from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” This problem led to the introduction of the Fibonacci numbers and the Fibonacci sequence, which will be discussed in further detail in section II. Today almost 800 years later there is a journal called the “Fibonacci Quarterly” which is devoted to studying mathematics related to the Fibonacci sequence. In the fourth section of Liber abbaci Leonardo discusses square roots. He utilized rational approximations and geometric constructions. Leonardo produced a second edition of Liber abbaci in 1228 in which he added new information and removed unusable information. Leonardo wrote his second book, Practica geometriae, in 1220. He dedicated this book to Dominicus Hispanus who was among the Holy Roman Emperor Fredrick II’s court. Dominicus had suggested that Fredrick meet Leonardo and challenge him to solve numerous mathematical problems. Leonardo accepted the challenge and solved the problems. He then listed the problems and solutions to the problems in his third book Flos. Practica geometriae consists largely of geometry problems and theorems. The theorems in this book were based on the combination of Euclid’s Book X and Leonard’s commentary, Elements, to Book X. Practica geometriae also included a wealth of information for surveyors such as how to calculate the height of tall objects using similar triangles. Leonardo called the last chapter of Practica geometriae, geometrical subtleties; he described this chapter as follows: “Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces…to complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle...
