Appear of the Earth and universal gravitation

Determination of dimensions of the Earth by the Picardy abbot

In 1660 the “Royal Society” is made up in London, with six years in advance on the Royal Académie of Sciences of Paris, founded in 1666 by Louis XIV on a proposal from its minister Colbert. Among the scientific discussions which take place in the Academy lately created, measurements of the Earth occupied a role of very first plan. It is with the abbot Jean Picard (1620 - 1682), one of the members of the Academy, which one owes the first really precise determination of the terrestrial ray R . It is the last determination of R based on the idea of a spherical Ground. It dates from the years 1668 with 1670 and can be summarized like as follows: Picardy measures an arc of Méridien between Sourdon, locality located in Picardy at the south of Amiens, and Malvoisine, in the southern suburbs of Paris. With this intention, it carries out a Triangulation by using - it is a first - a Théodolite provided with a reticle. It measures with great care a base between Villejuif and Juvisy-sur-Orge. By supposing the spherical Earth and by determining with the highhest possible degree of accuracy for the time the astronomical latitudes, it obtains for the length of an arc of meridian line of 1°, indicated by L (1°) in the continuation, the value L (1°) = 57060 measuring apparatuses. The ray R of the Earth which results from it is equal to (57060x360)/(2π) = 3,2693 million measuring apparatuses. The measuring apparatus used by Picard is that of Châtelet, or “Toise of Paris”. Until the adoption of the metric system in France, geodetic measurements length were reported to this measuring apparatus of Paris. Let us recall that a Toise is worth six feet, that a foot is worth twelve inches, and that an inch is worth twelve lines. The measuring rod is the “measuring apparatus of Châtelet”, outdistances separating two pins, or heels, sealed in a wall of the Châtelet old man, where the clothiers and other tradesmen were held to compare their rules of measurement. In 1799, one allotted a 1,949 m to him length, but it is not impossible that it varied in time, in consequence of the wear of the heels due to the permanent embedding of the rules to compare, so that this measuring apparatus was probably, according to Delambre, shorter towards 1670 than in 1792. In fact, before the adoption with the international scales of the meter like unit of length for the needs for geodesy, which was hardly thing easy to realize, most pleasant anarchy reigned in the field of the measures of length and measurements of derived surface and volume. Thus, the foot , used everywhere, is an inexhaustible mine of confusions. Let us quote, as example, some values (approximate): foot of Paris (0,3248 m), foot of the Rhine (or Leyde, 0,3138 m), foot of London (0,3048 m), foot of Bologna (0,3803 m), foot of North (0,3156 m), foot of Denmark (0,3139 m), foot of Sweden (0,2968 m), foot of Burgos (0,2786 m). This list is far from being exhaustive. In fact, in each State, the units of length varied from one province or one city to another.

At all events, the measurement of Picardy based on the measuring apparatus of Paris and converted into modern units roughly provides 111,25 kilometers for the length of an arc of meridian line of 1° and 6371,9 kilometers for the ray. Nominally, this last value deviates only from 0,014% of the currently allowed value R = 6371 km for the average ray equivolumetric, i.e. for the ray of a sphere whose volume would be that of the real Earth. To tell the truth, this quasi-perfect agreement is especially due to the fact that Picard operated with the Latitude S averages, where the distance to the center of the Earth is close to average.

Scientific progresses and techniques in second half of the XVIIe century

The year 1672 is an important date for astronomy and geodesy, because it corresponds to the completion of the construction of the Observatoire of Paris. Jean-Dominique Cassini (1625 - 1712) there “… was called by Roy to serve Its Majesty in the Academy which it has just established” and became director about it. In addition, the following year (i.e. in 1673) the French astronomer Jean Richer - envoy in 1672 with Cayenne to measure the Parallax there planet Mars, in.liaison.with the Picardy abbot and Cassini operating in Paris - made known that the length of a pendulum beating the second with Paris was to be shortened of 1 ¼ line (approximately 2,82 mm) to beat the second in Cayenne. This observation was going to be at the origin of the idea that the figure of the Earth cannot be spherical, but that it must be ellipsoidal. The goal of the measurement of the parallax of Mars, which had given place to the observation of Richer concerning the pendulum, was to fix the distance Ground-March at the time of the observation, the terrestrial ray being known with precision by lesrécentes mesuresde Picard. Thus one would obtain the scale of the solar system by the Third Law of Kepler. The parallax of Mars measured by Cassini, Picardy and Richer is 25" , implying for that of the Sun 9,5". These data make it possible to evaluate the distance Ground-Sun, i.e. the astronomical Unité, with (57060x360x360x3600)/(2πx2πx9,5) ≈ 7,098 X 10¹⁰ measuring apparatuses (or approximately 138 million kilometers). The allowed value currently for the parallax of Mars is 8,794". The measurement of 1673 thus underestimates the exact value of the astronomical unit from approximately 8,5%, because 1 U.A. is worth currently 149597,87 km. Taking into account the rather precarious instrumentation of the time, one can consider that the value of Cassini, Picard and Richer is not too bad.

In 1673 appears the “Horologium oscillatorium”, work in which Christian Huyghens (or Huygens, 1629 - 1695) described the complete mechanics of the pendulum. In particular, it definitively clears up there the question of the centrifugal force and the centripetal force in the circular motion (uniform). Like practical application, it makes use of the pendulum to return the walk of the regular clock and, by doing this, it invents the exhaust to maintain the oscillations. It should be noted that Huyghens had already appeared before skilful mechanic and optician like fine theorist and observer. Indeed, in March 1655, it had discovered Titan, largest satellite of Saturn and the Solar system, and it had solved the Saturn's ring in 1659. Huyghens became foreign member associated with the Royal Academy of Sciences as of its foundation and worked at the observatory of Paris while making use of glasses with very long focal distance conceived by itself.

In 1675, Huyghens exposes the principle of the spiral spring for the watches. Thus, after having acquired the glasses, astronomical geodesy now had the chronometers essential to its progress. Consequently, it missed nothing any more but mechanics and the mathematical tool to take a final rise. It is the English Isaac Newton (1643 - 1727) who will place in 1687 these tools theoretical at the disposal of the scientists. While waiting, it is in November 1675 which Olaf Rømer (1644 - 1710), Danish astronomer called by Picard at the Observatory of Paris, made its sensational measurement of the Speed of light, while being based on the delay of the eclipses of the satellites of Jupiter. It found c thus = 327000 km/s, value too large of 9% compared to the modern value. The same year 1675 saw the foundation of the Royal Observatory of Greenwich, with a few years of delay on that of Paris. The direction was entrusted by it to John Flamsteed (1646 - 1719).

Discussions around gravity

Isaac Newton (1643 - 1727) publishes his fundamental work, carrying the title mathematical Principes of natural philosophy (“Principia mathematica philosophiae naturalis”) in 1687. It poses the final foundations of modern physics there. It exposes to it its system of the world and shows the Lois of Kepler starting from the law of gravitation of the masses. Let us recall that according to this one, two unspecified mass points of the universe attract each other with a force which is inversely proportional to the square of the distance which separates them, and which the force acts along the direction which them joint. This law will be used henceforth basic with mechanics, the celestial mechanics, geodesy and gravimetry.

On the law of attraction of the bodies, the vaguest ideas and changeantes circulated before Newton, but this one was not the first to think only the action decreased with the distance like the reverse of the square. For Roger Bacon, all the remote actions are propagated in rectilinear rays, like the light. Johannes Kepler takes again this analogy. However, one knew since Euclide that the luminous intensity emitted by a source varies because reverse of the square of the distance to the source. In this optical analogy, the “virtus movens” (vertue moving) emanating from the Sun and acting on planets should follow the same law. However, with regard to dynamics, Kepler remains a peripatetician, i.e. a disciple of Aristote. Thus, for him the force is proportional at the speed and not to the rate of variation speed (with acceleration), as Newton will postulate it later. From its second law ( R v = constante), Kepler will thus draw the following erroneous conclusion: the virtus movens of the Sun on planets is inversely proportional to the distance from the Sun. To reconcile this law with the optical analogy, it supports that the light is spread on all sides in space, whereas the “virtus movens” acts only in the plan of the solar equator.

Later, Ismaël Boulliau (1605 - 1691) pushes until the end the optical analogy in its work “Astronomia Philolaïca”, published in 1645. It thus supports that the law of attraction is inversely proportional to the square of the distance. However, for Boulliau, attraction is normal with the radius vector, while for Newton it is central. In addition, Rene Descartes will be restricted to replace the “virtus movens” of Kepler by the drive of a Tourbillon éthéré. It is followed in that by Roberval, which is him also a follower of the theory of the swirls. Méritoirement, Giovanni Alfonso Borelli (1608 - 1679) explains why the planets do not fall on the Sun by evoking the example from the sling: it balances the “instinct” which has any planet to go towards the Sun by the “tendency” that has any body in rotation to move away from its center. For Borelli, this “screw repellens” (repelling power) is inversely proportional to the ray of the orbit. Robert Hooke, secretary of “Royal Society”, admits that attraction decrease with the distance. In 1672, it decides for the law of the reverse square, while being based on the analogy with optics. However, it is only in one writing gone back to 1674 and entitled “Year attempt to prove the annual motion off the Earth” (a test to prove the annual movement of the Earth) that he formulates the principle of the gravitation clearly. He writes indeed that “all the celestial bodies, without exception, exert a power of attraction or of gravity directed towards their center, in virtue of which not only they retain prevent their own parts and them from escaping, as we see that the fact the Earth, but still they attract also all the celestial bodies which are in the sphere of their activity. From where it follows, for example, that not only the Sun and the Moon act on the walk and the earthmoving, as the Earth acts on them, but that Mercure, Venus, Mars, Jupiter and Saturne also have, by their gravitational capacity, a considerable influence on the earthmoving, just as the Earth has powerful on the movement of these bodies.”

As it is seen, Hooke had formulated the first the law of the gravitation completely correctly, but it had not established it . To validate his assumption of the reverse square, Hooke should have known the laws of the centrifugal force. However, the statements of those were published by Huyghens only in 1673 in the form of thirteen proposals attached to its “Horologium oscillatorium”. In fact, Huyghens had written as of 1659 a treaty entitled “Of VI centrifuga” (On the centrifugal force), in which these laws, but this one was shown appeared only in 1703, in its posthumous works published by Volder and Fullenius. However, as of 1684, Sir Edmond Halley (1656 - 1742), friend of Newton, applies these theorems to the assumption of Hooke. By using the third law of Kepler, it finds the law of the reverse square.

This very brief presentation of the evolution of the ideas concerning gravific attraction before the publication of the “Principia” in 1687 watch in any case that the theory of the universal gravitation was not born spontaneously in the brilliant brain from Newton. Always it is that Newton is in possession, since 1666, of the laws of the uniform circular motion. By an analysis similar to that which Halley was to do, he formulates the law of attraction inversely proportional to the square of the distance, while being based on the third law of Kepler. Nevertheless, being undoubtedly more scrupulous than its precursors, Newton intends to subject this law to the control of the experiment. Also seeks he to check if the attraction exerted by the Earth on the Moon answers this law and if one can identify this attraction with terrestrial gravity, in order to establish the universal character of attraction. Knowing that the ray of the lunar orbit is worth approximately 60 terrestrial rays, the force which maintains the Moon on its orbit would be, under these conditions, 60 ² =3600 time weaker than gravity. “Serious” falling in freefall in the vicinity of terrestrial surface traverses in the first a second distance of 15 feet, or 180 inches. The Moon should thus fall towards the Earth at a rate of one twentieth from inch a second. However, knowing the period of revolution of the Moon and the dimension of his orbit, one can calculate his falling speed. With the value accepted in England in this time, Newton found only one twenty-third of inch a second. In front of this divergence, it renonça with its theory. They are only sixteen years later (in 1682) that he earlier learned during a meeting from “Royal Society” the value from the terrestrial ray determined by Picard in France a dozen from years. With the value that Picard gave for the ray of the Earth, Newton found that the falling speed of the Moon was well one twentieth of inch a second, value which confirmed its theory.

Among the proposals interesting the celestial mechanics and gravimetry, one finds in the “Principia mathematica” several theorems on the attraction of the spheres and other bodies. For example, Newton shows that the gravific attraction of a spherical body whose mass is distributed on isopycnic spherical layers is the same one as that of a mass point located at the center of the body and having the total mass of this one. Another important consequence of the theory of Newton, detailed as in the “Principia”, is as the Earth must be slightly flattened with the poles because of centrifugal force creates by the rotation of the ground on itself.

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