Petri Philomeni de Dacia In Algorismum vulgarem Johannis de Sacrobosco commentarius.
By: et praefatus est Maximillianus Curtze.
Contributors: Sacro Bosco, Joannes de, fl. 1230.; Curtze, E. L. W. M. 1837-1903.; YA Pamphlet Collection (Library of Congress)
Publisher: A.F. Host
Published: 1897
Place of publication: Hauniae = Copenhagen
Number of pages: 111
Subjects: Arithmetic -- Early works to 1900
Other titles: In Algorismum vulgarem Johannis de Sacrobosco.
First sentence: Omnia, quae a primaeva rerum origine processerunt, ratione numerorum formata sunt, et quemadmodum sunt, sic cognosci habent :  unde in universa rerum cognitione est ars numerandi cooperativa.
Description: Curtze, a German philologist and Gymnasium teacher, specialized in the history of mathematics. The present volume presents one of the better 13th century manuscripts, relating Johannes de Sacrobosco's famous text on the art of doing numerical calculations using the newly introduced Hindu-Arabic numerals.
The chapters  first begin by describing the new number format, using the digits from 1 to 9, as well as the new number zero, which is called 'cyfra',
and how these 9+1 digits are used within the new positional number system to express any integer, however large.  It then continues to describes the techniques of adding such numbers, subtracting them, multiplying them, dividing them, and finally how to extract their square as well as cubic roots.
 Sacrobosco's text was used as standard text throughout Europe for many centuries, at the various universities, and has been preserved in many libraries as handwritten copies with various dates. 

In the present volume, Curtze takes an extensive commentary  on Sacrobosco's text,  written by the 13th century Swedish monk, Petrus de Dacia,  and compares it with Sacrobosco's text. From this comparison certain conclusions may be drawn about how mathematical subjects were taught at European universities during the Middle Ages. Both manuscripts are here presented in their original Latin form. But Curtze's  introduction is written in German, and contains an extensive bibliography of the various manuscripts that have been preserved. Curtze was a good friend of the famous mathematician Georg Cantor, and Curtze's introduction begins by quoting Cantor's words - how he regards the matter. Curtze (1837-1903) is one of the better known 19th century historians of mathematics. For those who don't read Latin, the book is nevertheless valuable, because of the very informative 17 page introduction, that only presupposes a knowledge of German. Curtze was a graduate of the University of Greifswald in N.E. Germany.
Notes: Includes bibliographical references.
Latin text; introd. in German.
1. German intoduction by Maximillian Curtze = 17 pages.  
2. Iohannis de Sacrobosco algorismus vulgaris = 19 pages. (Latin).   N.B.  This text is also known under the title 'de Arte Numerandi'.  
3. Commentum Magistri Petri Philomeni de Dacia = 69 pages. (Latin).