of Mountaine and Dodson. He would thus, in case of tolerable success, effectually destroy the force of Halley's chief objection to the duality of the poles, by showing that two, when properly assumed, would give the declination under the same meridian partly east and partly west. "Comme je prouverai cela indubitablement dans la suite," he adds, in conclusion, "il me sera permis de regarder l'hypothèse de quatre poles magnétiques comme forte douteuse; et avant qu'on ait très évidemment prouvé, que deux poles magnétiques ne sont pas suffisans pour expliquer les phénomènes de la déclinaison magnétique, ce seroit contre les régles d'une bonne physique si on vouloit recourir à quatre poles."*

Euler's researches were entirely confined to the horizontal needle, and his hypothesis is simply this:-that the horizontal needle is a tangent to the circle passing through the place of observation and through the two points on the earth's surface at which the dipping needle becomes vertical, or at which the horizontal needle loses all directive tendency. At art. 55, of his first paper+, he thus expresses his principle, and his reasons for adopting it. "La solution de ce problème et des suivans est fondée sur ce principe, que la direction magnétique sur la terre suit toujours le petit cercle, qui passe par le lieu proposé et les deux poles magnétiques de la terre. On m'accordera bien ce principe à l'égard de la véritable direction magnétique, qui renferme ensemble l'inclinaison et la déclinaison; mais puis que la déclinaison dont il s'agit ici, se régle sur le plan vertical, qui passe par la direction magnétique, on en pourroit tirer quantité d'objections, dont la discussion menerait trop loin, et surpasseroit les bornes de notre connoissance. Mais on pourra en sorte fixer les idées, qui entrent ici en considération, que ces objections n'y ayent plus de prise. Si l'on plaçoit là les poles magnétiques de la terre, où l'axe magnétique traverse la surface de la terre, on seroit sans doute fort embarrassé; puis que la déclinaison n'y seroit plus indéterminée, à moins que l'arc magnétique ne passeroit par le centre de la terre. raison j'établirai les poles magnétiques de la terre là, où la véritable direction magnétique est verticale, de sorte que dans ces endroits il ne puisse y avoir question de la déclinaison; et ce sera dans ces points, où toutes les lignes Halleyennes doivent aboutir, de même qu'aux poles naturels de la terre. Or déterminant en sorte les poles magnétiques de la terre, sans s'embarrasser de l'arc véritable magnétique, les objections mentionnées n'empêcheront plus, qu'on n'accorde le principe établie, c'est à dire que partout l'aiguille aimantée dirige suivant la tangente du petit cercle tiré sur la surface de la terre par chaque lieu proposé, et les dits poles magnétiques, où l'inclination devient verticale."

Euler divided his paper into four general cases:

1. Where the poles are diametrically opposite:

Par cette

2. Where they are not diametrically opposite, but still in opposite meridians:

3. Where they are on the same meridian:

4. Where they are anyhow situated.

The first then he finds insufficient to furnish any approximation to the phenomena of terrestrial magnetism: though he finds in all, except the first, an answer to Halley's fundamental objection to the duality of + Ib., p. 200-1.

*Mém. de Berl., tom. xiii., p. 177.

the poles*. Now, that the law of force is known, it is easy to see that the projection of the dipping needle, upon the horizontal plane, cannot coincide with the tangent to a circle drawn through any two fixed points: and hence to ascertain that the descriptive hypothesis of Euler is incompatible with the necessary consequences of the true theory of magnetism. Euler, however, from not having any method by which, from observed positions of the needle, to determine the positions of the two points which he has called poles, was unable to do more than approach towards them by successive tentative operations. After all his attempts he found discrepancies: but these he attributed not so much to any error in his hypothesis, as in the difficulty of fixing the position of the poles, and the imperfection or inaccuracy of recorded observations. Had it occurred to him to seek the equation of condition that must subsist amongst the constants in the equations of their circles, that they may have common points of intersection, it is obvious that three good observations would have enabled him to calculate the actual positions of the corresponding magnetic poles: and hence he would at once have been able to bring his hypothesis to a direct and decisive testt.

So long as the poles had been considered to be only two, and diametrically opposite, the point at which the dipping needle would become vertical, and those towards which the Halleyan Lines ultimately tended, were points identical with those in which the axis of the magnet pierced the terrestrial surface. Euler's rejection of that symmetry of the distribution of magnetism led also to a separation of the systems of points, each of which had alike before answered to any definition of a pole. Some authors have subsequently fixed their attention upon one definition, and others on another, under almost every variety of aspect: yet still they all alike call the points defined in their respective ways by the common epithet, "pole." Much vagueness and confusion have arisen from this; and more especially as, for the most part, the term has been used without any previous definition at all, and the import of it left to be derived from the manner in which it is employed in the investigation. It is even, by many writers, used in more senses than one, that is, to designate points which are essentially distinct, except in the particular case of the diametral situation of the points in question. Some respectable authors have thereby not only led their readers into difficulty and error, but have even been led into singular mistakes on their own reasoning, in consequence of their mistaken condensation of two points essentially distinct, into one single point. Euler, however, very properly discriminates here between the points at which the needle is to be vertical, and those at which the magnetic axis intersects the surface of the earth: and from the first and well-established principles of magnetism, he is right in considering the point at which the horizontal needle loses its directive qualities, and those at which the dipping needle becomes vertical, as identical points. The contemplation of the magnetic curves (or currents),

* That is, with respect to the present state of the Halleyan Lines; but he thinks, that from the motion of these poles with respect to each other, each case might possibly be applicable to some past or future state of terrestrial magnetism.

+For the determination of such points, under such circmmstances, see our Note A, where the requisite equation of condition is investigated.

too, enabled him to see that if the needle ever could become vertical to the earth's surface, whilst the magnetic energy was unsymmetrically distributed with respect to the geometrical figure of the earth, the centres from which these currents emanated must lie within the earth, and at some considerable distance from the surface. Euler views this as a matter of course, not necessary to dwell upon, but either so familiar to his own mind, or perhaps to other philosophers also, as not to require to be enforced by a single reason, such as we should expect would be the case with a novel or unobvious doctrine. If, however, we except Halley's poles of his Terrella, which may be viewed as fixed deep below the surface of the outer shell of the earth, this is the first distinct view that we have met with of that consequence respecting the positions of the centres of magnetic force. The definition of the pole which is employed by Euler, is that which has most commonly prevailed since his time; but the methods of determining it, and its connexion with other points, from observation, have been mixed up with errors of various kinds, and errors too, in principle as well as in detail.

Subsequent researches led Euler to discover that his assumption respecting the needle being a tangent to the forementioned circle, was erroneous, and incapable of being made to accord with the phenomena completely. He, therefore, substituted the following, under the title of Corrections necessaires pour la Théorie de la Déclinaison Magnétique proposée dans le xiii. volume des Mémoires.

He supposes now that the true magnetic poles are at the surface of the earth, the chord joining which he calls the magnetic axis, and its middle the magnetic centre. Then a line drawn from the place of observation to the magnetic centre, being made to form the base of an isosceles triangle, one side of which being coincident with the magnetic axis, the other side will be the line of rest of the freely suspended or dipping needle. This hypothesis (and he actually calls it such; vid. vol. xxii., p. 227) fulfils certain conditions that were essential to any good theory. 1. It gives the needle the approximately accurate positions at the equator, the needle and axis being then parallel. 2. It fulfils the condition of the needle and axis forming a continuous line when at the poles. 3. It furnished two points at which the needle would be vertical. 4. It gave a series of positions, single for each place, and having a certain, and oftentimes pretty close, approach to the true position. Its defects are:

1. That it is inconsistent with the since discovered laws of magnetic action; but this, of course, ought not to be urged strongly here, since it only invalidates the hypothesis itself generally, and does not point out its peculiar discrepancies with phenomena.

2, That in all experiments with magnets we uniformly find the position of verticity to the magnetic axis (much more than of verticity to the circles, whose common chord that axis is), beyond the extremities of

*

It is extremely singular, after once entertaining the views he seems to have done in his former paper, as to the internal position of the poles, that he should have adopted this opinion of their superficial position. Possibly, indeed, he might not have intended to express any view at all on that subject in his former paper, and we might have interpreted him, therefore too liberally.

the magnet itself: whilst Euler's hypothesis furnishes positions between the extremities of the magnet. Vid. pr. vi., p. 233.

3. That it gives values of the inclination extremely remote from those furnished by observation.

4. That it does not furnish a magnetic equator differing from a great circle, and, consequently, in this respect, too, it is incompatible with observation. This, however, with one exception, is common to all the theories that have been proposed as to the position of the magnetic poles, ever since the true law of the variation of magnetic force has been accurately determined and generally understood.

5. That a curve, described by the continued intersections of the needle, according to his hypothesis, in any one plane passing through the magnetic axis, bears but little resemblance, as to actual form and curvature, at corresponding values of the abscissa, to the true magnetic curve.

By means of some lemmas in spherical trigonometry and this hypothesis, he proceeds to finds expressions, successively, for-the inclination of the needle, as it is referred to a tangent at the place of observation to the magnetic plane *—the inclination of the needle to the horizon-the declination of the needle from the geographical meridian— and one or two other lines and angles. These expressions are derived by a mixture of plain and spherical trigonometry, and his preceding lemmas (to which no exception can be taken, except that they are much less elegant and systematic than Euler's mathematical processes commonly are) and they are, for the most part, exceedingly complicated in their forms. He does not attempt to discuss the form of the Halleyan Lines under this hypothesis; and we doubt whether it would be possible to obtain an equation which would express them by means of these expressions and methods: but he proposes to substitute another class of lines, which he calls " magnetic routes," instead of them (p. 244), which he thus defines-Les routes magnétiques sont des lignes tirées sur la surface de la terre, dont des tangents marquent en chaque lieu le direction de la boussole and finds them to be less circles of the sphere whose centres are in the magnetic equator, and whose circumferences pass through the poles where the needle is vertical.

In a suggestion at the close, he purposes to consider the magnetic centre slightly removed from the middle of the magnetic axis, to make it

He calls them, very properly, magnetic meridians. They are the planes, passing through the true poles, and the several places of observation. In the language of the present day, the term, magnetic meridian, is exceedingly inappropriately used. By this term is commonly meant, the vertical great circle, passing through the axis of the needle. Various peculiar phenomena, have been imagined to take place in this plane, and it is a favourite position amongst meteorologists, in which to discover peculiar modifications of the Aurora Borealis ! It would be difficult to assign any reason, admitting the connexion between the causes of terrestrial magnetism, and the aurora, why a plane not passing through the centres of magnetic force, should be expected to have any peculiar claim to these remarkable peculiarities of phenomena, in preference to any other. The truth is, these peculiarities take place in every possible position; but, like the predictions of future events, rainy days, &c., when a dogma is once laid down the fulfilments are faithfully recorded, and duly dwelt upon, and as even rain a day or two before or after fulfils the wizard's prediction, so the phenomenon having an azimuth a few degrees east or west of this magnetic meridian, is easily by a little imagination, or a little fashionable philosophical "coaxing," tranferred to the plane itself: -the more extreme cases of exception being wisely forgotten!

agree with any slight discrepancy between the result of his hypothesis and the phenomena furnished by observation: but seeing that this circumstance would render the calculation much more difficult, he does not attempt to give, or even to suggest any details respecting the effect of this last named modification. From the complexity of the results already laid down, there could be little hope that the subject under this new aspect could be effectually treated in any one of its branches by means of the mathematical processes which he has laid down, or by others having any direct analogy to them.

Meanwhile, attempts to discover the true law of magnetic action were gradually working towards a decided proof of that law which is now established beyond all question-the inverse square of the distance. It appears to have been first distinctly proved by our countryman Michel, and published in his Treatise on Artificial Magnets, in 1750. He had the merit, too, of introducing the use of the Torsion Balance into the method of experimenting on the law of action of such forces—the instrument by which Coulomb, with singular address, fully and satisfactorily established the law. Lambert, however, in the volume (xxii.) to which we have referred for Euler's second paper, published a dissertation on the same subject. This paper displays that address both as a mathematician and an experimenter, for which Lambert was celebrated in a high degree but the method involved difficulties and fortuitous disturbances against which it was impossible to oppose the most consummate skill and discrimination. He, by a very ingenious and subtle mode of reasoning, established the true law of magnetic action, and showed a considerable approximation between its results, and the results of those experiments which he made*. They were repeated and varied by Dr. Robison, both as to circumstance and the form of the apparatus: and that truly eminent philosopher was led to results still more closely agreeing with theory than Lambert had obtained. Notwithstanding this, as he still thought the discrepancies were more than ought to have existed, he was too candid to urge that the law was established beyond the possibility of dispute. (Supplement to the third edition of the Encyclopædia Britannica, vol. ii.; and Mechanical Philosophy, vol. iv., p. 340.) He stood alone, however, in his estimate of the proof afforded by the experiments of Coulomb; and forms the solitary instance of doubt respecting the conclusiveness of that completely decisive course of experiment in finally deciding the question. Besides the memoirs of Coulomb himself, in the Mém. de l'Acad., the best view of them is given by Biot, in the third volume of his Traité de Physique, p. 66-70.

Though the magnetic dip had been very early discovered, yet it seems to have been uniformly neglected in discussions respecting terrestrial magnetism, with a few such exceptions as Euler, and Lambert, and Mayer. The variation seems to have been thought to be total effect of the cause, whatever that cause might be which produced it. The intensity of the magnetism, as manifested by the horizontal needle, was, indeed, observed with attention as to its daily changes, by Graham, nearly a century

* His determination of the position of repose of a needle, subjected to the influence of a magnet, will be considered hereafter.