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Auguste Bravais, from Lapland to Mont Blanc

Marie-Hélène Reynaud, with a Foreword by Dominique Luneau and Philippe Guionneau
[Auguste Bravais]

Foreword

Bravais is a name known – or at least heard of – by any scientist who, directly or indirectly, is interested in crystalline matter. Clearly, crystallographers are familiar with it but mainly with the associated founding concepts. Auguste Bravais (1811-1863) played a major role in establishing the rules governing the notion of crystal structure. The first theory of the periodic structure of crystals was the result of the work of René Just Haüy (1743-1822). This theory, stated from a mathematical point of view in 1801, proposes the description of crystals from a triperiodic assembly of parallelepipedic units called constituent molecules (molécules intégrantes). Widely commented on and considered as the foundation of modern crystallography, this initial theory faced many problems, including the merohedry of some crystals that do not have the symmetrical elements of the constituent parallelepiped. Auguste Bravais proposed a second theory that differs relatively little from Haüy's but which includes some essential adjustments. In essence, Bravais' theory (1848-1850) uses a polyhedral constitutive unit that does not fill space but follows a triperiodic pattern of translations. This is nowadays called the Bravais network. This major advance immediately opens the way to concepts fundamental to the daily life of the crystallographer, such as nodes, reticular planes, elementary cells or crystalline systems. The originality of Bravais' work, relatively difficult in its original version – let's face it – lies in the idea of structural units separated by a void. Although obvious to us crystallographers of the 21st century, this concept nevertheless appeared revolutionary in its context and it required the subsequent and remarkable intervention of many scientists to unravel it, improve upon it and disseminate it to all.

Thus, the 14 Bravais networks are among the first definitions given in a crystallography course – it should be noted that, although initial, these definitions nevertheless deserve particular attention because many pitfalls can mislead the novice student as well as the distracted or occasional researcher. Any worthy crystallography book includes a chapter or paragraph dedicated to Bravais' contribution. So this name is recognized by crystallographers, but is Auguste Bravais, the man, known? To answer this question, we asked a specialist, not of Bravais' scientific contribution but of the man, to enlighten us as much on the eclecticism of the work as on the hectic aspects of his life, which unfortunately ended in drama.

Marie-Hélène Reynaud, a Doctor of Letters, has devoted part of her time to studying the life of Auguste Bravais. In 1991, she published Auguste Bravais, de la Laponie au Mont-Blanc, a book prefaced by André Guinier, member of the Académie des Sciences (France). Marie-Hélène Reynaud is very active in promoting science, particularly in the city of Annonay located in the south-east part of France, where Auguste Bravais was born. The following article is the result of a conference given in his name at the National Congress of the Association Française de Cristallographie (AFC), which took place in July 2018 in Lyon. We believe that, in addition to the many pieces of information given about our scientist, many paths are being explored to further understand this extraordinary period in the 19th century for European science, particularly under the impetus of outstanding scientists. It should be noted in this regard that although René Just Haüy is included in the list of scientists honoured by France whose names are engraved on the first floor of the Eiffel Tower, Auguste Bravais is not. May Marie-Hélène Reynaud's article help to correct this omission!

Dominique Luneau (University of Lyon, France), Chair of AFC2018

Philippe Guionneau (University of Bordeaux, France), AFC President

The original article written by Marie-Hélène Reynaud is in French – we have attempted to translate it, making the choice to let some sentences such as the titles of the Bravais' memoirs remain in their original language.

Beginnings 

[portrait]As crystallographers you all know Bravais and his famous 14 crystal lattice types. The scientific work of Auguste Bravais (this was his full name) goes well beyond crystallography and may surprise us by its diversity, in these times of strong specialization. He was also a naturalist, a botanist, a geographer, a meteorologist, an optician etc. In all these fields, by his sagacious observations, he isolated and described phenomena or new facts; often, he was able to find a rational explanation. One must beware of the naïve idea that at that time discovery was easy because there was so little known. In fact, it was precisely because we started from a very low base and reliable documentation was very rare that even modest gains were of value. In this regard, the seminal work of Auguste Bravais in crystallography is astounding because of how, by simple means, he had found a property of the structures of crystals that then turned out to be a mathematical fact.[1]

Auguste Bravais was born on 23 August 1811 in Annonay, a city located south of Lyon in southern France, the same city as the Montgolfier brothers and Marc Seguin. He was the ninth child of Victor Bravais, a medical doctor and renowned botanist. Dr Victor Bravais had correspondence with several naturalists who sent him seeds and minerals. It was in his garden that the first French dahlias were grown, as vegetables. Auguste Bravais was one of those great scientists of the 19th century who were discovered, thanks to the French Revolution, and to which the development of science in France owes a great deal. In the first place was the creation of the Ecole Polytechnique, which he entered in 1829. The Polytechnique, or X as it was eventually called, combined research and high-level teaching; this was then totally novel. Could the young Bravais have made his innate gifts flourish, if he had not found at the Polytechnique such a scientific environment of unmatched level in the world at this time? Not only was Auguste Bravais educated in scientific culture there, but he could also be initiated in research by contact with the best people of the time.

Early career

Auguste Bravais graduated from the Polytechnique in 1831 and he decided then to make a career in the navy. Assigned to the brig “Le Loiret”, he went on campaigns on the coast of what was then the regency of Algiers. The crew's mission was to ensure the communication services between Algiers, Bône (Annaba) and Oran, while the population was hostile to the French occupation. Several other officers aboard “Le Loiret” were also scientists with a mission to identify the details of the Algerian coastline. At each stop, Auguste Bravais went in search of unknown insects, crustaceans, fish and plants not found in Europe. He sent them to Annonay, where his father and his brother Camille classified them. Some specimens were sent to the Ecole Polytechnique and to various scientists. The long days of sailing and night watching were opportunities for Auguste Bravais to engage in reflection and to write summaries of his discoveries.

[Fig. 1]Figure 1. Page extracted from the work on the symmetry of plant leaves by the Bravais brothers (1835).

His collaboration with his brother Louis was perfect. Together, they carried out an in-depth study of the constitution and the shape of plants, and in 1835 they presented to the Academy of Sciences a dissertation entitled "Essai géométrique sur la symétrie des feuilles curvisériées et rectisériées". They both studied the symmetry relations that can be presented, at various points of the stem, the insertions of the leaves and the organs that derive from them (Fig. 1). This work did not go unnoticed by the scientific community. A Swiss botanist from Candolle dedicated to the Bravais brothers a new member of the Bignoniaceae family that he called Bravisia. Bravais also engaged in mathematics and wrote two memoirs. In the first one, entitled Sur les méthodes employées dans les levés sous voiles, he drew attention to certain errors appearing on a nautical chart when sailing using the segmentation method, caused by a possible inaccuracy in the position of the three principal points, by omitting to correct the azimuthal bearing or by an error in angular observations. The second memoir was entitled Sur l'équilibre des corps flottants. These two studies formed the basis of the two PhD theses Bravais defended in 1837. His dissertation on the equilibrium of floating bodies particularly appealed to the Minister of Marine who subscribed to several copies to be deposited in libraries at ports, while at the same time the Academy placed Bravais' work into the Collection of Foreign Scholars.

Auguste Bravais was then at a turning point in his life. Aware of his value, the Minister of Marine decided to appoint him as a member of the scientific commission of the north, chaired by Mr Gaymard. On 23 April 1838, the Academy of Sciences gave its approval for an expedition aimed at finding the brig “La Lilloise”, which had disappeared in 1833, and also to engage in meteorological and physiological observations in Lapland and Spitsbergen. Aboard "La Recherche" they entered Bell-Sound Bay located at latitude 77° 30' north on 24 July 1838 to winter. Why choose the western shores of Norway and not elsewhere at the same latitude? In fact, the climate is milder there than in other places because the waters of the Gulf Stream warm the coast and prevent the fjord entrances from being obstructed by ice. There the scientific commission immediately made astronomical, physical and meteorological observations and calculations on the movements and the temperature of the sea. Auguste Bravais was the first to reach a difficult to reach peak subsequently named by the commission as “Pic Bravais”. He then initiated a fruitful collaboration with Dr Martins, a botanist. Together, they carried out botanical research, stored plants and wrote notes; they tried to make comparisons with the flora of the Alps and the Pyrenees. On the slopes of the mountains near Hammerfest, Bravais noticed two parallel and horizontal lines. Near Alten Fjord, similar shorelines surround a lake, well above sea level, suggesting that this lake was a bay long ago. His knowledge of botany and the presence of bladderwrack (Fucus vesiculosus) helped him in his determination of the height of the old banks. He quickly discovered that these lines are the traces of an old shoreline emerging as a result of a rise of the coast, the current shoreline being about 30 metres below. Bravais' observations were essentially geological. Whereas it was thought that the ocean level was dropping, Bravais proved that in fact it was the coast that had risen, demonstrating the rise of the Norwegian coast that some had suspected without being able to prove it mathematically.[2]

«La mobilité de l'écorce solide de notre globe s'est trouvée par là pleinement démontrée. On a pu dire que l'expression "ferme comme un roc", si on la prenait dans un sens trop absolu, ne serait que l'expression d'une illusion, et conclure que ce qu'il y a de moins instable autour de nous, c'est le niveau moyen de la mer.»

During this winter, the major work of Auguste Bravais was, no doubt, his study of the aurora borealis that can only be made near the Pole. In southern Europe, the aurora borealis is rarely seen and is of very short duration. Two or three times a year, well-informed observers perceive, in a northern direction, a glow that disappears rapidly. Its hues being those of twilight, novices often associate it with the reddish reflections that linger after the setting sun. If such errors are possible in our countries, in the north such confusion disappears and the aurora borealis is superb. It is like fireworks whose hues change at every moment. No aurora borealis looks like another; they vary constantly. All clear nights have had their aurora borealis; why not think that it is only because of bad sky conditions that no others are observed? Would it not be a daily phenomenon? Such a phenomenon allowed Bravais to conduct a masterful study that earned him the congratulations of all scientists.[2]

«Malgré les mouvements dont sont doués les arcs et les rayons de l’aurore boréale, il est évident qu'ils suivent le mouvement de rota­tion de la terre. L'aurore boréale est donc un phénomène atmosphérique, et non un phénomène cosmique. Canton, M. Becquerel et d'autres physiciens ont signalé la ressemblance qu'offrent les teintes rouges violacées de ce météore avec celles que déploie l'électricité en se mouvant dans le vide. Cette circonstance, jointe à l'action si souvent constatée de l'aurore boréale sur l'aiguille aimantée, a porté les physiciens à la ranger parmi les phénomènes électriques. Bravais a adopté cette opinion, dont un illustre physicien M. de la Rive a récemment vérifié l'exactitude par une magnifique expérience.»

To study the northern lights and come to this conclusion, Bravais invented new instruments adapted to his research, such as a new type of compass. These observations led physicists of the commission to these deductions: "The intensity of the horizontal force increases when dawn comes; then it diminishes and when it reaches the zenith, it is less than before the appearance of the dawn. As for the vertical intensity, it is lower during the northern lights. It only remains to find the explanation of these phenomena."

Bravais and his friends also indulged in more human studies. Having come into contact with Norwegians, Finns and Lapps, they decided to deepen their knowledge of these peoples, objects of many legends. Their physical appearance, behaviour and life differences required precise comparisons; physicists became anthropologists measuring skulls using a cephalometer. When the end of the expedition came, Bravais and Martins decided to return to France by land to complete their botanical studies, guided by a Lapp. Auguste, among other things, conducted very specific research on the growth of Scots pine: "Our goal was to discover the growth laws of Scots pine under different latitudes, from 50° to 70° parallel. This research seemed all the more interesting to us because, in no other country, the pine reached 70 m, and we could study its development on individual trees which are looked upon as the advanced sentinels of forest vegetation. Pine is distinguished by its eminent utility; alone in these icy regions, it can be used as lumber ...". Theory is not all that there is in the studies of Bravais, as practical applications and a better use of economic resources appeared in his conclusions. By planting these pines in the Hautes-Alpes and Isere, imports from Russia and Sweden would be avoided.

Recognition/honours of major work

Returning to France, Bravais wished to devote himself to the pursuit of his research and the exploitation of hundreds of pages of notes he had accumulated. The Minister of Marine entrusted him with the writing of the report of the expedition, thus freeing him from all active service. On 15 January 1841, at the age of 29, Auguste Bravais was made Chevalier in the Order of the Légion d’honneur; on 1 February, he was appointed professor of mathematics applied to astronomy at the Faculty of Sciences of Lyon where he joined his friend and Dean of the Faculty, Henri Tabareau.

As Lyon is close to Annonay, Auguste Bravais could now consider various expeditions with his brother Louis and his friend Charles Martins. In the summer of 1841, they left for Switzerland and the Faulhorn to compare their physical conditions with those of Lapland. They engaged in numerous barometric observations that they continued during the summer of 1842. Their experiments on the propagation of sound attracted many curious people. Small explosives (boxes) were used on the Faulhorn and on the banks of Lake Brienz located 2041 metres below. From each station, one could hear the noise of the explosions. The sight of the light being almost instantaneous, a watch made it possible to measure the delay of the noise and, thus, to determine the speed of propagation of sound. They found that differences appear with temperature variation: the colder it is, the lower the speed of sound.

[Fig. 2]Figure 2. Drawing of the panoramic view from Mont Blanc (Alps) by Auguste Bravais.

On 1 November 1843, Bravais was promoted to the grade of lieutenant (Navy). In the Chamber of Deputies, Arago quoted him: "as one of the officers who, by their knowledge, did the most honour to our navy, even comparing him, in his improvisation, to the geometers of antiquity." In 1844, Abel Villemain, then Minister of Public Instruction, and the physicist Claude Pouillet asked Bravais to carry out a scientific expedition to Mont Blanc. Since the ascent of Benedict de Saussure, 57 years earlier, almost everything remained to be discovered. In 1844, Auguste Bravais accompanied Charles Martins and Auguste Lepileur, a medical doctor, to climb Mont Blanc to renew and complete the observations of Horace Benedict de Saussure (Fig. 2). Comparing alpine areas to boreal lands was also part of the team's goals. The instruments carried for meteorological observations were numerous: barometers, thermometers suspended in the open air or sunk in the snow at different depths, a psychrometer that measured the relative humidity of the air, a pyrheliometer, an actinometer, a hygrometer, a compass, instruments for measuring the horizontal magnetic intensity, the inclination of a magnetic needle, electrical voltage, the boiling temperature of water, the hues of the sky, the transparency of the atmosphere ... . As soon as the theodolite was installed on the summit, Auguste Bravais began to determine the angles between the mountains, in order to establish a geodesic panorama. Such measurements are important for the development of accurate geographical maps; indeed, the knowledge of the angle between two vertices, and glimpses of a third, makes it possible to build a trigonometric network, based on the map:

"A culminating peak, like that of Mont Blanc, makes it possible to directly estimate the angular distance of the two invisible mountains simultaneously from any other point on the earth's surface ... . There is more: the angle of depression of [various] vertices below the horizontal line tangent to the summit of Mont Blanc, combined with the distance and the curvature of the earth, allowed him to calculate later in his office the relative height of these summits ... . Like de Saussure, we were struck by the disorder of the mountains rising south of Mont Blanc; the word chain is inapplicable to them, but that of groups suits them perfectly: one recognizes very well those of the Oisans or the Pelvoux, the Rousses, the Western Alps in between the Drac and the Arve rivers, Red needles above from Chamonix, and finally from Valais ... ".

[Fig. 3]Conical pendulum by Auguste Bravais.

The memoirs published by Auguste Bravais showed the seriousness of his research; sometimes complex for the novice, they are interesting for any scientist who has access to them. In 1845, Auguste Bravais was appointed at the Ecole Polytechnique as a Professor of physics and in the same year he received a very rare distinction as the King of Sweden raised him to the rank of Sword Knight of Sweden. Writing about his expeditions encouraged Bravais to go deeper into the discoveries made during his travels and to present condensed notes to the Académie des Sciences; the first dealt with the white rainbow (a rainbow produced by light reflected from the moon and sometimes called moonglow). For centuries, people had seen in the rainbow a sign of hope announcing better atmospheric conditions; some scientists had tried to deepen their knowledge, but were still far from completely understanding the meaning of this phenomenon. Bravais decided to unravel the mystery. Until then, no one had formulated a theory about the white rainbow. Bravais filled this gap.

"The purpose of this notice will have been achieved if I have shown in one of the most studied and best-known natural phenomena that an attentive observer can still find an ample harvest of important facts to collect. ... The rainbow is a phenomenon so well-known that it seems at first sight that it is hardly necessary to describe it ... . To put in evidence these details, to enumerate the peculiarities rarer than the rainbow can offer accidentally, to relate them to the laws of modern optics, this is the purpose of this notice.”

At the same time, Bravais presented to the Academy a memoir on the parahelion (or sun dogs) situated at the same height as the sun, and another more general one, entitled "Optical Phenomena to Frozen Clouds". The study that excited his colleagues was his presentation on halos. The name of Bravais is, alas, nowadays in the shadows; for his contemporaries, he was crowned with light, and the testimonials abound:[3]

"The halos are due to the refraction of light in the ice needles suspended in the middle of the atmosphere. A large number of physicists, among whom we must mention Huygens, Mariotte, Venturi, Cavendish, Fraunhofer, Young, Galle, Babinet etc. ... have studied the phenomenon of halos, but it is to Bravais that we owe 'the most complete study'."

In the eulogy he published on Bravais, Elie de Beaumont stated: "M. Bravais erased all doubts, by representing by the use of formulas, with the ingenious dexterity for which he had the secret, with a great simplicity, the progression of the reflected and refracted rays, and by deducing from this, the apparently most bizarre forms of the observed phenomena".

From 1845 to 1847, Auguste Bravais redoubled his activity, exploiting thousands of notes amassed during his expeditions. His fields of study shone by their variety: the physics of the globe, meteorology, terrestrial magnetism, research on the capillary action of mercury, astronomy, geometry, calculus of probabilities, and of course crystallography. The results of his research were the subject of many communications to members of the Academy or other learned societies. At times, Auguste Bravais suffered from the lack of a suitable instrument, and so he invented it, such as a polariscope.

The study of clouds with ice particles, halos and parahelions gradually led Auguste Bravais to his masterpiece: crystallography. In 1848, he presented his first memoir, On the geometrical properties of the assemblies of points regularly distributed in space.[4] In a second memoir, Crystallographic Studies, Auguste Bravais laid out his conclusions. His two studies constitute the high point of his publications. He remarkably complemented the essays of Romé de l'Isle and the studies of R. J. Haüy. Nowadays, many of our students do not know who is really the author of the famous postulate of Bravais, namely that if we take any point A of a crystalline substance, there exists an infinite set of points in the three directions of space, points around which the atomic arrangement is the same as around A, with an identical orientation, therefore the same atomic space. Auguste Bravais imagined a very simple method that made it possible to characterize the many modes of stratification. He could now apply the notion of symmetry to a crystalline medium. This highlighted 14 types of networks grouped under all possible symmetries. Through the nodes of these networks the symmetry elements of seven of the 32 symmetry classes pass. These seven systems have the orientations: cubic, tetragonal, hexagonal, rhombohedral, orthorhombic, monoclinic and triclinic. The two memoirs of Bravais were submitted to expert committees chaired by Cauchy. They did not hide their admiration; in their report they wrote that “Auguste Bravais has given new proofs of sagacity that he had already shown in other research”.

Elie de Beaumont continued: "In a second memoir entitled Crystallographic Studies, replacing empirical rules by theorems of geometry, M. Bravais deduces from his fundamental result all the formulas of crystallographers, with this marvelous facility which almost infallibly denotes the radical solution of the difficulties of the subject. I will confine myself to saying that in the second part of this memoir, ceasing to regard molecules as points and considering them as small bodies, which he calls atomic polyhedra, he studies and clarifies the relations existing between them and crystalline systems. He reduces to simple laws the hitherto almost mysterious phenomenon of hemihedry. M. Bravais shows that there could be 35 cases of hemihedry when only 11 had been discovered so far ... He was endowed, in fact, with an admirable facility for all kinds of intellectual work, and possessed the rare ability of being able to attend to the most varied subjects at the same time: hydrography, navigation, astronomy, atmospheric optics, physics proper, geometry, crystallography, pure analysis, natural sciences; one could almost say of him, despite the apparent opposition of words, that universality was his specialty".

Final endeavours

After these two memoirs on the assemblage of points and crystallography, Auguste Bravais continued to publish at a frantic pace; in five years, he wrote study after study; the list testifies to the diversity of studies of Bravais and their universality.[5] In a note on his work, he recalled his aims: "I will point out that all my works, even those of pure geometry, have always been written in order to arrive either at a practical application or at the observation of a natural phenomenon, in the hope of having the good fortune to discover the explanation of its cause. These remarks will be an answer to some people who accused me of being a geometer or a too pure analyst, and of not taking enough account of observation and experiment. I answer that geometry has never been in my hands but an instrument of which I have tried to best serve myself."

But fate soon burst on Bravais. In a few months his father and his brother Jules died, and also his only son was taken by typhoid. In early 1854, at 43 years old, Auguste Bravais was elected to the Academy of Sciences, to the chair previously occupied by Admiral Roussin, in the section of geography and navigation. To try to forget his mourning, he threw himself into his work. He wrote memoirs, prepared reports for the Academy, and stayed day and night at his desk: "He had work left: he applied himself to it relentlessly. He was stimulated by the desire to respond to your kind welcome (that of the Academicians) and by the advice of his friends who urged him to try to forget his pains. He completed many memoirs; he read us important reports, and perhaps he surpassed the measure of his strength. They soon began to betray him. Sleep was running away …"

Auguste was assiduous at the sessions of the academy; he presented many authoritative communications, but he derived no pleasure from them: "... His work was no longer successful. His memory was lacking; he could no longer find the ingenious ideas he had entrusted to memory without writing them down. He wished to put the finishing touches to a great memoir on mirages which was to complete his work on meteorological optics, and which, in his usual modesty, he called the least imperfect of his works; it caught him, it cut him off, it spoiled him and it understood ... What a new pain! The impossibility, where he was now to restore it. He felt the night overtaking that intelligence, once so bright and brilliant; he left the Ecole Polytechnique and we stopped seeing him among us."

Aware of his mental decline, Bravais could no longer teach. On 1 November 1855, the Ecole Polytechnique granted him leave. There is now no doubt that Auguste was suffering from a serious mental illness. Was this due to the series of fatalities that had affected him all at the same time or the after-effects of his military campaigns and their physically very demanding expeditions? It is difficult to dissociate the two causes; it seems that the first served as a trigger in a body weakened by excessive work. The ultimate testimony of the State was given to him on 3 September 1856: he was promoted to the rank of officer in the order of the Légion d’honneur.

[Fig. 4]Figure 3. Record of service of Auguste Bravais at the Polytechnic School.

On 21 April, Bravais sat for the last time in his chair at the Academy; on 9 November, he was removed from the Ecole Polytechnique. He was only 44 years old, and his wife was forced to assert his rights to retirement. From his file at the Ministry of War, it is surprising to see that his signature does not appear at the bottom of his records. His doctor said that "his state of insanity no longer allows him to check his conditions of service" (Fig. 3). He still had strength and preserved all the gentleness and the graceful expression of his countenance but his memory was lost without return. He no longer recognized objects or people. Some light came through this cruel night and gave rise to hopes that unfortunately soon vanished. At moments of lucidity, long periods of difficulties followed. In a fit of despair, he burnt all his papers, those studies that required years of experimentation and work. One afternoon when he came back from a walk, he saw his naval officer's uniform on a chair, and cried.

His agony lasted seven years and on 30 March 1863, Auguste Bravais died after a long descent into hell. His grave has disappeared for lack of maintenance and we have forgotten it little by little. Victor-Louis Goybet, his grand-nephew, explained it very well: "Scientists of pure science, such as Auguste Bravais, do not leave in the memory of people of intellectual half-culture the same memories as those who have left tangible results; their memory suffers from this”.[5]

Some Auguste Bravais publications

De la température de l'air à diverses hauteurs au-dessus du sol dans les contrées boréales.

Sur l'influence qu’exerce l'heure de la journée relativement à la mesure barométrique des hauteurs.

Formules pour représenter les variations périodiques des phé­nomènes météorologiques.

Description d'un nouveau polariscope et recherche sur les dou­bles réfractions peu énergiques.

Sur les systèmes dans lesquels les vibrations dextrogyres et lévogyres ne s'effectuent pas de la même manière.

Sur un halo observé.

Mémoire sur l'influence qu'exerce la rotation de la terre sur le mouvement d’un pendule à oscillations coniques.

Note sur la vitesse du son.

Note sur l'action qu'exerce un courant circulaire formant la base d'un cône sur une aiguille aimantée placée au sommet de ce cône.

Notice sur le nuage.

Recherche des erreurs de position des points d'un levé sous voiles.

Sur l'observation de la température de l'air.

Note sur le rapport géométrique qui lie le mouvement réel d'une étoile filante à son mouvement apparent.

References

[1] Preface by André Guinier for Marie-Hélène Reynaud in Auguste Bravais, de la Laponie au Mont Blanc, Annonay, 1991, 237 pp.

[2] Elie de Beaumont, Eloge historique d’Auguste Bravais, Paris, 1865, 65 pp.

[3] P. Poiré, Nouveau dictionnaire des sciences (1900).

[4] Auguste Bravais, Sur les propriétés géométriques des assemblages de points régulièrement distribués dans l’espace, CR de l’Académie des Sciences, 1848.

[5] Marie-Hélène Reynaud, Auguste Bravais, de la Laponie au Mont-Blanc, Annonay, 1991, 237 pp.

1 February 2019