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jusqu'à présent rien n'égale en exactitude les opérations géodésiques qui ont servi de fondement à notre système métrique;' and, lastly, an elaborate chapter is written on the measure of the earth, in which there is no more notice taken of the most correct of all trigonometrical surveys, carried on uniformly with great science and skill, and extremne public benefit, for 27 years, than if it had never commenced. This is rendered still more extraordinary by M. Biot's commendation of Messrs. Mason and Dixon's measure of a degree in Pennsylvania, though we will venture to say there is no respectable mathematician in Europe who is not aware of the extreme inaccuracy of the American results. Dr. Maskelyne, in the Philosophical Transactions for 1768, (from which the French authors obtaived their account of Mason and Dixon's belles opérations,') informs us, that Mr. Henry Cavendish
having mathematically investigated several rules for finding the attraction of the inequalities of the earth, has, upon probable suppositions of the distance and height of the Allegany mountains from the degree measured, and the depth and declivity of the Atlantic ocean, computed what alteration might be produced in the length of the degree, from the attraction of the said hills, and the defect of attraction of the Atlantic, and finds the degree may have been diminished from 60 to 100 toises from these causes. Yet this is the degree which our Gallic lovers of 'exactitude' prefer to any of those measured in England!
Our author has a diffuse though interesting chapter on atmospherical refractions, which is the more valuable as it is now
known that M. Lambert's theory, hitherto almost generally re• ceived, is erroneous. In this he traces the cause of several curious
phænomena which depend on variable refractions, and among others that which is known to their mariners under the name of mirage, and which the French army frequently observed in their expedition to Egypt.
• The surface of the ground of Lower Egypt is a vast plain, perfectly horizontal. Its uniformity is not otherwise broken than by some eminences, on which are situated the towns and villages, which, by such means, are secured from the inundations of the Nile. In the evening and morning the aspect of the country is such as comports with the real disposition and distance of objects; but when the surface of the earth becomes heated by the presence of the sun, the ground appears as though it were terminated at a certain distance by a general inundation. The villages beyond it appear like islands situated in the midst of a great lake. Under each village is seen its inverted image as distinctly as it would appear in water. In proportion as this apparent inundation is approached, its limits recede, the imaginary lake, which seemed to surround the villages, retires; lastly, it disappears entirely, and the illusion is reproduced by another town or village more distant. Thus,
as M. Monge, from whom I have borrowed this description, remarks, every thing concurs to complete an illusion which is sometimes cruel, especially in the desert, because it presents the image of water, at the time when it is most needed.'
The second book of this treatise is devoted to what is technically called the theory of the sun,' and is divided into eighteen chapters, occupying 342 pages. The distribution and arrangement of subjects will appear from the following enumeration. Proper motions of the stars, and the means of determining them; application to the sun, with the theory of its circular motion; calendar; manner of referring the position of the stars to the plane of the ecliptic; progressive diminution of the obliquity of the ecliptic; precession of the equinoxes; nutation; second approximation to the sun's motion, with the theory of its apparent elliptical motion; mode of determining the exact position of the solar ellipse upon the plane of the ecliptic, with the origin of mean time, &c.; exact determination of eccentricity from observations of the equation of the centre; use of equations of condition' for the simultaneous determination of the elements; construction of solar tables; inequality of solar days, and the equation of time; spots of the sun, their form, and rotation; inequality of days and seasons in different climates; temperature of the earth; hypothesis of the earth's annual motion; precession of the equinoxes considered as the effect of the displacing of the terrestrial equator ; use of the theory of the sun, and the motions of the equator, ecliptic, and equinoxes, in chronological researches, with some curious applications. This book contains much valuable matter, though not always exhibited in the best form.
In the fourth chapter there is a short but useful note on the method of determining the longitude and latitude of a heavenly body, the right ascension, declination, and obliquity of the ecliptic being given; as well as the method of solving the converse problem. Let w the obliquity of the ecliptic, d the declination of a star, or other body, a its right ascension, a its latitude, l its longitude; then the following formulæ are deduced from the principles of spherical astronomy : sin. N = sin. w cos. d sin. a + cos. w sin.d .
tan. d sin. o + sin. a cos. a tan. I = These two formulæ may be accommodated to the logarithmic calculus, by taking an auxiliary angle o such that tan.o = then exterminating sin. a from the first and tan. d, by means of the usual expressions for sines and cosines of sums and differences, there result
: for tan.d
cos. (+ o) sin, a = sin. d
sin. (v + )
tan. I = tan. a Again, to find the declination and right ascension the formulæ are similar, viz. sin. d = sin. w cos, a sin. I + cos. w sin. à .
tan, a sin. w + sin. I cos. a tan. a = Here, in like manner, taking a subsidiary angle, so that tan. po sin. 1 the resulting formulæ are,
cos. (** sin. d = sin, a
siu. (q? tan. a = tan, 1
sin. o' The angle of position S may be determined by either of the following theorems, viz. siv, a cos. a
sin. w cos. 1 sin. S =
or sin. S =
The preceding formulæ will answer for all positions of the stars, by making the sines, cosines, or tangents, positive or negative, according to the value of the arcs to which they correspond: they are very convenient in application, and, we think, preferable, on the whole, to the rules of Dr. Maskelyne for the same purpose, given in the first volume of Vince's Astronomy.
One of the most remarkable results to which the theory of attraction has led, is that of the Oscillation of all the irregularities of the planetary system within certain limits which they never pass. The variation in the obliquity of the ecliptic is an example of this kind; and M. Biot, in common with many other mathematicians, French and English, ascribes the discovery of this fact to M. Laplace, while, in truth, he has only the merit of affixing the last link to an interesting chain of deduction. Our countryman, Thomas Simpson, has the honour of forming the first ; for, in the resolution of some general problems in physical astronomy, in his Miscellaneous Tracts,' applying his results to the lunar orbit, he concludes, by showing that the effect of such terms or forces as are proportional to the cosine of the arch 2, is explicable by means of the cosines of that arch and of its multiples, (no less than the effects of the other terms that are proportional to the cosines of the multiples thereof,) a very important point is determined; for, since it appears thereby that no terms enter into the equation of the orbit but what by a regular increase and decrease do after a certain time return again to their former values, it is evident from thence that
the mean motion and the greatest quantities of the several equations undergo no change from gravity::--Tracts, p. 179.
The reasoning in the preceding quotation evidently applies to all that has been since done, and is, in fact, the source of every subsequent investigation. It was upon analogous principles that Frisi proved, in his third book De Gruvitate Universali Corporum, prop. 45, that the obliquity of the ecliptic can scarcely ever be more than a degree less than it is now, and that not in less than sixty centuries to come.' And, more generally still, M. Lagrange, employing the principles of Simpson, completed the discovery of the permanency of the whole system in a state but little different from what obtains at any assumed period of its existence; as well as traced the extent of the oscillations in many particular cases. His method has been thus developed - The law of the composition of forces enables us to express every action of the mutual forces of the sun and planets by the sines and cosines of circular arehes, which increase with an uniform motion. The nature of the circle shows, that the variation of the sines and cosines are proportional to the cosines and sinęs of the same arches. The variations of their squares, cubes, or other powers, are proportional to the sines or cosines of the double or triples, or other multiples of the same arches. Therefore, since the infinite serieses which express those actions of forces, and their variations, include only sines and cosines, with their powers and fluxions, it follows that all accumulated forces, and variations of forces, and variations of variations, through infinite orders, are still expressible by repeated sums of sines or cosines, corresponding to arches which are generated by going round and round the circle. These quantities, as every analyst knows, become alternately positive and negative; and therefore, in whatever way they are compounded by addition of themselves, or their multiples, or both, we must always arrive at a period after which they will be repeated with all their intermediate variations.'
Such, in brief, was the process, strictly conformable to the principles originally developed by Simpson, from which Lagrange proved, that the eccentricities of the planetary orbits, though varias ble, will never vanish entirely, nor exceed certain quantities; that the variation in the obliquity of the ecliptic, and every other apparent irregularity in the system, has its period and its limit. Hence, considering what was accomplished in succession by the three eminent geometers here mentioned, justice compels us to lower considerably the praise ascribed by M. Biot and others to Laplace for his discoveries in this department of physical astronomy. His merit consists in carrying their principles into the details. Thus, taking 1750 for the origin of any time t reckoned in years, the dis
tance antecedent to that date being reckoned negatively, and the time subsequent to it, positively, calling y the retrogradation of the equinoctial point on the fixed ecliptic, and V the obliquity of the equator from the fixed ecliptic, Laplace gives, in his Mécanique Céleste, the following formulæ expressed in the centesimal notation:
y=t. 155."5927 + 3.011019 + 4.25562 sin. (t.155."5927 + 95.90733) - 7.935308 cos. t. 99."1227 - 1.°757 2 sin. t.43."0446.
26.90812 0.° 36766 1.°81876 cos. (t 155."5927 + 95.90733)
+ 0.050827 cos. t. 43."0446 2.°84636 sin. t. 99."1227. If \' be the corresponding retrogradation of the equinoxes upon the moveable ecliptic, and V' the apparent obliquity of the equator from the movable ecliptic, then the theorems for any time whatever, reckoning from the epoch 1750, are,
+ = t. 155.5927. 19.42823 sin. t. 43."0446 + 6.°22038 sin. 24. 49."5613. v2 = 26.90812 – 1.003304 sin. t. 99."1227 – 0.973532 sin. *. 21."5223.
From these theorems, which have not, as yet, we believe, been published in any English work, it follows that, with regard to the obliquity of the equator from the fired ecliptic, its total change from the time t will be equal to the product of the annual acceleration into the half of t, that is to say, after the time t the obliquity V will become V + t. 0."00003037; while, for the annual change of the obliquity with respect to the moveable ecliptic, we have
1."6083-0."2486 sin t.43."0446 +3."2166 sin ? t. 49."5613 which, besides the terins proportional to the time, and to the powers of the time, contains the constant term – 1."6083, to which there is nothing analogous in the variations of obliquity with regard to the fixed ecliptic.
* The reason of this difference (says M. Biot) may be traced in the causes which produce the two phænomena. The attraction of the sun and moon, if they acted alone, would produce a constant precession equal to 155." 5927 (centes.) and would not change the obliquity of the equator from the ecliptic, which would then be fixed. But, by the effect of the planetary attraction, the true ecliptic is displaced in the heavens, and carries those two luminaries with it. Their action in consequence varies, and produces a small variation in the obliquity of the equator from the fixed ecliptic. This variation, at first insensible, becomes accelerated proportionably to the time, and the resulting absolute change of obliquity is therefore proportional to the square of the time. But, farther, the attraction of the planets which displaces the true ecliptic, inclines it also towards the fixed ecliptic. This other annual variation is at first constant, and its effect is proportional time. But the apparent obliquity which we observe is the difference of