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What the fecond Part
and this first part is terminated by the Methods of finding the Fluents of Auxional, 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.
The second part of the inverse Method of Fluxions, which treats of Auxional Quantities, including two or more variable Quantities, commences by shewing how to find the Fluents of such Auxional Quantities as require no previous Preparation; the Methods for knowing and distinguishing these Quantities or Equations; afterwards they pass to the Methods of finding the Fluents of Auxional Quantities, which have need of being prepared by some particular Operation, and as this Oper-compreation consists most commonly in separating the indeterminate Quantities, bends. 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 separating the variable Quantities in a proposed Equation, either by Multiplication, Division, or Transformations, being shewed their Application, first to homogeneous Equations, and after being taught how to conftru& these Equations in all Cases, 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 Auxional 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 shewn, first, that fluxional Equations of the third Order, have three Fluents of the second Order, but the last Fluent of a fuxionary Equation of any Order is simple; then the various Methods imagined by the most eminent Mathematicians for finding these Fluents, supposing the Fluxion of any one variable Quantity constant, are explained, and the Whole, in fine, terminated by the Application of this Doctrine to the Quadrature and Re&ification of Curves, Cubing of Solids, &c.
Such is the Plan of a Course of pure Mathematicks traced by New- Conclusion. ton, improved by Cotes, Bernoully, Euler, Clairaut, D'Alembert, M.Laurin, Simpson, Fontain, * &c. which serves as a Basis to the Instructions requisite 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.
PLAN of the System of the Pbysical and Moral World, including the
Instructions relative to young Noblemen and Gentlemen of Fortune.
PLAN of the System of the Physical World.
Nubem pellente mathefi,
Utility of TUDY in general is necessary to Mankind, and essentially contrithe Study of the Sy
butes to the Happiness of those who have experienced that a&ive Item of the Curiosity which induceth them to penetrate the Wonders of Nature. World,
It is, besides, a Preservative against the Disorders of the Passions; a
kind of Study therefore which elevates the Mind, which applies it Is a Preservative closely, consequently, which furnishes the most assured, arms against against the the Dangers we speak of, merits particular Distin&tion. « It is not Paffions.
“ sufficient, says Seneca, to know what we owe to our Country, to our
Family, to our Friends, and to ourselves, if we have not Strength of “ Mind to perform those Duties, it is not fufficient to establish Precepts, " we must remove Impediments, ut ad præcepta quæ damus pollit animus “ ire, folvendus eft. (Épift. 95.) Nothing answers better this purpose than the Application to the Study of the System of the World; the Wonders which are discovered captivate the Mind, and occupy it in a noble Manner; they elevate the Imagination, improve the Understanding, and satiate the Heart: The greatest Philosophers of Antiquity
have been of this Opinion. Pythagoras was accustomed to say, that Leads to Virtue.
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 so many Impediments to the Progress of our Reason, and the Improvement of our Mind. Plato held the Study of Nature in the highest Esteem ; he even goes so far as to say, that Eyes were given to Man to contemplate the Heavens :
To which alludes the following Passage of
Finxit in effigiem moderantum cuncia deorum,
The Poets who have illustrated Greece and Italy, and whose Works is celebratare now fure of Immortality, were perfeâly acquainted with the Hea
ed by the
Poets. 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 expressed a singular Admiration for this Science.
Ovid after having anounced in his Fasti, that he proposes celebrating the Principles on which the Division of the Roman Year is founded, enters on his Subject by the following pompous Elogium of the first Discoverers of the System of the World.
Claudian in the following Verses, celebrates Archimedes on his Invention of a Sphere admirably contrived to represent the celestial Motions.
Jupiter in parvo cum cerneret atbera vitro,
Rift, et ad superos talia difta dedit :
Jam meus in fragili luditur orbe labor.
Ecce Syracufus tranftulit Arte senex ;
Ei vivum certis motibus urget opus ;
Et fimulata novo Cyntbia menfe redit:
Gaudet, et bumana fidera mente regit.
Virgil seems desirous of renouncing all other Study, to contemplate the Wonders of Nature.
Me vero primum dulces ante omnia mufa,
GEOR. II. 475.
La Fontaine imitates the Regrets of Virgil in a masterly Manner, where he says,
Quand pourront les neuf saurs loin des cours et des villes,
Songe dun habitant du Mogol.
Voltaire, the first Poet of our Age, has testified in many Parts of his Works, his Taste for Astronomy, and his Efteem for Astronomers, whom he has celebrated in the finest Poetry. What he says of Newton is worthy of Attention.
Confidens du Tres Haut, Substances eternelles,
Parles ! du grand NEWTON n'etiez vous point jaloux. To which we can only oppose what Pope has said on the same Subje&t:
Nature and Nature's Laws lay hid in Night;
The great Geniuses of every Species have been surprized at the Indifference which Men few for the Spectacle of Nature. Talo puts Reflections in the Mouth of Rinaldo, which merit to be recited for the Instruction of those to whom the same 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,
Jerus. Cant. xviii. St. 12, 13.
The Knowledge of the System of the World has delivered us from Effects the Apprehensions which Ignorance occasions ; can we recal without which the
Ignorance Compassion, the Stupidity of those People, who believed that by making of the Syla great Noise when the Moon was eclipsed, this Goddess received Relief tem of the from her Sufferances, or that Eclipses were produced by Inchantments (a)?
Met. iv. 333
ed he Er
The Knowledge of the System of the World has dissipated the Errors of The KnowAstrology, by whose foolish Predictions Mankind had been so long abused. ledge of the
System The Adventure of 1186, should have covered with Shame the Attrologers the World of Europe; they were all, Christians, Jews and Arabians, united to has dillipat anounce, seven Y cars before, by Letters published throughout Europe, rors of a Conjunction of all the Planets, which would be attended with such Astrology. terrible Ravages, that a general Dissolution of Nature was much to be dreaded, so that nothing less than the End of the World was expected: this Year notwithstanding passed as others. But a hundred Lies, each as well attested, would not be sufficient 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,
their Understandings, unveil the Limits of Nature, and accustom them not to be terrified without Examination, and without Proof.
The Comets, as it is well known, were one of the great Objects of Terror which the Knowledge of the System of the World has, in fine,
(a) Seneca, Hipolit. 787. Tacit. Ann. Plutarch in Pericle, et de defectu Oraculorum.