Eratosthenes Batavus, de terræ ambitus vera quantitate a Willebrordo Snellio, Δια των εξ αποστηματων μετρουσων διοπτρων, suscitatus.

Leiden: George Abrahamsz van Maarssen for Jodocus Colster, 1617.

First edition of Snel’s most important work. This was “the foundation of modern geodesy” (Galileo Project) and of great significance in the history of navigation. “With the development of their oceanic commerce, the Dutch became very interested in the most accurate possible determination of the length of a degree of longitude. A professor of mathematics at the University of Leiden, Snel undertook the task using the method of triangulation first proposed by Gemma Frisius in 1533 but developed by Snel to such an extent that he may rightfully be called one of its founders. He measured the distance between Alkmaar and Bergen-op-Zoom, which lay approximately on the same meridian, and also the distance between the parallels of Alkmaar and Leiden, and from the mean of these two measurements calculated the length of a degree to be 352,347 feet, a more accurate reckoning than any previous attempt. Snel also solved the so-called recession problem for three points, a problem often named after him. The title of his book pays tribute to the Greek mathematician Eratosthenes, who was famous for his measurement of the circumference of the earth” (Norman). “His figure [for the length of a degree] showed the great error in the popular figure of 300,000 feet, which up to then was used by navigators. One navigator, at all events, was quick to see the advantage of using the lately calculated and more accurate figure, and used it upon his voyage to discover the North West Passage, in 1633. This was Captain Thomas James, after whom the bay to the south east of Hudson’s Bay was named. In the account of his ‘strange and dangerous’ voyage, as he described it, he related how, before leaving Bristol, he caused many small glasses to be made, whose part of time he knew with accuracy, and marked off the log like in accordance with Snellius’s measure of feet to one degree” (Hewson, A History of the Practice of Navigation (1963), p. 158).

Snel’s work, ‘The Dutch Eratosthenes, on the true size of the circumference of the earth, recalled from the grave by means of optical instruments according to measured distances,’ aimed to address both local and global issues. In his dedicatory letter to the States General, he pointed out that the question of the size of the earth was a very old one which had occupied many scientists. Moreover, the problem of determining the longitude of a ship at sea was urgent, and he proudly proclaimed his contribution to the solution of this problem: ‘I have tackled a problem the solution of which has always been desired by everyone, which has been tried very often, and which has also been made famous by the achievement of great men. I present here an accurate assessment of the size of the globe.’ This was relevant to navigation because one method of determining the position of a ship at sea was to estimate the distance sailed by the ship from its velocity, to convert this to an angular distance using the known length of a degree, and hence estimate the new position of the ship when the original position was known. The work also had a more local purpose: Snel surveyed a large part of Holland and the surrounding provinces which enabled the States General “beyond doubt, to register their home-country more accurately” than the Greeks, Romans or any other rulers. A final application, not mentioned in the dedicatory letter as it was less relevant for the dedicatees, was probably also important for Snel: the size of the earth was an important parameter in some astronomical calculations, for example, of solar distance.

“Snellius applied a triangulation, that is, a method to survey land by dividing it into triangles. He improved earlier efforts by Gemma Frisius and Tycho Brahe. Gemma Frisius explained the principles of triangulation for the first time in an appendix to his Cosmographicus liber Petri Apiani, published in 1533. The surveyor was to collect the data of the directions of different places from one place by means of a magnetic compass and a large circle, then travel to the next place and repeat the procedures. The distances between these places could be determined by walking and counting the steps. Later in the book, Gemma Frisius proposed to take the angles of the network instead of the directions, draw them on a map and calculate the required distances by using proportions. Thus no trigonometrical functions were used. Snellius would improve the precision of the method by calculating the sides of the triangles in the network by means of trigonometrical functions instead of measuring them on a map. There are no indications that Frisius actually carried out a substantial triangulation.

“The direct inspiration for Snellius’s endeavours was probably Tycho Brahe, who performed a triangulation in Denmark. Snellius knew Tycho personally. The latter used a combination of astronomical observations and angle measurements to interrelate the positions of a number of Danish localities. However, he did not actually calculate these positions. If he had, he would have noticed that his results were not very accurate …

“The Eratosthenes Batavus consists of two books. Its first book is devoted to a historical survey, Snellius shows himself a true humanist scholar here, knowing an extended range of sources and able to use them. He addressed a number of relevant issues, such as the shape of the earth, its location in the universe, and earlier endeavours to measure the earth. Snellius’s most famous predecessor was Eratosthenes of Cyrene (276-194 BC), who had computed the circumference of the earth on the basis of the distance and the difference in latitude between two localities in Egypt almost on the same meridian. Snellius devoted many pages to a precise explanation of this work, including many figures and calculations … Snellius also mentioned Hipparchus and Ptolemy and discussed the work of some Arab scholars and of Jean Fernel” (pp. 118-120).

The first step in Snel’s programme was to establish a unit of measurement (there were no standard units of measurement at the time). Snel proposed to use the Rhenish foot, and he included a picture of half a Rhenish foot in the book. This turned out to be problematic for, as Snel explained to his readers, the size of the paper changed in the process of printing, and indeed after printing the foot turned out to have the wrong measure, which necessitated a correction on the last page of the book.

Snel now used a surveyor’s chain to measure three base lines, two in the fields between Leiden and Zoeterwoude (one on the straight line between the towers of the two places and one perpendicular to it) and one between Wassenaar and Voorschoten (all three villages in the neighbourhood of Leiden). By measuring angles between the lines connecting the end points of the base lines and points in the two places nearby, he could calculate the distance between these points in Leiden and Zoeterwoude, and between those in Wassenaar and Voorschoten. The remainder of his observations were conducted from towers, which enabled him to cover larger distances. Snel thus established a triangulation network connecting Alkmaar and Bergen op Zoom. This was a schematic representation of a number of Dutch towns, interconnected by straight lines representing their distances.

The actual determination of the distances in the triangulation network was a huge task. Snel needed instruments that were both accurate and could survive transport. He used a semi-circle (diameter 32 feet, about 1.33 m) for measuring angular distances between towers and a quadrant of iron with a radius of over 52 feet for determining the polar altitudes. He used a smaller quadrant (25 feet) when determining the position of his base line between Leiden and Zoeterwoude. He made observations in Leiden, Alkmaar, Haarlem, Amsterdam, Utrecht, Gouda, Oudewater, The Hague, Zaltbommel, Breda, Willemstad, Dordrecht and Bergen op Zoom, taking all his measuring instruments with him. Fortunately, he did not have to make all these observations by himself. In 1615, he was accompanied by Erasmus and Casparus Sterrenberg, two young barons, who carried out part of the observations and calculations. Snel dedicated the second book of the Eratosthenes Batavus to them.

On the basis of the measured angles and the length of one side of the network (Leiden − The Hague), the directions and the lengths of all other sides could be calculated by simple trigonometry. He used his previous work to calculate the distance between lkmaar and Bergen op Zoom. He then had to find the geographical latitudes of Alkmaar, Bergen op Zoom and Leiden, which he did by measuring the height of the Pole Star. He also determined the azimuth (the direction in relation to the meridian) from his own house to the Leiden town hall and to The Hague and used it to calculate the azimuth Leiden − The Hague. This was necessary to orient his triangle network and thus to determine the difference in longitude between Alkmaar and Bergen op Zoom.

In connection to this, Snel solved a geometrical problem, called the Resection Problem, which gave him some lasting fame. Snel had to determine the distance of his house to three points in Leiden, the mutual positions of which were known. He considered his solution of the problem to be of no little importance, devoting a separate chapter to it and proudly announced his useful invention for surveying: ‘I have invented an elegant theorem for that problem, which can have a widespread application in our country from now on, because the distances between so many illustrious places have been registered with such precision’.

Snellius now knew the distance between Alkmaar and Bergen op Zoom and their relative positions (latitude and difference in longitude). One more datum was needed to reach the desired result: the value of π. Snel used the approximations of Viète, Romanus and Van Ceulen, which were accurate to many more digits than his measuring accuracy necessitated. Through a long series of calculations he finally arrived at the end of his quest: the length of one degree on the meridian of Alkmaar was 28,500 rods (107.33 km) (one rod equals 12 feet), and therefore the length of a meridian was 10,260,000 rods, about 38,639 km. This is about 3.65% less than the modern value.

Willebrord Snell (1580-1626) was the son of Rudolph Snell (1546-1613), the professor of mathematics at Leiden, whom he succeeded in 1613. In 1609 Willebrord published his first major work, a study of the conic sections of Apollonius of Perga. As a self-conscious citizen of the Dutch Republic, proud of his descent from the Batavans (the inhabitants of the Rhine delta in the Roman period), Snel titled the work Apollonius Batavus. He went on to publish a Latin translation of Simon Stevin’s Wisconstighe Ghedachtenissen, a commentary on Petrus Ramus’s Arithmetica, a treatise on the comet of 1618, and a book on navigation, Tiphys Batavus (1624) in which, among other things, he explained the mathematical theory of the rhumb line (loxodrome), the shortest distance between two points on earth as represented by the Mercator projection as a straight line, a theory lacking in Mercator’s explanation of his world map (Tiphys was the steersman of the Argonauts). Snel is best known to posterity for the discovery of ‘Snell’s law’, according to which the ratio of the sines of the angles of incidence and refraction is a constant. The manuscript in which he formulated this law was never published and is now lost; the law was first published by Descartes in his Discours (1637) (for which some accused Descartes of plagiarism). It now appears that Thomas Harriot preceded Snel in this discovery.

The Latin form Snellius is used by Dutch scholars, and was used by the English until the nineteenth century, when the rendition Snell appeared, and this is still used in the physics literature when referring to “Snell’s law’. Presumably the extra l arose by dropping the Latin ius ending.

Wheeler Gift 55; Honeyman 2864; Norman 1963; Waters, p. 424; DSB XII: 500. De Wreede, Willebrord Snellius (1580-1626): a Humanist Reshaping the Mathematical Sciences, Doctoral Thesis, University of Utrecht, 2007 (

4to (193 x 148 mm), pp. [xii], 263, [1], full-page text engraving, printer’s device on title-page with wreath enclosing quotation ‘O quam contempta res est homo, nisi supra humana se erexerit,’ decorative engraved initials, text in Latin with quotations and examples in Greek, errata on final page. Contemporary vellum, manuscript lettering along spine. Light damp stain to lower inner margin, a fine and unrepaired copy.

Item #5308

Price: $11,000.00