cal instruments, which have been constructed and divided by the most eminent artists. The best instruments in all the observatories of Europe, have been made by British artists, as Ramsden, Troughton, &c., and they are all graduated according to the sexagesimal division of the circle, therefore, supposing the change to take place, recommended by the French, it is manifest that all observations, made with these instruments, must be reduced to the centesimal division of the circle, before they can be used in calculation. All the latitudes and longitudes of places on the globe must likewise be changed, which change would render the different works on Geography, in a great measure, useless; or otherwise, those in the habit of making trigonometrical calculations, must be perpetually turning the old divisions of the circle into the new, or the new into the old. The same may be said of all the logarithmical tables of sines, tangents, &c., so that the change from the sexagesimal division to the centesimal is a matter of doubtful advantage, when balanced with the difficulties that must necessarily attend it.

CHAP. XXXIII.

MATHEMATICS,
Continued.

CONIC SECTIONS-Method of obtaining the several sections-History of the Science. Writers upon it-Apollonius-Pappus-WallisHamilton-Robertson-De la Hire-Simson-Boscovich-Newton -Vince-Jack-Trévigar-Steel, &c.-FLUXIONS-rules for finding the fluxions of quantities-application of fluxions-writers on fluxions-Newton-Rowe-Vince-Simpson-Maclaurin. Doctrine of Chances-Annuities, Insurance, &c.

cone.

CONIC Sections are such curve lines as are produced by the mutual intersection of a plane, and the surface of a solid The nature and properties of these figures were the subject of an extensive branch of the ancient geometry, and formed a speculation well adapted to the genius of the Greeks. In modern times the conic geometry is intimately connected with every part of the higher mathematics and natural philosophy. A knowledge of the great discoveries of the last century, cannot be attained without a familiar acquaintance with the figures and properties of the conic sections. These sections are derived from the different ways in which the solid cone is cut, by a plane passing through it ;

and they are, a triangle, a circle, an ellipse, a parabola, and an hyperbola. The last three of these, are peculiarly called Conic Sections, and the investigation of their nature and propérties, is generally denoted by the term "Conics." The mode of obtaining these sections is as follows:

(1.) If the cutting plane pass through the vertex of the cone, and any part of the base, the section will be a triangle. (2.) If the plane cut the cone parallel to the circular base, the section will be a circle, provided the cone be a right

one.

(3.) The section is a parabola when the cone is cut by a plane parallel to the side; or, when the cutting plane and the side of the cone make equal angles with the base.

(4.) The section is an ellipse, when the cone is cut obliquely through both sides, or when the plane is inclined to the base in a less angle than the side of the cone is.

(5.) The section is an hyperbola, when the cutting plane makes a greater angle with the base than the side of the cone makes, and if the plane be continued to cut the opposite cone, this latter section is called the opposite hyperbola to the former.

(6.) The vertices of any section are the points where the cutting plane meets the opposite sides of the cone. Hence, the ellipse and the opposite hyperbolas have each two vertices, but the parabola only one.

There is no work of antiquity which professedly treats of the history of conic sections, that has reached our time, and there is little to satisfy curiosity in this inquiry, excepting some incidental notices collected from different authors. The discovery of the curves, denominated the conic sections, is attributed to the philosophers of the school of Plato, or even to Plato himself. The theory of these curves probably grew up gradually from small beginnings, increasing in magnitude and importance, by the successive improvements of many geometricians. The history of the mathematics mentions two problems, famous in ancient times, and both of them so

difficult as to surpass the limits of plane geometry. These problems were the duplication of the cube, and the trisection of an angle; and there is no doubt but that the theory of the conic sections received great additions, and was enriched with many new properties, by the researches that were undertaken for resolving these problems. Two solutions of the former problem, derived from the conic sections, are preserved by Eutochius, in his commentary on the works of Archimedes, which are attributed to Menechmus. Some solutions of the latter problem, by means of the conic sections, are likewise extant in ancient authors, for which, science is thought to be indebted to the ingenuity of the followers of Plato. Hence it has been inferred, that great progress must have been made in investigating the properties of the conic sections before the time of Archimedes. This conclusion is confirmed by the writings of that celebrated mathematician, the best and most splendid edition of whose works was printed at the Oxford press in 1792. In these works many principal propositions are expressly said to have been demonstrated by preceding writers, and are spoken of as truths commonly known to mathematicians. Archimedes himself, perhaps the greatest genius of antiquity, and deserving to be ranked with Galileo and Newton, enriched the theory of the conic sections with many noble discoveries. After a lapse of two thousand years, the quadrature of the parabola is even yet the most remarkable instance in the science of geometry, of the exact equality of a curvilinear to a rectilineal space. To this discovery must be added, the determining of the proportions of the elliptic spaces to one another, and to the circle; and likewise the mensuration of the solids generated by the revolution of the conic sections about their axes.

We are principally indebted to the preservation of the writings of Apollonius, for a more perfect knowledge of the theories of the ancient geometricians, on conic sections. He was instructed in geometry in the school of Alexandria; and under the successors of Euclid, he there acquired that superior

skill in the science which distinguishes his writings. Besides his great work on conic sections, he published many smaller treatises, relating chiefly to geometrical analysis, which have all perished, and are known to us only by the account given of them in the seventh book of the collections of Pappus.

The treatise of Apollonius on the conic sections, is written in eight books, and it was a work in such high estimation among his contemporaries, that it obtained for him the title of "The great mathematician."

The four first books of the conics of Apollonius is the only part of that work that has come down to us in the original Greek. But in the year 1658, Borelli, passing through Florence, found an Arabic manuscript in the library of the Medici family, which he judged to be a translation of all the eight books of the Conics of Apollonius. Transported with joy, he had interest enough to prevail on the Duke of Tuscany to entrust him with the manuscript, which he carried to Rome, where he published a translation of it in 1661. The manuscript found by Borelli, was entitled "Apollonii Pergæi Libri Octo," and was at first supposed to be a complete translation of the work of the ancient geometrician; but on examination, it was found to contain the first seven books only. Two other Arabic translations of the conics of Apollonius, have since that period been discovered, but both these have the same defect as that found at Florence. Hence it is imagined, that since all the three manuscripts agree in wanting the eighth book, it was not in existence when the Arabic translations were made. It cannot be ascertained when the original of Apollonius's work disappeared, but it is certain that it was extant in the time of Pappus of Alexandria, as in his " ollectiones Mathematica," is given an account of the contents of the eight books, and he has even added the lemmas required for the demonstrations of the propositions which they contain: and this circumstance enabled Dr. Halley to annex to his edition of the conics of Apollonius, published in 1710, a restoration of the eighth book, executed with so