XI. A figure is that which is enclosed by one or more bounda. ries. XII. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines, drawn from a certain point within the figure to the circumference, equal to one another. XIII. There is therefore an essential difference between the circumference and the circle, the former being the boundary merely, and the latter the surface enclosed by it. Euclid however frequently employs the term circle, when the circumference only is meant. This is no doubt a violation of the very proper distinction which he has himself laid down, yet as it may be sufficient to have apprized the student here of this two-fold meaning of the term circle, there will be no necessity to alter the phraseology of the work in this respect. XIV. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XV. A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter. XVI. Rectilineal figures are those which are contained by straight lines only. The word "contained” which is preserved in this definition by all editors, had better be changed for enclosed, the word employed in the definition of figure. There is a precision in the term enclosed that contained wants. An angle is said to be contained by its sidero or the lines which form it, although these lines do not enclose any thing. A like change might be made in the definition immediately preceding, as also in definition XII. XVII. XVIII. XIX. Multilateral figures, or polygons, by more than four straight lines. XX. Of three-sided figures, an equilateral triangle is that wbich bas three equal sides. XXI. XXII. sides. XXIII. A right-angled triangle is that which has a right angle. XXIV. XXV. acute angles. XXVI. of four-sided figures, a square is that which bas all its sides equal, and all its angles right angles. This definition has been long known to be objectionable on the ground of its redundancy. More is said of a square than is absolutely necessary to characterize it, as will be seen in the 46th proposition of the first book. XXVII. right angles, but has not all its sides equal. XXVIII. but its angles are not right angles. XXIX. equal to one another, but all its sides are not XXX. All other four-sided figures besides these, are called Trapeziuns. XXXI. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. POSTULATES. I. Let it be granted, that a straight line may be drawn from any one point to any other point. II. That a terminated straight line may be produced to any length in a straight line. III. And that a circle may be described from any centre, at any distance from that centre. AXIOMS. I. Things which are equal to the same thing, are equal to one another. Il. Ill. IV. If equals be added to unequals, the wholes will be unequal. V. If equals be taken from unequals, the remainders will be unequal. VI. Things which are double of the same, are equal to one another. VII. Things wbich are halves of the same, are equal to one another. VIII. Magnitudes which coincide, or which may be conceived to coincide with one another, that is, wbich exactly fill the same space, are equal to one another. This axiom, referring to the most obvious test of geometrical equality, is scarcely comprehensive enough. Probably it would be an improvement to omit the words “that is, which exactly fill the same space;" as equality among ungles which do not fill space seems, by this condition, to be excluded. IX. X. XI. See Notes at the end. |