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DEF. 55. EXTREMES or BOUNDS, are the utmoft Limits of any thing.

So, the extremes of a Line are Points; the bounds or extremes of a Surface are Lines, (except the furface of a Sphere, which has no Bounds) and, the extremes of a Solid are Surfaces.

A Sphere has but one Surface; alfo, a Circle is bounded but by one Line, without beginning or end; confequently it has no Extremes.

Some Readers will, perhaps, think I have already deviated from the Plan propofed in the Preface, which was, to abridge the Elements of Euclid; but I hope they will find, that the time fpent on the Definitions (by way of Introduction) is not thrown away; having made feveral of them ufeful Leffons, as well as defined the Terms; and I am perfuaded, that the more perfectly they are understood, the progrefs, in the Subject they relate to, will be greatly facilitated. But, I muft inform the Reader, that it is not the length of the Definition itself, but the number of Terms I have defined, and the Notes to feveral, which have fwelled the bulk of them; having near twice the Number which fome Authors have, and, in my opinion, not one fuperfluous.

However, not to difcourage the young Student, I advise him not to burthen his memory too much at once; for there is not the leaft neceffity that he fhould retain them all, before he proceeds further; 'tis enough, at firft, to read them over with attention, and understand them clearly as he reads; they will foon become familiar to him, by frequent ufe, in the courfe of his ftudy of the Science.

I alfo particularly advife him, when he meets with any Term hereafter, of which he has not a clear Idea, to turn immediately back to the Definition of it; he may depend on it, that the time will not be entirely loft.

Several of the Terms which I have here defined, being frequently ufed in the Subject of Geometry, are abfolutely neceffary to be defined, but are not, properly fpeaking, elementary. Such are the 4th, the 9th, the 14th, 15th, 16th, and 18th; the 36th, the 41st, 43d, 44th, 47th, and 48th, and the last seven; which are chiefly operative or practical Terms. Though I must freely own, that I fee no reason why the 49th, 50th, 51ft, and 52d, are not as neceffry to be defined as the 53d and 54th: the 55th is made three feparate Definitions, in the 3d, 6th, and 13th of Eu

clid.

It is, or ought to be, the defign of every Author on any Science, to make his Book a perfect Tutor; confequently, no particular Term, made use of in that Science, and which is peculiar

to

to it, fhould be left undefined: for, we fhould fuppofe the Student to be entirely ignorant of all that relates to it. Can any thing be more abfurd, than to propofe bifecting a Line or Angle, or producing a Line, &c. to talk of Alternate Angles, &c. of Diagonals, Complements, Subtenfes, &c. without having first defined what is meant by them? I have often been furprized at the omiffion of the Definition of a Poligon, and the various kinds of Poligons; the forming, inferibing, and circumfcribing of which, is the principal fubject of the 4th Book. They are fre quently called by the Terms Pentagon, Hexagon, &c. which have never been properly, if at all, defined. In the 23d Definition they are, in general, called many-fided Figures; 'tis a ftrange ungeometrical Term, and never once made ufe of after, but other Terms are affumed.

Euclid, himself, has not defined a Parallelogram, that most useful and neceffary Figure. I admire Mr. Stone's Apology for that omiffion, as being unwilling to increase the Number, unneceffarily; and fays, that Euclid has fufficiently defined it in the 34th Propofition. Mr. Stone indeed has, but not Euclid; for why elfe has Keill defined it before the Propofition? But, if it be thought moft eligible to define that Term in the Propofition, where it is first made use of, why are not others fo defined? why are not all, as well as this?

At the fame time, Euclid has, in Mr. Stone's Opinion, given several unnecessary Definitions; viz. the 3d, 6th, and 13th of his Euclid, which I always thought fuperfluous; the 9th I think fo too, alfo the 20th. He, likewife, thinks the 18th, 26th, the 32d, and 33d, ufelefs. In refpect of the 26th (the 29th of this) I cannot fay it is effentially neceffary; yet, as Scalene Triangle is a general Term, including all that are neither Equilateral nor Ifofceles, I cannot think it redundant,

The 29th, of which he fays nothing, is really fuperfluous, viz. "when all the Angles are acute, it is called an acuteangled Triangle." I know of no properties peculiar to fuch an acute-angled Triangle, that is not common to every Triangle; for every Triangle has, neceffarily, two acute Angles. An equilateral Triangle is included in that Definition, and fo are the Ifofceles and Scalene, frequently.

Why is not a three-fided Figure (Def. 21.) called a Triangle, as it is always called afterwards? we might as well go on, from four to five or fifteen fided Figures. An Oblong (Def. 31.) for a Rectangle, is quite ungeometrical, and is never called fo afThe Rhombus and Rhomboides, 32. and 33. (Def. 36. of this) are, in a great meafure, ufelefs, being fully fignified in a Parallelogram, which alfo includes Squares and other Rectangles; all which, are only particular fpecies of Parallelograms; the Rhomboides including all unequal fided and acute angled,

ter.

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I can by no means agree with Mr. Stone, in thinking the 18th Definition, of a Semicircle, needlefs; I alfo think, the Radius as neceffary to be defined as the Diameter. In fpeaking of Angles in a Segment of a Circle, Prop. 31. of the 3d Book, the 12. of this, viz. "the Angle in a Semicircle is a Right one," &c.; fince there is a neceffity for calling it fomething, I cannot but think a Semicircle more elegant and expreffive than half a Circle, which does not confine it to any particular Shape or Figure; provided it has half the Area, it is half a Circle; whereas, the Semicircle (which undoubtedly means half a Circle) is always understood to be contained under a Diameter and half the Circumference.

Mr. Stone is fomewhat too dogmatical in his remarks. I cannot be of Opinion, that the manner in which Euclid has com→ piled his Elements, is the best that can poffibly be, because Euclid lived two thoufand Years ago. "Tis not to be imagined that the mathematical Sciences had arrived at their ne plus ultra in his Days, for it is notorious they were not; why then, should we fuppofe, the Elements of Geometry to be in their greatest perfection? I am not fo much wedded to antiquity, as to think them infallible; neither do I think that all Euclid's Definitions are neceffary, or that he has omitted none that are fo: Such, therefore, as are useless I have rejected, and have supplied their places with others, which I think more effential, and abfolutely neceffary to be defined.

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Poftulates are fundamental Principles in any Science; which, being plain and fimple in themfelves, may readily be granted; although, it is not poffible to give perfect and indifputable Demonftration.

The following are, therefore, requested to be granted, 1. That there may be a perfect Plane; and that it may be extended at pleasure.

2. That a Right Line may be drawn between any two given Points; i. e. from one Point to the other. 3. That a Right Line may be produced at pleafure; i. e. that a finite Line may be continued or lengthened. 4. That a Circle may be defcribed on any Center, and with any given Radius.

5. That one Figure may be applied to, or laid upon another Figure.

These Poftulates must be granted, at least in Idea, or all Geometry falls at once to the Ground; for, if there cannot be a Plane, a Right Line, or a Circle, the whole Elements of Geometry are to no purpofe; as it will be impoffible to form a Conftruction, whereby we may demonstrate the most eflential Properties of Figures in general, whether Plane Figures or Solids; and confequently, it no Demonftration can be given, there is an end of the Science, having no Data to build on.

In every mathematical or phyfical Science, there is a necefity for fome Data or firft Principles to be given, whereon to frame Hypothefes, in order to demonftrate the Theorems which follow; and the more fimple thofe Principles, are, the better; because there will be lefs room to difpute them. But, at the fame time, if they are difputed, they are the most difficult to demonftrate, for being the mott fimple; becaufe, there is no reverting back to any thing more fo; and confequently, there can be no Demonftration given. That the thing is fo, of itself, is fomewhat arbitrary, notwithstanding there is no poffibility of deny ing it; therefore, the more fimple the firft Principles are, the readier the affent will be given; and the Demonftrations, of the moft complex Propofitions, which follow after, will be cafier obtained and more firmly fupported; and confequently, the whole Science, which is built on thofe Principles, is more folid and permanent, and more fecurely established,

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Efore I proceed to practical Geometry, I think it proper, first to explain the Theory of Plane Angles; which I look on as a very neceffary Introduction.

In order to have a perfect and clear Idea of Plane Angles, and to determine their Quantity by a certain ftandard or measure, the Circumference of a Circle is fupposed to be divided into 360 equal parts, called Degrees (answering to the number of the Degrees on the Equator) it is evident that thofe divifions will be lefs in a fmall Circle than in a larger,

1. If from two Points in the Circumference of a Circle, as E and F, Lines are drawn to the Center, as EC and FC, there is made an Angle at the Center, C, which is greater or lefs, according to the number of Degrees on the Ark EDF, but G it will be the fame in a small Circle as in a large one, i. e. the lines will have the fame inclination to each other. . g.

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2. ADB is a Semicircle, whofe Center is C; the Arch AEDFB contains 180 deg. half 360, the whole Circumference.

From the middle point, D, of the Arch ADB, which is go deg, each way, from A and B, if CD be drawn, it will be perpendicular to AB; for ACD and DCB, arę

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