Tobias, one of the greatest astronomers of the eighteenth century, was born on the 17th of February 1723, at Marbach in the country of Württemberg. His father, who held the office of inspector of waters at Esslingen, was much occupied with hydraulic architecture, and early inspired him with a taste for the mathematical sciences and that of design. The latter talent, which is sufficiently rare amongst astronomers, proved useful to him upon more than one occasion, as may be seen by inspecting the volume containing his posthumous works. After the death of his father, young Mayer having no estate or fortune to depend upon, applied himself to teach mathematics, which he had learned by himself from the first books on the subject which fell into his hands. At the age of twenty, he studied the principles of artillery with a view to enter the service. In 1745 he published his Treatise on Curves for the Construction of Geometrical Problems, and in the same year his Mathematical Atlas, where all the parts of the science are represented in sixty plates. In 1746 he occupied himself with general geography; and having formed a connection with the astronomers Franz and Lowitz, he, like them, contributed to the establishment of the Cosmographical Society of Nuremberg, and inserted several interesting memoirs in the volume which that society published in 1750, under the title of Kosmographische Nachrichten und Sammlungen. Amongst these we would particularly notice his observations and calculations of the libration of the moon, a translation of which has been given by Lalande in the twentieth book of his Astronomie. The instruments which Mayer employed were very indifferent; but he conducted his observations with so much nicety and address, that he was enabled to determine, more exactly than had ever been done before, the elements which serve to indicate all the circumstances of that singular phenomenon, particularly the inclination of the lunar equator, or the position of the axis round which is operated the rotation of the moon. His method for calculating these elements had not all the geometrical rigour which he might have given to it, without rendering it either longer or more difficult; nevertheless, it had all the precision required. But this memoir, now so curious, is distinguished by a still more important novelty. It is the first wherein, for a problem which appeared not to require nor even to admit of more than three observations, there was imagined the method of conditional equations, which, instead of three observations strictly necessary, permits the employment of thousands, and conducts at once to the most certain or the most probable conclusions resulting from the totality of the observations; in fact, the errors which cannot be avoided, yet follow no certain law, are found to act each time in a different manner, and to correct themselves by mutual compensation. It is to this method that we are in a great measure indebted for the precision of the most modern astronomical tables; although it did not for some time attract the attention of astronomers, it is now generally employed; and it is thus that have been constructed from hundreds and even thousands of observations, the tables adopted by Lalande in the third edition of his Astronomie.

In 1757, Mayer went to establish himself at Göttingen, where he married, and was appointed director of the observatory, to which the king of England had presented a fine mural quadrant of six feet radius. This observatory was constructed on a tower forming part of the ancient enceinte of the walls of Göttingen. During the war of the seven years, the French troops had established, in the lower part of the tower, their powder magazine, the service of which was performed with very little precaution. Every evening, Mayer, with his lanthorn, crossed the lower story filled with powder, to ascend to his observatory. At the other extremity of the city the Saxons had also established the depot of their ammunition in a similar tower. One day a terrible explosion took place. It was the Saxon magazine, which, having taken fire, blew up, and destroyed seventy persons. But the astronomer of Göttingen, whose zeal for science rendered him intrepid, like the geometer of Syracuse in the midst of the horrors of war, remained unmoved by the shock, and calmly continued his observations. Mayer made the best use of the observatory for verifying the fundamental points of astronomy; namely, the refractions, the position of the stars, particularly of those of the zodiac, with which the planets were daily compared, and lastly, solar tables. His refractions differ but little from those of Bradley; his formula, apparently rather awkward, is at bottom the same with that of Bradley and Simpson, and only differs essentially in the manner of applying the thermometrical correction. His zodiacal catalogue consists of 998 stars, observed from four or five to twenty-five or twenty-six times; and these deserve all confidence. Others of less importance have been served only once or twice; and he himself states, that he does not answer for their accuracy within ten seconds. In the discourse which precedes his solar tables, he makes a declaration which does honour to his integrity. "In composing them," says he, "I had under my eyes those which the celebrated Lacaille had published in 1758, and of which he had the goodness to send me a copy. I soon perceived that it would be necessary to make very few changes, in order to accommodate them to the observations which I have made since the year 1756. I have therefore no intention of publishing what may be called new tables, but only in following the footsteps of this great astronomer, of making such small corrections as my own observations seemed to require." These changes are of two kinds. In the arguments of the inequalities, he had substituted the millesimal division of the circle for the sexagesimal; and this was an improvement convenient for calculators. With regard to the inequalities themselves, he had calculated them according to the theory. Lacaille had at first tried to deduce them from his observations; but, perceiving that the numbers which they gave differed very little from those which Clairaut had derived from his theory, he adopted the numbers of that illustrious geometer, who was also his friend. The difference, indeed, is small for the moon and for Jupiter; it is more sensible for Venus; and recent researches have proved that the equation of Mayer is too limited. The other change was much more considerable, and by no means fortunate. Mayer had augmented by twenty-seven seconds the secular motion of the sun. But in 1792 and 1800, it was found necessary to adopt, almost without modification, the motion as found by Lacaille. With regard to the inequality peculiar to the sun, Lacaille had very well determined it, such as it was, about 1755; and Mayer made no change therein.

The tables of the moon, which Mayer published in the acts of the academy of Göttingen, in 1755, were the first in which the errors never exceeded two minutes, whereas in the tables of Newton, Halley, and Cassini, they were from eight to ten minutes. Nevertheless, he had been obliged to construct his tables from about an hundred observations; so rare were observations at that period, or so difficult was it to procure them. He had profited by the theory of Euler, in which he made some fortunate changes; and he had sent these tables to London in 1755, in order to compete for the prize proposed by the Board of Longitude. They were there submitted to the judgment of Bradley, who attested that, in two hundred and thirty comparisons which he had made between them and as many observations then Mayer had never found an error which exceeded a minute and a half; and he admitted that a part of this error might be attributed to the observations with which they were compared. He concluded that these tables deserved the attention of the Board of Longitude; and he declared that this error, already so small, might be further diminished, inasmuch as, in eleven hundred observations newly calculated, it was reduced to less than a minute. The author, on his part, laboured incessantly to improve them, and at his death, in 1762, left a new copy, which his widow transmitted to London, where the tables obtained a recompense of £3000. The publication was confided to Maskelyne, and the impression had made considerable progress when a somewhat more complete copy was received, containing several slight improvements. This new copy was preceded by a memoir entitled Methodus Longitudinum Promota, in which Mayer recommended the distances of the moon from the sun or the stars, of which Lacaille and Maskelyne had already pointed out the advantages, and also gave a description of a new instrument for measuring those distances. To make allowance for the flattening of the earth in the calculations of the parallax, he had suppressed the corrections of the geometers, which rendered the operation long and uncertain, and, by a very simple and ingenious consideration, had reduced it to the same degree of simplicity as if the earth, instead of a spheroid, had been a sphere; and this method is now that generally adopted. Lastly, after an ample examination of the whole doctrine of Mayer, the Board of Longitude decided that a sum of £2000 should be added to that which the widow of the astronomer had already received. In the same memoir Mayer showed how he had constructed his tables, and also how they might be still further improved; and it is thus that, under the direction of Maskelyne, they were rendered more precise by Mason, who availed himself of twelve hundred observations made by Bradley. It is by the same means, with the help of the new theoretical researches of Laplace, that these tables have been successively improved by Bouvard, Burg, and Burckhardt. But whatever be the merit of the labours successively undertaken, and of those which may be attempted of new, it may always be said of the lunar tables what Mayer himself said of his solar tables, and of those of Lacaille: These are not precisely new tables, but the tables of Mayer, in which have been made certain small corrections necessary to adjust them to the observations.

The name of Mayer, so celebrated on account of his astronomical observations, became more so for another reason, thirty years after his death, and for an idea to which, in his lifetime, but little attention was paid. Whilst labouring to rectify the geography of Germany, he was unknown, having nothing but his genius, with very little money to procure an instrument with which he might measure the triangles which are the necessary foundation of a good map. By the principle of the indefinite multiplication of angles, and with only a board, a rule, a compass, and a line of chords, such as are contained in an ordinary case of mathematical instruments, he contrived to measure the angles with more precision than he could have obtained by means of the graphometers then in use. Of this invention he gave an account in the Memoires of Göttingen; but no one paid any attention thereto except Montucla, who in his Récréations Mathématiques, speaks of it as an idea simply ingenious, little foreseeing, however, all that might be derived from it. In considering only the idea itself; one would at first suppose that the errors in the observations might always be destroyed. There is something, however, to be deducted from this precision in practice; but the invention is not on that account the less remarkable and useful in all geodesical operations. Improved by Borda, it has been employed in the operations from which were deduced the figure of the earth, the mètre, and the new system of French measures. Mayer had given a primary application of his idea in the reflecting circle, which he proposed for all the operations of nautical astronomy; and this first improvement was applauded, but as yet no one dreamed of rendering it really useful. Borda, however, perceived that the instrument might be rendered at once more accurate and more convenient; he employed it, and his example taught mariners to appreciate its advantages. By one of those changes which are found so easy when they have been effected, Borda discovered that the instrument might be applied to the most delicate operations of astronomy; in measuring, by means of a circle with a radius of a foot and a half, the height of a star, and that too, with more precision than could be obtained with a quadrant of eight feet radius. The repeating circle was applied to all the operations connected with the measurement of degrees of the meridian. By means of this instrument the celestial arc included between the parallels of Dunkirk and Barcelona, or of Formentera, the most southern of the Balearic Isles, was accurately measured, the new arc of the polar circle determined, and every operation of the same kind in Germany and Italy performed by the French engineers.

Such is a view of what was accomplished by Mayer from the age of twenty-three to that of thirty-nine. A malady, chiefly characterised by languor, gradually undermined his strength, and consigned him to a premature grave. He died on the 20th February 1762, leaving two daughters and two sons, one of whom became a celebrated professor of physic. An edition of his works had been promised, but of these only one volume appeared in 1775, under the care of Lichtenberg, his friend and associate. This volume contains, 1. A method for determining more exactly the variations of the thermometer, with a formula for calculating the mean degree of heat in every latitude, and the seasons of the year in which occur the greatest heat and the greatest cold. 2. A memoir on the observations which he made with his mural arc of six feet, and the verifications to which he subjected that instrument. 3. An easy method for calculating eclipses of the sun, being at bottom the method of Kepler, which Lacaille also reproduced in his Leçons d'Astronomie. 4. A memoir on the affinity of the colours, in which he recognised only three primary colours, all the rest being obtained by different combinations of these. 5. His new catalogue of the stars, the work of two years, during which he experienced several interruptions, especially when, as already stated, his observatory was converted by the French into a powder magazine. 6. A memoir, followed by a catalogue of eighty stars, to which he assigned a peculiar motion, independently of the general motion of precession. The volume is terminated by a very beautiful map of the moon, designed according to the orthographic projection, from a lunar globe on which Mayer had painted the most remarkable points of the moon, with a list of a hundred and thirty-three spots, according to their longitudes and their latitudes. The second volume, the speedy publication of which was promised, has not yet appeared. The title of Mayer in the university of Göttingen was that of professor of economy, which is no doubt an odd denomination for an astronomer; but this is not the only instance in which learned bodies have sought to attach to them great men, without troubling themselves as to the particular form or manner in which this was effected. Mayer, however, did not give prelections on a science which he had not studied. The professor of economy gave lectures on mathematics, and on civil and military architecture. His éloge, pronounced at the academy by Kaestner, (Göttingen, 1762; in 4to.), is followed by a list of his works, which we shall here subjoin:

1. Description of a new Globe of the Moon; 2. Terrestrial Refractions; 3. Geographical Charts, including a Critical Chart of Germany, and a Map of Switzerland; 4. Description of a new Micrometer; 5. Observations on the Eclipse of the Sun in 1748; 6. Conjunctions of the Moon and the Stars observed in 1747 and 1748; 7. Proofs that the Moon has no atmosphere; 8. Motion of the Earth explained by a change in the direction of gravity; 9. Latitude of Nuremberg, and other Astronomical Observations; 10. Memoir on the Parallax of the Moon and its distance from the Earth, deduced from the length of a pendulum beating seconds; 11. On the Transmutation of rectilineal figures into triangles; 12. Invention of a Species of Painting of which the products may be multiplied; 13. Inclinations and Declinations of the Magnetic Needle, deduced from theory; 14. Inequalities of Jupiter.