ent Orders of Curves. Ellipfe, and Hyperbola are found, then it is easily fhewn that these The Conic Sections being finished, they pass to Curves of a fuperior The differ- Order, beginning by the Theory of multiple Points, of Points of Inflection, Points of contrary Inflection, of Serpentment, &c. Thefe Theories are founded partly upon the fimple algebraic Calculation, and partly on the direct Method of Fluxions. Then they are brought acquainted with the Theory of the Evolute and Cauftiques by Reflection and Refraction. They afterwards enter into a Detail of the Curves of different Orders, affigning their Claffes, Species, and principal Properties, treating more amply of the best known, as the Folium, the Conchoid, the Ciffoid, &c. Sublime The mechanic Curves follow the geometrical ones, beginning by the exponential Curves, which are a mean Species between the geometrical Curves and the mechanical ones; afterwards having laid down the general Principles of the Conftruction of mechanic Curves, by the Means of their fluxional Equations, and the Quadrature of Curves, they enter into the Detail of the best known, as the Spiral, the Quadratrice, the Cycloid, the Trochoid, &c. VI. Sublime Geometry comprehends the inverse Method of Fluxions, and Geometry. its Application to the Quadrature, and Rectification of Curves, the cubing of Solids, &c. Fluxional Quantities, involve one or more variable Quantities; the natural Divifion therefore of the inverse Method of Fluxions is into the Its Divifion. Method of finding the Fluents of fluxionary Quantities, containing one variable Quantity, or involving two or more variable Quantities; the Rule for finding the Fluents of fluxional Quantities of the most fimple Form, is laid down, then applied to different Cafes, which are more compofed, and the Difficulties which fome Times occur, and which embarrafs Beginners, are folved. What the first Part comprehends. These Researches prepare the Way for finding the Fluents of fluxional Binomials, and Trinomials, rational Fractions, and fuch fluxional Quantities as can be reduced to the Form of rational Fractions; from thence they pass to the Method of finding the Fluents of such fluxional Quantities which fuppofe the Rectification of the Ellipfe and Hyperbola, as well as the fluxional Quantities, whofe Fluents depend on the Quadrature of the Curves of the third Order; in fine, the Researches which Mr. Newton has given in his Quadrature of Curves, relative to the Quadrature of Curves whofe Equations are composed of three or four Terms; and this firft Part is terminated by the Methods of finding the Fluents. of fluxional, logarithmetical, and exponential Quantities, and thole which are affected with many Signs of Integration, and the various Methods of Approximation, for the Solution of Problems, which can be reduced to the Quadrature of Curves. The fecond Part of the inverse Method of Fluxions, which treats of fluxional Quantities, including two or more variable Quantities, commences by fhewing how to find the Fluents of fuch fluxional Quantities as require no previous Preparation; the Methods for knowing and diftinguishing these Quantities or Equations; afterwards they pafs to the Methods of finding the Fluents of fluxional Quantities, which have fecond Part need of being prepared by fome particular Operation, and as this Oper- compreation confifts most commonly in feparating the indeterminate Quantities, hends. after being taught how to construct differential Equations, in which the indeterminate Quantities are separated, they enter into the Detail of the different Methods for feparating the variable Quantities in a proposed Equation, either by Multiplication, Divifion, or Transformations, being fhewed their Application, first to homogeneous Equations, and after being taught how to conftruct these Equations in all Cafes, the Manner of reducing Equations to their Form is then explained. How the Method of indeterminate Co-efficients can be employed for finding the Fluents of fluxional Equations, including a certain Number of variable Quantities, and how by this Method, the Fluent can be determined by certain Conditions given of a fluxional Equation. Fluxional Quantities of different Orders follow; it is fhewn, first, that fluxional Equations of the third Order, have three Fluents of the fecond Order, but the last Fluent of a fluxionary Equation of any Order is fimple; then the various Methods imagined by the most eminent Mathematicians for finding thefe Fluents, fuppofing the Fluxion of any one variable Quantity conftant, are explained, and the Whole, in fine, terminated by the Application of this Doctrine to the Quadrature and Rectification of Curves, Cubing of Solids, &c. VII. Such is the Plan of a Course of pure Mathematicks traced by New- Conclufion. ton, improved by Cotes, Bernoully, Euler, Clairaut, D'Alembert, M'Laurin, Simpfon, Fontain, &c. which ferves as a Bafis to the Inftructions requifite to qualify Youth to appear with Dignity in the different Employments of Life, or to enable them in Time, to bring to Perfection the various Arts for which they are intended. * Quadratura curvarum, harmonia menfurarum, &c. Utility of the Study of the Syftem of the World. PLAN of the Syftem of the Phyfical and Moral World, including the S TUDY in general is necessary to Mankind, and effentially contributes to the Happiness of thofe who have experienced that active Curiofity which induceth them to penetrate the Wonders of Nature. It is, befides, a Prefervative against the Disorders of the Paffions; a kind of Study therefore which elevates the Mind, which applies it clofely, confequently, which furnishes the most affured, arms against against the the Dangers we fpeak of, merits particular Diftinction. Is a Prefervative Pailions. Leads to 66 "It is not "fufficient, fays Seneca, to know what we owe to our Country, to our Finxit in effigiem moderantum cuncta deorum, II. Poets. The Poets who have illustrated Greece and Italy, and whofe Works Is celebratare now fure of Immortality, were perfectly acquainted with the Hea- ed by the vens, and this Knowledge has been the Source of many Beauties in their Works: Homer, Hefiod, Aratus, among the Greeks: Horace, Virgil, Ovid, Lucretius, Manilius, Lucan, Claudian, among the Latins; make ufe of it in several Places, and have expreffed a fingular Admiration for this Science. Ovid after having anounced in his Fafti, that he propofes celebrating the Principles on which the Divifion of the Roman Year is founded, enters on his Subject by the following pompous Elogium of the first Discoverers of the Syftem of the World. Claudian in the following Verfes, celebrates Archimedes on his Inven tion of a Sphere admirably contrived to reprefent the celeftial Motions. Jupiter in parvo cum cerneret æthera vitro, Virgil feems defirous of renouncing all other Study, to contemplate the Wonders of Nature. Me vero primum dulces ante omnia mufæ, ... GEOR. II. 475. La Fontaine imitates the Regrets of Virgil in a masterly Manner, where he fays, Quand pourront les neuf fæurs loin des cours et des villes, Les divers mouvements inconnus à nos yeux, Les noms et les vertues de ces clartes errantes. Songe dun habitant du Mogol. Voltaire, the firft Poet of our Age, has teftified in many Parts of his Works, his Tafte for Aftronomy, and his Efteem for Aftronomers, whom he has celebrated in the finest Poetry. What he fays of Newton is worthy of Attention. Confidens du Tres Haut, Subftances eternelles, Qui parez de vos feux, qui couvrez des vos ailes, Parles! du grand NEWTON n'etiez vous point jaloux. To which we can only oppose what Pope has faid on the fame Subject: Nature and Nature's Laws lay hid in Night; The great Geniufes of every Species have been furprized at the Indifference which Men fhew for the Spectacle of Nature. Talo puts Tasso Reflections in the Mouth of Rinaldo, which merit to be recited for the Inftruction of those to whom the fame Reproach may be applied; it is at the Time when marching before Day towards Mount Olivet, he contemplates the Beauty of the Firmament. |