KELLY DE ROECK

 

ANCIENT EGYPT

 

The first step towards written numbers was taken in ancient Egypt when tally marks came into use probably 4000-5000 years ago.The priests and scribes took a step further by inventing a system of numerals which varied according to the size of the number. To report a total, they gave the individual numbers, and the number of each in the grand total.

Using these number-signs, the Egyptians could add, subtract, multiply and divide; but they had no special symbols for these operations, instead they gave a form of words describing what had to be done. Hieroglyphics were reserved for formal, official inscriptions (because they were too complex for ordinary purposes), and they make the picture writing we see in royal tombs and on temple walls. Numbers were seldom used in hieroglyphic writing. Scribes used a simplified version of hieroglyphic(pictorial), known as hieratic(symbolic), written in ink on papyrus.

Unfortunately, all that we know about Egyptian mathematics is preserved in, and has to be deduced from, no more than two papyrus rolls, a few fragments of papyrus and a scrap of incised leather. The most important of these rolls, bought by the Scottish antiquary A. Rhind in 1858 in Egypt, was compiled in the 16th century BC by the scribe Ahmes using the hieratic notation.

1.Egyptian numbers

The Rhind papyrus is a rich primary source of ancient Egyptian mathematics, containing 84 worked problems and describing the Egyptian methods of adding - substracting - multiplying and dividing with whole numbers and fractions, the solution of linear equations and the mesuration of simple areas and volumes.

Egyptian numbers were written from right to left. After the fashion of time, Ahmes did not use -or have- any signs for equals, plus or minus,... The Egyptians' system of numeration was based on the scale of ten, but they had no concept of place value. There was a simple repetition within each decade with separate symbols for 1, 10, 100 etc.

Egyptian numbers started at one and went up as far as a million.

In hieroglyphic 1 was symbolished by a paper leaf; 10 was a tie made by bending a leaf; 100 was what looks like a piece of rope; 1000 was a lotus flower. 10000 was a snake; 100000 was a tadpole and 1000000 was a scribe raising both hands above bis head.

But still Egyptian mathematics has long been devalued because it lacks a sign for zero and has no place-system. But the notion that different symbols should be used for different 'levels' of tens made the zero unnecessary and made it possible to write the numerals in any order. In a way, this system was simpler than ours.

2.Tables

Tables (calculation charts) were invented in Babylon, but Egyptian mathematics developped and perfected them in forms that were used and unchanged for millennia. The Rhind papyrus gives clear evidence of a two times table for addition, which could also be used for its complement, substraction.

There was also a table for unit fractions, so that long strings of fractions could be added routinely.

3.Multiplication

The Egyptians had no need to learn the ' times tables' for multiplication or division. The insight, they had, was that any whole number could be made up by adding selected terms from the binary series ; 1, 2, 4, 8, 16, 32, 64, 128, 256,...

Suppose for example, that an Egyptian scribe wanted to multiply 226 by 13. To get 13 we take 1, 4 and 8 (1+4+8=13) from the binary series. Then we multipy them by 256 and then we add them.

This means : 1 x 226 = 226

4 x 226 = 904

8 x 226 = 1808

226 x 13 = (226 x 8) + (226 x 4) + (226 x 1) = 1808 + 904 + 226 = 2938

This looks cumbersome to set out, but Egyptian scribes, used to such calculations, could presumably perform them very quickly, with minimal need to write things down.

4.Division

Perhaps the most dazzling mathematical insight of the Egyptians was that the four arithmetical processes are closely related. The previous calculation, for example, tells us not only that 226 multiplied by 13 equals 2938 but also that 2938 divided by 13 equals 226.

Using the binary series, exactly as for multiplication, an Egyptian scribe seeking to divide 256 by 17 would probably have consulted a division table(this is an official document on papyrus), and then written :

17    256
 1     15     

The modern equivalent would be :

1 x 17 = 17

2 x 17 = 34

4 x 17 = 68

8 x 17 = 136

256 = 136 + 68 + 34 + 17 + 1 and 256/17 = 8 + 4 + 2 + 1 + 1/17 = 15 +1/17

5.Fractions

Egyptian arithmetic was based on the operations of adding, doubling and halving; multiplication was achieved by successive doubling. Two reasons may be suggested for their great emphasis on fractions. First, in a society that did not use money, where transactions were carried out in kind, there was a need for accurate calculations with fractions, particularly in practical problems such as division of food, parcelling out land and mixing different ingredients for beer or bread. Second, division entails operating with fractions, and here we meet a distinctive feature of Egyptian arithmetic : numerical operations were achieved by successive doubling and/or halving. The process of halving in division often led to fractions. All fractions were reduced to sums of unit fractions (fractions of the form 1/n) The only exception was 2/3, which was accorded a special symbol.

When carrying out division, using unit fractions, it is often necessary to double a unit fraction (so to compute 2/n as a sum of unit fractions). No less than a third of the complete Rhind papyrus of a table for expressing fractions of the form 2/(2n+1), for values of n from 2 to 50, as sums of unit fractions, each with different denominator.

Ahmes wrote fractions as single numbers with a dot over the top. The dot was the hieratic equivalent of the hieroglyph for an open mouth referring to the original use of fractions : to divide out shares of food and drink.The modern equivalent of would be 1/5.

In the Rhind we find one of the earliest known algorithms in the history of mathematics : finding two-thirds of the fraction one-fifth. The algorithm of Ahmes says : multiply the denominator of the original fraction by two and write down the result. Then multiply the denominator of the original fraction by six and add the two results.

Following Ahmes' advise : 2/3 x 1/5 = 1/10 + 1/30 = 4/30 = 2/15

In modern notation : 2/3 x 1/5 = (1/2 + 1/6) x 1/5 = 1/10 + 1/30 = 2/15

6. Conclusion

Many of the calculations in the Rhind Papyrus evidently are practice exercices for young students. Although a large proportion of them are of a practical nature, in some places the scribe seems to have had puzzles or mathematical recreations in mind. Thus Problem 79 cites only "seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16807 hekats." It is presumed that the scribe was dealing with a problem, perhaps quite well known, in which in each of seven houses there are seven cats each of which eats seven mice, each of which would have eaten seven ears of grain. The problem evidently called not for the practical answer, which would be the number of houses, cats, mice, ears of spelt, and measures of grain.

Egyptian arithmetic was concerned with the most precise solution, not allowing to round off figures when dividing. Learning how to divide bread and bear as accurately as possible, as an academic exercise, was a useful preparation for more important calculations.

Knowing the algorithms, students could be prepared for really vital problems, such as: how do you place a narrow opening so that twice a year forever(on our 20 October and 20 February) the sun will shine on the face of Ramses II in the royal tomb at Abu Simbel ?

That brings us to another important function of all 'learned' mathematicians in ancient Egypt : calculating and organising the calender. This was linked to the ability to predict the beginning and duration of each of the three seasons recognised : annual flooding of the Nile delta, the period of seed-time and growth and the period of harvest.

Using the system described, the Egyptians could carry out the most extraordinary and complex calculations. Sophisticated calculations were needed to plan and build cities and monumental buildings, which are still a wonder in the modern world.

 

EARLY GREEK MATHEMATICS

 

Of what does the unique Greek contribution to maths consists ? We can identify three elements. The first is an insistence that all mathematical results must be established by deductive reasoning. Secondly the Greeks made mathematics abstract. And the third notable feature of Greek maths was their emphasis on geometry and the use of geometrical methods for solving problems.

The other side of the coin is the Greek failure to develop a symbolic notation of the kind needed to 0make real progress in algebra and in calculus. Nevertheless, the Greeks created first-class maths being inspired and motivated by their desire to understand the nature of reality. They valued and cultivated maths because they believed that the world was designed in accordance with mathematical laws. Most of the leading Greek mathematicians were also astronomers, and many applied their talents in the study of music, optics, mechanics or geography. Mathematics was twice blessed : it was valued both for its own sake and as the key to unlock the secrets of nature.

The Greeks of the classical age used the so-called Attic system of numeration, which resembled the primitive Egyptian number-language. It was eventually superseded by a new system known as the Ionic of Alexandrian. This system was still non- positional, but used 27 different numerals - the 24 letters of the Greek alphabet, supplemented by 3 obsolete letters - to represent the numbers 1,2,3,...,9 ; 10,20,30,...,90 ; 100,200,300,...,900. Intermediate numbers were written by combining 2 or 3 numerals : thus 109 was written as rq. Extra markings were used to denote numbers of 1000 and more. The scheme was reasonably servicable for integers, but quite inadequate for fractions. That is why the Babylonian place-value system was retained by most Hellenistic astronomers.

 

THE AGE OF PLATO

Plato is known to the world as the foremost Greek philosopher(c. 429 - c. 348 B.C.), student of Socrates, and teacher of Aristotle. He founded a school in Athens in a grove of trees named after the owner, Academos. From this name, Plato derived the name of his school, the Academy. The Academy was a center of learning for nearly a thousand years, claiming both commoners and kings as its students.

Plato is important in the history of mathematics largely for his role as inspirer and director of others, and perhaps to him is due the sharp distinction in ancient Greece between arithmetic(in the sense of the theory of numbers) and logic(the technique of computation). Plato regarded logistic as appropriate for the businessman and for the man of war, who "must learn the art of numbers or he will not know how to array his troops." The philosopher, on the other hand, must be an arithmetician "because he has to arise out of the sea of change and lay hold of true being." Moreover, Plato says in the Republic, "arithmetic has a very great and elevating effect, compelling the mind to reason about abstract number." So elevating are Plato's thoughts concerning numbers that they reach the realm of mysticism and apparent fantasy. In the last book of the Republic he refers to a number that he calls "the lord of better and worse births." There has been much speculation concerning this "Platonic number " and one theory is that it is the number

604 = 12960000 - important in Babylonian numerology and possibly transmitted to Plato through the Pythagoreans. In the Laws the number of citizens in the ideal state is given as 5040 ( that is, 1 x 2 x 3 x 4 x 5 x 6 x 7). This sometimes is referred to as the Platonic nuptial number, and various theories have been advanced to suggest what Plato had in mind.

As in arithmetic Plato saw a gulf separating the theoretical and computational aspects, so also in geometry he espoused the cause of pure mathematics as against the materialistic views of the artisan or technician. Plutarch, in his Life of Marcellus, speaks of Plato's indignation at the use of mechanical contrivances in geometry. Apparently Plato regarded such use as "the mere corruption and annihilation of the one good of geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence."

Plato's search for abstract perfection was a dominant theme. He conceived a spiritual world of abstract ideas an ideals - timeless, changeless an indestructible - over and above the imperfect and transitory world of matter as perceived by the senses. Plato believed that the thinking and rigor required by mathematics, particularly geometry, was indispensable to the study of philosophy. To him mathematics belongs, 'par excellence', to the ideal world and so should be studied by all who seek to lead society or to influence their fellow men. Hence the well-known motto commonly held to grace the entrance to Plato's Academy at Athens : 'Let no one ignorant of geometry enter here '.

Consequently, perhaps Plato may have been largely responsible for the prevalent restriction in Greek geometrical constructions to those that can be effected by straightedge and compasses alone.The reason for the limitation is not likely to have been the simplicity of the instruments used in constructing lines and circles, but rather the symmetry of the configurations.

Few specific mathematical contributions are attributed to Plato. A formula for Pythagorean triples (2n)2 + (n2 -1)2 = (n2 +1)2 , where n is any neutral number, bears Plato's name, but this is merely a slightly modified version of a result known to the Babylonians and the Pythagoreans.

 

EUCLID OF ALEXANDRIA

King Ptolemy I once asked Euclid whether there was any shorter way to a knowledge of geometry then by a study of the Elements, whereupon Euclid answered : "Highness, there is no royal road to geometry".

(Proclus Diadochus)

 

1.Author of the Elements

Among his early acts was the etablishment at Alexandria of a school or institute, known as the Museum, second to none in its day. As teachers at the school he called a band of leading scholars, among whom he was the author of the most fabulously successful textbook ever written : " The Elements (Stoichia)" of Euclid. This by far the most influential mathematical work ever written (since the invention of printing more than 1000 editions) is organized in 13 Books and contains no fewer than 467 propositions. In ' The Elements ' the results are presented as a sequence of propositions - either theorems to be proved, or problems to be constructed using straight-edge and compasses only.

Considering the fame of the author and of his best seller, remarkably little is known of Euclid's life. So obscure was his life that no birthplace is associated with his name. Although editions of the Elements often bore the identification of the author as Euclid of Megara and a portrait of Euclid of Megara often appears in histories of mathematics, this is a case of mistaken identity. The real Euclid of Megara was a student of Socrates and, although concerned with logic, was no more attracted to mathematics than was his teacher.Our Euclid, by contrast, is known as Euclid of Alexandria, for he was called there to teach mathematics. From the nature of his work, it is presumed that he had studied with students of Plato, if not at the Academy itself.

Euclid and the Elements are often regarded as synonymous : in reality the man was the author of about a dozen treatises covering widely varying topics, from optics, astronomy, music, and mechanics to a book on the conic sections. With the exception of the Spere of Autolycus surviving works by Euclid are the oldest Greek mathematical treatises extant.

2.Scope of Book I

Most of the propositions in Book I of the Elements are well known to anyone who has had a high school course in geometry. Included are the familiar theorems on congruence of triangles, on simple constructions by straightedge and compasses, on inequalities concerning angles and sides of a triangle, on properties of parallel lines and on parallelograms.It is to Euclid's credit that the Pythagorean theorem is immediately followed by a proof on the converse : if in a triangle the square on one of the sides is equal to the sum of the squares on the other two sides, the angle between these other two sides is a right angle.

3.Geometrical algebra

Book II is a short one, containing only fourteen propositions, is concerned, on the face of it, with geometrical algebra - with 'the application of areas' to prove algebraic identities such as :

a2 = (a - b)(a + b) + b2

We must remember that the Greeks had no symbolic algebra and did not recognize irrational numbers ; everything had to be proved geometrically. A number was thought of as a 'line segment', the product of two numbers as a rectangular area...not one of which plays any role in modern textbooks : yet in Euclid's day this book was of a great significance. The sharp discrepancy between ancient and modern views is easily explained; today we have symbolic algebra and trigonometry that have replaced the geometrical equivalents from Greece. For instance, Proposition I of Book II states that "If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments." This theorem, which asserts that :

AD(AP+PR+RB) = AD x AP + AD x PR + AD x RB is nothing more than a geometrical statement of one of the fundamental laws of arithmetic known today as the distributive law : a(b+c+d) = ab + ac + ad.

4. Number theory

In later Books (V and VI) we move to an entirely new level of mathematical sophistication : it expounds the Eudoxan theory of proportion. Herein we find also demonstrations of the commutative and associative laws for multiplication. Euclid shows how to divide a line in golden section and later moves to the construction of the regular pentagon.

Books VII, VIII, and IX are arithmetical in content, dealing with the properties of the natural numbers and their ratios. Of course the subject is treated geometrically, as is the case throughout the whole Elements. A number is regarded as a line segment and the arguments are presented verbally, with no symbolic aids.

Book IX is a curious mixture. It includes an elementary treatment of properties of odd and even numbers, which almost certainly goes back to the Pythagoreans. By contrast, some of the later propositions are of great importance and most elegantly proved. Thus for example, Prop.IX, 20 gives the well-known proof by contradiction that the number of primes is infinite, i.e. that there is no largest prime.

Books XI, XII and XIII deal with solid geometry.

Remarkable is : sometimes Euclid treats a subject twice over ( f.e. for the so-called Euclidean algorithm : first in Book VIII for numbers and later in Book X for magnitudes ). When the algorithm is applied to two positive integers, the process will always terminate : we eventually reach a remainder of zero. The previous remainder is the largest common factor. But for magnitudes : " if, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left over never measures the one before it, the magnitudes will be incommensurable". But returning to the numerical approach, in the Euclidean algorithm process, the ratio of the two initial numbers is uniquely defined by the finite sequence of quotients and can be encapsulated in a single expression being part of a continued fraction. Further study of the continued fractions leads to irrational numbers. Surprisingly, the simplest periodic continued fraction [1; 1,1,1,...] gives us the golden ratio ! Truncating a continued fraction at successive terms results in a sequence of increasingly accurate rational approximations. Later some mathematicians doubted the emphasis on the Euclidean algorithm as a working tool but the fact that the Greeks were able to compute rational approximations is never contradicted.

 

Bibliography

"The Story of Numbers" by John McLeish

"The Crest of the Peacock" by George G. Joseph

"An introduction to the history of mathematics" by Howard Eves

"Makers of mathematics" by Stuart Hollingdale

"A history of mathematics" by C.B. Boyer and U.C.Merzbach

"Classic MATH : History Topics for the Classroom" by Art Johnson

Back to OLVP page