curvilinear motion, with the investigations of the ancients on the conic sections, and the beautiful discoveries which Huygens had published according to this method. Geometrical synthesis has besides the property of never losing sight of its object, and of enlightening the whole path which leads from the first axioms to their last consequences, while algebraic analysis soon makes us forget the principal object, to occupy ourselves with abstract combinations, and only brings us back to it at the end. But in thus quitting the object of investigation, after having assumed what is indispensably necessary to arrive at the required result, by directing all our attention to the operations of analysis, and reserving all our forces to overcome the difficulties which present themselves, we are conducted by the universality of this method,-by the inestimable advantage of thus transferring the train of reasoning in mechanical questions,-to results often inaccessible to synthesis. The theory of the system of the world offers a great number of examples of this power of analysis which give it a degree of perfection it would never have acquired had no other path been followed than that traced by Newton. Such is the fecundity of analysis, that if we translate particular truths into this universal language, we shall find a number of new and unexpected truths arise merely from the form of expression. No language is so susceptible of the elegance which arises from the development of a long train of expressions connected with each other, and all flowing from the same fundamental idea. lysis unites to all these advantages that of being ever able to conduct us to the most simple methods. Nothing more is requisite than to apply it in a convenient manner by a judicious choice of unknown quantities, and by giving to the results the form most easily reducible to geometrical construction, or to numerical calculation. The geometricians of the present century, convinced of its superiority, have peculiarly applied themselves to extend its domain, and enlarge its boundaries.

Geometrical considerations, however, ought not to be abandoned; they are of the greatest utility in the arts. Besides which it is curious to imagine the different results of analysis represented in space; and reciprocally, to read all the affections of lines and surfaces, and all the variations in the motions of bodies, in the equations which express them. This approximation of geometry and analysis, diffuses a new light over the sciences; the intellectual

operations of the latter, rendered perceptible by the images of the former, are more easy to comprehend, and more interesting to pursue; and when observation realizes these images, and transforms these geometrical results into laws of nature, and when both, embracing the whole universe, display to our contemplation its present and future state, the view of so sublime a spectacle, presents to us one of the most noble pleasures that have been kept in reserve for mankind.

Nearly fifty years have now passed away since the discovery of the theory of gravitation, without any remarkable addition to it. All this time has been necessary to render so great a truth generally understood, and to surmount the obstacles opposed to it by the system of vortices, and the authority of geometricians contemporary with Newton, who combatted it perhaps from vanity, but who nevertheless accelerated its progress by their labours on infinitesmal analysis.

At length astronomers have conceived the fortunate idea of applying this analysis to the celestial motions by reducing them to differential equations, which they have rigorously integrated, or by: converging approximation. They have thus explained by the law of gravitation all the known phænomena of the system of the world, and have given an unhoped for precision to astronomical tables. It has been necessary, for this object, to bring to perfection at once mechanics, optics, and analysis, which principally owe their rapid improvements to their being necessary to the purposes of physical astronomy. It might be rendered yet more correct and simple, but posterity will no doubt see with gratitude that the geometers of the present century have transmitted no astronomical phenomenon of which they have not determined the cause and the law.

Justice to France requires us to observe, that if England have had the advantage of giving birth to the discovery of universal gravitation, it is principally to the French geometers, and to the patronage of the Academy of Sciences, that we are indebted for numerous developments of this discovery, and the revolution which it has pro.. duced in astronomy.

CHAP. XVIII.

OBSERVATIONS ON THE PLANET VENUS,
BY DR. HERSCHEL.

THE planet Venus is an object that has long engaged my particular attention. A series of observations on it, which I began in April 17.77, has been continued down to the present time. My first view, when I engaged in the pursuit, was to ascertain the diurnal rotation of this planet, which, from the contradictory accounts of Cassini and Bianchini, the former of which states it at 23 hours, while the latter makes it 24 days, appeared to remain unknown, as to its real duration: for the observations of these gentlemen, how widely different soever with regard to time, can leave no doubt but that this. planet actually has a motion on its axis.

The next object was the atmosphere of Venus; of the existence, of which also, after a few months observations, I could not enter, tain the least doubt. The investigation of the real diameter was the 3d object I had in view. To which may be added, in the last place, an attention to the construction of the planet, with regard to permanent appearances; such as might be occasioned by, or ascribed to, seas, continents, or mountains.

The result of my observations would have been communicated long ago, if I had not still flattered myself with the hopes, of some better success, concerning the diurnal motion of Venus; which, on account of the density of the atmosphere of this planet, has still eluded my constant attention, as far as concerns its period and direction. Even at this present time I should hesitate to give the following extract from my journals, if it did not seem incumbent on me to examine by what accident I came to overlook mountains, in this planet, which are said to be of such enormous height, as to exceed 4, 5, and even 6 times the perpendicular elevation of Cimboraço, the highest of our mountains!

The same paper, which contains the lines I have quoted, gives us

* See Phil. Trans. for 1792, part 2, page 337.

likewise many `extraordinary accounts, equally wonderful; such as hints of the various and singular properties of the atmosphere of Saturn. A ragged margin in Venus, resembling the uneven border of the moon, as it appears to a power magnifying from 1 to 4t. One cusp of Venus appearing pointed, and the other blunt, owing to the shadow of some mountain. Flat spherical forms conspicuous on Saturns. All which being things of which I have never taken any notice, it will not be amiss to show, by what follows, that neither want of attention, nor a deficiency of intruments, could oc casion my not perceiving these mountains of more than 23 miles in height; this jagged border of Venus; and these flat spherical forms on Saturn.

Indeed with regard to Saturn, I cannot hesitate a single moment to say, that had any such things as flat spherical forms existed, they could not possibly have escaped my notice, in the numberless observations with 7, 10, 20, and 40-feet reflectors, which I have so often directed to that planet. However, if the gentleman who has seen the mountains in Venus, has made observations on flat spherical forms on Saturn, it is to be regretted that he has not attended to the revolution of this planet on its axis, which could not remain an hour unknown to him when he saw these forms. Last night, May 31, 1793, for instance, I saw two small dark spots on Jupiter; I shall not call them flat spherical forms, because their flatness, as well as their sphericity, must be hypothetical; indeed these two terms seem to me to contradict each other. These were evidently removed, in less than an hour, in such a manner as to point out, very nearly, the direction and quantity of the rotation of this planet.

Before I remark on the rest of the extraordinary relations abovementioned, I will give a short extract of my observations on Venus, with such deductions as it seems to me that we are authorized to make from them.T hus,

April 17, 1777, the disk of Venus was exceedingly well defined,

See Phil. Trans. for 1792, part 2, page 309.

+ Ibidem, page 310.

+ p. 312.

§ p. 396.

The height of Chimbo-raço, according to Mr. Condamine, is 3200 French toises: and the English mile, by Mr. De Lalande, measures 830. If the mountains in Venus exceed Chambo-raço six times in perpendicular elevation, they must be more than 23 miles in height.

distinct, and bright, but no spot was visible by which I could judge, of her diurnal motion. The same telescope shows the spots on Mars extremely well. 7-feet reflector.

April 26, 1777. The disk well defined, and bright, but no spot. 10-feet reflector.

In this manner Dr. H. sets down a number of similar observations; whence he infers, that Venus has a motion on an axis; and that she has an atmosphere he considers evident, from the changes he took notice of, which could not be on the solid body of the planet.

Then follow many other observations on the same, with some on the diameter of Venus. After all, Dr. H. adds, a very few evident results may be drawn from the foregoing observations.

With regard to the rotation of Venus on an axis, it appears that we may be assured of this planet's having a diurnal motion, and though the real time of it is still subject to considerable doubts, it can hardly be so slow as 24 days. Its direction, or rather the position of the axis of Venus, is involved in still greater uncertainty.

The atmosphere of Venus is probably very considerable; which appears not only from the changes that have been observed in the faint spots on its surface, but may also be inferred from the illumi nation of the cusps, when this planet is near its inferior conjunction; where the enlightened ends of the horns reach far beyond a semicircle. I must here take notice, that the author we have before quoted on this subject, has the merit of being the first who has pointed out this inference, but he has overlooked the penumbra arising from the diameter of the sun; which has certainly a cousiderable share in the effect of the extended illumination, and in his angle of 15° 19′ will amount to more than 1° 11′ 47′′.6. His measures are also defective; as probably the mirror of his 7-feet reflector, which was a very excellent one, was by that time considerably tarnished, and had lost much of the light necessary to show the extent of the cusps in their full brillianey.

-- I do not give the calculations I have made of the extent of the twilight of Venus, because my measures were not so satisfactory to myself as I wish them to be; nor so near the conjunction as we may hereafter obtain them; neither were they sufficiently repeated. My computations, however, when compared to those given in the paper on the atmosphere of Venus, show sufficiently that it is of much greater extent, or refractive power, than has been computed in that