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INTRODUCTION

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GEOMETR

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Ontaining a full Definition of the Terms peculiar to, or made use of in that Science, with explanatory Notes and Remarks, where it is neceffary to illuftrate or enlarge. Likewife a fhort Theory of Plane Angles; in which they are more fully explained than in any other Work of the kind, that I have feen. Indeed, moft Authors in Geometry are entirely deficient in that refpect; for want of which, the young Geometrician is frequently at a lofs, to conceive a clear Idea of Angles. I have therefore, been explicit on that Head.

It also contains an Explanation of all the Abbreviations made ufe of in this Work; many of which are explained in English Grammars, and other fchool Books, and ought to be known to every English Reader; yet, as I know that their true fignifications are not fo generally understood, it may not be thought impertinent, or fuperfluous here.

Nor have I omitted any thing, that is neceffary to elucidate the Subject I am about to treat on; at leaft, I think I have not; for, in the courfe of my own ftudy of it, and in teaching others, I have been enabled to difcover what is needful to moft Capacities; and if I have any pretence to merit in this Work, it is chiefly on that account; in rendering an intricate, yet generally ufeful and most neceflary Science, attainable to ordinary Capacities; and, at the fame time, I hope, not exceptionable to thofe of acuter talents.

I flatter myfelf that it will not be lefs acceptable to any, for being eafy to be attained, to write only for Proficients, is to little purpofe. By fuch I may, in fome cafes, be thought rather

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rather prolix; yet, I prefume, not tedious; a repetition is fometimes neceffary to young Minds, and is more agreeable,

in general, than turning back, which they are too frequently obliged to do; it being impoffible to retain, in memory, all that is paffed over on the first perusal.

GEOMETRY, according to its original derivation, fignifies to measure the Earth. It is a Science which contemplates continued Quantity, Extenfion or Magnitude, abstractedly confidered; it teaches the nature and properties of Lines, Angles, Figures, Surfaces and Solids.

Geometry is in two parts, fpeculative and practical; the first demonstrates the properties of Right Lines, Figures, &c. in fpeculation; from which is deduced the practical part for various ufes, for the benefit of mankind, in mechanic. Arts, &c.

Euclid has judiciously divided the Subject into Books or Sections; each of which, treats of different Figures, or different properties of Figures, the power of Lines, Proportion, &c. which fome Authors have thought proper to deviate from, though without any juftifiable reafon for fo doing.

It treats, in the firft, third, fourth, and fixth Books, of Plane Figures, and thence is called Plane Geometry; and afterwards, in the 11th and 12th of Euclid, the 7th and 8th of thefe Elements, of Planes and Solids.

Each Book contains fundry Propofitions; from which, are deduced Corollaries, Scolia, &c. the fignification of all which I shall firft beg leave to explain or define; and then proceed to the Definitions of the more effential Terms, which are the Subject of Geometry.

A DEFINITION is the defining or explaining the full fignification of any Term or particular Word, peculiar to, or made ufe of, in that Science of which we are about

to treat.

A PROPO

A PROPOSITION is either a Theorem, propofed to be proved or demonftrated, contemplatively; or, it proposes fomething to be done, problematically or mechanically.

A CONVERSE PROPOSITION is the contrary of the other; that, which in the foregoing was the Conclufion, drawn from the Premifes of it.

e. g. If a Triangle have two equal Sides, the Angles which they fubtend are alfo equal; the Converfe is, that if a Triangle have two equal Angles, the Sides fubtending them are also equal.

A THEOREM is a fpeculative Propofition; a declaration of certain properties, equality, or other proportion relative to Quantity, or Figure, mathematically confidered,

A PROBLEM, is a Propofition which propofes fomething to be done, practically or mechanically.

An AXIOM is a felf evident Propofition, which does not require to be demonftrated. See Axioms, Book 1. El.

A LEMMA is a Propofition, as it were by the bye, or out of the way, which ferves, previously, to prepare the way for the more eafily comprehending the Demonftration of the following Propofition.

I do not make ufe of Lemmas in this Work, as fome geometrical Authors do; for if there be a neceffity for a Lemma, I fee no reason why that Lemma is not as much a Propofition as any other. The 7th and 16th Propofitions of the first Book of Euclid, may be called Lemmas, for they are certainly redundant Propofitions. In other mathematical Works, Lemmas are frequently neceffary, but, in Geometry, they are quite inconfiftent.

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A COROLLARY is a neceffary confequence deducible from fome Propofition, already demonstrated.

A SCHOLIUM is a remark, or ufeful leffon derived from the preceding Propofition.

A POSTULATE is a petition or requeft which is required to be granted. See Poftulates, p. 21.

HYPOTHESIS. Whatever is fuppofed or premised, in a Propofition, is called the Hypothefis or Premises of it from which fome certain Confequence is deduced, as affirmed, and afterwards demonftrated, called the Thefis or Affirmation.

e. g. If a Right Line, cutting two Right Lines, makes equal Angles with them both, thofe lines are parallel. Here, the Hypothefis is, if the Angles are equal; and, the Confequence, that the Lines are parallel.

SUPPOSITION.

In demonftrating fome Theorems, it is neceffary to have recourfe, frequently, to fuppose such and fuch things, which are not fo in reality; by the abfurdity of the confequences, arifing from such a supposition, a conclufion is drawn, and the Demonftration is made evidently to appear.

Such kind of Demonftration is called reductio ad abfurdum, i. e. proving it to be abfurd, or impoffible to be on that fuppofition; which, not being direct and pofitive, is, to many, very unfatisfactory; yet, if rightly confidered, is full, though not direct Demonftration.

CONSTRUCTION, is the contriving or difpofing, geometrically, Lines and Figures, neceffary for making the Demonftration appear, clear and confpicuous; and muft always

always be made of fuch Figures or Lines as are already well understood; the Properties, of which, being previously demonftrated.

DEMONSTRATION. When any thing is propofed, or affirmed in a Propofition, the Cafe is first stated and prepared, by drawing fuch lines, or forming fuch a Conftruction as is neceffary; and it is afterwards demonftrated; that is, the truth of the Affertion is made to appear, obvious, and without the leaft doubt remaining; the performance of operation of which is called the Demonftration.

The three last Terms being common words in the English Tongue, may, by fome, be thought impertinent; but, notwithftanding the common acceptation of them is almost univerfal, yet the application of them in Geometry requires to be explained."

DEFINITION S.

Of the effential and operative TERMS.

The Terms of Art, to be defined in any Science, are Names, arbitrarily given, by the first Authors, or others, to certain Symbols, Figures, Marks or Characters, poffeffing certain properties or relations, in refpect of figure, pofition, fituation, &c.

The operative Terms are, generally, technical Words, peculiar to that Science, though perhaps applicable to others; which are not of common ufe in Language, or have a different fignifica tion.

The following Definitions are frequently referred to, hereafter, for illuftratien or proof of what is advanced in the Propofitions. When any Figure, &c. which we are contemplating, is found to poffefs fuch or fuch properties, we affirm it to be fuch a Figure, as anfwers to them; or, in contemplating any Figure, given in the Premifes, we affirm that it has fuch or fuch properties, arbitrarily, by the Definition of it; and therefore, it requires no other Demonftration.

DEF.

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