www.fransvanschooten.nl

 

GeoGebra

Animations in GeoGebra support the presentation of the calculations. One of them gives students the opportunity to design a fortress themselves.

  • GeoGebra Animation: Build Your Own Fortress

    You can draw a fortress of your own to investigate the relationships between sides and angles.

  • GeoGebra Animation: Comparison of Profiles

    You can use this animation to compare fortifications schemes of eminent engineers

  • GeoGebra Animation: Comparison of Forms and Dimensions

    You can use this animation to compare more than 200 drawings of fortifications

 

Fortification as a context for the sine, cosine and tangent rules

Euclides is the Dutch journal for mathematics teachers. Euclides 96-6 contains my article on fortification in the sixteenth and seventeenth centuries as a source for new trigonometry assignments.

Euclides 96-6 (in Dutch)

article (in Dutch)

 

Saillant 2021-4

The foundation Menno van Coehoorn ppublishes four times a year the journal Saillant. Saillant 2021-4 has my contribution Onderwijs in de vestingbouw in de zeventiende eeuw: cursussen en manuscripten uit de generatie Van Schooten. Before, I wrote articles on this topic in Euclides and Wiskunde & Onderwijs.

PDF

More publications in Dutch

Menno van Coehoorn

 

Comparison of calculation schemes

This webpage contains a graphical overview of 17th-century fortification calculation schemes in the Netherlands. The table presents an overview. Below the table, there are small chapters with drawings and short explanations. The aim is to find out what the significance is of the differences in these calculation schemes. An important outcome is that different types of fortifications are calculated. In one type, the length of the line of defense is key, and in the other the distance between the points of the two bastions.

  1. Small Royal (or Petit Royale) with a distance between the points of the two bastions is 60 rods (1 rod ≈ 3,767 meters)
  2. Mean Royal (or Moyen Royale) with a distance between the points of the two bastions larger than 60 rods, but the length of the line of defense is 60 less than rods (226 meters)
  3. Great Royal (or Grande Royale) with a length of the line of defense is 60 rods (226 meters)

In the early seventeenth century, 60 rods were considered the maximum length of a musket shot that was effective to kill a person.

Books and manuscripts of more than twenty authors were compared. All figures on this webpage have been taken from the original sources. For the sake of readability, the letters have been replaced by a modern font. Animations in GeoGebra support the presentation of these calculations. The list of 17th-century books on fortification is much longer. In the second half of that century, Brueil, Mallet, van Coehoorn and Fournier wrote important contributions. This overview concentrates on the first half of the 17th-century.

  • Table of Calculation Schemes

    The table presents an overview of the way in which authors labeled letters to points in their drawings.

  • Summary of Input Parameters

    The table presents an overview of the input parameters of the various designs. Designs are grouped by the rule of the bastion angle.

    • the bastion angle is obtuse (as Stevin)
    • the bastion angle is a right angle (as Errard and De Ville)
    • the bastion angle is 15° more than half of the polygon angle (as Marolois, Henrion, De Ville, Schooten Sr, Goldmann, Ruse, Kechelius, Pieter van Schooten, Anonymus Lombaerde) (Melder approximates this rule))
    • the bastion angle is 20° more than half of the polygon angle (as Freitag) (Hondius approximates this rule)
    • the bastion angle is 30° more than one-thirds of the polygon angle (as Kinckhuysen, Pieter van Schooten, anonymus Tresoar, anonymus KB, Cuijck van Meteren, and Huygens)
    • the bastion angle is two-thirds of the polygon angle (as Metius and Cellarius)
  • Overview of Primary and Secondary Sources

    My website has a long list of primary and secondary sources. Many of them are available online.

  • GeoGebra Animation: Build Your Own Fortress

    You can draw a fortress of your own to investigate the relationships between sides and angles.

  • GeoGebra Animation: Comparison of Profiles

    You can use this animation to compare fortifications schemes of eminent engineers

  • GeoGebra Animation: Comparison of Forms and Dimensions

    You can use this animation to compare more than 200 drawings of fortifications

 

 

Real life example

The fortress Bourtange on the border of the Dutch Republic and Germany is a fine example of a fortress with a 60-rod (226 meters) distance between the points of the two bastions. According to GoogleMaps, the distance between two points of the wet ditch in front of the bastions is 220 meters. So, a soldier at the tip of the bastion could kill any person on the other side of the wet moat with his musket. These videos help to understand the meaning of the drawings.

YouTube: short video

YouTube: long video

official website Bourtange

 

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Small, Mean, and Great Royal

The survey distinguishes three types of fortification. This classification is common in all the books after 1600.

  1. (+) Small Royal (or Petit Royale) with a distance between the points of the two bastions is 60 rods (1 rod ≈ 3,767 meters)
  2. (++) Mean Royal (or Moyen Royale) with a distance between the points of the two bastions larger than 60 rods, but the length of the line of defense is 60 less than rods (226 meters)
  3. (+++) Great Royal (or Grande Royale) with a length of the line of defense is 60 rods (226 meters)

Most drawings are expressed in Rhineland rods, an old measure of length. A Rhineland rod is the equivalent of 3.767 meters. A distance of 60 rods is therefore 226 meters. The Rhineland rod was usually divided into 12 feet. One foot is then 31.39 cm long. Some authors divided a rod into ten feet, in accordance with the new decimal approach. The French toise is six feet and about 1,80 meters. For sake of simplicity, a toise is almost half a rod. Three toises are as long as one verge. So one verge is also approximately one rod.

  • GeoGebra Animation: Comparison of Forms and Dimensions

    You can use this animation to compare more than 200 drawings of fortifications

 

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Bastion Angles

The survey distinguishes various formulas for the calculation of the size of the bastion angle

  1. the bastion angle is obtuse (as Stevin)
  2. the bastion angle is a right angle (as Errard and De Ville)
  3. the bastion angle is 15° more than half of the polygon angle (as Marolois, Henrion, De Ville, Schooten Sr, Goldmann, Ruse, Kechelius, Pieter van Schooten, Anonymus Lombaerde) (Melder approximates this rule))
  4. the bastion angle is 20° more than half of the polygon angle (as Freitag) (Hondius approximates this rule)
  5. the bastion angle is 30° more than one-thirds of the polygon angle (as Kinckhuysen, Pieter van Schooten, anonymus Tresoar, anonymus KB, Cuijck van Meteren, and Huygens)
  6. the bastion angle is two-thirds of the polygon angle (as Metius and Cellarius)

My animations give an idea of the differences between these rules.

  • GeoGebra Animation: Comparison of Profiles

    You can use this animation to compare fortifications schemes of eminent engineers

  • GeoGebra Animation: Comparison of Forms and Dimensions

    You can use this animation to compare more than 200 drawings of fortifications

4

The next table shows the bastion angles for the above mentioned rules.







number of bastionsright-angledhalf + 15°half + 20°thirds + 30°two-thirds






4not possible60°65°60°75°
590°69°74°66°87°
690°75°80°70°90°
790°79,3°84,3°72,9°90°
890°82,5°87,5°75°90°
990°85°90°76,7°90°
1090°87°90°78°90°

The next table shows the enfilade angles for the above mentioned rules.







number of bastionsright-angledhalf + 15°half + 20°thirds + 30°two-thirds






4not possible15°12,5°15°7,5°
519,5°17°21°10,5°
615°22,5°20°25°15°
719,3°24,6°22,1°27,9°19,3°
822,5°26,3°23,8°30°22,5°
925°27,5°25°31,7°25°
1027°28,5°27°33°27°

 

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Table of Calculation Schemes with Lettering

In our publication, we state that there is hardly any common logic in the lettering of the drawings. The table presents an overview of the way in which authors labeled letters to points in their drawings. Every calculation scheme has inputs and outputs. Inputs are marked by coloring the cell, and by underlining the letters in a bold face type. My GeoGebra animations provides a comparison of shapes and dimensions.

1.2

TIP: Hover the mouse over the link with the name of the author for a view of the drawing and additional information.












Author
Small, Mean, or Great Royal
Formula Bastion Angle

square
pent.
hex.
Centre
Polygon
Side
Polygon
CurtainDistance
between
points
bastion
FlankFaceGorgeCapitalLine
of
defense











  • Stevin (+++) (obtuse)

    In Stevin's hexagonal design, the distance between the points of the two bastions is 1237 feet (about 100 rods or 3800 meters). The bastion angle is obtuse. The length of the line of defense is 968 feet (about 81 rods or 300 meters). This design is larger than a Great Royal.

    Simon Stevin: Sterctenbovwing 1594
(6)ABCHKYZHMMZBHBZTY
  • Errard (++) (90°)

    In Errard's hexagonal design, the distance between the points of the two bastions is 130 toises (65 rods or 245 meters). The bastion angle is right. The length of the line of defense is 100 toises (50 rods or 188 meters). This design is a Mean Royal.

    Jean Errard: Fortification réduite en art et démontrée 1600
(6)P-FIBCFGBGFHBHBI
  • Marolois (+) (½+15°)

    In the pentagonal fortress of Marolois, the distance between the points of the two bastions is 63 verges (63 rods or 237 meters). The bastion angle is 69°, which is 15° more than half of the polygon angle. The length of the line of defense is 47 verges (47 rods or 177 meters). So, this is a little bit more than a Small Royal fortification.

    Samuel Marolois : Fortification ou architectvre militaire 1614
(5)-AEBDKLBFFKABAKBL
  • Marolois (++) (½+15°)

    In the hexagonal design of Marolois, the distance between the points of the two bastions is 70 rods (264 meters). The bastion angle is 69°, which is 15° more than half of the polygon angle. The length of the line of defense is 53 rods (200 meters). This is a Mean Royal fort.

    Samuel Marolois : Fortification ou architectvre militaire 1614
(6)-AIBHDPBCCDABADBP
  • Henrion (+++) (½+15°)

    Depending upon the number of bastions, the distance between the points of the two bastions is around 160 toises (80 rods or 300 meters). The bastion angle is 15° more than half of the polygon angle. The length of the line of defense is around 120 toises (60 rods or 226 meters). So, this is a Great Royal design.

    Pentagonal design
    Didier Henrion: Construire les Fortifications pratiquess aux Pays Bas 1621
(5)MABCDHLCEEHACAHDH
  • Henrion (2) (+++) (½+15°)

    For a hexagon, the distance between the points of the two bastions AB is around 149 toises (74 rods or 280 meters). The bastion angle is 15° more than half of the polygon angle. The length of the line of defense AI is around 114 toises (57 rods or 215 meters). This is the design of a Great Royal.

    Didier Henrion: Construire les Fortifications pratiquess aux Pays Bas 1621
(6)-NOGIABDGADGNANAI
  • Schooten Sr BPL1013 (+) (½+15°)

    In BPL 1013, Frans van Schooten Sr did not mention computational results. He assumes that the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 15° more than half of the polygon angle. So, the design is a Small Royal.

    Frans van Schooten Sr: BPL 1013 1622
(4)---------
  • Hondius (+++) (½+20°)

    For a pentagon, the distance between the points of the two bastions is 93 rods (350 meters). The bastion angle is almost 20° more than half of the polygon angle. The length of the line of defense is 67 rods (254 meters). So, the design is a Great Royal.

    Henricus Hondius, Korte beschrijvinge, ende afbeeldinge van de generale regelen der fortificatie 1624
(5)---------
  • Metius (++) (⅔)

    In the hexagonal design of Metius, the distance between the points of the two bastions is 65 rods (245 meters). The bastion angle is two-thirds of the polygon angle, but 90° at most. The length of the line of defense is 47 rods (177 meters). So, this is Mean Royal design.

    Adriaan Metius: Fortificatie ofte stercken-bouwinghe 1626
(5)KMNABGHBDDHBOHOAH
  • De Ville (+++) (90°)

    In the hexagonal design of De Ville, the distance between the points of the two bastions is 240 steps (80 rods or 300 meters). The bastion angle is a right angle. The line of defense is 180 steps (60 rods or 226 meters). So, this is a Great Royal design.

    Antoine de Ville: Les fortifications du chevalier Antoine de Ville 1628
(6)SHRKQ-MQAMHQAHAK
  • Freitag (+++) (½+20°)

    In the pentagonal design of Freitag, the distance between the points of the two bastions is 82 rods (310 meters). The bastion angle is 20° more than half of the polygon angle. The length of the line of defense is 61 rods (230 meters). So, this is a Great Royal design.

    Adam Freitag: Architectura militaris nova et aucta 1635
(5)LKOABHPBDCHAKHKBH
  • Kinckhuysen (+) (⅓+30°)

    In the pentagonal design of Kinckhuysen, the distance between the points of the two bastions is 62 rods (234 meters). The bastion angle is is 30° more than one-thirds of the polygon angle. The length of the line of defense is 46 rods (173 meters). So, this is a Small Royal design.

    Gerard Kinckhuysen: De Theorie der Fortificatie ofte Sterck Bovwinge 1640
(5)LABCH-CDDEACAEEH
  • Cellarius (+++) (⅔)

    The distance between the points of the two bastions is around 80 rods (300 meters). The bastion angle is two-thirds of the polygon angle. The length of the line of defense is around 60 rods (226 meters). So, this is a Great Royal design.

    Andreas Cellarius: Architectvra Militaris 1645
(5)ABHCGMSCIIMBCBMCS
  • Goldmann (+++) (½+15°)

    In the pentagonal design of Goldmann, the distance between the points of the two bastions is around 77 rods (290 meters). The bastion angle is 15° more than half of the polygon angle The length of the line of defense is around 60 rods (226 meters). So, this is a Great Royal design.

    Nicolaus Goldmann: La nouvelle fortification 1645
(5)ZabSTAXRSARaSaAAT
  • Melder (+++) (± ½+15°)

    In the pentagonal design of Melder, the distance between the points of the two bastions is around 85 rods (320 meters). The bastion angle is about 15° more than half of the polygon angle. The length of the line of defense is around 63 rods (238 meters). Therefore, this design is a Great Royal.

    Gerard Melder: Korte en klare instructie van regulare en irregulare fortificatie 1651
(5)ABEGHCDHICIBHBCCG
  • Ruse (+++) (± ½+15°)

    The distance between the points of the two bastions is around 80 rods (300 meters). The bastion angle is about 15° more than half of the polygon angle. The length of the line of defense is 59 rods (222 meters). So, this is a Great Royal.

    Henrik Ruse: Versterckte Vesting 1654
(4)MCDGHABHIAICHACBH
  • Kechelius (+++) (½+15°)

    In the pentagonal design of Kechelius, the distance between the points of the two bastions is around 81 rods (306 meters). The bastion angle is 15° more than half of the polygon angle. The length of the line of defense is 61 rods (230 meters). So, this is a Great Royal design.

    Samuel Kechelius: BPL 1351 1655
(5)ABCDEFGDKFKBDBFEF
  • Pieter v Sch. BPL 1993 (+) (⅓+30°)

    In this pentagonal design, the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 15° more than half of the polygon angle. The length of the line of defense is 45 rods (170 meters). This is typically a Small Royal design.

    Pieter van Schooten: BPL 1993date unknown
(5)MABDOCIEOCEAOACCD
  • Pieter v Sch. HS441 (+) (⅓+30°)

    In this pentagonal design, the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 30° more than one-thirds of the polygon angle. The length of the line of defense is 45 rods (170 meters). This is typically a Small Royal design.

    Pieter van Schooten: HS 441 1656
(5)OCDEFABEGAGCEACAF
  • Tresoar (+) (⅓+30°)

    In this pentagonal design, the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 30° more than one-thirds of the polygon angle. The length of the line of defense is 45 rods (170 meters). This is typically a Small Royal design.

    Anonymus (Tresoar): anonymus date unknown
(5)OCDEFABEGAGCEACAF
  • KW1900A242 (+) (⅓+30°)

    In this pentagonal design, the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 30° more than one-thirds of the polygon angle. The length of the line of defense is 45 rods (170 meters). This is typically a Small Royal fortification

    Anonymus (Koninklijke Bibliotheek): KW1900A242 date unknown
(5)OCDEFABEGAGCEACAF
  • Huygens HUG16 (+) (⅓+30°)

    In this pentagonal design, the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 30° more than one-thirds of the polygon angle. The length of the line of defense is 45 rods (170 meters). This is typically a Small Royal design.

    Christiaan Huygens: HUG 16 ±1650
(5)OCDGHABGKAKCGACAH
  • Cuyck van Meteren (+) (⅓+30°)

    In this pentagonal design, the distance between the points of the two bastions of a hexagon is 60 rods (226 meters). The bastion angle is 30° more than one-thirds of the polygon angle. The length of the line of defense is 45 rods (170 meters). So, this is a Small Royal design.

    Adriaan Cuijck van Meteren: BPL 3457 date unknown
(5)LKOABHPBDCHAKHKBH
  • Lombaerde (14) (+++) (½+15°)

    In this pentagonal design, tThe distance between the points of the two bastions is 81 rods (306 meters). The bastion angle is 15° more than half of the polygon angle. The length of the line of defense is 60 rods (226 meters). This is typically a Great Royal design.

    Anonymus (Lombaerde): date unknown
(5)OABDGCMDECEADACCG
  • Lombaerde (15) (+) (½+15°)

    In this pentagonal design, the distance between the points of the two bastions is 60 rods (226 meters). The bastion angle is 15° more than half of the polygon angle. The length of the line of defense is 45 rods (170 meters). This is typically a Small Royal design.

    Anonymus (Lombaerde): date unknown
(5)OABDGCMDECEADACCG

 

 

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Line of Defense

Especiallly in the Dutch approach, there are two different lines of defense with different names. The International Fortress Council does not mention the English word. Old books suggest that the line starting in the flank angle is called "Line of Defense" (line j in the figure) and that the extended line of the face meeting at the curtain is called "Line of Defense Flanking" or "Line of Defense Razant" (line k in the figure). A special feature is the "second flank" (line s in the figure) which is that part of the curtain where defenders can shoot parallel to the razing line of defense.

The second flank is about one-thirds to half of the curtain in the Dutch Small Royal design, especially when the bastion angle has to be 30° more than one-thirds of the polygon angle. Also the Great Royal designs of Goldmann and manuscript Lombaerde have a large second flank. According to Freitag (1/2 + 20°), the second flank is about one-thirds of the curtain. In the late 17th century, French authors called this way of fortification the Dutch design.

International Fortress Council

2.2

© Chris Weber

1.2

© Chris Weber

 

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Summary Input Parameters

The next table gives an overview of the input parameters of the above mentioned calculation schemes. Designs are grouped by the rule of the bastion angle, for example two-thirds of the polygon angle or 30° moree thanone-thirds of the polygon angle. A (+) indicates a Small Royal and a (+++) indicates a Great Royal. All lengths are expressed in rods or as a ratio. For instance, if the column capital says (1/3) f, it means that the length of the capital (c) is one-thirds of polygon side (f).


rule bastion angleAuthorCurtain
(a)
Face
(b)
Capital
(c)
Flank
(d)
Gorge
(e)
Side
Polygon
(f)
Distance
between
points
bastion
(g)
Line
of
defense
(h)

obtuseStevin (+++)

90°Errard (++)(b)=(c)8(d)=(e)
De Ville (+++)101/6 (f)60

2/3Metius (++)253/10 (a)4/10 (a)
Marolois B (+++)36246/5 (d)
Cellarius (+++)36249

1/2+15°Marolois A (+++)36246/5 (d)
Henrion (+++)36249
Goldmann (+++)40207
Melder (+++)36231012
Kechelius (+++)36246/5 (d)
Pieter BPL 1993 (+)1/3 (f)1/5 (f)60
Lombaerde A (+)1/3 (f)1/5 (f)60
Lombaerde B (+++)1/3 (f)1/5 (f)60

1/2+20°Hondius (+++)30288
Freitag (+++)36247

1/3+30°Frans van Schooten Sr (+)1/3 (g)60
Kinckhuysen (+)3/5 (f)1/3 (f)3/5 (e)1/5 (f)50
Pieter HS 441 (+)1/3 (f)1/5 (f)60
Tresoar (+)1/3 (f)1/5 (f)60
KW1900A242 (+)1/3 (f)1/5 (f)60
Huygens (+)1/3 (f)1/5 (f)60
Cuijck van Meteren (+)1/3 (f)1/5 (f)60

French authors calculated in toises, and others in feets. In this table, everything has been converted to rods.

Marolois presented many calculations as exercises, but also presented two tables for the rules ½+15° and ⅔. Freitag presented many calculations and many tables for both the rules ½+15° and the ½+20°.

  1. Capital is expressed as one-thirds of the side of the polygon
  2. Gorge is expressed as ratio six to five of the gorge to the flank or as one-fifths of the side of the polygon
  3. Freitag's flank is increasing from 6 to 14 in the Great Royal.
  4. Frans van Schooten did not provide calculation schemes. On one folio he made the remark that the face should be one-thirds or one-quarter of the distance between the points of two bastions.
  5. If the distance between the points of the two bastions is 60 rods, the design is a small royale.

 

 

Europe's Star Cities, marvels of Renaissance engineering

CNN paid attention to the star-shaped fortifications in Europe on 21 January 2021. Especially the Naarden fortress attracts attention. "The Netherlands is a prime destination for those interested in "star city" architecture and Naarden is perhaps the most impressive."

CNN

 

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Simon Stevin (1548-1620)

In 1594, Simon Stevin wrote his book Sterctenbovwing, including a detailed description of a hexagonal fortification. It is one of the first books to mention all measurements in detail. I made a reconstruction as a 3D printed model. My GeoGebra animation does a step-by-step construction of a fortress in the same sequence as Stevin did.

2

Stevin mentions all lengths of all sides. He does not show his calculations. There is no trace of trigonometry.

  • hexagon BCDEFG with sides 1000
  • gorge BH is 182
  • flank HM is 140
  • curtain HK is 636
  • etc...

The bastion angle is obtuse. The distance between the points of the two bastions is 1237 feet (about 100 rods or 3800 meters). The length of the line of defense is 968 feet (about 81 rods or 300 meters). This design is larger than a Great Royal.

 
back to the Table of Calculation Schemes with Lettering
 

Stevin mentions all length of all sides. He does not show his calculations. There is no trace of trigonometry.

  • hexagon BCDEFG with sides 1000
  • gorge BH is 182
  • flank HM is 140
  • curtain HK is 636
  • etc...

The bastion angle is obtuse. The distance between the points of the two bastions is 1237 feet (about 100 rods or 3800 meters). The length of the line of defense is 968 feet (about 81 rods or 300 meters). This design is larger than a Great Royal.

 
back to the Table of Calculation Schemes with Lettering
 

Sources for Simon Stevin:

 

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Jean Errard (1554-1610)

In 1600, Jean Errard wrote his book Fortification réduite en art et démontrée. His design has a unique definition of the gorge. Most authors treat the gorge as the prolongation of the curtain in order to simplify calculations. As a result, the bastion is rather a six-sided area, following the lines of the polygon. Errard, however, draws a five-sided bastion in which the gorges are perpendicular to the capital.

2

Errard started with given:

  • bastion angle B is right
  • defense angle BIF is at least 15°
  • flank FG is at least 16 toises (8 rods or 30 meters)
  • gorge FH has the same length as flank FG
  • capital BH has the same length as face BG

This is enough information to calculate the dimensions of his proposal; approximately:

  • face BG is 38 toises (19 rods or 72 meters)
  • the length of the line of defense BI is 100 toises (50 rods or 188 meters)
  • the distance between the points of the two bastions BC is 130 toises (65 rods or 245 meters)

This design is a Mean Royal.

 
back to the Table of Calculation Schemes with Lettering
 

Errard started with given:

  • bastion angle B is right
  • defense angle BIF is at least 15°
  • flank FG is at least 16 toises (8 rods or 30 meters)
  • gorge FH has the same length as flank FG
  • capital BH has the same length as face BG

This is enough information to calculate the dimensions of his proposal; approximately:

  • face BG is 38 toises (19 rods or 72 meters)
  • the length of the line of defense BI is 100 toises (50 rods or 188 meters)
  • the distance between the points of the two bastions BC is 130 toises (65 rods or 245 meters)

This design is a Mean Royal.

 
back to the Table of Calculation Schemes with Lettering

Sources for Jean Errard:

list of sources on my website

Rara: Fortification réduite en art et démontrée

Wikipedia: Errard

GeoGebra Animations: Calculation Scheme Jean Errard

 

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Samuel Marolois (1572-1627)

In 1615, Samuel Marolois wrote his book Fortification ou architectvre militaire. After 1627 new editions appeared with additions by Frans van Schooten Sr and Albert Girard.

Marolois gave solutions to many exercises. His book has two tables. One table describes the fortification system where the bastion angle is 15° more than half of the polygon angle. The other table deals with the situation where the bastion angle is two-thirds of the polygon angle.

 
back to the Table of Calculation Schemes with Lettering

2

Marolois started exercise number 7, a pentagonal design, with given:

  • bastion angle K is 69°, which is 15° more than half of the polygon angle A.
  • distance between points of two bastions KL is 63 verges (63 rods or 237 meters)
  • surface GK is two-sevenths of the distance between points of two bastions
  • capital AK is two-sevenths of the distance between points of two bastions
  • flank angle is 90°

Marolois computed all lengths, angles, and distances, for instance, approximately:

  • curtain BD is 27 verges (27 rods or 102 meters)
  • capital AK is 18 verges (18 rods or 102 meters)
  • gorge AB is 7 verges (7 rods or 26 meters)
  • flank BF is 8 verges (8 rods or 30 meters)
  • face FK is 19 verges (19 rods or 72 meters)
  • line of defense DK is 47 verges (47 rods or 177 meters)

So, this is a Small Royal with a 60 rods distance between the points of the two bastions.

 
back to the Table of Calculation Schemes with Lettering
 

Marolois started exercise number 7, a pentagonal design, with given:

  • bastion angle K is 69°, which is 15° more than half of the polygon angle A.
  • distance between points of two bastions KL is 63 verges (63 rods or 237 meters)
  • surface GK is two-sevenths of the distance between points of two bastions
  • capital AK is two-sevenths of the distance between points of two bastions
  • flank angle is 90°

Marolois computed all lengths, angles, and distances, for instance, approximately:

  • curtain BD is 27 verges (27 rods or 102 meters)
  • capital AK is 18 verges (18 rods or 102 meters)
  • gorge AB is 7 verges (7 rods or 26 meters)
  • flank BF is 8 verges (8 rods or 30 meters)
  • face FK is 19 verges (19 rods or 72 meters)
  • line of defense DK is 47 verges (47 rods or 177 meters)

So, this is a Small Royal with a 60 rods distance between the points of the two bastions.

 
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2

Marolois started exercise number 50, a pentagonal design, with given:

  • bastion angle A is 72°, which is two-thirds of the polygon angle D.
  • face AI is 24 verges (24 rods or 90 meters)
  • ratio between face AI and curtain FG = 2:3
  • flank angle F is 90°
  • internal angle FDI is 40°

Marolois forgot to specify the size of the internal angle FDI in 1615, but Frans van Schooten corrected this omission in the 1627 edition. The ratio between flank and gorge is 5 : 6. In another exercise, he specified the length of the prolonged flank FN.

Marolois computed all lengths, angles, and distances, for instance, approximately:

  • curtain FG is 36 verges (36 rods or 90 meters)
  • flank FI is 9 verges (9 rods or 34 meters)
  • capital AD is 20 verges (20 rods or 75 meters)
  • the length of the line of defense AG is 61 verges (61 rods or 230 meters)
  • the distance between the points of the two bastions AC is 82 verges (82 rods or 309 meters)

So, this is a Great Royal fortification with a 60 rods line of defense.

 
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Marolois started exercise number 50, a pentagonal design, with given:

  • bastion angle A is 72°, which is two-thirds of the polygon angle D.
  • face AI is 24 verges (24 rods or 90 meters)
  • ratio between face AI and curtain FG = 2:3
  • flank angle F is 90°
  • internal angle FDI is 40°

Marolois forgot to specify the size of the internal angle FDI in 1615, but Frans van Schooten corrected this omission in the 1627 edition. The ratio between flank and gorge is 5 : 6. In another exercise, he specified the length of the prolonged flank FN.

Marolois computed all lengths, angles, and distances, for instance, approximately:

  • curtain FG is 36 verges (36 rods or 90 meters)
  • flank FI is 9 verges (9 rods or 34 meters)
  • capital AD is 20 verges (20 rods or 75 meters)
  • the length of the line of defense AG is 61 verges (61 rods or 230 meters)
  • the distance between the points of the two bastions AC is 82 verges (82 rods or 309 meters)

So, this is a Great Royal fortification with a 60 rods line of defense.

 
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2

In this exercise, Marolois described a hexagonal Mean Royal with given:

  • distance between the points of the two bastions D and P is 70 rods (264 meters)
  • bastion angle K is 75°, which is 15° more than half of the polygon angle
  • face CD is two-thirds of the length of the curtain BH
  • Ratio between face CD and flank BC is 5:2
  • flank angle F is 90°

Marolois computed all lengths, angles, and distances, for instance, approximately:

  • curtain BH is 31 rods (17 meters)
  • face CD is 21 rods (79 meters)
  • flank BC is 8 rods (30 meters)
  • gorge AB is 10 rods (38 meters)
  • capital AD is 19 rods (72 meters)
  • the length of the line of defense DH is 53 rods (200 meters)

So, this is a Mean Royal fortification.

 
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In this exercise, Marolois described a hexagonal Mean Royal with given:

  • distance between the points of the two bastions D and P is 70 rods (264 meters)
  • bastion angle K is 75°, which is 15° more than half of the polygon angle
  • face CD is two-thirds of the length of the curtain BH
  • Ratio between face CD and flank BC is 5:2
  • flank angle F is 90°

Marolois computed all lengths, angles, and distances, for instance, approximately:

  • curtain BH is 31 rods (17 meters)
  • face CD is 21 rods (79 meters)
  • flank BC is 8 rods (30 meters)
  • gorge AB is 10 rods (38 meters)
  • capital AD is 19 rods (72 meters)
  • the length of the line of defense DH is 53 rods (200 meters)

So, this is a Mean Royal fortification.

 
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Sources for Samuel Marolois:

list of sources on my website

Rara: Fortification ou architectvre militaire

Wikipedia: Samuel Marolois

GeoGebra Animations: Calculation Scheme Marolois

 

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Didier Henrion (-1632)

In 1621, a book Construire les Fortifications pratiquess aux Pays Bas has been published under the name of Henrion.

2

Henrion presented given:

  • bastion angle H is 15° more than half of the polygon angle A
  • curtain CD is 72 toises (36 rods or 135 meters)
  • flank CE is 18 toises (9 rods or 34 meters)
  • face EH is 48 toises (24 rods or 90 meters)
  • flank angle C is 90°

Henrion does not show his computations but delivers a table with results. Depending upon the number of bastions, the distance between the points of the two bastions HL is around 160 toises (80 rods). The length of the line of defense DH is around 120 toises (60 rods). This is the design of a Great Royal.

 
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For a pentagon, Henrion presented given:

  • bastion angle H is 15° more than half of the polygon angle A
  • curtain CD is 72 toises (36 rods or 135 meters)
  • flank CE is 18 toises (9 rods or 34 meters)
  • face EH is 48 toises (24 rods or 90 meters)
  • flank angle C is 90°

Henrion does not show his computations but delivers a table with results. Depending upon the number of bastions, the distance between the points of the two bastions HL is around 160 toises (80 rods). The length of the line of defense DH is around 120 toises (60 rods). This is the design of a Great Royal.

 
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Henrion also describes one of the design proposals of Marolois, but his lettering is different. At first sight, the shape of the bastion is the same, but the angles are different.

2

For a hexagon, Henrion presents given:

  • bastion angle A is 15° more than half of the polygon angle
  • curtain GI is 64 toises (32 rods or 120 meters)
  • flank DG is 18 toises (9 rods or 34 meters)
  • face AD is 48 toises (24 rods or 90 meters)
  • flank angle C is 90°
  • internal angle DNG is 40° (like Marolois, the ratio between flank and gorge is 5 : 6.)

Henrion shows parts of his computations and some results. For a hexagon, the distance between the points of the two bastions AB is around 149 toises (74 rods). The length of the line of defense AI is around 114 toises (57 rods). This is the design of a Great Royal.

 
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Henrion presents given:

  • bastion angle A is 15° more than half of the polygon angle
  • curtain GI is 64 toises (32 rods or 120 meters)
  • flank DG is 18 toises (9 rods or 34 meters)
  • face AD is 48 toises (24 rods or 90 meters)
  • flank angle C is 90°
  • internal angle DNG is 40° (like Marolois, the ratio between flank and gorge is 5 : 6.)

Henrion shows parts of his computations and some results. For a hexagon, the distance between the points of the two bastions AB is around 149 toises (74 rods). The length of the line of defense AI is around 114 toises (57 rods). This is the design of a Great Royal.

 
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Sources for Didier Henrion:

 

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Frans van Schooten Sr (1581-1645)

The university library of Leiden owns manuscript BPL1013 Uitgewerkte voorstellen van theoretische en toegepaste meetkunde, opgehelderd door net geconstrueerde teekeningen BPL 1013 , including a book on fortification. This book contains beautiful coloured drawings but there are no pages with computations at all. He should have computations or tables, but not in this manuscript.

2

According to Frans van Schooten Sr, the bastion angle is 15° more than half of the polygon angle. He presents given:

  • bastion angle A is 60°
  • face AB is 18 rods, being 3/10 of the distance between the points of the bastions

In BPL 1013, Frans van Schooten Sr did not mention computational results. His table of angles contains only one line. The drawing mentions alternatives too: face AB is 15 rods, being 1/4 of the distance between the points of the bastions.

Both ways, the distance between the points of the two bastions is 60 rods. So, the design is a Small Royal.

 
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Frans van Schooten Sr presents given:

  • bastion angle A is 60°
  • face AB is 18 rods, being 3/10 of the distance between the points of the bastions

In BPL 1013, Frans van Schooten Sr did not mention computational results. His table of angles contains only one line. The drawing mentions alternatives too: face AB is 15 rods, being 1/4 of the distance between the points of the bastions.

Both ways, the distance between the points of the two bastions is 60 rods. So, the design is a Small Royal.

 
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Sources for Frans van Schooten Senior:

 

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Hendrik Hondius (1573-1650)

In 1624, Hendrik Hondius wrote Korte beschrijvinge, ende af-beeldinge van de generale regelen der fortificatie.

2

Hondius has given:

  • curtain is 360 feets (30 rods or 113 meters) for a square and 540 feet (45 rods or 170 meters) for an octagon
  • gorge is 170 feets (14 rods or 53 meters) for a square and 230 feet (19 rods or 72 meters) for an octagon
  • flank is 100 feets (8 rods or 31 meters) for a square and 140 feet (12 rods or 44 meters) for an octagon
  • face is 335 feets (28 rods or 105 meters) for a square and 420 feet (35 rods or 132 meters) for an octagon
  • flank angle is 90°

This way, approximately,

  • bastion angle is 20° more than half of the polygon angle.
  • capital line is 225 feets (19 rods or 71 meters) for a square and 366 feet (30 rods or 115 meters) for an octagon
  • the distance between the points of the two bastions is 1020 feets (85 rods or 320 meters) for a square and 1280 feet (107 rods or 402 meters) for an octagon
  • the length of the line of defense is 708 feets (59 rods or 222 meters) for a square and 971 feet (8 rods or 305 meters) for an octagon

So, this is a Great Royal design.

 
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Hondius has given:

  • curtain is 360 feets (30 rods or 113 meters) for a square and 540 feet (45 rods or 170 meters) for an octagon
  • gorge is 170 feets (14 rods or 53 meters) for a square and 230 feet (19 rods or 72 meters) for an octagon
  • flank is 100 feets (8 rods or 31 meters) for a square and 140 feet (12 rods or 44 meters) for an octagon
  • face is 335 feets (28 rods or 105 meters) for a square and 420 feet (35 rods or 132 meters) for an octagon
  • flank angle is 90°

This way, approximately,

  • bastion angle is 20° more than half of the polygon angle.
  • capital line is 225 feets (19 rods or 71 meters) for a square and 366 feet (30 rods or 115 meters) for an octagon
  • the distance between the points of the two bastions is 1020 feets (85 rods or 320 meters) for a square and 1280 feet (107 rods or 402 meters) for an octagon
  • the length of the line of defense is 708 feets (59 rods or 222 meters) for a square and 971 feet (8 rods or 305 meters) for an octagon

So, this is a Great Royal design.

 
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Sources for Hendrik Hondius:

list of sources on my website

Google Books: Korte beschrijvinge, ende afbeeldinge van de generale regelen der fortificatie ....

Wikipedia: Hendrik Hondius

 

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Adriaan Metius (1571-1635)

In 1626, Adriaan Metius wrote Fortificatie ofte stercken-bouwinghe. Metius did not have one sytem of his own. He wrote down from his fathers notebooks some alternatives.

2

Metius opens with given:

  • bastion angle G is two-thirds of the polygon angle but does not exceed 90°
  • curtain AB is 25 rods (94 meters)
  • flank BD is 3/10 of the length of the curtain
  • gorge BO is 4/10 of the length of the curtain
  • flank angle A is 90°

This way, approximately,

  • face DH is 21 rods (79 meters)
  • flank BD is 7,5 rods (28 meters)
  • capital HO is 17 rods (64 meters)
  • the distance between the points of the two bastions GH is 65 rods (245 meters)
  • the length of the line of defense BG is 47 rods (177 meters)

So, this is Mean Royal design.

 
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Metius opens with given:

  • bastion angle G is two-thirds of the polygon angle but does not exceed 90°
  • curtain AB is 25 rods (94 meters)
  • flank BD is 3/10 of the length of the curtain
  • gorge AN is 4/10 of the length of the curtain
  • flank angle A is 90°

This way, approximately,

  • face DH is 21 rods (79 meters)
  • flank BD is 7,5 rods (28 meters)
  • capital HO is 17 rods (64 meters)
  • the distance between the points of the two bastions GH is 65 rods (245 meters)
  • the length of the line of defense BG is 47 rods (177 meters)

So, this is Mean Royal design.

 
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Sources for Adriaan Metius:

list of sources on my website

Google Books: Fortificatie ofte stercken-bouwinghe

Wikipedia: Adriaan Metius

Student Assignment Metius (Dutch)

GeoGebra Animations: Solution of the student assignment

 

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Antoine de Ville (1596-1656)

In 1628, Antoine de Ville wrote Les fortifications du chevalier Antoine de Ville, contenans la manière de fortifier toute sorte de places... avec l'ataque et les moyens de prendre les places... plus la défense... .

2

For a hexagon, Antoine de Ville provides given:

  • bastion angle A is 90°
  • polygon side HR is 180 steps (approximately 60 rods or 226 meters)
  • flank MQ is 30 steps
  • gorge HQ is one-sixth of the polygon side

After applying the sine rule and the tangent rule, de Ville obtains the lengths of all sides.

  • curtain KQ is 120 steps
  • face AM is 58 steps
  • capital AH is 52 steps
  • line of defense AK is 180 steps (approximately 60 rods or 226 meters)
  • distance between the points of the two bastions is 240 steps (approximately 80 rods or 301 meters)

So, this is Great Royal design.

 
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For a hexagon, Antoine de Ville provides given:

  • bastion angle A is 90°
  • polygon side HR is 180 steps (approximately 60 rods or 226 meters)
  • flank MQ is 30 steps
  • gorge HQ is one-sixth of the polygon side

After applying the sine rule and the tangent rule, de Ville obtains the lengths of all sides.

  • curtain KQ is 120 steps
  • face AM is 58 steps
  • capital AH is 52 steps
  • line of defense AK is 180 steps (approximately 60 rods or 226 meters)
  • distance between the points of the two bastions is 240 steps (approximately 80 rods or 301 meters)

So, this is Great Royal design.

 
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Adam Freitag (1608-1650)

In 1631, Adam Freitag wrote his book Architectura militaris nova et aucta, oder newe vermehrte Fortification, von Regular Vestungen, von Irregular Vestungen und Aussen Wercken, von praxi offensiva und defensiva: auff die neweste niederländische Praxin gerichtet und beschrieben / durch Adamum Freitag.

Freitag proposes two different systems:

  1. bastion angle is 15° more than half of the polygon angle (like Marolois)
  2. bastion angle is 20° more than half of the polygon angle

2

For the bastion angle being 20° more than half of the polygon angle, Freitag supposed given:

  • bastion angle H is 20° more than half of the polygon angle K
  • curtain AB is 36 rods (136 meters)
  • face CH is 24 rods (90 meters)
  • flank BD is 6 rods for a square, 7 rods for a pentagon, 8 rods for a hexagon, etc...
  • flank angle A is 90°

This information is sufficient to compute all the lengths. For a pentagon or hexagon:

  • gorge AK is 13 rods (49 meters)
  • capital HK is 17 or 18 rods (64 or 68 meters)
  • the distance between the points of the two bastions HP is 82 rods (309 meters)
  • the length of the line of defense BH is 61 rods (230 meters)

Therefore, his fort is a Great Royal.

 
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For the bastion angle being 20° more than half of the polygon angle, Freitag supposed given:

  • bastion angle H is 20° more than half of the polygon angle K
  • curtain AB is 36 rods (136 meters)
  • face CH is 24 rods (90 meters)
  • flank BD is 6 rods for a square, 7 rods for a pentagon, 8 rods for a hexagon, etc...
  • flank angle A is 90°

This information is sufficient to compute all the lengths. For a pentagon or hexagon:

  • gorge AK is 13 rods (49 meters)
  • capital HK is 17 or 18 rods (64 or 68 meters)
  • the distance between the points of the two bastions HP is 82 rods (309 meters)
  • the length of the line of defense BH is 61 rods (230 meters)

Therefore, his fort is a Great Royal.

 
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Sources for Adam Freitag:

list of sources on my website

Rara: Architectura militaris nova et aucta ...

Wikipedia Adam Freitag

GeoGebra Animations: Fortification in the Netherlands in the 17th-century

 

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Gerard Kinckhuysen (1625-1666)

The university library of Leiden owns manuscripts of Gerard Kinckhuysen. One of them is called De Theorie der Fortificatie ofte Sterck Bovwinge and is dated 1640.

2

Kinckhuysen started his exercise with given:

  • polygon side AB depends upon the shape:
    • quadrangle: 600 feet
    • pentagon: 540 feet
    • hexagon: 600 feet
    • heptagon: 640 feet
    • octagon: 600 feet
  • curtain CH is three-fifth of the polygon side AB
  • gorge AC is one-fifth of the polygon side AB
  • capital AE is one-thirds of the polygon side AB
  • flank CD is three-fifth of the gorge AC

Note that, the bastion angle E is almost 30° more than one-thirds of the polygon angle.

  • the distance between the points of the two bastions is always 60 rods
  • the length of the line of defense is 45 or 46 rods depending upon the number of bastions.

Therefore, his fort is a Small Royal design.

 
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Kinckhuysen started his exercise with given:

  • polygon side AB depends upon the shape:
    • quadrangle: 600 feet
    • pentagon: 540 feet
    • hexagon: 600 feet
    • heptagon: 640 feet
    • octagon: 600 feet
  • curtain CH is three-fifth of the polygon side AB
  • gorge AC is one-fifth of the polygon side AB
  • capital AE is one-thirds of the polygon side AB
  • flank CD is three-fifth of the gorge AC

Note that, the bastion angle E is almost 30° more than one-thirds of the polygon angle.

  • the distance between the points of the two bastions is always 60 rods
  • the length of the line of defense is 45 or 46 rods depending upon the number of bastions.

Therefore, his fort is a Small Royal design.

 
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Sources for Gerard Kinckhuysen:

 

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Andreas Cellarius (1595-1665)

In 1654, Andreas Cellarius wrote his book Architectvra Militaris, Oder Gründtliche Vnderweisung der heuttiges tages so wohl in Niederlandt als andern örttern gebräuchlichen Fortification Oder Vestungsbau.

2

Cellarius started with given:

  • bastion angle M is two-thirds of the polygon angle but does not exceed 90°
  • curtain CG is 36 rods (136 meters)
  • face IM is 24 rods (two-thirds of the curtain)
  • flank CI is 9 rods (quadrangle), 10 rods (pentagon), 11 rods (hexagon), 12 rods (heptagon and others)
  • flank angle C is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. Depending upon the number of bastions:

  • the distance between the points of the two bastions MS is around 80 rods
  • the length of the line of defense CS is around 60 rods

Therefore, this design is a Great Royal.

 
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Cellarius started with given:

  • bastion angle M is two-thirds of the polygon angle but does not exceed 90°
  • curtain CG is 36 rods (136 meters)
  • face IM is 24 rods (two-thirds of the curtain)
  • flank CI is 9 rods (quadrangle), 10 rods (pentagon), 11 rods (hexagon), 12 rods (heptagon and others)
  • flank angle C is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. Depending upon the number of bastions:

  • the distance between the points of the two bastions MS is around 80 rods
  • the length of the line of defense CS is around 60 rods

Therefore, this design is a Great Royal.

 
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Sources for Andreas Cellarius:

list of sources on my website

Munchen: Architectvra Militaris, ...

Wikipedia Andreas Cellarius

Student Assignment Cellarius (Dutch)

GeoGebra Animations: Solution of the student assignment

 

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Nicolaus Goldmann (1611-1665)

In 1645, Nicolaus Goldmann wrote his book La nouvelle fortification .

2

Goldmann started with given:

  • bastion angle A is 15° more than half of the polygon angle
  • curtain ST is 480 feet
  • flank RS is 80 feet
  • face AR is 240 feet
  • flank angle S is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. Depending upon the number of bastions:

  • the distance between the points of the two bastions AX is around 77 rods
  • the length of the line of defense AT is around 60 rods

Therefore, this design is a Great Royal.

 
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Goldmann started with given:

  • bastion angle A is 15° more than half of the polygon angle
  • curtain ST is 480 feet
  • flank RS is 80 feet
  • face AR is 240 feet
  • flank angle S is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. Depending upon the number of bastions:

  • the distance between the points of the two bastions AX is around 77 rods
  • the length of the line of defense AT is around 60 rods

Therefore, this design is a Great Royal.

 
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Sources for Nicolaus Goldmann:

list of sources on my website

Rara:La nouvelle fortification

Wikipedia: Nicolaus Goldmann

Student Assignment: Goldmann (Dutch)

GeoGebra Animations: Solution of the student assignment

 

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Gerard Melder (?)

In 1658, Gerard Melder, published his book Korte en klare instructie van regulare en irregulare fortificatie, met hare buyten-wercken ...: Met een korte weder-legginge der sustenu van de heer Henrick Ruse, over de hedendaagsche fortificatie.. Melder, like Errard, drew a real gorge GO perpendicular to capital line AmED. His aim is to have a line of defense CG of about 60 rods.

2

Melder started his proposal with given:

  • curtain GH is 36 rods (136 meters)
  • flank HI is 10 rods
  • gorge BH is 12 rods
  • capital BC is 23 rods

This is sufficient information to calculate the size of the angles and the length of the sides. Depending upon the number of bastions:

  • face CI is around 26 rods
  • the distance between the points of the two bastions CD is around 85 rods
  • the length of the line of defense CG is around 63 rods
  • the bastion angle is about 15° more than half of the polygon angle

Therefore, this design is a Great Royal.

 
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Melder started his proposal with given:

  • curtain GH is 36 rods (136 meters)
  • flank HI is 10 rods
  • gorge BH is 12 rods
  • capital BC is 23 rods

This is sufficient information to calculate the size of the angles and the length of the sides. Depending upon the number of bastions:

  • face CI is around 26 rods
  • the distance between the points of the two bastions CD is around 85 rods
  • the length of the line of defense DH is around 63 rods
  • the bastion angle is about 15° more than half of the polygon angle

Therefore, this design is a Great Royal.

 
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Sources for Gerard Melder:

 

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Henrik Ruse

In 1654, Ruse wrote his Versterckte Vesting.

2

Ruse specified dimensions without a calculation scheme

  • curtain HG is 34,97 rods
  • face AI is 23,31 rods
  • gorge CH is 8,96 rods
  • capital CA is 19,16 rods
  • flank HI is 7,52 rods
  • prolonged flank HL is 6,77 rods
  • polygon side CD is 52,88 rods
  • line of defense BH is 59,05 rods
  • the bastion angle is about 15° more than half of the polygon angle

Therefore, this design is a Great Royal.

Note that the bastion angle of this quadrangle is almost 60°.

 
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Ruse specified dimensions without a calculation scheme

  • curtain HG is 34,97 rods
  • face AI is 23,31 rods
  • gorge CH is 8,96 rods
  • capital CA is 19,16 rods
  • flank HI is 7,52 rods
  • prolonged flank HL is 6,77 rods
  • polygon side CD is 52,88 rods
  • line of defense BH is 59,05 rods
  • the bastion angle is about 15° more than half of the polygon angle

Therefore, this design is a Great Royal.

Note that the bastion angle of this quadrangle is almost 60°.

 
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Sources for Henrik Ruse:

 

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Samuel Kechelius

In 1655, Kechelius wrote a manuscript on request of his pupil Joos Crommeling, including a chapter on fortification. Kechelius distinguished between Grand Royale (length of the line of defense is more than 60 rods), the Petit Royale (distance between the points of the two bastions is 60 rods or less) and the Medium Royale.

2

Kechelius specified dimensions with an extensive calculation scheme

  • bastion angle F is 15° more than half of the polygon angle
  • internal angle DBK is 40° (as Marolois, the ratio between flank and gorge is 5 : 6.)
  • curtain DE is 36 rods (136 meters)
  • face FK is 24 rods (90 meters)

This information is sufficient to compute all the lengths. For a pentagon:

  • flank DK is 9 rods
  • gorge BD is 10 rods
  • capital BF is 21 rods
  • the length of the line of defense EF is 61 rods
  • distance between the points of the two bastions F and G is 81 rods

So, this is a Great Royal fortification with a 60 rods line of defense.

 
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Kechelius specified dimensions with an extensive calculation scheme

  • bastion angle F is 15° more than half of the polygon angle
  • internal angle DBK is 40° (as Marolois, the ratio between flank and gorge is 5 : 6.)
  • curtain DE is 36 rods (136 meters)
  • face FK is 24 rods (90 meters)

This information is sufficient to compute all the lengths. For a pentagon:

  • flank DK is 9 rods
  • gorge BD is 10 rods
  • capital BF is 21 rods
  • the length of the line of defense EF is 61 rods
  • distance between the points of the two bastions F and G is 81 rods

So, this is a Great Royal fortification with a 60 rods line of defense.

 
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Sources for Samuel Kechelius:

list of sources on my website

Leiden: BPL 1351: S.C. Kechelius van Hollensteyn, "Beginsel der geometrie". Manu Joos Crommeling (belgice, latine)

DBNL: Nieuw Nederlandsch biografisch woordenboek

GeoGebra Animations

 

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Pieter van Schooten (1634-1679)

The university of Leiden owns manuscript BPL 1993 Petrus van Schooten, Versterkingskunst. (belgice) of Pieter van Schooten. The university of Groningen owns manuscript HS 441 Arithmétique ou l'art de cyfrer en andere teksten from Pieter van Schooten, dated 1655 or 1656. It is of interest that Pieter van Schooten used both calculation schemes for the bastion angle. In BPL 1993, he presented a table using the rule that the bastion angle is 15° more than half of the polygon angle, but in the calculation example, he used the rule that the bastion angle is 30° more than one-thirds of the polygon angle, as in HS 441.

2

In BPL 1993, Pieter van Schooten described a Small Royal with given:

  • bastion angle C is 15° more than half of the polygon angle
  • flank angle O is 90°
  • distance between the points of the two bastions CI is 60 rods
  • capital AC is one-thirds of polygon side AB
  • gorge AO is one-fifth of polygon side AB

As a result, for a pentagon or hexagon, the dimensions are about:

  • capital AC is 15 rods
  • face CE is 18 rods
  • flank EO is 6 rods
  • curtain DO is 26 rods
  • the length of the line of defense CD is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
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In BPL 1993, Pieter van Schooten described a Petit Royal with given:

  • bastion angle C is 15° more than half of the polygon angle
  • flank angle O is 90°
  • distance between the points of the two bastions CI is 60 rods
  • capital AC is one-thirds of polygon side AB
  • gorge AO is one-fifth of polygon side AB

As a result, for a pentagon or hexagon, the dimensions are about:

  • capital AC is 15 rods
  • face CE is 18 rods
  • flank EO is 6 rods
  • curtain DO is 26 rods
  • the length of the line of defense CD is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
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2

In HS 441, Pieter van Schooten described a pentagonal Small Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle D is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side AB
  • gorge CE is one-fifth of polygon side AB

For a pentagon, the dimensions are about:

  • capital AC is 14 rods
  • face AG is 18 rods
  • flank EG is 5 rods
  • curtain EF is 26 rods
  • gorge CE is 9 rods
  • the length of the line of defense AF is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
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In HS 441, Pieter van Schooten described a Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle D is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side AB
  • gorge CE is one-fifth of polygon side AB

For a pentagon, the dimensions are about:

  • capital AC is 14 rods
  • face AG is 18 rods
  • flank EG is 5 rods
  • curtain EF is 26 rods
  • gorge CE is 9 rods
  • the length of the line of defense AF is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
back to the Table of Calculation Schemes with Lettering

Sources for Pieter van Schooten:

list of sources on my website

Wikipedia: Petrus van Schooten

Leiden: BPL 1993 Petrus van Schooten, Versterkingskunst. (belgice)

Student Assignment Pieter van Schooten(Dutch)

GeoGebra Animations: Solution of the student assignment BPL 1993

Groningen: HS 441 Arithmétique ou l'art de cyfrer en andere teksten

Student Assignment Pieter van Schooten (Dutch)

GeoGebra Animations: Calculation Scheme HS441

 

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Anonymus Tresoar

Tresoar Leeuwarden owns a manuscript with the title Fortificatie defensive ende offensive, als mede 't rangeeren der Bataillen, geinventeert door den E. Professor Frans van Schooten ... .

2

This manuscript describes a pentagonal Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle E is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side CD
  • gorge CE is one-fifth of polygon side CD

The dimensions of this pentagonal fortress are about:

  • curtain EF is 26 rods
  • flank EG is 5 rods
  • face AG is 18 rods
  • second flank FP is 13 rods
  • gorge CE is 9 rods
  • capital AC is 15 rods
  • the length of the line of defense AF is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
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This manuscript describes a pentagonal Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle E is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side CD
  • gorge CE is one-fifth of polygon side CD

The dimensions of this pentagonal fortress are about:

  • curtain EF is 26 rods
  • flank EG is 5 rods
  • face AG is 18 rods
  • second flank FP is 13 rods
  • gorge CE is 9 rods
  • capital AC is 15 rods
  • the length of the line of defense AF is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
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Sources for manuscript Tresoar:

 

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Anonymus KB

The Koninklijke Bibliotheek (The Netherlands) owns anonymus manuscript KW1900A242 Architectura militaris. The lettering of the points is the same as he lettering of the Tresoar manuscript.

2

Page 11 verso describes a pentagonal Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle F is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side CD
  • gorge CE is one-fifth of polygon side CD

The dimensions of this pentagonal fortress are about:

  • curtain EF is 26 rods
  • flank EG is 5 rods
  • face AG is 18 rods
  • second flank FP is 13 rods
  • gorge CE is 9 rods
  • capital AC is 15 rods
  • the length of the line of defense AF is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

Note that page 12 verso contains a table where the bastion angle is 15° more than half of the polygon angle.

 
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Page 11 verso describes a pentagonal Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle F is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side CD
  • gorge CE is one-fifth of polygon side CD

The dimensions of this pentagonal fortress are about:

  • curtain EF is 26 rods
  • flank EG is 5 rods
  • face AG is 18 rods
  • second flank FP is 13 rods
  • gorge CE is 9 rods
  • capital AC is 15 rods
  • the length of the line of defense AF is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

Note that page 12 verso contains a table where the bastion angle is 15° more than half of the polygon angle.

 
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Sources for manuscript Koninklijke Bibliotheek:

list of sources on my website

Koninklijke Bibliotheek: Architectura militaris (±17xx)

GeoGebra Animations: Fortification in the Netherlands in the 17th-century

 

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Christiaan Huygens (1629-1695)

The university of Leiden owns manuscript HUG16 Van Sterckten Bouwen.

2

Page 28 verso describes a Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle G is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side CD
  • gorge CG is one-fifth of polygon side CD

The results are exactly the same as for manuscript HS 441 Pieter van Schooten, anonymus manuscript Tresoar, anonymus manuscript KW1900A242, and manuscript Cuijck van Meteren.

 
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Page 28 verso describes a Petit Royal with given:

  • bastion angle A is 30° more than one-thirds of the polygon angle
  • flank angle G is 90°
  • distance between the points of the two bastions AB is 60 rods
  • capital AC is one-thirds of polygon side CD
  • gorge CG is one-fifth of polygon side CD

The results are exactly the same as for manuscript HS 441 Pieter van Schooten, anonymus manuscript Tresoar, anonymus manuscript KW1900A242, and manuscript Cuijck van Meteren.

 
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Sources for Christiaan Huygens:

list of sources on my website

Leiden: Fortification réduite en art et démontrée (1600)

Wikipedia

GeoGebra Animations

 

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Adriaan Cuijck van Meteren

The university of Leiden owns manuscript BPL 3457 Fortificatie that belonged to Adriaan Cuyck van Meteren.

2

The calculation starts with given:

  • bastion angle H is 30° more than one-thirds of the polygon angle
  • distance between the points of the two bastions HP is 60 rods
  • capital HK is one-thirds of polygon side KO
  • gorge AK is one-fifth of polygon side KO
  • flank angle A is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. For a pentagon, the approximate dimensions are:

  • curtain AB is 26 rods
  • flank AC is 5 rods
  • face CH is 18 rods
  • capital HK is 14 rods
  • gorge AK is 9 rods
  • second flank BF is 13 rods
  • line of defense BH is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions

 
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The calculation starts with given:

  • bastion angle H is 30° more than one-thirds of the polygon angle
  • distance between the points of the two bastions HP is 60 rods
  • capital HK is one-thirds of polygon side KO
  • gorge AK is one-fifth of polygon side KO
  • flank angle A is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. For a pentagon, the approximate dimensions are:

  • curtain AB is 26 rods
  • flank AC is 5 rods
  • face CH is 18 rods
  • capital HK is 14 rods
  • gorge AK is 9 rods
  • second flank BF is 13 rods
  • line of defense BH is 45 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions

 
back to the Table of Calculation Schemes with Lettering

Sources for Adriaan Cuijck van Meteren:

list of sources on my website

Wikipedia

GeoGebra Animations

 

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Anonymus Lombaerde

Anonymus manuscript Lombaerde shows two calculation schemes.

  1. folio 14: (Great Royal) line of defense is 60 rods
  2. folio 15: (Small Royal) distance between the points of the two bastions is 60 rods

2

The calculation starts with given:

  • bastion angle C is 15° more than half the polygon angle
  • line of defense CG is 60 rods
  • capital AC is one-third of polygon side AB
  • gorge AD is one-fifth of polygon side AB
  • flank angle D is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. The approximate dimensions are:

  • curtain DG is 35 rods (132 meters)
  • flank DE is 7 rods
  • face CE is 24 rods
  • capital AC is 19 rods
  • gorge AD is 12 rods
  • second flank DF is 34 rods
  • distance between points of two bastions CM is 80 rods

 
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The calculation starts with given:

  • bastion angle C is 15° more than half the polygon angle
  • line of defense CG is 60 rods
  • capital AC is one-third of polygon side AB
  • gorge AD is one-fifth of polygon side AB
  • flank angle D is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. The approximate dimensions are:

  • curtain DG is 35 rods (132 meters)
  • flank DE is 7 rods
  • face CE is 24 rods
  • capital AC is 19 rods
  • gorge AD is 12 rods
  • second flank DF is 34 rods
  • distance between points of two bastions CM is 80 rods

 
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2

The calculation of the Small Royal starts with given:

  • bastion angle C is 15° more than half the polygon angle
  • distance between points of two bastions CM is 60 rods
  • capital AC is one-third of polygon side AB
  • gorge AD is one-fifth of polygon side AB
  • flank angle D is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. The approximate dimensions are:

  • curtain DG is 26 rods
  • flank DE is 6 rods
  • face CE is 18 rods
  • capital AC is 14 rods
  • gorge AD is 9 rods
  • second flank DF is 10 rods
  • length of line of defense CG is 44 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
back to the Table of Calculation Schemes with Lettering
 

The calculation of the Small Royal starts with given:

  • bastion angle C is 15° more than half the polygon angle
  • distance between points of two bastions CM is 60 rods
  • capital AC is one-third of polygon side AB
  • gorge AD is one-fifth of polygon side AB
  • flank angle D is 90°

This is sufficient information to calculate the size of the angles and the length of the sides. The approximate dimensions are:

  • curtain DG is 26 rods
  • flank DE is 6 rods
  • face CE is 18 rods
  • capital AC is 14 rods
  • gorge AD is 9 rods
  • second flank DF is 10 rods
  • length of line of defense CG is 44 rods

This is typical a Small Royal fortification with a 60 rod distance between the points of the two bastions.

 
back to the Table of Calculation Schemes with Lettering

Sources for manuscript Lombaerde:

list of sources on my website

GeoGebra Animations Fortification in the Netherlands in the 17th-century

 

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