How to be a Surveyor

The Dutch made use of every bit of land that they were able to reclaim from the water. Because the owners of this land all had to cooperate with each other to keep the interconnected water drainage system working, they could not be distracted by petty boundary disputes. The Dutch were at the same time developing a strong legal system to enforce deeds and contracts.

The property system relied on the accuracy of the surveyors' observations and the validity of their mathematical calculations. They needed skills in arithmetic and geometry that were not taught at universities. Surveyors (landmeters) had to be admitted to practice by the Court of Holland after their skills were certified by oral examination. Candidates were self-taught, but they may have apprenticed with an admitted surveyor.

The image on the right comes from the cover of later editions of Practijck des lantmetens (Practice of Surveying) published in 1600 by two surveyors, J. Sems and J. P. Dou. It seems to have been the most prominent of such textbooks, and was perhaps studied by Leeuwenhoek.

The image below left is the cover from the first edition of Practijck des lantmetens. The other two illustrations come from the text. Click on thumbnails for larger sizes.

The instruments used in Leeuwenhoek's time had been used for hundreds of years:
  • a rope or cord or (after 1500) a chain, made of short links, to measure distances on the ground (Note the chain in the lower left corner of the leftmost image above.)
  • a level, also called a surveyor's cross, to determine height or elevation differences
  • a theodolite, set on a tripod, to measure angles

These same instruments were used until the end of the 20th century. The instruments were not perfect, but they were good enough for the legal system at the time.

Until the 16th century, land surveyors commonly used a dioptra like the drawing of Heron of Alexandria's (below left), who lived at the same time as Jesus of Nazareth. Usually on a tripod, it pointed at a distant object and then measured the horizontal and vertical angles. A series of these measurements from different locations created topographic precision that helped build the Roman aqueducts across valleys and through mountains.

The pointing was done by the unaided human eye until the English surveyor and author Leonard Digges (1520 - 1559) mounted a sighting device on one of these dioptra and made what he called a theodolite (below right).

While one lens had been used since the 1300's to correct human vision, it was not until the 1500's that people like Leonard Digges (c.1515–c.1559) realized the power of two lenses working together. In the surveying textbook Pantometria (1591, published posthumously by his son Sir Thomas), Digges referred to his "topographical instrument" as "perspective glasses" and "proportional glasses". Sir Thomas's claim that the glasses let his father see tiny details at a distance of seven miles is probably not true. Leonard may have made a rudimentary reflecting telescope that didn't work very well but demonstrated the principle.

The illustration on the left from Pantometria shows the practical uses of Digges' techniques, especially for warfare. Click for larger size.

It wasn't long before the spyglass greatly extended the range and accuracy of finger pointing, and many early surveyors added a compass, too.


Instruction books

Practijck des lantmetens

The principles of the arithmetic and the examples of the practical applications would have been available in several books that Leeuwenhoek could have studied, foremost among them Sems and Dou's, Practijck des lantmetens. Dou published another book in 1612, Tractaet vant maken ende Gebruycken eens nieu gheordonneer­den Mathematischen Instruments (Tract for making and using newly decreed mathematical instruments). In it, he introduced his newly invented instrument (below right) that was sometimes called the circle of Dou or more commonly the Holland circle, which became standard equipment for surveyors for the next century.

Manuale arithmeticae & geometrice practicae

 Another popular book was Metius's Manuale arithmeticae & geometrice practicae (Manual of arithmetical and geometric practice), 1633.

Its subtitle:

Manuale arithmeticae & geometrice practicae: in het welcke beneffens de stock-rekeninge ofte rabdologia I. Nepperi, cortelijck ende duydelijck t'ghene den landtmeters ende ingenieurs, nopende het landt-meten ende stercktenbouwen nootwendich is, wort geleert ende exemplaerlijck aengewesen

"Nepperi" refer's to Table of Logarithmes (1616) and the calculating tool in Rabdologie (1617) published by Englishman John Napier (1550-1617). They were a huge contribution to the effort to cope with the large quantities inherent in astronomical calculations.

The first half of Metius's book is arithmetic: adding, subtracting, and multiplying. The arithmetic in these books is what today we expect teenagers to learn in a high school geometry course. The second part applies the math to irregular two-dimensional shapes -- what we call plane geometry (above left). In as much detail, it explains three-dimensional shapes -- solid geometry. It has examples like the one below for finding the area inside a curved, irregular space, say, for example, a farm between two winding creeks. It also has examples like the one on the wine gauger page for finding the volume inside regular shapes like cones, solid rectangles, and even curved rectangles like barrels.

Hondert Geometische Questien

The third book was Sybrandt Hansz. Cardinael's Hondert Geometische Questien (A Hundred Geometrical Problems) published in 1612. The only two of his examples that are not geometric shapes are those on pages 17 and 21 on the right sidebar. They show how to measure the height of a tower.