Torsion siege engine

Sketch of an onager, a type of torsion siege engine

A torsion siege engine is a type of artillery that utilizes torsion to launch projectiles. They were initially developed by the ancient Greeks, specifically Philip II of Macedon and Alexander the Great, and used through the Middle Ages until the development of gunpowder artillery in the 14th century rendered them obsolete.

History

Greek

Modern depiction of a Hellenistic artillery tower equipped with torsion ballistae

Preceding the development of torsion siege engines were tension siege engines that had existed since at least the beginning of the 4th century BC, most notably the gastraphetes in Heron of Alexandria’s Belopoeica that was probably invented in Syracuse by Dionysius the Elder.[1] Though simple torsion devices could have been developed earlier, the first extant evidence of a torsion siege engine comes from the Chalcotheca, the arsenal on the Acropolis in Athens, and dates to c. 338 - 326 BC. It lists the building's inventory that included torsion catapults and its components such as hair springs, catapult bases, and bolts.[2] The transition from tension machines to torsion machines is a mystery,[3] though E.W. Marsden speculates that a reasonable transition would involve the recognition of the properties of sinew in previously existing tension devices and other bows. Furthermore, traditional historiography puts the speculative date of the invention of two-armed torsion machines during the reign of Philip II of Macedon circa 340 BC, which is not unreasonable given the earliest surviving evidence of siege engines stated above.[4]

The machines quickly spread throughout the ancient Mediterranean, with schools and contests emerging at the end of the 4th century BC that promoted the refinement of machine design.[5] They were so popular in ancient Greece and Rome that competitions were often held. Students from Samos, Ceos, Cyanae, and especially Rhodes were highly sought after by military leaders for their catapult construction.[6] Torsion machines in particular were used heavily in military campaigns. Philip V of Macedon, for example, used torsion engines during his campaigns in 219-218 BC, including 150 sharp-throwers and 25 stone-throwers.[7] Scipio Africanus confiscated 120 large catapults, 281 small catapults, 75 ballistae, and a great number of scorpions after he captured New Carthage in 209 BC.[8]

Roman

Roman carroballista on Trajan's Column

The Romans obtained their knowledge of artillery from the Greeks. In ancient Roman tradition, women were supposed to have given up their hair for use in catapults, which has a later example in Carthage in 148-146 BC.[9] Torsion artillery, especially ballistae came into heavy usage during the First Punic War and was so common by the Second Punic War that Plautus remarked in the Captivi that “Meus est ballista pugnus, cubitus catapulta est mihi” (“The ballista is my fist, the catapult is my elbow").[10]

By 100 AD, the Romans had begun to permanently mount artillery, whereas previously machines had traveled largely disassembled in carts.[11] Romans made the Greek ballista more portable, calling the hand-held version manuballista and the cart-mounted type carroballista. They also made use of a one armed torsion stone-projector named the onager.[12] The earliest extant evidence of the carroballista is on Trajan’s Column. Between 100 and 300 AD, every Roman legion had a battery of ten onagers and 55 cheiroballistae hauled by teams of mules. After this, there were legionaries called ballistarii whose exclusive purpose was to produce, move, and maintain catapults.[13]

Medieval

In later antiquity the onager began to replace the more complicated two-armed devices.[14] The Greeks and Romans, with advanced methods of military supply and armament, were able to readily produce the many pieces needed to build a ballista. In the later 4th and 5th centuries as these administrative structures began to decline, simpler devices became preferable because the technical skills needed to produce more complex machines no longer remained. Vegetius, Ammianus Marcellinus, and the anonymous "De rebus bellicis" are our first and most descriptive sources on torsion machines, all writing in the 4th century AD.[15] A little later, in the 6th century, Procopius provides his description of torsion devices. All use the term ballistae and provide descriptions similar to those of the ancients.[16]

The evidence for specific medieval engines in scarce. There are citations of Arabs, Franks, and Saxons using ballistae, but because of the mutability of the terms (see Terminology below), it is uncertain whether torsion machines were indicated.[17] One good example is the siege of Paris in 885-886, in which Rollo pitted his forces against Charles the Fat, at one point impaling seven Danes at once with a bolt from a funda.[18] What is uncertain is whether the machine was tension or torsion powered. In another example, the diminutive Latin word manga/mangana is used by William of Tyre and Willam the Breton to indicate small stone-throwing engines, though again it is unclear whether they were torsion-powered.[19]

Jacques de Vitry mentions "cum cornu" ("with horns") in 1143 whilst referencing siege engines, which could indicate double arms made of horn required by a torsion machine (though it could just as likely be a tension device).[20] The best medieval source is a 12th-century treatise by Mardi ibn Ali al-Tarsusi. The account is highly detailed, if incredibly dense.[21] It describes a single-armed torsion machine on a triangular frame that could hurl 50 lbs. stones. Additionally, Persian double-armed devices similar to ancient Greek design are also described. The major problem with this source, however, is that most of the illustrations show trebuchets, not onagers or other torsion machines.[22] Also by the 12th century, siege engines were used in batteries, often consisting of large numbers of torsion devices, as in Philip Augustus’ siege of Chinon in 1205 during which he collected 400 cords for petrariae.[23] These batteries were gradually replaced with trebuchets and early gunpowder machines.

Historical continuity

There has been some scholarly debate over the use of torsion siege engines.[24] Beginning in the mid-19th century, Guillaume Defour and Louis-Napoléon Bonaparte definitively claimed that torsion siege engines were replaced by trebuchets, tension machines, and counterweight machines early in the Middle Ages because the requisite supplies needed to build the sinew skein and metal support pieces were too difficult to obtain in comparison to the materials needed for tension and counterweight machines.[25] Opposition to this viewpoint appeared later in the 19th century, when General Köhler argued that torsion machines were used throughout the Middle Ages.[26] Scholarly views become more complex at this point, with Rudolf Schneider arguing that the loss of classical knowledge in the early Middle Ages prevented ancient siege engines from being reproduced,[27] while Kalervo Huuri argued that one-armed torsion machines, such as the Roman onager, may have been used in the Medieval Mediterranean, though there was no evidence of two armed machines, such as the ballista, in this view.[28] Much more recently, Randall Rogers and Bernard Bachrach have argued that the lack of evidence regarding torsion siege engines in the Middle Ages does not provide enough proof that they were not used, especially considering that the narrative accounts of these machines almost always do not provide enough information to definitively identify the type of device being described, even with illustrations.[29]

Rogers and Bachrach seem to be the norm today, especially as medieval studies has become less centered on Western Europe. Torsion powered arrow throwers were used throughout the Byzantine Empire at least through the 11th century, and existed in western Europe up through the 14th century as the espringal, as well as in the Muslism World as the ziyar .[30] This is only for two-armed arrow-firing machines, though. Onagers and two-armed stone throwers are still up for modern debate. Konstantin Nossov argues that “beam-sling” stone-throwers that were the precursors of proper trebuchets largely replaced torsion stone-throwers by the 9th century.[31] Tracey Rihill argues that contrary to the literary evidence, one-armed machines predated or at least were concurrent with two-armed machines because they were conceptually and constructionally simpler.[32]

Construction

Reproductions of ancient Greek artillery, including catapults such as the polybolos (to the left in the foreground) and a large, early crossbow known as the gastraphetes (mounted on the wall in the background)

Design

In early designs, machines were made with square wooden frames with holes drilled in the top and bottom through which a skein was threaded, wrapped around wooden levers that spanned the holes, enabling the adjustment of tension.[33] The problem with this design is that when increasing the tension of the skein, turning the lever became nigh impossible because of the friction caused by the contact made between the wood of the lever and the wood of the frame.[34] This problem was solved simply with the addition of metal washers inserted in the holes of the frames and fastened either with tenons or rims which enabled greater control over the machine’s tension and the maximization of its power without sacrificing the integrity of the frame.[35] Further design modifications that became standard include combining the two separate spring frames into a single unit to increase durability and stability, the addition of a padded heel block to stop the recoil of the machine,[36] the development of formulae to determine the appropriate engine size (see Construction & Measurements below), and a ratcheting trigger mechanism that made it quicker to fire the machine.[37] Marsden suggests that all of these initial developments occurred in fairly rapid succession, potentially over the span of just a few decades, because the deficiencies in design were fairly obvious problems. Thereon, a gradual refinement over the succeeding centuries provided the adjustments given in the chart below. Marsden’s description of torsion machine development follows the general course that Heron of Alexandria lays out, but the Greek writer does not give any dates, either. Marsden’s chart below gives his best approximations of the dates of machine development.

Machine Type[38] Main Improvement Authority Date
Mark I, arrow-firer pair of simple spring-frames and wrapped-above-torsion-springs Heron c. 350 BC
Mark II, arrow-firer spring-frames with holes Heron before 340 BC
Mark III, arrow-firer usage of washers Heron after 340 BC
Mark IIIa, arrow-firer increased angle between the extreme positions of the arms Philon before 334 BC
Mark IIIb, stone-projector increased angle between the extreme positions of the arms Philon b/t 334 & 331 BC
Mark IVa, arrow-firer built according to formula for arrow-firers Heron/Philon c. 270 BC
Mark IVb, stone-projector built according to formula for stone-projectors Heron/Philon c. 270 BC
Modified Mark IVa, arrow firer curved arms Vitruvius c. 150 BC
Mark Va, arrow-firer oval washers Vitruvius c. 60 BC
Mark Vb, stone-projector oval washers Vitruvius c. 60 BC
cheiroballista all-metal frames, arch-shaped sighting device, an even larger angle between the extreme positions of the arms Trajan’s Column c. 100 AD

Only a few specific designs of torsion catapults are known from ancient and medieval history.[39] The materials used are just as vague, other than stating wood or metal were used as building materials. The skein that comprised the spring, on the other hand, has been cited specifically as made of both animal sinew and hair, either women’s and horse.[40] Heron and Vegetius consider sinew to be better, but Vitruvius cites women’s hair as preferable.[41] The preferred type of sinews came from the feet of deer (assumedly achilles tendons because they were longest) and the necks of oxen (strong from constant yoking).[42] How it was made into a rope is not known, though J.G. Landels argues it was likely frayed on the ends, then woven together.[43] The ropes, either hair or sinew were treated with olive oil and animal grease/fat to preserve its elasticity.[44] Landels additionally argues that the energy-storing capacity of sinew is much greater than a wooden beam or bow, especially considering that wood’s performance in tension devices is severely affected by temperatures above 77 degrees Fahrenheit, which was not uncommon in a Mediterranean climate.[45]

Measurements

Two general formulas were used in determining the size of the machine and the projectile it throws. The first is to determine the length of the bolt for a sharp-thrower, given as d = x / 9, where d is the diameter of the hole in the frame where the skein was threaded and x is the length of the bolt to be thrown. The second formula is for a stone thrower, given as d = (1.1)100m^(1/3), where d is the diameter of the hole in the frame where the skein was threaded and m is the weight of the stone. The reason for the development of these formulas is to maximize the potential energy of the skein. If it was too long, the machine could not be used at its full capacity. Furthermore, if it was too short, the skein produced a high amount of internal friction that would reduce the durability of the machine. Finally, being able to accurately determine the diameter of the frame’s holes prevented the sinews and fibers of the skein from being damaged by the wood of the frame.[46] Once these initial measurements were made, corollary formulae could be used to determine the dimensions of the rest of the machines. A couple of examples below serve to illustrate this:

Length/Weight of Missile[47] Diameter of torsion spring Height of torsion spring Machine length Machine width
31 cm 3.4 cm 22.1 cm Hand-held Hand-held
54 cm 5.6 cm 36.4 cm 1.4 0.8 m
54 cm 6.0 cm 39.0 cm 1.5 m 0.9 m
69 cm 7.5 cm 48.8 cm 1.9 m 1.1 m
77 cm 8.3 cm 54.0 cm 2.1 m 1.2 m
77 cm 8.4 cm 54.6 cm 2.1 m 1.2 m
123 cm 13.6 cm 88.4 cm 3.4 m 1.9 m
10 minas 21.2 cm 1.91 m 6.4 m 3.2 m
15 minas 24.3 cm 2.19 m 7.3 m 3.6 m
20 minas 26.8 cm 2.41 m 8.0 m 4.0 m
30 minas 30.7 cm 2.76 m 9.2 m 4.6 m
50 minas 36.3 cm 3.27 m 10.9 m 5.4 m
1 talent 38.4 cm 3.46 m 11.5 m 5.8 m
2 talents 48.6 cm 4.37 m 14.6 m 7.3 m

d is measured in dactyls , and 1 dactyl = 1.93 cm

m is measured in minas, and 1 mina = 437 g

1 talent = 26 kg

Effective use

No definitive results have been obtained through documentation or experiment that can accurately verify claims made in manuscripts concerning the range and damaging capabilities of torsion machines.[48] The only way to do so would be to construct a whole range of full-scale devices using period techniques and supplies to test the legitimacy of individual design specifications and their effectiveness of their power. Kelly DeVries and Serafina Cuomo claim torsion engines needed to be about 150 meters or closer to their target to be effective, though this is based on literary evidence, too.[49] Athenaeus Mechanicus cites a three-span catapult that could propel a shot 700 yards.[50] Josephus cites an engine that could hurl a stone ball 400 yards or more, and Marsden claims that most engines were probably effective up to the distance cited by Josephus, with more powerful machines capable of going farther.[51]

The obvious disadvantage to any device powered primarily by animal tissue is that they had the potential to deteriorate rapidly and be severely affected by changing weather. Another issue was that the rough surface of the wooden frames could easily damage the sinew of the skein, and on the other hand the force of the tension provided by the skein could potentially damage the wooden frame. The solution was to place washers inside the holes of the frame through which the skein was threaded. This prevented damage to the skein, increased the structural integrity of the frame, and allowed engineers to precisely adjust tension levels using evenly spaced holes on the outer rim of the washers.[52] The skein itself could be made out of human or animal hair, but it was most commonly made out of animal sinew, which Heron cites specifically.[53] Life of the sinew has been estimated to be about eight to ten years, which made them expensive to maintain.[54]

What is known is that they were used to provide covering fire while the attacking army was assaulting a fortification, filling in a ditch, and bringing other siege engines up to walls.[55] Jim Bradbury goes so far as to claim torsion engines were only useful against personnel, primarily because medieval torsion devices were not powerful enough to batter down walls.[56]

Archaeological evidence

Archaeological evidence for catapults, especially torsion devices, is rare. It is easy to see how stones from stone-throwers could survive, but organic sinews and wooden frames quickly deteriorate if left unattended. Usual remains include the all-important washers, as well as other metal supporting pieces, such as counterplates and trigger mechanisms. Still, the first major evidence of ancient or medieval catapults was found in 1912 in Ampurias.[57] It was not until 1968-1969 that new catapult finds were discovered at Gornea and Orşova, then again in 1972 in Hatra, with more frequent discoveries thereafter.

Stone projectiles

The sites below contained stone projectiles ranging in size from 10-90 minas (c. 4.5–39 kg).[58]

Catapult remains

NOTE: This list is not meant to be comprehensive. It is meant to show the widespread use of catapults in the Western world.[59]

Location Frame Material Date Washer amt. & avg. diameter (mm)
Ampurias (Spain) Wood c. 100 BC 4 x 81
Auerberg (Germany) Wood c. 75 AD 1 x 88
Azaila #1 (Spain) Wood c. 80 BC 1 x 94
Azaila #2 Wood c. 80 BC 1 x 94 (est. from frame remains)
Azaila #3 Wood c. 80 BC 1 x 100 (est. from counter-plate)
Bath (UK) Wood c. 100 AD 1 x 38
Caminreal (Spain) Wood c. 75 BC 4 x 84
Cremona #1 (Italy) Wood c. 69 AD 4 x 73
Cremona #2 Wood c. 69 AD 4 x 89
Elginhaugh (UK) Wood c. 90 AD 1 x 35 (ratchet found, too)
Ephyra #1 (Greece) Wood c. 169 BC 2 x 84
Ephyra #2 Wood c. 169 BC 3 x 83
Ephyra #3 Wood c. 169 BC 4 x 136
Ephyra #4 Wood c. 169 BC 4 x 60
Ephyra #5 Wood c. 167 BC 2 x 75
Ephyra #6 Wood c. 167 BC 1 x 34
Ephyra #7 Wood c. 167 BC 2 x 56
Gornea #1 (Romania) Metal c. 380 AD 2 x 54
Gornea #2 Metal c. 380 AD 2 x 59
Gornea #3 Metal c. 380 AD 2 x 54
Hatra #1 (Iraq) Wood c. 241 AD 3 x 160
Hatra #2 Wood c. 241 AD
Lyon (France) Metal c. 197 AD 2 x 75
Mahdia #1 (Tunisia) Wood c. 60 BC 2 x 94
Mahdia #2 Wood c. 60 BC 1 x 72
Mahdia #3 Wood c. 60 BC 1 x 45
Orşova (Romania) Metal c. 380 AD 2 x 79
Pergamon (Turkey) Wood c. 2nd century BC 1 x 60 (mystery bracing also found)
Pityous (Georgia) Wood c. 4th century AD 1 x 84
Sala Metal c. 4th century AD c. 80 (cast in one piece)
Sounion (Greece) Wood c. 260 BC 130 (lost)
Tanais (Ukraine) Unknown c. 50 BC?
Volubilis #1 (Morocco) Wood c. 2nd-3rd century AD 1 x 41
Volubilis #2 Wood c. 2nd-3rd century AD 1 x 44
Xanten (Germany) Wood c. 1st century AD 4 x c. 40 (diameter estimated from frame)

Literary evidence

The literary examples of torsion machines are too numerous to cite here. Below are a few well-known examples to provide a general perspective held by contemporaries.

Examples

Diodorus of Sicily, History, 14.42.1, 43.3., 50.4, c. 30 - 60 BC

"As a matter of fact, the catapult was invented at this time [399 BC] in Syracuse, for the greatest technical minds from all over had been assembled in one place...The Syracusans killed many of their enemies by shooting them from the land with catapults that shot sharp-pointed missiles. In fact this piece of artillery caused great consternation, since it had not been known before this time."[60]

Josephus, The Wars of the Jews, 67 AD

"The force with which these weapons threw stones and darts was such that a single projectile ran through a row of men, and the momentum of the stone hurled by the engine carried away battlements and knocked off corners of towers. There is in fact no body of men so strong that it cannot be laid low to the last rank by the impact of these huge stones...Getting in the line of fire, one of the men standing near Josephus [the commander of Jotapata, not the historian] on the rampart had his head knocked off by a stone, his skull being flung like a pebble from a sling more than 600 meters; and when a pregnant woman on leaving her house at daybreak was struck in the belly, the unborn child was carried away 100 meters."[61]

Procopius, The Wars of Justinian, 537-538 AD

"...at the Salerian Gate a Goth of goodly statue and a capable warrior, wearing a corselet and having a helmet on his head, a man who was of no mean station in the Gothic nation...was hit by a missile from an engine which was on a tower at this left. And passing through the corselet and the body of the man, the missile sank more than half its length into the tree, and pinning him to the spot where it entered the tree, it suspended him there a corpse."[62]

Images

Manuscripts

  1. Espringal from the anonymous Romance of Alexander, c. 14th century, MS Bodleian 264.
  2. Espringal from De re militari by Roberto Valturio, 1455.
  3. Mangonel from BL Royal 19 D I, f.111.
  4. Onager from Walter de Milemete's De nobilitatibus, sapientiis, et prudentiis regum, 1326.]

Iconography

  1. Cheiroballista behind fortifications, Trajan's Column, 1st century AD
  2. Cheiroballista, mounted on wall, Trajan's Column.
  3. Cheiroballista hauled by horse, Trajan's Column.
  4. Bronze Washers from the Amparius catatpult, cited in Schramm.

Diagrams

One-Armed Machines
  1. Catapult with bucket.
  2. Catapult with sling.
  3. Onager.
Two-Armed Machines
  1. Ballista.
  2. Euthytonon.
  3. Euthytonon range of movement.
  4. Oxybolos.
  5. Palintonon.
  6. Palintonon side view.
  7. Scorpion.
  8. Stone Thrower.

Reproductions

One-Armed Machines
  1. Catapult at the Stratford Armouries, Warwickshire, England.
  2. Onager at Felsenburg Neurathen, Saxony.
Two-Armed Machine
  1. Ballista at Caerphilly Castle, Wales.
  2. Ballista at Warwick Castle, England.
  3. Cheiroballista.
  4. Espringal side view and rear view.
  5. Polybolos & cheiroballista. Arsenal of ancient mechanical artillery in the Saalburg, Germany. Reconstructions made by the German engineer Erwin Schramm (1856-1935) in 1912.
  6. Roman Ballista in the Hecht Museum, Haifa.
  7. Roman Ballista.
  8. Zayir at Trebuchet Park, Albarracín, Spain.

Terminology

There is controversy over the terminology used to describe siege engines of every kind, including torsion machines. It is frustrating to scholars because the manuscripts are both vague in their descriptions of the machines and inconsistent in their usage of the terms. Additionally, in those few instances where torsion engines are identifiable, it is never certain which specific type of machine is being cited. Some scholars argue this abundance of terms indicates that torsion devices were in widespread use during the Middle Ages, though others argue that it is this very confusion about machine terminology that proves the few ancient texts that survived in the Latin West did not provide adequate information for the continuation of ancient torsion machines.[63] The list below provides terms that have been found in reference to torsion engines in the ancient and medieval eras, but their specific definitions are largely inconclusive.[64]

algarradas ("bull headed") fonevola ("volatile spring"?) oxybolos (“sharp thrower”)
ballista funa (thong of a sling) palestra ("stake caster"?)
ballista fulminalis ("lightning ballista") fundibula (sling) palintonos (“fold back spring”)
brigoles lithobolos (“stone thrower”) pararia (lit. "the equalizer")
calibres katapeltes patera
carroballista (see cheiroballista) machina (“machine”) paterells
catapulta (“shield breaker”) mangana peralia
chaabla mangonellus (see mangana) petraria
chatcotonus (“bronze spring”) mangonon (see mangana) petrobolos ("stone thrower")
cheiroballista (“hand ballista”) manjanîq polybolos (“multi-thrower”)
cum cornu (“with horn”) manuballista (“hand ballista”) scorpio
espringal monagkon tormentum
euthytonos (“straight-spring”) onager (“wild ass”) ziyar, qaws al-ziyar

Notes

  1. ↑ Marsden, Historical Development, 5,16,66; Chevedden, 134.
  2. ↑ Marsden, Historical Development, 56-57; Rihill, 79; Nossov, 133.
  3. ↑ Marsden, Historical Development, 17.
  4. ↑ DeVries & Smith, 42.
  5. ↑ Marsden, Historical Development, 73-74.
  6. ↑ DeVries, 130.
  7. ↑ Marsden, Historical Developments, 77.
  8. ↑ Livy, 26.47.5-6 .
  9. ↑ Vergil, Aeneid, XI.1-99,597-647 ; Vegetius, De Re Militari, IV.9; Marsden, Historical Development, 83.
  10. ↑ Plautus, Captivi, 796
  11. ↑ Marsden, Historical Development, 164.
  12. ↑ DeVries, 130-131
  13. ↑ Nossov
  14. ↑ Landels, 132; Chevedden, 137.
  15. ↑ Chevedden, 138-139, 152-158.
  16. ↑ Chevedden, 160-162.
  17. ↑ Bradbury, 251.
  18. ↑ Abbo Cernuus, Bella Parisiacae urbis; Bradbury, 252.
  19. ↑ Bradbury, 252.
  20. ↑ Bradbury, 254.
  21. ↑ Bradbury, 255.
  22. ↑ Bradbury, 256.
  23. ↑ Joinville, Life of St. Louis; Bradbury 257 .
  24. ↑ Chevedden, 131-150; Bradbury, 250-270.
  25. ↑ Dufour, 97,99; Bonaparte, 26.
  26. ↑ Köhler, 139-211
  27. ↑ Schneider, 10-16.
  28. ↑ Huuri, 51-63, 212-214.
  29. ↑ Rogers, 254-273.
  30. ↑ Bradbury, 256-257; Hacker, 43.
  31. ↑ Nossov, 155.
  32. ↑ Rihill,63,76
  33. ↑ Marsden, Historical Development, 19.
  34. ↑ Heron, W96.
  35. ↑ Marsden, Historical Development, 19-20; DeVries, 129.
  36. ↑ Landels, 117
  37. ↑ Marsden, Historical Development, 24-34.
  38. ↑ Marsden, Historical Development, 43; Marsden, Tehnical Treatises, 270; Nossov, 148.
  39. ↑ Rihill,21.
  40. ↑ Marsden, Historical Development, 87.
  41. ↑ Heron, W 110; Vegetius, IV.9; Vitruvius, X.11.2.
  42. ↑ Landels, 108.
  43. ↑ Landels, 109.
  44. ↑ Landels, 111
  45. ↑ Landels, 106.
  46. ↑ Philon, 53-54; Vitruvius, X.10-11; Marsden, Historical Development, 25-26; Nossov, 136-137; Landels, 120-121; Reinschmidt, 1247.
  47. ↑ Marsden, Historical Development, 44-47; Marsden, Technical Treatises, 266-269; Nossov, 139-140. Similar tables can also be found in Rihill, 290-292.
  48. ↑ Marsden, Historical Development, 86.
  49. ↑ Cuomo, 771; DeVries, 131.
  50. ↑ Marsden, Historical Development, 88.
  51. ↑ Marsden, Historical Development, 91-92; Johnson, 79.
  52. ↑ Landels, 112; Nossov, 142, 147.
  53. ↑ Heron, W83; Marsden, Technical Treatises, 24-25; Marsden, Historical Development, 17; Rihill, 76.
  54. ↑ Johnson, 79; DeVries, 132.
  55. ↑ Nossov, 153; Landels, 123; Hacker, 45.
  56. ↑ Bradbury, 250, 255.
  57. ↑ Baatz, 1-2.
  58. ↑ Nossov, 137-142; Marsden, Historical Development, 79.
  59. ↑ Rihill, 295-296; Baatz, 1-17.
  60. ↑ Humphrey, et al., 566
  61. ↑ DeVries, 131.
  62. ↑ DeVries, 132.
  63. ↑ Hacker, 41.
  64. ↑ Bradbury, 251,254; Hacker, 41; Nossov, 133, 155; Ammianus, 23.4.1-7; Tarver, 143.

Bibliography

Primary Sources

(see also External Links below)

Secondary sources
Ammianus Marcellinus
Athenaeus Mechanicus
De rebus bellicis
Heron of Alexandria
Philon of Byzantium
Procopius
Vegetius
Vitruvius
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