MAGNITUDE
, any thing made up of parts locally extended, or continued; or that has several dimensions; as a line, surface, solid, &c. Quantity is often used as synonymous with Magnitude. See Quantity.
Geometrical Magnitudes, are usually, and most properly, considered as generated or produced by motion; as lines by the motion of points, surfaces by the motion of lines, and solíds by the motion of surfaces.
Apparent Magnitude, is that which is measured by the optic or visual angle, intercepted between rays drawn from its extremes to the centre of the pupil of the eye. It is a fundamental maxim in optics, that whatever things are seen under the same or equal angles, appear equal; and vice versa.—The apparent Magnitudes of an object at different distances, are in a ratio less than that of their distances reciprocally.
The apparent Magnitudes of the two great luminaries, the sun and moon, at rising and setting, are a phenomenon that has greatly embarrassed the modern philosophers. According to the ordinary laws of vision, they should appear the least when nearest the horizon, being then farthest from the eye; and yet it is found that the contrary is true in fact. Thus, it is well known that the mean apparent diameter of the moon, at her greatest height in the meridian, is nearly 31′ in round numbers, subtending then an angle of that quantity as measured by any instrument. But, being viewed when she rises or sets, she seems to the eye as two or three times as large as before; and yet when measured by the instrument, her diameter is not found increased at all.
Ptolomy, in his Almagest, lib. 1, cap. 3, taking for granted, that the angle subtended by the moon was really increased, ascribed the increase to a refraction of the rays by vapours, which actually enlarge the angle under which the moon appears; just as the angle is enlarged by which an object is seen from under water: and his commentator Theon explains distinctly how the dilatation of the angle in the object immersed in water is caused. But it being afterwards discovered, that there is no alteration in the angle, another solution was started by the Arab Alhazen, which was followed and improved by Bacon, Vitello, Kepler, Peckham, and others. According to Alhazen, the sight apprehends the surface of the heavens as flat, and judges of the stars as it would of ordinary visible objects extended upon a wide plain; the eye sees then under equal angles indeed, but withal perceives a difference in their| distances, and (on account of the semidiameter of the earth, which is interposed in one case, and not in the other) it is hence induced to judge those that appear more remote to be greater. Some farther improvement was made in this explanation by Mr. Hobbes, though he fell into some mistakes in his application of geometry to this subject: for he observes, that this deception operates gradually from the zenith to the horizon; and that if the apparent arch of the sky be divided into any number of equal parts, those parts, in descending towards the horizon, will subtend an angle that is gradually less and less. And he was the first who expressly considered the vaulted appearance of the sky as a real portion of a circle.
Des Cartes, and from him Dr. Wallis, and most other authors, account for the appearance of a different distance under the same angle, from the long series of objects interposed between the eye and the extremity of the sensible horizon; which makes us imagine it more remote than when in the meridian, where the eye sees nothing in the way between the object and itself. This idea of a great distance makes us imagine the luminary the larger; for an object being seen under any certain angle, and believed at the same time very remote, we naturally judge it must be very large, to appear under such an angle at such a distance. And thus a pure judgment of the mind makes us see the sun, or the moon, larger in the horizon than in the meridian; notwithstanding their diameters measured by any instrument are really less in the former situation than the latter.
James Gregory, in his Geom. Pars Universalis, pa. 141, subscribes to this opinion: Father Mallebranche also, in the first book of his Recherche de la Verité, has explained this phenomenon almost in the expression of Des Cartes: and Huygens, in his Treatise on the Parhelia, translated by Dr. Smith, Optics, art. 536, has approved, and very clearly illustrated, the received opinion. The cause of this fallacy, says he, in short, is this; that we think the sun, or any thing else in the heavens, farther from us when it is near the horizon, than when it approaches towards the vertex, because we imagine every thing in the air that appears near the vertex to be farther from us than the clouds that fly over our heads; whereas, on the other hand, we are used to observe a large extent of land lying between us and the objects near the horizon, at the farther end of which the convexity of the sky begins to appear; which therefore, with the objects that appear in it, are usually imagined to be much farther from us. Now when two objects of equal magnitude appear under the same angle, we always judge that object to be larger which we think is remoter. And this, according to them, is the true cause of the deception in question. It is really astonishing that an hypothesis so palpably false should ever be held and maintained by such eminent men; for it is daily feen that the moon or sun, when near the horizon, very suddenly change their magnitude, as they ascend or descend, though all the intervening objects are seen just as before; and that the luminary appears largest of all when fewest objects appear on the earth, as in a thick fog or mist. It is no wonder therefore that other reasons have been assigned for this remarkable phenomenon.
Accordingly Gassendus was of opinion, that this effect arises from hence; that the pupil of the eye, being always more open as the place is more dark, as in the morning and evening, when the light is less, and besides the earth being then covered with gross vapours, through a longer column of which the rays must pass to reach the horizon; the image of the luminary enters the eye at a greater angle, and is really painted there larger than when the luminary is higher. See APPARENT Diameter and Magnitude.
F. Gouge advances another hypothesis, which is, that when the luminaries are in the horizon, the proximity of the earth, and the gross vapours with which they then appear enveloped, have the same effect with regard to us, as a wall, or other dense body, placed behind a column; which in that case appears larger than when insulated, and encompassed on all sides with an illuminated air.
The commonly received opinion has been disputed, not only by F. Gouge, who observes, Acad. Sci. 1700, pa. 11, that the horizontal moon appears equally large across the sea, where there are no objects to produce the effect ascribed to them; but also by Mr. Molyneux, who says, Philos. Trans. abr. vol. 1, pa. 221, that if this hypothesis be true, we may at any time increase the apparent magnitude of the moon, even in the meridian; for, in order to divide the space between it and the eye, we need only to look at it behind a cluster of chimneys, the ridge of a hill, or the top of a house, &c. He makes also the same observation with F. Gouge, above mentioned, and farther observes, that when the height of all the intermediate objects is cut off; by looking through a tube, the imagination is not helped, and yet the moon seems still as large as before. However, Mr. Molyneux advances no hypothesis of his own.
Bishop Berkley supposed, that the moon appears larger near the horizon, because she then appears fainter, and her beams affect the eye less. And Mr. Robins has recited some other opinions on this subject, Math. Tracts, vol. 2, pa. 242.
Dr. Desaguliers has illustrated the doctrine of the horizontal moon, Philos. Trans. abr. vol. 8, pa. 130, upon the supposition of our imagining the visible heavens to be only a small portion of a spherical surface, and consequently supposing the moon to be farther from us in the horizon than near the zenith; and by several ingenious contrivances he demonstrated how liable we are to such deceptions. The same idea is pursued still farther by Dr. Smith, in his Optics, where he determines that, the centre of the apparent spherical segment of the sky lying much below the eye, or the horizon, the apparent distance of its parts near the horizon was about 3 or 4 times greater than the apparent distance of its parts over head; from which reason it is, he infers, that the moon always appears the larger as she is lower, and also that we always think the height of a celestial object to be more than it really is. Thus, he determined, by measuring the actual height of some of the heavenly bodies, when to his eye they seemed to be half way between the horizon and the zenith; that their real altitude was then only 23°: when the sun was about 30° high, the upper always appeared less than the under; and he thought that it was constantly greater when the sun was 18° or 20° high. Mr. Robins, in| his Tracts, vol. 2, pa. 245, shews how to determine the apparent concavity of the sky in a more accurate and geometrical manner; by which it appears, that if the altitude of any of the heavenly bodies be 20°, at the time when it seems to be half way between the horizon and the zenith, <*>he horizontal distance will be hardly less than 4 times the perpendicular distance; but if that altitude be 28°, it will be little more than 2 and a half.
Dr. Smith, having determined the apparent figure of the sky, thus applies it to explain the phenomenon of the horizontal moon, and other similar appearances in the heavens. Suppose the are ABC to re- present that apparent concavity; then the diameter of the sun and moon would seem to be greater in the horizon than at any altitude, measured by the angle AOB, in the ratio of its apparent distances, AO, BO. The numbers that express these proportions he reduced into the annexed table, answering to the corresponding altitudes of the sun or
| The alt. of the sun or moon in degrees. | Apparent diameters or distances. |
| 00 | 100 |
| 15 | 68 |
| 30 | 50 |
| 45 | 40 |
| 60 | 34 |
| 75 | 31 |
| 90 | 30 |
