Diophantus is known as the father of algebra. Roughly five centuries after Euclid’s era, he solved hundreds of algebraic equations in his great work Arithmetica, and was the first person to use algebraic notation and symbolism.
Today we usually indicate the unknown quantity in algebraic equations with the letter x. In the oldest copies of Arithmetica the unknown quantity is indicated by a character similar to an accented Greek letter sigma: ς’.
Arithmetica inspired some of the world’s greatest mathematicians including Leonhard Euler and Pierre de Fermat to make significant new discoveries.
The Life of Diophantus
Diophantus (pronounced dy-o-Fant-us) flourished during the third century AD in the Greco-Roman city of Alexandria in Egypt.
Like other educated people in the Eastern Mediterranean at that time he was a Greek speaker. We do not know what he looked like. The years of his birth and death are highly uncertain.
The little we know about Diophantus’ life comes from a word puzzle reputed to be his epitaph. Presumably it was written by a friend who knew Diophantus’ life story and who wished to give him a fittingly algebraic memorial. The epitaph is known to us through the Greek author Metrodorus who recorded it in his anthology of puzzles in about the sixth century.
A 1941 translation of the epitaph by Ivor Thomas says:
This tomb holds Diophantus. Ah, what a marvel.
And the tomb tells scientifically the measure of his life.
God vouchsafed that he should be a boy for the sixth part of his life;
When a twelfth was added, his cheeks acquired a beard;
He kindled for him the light of marriage after a seventh,
And in the fifth year after his marriage He granted him a son.
Alas! late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him.
After consoling his grief by this science of numbers for four years, he reached the end of his life.
We can solve the epitaph as an algebraic equation:
And we find x = 84, from which it follows:
Diophantus’ boyhood lasted 14 years.
When he was 21, his beard grew.
He married at 33.
His son was born when Diophantus was 38.
His son died age 42, when Diophantus was 80.
Diophantus died at age 84.
Lifetimes of Selected Greek Mathematicians
A Brief History of Algebra before Diophantus
Algebra has a long history. Between 2000 – 1600 BC the Babylonians produced rather sophisticated algebra some of which survives on clay tablets. Babylon’s mathematicians were not concerned with exact numerical solutions to problems – they were happy with good approximations from reference tables they compiled. They could solve quadratic equations using geometric drawings of areas and lengths of squares.
The Rhind Mathematical Papyrus dating to about 1550 BC contains Ancient Egyptian algebra, such as: What must the number (1 + ½ + ¼) be multiplied by to give the answer 10?
The Nine Chapters on the Mathematical Art from China was composed in stages over a timespan possibly stretching between 1000 BC – 200 AD. In the eighth chapter, agricultural problems give rise to linear equations solved using rows of numbers similar to matrices.
Modern historians of mathematics sometimes wrangle over Book 2 of Euclid’s Elements from about 300 BC, debating whether it contains algebra written in geometric language. Certainly the great 11th century Persian mathematician Omar Khayyam had no doubts:
“Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by Propositions 5 and 6 of Book 2 of Euclid’s Elements.”
Khayyam was arguing for algebra to be seen as a legitimate branch of mathematics. Many mathematicians felt uneasy about algebra because it lacked the compelling logical rigor Euclid’s Elements had brought to geometry. Khayyam’s work with cubic equations gave him certainty that algebra and geometry are linked.
The Mathematics of Diophantus
Introduction to Arithmetica
Diophantus tells us at the beginning of his classic work Arithmetica that he has written it as a textbook to help his friend Dionysius (and others presumably) to solve mathematics problems. Arithmetica tackles the construction and solution of equations to find one or more unknowns. All copies of Arithmetica in Diophantus’ time were handwritten. Copies were made by scribes for over a thousand years until the first copies were printed in Europe.
“Knowing, my most esteemed friend Dionysius, that you are anxious to learn how to investigate problems in numbers, I have tried, beginning from the foundations on which the science is built up, to set forth to you the nature and power subsisting in numbers.
“Perhaps the subject will appear rather difficult, because is not yet familiar (beginners are, as a rule, too ready to despair of success); but you, with the impulse of your enthusiasm and the benefit of my teaching, will find it easy to master; for eagerness to learn, when seconded by instruction, ensures rapid progress.”
Arithmetica may have been conceived in a similar way to Euclid’s Elements. Euclid compiled and, where necessary, improved on the work of mathematicians such as Eudoxus and the Pythagoreans. Any other books dating from classical times with similar themes to Arithmetica have been lost.
Volumes of Arithmetica
Diophantus composed Arithmetica in thirteen volumes of which six survived in Greek. Four exist as Arabic translations.
- Volumes 1, 2, and 3 survive in Greek from Byzantium.
- Volumes 4, 5, 6, and 7 exist as Arabic translations of Greek from Baghdad. They were translated in the ninth century by the Byzantine Christian scholar Kostas Luka known in Arabic as Qusta ibn Luqa. The books were discovered in 1971 in Meshed, Iran, where they had been misfiled for centuries in the Astan Quds Library as the work of Qusta ibn Luqa rather than Diophantus.
- Three other volumes of Arithmetica exist in Greek, but their volume numbers are uncertain – they could be any three of volumes 8, 9, 10, 11, 12, or 13.
The Arabic translations contain more commentary on the solutions than the Greek versions. It is possible that the Arabic editions were copied from the lost edition adapted by Hypatia for the students at her school.
The Mathematics in Arithmetica
Diophantus begins with definitions and rules. For example, he defines the results of multiplication of quantities with different signs and tells his readers he will indicate subtraction with a symbol. He says:
“A minus multiplied by a minus makes a plus; a minus multiplied by a plus makes a minus; and the sign of a minus is a truncated Ψ turned upside down, thus .”
Book 1, Problem 1
The Problem
In the very first problem in the very first book of Arithmetica Diophantus asks his readers to divide a given number into two numbers that have a given difference.
The number he gives his readers is 100 and the given difference is 40.
The Solution
Diophantus writes [we use modern notation]:
2x + 40 = 100
Hence x = 30
Therefore the two numbers are 30 and 70.
Things Get Harder
Diophantus continues in this way, describing hundreds of problems which he translates into solvable equations. The level of difficulty rises as he introduces quadratics, cubics, and equations in higher powers of x.
Quadratics, Cubics, and More
At high school, we learn about equations of general form ax2 + bx + c = 0; these are quadratic equations.
Cubic equations are of general form ax3 + bx2 + cx + d = 0. Cubic equations are harder to solve than quadratics.
Diophantus ignores negative and irrational solutions to equations.
A major criticism of Diophantus is that he rarely offers his readers general methods to solve a particular class of problem such as quadratics.
The level of mathematical sophistication in Arithmetica is sometimes disconcertingly high. It required the genius of 18th century mathematician Joseph Lagrange to finally prove that every number can be written as the sum of four squares – a result Diophantus was well aware of.
Diophantine Equations
In Arithmetica, Diophantus launched the study of indeterminate equations – these are polynomial equations in which the number of unknowns exceeds the number of equations given.
The solutions to Diophantus’ indeterminate equations were always positive rational numbers. (Diophantus was interested only in single number solutions, so he did not, for example, seek two numbers as solutions to quadratic equations.)
Nowadays we define a Diophantine Equation as an indeterminate equation whose solutions must be integers.
Consider this Diophantine Equation:
If n=2, we have Pythagoras’s theorem, which has an infinite number of whole number solutions, the most famous example of which is the 3-4-5 triangle: x=3, y=4, z=5.
Fermat’s Last Theorem claims that if n is a whole number bigger than 2, the equation has no whole number solutions for x, y and z. Fermat claimed to have proved this for all values of n, but famously said the margin of his book was too small to write his proof. Not surprisingly, the book was Diophantus’ Arithmetica.
After Arithmetica
About four centuries after Diophantus wrote Arithmetica, the great seventh century Indian mathematician Brahmagupta found the general solution for linear and quadratic equations.
After another two centuries, the great Persian mathematician al-Khwarizmi presented systematic solutions of linear and quadratic equations. This was soon followed by Qusta ibn Luqa’s ninth century translation of Arithmetica into Arabic.
Brahmagupta and al-Khwarizmi’s works were less ambitious than Diophantus’ in that they dealt with equations in x and x2. Diophantus frequently dealt with cubic and higher power equations, up to x9. Also, neither of them used the symbolic algebra Diophantus had pioneered. Brahmagupta and al-Khwarizmi’s crucial contribution is the concept of the general solution of an equation. General solutions are not offered in any of the surviving books of Arithmetica.
The fusion of al-Khwarizmi’s highly systematic algebra with Diophantus’ intriguing problems and solutions led to a great flowering of algebra in Persia and the Islamic world. In the eleventh century, Omar Khayyam showed how the intersections of conic sections such as parabolas and circles can yield geometric solutions of cubic equations.
Following the Renaissance, European mathematicians of the highest rank were captivated by the mathematics of Arithmetica.
In 1535, Niccolo Tartaglia found general solutions for all cubic equations.
In the 1600s, Arithmetica inspired a great number of Pierre de Fermat’s new ideas. He worked on Arithmetica for pleasure, much as a modern person might work on a crossword puzzle or a game of Sudoku. When new ideas came to him, he scribbled them in the margin of the book. These ideas, including Fermat’s Last Theorem, transformed number theory.
In the 1700s, about 1,500 years after Diophantus wrote Arithmetica Leonhard Euler took great delight and inspiration from attacking its trickier problems. Euler’s words provide us with a fitting final tribute to Diophantus:
“Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats… Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use today; nay, we are obliged to admit that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Diophantus.”
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Further Reading
Sir Thomas Heath
Diophantus of Alexandria; A study in the history of Greek algebra
Cambridge University Press, 1910
Morris Kline
Mathematical Thought from Ancient to Modern Times
Oxford University Press, New York, 1972
Norbert Schappacher
Diophantus of Alexandria : a Text and its History