Around 1460, the Flemish painter Justus van Gent arrived in Urbino with a mandate to develop a serie of 28 portraits of the most outstanding personalities of culture, literature and politics. The pictures had to decorated the walls of the new library of the city. The portraits included names as Dante, Ptolemy, the cardinal Bessarione,… and our Euclid of Alexandria. Since this is my first post on this blog, I thought that we had to get started in this virtual world talking about the great mathematician Euclid.
Euclid was a Greek mathematician who was active in Alexandria around the 3rd century BC. We can read in Proclus’ Commentary on Euclid that:
“Not much younger than those (sc. Hermotimus of Colophon and Philippus of Medma) is Euclid, who put together the Elements, collecting many of Eudoxus’ theorems, perfecting many of Theaetetus’ and also bringing to irrefutable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid; and further they say that Ptolemy once asked him if there was in geometry any shorter way than that of the Elements, and he replied that there was no royal road to geometry. He is then younger than the pupils of Plato, but older that Erathostenes and Archimedes, the latter having been contemporaries, as Erathostenes somewhere says.
The Elements were his great work. Euclid was aware of all the mathematical knowledge of his time and he decided to put all these results in order so that he could built all the Geometry from a limited number of definitions and postulates. There were more Elements before Euclid but none of them can be compared with his thirteen books about 2D and 3D Geometry, Number Theory and Arithmetic. He also wrote other works as the Data, the Catoptrics, the Conics, the Porisms, On the Division of the Figures,… but the main thing which makes the Elements more important than the others is that nobody after him wrote any mathematical text without reading them. As a small taste of this great work I’m going to summarize the first book which begins with 23 initial definitions:
- A point is that of which there is no part.
- And a line is a length without breadth.
- And the extremities of a line are points.
- A straight-line is any one which lies evenly with points on itself.
- And a surface is that which has length and breadth only.
- And the extremities of a surface are lines.
- A plane surface is any one which lies evenly with the straight-lines on itself.
- And a plane angle is the inclination of the lines to one another, when two lines in a plane meet one another, and are not lying in a straight-line.
- And when the lines containing the angle are straight then the angle is called rectilinear.
- And when a straight-line stood upon another straight-line makes adjacent angles which are equal to one another, each of the equal angles is a right-angle, and the former straight-line is called a perpendicular to that upon which it stands.
- An obtuse angle is one greater than a right-angle.
- And an acute angle is one less than a right-angle.
- A boundary is that which is the extremity of something.
- A ﬁgure is that which is contained by some boundary or boundaries.
- A circle is a plane ﬁgure contained by a single line [which is called a circumference], such that all of the straight-lines radiating towards [the circumference] from one point amongst those lying inside the ﬁgure are equal to one another.
- And the point is called the center of the circle.
- And a diameter of the circle is any straight-line, being drawn through the center, and terminated in each direction by the circumference of the circle. And any such straight-line also cuts the circle in half.
- And a semi-circle is the ﬁgure contained by the diameter and the circumference cuts off by it. And the center of the semi-circle is the same point as the center of the circle.
- Rectilinear ﬁgures are those ﬁgures contained by straight-lines: trilateral ﬁgures being those contained by three straight-lines, quadrilateral by four, and multilateral by more than four.
- And of the trilateral ﬁgures: an equilateral triangle is that having three equal sides, an isosceles triangle that having only two equal sides, and a scalene triangle that having three unequal sides.
- And further of the trilateral ﬁgures: a right-angled triangle is that having a right-angle, an obtuse-angled triangle that having an obtuse angle, and an acute-angled triangle that having three acute angles.
- And of the quadrilateral ﬁgures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to one another which is neither right-angled nor equilateral. And let quadrilateral ﬁgures besides these be called trapezium.
- Parallel lines are straight-lines which, being in the same plane, and being produced to inﬁnity in each direction, meet with one another in neither of these directions.
These 23 definitions are followed by the 5 postulates which have to be the basis of all the Geometry: nothing can be done without them:
- Let it have been postulated to draw a straight-line from any point to any point.
- And to produce a ﬁnite straight-line continuously in a straight-line.
- And to draw a circle with any center and radius.
- And that all right-angles are equal to one another.
- And that if a straight-line falling across two other straight-lines makes internal angles on the same side of itself whose sum is less than two right-angles, then the two other straight-lines, being produced to inﬁnity, meet on that side of the original straight-line that the sum of the internal angles is less than two right-angles (and do not meet on the other side).
Finally, Euclid gives five common logic notions with which he will be able to use his postulates:
- Things equal to the same thing are also equal to one another.
- And if equal things are added to equal things then the wholes are equal.
- And if equal things are subtracted from equal things then the remainders are equal.
- And things coinciding with one another are equal to one another.
- And the whole is greater than the part.
The propositions of Book I fall into three distinct groups. The first consists in 26 propositions dealing with triangles without using the fifth postulate. For example, proposition 1 gives the construction of an equilateral triangle from a given straight segment. Euclid uses the third postulate to construct two respectively equal circles with center in both extremities of the segment and joins these extremities with the intersection point of the two circles. Then, he demonstrates the equality of the three sides using the common notions. The second group of 5 propositions deals with the theory of parallels and the last 15 propositions deals with parallelograms, squares and triangles with reference to their areas. The last proposition is the well-known Pythagoras theorem.
So if you go to Urbino, you will have to visit the Natonal Gallery of Marche to find this portrait painted by Justus van Gent.