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 be said to employ symbols in a systematic way. Leonardo of Pisa possessed no algebraic symbolism. Like the Arabs, he expressed the relations of magnitudes to each other by lines or in words. But in the mathematical writings of the monk Luca Pacioli (also called Lucas de Burgo sepulchri) symbols began to appear. They consisted merely in abbreviations of Italian words, such as p for piu (more), m for meno (less), co for cosa (the thing or unknown quantity). "Our present notation has arisen by almost insensible degrees as convenience suggested different marks of abbreviation to different authors; and that perfect symbolic language which addresses itself solely to the eye, and enables us to take in at a glance the most complicated relations of quantity, is the result of a of small improvements."[23]

We shall now mention a few authors who lived during the thirteenth and fourteenth and the first half of the fifteenth centuries. About the time of Leonardo of Pisa (1200 A.D.), lived the German monk Jordanus Nemorarius, who wrote a once famous work on the properties of numbers (1496), modelled after the arithmetic of Boethius. The most trifling numeral properties are treated with nauseating pedantry and prolixity. A practical arithmetic based on the Hindoo notation was also written by him. John Halifax (Sacro Bosco, died 1256) taught in Paris and made an extract from the Almagest containing only the most elementary parts of that work. This extract was for nearly 400 years a work of great popularity and standard authority. Other prominent writers are Albertus Magnus and George Purbach in Germany, and Roger Bacon in England. It appears that here and there some of our modern ideas were anticipated by writers of the Middle Ages. Thus, Nicole Oresme, a bishop in Normandy (died 1382), first conceived a notation of fractional powers, afterwards re-discovered by Stevinus, and gave rules for operating with them.