The origin of geometry

Speaking of the origin of geometry in the present world cycle, Aristotle tells us that the same ideas have repeatedly come to men at various periods of the universe (universe is here refere at the men’s universe or culture or world).

It is not, he goes on to say, in our time or in the time of those known to us that the sciences have first arisen, but they have appeared and again disappeared, and will continue to appear and to disappear, in various cycles, of which the number both past and future is countless.

But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement.

For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person’s land to disappear.

Yet, it should occasion no surprise that the discovery both of this science and of the other sciences proceeded from utility, since everything that is in the process of becoming advances from the imperfect to the perfect.

The progress, then, from sense perception to reason and from reason to understanding is a natural one.

And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated.

The beta of Eratosthenes

Eratosthenes was born in Cyrene which is now in Libya in North Africa. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens.

The library at Alexandria was planned by Ptolemy I Soter and the project came to fruition under his son Ptolemy II Philadelphus. The library was based on copies of the works in the library of Aristotle. Ptolemy II Philadelphus appointed one of Eratosthenes’ teachers Callimachus as the second librarian. When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator. On the death of Callimachus in about 240 BC, Eratosthenes became the third librarian at Alexandria, in the library in a temple of the Muses called the Mouseion. The library is said to have contained hundreds of thousands of papyrus and vellum scrolls.

Despite being a leading all-round scholar, Eratosthenes was considered to fall short of the highest rank:

[Eratosthenes] was, indeed, recognised by his contemporaries as a man of great distinction in all branches of knowledge, though in each subject he just fell short of the highest place. On the latter ground he was called Beta, and another nickname applied to him, Pentathlos, has the same implication, representing as it does an all-round athlete who was not the first runner or wrestler but took the second prize in these contests as well as others.

Certainly this is a harsh nickname to give to a man whose accomplishments in many different areas are remembered today not only as historically important but, remarkably in many cases, still providing a basis for modern scientific methods.

One of the important works of Eratosthenes was Platonicus which dealt with the mathematics which underlie Plato’s philosophy. This work was heavily used by Theon of Smyrna when he wrote Expositio rerum mathematicarum and, although Platonicus is now lost, Theon of Smyrna tells us that Eratosthenes’ work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music.

One rather surprising source of information concerning Eratosthenes is from a forged letter. In his commentary on Proposition 1 of Archimedes’ Sphere and cylinder Book II, Eutocius reproduces a letter reputed to have been written by Eratosthenes to Ptolemy III Euergetes. The letter describes the history of the problem of the duplication of the cube and, in particular, it describes a mechanical device invented by Eratosthenes to find line segments x and y so that, for given segments a and b,

a : x = x : y = y : b.

By the famous result of Hippocrates it was known that solving the problem of finding two mean proportionals between a number and its double was equivalent to solving the problem of duplicating the cube. Although the letter is a forgery, parts of it are taken from Eratosthenes’ own writing. The letter, which occupies an important place in the history of mathematics, is discussed in detail. An original Arabic text of this letter was once kept in the library of the St Joseph University in Beirut. However it has now vanished and the details given in come from photographs taken of the letter before its disappearance.

Other details of what Eratosthenes wrote in Platonicus are given by Theon of Smyrna. In particular he described there the history of the problem of duplicating the cube:

… when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an alter of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.

Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube:

If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging. Do not thou seek to do the difficult business of Archytas’s cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form of lines as is described by the god-fearing Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of means, beginning from a small base. Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands. Thus may it be, and let any one who sees this offering say “This is the gift of Eratosthenes of Cyrene”.

Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the ‘Sieve of Eratosthenes’ which, in modified form, is still an important tool in number theory research. The sieve appears in the Introduction to arithmetic by Nicomedes.

Another book written by Eratosthenes was On means and, although it is now lost, it is mentioned by Pappus as one of the great books of geometry. In the field of geodesy, however, Eratosthenes will always be remembered for his measurements of the Earth.

Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. Details were given in his treatise On the measurement of the Earth which is now lost. However, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo. Eratosthenes compared the noon shadow at midsummer between Syene (now Aswan on the Nile in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia.

Of course how accurate this value is depends on the length of the stadium and scholars have argued over this for a long time. The article [11] discusses the various values scholars have given for the stadium. It is certainly true that Eratosthenes obtained a good result, even a remarkable result if one takes 157.2 metres for the stadium as some have deduced from values given by Pliny. It is less good if 166.7 metres was the value used by Eratosthenes. Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured the tilt of the Earth’s axis with great accuracy obtaining the value of 11/83 of 180°, namely 23° 51′ 15″.

The value 11/83 has fascinated historians of mathematics. Perhaps the most commonly held view is that the value 11/83 is due to Ptolemy and not to Eratosthenes. Some argues that Eratosthenes used 24° and that 11/83 of 180° was a refinement due to Ptolemy. Others agrees with attributing 11/83 to Ptolemy although he believes that Eratosthenes used the value 2/15 of 180°. Eratosthenes made many other major contributions to the progress of science. He worked out a calendar that included leap years, and he laid the foundations of a systematic chronography of the world when he tried to give the dates of literary and political events from the time of the siege of Troy. He is also said to have compiled a star catalogue containing 675 stars.

Eratosthenes made major contributions to geography. He sketched, quite accurately, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He also suggested that lakes were the source of the river. A study of the Nile had been made by many scholars before Eratosthenes and they had attempted to explain the rather strange behaviour of the river, but most like Thales were quite wrong in their explanations. Eratosthenes was the first to give what is essentially the correct answer when he suggested that heavy rains sometimes fell in regions near the source of the river and that these would explain the flooding lower down the river. Another contribution that Eratosthenes made to geography was his description of the region “Eudaimon Arabia”, now the Yemen, as inhabited by four different races. The situation was somewhat more complicated than that proposed by Eratosthenes, but today the names for the races proposed by Eratosthenes, namely Minaeans, Sabaeans, Qatabanians, and Hadramites, are still used.

Eratosthenes writings include the poem Hermes, inspired by astronomy, as well as literary works on the theatre and on ethics which was a favourite topic of the Greeks. Eratosthenes is said to have became blind in old age and it has been claimed that he committed suicide by starvation.

Making combinations

Combinatorics is namely a branch of pure mathematics. In this sense it deals with the study of discrete and often finite objects. And it relates to numerus of other areas of mathematics; such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics.

In one sense combinatorics is the science of counting. Because, some aspects of combinatorics include the “counting” of objects satisfying certain criteria like enumerative combinatorics. It decides when the criteria can be met, and constructing and analyzing objects meeting the criteria as in combinatorial designs and matroid theory, finding “largest”, “smallest”, or “optimal” objects extremal combinatorics and combinatorial optimization, and finding algebraic structures these objects may have, algebraic combinatorics.

This is the area of mathematics which serves  to study families of sets, usually finite, with certain characteristic arrangements of their elements or subsets. And ask what combinations are possible, and how many there are. This includes numerous quite elementary topics, such as enumerating all possible permutations or combinations of a finite set.

Consequently, it is difficult to show all the topics with which a person new to combinatorics might come into contact. Moreover, because of the approachable nature of the subject, combinatorics is often presented with other fields; such as elementary probability, elementary number theory, and so on, to the exclusion of the more significant aspects of the subject.

These include more sophisticated methods of counting sets. For instance, the cardinalities of sequences of sets are often arranged into power series to form the generating functions, which can then be analyzed using techniques of analysis. Since many counting procedures involve the binomial coefficients, it is not surprising to see the hypergeometric functions appear frequently in this regard.

In some cases the enumeration is asymptotic, for instance, the estimates for the number of partitions of an integer. In several cases, the counting can be done in a purely synthetic manner using the “umbral calculus”. Combinatorial arguments determining coefficients can be used to deduce identities among functions, particularly between infinite sums or products, such as some of the famous Ramanujan identities.

A non-enumerative branch of combinatorics is the study of designs. This design theory is properly a study of combinatorial designs.  Sets and their subsets arranged into some particularly symmetric or asymmetric pattern, which are collections of subsets with certain intersection properties. Perhaps most familiar are the Latin squares, arrangements of elements into a rectangular array with no repeats in any row or column. Block designs are combinatorial designs of a special type. This area is one one oldest parts of combinatorics, such as in Kirkman’s schoolgirl problem proposed in 1850.

Also famous is the Fano plane, seven points falling into seven “lines”, each with three points. This suggests that connection exists with finite geometries. Suitably axiomatized, these tend to look like geometries over finite fields, although finite planes are much more flexible. The solution of the problem is a special case of Steiner system, which play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics. Matroids may be viewed as generalized geometries; they are included here as well. Of course graphs themselves are designs, although it is only the most regular graphs which are included in these discussions.

Oh the other hand, A graph is a set V of vertices and a set E of edges —pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing binary relations on sets, which is clearly a broad topic. Graphs are basic objects in combinatorics, though it should be noted that while there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects.  Among the topics of interest are topological properties such as connectivity and planarity, counting problems (how many graphs of a certain type?); coloring problems (recognizing bipartite graphs, the Four-Color Theorem); paths, cycles, and distances in graphs (can one cross the Köningsberg bridges exactly once each?).

There is a significant number of graph-theoretic topics which are the object of complexity studies in computation (e.g. the Traveling Salesman problem, sorting algorithms, the graph-isomorphism problem). The theory also extends to directed, labeled, or multiply-connected graphs. As you can see, the questions range from counting (e.g. the number of graphs on n vertices with k edges) to structural (e.g. which graphs contain Hamiltonian cycles) to algebraic questions (e.g. given a graph G and two numbers x and y, does the Tutte polynomial TG(x,y) have a combinatorial interpretation?).

Finally, it is important to assert that algebraic tools are used in a number of ways in combinatorics. For instance, when incidence matrices can be associated to graphs, or symmetry groups can be associated to block designs, and so on. Particularly common in the study of strongly regular graphs are association schemes.

A particular algebraic topic of interest to combinatorialists is the study of Young tableaux, closely connected to the symmetric groups (enumerating, for example, their representations). Codes (in the sense of coding theory) may be considered part of combinatorics, particularly the construction of nonlinear codes.