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Mathematical Treasure: Petri’s Reckoning and Cypher Book

Author(s): 
Frank J. Swetz (The Pennsylvania State University)

Nicolaus Petri (fl. 1567-1603) was a Dutch mathematician and astronomer. His Practicque om te leeren reeckenen, cypheren, ende boeckhouwen ... (1635 edition, first published in 1583) was a manual on reckoning, bookkeeping, simple algebra, and both plane and spherical geometry. The title page plate features a portrait of the author designating him as a man of science and learning. The scene is littered with scientific paraphernalia and activity: Petri holds a globe and dividers; a suspended polyhedron, a symbol of proficiency in geometry, hangs above his right shoulder; similarly, the armillary sphere over his left shoulder indicates excellence in astronomy. The motto above his head, “Man proposes, God disposes,” marks his humility. This portrait now resides in the Arthur M. Sackler Museum of Art at Harvard University, Cambridge, Massachusetts.

Title page from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The first page of the table of contents list topics of contemporary importance: the Rule of Three, interest, allegation, etc.

Table of Contents from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The introduction to arithmetic and its operations is rather brief and concise. Numeration and the designation of numbers on a counting table is the initial topic, followed by addition, subtraction, multiplication, division, progressions, and finally a series of rules and applications relevant to merchants.

Table from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

Folio 2 from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

Addition example from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The multiplication algorithm Petri taught is the downward form, which is still popular today.

Multiplication examples from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

More multiplication examples from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

On folio 48, the reader finds a discussion and demonstration of the “Converse Rule of Three.” In modern form, the problem given at the bottom of the first page below can be read as “32 is to 6 as 48 is to x” and x is found to be 9.

Converse Rule of Three example from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

More Converse Rule of Three examples from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

Introducing the concept of a cubic equation on folio 166, Petri constructed such an equation:  \[(x + 2)^3  = (x + 2)(x + 2)(x +2) = x^3 + 6x^2 +12x + 8.\]

Polynomial multiplication example from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The section on geometry is initiated with the discussion of a 9-12-15 (3-4-5) right triangle.

Folio 189 from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

Among the given geometry problems is the situation of an inscribed triangle with sides of lengths 40, 28, and 32, where the diameter of the inscribing circle is required. Note the use of construction lines in obtaining the solution.

Diagram of inscribed triangle from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

A section on measurement employing a quadrant contains the example of finding the depth of a well.

Diagram of well from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The discussion of solid geometry includes a discussion of the triangular pyramid and its truncated form.

Pyramid diagrams from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The latter section of the work concerns spherical geometry.

Spherical geometry from 1635 edition of Practicque om te leeren reeckenen by Nicolaus Petri

The images above were obtained through the courtesy of the University of California Libraries. The work may be viewed in its entirety in the Internet Archive.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Petri’s Reckoning and Cypher Book," Convergence (July 2017)

Dummy View - NOT TO BE DELETED