On the Variations of the Terrestrial Magnetism in different Latitudes.

By Messieurs Humboldt and Biot.

Read by M. Biot, in the Mathematical and Physical Class of the French National Institute 26th Frimaire, An 13. (17th December 1804.)

From the Journal de Physique, Frimaire, An 13.

An inquiry into the laws of terrestrial magnetism is no doubt one of the most important questions that philosophers can propose.

The observations already made on this subject have discovered phaenomena so curious, that one cannot help endeavouring to solve the difficulties they present; but notwithstanding the efforts hitherto employed, it must be confessed that we are absolutely unacquainted with the causes of them.

It was difficult to obtain on this point any precise knowledge at a time when the construction of the compass was still imperfect; and so little time has elapsed since the discoveries of M.

Coulomb have taught us to render them completely exact, it needs excite no astonishment that so few facts in the observations of travellers have been found worthy of confidence.

The expedition which M. Humboldt has terminated has procured for this part of philosophy a collection no less valuable than those with which he has enriched the other branches of human knowledge.

Furnished with an excellent dipping-needle, constructed by Le Noir on the principles of Berda, M. Humboldt has made more than three hundred observations on the inclination of the magnet, and on the intensity of the magnetic force in that part of America which he traversed.

By adding to these results those which he had already obtained in Europe before his departure, we shall have for the first time a series of correct facts on the variation of the magnetic forces in the northern part of the globe, and in some points of its southern part.

The friendship which M. Humboldt has testified for me since his return having given me an opportunity of communicating to him some experiments on this subject, which I made this year in the Alps, he immediately offered to unite his to mine in a memoir.

But if friendship and a desire of making known new facts induced me to accept this offer of M. Humboldt, justice forbids me to take advantage of it to his prejudice; and I must here declare, that a very small part of it belongs to me.

To place in order the facts and consequences which may be deduced from them, it is necessary to consider the action of terrestrial magnetism under different points of view, corresponding to the different classes of the phaenomena which it produces.

If we consider it first in general, we find that it acts on the whole surface of the globe, and that it extends beyond it.

This last fact, which was doubted, has been lately proved by one of us, and particularly by our friend M. Gay- Lussac, during his two aerostatic voyages.

And if these observations, made with all the care possible, have not shown the least sensible diminution in the intensity of the magnetic force, at the greatest height to which man can attain, we have a right to conclude that this force extends to an indefinite distance from the earth, where it decreases, perhaps, in a very rapid manner, but which at present is unknown to us.

If we now consider magnetism at the surface even of the earth, we shall find three grand classes of phaenomena, which it is necessary to study separately, in order to have a complete knowledge of its mode of action.

These phaenomena are; the declination of the magnetic needle, its inclination, and the intensity of the magnetic force, considered either comparatively in different places or in themselves, paying attention to the variations which they experience.

It is thus that, after having discovered the action of gravity as a central force, its variation, resulting from the figure of the earth, was afterwards ascertained in different latitudes.

The declination of the magnetic needle appears to be that phaenomenon which hitherto has more particularly fixed the attention of philosophers, on account, no doubt, of the assistance which they hoped to derive from it in determining the longitude; but when it was known that the declination changes in the same place, in the course of time, when its diurnal variations were remarked, and its irregular traversing, occasioned by different meteors; in a word, the difficulty of observing it at sea, within one degree nearly, it was necessary to abandon that hope, and to consider the cause of these phaenomena as much more complex and abstruse than had been at first imagined.

In regard to the intensity of the magnetic force in different parts of the earth, it has never yet been measured in a comparative manner.

The observations of M. Humboldt on this subject have discovered a very remarkable phaenomenon; it is the variation of the intensity in different latitudes, and its increase proceeding from the equator to the poles.

The compass, indeed, which at the departure of M.

Humboldt gave at Paris 245 oscillations in 10 minutes, gave no more in Peru than 211, and it constantly varied in the same direction; that is to say, the number of the oscillations always decreased in approaching the equator, and always increased in advancing towards the north.

These differences cannot be ascribed to a diminution of force in the magnetism of the compass, nor can we suppose that it is weakened by the effect of time and of heat; for, after three years' residence in the warmest countries of the earth, the same compass gave again in Mexico oscillations as rapid as at Paris.

There is no reason, either, to doubt the justness of M. Humboldt's observations, for he often observed the oscillations in the vertical plane perpendicular to that meridian; but by decomposing the magnetic force in the latter plane, and comparing it with its total action, which is exercised in the former, we may from these data calculate its direction, and consequently the direction of the needle.

This inclination, thus calculated, is found always conformable to that which M. Humboldt observed directly.

When he made his experiments, however, he could not foresee that they would be subjected to this proof by which M. La Place verified them.

Let HOC (plate V. fig. 1.) be the plane of the magnetic meridian passing through the vertical OC; let OL be the direction of the needle situated in that plane, and OH a horizontal.

The angle LOH will be the inclination of the needle, which we shall denote by I. If F represent the total magnetic force which acts in the direction OL, the part of this force, which acts according to OC, will be F sine of I: but the magnetic forces which determine the oscillations of the needle in any plane, are to each other as the squares of the oscillations made in the same time.

If we denote then by M, the number of the oscillations made in 10' of time in the magnetic meridian, and by P, the number of oscillations made also in 10', in the perpendicular plane, we shall have the following proportion.

[Formel] from which we deduce [Formel] The inclination then may be calculated by this formula, when we have oscillations made in the two planes.

In like manner, by making a needle oscillate successively in several vertical planes, we might determine the direction of the magnetic meridian.

As the justness of these observations cannot be contested, we must allow also the truth of the result which they indicate, and which is the increase of the magnetic force proceeding from the equator to the poles.

To follow these results with more facility it will be proper to set out from a fixed term, and it appears natural to make choice for that purpose of the points where the inclination of the magnetic needle is null, because they seem to indicate the places where the opposite action of the two terrestrial hemispheres is equal.

The series of these points forms on the surface of the earth a curved line which differs very sensibly from the terrestrial equator, from which it deviates to the south in the Atlantic Ocean and to the north in the South Sea.

This curve has been called the magnetic equator , from its analogy to the terrestrial equator, though it is not yet known whether it forms exactly a great circle of the earth.

We shall examine this question hereafter; at present it will be sufficient to say, that M. Humboldt found this equator in Peru about 7·7963° (7° 1') of south latitude, which places it, for that part of the earth, nearly in the spot where Wilke and Lemonnier had fixed it.

The places situated to the north of that point may be divided into four zones; the three first of which, being nearer the equator, are about 4·5° (4°) of breadth in latitude; while the latter, more extensive and more variable, is 16° (14°).

So that the system of these zones extends in America from the magnetic equator to 25·5556° (23°) of north latitude, and comprehends in longitude an interval of about 56° (50°).

The first zone extends from 7·7963° (7° 1") of south latitude to 3·22° (2° 54').

The mean number of the oscillations of the needle in the magnetic meridian in 10' of time is there, 211·9: no observation gives less than 211, or more than 214. From M. Humboldt's observations one might form a similar zone on the south side of the magnetic equator, which would give the same results.

The second zone extends from 2·4630° (2° 13') of south latitude to 3·61° (3° 15') of north latitude.

The mean term of the oscillations is there, 217·9: they are never below 220, or above 226.

The fourth zone, broader than the other two, extends from 10·2778° (9° 15') to 25·7037° (23° 8') of north latitude.

Its mean term is 237: it never presents any observation below 229, or above 240.

We are unacquainted, in regard to this part of the earth, with the intensity of the magnetic force beyond the latitude of 26° (23°) north; and on the other hand, in Europe, where we have observations made in high latitudes, we have none in the neighbourhood of the equator: but we will not venture to compare these two classes of observations, which may belong to different systems of forces, as will be mentioned hereafter.

However, the only comparison of results, collected in America by M. Humboldt, appears to us to establish with certainty the increase of the magnetic force from the equator to the poles; and, without wishing to connect them too closely with the experiments made in Europe, we must remark, that the latter accord so far also with the preceding as to indicate the phaenomenon.

If we have thus divided the observations into zones parallel to the equator, it is in order that we may more easily show the truth of the fact which results from them, and in particular to render the demonstration independent of those small anomalies which are inevitably mixed with these results.

Though these anomalies are very trifling, they are, however, so sensible, and so frequently occur, that they cannot be ascribed entirely to errors in the observations.

It appears more natural to ascribe them to the influence of local circumstances, and the particular attractions exercised by collections of ferrugineous matters, chains of mountains, or by the large masses of the continents.

One of us, indeed, having this summer carried to the Alps the magnetic needle employed in one of his late aerial excursions, he found that its tendency to return to the magnetic meridian was constantly stronger in these mountains than it was at Paris before his departure, and than it has been found since his return.

This needle, which made at Paris 83·9° in 10 minutes of time, has varied in the following manner in the different places to which it was carried:

Places of observation.

Number of oscillations in ten minutes of time.

Paris before his departure - 83·9 Turin - - - 87·2 On Mount Genevre - 88·2 Grenoble - - - 87·4 Lyons - - - 87·3 Geneva - - - 86·5 Dijon - - - 84·5 Paris, on his return - 83·9

These experiments were made with the greatest care, conjointly with excellent observers, and always employing the same watch verified by small pendulums, and taking the mean terms between several serieses of observations, which always differed very little from each other.

It appears thence to result that the action of the Alps has a sensible influence on the intensity of the magnetic force. M. Humboldt observed analogous effects at the bottom of the Pyrenees; for example, at Perpignan. It is not improbable that they arose from the mass of these mountains, or the ferrugineous matters contained in them; but whatever may be the cause, it is seen by these examples that the general action of terrestrial magnetism is sensibly modified by local circumstances, the differences of which may be perceived in places very little distant from each other.

This truth will be further confirmed by the rest of this memoir.

It is to causes of this kind, no doubt, that we must ascribe the diminution of the magnetic forces observed in some mountains; a diminution which, on the first view, might appear contrary to the results obtained during the last aerial voyages.

This conjecture is supported by several observations of M. Humboldt. By making his needle to oscillate on the mountain of Guadaloupe, which rises 676 metres (338 toises) above Santa-Fe, he found it in 10 minutes of time give two oscillations less than in the plain.

At Silla, near Caracas, at the height of 2632 metres (1316 toises) above the coast, the diminution went so far as five oscillations; and, on the other hand, on the volcano of Antisana, at the height of 4934 metres (2467 toises), the number of oscillations in 10 minutes was 230; though at Quito it was only 218: which indicates an increase of intensity. I observed, indeed, a similar effect on the summit of Mount Genevre, at the height of 1600 or 1800 metres (8 or 900 toises), as may be seen by the numbers which I have already given; and it was on this mountain that I found the greatest intensity of the magnetic force.

I saw on the hill of La Superga, in the neighbourhood of Turin, an example of these variations equally striking.

Observing, with Vassali, on this hill, at the elevation of about 600 metres (300 toises), we found 87 oscillations in 10 minutes of time.

On the side of the hill we had 88·8 oscillations; and at the bottom, on the bank of the Po, we obtained 87·3.

Though these results approach very near to each other, their difference is, however, sensible, and fully shows that their small variations must be considered as slight anomalies produced by local circumstances.

This examination leads us to consider the intensity of magnetism on the different points of the surface of the globe, as subject to two sorts of differences.

One kind are general:

they depend merely on the situation of the places in regard to the magnetic equator, and belong to a general phaenomenon, which is the increase of the intensity of the magnetic forces in proportion as we remove from the equator: the other kind of variations, which are much smaller and altogether irregular, seem to depend entirely on local circumstances, and modify either more or less the general results.

If we consider terrestrial magnetism as the effect of an attractive force inherent in all the material particles of the globe, or only in some of these particles, which we are far from determining, the general law will be, the total result of the system of attraction of all the particles, and the small anomalies will be produced by the particular attractions of the partial systems of the magnetic moleculae diffused irregularly around each point; attractions rendered more sensible by the diminution of distance.

It now remains to consider the inclination of the magnetic needle in regard to the horizontal plane.

It has long been known that this inclination is not every where the same: in the northern hemisphere the needle inclines towards the north; in the southern towards the south; the places where it becomes horizontal form the magnetic equator; and those where the inclination is equal, but not null, form on each side of that equator curved lines, to which the name of magnetic parallels has been given from their analogy to the terrestrial parallels.

One may see in several works, and particularly in that of Lemonnier, entitled Lois du Magnetism, the figure of these parallels and their disposition on the face of the earth.

It evidently results from this disposition that the inclination increases in proportion as we recede from the magnetic equator; but the law which it follows in its increase has not yet, as far as appears to us, been given.

To ascertain this law, however, would be of great utility; for the inclination seems to be the most constant of all the magnetic phaenomena, and it exhibits much fewer anomalies than the intensity.

Besides, if any rule, well confirmed, could be discovered on this subject, it might be employed with advantage at sea to determine the latitude when the weather does not admit an observation of the sun; which is the case in various places during the greater part of the year.

We have some reason to expect this application when we see the delicacy of that indication in the observations of M. Humboldt, where we find 0·65° (35' 6") of difference between two towns so near each other as Nismes and Montpellier.

These motives have induced us to study with great interest the series of observations made by M. Humboldt in regard to the inclination; and it appears to us that they may be represented very exactly by a mathematical hypothesis; to which we are far from attaching any reality in itself, but which we offer merely as a commodious and sure mode of connecting the results.

To discover this law, we must first exactly determine the position of the magnetic equator, which is as an intermediate line between the northern and the southern inclinations.

For this purpose we have the advantage of being able to compare two direct observations; one of Lapeyrouse, and the other of M. Humboldt.

The former found the magnetic equator on the coasts of Brasil at 12·1666° (10° 57') of south latitude, and 28·2407° (25° 25') of west longitude, counted from the meridian of Paris.

The latter found the same equator in Peru at 7·7963° (7° 1') of south latitude, and 89·6481° (80° 41') of west longitude, also reckoned from the same meridian.

These data are sufficient to calculate the position of the magnetic equator, supposing it to be a great circle of the terrestrial sphere; an hypothesis which appears to be conformable to observations.

The inclination of this plane to the terrestrial equator is thus found to be equal to 11·0247° (10° 58' 56"), and its occidental node on that equator is at 133·3719° (120° 2' 5") west from Paris, which places it a little beyond the continent of America, near the Gallipagos, in the South Sea; the other node is at 66·6281° (59° 57' 55") to the east of Paris, which places it in the Indian Seas.

To calculate this position let NEE' (Plate V. fig. 2.) be the terrestrial equator; NHL the magnetic equator, supposed also to be a great circle; and HL the two points of that equator, observed by Messrs. Humboldt and Lapeyrouse.

The latitudes HE, LE', and the arc EE', which is the difference of longitude of these two points, is known: consequently, if we suppose HE = b, LE' = b', EE' = v, EN = x, and the angle ENH = y, we shall have two spherical triangles NEH, NE'L, which will give the two following equations:

[Formel] from which we deduce [Formel] and developing [Formel] Let us now take an auxiliary angle ph, so that we may have [Formel] and we shall have [Formel] .

By these equations we may find x, and then y, by any of the first two.

We do not give this determination as rigorously exact: some corrections might no doubt be made to it, had we a greater number of observations equally precise; but we are of opinion that these corrections would be very small; and it will be seen hereafter that, independently of the confidence which the two observations we have employed deserve, we have other reasons for entertaining this opinion.

Since this memoir was read, we have collected new information which confirms these first results.

Lapeyrouse, after having doubled Cape Horn, fell in a second time with the magnetic equator in 18' north lat. and 119° 7' of longitude west from Paris.

He was therefore very near the node of the magnetic equator, such as we have deduced it from observations.

This fact establishes in a positive manner two important consequences: first, that the preceding determinations require only very slight corrections; and the second, that the magnetic equator is really a great circle of the earth, if not exactly at least very nearly.--Note of the Authors.

It is very remarkable that this determination of the magnetic equator agrees almost perfectly with that given long ago by Wilke and Lemonnier.

The latter in particular, who for want of direct observations had discussed a great number of corresponding observations, indicates the magnetic equator in Peru towards 7° [Formel] of south latitude; and M.

Humboldt found it in the same place at 7·7963° (7° 1'); besides, Lemonnier's chart, as well as that of M. Wilke, indicates for the inclination of the magnetic meridian 12·22° (about 11°), and they place the node about 155° 56' (140°) of west longitude, reckoned from the meridian of Paris.

Can it be by chance, then, that these elements, found more than 40 years ago, should accord so well with ours founded on recent observations? or does the inclination of the magnetic equator experience only very small variations, while all the other symptoms of terrestrial magnetism change so rapidly?

We should not be far from admitting the latter opinion, when we consider that the inclination of the magnetic needle has changed at Paris 3° during 60 years since it has been observed; and that at London, according to the observations of Mr. Graham, it has not changed 2° in 200 years; while the declination has varied more than 20° in the same interval, and has passed from east to west: but, on the other hand, the observation of the inclination is so difficult to be made with exactness, and it is so short a time since the art of measuring it with precision was known, that it is perhaps more prudent to abstain from any premature opinion on phaenomena the cause of which is totally unknown to us.

To employ the other observations of M. Humboldt in regard to the inclination, I first reduced the terrestrial latitudes and longitudes reckoned from the magnetic equator.

The latter, being reckoned from the node of that equator in the South Sea, I could first perceive by these calculations that the position of that plane determined by our preceding researches was pretty exact; for some of the places, such as Santa-Fe and Javita, where M. Humboldt observed inclinations almost equal, were found nearly on the magnetic parallel, though distant from each other more than 6·6666° (6°) in longitude.

This confirms what we have already said, that the magnetic equator is sensibly a great circle of the earth.--Note of the Authors.

When these reductions were made, I endeavoured to represent the signs of the inclinations observed, and to leave as little to chance as possible.

I first tried a mathematical hypothesis conformable enough to the idea which has hitherto been entertained in regard to terrestrial magnetism.

I have supposed in the axis of the magnetic equator, and at an equal distance from the centre of the earth, two centres of attractive forces, the one austral and the other boreal, in such a manner as to represent the two opposite magnetic poles of the earth: I then calculated the effect which ought to result from the action of these centres in any point of the surface of the earth, making their attractive force reciprocally vary as the square of the distance; and in this manner I obtained the direction of the result of their forces, which ought to be that also of the magnetic needle in that latitude.

[To be continued.]

On the Variations of the Terrestrial Magnetism in different Latitudes.

By Messrs. Humboldt and Biot.

Read by M. Biot in the Mathematical and Physical Class of the French National Institute 26th Frimaire, An 13. (17th December 1804.)

[Concluded from p. 257.]

The calculation is as follows:--I suppose that the point B (fig. 3.) is the north magnetic pole of the earth, and that the point A is the south magnetic pole: I suppose also that there is in the point M, at the surface of the earth, a molecula of the austral fluid which is attracted by B and repelled by A in the inverse ratio of the square of the distance; and I require what will be the direction of the power resulting from these two forces acting on that molecula.

It is evident that this direction will be that also which would be assumed in the point M by the needle of a compass freely suspended: for, in consequence of the smallness of the needle in comparison of the radius of the earth, the lines drawn from its points to one centre, B or A, may be considered as parallel, especially if the points A and B are near the centre of the earth; which is the case with nature, as may be seen.

I shall first suppose that the earth has a spherical figure, and that the two poles A and B are equal in force; I shall then examine how far the latter supposition agrees with the results observed.

Let AM then = D', BM = D, CP = x; PM = y, the angle MCP = u, CA = CB = a, and I shall make a = Kr; r being equal to the radius of the earth, and K a constant but indeterminate quantity.

Let X, Y, also be the forces which attract M in a direction parallel to the axes of the co-ordinates, and b the angle which the resulting force makes with the axis ABC.

We shall first have the following equations, in which F is the magnetic force, at a distance equal to unity.

[Formel] ;

[Formel]

[Formel] ;

[Formel] ,

or by putting for the cosines their values:

[Formel]

[Formel] ;

and as we have

[Formel] ,

we shall have also

[Formel] ;

and putting for x, y, and a, their values, r cos. u; r sin. u; Kr;

[Formel] ;

[Formel] ;

[Formel] ;

which gives the system of the two equations,

[Formel]

[Formel] .

These equations determine the direction of the magnetic needle in regard to each point M, the distance of which from the magnetic equator is known; but it is seen that this direction depends on the quantity K, which represents the distance of the magnetic centres from the centre of the earth: this distance being expressed in parts of the terrestrial radius, we must therefore first determine this quantity from observations.

To do it in the manner of approximation, and thus acquire a first idea of the value of K, I have chosen an observation made by M. Humboldt at Carichana in 7·2978° (6° 34' 5") of north latitude counted from the terrestrial equator, and 78·111° (70° 18') west longitude reckoned from the meridian of Paris; which gives 16·526° (14° 52' 25") of latitude counted from the magnetic equator, and 53·7390° (48° 21' 53") of west longitude proceeding from the node formed by that equator with the equator of the earth.

The inclination of the magnetic needle was observed in that place by M. Humboldt in the month of Messidor, year 8, and found to be equal to 33·78° of the centigrade division.

A comparison of this result, with the other observations of M. Humboldt, shows that it may be indeed considered as agreeing to that latitude.

All the measures of inclination which I have given in this memoir will be expressed, like those of M. Humboldt, in decimal parts of a quadrant.

To make use of it I have successively given to K different values in the formula: I have calculated the inclinations resulting from that latitude; and, comparing these results with that which M. Humboldt really observed, the progress of the errors naturally led me to the most proper supposition.

The following is a table of these trials:

Values of K. Inclinations of the Needle.

Errors. K=1 7·73° 26·04° K=0·6 18·80 14·97 K=0·5 22·04 11·73 K=0·2 29·38 4·39 K=0·1 30·64 3·13 K=0·01 31·04 2·73 K=0·001 31·07 2·7

The first value of K would place the centre of the magnetic forces at the surface of the earth and the poles of the magnetic equator.

It is seen that this supposition cannot be admitted, because it would give an increase of inclination much less rapid than that indicated by observations.

The case is the same with the following results, which place the centres of action on the terrestrial radius at different distances from the centre of the earth: but it is seen also, in general, that they approach more and more to the truth in proportion as this distance becomes less; which evidently shows that the two centres of action of the magnetic forces are situated near the centre of the earth.

All the other observations of M.

Humboldt would also lead to the same consequence.

The most proper supposition would be to make K null, or so small that it would be needless to pay attention to it; which amounts to the same thing as to consider the two centres of action placed, as we may say, in the same molecula.

The result, indeed, obtained in this manner is the most exact of all; it is equal to 31·0843°: this value is still a little less than that which M. Humboldt observed, and the difference is equal to 2·69; but it must be considered also that the formula from which we derive these values supposes the position of the magnetic equator to be perfectly determined; but it may not be so with the utmost exactness, according to the only two observations of Lapeyrouse and Humboldt, which we have employed.

It is therefore by studying the progress of the formula, and comparing it with the observations, that we are able to appreciate it justly; after which we may think of remedying the small errors with which it may be accompanied.

To obtain the result I have here mentioned, and which is, as it were, the limit of all those which may be obtained by giving to K different values, it is to be remarked that the quantity

[Formel]

or

[Formel]

becomes [Formel] when K is null, but by applying to it the methods of known quantities it will be found that its value in this supposition is really determinate and equal to [Formel] .

By substituting this in the formula we shall have

[Formel]

an equation which may be reduced to this form:

[Formel] ;

which will easily give the value of b: and when this value is known we shall have the inclination I, by the following formula:

[Formel] ,

which will serve throughout the whole extent of the two hemispheres.

From the progress I have traced out it is seen that the preceding formula is not merely an empyric construction of observations; on the contrary, it is totally independent, and only supposes the inclination of the magnetic needle to be produced by a magnet, infinitely small, placed in the centre of the terrestrial surface; but by calculating from this formula the inclination for the different latitudes, I have found precisely the same numbers as M. Humboldt observed either in Europe or in America: and it is not his observations only that are represented in this manner; but those which have been made in Russia, and at Kola in Lapland, during the last transit of Venus, are also comprehended under the same law.

This is proved by the table annexed to this memoir, in which I have calculated the observations of Mallet and Pictet, with a part of those of M. Humboldt, which I took at random, but, however, in such a manner as to include all the rest in the intervals.

It is seen that the results of the formula deviate very little from the observations; but these differences may be rendered still smaller.

By examining, indeed, the progress of the errors, it is seen that the numbers given by calculation are a little too small in America for the low latitudes, and a little too great for the high latitudes; which shows that the whole may be allowed, with some slight modifications, either by changing, however little, the node and inclination of the magnetic equator, which two observations cannot determine with the utmost exactness, or by displacing ever so little our small magnet, leaving, however, its centre in the plane of the magnetic equator, and placing it in such a manner that it shall be a little nearer America than Europe. It is by the observations themselves, when we shall have a greater number, that we must be guided in these small corrections.

In a word, it must not be expected that we can represent in a rigorous manner, by a mathematical law, all the inclinations observed; for the phaenomenon of the inclination, though more regular than the other magnetic effects, is not free from some anomalies: this may be easily seen on constructing the curve given by the observations themselves.

Thus, for example, the inclination observed at Popayan is 0° 10' greater than at St.

Carlos del Rio Negro, though the magnetic latitude of the latter is 0·6852° (3° 7') greater.

The case is the same with observations made at Javita and Santa-Fe.

Other anomalies are discovered in the comparative progress of the observations and formula.

This is the case in regard to Carichana, St. Thomas de la Guyane, and Carthagena.

The increase of the inclination from the first to the second of these points is by no means in harmony with the increase from the second to the third; and if we compare together the intensities observed in these different places, the anomalies they exhibit announce in some measure those which the inclination ought to experience.

The cause of these anomalies becomes evident from what has been already remarked; they are merely the effect of local circumstances, and arise from the small systems of attraction by which the general phaenomena are modified.

This must be sensible in particular for that part of America which M. Humboldt travelled over, and which is traversed throughout its whole length by the grand chain of the cordillera of the Andes.

It is also in these places that the most considerable differences exist.

Popayan, for example, is situated near the volcanoes of Sotara and Pourace; it is joined to basaltic mountains abounding with magnetic iron.

Near Sulumito, to the east of Popayan, these basaltic columns have very striking poles: in like manner Mexico is situated at the height of 1160 toises on the ridge of the grand cordillera of Lenschtitlan:

the ground there is covered with porous basaltes and amygdaloids, which are almost all charged with magnetic iron.

Must not all these causes have a sensible influence on the inclination of the magnetic needle; and must not the different dispositions of the ferruginous masses, or their change of state, in consequence of the action of nature, produce also variations? M.

Humboldt made on this point a decisive observation: the earthquake of the 4th of November 1799 changed at Cumana the inclination of the needle.

On the 1st of November it was 43° 65'; on the 7th it was only 42° 75'; and ten months after it returned to 42° 85': but it did not regain its former value; the intensity of the magnetic force was not changed by the effect of this earthquake.

It is proved, then, by these observations, that local circumstances may have on the inclination a sensible influence; and this influence is remarked in the countries traversed by M. Humboldt .

We can observe that the anomalies are sensible in particular in the islands.--Note by the Authors of the Memoir.

It appears, therefore, that the mathematical hypothesis which we have employed really expresses the law of nature at least to the north of the magnetic equator; for, though the first results observed towards the south seem to bend to it also, the uncertainty under which we are in regard to the true cause of these phaenomena must stop our conjectures, and prevent us from extending too far the consequences of the laws which we observe.

Since this memoir was read, we can advance something more positive.

Observations made at the Cape of Good Hope, Cape Horn, and New Holland, by different navigators, are very exactly represented by our formula; and it follows, that it extends also to the austral hemisphere.

We hope soon to have numerous and very exact observations on the inclination of the needle in that part of the earth.

But we have thought it our duty to add to our table such results as relate to it, and which we have been able to procure.

We have inserted also two observations on the intensity, made with great care by M. Rossel, during the expedition of d'Entrecasteaux, which are very important, as they prove that the terrestrial magnetic force increases also in the austral hemisphere in proportion as one removes from the equator.--Note by the Authors of the Memoir.

From the preceding results, we may calculate the points where the axis of the magnetic equator pierces the terrestrial surface; for their latitudes are equal to the complements of the obliquity of that equator, and their meridian is at 100° of longitude from its nodes.

The north magnetic pole is found also at 97·7975° (79° 1' 4") of north latitude, and at 33·3719° (30° 2' 5") of longitude west from Paris, which places it to the north of America.

The other magnetic pole, symmetric to the preceding, is situated in the same latitude south, and at 66·6281° (149° 67' 55") of longitude east from Paris, which places it amidst the eternal ice: indications entirely analogous to those of Wilke and Lemonnier.

If we could reach these poles, the compass would be seen vertical; but if any confidence can be placed in the law which we have discovered, this would be the only difference which would be observed in regard to the inclination, and we should be still as far distant as in Europe from the real centres which produce it.

This result might appear to be of such a nature as to diminish the interest one might have in visiting these horrid regions, had we not also the hope of discovering there new phaenomena in regard to the intensity of the magnetic force, and the influence of meteors.

These consequences do not entirely accord with the opinion pretty generally received, and which ascribes the increase of the magnetic effects towards the north to the great quantity of iron dispersed throughout these regions; but it appears to us that this opinion is not agreeable to the truth.

The cordillera of the Andes contains an enormous quantity of magnetic iron:

the native iron of Chaco, that problematic mass analogous to that of Pallas, and those of Xacateras in Mexico, is found even under the tropics.

We may now add to the preceding considerations this decisive fact, that the intensity also increases when one approaches the south pole.--Note by the Authors of the Memoir.

On seeing the inclinations of the compass so exactly represented in our hypothesis, we endeavoured to discover whether it could he applied also to the intensities observed by M. Humboldt; but we found that it did not apply.

It gives, indeed, an increase of the magnetic forces from the equator to the pole; but this increase, which at first is too slow, becomes afterwards too rapid: I have not yet been able to try whether the small displacement of the terrestrial magnet will contribute towards representing them better: but it must be remarked, that the series of the intensities is extremely whimsical, and contains an infinite number of anomalies; so that local phaenomena may have on this phaenomenon a much more sensible influence than on the inclination.

On reviewing the results which we have given in this memoir, it is seen that we have first determined the position of the magnetic equator by direct observations, which had never been done before; we have then proved that the magnetic force increases in proceeding from that equator to the poles: in the last place, we have given a mathematical hypothesis, which when reduced to a formula satisfies all the inclinations hitherto observed.

Supposing, as we have done in this formula, the small corrections of which it is susceptible, its utility becomes evident, either for making known, in the course of time, the variations which may take place in the action of the terrestrial magnetism, or to ascertain or even foresee the value of the inclination, which in a great many cases is of great importance.

For example, near the magnetic equator, the increase or diminution of the inclination will indicate to a vessel on a voyage whether she has gained or lost in latitude by currents.

This knowledge of the latitude is sometimes as important as that of longitude.

On the coasts of Peru, for example, the currents tend from Chiloe to the north and north-east with such force, that one may go from Lima to Guayaquil in three or four days, and two, three, and sometimes five months are necessary to return.

It is consequently of the greatest importance for vessels coming from Chili which stretch along the coast of Peru, to know their latitude.

If they go beyond the port to which they are bound they must work to the southward, and every day's progress requires often a month of return.

Unfortunately, the fogs which prevail during four or five months on the coasts of Peru prevent navigators from distinguishing the form of the coast: nothing is seen but the summits of the Andes, and that of the peaks which rise above that stratum of vapours; but the figure of it is so uniform that pilots fall into mistakes.

They often remain twelve or fifteen days without seeing the sun or the stars, and during that interval they come to anchor, being afraid of overshooting their port:

but if we suppose that the inclination of the magnetic needle in the ports to the south of Lima is known, for example at Chancay, Huaura, and Santa, the dipping needle will show whether it be, in regard to Lima, to the south or the north.

It will show at the same time opposite what point of the coast a vessel is; and this indication will be attended with more exactness than one could hope for, because in these seas the inclination varies with extraordinary rapidity. M. Humboldt, to whom we are indebted for these remarks, observed in these seas the following values:

Places.

South Latitudes.

Inclinations.

Huancey - 10° 4' - 6,80° Huaura - 11 3 - 9,00 Chancay - 1133 - 10,35

These observations prove that the error of three or four degrees in the inclination in these seas would produce but a degree of error in latitude; and, on account of the tranquillity of the Pacific Ocean, the inclination may be observed to within a degree nearly.

Frequent instances of such results may be seen in books of voyages.

In like manner, if one knew exactly the inclination at the mouth of the Rio de la Plata, it would be very useful to navigators, who, when the Pamperos blow, remain fifteen or eighteen days without seeing the heavenly bodies, and go on different tacks for fear of losing the parallel of the mouth of that river.

In a word, the inclination may indicate also the longitude in these seas: and this method may be employed when others fail.

A vessel which sails there in the direction of a parallel could not find its longitude either by a chronometer or the declination of Halley, unless a star could be seen in order to take an horary angle or the magnetic azimuth.

The dipping needle, then, throws light on the longitude amidst the thickest fogs.

We point out this method as one of those which have only a local application; but hitherto little attention has been paid to it.

These ideas may be extended and rectified by able navigators.

In general, if the inclination of the needle, and the law we have tried to establish, could be depended on, to observe the inclination and the terrestrial latitude would be sufficient to determine also the longitude: but we have not yet examined the extent of the errors of which this method may be susceptible, and consequently we confine ourselves to a mere indication of it.

The phaenomenon of the inclination has in maritime observations a particular and very remarkable advantage, namely, that of not being subject to those great progressive variations which affect the declination.

Without repeating what we have already said above on the supposed constancy of this phaenomenon, it may be remarked that our formula even affords a new proof that it may comprehend in the same law the observations made thirty-six years ago in Lapland, those which Lacaille brought back in 1751 from the Cape of Good Hope, and those which M. Humboldt has lately made in America.

In short, when we tried to represent the inclinations in different latitudes by the supposition of a magnet infinitely small, very near the centre of the earth and perpendicular to the magnetic equator, we did not pretend to consider that hypothesis as any thing real, but only as a mathematical abstraction useful to connect the results, and proper to ascertain in future whether any changes exist.

In regard to the declination and intensity, we freely confess that we are entirely unacquainted with their laws or their causes; and if any philosopher is so fortunate as to bring them to one principle, which explains at the same time the variations of the inclination, it will no doubt be one of the greatest discoveries ever made.

But this research, exceedingly difficult, requires perhaps before it be attempted more observations, and in particular more precise observations than have hitherto been collected.

For this reason we thought we might present to the class the preceding researches, imperfect as they are, begging it to receive them with indulgence.

Should we be so happy as to find that our results appear of any utility, we propose to unite all the exact observations which have been made on this subject, in order to give the utmost degree of precision to the law we have discovered.

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NORTHERN MAGNETIC HEMISPHERE.

Names of the Observers.

Places of Observation.

Latitudes reckoned from the Terrestrial Equator.

Longitudes reckoned from the Meridian of Paris.

Latitudes reckoned from the Magnetic Equator.

East Longitudes reckoned from the Node of the Magnetic Equator in the South Sea.

Number of Oscillations in 10' of Time.

Inclinations given by Theory Inclinations given by direct Observation.

Differences in Degrees of the Centesimal Division.

Old Division.

Old Division.

Old Division.

Old Division.

Centig.

Division.

Centig.

Division.

Humboldt - Magnetic equator in Peru - - - 7° 1' 0" S 80° 41' 0" W 0° 0' 0" 40° 17' 56" 211 0,000 0,00 0,00° Lapeyrouse Magnetic equator at sea between Brazil and the Island of Ascension - - - - 10 57 0 25 25 0 0 0 0 95 33 56 - - - 0,000 0,00 0,00 Humboldt - Tompenda - - 5 31 4 80 27 0 1 30 54 39 52 51 213 3,3642 3,55 0,1858 The same - Loxa - - - - 4 0 0 81 12 0 2 54 27 38 55 0 212 6,440 6,00 + 0,44 The same - Cuenca - - - 2 54 9 80 43 0 4 3 44 39 13 52 214 8,97 9,35 -- 0,38 The same - Quito - - - - 3 13 17 80 15 0 6 46 59 39 17 52 218 14,87 14,85 + 0,02 The same - St. Antonio - - 0 0 0 80 12 0 7 0 53 39 18 52 220 15,29 16,02 -- 0,73 The same - Popayan - - - 2 24 33 N 7845 0 9 36 16 40 24 27 223 - - - - 23,20 - - - - The same - St.

Carlos del Rio Negro - - - 1 52 4 70 10 0 10 13 14 49 6 35 216 22,0278 23,10 -- 1,0722 The same - Javita - - - - 2 49 0 70 30 0 11 7 40 48 39 6 218 23,87 27,00 -- 3,13 The same - Esmeralda - - 3 13 26 68 38 0 11 45 45 50 29 15 217 - - - - 28,85 - - - - The same - Santa Fe di Bagota 4 36 5 76 37 0 12 5 13 42 17 13 226 25,76 26,97 -- 1,21 The same - Carichana - - 6 34 5 70 18 0 14 52 25 48 21 53 227 31,08 33,77 -- 2,69 The same - St. Thomas of Guyana - - 8 8 24 66 26 0 16 54 18 52 7 26 222 34,77 39,00 -- 4,23 The same - Carthagena - - 10 25 57 78 2 0 17 38 43 39 55 13 240 36,07 39,17 -- 3,10 The same - Mexico - - - 19 26 2 101 22 0 22 35 14 14 36 41 242 44,87 46,85 -- 1,98 De Rossel in 1791 Humboldt in 1799 St. Croix in Teneriffe - - - - 28 28 30 18 37 0 39 12 40 72 0 26 238 64,9975 69,85 -- 4,35 The same - Atlantic Ocean - 38 52 0 16 20 0 49 28 22 106 30 10 242 74,29 75,76 -- 1,47 The same - Paris - - - - 48 50 15 0 0 0 57 57 0 128 22 47 245 80,69 77,62 + 3,07 Euler junior Petersburgh in 1755 - - - 59 56 23 27 58 0 E 64 41 0 173 30 25 - - - 85,21 81,67 + 3,54 Mallet - - Kola, in Lapland, in 1769 - - 68 52 30 30 40 30 E 71 44 36 179 9 29 - - - 89,59 86,39 + 3,20 Lord Mulgrave In an island near Spitzbergen in 1773 - - - 79 50 00 7 38 00 E 83 9 50 127 40 5 - - - 96,1882 91,1111 + 5,0771

SOUTHERN MAGNETIC HEMISPHERE.

Names of the Observers.

Places of Observation.

Latitudes reckoned from the Terrestrial Equator.

Longitudes reckoned from the Meridian of Paris.

South Latitudes reckoned from the Magnetic Equator.

East Longitudes reckoned from the Node of the Magnetic Equator in the South Sea.

Inclinations given by Theory Inclinations given by direct Observation.

Differences in Degrees of the Centesimal Division.

Number of Oscillations in 10' of Time.

Old Division.

Old Division.

Old Division.

Old Division.

Centig.

Division.

Centig.

Division.

Humboldt - Lima - - - 12° 2' 31" S 79° 33' 0" W 4° 48' 36" S 41° 42' 51" 10,6145 11,10 0,4855° 219 Derossel, expedition of Entrecasteaux Sourabaya in the Isle of Java - 7 14 23 110 21 28 E 15 37 22 228 56 50 32,4660 28,5185 + 3,9475 204 Bayli in 1775 Cape of Good Hope 33 55 30 16 10 0 E 26 15 34 131 38 53 49,58 47,78 + 1,8 - - - Lapeyrouse Bay of Talcaguara 36 42 0 75 53 0 W 28 42 14 49 0 5 52,8889 55,555 -- 2,6667 - - - The same - In sight of the Isle of Patagonians 52 21 26 69 38 0 W 44 30 3 57 13 52 70,04 68,89 + 1,15 - - - Derossel, expedition of Entrecasteaux New Holland - 43 34 30 144 36 33 W 54 12 43 263 21 18 78,7037 77,9667 -- 0,737 265

The results comprehended in this table extend from 38° 55' to 263° 21' 18" east longitude, reckoned from the node of the magnetic equator in the South Sea: consequently, they comprehend more than 224°, and their agreement shows that in this extent the magnetic equator is sensibly a great circle of the earth.

We have not yet calculated the observations for the 36 degrees of longitude which would complete the contour of that equator.--For the southern hemisphere we have given the observations, made with great care, by M. Derossel, during the expedition of d'Entrecasteaux.

It results from them that the intensity of the terrestrial magnetism increases also in that hemisphere, when one removes from the magnetic equator.

The inclination observed by M. Derossel at Teneriffe being exactly the same as that observed by M.

Humboldt eight years after, this agreement has allowed us to render comparable the results obtained by these philosophers in regard to the intensity: for this purpose we have multiplied the results of M. Derossel, by giving the numbers which he and M. Humboldt observed at Teneriffe.

The result of this calculation will be found in the column of oscillations.

It is there again seen that this phaenomenon is very much modified by local circumstances, and incomparably more than the inclination.

The increase of the intensity deduced from the observations of M.

Humboldt is less than that which would result from our hypothesis; and that given by the observations of M.

Derossel is too great: which proves that nothing can be determined in regard to the law of this increase.

The influence of local circumstances on the inclination is particularly sensible in the isles.

The declination and intensity of the magnetic forces experience there also sensible anomalies.

This fact is indicated by several observations, and in particular by those which M. Derossel made at Sourabaya in the Island of Java.

In a word, by comparing the results of our formula with the observations of different navigators, the latter must be examined with critical accuracy, and must not be admitted but when they agree with each other and with those of other navigators: without this precaution we should every moment fall into great errors, occasioned by the incoherency of the results: besides, we present the preceding only as a first approximation.

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