What is Euclid's Elements?
The Elements is a very famous book on mathematics written by the Greek mathematician Euclid living in Alexandria, Egypt around 300 BC.
There are very few books that have had as much impact on the history of mathematics as did Euclid's Elements.
In recent history, the greatest breakthroughs in mathematics are usually first announced in papers published in mathematical journals. (For example, Alan Turing's publication of his famous paper appeared in the Proceedings of the London Mathematical Society.) The earliest scientific journal that is mentioned in Florian Cajori's A History of Mathematics is the French Journal des Savans founded in 1665.
Before that, the greatest mathematical discoveries were shared through correspondences between mathematicians. (For example, the mathematical field of Probability Theory originated in the letters exchanged between Blaise Pascal and Pierre de Fermat in 1654. The letters between these two Frenchmen were written mostly in French, except for those portions where it was easier to more clearly explain some concepts in Latin.)
Of course, throughout all of history there were many books written on the subject of mathematics; at first they were written on parchment or papyrus, and later on paper. Today, you can find hundreds of books on nearly every field of mathematics.
However, in all of history there are only a few books that have had a truly significant impact on the history of mathematics. There may be different opinions as to which books should be included in this list, but here, in chronological order, is my own personal list of the greatest books on mathematics:
The Elements written by Εὐκλείδης (Euclid) around 300 BC in Alexandria, Egypt.
More will be said about this 13-volume book on this webpage below.
Arithmetica written by Διόφαντος ὁ Ἀλεξανδρεύς (Diophantus of Alexandria) during the 3rd century AD in Alexandria, Egypt.
This book is about solving equations and at the same time requiring that the answers be whole numbers. Equations of this type became known as Diophantine Equations.
In the year 1900, the famous mathematician David Hilbert gave an address in which he presented a list of mathematical problems that he felt should be the focus of attention for mathematicians in the twentieth century. Problem 10 on this this list was to devise some sort of method that could determine whether a Diophantine Equation had a solution. This problem became known as the Entscheidungsproblem.
It was the Entscheidungsproblem that was one of the major motivators that led Alan Turing to the invention of the Turing Machine.
الكتاب المختصر في حساب الجبر والمقابلة (Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala) written by محمد بن موسى خوارزمی (Muḥammad ibn Mūsā al-Khwārizmī) around 820 AD in Baghdad.
This was the first book on Algebra.
Algebra was first introduced to Europe by Leonardo Pisano (better known as Fibonacci), who was born in Italy, educated in North Africa, and returned to Pisa around 1200 AD. He also brought Arabic Numbers to Europe, which were much easier to work with than the Roman Numerals that were used in Europe up to this time. (Actually, what we call Arabic Numbers originally came from India.)
De revolutionibus orbium coelestium written by Mikołaj Kopernik (Nicolaus Copernicus) in 1543 in Frombork, Poland.
This was the book that, for the first time, convincingly explained that it was the earth that moved around the sun, rather than the sun around the earth.
It took over 30 years to write the six volumes of this book, and it was published just before Copernicus died in 1543.
Actually, this book was a monumental breakthrough in astronomy, rather than mathematics, but it sure had a heck of a lot of trigonometry in it!
Philosophiæ Naturalis Principia Mathematica written by Sir Isaac Newton in 1687 in Cambridge, England.
This book explains the laws of motion and the laws of universal gravitation, and is therefore primarily viewed as a remarkable achievement in Physics. However, the explanations are done using the Calculus, a new branch of Mathematics that Newton invented specifically for this purpose.
With the invention of the Calculus, advances in science and engineering took off at a rate as never before in history.
Principia Mathematica written by Alfred North Whitehead and Bertrand Russell in 1910 - 1913 in Cambridge, England.
The purpose of this three-volume book was to place mathematics on a firm logical foundation and to eventually perhaps even prove that mathematics is consistent, complete, and decidable. Doing so would answer the questions of the Entscheidungsproblem.
The book itself did not resolve the Entscheidungsproblem. However, it should not be viewed as a failure. In fact, it turned out to be a great success, because it gave Kurt Gödel, in 1931, exactly the tools he needed to prove that if we believe that mathematics is consistent, then it cannot be complete.
The Gödel's Incompleteness Theorem settled part of the Entscheidungsproblem, but not in any way that anyone expected.
The last part of the Entscheidungsproblem was settled in 1936 when Alonzo Church proved that mathematics is also undecidable. Alan Turing came to the same conclusion that same year using his Turing Machine.
The six books listed above are, in my opinion, the greatest books in mathematics. For a second opinion, you can take a look at the list by Robert Talbert. You might also compare it with the list by Mortimer Adler of the world's greatest books as shown on a webpage in Wikipedia.
My guess is that Euclid's book The Elements would appear on anybody's list of the world's greatest books in mathematics.
It seems that Euclid brought together into one book all of the mathematical knowledge that was known in his day. Many people think that this is strictly a book on Geometry, but it contains other mathematics as well. For example, it contains the proof of the fact that the square of two is not rational; in other words, there is no fraction (i.e., there is no ratio of two whole numbers) that can equal the square root of two.
Euclid starts his book by first giving definitions (of point, line, etc.), and then listing Axioms. Axioms are statements that are so self-evident that they are obviously true without proof. Axioms that pertain specifically to Geometry are called Postulates.
Then, using logical rules of inference, Euclid proves theorems based on the Axioms and other Theorems that were already proved.
Some of the Theorems that are proved in The Elements have been known for centuries even before Euclid wrote his famous book. For example, Pythagoras first proved the Pythagorean Theorem sometime in the 6th century BC.
Of course, the Pythagorean Theorem is one of the greatest theorems in the history of mathematics. Special cases of this theorem were known to be true for thousands of years before Pythagoras; for example, the ancient Egyptians "knew" that a triangle with sides of 3, 4, and 5 had two sides that were perpendicular, and used that fact in their architecture and engineering. But it was only after Pythagoras gave a proof of that fact that people began to understand why it was true.
In his 13-Part TV Documentary Ascent of Man, Jacob Bronowski explains that the proof of the Pythagorean Theorem was a great moment in history, not only because it was a significant discovery in mathematics, but more importantly, because it was a great revelation of a new way of thinking. It was truly an important step in the Ascent of Man!
The very fact that, the proof that square root of 2 is not rational, and the proof of the Pythagorean Theorem, both appear in the same book carries a significance of its own.
For thousands of years, man felt that the four basic operations of arithmetic (addition, subtraction, multiplication, and division) were sufficient to express the value of any possible number. If he needed to measure the length of a line, and if it wasn't a whole number of whatever units he used, then all he needed to do was draw enough equal subdivisions on his ruler, and he could always get the precise length that way.
When the ancient Greeks first realized that √2 does not have the value of any fraction, then they strongly felt that something like this simply does not exist.
Oh sure! The √2 may be something interesting to think about, but it couldn't possibly exist because there is no fraction equal to it. It couldn't possibly be the answer to any real-life story problem. It couldn't possibly be the length of any line, because everybody knows that the length of any line can always be expressed as a whole number or a fraction. Right?
However, there is nothing to prevent you from drawing a horizontal line of length 1. Then, there is also nothing to prevent you from drawing a vertical line of length 1 at one endpoint of your horizontal line. Finally, there is nothing to prevent you from connecting the two remaining endpoints with a straight line. So, what is the length of the hypotenuse of the right triangle you just drew?
Well, according to the Pythagorean Theorem, the length of the hypotenuse is √2. But, how can that be if everybody in the whole world knows that √2 is not a number?!
The only way out of this dilemma is to accept the fact that √2 is a real number. There are real numbers out there in this world that cannot be written as whole numbers or fractions!!!
Man started to realize that, by using logic and mathematics alone, he could discover truths that were unimaginable!
The Euclidean Algorithm, a method for finding the Greatest Common Divisor, is also included in Euclid's Elements. This is the first algorithm that was written down anywhere.
The Turing Machine EUC uses the Euclidean Algorithm. EUC is described in Sir Roger Penrose's book The Emperor's New Mind and is also one of the sample Turing Machines in the Excel file that can be downloaded from the Home Page of this website.
The Elements written by Εὐκλείδης (Euclid) around 300 BC in Alexandria, Egypt.
More will be said about this 13-volume book on this webpage below.
Arithmetica written by Διόφαντος ὁ Ἀλεξανδρεύς (Diophantus of Alexandria) during the 3rd century AD in Alexandria, Egypt.
This book is about solving equations and at the same time requiring that the answers be whole numbers. Equations of this type became known as Diophantine Equations.
In the year 1900, the famous mathematician David Hilbert gave an address in which he presented a list of mathematical problems that he felt should be the focus of attention for mathematicians in the twentieth century. Problem 10 on this this list was to devise some sort of method that could determine whether a Diophantine Equation had a solution. This problem became known as the Entscheidungsproblem.
It was the Entscheidungsproblem that was one of the major motivators that led Alan Turing to the invention of the Turing Machine.
الكتاب المختصر في حساب الجبر والمقابلة (Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala) written by محمد بن موسى خوارزمی (Muḥammad ibn Mūsā al-Khwārizmī) around 820 AD in Baghdad.
This was the first book on Algebra.
Algebra was first introduced to Europe by Leonardo Pisano (better known as Fibonacci), who was born in Italy, educated in North Africa, and returned to Pisa around 1200 AD. He also brought Arabic Numbers to Europe, which were much easier to work with than the Roman Numerals that were used in Europe up to this time. (Actually, what we call Arabic Numbers originally came from India.)
De revolutionibus orbium coelestium written by Mikołaj Kopernik (Nicolaus Copernicus) in 1543 in Frombork, Poland.
This was the book that, for the first time, convincingly explained that it was the earth that moved around the sun, rather than the sun around the earth.
It took over 30 years to write the six volumes of this book, and it was published just before Copernicus died in 1543.
Actually, this book was a monumental breakthrough in astronomy, rather than mathematics, but it sure had a heck of a lot of trigonometry in it!
Philosophiæ Naturalis Principia Mathematica written by Sir Isaac Newton in 1687 in Cambridge, England.
This book explains the laws of motion and the laws of universal gravitation, and is therefore primarily viewed as a remarkable achievement in Physics. However, the explanations are done using the Calculus, a new branch of Mathematics that Newton invented specifically for this purpose.
With the invention of the Calculus, advances in science and engineering took off at a rate as never before in history.
Principia Mathematica written by Alfred North Whitehead and Bertrand Russell in 1910 - 1913 in Cambridge, England.
The purpose of this three-volume book was to place mathematics on a firm logical foundation and to eventually perhaps even prove that mathematics is consistent, complete, and decidable. Doing so would answer the questions of the Entscheidungsproblem.
The book itself did not resolve the Entscheidungsproblem. However, it should not be viewed as a failure. In fact, it turned out to be a great success, because it gave Kurt Gödel, in 1931, exactly the tools he needed to prove that if we believe that mathematics is consistent, then it cannot be complete.
The Gödel's Incompleteness Theorem settled part of the Entscheidungsproblem, but not in any way that anyone expected.
The last part of the Entscheidungsproblem was settled in 1936 when Alonzo Church proved that mathematics is also undecidable. Alan Turing came to the same conclusion that same year using his Turing Machine.
For thousands of years, man felt that the four basic operations of arithmetic (addition, subtraction, multiplication, and division) were sufficient to express the value of any possible number. If he needed to measure the length of a line, and if it wasn't a whole number of whatever units he used, then all he needed to do was draw enough equal subdivisions on his ruler, and he could always get the precise length that way.
When the ancient Greeks first realized that √2 does not have the value of any fraction, then they strongly felt that something like this simply does not exist.
Oh sure! The √2 may be something interesting to think about, but it couldn't possibly exist because there is no fraction equal to it. It couldn't possibly be the answer to any real-life story problem. It couldn't possibly be the length of any line, because everybody knows that the length of any line can always be expressed as a whole number or a fraction. Right?
However, there is nothing to prevent you from drawing a horizontal line of length 1. Then, there is also nothing to prevent you from drawing a vertical line of length 1 at one endpoint of your horizontal line. Finally, there is nothing to prevent you from connecting the two remaining endpoints with a straight line. So, what is the length of the hypotenuse of the right triangle you just drew?
Well, according to the Pythagorean Theorem, the length of the hypotenuse is √2. But, how can that be if everybody in the whole world knows that √2 is not a number?!
The only way out of this dilemma is to accept the fact that √2 is a real number. There are real numbers out there in this world that cannot be written as whole numbers or fractions!!!
Version 1.0 -- July 26, 2022