Gemma Frisius's Arithmeticae Methodus Facilis

An example of the use of double false position to solve a problem in two unknowns found in the Arithmeticae Practicae Methodus Facilis (1540), by Gemma Frisius (originally Regnier Gemma) (1508-1555).  Gemma Frisius was best known for his work in astronomy and map-making; he worked closely with Gerardus Mercator in making an early globe.  He also suggested a method for determining longitude at sea.  


Gemma Frisius's Arithmeticae Methodus Facilis

Antichissimo di Algorismo

One of two illustrations from the fourteenth century Italian codex,  Antichissimo di Algorismo. This is one of many algorisms written at this time. They were arithmetics designed to introduce the Hindu-Arabic numerals, their operational algorithms and demonstrate their use in problem solving. The majority of the problems considered in this codex are commercial in nature. A few might be categorized as “recreational problems”. A special feature of this codex is that it contains 42 illustrations, many of which supplement problems. 

The illustration on folio 60 presents the situation where three couples wish to cross a stream. The small boat they have will only accommodate two persons at a time. How can they all get to the other shore if no man is to cross with another’s wife? This is a variation of the  puzzle-type “River Crossing Problem” that has been posed over the centuries in many guises. 


Antichissimo di Algorismo

German protractor

Brass protractor from about 1700 of German manufacture. Its base plate contains some Baroque decoration. Note its similarity to a present day student protractor. 


German protractor

Simon Jacob's Rechenbuch

This is the title page of the "New and Fully Revised" Rechenbuch of Simon Jacob (d. 1564), one of the best-known Rechenmeisters of the sixteenth century.  The book was first published in 1560, but this illustration is from the 1565 edition.  


Simon Jacob's Rechenbuch

English tally sticks

Notched pieces of wood or bone were used by many ancient peoples to record numbers. The most common type of these “tally sticks” was made of wood. Tally sticks served as records or receipts for financial transactions such as the payment of taxes, debts and fines. From the 12th century onward tally sticks were officially employed by the Exchequer of England to collect the King’s taxes. Local sheriffs were given the task of actually collecting the taxes. The depth and series of notches on these sticks represented the value of the transaction. In recording a debt, wooden sticks were often split horizontally into two parts: the lender receiving one part, the stock; and the debtor, the other part, the foil. This box contains sticks that date from the year 1296 and were found in the Chapel of the Pyx, Westminster Abbey in 1808. England abolished the use of tally sticks in  1826.The accumulation of tally sticks in the Office of the Exchequer were burned in 1834 resulting in a fire that destroyed the Parliament Building. 

English tally sticks