Obfervations made on Pallas by different Aftronomers. I. Obfervations of Dr. Olbers, the Discoverer of Pallas, in which the Field of the Telescope was employed as Circular Micrometer. " 184 46 36 11 52 50 } 30 8 03 17 April 1 8 00 04 2 7 56 55 Mean time right Afcen- declination fion of Pal- of Pallas 184 56 49 11 33 OC 184 36 22 12 13 48 C. d. t. x 673,674. 184 15 38 12 54 25 No. 225 Bode's Catalogue. 184 05 07 13 14 28 138 G. Magn. La Lande's Histoire Celeft. Compared with Stars. No. 20 Virg. Zach. 3 800 37 48 01 08 183 44 40 13 53 OC No. 143, Bode. 58 32 36 183 36 38 14 11 OC 6 8 16 00 183 25 31 14 30 217 7 8 33 05 183 16 26 14 47 25 S No.109, Bode. 9 8 18 20 182 58 27 15 20 52 No. 109 and 111, Bode. 10 8 46 40 182 49 34 15 37 26 No. 111, Bode. 11 8 06 28 182 41 21 15 53 53 No. III, Bode; stormy weather. 12 8 19 00 182 33 28 16 09 No. 111, Bode. 13 8 33 59 182 25 43 16 24 35 No. 87 and 114, cloudy. 14 8 28 20 182 18 28 16 39 15 No 114. 17 10 11 35 17 13 17 49 18 8 26 21 181 56 25 17 22 05 From this time Pallas was compared with 181 55 40 17 23 3cftars, which approach Ceres very near in a 181 50 40 17 33 08 parallel, a catalogue of which is to be found 19 11 16 07 181 43 45 17 47 35 in Colonel Zach's Journal for April, &c. 20 13 25 55 181 38 16 18 00 05 21 12 18 33 181 33 01 18 11 29 23 9 41 02 181 23 59 18 32 11 26 12 37 20 181 11 25 19 02 38 27 12 07 40 181 08 19 19 12 02 II. Meridian Observations of Pallas, made with the Mural quadrant, by Dr. Seyffer, Profeffor, and Director of th 28 11 44 11 181 05 35 19 19 52 Royal Obfervatory, at Göttingen. 29 12 03 10 181 03 15 19 27 57 30 12 03 25 181 01 10 19 35 37 12 27 15 180 59 18 19 43 31 2 11 35 20 180 58 03 19 50 25 5 11 02 35 180 56 06 20 08 59 1802. 7 11 20 27 180 56 40 20 19 38 April 6 11 15 43,278; 183 25 06,0; 14 31 37,0 7 11 11 09,684; 183 15 39,2; 14 49 05,4 23 10 00 49,07; 181 23 50,25; 18 32 09,9 27 9 44 05,601; May 8 900 03,892; 181 08 50,3; 19 10 49.5 16 9 29 54,2625; 181 16 36,0; 20 24 30,0 2051009 * The best observation has been made with an excellent four-feet achromatic telescope of Dollo the day-light being too ftrong for the mural quadrant. Profe far Seyffer, on comparing his Obfervations with the nervest Elements of Dr. Gaufs, found them to agree as follows : From the III. Elements for the Orbit of Pallas. Calculated right Ajcenfions. 1 Calculated Declinations. Meridian Obfervations of Pallos, made at the Obfervatory of Seeberg, near Gotha, by Colonel Baron Zach. Mean Time Apparent right Apparent Deat Seeberg. Afcenfion of clination of Pal- IV. Meridian Observations, made by Profeffor Bode, of Pallas. 1 las, North. 4 11 24 51,9 183 44 6,6 13 54 52,0 Berlin. VI. Obfervations of Pallas, made at the Brera Obfer-VII. Meridian Obfervations of Pallas, made at the vatory, in Milan, on an equatorial Sector, by Sig-Objervatory at Cracau, in Poland, by Profi nor Oriani. Sniadefki. Thus far the obfervations of the new celestial body, Pallas, are published, but we shall not omit communicating, in future numbers of this Journal, any new observations and discoveries relative to the nature of so remarkable a body as this appears to be " among the radiant orbs, that more than deck, that animate the sky, the life-infufing funs of other worlds." For the Monthly Magazine. A SKETCH of the HISTORY of PURE MATHEMATICS, translated from "Traivé Elementaire de Mathematiques Pures, par LEMOINE, Profeffeur de Mathema tiques et de Physique, &c. 93. TH ALGEBRA. 3 [Concluded from page 24.] HE methods which Newton difcovered, and which enabled him to investigate all the great questions in mechanics and astronomy, were for fome time a hidden treasure, of which he was the fole proprietor. And it is fingular, that the English geometricians knew nothing of the new calculi, except what they collected from the pieces which Leibnitz inferted in the Atta Eruditorum of Leipfic. Nor were the germs of the differential and the integral calculus, there depofited, immediately developed, even on the Conti nent; and the excellence of the new invention was not for several years under. tood. Fames Bernoulli was the first ge * James Bernoulli, who was born at Bafil in 1654, was originally intended for pursuits very different from those of the mathema tics; but his inclination prevailed against the appofition of his relations, and he was his own preceptor. After having travelled, he returned into his own country, where he was ometrician whose eyes were opened, and who began to second the efforts of Leibnitz. The Infinitesimal Calculus, concerning Cernit which he (Bernoulli) published an Effay, in the Leipfic Acts for 1691, foon became, in his hands, a penetrating instrument, which he handled with great dexterity. He used it in analysing the most delicate problems in geometry and mechanics. When deeply reflecting on the properties of curve lines, he found, by the way, that the evolute of the logarithmic spiral is a logarithmic spiral equal to the first, and differing from it only in pofiDelighted with this discovery, James Bernoulli defired that the memory of it might be perpetuated by defcribing on his tomb a logarithmic spiral, with these words: Eadem mutata refurgo. tion. 94. John Bernoulli did not linger be appointed Profeffor of the Mathematics in the University of Bafil. He died on the 16th of August, 1705. He was flow but fure in his advances in the sciences; and he gave none of his pieces to the public till he had repeat edly revised and examined them. * John Bernoulli was born at Bafil, in the year 1667, and died in the fame town, in 1748. He was successively a Profeffor of the Mathematics at Groningen and in his native country. His brother was his preceptor; who, wishing to preferve the tone of fuperiority which his greater age, and the quality of hild his brother in this glorious career. Like James he participated in the folution of the finest problems which were agitated among the geometricians of that period. He proposed several himself, and Keill had fome reafon to repent of having called forth his powers. In 1698, John Bernoulli published the rules and the use of the exponential calculus, which Leibnitz and he had invented, each in his turn; and to the geometrician of Bafil France is indebted for her first knowledge of the new calculus. He made a journey to Paris, in 1691, when he became acquainted with L'Hospital, initiated him in the new geometry, and for his use he compofed his Leçons de Calcul Differentiel et de Calcul Integral (Lectures on the Differential and the Integral Calculi). The care of Bernoulli was not loft; for L'Hopital foon became one of the first geometricians in Europe. The work which he (L'Hopital) published under the title of Ana. lyse des Infiniment Petits (the Analysis of Infinitefimals) was received with universal applaufe*. of preceptor, conferred on him; and the younger brother forgetting the obligations of gratitude; an open rupture was the confequence, and their sharp disputes were only terminated by the death of james. The in. finitefimal geometry, however, was perhaps as much promoted by the illustrious Bernoullis as by Leibnitz himself. They were both geniuses of the first order, and it would be difficult to fettle the point of pre-eminence between them. • The Marquis de l'Hopital or Hospital, who was born in 1661, had in his childhood an extreme pallion and decided talents for the mathematics. Scarcely had he attained his fifteenth year, when he gave proofs of his fagacity, by the solution of some very difficult problems. He ferved some time in the army, but the weakness of his fight obliged him to abandon a profession in which he never could have fignalized himself. The mathematics then took entire poffeflion of his mind; and L'Hofpital faw himself placed nearly on a level with Newton, Leibnitz, and the Bernoullis. He was carried off by an apoplexy, in February, 1704.. Note by the Translator - The Marquis de L'Hofpital's excellent Analyse des Infiniment Petits contains only the Differential Calculus, or what we call the direct method of fluxions; for, when the author was proceeding to the integral calculus, or inverse method of fluxLons, Leibnitz wrote him, that he was about to publish a work, De Scientia Infiniti, which would comprise that doctrine. The Marquis, in consequence, modeftly defifted, and Leibanta never published his intended perforMance, any more than his Analysis Situs, and 95. It is the lot of all great inventors to be opposed by contradiction. The some other works which he promised to the world. Thus was the public deprived of the second part of the Analyse des Infiniment Petits, which, it is fair to fuppofe, would have been as well executed as the firft. The Marquis was, undoubtedly, a great mathematical genius; but he enjoyed otium cum dignitate leifure and fortune, and, as our author tells us, had for his preceptor one of the greatest mathematicians in Europe, who wrote a book (The New Calculus) purpofely for his ufe. What then are we to think of the Scotch gardener, Stone, who, having been only taught his alphabet, penetrated, by mere dint of genius and folitary. study at his leisure hours, into all the arcana of the higher geometry, began where the Marquis left off, and completed the most arduous part of the plan, which, as we have feen, the great author was prevented from executing? And what are we to think of the Leicestershire weaver, Simpfon, who, with little more original instruction than Stone, and no other help than the joint work of him and L'Hospital, just mentioned, fat on his loom, and wrote a still better book. Above all, what must we think of Saunderson, who, with "wisdom from one entrance quite shut out," and labouring under many other disadvantages, wrote, or rather dictated, ably on some of the most abstruse parts of the mathematics; and, without any idea of light or colours, lectured learnedly on optics!-Vide Wolfi Elem. Math. Univ. tom. v. p. 60; Saverion's Dict. Math. et Phys. Art. Calcul. Integral; T. Simpson's Life in the Biograph. Dict.; Stone's Life, prefixed to his Euclid, by his learned countryman, the Chevalier Ramfay; and Saunderson's Life, in the 4to edition of his Algebra. I cannot help thinking, that our author should have taken some short notice of these aftonishing phenomena of genius, particularly as all the three, but especially Simpson, had rights to be confidered as inventors. Nor, in my humble opinion, should he have neglected to name Matthew Stewart and (Glasgow) Simfon, as diftinguished restorers and cultiva. tors of the ancient geometry; or M'Laurin, to whom the method of fluxions, sometimes called the modern geometry, owes its fecu rity from all future metaphyfical affailants, unless we can suppose, that some more formidable one than the very acute Bishop of Cloyne should make a second attempt to fap its immoveable foundations. Having mentioned that great mathematician, virtuous citizen, and amiable man, it may not be amiss if, like our author, who has given us the end of Newton's Epitaph, I infert a fimilar extract from the equally admired one of his friend M'Laurin, which I copied in 1786, from his monument, in the Grey Friar's Church-yard, Edinburgh, It is faid to have 96. The dispute respecting the invention of the new analysis had kindled a war of problems between the English mathematicians and John Bernoulli, who supported the cause of Leibnitz. It was a curious spectacle to behold the disciples of Newton ranged on one fide, and on the other John Bernoulli making head against them, and, like Horatius Cocles, futtain ing alone all the efforts of the British army. Taylort diftinguished himself above all the defenders of Newton. He folved most of the problems proposed, and to him we owe the first attempt to apply the been written by the celebrated Dr. Johnson, and is certainly worthy of his taste in Latin compofition; H. L. P. F. Non ut nomine paterno confulat; It is impoffible, at least for me, to do juftice to this elegant and nervous sentence in English, but the following tranflation may erve to convey some idea of it: "His son erected this monument, not to perpetuate his father's name, for it needs no fuch aid; but that, in this valley of tears," where Fear and Sorrow hold their reign, mortals might receive some confolation; for, let them study his works, and be inspired with the belief, that the capacious mind, which " grafped such fublime fyftems," survived the perishing body." * Peter Varignon, born at Caen, in 1654, studied the mathematics profoundly; and his fuccess procured him a profeffor's chair in Mazarin College, in which he has been fuc. ceeded by justly celebrated geometricians. Varignon died, almost suddenly, in 1722. To great learning, he joined much facility of Jabour; and no mathematician knew better how to generalize the methods of investiga tion. + Taylor, the scholar of Newton, was born in 1690, and died in 1734. new analysis to Finite Differences, which Nicole unfolded and improved. 97. Among other geometricians who issued from the school of Bafil, Hermannts Daniel Bernoullit, and Eulers, shewed themselves worthy of the masters under whom they were formed. 98. Other geometricians also signalized themselves by their labours and their dif coveries. Ifchirnbaufen||, who made him. * Francis Nicole was born at Paris, in 1683, and finished his course in 1758. His talents for the mathematics shine in the works which he left behind him. + James Hermann was born at Bafil, in 1680, and studied under James Bernoulli Peter the Great called him to Petersburg, where he was Profeffor of the Mathematics till 1722, when he returned to his native country, where he died in 1734. He vigoroufly defended the principles of the Differential Calculus. I Daniel Bernoulli, the fon and the pupil of John Bernoulli, was born at Groningen, in February, 1700. He travelled into Italy and into Russia, in which last country the court of Petersburg in vain endeavoured to retain him. He chose rather to occupy a professor's chair in the University of Bafil, and there it was that academic crowns were accumulated on his head. He died in 1782. His elder brother, Nicholas Bernoulli, had foared rapidly to the highest regions of geometry, when the hand of death arrested him, at twenty-feven years of age.. § Leonard Euler was born at Bafil, in 1707. An irrefistible attraction early urged him to the mathematics. Having been called to Petersburgh he foon enriched the Academical Collections of that metropolis with a great number of memoirs. In 1741, the King of Pruffia invited him to Berlin, where he lived several years with that monarch. On his return to Petersburg, he was attacked with a violent diforder, which deprived him of fight. But that misfortune did not abate the activity and fecundity of his genius, and he continued his labours till his death, which took place in 1783. All the works of Euler bear the stamp of genius, and marks of the most profourd knowledge. || Ernfroy Walter de Tschirnhaufen was defcended of an ancient family, and was. born in 1651, at Killingswald, a manor belonging to his father, in Lufatia. He made some campaigns in the Dutch service, and about the year 1672 ented on his travels. He visited Paris, for the third time, in 1682, and was numbered among the members of the Academy of Sciences. After his return to his native country, he made ftudy his principal occupation. He died in the end of the year 1708. Abraham 1 |