**A Tale of Family History, Calculators, and the Father of the Computer**

One of the things that I love about family history are the small stories you learn along the way. Once you sift through all the farmhands and servants, you may occasionally come across a person who seems a bit more interesting. Suddenly, the focus is no longer about various lineages and families evolving across centuries. Instead, you find yourself looking through a window into the life of one individual. The questions are no longer just `Who is this person related to?’ or `Where did this person live?’, but also `What was it like to live in this period?’ and `Where did this person fit in society?’. Being able to ask these questions allows me to have a much deeper connection to the research, which can otherwise feel like an exercise in spreadsheets and convoluted filing systems. And I think that is pretty cool.

Now, throughout my historical research, these types of individuals are mostly based in the armed forces. This might just be because of the historically working-class nature of most of my family, or maybe because joining the armed forces is a life-altering decision that also happens to have a decent paper trail. Either way, it was a pleasant surprise to recently come across my great(x5) granduncle Joseph Clement, an engineer based in Southwark, London during the early 1800s. Though I had never heard of Joseph before, I soon discovered that he was involved in one of the most influential mathematical developments of the Victorian era. And, as I found out, it all starts with the machines that occupy most of our lives.

__The Machine Revolution__

It is sometimes difficult to appreciate what life was like before machines. By 2025, the World Economic Forum predicts that half of all work tasks will be handled by machines, and nowadays the average person always has access to a digital calculator in their phones. But this is still a relatively recent development; the first all-electric calculators were released in the late 1950s, but portable calculators were not commercially available until the 1970s. I think it is interesting that the public understanding of calculating tools includes the abacus (originating before 2300 BCE) and electronic calculators, but there is a gaping chasm between these two technologies that seems to be often forgotten.

When I try to think of a mechanical calculator the first thing that comes to mind is an old-fashioned till, or cash register in the US, partly due to watching a significant amount of the British sitcom *Open All Hours* as a child. The first mechanical tills were simple examples of a calculator, since they usually only added numbers, and were first patented in 1883.

Along with the *Arithmometer* and *Comptometer*, which both entered production in the mid-to-late 1800s, mechanical tills helped to popularise the use of calculating machines in businesses. Over time, these machines expanded their capabilities, allowing for mathematical operations such as subtraction, multiplication, and division. But it all started with a simple adding machine.

__It’s all adding up now__

Although calculating machines became commonplace in the late 1800s, mathematicians had been developing this technology for centuries prior. In 1642, 18-year-old Blaise Pascal was assisting his father in supervising the taxes of Rouen. To reduce his workload, Pascal designed a device known as the *Pascaline* that could add or subtract two numbers directly. This machine used a system of spoked wheels and was notable for being able to `carry the one’ across different digits.

Gottfried Leibniz, of biscuit fame, continued this work by completing his `*stepped reckoner*’ in 1694. This device was supposed to add and subtract two numbers directly as well as being able to perform long multiplication and long division. Unfortunately, the machine proved to be unreliable and never progressed past the prototype stage. The operating mechanism, called the *Leibniz wheel*, continued to be used in calculating machines for the next 200 years, however.

During the 1700s, various new calculating machines were designed based off the work of Pascal and Leibniz. These machines often tried to reliably multiply and divide two numbers directly, and often failed. Part of the reason for this is that multiplication and division are hard. As numbers get larger, trying to multiply becomes a laborious process that we have developed tricks to speed up.

Imagine how you would solve 62 **x** 47, and then imagine how you would instruct a machine to solve the same problem. You might think to use long multiplication, box multiplication, or maybe some other method to find that 62 **x** 47 **=** 2914. In every case, however, you are probably using those multiplication tables that you had to memorise as a child, something that calculating machines do not have access to. We now have multiple ways to speed up this process for machines, including the use of a binary number system, but this was still a glaring problem for the calculating machines of the time.

Conveniently, an alternate way to multiply and divide numbers had been made possible in the 1600s, thanks to the discovery of logarithms. Logarithms can be defined for various base numbers, commonly 10, 2, or the exponential constant *e*, but they all share the same property that

log(a) **+** log(b) **=** log(a **x** b).

Then, you can calculate (a **x*** *b) by first adding the logarithms of a and b and then finding the number whose logarithm equals that addition. Let’s take the example of 62 **x** 47 again, and I will use the “common” logarithm (which just means base 10). Looking up the values for log(62) and log(47), I find that

log(62) **=** 1.79329, log(47) **=** 1.6721,

and so,

log(62 **x** 47) **=** 1.79329 **+** 1.6721 **=** 3.46449.

Similarly, I can look up what number has a logarithm of 3.46449, and I discover that

log(2914) **=** 3.46449.

So, if I can look up possible logarithm values, I can calculate (a **x** b) without ever actually doing multiplication, which could then be automated by a mechanical calculator of the time. Extensive tables of logarithms were produced by hand and proved exceedingly helpful for engineering and navigation, since calculations required a great degree of accuracy. However, these tables were known to contain errors, which could prove deadly for sailors. And so, in stepped Charles Babbage.

__Babbage makes the difference__

Charles Babbage was a busy man. He reformed the British post system, invented the cowcatcher, and publicly crusaded against children rolling a hoop down the street. He was also a founding member of the Royal Astronomical Society, who had taken an interest in the problems regarding the accuracy of logarithm tables. Babbage argued that he could design a machine that would automate the production of these tables, and that this would ensure their accuracy.

His pitch was successful, and the British government provided him with £1700 (a little under £100,000 in today’s money) to begin work on his *`difference engine’*. The difference engine was ground-breaking in its complexity compared to other mechanical calculators; it had storage, where data was held for processing, and was designed to take up an entire room. Importantly, it would stamp its results into soft metal, which could be used for printing. This removed any possibility of errors in copying the results during typesetting, which Babbage saw as the main culprit for any errors in hand-written tables.

The engine used the principle of divided differences, hence its name, to compute values of a polynomial. To see this principle in action, let’s think about an example. Take a racing car driving down a straight racetrack; we don’t how fast the racing car is going, but we’ll assume that they are accelerating at a constant rate. The question is: if we know where the car is at certain times, can we predict where it will be in the future?

First, let's solve the problem by hand. By assuming that the car is accelerating at a constant rate, we can write down a formula for the distance s in terms of the time t, acceleration a, and initial speed u. The formula looks like this:

s **=** u **x**** **t

**+**(1/2)

**x**a

**x**

*t^2,*

where t^2 means t squared; this can be derived from calculus, though that isn’t important here. So far, we have a formula for the distance s, but we need to know how fast the car is accelerating and its initial speed. Let’s say we now measure the car’s distance at certain times, and we find that it has travelled 4 meters after one second and 12 meters after two seconds. Then, we could use these values to solve for u and a:

t **=** 1, 4 **=** u **+** (1/2) **x** a,

t **=** 2, 12 **=** 2 **x** u **+** 2 **x*** *a,

and we find that a=4 and u=2. This means that

s **=** 2 **x** t **+** 2 **x*** *t^2,

and so, the car will have travelled 24 meters after 3 seconds and 40 meters after 4 seconds. This is a reasonably straight-forward solution, but it required us to solve two simultaneous equations using some mathematical logic. So how do you get a machine to solve this problem?

To solve this problem, we can use a technique known as finite difference. We begin by defining the first difference d1( t ) **=** s( t **+ **1 ) **-** s( t ), which is effectively the change in distance travelled in the next second, and the second difference d2( t ) **=** d1( t **+ **1 ) **-** d1( t ), which is effectively the change in average speed in the next second. If we return to the general formula for the distance s in terms of time t, acceleration a, and initial speed u,

s(t) **=** u **x** t **+** (1/2) **x** a **x** t^2,

we note that,

d1(t) **=** u **+** a **x** t **+** (1/2) **x** a, d2(t) **=** a.

The important thing here is that d2(t) is constant, i.e. it doesn’t depend on the time t. This is true for all quadratic polynomials (that is, where t^2 is the highest power of t). In fact, for any polynomial where t^m is the highest power, the m-th difference is constant; this is the basis behind Babbage’s design for the difference machine.

We can write down a table for the distances and first and second differences for our earlier hypothetical measurements,

t s(t) d1(t) d2(t)

------------------------------------------------

0 0 4 4

1 4 8

2 12

Then, since the fourth column is constant, we can start to fill in the missing values recursively. This means that, by adding each cell to its right neighbour we can compute the value of the cell below it. So, d1(2) **=** d1(1) **+** d2(1) **=** 8 **+** 4 **=** 12, and so on:

t s(t) d1(t) d2(t)

-------------------------------------------------------------------------------------------

0 0 4 4

1 4 8 4

2 12 (8+4) = 12 4

3 (12+12) = 12 (12+4) = 16 4

4 (24+16) = 40 (16+4) = 20 4

5 (40+20) = 60 (20+4) = 24 4

which we can check with our previous answer that we found by hand. Hence, a machine with (m+2) columns can accurately tabulate the values of any polynomial up to a power of m, given that you have (m+1) initial values. Many more complicated functions (such as logarithms and trigonometric functions) can be approximated by polynomials, and so Babbage’s design appeared to be the solution to the error-filled logarithm tables that plagued British sailors. However, things are never that quite that simple.

__You just can’t get the staff these days!__

Much like with the mechanical calculators of the 1600s, the construction of the difference engine was hampered by the engineering capabilities of the time. Babbage was very serious about the project: he built a dust-proof environment to test the machine, set up a fire-proof workshop, and hired master machinist Joseph Clement (remember him from earlier?).

Westmorland-born, Clement was famous in London at the time for his high-precision tools, including lathes and planers. By 1832, Babbage and Clement produced a working model of one-seventh of the full engine, which could compute second-order differences with up to six-digit numbers. This portion of the engine consisted of about 2,000 parts and can still be seen at the Science Museum in London.

However, in 1833 Babbage and Clement had a falling-out – supposedly Babbage thought that Clement was using the projects funds to improve his own workshop, while Clement refused to continue his work unless Babbage paid for the tools required to build the engines parts. Whatever the reason, work on the engine was suspended and the project was abandoned in 1842; in this time, £17,000 (just under £1 million in today’s money) of governmental funds had been invested into the project, and a working engine was still no closer. It is fair to say that, in the eyes of the British Government, Babbage’s difference engine was a complete and utter failure.

__Rise of the Machines__

But of course, Babbage’s story does not end there. During the difference engine project, Babbage began work on a more general design, known as the *analytical engine*. This machine would be provided with initial data and programs via punched cards and could then perform any number of arithmetic tasks. The logical structure of the analytical engine is essentially the same as modern-day computers, but the significance of this work was lost on many scientists at the time.

One person who saw the potential in this machine was Ada Lovelace, a friend of Babbage’s who had been inspired by the difference engine prototype built by Clement. In 1843, Lovelace translated an article by Italian mathematician Luigi Menabrea on the analytical engine and, with it, attached a set of her own notes. These notes almost doubled the article itself and included a detailed algorithm for the engine to calculate Bernoulli numbers. This algorithm is often seen as the first computer program, and Lovelace as the first computer programmer.

While Babbage focused on the capabilities of machines to calculate functions and crunch numbers, Lovelace saw the potential far beyond this. She observed that machines may be able to compute things other than numbers, if those things satisfied mathematical rules. In an age where machines determine large portions of everyday society, it is hard to disagree with her.

And so, that is where my research ends for now. I began with the life of Joseph Clement, a small cog in a machine that spent part of his life building small cogs for a machine. But in that life, I saw the efforts of multiple scientists (such as Pascal, Leibniz, and the engineers of the early 1800s) converge in one of the most ambitious feats of engineering of its time. And though the project was a failure, its influence, thanks to Charles Babbage and Ada Lovelace, can still be seen in computers to this day. Hundreds of years of mathematics and engineering distilled into one story. And, as I said, I think that is pretty cool.

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