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Method of


The Synthe

tick Method

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fimple Ele

ther than the


As to the Method of teaching Mathematicks, the fynthetic Method
teaching Ma being neceflary to difcover 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 ferious Attention,
stand in Need of having their Imagination helped by sensible Objects,
fuch as Figures, and by a certain Detail in the Demonftrations, 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 defcribing a vast many
Lines and Arches, whofe Pofition and Angles are carefully to be ob-
served, and by drawing from these Operations a great Number of in-
cidental Propofitions which are so many Acceffaries to the Subject; and
very few having Courage enough, or even are capable of fo earnest an
Application as is neceffary to follow the Thread of fuch complicated
Demonstrations: afterwards a Method more eafy and lefs fatiguing to
the Attention is pursued. This Method is the analitic Art, the inge-
nious Artifice of reducing Problems to the moft fimple and easiest
Calculations that the Question proposed can admit of; it is the uni-
verfal Key of Mathematicks, and has opened the Door to a great Num-
ber of Perfons, to whom it would be ever fhut, without its Help; by
its Means, Art fupplies Genius, and Genius, aided by Art fo useful,
has had Succeffes 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 diftributed in different Orders, Claffes, Genders,
and Species, which as in an Arsenal, where Arms are properly arrang-
ed, puts us in a State of chufing readily thofe which ferve in the Re-
folution of a Problem propofed, either in Mathematicks, Aftronomy,
tick Method Opticks, &c. It is it which has conducted the great Sir Ifaac Newton
is the Key of to the wonderful Difcoveries he has made, and enabled the Men of
tical Difcove Genius, who have come after him, to improve them. The Method of


The Anali-

all mathema


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

(a) It is for thefe Reasons that in all the public mathematical Schools established in England,
Scotland, &c. the Masters commence their Courses by the Elements of Geometry; we fhall
only inftance that of Edinburgh, where a hundred young Gentlemen attend from the first of
November to the first of Auguft, and are divided into five Claffes, in each of which the Master
employs a full Hour every Day. In the first or lowest Clafs, he teaches the first fix Books of
Euclid's Elements, plain Trigonometry, practical Geometry, the Elements of Fortification, and
an Introduction to Algebra. The fecond Class studies Algebra, the 11th and 12th Books of
Euclid, fpherical Trigonometry, conic Sections, and the general Principles of Aftronomy. The
third Class goes on in Aftronomy and Perspective, read a Part of Sir Ifaac Newton's Principia,
and have a Course of Experiments for illustrating them, performed and explained to them: the
Master afterwards reads and demonftrates the Elemen s of Fluxions. Tho e in the fourth Clafs
read a Syftem 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

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ing the Art of finding Magnitudes infinitely small, which are the Elements of finite Magnitudes; the fecond the Art of finding again, by the Means of Magnitudes infinitely fmall, the finite Quantities to which they belong; the first as it were refolves a Quantity, the last restores it to its first State; but what one refolves, the other does not always reinftate, and it is only by analitic Artifices that it has been brought to any Degree of Perfection, and perhaps, in Time, will be rendered univerfal, and at the fame Time more fimple. What cannot we expect, in this Refpect, from the united and conftant Application of the firft Mathematicians in Europe, who, not content to make use of this fublime Art, in all their Discoveries, have perfected the Art itfelf, and continue fo to do.


This Method has also the Advantage of Clearnefs 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 fuffices to raise a Perfon, Evidenc of fome Talents, above these Geniufes who were the Admiration of and Brevity. Antiquity; and we have seen a young Man of Sixteen, publish a Work, (Traite des Courbes à double Courbure par Clairaut) that Archimedes would have wifhed to have compofed at the End of his Days. The Teacher of Mathematicks, therefore, fhould be acquainted with the different Pieces upon the analitic Art, difperfed 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 clofe 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 fhort Time, not only to understand the Writings of the most eminent Mathematicians, but, reflecting on their Method of Proceeding, to make Discoveries honourable to themfelves and useful to the Public.



Arithmetick comprehends the Art of Numbering and Algebra, confe- How Arith quently is diftinguished into particular and univerfal Arithmetick, because metick nu the Demonstrations which are made by Algebra are general, and nothing meral and can be proved by Numbers but by Induction. 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 Divifion are deduced. The Explication of the Signs and Symbols ufed in Algebra follow, and the Method of reducing, adding, fubtracting, 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


Manner of comparing them, and their Operations, are carefully unfolded. The Compofition and Refolution 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 Compofition and Refolution of Quantities being finished, the Doctrine of Equations prefents itfelf next, where The Art of their Genefis, the Nature and Number of their Roots, the different folving Eqa Reductions and Transformations that are in Ufe, the Manner of folving them, and the Rules imagined for this Purpose, fuch 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 InveftiThe 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 de


termining the Degrees of Probability in civil, moral and political Enquiries are difclofed. Ars cujus Ufus et Neceffitas ita univerfale eft, ut fine illa, nec Sapientia Philofopbi, 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.


Divifion of GEOMETRY is divided into ELEMENTARY, TRANSCENDENTAL Geometry

into Elemen

tary, Tran


To open to Youth an accurate and eafy Method for acquiring a fcendentel Knowledge of the Elements of Geometry, all the Propofitions in Euclid (a) in the Order they are found in the best Editions, are retained with

and Sublime.


(a) Perfpicuity in the Method and Form of Reafoning, 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 Defchales, but of the Original, handed down to us by Antiquity. His Demonftrations being conducted with the most exprefs Design of reducing "the Principles affumed to the feweft Number, and most evident that might be, and in a Me"thod the most natural, as it is the most conducive towards a juft and complete Comprehenfion " of the Subject, by beginning with such Particulars as are most easily conceived, and flow most "readily from the Principles laid down; thence by gradually proceeding to fuch as are more ob→ "fcure, 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 fame Time their Diflike to the new modelled Elements that at prefent every where abound; and to the illiberal and mechanic Methods of teaching those most perfect Arts; which is to be hoped, will never be countenanced in the Public Schools in England and Scotland, &c.

Order in which the

all poffible Attention, as alfo the Form, Connection and Accuracy of his Demonstrations. The effential Parts of his Propofitions being fet Methodical forth with all the Clearnefs imaginable, the Senfe of his Reasoning are explained and placed in fo advantageous a Light, that the Eye the leaft Elements of attentive may perceive them. To render these Elements still more eafy, Euclid are the different Operations and Arguments effential to a good Demonftra- digefted. tion, are distinguished in feveral feparate Articles; and as Beginners, in order to make a Progrefs in the Study of Mathematicks, fhould 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 ferve to establish a new Truth; in fine, to difcover all the intrinsical Parts of a Demonstration, which it being impoffible for them to do without knowing what enters into the Compofition of a Theorem and Problem, First, The Preparation and Demonstration are distinguished from each other. Secondly, The Propofition being set down, what is fupposed in this Propofition is made known under the Title of Hypothefis, and what is affirmed, under that of Thefis. Thirdly, All the Operations necessary to make known Truths, ferve as a Proof to an unknown one, are ranged in feparate Articles. Fourthly, The Foundation of each Propofition relative to the Figure, which forms the Minor of the Argument, are made known by Citations, and a marginal Citation recalls the Truths already demonftrated, 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.


dental Geometry.

Transcendental Geometry prefupposes the algebraic Calulation; it com- Transcenmences by the Solution of the Problems of the second Degree by Means of the Right-line and Circle: This Theory produces important and curious Remarks upon the pofitive and negative Roots, upon the Pofition of the Lines which exprefs them, upon the different Solutions that a Problem is fufceptible of; from thence they pass to the general Principles In what it of the Application of Algebra to curve Lines, which confift, First, confifts. In explaining how the Relation between the Ordinates and Abciffes of a Curve is represented by an Equation. Secondly, How by folving this Equation we difcover the Courfe of the Curve, its different Branches, and its Afymptots. Thirdly, The Manner of finding by the dire& 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 confider them as Lines of the fecond Order, to divide them into of treating their Species. When the moft fimple Equations of the Parabola, tions.

Conic Sec

ent Orders

of Curves.

Ellipfe, and Hyperbola are found, then it is eafily fhewn 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 pafs 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, & Thefe Theories are founded partly upon the fimple algebraic Calculation, and partly on the direct Method of Fluxions. Then they are brought acquainted with the Theory of the Evolute and Cauftiques by Reflection and Refraction. They afterwards enter into a Detail of the Curves of different Orders, affigning their Claffes, Species, and principal Properties, treating more amply of the best known, as the Folium, the Conchoid, the Ciffoid, c.


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 Conftruction 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.


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 Divifion therefore of the inverfe 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 Cafes, which are more compofed, and the Difficulties which fome Times occur, and which embarrass Beginners, are folved.

What the firft Part


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 fuppofe the Rectification of the Ellipfe and Hyperbola, as well as the fluxional Quantities, whofe 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 whofe Equations are compofed of three or four Terms;

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