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V.

PLAN of the naval Art, including the Inftructions relative to Ship-Buiders, Sea-Officers, and to all thofe concerned in the Business of the Sea.

VI.

of this King

PLAN of a School of Mechanic Arts, where all Artists, fuch as Architects, Painters, Sculptors, Engravers, Clock-makers, &c. receive The Youth the Inftructions in Geometry, Perfpective, Staticks, Dynamicks, Phyficks, &c. which fuit their refpective Profeffions, and may contribute to improve their Tafte and their Talents.

Those PLANS have convinced the Noblemen and Gentlemen of Fortune of this Kingdom, that their Children, and in general, the Youth of this Country, were deftitute of the most important Means of Instruction, and would ever be deftitute of them, until they had refolved that Men of Genius and Education fhould be encouraged to appear as Teachers.

PLAN of a Course of pure Mathematicks, abfolutely neceffary for the right understanding any Branches of practical Mathematicks in their Application to geographical, nautical, mechanical, commercial, and military Inquiries.

Vix quicquam in univerfa Mathefi ita dificile aut arduum occurrere poffe, quo non inoffenfo Pede per banc Methodum penetrare liceat.

P

I.

URE Mathematicks comprehend Arithmetick, and Geometry.
Practical Mathematicks, their Application to particular Objects,

dom deftitute of the most important Means of Inftructi

on.

as the Laws of Equilibrium, and Motion of folid and fluid Bodies, the Motion of the heavenly Bodies, &c. they extend to all Branches of Mathemahuman Knowledge, and ftrengthening our intellectual Powers, by form- ticks ing in the Mind an Habit of Thinking closely, and Reasoning accurately, ferve to bring to Perfection, with an entire Certitude, all Arts which Man can acquire by his Reason alone. It is therefore of the highest. Importance, that the Youth of this Country should be methodically brought acquainted with a Course of pure Mathematicks, to serve as an Introduction to fuch Branches of Knowledge as are requifite to qualify them for their future Stations in Life. The Noblemen and Gentlemen of Fortune, therefore, have unanimously refolved, that such a Course should be given on the most approved Plan, in the DRAWING SCHOOL eftablifhed under their Inspection, by a Perfon, who, on account of the Readiness and Knowledge he has acquired in these Matters, during the many Years that he has made them his principal Occupation, is qualified for making the Entry to those abftrufe Sciences, acceffable to the meanest Capacity.

The proper Age to commence this Course is 14.

Method of

thematics.

The Synthe

'tick Method

fhould not

extend fur

ther than the

fimple Ele

II.

As to the Method of teaching Mathematicks, the fynthetic Method teaching Ma being neceffary 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 fenfible Objects, fuch as Figures, and by a certain Detail in the Demonstrations, is followed in the Elements (a). But as this Method, when applied to any other Research, attains its Point, but after many Windings and perplexing Circuits, viz. by multiplying Figures, by defcribing a vast many Lines and Arches, whofe Pofition and Angles are carefully to be ob ferved, and by drawing from these Operations a great Number of incidental 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 ingenious Artifice of reducing Problems to the moft fimple and easiest Calculations that the Question propofed can admit of; it is the univerfal Key of Mathematicks, and has opened the Door to a great Number 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 unfolded, and have been diftributed in different Orders, Claffes, Genders, and Species, which as in an Arsenal, where Arms are properly arranged, puts us in a State of chufing readily those which ferve in the Refolution 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

ments.

The Anali

all mathema

ries.

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 eftablished in England, Scotland, &c. the Mafters commence their Courses by the Elements of Geometry; we shall 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 Class, 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, spherical 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 Courfe of Experiments for illuftrating them, performed and explained to them: the Mafter afterwards reads and demonstrates the Elemen:s of Fluxions. Tho.e in the fourth Clafs read a Syftem of Fluxions, the Doctrine of Chances, and the reft of Newton's Principia, with the Improvements they have received from the united Efforts of the firft Mathematicians of Europe.

ing the Art of finding Magnitudes infinitely fmall, 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 first Mathematicians in Europe, who, not content to make ufe of this fublime Art, in all their Difcoveries, have perfected the Art itfelf, and continue fo to do.

Clearnefs,

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, Evidence, of fome Talents, above these Geniuses 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, 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 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 themselves and useful to the Public.

III.

treated.

Arithmetick comprehends the Art of Numbering and Algebra, confe- How Arith quently is diftinguished into particular and universal Arithmetick, becaufe 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 Divifion are deduced. The Explication of the Signs and Symbols used 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

The Art of folving Eqations.

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 imperfe& 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 itself next, where their Genefis, the Nature and Number of their Roots, the different Reductions and Transformations that are in Ufe, the Manner of folving them, and the Rules imagined for this Purpose, fuch as Tranfpofition, Multiplication, Division, 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 Reverfion, 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

Chance.

termining the Degrees of Probability in civil, moral and political Enqui ries 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.

IV.

Divifion of GEOMETRY is divided into ELEMENTARY, TRANSCENDENTAL Geometry into Elemen

tary, Tran

and SUBLIME.

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 Su

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 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 fewest 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 fuch 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 exprefs'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.

66

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 Reafoning are Order in explained and placed in fo advantageous a Light, that the Eye the leaft Elements of attentive may perceive them. To render thefe Elements ftill more eafy, Euclid are the different Operations and Arguments effential to a good Demonftra- digefted. tion, are distinguished in several separate 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 intrinfical Parts of a Demonftration, which it being impoffible. for them to do without knowing what enters into the Compofition of a Theorem and Problem, Firft, The Preparation and Demonftration are diftinguished from each other. Secondly, The Propofition being fet down, what is fuppofed in this Propofition is made known under the Title of Hypothefis, and what is affirmed, under that of Thefis. Thirdly, All the Operations neceffary 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 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 Reafoning.

V.

metry.

Transcendental Geometry prefuppofes the algebraic Calulation; it com- Tranfcenmences by the Solution of the Problems of the fecond Degree by Means of dental Geothe 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 pafs to the general Principles In what it of the Application of Algebra to curve Lines, which confift, Firft, confifts. In explaining how the Relation between the Ordinates and Abciffes of a Curve is reprefented 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 direct Method of Fluxions, the Tangents, the Points of Maxima, and Minima. Fourthly, How the Areas of Curves are found by the inverfe 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

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