The Mathematics of Renaissance Europe

The 15th and 16th centuries in Europe are often referred to as the Renaissance. The word “renaissance” means “rebirth” and describes the renewed interest in intellectual pursuits which characterized the period. This interest focused on literature, architecture, sculpture, painting, and the revival of ancient Greek and Roman cultures. Advances in science and mathematics came along with a growing appreciation for learning in general.

In the 15th century Europeans began to develop the art of navigation and to travel to distant continents, bringing their culture with them. By the end of the 16th century, Jesuit schools had been established all over the world. As a result European mathematics was taught and studied everywhere, and eventually became the dominant form of mathematics worldwide.

As European sailors began traveling to other continents, solving the technical problems of navigation became increasingly important. Successful long-range navigation depended on understanding astronomy and spherical geometry. Astrology was also an important part of the culture of this period, and the making star charts depended on knowing trigonometry. For these reasons, trigonometry was one of the major themes of Renaissance mathematics.

(RG) Which book did for trigonometry what Euclid’s Elements did for geometry? Who wrote it? Most of this book deals with “solving” a triangle. What does it mean to “solve” a triangle?

Proposition 11.1 is an example of a result contained in On Triangles. What does it claim is possible?

Proposition 11.1: Given a triangle with three unequal sides, and the perpendicular to one of the sides, the lengths that the perpendicular divides the base can be determined.

Rule 11.1 gives a procedure for doing what Regiomantus claimed he could do in Proposition 11.1.

Rule 11.1: Multiply the sum of the two sides by the difference of the two sides, and divide by the base. This gives the difference between the segments. Subtract this from the base, or the base from this amount; half the remainder is the shorter side. (DQ) Do section 11.1 exercise 1 using the method Rule 11.1. How can you check that your answer is correct?

The Renaissance also saw a growing interest in arithmetic and algebra. With the rise of the merchant class, more people found that they needed to be able to compute. Since algebra was seen as generalized arithmetic, it was natural for scholars to move from arithmetic to algebra as they delved deeper into their studies.

An important advance in algebra made during the Renaissance period was the general solution of certain cubic equations.

What is a “cubic equation”?

An important advance in algebra made during the Renaissance period was the general solution of certain cubic equations.

A cubic equation has the form ax3 + bx2 + cx + d = 0. Since Renaissance mathematicians were still not comfortable using negative numbers, they would write a cubic equations such as x3 + 2x – 1 = 0 (with a = 1, b = 0, c = 2 and d = –1) as x3 + 2x = 1.

What does it mean to have a “general solution” to an equation?

(RG) Who was Tartaglia? For what is he known?

(RG) Why is Tartaglia’s procedure of solving cubic equations known as “Cardano’s method”?

Rule 11.3 gives Cardano’s method for solving equations of the form “the cube and things equal the number” (x3 + px = q).

(DQ) What formula for x (in terms of p and q) does this rule imply?

Rule 11.3: Cube one third the number of things. To this add the square of half the number, and take the square root of the whole. Work with this twice: first add half the number, and second subtract half the number. This will give you a binomial and an apotome, respectively. Find the cube root of the binomial, and subtract from it the cube root of the apotome, and the result is the value of the thing.

The way Cardano solved the equation x3 + px = q was to write q as a difference of cubes (q = u3 – v3) and p as three times the product of the sides of the cubes (p = 3uv). When he did this, he could use geometry to prove that x was the difference in the sides (x = u – v ).

The way Cardano solved the equation x3 + px = q was to write q as a difference of cubes (q = u3 – v3) and p as three times the product of the sides of the cubes (p = 3uv). When he did this, he could use geometry to prove that x was the difference in the sides (x = u – v ).

What did the types of cubic equations that Cardano could solve have in common?

The way Cardano solved the equation x3 + px = q was to write q as a difference of cubes (q = u3 – v3) and p as three times the product of the sides of the cubes (p = 3uv). When he did this, he could use geometry to prove that x was the difference in the sides (x = u – v ).

What did the types of cubic equations that Cardano could solve have in common?

They didn’t have an x2 term. Cardano solved cubic equations that did have a quadratic term by first converting them to cubic equations that didn’t.

The equation x3 = 6x2 + 100, for example, can be converted into a cubic equation with no quadratic term by making the substitution x = y + 2.

What happens when we make this substitution?

Why did we choose y + 2?

How do we obtain an answer to the original equation from the new one?

An argument started between Cardano and Tartaglia because Cardano wanted to publish the Ferrari’s solution to the fourth degree equation x4 + 6x2 + 36 = 60x.

The solution involves transforming the equation so that both sides are perfect squares. Ferrari did this so that he could take the square root of both sides and be left with a quadratic equation which he could solve.

Along the way a cubic equation also had to be solved. For this reason Cardano needed to publish Tartaglia’s secret method which he had promised never to reveal.

Cardano’s methods often led to solutions that required taking the square root of a negative number. He called such square roots “impossible roots”.

(RQ) How did he deal with them?

(RG) According to the author of our textbook, “the practical value of solutions to the cubic and biquadratic equation was nonexistent.” Why? If there was no practical value in the solutions, then why did people seek (and fight over) them?

Cardano was concerned about general solutions to cubic equations, not because they had practical value, but rather because he was attracted by the challenge of doing something so many people considered difficult, or even impossible.

If he needed the solution to a cubic equation for a practical purpose, he could have used one of the various known methods for finding approximate solutions to cubic equations. Such approximate solutions could be made accurate as desired.

Cardano included two methods for obtaining approximate solutions to cubic equations in The Great Art. Our textbook discusses on such method is now called the “method of secants”. It is based on the technique of double false position.

(RG) Who was Viete? What “revolution in mathematical notation” did he start? Why was it important? (DQ) Rewrite Rule 11.6 using modern notation. Why is this rule labeled the “Quadratic Formula”?

Algebraic notation as we use it today was developed during the Renaissance. Until this point in the history of mathematics, algebra problems were either presented verbally or described using a combination of words and abbreviations (in the manner of Diophantus).

The fact that it took so long to develop algebraic notation is a testament to its conceptual difficulty. Many high school students have a hard time making sense of the symbols used in their algebra classes. Mathematicians did algebra verbally for hundreds of years before the procedures became concrete enough for people to meaningfully describe them using abstract labels. Any concept that look a long time to develop historically is typically difficult for students as well.