return to Vignettes of Ancient Mathematics

Definitions
Postulates
Common Notions
Propositions


Definitions Brief comments
1 A point is that for which there is no part,
2 and a line is breadthless length,
3 and limits of a line are points. Is this a separate definition of point (as in Aristotle) or a property of lines? Even if it is a definition of ‘limit of a line’, then it is still making a substantial claim about lines.
4 A straight line is whatever lies equally with the points on it, The meaning of this definition is very unclear.  It is common to translate 'ex isou', here 'equally' as 'evenly', cf. def. 7.
5 and a surface is what only has length and breadth,
6 and lines are limits of a surface. Is this a separate definition of lines (as in Aristotle), or a property of surfaces? Cf. def. 3.
7 A plane surface is whatever lies equally with the lines on it, Cf. def. 4 and comment.
8 and a plane angle is the inclination on a plane of two lines touching one another and with the lines not lying on a straight line with respect to one another, Note that by the definition any curved line will make an angle at any point on it. Since angles between curved lines play no role in the Elements, this is not harmful to the content of the book.
9 and whenever the lines enclosing the angle are straight the angle is called ‘rectilinear’. The angle of a straight-line and circular-arc is treated at III 16. Otherwise, all angles in the Elements are rectilinear. Cf. Aristotle's version of Elements I 5 at Prior Analytics I 24.
10 and whenever a straight-line stood on a straight-line makes its successive angles equal to one another, each of the equal angles is right and the one standing upon it is called perpendicular to the line it stands upon.
11 Obtuse angle is that larger than a right-angle,
12 and acute angle is that smaller than a right-angle.
13 A boundary is what is a limit of something. This notion of a boundary is only used in Def. 14.
14 A figure is what's enclosed by some boundary or some boundaries.
15 A circle is a plane figure enclosed by one line [which is called circular-arc], such all the straight-lines falling upon it from one point among those lying within the figure [to the circular-arc of the circle] are equal to one another, 1. If one were to think of this definition as providing a construction, the construction would be pointwise.  In Euclid, this is inappropriate; however, did anyone ever think of this definition in this way?
2.  Whether or not the phases in square brackets are part of Euclid's text, the word which we normally translate as 'circumference' is often better translated as 'circular arc', as in def. 18. This practice is followed throughout these web pages.
16 and the point is called ‘center of the circle’,
17 and a diameter of the circle is a certain straight-line drawn through the center and limited on both sides by the circular-arc of the circle, which also bisects the circle. Are these two definitions conflated into one?
18 and a semicircle is the figure enclosed by the diameter and the circular-arc cut off by it, 
18a and a center of the semicircle is the same as what is also center of the circle.
19 Rectilinear figures are those enclosed by straight-lines, trilaterals those by three, quadrilaterals those by four, multilaterals those enclosed by more than four straight-lines, 'Trilateral' is only used three times in Euclid, here and in defs. 20 and 21.  ‘Multilateral’ is used only once.  Hence, Euclid is careful to define terms corresponding to the names, even though he will use ‘triangle’, even in naming the species of trilaterals in 20 and 21, and ‘polygon’ in books vi and xii (as well as the Catoprics). Neither ‘pentagon’ nor any more sided figure gets defined, although pentagons, hexagons, and 15-gons are the subject of IV.
20 and of trilateral figures, equilateral triangle is that which has three equal sides, isosceles that which has only two equal sides, and scalene that which has three unequal sides. Note that equilateral triangles are not isosceles.  There are three species of trilaterals.
21 And, furthermore, of trilateral figures a right-angled triangle is one having a right angle, and obtuse-angled one having an obtuse angle, and acute-angled one having the three angles acute. As in def. 20, the three species are distinct.
22 and of quadrilateral figures, a square is that which is both equilateral and right-angled, and an oblong is that which is right-angled but not equilateral, and a rhombus is that which is equilateral but not right-angled, and rhomboid is one having opposite sides and angles equal to one another but which is neither equilateral nor right-angled, and let any quadrilaterals besides these be called 'trapezia'. Square is called 'tetragonon' (four-angle [figure]).  Here too (cf. defs. 20 and 21) the classifications are exclusive and include all possible quadrilaterals by using 'trapezion' as a catchall.  British English follows Proclus (170-1) in treating trapezia as figures with two parallel lines and trapezoids as the having not parallel sides (in America we reverse the terminology).  Euclid only uses the term in I 34, where the two trapezia happen to have two parallel lines.  There are no texts independent of Proclus or his sources giving an account of the two terms.

Rhomboids, however, will turn out to be parallelograms, a term which Euclid introduces without definition in the statement of the theorem at I 34.  Obviously, the term cannot be introduced before he defines parallel lines.  Neither rhombus nor rhomboid appear again in Euclid.

23 Parallels are any straight-lines which, being on the same plane and being extended infinitely towards both sides fall-together on neither side.
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Postulates
1 Let it be postulated that from any point to any point to draw a straight line.
2 and to extend a finite straight line continuously in a straight line
3 and with any center and distance to describe a circle,
4 and that all right angles are equal to one another,
5 and that if a straight-line falling into two straight-lines makes the angles that are interior and on the same sides smaller than two right angles, then on being eztended infinitely the two lines fall-together on the sides where the angles are smaller than two right-angles.

Common Notions
1 Things equal to the same are also equal to one another.
2 and if equals are added to equals the wholes are equal.
3 and if equals are taken away from equals, the remainders are equal.
4 [and if unequals are added to equals, the wholes are unequal.]
5 [and the doubles are of same are equal to one another.]
6 [and the halves are of same are equal to one another.]
7 and things which fit onto one another are equal to one another.
8 and the whole is larger than the part.
9 [and two straight-lines do not enclose a region.]

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Prop. 1 (problem): On the given straight finite straight-line to construct an equilateral triangle.

Prop. 2 (problem): To position at the given point a straight-line equal to the given line.

Prop. 3 (problem): With two given, unequal straight-lines to take away from the larger a straight-line equal to the smaller.

Prop. 4 (theorem): If two triangles have two sides respectively equal to two sides, and the angle contained by the straight lines equal to the angle, then they will have the base equal to the base and the triangle will be equal to the triangle, and the remaining angles will be respectively equal to the remaining angles which the equal sides subtend.

Prop. 5 (theorem): The angles at the base of isosceles triangles are equal to one another, and when the equal sides are extended the angles under the base will be equal to one another.

Prop. 6 (theorem): If two angles of a triangle are equal to one another, then the sides subtending the equal angles will be equal to one another.

Prop. 7 (theorem): On the same straight-line two different straight-lines respectively equal to the same straight-lines will not be constructed at one and another point on the same sides while having the same limits as the initial lines.

Prop. 8 (theorem): If two triangles have the two sides equal to the two sides respectively, but also have the base equal to the base, then they will have the angle that’s enclosed by the equal straight lines equal to the angle.

Prop. 9 (theorem): To bisect the given rectilinear angle.

Prop. 10 (problem): To bisect the given finite straight-line.

Prop. 11 (problem): To draw a straight line at right angles to the given straight-line from the given point on it.

Prop. 12 (problem): To draw a perpendicular straight line to the given infinite straight-line from the given point, which is not one it.

Prop. 13 (theorem): If a straight-line stood against a straight-line makes angles, it will make either two right-angles or equals to two right angles.

Prop. 14 (theorem): If at some line and the point on it two straight-lines, not positioned on the same sides, make successive angles equal to two right-angles, the straight-lines will be on a straight-line with one another.

Prop. 15 (theorem): If two straight-lines cut one another, they make the angles at the vertex equal to one another.

Prop. 16 (theorem): Upon one of the sides of any triangle being extended, the external angle is larger than each of the interior and opposite angles.

Prop. 17 (theorem): The two angles of any triangle, taken in any way, are smaller than two right-angles.

Prop. 18 (theorem): The larger side of any triangle subtends the larger angle

Prop. 19 (theorem): Subtending the larger angle of any triangle is the larger side. Let there be a triangle,

Prop. 20 (theorem): The two sides of any triangle, taken in any way, are larger than the remaining.

Prop. 21 (theorem): If two straight-lines are constructed within a triangle from the limits on one of the sides, the constructed straight-lines will be smaller than the remaining two sides of the triangle and enclose a larger angle.

Prop. 22 (problem): To construct a triangle from three straight-lines which are equal to three, the given straight-lines.

Prop. 23 (problem): To construct at the given straight-line and the point on it an angle equal to the given rectilinear angle.

Prop. 24 (theorem): If two triangles have the two sides respectively equal to the two sides but have the angle enclosed by the equal straight-lines larger than the angle, then they will have the base larger thn the base.

Prop. 25 (theorem): If two triangles have the two sides respectively equal two sides and the base larger than the base, then they will have the angle that’s enclosed by the equal sides larger than the angle.

Prop. 26 (theorem): If two triangles have two angles respectively equal to two angles and one side equal to one side, whether that at the equal angles or that subtending one of the equal angles, it will also have the remaining sides respectively equal to the remaining sides and the remaining angle to the remaining angle.

Prop. 27 (theorem): If a straight-line falling into two straight-lines makes alternate angles equal to one another, the straight-lines will be parallel to one another.

Prop. 28 (theorem): If a straight-line falling into two straight-lines makes the external angle equal to the angle that’s internal and opposite and on the same sides or the angles that are interior and on the same sides equal to two right-angles, the straight-lines will be parallel to one another..

Prop. 29 (theorem): The straight-line falling into parallel straight-lines makes the alternate angles equal to one another and the external angle equal to the angle that’s internal and opposite and on the same sides and the angles that are interior and on the same sides equal to two right-angles..

Prop. 30 (theorem): The parallels to the same straight-line are also parallel to one another.

Prop. 31 (problem): To draw a straight line through the given point parallel to the given line.

Prop. 32 (theorem): Of every triangle, when one of the sides is extended the external angle is equal to the two interior and opposite angles and the three interior angles of the triangle are equal to two right angles.

Prop. 33 (theorem): The opposite sides and angles of parallelogram regions are equal to one another, and the diameters bisect them

Prop. 34 (theorem): Parallelograms that are on the same base and in the same parallels are equal to one another.

Prop. 35 (theorem): Parallelograms that are on the same base and in the same parallels are equal to one another.

Prop. 36 (theorem): Parallelograms that are on equal bases and are in the same parallels are equal to one another.

Prop. 37 (theorem): Triangles that are on the same base and are in the same parallels are equal to one another.

Prop. 38 (theorem): Triangles that are on equal bases and are in the same parallels are equal to one another.

Prop. 39 (theorem): Equal triangles that are on the same bases and are on the same sides are also in the same parallels.

Prop. 40 (theorem):Equal triangles that are on equal bases and are on the same sides are also in the same parallels.

Prop. 41 (theorem): If a parallelogram has a base that’s the same for a triangle and is in the same parallels, the parallelogram is double the triangle.

Prop. 42 (problem): To construct a parallelogram equal to the given triangle in the given rectilinear angle.

Prop. 43 (theorem): For every parallelogram, the complements of the parallelograms about the diameter are equal to one another.

Prop. 44 (problem): To apply along the given straight-line a parallelogram equal to the given triangle in the given rectilinear angle.

Prop. 45 (problem): To construct a parallelogram equal to the given rectilinear-figure in the given rectilinear angle.

Prop. 46 (problem): From the given straight-line, to describe up a square.

Prop. 47 (theorem): In right-angled triangles the square from the side subtending the right angle is equal to the squares from the sides containing the right angle.

Prop. 48 (theorem): If the square from one of the sides of a triangle is equal is equal to squares from the remaining two sides of the triangle, the angle enclosed by the remaining sides of the triangle is right. .

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