Ancient Science: From Effects to Ballistics Parameters ^†

Abstract

A well-equipped legionary army prepared to lay siege to Pompeii. Among the weapons deployed along the northern stretch of the city walls were battering rams and mobile siege towers equipped with ballistae and scorpions. The impact marks from Republican-era stone balls and dart tips remain visible today between the Vesuvio and Ercolano Gates. In 2002 and 2016, the authors surveyed significant cavities using both direct and indirect methods. The collected data were then used to calculate the volume of fractured stone material. Given the hardness of the wall ashlars, ballistic parameters were quantified based on Hellenistic treatises. The results make it possible to derive dimensions for reconstructing artillery calibrated to the observed effects.

1. Introduction

In the spring of 89 BC, Sulla’s army laid siege to Pompeii, initiating its attack from the northern section of the city walls, an impressive defensive wall punctuated by three towers between the Vesuvio and the Ercolano Gates. From the edge of the moat to the slopes of the volcano, a wide plain extends, an ideal area for offensive and defensive operations [1].

This segment of the perimeter fortification is undoubtedly the most suitable for the deployment of siege machines. Moreover, it is due to this evident vulnerability that its defensive potential was increased over time, until the subsequent insertion of towers [2], armed with many ballistic artillery pieces, distributed on three levels, as the embrasures testify. Sulla’s troops carefully prepared for the operations, bringing in several siege towers, or helepolis, many ballistae and catapults (known as scorpions) with their ammunition, placing them in protected emplacements. Having completed the preparations, at dawn on a clear summer’s day, the order was given to open fire simultaneously. The signal was also understood by the defenders on the walls: “[...] preceded by menacing screeching sounds, a hail of stone balls crashed against the battlements, crushing them together with the wooden shields” [3] (III, 7.243). A hail of darts, through the many breaches, pierced all those paralyzed by the uproar and violence of the scene they were still observing.

The impacts, still visible today between the gates, evoke, in a vivid and tangible manner, the violence of the scenes reconstructed based on historical accounts [4] (II, 75), not only ancient sources. According to Amedeo Maiuri [5] (pp. 145–160), the wide walkway (approximately 12 m) was protected by a double battlement (Figure 1), higher on the inner side; a defense that was not exceptional in Hellenic military architecture as confirmed by Philo of Byzantium.

Traces of damage recognized as “imprints” caused by the impacts of projectiles launched by Sulla’s artillery [5] (pp. 280–286), [6] (pp. 110–111) are still visible today along the stretch between Porta Vesuvio and Porta Ercolano (Figure 2).

Considering what we can currently observe, we explore the opportunity to derive certified data from the cavities produced by stone throwers and dart throwers on stone ashlars. Based on the scientific knowledge of the time, certified parameters of external ballistics will be functional to the proportioning of artillery, reconstructed based on the effects detected.

2. Materials and Methods

The analysis is based on the possibility of describing in detail the geometry of the cavities caused by stone throwers or dart launchers [5,6]. Knowing the density of the stone blocks that constitute the extrados of the perimeter, the data collected will allow us to calculate the volumes of the crushed material. Basic notions of physics will allow us to quantify the kinetic energy necessary to cause the observed damage; in other words, the deformation work that allowed the penetration of stones and darts, pulverizing the material of the wall blocks. From the mechanical resistance of the impacted material, it will be possible to inversely trace the velocity of the projectiles and thus the terminal ballistics parameters. Tracing the effects back to the causes is the premise for the certified reconstruction of ballistae and scorpions used during the Sullan siege.

The connection between projectile properties and the proportioning module of siege engines was well known to the ancients, who, based on the dimensions of the hole in which the elastic bundle was tightened, calibrated ballistae and scorpions, predicting their effects: impact speed, trajectories and distances from the target were obtained based on repeated experimental tests [7] (p. 187). In the progress reports published in the Memoirs of the American Academy in Rome, Van Buren describes the anthropic traces, distinguishing them into large, medium and small [8]. It is not clear today whether the small imprints were more evident in the past; they were certainly observable before the restorations and the degradation to which the walls were subjected over the last century. The spheroidal imprints, however, remain unmistakable in shape and size, which the scholar has related to the large and medium-sized smooth stones found on site [8].

To start the investigation from the effects, it was deemed logical to begin with the survey of the large-diameter cavities that were safer to identify and measure. At the beginning of the millennium, the Superintendency of the Archaeological Park of Pompeii was asked for the required authorization to take silicone molds of the cavities of larger diameter, the most suitable to provide reliable volumes. The molds were removed in early February 2002; two layers of silicone resin were applied by the technicians one day apart and then filled with scagliola (Figure 3).

We now know how invasive the technique was—pigments from the plaster were removed with the molds, according to some of those present during the survey. Currently, it appears that reddish pigments can be discerned inside small and large indentations, but only a technical analysis by experts in the field could reliably confirm this (Figure 4).

Non-contact acquisition techniques promise to solve some of the problems encountered. Active or passive sensors provide reality-based acquisitions; subsequent processing creates models with inverted normals, reliable copies suitable for documenting the state of the art for future reference, offering irrefutable shared and interoperable documents in case of new natural disasters or human damage. The models can be used for research workflows, with localized analyses aimed at mechanical characterization and action management, used as a tool for scientific investigation and verification. When used as an access interface to a flexible and updatable information complex, the same 3D models can be adapted to different user levels, offering a complete integration of thematic knowledge. Reading the tangible and invisible signs, adopting a “cognitive approach” common to tactile and optical experiences, is in any case useful to retrace historical and archaeological data in order to approach the steps necessary to “build well-designed machines” as the Hero of Alexandria pointed out [7] (p. 193).

Without forgetting the involvement of the senses and emotions linked to the experience of the survey, the calibration formulas in the ancients’ drawings are based on the proportional laws used by Philo and Heron. Standardizing data was an indispensable necessity on the battlefield to allow the rapid replacement of components in the event of breakage or damage, but above all to predict the lethality of the shots based on the motion of the projectile. The trajectory of a ballista’s projectile, as with a catapult, is a parabola, whose flight path is divided approximately into three segments: the first is a straight shot, equal to about one-third of the entire parabola, characterized by an almost rectilinear trajectory. The second segment follows a large radius curve that directs it downwards. The third segment runs from that variation until the ground impact, which occurs after a tighter curve of the trajectory [9].

Direct, close observation of the large diameter craters identified along the Pompeian walls revealed a deep penetration into the stone ashlars of approximately 120 mm. The trajectory appears orthogonal according to the shape of the nearly “cylindrical-semispherical” trace. The direct shot, perpendicular to the extrados of the defensive wall, from a ballistic point of view, retains maximum residual kinetic energy. Assuming the distance of the target to be one hundred meters, one can conclude that the ballista artillery was placed on the ground in front of the moat and probably at the same level of the wall, or just below it, in order to hit the protective shields erected on it.

3. Result: Calibration According to Philo and Heron

The stones, being thrown along a sort of launch channel, could have been river pebbles. However, instead, the projectiles used by Sulla’s ballistae in Pompeii appear so precisely worked and of a diameter classifiable in categories as described in detail by the archaeologist Van Buren. Different hypotheses have been put forward. In any case, none of them justify the standardization of diameters if the tactical reasons mentioned above are excluded—replacing broken or damaged components and foreseeing the lethality of machines with the same caliber.

All the treatise writers directly refer to the proportional relationship linking the diameter of the hole through which the “tighteners” of the twisted bundles pass with the aid of the arms used for torsion. However, no one has quantified the measurements, so reconstructions have always been based on data from the few finds discovered and identified as modioli of elastic torsion machines. Instead, we would like to adopt the proportional criterion recalled by Vitruvius (De Arch., X, 10–13, 19 AD) for the measurements of the bundles derived from the study of the effects found along the walls of Pompeii. Vitruvius offers the correspondence between the diameters of the modioli measured in Roman fingers and the weights of the projectiles, expressed in Roman pounds. By knowing the volume of the crushed material, the negative molds of the cavities detected, and the density of the stone material, and, therefore, the relative “breaking strength”, it will be possible to reverse engineer the impact dynamics and thereby advance in the interpretation of the marks detected. Assuming that the stone density of Vesuvian basalt and therefore the elasticity coefficient Poisson’s ratio of 0.20–0.25 [10], it will be possible to proceed to retrace the criterion formalized by ancient science and proceed further.

The weight of a spherical projectile with a diameter of 140 mm (r = 70 mm) has already been calculated, showing that a slight variation in diameter measurement significantly affects its weight and, therefore, the calibration value of the device.

Sphere volume with radius 70 mm 4/3πR3 with density 2.5 = 3.6 kg

Sphere volume with radius 80 mm 4/3πR3 with density 2.5 = 5.5 kg.

Regarding the exact evaluation of the launch tension of the arms of ballistae and scorpions, which influences the initial and final speed of the projectile, it is useful to recall that they were drawn back by a similar loading angle, almost always coinciding with the end of their rotation (90° for the arms oriented in the correct direction—therefore called eutitone). Throwing stones of equal weight would have guaranteed identical throws; the fraction of the drawn shot would have been analogous, as is usually used for aimed shots [11].

The same criterion is theoretically applicable to the launch of darts; once the target had been agreed upon, it was necessary to concentrate shots on it using the same launch variables, such as the loading angle of the arms, the length of the dart, and its weight and, obviously, the identical inclination of the scorpion catapult. Both for stone-ball launchers and for dart launchers, aiming devices were used, but were never extremely precise.

Given this calculation, the terminal ballistic parameters are as follows:

• For spheroidal projectiles, based on the formula of Philo of Byzantium [7] (p. 186), the diameter of the hole, measured in fingers, drilled vertically above and below the frame to allow the fiber bundle to pass through, D, is related to the cube root (√³) of the weight of the projectile (a stone carefully polished and standardized in diameter (p) measured in drachmas:

D = 1.1 √³ p

(1)

• For the supposed truncated-pyramidal dart tips, which were the most widespread, the calculation is based on the formula reported by Heron (1st century AD). The diameter of the “stretcher” must be proportional to one-ninth of the length of the dart [7] (p. 192).

D = L/9

(2)

3.1. Spheroidal Traces

For the spheroidal traces (Figure 5) attributed to stones, we have the following:

1.

BallisticData

Crater diameter Ø 140 mm;	(3)
r (radius) = 70 mm; b (ball penetration) = 50 mm;
Normal penetration: 120 mm;
Sphere volume: V = 4/3 πr3; for r = 70 mm, V = 1436 cm3;
Lava stone density: 2.8 g/cm3;
Weight of the spherical stone for r = 70 mm, p ≈ 4.0 kg;
Distance from initial point: 100 m;
Initial velocity = X0;
Residual impact velocity = X1.

2.

Diameter of the bundle

Philo’s formula for a spherical projectile: D = 1.1 √3 p;	(4)
D (diameter in Roman fingers): 1 Roman finger ≈ 19 mm;
p (weight of the stone-ball in Attic drachmas): 1 drachma ≈ 6 g;
conversion of the weight from kg to drachmas: 4 kg ≈ 666.6 drachmas, from which
D = 1.1 √3 666; D = 1.1 × 8.73; D ≈ 9.6 Roman fingers;
conversion of the diameter from Roman fingers to millimeters;
D = 9.6 fingers ≈ 182 mm;
Diameter of the bundle = module ≈ Ø 180 mm.

3.

Impact velocity

E (kinetic energy) = L (deformation work); L = Force * displacement, from which	(5)
L = GR (estimated breaking stress) * A (area of the spherical impact crater) × b (displacement); L = GR * V (volume of crushed material) becomes
L= π r2 b + 1/2 * 4/3 πr3 = 1488 cm3.
Setting GR = 100 kg /cm2 ≈ 1000 N/cm2, the work performed by the impact force at its end will be
L = 1000 N/cm2 * 1488 cm3 = 14,880 N*m.
If all the kinetic energy possessed by the spherical stone is transformed into deformation work of the impact block, we have
E = 1/2 m v2 with
m (mass in kg) and v (speed in m/s), from which 14,880 N*m.
Assuming that the weight of the spheroidal stone is 4 kg and that the work is equal to the kinetic energy, we have
L = E = 1/2 * 4.0 kg v2 (kg*m2/s2). Solving with respect to v,
v2 = 14,880 (m2/s2)
v = 86.3 m/s.

In the analysis of the motion of the projectile in the air, other types of aerodynamic resistance are not considered.

3.2. Quadrangular-Shaped Traces

For the pyramid-shaped traces (Figure 6) caused by iron bolts, we have as follows:

4.

Ballistic Data

Crater side: 30 mm;	(6)
Normal penetration: 30 mm, unpenetrated part 10 mm square pyramid;
Volume V = l2 × h/3;
Tip volume l2 × h/3 = 3 × 3 × 4/3 equal to 12 cm3;
Iron density: 7.8 g/cm3;
Weight of the dart tip: 93 g;
Weight of the entire dart ≈ 150 g;
Distance from the initial point: 100 m
Initial velocity = X0;
Residual impact velocity = X1.

5.

Diameter of the Bundle

Heron’s formula: D = 1/9 L;	(7)
D (diameter in Roman fingers), 1 finger ≈ 19 mm;
Length of the Philo’s dart: 481 mm;
Diameter of the scorpion bundle: D = 7.7 module.

6.

Impact velocity

E (kinetic energy) = L (deformation work); L = Force × displacement.	(8)
From which
L = GR (estimated breaking stress) * A (area of the impact crater of the dart) * the displacement; L = GR × V (volume of crushed material) becomes the following:
Setting: GR = 100 kg/cm2 ≈ 1000 N/cm2;
V (pyramid frustum) = l2 h/3 = 9 cm3;
L = 1000 N/cm2 * 9 cm3 = 90,000 N*cm = 900 N*m.
If all the kinetic energy possessed by the dart is transformed into work of deformation of the impact block, we have the following:
E = 1/2 m v2 with
m (mass in kg) and v (speed in m/s).
Assuming that the weight of the dart is 0.15 kg and that the work is equal to the kinetic energy, we have
L = E = 900 N*m = 1/2 * 0.15 v2 (kg*m2/s2). Solving with respect to v, we have
v2 = 900/0.5/0.15 =12,000 (m2/s2):
v = 109 m/s

When analyzing the motion of the projectile in the air, other types of aerodynamic resistance are not considered.

Comparing the data, the difference in work (material breakage) caused by iron tips, and rarely bronze, compared to the damage from spherical stone projectiles is evident. The pressure is halved when the impact area is doubled. The approach should therefore be optimized by analyzing the diversity in detail. However, we do not have indications that can provide us with reliable criteria and methods in the light of ancient science. Furthermore, the calibrations described above ignore the presence and significant thickness of the plaster, present at the time of the siege according to some scholars, and of a non-negligible thickness since the Roman plaster itself is several centimeters thick and in this specific case used to hide the fillings resulting from the construction of the towers. The depth detected should therefore be increased by an estimate that, with the data acquired to date, appears arbitrary depending on the thickness and consistency of the mixture with lime and pozzolana. Post-siege evidence is the existence of plaster along the “murum et plumam” [14] (p. 151), the fake rustication painted in the background of the portrayed “Fight between Pompeians and Nucerians,” which took place in 59 AD (Figure 7) [15].

This criterion, if verified through advanced surveying and reverse-engineering techniques, will allow both virtual and real prototypes to be built in the near future. The module of the machines will be calibrated to the ballistic effects attributed to 1st-century BC artillery and scientifically verifiable. This procedure will distinguish it from other existing prototypes. In fact, various machines have been reconstructed on the basis of the critical interpretation of Greek texts of the module of the ballistae by Vitruvius in Book X of De Architectura. Reliable reconstructions have been obtained by applying the criteria and principles based on archaeological findings.

4. Discussion

The spherical impact craters, as widely acknowledged, testify to the use of stone balls of 120–140 mm in diameter. Inside the city, during its excavations, however, numerous rounded stone balls were found, measuring about 270 mm in diameter, a size which corresponds to a weight of about 26 kg, a value coinciding with the ancient unit of weight called “talent” (equal to 26 kg) in the projectiles already mentioned in the historical accounts by Flavius Josephus. At least two questions follow: (i) why were only 140 mm balls used to hit the walls while 270 mm balls were used to hit the inside of the city? (ii) How was it possible for the 270 mm balls to reach over the walls knowing full well that the ballistae barely reached a trajectory grazing the top of the walls resulting in weak and oblique impacts, of little destructive effectiveness, as some of the clearly visible traces testify?

Regarding the first question, it must be assumed that, due to the lower weight of the corresponding ball, its initial and residual speed were higher, with greater devastating effects, also due to the higher pressure that such a ball produced on the planks upon impact, thus requiring a shorter loading time for the relative ballistae. However, there was a second type of elastic artillery: although the criterion informing the elastic torsion machines was, in fact, the same, the performance and structural architecture of the main throwing machines were not. No image supports the existence of a singular stone thrower with a very parabolic shot, remembered by the Greeks as monoancon (single arm) and by the Romans as onager. However, there is no lack of mentions and traces of its existence, starting with Philo of Byzantium [16] (p. 91–10).

Even Apollodorus of Damascus, who lived in the 1st century AD, Trajan’s long-time military engineer, alludes to the monoancon [17], and an equally laconic and incomplete mention is found in the histories of Ammianus Marcellinus, who lived in the 4th century AD. He was also substantially contemporary with the Flavius Vegetius Renatus, who lived between the 4th and 5th century AD, who mentioned it thus: “For the same purpose, they make very large stones weighing a talent [stone ball of 26 kg, approximately Ø 270 mm] … are launched by means of monoancon“ [18] (IV, chap. 22). From what has been cited, one should conclude that monoancon machines did not exist before the very laconic mention by Philo of Byzantium, that is, before the 3rd century BC. But a strange device used in gymnastic races, the hysplex (ύσπληξ), belies this conclusion. The device in question was invented, or rather, co-opted, by Greek engineers to frustrate the asynchronous start of athletes engaged in running races [19].

It appeared in 344–343 BC as attested by a contemporary vase painting, the architectural remains of three stadia located in the north-eastern Peloponnese and, above all, by the discovery of two Hellenistic inscriptions from Delos. This evidence, combined with the hypothesis that the operating mechanism of such a device could have been influenced by the recent invention of the onager, allowed its reconstruction. The hysplex consisted of a pair of ropes stretched horizontally between two vertical arms, to which they were tied, respectively, in front of the waist and knees of the runners. The bases of the arms were instead secured in a coil of twisted rope, so that by releasing the restraints the arms would spring towards the ground in a flash, allowing the athletes to start. The device, whose close resemblance to the motion and the conception of the monoancon, attests to its existence a few years before the middle of the 4th century BC, perhaps at the court of Philip II of Macedonia. The machine, moreover, was much simpler to build than the ballistae, as there was no need to balance the arms.

But from a ballistic point of view, what was the advantage of this machine in sieges? The ball thrown by its single arm followed a very high parabolic trajectory, with a vertex placed at a height of over 50 m, compared to the plane of the machine’s position. Therefore, as the heavy ball ascended, it lost speed and once it reached the vertex its ascending speed was zero. From that moment, it began to fall with a progressive increase in speed, which, therefore, neglecting the air resistance on impact with the ground, had the same initial speed, and was therefore able to penetrate through roofs thanks to its weight. The effect, especially during night-time shooting and with incendiary projectiles, was terrifying, hitting civilian homes. This is a hypothesis that deserves to be properly discussed.

5. Conclusions

The procedure supporting the calculated data is based on the mathematical description of damage and the knowledge handed down from Greek and Roman treatises. The results were partially tested using reconstructions produced by Flavio Russo’s Archeotecnica for public and private institutions (Figure 8). If the criterion is properly validated, using advanced detection techniques and reverse engineering processes, it will be possible, in the near future, to build (im)material twins [20] of “certified” machines. In fact, the process of quantifying the terminal ballistic parameters makes it possible to determine the module diameter, on the basis of which the weapon’s morphology will be proportionate and thus, inductively, the machine’s power can be inferred.