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G Ε Ο Μ
YOntaining a full Definition of the Terms peculiar to, or
made use of in that Science, with explanatory Notes and Remarks, where it is necessary to illustrate or enlarge.
Likewise a short Theory of Plane Angles; in which they are more fully explained than in any other Work of the kind, that I have seen. Indeed, most Authors in Geometry are entirely deficient in that respect; for want of which, the young
Geometrician is frequently at a loss, 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 use of in this Work; many of which are explained in English Grammars, and other school Books, and ought to be known to every English Reader ; yet, as I know that their true fignifications are not so generally understood, it may not be thought impertinent, or fuperfluous here.
Nor have I omitted any thing, that is necessary to elucidate the Subject I am about to treat on; at least, I think I have not; for, in the course of my own ftudy of it, and in teaching others, I have been enabled to discover what is needful to most Capacities; and if I have any pretence to merit in this work, it is chiefly on that account; in rendering an intricate, yet generally useful and most necessary Science, attainable to ordinary Capacities; and, at the same time, I hope, not exceptionable to those of acuter talents.
I faiter myself that it will not be less acceptable to any, for being easy to be attained, to write only for Proficients, is to little purpose. By such I may, in some cases, be thought
rather prolix; yet, I presume, not tedious; a repetition is sometimes necessary to young Minds, and is more agreeable, in general, than turning back, which they are too frequentJy obliged to do ; it being impossible to retain, in memory, all that is passed over on the first perusal.
GEOMETRY, according to its original derivation, fignifies to measure the Earth. It is a Science which contemplates continued Quantity, Extension or Magnitude, abstractedly considered; it teaches the nature and properties of Lines, Angles, Figures, Surfaces and Solids.
Geometry is in two parts, speculative and practical; the first demonstrates the properties of Right Lines, Figures, &c. in speculation ; from which is deduced the practical part for various uses, 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 some Authors have thought proper to deviate from, though without any justifiable reason for so doing.
It treats, in the first, third, fourth, and fixth Books, of Plane Figures, and thence is called Plane Geometry; and afterwards, in the rith and 12th of Euclid, the 7th and 8th of these Elements, of Planes and Solids.
Each Book contains sundry Propofitions; from which, are deduced Corollaries, Scolia, &c. the signification of all which I shall first beg leave to explain or define; and then proceed to the Definitions of the more essential Terms, which are the Subject of Geometry.
A DEFINITION is the defining or explaining the full signification of any Term or particular Word, peculiar to, or made use of, in that Science of which we are about to treat,
A PROPOSITION is either a Theorem, proposed to be proved or demonstrated, contemplatively; or, it proposes something to be done, problematically or mechanically.
A CONVERSE PROPOSITION is the contrary of the other; that, which in the foregoing was the Conclusion, drawn from the Premises of it.
e. g. If a Triangle have two equal Sides, the Angles which they subtend are also equal; the Converse is, that if a Triangle have two equal Angles, the Sides subtending them har are also equal.
A THEOREM is a speculative Proposition; a declaration of certain properties, equality, or other proportion relative to Quantity, or Figure, mathematically considered.
A PROBLEM, is a Proposition which proposes fomething to be done, practically or mechanically.
An AXIOM is a self evident Proposition, which does not require to be demonstrated. See Axioms, Book 1. El.
A LEMMA is a Proposition, as it were by the bye, or out of the way, which serves, previously, to prepare the way for the more easily comprehending the Demonstration of the following Proposition.
I do not make use of Lemmas in this work, as some geometrical Authors do; for if there be a necessity for a Lemma, I see no reason why that Lemma is not as much a Proposition as any other. The gth and 16th Propositions of the firit Book of Euclid, may be called Lemmas, for they are certainly redundant Propositions. In other ma. thematical Works, Lemmas are frequently necessary, but, in Geometry, they are quite inconsistent.
A COROLLARY is a necessary consequence deducible from some Propofition, already demonstrated.
A SCHOLIUM is a remark, or useful lesson derived from the preceding Proposition.
A POSTULATE is a petition or request which is required to be granted. See Postulates, p. 21.
HYPOTHESIS. Whatever is supposed or premised, in a Proposition, is called the Hypothesis or Premises of it; from which some certain Consequence is deduced, as affirmed, and afterwards demonstrated, called the Thesis or Affirmation.
e. g. If a Right Line, cutting two Right Lines, makes equal Angles with them both, those lines are parallel,
Here, the Hypothesis is, if the Angles are equal ; and, the Consequence, that the Lines are parallel.
SUPPOSITION. In demonstrating fome Theorems, it is necessary to have recourse, frequently, to suppose such and such things, which are not so in reality ; by the absurdity of the consequences, arising from such a supposition, a conclusion is drawn, and the Demonstration is made evidently to appear.
Such kind of Demonstration is called reductio ad abfurdum, i. e. proving it to be absurd, or impossible to be on that supposition; which, not being direct and positive, is, to many, very unsatisfactory; yet, if rightly considered, is full, though not direct Demonstration.
CONSTRUCTION, is the contriving or disposing, geometrically, Lines and Figures, necessary for making the Demonstration appear, clear and conspicuous; and must
always be made of such Figures or Lines as are already well understood; the Properties, of which, being previously demonstrated.
DEMONSTRATION. When any thing is proposed, or affirmed in a Propofition, the Case is first stated and prepared, by drawing such lines, or forming such a Construction as is necessary; and it is afterwards demonstrated ; that is, the truth of the Affertion is made to appear, obvious, and without the least doubt remaining; the performance or operation of which is called the Demonstration.
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.
Of the effential and operative T E R M S.
The Terms of Art, to be defined in any Science, are Names, arbitrarily given, by the fift Authors, or others, to certain Symbols, Figures, Marks or Characters, poffeiling certain properties or relations, in respect of figure, polition, situation, &c.
The operative Terms are, generally, technical Words, peculiar to that Science, though perhaps applicable to others; which are not of common use in Language, or have a different lignification.
The following Definitions are frequently referred to, hereafter, for illustratien or proof of what is advanced in the Propositions.
When any Figure, &c. which we are contemplating, is found to possess such or such properties, we afirm it to be such a Figure, as answers to them; or, in contemplating any Figure, given in the Premises, we aflirm that it has iuch or such propei ties, arbitrarily, by the Definition of it; and therefore, it requires 119 other Demonitration,