Abraham bar Hiyya is famed for his book Hibbur ha-Meshihah ve-ha-Tishboret (Treatise on Measurement and Calculation), translated into Latin by Plato of Tivoli as Liber embadorum in 1145. This book is the earliest Arab algebra written in Europe. It contains the complete solution of the general quadratic and is the first text in Europe to give such a solution. Rather strangely, however, 1145 was also the year that al-Khwarizmi's algebra book was translated by Robert of Chester so Abraham bar Hiyya's work was rapidly joined by a second text giving the complete solution to the general quadratic equation.
It is interesting to see the areas of mathematics and the mathematicians with which Abraham was familiar. Of course he knew geometry through the works of Euclid, but he also knew the contributions to geometry from other Greek texts such as Theodosius's Sphaerics in three books, On the Moving Sphere which is a work on the geometry of the sphere by Autolycus, Apollonius's Conics, and the later contributions by Heron of Alexandria and Menelaus of Alexandria. Abraham had also studied some of the important works on algebra by Arab mathematicians, in particular al-Khwarizmi and al-Karaji.
Among other texts written by Abraham bar Hiyya was Yesod ha-Tebunah u-Migdal ha-Emunah (The Foundation of Understanding and the Tower of Faith). This work is an encyclopaedia of mathematics, astronomy, optics and music. It is the first encyclopaedia in the Hebrew language.
Abraham also wrote a number of texts on astronomy; in particular he wrote on the form of the Earth and the calculation of the paths of the stars on the celestial sphere. His book Tables of the Prince refers to the tables of al-Battani while Abraham's treatise Sefer ha-Ibbur (Book of Intercalation), written in 1122-23, is the first Hebrew work devoted exclusively to a study of the calendar.
In the philosophical treatise Hegyon ha-Nefesh ha-Azuva (Meditation of the Sad Soul) Abraham deals with the nature of good and evil and ethics. Megillat ha-Megalleh (Scroll of the Revealer) outlines Abraham's view of history based on astrology. It claims to forecast the messianic future.
Perhaps one of the most important features of Abraham bar Hiyya's work is the fact that it appears to have stimulated an interest in Arabic mathematics and, together with the work of Abraham ibn Ezra, marks the beginning of Hebrew scholarly study of mathematics. As the author of  writes:-
The major part of the mathematical 'classics' in Hebrew were translated from Arabic between the second third of the thirteenth century and the first third of the fourteenth century, within the northern littoral of the western Mediterranean. This movement occurred after the original works by Abraham bar Hiyya and Abraham ibn Ezra became available to a wide readership.It is rather difficult to place Abraham bar Hiyya in the development of mathematics since in most respects he did not fit nicely into one culture but spanned several. It may indeed be for just that reason that he is important since he produced a cross-fertilisation of ideas between these cultures. As Levey (the author of ) writes in , Abraham:-
... did not definitely belong definitely to one mathematical group. He spent most of his life in Barcelona, an area of both Arab and Christian learning, and was active in translating the masterpieces of Arab science. ... he deplored the lack of knowledge of Arab science and language among the people of Provence. He wrote his own works in Hebrew, but he helped translate ... works into Latin....
Article by: J J O'Connor and E F Robertson
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