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The Dawn of Machine Calculation 300 Years of Computing: From Pascal to Zuse Long before relays, before vacuum tubes, and centuries before transistors, the idea of a machine that could calculate was taking shape. The journey from the first mechanical calculators of the 1600s to the relay computers of the 1940s is a story of slow, steady evolution. Each generation of machines solved a different piece of the puzzle —user interaction, programmability, precision, speed, and reliability. Early Mechanical Calculators: The First Steps Towards Computers The story begins in 1623 with Wilhelm Schickard, a German astronomer who built what is now recognized as the first mechanical calculator. His “Calculating Clock” used gears and rotating drums to add and subtract, and it assisted Johannes Kepler with astronomical tables. Although Schickard’s machine was lost to history for centuries, it introduced a crucial idea: arithmetic could be mechanized. Two decades later, in 1642, Blaise Pascal (French) created the Pascaline, a robust adding machine built from precision gears and carry mechanisms. Unlike Schickard’s device, Pascal’s machines survived, were manufactured in quantity, and were demonstrated widely. He designed the machine to add and subtract two numbers and to perform multiplication and division through repeated addition or subtraction. The Pascaline could handle between 5 and 10 digits, depending on the specific model. The Pascaline: 6-digits, CNAM Museum, Paris Early Mechanical Calculators: The First Steps Towards Computers The next major leap came from Gottfried Wilhelm Leibniz in 1672 (Germany). His Stepped Reckoner introduced the stepped drum (later known as the Leibniz wheel), a mechanical innovation that allowed a machine to multiply and divide automatically. Leibniz also articulated a deeper vision: reasoning could be reduced to mechanical operations. His ideas included Boolean logic concepts and digital computation 175 years before George Boole’s work. Important too was the binary work of English scientist Thomas Harriot (1560-1621), before Leibniz. Leibniz’ machine was the first calculator capable of performing all four basic arithmetic operations—addition, subtraction, multiplication, and division. In 1923, the Deutsches Museum, Germany, commissioned a fully functional reconstruction based on Leibniz’s surviving drawings and descriptions. The machine has an 8-digit input section and a 16-digit accumulator (where the results appeared). Historical drawings also show that a 12-digit version may have existed. Replica of Leibniz's stepped reckoner in the Deutsches Museum, Germany Of note; Leibniz proposed a different machine (he never built it) that worked only in binary. It used rolling marbles, gates and registers to perform arithmetic using base-2. See Binary calculator using marbles for an excellent video based on Leibniz’ concepts. These early machines were limited, slow, and delicate, but they established the essential principle that calculation could be performed by a machine following deterministic rules. [Ifrah] Industrial‑Era Calculators: Scaling Up Computation One of the most famous machines developed was the Difference Engine (DE) by Charles Babbage in the 1820s. This was an automatic mechanical calculator engineered to tabulate polynomial functions using intricate gears and the method of finite differences. Although manufacturing limitations and funding disputes prevented completion during Babbage's lifetime, modern reconstructions have proven his visionary design was perfectly functional. Every level of its operation—from its internal state representation to its arithmetic logic and error-correction mechanisms—operated entirely on discrete, integer states (specifically decimal, base-10). Replica Difference Engine No.2, designed by Charles Babbage, Science Museum, London, completed June 1991 By the 1800s and early 1900s, mechanical calculators had become industrial tools. Companies like Burroughs, Monroe, and Marchant built rugged, crank‑driven machines capable of handling business arithmetic. These devices were faster and more reliable, but they were still fundamentally mechanical: gears, cams, levers, locks, and springs. They computed, but they could not decide. They lacked programmability, conditional logic, and the ability to store and manipulate symbolic information. The conceptual leap from “calculator” to “computer” required a new kind of switching element—something that could represent binary states (0,1), switch rapidly, and be wired into arbitrary logical structures. [Williams] The Relay: The First True Digital Logic Element Edward Davy (English, 1806-1885) built an “electric relay” to repeat telegraph signals. Davy invented the relay to “renew the current” on long-distance telegraph lines and to steer telegraph signals to different endpoints. Telephone switching systems likewise relied on relays to steer signals and make logical decisions based on the number dialed. By the early 20th century, telephone exchanges used millions of electromechanical relays to route calls. Relay characteristics: Two states (binary): 1/0, true/false, on/off, high/low, +volts/0_volts, per contact pair Capable of moving several contact arms to independently switch circuits Electrically controlled – a digital switch with many practical contact arrangements Capable of implementing logic (AND, OR, XOR, NOT gates) Capable of storing logic state Reliable and mass produced Western Electric Company produced about 100 million relays between 1920 to 1948 for exchange switching. So they knew more about relay manufacturing and reliability than anyone. View this short video (one contact pair) to get a feel for relay operations. The Big Three Switches The relay was the first switch but what about the vacuum tube (1906) and transistor (1948)? All three can change state from 0/1 or 1/0. However, a single relay can support many linked contact forms (on/off, off/on, form-C, and more) per device and can do far more logic work than one vacuum tube or one transistor. However, tubes and transistors can do more 0/1 switching work per second. See The History of the Telephone Relay for more insights. The first machine to compute using vacuum tubes was Atanasoff and Berry (the ABC, in 1942). Other tube-based systems followed, for example: the Colossus (Tommy Flowers at Bletchley Park, UK, 1943) and the ENIAC (1945). One version of a full 1-bit binary adder uses 9 Triode tubes for the gate switching whereas the same function can be done using only 2 multi-contact relays. More on this later. In 1954, Bell Labs built the TRADIC (TRAnsistor DIgital Computer) and it was the first computing machine to eliminate vacuum tubes and relays. In 2026, for a single, monolithic die, ~104 billion transistors (about 26 billion logic gates) is the state of the art. This is a component in the NVIDIA/TSMC Blackwell B200, a data‑center AI GPU. Famously, the transistor was invented at Bell Labs (AT&T). It should be no surprise that telephone engineers were looking for a relay replacement (something faster, smaller, and more reliable). After all, "Necessity is the mother of invention". A trio of Bell scientists John Bardeen, Walter Brattain, and William Shockley invented the transistor in December of 1947. All three received the Nobel Prize in Physics in 1956. The First Binary Adder using Relays George Stibitz (mathematician at Bell Labs) is recognized as the father of the binary adder using relays. In his kitchen in 1937 he built the Model K relay adder (using two telephone exchange relays) with two input switches, A and B (0 and 1 inputs), and 2 outputs Sum and Carry_out (lamps). This was the first relay-based binary adder, albeit very rudimentary. Of note, his concept model was only a “half adder”. This means that there was no “Carry_in” terminal only and A and B keys. Two of these adders can be easily combined to make a “full adder” with 3 inputs and 2 outputs. Check out an interactive adder demo here (based on Boolean logic gates). Keep In mind that the relay was invented 100 years earlier and it took a PhD mathematician to conceive of 2 relays doing binary math. This seems obvious today but was a gigantic intellectual leap in 1937. See Endnote A. This experiment started a fire at AT&T, and Stibitz was effectively the “chief engineer” on the Model I to Model VI relay calculators/computers over the next many years (discussed below). Model K, 1-bit binary adder by Stibitz (Computer History Museum) Konrad Zuse and the First Relay Computers In the 1930s, German engineer Konrad Zuse (with colleague Helmut Schreye) recognized that relays could be wired into networks that performed logical operations. He built a series of groundbreaking machines: Z1 (1936) – all mechanical logic/memory, proof of concept, unreliable Z2 (1939) – hybrid design, 600 relays for the ALU but using mechanical memory Z3 (1941) – fully relay‑based, ~2,600 relays, 22-bit floating‑point arithmetic, program on punched film. The 22 bits were split as: 14 mantissa, 7 exponent, 1 sign. The Z1/Z2 designs were powered by an electric motor. Through a system of cams, direction-changer levers, and pushrods located in the "basement" (lower layer) of the machine, rotational motor motion was converted into horizontal and vertical reciprocating motion. The machines used binary logic gates and were implemented with flat, sliding sheet-metal plates, pins, and guide slots. Very clever! The Z3 is widely regarded as the first working, programmable, digital computer. It proved that relays could implement not just arithmetic but general computation. Interestingly, it did not support a dedicated conditional branch instruction (If this, then jump there) yet is regarded as a computer by most experts. Computer scientist Raúl Rojas (1998) proved that the Z3 is theoretically Turing-complete, hence a computer not just a calculator [ Rojas]. There have been 4 major reconstructions of the original Z3. The Z3 1-bit full relay adder (5 relays) was not based on Stibitz design but a variation with some possible help from Helmut Schreye. See Endnote B. In his autobiography, The Computer – My Life (1993), Zuse said, "A relay computer doesn't just calculate; it works. It has a rhythm, a cadence, and when hundreds of relays flip in harmony, you can hear the logic happening ." The three Figures below show the Zuse Z3, all-relay computer. Zuse Z3 replica on display at Deutsches Museum in Munich The image above is from a 2016 article by Horst Zuse, son of Konrad Zuse. Note the blazing clock speed of 5.33 Hz. https://www.zuse.de Other Relay-based Machines The years 1940–1947 were the golden age of electromechanical machines. A calculator executes a fixed sequence of operations to evaluate mathematical expressions, whereas a computer uses conditional branching to dynamically alter its own execution path based on intermediate data. About the same time as Zuse, Bell Labs employees (George Stibitz and Sam Williams took the lead working with 20 other engineers) began building relay calculators then relay computers. Below are some notable models. The Model I was not programmable in the same fashion as the Zuse Z3, they are in different categories. Bell Labs Model I Complex Number Calculator ( 1940) – 450 relays and eight, 100-point, crossbar switches for memory 8 decimal places of precision, used a modified form of BCD arithmetic. Bell Labs Model II, Relay Calculator (1943) 493 relays, crossbar memory, operational until 1961 Bell Labs Model V Computer (1944) – 9,000 relays Harvard’s Model II Computer (1947) – Fully relay based with approximately 13,000 high-speed relays. Built by IBM as a partner. 10 digit floating point numbers, 8 additions per second The only mechanical parts were the peripheral input/output devices These models demonstrated that relay computers could be scaled, maintained, and operated reliably for long periods. All Bell Labs models were constructed with “telephone relays” designed by Bell Labs and manufactured by Western Electric Co, both subsidiaries of AT&T [Williams, pg 224] . Bell Labs Complex Number Calculator, Model Mark I, 1940 Photo archive, Lucent Technologies The Bridge to Electronic Computing Relay computers were slow by modern standards (about 5 to 15 additions per second) but they were the first machines to embody the full architecture of digital computers. Relay logic provided the conceptual blueprint that vacuum tubes, transistors, and integrated circuits would later accelerate by many orders of magnitude. Can Artificial Intelligence (AI) be implemented using only electromechanical relays? Check out a discussion on this here and the history of the relay here . Conclusion: A 300‑Year Journey to the First Computers From Pascal’s wheels to Babbage’s DE, to Stibitz’ K1 relay adder, to the Zuse Z3, the evolution of computing was a slow accumulation of breakthrough ideas: Mechanizing arithmetic Represent information symbolically Implement logic using electro-magnetics Store and manipulate state Build programmable systems The relay was the first true digital switch. A device that leapfrogged centuries of mechanical ingenuity and helped launch the dawn of the computer age. For more information related to these topics including a list of DIY relay computers, check out the TOC pages from the main menu above. Endnotes A : Relay math was not completely foreign to Stibitz. He worked for Bell Labs (AT&T) and was certainly aware of how relays were used to count dial pulses from a caller. The Panel telephone exchange had a wonderful example of a relay-based digit counter using a so-called 1/2/4/5/Z code (1927). It wasn’t binary but a step in the right direction. See Fig 9 in the companion website Calling315.com for a description of how this worked. B . The original Z3 (destroyed in December of 1943, WW2) used 5 relays for each full adder circuit, or 6 with a "sum" relay output. The 1961 Z3 reconstruction by Konrad Zuse uses a 2 relay 1-bit full adder. Both the 5 and 2 relay circuits are described by [Zuse] , page 198. References Ifrah, Georges. The Universal History of Computing: From the Abacus to the Quantum Computer. Wiley, 2001. Kovalick, Al. https://www.calling315.com/history-of-the-telephone-relay Rojas, R., How to make Zuse's Z3 a universal compute r, IEEE Annals of the History of Computing ( Volume: 20, Issue: 3, July-Sept. 1998) Williams, Michael R. A History of Computing Technology. 2nd ed., IEEE Computer Society Press, 1997. Zuse, Konrad. The Computer – My Life . Springer, 1993.

The Building Blocks In the early days of computing, two foundational models emerged: Aiken’s Harvard architecture (1944) and von Neumann’s architecture (1945). Over time, engineers refined and extended both ideas, and the machines we use today blend elements of each. Modern CPUs expose a unified memory model like von Neumann envisioned, yet internally they borrow Harvard’s separation of instruction and data pathways to boost speed and parallelism. The result is a hybrid design that traces its lineage directly back to those two original approaches. There are ample resources explaining basic computer organization, so it won’t be covered here. The focus here is the implementation stack . What is this? Figure 1 shows a hierarchy of five implementation layers. Each layer provides the primitives needed to construct the next. In this case the “top” is the most fundamental element that all lower layers depend on. The coverage is more about the building blocks (or nuts and bolts) and less about the architect's floorplans. Decoupling the two allows an architectural model to endure across vastly different hardware designs, with software only entering the hierarchy at layer 5. Fig 1, A computer implementation stack Although this site focuses on relay-based computing, the following discussion examines three logic-switch technologies fundamental to computer operation: relays, vacuum tubes, and transistors. Each independently played an important role in the history of computing. Understanding these 3 switching devices in relation to each other provides a wider perspective than just how relays were used. The Figure 1 model is deliberately kept simple. Let’s examine each layer. The Stack Layer 1: The Switch Nothing computes or decides without the active (powered) on/off binary switch. This is not a hand operated switch. Rather it’s a 3-port device with a control port(s) that opens and closes a switch(s). Figure 2 shows the concept diagram of this type of switch. For this example, the circuit performs an inverter action, a HIGH In closes the switch and produces a LOW out (0V). When the input is LOW (0V) the switch opens and the Output is HIGH (+V). The control port is typically low current, and this causes switching action of a higher current/voltage if needed. There are many ways to wire switches and Fig 2 is just one example. The Rload resistor limits the current when the switch is on and other variations may not have this resistor. Fig. 2. Conceptual electrically controlled switch Figure 3 shows the Relay, Vacuum tube, and Transistor. Each can be modeled as a binary switch by the Figure 2 concept diagram. These three are the most important electrically controlled switching devices ever invented. Importantly, all three in this figure were manufactured by Western Electric with the designs by Bell Laboratories (both AT&T). There are more exotic experimental switches, but our focus is on the three well established types in Fig 3. See Appendix A . Fig 3, Three primary switching devices from Western Electric U650 Relay (1920s-50s), 300B Tube (1938), 2N67 Transistor (1953) With 60 years of global dominance, Bell Labs was the premiere industrial research house worldwide from 1925 to ~1985. That’s an extraordinary run — no other industrial research lab has maintained world leadership for anything close to six decades. Lab scientists were awarded ten Nobel Prizes in physics and chemistry. First, they defined the state of the art for relay design (telephone exchanges) for ~50 years. Siemens (Germany), Erickson (Sweden) and Automatic Electric Company (US) were also major relay innovators. They did not invent the relay; that was done in 1838 by Edward Davy (England). The Labs invented or co‑invented major tube families. Most famously, Bell Labs scientists invented the transistor in 1948. Telephone systems have a need for route switching and voice amplification, hence Bell Lab’s concentrated innovation efforts in these areas. Each switching-type generation improved important metrics – speed, size, power, cost, other. Check out the History of the Relay for a deeper dive. Besides on/off switching, tubes and transistors can also linearly amplify analog signals with proper circuit design. Relays can’t linearly amplify analog signals. Interestingly, in 2026 Western Electric (new owners) still handmake vacuum tubes (model 300B) for high-end audio amplifiers. Layer 2: Primitive Gates If you are unfamiliar with logic gates it will help to review this explainer first (with an interactive explorer). The most fundamental elements are the 6 logic gate types at layer 2 in Fig 1. The NAND and NOR gates are Negated AND & Negated OR, respectively. Inversion (also called negation) turns 0 to 1 and 1 to 0. All digital operations can be performed using only these primitives or clever logically equivalent circuits. Each of these gates are implemented using switches. Figure 4 shows three example gates with supply (+V ) voltage values of +12V, +350V, and +6V, respectively. Why are they so different? The supply voltages are selected based on the individual circuit design needs. Important for us, regardless of the actual voltage, a logic HIGH output is +V and a logic LOW output is at ground potential, zero volts. Fig 4, relay, tube and transistor logic circuit examples For the relay AND gate, if A and B are HIGH their respective relays engage and the output is HIGH (+12V in this case), else LOW. Without describing how resistors, tubes or transistors work (see Appendix A), the following gives a high-level explanation for how, for example, NOR and OR logic gates operate. For the tube based NOR, when A and B are LOW (A = 0, B = 0), neither tube conducts electricity. Because no current flows through the pull-up resistor the output voltage stays at +350V, which represents a HIGH (1) state. If A or B = 1 then the output is LOW (0V) because one or both tubes are conducting. For the transistor OR circuit, if either input is HIGH (respective transistor conducts) the output is HIGH (~6V). There are many variations of these circuits for each gate type. Importantly, using only a voltage source, switches, wires and passive resistors all logic gate types can be built. Some modern designs use semiconductor diodes to assist. The AND gate and OR gate can be built using only diodes and resistors (DRL gates) and a switch is not strictly needed. This design approach has limits due to signal loss but it was used in some early designs. The IBM 608 (1957), an all-transistorized computer, used diode-resistor logic for AND/OR gating, with transistors only for inversion and level restoration. In contrast, the Colossus Mark 1 (England) [3] used about 1,500 vacuum tubes and no relays. Layer 3: Combinational Logic Circuits With gates available as building blocks, we can construct complex logic circuits. Figure 5 is an example of a “multiplexer” (Mux: k-in, 1-out), a commonly used module for digital designers. There are 4 data inputs and 2 selector inputs. The idea is for the 2 selector values to route one of the 4 inputs to the single output, Y. This is a chooser circuit and found often in computer design. See the Truth Table to see all four routing combinations. For example, S1 S0 = 1 1 routes only I3 to the output. Notice it uses 3 primitive gates – AND, OR and the Inverter. No matter how complex, all digital circuits can be made from the primitive gates. There are clever circuit shortcuts for some designs but in the end, all circuits use the primitives in one way or another. It’s easy to imagine this circuit being packaged with 4+2 inputs, 1 output and 2 pins for power. There are 500+ different transistor-based modules available today. Search for the 74LS153 Mux module as an example. These are like Lego blocks for DIY computer builders. There are relay and tube equivalent Mux implementations, but they are not described here. Fig 5, Building a multiplexer using gate primitives Another important element is bit memory. Fig 6 shows a latching relay circuit. In a computer, memory is needed for both program and raw data storage. In the figure, there is a push button to engage the relay (it stays on once its contact closes) and a reset button to disengage it. There may be other contacts if needed. Using a relay to store bits is expensive! 1,000 bits need 1,000 relays. So, engineers looked for more efficient ways to store bits. The first relay-based calculator from Bell Labs used 100-point crossbar switches to store ten decimal digits per switch. See examples here on a companion site dedicated to relay-based telephone systems. Also, some early relay (mostly) computers use short term “1-bit capacitor memory”, some used special mechanically latched relays, and some modern versions use semiconductor memory (often NAND based). Fig 6, Relay latch circuit for 1 bit Layer 4: Hardware Based Math One of the most powerful logic modules is the 1-bit full adder. Experiment with an interactive single 1-bit adder here . Cascading cells enables composite adders of any length in hardware (4, 8, 16, 32 bits,…). See Fig 7 for a hardware 4-bit adder example. A and B each are 4 input bits, S (Sum) are the 4 output bits. Note that the Carry_out bit of each adder is connected to the Carry_in bit of the next adder in the chain. The final Carry_out bit may be forwarded to another adder chain to increase the effective adder bit length beyond 4 bits. Search for the transistorized 74LS83 4-bit adder module to learn more. The R30 cascades six, 1-bit, relay-based full adders, resulting in a 6-bit adder. Each 1-bit adder needs only two relays. The adder is described elsewhere TBD on this site. Th R30's six-bit adder is cascaded (in 6 passes), via the microcontroller, to make a full 36-bit adder and used for all math operations. Fig 7, Hardware 4-bit integer adder made of four small 1-bit adders What about hardware multipliers and dividers? These are also made of one or more n-bit adders but with additional control logic to implement the methods that we all learned in primary school. Here is the systematic method for unsigned integer multiplication: Bit Check: Look at the multiplier one bit at a time, from right to left. Add if 1, skip if 0 : If the bit is a 1, add the multiplicand into a running total (the accumulator). If it is a 0, add nothing. Shift : Slide the multiplicand one bit left, set the new LSB with a zero, and move to the next bit. Repeat : A simple controller repeats this cycle until every bit is checked and the final total is ready With an effort to manage the multiplier and multiplicand exponents (adding exponents is equivalent to multiplying), this method can be extended to signed floating point numbers. The R30 uses the methods above to multiply and the non-restoring method to divide. The R30 supports 32b floating point math with a dynamic range of (+-) 1.4×E-38 to 3.4E+38 based on its 36-bit adder discussed above. Its math accuracy conforms to IEEE 754 Standard for Floating-Point Arithmetic . See the explainer on relay adders here TBD. Many vendors make Floating Point Math accelerators. These are digital chips that can “bolt on” to CPU hardware chips to improve calculation speeds (not expanded on here). Note too that there are many methods available to speed up hardware calcs (i.e. Booth’s algorithm for multiply) but these are beyond our coverage although the R30 uses Booth. Layer 5: The Execution Layer Layers 1 through 4 provide the nuts and bolts —switches, gates, decoders, adders, registers, storage and more. But none of these components can decide what to do on their own. Layer 5 provides the conductor: the control unit, instruction decoding logic, and sequencing hardware (all based on logic gates) that orchestrate data movement across the machine. Of course, these four layers appear as “black boxes” to Layer 5 execution. So, it’s possible to design a hybrid computer using combos of the switches in Fig 2. For example, the IBM SSEC (Selective Sequence Electronic Calculator, 1948) is one of the largest switch hybrids ever built. It combined 12,500 vacuum tubes (for the arithmetic unit and high-speed control) with 21,400 electromechanical relays used for intermediate memory registers and sequence control [1]. There are no known hybrids that use all three switch types. Another example is the famous ABC “first vacuum tube” computer (1939-42). It was more a calculator than a computer. Nonetheless it used tubes for all math and relays for program execution. See Endnote A. Fig 8, R30 program and output listings for solving a quadratic equation Execution centers on the classic instruction cycle: Fetch, Decode, and Execute. A program counter (PC) holds the address of the next command, pointing to a location in memory (whether film, paper tape, electromagnetic store, or modern semiconductor memory). During the fetch phase, this instruction is loaded into an Instruction Register. Next, decoding logic inspects the bit pattern (the opcode). Rather than running math directly, the decoder is essentially a traffic director: it asserts specific control lines that open multiplexers, enable register output onto shared buses, and tell the Layer 4 math units whether to add, invert, or shift. For the famous Zuse Z3 relay computer, about 25% of all the relays (plus some electromechanical stepping switches) were used for program execution. See Endnote B. Layer 5 also manages the boundary between the internal machine state and the outside world via Input/Output (I/O). Using a microcontroller to manage all user I/O (WiFi connected tablet interface for the R30) is very practical and saves many relays. See Figure 8 for a R30 program listing and outputs for computing the two roots of the quadratic equation x*x + 5x + 6 = 0. Appendix A Tubes and Transistors The Vacuum Tube There are many excellent web resources on the operations of the vacuum tube and the transistor. This Appendix gives a tip-of -the-iceberg view of these important devices, especially functioning as binary switches. In 1904, John Ambrose Fleming (British) patented the first vacuum tube diode, called the Fleming Valve . He realized that the Edison effect — one‑way electron flow inside a heated vacuum bulb — could be used as a radio-wave detector (passes one signal polarity, blocks the other). Edison himself had not recognized any practical use for it. Fleming’s diode became the first practical electronic component, enabling reliable detection of wireless signals and effectively starting the electronics industry. In the figure below, if the "plate" element on the right side of the tube is positive in charge, it attracts the electrons from the heated filament. If the plate has a negative voltage the plate repels electrons. A vacuum provides a collision‑free path, allowing electrons to reach the plate more effectively. Fleming's invention is not a full on/off switch but very close to it. The Fleming Valve Vacuum Tube Lee de Forest’s invention — the Audion (1906) — was directly built on Fleming’s valve. De Forest began experimenting with the Edison effect and Fleming’s diode, trying to improve radio detection. He added a third electrode, the grid, creating the first triode (3 element) tube. The Audion became the foundation of radio broadcasting, long‑distance telephony, and eventually early computers. Adding a grid (the control element in Figure 2 above) to the tube allowed it to function as an on/off binary switch or linear amplifier depending on how the tube was wired into the circuit. The grid (see image below) controls the flow of electrons from the cathode to the plate, allowing amplification, switching, and modulation. Without the grid, a tube is just a diode -- a Fleming Valve. By applying varying voltages to the grid element the electron flow is modulated as needed from binary on/off to varying in a linear fashion and causing amplification of audio and radio signals. Finally a new on/off electrical switch that is much faster than a relay! The Lee de Forest Triode Vacuum Tube Source: hamradioschool.com AT&T (through Western Electric and what later became Bell Labs) bought the rights to De Forest’s Audion, and this purchase is what allowed them to industrialize and radically improve the triode. AT&T saw the potential for long‑distance telephone amplifiers and negotiated rights to use and develop the Audion. Over time the triode was used as a binary switch but this was not AT&T's original use case. The first telephone call from New York to San Francisco in 1915 used three vacuum‑tube repeaters along the 3,400 mile (copper wire length) route. Surprisingly, before the tube AT&T used mechanical amplifiers (repeaters) for weak telephone signals. Hundreds of model 6C8G "dual triode" (RCA or Sylvania likely) tubes were used in the original ABC tube computer (1942). The Transistor The transistor was invented (there were two versions) in late 1947-8 at Bell Telephone Laboratories by John Bardeen, Walter Brattain, and William Shockley. Their breakthroughs created the first solid-state device capable of amplifying and switching electrical signals. Several names were proposed and one was semiconductor triode [2], a nod to the triode tube. In the end, "transfer resistor" or transistor was chosen. One way to understand a transistor is to picture a water faucet and its valve. Instead of water, electricity flows through it. Turning the handle is equivalent to applying a tiny trickle of electricity to the B (base) terminal causing electricity to flow between the C and E terminals. There is a similarity between the tube and transistor, each have 3 terminals and one controls the flow of current between the other two. Tube: cathode → grid → plate Transistor: emitter → base → collector Both use a small input signal at the control terminal (grid/base) to modulate a larger current between the other two terminals. One type of transistor (NPN) is made of layers of the element Silicon with tiny amounts of Phosphorus and Boron. Applying a small forward voltage to the base lowers the internal energy barrier at the base-emitter junction, allowing electrons to flow between the collector and the emitter. The figure below shows a working circuit to supply current to a load (lamp, other) and this is controlled by a low current control switch. The resistor limits the base current to a small value. When the control switch is closed the transistor "turns on" and current flows into the load (as in the figure below). When the control switch is off (open), the transistor "turns off" and there is no load current. Importantly the load current can be ~100x more than the control base current so the transistor is an amplifier in a similar way to how tubes amplify [Endnote C]. Relays can't be wired to linearly amplify a signal whereas a tube and transistor can. Vitally, all three devices can operate as binary switches. Faucet and Transistor relationship illustration Transistor circuit and its equivalence with the control switch =closed Transistors are far smaller, more rugged, and draw significantly less power than vacuum tubes because they operate without fragile glass envelopes or power-hungry heater filaments. They also turn on instantly, generate minimal heat, and last decades without the inevitable burnout and degradation that plagued vacuum tubes. Hence, transistors dominate our digital world co-existing with some specialty vacuum tubes for audio and relays for power switching. How many transistors were produced in 2025? There is no official count. However, extending the reasoning of industry analyst Jim Handy [4], an order-of-magnitude estimate is about 10E23 (about a trillion trillion) transistors, with almost all a part of integrated circuits. For example, a single Nvidia Blackwell B200 die has ~104 billion transistors implementing about .7 million adders of various lengths. We've come a long way since the George Stibitz 2-relay adder in 1937. References 1. Bashe, C. J., Johnson, L. R., Palmer, J. H., & Pugh, E. W. (1986). IBM's Early Computers . MIT Press. 2. Gertner, Jon, The Idea Factory, Bell Labs and the Great Age of American Innovation, 2012, page 98 3. Randell, Brian (ed.). The Origins of Digital Computers: Selected Papers . Springer, 1973. Contains Flowers’ original post‑war paper describing Colossus. 4. Computer History Museum --https://tinyurl.com/ycfermse Endnotes Endnote A: The control system of the ABC did not use vacuum tubes for decoding because it was not a stored-program or tape-programmed computer. It ran a fixed algorithm (Gaussian elimination to solve simultaneous linear equations) governed by approximately 30 telephone relays. The vacuum tubes outside the adders served as amplifiers for the capacitor memory drums (about 30 tubes). The arithmetic logic functions were fully electronic, implemented with vacuum tubes. Based on materials in: Alice R. Burks and Arthur W. Burks: The First Electronic Computer: The Atanasoff Story (University of Michigan Press, 1988). Endnote B : Dr. Raúl Rojas reverse-engineered the relay logic schematics of the Z3 computer by Konrad Zuse. He wrote the paper: "Konrad Zuse’s Legacy: The Architecture of the Z1 and Z3 " (IEEE Annals of the History of Computing, Vol. 19, No. 2, 1997). In Section 3 ("The Control Unit"), Rojas notes: The opcodes read from the 8-track punched film were parsed by a decoding tree of relays requiring minimal hardware (fewer than 40 relays for instruction decoding). The execution sequencers (handling iterative shifts for multiplication and square roots) used rotary stepping switches and interlock chains, leaving data storage and addition related operations to consume over 75% of the machine's physical relay inventory. Endnote C: Tubes and transistors amplify in the sense that the control signal input (grid or base terminal) controls a much larger current. For a transistor, this is done in a linear fashion such that Current_out = K*Control_current, where K is the amplifying factor. A tube is a voltage‑controlled current source, different from how a transistor works. The Current_out= Function(grid voltage). The input-output transfer curve is not linear and beyond the scope here. With a relay the relationship is different. The control current engages the electromagnetic coil to operate one or more switch contacts. The switch current is independent of the coil current and can be as high or low as needed based on the contacts ability to handle the current.

Work in Progress.. Stay Tuned (calling315.com companion site) Your Home for Relay Computing Before vacuum tubes or transistors (chips), engineers used electromechanical relays to build digital logic circuits. Using Boolean logic and relay switching, pioneers like Claude Shannon, George Stibitz, and Konrad Zuse launched the computing revolution in the late 1930’s. Building a relay-based computer is more than a fun retro-engineering challenge. It is a hands-on masterclass showing how the essential on/off switch underpins all forms of digital computation. For a backgrounder, select "From Pascal to Zuse" from the main menu above. Select "R30 Computer" to get an overview of a DIY relay-based computer. The "Table of Contents" has links to explainers covering binary math, logic, relays, several interactive demos and historical perspectives.

Relay Basics Introduction A relay is a remotely controlled switch that can open or close (0/1) contacts when sufficient energy is supplied to its coil of wound wire. The coil becomes an electromagnet when energized. Before the vacuum tube, transistor and integrated circuit, relays were the perfect on-off switch to build logical circuits using electricity. By far, telephone switching consumed nearly all relay use (some for telegraph) from the 1880's until the late 1970's. Relays made for telephone switching by Siemens & Halske in Germany were used by Konrad Zuse for his Z3 relay-based computer in 1941 [Rojas]. In theory, relays could be the foundational switch to implement a (very very slow) AI inference engine (i.e. ChatGPT, Claude). In Fig 1, the relay is an unoperated state (A) and an operated state for (B). This relay type was called a flat spring relay based on its design. The flexible arm (armature) moves to make or break contacts depending on if the core is energized or not. Figures 1 and 2 below are from [Keister], the classic book on relay applications as used by the Bell System and others. Fig 1, The basic flat spring relay-- off and on The following animation outlines the basics of relay operations. Fig 2 displays two classic relay form factors from AT&T. The left one is a U-type flat spring relay, while the right one represents the cylinder style 200 series. This second style was popular for Step-by-Step exchange switches worldwide. The 3000 type (no image) was designed by engineers Ray and Biddlecombe, of the British GPO, in cooperation with many providers and contractors. It was adopted as a standard in 1932 [POEEJ-2]. Fig 2, Typical General-Purpose relays In 1952 the Bell System introduced a new type of relay, the wire-spring type. This type was Western Electric’s major leap forward over the older flat‑spring relay, and it added better reliability, manufacturability, choice, and easier maintenance. Many millions were made annualy. See Figure 3 and the short video below of an AF-22 relay following the dial pulses from a vintage rotary telephone dial. Fig 3, AJ-9 type wire-spring relay (author's collection) How many telephone relays were made from the 1920's until 1975? There is no published worldwide value. But we can estimate from available information. For one data point, from 1910 until 1952 about 320 million were manufactured by Western Electric Company according the Bell System records [Kovalick]. There were about 360 million telephones (including extension and PBX phones), dial by far, worldwide in 1975. These were served by 240 million access lines. This is from AT&T’s annual publication, The World’s Telephones. There is an important metric of about 7 relays per subscriber line, factoring in the many thousands of relays for the route switching per central office [Kovalick]. So, from this we can roughly estimate that about 240*7 = 1.7 billion relays were installed worldwide for telephone switching in 1975. Hence, the total amount manufactured is somewhat greater given the 30-40 year lifetime per relay. The total count manufactured is likely in the range 2+ billion relays from 1920-1975. Relay circuits With the basics of relays explained, let's look at combinations of wired relays. Need a multiplexer, demux, bit shifter, comparator, flip flop, adder, ...? Wiring relays together creates any logical circuit desired. Of course, relay-count, power, speed and other factors will limit some functions but this is where having a variety of relay coil types, operate speeds, and contact forms is of value. The Bell System manufactured 1000's of relay types to give engineers a wide choice of design options. Fig 4 is an example 4 relays (A-D) wired to create a 1-in, 16-out multiplexer. There are 16 combinations of on/off for these 4 relays. This kind of circuit is used to route the sole input terminal on the left to one of 16 outputs on the right. The ABCD terminal on the bottom right connects to the input terminal only if relays A,B,C,D are all engaged. Each relay has a different contact configuration. Hence, having a large catalog of relay choices, a relay bible, is essential for designers. For relay D, the Western Electric (WE) Wire Spring relay model AJ21 has 8 "early break then make" contacts (sometimes called a EBM, form-C or SPDT contact) fits the circuit design needs. Ideally, a different relay would be chosen with one "early break then make" contact for relay A. Most WE telephone relays operated (coil) with 48 volts DC but there were exceptions. Check out my Youtube page for ~30 videos covering a wide range of switching device demos. The channel is dedicated to the electromechanical devices (1880-1975) used to make telephone calls over different exchange types. Included are many relays, both 1D and 2D linear and rotational talking-path switches, sequence switches, and more. Fig 4, The basic 4-relay transfer tree [Keister] References Britannica Yearbook Of Science And The Future 1973 Keister, Ritchie and Washburn; Members of Bell Laboratories, “The Design of Switching Circuits ”, 1951 Kovalick, Al, https://www.calling315.com/relay-logic Fig 11 and https://www.calling315.com/history-of-the-telephone-relay , Endnote B POEEJ-2: The Introduction of the Standard Telephone Relay, R. W. Palmer, The Post Office Electrical Engineers' Journal, April 1934. Rojas, Paul, The Design Principles of Konrad Zuse’s Mechanical Computers, 2015

Exploring the Technical Aspects of Relay Computers The Building Blocks Binary Number Explorer TBD4 History of the Relay Historical and Contemporary Studies Digital Logic Gates Relay Basics TBD5 DIY Modern Relay Computers Interactive 1-bit Adder Relay-based Adders TBD6 Electromechanical Telephone Exchanges

WIP -- much more to add... Technical Specifications Architectural Core 32-bit electromechanical logic architecture A comprehensive breakdown of the R30's architecture and design philosophy. Switching Logic All binary math is done using telephone relays from the 1940's Power & Voltage Operating voltage: 48V DC. Designed for stable, low-noise power supply integration. Physical Dimensions TBD Design Philosophy The R30 is built to be a masterclass in hybrid electromechanical engineering, emphasizing precision and educational value over speed or "all relay" correctness.

Modern Day Relay-based Computers A small group of DIY inventors have built relay computers and calculators. This section does not cover every known system. However, it highlights a few notable projects by dedicated experimenters. In general, “relay computers” fall into three broad classes (used here to add clarity). These are: Pure relay-only for all functions (~300 to ~2K relays). Some have capacitor charge memory and may use diodes for some logic gating. Mostly relays with some diodes for logic steering and semiconductor RAM/ROM memory. It takes 1,000 relays to store 1,000 bits so using semiconductor memory offers a huge saving in size, power and cost. Hybrid designs. A microcontroller (Arduino, Raspberry PI, ESP32, …) assists with user interface (ex: tablet with WiFi), storage, some program control with relays for all math. Enables lightweight, portable, low power, 32b floating point, with excellent user interface. Some designs use only one relay type repeatedly (say, SPDT) while others use SPDT, DPDT and even more complex contact forms in the same machine. The following summaries are based on my current understanding of these machines. If you see an error please contact me and I will update the description. All images are taken from the inventor's respective published sites. The R30 Electromechanical computer (Class 3) is a DIY project is covered here . The famous Z3 computer was reconstucted by experts and a coverage follows the DIY inventers. Review of some Noteworthy DIY Relay Computers 1- - Jeroen Brinkman (Netherlands), MERCIA all Relay Computer, 2016. ~2,000 relays. 10 bits instruction width, 1K storage, integer math. No semiconductors except for diodes. One relay type (SPDT) is repeatedly used for all logic circuits. Capacitors are used for 1/0 storage and refreshed if needed. This is a Class 1 design and an amazing effort to show all the key aspects of relay computing. It is strictly a fixed-point/integer 10-bit machine. Any fractional or extended-precision arithmetic could be implemented in software/microcode. MERCIA relay computer 2 -- Simon Winder (UK), 8 bits, floating point, 480 relays. Computes square roots only. Note the value in the Nixie tubes below, the square root of 2. Simon started the project in 2009 and finished it in 2014. Digit entry uses a telephone dial. This is a Class 1 design. See a video here . 3 —Dr. Harry Porter’s relay computer. 2015, 415 relays, Data Bus (8 bits), Address Bus (16-bits). Uses the Konrad Zuse Z3 relay adder circuit. Uses semiconductor SRAM for program storage to reduce relay use so falls into a Class 2 design. It supports 8-bit integer math. https://web.cecs.pdx.edu/~harry/Relay/ https://www.youtube.com/watch?v=n3wPBcmSb2U 4 —DipDot, 2026 (YouTube). A DIY relay computer, evolving story. It appears that this is a Class 1 relay computer. Initially, the designer used semiconductor static RAM then later transitioned to a period-accurate 1940s-style Capacitor-Diode Memory . ~450 DPDT relays fully populated. Very clean design with many custom PC boards. He uses an Arduino to assist in testing new designs quickly, but this is not needed for real-time math calculations. Uses 8-bit integer math with a 16-bit address bus for storage access. https://www.youtube.com/watch?v=Jy2j2jv5MPI 5 -- Paul Law's architecture is heavily inspired by Dr. Harry Porter's Relay Computer. Law’s machine reportedly has about 400 relays. So, it supports 8-bit integer math. Pragmatically, it uses semiconductor memory and transistor arrays to interface between SRAM and the relays. It uses a 5-relay per bit, full adder design. It was designed for portability and has the feeling of the MITS Altair 8800 computer (1975) user control panel. It is packaged beautifully and portable. Because the hardware ALU can only operate on 8 bits at a time, calculations like division or long addition are chained together using multi-byte software algorithms (e.g., repeating 8-bit addition with carry). In a Youtube video Paul shows an example of calculating PI to 20 decimal places. He needs an external controller (PC or other) that “assembles” a complex math routine into 8-bit operations that are loaded into the machine’s semiconductor ROM. So, without external assembly of microcode this is a Class 2 machine. For complex math operations this looks like a Class 3 machine. https://youtube.com/shorts/UN4LOo79pjg?si=BHGuPcSv5SLtqjzi Paul Law’s relay computer, front side Z3 Reconstructions by Experts There is a special class of "DIY" relay computer that is built by experts from universities and museums. Outstanding examples are the four versions of the Z3 relay computer originally by Konrad Zuse. For more info on the original Z3 (1941) see the article on this site - From Pascal to Zuse . Of the four, two are practical replicas and two are hybrid machine reconstructions. In 2027 each of these are in museums in Germany and Switzerland. The first replica was built by Konrad Zuse in 1961 for the Deutsches Museum in Munich. This is a Class 1 all-relay machine with electromechanical stepping switches for memory. Because the original machine was destroyed by wartime bombing in 1943, Konrad Zuse recreated his masterpiece entirely from memory and surviving documents to act as an historically accurate, structurally identical clone. Pictures of this version are often considered the original since there are no images remaining of the 1941 version. A second major reconstruction (Z3r) was engineered and built by Konrad Zuse’s eldest son, Professor Dr. Horst Zuse, between 2008 and 2010. Permanently housed at the Deutsches Technikmuseum in Berlin, this version deliberately diverges from the original Z3 blueprint for educational purposes. Horst added diagnostic LEDs to the relay blocks so live museum audiences could physically trace the movement of bits. He made other changes too for demo and simplification purposes. It uses a Siemens Simatic PLC (S7) electronic controller to assist in stepping through program sequences. This is a Class 3 machine. See a video of Horst explaining, in German and English, his version of the Z3. Two other versions were build by Dr. Raúl Rojas and Dr. Christof Traber respectively. See below. Snapshot of Horst Zuse in front of his version of the Z3, 2022 at the Museum The four reconstructions of the Z3: 1. 1961 (Zuse KG / Konrad Zuse): Built by Zuse's own company to serve as technical evidence for patent claims and historical priority. It reproduced the logical operation using newer, smaller relays in a single memory cabinet configuration and is on permanent display at the Deutsches Museum in Munich. This uses the very clever 2 relay full adder circuit compared to the 5 relay 1-bit adder design for the original Z3. 2. 2001 (Dr. Raúl Rojas / Free University of Berlin): An educational, hybrid reconstruction using relays for the arithmetic and memory units combined with digital components and a PC for user I/O. It is housed at the Konrad Zuse Museum in Hünfeld, Germany. 3. 2010 (Dr. Horst Zuse's Z3r): Built at approximately full historical scale using around 2,500 modern industrial relays to mark the centenary of Konrad Zuse’s birth. It features LED indicators and manual single-bit stepping controls, and is located at the German Deutsches Technikmuseum in Berlin. 4. 2021–2024 (Dr. Christof Traber): A fully operational electromechanical reconstruction based on the 1941 patent application, using modern relays, stepping switches, and relay-chain clock generation. It is housed and regularly demonstrated at the Enter Technikwelt Solothurn in Derendingen, Switzerland. The 2001 reconstruction of the Z3 by Raul Rojas and team. Modern relays were interconnected using circuit boards. The console and punched tape were simulated by means of a modern display. Photograph by R. Rojas. Christof Traber describing the Z3 replica at Enter Technikwelt Solothurn circa 2025

Binary Numbers Explorer

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Digital Logic Gates Before there were physical logic gates, there was the math. In 1847, English mathematician George Boole published The Mathematical Analysis of Logic. He invented Boolean Algebra - a system where logical propositions could be evaluated using only two states: True (1) and False (0) . He formally defined the fundamental logical operations, AND, OR, XOR (eXclusive OR) and NOT, but using other names. Each performs some operation based on the 1/0 state of its inputs. By wiring copies of these four basic gate types, we can construct any digital computing device. The four gates are: A and B are the inputs, either 1 or 0 for either. C is the output. Importantly, all digital gates are internally built from combinations of on/off switches. By definition; AND: A and B must both be 1 for C=1 OR: A or B must be 1 for C=1 XOR: Output C= 1 only when its two inputs are different . So, C=0 when A=B. INV: The output is the inversion of the input. 1 becomes 0 , 0 becomes 1 Experiment with the Digital Logic Evaluator below to see how each gate works. Keep in mind that all computer hardware operations are made from these gate types or logically equivalent circuits. The power of the inverter Augustus De Morgan, a 19th-century British mathematician proved (De Morgan’s Law) that NAND (AND with Inverter on output) gates alone can build any logic circuit. The same holds true for a NOR (OR with Inverter on output). This means that the four basic gates above can each be made using some wired combo of NANDs only (or NORs only). For example, here is the equivalent of a standalone XOR gate made with 4 NANDs. Why is this important? One gate type can be reapplied to make the most sophisticated computers imaginable. Is the NAND (or NOR) only approach the most efficient way to build any gate type? Not usually, but the concept is powerful. Also, without the inverter operation (needed for the XOR no matter how it is built), there would be no digital adders -- no computers. An inverter must be built using an active device like a relay or vacuum tube or transistor. On the other hand, AND and OR gates can be constructed using passive resistors and diodes only. In 1921 Emil Post, a mathematics professor, proved that a system of logic gates is only capable of performing any mathematical operations if it includes a way to flip bits, such as a NOT gate. Without this ability to invert values, a system relies entirely on gates whose outputs can never decrease when inputs increase, making general computation impossible. A relay is a natural inverter, it deserves our respect.

Interactive 1-bit Full Adder Addition is the fundamental arithmetic operation in digital computers. Just like elementary school decimal addition—where adding 9 + 3 produces a sum of 2 and a carry-out of 1 —computers add digits column by column, passing carries to the next position. While humans calculate in base-10, computers operate in binary (0 and 1). Check out the Binary Number Explorer for some interactive fun. The standard building block for multi-bit addition is the 1-bit Full Adder. It takes three 1-bit inputs: A and B (the bits to add) Carry_in (the Carry_out bit received from the previous column) The adder processes these inputs to produce two outputs: Sum (the result bit for the current column) Carry_out (the carry bit passed to the next column as Carry_in) Expressed as an equation: A + B + Carry_in = 2 x Carry_out + Sum The classic adder truth table below shows all possible input and output values. With 3 inputs there are 8 possible sets of outputs. The "Dec" column is the decimal equivalent of the 2 binary outputs. See for example the last row where all the inputs add to decimal 3. The adder below is the classic 1-bit form made of AND, OR and Exclusive OR gates. Learn more about these gates here . This basic block can be replicated and chained to create any length of adder (8, 16, 32, 64,...bits). The R30 relay computer uses a 2-relay, 1-bit adder, six times to create a 6-bit adder. The two-relay adder is described elsewhere on this site and is equivalent to the diagram below in logic operation. Importantly, there are many variations of this circuit so don't get confused if you see something called a "Full Binary Adder" and the design looks different. All methods are equivalent logically with 3 inputs and 2 outputs. To see the adder in operation, experiment with the demo below. Try different inputs and see the resultant outputs.