Standing on the Shoulders of Giants

I had always associated the phrase “If I have seen further, it is by standing on the shoulders of giants” with Sir Isaac Newton. (Letter to Hooke, \(1676\)).  That is, we build our knowledge on previous work done by others and it would be arrogant to claim it all for oneself.

Only ignorance would deny proven work that has gone before.

In fact, the phrase is much older.  Here it is at Chartres Cathedral (circa \(1215\)); Matthew, Mark, Luke and John are standing on the shoulders of Isaiah, Jeremiah, Ezekiel and Daniel.

OldTrout, Going Postal

Or go back to the Ancient Greeks, Cedalion and Orion.

OldTrout, Going Postal


Submission/subjugation (islam) is trying to claim that they invented many things including algebra.  It is true that the etymology of the word algebra is derived from the arabic for restoration (including broken bones). It comes from a text written by Al-Khwarizmi in \(830\, A.D.\)  However, algebra is much older than this text.

What is elementary algebra?  One learns to count, then add, subtract, multiply and divide (arithmetic).  Division lead to fractions and subtraction leads to zero and negative numbers.   Algebra is the next step; how may we quantify unknown numbers?  What is \(x\)?   We begin in school with linear and quadratic equations.

The origins of algebra can be traced back to the Babylonians and Ancient Egyptians, circa \(2,000\, – 1,800\, B.C.\)  They certainly knew about Pythagorean Triples, such as \((3,4,5)\). That is, \(a² + b² = c²\).  The next solution in whole numbers is \((5,12,13)\). They left no proofs and it is debatable as to whether they used algebraic methods or sought numerical solutions via trial and error.  The Plimpton 322 tablet showing a list of Pythagorean Triples : –

OldTrout, Going Postal


The Babylonians counted in base \(60 \,(3\times 4\times 5)\) and used a positional place-number system leaving an empty space for unrepresented numbers. They were able to represent \(√2\) to six decimal places.  The Yale 7289 tablet : –


OldTrout, Going Postal


The Pythagorean school of mathematics (beginning in \(6th\) century \(B.C.\)) treated the study of polygonal numbers and geometry almost as a mystical cult.  So much so, that the tetractys was worshipped and irrational numbers (such as \(√2\)) were resisted.  However, they did provide a rigorous geometric proof of \(a² + b² = c²\)  for the counting numbers.


OldTrout, Going Postal

Diophantus wrote Arithmetica in the \(3rd\) century \(A.D.\) and extends mathematics from numbers and geometry to algebra.  Of the thirteen original books, only six have survived.  Much of the work is devoted to solutions of quadratic equations.  He considers three classes

\(ax² + bx = c\)


\(ax² = bx + c\)


\(ax² + c = bx\)


where the coefficents \((a, b\) and \(c)\) and solutions for \(x\) are positive numbers.

Brahmagupta of India writes the rules for arithmetic for the negative numbers and defines zero as a number in its own right in \(628\, A.D.\)

\(a \,- a = 0\)


He also writes on linear and quadratic equations using these rules and explicitly describes the formula for the quadratic equation.  He has extended mathematics from positive numbers to negative numbers and zero and extended algebra to include them.

Al-Khwarizmi writes his texts on Indian numerals and algebra.  He is described as Persian.  His name derives from a place in current day Uzbekistan, which was a part of Ancient Persia.  He may have been Zoroastrian.  The text on the Indian numerals \(1\, – 9\) and \(0\) and the decimal positional place-number system is all derived from Sanskrit texts.  (Fibonacci introduced the system to Europe in \(1202\)).  In his text on algebra he does not make use of negative numbers or zero; he rejects such solutions.  There has been no extension of the knowledge given to us by Brahmagupta.

They would tell us that he then invents algebra.  He has restored and balanced it!

He stood on the shoulders of giants and saw very little.  We have since progressed far beyond that.

Everything is taken to the left-hand side of the equation and set to zero in the general form.

\(ax² + bx + c = 0\)


Now we may discriminate and do other useful things.

“… the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation (see History of algebra) found in the Greek Arithmetica or in Brahmagupta’s work. Even the numbers were written out in words rather than symbols! “

— Carl B. Boyer, A History of Mathematics

I have made no mention of Chinese mathematics which developed independently.

© OldTrout 2017