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THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[September, 



penetrated. Jupiter, on the otlier hand, that globe by the side of 

 wliicli (lur's is so trifling, would have gone by an inverse march and 

 involved itself in the incandescent matter of the sun ; men would at 

 last have seen the moon throw itself on the earth. Nothing doubtful 

 or systematic entered into these sinister forebodings. Uncertainty 

 could only affect the precise dates of the catastrophes. It was how- 

 ever known that they would be very far off': so that neither technical 

 dissertations nor the animated descriptions of certain poets interested 

 the public. It was not so wilh learned societies. There they viewed 

 with regret our planetary system on the road to ruin. The Academy 

 of Sciences called the attention of mathematicians of all countries to 

 these threatening perturbations. Euler and Lagrange entered the 

 arena. Never did their mathematical genius throw brighter lustre; 

 however the question remained undetermined. The inutility of such 

 efforts seemed to leave room for resignation only, when from two ob- 

 scure corners, contemned by analytical theory, the author of the Me- 

 canique Celtsle, clearly raised the laws of those great phenomena. The 

 varying velocity of Jupiter, Saturn and the Moon had thencefortli 

 evident physical causes, and returned to the category of common per- 

 turbation, periodical and dependent on gravity, while the so much 

 dreaded changes in the dimensions of orbits became a simple oscilla- 

 tion, kept within very narrow limits, in tine by the almightiness of a 

 mathematical formula, the material world was made firm on its base. 



I cannot leave this subject without at least naming the elements of 

 our solar system, on which depend the variations of speed, of the 

 Moon, Jupiter and Saturn, so long unexplained. The bulk of the 

 movements of the earth around the sun is effected in an ellipse of 

 which the form on account of perturbations is not always the same. 

 Those changes of form are periodical ; sometimes the curve without 

 ceasing to be elliptical, approaches the circular, and sometimes departs 

 from it, according to the oldest observations the eccentricity of the 

 terrestrial orbit has diminished from year to year ; hereafter and later 

 it will increase within the same limits, and according to the same 

 laws. Now Laplace has proved that the mean circul.itory speed of 

 the moon around the earth is connected with the form of the ellipse 

 described by the earth around the sun ; that a diminution in the eccen- 

 tricity of this ellipse inevitably produces an augmentation in the speed 

 of our satellite and reciprocally ; and in hue that this cause is enough 

 to account numerically for the acceleration in its course, which the 

 moon has exhibited from the earliest times down to our epoch. 



The origin of the inequalities of speed in Jupiter and Saturn will, 

 1 hope, be as easy to conceive. Mathematical analysis has not suc- 

 ceeded in representing by finite terms the value of the disturbances 

 which each planet encounters in its orbit by the action of all the 

 others. This value exhibits itself in the present slate of science under 

 the form of an indefinite series of terms, which rapidly diminish in 

 extent as they are removed from the first term. In calculation we 

 neglect those of the terms, which by their rank, correspond with quan- 

 tities below errors of observation, but there are cases where the rank 

 in the series, does not alone determine whether a term may be great 

 or small ; certain numerical relations between the primitive elements 

 of the disturbing and disturbed planets may give to terms, generally 

 negligible, sensible values. This case occurs in the perturbations of 

 Saturn originating with Jupiter, and in the perturbations of Jupiter 

 originating wilh Saturn. There exists between the mean velocities 

 of these two large planets, commensurable simple relations ; hve times 

 the velocity of Saturn very nearly equals twice the velocity of Jupiter, 

 terms which without this circumstance, would have been very little, 

 acquire considerable value. Thence results in the movements of the 

 two stars, inequalities of a long period, perturbations, the complete 

 development of which requires mure than "JUU years, and which won- 

 derfully represent all the contradictions disclosed by observers. Are 

 we not surprised to find in the commensurability of the movements 

 of the two planets so influential a perturbing cause, and to find it de- 

 pendent on this numerical relation; "five times the movement of 

 Saturn is nearly equal to twice the movement of Jupiter," the defini- 

 tive solution of an immense difliculty which the genius of Euler had 

 not been able to overcome, and whicli left it in doubt whether univer- 

 sal gravitation was suflicient to explain the phenomena of the firma- 

 ment? The delicacy of the conception and its results, are in this case 

 equally worthy of admiration. 



We have just explained how Laplace demonstrated that the solar 

 system can only sustain slight periodical oscillations around a certain 

 mean state. Let us now see, in what manner he succeeded in deter- 

 mining the absolute dimensions of the orbits. What is the distance 

 ot the sun from the earth? No scientific question has occupied men 

 more. Mathematically speaking nolhing is more simple; it is enough 

 as in surveying to take from the ends of a known base visual lines to 

 the inaccessible object ; the rest is an elementary calculation. Un- 

 lortunately in the case of the sun the distance is great, and the bases 



which may be measured on the earth, are very small. In such case 

 slight errors of sight exercise enormous influence over the results. 

 In the beginning of the last century Halley remarked that certain in- 

 terpositions of Venus between the earth and the sun, or to employ a 

 consecrated expression, the passages or transits of the planet over the 

 solar disc, would supply every observatory with the indirect means of 

 fixing the position of the visual ray, much superior in exactness to the 

 most perfect direct methods. Such was the occasion in 1761 and 

 1769 of the scientific voyages in which, without speaking of Europe, 

 France was represented in the Isle of Rodriguez by Pingrc, in St. Do- 

 mingo by Fleurieu, in California by the Abbe Chappe, and at Pondi- 

 cherry by Legentil. At the same time England sent out Maskelyne 

 to St. Helena, Wallis to Hudson's Bay, Mason to the Cape of Good 

 Hope,Capt.Cook to Tahiti, &c. The observations in the Southern hemi- 

 sphere, compared with those in Europe, and particularly with the ob- 

 servations, which Father Hell, a famous Austrian astronomer, went to 

 make at Wardhuus, in Lapland, gave for the distance of the sun, the 

 result which has since figured in all the treatises on astronomy and 

 navigation. No government hesitated to furnish learned societies 

 with the means, at whatever cost, of suitably establishing their ob- 

 servers in the most distant regions. We have already remarked that 

 the determination of projected distance appeared imperiously to re- 

 quire a great base, and that small bases would not have sufficed. La- 

 place then solved this problem numerically without any sort of base ; 

 he deduced the distance of the sun, from observations of the moon, 

 made in a single and the same place. 



The sun is the cause of perturbations to our satellite, which evi- 

 dently depend on the distance of that immense inflamed globe from 

 the earth. Who does not see that these perturbations would diminish 

 if the distance augmented, and on the other hand, would increase if 

 the distance diminished; that distance in fact regulates the greatness 

 of them. Observation gives the numerical value of these perturba- 

 tions ; theory on the other hand developes the general mathematical 

 relation which connects them with the solar distance and other known 

 elements. When we have reached this term, the determination of 

 the mean radius of the terrestrial orbit becomes one of the easiest 

 algebraic operations. Such is the happy combination by means of 

 which Laplace solved the great and celebrated problem of the parallax; 

 thus did this ingenious mathematician find for the mean distance of 

 the sun from the earth, expressed in radii of the earthly globe, a num- 

 ber little different from that which had been deduced from so many 

 laborious and costly voyages. According to the opinion of very com- 

 petent judges, it might perhaps be that the result of the indirect 

 method was worthy of the preference. 



The movements of the moon were to our great geometer a fertile 

 mine. His penetrating gaze knew how to find out their unknown 

 treasures. He cleared them from all that hid them from vulgar eyes, 

 with a skill and constancy equally worthy of admiration. We shall 

 be excused for quoting a new example. The earth governs the 

 moon in its course. The earth is flattened. A flattened body does 

 not attract like a sphere. There must therefore be in the movement, 

 we had almost said, in the allure of the moon, a sort of impress of the 

 terrestrial flatness. Such was at the first blush the thought of Laplace. 

 It remained to be determined, and in that lay aU the difliculty, whe- 

 ther the characteristic trails which the flattening of the earth would 

 communicate to our satellite, were sensible enough, apparent enough 

 not to be confounded with errors of observation; it was also requisite 

 to find the general formula of this kind of perturbation, in order to be 

 able, as in the case of the solar parallax, to extricate what was un- 

 known. The ardour and analytical power of Laplace surmounted all 

 these obstacles. At the close of a task which had exacted infinite at- 

 tention, the great geometer found in the lunar movement, two per- 

 turbations, clear and characteristic, depending on the terrestrial flat- 

 tening. The former aft'ected the portion of the movement of our 

 satellite, which is particularly measured by the instrument known in 

 our observatories under the name of the meridian lunette; the second, 

 developing itself nearly in a north and south direction, could only be 

 manifested in observations by a second instrument, the mural circle. 

 These two inequalities of very diflerent values, measured with two 

 instruments entirely distinct, connected with the cause which produced 

 them by the most different analytical combinations, have however led 

 to the same flattening. The flattening thus deduced from the move- 

 ments of the moon is not, it must be well understood, the particular 

 flattening corresponding with such or such country; the flattening ob- 

 served in France, England, Italy, Lapland, North America, India or 

 the Cape ; for the earth having suffered at various times and in various 

 places, considerable elevations, the primitive regularity of its curve 

 has been notably disturbed ; the moon, and that it is which renders 

 the result inappreciable, should give and has effectively given the 

 general flattening of the globe, a sort of mean between the various de- 



