Analog Computers Vs Digital Computers: A Short History

You can find the Veritasium channel here.

What is an analog computer and how is it different than a digital computer? Where did digital computers get their name? What role did computers play in World War II? Veritasium covers a lot of ground in the 20-minute video whose transcription appears below.

How Computers Were Invented

“In 1901, this ancient Greek artifact was discovered in a shipwreck off the island of Antikythera.

3D x-ray scans have revealed it contains 37 interlocking bronze gears, allowing it to model the motions of the sun and moon, and predict eclipses decades in advance.

Constructed around 100 or 200 BC, the Antikythera mechanism represents a sophisticated early computer the likes of which would not be seen again for at least a thousand years.

Now, of course, this computer didn’t work like modern digital computers. It works by analogy. The gears were constructed in such a way that the motions of certain dials are analogous to the motion of the sun and moon. It is an analog computer.

Here is a simple analog computer for adding two numbers together. If you turn the black wheel some amount and then turn this white wheel a different amount, the gray wheel shows the sum of the two rotations.

In contrast, this is a digital mechanical computer where you can add two single-bit numbers. So zero plus zero equals zero. Zero plus one equals one. And one plus one equals two.

These two devices illustrate the differences between analog and digital computers. Analog computers have a continuous range of inputs and outputs, whereas digital-only works with discrete values. With analog computers, the quantities of interest are actually represented by something physical, like the amount a wheel has turned whereas digital computers work on symbols like zeros and ones.

If the answer is, say, two, there is nothing in the computer that is ‘twice as much’ as a one. In analog computers, there is. For thousands of years, people used analog devices like the Antikythera mechanism or slide rules, alongside digital devices like abacuses.

And up until the 1960s, the most powerful computers on the planet were actually analog. Digital computers exploded onto the scene with the advent of solid-state transistors. And now, almost everything is digital.

Most people have never even heard of analog computers. But today, that may all be changing. Moore’s Law, the idea that you can double the number of transistors on a chip every two years, it’s reaching its limit, because transistors are nearly the same size as atoms.

Simultaneously, advancements in machine learning are straining the capabilities of digital computers. The solution to these challenges may well be a new generation of analog computers.

One of the most important problems humans have faced for millennia is predicting the tides. Napoleon and his men nearly died crossing the Red Sea due to a miscalculation of the rising tide. And sailors routinely needed to know the tides to bring their ships into port without running aground.

Most coastal locations on earth experience two high and two low tides per day, but their exact timing varies as does their magnitude. And this is partly caused by local factors like the depth of the sea bed and the shape of the shoreline.

In the late 1700s, to describe the tidal flow of the oceans, Pierre-Simon Laplace derived a set of complicated differential equations. They had no analytical solution, so at that time they were basically useless. But in the process of deriving his equations, Laplace made a key finding: tides are driven at only a few specific astronomical frequencies, including the moon, the sun, and the eccentricity of the lunar orbit.

Each one of these factors contributes a sine wave of a particular amplitude and phase to the total tide curve. If someone could figure out how to correctly combine these frequency components, well the tides could finally be predicted.

It took nearly a century but in the 1860s, William Thompson, later Lord Kelvin, took up the challenge. Having completed several voyages to lay the first transatlantic telegraph cable, he developed a fascination with the sea.

And subsequently, he threw his full scientific effort into measuring and predicting the tides. Tide gauges at that time used a buoy to record the height of the sea onto a paper roll. Kelvin set out to determine how sine waves, with the frequencies identified by Laplace, could add together to produce the observed tidal curve.

The key was to apply the work of French mathematician, Joseph Fourier, who had shown how to decompose any function into a sum of sine waves. Most English scientists were skeptical of the work, but Thompson was enthralled by it. His first paper, published at 17, was a defense of Fourier. While it was straightforward to apply Fourier’s analysis to tidal curves, the computation required was enormous.

How Analog Computers Used–And Were Developed With-Math And Computation

First, divide the tide curve up into short time intervals. And for each interval, multiply the tide level by a sine wave with the frequency of interest. Add up the area of all these rectangles, and divide by the total time. And this gives you a single coefficient–the amplitude of the sine wave with this frequency.

Then you have to repeat the process for a cosine function with the same frequency. Kelvin found that to make accurate predictions, he actually needed 10 different frequency components. So that is a lot of multiplication and addition to characterize the tides at just one location.

For each additional location, you have to perform this analysis all over again. And this is only half the problem. Once you have the amplitudes and phases of the sine functions, you have to add them up to predict the future tides.

Lord Kelvin spent years analyzing and predicting tides by hand. Then he had a stroke of inspiration. Could you design a machine to carry out these calculations automatically? In Kelvin’s words to “substitute brass for brains,” the resulting analog computers were in use for nearly a century. They even played a critical role in the outcome of World War II.

Kelvin started with the problem of prediction, adding the sine waves together, given you know their amplitudes and phases. He knew he could create sinusoidal motion with a device called a scotch yoke. It extracts one dimension from uniform circular motion.

But to make a tide prediction, he needed a way to combine 10 sine waves together. He needed a mechanical analog for addition. Stuck on this problem in 1872, Kelvin boarded a train for a meeting with the main sponsor of his tidal research, the British Association. On the train, Kelvin bumped into a friend, inventor Beauchamp Tower, to whom he explained his dilemma.

Towers suggested he used Wheatstone’s plan of a chain passing around a number of pulleys. And this was exactly the addition mechanism Kelvin was looking for. By attaching a pulley to each scotch yoke and running a weighted cord around them, he could mechanically add all of their contributions at once.

He scribbled down the entire plan for this predictor machine by the end of the train ride. He pitched it to the British Association, and secured funding to build it all before he returned home.

If you knew the relative contributions of different frequency components, Kelvin now had a machine to automate the tedious task of predicting future tides. This was a great leap forward. Four hours of cranking the handle yielded a full year of tidal predictions.

But for many years, the harder half of the problem was still done by hand, breaking apart an existing tide curve into its component frequencies. To automate this step, Kelvin needed a machine capable of multiplying the tide curve times the sine wave and then taking its integral.

What would such a device even look like?

With his older brother, James Thompson, Kelvin came up with a mechanical integrator. It consists of a ball on a rotating disk. Due to the rotation of the disk, the further the ball is from the center, the faster it spins.

If the ball is at the very center of the disk, it doesn’t turn at all. And if it’s on the left side, it turns in the opposite direction. Now the motion of the ball is converted into an output via a roller, which moves a pen up or down on the output graph paper.

So the way it works is you trace the function you want to integrate with a stylus, and the stylus controls the position of the ball on the disk and hence its speed of rotation. This is transferred through the roller to the output, which plots the integral of the original function.

Now to decompose a tide curve, we don’t just wanna integrate the function. We first wanna multiply it by a sine wave of a particular frequency. And the way to do this is to make the disk rotate back and forth at that specific frequency.

Now the rotation of the ball depends not only on where it is on the disk, but also on how the disc is turning at an instant. You trace the tide curve with the stylus, which moves the ball back and forth on the oscillating disk, and the roller sums up the integral of the tide curve times the sine wave.

Simply divide by the total time to get the coefficient. Several of these ball and disk integrators can be connected in parallel with each disk oscillating at a different frequency to calculate the coefficients for multiple frequency components at the same time.

Kelvin’s analog computers revolutionized our ability to predict tides. Tidal curves from anywhere in the world could be turned into a set of sinusoidal coefficients using the ball and disk harmonic analyzer. And the resulting sinusoids could be added together to predict the future tides using his scotch yoke pulley machine.

How Computers Helped The Allies In World War II

Derek Kelvin’s harmonic analyzers were the basis for a landmark analog computer called the differential analyzer, and his tide predicting machines were used well into the 1960s. In fact, they were later overhauled to include 26 frequency components and used to plan the Allied invasion on D-Day.

The Germans expected any invasion to come at high tide to minimize the time Allied soldiers would be exposed on the beaches. So they installed millions of obstacles that would be underwater by mid-tide, many with explosive mines attached.

But the allies spotted the obstacles and change tack. Instead, they plan to begin the invasion at low tide. This would allow demolition teams to first clear channels through the obstacles, then the main forces could come through those gaps as the water rose.

This would also give landing craft enough time to depart without getting beached. The low water times were different at the five landing beaches by over an hour, so the invasion times were staggered according to the tide predictions.

This wasn’t the only use of analog computers in World War II. Dive bomber aircraft would plummet out of the sky directly toward their targets at up to an 80-degree angle, and their rapid descents made them very difficult to shoot down. So the U.S. began searching for devices to automatically aim guns at dive bombers.

Most of the proposed solutions fell into one of two categories. Some were analog machines like Lord Kelvin’s. Others were essentially fast calculators. Mechanical calculating machines like the abacus had been around for millennia, but they were far too slow to respond to dive bombers.

These new calculating machines sped things up by using electrical pulses. The committee considered naming these devices after the pulses they used. But member George Stibitz proposed a more general name: ‘digital’ because these machines operated on numbers themselves (or digits). And this is the origin of the term digital computer.

But digital would have to wait. Of all the proposals and innovative analog machine from David Parkinson won out. At Bell Labs in New York, Parkinson had been working on a device to chart telephone data called an automatic level recorder.

It used a variable resistor called a potentiometer to control the motion of a pen. One night, after hearing reports of the harrowing allied evacuation of Dunkirk, Parkinson had a dream that he was on the front lines.” I found myself in a gun pit with an anti-aircraft gun crew.

“A gun there was firing occasionally, and the impressive thing was that every shot brought down an airplane. After three or four shots, one of the men in the crew smiled at me and beckoned me to come closer to the gun. When I drew near, he pointed to the exposed end of the left trunnion. Mounted there was the control potentiometer of my level recorder.”

When he woke up, Parkinson realized the device he was building to control a pen could be scaled up to control an anti-aircraft gun. He shared this idea with his supervisor, and after receiving approval from the military, they set out to make Parkinson’s dream a reality.

Researchers at Bell Labs had recently invented an analog electrical device called an operational amplifier or op-amp. It could perform mathematical operations with voltages, like addition and multiplication. They used these op-amps to create an analog computer that could solve the ballistics equations for anti-aircraft guns.

Using radar and optical sites to obtain the speed, altitude, and direction of enemy planes, the M9 Gun Director, as the computer was called, could rapidly calculate the correct trajectory and few setting. Potentiometers were used to ascertain the direction the gun was pointing. This was not the first electric analog computer, but it was an important one.

In World War I, It took an average of 17,000 rounds to take down a single airplane. In 1943, after the invention of the M9, it took an average of only 90. During the war, the U.S. invested big in analog computers. If you break down their total military budget, the third largest single expense was the development and production of an incredibly complex mechanical analog computer called the Norden bombsight.

Unfortunately, they didn’t get their money’s worth. Designed by the eccentric Dutch engineer, Carl Norden, the Norden bombsight was built to enable high precision airborne bombing. It implemented 64 different simultaneous algorithms, including one that compensated for the rotation of the earth as the bomb fell.

The Norden was one of the most closely guarded secrets of the war. To prevent the technology from falling into enemy hands, American bombardiers carried handguns specifically to destroy it in the event of a crash. But despite its hype and funding, the Norden didn’t work as advertised.

With over 2,000 fine parts, it required extreme precision to manufacture. The problem with analog computers is that the physical device is a model for the real world. So any inaccuracy in the components translates into inaccuracy of the computation.

And since there will always be some slop in the connections between parts, if you run the same calculation twice, you won’t get the exact same answer. In the American campaign against Japan, bomber crews using the bombsite were unable to destroy critical Japanese war infrastructure. And ultimately, the U.S. abandoned its precision bombing approach, and instead blanketed whole Japanese cities in napalm.

As the war progressed, digital computers gained traction. The digital and electronic Colossus machines of Bletchley Park in the UK were critical to breaking German codes. In the United States, the military invested in an enormously complex and expensive digital machine, known as ENIAC. It was designed to speed up the calculation of land artillery firing tables. At the time, these were computed using differential analyzers, the analog mechanical computers based on Kelvin’s harmonic analyzer.

Although not finished until after the war, ENIAC demonstrated the power of digital computers. It’s considered by many to be the first modern computer. What really opened the door to this digital revolution was the discovery made by Claude Shannon in his 1936 master’s thesis.

He showed that any numerical operation can be carried out using the basic building blocks of Boolean algebra: Two values, true or false, also notated as one or zero, and three operations and, or, and not.

The Advantages Of Digital Computers Over Analog Computers

This makes digital computers the ideal versatile computing machines. In contrast, each analog computer is an analog for only one type of problem. Furthermore, since digital computers operate on ones and zeros, they are more resilient in the face of noise. It would take a large error to mistake a one for a zero or vice versa.

Whereas, even small errors in analog computers can grow and ultimately swamp the signal. So these days, everything is digital. Our phones, computers, and internet data centers, even TV and radio is now being broadcast as digital.

The advantages are obvious. Since digital devices operate on symbols, usually zeros and ones, they provide exact answers. And repeat the calculation, and you get the same result. They are oblivious to noise.

Plus, since only a few components are required to perform virtually any computation, those components have been miniaturized and optimized, making digital computers the ideal universal computing machines. So you would think analog computers would be long gone, a relic of the distant past. But, analog may now be making a comeback. There are startups actively working on analog computers.

Why is this happening? What could be the benefit of analog? I wanted to put all of these into one video. But the story is too good to bury 20 minutes in, so that is coming up in part two.”