Diophantus

diophantus

Summary: Diophantus is described as the beginning of modern algebra as it is known. What of the man and his worth to the world of mathematics?

 Look at his life and work.

 The life of Diophantus seems to be extremely vague in knowing when he lived or even as to his life biography wise. Often the life of an individual is easy to place in some context, but his seems to extend from scarps of information here and there that passed down from generation to generation.

There is some consensus that his life was anywhere from 250 to upwards of 350 AD. An argument could be given that he had some relation to BC around 150, as he writes a letter where he makes references to Hypsicles and his work with polygonal numbers.

The later years of his life being given credence would be based upon words that Theon of Alexandria gives using a definition from Diophantus and produced in 350 AD. Another aspect that places this ancient mathematician in the years before 350 AD from the works of Michael Psellus (11th Century) and is from Thomas Heath’s book Diophantus of Alexandria: A Study in the History of Greek Algebra.

 Diophantus dealt with [Egyptian arithmetic] more accurately, but the very learned Anatolius collected the most essential parts of the doctrine as stated by Diophantus in a different way and in the most succinct form, dedicating his work to Diophantus.

But as other bits of trivia have more bearing on the AD timeline, placing him in 250 AD seems more appropriate based upon the known assumptions. Some of that information for listing 250 AD is that Diophantus worked with a scholar by the name of Anatolius. Their work involved the dealing with “the Egyptian method of reckoning”. Of course, this idea has merit if the Anatolius mentioned is the teacher of mathematics that lived in Laodicea. This writer and teacher would have given conviction to the third century life span of Diophantus.

One scholar, Wilbur Knorr, speculates that the Diophantus being mentioned could be a different individual. He is not so convinced about the Anatolius connection. Knorr surmises that the two did not work together because Anatolius was a master of Greek mathematics and Diophantus’s area of expertise was the math of Egypt.

In referencing the work of Psellus, Knorr states,

But one immediately suspects something is amiss: it seems peculiar that someone would compile an abridgement of another man’s work and then dedicate it to him, while the qualification “in a different way”, in itself vacuous, ought to be redundant, in view of the terms “most essential” and “most succinct”.

In this reflection, an argument can be made that indeed it might have been a separate Diophantus that spent a period of collaboration with Anatolius. Admittedly Knorr’s main disagreement stems from a grammatical application by Psellus; however, that does not preclude the possible accuracy of his assertion either. It comes down to the very thoughts of Knorr who does allude to the idea that more research needed to be considered for a more exact positioning of the lifetime of Diophantus.

Another area of interest in dealing with Diophantus is in considering his age of when he passed away. Thomas Heath in his book, Diophantus of Alexandria: A Study in the History of Greek Algebra, writes about an epigram that was apparently composed in reference to the time of Diophantus’s left span. The Metrodorus composition says,

“Here lies Diophantus.” The wonder behold- Through art algebraic, the stone tells how old: “God gave him his boyhood one-sixth of his life, One-twelfth more as youth while whiskers grew rife; And then yet one-seventh eve marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage Met fate at just half his dad’s final age. Four years yet his studies gave solace from grief; Then leaving scenes earthly he, too found relief.”

 The epigram basically places the mathematician’s age at 84. Through its poetic approach and descriptive flair, Metrodorus left a unique marker for the generations to come about the man.

Diophantus, most common work seems to be his publication called Arithmetica. Since he is described as the founder, or father, of algebra; his work with algebraic solutions is fitting for this book. In the book he works with a group of 130 problems. This collective looked at indeterminate, as well as determinate, equations. In the determinate or realized equations, Diophantus articulates numerical possibilities.

As to the area of indeterminate equations, a way of dealing with finding their solutions was defined as The Diophantine Analysis. The way it worked is that only solutions had to be figured for whole numbers, or integers, only. There could not be a focus on fractions or decimals.

The Arithmetica work appears to have been mostly lost in the annals of history. Arabic translations of the books are the only known pieces to have survived and these come from only six of the original manuscripts. Thomas Heath refers to this in stating,

The missing books were evidently lost at a very early date. Paul Tannery suggests that Hypatia’s commentary extended only to the first six books, and that she left untouched the remaining seven, which, partly as a consequence, were first forgotten and then lost.

The idea that there were 13 books of this work came from Diophantus himself with the first publication.

Some argument also appears to be in the finding another book of the original 13. Someone located in The Holy Shrine library in Iran, a manuscript in Arabic. This manuscript is a supposed translation of Diophantus’s missing books four through seven, thus adding the book after the known six. However, a historian looking at these works came up with a different assumption as to their authenticity. He does not doubt that reflect the work of Diophantus in the message; but felt the work actually was of someone after Diophantus doing a study of his books and was simply making notes of the content. As the historian reflected, the problem was in,

…the great difference between the Greek books of Diophantus’s Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books…

One math scholar, Gow, considers Diophantus’s book to be one solution focused if nothing else. He thought Diophantus’s book was meant to offer numerous and general solutions to algebraic equations. The purpose was to expound upon the idea that each equation should be part of a set of rules that work to make a complete answer and that any equation should fit in the rule. But Gow goes on to theorize that Diophantus actually only offers one solution when others were possible. In his assessment, Diophantus knew his material, but he manipulated the result to match his assumption of the solution. In essence, the mathematician would go as far as he could to simplify the actual equation and work to solve it. He didn’t strive to work the problem as it appeared, but to make it work to his ability to solve it.

Other scholars that studied the work of Diophantus have come with various assessments of Arithmetica. These come from Heath’s work on Diophantus. One man named Nesselman seemed to believe in a closely agreeable position with that of Gow. He surmised Diophantus’s book properly handled dwelling on the solution more than the method of the problem.

Another scholar from Heath’s book, Hankel, deigned to subscribe to Diophantus as being thorough in his approach to the problems. He felt that Diophantus wanted to reach the outcome more than deliver his method how the outcome came about. As he put it as found in Heath’s book,

“…shrewd, clever, quick-sighted, indefatigable, but [who] does not penetrate thoroughly or deeply into the root of the matter. As his problems seem framed in obedience to no obvious scientific necessity, but often only for the sake of the solution, the solution itself also lacks completeness and deeper signification”.

Hankel made no bones that Diophantus was an intelligent person but failed to grasp the need to showcase the how with the what.

Interestingly, another scholar from Heath’s treatise,named Euler, gave more credit to Diophantus then did Hankel. Euler gives credit to Diophantus for making extra work of figuring out the result and delivering to the reader the method involved. Euler reminds everyone that in the time of Diophantus many equations were written out as stories rather than having a means to represent numbers in the equation. He felt Diophantus made attempts at delivering the problem to be figured out by offering letters to represent the unknown aspect of the equation. (This could explain Nesselman’s view of the mathematician just simplifying things.) Euler’s summation renders a clear picture of Diophantus’s approach matching fairly close to contemporary handling of algebra by the use of letters as well. In essence, Diophantus was making a foundation for how algebra could be dealt with that follows modern methods…rather predates modern methods.

Looking at the Books of Arithmetica

Book 1 appears to be a simple approach to looking at algebra. The book contains easily understood examples of how solving algebraic problems could be handled. Basically, this book is an opening call to what is to follow in the other books (realizing only that six books were known to have remained).

As mentioned before about indeterminates (more than one solution is possible), Book II and on follow the deeper work. Also Diophantus placed into the books the concept that equations can be reduced to easier expression and that the result will be a rational realization of the problem. This means the result will extensibly be in the form of a numerical square or even a cube. Remembering that Diophantus did not believe in negative numbers, or zero for that matter; the result must imply a positive, rational numerical entity.

Book II begins illustrating general methods. For example, in Book II, three problems give these details:

  • A square number as the sum of two rational numbers squared.
  • Any given number, not squared, as the sum of two other squared numbers.
  • Two squared numbers offering a difference of any rational number.

His book also gives rise to what an integer number is in his view. He did not prescribe to a belief in negative numbers, so integer use had to never give an allusion to negative possibilities. Although he did not prescribe to the concept, he was willing to acknowledge that different methods could exist which obviously would preclude his concept of below zero integers. The case for number theory.

In Books IV through VI, Diophantus gives the case for deeper degrees of equations having simplified methods for their being solved. His desire is that the reader of his books will practice the methods of their own volition to prove his points. So it seems that teaching methods was a big reason for the concept of the books of this collection. Diophantus was declaring in Arithmetica that understanding and solving algebra equations should not be difficult, but made easier for the reader to practice themselves. Book VI also put into play right-angled triangles. These were looked at falling under certain situations for being handled.

Final Look

Diophantus is described as having written a couple of other works. Much of their content seems to be aspects of his major work, Arithmetica. This book brings an identifiable identity to a man that seems vaporous in extent to his actual life. His work personifies much of his thought; yet obscurity shrouds knowledge of the man himself.

Although the scholars have various interpretations of Diophantus’s work and its importance; there is no denying the legacy of this man Diophantus. Obviously he was a man that had dreams and longings. In the epigram that depicted his age, Diophantus had been married and had been a father. To what extent he was involved in the family is simple conjecture. But in the realm of mathematics, his importance is not minimized.

Maybe the man did not go as involved as some mathematicians would have preferred; but the truth is he did what he knew and sought to be as complete as he knew to be.

Diophantus lived a life that has no data to determine when he lived. But to deny him his place in the world of mathematics is not possible. This “father of algebra” definitely added to the discipline of mathematics.

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