quod pungit: then, by an eafy derivation, any mark made by that sharp thing; and this is the meaning of point in geometry. Perhaps the difficulty had never occurred, if instead of point, the word dot, or even mark, onμelov, had been always used in its σημείον, place, and there had been no attempt at definition. For a point is only to mark the place whence a line is to begin, or where it is to end : γραμμῆς δέ πέρατα σημεία. DEF. III. Thus in a circle it marks the fpot within the figure, from which all ftraight lines drawn to the circumference, are equal. Now one would make such a mark as small as poffible, provided it be ftill diftinct, that the length of lines and their meetings and interfections may appear plainly, and from this effect of convenience has arisen the phrase that is supposed to defcribe its effence; that it is without parts. This idea has nothing to do with the rea foning; foning; all that is neceffary is, that the aot or mark should take up no fenfible part of the line, in order that the diagram may be diftinct. Points then are only fubfervient to the convenience of construction. The next definition, after this explanation of the first, will present no difficulty. A line is length without breadth. Γραμμη δε, μηκος απλάζες. Draw your lines as narrow as you conveniently can, your diagrams will be the clearer; but you cannot, and you need not, conceive length without breadth. DEFINITION III. The extremity of a line are points. D DEFI A ftraight line is that which lies evenly between its extreme points. Ευθεία γραμμη εςιν, ήτις εξισε τοις εφ' εαυτης σημείοις κείται. The impoffibility of defining a word expreffive of a fimple perception is well known. The definition of a complex term confists merely in the enumeration of the fimple ideas, for which it ftands. The only way of rendering the meaning of a fimple term intelligible, is to exhibit the object of which it is the fign; or, if you please, fome fenfible reprefentation of that object. A ftraight line therefore must be fhewn; and by drawing a crooked one at the fame time, it will be perfectly underftood, if any one require an explanation. All definitions must have some term, equal ly ly requiring a definition with that defined, as ε108, evenly, upon an equality. The definitions of a furface, and a plane surface, must in like manner be made intelligible by an appeal to the fenfes. By putting a straight rule along different furfaces, it will appear whether they are plane or otherwise. DEFINITION X. When a ftraight line ftanding on another ftraight line makes the adjacent angles equal to one another, each of the angles is called a right angle, and the line which ftands on the other is called a perpendicular to it. Here we have an appeal made to the fenfes, which alone can inform us what is the expansion between lines meeting at a point, or what is their inclination or bending towards each other. The eye can pretty well determine, whether the meeting lines are more inclined to each other on one fide than the other, i. e. the equality or inequality of the adjacent angles. This measure of the eye would not be fufficiently exact to fatisfy us that the angles are equal; we must obtain a measure by real or imagined fuper-pofition, as we do in one particular cafe, by applying the refult of the demonstration of the eighth, to the eleventh propofition of Book 1. But the term right probably preceded the application of any fuch exact measure; and I fhould conjecture, that it might be deduced from the meeting of lines or of bodies, that feemed to ftand perfectly erect and not to bend towards each other; in the case of lines, ftraight and right are perfectly fynonymous; and an angle formed by the meeting of ftraight lines, ftanding right or upright, with respect to each other, would be |