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and this first Part is terminated by the Methods of finding the Fluents of fluxional, logarithmetical, and exponential Quantities, and those 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.

fecond

The fecond Part of the inverfe 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 distinguishing these Quantities or Equations; afterwards they pafs to the Methods of finding the Fluents of fluxional Quantities, which have art need of being prepared by fome particular Operation, and as this Oper- compreation confists 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, firft, that fluxional Equations. of the third Order, have three Fluents of the fecond Order, but the laft Fluent of a fluxionary Equation of any Order is fimple; then the various Methods imagined by the most eminent Mathematicians for finding these 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, Simpson, Fontain, * &c. which serves 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
Inftructions relative to young Noblemen and Gentlemen of Fortune.

ST

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TUDY in general is neceffary to Mankind, and effentially contributes to the Happiness of those 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 closely, confequently, which furnishes the most affured, arms against against the the Dangers we speak of, merits particular Diftin&tion. "It is not

Is a Prefervative

Paffions.

Leads to
Virtue.

"fufficient, fays Seneca, to know what we owe to our Country, to our

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Family, to our Friends, and to ourselves, if we have not Strength of "Mind to perform thofe Duties, it is not fufficient to establish Precepts, "we must remove Impediments, ut ad præcepta quæ damus poffit animus "ire, folvendus eft. (Epift. 95.) Nothing anfwers better this Purpose than the Application to the Study of the System of the World; the Wonders which are difcovered captivate the Mind, and occupy it in a noble Manner; they elevate the Imagination, improve the Understanding, and fatiate the Heart: The greatest Philofophers of Antiquity have been of this Opinion. Pythagoras was accustomed to say, that Men should have but two Studies, that of Nature, to enlighten their Understandings, and of Virtue to regulate their Hearts; in effect to become virtuous, not through Weakness but by Principle, we must be able to reflect and think closely; we must by Dint of Study be delivered from Prejudices which makes us err in our Judgments, and which are fo many Impediments to the Progrefs of our Reason, and the Improvement of our Mind. Plato held the Study of Nature in the highest Efteem; he even goes fo far as to fay, that Eyes were given to Man to contemplate the Heavens: To which alludes the following Passage of Ovid.

Finxit in effigiem moderantum cunca deorum,
Pronaque cum fpectant animalia cetera terram,
Os bomini fublime dedit, cœlumque tueri
Juffit, et erectos ad fidera tollere vultus.

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 use of it in several Places, and have expreffed a fingular Admiration for this Science.

Ovid after having anounced in his Fafti, that he proposes 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.

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Claudian in the following Verfes, celebrates Archimedes on his Invention of a Sphere admirably contrived to represent the celestial Motions.

Jupiter in parvo cum cerneret æthera vitro,
Rifit, et ad fuperos talia dicta dedit :
Huccine mortalis progressa potentia curæ !
Jam meus in fragili luditur orbe labor.
Jura poli, rerumque fidem legefque deorum
Ecce Syracufius tranftulit Arte fenex;
Inclufus Variis famulatur fpiritus aftris,
Et vivum certis motibus urget opus;
Percurrit proprium mentitus fignifer, annum,
Et fimulata novo Cynthia menfe redit :
Famque fuum volvens audax induftria mundum
Gaudet, et bumana fidera mente regit.

Virgil feems defirous of renouncing all other Study, to contemplate the Wonders of Nature.

Me vero primum dulces ante omnia mufa,
Quarum facra fero ingenti percuffus amore,
Accipiant, cælique vias et fydera monftrent
Defectus folis varios, lunæque labores,
Unde tremor terris, qua vi maria alta tumefcant
Objicibus ruptis, rurfufque in feipfa refidant,
Quid tantum oceano properent fe tingere foles
Hyberni, vel que tardis mora noctibus obftet.
Felix qui potuit rerum congnofcere caufas.

GEOR. II. 475.

La Fontaine imitates the Regrets of Virgil in a masterly Manner, where he says,

Quand pourront les neuf fæurs loin des cours et des villes,
M'occuper tout entier, et m' apprendre des cieux

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,
Le trone ou votre maitre eft affis parmis vous:

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;

God faid, Let Newton be, and all was Light.

The great Geniuses of every Species have been furprized at the Indifference which Men fhew for the Spectacle of Nature. Tafo puts Reflections in the Mouth of Rinaldo, which merit to be recited for the Inftruction of thofe 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.

Con gli occhi alzati contemplando intorno,
Quinci notturne e quindi matutine
Bellezze, incorruptibili e divine;
Fra fe fteffo penfava, o quanto belle
Luci, il tempio celefte in fe raguna!
Ha il fuo gran carro il de, l'aurata ftelle
Spiega la notte, e l'argentata Luna;
Ma non è chi vagbeggi o quefta, o quelle;
E miriam noi torbida luce e bruna,
Ch'un girar d'occhi, un balenar di rifo
Scopre in breve confin di fragil vifo.

III.

JERUS. Cant. xviii. St. 12, 13.

The Knowledge of the Syftem of the World has delivered us from Effects the Apprehenfions which Ignorance occafions; can we recal without which the Ignorance Compaffion, the Stupidity of those People, who believed that by making of the Syfa great Noife when the Moon was eclipfed, this Goddefs received Relief tem of the from her Sufferances, or that Eclipfes were produced by Inchantments (a)?

Cum fruftra refonant Era auxiliaria Luna.
Cantus et e Curru Lunam deducere tentant,
Et faceret fi non Æra repulsa sonent.

Met. iv. 333.

Tib. El. 8.

World has

produced.

Syftem of

ed the Er

The Knowledge of the Syftem of the World has diffipated the Errors of The KnowAftrology, by whofe foolish Predictions Mankind had been fo long abufed. ledge of the The Adventure of 1186, should have covered with Shame the Aftrologers the World of Europe; they were all, Chriftians, Jews and Arabians, united to has diffipat anounce, feven Years before, by Letters published throughout Europe, rors of a Conjunction of all the Planets, which would be attended with fuch Aftrology. terrible Ravages, that a general Diffolution of Nature was much to be dreaded, fo that nothing lefs than the End of the World was expected: this Year notwithstanding paffed as others. But a hundred Lies, each as well attested, would not be fufficient to wain ignorant and credulous Men from the Prejudices of their Infancy. It was necessary that a Spirit of Philosophy, and Research, should spread itself among Mankind, open their Understandings, unveil the Limits of Nature, and accuftom them not to be terrified without Examination, and without Proof.

IV.

The Comets, as it is well known, were one of the great Objects of Terror which the Knowledge of the Syftem of the World has, in fine,

(a) Seneca, Hipolit. 787. Tacit. Ann, Plutarch in Pericle, et de defectu Oraculorum.

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