pher, who afcribes the origin of this fcience to him; and who afferts, without fufficient evidence and even in contradiction to Archimedes's own acknowlegement, that Apollonius availed himself of what he had written, and published the work of Archimedes as his own. It would lead us far beyond our proper limits to enumerate the various difcoveries, befides the quadrature of the parabola, which occur in the writings of this antient mathematician. They are recorded in his works, to which every one may have access; and our author has done ample justice to his merit. It is moft probable that this fcience, like many others, was gradually augmented and improved; and that each of those antient mathematicians, whofe names we have mentioned, and others whofe writings are loft, contributed to advance it to the state in which Apollonius found it. It has been commonly afferted, and very generally believed, that the terms parabola, ellipfe, and hyperbola, were first introduced by Apollonius our author controverts this pofition: -but we ftill incline to adopt the opinion of those who think that, though the appellations of parabola and ellipfe occur in the works of Archimedes, they were inferted after the time of Apollonius. They are found fo feldom, and the periphrafis of the fections of right-angled, of acute-angled, and of obtufe-angled cones is fo generally used, when it is natural to fuppofe that the other more concife appellations would have been substituted for them if they had been known, that we are difpofed to acquiefce in the fentiments of Dr. Wallis, (fee his Works, vol. I. p. 293.) and of others who afcribe the origin of them to Apollonius. In the fecond chapter of his Appendix, our author proceeds to describe the different methods by which writers on this subject have inveftigated the principal properties of the various fections of the cone. Some have deduced them from the defcription of the feveral curves on a plane: others have confidered them as they refult from the fection of the cone itself. This latter method Mr. R. very juftly prefers. The antients alfo feem to have adopted it. Those who preceded Apollonius used only the right cone; and, allowing no other method of cutting it befides that which fuppofes the interfecting plane to be perpendicular to one of its fides, they were under a neceffity of having recourfe to three different cones, viz. thofe whofe vertical angles are right, acute, and obtufe, in order to obtain the curves that are now denominated the parabola, ellipfe, and hyperbola. Apollonius first fhewed that the three curves might be deduced from the fame cone, either right or fcalene, by merely varying the inclination of the interfecting plane with refpect to one of its fides. This was a very important and ufeful discovery, and gradually led to the extenfion of this science, 13 fcience, and to the eafy inveftigation of the many properties of the feveral curves. Apollonius was born at Perga in Pamphylia, and lived in the time of Ptolemy Evergetes, king of Egypt, whose reign commenced in the year 247 before Chrift. He was therefore about 40 years later than Archimedes. He learned geometry of one who was taught by Euclid himself; and he published eight books on the conic fections, four of which remain in the original Greek. The other four were loft for many ages, but three of them were recovered by means of Arabian manuscripts; so that there are now seven books extant. Dr. Halley has published these, with a Latin tranflation, in his valuable edition of Apollonius's Conics, printed at Oxford in 1710; and he has attempted to supply the eighth book; concerning which he fays. (fee his preface, p. 3.) that, if it does not perfectly agree with the original, it is not very different from it. So highly esteemed was Apollonius's treatife among his contemporaries, that he was denominated, on account of it," the great Geometer." How much it was valued by the Greeks appears by the commentaries of Pappus, Hypatia, Serenus, and Eutocius; nor was it in lefs efteem among the Arabians and Perfians.-The first perfon in later times, who directed any particular attention to the science of conic fections, was Mydorgius, who publifhed two books on the subject at Paris in 1631, and two other books in 1641. It was his intention to have added four other books, but it does not appear that he ever completed his plan. De la Hire, Regius Profeffor of Mathematics at Paris, was the next writer who diftinguifhed himself by his labours in this department of science. His Commentarii de Sectionibus Conicis were published at three different periods, viz. in 1673, 1679, and 1685. The laft edition was his principal work, and is divided into nine books. The general principle, on which his whole system is founded, is demonstrated in the 4th propofition of the 2d book. It is thisthat all parallel right lines, howfoever drawn and terminated on both fides, either by a single section or by opposite fections, are bifected by a right line, which is called the diameter of the fection of these parallels.-James Milnes, A. M. in a work entitled Sectionum Conicarum Elementa Nova Methodo demonftrata, and published at Oxford in 1702, availed himfelf of the trea tife of De la Hire, though he differs from him and other writers in his method of deducing the primary properties of the curves. The general principles which he adopts are demonstrated, in all the sections, by means of the afymptotes of an hyperbola.-Of all the writers, who derive the fundamental properties of the feveral fections from the cone, our author gives the preference to Dr. Hamilton; of whofe excellent treatife he has made very confiderable confiderable use, without introducing any alterations in the primary propofitions, befides those that were thought neceffary for adapting them to the apprehenfion of learners. The method which Dr. Hamilton adopts was first propofed by Guarinus, and published at Turin in 1771; and the propositions which illuftrate it were recited in Jones's Synopfis Palmariorum Mathefeos, publifhed at London in 1706. Our author, however, acquits Dr. Hamilton of plagiarifm, and confiders him as no lefs an original discoverer than Guarinus, who does not seem to have perceived the extenfive application and ufe of the principles which he had discovered. The first perfon, who deduced the primary properties of the conic fections from a description of the curves on a plane, was Dr. Wallis, in a treatife publifhed at Oxford in 1655, and reprinted in the 1ft volume of the collection of his works, p. 291 -354. The firft part of this treatife inveftigates fome of the principal properties of the curves from a view of them, as fections of the cone. The fecond part comprehends an illuf. tration of the new method which he propofes of deducing their properties from the fundamental equation of each curve, as it is defcribed on a plane. The fundamental equation expreffes in algebraic terms the primary property of each curve, or that from which its appropriate name was deduced by Apollonius; and from thefe equations refpectively Dr. Wallis inveftigates, by an analytical procefs, the other principal affections of the curves. De Chales, in his Curfus Mathematicus, published at Lyons in 1674, purfues a fimilar method, and affumes the equations, expreffing the relation between the abfciffes of the diameter and their correfponding ordinates, as definitions of the curves; and from thefe principles he inveftigates the other properties by a method more geometrical than that of Dr. Wallis. In this connection Mr. R. refers to a treatise of the famous John de Witt, published at Amfterdam in 1659, and intitled Elementa Linearum Conicarum; in which he propofes, by a variety of lines and by a very complicated motion of them, not at all adapted to the conception of learners, to defcribe the feveral curves on a plane. This work, executed by the ingenious writer at the age of 23, does great honour to his abilities but his method of conftructing the curves, and of deducing their feveral properties, is fo abftrufe as to afford Jittle advantage to thofe who are not proficients in this fcience. De la Hire, in his Nouveaux Elemens des Sections Caniques, pubJifhed at Paris in 1679, fupplied the defects of De Witt's treatife, and, pursuing the general principles fuggefted by that writer, rendered them more intelligible, and more capable of general application. He confiders each curve as defcribed on 5 a plane: a plane: but his method of actually defcribing it, and of inveftigating its properties, is much more fimple and easy than that of De Witt. In defcribing the parabola, he ufes two equal lines, meeting in the fame point of the curve, one of which is drawn to the focus and the other at right angles to the directrix. The principles which he adopts for defcribing the ellipfe and hyperbola are well-known properties of thefe curves; viz. that in the former the fum, and in the latter the difference of two lines, drawn from the foci to any point in the curve, will be equal to the tranfverfe axis. From thefe plain and easy methods of construction, he deduces the primary affections of the curves. Many of our most approved writers have adopted his method. In the third chapter, Mr. R. recites feveral difcoveries and improvements both of the antients and moderns, relating to the axes, foci, directrices, afymptotes of the hyperbola, fimilar fections, the quadrature of the fections, ofculatory circles, and the defcription of the fections on a plane, which he had not noticed in the former chapters. In this part of the appendix, he has taken occafion to pay a just tribute of respect to those who have enlarged our acquaintance with the properties of the conic fections but for farther particulars we must refer to the work itself, which the mathematical reader will perufe with pleasure and advantage. ART. V. The Poems of Baron Haller. Tranflated into English by Mrs. Howorth. 12mo. 2s. 6d. fewed. Bell, Oxford-street. THE HE merit of Haller as a phyfiologift has placed him high among the benefactors of his fpecies. An almoft fuperftitious goodness endeared him to his neighbours. It is interefting to study the minds of fuch men in their moments of relaxation, and to contemplate the occupations of their leifure. Although poetry was with him a fecondary purfuit, and although he was very eminent in this walk only while the fectators of the Mufes were few in Germany, yet his productions are far from wanting that intereft which great powers of language, a knowlege of nature the moft varied and accurate, and a warm moral zeal, cannot fail to beftow. They farther recommend themfelves to the English reader by an induftrious refemblance to thofe of Pope, whofe didactic works were Haller's favourite ftudy.-The volume before us contains the principal but not all the poems of Haller; an elegy on the death of his fecond wife, feveral fables, fome inferiptions, and other compofitions of no great importance, being omitted. Four of the pieces are given in verfe, and the remainder in profe: the latter appear to us the most successfully rendered. As As the Doris paffes for the most beautiful, and the Alps for the noft fublime, of Haller's poems, we fhall infert a fragment of each. The opening of the first of these poems, rendered literally, would run thus: "The light of day is grown dim: the purple, that fparkled in the west, fades to a fallow grey. The moon lifts her filver horns, the cool night ftrews her poppy-kernels, and flakes the thirsty world with dew. Come, Doris, come to yon beechtrees; let us vifit the filent glade, where nothing ftirs; fave when the amorous breath of Zephyr animates the weak leaves of the boughs, and beckons thee," &c. These lines are thus elegantly paraphrafed by the translatrefs: Now falls the fplendour of the day! In the weft a vapour grey Succeeds the clouds of glowing red Come forth, O Doris, lovely maid! Soft Zephyrus' careffing gale Calls us to this hidden vale, Where its breathings, full of love, Softly through the light leaves move.' Surely, however, the exquifite Wo nichts fich regt als och und du of the original fhould not have been wholly palled over. The following picture is from the Alps: An old man, whofe venerable looks add an Intereft to all he atters, defcribes the battles he has feen, counts the colours which were borne away, marks the trenches where the enemy retired, and repeats the name of each feveral engagement. Our grandfathers formerly bore witnefs to his valour: the weight of a whole century has bowed down his body, and elevated his foul: he is the living image of his ancestors, whofe arms wielded thunder, and who bore their God in their bofoms. The young men liften to him with aftonishment, and difcover by their geftures a noble emulation even to furpass his deeds. • That man, alike venerable for his age, is the living law and the oracle of the people. He makes former occurrences pafs in review before their eyes; and fhews them degenerate nations prefenting their necks to the yoke, and the vain fplendour of courts devouring the fubfiftence of the people. He defcribes the brave Teil trampling under foot the oppreffive fceptre, whofe fway is fill acknowledged by half Europe. Defpotifm," cries this rural philofopher, is the parent |