thematics.

11.

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 fuiting
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-

Method

simple Ele

ments.

served, and by drawing from these Operations a great Number of in-
The Synthe cidental Propositions which are so many Accessaries to the Subject; and
hould not very few having Courage enough, or even are capable of so earnest an
extend fure Application as is necessary to follow the Thread of such complicated
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 so 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, Astronomy,

The Anali-

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
tical Discove 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-

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 Perfection, and perhaps, in Time, will be rendered universal, and at the same Time more simple. What cannot we expect, in this Respect, from the united and constant Application of the first Mathematicians in Europe, who, not content to make use of this fublime Art, in all their Discoveries, have perfected the Art itself, and continue so to do.

Evidence,

This Method has also the Advantage of Clearness and Evidence, and Has the Adthe Brevity that accompanies it every where does not require too ftrong vantage of an Attention. A few Years moderate Study suffices to raise a Perfon, Clearnefs, of fome Talents, above these Geniuses who were the Admiration of and Brevity. Antiquity; and we have feen a young Man of Sixteen, publish a Work, ('Traite des Courbes à double Courbure par Clairaut) that Archimedes 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 most 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 Distaste for a Science they are defirous 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.

111.

Arithmetick comprehends the Art of Numbering and Algebra, confequently is distinguished into particular and universal Arithmetick, because metick nuthe Demonstrations which are made by Algebra are general, and nothing meral and can be proved by Numbers but by Induction. The Nature and Forma- fpecious is 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, fubtracting, 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

The Art of folving Eqations.

Manner of comparing them, and their Operations, are carefully unfolded. The Compofition and Resolution of Quantities comes after, including the Method of raifing Quantities to any Power, extracting of Roots, the Manner of performing upon the Roots of imperfect Powers, radical or incommenfurable Quantities, the various Operations of which they are fufceptible. The Composition and Resolution of Quantities being finished, the Doctrine of Equations presents itself next, where their Genefis, the Nature and Number of their Roots, the different Reductions and Transformations that are in Use, the Manner of folving them, and the Rules imagined for this Purpose, such as Tranfpofition, Multiplication, Divifion, Substitution, and the Extraction of their Roots, are accurately treated. After having confidered Quantities in themselves, it remains to examine their Relations; this Doctrine comprehends arithmetical and geometrical Ratios, Proportions and Progreffions: The Theory of Series follow, where their Formation, Methods for difcovering their Convergency, or Divergency, the Operations of which they are fufceptible, their Reversion, Summation, their Ufe in the InvestiThe Nature gation of the Roots of Equations, Conftruction of Logarithms, &c. are and Laws of taught. In fine, the Art of Combinations, and its Application for determining the Degrees of Probability in civil, moral and political Enquiries are difclosed. Ars cujus Ufus et Neceffitas ita univerfale eft, ut fine illa, nec Sapientia Philofophi, nec Hiftorici Exactitudo, nec Medici Dexteritas, aut Politici Prudentia, confiftere queat. Omnis enim borum Labor in conjectando, et omnis Conjectura in Trutinandis Caufarum Complexionibus aut Combinationibus verfatur.

Chance.

IV.

Divifion of GEOMETRY is divided into ELEMENTARY, TRANSCENDENTAL and SUBLIME.

Geometry into Elemen tary, TranTo open to Youth an accurate and easy Method for acquiring a scendentel Knowledge of the Elements of Geometry, all the Propositions in Euclid (a) in the Order they are found in the best Editions, are retained with

Order in which the

all possible Attention, as also the Form, Connection 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 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 Demonftra- digested. tion, are diftinguished 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 just 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 Geothe Right-line and Circle: This Theory produces important and curious metry. 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 consist, First, confifts. In explaining how the Relation between the Ordinates and Abcisses of a Curve is represented by an Equation. Secondly, How by folving this Equation we discover the Course of the Curve, its different Branches, and its Asymptots. Thirdly, The Manner of finding by the direct 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 Sections 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 their Species. When the most simple Equations of the Parabola, tions.

Conic Sec

ent Orders of Curves.

Ellipfe, and Hyperbola are found, then it is easily shewn that these
Curves are generated in the Cone. The Conic Sections are terminated
by the Solution of the Problems of the third and fourth Degree, by the
Means of these Curves.

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. These Theories are founded partly upon the simple algebraic Calculation, and partly on the direct Method of Fluxions. Then they are brought acquainted with the Theory of the Evolute and Caustiques by Reflection and Refraction. They afterwards enter into a Detail of the Curves of different Orders, afsigning their Classes, 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 Construction 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 Division 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 simple Form, is laid down, then applied to different Cases, which are more composed, and the Difficulties which some Times occur, and which embarrass Beginners, are folved.

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 fuch fluxional Quantities which suppose the Rectification of the Ellipse and Hyperbola, as well as the fluxional 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;