards on the flame, thereby causing a more perfect combustion and the evolution of more light.

[38] That the form of a flame is necessarily conoidal, and that
its height is determined by the relation subsisting between its
diameter and the continually varying velocities of the currents
of gas and air, may be easily shewn; and the combustion of each
annular film of the stream of gas from the wick can take place only
at a level determined by, and continually varying with, the ratio
of the velocities of the streams of gas and air. I am unwilling to
offer this explanation in my own words, when those of M. Peclet,
in his excellent work, Traité de l’Eclairage, are at hand,--“Let
us conceive,” says he, “a very thin film or layer of inflammable
gas placed horizontally, and which rises into the air parallel to
itself, with a uniform motion. We shall suppose that it cannot be
burnt, except at its circumference, and that the top and bottom of
the film are, by some means, preserved from combustion (they are so
preserved in ordinary flames, by the films which precede and follow
them). If the circumference is at a high enough temperature it will
burn; at each instant the film or layer of air, which has assisted
the combustion and also the products of that combustion, being very
hot, will rise very rapidly, and will make room for other layers or
films of air, which will rise in their turn; and as the diameter
of the film of gas is continually diminishing, it is obvious that
its combustion will offer the appearance of a series of circles
continually growing smaller, and terminating at length in a point.
If we trace in thought the series of circles which the combustion
has successively developed, we shall form a cone whose length will
depend on the ratio of the velocities of the films of gas and of
air which escape after combustion. If, for example, the velocity
of the current of air were very great, compared to the velocity of
the cylinder of gas, the entire combustion would take place, while
the film of gas passes over a very small space; and the cone formed
by the succession of luminous circles would, consequently, be very
short. If, on the contrary, there were but a very small difference
between these velocities, the luminous circles would only appear
at considerable intervals from each other; for the air which had
served for combustion, being unable to feed it longer, the surface
of the cylinder could not become luminous until the difference of
velocity had freed it from the air which had served for the preceding
combustion. If, then, we imagine a set of similar films succeeding
each other, each of them would give rise to the same series of
coloured rings; and as there would be a film in each section of the
cone in a state of combustion at the same instant of time, the cone
would, of course, appear luminous throughout its height.”--Peclet,
_Traité de l’Eclairage_, p. 51.

Great as the improvement of ARGAND undoubtedly was, the value of the lamp alone as a means for the illumination of lighthouses must be regarded as comparatively small. The primary object of a lighthouse is to give early notice to the mariner of his approach to the coast, and it is therefore necessary that the light be of such a kind that it may be seen at a great distance. Every one is practically acquainted with the fact that the rays proceed in all directions from a luminous body in straight lines; and if we could obtain a ball equally luminous in every part of its surface, it would give an equal share of light to every part of the inner surface of a hollow sphere, whose centre coincided with the centre of the ball. Again, if an opaque body were placed between the luminous ball and the hollow sphere, the part opposite that body would be deprived of the light by the interception of the rays, and no light would emerge from a hole bored in that part of the surface of the hollow sphere. The bearing of these facts is obvious; and no one can fail to perceive that in the case of a lighthouse illuminated by a single unassisted burner, a seaman could only receive the benefit of that small portion of light which emerges from the lamp in a line joining his eye and the centre of the flame. The other rays would be occupied partly in making the light visible in other parts of the horizon, and but a very small portion of them would be usefully employed for that purpose, while all the rest would be lost by escaping upwards into the sky, or downwards below the plane in which seamen can see a lighthouse. This state of matters would be little improved by increasing the number of burners, as the effective part of the light would only be augmented by the addition of an equally trifling portion of light from each burner. The small pencils of rays thus meeting at the eye of a distant observer, would form a very minute fraction of the whole quantity of light uselessly escaping above and below the horizon, and also at the back of each flame; and the wasteful expenditure of light would be enormous. By such a method no practically efficient sea-light could ever be obtained.

CATOPTRIC[39] SYSTEM OF LIGHTS.

[39] From the Greek κατοπτρον, a _mirror_; a compound of κατα,
_opposite to_, and ὂπτομαι, _I see_.

For those defects a simple remedy is found in the well known power possessed by most bodies, of _reflecting_ or throwing back from them the light which falls upon them. This property is not possessed by all reflecting bodies in an equal degree, some absorbing more and some less of the incident light. Perhaps the earliest attempts to apply this property as a corrective for the direction of the rays from a Lighthouse, would be confined to placing plane mirrors behind each lamp; yet this would prove but a partial remedy, as it would still leave the greater part of the light to stray above and below the proper direction. Hollow mirrors of a spherical form might next be tried; and if properly placed with reference to the flame, would constitute a very great improvement in lighthouse illumination. But those steps in the march of improvement are more imaginary than real; and I am not aware of any well authenticated records of such gradual attempts having preceded the adoption of the right mode of applying reflection as a means of rectifying the direction of the rays emerging from a lighthouse. There is, on the contrary, distinct evidence that the impulse given by ARGAND’S invention, led to an immediate adoption of the most perfect form of reflecting instruments.

~Application of Paraboloïdal Mirrors into Lighthouses.~

The name of the inventor of paraboloïdal mirrors and the date of their first application to Lighthouses, have not been accurately ascertained. The earliest notice which I have been able to find, is that by Mr WILLIAM HUTCHINSON, the pious and intelligent author of a quarto volume on “Practical Seamanship” (published at Liverpool in 1791), who notices (at p. 93) the erection of the four lights at Bidstone and Hoylake, in the year 1763, and describes large parabolic moulds, fashioned of wood and lined with mirror-glass, and smaller ones of polished tin-plate, as in use in those Lighthouses. Mr HUTCHINSON seems to have understood the nature, properties, and defects of the instruments which he describes, and has shewn a good acquaintance with many of the most important circumstances to be attended to in the illumination of Lighthouses. Many claims to inventions rest on more slender grounds than might be found in Mr HUTCHINSON’S book for concluding him to have first invented the paraboloïdal mirror and applied it to use in a Lighthouse;[40] but, in the absence of any statement as to the date when the mirrors were really adopted, the merit of the improvement must, in justice, be awarded to others.

[40] Mr HUTCHINSON seems also (“Practical Seamanship,” p. 198) to
have tried speculum metal as a material for Lighthouse reflectors.

M. TEULERE, a member of the Royal Corps of Engineers of Bridges and Roads in France, is, by some, considered the first who hinted at the advantages of paraboloïdal reflectors; and he is said, in a memoir dated the 26th June 1783, to have proposed their combination with Argand lamps, ranged on a revolving frame, for the Corduan Lighthouse. Whatever foundation there may be for the claim of M. TEULERE, certain it is that this plan was actually carried into effect at Corduan, under the directions of the CHEVALIER BORDA; and to him is generally awarded the merit of having conceived the idea of applying paraboloïdal mirrors to lighthouses. These were most important steps in the improvement of lighthouses, as not only the power of the lights was thus greatly increased, but the introduction of a revolving frame proved a valuable source of differences in the appearance of lights, and, in this way, has since been the means of greatly extending their utility. The exact date of the change on the light of the Corduan is not known; but as it was made by LENOIR, the same young artist to whom BORDA, about the year 1780, entrusted the construction of his reflecting circle, it has been conjectured by some that the improvement of the light was made about the same time. The reflectors were formed of sheet-copper, plated with silver, and had a double ordinate of 31 French inches. It was not long before these improvements were adopted in England, by the Trinity-House of London, who sent a deputation to France to inquire into their nature. In Scotland, one of the first acts of the Northern Lights Board in 1786, was to substitute reflectors in the room of the coal-light then in use at the Isle of May in the Frith of Forth, which, along with the light on the Cumbrae Isle in the Frith of Clyde, had, till that period, been the only beacons on the Scotch coast. The first reflectors employed in Scotland were formed of _facets_ of mirror glass, placed in hollow paraboloïdal moulds of plaster, according to the designs of the late Mr THOMAS SMITH, the Engineer of the Board, who (as appears from the article _Reflector_, in the Supplement to the third edition of the Encyclopædia Britannica) was not aware of what had been done in France, and had himself conceived the idea of this combination. The same system was also adopted in Ireland; and in time, variously modified, it became general wherever lighthouses are known.

~Reflection.~

To enable us to enter on the subject of the proper forms of reflectors, we must glance very briefly at the _laws of reflection_. Those laws are two in number. _1st_, The ray which falls on a reflecting surface, called the _incident_ ray, and the ray which leaves the reflector, called the _reflected_ ray, are always in one _plane_, which plane is perpendicular to the _reflecting surface_. _2d_, The angle which the _reflected_ ray makes with the reflector is always equal to the angle which the _incident_ ray makes with it, or, in other words, the angle of _incidence_ is equal to the angle of _reflection_.[41]

[41] This will be more readily understood by referring to the
accompanying figure (No. 22), in which CDEF is the reflecting
surface; GHOKI the plane of reflection perpendicular to that surface;
BO a line perpendicular or _normal_ to the surface CDEF; and AO the
incident ray. Then if in the plane GHOKI, the angle BOI be made equal
to AOB, OA′ is the reflected ray; BOG is then the angle of incidence;
and BOI the angle of reflection. GOH and IOK, which are the
complements of those angles, are, indeed, more strictly speaking, the
angles of incidence and reflection; but in cases where the reflecting
surface is curved, it is more convenient to refer the angles to the
normal BO.

It would lead to prolixity altogether superfluous in this place, to explain, in a rigorous manner, the effects produced by various reflecting surfaces on the direction of the rays incident on them; as any one who comprehends the laws of reflection just enumerated, may easily satisfy himself of the following truths: _1st_, That a plane mirror makes no change on the divergence of the rays, but merely causes them to emerge from its surface in the same direction as if they had come from a point as much behind the mirror as the luminous body lies in front of it. _2d_, A convex reflecting surface increases divergence, and disperses the rays in the same manner as if they had come directly from a point behind it, whose distance from the mirror increases with the distance of the luminous body from its surface, and diminishes with the degree of convexity of the mirror. _3d_, A concave surface diminishes the divergence of the rays incident upon it from a point between the surface and its centre of curvature; the distance of the point in which the reflected rays converge diminishing as the distance of the radiant point or the concavity of the mirror is increased. It is obvious, therefore, that concave mirrors are those which are required to produce a correction of the path of the rays, so as to apply them to most advantage in a lighthouse, the object to be attained being that of throwing the greatest amount of light towards given points in the horizon, and collecting the divergent rays, which, as we have already seen, are scattered above and below it.

To simplify our view of this matter, I shall, in the first place, suppose that the object to be attained is to throw the whole rays of a single lamp, with an infinitely small flame, to a given mathematical point at a moderate distance; and, as this is a case which can hardly occur in the practice of Lighthouse illumination, I content myself with observing that this object may be attained _approximately_ by placing the lamp in front of a spherical mirror at any distance greater than half the radius of the curve surface, or _accurately_ by placing it in one focus of an elliptical mirror; in all those cases the rays would meet in the opposite, or, as they are termed, _conjugate foci_. Let us next suppose that our object is to illuminate, by means of a mathematical point of light, a small circular space on the horizon equal in diameter to the mirror employed; this object will be rigorously attained only by placing the light in the focus of a paraboloïdal reflector. The same object may be approximately attained by placing the light in a spherical mirror, at a point _half-way_ between the centre of curvature and the surface of the mirror, provided the surface of the mirror shall subtend only a small angle at the centre of curvature. The paraboloïdal mirror, on the contrary, has the property of converging to the focus parallel rays falling upon every point of its surface, however extended it may be.

~Paraboloïdal Mirrors.~

Any one practically acquainted with this subject, must at once perceive that the paraboloïdal mirror completely fulfils one great object required in a lighthouse; and to render this more obvious to the general reader, I shall, for the present, confine my remarks to the case of those lighthouses which exhibit to the mariner in every part of the horizon, pencils of light at certain intervals of time, separated by periods of darkness, reserving the consideration of Lights which are continually in sight all round the horizon or over a given portion of it, for a subsequent part of these Notes. In doing this, I am aware that I may appear to be departing from the strict order of investigation, by suddenly introducing the idea of motion; but a little consideration will, I think, satisfy the reader that this is, in reality, the more convenient mode of treating the subject. Let us suppose, then, that our object is to give occasional flashes of light, separated by intervals of darkness, to seamen in various azimuths and at various distances from a lighthouse. It is obvious that this may be most efficiently done by causing concave mirrors, which collect the rays from lamps placed in them and thereby increase the light in front of the mirror, to revolve round a vertical axis with a velocity suited to produce the required number of flashes in a given time. The paraboloïdal mirror is best adapted for producing this effect, for the following reasons: _1st_, Because it alone produces a rigorous parallelism of all rays proceeding from its focus, and falling upon any point of its surface, however distant the point of reflection from that focus, or however far _in front_ of it. _2d_, Because it therefore embraces in its action the greatest number of the whole rays coming from the focus, and, _cæteris paribus_, will produce the strongest light. _3d_, Because the _theoretical_ object to be attained is to make those flashes equally powerful at any distance, an effect which would be rigorously fulfilled by placing an infinitely small flame in a perfect paraboloïdal mirror. And, _4th_, Because, although absolute equality of luminousness at any distance is not attainable, and, in practice, is inconsistent with other conditions required in a useful light, we still, by using the parabolic mirror, make the nearest approach to parallelism of the reflected rays, and consequently obtain the strongest light which is consistent with a due regard to a certain duration of the flash on the eye of a distant observer, which is measured by the angle of the luminous cone projected to the horizon.

Having thus so far anticipated what some might think would more naturally have occurred in a subsequent part of these Notes, I return to a more detailed consideration of the parabola itself, and its product, the paraboloïdal mirror. I content myself, however, with describing the parabola, by that property which peculiarly adapts it to the purposes of a lighthouse. The parabola, then, is a curve of the second order, obtained by cutting a cone in a plane parallel to one side, which possesses this remarkable property, _that a line drawn from the focus to any point in the curve, makes, with a tangent at that point, an angle equal to that which a line parallel to the axis of the curve makes with that tangent_.[42]

[42] See third corollary to Proposition III. of Wallace’s Conic
Sections, which shews that a tangent to the parabola makes equal
angles with the diameter which passes through the point of contact
and a straight line drawn from that point to the focus. The curve
may be traced in two different ways, both dependent on the property,
_that the distance of any point in the parabola from the focus is
equal to its distance from the directrix_.

To draw the curve mechanically (fig. 23), let F be the focus, MF
the focal distance (chosen at pleasure according to rules which I
shall afterwards notice), KMX is the axis, and AB the directrix (the
dotted line _f_ F _e_, bounded by the curve at either end, would then
be the _parameter_ or _latus rectum_). Place the edge of the straight
ruler AKHB along the directrix; and let LHB be a square ruler which
may slide along the fixed ruler AKHB, so that the edge HL may be
constantly perpendicular to AB, or parallel to MX, the axis; let LDF
be a string equal in length to HL, and having one end fixed in F, and
the other at L, a point in the sliding square. Then if the string be
stretched by a pencil D, so as to keep the part DL close to the edge
of the square, and if at the same time the square be gently pushed
along the line AB, the point D will be forced to move along the
edge LH of the square, and will trace out a curve which will be the
required parabola. This is obvious from the consideration, that the
string LDF being equal in length to LH, and LD being common to both,
the remainder DF must be equal to the remainder DH, so that the point
which traces the curve being equidistant from the directrix and the
focus must, in terms of the above definition, describe a parabola.

In the second place, the same property, as already stated, furnishes
us with the means of tracing the curve by finding successive points
therein. Draw a line _a b_ perpendicular to the axis OX, and the
position in this line, of a point _p_ through which the curve passes,
is easily found thus: Describe from F the focus as a centre with a
radius equal to the perpendicular distance O _d_ of the line _a b_
from the directrix AB, a circle cutting the line _a b_ in two points
_p_ and _p′_; then both these points are in the curve. By repeating
the same process, any number of points in the curve may be obtained.

Lastly, from the equation to the curve, the length _y_ of any
ordinate may be computed, in terms of _m_ its principal focal
distance, and x its abscissa, by the simple expression,--

_y_ = √(4 _m_ _x_).

It is easy to see, that if this curve revolve about its axis, it will generate a parabolic conoid, which we may conceive to be concave or convex, as we please. If the surface be concave, we obtain the mirror of which we are in search; for every principal section, or that passing through the axis of such a mirror, will necessarily possess the same properties as that of the plane curve, and will each have a focus meeting in one and the same point; the union of all these sections will therefore form a mirror capable of reflecting, in a direction parallel to the axis and to each other, all the rays of light which fall on its surface.

~Divergence of Paraboloïdal Mirrors.~

We have already seen that a perfect paraboloïdal mirror, with a point of light infinitely small placed in the focus, would project a beam equally intense at any distance, every transverse section of which would be of the same superficial extent. In practice, these conditions can never be rigorously fulfilled. No perfect instrument can come from the hands of man, and every mirror must of necessity possess many defects. To obtain a true mathematical point of light is also impossible; and for the purposes of a lighthouse, it would be completely useless, as will appear from the following simple considerations. Let us suppose that a true paraboloïdal mirror, having a double ordinate or space of two feet, and illuminated by a point of light, projects a truly cylindric beam of light to the horizon, and that it revolves horizontally round a vertical axis, with such a velocity as to cause the beam to pass over the eye of an observer stationed at the distance of 100 feet in one second of time, and we shall find that another observer, at a distance of 15 miles from the mirror, would not see the light at all, although of equal size, because its velocity at that distance would be so great as only to be present to his eye for ¹⁄₇₉₂d of a second, a space of time far too short to make a perceptible impression on the eye of a distant observer. This is no mere hypothesis unsupported by facts; for I shall have occasion, in another part of these Notes, to describe certain experiments, by which it was ascertained that a beam of light emerging from a lens, and passing over the eye of an observer at 14 miles distance, in a space of time equal to ¹⁄₁₆₆th of a second, became altogether invisible at that distance.

For this evil, happily a very simple and efficient remedy may be found in what may be said to constitute a _theoretical_ defect in the combination of the Argand burner with the reflector. The burner, instead of being a mathematical point, has generally a diameter of about one inch, and a ray proceeding from the edge of the flame to any point on the surface of the mirror, makes with the line joining that point and the principal focus an angle which, being repeated by reflection, gives the effective divergence of _each_ side of the mirror at that point.[43]

[43] This is easily understood by reference to the accompanying
figure (No. 25.), in which AOB is a central section of a paraboloïdal
mirror.

PF = distance from the focus F to a point in the curve P, and PG a
tangent drawn from P to the surface of the flame at G;

FG = radius of the wick or flame;

and GPF = G′PF′ = divergence of one side of mirror, and consequently
2 GPF = the whole effective divergence of the mirror at that cross
section.

GF
Now sin GPF = --
PF

or the sine of the divergence from each point

Radius of flame.
= -------------------------------------------
Distance from focus to point of reflection.

It is obvious that this quantity which varies _inversely_ with the
distance of the reflecting surface from the focus, is greatest
at the vertex of the curve, and least at the sides or edges of
the paraboloïd. The most useful part of the light, or that which
conduces to the strongest part of the flash in a revolving light,
is that which is derived from the cone of rays which is bounded by
the limits of this _minimum_ divergence; for the faint light which
first reaches the eye of a distant observer, in the revolution of a
reflector, is not that which is reflected by the sides or edges, as
might at first be supposed, but proceeds from the centre. The light,
in fact, gradually increases in power in proportion as additional
rays of reflected light are brought to bear on the observer’s eye,
until, last of all, the extreme edge of the mirror adds its effect.
The light continues in its best state until the opposite limit of
minimum divergence has been reached, when it begins gradually to
decline, receding from the margin of the mirror towards the centre,
and, having at length reached the limit of its maximum divergence,
it finally disappears at the centre. The increase and decline of the
power of a mirror in the course of its movement round the circle of
the lantern, as seen by a distant observer, will, therefore, in all
its different states, be measured by the areas of a series of circles
described from its focus, with radii equal to the distance of the
focus from the point of the mirror which reflects to the observer’s
eye the extreme ray which can reach him in any given position of
the mirror. This will be more easily understood by referring to the
accompanying diagram, Fig. 26, in which _e a e′_ is the principal
section of a paraboloïdal mirror, F its focus, αFA its axis, and FK
the radius of the flame. If the reflector revolve round a vertical
axis at O, an observer placed in front of it (at a distance so great
that the subtense of the mirror’s width would be small enough to
allow us safely to consider the lines drawn from _e_ and _e′_ to
his eye as parallel), would receive the first ray of light in the
direction _a_ D, as reflected at _a_, from a single point on the
edge of the flame (where a tangent to the flame would pass through
_a_); and conversely he would lose the last ray at D′, as reflected
at _a_, from a single point on the opposite margin of the flame; and
hence, as above, the greatest divergence is measured by the angle
which the flame subtends at the vertex a of the mirror, being the
sum of the angles _α_ and _α′_. We shall next suppose the mirror to
move a little, so that the observer may receive at G a ray of light
from some other point in the flame which is reflected at _b_; while
another ray from an opposite point reflected at _b′_ would be seen
in the parallel direction _b′_ G′, thus indicating the boundary of
a circular portion of the mirror _b a b′_, the whole of which would
reflect light to the distant observer’s eye. Again, let us suppose
a ray to come from another part of the flame, and be reflected at
the mirror’s edge _e_ into the direction _e_ H, and another from
the opposite side of the flame to be reflected at its opposite edge
_e′_, into the direction _e′_ G″, and we obtain the full effect of
the whole reflecting surface, which will continue unabated until
the mirror in the course of its revolution shall reflect at _e′_ to
the observer’s eye, a ray from a point in the margin of the flame
(through which a tangent drawn from _e′_ to the flame would pass)
in such a direction, that the angle which it makes with the axis of
the mirror is equal to that subtended by the radius of the flame at
the distance F _e_ or F _e′_. After this the light would recede from
the edges of the mirror in the same gradual manner, until it should
vanish in the direction _a_ D′, which is the opposite limit of the
extreme divergence of the instrument. In the above explanation, I
have confined myself simply to the effects of the outer ring of the
flame, which is the source of divergence; but I need not remind the
reader that every portion of the flame radiates light, which, being
reflected, conduces to the effect. Some rays also are passing from
the opposite sides of the flame through the true focus, so as to be
normally reflected in lines parallel to its axis. The solid lines
in the diagram shew the theoretical reflection of rays proceeding
from F to _b_, _b′_, _e_, _e′_, where they are diverted into the
directions _b_ B, _b′_ B′, _e_ E, and _e′_ E′; and by contrast with
the dotted lines, serve to render more perceptible the path of the
divergent rays which come from the edge of the flame. The Greek
letters indicate the angles of divergence, and point out their
relations to each other on either side of the mirror. The arcs of
greatest and least divergence are marked in the diagram. This subject
will be found treated less directly, but, certainly, more concisely
and neatly, by Mr W. H. BARLOW, in a paper on the Illumination of
Lighthouses in the London Transactions for 1837, p. 218.

It is still more obvious that a perfect paraboloïdal figure, and a luminous _point_ mathematically true, would render the illumination of the whole horizon by means of a fixed light _impossible_; and it is only from the divergence caused by the size of the flame which is substituted for the _point_, that we are enabled to render even revolving lights practically useful. But for this aberration, the slowest revolution in a revolving light would be inconsistent with a continued observable series, such as the practical seamen could follow, and would, as we have seen, render the flashes of a revolving light too transient for any useful purpose; whilst fixed lights, being visible in the azimuths only in which the mirrors are placed, would, over the greater part of the distant horizon, be altogether invisible. The size of the flame, therefore, which is placed in the focus of a paraboloïdal mirror, when taken in connexion with the form of the mirror itself, leads to those important modifications in the paths of the rays and the form of the resultant beam of light, which have rendered the catoptric system of lights so great a benefit to the benighted seamen.

In order to obtain a mirror capable of producing a given divergence of the reflected beam, therefore, we must proportion its focal distance to the diameter of the flame in such a manner, that the sine of _one-half_ of the whole effective divergence of the mirror, may be equal to the _quotient of the radius of the flame, divided by the distance of a given point on the surface of the mirror from the focus_. The best proportions for paraboloïdal mirrors depend on the objects which they are meant to attain. Those which are intended to give great divergence to the resultant beams, as in fixed lights, capable of illuminating the whole horizon at one time, should have a short focal distance; while those mirrors which are designed to produce a nearer approach to parallelism (as in the case of revolving lights which illuminate but a few degrees of the horizon at any one instant of time), will have the opposite form. Those two objects may, no doubt, be attained with the same mirror, by increasing or diminishing the size of the burner; but that is by no means desirable, as any change on the size of a burner, which is found to be the best in other respects, must be considered as to some extent disadvantageous.