II. a Method of As to the Method of teaching Mathematicks, the synthetic Method teaching Ma being necessary to discover the principal Properties of geometrical Figures, which cannot be rightly deduced but from their Formation, and suiting Beginners, who, little accustomed to what demands a serious Attention, stand in Need of having their Imagination helped by sensible Objects, such as Figures, and by a certain Detail in the Demonstrations, is fol- lowed in the Elements (a). But as this Method, when applied to any other Research, attains its Point, but after many Windings and per- plexing Circuits, viz. by multiplying Figures, by describing a vast many Lines and Arches, whose Position and Angles are carefully to be ob- served, and by drawing from these Operations a great Number of in- tick Method cidental Propositions which are so many Accessaries to the Subje&; and very few having Courage enough, or even extend furie Application as is necessary to follow the Thread of such complicated simple Ele Demonstrations : afterwards a Method more easy and less fatiguing to the Attention is pursued. This Method is the analitic Art, the inge- nious Artifice of reducing Problems to the most simple and easiest Calculations that the Question proposed can admit of; it is the uni- versal Key of Mathematicks, and has opened the Door to a great Num- ber of Persons, to whom it would be ever shut, without its Help; by its Means, Art supplies Genius, and Genius, aided by Art fo useful, has had Successes that it would never have obtained by its own Force alone; it is by it that the Theory of curve Lines have been unfold- ed, and have been distributed in different Orders, Classes, Genders, and Species, which as in an Arsenal, where Arms are properly arrang- ed, puts us in a State of chusing readily those which serve in the Re- solution of a Problem proposed, either in Mathematicks, Aftronomy, tick Method Opticks, &c. It is it which has conducted the great Sir Isaac Newton is the Key of to the wonderful Discoveries he has made, and enabled the Men of aid mathéma Genius, who have come after him, to improve them. The Method of Fluxions, both direct and inverse, is only an Extention of it, the first be- ries. (a) It is for these Reasons that in all the public mathematical Schools established in England, S ing the Art of finding Magnitudes infinitely small, which are the Elements of finite Magnitudes; the second the Art of finding again, by the Means of Magnitudes infinitely small, the finite Quantities to which they belong; the first as it were resolves a Quantity, the last restores it to its first State ; but what one resolves, the other does not always reinstate, and it is only by analitic Artifices that it has been brought to any Degree of Perfe&ion, and perhaps, in Time, will be rendered universal, and at the same Time more simple. What cannot we expea, in this Respe&, from the united and constant Application of the first Mathematicians in Europe, who, not content to make use of this sublime Art, in all their Discoveries, have perfected the Art itself, and continue so to do. This Method has also the Advantage of Clearness and Evidence, and Has the Adthe Brevity that accompanies it every where does not require too strong vantage of an Attention. A few Years moderate Study suffices to raise a Person, Evidence; , of some Talents, above these Geniuses who were the Admiration of and Brevicy. Antiquity; and we have seen a young Man of Sixteen, publish a Work, ('Traité des Courbes à double Courbure par Clairaut) that Arcbimedes would have wished to have composed at the End of his Days. The Teacher of Mathematicks, therefore, should be acquainted with the different Pieces upon the analitic Art, dispersed in the Works of the moft eminent Mathematicians, make a judicious Choice of the most general and essential Methods, and lead his Pupils, as it were, by the Hand, in the intricate Roads of the Labyrinth of Calculation; that by this Means Beginners, exempted from that close Attention of Mind, which would give them a Diftaste for a Science they are desirous to attain, and methodically brought acquainted with all its preliminary Principles, might be enabled in a short Time, not only to understand the Writings of the most eminent Mathematicians, but, reflecting on their Method of Proceeding, to make Discoveries honourable to themselves and useful to the Public. III. How Arith quently is distinguished into particular and universal Arithmetick, because metick nuthe Demonstrations which are made by Algebra are general, and nothing meral and Specious is can be proved by Numbers but by Indu&ion. The Nature and Forma- pested. tion of Numbers are clearly stated, from whence the Manner of performing the principal Operations, as Addition, Subtraction, Multiplication and Division are deduced. The Explication of the Signs and Symbols used in Algebra follow, and the Method of reducing, adding, subtra&ing, multiplying, dividing, algebraic Quantities simple and compound. This prepares the Way for the Theory of vulgar, algebraical, and decimal Fractions, where the Nature, Value, Man a Manner of comparing them, and their Operations, are carefully unfolded. The Compofirion and Resolution of Quantities comes after, including the Method of raifing Quantities to any Power, extra&ting of Roots, the Manner of performing upon the Roots of imperte& Powers, radical or incommensurable Quantities, the various Operations of which they are fasceptible. "The Composition and Resolution of Quantities being finished, the Doctrine of Equations presents itself next, where The Art of folving Eqa their Genefis, the Nature and Number of their Roots, the different tions. Reductions and Transformations that are in Ule, the Manner of solving them, and the Rules imagined for this Purpose, such as Transpofition, Multiplication, Divifion, Substitution, and the Extraction of their Roots, are accurately treated. After having considered Quantities in themselves, it remains to examine their Relations; this Doctrine comprehends arithmetical and geometrical Ratios, Proportions and Progressions : The Theory of Series follow, where their Formation, Methods for discovering their Convergency, or Divergency, the Operations of which they are susceptible, their Reversion, Summation, their Use in the InvestiThe Nature gation of the Roots of Equations, Conftru&tion of Logarithms, &c. are and Laws of taught. In fine, the Art of Combinations, and its Application for deChance. termining the Degrees of Probability in civil, moral and political Enqui ries are disclosed. Ars cujus Ufus et Necesitas ita universale eft, ut fine illa, nec Sapientia Philofophi, nec Hiftorici Exatlitudo, nec Medici Dexteritas, aut Politici Prudentia, confiftere queat. Omnis enim horum Labor in conjectando, et omnis Conjectura in Trutinandis Causarum Complexionibus aut Combinationibus versatur. IV. Division of GEOMETRY is divided into ELEMENTARY, TRANSCENDENTAL Geometry and SUBLIME. into Elemen tary, Tran To open to Youth an accurate and easy Method for acquiring a scendentel Knowledge of the Elements of Geometry, all the Propositions in Euclid and Su (a) in the Order they are found in the best Editions, are retained with blime. (a) " Perfpicuity in the Method and Form of Reasoning, is the peculiar Characteristic of “ Euclid's Elements , not as interpolated by Campanus and Clavius, anatomised by Herigone and “ Barrow, or depraved by Tacquet and Deschales, but of the Original, handed down to us by “ Antiquity. His Demonstrations being conducted with the most express Design of reducing “ the Principles assumed to the fewest Number, and most evident that might be, and in a Mes " thod the most natural, as it is the most conducive towards a just and complete Comprehenfion “ of the Subject, by beginning with such Particulars as are most easily conceived, and Aow most " readily from the Principles laid down; thence by gradually proceeding to such as are more ob" scure, and require a longer Chain of Argument, and have therefore been regarded in all Ages, as the most perfect in their Kind.” Such is the Judgment of the ROYAL SOCIETY, who have express’d at the same Time their Dillike to the new modelled Elements that at present every where abound; and to the illiberal and mechanic Methods of teaching those most perfect Arrs; which is to be hoped, will never be countenanced in the Public Schools in England and Scota land, &c. all possible Attention, as also the Form, Conne&tion and Accuracy of his Demonstrations. The essential Parts of his Propositions being set Methodical forth with all the Clearness imaginable, the Sense of his Reasoning are Order in explained and placed in so advantageous a Light, that the Eye the least Elements of attentive may perceive them. To render these Elements still more easy, Euclid are the different Operations and Arguments essential to a good Demonstra- digested. tion, are distinguished in several separate Articles; and as Beginners, in order to make a Progress in the Study of Mathematicks, should apply themselves chiefly to discover the Connection and Relation of the different Propofitions, to form a juft Idea of the Number and Qualities of the Arguments, which serve to establish a new Truth; in fine, to difcover all the intrinsical Parts of a Demonstration, which it being impossible for them to do without knowing what enters into the Composition of a Theorem and Problem, First, The Preparation and Demonstration are distinguished from each other. Secondly, The Proposition being set down, what is supposed in this Proposition is made known under the Title of Hypothesis, and what is affirmed, under that of Thesis. Thirdly, All the Operations necessary to make known Truths, serve as a Proof to an unknown one, are ranged in separate Articles. Fourthly, The Foundation of each Proposition relative to the Figure, which forms the Minor of the Argument, are made known by Citations, and a marginal Citation recalls the Truths already demonstrated, which is the Major : In one Word, nothing is omitted which may fix the Attention of Beginners, make them perceive the Chain, and teach them to follow the Thread of geometrical Reasoning. V. Transcendental Geometry presupposes the algebraic Calulation; it com- Transcenmences by the Solution of the Problems of the second Degree by Means of dental Geo metry. the Right-line and Circle : This Theory produces important and curious Remarks upon the positive and negative Roots, upon the Position of the Lines which express them, upon the different Solutions that a Problem is susceptible of; from thence they pass to the general Principles In what it of the Application of Algebra to curve Lines, which confift, First, consists. In explaining how the Relation between the Ordinates and Abcisses of a Curve is represented by an Equation. Secondly, How by solving this Equation we discover the Course of the Curve, its different Branches, and its Afymptots. Thirdly, The Manner of finding by the direa Method of Fluxions, the Tangents, the Points of Maxima, and Minima. Fourthly, How the Areas of Curves are found by the inverse Method of Fluxions. The Conic Se&ions follow; the best Method of treating them is to Best Method consider them as Lines of the second Order, to divide them into of treating Conic Sectheir Species. When the most simple Equations of the Parabola, tions. I Ellipse, and Hyperbola are found, then it is easily shewn that these The Conic Sections being finished, they pass to Curves of a superior The differ- Order, beginning by the Theory of multiple Points, of Points of Inflecent Orders of Curves. tion, Points of contrary Inflection, of Serpentment, &e. These Theo ries are founded partly upon the simple algebraic Calculation, and partly on the dire&t Method of Fluxions. Then they are brought acquainted with the Theory of the Evolute and Caustiques by Reflexion and Refraction. They afterwards enter into a Detail of the Curves of different Orders, assigning their Classes, Species, and principal Properties, treating more amply of the best known, as the Folium, the Conchoid, the Cifloid, Esc. 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 Construction of mechanic Curves, by the Meads. 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 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 Division therefore of the inverse Method of Fluxions is into the Its Division. Method of finding the Fluents of Auxionary Quantities, containing one variable Quantity, or involving two or more variable Quantities; the Rule for finding the Fluents of Auxional Quantities of the most simple Form, is laid down, then applied to different Cases, which are more composed, and the Difficulties which some Times occur, and which em barrass Beginners, are solved. What the These Researches prepare the way for finding the Fluents of Auxional first Part Binomials, and Trinomials, rational Fractions, and such fluxional Quanhends. tities as can be reduced to the Form of rational Fra&tions ; from thence they pass to the Method of finding the Fluents of such Auxional Quantities which suppose the Re&tification of the Ellipse and Hyperbola, as well as the Auxional Quantities, whose 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 whose Equations are composed of three or four Terms; compre |