De Arithmetische en Geometrische fondamenten … In veele verscheydene constighe question, soo geometrice door linien, als arithmetice door irrationale ghetallen, cock door den regel coss, ende de tafelen sinuum ghesolveert.

Leiden: Joost van Colster and Jacob Marcus, 1615.

First edition of this rare work, containing the best approximation to the value of π achieved at that time. “Van Ceulen was an indefatigable computer and concentrated on the computation of π, sometimes called Ludolph’s number … He became acquainted with Archimedes’ The measurement of the circle … Van Ceulen began to work in the spirit of Archimedes, computing the sides of more regular polygons inscribed within and circumscribed about a circle than Archimedes had and inventing a special short division for such computation … In his Arithmetische en Geometrische fondamenten (1615), published by his widow, he reached thirty-three decimal places, always enclosing π between an upper and a lower limit” (DSB). Originally from Germany, Ceulen (1540-1610) spent most of his life in Holland and was one of the most important Dutch mathematicians of his time. His work on the calculation of π shows him to have been “as expert in trigonometry as his contemporary Viète. In 1595 the two men competed in the solution of a forty-fifth degree equation proposed by Van Roomen in his Ideae mathematicae (1593) and recognized its relation to the expression of sin 45A in terms of sin A” (ibid.). Van Ceulen’s influence continued through his pupil Willebrord Snel. “In his Cyclometricus (1621), [Snel] published Van Ceulen’s final triumph: π to thirty-five decimal places. This was inscribed on his tombstone in the Pieterskerk in Leiden” (ibid.). The present copy of the Arithmetische is a rare variant issue, with a woodcut geometrical device on the title replacing the engraved portrait found in most other copies (no priority established). Snel published a Latin translation of the present work later in 1615, but it was hurriedly printed so as to be ready for the Frankfurt book fair and is full of mistakes. ABPC/RBH lists the sale of four copies of this Dutch first edition since 1981: Van de Wiele, 2013, €1,612; Zisska & Kistner, 2007, €1,955; Zisska & Kistner, 2004, €1,955; Sotheby’s, 2002, £611.

“A few years after Ludolph van Ceulen’s death in 1610, his widow and some other heirs had decided to publish part of his manuscripts, which were written in Dutch. They would indeed appear in 1615 under the title Arithmetische en Geometrische Fondamenten … Although there is no reason to doubt the sincerity of the wish of Adriana Simons, Van Ceulen’s widow, to enhance her deceased husband’s reputation by making his work known, she was certainly driven by financial motives as well. She even had the Fondamenten printed with three different dedicatory letters: to Count Ernest of Nassau and the States of Gelderland, to Count Maurice of Nassau and the States of Holland and West-Friesland, and to the Admiralties of Holland and West-Friesland, apparently determined to gain as much as possible from the book by addressing different groups of potential patrons …

“Ludolph van Ceulen’s De Arithmetische en Geometrische Fondamenten (Arithmetical and Geometrical Foundations) is a rich work, collecting much of the standard fare of the period in the fields of arithmetic and geometry, but also containing some innovations. Among these is an original introduction of the use of numbers in geometry. Other topics include basic arithmetic and calculations with roots (Book 1), a summary of results from the Elements (Book 2), and propositions and problems involving (regular) polygons and circles, such as triangle division problems, the construction of a cyclic quadrilateral, and calculations with in- and circumscribed polygons (Books 3–6). Some of these problems are solved with the aid of numbers, while others are solved using trigonometry or algebra. Van Ceulen often informed his readers about the genesis of a problem and its solution, which gives us a better insight into the mathematical practice of the period” (de Wreede)

One of the most interesting topics treated in van Ceulen’s De Arithmetische was the use of numbers in geometry. This was a controversial subject due to the difficulties that Ceulen and his contemporaries experienced in assigning a number to a line segment or to an area. “In summary, some of the difficulties relevant in this period were:

  1. Numbers were traditionally absent from geometry, which meant that criteria for the exactness of operations involving them had not yet been developed.
  2. There existed no natural candidate for the unit in geometry, that is, the line segment with the same function as the number 1.
  1. In arithmetic, dimensions are absent. For this reason as many numbers as one wishes can be multiplied, and the product is still a number. On the other hand, all geometrical magnitudes have a dimension. The nearest operation to multiplication in arithmetic is rectangle formation in geometry, yet this latter increases the dimension of the constructed object by one every time it is applied. Because an object with more than three dimensions was unthinkable in classical geometry, geometrical magnitudes could not be multiplied without problems of interpretation.
  1. The incommensurability problem. Only if two line segments are commensurable, which means that a magnitude exists of which both are multiples, can both of them be expressed by a rational number. It was not clear how the relationship between incommensurable geometrical magnitudes could be described by numbers.
  1. Good proof techniques were lacking in arithmetic, partly due to the absence of the concept of an indeterminate number.

“Even though these difficulties were seen as obstacles, some mathematicians felt the need to explore both the similarities between arithmetic and geometry and the suitability of algebraic techniques for geometry in their search for an improvement of the methods of geometrical problem solving. Some authors—in particular Adrianus Romanus and François Viète—focused on a general science (called Mathesis Universalis according to the former) of (abstract) magnitude, encompassing both numbers and geometrical magnitudes. These magnitudes were dealt with by means of proportions and algebra … Another approach to bridging the gap between numbers and geometrical magnitudes was chosen by Simon Stevin, [who] extended the domain of numbers fundamentally, claiming that ‘number is that by which the quantity of everything is expressed,’ in this way suggesting that ‘number essentially belongs in the nature of magnitude.’ This … was closer to Van Ceulen’s practice …

“A fine example of Van Ceulen’s … actual use of numbers in geometry is found in the section of the Fundamenta in which [he] introduce[s] the four elementary operations applied to line segments of which the length is expressible in numbers. Numbers are … integers, fractions, square roots, and nested roots, all positive. If a unit line segment is given, line segments of all these lengths are constructible with ruler and compass alone. In this section of the Fondamenten, Van Ceulen explained the reverse; that is, how to construct a unit length on the basis of a line segment with its length given as a number … Van Ceulen had a creative approach to the problem of linking line segments and numbers expressing their measure. This extension of classical geometry demanded some arithmetical ability, because the properties of numbers (e.g., being square) were involved. Some examples will clarify this. He first explained how two line segments had to be added.

Problem 1: Given a line segment AB of which it is given that the length is √28, it is required to construct a line segment AC of length √28 + 4.

“It has to be noted that this problem is not trivial. It would have been if it had been an arithmetical problem; then the sum of the two numbers and 4 would have been √28 + 4. In a practical context, this sum could have been approximated by a decimal fraction by means of a standard algorithm. If two line segments had been given, it would also have been easy to add them geometrically. However, in this case the solution to the problem is a constructed line segment, yet it falls beyond the reach of traditional Euclidean geometry because it involves numbers. The unit that has to be constructed on the basis of the given line segment is defined locally, which means that its length can be different in every problem …

“[The next problem] is the multiplication of two line segments …

Problem 2: Given a line segment a of which it is given that the length is √19 and a line segment b of which it is given that the length is 3, it is required to construct a line segment of length 3√19 …

“This construction contains no problematic aspects. It does not depend on the actual values of the numbers and is therefore more general than Van Ceulen’s construction of addition. This example could even be solved in a simpler way, because 3 is an integer; just joining three copies of would give the answer. However, the formulation of the task deserves a closer scrutiny. Van Ceulen first asked which size a rectangle contained by the two given line segments would have, and then ‘how long that product would be, that is, geometrically to find a line the number of which would be equal to the product.’ He did not state that the product of two line segments was a line segment, but through the detour of the measures of the line segments, and of the areas, he seemed to imply this and Snellius indeed interpreted it in this way. In his formulation, Van Ceulen was not distinguishing explicitly between a product of two numbers, a line segment, and a rectangle, which belong to different categories of magnitudes due to their different numbers of dimensions (resp. 0, 1, and 2)” (ibid.).

This example shows Ceulen to have been well ahead of his time in his willingness to mix quantities of different numbers of dimensions. In his translation, Snel expressed his alarm at “the possible implication of these words by Van Ceulen and disassociated himself publicly from them. In a long note following Van Ceulen’s second example of a multiplication, he wrote that the geometrical operation of forming a rectangle was analogous to the arithmetical operation of multiplication, and application of a parallelogram to a line equivalent to division. However, this analogy certainly did not mean [in Snel’s view] that the domains of geometry and arithmetic overlapped” (ibid.).

Bierens de Haan 839; DSB III: 181; Parkinson p. 56. De Wreede, ‘A dialogue on the use of arithmetic in geometry: Van Ceulen’s and Snellius’s Fundamenta Arithmetica et Geometrica,’ Historia Mathematica 37 (2010), pp. 376-402.



Folio (295 x 197 mm), pp. [iv], 271, [1:blank]. Contemporary flexible vellum. A very good, untouched copy.

Item #3273

Price: $4,850.00

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