Classical Greek Geometry - 2

The most famous of Euclid’s works is his ElementsIt consists of thirteen

books (a book here means a long chapter). The first book contains:

(a)  the definitions, or explanations of the terms used in the text.

(b)  the postulates, which limit the instruments to be used in constructions to the ruler and compass.

(c)  the common notions or axioms, the fundamental principles from which the theorems or propositions are deduced. The axioms are taken as given, and are not provable.

The propositions of Book 1 deal with rectilineal figures (figures bounded by straight line segments), mainly the triangle and parallelogram. Book 1 concludes with the theorem of Pythagoras (Proposition 47) and its converse (Proposition 48).

Book 2 is concerned with rectangular parallelograms contained by segments of straight lines, and their relation to certain squares.

Book 3 discusses properties of circles.

Book 4 contains no theorems. It is concerned with constructions, such as how to inscribe a circle inside a triangle, or circumscribe a circle about a triangle. There is a method for constructing a regular pentagon (important to the Pythagoreans).

The first four books build successively on the earlier books.

Book 5 is unusual, as it is independent of the first four books. It contains the theory of proportion, applied not only to geometrical quantities, such as lines, angles, areas, etc, but also to arithmetical quantities. The doctrine of proportion in Euclid has been much investigated, especially as it relates to the concepts of magnitude and number. Roughly speaking, as we observed earlier, early Greek mathematicians had considered all numbersto be rational (i.e. fractions) and could then compare them in size by simple proportion.

The Pythagoreans discovered through the theorem of Pythagoras that not all numbers

√ are rational, for example, 2 is not rational. This led to a knowledge of the existence of incommensurable magnitudes, causing a crisis in mathematics. Euclid’s fifth book is

considered to be difficult to follow, because of the inherent complexity of the subject.

Book 6 contains applications of the theory of proportion, mainly to rectilineal figures.

Books 7, 8 and 9 are devoted to arithmetic and properties of numbers. The last proposition of Book 9 gives a construction of even perfect numbers, a construction that Euler showed in the 18th century to give all even perfect numbers.

Book 10 is devoted to consideration of commensurable and incommensurable magnitudes, and it is shown there that the diagonal and side of a square are incommensurable, i.e. that is irrational.

Books 11, 12 and 13 are devoted to solid geometry. Book 11 is mainly concerned with planes and solid angles. It provides the theory necessary to prove that there are at most five regular solids. Book 12 shows that, given spheres of radius r and r1, the ratio of their volumes equals the ratio of r3 to r13–in other words, there is a constant k such that the volume of a sphere of radius r equals kr3. In fact, as we now know, .

The tool for this result is the method of exhaustion, used to approximate area and volume by successive subdivision. The method is due to Eudoxus, whom we mentioned in connection with Plato’s Academy, and it was much used later by Archimedes in volume calculation. It is a forerunner of the methods of the integral calculus, developed in the 17th century.

Book 13 finishes with the theory of the five regular or Platonic solids, and considers in particular how these solids may be constructed inside spheres. Some commentators have believed that the whole of the Elements was designed to provide the theory necessary for the classification of such solids.

We do not know for certain how much of the Elements was original with Euclid, or how much derived from earlier work. We have to rely on the evidence offered by Pappus and Proclus. It is claimed that the theorem of Pythagoras on right angled triangles was already known (although not necessarily proved by Pythagoras), that the subject matter of plane geometry was already restricted to the straight line and circle, and that the regular solids and their properties were already known (around the time of Plato).

Euclid’s Elements was much used as a school and university textbook, especially the first six books, and books eleven and twelve. In the latter half of the nineteenth century, attempts were made to revise the way in which geometry was taught. In particular, it was considered necessary to depart from the order in which Euclid presented his material, as this was held to be arbitrary and unsystematic. People also objected to the frequent use of proof by reductio ad absurdum–assuming the opposite of what was required to be proved.

Let us now examine the opening of Book 1 and comment on Euclid’s working method. There are 23 definitions.

1.  A point is that which has no part.

2.  A line is breadthless length.

3.  The extremities of a line are points.

4.  A straight line is a line which lies evenly with the points on itself.

5.  A surface is that which has length and breadth only.

6.  The extremities of a surface are lines.

7.  A plane surface is a surface which lies evenly with the straight lines on itself.

8.  A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

9.  And when the lines containing the angle are straight, the angle is called rectilineal.

10.  When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

11.  An obtuse angle is an angle greater than a right angle.

12.  An acute angle is an angle less than a right angle.

13.  A boundary is that which is an extremity of anything.

14.  A figure is that which is contained by any boundary or boundaries.

15.  A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;

16.  and the point is called the centre of the circle.

17.  A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

18.  A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

19.  Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20.  Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

21.  Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

22.  Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

23.  Parallel straight lines are straight lines which being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

The definitions are followed by five postulates.

Let the following be postulated.

1.  To draw a straight line from any point to any point.

2.  To produce a finite straight line continuously in a straight line.

3.  To describe a circle with any centre and distance.

4.  That all right angles are equal to one another.

5.  That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Finally, there are five common notions (or axioms).

1.  Things which are equal to the same thing are also equal to one another.

2.  If equals be added to equals, the wholes are equal.

3.  If equals be subtracted from equals, the remainders are equal.

4.  Things that coincide with one another are equal to one another.

5.  The whole is greater than the part.

Euclid’s definitions, postulates and common notions show the influence of Aristotle’s discussion of these topics. On the other hand, Aristotle shows no awareness of Euclid, and in one of his works he even gives a pre-Euclidean proof of the theorem that the base angles of an isosceles triangle are equal. This strongly suggests that Euclid was active after the death of Aristotle (recall that we do not know Euclid’s exact dates).

Regarding the definitions, various commentators have observed that certain are inadequate. For example, the definition of a plane surface is inadequate, and further postulates must be added to obtain its basic properties. This is important, as much of Euclid’s geometry is studied in a plane. In the definition of a straight line, we may wonder what is exactly meant by evenly. Amended definitions have been suggested, such as

•  a plane surface is one which divides space into two halves, identical in all respects except position.

•  a straight line is one which divides a plane into two halves, identical in all respects except position.

Modern editions of the Elements have expanded the number of definitions to 35, reduced the postulates to three, and given as many as 12 axioms.

The most notorious of the definitions and postulates is the fifth postulate, also known as the parallel postulate. The postulate asserts that, if a straight line cuts two other straight lines AB and CD in points P and Q so that the sum of the angles BPQ (β) and DQP (α) is less than two right angles, AB and CD will meet on the same side of PQ as α and β, that is, they will meet if produced beyond B and D.

There was a feeling in antiquity that the fifth postulate could be proved, and erroneous proofs were offered, resting on some assumption or other that could not be justified. In the 18th century, the Italian mathematician Saccheri and the German mathematician Lambert looked at what could be achieved in geometry if the parallel postulate is not assumed. They argued that if the parallel postulate, or some equivalent, is not assumed, we cannot prove that the sum of the angles in a triangle is two right angles (this is Proposition 32 of Book 1 of Euclid, the theorem attributed to the Pythagoreans). Furthermore, if a quadrilateral is drawn, in which three of its four angles are right angles, we cannot conclude that the fourth angle is also right. The fourth angle may be acute, or right, or obtuse, and they proved that whatever occurs in one case is true in all cases. They then

inferred that there are three conceivable types of geometry, in two of which the parallel postulate is not true, and in the third it is true. Thanks to the work of the 19th century mathematicians Bolyai, Gauss and Lobachevski, we now know that all three possibilities can occur in suitable models of space. Thus the fifth postulate is an unprovable assumption defining a particular type of geometry of space–the Euclidean geometry. The other two models of geometry, in which the parallel postulate is abandoned, define so-called non-Euclidean geometries.

We have mentioned the importance of the five regular solids in Greek mathematics, especially in Plato’s cosmology. Euclid investigates the regular solids in Books 11 and 13 of the Elements. The notion of a solid angle plays an important role in his analysis. We will briefly discuss solid angles and state without proof one of the main theorems relating to them. We will then show how to deduce that there are at most five regular solids.

When three or more planes taken in order intersect, each with the next, in lines which meet at a point, they are said to form a solid angle. The point of concurrence is called the vertex; the lines of intersection of consecutive planes are called the edges of the solid angle; the angles between consecutive planes are called its dihedral angles; the plane angles formed by consecutive edges are called its face angles.

A solid angle formed by three concurrent planes is said to be trihedral; if it is formed by more than three concurrent planes, it is said to be polyhedral.

The following two important results are proved in Book 11 of the Elements.

Proposition 20. If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one. (In more modern language, in a trihedral angle, the sum of any two face angles is greater than the third.)

Proposition 21. Any solid angle is contained by plane angles less than four right angles. (Modern form: in a convex solid angle, the sum of the face angles is less than four right angles.) A solid angle is convex if any point lying in the interior of the angle can be joined to any other point lying in the interior of the angle by a line segment also lying in the interior of the angle.

It should be noted that, although Euclid enunciates Proposition 21 for any solid angle, he only proves it for trihedral angles. He also neglects to include any mention of restriction to

convex solid angles. Modern proofs of Proposition 21 use Proposition 20 as a preliminary result.

A polyhedron is a solid bounded by plane faces. A regular polyhedron has faces which are equal regular polygons. Suppose a regular polygon has n ≥ 3 edges. It is straightforward to see that each edge subtends an angle equal toright angles (we measure angles, as the Greeks did, in terms of right angles). It follows easily that the angle formed at each vertex of the polygon by any two adjacent edges is

We can now desribe the possibilities for the faces of a regular polyhedron.

Let P be a regular polyhedron whose faces are regular polygons of n sides. Suppose that at each vertex of P, r faces meet (r is the same for each vertex, by regularity). At each vertex, a convex solid angle is thus formed by r concurrent planes (where we must have r ≥ 3). The face angles of each of these solid angles are all equal toright

angles and the sum of the face angles of the solid angle is thus

right angles.

By Proposition 21 above, we obtain

and hence

Since necessarily 3 ≤ n, we obtain 3(r - 2) < 2r and hence r < 6. It follows therefore that r = 3, r = 4 or r = 5 and we go through each value in turn. If r = 3, we obtain n < 6. Thus we can have n = 3, n = 4 or n = 5 in this case. These three solutions correspond to three faces meeting at each vertex, the faces being equilateral triangles, squares, or regular pentagons. The corresponding solids are the tetrahedron, cube, and dodecahedron. If r = 4, we obtain n < 4. It follows that n = 3 is the only solution. This corresponds to four faces meeting at each vertex, the faces being equilateral triangles, and the solid is an octahedron. If r = 5, we obtain. This leaves only the solution n = 3.

Here, five faces meet at each vertex, the faces being equilateral triangles, and the solid an icosahedron. Hence there are exactly five solutions and each corresponds to a different solid.

Euclid’s Elements describes the achievements of Greek geometry over the three hundred years or so since Thales began studying the subject. There were, however, three famous problems that exercised the minds of Greek mathematicians over many centuries and never admitted any satisfactory solution. These problems were:

•  to square the circle

•  to construct a cube whose volume is twice that of a given cube

•  to trisect any given angle.

It was assumed that, ideally, any solution of a given problem would be accomplished within the framework of Euclidean geometry, namely, by using a straight edge and compass. The first problem, to square the circle, is as follows: to construct a square, by geometrical methods, whose area equals that of a given circle. Thus, if we take a circle of radius 1 unit, we know now that its area is π square units and we therefore need to construct a square whose sides have lengthunits. At first, the value of π was not known accurately to Greek mathematicians. Later, Archimedes realized that π could be approximated to any degree of accuracy, by a process of subdivision similar to the methods of integral calculus. He obtained the following estimate for π:


This did not solve the squaring the circle problem, as there was still no geometric method for constructing a line of lengthunits. In 1882, the German mathematician Lindemann proved that the number π is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients. This finally laid the Greek problem to rest, as the only numbers that can be constructed by the methods of Greek geometry are algebraic numbers of a special type (lying in iterated quadratic extensions of the rational numbers), and these numbers are certainly not transcendental.

The second problem, also known as the problem of duplicating the cube, involves constructing a line of length equal to the cube root of two,, since a cube whose sides have length will have volume equal to 2 cubic units. Work of Gauss in the early

19th century showed that a lengthcannot be obtained by straight edge and compass, because determines a degree 3 extension of the rationals, and hence this number

does not lie in an iterated quadratic extension. Already Archytas, whom we mentioned in connection with Plato, proposed a solution to the duplication problem using a cylinder, but

Plato held that such methods were contrary to the spirit of geometry. Later, Menaechmus devised another method, using special curves that cannot be constructed by Euclidean means.

Concerning the third problem, it is well known that one can bisect an angle using a straight edge and compass. One can then divide an angle into four, eight, etc, equal parts. However, no construction for trisecting an angle using a straight edge and compass was ever discovered. It was later shown as a consequence of the work of Gauss that it is impossible to trisect a general angle using only a straight edge and compass. The proof lies within the theory of abstract fields, as do the first and second problems previously described. It is possible to trisect an angle using an infinite sequence of bisections of related angles, but this is inadmissible as a solution. Nonetheless, the idea allows the possibility of obtaining an approximation to the trisection of an angle to any degree of accuracy using only a finite number of bisections.

It is of some interest to know how the Elements were transmitted to later generations. In the 4th century CE, Theon of Alexandria edited a version of the work which has served as the basis for virtually all subsequent editions of the Elements. Theon’s daughter, Hypatia, was herself a celebrated geometer, but, as a pagan, she suffered the indignity of being murdered by a Christian mob in Alexandria in 415. The earliest extant manuscript of a Greek version of the Elements dates from 888, and is located in Oxford. In the early 7th century, around 620-630, the creation of Islam and the subsequent expansion of Arab conquerors into much of North Africa and Asia led to the demise of the greater part of the Byzantine Empire, which was the inheritor of the Greek tradition, and of the remains of the Western Roman Empire. The Arabs came into contact with Greek learning, especially in Alexandria. Initially, they were without intellectual interest in this unfamiliar knowledge, rather as the Romans had felt with regard to their Greek predecessors, but by 750, the Arabs began to absorb the Greek legacy of mathematics and astronomy. A centre of learning was established in Baghdad in the early 9th century, under the support of a series of enlightened caliphs. Here, translations into Arabic were made of Greek classics, such as Euclid’s Elements. Without these translations, many of the most famous and original of the Greek treatises might have been lost during periods of political confusion, barbarian invasion and intellectual sterility in Europe.

Knowledge of the Greek version of Euclid was lost in Western Europe for many centuries. Eventually, in the early years of the twelth century, an English monk, Athelhard

of Bath (also known as Adelhard) travelled in Spain, Greece, Asia Minor and Egypt, where he gained a knowledge of Arabic. Probably working from a Spanish source, he translated the Elements from Arabic into Latin. His translation served as the basis for virtually all 15th and 16th century translations of Euclid’s work into the vernacular languages of Europe. The earliest printed edition of the Elements was produced in Venice in 1482, and in the next century, printed editions in most modern European languages appeared. Scholarly attempts to obtain as accurate a version as possible of the Elements began in the 18th century, using manuscript copies of the work available in European libraries (such as the Oxford manuscript). This process culminated in the work of the Danish mathematician Johan Heiberg, who produced a definitive version of the text in parallel Greek and Latin forms, in a book published in 1883.

We mentioned earlier that much of our knowledge of ancient Greek geometry derived from the writing of Proclus. As he plays a significant role in the history of mathematics, we will provide some information about him. Proclus (410-485 CE) was born in Constantinople. He studied in Alexandria and then decided to devote his life to philosophy. He therefore moved to Athens, where he became a member of the revived Platonic Academy there. He eventually became a teacher and then head of the Academy.

Proclus was the last great representative of the philosophical movement known as Neoplatonism, which had begun about two centuries earlier under the influence of Plotinus. Proclus aimed to perfect Neoplatonism by systematizing and extending the views of his predecessors, and showing how all was derived from the teaching of Plato. He collected information about a variety of Greek cultural activities, such as religion, literature, science and philosophy. Many of his writings are lost, but what has survived gives us one of our best sources of information on this last stage of Greek culture, and also on earlier achievements.

Proclus believed that one needed a thorough grounding in logic, mathematics and natural science to understand philosophy. His Commentary on the first book of Euclid’s Elements has been transmitted to us. It was probably written as part of his work in the Academy. He had a detailed knowledge of Greek mathematics from Thales to his own time, and he provides us with one of the best references to Euclid and his predecessors and successors. He supposedly was able to use a history of geometry due to Eudemus of Rhodes. Eudemus lived a little before Euclid, and he provides information about the predecessors of Euclid. Eudemus’s work is now lost, and we must take it on trust that

Proclus had access to this source, seven hundred years after it was written (perhaps a manuscript copy had been retained in the Alexandria until the time of Proclus).

Proclus’s Commentary is tinged with the mysticism of the Neoplatonic school, but it contains worthwhile remarks on the definitions, postulates, axioms and propositions of the first book.

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