Rechnung nach der lenge, auff den Linihen und Feder. Darzu forteil und behendigkeit durch die Proportiones Practica genant. Mit grüntlichem unterricht des visierens.

Leipzig: Jakob Bärwald, 1550.

First edition, rare, of Riese’s last, and most comprehensive, arithmetic text. “Riese (1492-1559), while not the first Rechenmeister to publish an arithmetic book in Germany, was by far the most famous and influential. His works went through at least one hundred editions and were used in schools for over a century. They were the main force behind the replacement of the old methods using the table abacus (auff der Linien) by the new use of the pen (auff Federn)” (Tomash & Williams). “A comprehensive work, which far surpassed his books written at Erfurt, especially in the number of examples. Most of the work had been completed by 1525; but it was not published until 1550, after Elector Maurice of Saxony had advanced the printing costs. Because the expense was so great, the book was reprinted only once, in 1616. It represents the culmination of Riese’s work, and is the best exponent of the practical arithmetic of the middle of the century in Germany” (Smith). ‘Line reckoning’ was a legacy from Roman times: several parallel lines are drawn on a board associated with powers of 10 (1, 10, 100, …); a given number is represented by placing the appropriate number of counters on each line; two numbers are added by amalgamating their counters, moving groups of 10 counters from each row to the one above; subtraction is similar. Printed texts such as the present work clearly demonstrated the superiority of the use of Hindu-Arabic numerals (use of the pen), especially for multiplication and division. The present work “contains material on elementary arithmetic done both on the table abacus and with Hindu-Arabic numerals, but unlike his approach in his other arithmetic book, here he assumes some knowledge of simple operations—for example, he does not bother to give a multiplication table. It contains a section on gauging in which there is a discussion of roots of numbers. Riese’s presentation of the table of roots has often been cited as a precursor to decimal fractions—however, it lacks the use of the decimal point. The title page contains an impressive portrait of a full-bearded Riese” (Tomash-Williams). The book “contains four sections: ‘Auff den Linihen’, on counter reckoning; ‘Auff der Feder’, on the common algorism; ‘Practica’, on solving problems through the use of proportional parts; and ‘Visieren’, on gauging” (Norman). “Riese did more than any previous author to spread knowledge of arithmetic, the branch of mathematics most useful in arts and trade. He was a pioneer in the use of Indian numerals. Riese soon became synonymous with “arithmetic”; to this day, ‘nach Adam Riese’ signifies the accuracy of a calculation” (DSB).

The son of Contz and Eva Riese, Adam (who always signed himself simply ‘Risz’ or ‘Ries’) came from a wealthy family. Little is known about his youth and nothing about his education. In 1509 he was at Zwickau, where his younger brother Conradus was attending the famous Latin school, and in 1515 he was living in Annaberg, a mining town. Ries finally settled at Erfurt in 1518, working there until 1522 or 1523 as a Rechenmeister. He benefited greatly from his contact with the university humanists, who gathered at the house of Georg Sturtz, a rich physician from Annaberg. Ries wrote his first two books while at Erfurt: Rechnung auff der linihen (1518), of which no copy of the first edition is known, and Rechenung auff der linihen vnd federn (1522), which had gone through more than 108 editions by 1656 …

“In 1525 Ries married Anna Leuber, by whom he had eight children. He then purchased his own home and became a citizen of Annaberg. He held important positions in the ducal mining administration: Rezesssschreiber (recorder of mine yields, from 1525), Gegenschreiber (recorder of ownership of mining shares from 1532), and Zehnter auf dem Geyer (calculator of ducal tithes, 1533–1539). While fulfilling his official responsibilities he still found time to continue teaching arithmetic. He ran a highly regarded school, and improved and revised his books. During this period he wrote a comprehensive work, Rechenung nach der lenge, auff den Linihen vnd Feder [the offered work], which far surpassed his books written at Erfurt, especially in the number of examples. Most of the work had been completed by 1525; but it was not published until 1550, after Elector Maurice of Saxony had advanced the printing costs. Because the expense was so great, the book was reprinted only once, in 1616.

“The year 1539 was decisive for Ries. Duke Georg, an intransigent defender of Catholicism, was succeeded by his brother Heinrich, who favored the Lutherans. The change in rulers ended the religious troubles with which Annaberg, like so many other German cities, had been aficted. In the same year Ries received the title ‘Churfürstlich Sächsischer Hofarithmeticus’ … In all his arithmetic books (but with greatest detail in the one of 1550) Ries described how the computations were done, both on the abacus and with the new Indian methods. He employed the rule of three to solve many problems encountered in everyday life. While asserting that he had found ‘proper instruction in only a few places’ in the arithmetic of his predecessors Ries failed to set forth the logical foundations of the subject. Instead, he simply presented formulas with the command ‘Do it this way.’

“Ries did, however, furnish the student with a great number of exercises. The steps to be followed were presented in detail, and the reader could check the correctness of answers by following the procedure used to obtain them. Ries surpassed his predecessors in the presentation of his material: it was clear and orderly, and proceeded methodically from the simpler to the more difficult.

“Besides the section on gauging, the Rechenung nach der lenge contains an extensive section entitled ‘Practica,’ in which Ries solves problems according to the ‘Welsch practice’ through the use of proportional parts [the method originated in northern Italy, residents of which were referred to as ‘Welsch’ by Germans]. In addition he treats problems taken from recreational mathematics, solving them according to the regula falsi. Particularly noteworthy is the fact that in his table of square roots the fractions are repeated in a manner that prepared the way for the use of decimal fractions …

“It is not known how Ries learned Latin. While in the Erzgebirge he gained a thorough knowledge of mining and of mining problems that lend themselves to computation. At Erfurt he obtained the mathematics books of Widman, Köbel, and Grammateus, and he also saw the book from which Widman had taken his examples, which ultimately stem from the Algorismus Ratisbonensis [The earliest practical arithmetic text employing Hindu-Arabic numerals north of the Alps]” (ibid.).

Tomash & Williams R93; Hoock & Jeannin R7.29; Norman 1834; Smith, Rara arithmetica, pp.250-252; USTC 690004; VD16 R2415

4to (207 x 165 mm), ff. [iv], 196 (early, possible contemporary, manuscript annotations and arithmetical calculations to rear end leaves, some soiling). Contemporary blind-stamped pigskin with brass corner-pieces and clasps. A very good, completely genuine copy without restoration.

Item #5199

Price: $25,000.00

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