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II.

thematics.

hould not

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-
The Synthe cidental Propositions which are so many Accessaries to the Subje&; and

very few having Courage enough, or even are capable of so earnest an
ther than the
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 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-
The Analisolution of a Problem proposed, either in Mathematicks, Aftronomy,
tick Method Opticks, &c. It is it which has condu&ted the great Sir Isaac Newton
is the Key of to the wonderful Discoveries he has made, and enabled the Men of
all mathéma
tical Discove

Genius, who have come after him, to improve them. The Method of
ries.

Fluxions, both direct and inverse, is only an Extention of it, the first be-

ments.

(a) It is for these Reasons that in all the public mathematical Schools established in England,
Scotland, &c. the Masters commence their Courses by the Elements of Geometry; we shall
only instance that of Edinburgh, where a hundred young Gentlemen attend from the first of
November to the first of August, and are divided into five Classes, in each of which the Master
employs a full Hour every Day. In the first or lowest Class, he teaches the first six Books of
Euclid's Elements, plain Trigonometry, practical Geometry, the Elements of Fortification, and
an Introduction to Ålgebra. The second Clase studies Algebra, the uth and 12th Books of
Euclid, spherical Trigonometry, conic Sections, and the general Principles of Astronomy. The
third Class goes on in Aftronomy and Perspective, read a Pori of Sir Isaac Newton's Principia,
and have a course of Experiments for illustratiog them, performed and explained to them : the
Master afterwards reads and demonstrates the Elemen.s of Fluxions. Those in the fourth Class
read a System of Fluxions, the Doctrine of Chances, and the rest of Newton's Principia, with
the Improvements they have received from the united Efforts of the first Mathematicians of

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 firit State ; but what one resolves, the other does not always reinstate, and it is only by analitic Artitices that it has been brought to any Degree of Perfeaion, and perhaps, in Time, will be rendered universal, and at the same Time more simple. What cannot we expea, in this Respea, 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 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 effential 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. Arithmetick comprehends the Art of Numbering and Algebra, consequently 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 Indu&ion. The Nature and Forma specious 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, subtra&ing, multiplying, dividing, algebraic Quantities fimple and compound. This prepares the Way for the Theory of vulgar, algebraical, and decimal Fractions, where the Nature, Value, Man

How Arith

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 imperfe& Powers, radical or incommenfurable Quantities, the various Operations of which they are fasceptible. The Composition and Resolution of Quantities

being finished, the Doctrine of Equations prelents 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 Use, the Manner of solving
them, and the Rules imagined for this purpose, such as Transposition,
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 arith-
metical and geometrical Ratios, Proportions and Progressions :
Theory of Series follow, where their Formation, Methods for discover-
ing 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&ion 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 borum Labor in conjectando, et omnis Conjectura in Trutinandis Causarum Complexionibus aut Combinationibus verfatur.

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 Sublime.

(a) in the Order they are found in the best Editions, are retained with

(a) “ Perspicuity 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 Me “ 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 mort 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 Dislike to the new modelled Elements that at present every where abound; and to the illiberal and mechanic Methods of teaching those most perfect Ares; which is to be hoped, will never be countenanced in the Public Schools in England and Scot:

all possible Attention, as also the Form, Conne&ion 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.

conGifts.

Transcendental Geometry presupposes the algebraic Calulation; it com- Tranfcenmences 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 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, 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 dire&t 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 their Species. When the most simple Equations of the Parabola, tions.

Conic Sec

Ellipse, 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 Se&ions 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 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 Refra&ion. 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 Cisloid, Esc.

The mechanic Curves follow the geometrical ones, beginning by theexponential 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 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 fluxionary 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 Fra&ions, and such fuxional Quanhends. tities as can be reduced to the Form of rational Fra&ions; from thence

they pass to the Method of finding the Fluents of such Auxional Quantities which suppose the Re&ification 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;

compres

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