We return to *Math Through the Ages: A Gentle History for Teachers and Others* *(Expanded Edition)*, by William Berlinghoff and Fernando Gouvêa, published in 2004 by Oxton House Publishers and the Mathematical Association of America. In this post I’ll continue my synopsis of its thumbnail sketch of the history of math. After that part, the book gives a series of twenty-five deeper sketches about topics from this narrative. I may leave that for the curious, but they may inspire me to post a few times as well.

We’ll take the subject’s development as described by our authors from the fall of Rome, at which original mathematics in Europe stalled, through Medieval times and up to the dawn of the Age of Enlightenment, setting the stage for the advent of calculus.

- The authors continue to emphasize that mathematics is guiding everyday life:

“Of course, throughout this period [400 – 900 A.D.] people were still building, buying, selling, taxing, and surveying, so the subscientific tradition certainly persisted in all of these areas.”

- We now begin an account of Medieval mathematics in the subcontinent of India:

“During this quiet period in Europe and North Africa, the mathematical tradition of India grew and flourished….It is likely that this tradition received some influence from the late Babylonian astronomers, and it is certain that Indian scholars knew some of the Greek astronomical texts. Astronomy was, in fact, one of the main reasons for the study of mathematics in India.”

- There are some names worth knowing here:

“As in the case of Greek mathematics, there are only a small handful of mathematicians whose names we know and whose texts we can study. The earliest of these is Aryabhata, who did his mathematical work early in the 6th century A.D. In the 7th century, the most important mathematicians are Brahmagupta and Bhaskara [I], who were among the first people to recognize and work with negative quantities… Probably the most important mathematician of medieval India was another Bhaskara [II], who lived in the 12th century.”

- Of course, there is the Indian innovation in math that we take utterly for granted:

“The most famous invention of the Indian mathematicians is their decimal numeration system….The history of this momentous step is obscure. It seems likely that there was some influence from China, where a decimal counting board was used. In any case, by the year 600, Indian mathematicians were using a place-value system based on powers of ten. They had also developed methods for doing arithmetic with such numbers….It quickly spread to other countries. A manuscript written in Syria in 662 mentions this new method of calculation. There is evidence that the system was used in Cambodia and in China soon after. By the 9th century, the new system of numeration was known in Baghdad, and from there it was transmitted to Europe.”

- The Indian tradition is also a big step toward our modern conception of the beloved subject of trigonometry:

“Greek astronomers had invented trigonometry to help them describe the motion of planets and stars. The Indian astronomers probably learned this theory from Hipparchus, a predecessor of Ptolemy. Greek trigonometry revolved around the notion of the

*chord*of an angle….It turns out, however, that in many cases the right segment to consider is not the chord, but rather half the chord of twice the angle. So the Indian mathematicians gave this segment its own name. they called it a “half-chord.” This name was mistranslated into Latin (via the Arabic) as “sinus,” giving rise to our modern*sine*of α [alpha].” - Indian mathematics also put mankind on track to discover what we today call algebra:

“Indian mathematicians were also interested in algebra and in some aspects of combinatorics. they had methods for computing square and cube root. They knew how to compute the sum of an arithmetic progression. They handled quadratic equations using essentially the same formula we use today except they expressed it in words….In addition to equations in one variable, the Indian mathematicians studied equations in several variables… Later, Bhaskara II … described a method that will find a solution of

*nx*in whole numbers whenever such a solution exists. Problems of this type are difficult and the Indian achievement in this area is quite impressive.”^{2 }+ b = y^{2}

Obviously, all of this knowledge made its way to Europe, but it was not a direct trip from central India to Western Europe.

- When the Islamic conquest period concluded with the advent of the Abbasid Empire, scholarship flourished for a while in Baghdad.t is important to note that Arabic mathematicians are those who did their work in Arabic. Plenty were not Arab ethnically nor Muslim in religion. Arabic, like Greek before and Latin later, was simply the common language of scholarship:

“[The Abbasid caliphs] founded the

*House of Wisdom*, a kind of academy of science, and began to gather together scholarly manuscripts in Greek and Sanskrit and scholars who could read and understand them….One of the first Greek texts to be translated was, of course, Euclid’s*Elements*. It had a huge impact. Once they had learned and absorbed the Euclidean approach, the Arabic mathematicians adopted it wholeheartedly.” - The Arabic tradition produced some important names and words that we encounter in math and science every day, such as “algorithm” and “algebra”:

“Muhammad Ibn Musa Al-Khwarizmi was one of the earliest Arabic mathematicians to make an enduring name for himself. his name indicates that he was from Khwarizm, a town (currently called Khiva) in what is now Uzbekistan…One [of his books] was an explanation of the decimal place value system for writing numbers and doing arithmetic, which he said came from India….Also by Al-Kwarizmi was the book of ‘al-jabr w’al-muqabala,’ which means something like ‘restoration and compensation.’…When this book was later translated into Latin, ‘al-jabr’ became ‘algebra.'”

- Many mathematicians are actually polymaths — people with multiple intellectual and academic talents — and one of the Arabic mathematicians is a fine example:

“One of the most famous Arabic mathematicians was ‘Umar Al-Khayammi, known in the West as Omar Khayyam. He lived approximately from 1048 to 1131. Nowadays mostly known as a poet, in his day he was also famous as a mathematician, scientist, and philosopher. One of the goals of Al-Khayammi’s algebra book was to find a way to solve equations of degree 3….The challenge he laid down … was to be taken up by the Italian algebraists many centuries later.”

- Many of my students would agree with the Arabic mathematicians on certain points:

“To the Arabic mathematicians, only positive numbers made sense.”

- The Arabic mathematicians were also obsessed with astronomy and trigonometry, but they investigated other areas more deeply:

“Trigonometry was a major concern, mostly because of its applications to astronomy. The work on trigonometry led inevitably to work on approximate solutions of equations. A particularly notable instance of this is a method for approximating the

*n*th root of a number, developed by Al-Kashi in the 14th century. Combinatorics also shows up in the Arabic tradition. They knew at least the first few rows of what we now call “Pascal’s Triangle,’ and they understood both the connection with*(a + b)*andthe combinatorial interpretation of these numbers. Stimulated by Euclid and Diophantus, they also did some work in number theory.”^{n} - As in other places in this thumbnail narrative, the authors emphasize that mathematics is developing in the work of more anonymous people, in this case, artists who were prevented by Islam from depicting the human body:

“Finally, it is important to mention that practical mathematics was also advancing….Buildings were decorated by repetitions of a simple basic motif. This kind of decoration requires some level of forethought, because not all shapes can be repeated in such a way as to cove a plane surface. Deciding what sorts of shapes can be used in this way is really a mathematical question, linked both to the study of plane tilings and the mathematical theory of symmetry.”

At this point, the authors rewind the time machine a few centuries and discuss European scholarship and the transition from the intellectual doldrums of the post-Roman period to a more active Medieval period.

- Our first inklings of a Western mathematical Renaissance come from a decidedly curious French cleric:

“Once [Medieval European] people became interested in mathematics, where could they go to learn more? The obvious thing was to go to places under Islamic control, of which the most accessible was Spain. Gerbert d’Aurillac (945-1003), later to be Pope Sylvester II, is an example.”

- Yes, yes he is: Gerbert was a proto-humanist, three hundred years before his time, not only learning advanced math and astronomy, but also reading Plato and Aristotle at a Spanish monastery.

“Gerbert visited Spain to learn mathematics, then reorganized the cathedral school at Rheims, France. He reintroduced the study of arithmetic and geometry, taught students to use the counting board, and even used the Hindu-Arabic numerals (but not, it seems, the full place-value system).”

- We know that it was during these Medieval times that the Europeans began to establish universities, and these began with what they could resurrect of the Greeks:

“Aristotle’s work did have a great impact, however. His work on the theory of motion led a few scholars at Oxford and paris to think about

*kinematics*, the study of moving objects. Perhaps the greatest of these scholars was Nicole Oresme (1320 – 1382). Oresme worked on the theory of ratios and on several aspects of kinematics, but his most impressive contribution is a graphical method for representing changing quantities that anticipates the modern idea of graphing a function.” - The Italians got a nice jump start on the Renaissance, in my opinion, because a young man learning his father’s mercantile business decided to publish what he learned from Arabic merchants:

“Leonardo of Pisa (c. 1170-1240) was the sone of a trader. Traveling with his father, he learned quite a bit of Arabic mathematics. In his books, Leonardo explained and extended what he had learned. His first book was

*Liber Abbaci*(‘Book of Calculation’). It started by explaining Hindu-Arabic numeration and went on to consider a wide array of problems…” - The Italians kept at the math long after Leonardo of Pisa, whom we know as “Fibonacci,” was gone:

“As the Italian merchants developed their businesses, they had more and more need of calculation. The Italian “abbacists” tried to meet this need by writing books on arithmetic and algebra….The culmination of this tradition was the work of Luca Pacioli (1445-1517), whose

*Summa de Aritmetica, Geometria, Proportione e Proportionalita*was a huge compendium of practical mathematics, from everyday arithmetic to double-entry bookkeeping.”The tradition of the ten-pound math textbook obviously began here.

At this point the authors guide us into the Renaissance of math, in which a primary focus is marrying algebra to geometry to create the algebra content most American students endure (some even learn it, bless them) in secondary school and college, a process that takes us to the Age of Enlightenment (or Age of Calculus, as we call it in the math world).

- Navigation and astronomy are still driving math as we begin this phase, but when we complete it, mathematics will have a momentum separate from these, perhaps for the first time in its history:

“Long-range navigation depends on astronomy and on a good understanding of the geometry of the sphere; this helped propel trigonometry to the center of attention….In parallel with the intense study of navigation, astronomy, and trigonometry, there was also growing interest in arithmetic and algebra. with the rise of the merchant class, more people found that they needed to be able to compute.”

- Math begins to take more familiar forms as this plays out:

“A great many new ideas were introduced into trigonometry at this time. The list of trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) became standardized. New formulas and new applications were discovered.”

- New techniques in art demanded more mathematical prowess as well:

“Somewhat related to all of this was the discovery of perspective by Italian artists. Figuring out how to draw a picture that gave the impression of three-dimensionality was quite difficult. the rules for how to do it have real mathematical content….Some [Renaissance artists], such as Albrecht Dürer, were quite sophisticated in their understanding of the geometry involved. In fact, Dürer wrote the first printed work dealing with higher plane curves, and his investigation of perspective and proportion is reflected both in his paintings and in the artistic work of his contemporaries.”

- And now, the Italians are ready to take on the challenge of Omar Khayyam: the general solution to the cubic equation:

“The crucial breakthrough was made in Italy, first by Scipione del Ferro (1465-1526) and then by Tartaglia (1500-1557)….Both men kept their solutions secret, because at this time scholars were mostly supported by rich patrons and had to earn their jobs by defeating other scholars in public competitions.”

- No Internet? No problem. You just need someone who decides it’s okay to share information and put in in a printed book (the Internet of the Renaissance):

“In the case of the cubic, this pattern was broken by Girolama Cardano (1501-1576)….Cardano was able to generalize [Tartaglia’s method] to a way of solving any cubic equation. Feeling that he had actually made a contribution of his own, Cardano decided that he was no longer bound by his promise of secrecy [to Tartaglia]. He wrote a book called

*Ars Magna,*“The Great Art,” [my note: we mathematicians are not humble people]. This scholarsly treatise (written in Latin) gave a complete account, with elaborate geometric proofs, of how to solve cubic equations.” - Remember, all this time mathematicians are managing the intellectual complexity of communicating math by writing it out using words. There were no variables, until:

“Algebra began to look more like it does today towards the end of the 16th century, in the hands of Francois Viète (1540-1603). Among many other things, Viète worked for the French court as a cryptographer, a code-breaker who deciphered intercepted secret messages. This may be what led him to one of his most important innovations; the notion that one could use letters to stand for numbers in equations.”

- Two mathematicians capped this process of making algebra a mature mathematical science and setting the stage for the emergence of calculus, Rene Descartes and Pierre de Fermat:

“Three innovations from this period were to be extremely important. First, the fact that no one could figure out how to solve the general quintic (fifth-degree) equation led algebraists to start asking deeper questions. Slowly, a theory about polynomials and their roots evolved. Second, Descartes and Pierre de Fermat (1601-1665) linked algebra and geometry, inventing what we now call “coordinate geometry.”… Third, Fermat introduced a whole new category of algebraic problems. These were related to the work of the Greek mathematician Diophantus, but went far beyond his work. Specifically, Fermat began asking “questions about numbers,” by which me meant

*whole*numbers…. Unfortunately, for a long time, Fermat was alone in finding these questions interesting.”I feel your pain, Pete. I really do.

Well, that takes us from the Fall of Rome to the Dawn of Calculus, an intellectual journey of more than twelve centuries.

The next installment will wrap up my synopsis of the authors’ historical narrative. I will try to summarize how we got from high school algebra to an age of magic, where mathematics only a very curious elite understands makes our world work. The rest are at least dependent on that technology and, in too many cases, addicted to it. But that is another story in itself, I suppose.