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CONTENTS OF No. XXII.
This Day is Publised,
CONSISTING OF NEARLY
Including the valuable CLASSICAL Library of the late Profeffor
Hensler of Kiel, in HOLSTEIN.
On fale at the Shop of
EDINBURGH. JAN. 1. 1808.
art. Í. Traité de Méchanique Céleste. Par-P.S: La Place; Mema
bre de l'Institut National de France, et du Bureau des Longia tudes. Paris. Vol. I. An 7. Vol. 3. & 4.. 1805:
A STRONOMY is distinguished by several great and striking cha:
racters, which place it decidedly at the head of the physical sciences. The objects which it treats of, cannot fail to impart to it a degree of their own magnificence and splendour; while their distance, their magnitude, the steadiness and regularity of their movements, deeply impress the imagination, and afford a noble exercise to the understanding. Add to this, that the history of astronomy is that which is best marked out in the progress of human knowledge.' Through the darkness of the early ages, we perceive the truths of this science shining as it were by their own light, and scattering some rays around them, that serve to discover a few definite objects amid the confusion of ancient traa dition,-a few fixed points amid the uncertainty of Greek, Egypa tian, or even Hindoo mythology. But what distinguishes astronomy the most, is the perfect explanation which it gives of the celestial phenomena. This explanation is so complete, that there is not any fact concerning the motions of the heavenly bodies, from the greatest to the least, which is not reducible to one single law--the mutual gravitation of all bodies to one another, with forces that are directly as the masses of the bodies, and inversely as the squares of their distances. On this principle Sir Isaac Newton long ago accounted for all the great motions in our system ; and, on the same principle, his successors, after near á century of the most ingenious and elaborate investigation, have exo plained all the rest. The work before us brings those explanation's into-one view, and deduces them from the first principles of me. VOL. XI. NO, 22.
chanics. chanics. It is not willingly that we have suffered so much time to elapse without laying before our readers an analysis of a work the most important; without doubt, that has distinguished the conclusion of the last or the commencement of the present century." But the book is still, in some respects, incomplete, and a his-" torical volume is yet wanting, which, had we been in possession of it, would have very much facilitated the task that we have now : undertaken to perform. We know not whether this volume is ac. tually published. In the present state of Europe it may be a long time before it can find its way to this country; and, in the mean time, our duty seems to require that an account of the four: volumes, which we possess, should no longer be withheld from the public.
Though the integral calculus, as it was left by the first invent- 3 ors and their contemporaries, was a very powerful instrument of investigation, it required many improvements to fit it for extending the philosophy of Newton to its utmost limits.' A brief e-: numeration of the principal improvements which it has actually received in the last seventy or eighty years, will very much assist us in appreciating the merit of the work which is now before ,us.
1. Descartes is celebrated for having applied algebra to geometry; and Euler hardly deserves less credit for having applied the same science to trigonometry. Though we ascribe the invention. of this calculus to Euler, we are aware that the first attempt toward it was made by a mathematician of far inferior note, Christian Mayer, who, in the Petersburgh Commentaries for 1727, published a paper on analytical trigonometry. In that memoir, the geometrical theorems, which serve as the basis of this new species of arithmetic, are pointed out; but the extension of the method, the introduction of a convenient notation, and of a peculiar algorithm, are the work of Euler. By means of these, the sines and cosines of arches are multiplied into one another, and raised to any power, with a simplicity unknown in any other part of algebra, being expressed by the sines and cosines of multiple arches, of one dimension only, or of no higher power than the first. It is incredible of how great advantage this method has proved in all the parts of the higher geometry, but more espea cially in the researches of physical astronomy. As what we observe in the heavens is nothing but angular position, so if we would compare the result of our reasonings concerning the action of the heavenly bodies, with observations made on the surface of . the earth, we must express those results in terms of the angles observed, or the quantities dependent on them, such as sines, tangents, &c. It is evident that a calculus which teaches how this is to be accomplished, must be of the greatest value to the as