sis be as 100 to 1, an addition of one fifty thousandth part of its velocity would produce the same effect.* It is evident, therefore, that the returns of comets to the sun, are not only liable to great variations in respect to time, in consequence of the actions of the planets, but that these actions may be so considerable, especially on comets of very eccentric orbits, as to cause them never

to return.

Some astronomers have ventured to predict the returns of comets on the principle of their uniformity, as to the times of their revolutions, with as much certainty, and attention to minute accuracy, as are due to deductions founded on the strictest principles of the mathematics: But the only instance of such predictions being verified in any degree by events, is that of Dr. Halley, in respect to the comet of 75 years. This comet having appeared, several times, at nearly equal intervals, induced this astronomer to hazard a conjecture, that it would again return in the year 1758. The comet did indeed appear, at a time not differing more than 12 or 14 months from that predicted; but this variation of time, in respect to different revolutions of this comet, whose orbit has so little eccentricity, is a fact corroborating the statement I have made, of the effects which might be produced by the attractions of the planets on comets whose orbits are very eccentric, such as those of 1680, 1769, and many others, which have been observed during the two last centuries.

The identity of the comets of 1582, 1661, has been considered as certain; and its return in 1789 or 1790, at an interval of 129 years, was predicted to the minuteness of hours and minutes, by a celebrated astronomer, now living. The comet, however, did not at that time appear, nor has it been observed since. I am far from thinking that this, or any of the comets, have been known to make more than one revolution about the sun,

*See Prop. 16, and Corollaries, of Book I. of Newton's Principia, where it is demonstrated, that the velocities requisite for bodies moring in different conic sections, the focal distance of the vertex being the same, is in the sub-duplicate ratio of their principal para

meters.

except that of 75 years, whose returns appear to have been observed by astronomers, several times in succession; yet admitting those of 1532 and 1661 to be one and the same comet, it is evident, that its periodical time must have varied from that of its preceding revolution; and if this be the case, it furnishes another fact illustrative of the theory which I have advanced.

Dr. Halley, in his cometography, prompted, undoubtedly, by that enthusiasm which he felt for scientific improvement, says, that "time will reveal to posterity all the mysteries of comets," or in words to this effect. But when the causes, which retard the progress of this branch of science, are duly considered, few will hesitate to adopt the contrary opinion, that ages will pass away before mankind can attain to much more knowledge of the comets, and that the periods of many will ever remain a problem, above human research and investigation. Cincinnati, Jan. 24, 1808.

No. VIII.

OF THE FIGURE OF THE EARTH.

BY COL. JARED MANSFIELD,

SURVEYOR-GENERAL OF THE UNITED STATES.

TH

HE celebrated question concerning the true figure of the earth, so much agitated by rival philosophers of the last century, is one of the many in astronomy and physics, the solution of which is almost wholly dependent on the mathematics. It is true indeed, that physical considerations of the nature of gravity, and the rotary motion of the earth, first suggested to Newton the idea that its figure must necessarily differ from that of a perfect sphere or globe. This sagacious philosopher and mathematician was likewise enabled, by the use of his own sublime geometry, to determine a priori, whatever is required in this problem, with a wonderful degree of precision. But the physical principles of Newton had not yet been verified by a sufficient number of experiments and observations; and the method of Induction on which they were founded, must ever be inferior in evidence, to the pure results of the mathematics.

In order, therefore, to a complete and satisfactory solution of this problem, as well as for an investigation of the principles and conclusions of Newton, it was necessary to have recourse to an actual mensuration of the earth, both in respect to magnitude and figure. The

first of these, viz. the magnitude of the earth, on the supposition of its entire sphericity, or globular figure, is easily determined. It is only requisite that the whole, or some given part of one of its great circles, be ascertained according to known measures. With this view, the arch of the meridian has been selected, as best adapted to celestial observations. This work, for nautical and astronomical purposes, has been performed long since by Picard, Norwood, and others. The more general question of the earth's figure, which necessarily involves that of its magnitude, is of a different nature; and though not difficult to those who are well versed in the higher geometry, is considerably remote from ordinary investigations. Its analysis affords an illustrious instance of the utility of those abstract mathematical speculations, which we have partly derived from the Greeks; but for which we are chiefly indebted to the moderns, viz. Des Cartes, Huygens, Clairaut, the Bernouillis, D'Alem, bert, Euler and Newton.

The question may be propounded in general terms, thus: To determine in any curve, but more particularly in the conic sections, the dimensions of that curve; or the principal lines which regulate it, the diameter of the Osculatory circle, in two or more points of the curve being given.

The Osculatory circle, or circle of curvature of any curve, is that which not only touches the curve in a point, but so nearly coincides with it, that no other circle can be drawn between them. The curvature of the curve, and circle, in that point, is therefore considered as the same. As this curvature, however, in all curves, the the circle only excepted, is perpetually varying; it can be considered the same no where but in the very point of Osculation, or very near it. The measure then of a small portion of the curve at or near this point, may be obtained from the corresponding portion of the circle, and vice verse, that of the circle from a portion of the

curve.

The osculatory circle of any two points of the meridian of the earth, be the curve of any kind whatever, may be found by the mensuration of a small portion of it, at those two points corresponding to any small arc, or am

plitude; or by the distance of lines perpendicular to the tangents in those points, whose intersection constitutes a small known angle, suppose of one degree. A degree of this circle being known, the circle itself is known; and if this be known in two or more points of the curve, the dimensions of the figure, viz. the ratios of the axes, ordinates, parameters, &c. may be found.

With a view to the foregoing process, mathematicans, in order to determine the figure of the earth, directed the measurement of a degree to be made in two or more distant parts of the meridian; where, supposing the figure elliptical, the curvature must necessarily have a perceptible difference. If these requisites could be obtained accurately, the conclusions respecting the form of the earth were considered as incontrovertible as any propositions of Euclid; and as ultimately decisive of the dispute which had been, for a long time, maintained on this subject. For Cassini and his followers had opposed the deductions of Newton, wholly on the ground that the measure of a degree of the meridian near the pole, would be found less than that of one near the equator; which opinion he was led into from a comparison of the lengths of the arches, which had been imperfectly measured by Snellius, Picard, Musschenbroek, and others.

The Newtonians, on the other hand, maintained that these measures were not sufficiently accurate, or properly adapted to the determination of this question; but if an exact mensuration could be made of the length of a degree of the meridian near the pole, and also at or near the equator, that all physical arguments, which in themselves are merely probable or hypothetical, must yield to the certain and demonstrable conclusions of the mathematics. For, if the measure of a degree at or near the pole, should be found less than one at or near the equator, the axis of the earth must necessarily be longer than a diameter of the equator; and on the contrary, if the length of a degree at or near the pole should be greater than one at or near the equator, the equatorial diameter of the earth must necessarily be more extended than its axis. These deductions, though not ob P