There are two things which research has discovered which are controversial. One is a marginal note written on a manuscript which states:-
This is enough to say for the instruction of the students of Toulouse.This was, for quite some time, thought to have been written by Jordanus so proving that he taught at Toulouse. However Thomson, in , discounts the manuscript as being written by Jordanus and in this case there is no reason to associate him with Toulouse.
The second controversy among scholars regarding Jordanus is whether Jordanus de Nemore and Jordanus de Saxonia are the same person. It is known that Jordanus de Saxonia was the first successor to Saint Dominic as the Grand Master of the Dominicans. Jordanus de Saxonia:-
... was renowned in Paris for his secular knowledge particularly in mathematics and, it is said, wrote two very useful books ...This is fair evidence that the two Jordanus's are the same person but it is far from conclusive. Hughes (see for example ) does not believe that the two are the same.
What we do know of Jordanus is his works, many of which have survived. These show that he was someone of considerable importance in the development of mathematics and science. Hughes writes :-
Jordanus de Nemore was, and is, recognised as one of the most prestigious natural philosophers of the thirteenth century. His activities encompassed the field of mathematical physics. In particular, he laid the foundation for the entire area of medieval statics. At a more elementary level, his mathematical works on arithmetic, both logistic and specious, and algebra were copied and printed many times, well into the sixteenth century.Jordanus was the first to correctly formulate the law of the inclined plane. He wrote several books on arithmetic, algebra, geometry and astronomy. He also used letters to replace numbers and was able to state general algebraic theorems but this early use of algebraic notation was not used by subsequent writers. Let us look in more details at his mathematical work.
There are six mathematical treatises written by Jordanus. The Demonstratio de algorismo gives a practical explanation of the Arabic number system. It deals only with integers and their uses while, on the other hand, the text Demonstratio de minutiis deals with fractions. A theoretical work on arithmetic which became the standard source of Middle Ages texts is De elementis arithmeticae artis . Geometry is developed in the work Liber phylotegni de triangulis which is an excellent example of a Middle Ages Latin geometry text. The Demonstratio de plana spera is a specialised work on geometry which studies stereographic projection. Perhaps the most impressive of all is the De numeris datis which is the first advanced algebra to be written in Europe after Diophantus. It :-
... is recognised as the first advanced algebra composed in western Europe.In De numeris datis Jordanus gives results on solving quadratic equations similar to those given by al-Khwarizmi except general forms are given rather than the numerical examples of the earlier text. The proofs given by Jordanus, like those of al-Khwarizmi, are by the method of completing the square. Let us give an example of one of the problems (using the translation given in ):-
If a given number is separated into two parts such that the product of the parts is known, then each of the parts can be found.The solution illustrates the use of letters by Jordanus:-
Let the given number a be separated into x and y so that the product of x and y is given as b. Moreover, let the square of the sum of x and y be e, and the quadruple of b be f. Subtract this from e to get g, which will then be the square of the difference of x and y. Take the square root of g, call it h. Then h is also the difference of x and y. Since h is known, then x and y can be found.What Jordanus has done, of course, is to use the fact that (x - y)2 = (x + y)2 - 4xy. After the general result he then gives a numerical example of the method he has just explained:-
The mechanics of this is easily done. For example, separate 10 into two numbers whose product is 21. The quadruple of this is 84, which subtracted from the square of 10, namely 100, yields 16. Now 4 is the root of this and also the difference of the two parts. Subtracting this from 10 to get 6, which halved yields 3, the lesser part; and the greater is 7.In astronomy Jordanus used letters to denote the magnitudes of stars (not unrelated to his use of letters for algebraic notation). He wrote a treatise on mathematical astronomy called Planisphaerium as well as Tractatus de Sphaera. On statics he wrote De ratione ponderis which contains results such as:-
If the arms of a balance are proportional to the weights suspended, in such manner that the heavier weight is suspended from the shorter arm, the weights will have equal positional gravity.We can compare this result on statics with a result from De numeris datis which illustrates a possible motivation for the algebraic results in the latter text:-
If the ratio of the two parts of a given number is known, then each of them can be found.Jordanus visited the Holy Land and, on the return journey, he lost his life at sea.
Article by: J J O'Connor and E F Robertson