PETER, an eminent French mathematician, was born in 1698. His father was king's professor of hydrography at Croisic in Lower Brittany, one of the best hydrographers of his time, and author of an excellent treatise on navigation. Young Bouguer was bred to mathematics from his infancy, and made rapid progress in that science. At an early age he was appointed to succeed his father in the chair of professor of hydrography, after having undergone a strict examination in mathematics, so as completely to satisfy his examiners. In 1727 he gained the prize given by the Academy of Sciences of Paris for his paper "On the best manner of forming and distributing the masts of ships." He got two other prizes from the academy in the course of four years: the one was bestowed on him for his dissertation "On the best method of observing the altitude of stars at sea;" the other for his paper "On the best method of observing the variation of the compass at sea." These papers are published in the Prix de l'Académie des Sciences. In 1729 he published a work entitled Essai d'Optique sur la Gradation de la Lumière, the object of which is to define the quantity of light which is lost by passing through a given extent of the atmosphere. He finds the light of the sun to be 300 times more intense than that of the moon.

He was soon after made professor of hydrography at Havre, whereby he had the advantage of being nearer Paris than before; and he was chosen associate geometer of the Academy of Sciences, an office which did not require residence in Paris. In this office he was the successor of Mauupertuis. Afterwards he was promoted in the academy to the place of pensioned astronomer, and came to reside in Paris.

It was resolved in France to send an expedition to South America for the purpose of measuring a degree of the meridian near the equator. From that measurement, compared with the length of a degree of the meridian in other latitudes, the deviation from sphericity in the figure of the earth might be known. The academy made choice of four of its members to proceed on this voyage; they were Godin, Bouguer, and De la Condamine, for the geodetical operation, and the younger Jussieu for observations in natural history. Bouguer and his fellow-travellers sailed from La Rochelle in 1735, and it was ten years before he returned to France. The account of his operations during the expedition is given by him in the Memoirs of the Academy of Sciences, 1744, and in a separate work entitled La Figure de la Terre déterminée par les Observations de MM. Bouguer et De la Condamine. There is like- wise an account of this expedition published by Don George Juan and Don Antonio de Ulloa, two scientific naval officers, who accompanied the expedition by order of the Spanish government. The length of a portion of the meridian was measured on the ground by means of a base and a set of triangles. Then, by observing the altitude of the star Orion which passed near the zenith simultaneously at the two ends of the meridian line that had been measured, that line was found to contain $3^\circ 7'$ of latitude. A star near the zenith was employed, to the end that the observation might not be affected by refraction; Orion passed the meridian in the zenith near the middle of the line measured, so that the distance of that star south of the zenith of the northern extremity of the line was $1^\circ 25' 46''$; and its distance north of the zenith of the southern extremity of the line was $1^\circ 41' 13''$, the sum of these two numbers making $3^\circ 7'$. The altitude was taken by zenith sectors of a long radius. The ground on which these operations were performed was elevated 12,000 feet above the level of the sea, and 4200 feet above the neighbouring city of Quito, and situated in a plain extending from north to south, between the two ridges of the Cordillera.

The northern extremity of the arc was on the equator. The length of the degree resulting was 56,767 toises; but this was the degree of a curve circumscribed round the earth at the height of 12,000 feet above the level of the sea; and the length of the degree at the level of the sea deduced from this, with some other corrections, is 56,758 toises. This length of the degree of the meridian at the equator was compared with the degree of the meridian measured in France, with the degree measured in Lapland, and with the degree of longitude deduced in the south of France. From this comparison it was concluded that the equatorial diameter of the earth is to the polar diameter as 179 to 178, and that the equatorial radius of the earth was about eight leagues longer than the polar. Since the time of Bouguer, degrees have been measured in different climates with more accurate instruments than he possessed; but the precise proportion of the equatorial and polar diameters of the earth is not yet finally ascertained. Bouguer makes the excess of the equatorial diameter above the polar to be $\frac{1}{175}$; Sir Isaac Newton made it $\frac{1}{175}$; Laplace, calculating from the lunar motion, $\frac{1}{175}$; Melanderhelm and Svanberg, from a degree measured anew in Lapland in 1783, compared with the degree measured in the province of Quito, $\frac{1}{175}$. Bouguer found the seconds pendulum $\frac{100}{100}$ of a line shorter at the summit of Pichincia than at the level of the sea; that is, the force of gravity was less by one 1200th part at that elevation.

He made some observations on the limit of perpetual snow, a subject which has been elucidated since his time by the researches of Humboldt, Von Buch, Wahlenberg, and others. At the equator the limit of perpetual snow is at 14,760 feet above the sea, a height equal to that of Mont Blanc. In Mexico, in the latitude of $19^\circ 20'$, it is at 13,800 feet, according to Humboldt. In latitude $28^\circ 15'$, where the Peak of Teneriffe is situated, it is supposed to be 11,700 feet; the Peak is only 11,454 feet, and has no perennial snow. On Etna, in latitude $37^\circ 30'$, the edge of the perennial snow is at the height of 9000 feet. On Mount Caucasus, in latitude $42^\circ 30'$, the limit is at 9900 feet; whilst on the Pyrenees, in latitude $42^\circ 45'$, it descends to 8400 above the sea; and on the Swiss Alps, in latitude $46^\circ$, to 8200 feet. In Iceland, in latitude $65^\circ$, the edge of the perennial snow is at the perpendicular height of 2892 feet from the sea. In Lapland, in latitude $67^\circ$, where the summers are warmer than in Iceland, though the winters are colder, the perennial snow does not descend so low, attaining only to 3300 French feet from the sea, as Von Buch and Wahlenberg ascertained by barometrical observations. When the latitudes are the same, a solitary Bouguer mountain will have the edge of the perennial snow higher than a mountain surrounded by others, on account of the warm winds from the neighbouring plains. A mountain in an inland situation will have the border of the perennial snow higher than a mountain in the same latitude, and situated in an island, the summers which reduce the limits of the snow being warmer in the inland situation. When the mass of perennial snow is large, glaciers are formed which descend below the limit of perennial snow. Chimborazo has 5400 feet of its height covered with perpetual snow, according to Humboldt. Bouguer thought he could perceive that the clouds do not ascend higher than 2400 feet above the summit of Chimborazo. If there were mountains whose height reached beyond the greatest height to which the clouds attain, all the part of the mountain above the region of the clouds would be free from snow, although exposed to intense cold. On Bouguer's supposition of the height to which the clouds ascend, the upper limit of snow at Chimborazo would be at the height of 22,200 feet above the sea; and the distance between the upper limit of snow and the lower limit would be there about 7800 feet.

Bouguer, whilst he was at the equator, made observations to ascertain the obliquity of the ecliptic, which he found to be $23^\circ 28' 28''$. He also made some experiments on the deviation of the plumb-line from the vertical, occasioned by the attraction of a neighbouring mountain, a phenomenon afterwards investigated by Dr Maskelyne on the mountain Scheballien.

The number of Bouguer's papers contained in the printed Memoirs of the Academy of Sciences, is a proof of the assiduity with which he performed his duty in the academy. His Heliotometer is described in the Memoirs of the Academy for 1748. It is an object-glass micrometer, and its essential parts consist of an astronomical dioptric telescope, with two object-glasses of the same focal length placed side by side. When this instrument is directed to the sun, each object-glass gives an image of that luminary; and the object-glasses are so placed that the limbs of the two images touch when the diameter of the sun is greatest, and when the diameter is less there is an interval between the limbs of the two images.

Some experimenters maintained that the plumb-line had a diurnal oscillation; Bouguer showed that it remains at rest. He employed for this purpose a telescope, attached to the end of a chain 187 feet long, suspended within the dome of the church of the Hospital of Invalids at Paris: the telescope was directed to a distant mark, so that any motion in this long pendulous system might be seen by the deviation of the wires of the telescope from the mark. The particulars of this experiment are to be found in the Mém. de l'Académie des Sciences, 1754.

In the volume for 1789 and 1749 there are papers of his on the astronomical refraction in the torrid zone, particularly in cases where the star is seen at more than 90° from the zenith, in consequence of the observer being in a high situation. In the volume for 1747, he proposed a log of a new construction for measuring a ship's way.

In the same collection there are papers of his on the length of the pendulum, on the form of the prow which suffers least resistance in passing through the water, and on a variety of other subjects. He bestowed great pains on his works, and his health at length became impaired by a sedentary life, and too constant application to scientific pursuits. He died in 1758, aged sixty. His disposition was naturally mild, and the dissensions that arose between him and his fellow-traveller De la Condamine caused him great vexation. He was impressed, from his earliest years, with a conviction of the truths of Christi- Bouhours. By economy he had acquired a moderate fortune, a part of which he bequeathed to the poor.

The following is a list of his principal works: Traité d'Optique sur la Gradation de la Lumière, 1729 and 1760; Entretiens sur la Cause de l'Inclinaison des Orbes des Planètes, 1734; another edition in 1749. Traité de Novire, de sa construction, et de ses mouvements, 1746, 4to. La Figure de la Terre déterminée, par les Observations de Mess. Bouguer et de la Condamine, envoyés par ordre du Roy au Pérou; par M. Bouguer, 1749, 4to. Nouveau Traité de Navigation, contenant la Théorie et la Pratique du Pilotage, 1753. A new edition by De la Caille, 1761. Solution des Principaux Problèmes sur la Manœuvre des Vaisseaux, 1757. Opérations faites pour la Verification du Degré du Méridien entre Paris et Amiens; par Mess. Bouguer, Camus, Cassini, et Pingré, 1757.

After his return from South America he was editor of the Journal des Savans. Some of his papers in the Memoirs of the Academy of Sciences have been mentioned in this article; his Éloge is contained in the volume for 1758.