Part II: Math Class in the Late Medieval and Renaissance Eras
As you have struggled with a complex math problem, have you ever considered how someone first figured out how to solve it? Who in history came up with the idea of isolating variables to solve equations?
A Grade Ahead challenges our students to find solutions to real-world problems that get increasingly complicated as students move through the grades. For example, our older math students learn to solve quadratic equations, which can be used to calculate everything from the relationships between costs and profits to the trajectories of rockets.
We were curious about the origins of the methods we teach A Grade Ahead students, so we began this series to understand why we do what we do in math class based on all that’s occurred in history. In Part II, we will look at the innovations introduced by late Medieval and Renaissance mathematicians.
Is your child interested in math enrichment? Call or visit an A Grade Ahead Academy near you, or take a free assessment and get started today!
Business and the Opening of the Western Mind
Until around the 1200s, math was for intellectuals and philosophers with few practical applications. Then, Italian merchants began to engage in more complex financial transactions using promissory notes, letters of credit, and bills of exchange. This also meant that they needed to find ways to easily calculate interest rates. These transactions required new and more accurate bookkeeping techniques.
So, mathematicians had to find practical applications for what were once abstract concepts. A Grade Ahead students similarly find solutions to real world problems, especially during May when we prepare them to move A Grade Ahead.
Among the merchants trading in North Africa and along the Mediterranean coast was a young Italian named Leonardo Pisano, known to us as Fibonacci. You may know him from the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, and so on). He studied the ancient Greek and Islamic mathematicians as a young person, and, in 1201, he wrote Liber Abbaci, which means “Book of the Abacus.”
Although this book did not really add to our understanding of algebra or math, it did introduce Hindu-Arabic numerals to Europeans. Those who used them became known as “abacists.” Many of the abacists began to teach the sons of the merchants this “new” math, which made it much easier to work out the new bookkeeping techniques. Our A Grade Ahead students may find it a little odd to call these numbers “new,” since they use them all the time, but they were new to so many Europeans in 1201!
Notating Unknowns
In the fourteenth century (1300s), as the Renaissance began to emerge in Italy, the abacists began to use abbreviations to represent unknown values or just “unknowns.” But until Nicolas Chuquet, a French mathematician, wrote Triparty en la science des nombres (“The Science of Numbers in Three Parts”) in 1484, European understandings of math and algebra remained limited.
Chuquet was the first European to use more flexible notations for exponents and to use negative numbers. He denied that they could be solutions to certain problems, though, which slowed European progress. At this time, mathematicians would accept and work with negative numbers in one context but reject them in another similar context.
In the fifteenth century (1400s), German mathematicians, called “cossists,” also added more symbols and abbreviations to our understanding of math, but there was still no consensus about which symbols to use or how to solve the problems. So, math was still approaching the methods A Grade Ahead teaches our students, but it was not there yet.
Cubic and Quartic Equations and the Appreciation of Complex Numbers
Enter Girolamo Cardano, an Italian physician and writer. In his 1545 work, Ars Magna, or “Great Work,” Cardano relied on Islamic practices, including coming up with one solution for every possible case, and then used Euclid’s ancient propositions in Elements to justify his procedures. He also avoided using symbols in his manipulations of the equations as well as negative numbers for coefficients, continuing Greek traditions of geometric justifications.
In general, Cardano and his students contemplated third-degree equations, like x3 + 2x = 15, and fourth-degree equations, like
2x4 + 2x = 36, but they came up with multiple solutions for each equation using various possible positive numbers only. They might consider a negative number, but they thought of that as a “false” solution. So, while they were able conceive of some general rules for solving these equations, they were not solving them as A Grade Ahead students do today.
Ultimately, Cardano started to think about more complex numbers, including the square root of a negative number, but the idea unsettled him. He considered it “as refined as it was useless.”
François Viète’s Formal Equations
In 1591, a French mathematician named François Viète published an incredibly important work, In artem analyticam isagoge, or “Introduction to the Analytic Art.” This book used vowels to represent unknowns and consonants to represent knowns in equations. Among other things, this allowed mathematicians to more clearly analyze the relationships between the solutions and the values of the coefficients in the original equations. This was the first time mathematicians began to systematically solve equations rather than consider a variety of solutions.
In particular, Viète began to establish rules for solving equations that ultimately laid the foundation for how A Grade Ahead students learn to isolate variables to solve equations today. His new innovations essentially allowed us to manipulate equations, but he still was reluctant to use negative numbers or zeros for coefficients, which meant that multiple solutions were still found for many types of equations.
Simon Stevin’s De Thiende (1585; The Art of Tenths)
Flemish mathematician Simon Stevin published a manual, De Thiende, to teach people how to complete operations with decimal fractions. This was the first time that no distinction was made between numbers and magnitude. Stevin also declared that 1 was a number as are the roots of any numbers. Moreover, he explained that a single idea of number, expressed as a decimal fraction, could be used in a variety of real-world contexts, from land surveying to finances. In spite of this, mathematicians remained limited in their understanding of complex numbers.
Debates continued throughout the 1600s, but in the 1700s a new foundation of understanding emerged in terms of solving equations. This was at the birth of the Enlightenment, which changed how we understand the world around us.
If your child wonders about the world around us, look into A Grade Ahead’s science curriculum! Also, don’t forget that summer is swiftly approaching! See if an A Grade Ahead near you is offering Mathventure Jr., Mathventure, or one of our other Enrichment Camps in your area! For more information, check out this blog post.
What do you think? What is your favorite math topic? Have you studied the history of mathematics? Which mathematician do you find most interesting? We would love to hear from you in the comments! And stay tuned! In the next part of this series, we’ll look at the Scientific Revolution, the Enlightenment, and the birth of calculus.
Author: Susanna Robbins, Teacher and Franchise Assistant at A Grade Ahead
