There areĪnd, furthermore, of trilateral figures a right-angled triangle Note that equilateral triangles are not isosceles. Has three equal sides, isosceles that which has only two equal sides,Īnd scalene that which has three unequal sides. Neither ‘pentagon’ nor any more sided figure gets defined, although pentagons, hexagons, and 15-gons are the subject of IV.Īnd of trilateral figures, equilateral triangle is that which In books vi and xii (as well as the Catoprics). ‘triangle’, even in naming the species of trilaterals in 20 and 21, and ‘polygon’ Hence, Euclid isĬareful to define terms corresponding to the names, even though he will use 'Trilateral' is only used three times in Euclid, here and in defs.Ģ0 and 21. Those by three, quadrilaterals those by four, multilaterals those enclosed by more than four straight-lines, Rectilinear figures are those enclosed by straight-lines, trilaterals

Of the circle, which also bisects the circle.Īre these two definitions conflated into one?Īnd a semicircle is the figure enclosed by the diameter andĪnd a center of the semicircle is the same as what is also Through the center and limited on both sides by the circular-arc This practice is followed throughout these web pages.Īnd the point is called ‘ center of the circle’,Īnd a diameter of the circle is a certain straight-line drawn Text, the word which we normally translate as 'circumference' is often better Whether or not the phases in square brackets are part of Euclid's However, did anyone ever think of this definition in this way?Ģ. If one were to think of this definition as providing a construction, 14.Ī figure is what's enclosed by some boundary or some boundaries.Ī circle is a plane figure enclosed by one line, such all the straight-lines falling upon it from one pointĪmong those lying within the figure areġ. This notion of a boundary is only used in Def. Obtuse angle is that larger than a right-angle,Īnd acute angle is that smaller than a right-angle.Ī boundary is what is a limit of something. Standing upon it is called perpendicular to the line it stands upon. Aristotle's version of Elements I 5 at Prior Analytics I 24.Īnd whenever a straight-line stood on a straight-line makes its successiveĪngles equal to one another, each of the equal angles is right and the one Otherwise, all angles in the Elements are rectilinear. The angle of a straight-line and circular-arc is treated at III 16. Since angles between curved lines play no role in the Elements, this is not harmful to the content of the book.Īnd whenever the lines enclosing the angle are straight the angle Note that by the definition any curved line will make an angle at any point on it.

Touching one another and with the lines not lying on a straight line with respect to one another, Is this a separate definition of lines (as in Aristotle), or a propertyĪ plane surface is whatever lies equally with the lines onĪnd a plane angle is the inclination on a plane of two lines 7.Īnd a surface is what only has length and breadth, To translate 'ex isou', here 'equally' as 'evenly', cf. The meaning of this definition is very unclear. Of lines? Even if it is a definition of ‘limit of a line’, then it is still making a substantial claim about lines.Ī straight line is whatever lies equally with the points on Is this a separate definition of point (as in Aristotle) or a property A point is that for which there is no part,

Common notion definition