The engineering

The model, written in Python for this project, treats a trebuchet as rigid parts joined by pins and derives their equations of motion by Lagrange’s method. It covers six layouts: traction, a fixed or hinged counterweight on a fixed or wheeled frame, and the floating arm, whose weight drops straight down in guides while its axle rolls forward on a rail. Friction in each bearing follows the load on it, and the stone flies with an assumed drag coefficient of 0.40.

A throw is integrated in four stages. The stone slides along the trough while the arm begins to turn, held on the trough by a constraint until its upward push reaches zero. It then swings out in the sling, which must stay taut. Release comes when the sling’s ring slides off the prong, or at the ideal instant for range when the model stands in for a crew that has tuned its prong. Then the stone flies through the air to the target’s level, and the machine swings on without it for another 1.5 s so that its loads after release are recorded too.

The algebra was done by computer, so no equation was typed by hand, and the same generated code drives the Python model and the Proving Ground. For a hinged machine at rest in the trough the arm’s starting acceleration matches Horsdal’s closed form: 5.78 rad/s² against his 5.79 on his 19.1 t design. Without losses the model conserves energy to one part in ten billion.

Every angle is measured from the downward vertical and is positive in the throwing sense, so 0° hangs straight down and 180° points straight up. A link at angle a points along d(a) = (−sin a, −cos a). The Jerusalem 1187 design is cocked at θ₀ = 26°, its long arm pointing down and back.

With the axle at P = (x, H), the tip S, the short-arm end K, the counterweight C and the stone p lie at

For the hinged counterweight on a fixed frame the kinetic energy T and potential energy V are

With ℒ = T − V, each coordinate q_j obeys d/dt (∂ℒ/∂q̇_j) − ∂ℒ/∂q_j = Q_j, where Q_j collects the losses and any crew force. Together the equations take the form M(q) q̈ + h(q, q̇) = Q, with M the mass matrix. What drives the throw is the moment (M L₂ − m_b L_CM − m L₁) g sin θ: the counterweight’s turning moment less those of the beam and the stone.

A fixed counterweight is the hinged one with L₃ = 0 and ψ = θ. On wheels, the frame’s displacement x enters only through speeds, so the frame rolls to keep the centre of mass from moving sideways and the counterweight falls more nearly straight. In the floating arm the pin fixes the axle at P = (−L₂ sin θ, H), so the counterweight drops straight down.

Release by prong: a ring at the free end of the sling sits on a prong at the tip, and slides off once the sling-to-arm angle δ = φ − θ reaches δ_r = β − 90° + arctan μ_p. A larger prong angle β releases later. Friction at each pin is a torque equal to the friction coefficient times the pin’s radius times the force on the pin. The pin forces depend on the accelerations, which depend on the friction, so each evaluation goes round that loop three times; more passes change the range by 4 × 10⁻⁶ m.

Efficiency compares the stone’s energy at release with what the counterweight can give up:

Δh_CW is the drop of the counterweight’s centre of mass from the cocked position to the lowest point it could reach. Energy left in the machine at release counts as a loss, and the beam’s weight is not netted off, so a heavy long arm is penalised. For the 1187 design that gives 72 kJ ÷ 96 kJ = 0.75; for the Optimal, 129 kJ ÷ 142 kJ = 0.91.

Two other measures appear for comparison with the literature. Horsdal’s efficiency adds the stone’s height at release and nets off the beam’s weight, so his 91% design scores 0.76 on ours. Siano’s range efficiency, which Saimre also uses, compares the range with that of an ideal machine putting all the counterweight’s energy into a 45° throw in a vacuum: η_R = R ÷ 2(M/m) Δh_CW. Horsdal found an apparent consensus in the literature that the stone could receive at most about 70% of the energy the counterweight gave up, and his own optimised hinged designs put more than 90% into the stone in calculations without friction or air.

Where the rest goes: in a throw of the 1187 design the bearings take about 3 kJ and the air about 1 kJ on the way, and about 20 kJ stays in the swinging beam and counterweight after the stone has left. On the Aarhus laboratory engine friction and air take about 5%. For both designs the energy books balance to one part in a billion.

In flight the stone is a point mass with quadratic drag in still air: F_D = −½ ρ_a C_d (πd²/4) |v| v, acting in the machine as well as in flight, with the pouch counted at the stone’s own area. The 1187 design’s stone would fly 741 m in a vacuum and flies 580 m in air, so air costs about 22%; the faster Optimal stone loses 32%, because drag grows with the square of the speed.

C_d is uncertain by a factor of two or more. Stones of 0.08 to 0.4 m at 20 to 66 m/s fly at Reynolds numbers of about 10⁵ to 2 × 10⁶, across the drag crisis, where a sphere’s drag falls sharply as the air at its surface turns turbulent. A smooth sphere’s drag coefficient is about 0.4 up to a Reynolds number of 2 × 10⁵ and about 0.09 by 4 × 10⁵, and roughness brings the crisis on earlier and holds the drag beyond it at up to about 0.4. The model uses 0.40 for rough stone. Horsdal assumed 0.25 in one paper, and he and his colleagues about 0.5 in another. Around the hinged optimum a coefficient of 0.10 gives 698 m and 0.50 gives 550 m, against 580 m at 0.40.

Air density is 1.14 kg/m³, the standard atmosphere at an assumed 750 to 800 m for Jerusalem. Ancient numbers are reports: Josephus’s talent at two stades would need 55 kJ at launch and land with 41 kJ if it were true, and the arithmetic says what the report would mean, not whether it happened.

The scoreboard is range divided by beam length, R* = R ÷ (L₁ + L₂). Built to the same proportions, a trebuchet twice the size throws twice as far in a vacuum, because its equations of motion, written in proportions, do not depend on size. Horsdal’s scaled engines from 1.2 to 10 m all throw 33.7 times the length of their beam. Philo of Byzantium sized every part of a torsion catapult from one module so that its proportions held at all sizes, and R* rests on the same idea.

Shrink Horsdal’s 19.1 t design to a quarter of its size and it throws 112 m from a 1.8 m beam; double it and it throws 900 m from a 14.7 m beam; in a vacuum both score 61.3. With air a bigger stone keeps more of its speed, so R* in air rises slowly with size, from 52.4 at a quarter size to 56.4 at double. R*_ideal, computed in a vacuum without friction and with the ideal release, ranks proportions; R*_ref adds drag, friction, a prong release and a beam sized to its own loads at the reference scale, and decides every feature.

Horsdal divides range by the long arm alone. We use the whole beam because the short arm is part of the machine’s size and cost. Options are compared only at equal M/m, because R* rises with it: from 20.4 at M/m 25 to 113.9 at 400 for R*_ideal. Strength does not scale: twice the size puts twice the stress in the same beam, so materials are judged at one reference size.

Both designs are sized at one reference scale. The beam is 10 m long, inside the 8 to 12 m that Viollet-le-Duc gives for a trebuchet’s beam and the 10 to 12 m of Finó’s working calculation. The stone weighs 20 kg, a middle class among the catapult stones excavated at Arsuf. The counterweight is 3,000 kg, so M/m is 150, between Hansen’s 133 and the 191 of Horsdal’s reference design.

The designs are compared in two builds. The period build uses what a twelfth-century engineer could make: a tapered white ash beam, greased iron journals in hardwood (μ 0.2 on a 7.5 cm journal), a hemp sling worked at an eighth of its breaking load, a rubble-filled box and a hinge limited to 5 Mg for a timber box on iron straps. The modern build uses a carbon-fibre tube, rolling bearings (μ 0.0015), an HMPE sling at a fifth of its breaking load, a steel block and an 8 Mg pin. Both share a drag coefficient of 0.40, a stone density of 2,600 kg/m³, air at 1.14 kg/m³ and a trough with no friction.

Timber is worked at 0.375 of the clear-wood bending strength: 0.75 for the exclusion limit times 0.5 for hand-picked logs, with the usual allowance for impact loads held in reserve against fatigue over hundreds of shots. White ash breaks at 106 MPa in bending and is worked at 39.8 MPa. Steel and aluminium are worked at half their yield strength, wrought iron at a quarter of its ultimate strength, and the carbon-fibre laminate at 290 MPa. Per kilogram, ash and the straight softwoods make better beams than oak: at the 1187 geometry a white pine beam throws 594 m, Douglas fir 587, ash and elm 580 and white oak 566. Ash was kept for its toughness under repeated shock and its precedent in the documented reconstruction arms.

Aarhus (+13%), Nykøbing (+7%) and Visby (−4%) agree with the model as built, which overshoots laboratory throws by 10 to 15%. Varbola’s claimed 120 m, against 178 m with the rule’s 286 kg beam, would need a beam near 400 kg or a poorly tuned release. Black Widow’s single 270 m shot lies beyond all 80 sampled machines. As printed, Favé’s weights cannot swing a beam heavier than 254 kg, and a short arm of about 1.1 m would reproduce his 175 m.

Walked to the optimum one change at a time, Hansen’s machine gains 16.4 from an ash beam sized to its loads, 14.3 from raising the frame and only 1.5 from re-proportioning: the beam supplies 51% of the gap and the frame height 44%. Across all five machines the beam supplies 44 to 94% and the frame height 15 to 44%. The beam’s share is only as firm as the guess: at half the assumed weight its step shrinks from 16.4 to 4.5.

Held to Hansen’s frame height, an axle at 0.45 of the beam’s length, the best machine the optimiser finds has an arm ratio of 4.7, a hanger 1.0 times the short arm, a sling 0.76 times the long arm and a start 57° from the vertical. Hansen’s has 4.9, 1.1, 0.85 and 58°, and with the same ash beam it throws within 4% of that best machine. The low frame itself costs a quarter of the range. Hansen’s machine already had a hinged counterweight and a prong release; on its proportions a fixed weight would cost 14%, a prong set 10° early 5% and dry wooden bearings 3%.

On the north the sappers had to cross a rock-hewn moat 14 to 20 m wide and more than 9 m deep, then pass a forewall 3.2 to 4.5 m thick standing 8 to 10 m in front of the main wall. The main wall is unmeasured, so the working values of 11 m high (9 to 15 m) and 2.5 m thick (2.0 to 3.2 m) lean on walls of its age such as Pisa’s, which have a weak core and average 11 m by 2.2 m. It is taken to be faced in meleke, the Old City’s main building stone, which is light (1,976 kg/m³) and weak, crushing at 11 to 16 MPa.

No source says how far back the engines stood. Saladin’s tent was within reach of the defenders’ weapons and his engines worked behind screens. Shooting a later steel siege crossbow, Ralph Payne-Gallwey reached 421 m (460 yards). Saladin’s engines were most probably traction machines, since the counterweight was only then appearing, and these threw 100 m or more. The stand-off, the distance from engine to wall, is taken as 70 to 150 m, a judgement rather than a measurement.

From out of range the 1187 design mostly misses: a stone coming down at 48° drops past the wall’s 11 m while travelling only about 10 m forward, and small differences in the stones and in the moment of release shift the landing point by 13 to 16 m, so only 27 to 30% of these shots strike the face; about 40% fly over into the city and the rest fall short. Moved in to the stand-off, with its sling and release re-tuned, it throws 108 kg to 98 m or 75 kg to 165 m, striking with 68 to 69 kJ, and 68% of the 108 kg stones hit the face. Released late, a 20 kg stone flies almost flat and hits 99.5% of the time with 71 kJ. The sling, not the beam, is the weak point: it pulls 28% harder at 108 kg and 64% harder in the flat shot.

The verdict leans on a model that runs 10 to 15% long, a borrowed wall, a judged stand-off and no tested link between impact energy and a breach. What holds it up is that the chroniclers’ sappers and the comparison with gunpowder agree: it was gunpowder that forced builders to thicken their walls, and a soft inner core like Pisa’s makes impact damage shallower but wider, so a clean hole is harder to open. Fortresses of the crusade era fell most often to mining.

What the 1187 design could have done at Jerusalem’s wall. Ranges are model values. Shares are from 200 simulated shots per setting aimed 8 m up the face, without wind. Model runs P7-R12, P8-R05 and P8-R06.

The Proving Ground runs a JavaScript port of the same equations, line by line. Checked against the Python model on 45 machines across all six layouts, and again on 52 after the designs, the historical machines and the Jerusalem settings were added, the largest range difference was 8.8 × 10⁻⁷ %, under a millionth of a per cent; release speed, efficiency, impact energy and peak loads agree to within 3 × 10⁻⁵ %. The two programs use different numerical solvers, so this tests the equations and the release logic, not one headline number.

Its simple mode changes six things on whatever machine is loaded: the counterweight, the stone, the arm ratio, the sling, the axle height and the release. The engineer mode opens every input in the model, in the symbols of the equations above, with six release modes and switches for the layout. The throw plays in three dimensions, drawn from the positions the physics gives, with a side view to scale beside it, and tabs plot the flight and energy, the loads, a sweep of one input, and a comparison of up to three machines.

What it leaves out: the beam, the frame and the sling are rigid, so its ranges share the model’s 10 to 15% optimism; the traction crew is fitted to one crew size; the stone’s drag is uncertain near the drag crisis; and it sizes no parts, so a machine that throws far on screen may not survive in timber. A sweep leaves the sling and prong untouched, so its curves show tendencies, not optima. It gives no numbers to quote: every number on this site comes from a model run with an identifier.