The rhombicuboctahedron

Dear Ones,

Now that the end of the summer semester is upon us, I thought to share a bit of Mathematics history as I have known it. During my youth, Mathematics was a most popular science. Mathematicians were revered, and renowned in all kingdoms. Read on, I urge you! And share with me your experiences with Mathematics in this modern, busy world of yours!

Algebra in the Renaissance

The general cultural movement of the renaissance in Europe had a profound impact also on the mathematics of the time. Italy was especially impacted.

The Italian merchants of the time traveled widely throughout the East, bringing goods back in hopes of making a profit. They needed little by way of mathematics. Only the elementary needs of finance were required.

• determination of costs
• determination of revenues

After the crusades, the commercial revolution changed this system. New technologies in ship building and safety on the seas allows the single merchant to become a shipping magnate. These sedentary merchants could remain at home and hire others to make the journeys.

This allowed and required them to make deals, and finance capital, arrange letters of credit, create bills of exchange, and make interest calculations.

Double-entry bookkeeping began as a way of tracking the continuous flow of goods and money. The economy of barter was slowly replaced by the economy of money we have today.

Needing more mathematics, they inspired the emergence of a new class of mathematician called abacist, who wrote the texts from which they taught the necessary mathematics to the sons of merchants in schools created for this purpose. There are hundreds of different ones still in existance. (Compare quadrivium (arithmetic, geometry, music, astronomy. Compare trivium: (grammar, rhetoric, and dialectics).

The Italian Abacists

The Italian abacists of the 14th century were instrumental in teaching the merchants the “new” Hindu-Arabic decimal place-value system and the algorithms for using it. There was formidable resistance to this system, in Italy and most of Europe.

These abacists had thoroughly studied arabic mathematics, which emphasized algebraic methods.

In fact, for many years Roman numerals were used to keep account ledgers. The old system of counting boards required the board plus a bag of counters. The new system required only pen and paper. By and by, as with new technologies in general, the superior Hindu-Arabic system won out.

Note. “Believe it or not” ….The decreasing costs and availability of paper was a factor in this.

Mathematical Texts

Mathematical texts were mostly practical, teaching only those problems young merchants would need in carrying out daily transactions. Problems and their solutions were described in detail, with all steps fully described.

Besides the business problems required for their profession there were also recreational problems. There were problems in geometry, elementary number theory, the calendar, and astrology.

The texts did not dwell on problems without a solution. Therefore, some student-teacher interaction would accompany the learning.

During the 14th and 15th centuries the abacist extended the Islamic methods by introducing abbreviations and symbolisms, developing new methods for dealing with complex algebraic problems.

Perhaps most important were the lessons learned in the use of algebra to solve practical problems.

• Example. The gold florin is worth 5 lire, 12 soldi, 6 denarii in Lucca. How much (in terms of gold florins) are 13 soldi, 9 denarii worth. [One was given the relative worth of the soldi, denarii and lire.]
• Example. A field is 150 feet long. A dog stands at one corner and a hare at the other. The dog leaps 9 feet in each leap while the hare leaps 7. In how many feet and leaps with the dog catch the hare. [Assume leaps are made consecutively in the same time.]

Partly because of this practical need for mathematics the new direction of
mathematics was toward algebraic methods.

New Algebraic Techniques

Unlike Islamic algebra, which was entirely rhetorical, the abacists allowed the use of symbols for unknowns. Standard words were:

ß

From Antonio de’ Mazzinhi (1353-1383), known for his cleverness in solving algebraic problems, we have the example. “ Find two numbers such that multiplying one by the other makes 8 and the sum of their squares is 27.”

The solution begins by supposing that the first number is un cosa meno la radice d’alchuna quantità ( a thing minus the root of some quantity) while the second number equals una cosa più la radice d’alchuna quantità (another thing plus the root of some quantity. We have

Answer: . Solve the problem

Higher Degree Equations

Another innovation of the abacists was their extention of the Islamic quadratic solving techniques to higher order equations.

Of course, each text began with the standard six type of quadratics as described by . But many went further.

Maestro Dardi of Pisa in a 1344 work extended this list to 198 types of equations of degree up to four, some involving radicals. He gave an example of how to solve a particular cubic equation, but the methods would not generalize.

Another mathematician of this age was Luca Pacioli (1445-1517), who was reknown for his teaching. Luca Pacioli was born in 1445 in Borgo San Sepolcro, a small Tuscan town and belonged, being the son of Bartholomeus Pacioli, to a middle class family. His first teacher was no less a person than the painter Piero della Francesca, who, typically for Italian Humanism, masterfully connected mathematics, science and art. In 1464 Luca Pacioli became employed as a private teacher by a rich Venetian merchant by the name of Ailtonio de Rompiasi. Together with Rompiasi’s sons he attended the lectures of the mathematician Domenico Bragadino in the Scuolo di Rialto, a school of great importance for the history of Aristotelianism.

Tags: arithmetic geometry, bills of exchange, dialectics, double entry bookkeeping, History of Mathematics, Middle Ages, popular science, rhombicuboctahedron, summer semester