in the same manner. This, notwithstanding the height of the tree, is easily accomplished. The climber provides himself with a tough woodbine hoop, the circumference of which embraces the tree and his body, but with so much space intervening, as permits him to lean back at arms-length from the tree, thus enabling him to fix his feet firmly against the knobs. In this way, by jerking the hoop upwards, he ascends very quickly.

The wine is always extracted from the male tree; the female, which bears the nuts, being too valuable to use in that way. The nut is nearly of the size and figure of the walnut. Each tree produces three or four bunches, which are sometimes so large that a single cluster has been known to weigh above 100 pounds.

(To be continued.)

ART. XIV. On the Method of finding the Dip or Depression of the Horizon. By ADAM ANDERSON, Esq. A. M.

F. R. S. E.

THE altitude of a heavenly object, as it is obtained by ob

servation, being an arch of a vertical circle intercepted between the object and the apparent horizon, it is evident that this arch must be greater as the eye of the observer is raised above the plane of the horizon. Let A, Plate VI. Fig. 4. be any place on the surface of the earth, b AB the sensible horizon of that place, A' a point directly vertical to A, and b'A'B' the corre sponding horizontal plane; then C being the centre of the earth, if O be an object at a great distance, the angle B'A'O, will not differ sensibly from the angle BAO, the apparent altitude at A; but at A' the apparent altitude will not be the angle B'A'O. but the angle OA'E, which differs from the former by the angle B'A'E. This difference, which varies with the altitude of the observer, is called the Dip of the horizon, and is equal to the angle at the centre ACE. We shall compute the magnitude of it for a particular altitude in feet, and then give a general expression, by which it may be determined with sufficient accuracy in all other cases.

Let r denote the radius of the earth, corresponding to the mean parallel of 45°, in which case it is 20972190 feet; then, if h be the height of the observer, in feet, above the surface of the ocean, and t the tangent at the point A'.

t=√DA'xAA′ = √ 2 r h + h2.

But if the arch AE, which, from what we have shewn, must measure the dip, be represented by D, then, from the well known expression for an arch, in terms of its tangent,

If t be expressed in terms of the radius, considered as unity, then h must be divided by 20972190, by which means it will become a very small fraction, and it will be quite unnecessary to retain more than two terms of the above series. We shall thus

This expression will give the dip in minutes, if it be multiplied by 3437.75, the number of minutes equal to the radius, when the circumference is 360°; but before it can be used for practical purposes, it must be corrected for atmospherical refraction. At present it is sufficient to state, that, according to La Place and De Lambre, the effect of refraction, in the ordinary condition of the atmosphere, is to increase angles of elevation observed near the surface of the earth, 1 of the intercepted terrestrial arch between the object and the place of the observer. Hence the angle AA'E must be multiplied by 18 to reduce it to its proper magnitude divested of refraction. We thus obtain

8

')'{

100

10

1 + 2h + h2

{ 1 +

3

For elevations not exceeding 400 feet, it will be sufficiently

correct to give this expression the form

D=3183√2h+h', or even D

3183 √2 h.

Thus if h 25 feet, then h =

25 20972190

= =·00000119205.

And D 31830000023841 = 3183 × ·00154 = 4′ ·9.

It appears by the approximated expression, D=3183 2h, that the dip for different elevations above the level of the sea, is nearly proportional to the square root of the height; and since we have found it 4.9 for 25 feet, we obtain for any other height in feet H,

Hence the dip in minutes is very nearly equal to the square root of the altitude in feet. This method of finding the dip, the simplest, we believe, that has yet been given, will seldom differ in result above four or five seconds from the most rigid calculation, and, considering the changeable nature of the height of the observer at sea, may be regarded as sufficiently correct for all practical purposes. We deduce from ti the following practical rule for the calculation of the dip: Take half the logarithm of the height of the eye of the observer in feet, and it will be the logarithm of the dip in minutes.

EXAMPLE.-What is the dip of the horizon for an observation taken at the height of 45 feet above the ocean?

Hence the Dip is 6'.7, or 6'.42′′.

The relation deduced in the preceding investigation, is one of the few coincidences among the objects of science, which are readily impressed on the mind, by their singularity and simplicity. Those who are acquainted with the allowance for the depression of the earth's surface below the tangent at a particular point, will at once recognize a similarity in point of simplicity between the above expression for dip, and the usual correction for curvature, in conducting the operations of levelling. In that case, the tangent line, or the direction indicated by an accurate level, is elevated above the true level, by a quantity expressed in feet, which is obtained by taking two-thirds of the square of the distance in miles. The relation between the height of the eye and the dip is still more curious, and not less simple; but the singularity of the coincidence, in the two cases, is the more remarkable, as they both depend on the particular magnitude of the earth, and the accidental length of the English foot.

ART. XV.-On certain remarkable Instances of deviation from NEWTON's Scale in the Tints developed by Crystals with one Axis of Double Refraction, on exposure to Polarized Light. By J. F. W. HERSCHEL, A. M. F.R. S. LOND. & EDIN. and of the CAMB. Phil. Soc. *

THE discovery of crystals which possess Two axes of double

refraction, which we owe to Dr Brewster, is, perhaps, the greatest step which has been made in Physical Optics since the discovery of double refraction itself by Bartholin, and its reference to an axis by Huygens. It has opened new views on the structure of crystals, and will, in all probability, be the means of leading us to a more intimate knowledge of the nature and laws of those forces, by which the ultimate particles of matter act on light and on each other. When we reflect on the situation of these axes in different crystallized media, we cannot fail to be struck by the variety of the angles they include, and of the positions they hold, with respect to the prominent lines or axes of symmetry of the primitive molecules, and the question immediately suggests itself, What are the circumstances which determine their position in the interior of a crystal?

It seems to have been all along taken for granted, that whatever these circumstances may be, the nature of the ray must at least be a matter of indifference; in other words, that a red and a violet ray similarly polarized, and incident in the same direction on the same point of a doubly refracting surface, will either both undergo, or both not undergo, a separation into two pencils, without any distinction arising from the place of the ray in the prismatic spectrum. Were this the case, the two axes would be fixed lines within the primitive form, absolutely determined by the nature of the body, as much so as the lines which bound the primitive form itself, and any attempt to substitute for them hypothetical axes, coinciding with remarkable lines in the latter figure, however ingeniously devised, must be regarded as mere speculation. The fact, however, is other

From the Memoirs of the Cambridge Philosophical Society, vol. i., which will soon be published. This paper was read on the 1st May 1820.

wise. In a paper recently presented to the Royal Society, I have shewn that the axes of double refraction in one and the same crystal differ in their position according to the colour of the intromitted ray, a violet ray being separated into two pencils, when incident in the same direction in which a red one would be refracted singly. This remarkable fact, which is almost universal in crystals with two axes, places the question in a very different light. It appears that the nature of the ray, as well as that of the medium, has its share in determining the position of the axes, and that the intensity of the action of the medium on the ray is one of the elements involved in this problem. Now, it is hardly possible to conceive the neutral axis of a crystal otherwise than as a position of equilibrium, or direction in which the axis of translation of a luminous molecule (if such exist) must be placed, that certain forces may act in opposition, and balance one another; but since forces which balance will likewise counteract each other when increased or diminished all in the same ratio, it follows that the partial or elementary forces so held in equilibrium do not observe the law of propor tionality, when the colour of the incident ray varies. If we suppose, then, with Dr Brewster, that these partial forces emanate from certain fixed axes coincident with remarkable lines in 'the primitive form, it will follow that each separate axis has a peculiar specific law, which regulates the intensity of its action on the differently coloured rays, and that each axis, supposing the others not to interfere with it, would exhibit separately a set of circular rings, of which the tints would manifest a more or less marked deviation from the Newtonian Scale of Colours, as displayed by their uncrystallized laminæ,

This view of the subject will be remarkably supported by the facts about to be described, by which it will appear, that among crystals with one axis only, there exists the greatest, I might almost say the most capricious diversity in this respect, and that probably no two crystals, either with one or two axes, have the same scale of action, or polarize the differently coloured rays with an energy varying according to the same law precisely.

To this it may be objected, that from the result of a most elaborate examination of the colours exhibited by sulphate of lime, rock-crystal, and mica, M. Biot has concluded that they follow