## Luca Pacioli (1445 - 1517)

Born in a small town in the centre of Italy, Pacioli lead a nomadic life, living in all the great Italian cities of the day and teaching mathematics at a wide range of Universities.

His most famous book on mathematics was published in 1494 and contained a summary of all the known mathematics of the day borrowing freely from Euclid, Boethius and Fibonacci.

The book showed little by way of original ideas but it was to greatly influence those who followed him and provided a basis for the major mathematical advances which were to come in Europe.

Mathematics was rarely formally taught in universities - future mathematicians looked to books such as Pacioli’s for their training and motivation.

The book is also interesting for being the first published account of double-entry bookeeping. This form of accountancy had been used in Venice for a while, but Pacioli was the first to write it down as a systematic system. He is now known as the father of accountancy.

## Pacioli’s mathematics

Pacioli wrote another book which is of interest to us. This was concerned with the ‘golden ratio’. This ratio provides the dimensions of a rectangle that seems to have some sort of fundamental appeal - it provides the focal point of many masterpieces in art, the dimensions of important buildings, and even the ratio between such things as the growth characters of plants and animals.

 The golden ratio is the ratio between the sides of a golden rectangle in which, {b\over{a}}={{\sqrt{5}-1}\over{2}}

There is a connection between this ratio and Fibonacci numbers - it is the ratio between successive terms of the sequence that they aspire to. In turn, the number of petals in a sunflower are Fibonacci numbers as are the dimensions of the spirals found in a Nautilus shell. The golden mean is the common thread that binds together masterpieces in art and architecture, and the way that nature orders growth.

Much used in art and architecture it is likely the book was inspired by Pacioli’s great friend Leonardo da Vinci. The book has high claims to be the best illustrated mathematics book of all time - Leonardo Da Vinci provided the illustrations.

Another of Pacioli’s interests was magic squares - arrays of the first whole numbers that have the same total in any row, column or main diagonal. For example a 3×3 magic square containing the whole numbers, 1,2,3 …9 is:

The magic constant of any nxn magic square - the total of the rows, columns and diagonals sum to - is,

{n(n^2 + 1})\over{2}