Frühe mechanische Rechner der Artillerie und Fliegerabwehr: Automatisierte Rechner für Geschossflugbahnen

Complete English translation of the original German-language document (24 pages).

Early Mechanical Computers of the Artillery and Anti-Aircraft Defense: Automated Computers for Projectile Trajectories

Overview

Hitting an aircraft in mid-flight with guns is extremely demanding. The projectiles of earlier heavy anti-aircraft guns fly for 10 to 15 seconds, and for larger calibers perhaps 20 seconds — which means the curvature of the trajectory must be precisely known. The trajectory is stored in the cam bodies of the fire-control directors, which calculate the point of impact — but beforehand the trajectory curve must first be known, that is, calculated in detail.

Manual calculations for complete projectile trajectories represent a very large amount of work. The entire trajectory must be pieced together computationally in small time steps. If the intervals are larger, the calculation proceeds more quickly but everything becomes less accurate — and vice versa.

The calculations can be carried out mechanically. The variables (position, velocity, direction of flight, etc.) are interrelated, influence one another, and are continuously updated. A rotating shaft represents, for example, the advancing flight time of the projectile, and all further quantities are updated at the appropriate scale. The intermediate results are read off, photographed, or continuously recorded as a curve.

In a typical classroom exercise (the parabolic throw), it is first shown how the method of operation works in principle. Then the summation gears of that era are introduced, with which mechanical summation is achieved: using rotating discs, balls, cones, etc.

In the main section, two concrete mechanical devices for the automatic computation of trajectories are presented:

P. Füsgen designed a trajectory computing device as part of his dissertation, presented in 1937 (at the firm Rheinmetall-Borsig). It is possible that the idea remained on paper.
P. Curti presented ideas for a trajectory integraph in 1937 (later developed further with F. Dubois). A prototype was built by Haag-Streit. In 1948 the device was acquired by the Swiss Army, in a realization by the firm Amsler & Cie. SH. The system was in use at the KTA (Armaments and Ordnance Department) for many years, was rebuilt and maintained; later, extensions for rocket projectiles were sketched out. From 1954 (plans) / 1956 (delivery) the integraph was built in a second form, with newly designed ball integrators and with a substantially altered functional diagram.

Appendix I shows how the muzzle velocity v₀ of projectiles was measured in the pre-electronic age. Without precise knowledge of v₀, no trajectory can be calculated. >> Page 18. Appendix II gives a chronology of the development of the Curti device, including notes on the redesign of the device in 1954/55. >> Page 19. Appendix III describes matters that remain unclear. >> Page 22.

The author: André Masson, CH-4900 Langenthal

May 2017

The Oblique Throw — Recollections of School Instruction under Hans Bützberger

Secondary School Langenthal, ca. 1960

The basic rule is: every second, the velocity of a stone thrown obliquely upward increases by 10 m/s downward, as a result of gravity. One throws the stone, and at each new position after each second adds a velocity of 10 m/s downward — and so forth, until the stone touches the ground again.

A sequence of velocities reached at the end of the one-second intervals is obtained. Since the path always runs parallel to the velocity, the resulting path can also be regarded as a trajectory, as a view of position. Everything would now need to be repeated with finer intervals: that would give more work but fewer corners. And air resistance would also have to be taken into account, if only one knew how…

In each interval (e.g., one second) always the same operation: new velocity = old velocity plus 10 m/s downward added. This produces a series of parallelograms.

With ever finer intervals everything would become ever more accurate — that gives ever more work. Instead of working with graphical parallelograms, one can also work with columns of numbers.

Continuous Additions — A Perpetual Patchwork

Exactly as in the classroom exercise, a “proper” calculation of the projectile trajectory also proceeds. Computing an entire trajectory by hand means hours of work. Since everything must be as accurate as possible, the intervals of the individual calculation steps must be kept small. In addition to the lengthy work, the risk of unavoidable errors also increases with many calculation steps — and the errors propagate ever further, remaining present in all subsequent steps.

Figures on the manual computational effort have been found. The following figures presumably refer to a trajectory “down to the ground again,” as is normal in artillery. In anti-aircraft work, only a short initial segment of the path would be needed — but when the computation is compared with actual test shots for the purpose of adjusting air resistance and verifying accuracy, the entire trajectory must still be computed all the way back to the ground. The calculation intervals are not all equal in size — they are smallest around the speed of sound.

1936, Prof. Cranz: For a “normal trajectory,” a computation time of approximately 12 hours is required (at an accuracy in range of 4%).
1937, Dr. E. Pflanz, Stuttgart, presents a manual semi-graphical approximate solution requiring a computation time for a normal trajectory of 3 to 4 hours (compared with other methods that require 10 to 20 hours to achieve the same accuracy). Individual intermediate steps are no longer calculated but instead measured off with dividers from special auxiliary curves.
1938, Ordnance Testing Section, Thun, during the study of the trajectory computing device by P. Füsgen, described further below:

“An exact trajectory calculation (by hand) requires on average 45 working hours. A complete trajectory chart for an anti-aircraft gun for one charge and one type of projectile requires the computation of: 6 trajectories for determining the form value, 9 trajectories for the trajectory chart at standard air density, 5 trajectories for determining correction values — totaling 20 trajectories, corresponding to 900 working hours, that is, more than a month for 4 qualified personnel.” (The strong desire to acquire a computing machine may possibly have pushed the figures for computation time toward the upper limit…). Printed in the trajectory chart of the 7.5 cm Flab Kan 38 are 16 trajectories.

In any case it is clearly evident what an enormous simplification a mechanical, automatic calculation of trajectories would represent! The goal, idea, and hope: one sets all values correctly (temperature, barometer, elevation angle, initial velocity, projectile shape, etc.), then the calculations should proceed automatically — continuously, without any manual intervention, and with infinitely fine calculation steps. One only needs to write down the finished intermediate values, possibly photograph them, possibly have the entire trajectory curve drawn automatically.

Historical Summation Gears — In Various Structural Forms

Here various gear forms are presented that “continuously sum.” In this process the magnitude of the quantity repeatedly added may remain constant (the output will increase linearly) or continuously and continuously change (the integrated quantity is then available at the output). That a rotation increases linearly with time is the normal case (integration over time), but need not be so. In the drum gear, time corresponds to a linear movement, while the rotation angle of the drum represents the output — the reverse of the usual.

Friction-wheel gear: The large disc rotates, for example, at constant speed; the small friction wheel experiences a different speed depending on the distance from the center. The angle traversed by the small wheel corresponds to the distance from the center of rotation, summed over time: distance = velocity times time. (From Ref. 7.)

Friction-wheel gear: Certainly two, possibly three small friction-wheel gears in a gun sight. One rotating disc sits entirely at the top horizontally, the second to the left vertically. By means of a lead screw, the friction roller is displaced. The large disc will be less than 3 cm in diameter. The mathematical task in the mechanical computing sight is unknown. Aviation and Anti-Aircraft Museum Dübendorf. The optical sight at the rear does not belong to it.

Cone gear: The friction wheel in the foreground is displaced by the hand wheel at lower right; control or reading with the magnifier at the long scale with vernier. The cone circumference increases the further the friction wheel moves to the left. If the cone runs at constant speed, a variable, finely adjustable speed is produced at the friction wheel. Built into a telescope in the watch museum in La-Chaux-de-Fonds. Estimated cone length approximately 30 cm.

Spherical cap gear: The friction wheel at the top remains always at the same location; the rotating spherical cap beneath it is rotated away toward the viewer. This is said to be constructionally simpler than a clean longitudinal displacement of the friction wheel. The circumference of the spherical cap at the point of contact does not increase linearly with the rotation angle. Module with two spherical caps, from Ref. 5, German angular fire-control device WIKOG 9 SH; this form was also used in the KdoGt 40 of the Wehrmacht.

Regarding the cone gear in the telescope: It is installed in a meridian instrument that can only rotate up/down, i.e., about the horizontal axis (and always remains fixed pointing toward the south). The gear therefore cannot compensate for Earth’s rotation as with a normal telescope. However, it is possible that only the vertical velocity of the stars is compensated — perhaps this offers advantages for time determination when the star passes through the field of view exactly horizontally at the meridian.

Ball integrator after Amsler: Two friction wheels (input, here time, and output) are led at right angles onto a freely rotatable ball whose axis of rotation is not yet determined. The circumference at the location of the friction wheels depends on the position of the rotation axis. A third roller rests on the ball, neither driven nor driving; its position (angle) determines the rotation axis of the ball, and thus the ratio of the speeds of the other two rollers. Used in the Curti-Dubois-Amsler integraph.

Other Amsler ball integrators have also been described, with different sequences of input and output. The sketches here correspond to the integrators as they were used in the ballistic integraph of Curti / Dubois (plans with these integrators from 1942, construction 1948; see the chronology in Appendix 2, page 19).

Drum gear: The small wheel with a sharp blade is pressed against the drum with an obliquely set axis, simultaneously displaced along the drum axis. The obliqueness of the wheel axis is the input variable; the rotation angle of the drum executed so far is the output quantity. The drum produces no force for subsequent gears; the drum angle must previously be “servo-tracked” electrically, i.e., electrically copied. Used in the Füsgen dissertation.

Internal ball integrator: Servo control of the hand wheel for the azimuth angle of the fire-control director KdoGt Gamma-Hasler. The electric motor 128 rotates the spherical cap at constant speed. A deflection of the hand wheel 1 adjusts via gear wheels the obliqueness of the two internal sensing rollers. Their rotation is compared in a differential gear; the difference in rotation rates exits the housing to differential gear 120, from there further as a finely adjustable rotation of the entire fire-control director. Realized in this way from the 1943 model onward. At the very bottom, fixed ground, connected to the base plate. Image: diagram of KdoGt 43/50R at the Aviation and Anti-Aircraft Museum Dübendorf.

In all of these summation gears, one input channel often corresponds to time and is represented by a steady, uniform motion. The second input is the quantity (fixed or also variable) that is to be summed, i.e., integrated, over time. This second input is represented in the individual gear forms as follows:

Gear type	Second input representation
Friction-wheel gear	Linear distance of the friction roller from the center of the disc
Cone gear	Linear distance of the friction roller from the apex of the cone
Spherical cap gear	Tilting of the rotating spherical surface by an angle
Ball integrator after Amsler	Setting of the ball axis with the aid of the guide roller — an angle
Drum gear	Setting of the oblique angle of the blade wheel — an angle
Internal ball integrator	Setting of the asymmetry of the two sensing rollers — an angle

The Trajectory Computing Device after Peter Füsgen, VDI (Dissertation, TH Aachen, 1937)

Peter Füsgen worked at the firm Rheinmetall-Borsig, Düsseldorf (armaments division).
The device was conceived as part of the dissertation but was possibly never actually built by 1937. Whether it was ever later constructed and put into service in this or a similar form is not known. That nothing had yet been constructed up to the time of the dissertation can be inferred from individual words: the description deviates in individual details from the “planned execution.” Thus (in contrast to the drawn diagram), ball bearings are “provided for” throughout (Diss. p. 14).
Mathematics Professor Josef Heinhold (Munich) does not know, even in 1943, whether the Füsgen device ever proved itself in practice.
A picture (3 MB) of the overall diagram can be supplied separately; it may be more useful (for enlargements, taking parts from it, etc.) when not embedded in a Word or PDF format. Here only partial excerpts are shown.
The interconnection of the parts and the feedback of the variables does not come across well in the Füsgen device. It appears to be the same as in the device described below from Amsler in Schaffhausen. There, clear pictures are available (see below, page 13). Only the air resistance section differs significantly between the two installations.

Construction

The device is mounted on four levels. In the front part the device rests stationary on the table (levels 1 to 3); in the rear part the gears on levels 2 to 4 are built on a movable, motor-driven carriage that travels from left to right past the stationary part.

Level 1 (stationary part only): Time motor, moves the entire carriage. Various start-stop devices.
Level 2: Integration drums for x and for y. The drums and the reading of the values (coarse and fine range) are on the stationary part; the controls for the oblique positioning of the small wheels that influence the integrators are all on the traveling carriage.
Level 3: Integration drums for v and angle φ. Distribution as above.
Level 4: Air resistance calculator, on the carriage only. Drums for storing air resistance and air density. Setting of the projectile shape. The stationary part has nothing on this uppermost level.

In addition, the stationary part has two hand levers for manual operation: start, stop, setting of initial values, releasing or locking drum brakes, lifting the small rollers, etc. — all of this, in a sense, outside the computing process.

The four drums drive the easy-running readout displays for the instantaneous values, each with coarse and fine range. Where the outputs of the integrators are used mechanically, a power amplifier is needed, i.e., an electrical servo-tracking of the values: the output of the three drums for y, v, φ controls fine electrical contacts that command an electric motor (forward and backward), whose driven shaft follows the drum output exactly at all times.

The following is a picture of the upper levels 4 (air resistance calculator) and 3 (control of the variables v and φ, velocity and flight angle):

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The uppermost level 4 is located entirely on the traveling carriage. At level 3, the integration drums for v and φ, their servo motors (outside), and their readout units (inside) are on the stationary part of the installation. The air resistance drum at the top depends on the projectile shape and must be exchanged for a different caliber; it rotates according to the projectile velocity v. The lower drum, oriented in the direction of travel, gives the instantaneous air density — it rotates according to the height y. Vertical shafts to level 4: left v, signal leads upward; right shorter — air resistance, downward; right longer — y, the signal comes up from the y integration drum on level 2 and is motor-amplified.

The air resistance calculator on the uppermost level is discussed in more detail further below.

The task of the gears next to / behind the integration drums (at the same level) is to correctly adjust the oblique position of the small wheels that are in contact with the drums and roll on them. As the rear part of the device travels past with increasing computing time, i.e., projectile flight time, the wheels rotate the drums to the correct degree, so that the rotation of the drums at all times corresponds to the current value of the variable.

Up to an oblique angle of the wheels of 40° a clean rotation of the drum should still be possible. As a precaution, in the Füsgen machine the wheels are never deflected by more than 30° from the direction of travel.

Time scale:

200 mm total carriage travel corresponds to 2 seconds projectile flight time
6 seconds computing time corresponds to 2 seconds projectile flight time
33.33 mm/sec corresponds to the carriage travel speed

Lower levels 3, 2, 1: see diagram on next page

The levels 3, 2, and 1. At the very bottom far left: electric motor drives the moving carriage by means of the rack gear just below the long dark coil spring. The four integration drums sit on the stationary part; everything behind them is on the movable carriage. At the far left and far right outside are two hand levers, with which the installation is controlled by means of cam discs: release of the variables for setting the initial values, brakes for the drums, lifting of the oblique small rollers from the drums (the small rollers travel with the carriage), return of the carriage with frozen computed values for the next pass, etc. Behind the left lever the front guide of the carriage is visible; there is also one at the rear. Estimated width of the entire installation: approximately 1.2 m; height: approximately 1.5 m (estimated from the 20 cm carriage travel).

Data Output

All intermediate values of the computed trajectory (x, y, v, φ, t) must be read visually at the double-wheel scales with coarse and fine range and immediately written down by hand. This applies for the entire trajectory from beginning to end. Presumably the machine must be stopped each time and then restarted after the interruption. No trajectory is drawn with a stylus on paper. In this sense the Füsgen device is clearly inferior to the Amsler integraph (see further below). — Drawing the computed trajectory on paper would actually be only a small additional detail.

Data Ranges and Scales

All four integration drums rotate multiple times to run through one trajectory. Thus approximately the entire circumference of the flight-angle drum (φ) corresponds to only three degrees — meaning that if the shot from an anti-aircraft gun is initially fired at 60° to the horizontal, the drum must rotate 20 full turns to reach the point of culmination where the projectile is traveling horizontally, then approximately 20 more turns in the same direction until the projectile returns to the ground (for illustration purposes — the trajectory is symmetrical only without air resistance). To cover a horizontal distance of 10 km, the x-drum must rotate 10 full turns, etc.

Mechanical Realization of Time

When the carriage has traveled the full drum length, all drums must be braked, the integration wheels with blocked oblique position lifted from the drums, the time motor decoupled, the carriage then pulled back to the starting position by spring force, coupled again to the motor, and once all locks on the integration drums are released, the next pass can be made. One pass (travel of 20 cm) corresponds to 2 seconds of projectile flight time. For artillery: partly 20 to 30 passes!!

Details and Particulars: The Acceleration Influences the Velocity

First, a relatively simple actuation of a small roller that serves the velocity drum (lower right in the excerpt). The short vertical coil spring slightly to the right of the image center pulls the tiltable fork lever up from behind, so that the small wheel is pressed against the drum. When the vertical rod (to the right of “1:25”) pulls downward, the small roller lifts off — without losing the angle information (since the gear wheels above the small wheel remain engaged).

The velocity (drum at lower right) is updated by continuously summing the acceleration. The v-motor, cut off at lower left, copies the position of the v-drum and leads it via the vertical rod all the way up to the air resistance calculator. The slightly twisted small roller is moved to the right with the carriage; therefore the drum rotates. The tangent of the angle at the small roller must be proportional to the acceleration. Added to the push rod at the very rear is the total acceleration in the direction of flight: the gravitational component plus air resistance. The pin in the hollow guide rail ensures that the tangent of the rotation angle to the small roller corresponds precisely to the acceleration (triangle at top).

The Velocity Determines the Distance

Now it becomes somewhat more complicated — one wishes to do the machine justice! In the image below, the x-drum (horizontal distance of the projectile) with the x-readout in km and in m. The x-drum must continuously sum the instantaneous x-component of velocity over time. For illustration with wrong numbers: 5 m/s gives a distance increment of 0.5 m every 0.1 seconds — this is summed over all seconds while the velocity itself may be continuously decreasing.

To the right of center, the φ-shaft 3 brings from above the instantaneous inclination angle φ of the trajectory. Using a rotating pin, two components are generated in the x- and y-directions — one component goes to the left toward the x-calculator, one to the right toward the y-calculator. This first gives only sin/cos from the unit length; the instantaneous velocity v must still be multiplied in by the 5 guide rods in the left third of the image.

At lower left, the large wheel brings in the velocity from above, thereby displacing the lowest of the 3 pins in the five-rod structure. The entire rod mechanism enables multiplication of two continuously changing variables. The long oblique rod going to the left rear is directly coupled to the required inclination of the small roller, i.e., the tangent of this angle = vx.

The rod structure is not easy to understand. The construction resembles a known form of multiplication (see also in the air resistance section, uppermost level, page 15), but is more complicated here. This will be related to the fact that not the product v · cos φ (i.e., the x-component of velocity) is to make up the inclination of the small roller, but rather that the tangent of the required inclination angle must correspond to the value v · cos φ. Often geometric auxiliary constructions are drawn into the plan above the partial gears, and also explained in the text. With some effort it could be confirmed that the rods perform the required task. — The multiplication with rods is based on similar triangles; a proportionality equation between corresponding sides (a : b = c : d) can also be written as a multiplication: a · d = b · c.

How Accurate Were the Trajectory Calculations?

What is primarily imprecisely known is the air resistance (braking force) on the projectile at known velocity and air density, as well as air density and temperature as functions of height.

It is not easy to read into the details and evaluate everything correctly. The author has tried, with limited success. It gives the impression that there were various “schools” for treating air resistance, and it is obvious that a great deal was copied from each other and taken over from other countries. In part, exactly the same numerical values and exactly the same completely arbitrary interval limits for velocity that came from the Academy of the Russian Tsar in St. Petersburg around 1895 were still circulating 40 years later (1937) in Nazi Germany — great caution, verging on distrust, is warranted!

More detail on this is found in the earlier work in this series: “Calculating with Steel Bodies.”

Regarding the Füsgen machine, the Ordnance Testing Section in Thun established in March 1938 that air resistance may not be assumed as Füsgen does — as a constant c (incorporating air density, caliber, etc.) times a function of projectile velocity — but must instead be written as c times a function of the Mach number. At greater altitudes air pressure drops, but temperature also falls, and therefore the speed of sound is not the same everywhere. It would be necessary additionally to be able to assume an arbitrary temperature gradient at each altitude and input it into the machine. This is how it was done in the Amsler integraph (see further below).

Despite these improvement requests, the ordnance testing specialists from Thun wrote on 29 March 1938: “Of the ballistic computing machines known so far, the one by Füsgen is the first of which we believe that it is practically realizable.”

This again indicates that the device was only conceived, not built, up to 1937 or 1938.

Predecessor Devices

P. Füsgen mentions four predecessor devices or predecessor ideas, without providing precise details. Apparently, air density as a function of height was usually left constant (or taken as a first approximation, with further approximations then needed):

L. Filloux: Drawing of hodograph v(φ) and trajectory y(x), 1908. Whether the described device was ever executed, Füsgen does not know.
L. Jacob: The device was actually executed; draws the hodograph v(φ), 1911, Paris.
E. Pascal: Draws a modified hodograph: log(v) against −sin(φ). There exists a picture of an executed device, 1914, Naples.
A. Perrin: Velocity and deceleration are resolved into rectangular coordinates. Planimeter rollers; Füsgen suspects difficulties with friction-free displacement of these rollers parallel to the axis. The device produces no graphical records. 1922, France. No information on whether the device was ever realized.

The Trajectory Integraph after the Curti-Dubois-Amsler System, called “Mariandl”

Construction

The computer consists of three tables standing side by side, each approximately 1.30 m wide. On the left-hand table is the entire air resistance calculator; on the middle table, as the core piece, are the four integrators (= summation units, later rebuilt to five) as well as a drawing device for the curve v(φ) called the hodograph; and on the right-hand table, the rectangular recording of the trajectory on paper takes place, i.e., y(x), height as a function of horizontal distance.

The mechanical operations are precise — and at the same time sensitive. The ball integrators with their fine rollers (cf. page 4) do not generate enough force to pass on rotation values and to drive a further gear at the next table. Force, slippage, and precision are incompatible with one another. For this reason sensitive values are immediately “electrically servo-tracked,” i.e., by means of fine contacts an electric motor is controlled such that a forcibly driven rotating shaft always occupies exactly the same position as the previously determined quantity. The motor then has sufficient force to drive the subsequent stages (even against some resistance).

The totality of the three tables, with the air resistance calculator on the left. In the middle, the four integrators. At the very rear, the normal recording of the trajectory y(x) — not discussed further here, as it is rather without puzzles or wonders. To the right above the integrators, the camera that captures the intermediate values on film. Beneath the tables there are additional rotating shafts and gears. (Images from Ref. 2, photographs of contemporary paper photographs from the HAM Foundation, Thun.)

In the middle, the four ball integrators, surrounding the somewhat abstract recording of the curve v(φ) on paper, called the hodograph. The four integrators, arranged from upper left to lower right, update (clockwise): velocity (here designated w), x-coordinate, elevation angle φ, and height y. All integrators sense the same time (rotating shaft from the central motor at top). The angle φ is also needed in every other integrator. Roughly in the center: the decomposition of w into horizontal (for x) and vertical (for y). On the left is the air resistance mechanism, with barometric pressure (right) and Mach number (left), including a template that must be newly computed for each projectile. At lower left: the temperature at the projectile’s location is computed from an adjustable gradient (e.g., 0.5 °C/100 m). At the very bottom, five variables for the camera. Plan drawn: 12 August 1948.

The camera captures in the same image the current numerical values for five variables: t, φ, v, x, y. The camera can be controlled to take a picture at fixed time intervals (every 0.1 to 1 second), or after every 100 m or 500 m in the x- or y-direction. Time markers can also be set on both paper curves.

Roller chain technology and traction bands (left) — in combination with electron tubes. In the foreground the exchangeable template, which gives for each projectile type the velocity dependence of air resistance. From the right, a fine wheel scans the template — the servo motor (to the right behind it, bright) amplifies the read value and passes it via the fine shaft to the left.

So that the fine contacts of the electrical servo-tracking are not traversed by the powerful motor current, electron tubes are interposed. The mechanical contacts go to the grids of the tubes, and the tubes then switch the currents for the motors (two tubes are needed for the forward/backward directions). On the left at the rear are 12 electron tubes.

Images 956, 973: Electron tubes already in use from 1948 onward. The y-output requires double power and therefore has two motors in tandem, “each with its own pair of electron tubes” — giving a total of 12 tubes, as illustrated.

Diagram of the computing devices in the Curti integraph. The four integrators are at the corners of a square and update (clockwise) the velocity (here designated w), the x-coordinate, the height y, and the angle of the projectile trajectory φ from the horizontal. All integrators sense the same time (rotating shaft from the central motor at top). The angle φ is also needed in every other integrator. Around the center: the decomposition of w into horizontal (for x) and vertical (for y). On the left is the air resistance mechanism, with barometric pressure (right) and Mach number (left), including a template that must be newly computed for each projectile. At lower left: the temperature at the projectile’s location is computed from an adjustable gradient (e.g., 0.5 °C/100 m). At the very bottom, five variables for the camera. Plan drawn: 12 August 1948.

(Blue: diazo copy)

Air Density, Air Resistance

The assumptions regarding air resistance are of great importance in the calculation of a trajectory. And it is precisely here that great uncertainties exist… how is one to obtain precise information in the pre-electronic age about the braking of a supersonic projectile? (Cf. also pp. 22/23.) Regarding the treatment of air resistance, the two devices — Füsgen / Amsler — differ significantly. It is impossible here to discuss all the uncertainties in connection with air resistance in detail, but the principal differences between the two devices will be shown.

	Füsgen	Curti-Dubois-Amsler
Set once by hand	Projectile shape (combination of various factors)	Template for form factor as function of Mach number; temperature gradient: …°C/100 m
Input from variables	v, y (i.e., velocity and height)	v, y
Calculated inside the air resistance computer	Air density (from height y)	Absolute temperature (from y and temp. gradient). Barometric pressure (from y). Speed of sound s (from temperature). Mach number (from s and v)
Output	Air resistance	Air resistance

The mechanism of the air resistance calculator of the Füsgen installation can be understood in its basic outlines. The tradition of the time held that the air resistance of a flying projectile should be conceived as the product of three factors:

Total braking deceleration = form factor × velocity factor × density factor

Form factor: depends only on the shape of the projectile: projectile cross-sectional area times correction value times form value divided by projectile mass (roughly… pages of explanations). Such a product is to be set on the Füsgen device by the hand wheel at the “projectile scale.”
Velocity factor: a function of velocity, often written as K(v) · v², because it had long been historically suspected that air resistance increases with the square of velocity. This is only accurate at lower velocities; as the speed of sound is approached, air resistance increases markedly — meaning K(v) rises toward the speed of sound, then remains approximately constant at still higher velocities.
Density factor: direct ratio of air density at the current flight altitude to normal air density, i.e., braking force is proportional to air density. Higher up in the trajectory, air braking is less — but for all projectile shapes and at all velocities, half the air density should also give only half the braking effect.

In Curti-Dubois-Amsler, on the other hand, the ratio of barometric pressure to normal pressure is taken, which does not give exactly the same result as for air density. According to the tabulated standard atmosphere, air pressure decreases by half from sea level to 5,500 m, whereas one must ascend to 6,650 m before air density is only half as great. Temperature also changes with altitude.

How well an approximation of these three mutually independent factors corresponds to reality remains open for now. Perhaps the approximations also serve only the goal of keeping the trajectory calculation within a somewhat acceptable framework — i.e., making it feasible at all.

The KTA (Ordnance Testing Section, Thun) does not compute the velocity factor using a function of velocity F(v), but rather F(M), i.e., according to this view air resistance should depend on the Mach number of the projectile (a multiple of the speed of sound). Since temperature decreases with altitude, the speed of sound also decreases slightly with altitude (typical 3% decrease at 3,000 m ascent), so that Mach number 1 is reached at higher altitude at a lower velocity.

Now follows the description of the air resistance calculator of the Füsgen installation, based on the original diagram, in which certain variables are additionally labeled:

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Uppermost level of the Füsgen device with the calculator for aerodynamic braking. Inputs: instantaneous flight height y from lower right, and instantaneous velocity from lower left. These values are each fed to a rotatable drum — for air density at the corresponding altitude, and for air resistance (without form factor) respectively. Output: completed aerodynamic braking, i.e., deceleration, to which the gravitational acceleration in the direction of flight is later added (in the differential gear further below, cut off in the image): at a steep trajectory the weight brakes or accelerates more strongly than at a flat one.

Calculation of instantaneous air resistance:

From the height, the local air density is to be determined.
From the velocity, the “standard resistance” is determined.
By hand, the projectile shape is set at the adjustment knob.
Multiplication of three factors.

The current velocity of the projectile rotates the air resistance drum so that the reading stylus at the left rear transfers the corresponding “tabulated” air resistance value to the linkage. Pin Z1 thereby adjusts the obliqueness of the diagonal rail. Using similar triangles D4 and D5 immediately below the air resistance drum, the product of the air resistance (without projectile form) and the form factor is formed and channeled downward via pin Z2 (the proportionality in the similar triangle can also be written as a product). With the aid of two movable guide rails further below, which always hold exactly the same direction, the air resistance value is now reduced to the instantaneous air density according to flight height — pin Z3 displaces itself according to the air density and thereby passes the finished final value of air resistance to a translation linkage. At the end, this value is converted by a gear wheel (below the projectile scale) into a rotation that is led downward.

Cut off in the image: a differential gear adds the projectile weight times sin φ, which (depending on the inclination of the projectile trajectory) also contributes to braking the projectile, or, at negative angle φ, accelerates it again once the projectile has exceeded its maximum height.

Air Resistance in the Curti-Dubois-Amsler Installation, called “Mariandl”

Here somewhat more effort is expended: from the projectile’s height, the barometric pressure is determined as well as the air temperature, where the rate of temperature decrease with height can be chosen. From this, the speed of sound is determined (varies as the square root of temperature), and from it, the Mach number of the projectile. A spiral sheet-metal template, which must be newly inserted for each projectile type being computed, is rotated by the Mach number and delivers the velocity-dependent component of air resistance. This is then converted to the air pressure at the current altitude — and all of this is continuously updated throughout the entire projectile flight path.

The output of the calculator is the deceleration due to air resistance, which is then further combined with a component of the projectile weight (depending on the steepness of the trajectory). All of this goes to the velocity integrator and allows the velocity to increase or decrease to the correct degree, just as in the Füsgen device.

The air resistance calculator occupies the entire left-hand table.

Air resistance calculator of “Mariandl” as diagram and as manufactured in metal. Centrally at top, the template for reading out the velocity dependence; rotation angle approximately 360° plus 90°; pickup from the left, then immediately electrical servo-tracking of the value. Left half: determination of Mach number; right half: barometric pressure as a function of flight height, proportional conversion of air resistance to this pressure. Along the lower edge: various manual settings (partly possible during computation) and selection of the rate of temperature decrease with altitude (dark box at lower edge, temperature readout above it). Bright left of center: servo motor for reading the template F(M).

(Images from Ref. 2, HAM Foundation, Thun.)

s = speed of sound, q = deceleration due to air resistance. B₀, By = barometric pressure at standard altitude, at current altitude.

Left diagram, 1942: At lower right, the current flight height y emerges from the integrator and drives the large barometric pressure drum with milled pressure information (here still without temperature influence). The y-shaft continues to the adjustable temperature decrease with altitude, then to the 1/sy-spiral on which the speed of sound is stored. From above, the velocity arrives, whereby the position of the pin in the middle of the left oblique lever is determined. Its left-right position gives the Mach number, which rotates the spiral template (called “the heart”). Air resistance read from “the heart” is led to the right oblique lever and determines the required q-value (horizontal position of the q-carriage), which is passed further to the velocity integrator. The barometric pressure from the spiral groove determines the angle of the oblique lever.

Right metal version, 1948: The y-shaft is not visible; according to the 1948 plan, it now runs beneath the table. The 1942 diagram still shows it differently. The speed-of-sound spiral, far left, is only very small but barely visible. At lower right are three readout scales: constant setting for standard pressure with hand wheel / larger scale — barometer at flight height y / y (and below to the right a hand wheel for barometer adjustment). Around the outside, on both oblique levers, are poorly visible scales for Mach number, v, air resistance as a function of Mach number, final air resistance q (i.e., reduced to current air pressure).

Installations by Vannevar Bush (USA — Brief Mention Only, without Details)

The two trajectory computers discussed were built precisely for their purpose — i.e., exactly tailored to measure. A different application calls for a different machine.

Vannevar Bush (1890–1974) built similar mechanical installations in the USA in which reprogramming was possible for new tasks. There were a number of input units, initial conditions, then adders, integrators, plotters, etc., that could be coupled with one another as needed by means of long rods (lying crosswise).

A first device, the “Differential Analyzer,” was built from 1928 to 1932 using purely mechanical means, as discussed here, but considerably larger. It had mechanical torque amplifiers instead of electrical servo drives.

Images: https://ub.fnwi.uva.nl/computermuseum//vbush_tbl.html and http://www.mit.edu/~klund/analyzer/

A second device from 1942 was already equipped with 2,000 electron tubes and reportedly weighed 100 tons. Name: RDA2 (Rockefeller Differential Analyzer No. 2). The device had 18 integrators and an accuracy to 1 per mille.

http://ethw.org/Differential_Analyzers

Th. Erismann of the firm Amsler in Schaffhausen knew of the Bush devices. He judges them as cumbersome, since the conversion to a new purpose would be very laborious. A purpose-built installation dedicated to exactly one task would have great operational advantages. This is said, however, by the Amsler specialist who wishes to sell his own devices (found in: offprint for the presentation of the new Amsler installation with seven integrators from 1955, see Ref. 2; unfortunately the place of publication was cut off in the photograph, possibly NZZ 12.9.1956; see images 1056–1062. Image collection: cf. p. 21).

The author of this work attended a lecture by Th. Erismann on integrating systems at ETH in the summer semester of 1967. Trademark: bow tie. Later, Erismann became Director of EMPA.

Appendix 1: Determination of the Initial Velocity of the Projectile

Various, mostly complicated texts are found on the Le Boulengé device, which for decades in most countries measured the muzzle velocity of artillery pieces; illustrations are rare. By means of the fall of two metal pieces released in quick succession, the time between the two release points can be determined.

(Image sources no longer known — Internet)

Using the time difference of falling metal rods, it is measured how long the shot takes from the first wire grid to the second.

There are minor printing errors: three times the creation of notch 1 is explained. A normal shot produces notch 3. Separating both circles from each other gives notch 2. Without any fall at all: notch 1.

Le Boulengé apparatus with coils: Using long rods and ropes, both coils are attached at a suitable distance from the gun muzzle. (Image from Ref. 3. Coil tests by P. Curti, Ref. 3, estimated at ca. 1926.)

Normally, wires stretched in a frame were shot through. These installations appear to have functioned reliably over many decades. During the Second World War, mobile units in Germany followed the anti-aircraft positions in rotation and tested all the guns one by one. After many shots the barrel is worn and the muzzle velocity declines.

The fire-control director must know how high the initial velocity is. However, only a single muzzle velocity can be set for all guns of a battery.

First Le Boulengé experiments from 1863, used until the Second World War. From approximately 1955 to 1964, the Swiss anti-aircraft forces used a modern electronic v₀ determination, even ahead of the artillery. Measurement base: two coils at a distance of 1 m, directly in front of the barrel; no wire is torn. In some cases the projectiles had to be pre-magnetized. See image below. The Boulengé device was used sporadically in Switzerland until 1974.

*) Found with the 12.8 cm Flak in Germany: The barrel life was 1,000 rounds, but could be extended to 2,000 rounds over the course of the war. (“Could” or “had to” be extended??) The heavy 7.5 cm Flab (anti-aircraft) cannons were already fitted with a fixed measurement base to measure the muzzle velocity of the projectiles. Measurement base: 1 m. Trials in 1954, introduction before 1960. In an electronic counter, the flight time through the measurement distance was displayed with an array of glow lamps. In four decades, one had to manually note the binary values, add everything up, then look up the value for the muzzle velocity in a table. Oerlikon brochure ca. 1958; correspondence, discussions, and measurements regarding the choice of systems ca. 1954–1960 are held in the Federal Archives. The contemporary witness U.W., who completed his military service with the 7.5 cm guns in 1962, recalls the measurements using the v0 measurement base. They were routine, i.e., not intended only for trials or individual actions. From around 1941 there was a v0 measuring instrument of the AFIF device (ETH) that measured v0 using photomultipliers alongside the projectile flight path with a 2 m base — more for research purposes, not for the troops. The Flab was somewhat ahead of the artillery with the v0 measurement base. Image from the Federal Archives, search under: Comparative measurements with the v0 measuring systems Weibel and Oerlikon.

Appendix 2

Chronology — Development of the Curti-Dubois-Amsler System

Italic: Other computers of the period

1937 Curti sends documents on his integraph to a major and looks forward to hearing his opinion on the occasion of a visit. Ref. 2. Air resistance is solved differently (using “rods”) than in the later realized device. Images 1070–1076. “Rolling blade and integrating carriage” are mentioned, which were “known from the Coradi integraph”! These would thus be the same integrators as in the Füsgen device.

1940 An early form of the computer is illustrated in Ref. 3, as “Trajectory Integraph Curti 1940, designer Dr. Francis Dubois. Manufactured by Haag-Streit, Bern” (thus not yet at Amsler SH). In the clear photograph, only three integrators are visible around the drawing area for the hodograph — specifically the classical friction-wheel integrators with flat disc (thus no ball integrators yet). The fourth, indispensable integrator is not visible in the image. The drawing board y(x) is visible, but of the air-resistance computer in its later form only early beginnings are discernible. The resemblance to the later form is striking; the Füsgen apparatus certainly did not inspire this design, nor vice versa.

In Curti’s book of 1945, the four “ball integrators” are already mentioned and the schematic is drawn as in the final form. A clue regarding the missing integrator of the prototype can be found there: Curti describes a possibility to build two integrators at once using a single turntable, with small rollers pressing from above and below simultaneously.

March 1943 The KTA (War Technology Department) requires, among other things, that the trajectory computer should be able to process v0 up to 1,500 m/s (very high! The 7.5 cm Flab cannon achieves v0 = 805 m/s against aerial targets, 840 m/s against tanks).

May 1944 The requirements of the EMD / KTA are assessed by the firm Amsler. Some things cannot be done (great altitudes above 14,000 m a.s.l., low barometric pressure down to 100 mm Hg). By dividing the range into two, velocity ranges of up to 2,000 m/s are made possible — beyond all values achievable at the time. A contract is about to be concluded (images 1097–1103).

1948 The device is built by the firm Amsler AG (according to information from the HAM Foundation, Ref. 6). Possibly earlier?? The plans were also already drawn in detail in 1942. A detailed operating manual dates from 1948.

From 1950 (at the time as an idea): A fifth integrator is added: v (velocity) and air resistance respectively are divided into two numerical ranges; this is considered advantageous (proposal of the KTA, images 1047–1049, 1053).

1953, 1955 Computation work by R. Sänger, Prof. ETH / W. Roth for projectile trajectories on the rented digital relay computer at ETH, the Zuse Z4 (approx. 900 hours, one of the largest cost items). Table of computational assignments: H. Bruderer. The Z4 operated with approx. 2,200 relays and a mechanical memory capable of storing 64 numbers. Program loops were realized by glued-together perforated film strips.

1954/55 Development of a new trajectory integration system by the firm Amsler. The ball integrators are newly designed, and all the mathematical connections rebuilt (image next page). It now has seven integrators, as well as new reading heads that optically and automatically follow function curves drawn on paper. Included in Ref. 2. (Images 922–947, 1056–1062). Multiplication is no longer performed using similar triangles — instead, the logarithm of both variables is formed, added, then anti-logarithmized, all using function tables and optical curve-following. There are still no electrical integrators; everything there is mechanical.

1956 The first unit of the successor type is accepted by the Spanish Navy. The calculation of rocket trajectories is already incorporated here — these must be unguided rockets, whose engines are very well known (thrust, mass reduction).

1956 Prof. Stiefel shows an illustration of the new Amsler integraph, with the brief note that the section for firing trials in Thun uses this device (hitherto the only known reference). Source: E. Stiefel: “Rechenautomaten im Dienste der Kriegstechnik” [Computers in the service of military technology], ASMZ, Vol. 122, 1956, No. 11.

1957 Analog computing device AR2 (tube-based), manufacturer Göttinger AG, manual by Prof. Erwin Engeler, with electrical integrators. To be found via H. Bruderer.

1955, 1956, 1958 Integration system 55, analog computer for fire control device 56 (proto-Fledermaus), integration system 58, all by Contraves AG. Digital computers: 1956 ERMETH put into operation at ETH, 1958 ZEBRA at EPUL, tube-based computers.

1959 Order from the KTA to the firm Amsler for the revision of the old five ball integrators, no later than August 1959 — the 1948 system is certainly still in use. Certain components in the drive of integrators 1 to 4 are to be “of the same type as in the multipurpose system,”

evidently there is yet another entirely different device alongside the ballistic integraph (possibly a freely programmable device?). Cf. images 1022, 1023, letter KTA to Amsler, September 1958.

Schematic of the newly designed Amsler system, drawn in 1954.

In all sketches and constructed systems from 1937 to 1948, a unified time drive led to all integrators; this is no longer the case. There are now 7 integrators instead of the original 4, as well as many “function tables” where pre-drawn mathematical functions are read optically from paper rolls.

From Ref. 2

Nonlinear functions are drawn with a precise ink line on the paper tape and read out by two optical scanning heads. The function value sweeps over the paper width multiple times — on overflow at the top, the curve begins again immediately from the bottom edge of the paper strip. For this reason there are two heads, so that one is always ready in the correct position. A light spot of approximately 0.15 mm is read, and the control keeps the function head always tracking along the boundary between the white paper and one edge of the black ink line. Paper width 50 mm, force-fit perforation. Optics, photocell, amplifier, servomotor. Light modulation for insensitivity to daylight and lamp light. From Ref. 2

Image numbers refer to a larger collection of photographs from various archives and museums, totaling more than 1,200 images and additional documents. The photographs cover all topics treated in this series (Mechanical computing devices of anti-aircraft defense). If all goes well, the images will eventually be available for further research in the library at Guisanplatz in Bern and in the Aviation-Flab Museum in Dübendorf.

Appendix 3

Unclear and Unresolved Matters

Hodograph: Already in the first discernible ideas of P. Curti from 1937, the recording of the hodograph is provided for. But why, exactly — what purpose does it serve?

The hodograph is the graphical representation of the velocity of the projectile (distance from the origin) as a function of the angle at which the projectile is moving (angle relative to the horizontal). Example: parabolic throw in a vacuum, launch angle of the stone at 45°, v0 = 50 m/s. At 45°, a distance (at any convenient scale) of 50 m/s is to be plotted = starting point. After launch the angle continuously decreases, down to the apex. At the top, the stone still has a velocity of 50 m/s times cos 45° and flies horizontally — the corresponding point in the hodograph lies vertically below the starting point. The hodograph of the parabolic throw (connecting all v-arrowheads) is a vertical line downward from the starting point. In flight with air resistance, the hodograph approaches the origin more closely (images 1065, 1067).

It is not clear what the hodograph is used for, once the desired projectile trajectory has already been fully computed. Idea, conjecture: After the computation of individual projectile trajectories, there will be much work to determine the intermediate trajectories printed in the trajectory tables, which were never actually computed. Perhaps the hodograph offers advantages in determining these additional, merely interpolated or estimated trajectories. Possibly these intermediate trajectories can be determined better with the aid of neighboring hodographs than from the trajectory curves y(x) alone. This is merely a conjecture made out of necessity!

Air resistance: The printed trajectory tables are accurate only if the air resistance of the projectiles at all possible velocities is precisely known. The air resistance can apparently be determined from a small number of computed and also fired trajectories — unfortunately it is not clear how this was done at the time.

Here is a conjecture based on imagination and few sources. In Ref. 9, p. 336, there is an illustration that can be interpreted as follows: on the rear tripod stands the camera, which during night exposures keeps the shutter continuously open and records the trajectory of a tracer round. On the front tripod, a disc rotates at a known speed and periodically interrupts the optical path. By measuring and rectifying the image, the velocity of the projectile along the entire trajectory section is obtained, from which the deceleration and braking force follow. However, the rotation of the disc must be known precisely, as must the distances to the projectile trajectory.

Ref. 9, Figure 8, p. 336, with the following caption:

From left to right: Maurice Garnier, Louis Fort, and Gustave Lyon with measuring instruments, undated. From [Patard 1930, p. 275]

Interpretation: In the upper right of the gap, the lens of the camera is visible, standing on the main tripod. In front of it a shutter; the electrical cables enable rapid rotation of the disc. The luminous trail of the projectile is thereby chopped — divided into segments of constant time duration. The distances can be determined from the photograph, i.e., v can be determined everywhere.

Switzerland from 1940: Ballistic chamber Wild; a pair of electrically synchronized cameras makes multiple exposures of the tracer projectile, with stereo evaluation. Viewing direction in the direction of fire.

Sources

Peter Füsgen: Flugbahn-Rechengerät [Trajectory computing device]. Doctoral dissertation for the degree of Doctor-Engineer, Technische Hochschule Aachen, 1937. The work is held in the library of ETH Zurich, as well as in the documents on the Curti/Dubois integraph at the HAM Foundation, Thun.
Documents on the trajectory integraph known as “Mariandl” by P. Curti and F. Dubois; ideas from ca. 1937 to 1959; several boxes of mixed documents, correspondence, plans, photographs, etc., at the HAM Foundation, Thun.
Paul Curti: Einführung in die äussere Ballistik [Introduction to exterior ballistics], 1945. Publisher Huber & Co., Frauenfeld.
P. Curti, F. Dubois: Die mechanische Lösung des ausserballistischen Hauptproblems [The mechanical solution of the main problem of exterior ballistics]. In: Schweizerische Bauzeitung, 67, 1949, pp. 52–54. http://dx.doi.org/10.5169/seals-83990
Jenaer Jahrbuch zur Technik- und Industriegeschichte [Jena Yearbook on the History of Technology and Industry], Vol. 11, 2008. Published by Verein Technikgeschichte in Jena e.V.
The Curti-Dubois-Amsler integraph has been restored by the HAM Foundation; see the HAM Foundation Bulletin 3/10, from page 20: Antonin Tarchini, Tim Hellstern, Henri Habegger: Konservierungs- und Restaurierungsarbeiten an einem Ballistik-Integraphen von 1948 [Conservation and restoration work on a ballistics integraph from 1948]. Addresses change continuously — search under http://www.armeemuseum.ch, then manually search for Bulletin VSAM: No. 3, 2010.
Senior Engineer Alfred Kuhlenkamp (Wehrmacht): Flak-Kommandogeräte [Anti-aircraft fire control devices], VDI-Verlag, 1943.
Eugen Willerding: Die mathematische Theorie ballistischer Kurven [The mathematical theory of ballistic curves], 2017. Intensive mathematical treatises on all kinds of trajectories with air drag, including rockets and satellites. Demanding! Available in full text online at: www.eugen-willerding.de
(Ballistics 25 years earlier, World War I) David Aubin: I’m just a mathematician. Why and how mathematicians collaborated with military mathematicians and ballisticians at Gâvre, 2010. Ballistics in the French military Gâvre Commission, before/during World War I. 44 pages. http://hal.upmc.fr/hal-00639895v2 (click “Fichier” there), or: https://hal-univ-diderot.archives-ouvertes.fr/hal-00639895/document

p. 322, 323: ever finer, ever more absurd approximations of the “formula” for air resistance. p. 329, 330, conditions before the serious work of the mathematicians at the start of WWI: there must have been great discrepancies between the prepared range tables and the observations of the troops:

Some artillery units lost confidence in the tables computed by theoretical means and corrected them on their own by experimental means. “Thus instead of having a single range table carefully established in the rearlines, there were many built with the help of a very large number of shots carried out in lousy experimental conditions.”

The electrical engineer Hippolyte Parodi, … underscored that he had first become aware of the initial insufficiency of firing tables while he was fighting on the front: “When I was called to head the Service de balistique et de préparation des tables de tir, I had long been aware, according to the shots I had taken or controlled on the front, that the near totality of firing tables in use in the Army were clearly false and that they had been established … through ‘archaic,’ inexact and simplified methods.”

The tables, Parodi went on, were not only wrong; they were also inconsistent and self-contradictory. Some projectile might, for example, have, for the same line of departure, a greater range for a smaller initial velocity.

pp. 334–339 give an idea of the effort the mathematicians had to expend ca. 1915–1917 to bring computed trajectories to a practicable accuracy. Shot observations and surveys are mentioned briefly but unfortunately incomprehensibly, p. 338. Nowhere in this text are mechanical computational aids mentioned (Füsgen, this work p. 11). Ref. 9 has an extensive bibliography.

https://hal.archives-ouvertes.fr/hal-01187210/document (Dominique Tournès, 2013, Europ. Math. Soc.) Overview of historical attempts to adapt the mathematics to the trajectory — and vice versa. “Ballistics was the laboratory for the development of mathematics!”

Regarding Refs. 3, 4: Curti Paul, 1882–1965, was Ordinary Professor for Military Sciences at ETH from 1927 to 1947. He was presumably the driving force behind the construction of the ballistic integraph, perceiving the military and ballistic necessities. Dr. F. Dubois translated everything into an engineeringly realizable mechanical form and planned the air-resistance section. The machine factory Alfred J. Amsler & Co. in Schaffhausen built the device, probably as a one-off piece (?).

Regarding Ref. 4: A fine passage (p. 54) is reproduced here verbatim, from the year 1949:

“For very high velocities above 12,000 m/s, the projectile no longer falls back to earth but describes its path around the central body as a satellite. For this case we have devised, following the same principles, a planetary integraph. Since, however, astronomers can calculate the planetary orbits, this initially has little prospect of realization; it becomes interesting only when considering perturbations, where the first satellite is disturbed by a second and the second by the first. As soon as, in this ‘three-body problem,’ the masses of the satellites can no longer be neglected relative to the mass of the central body, the mechanical solution is appropriate.”

Small linguistic precision: with cannons, no satellite can be shot into orbit — it would have to return to the firing point, i.e., pass through the Earth. The three-body problem remains interesting nonetheless!

Author: André Masson, Langenthal, Switzerland

February 2017 – May 2017

This is the ninth work on mechanical computers of anti-aircraft defense at the time of the Second World War.

First work: Fire control device SPERRY
Second work: Fire control device GAMMA-JUHASZ-HASLER
Third work: Various anti-aircraft devices: range-finding, control and training devices
Fourth work: Computing with shaped solids
Fifth work: Fire control device for the 34 mm cannon (angular velocity device)
Sixth work: Early CONTRAVES devices: Stereomat, Verograph, Oionoskop
Seventh work: Sound ranging: Elascop and Orthognom
Eighth work: Curved-flight computers

Frühe mechanische Rechner der Artillerie und Fliegerabwehr: Automatisierte Rechner für Geschossflugbahnen

Early Mechanical Computers of the Artillery and Anti-Aircraft Defense: Automated Computers for Projectile Trajectories

Overview

The Oblique Throw — Recollections of School Instruction under Hans Bützberger

Continuous Additions — A Perpetual Patchwork

Historical Summation Gears — In Various Structural Forms

The Trajectory Computing Device after Peter Füsgen, VDI (Dissertation, TH Aachen, 1937)

Construction

Data Output

Data Ranges and Scales

Mechanical Realization of Time

Details and Particulars: The Acceleration Influences the Velocity

The Velocity Determines the Distance

How Accurate Were the Trajectory Calculations?

Predecessor Devices

The Trajectory Integraph after the Curti-Dubois-Amsler System, called “Mariandl”

Construction

Interconnection: How the Variables Are Related to Each Other

Air Density, Air Resistance

Air Resistance in the Curti-Dubois-Amsler Installation, called “Mariandl”

Installations by Vannevar Bush (USA — Brief Mention Only, without Details)

Appendix 1: Determination of the Initial Velocity of the Projectile

Appendix 2

Chronology — Development of the Curti-Dubois-Amsler System

Appendix 3

Unclear and Unresolved Matters

Sources