A History of Mathematics From Mesopotamia to Modernity

Understanding the‘ScientificRevolution’ 143

‘algebra’ of the al-Khw ̄arizm ̄i kind and the extraction of square and cube roots were added. The

problems addressed were pseudo-practical and generally solved by methods of false position which

could be traced back to pre-Greek times:

1. A tree is 1/3 and 1/4 underground and above ground it is 30 braccia. I want to know how
   long it is altogether?


2. A man had a denaro and another came to him and he asked, ‘I have one denaro. How much do
   you have?’ And he replied as follows, ‘I have so much that with the same amount and with one
   half of what I have and with a quarter and with your denaro it would be 100.’ How much did
   he have?


3. How much does 87 gold florins 35 s. 6 d. earn in 2 years 7 months and 15 days at 10 per cent
   simple interest?
   [I assume, but I may be wrong, that 12 d. make 1 s. and 20 s. make one gold florin. At any rate, the introduction
   of interest—which the Church condemned, and merchants used various devices to disguise—is a novelty in this
   mathematics, if in other respects it looks rather like the third dynasty of Ur.] (van Egmond 1980, pp. 22–3)

These questions (more are quoted in van Egmond’s book) make clear the new input of merchants’
needs into mathematics; but also (in my view) it was not so much for ‘advanced’ mathematics as
for facility in training. Again the parallel with Ur III comes to mind. The ‘abbacus schools’ have
come recently into prominence as a ‘lowlier’ form of mathematics than that of the universities; but
it may be that claims for their influence on the major subsequent developments are overstated.

Exercise 3.(a) Do questions 1 and 2 by the method of false position. How do you think you should

approach question 3? (b) Assuming 240 pence to the pound, prove the neat calculation rule (from a

problem in BL Add.MS): If the rate of simple interest is x pence per pound per month, then the annual rate

is 5 x per cent (that is, 5 x pounds for every 100).

6. Tartaglia and his friends

Let no man who is not a Mathematician read the elements of my work. (Leonardo 2004, vol. 1, opening admonitions)

It is around 1500 that the various developments sketched so far come together; the dividing line

between university and informal mathematics is, at least to some extent, broken down; and the

whole pattern of change becomes rather complex and difficult to classify. [For example, I shall

omit completely (a) the very important subject of the effect of painting and perspective, which

I recommend you to research if you are at all interested^7 ; see Rotman (1987) and Field (1997);

(b) trigonometry, an import from the Islamic world which was both theoretically and practically

important.] Simplifying, we can trace two major threads: a rapid development in algebra and the

general idea of ‘number’ on the one hand, and (later) the beginnings of a use of the infinitely small.

Both are associated with the continuing problem of the Greek tradition; and in both cases we can

see two important simultaneous and competing developments:

1. An increased familiarity with the works of the Greeks (including Archimedes in particular)
   through translation;


2. To open with a quote from Leonardo might seem, in contrast, to foreground painting; but Leonardo was interested in so many
   other practical pursuits that he can be considered rather as an example of the ‘new model’ of interested artisan.