A few miles down the Tigris from Baghdad, a great arch stands alone in the countryside. It is the Taq Kasra, one of the last remaining structures of the ancient Persian city of Ctesiphon. Its origins are lost to history.

Its local name suggests it was built for the Sasanian king of kings, Khosrow I, to commemorate a victory over the Eastern Roman Empire in the 6th century. Other sources date it as far back as the reign of Shapur, who defeated the Roman emperor Valerian in 250. However long ago, in the millennia since the arch was built, histories were lost, empires rose and fell and still it stood, far taller and more graceful than anything built until the modern era. Modern visitors can only stand and wonder how the early engineers who designed the structure planned its shape. And how they explained their unique and revolutionary idea to the king of kings himself.

One thing those ancient engineers did not do is use anything like modern mathematics or calculus. These approaches were not invented until the era of Leibniz and Hooke, at least a thousand years later. But the arch’s shape perhaps tells the story for us. It is an inverted catenary and so mirrors the curve formed by a chain hanging between two points. As this shape is determined by the action of gravity, it may have helped them develop a form that could stand strong through the ages.

It is possible those early engineers literally hung a chain from posts, moving them closer and further apart until they settled on a shape they liked, and then building a scale model to show how the arch would stand.

Building on tradition

More than three thousand miles away, another catenary, the Clifton Suspension Bridge, spans another river, The Avon in the UK. Unlike those unnamed Sasanians, the designer of this bridge, Isambard Kingdom Brunel, had the advantage of new tools. He had the calculus developed by Leibniz and Hooke, and he had paper.

Brunel was doing engineering calculations with nothing more than paper, a quill pen presumably, and a pot of ink. His original calculation notebooks are still lovingly preserved at the University of Bristol. He combined the tools available to him to match the grace of the Taq Kusra.

“What strikes me about his notebooks was he built these amazing feats of engineering such as the Clifton suspension bridge, using nothing more than the diagrams he drew on paper and calculations he did in his notebooks,” says Samir Khan, a Product Manager at Maplesoft, responsible for a new piece of software known as Maple Flow. “So, the Clifton Suspension Bridge – I’ve seen the original calculation notebooks for that – it’s a catenary bridge. So there’s a chain, which is the catenary, and there are the rods which connect the catenary to the actual bridge.

“To work out the length of the catenary chain and the height of the rods, he actually did calculus, it’s right there in his original calculation notebooks. So he derived the equations of a catenary using consequent calculus, he then substituted in numerical values to calculate the length of the chain and the height of the rods. And he was doing this back in the 1850s.”

Throughout history, engineers, mathematicians, and all those who work with numbers have invented tools to speed their work. The tools and theories they use feed off each other. A new tool makes it possible to work with numbers in new ways, and the theories produced from that more efficient mental labour allows for the development of new tools.

Start with what you know

One of the earliest tools for working with numbers was developed by another empire of the Middle Eastern fertile crescent, the Sumerians, as much as five millennia ago. The approach to counting they developed is still used around the world, every minute of the day. By tapping your thumb against each of three bones on the four fingers of the same hand, you can easily keep track of a count up to twelve. Folding the fingers and thumb of the other hand lets you keep track of multiples of 12, up to a total of 60.

Those early agricultural empires were concerned with fractions and seasons, and a number system made up of multiples of twelves and fives is easy to divide in many ways. It is so useful, that today, the base 60 number system derived from that counting system remains in use, in our seconds, minutes, hours and months.

Getting out of hand

But how can bureaucrats ensure an army of tens of thousands can be fed for a years-long campaign? Or a landowner calculate how many cabbages can be grown on a vast estate, or how many ships will be needed, on what days, to transport them at a steady rate to merchants in cities across a kingdom?

When people needed to count with more precision to much higher numbers, the human body gave way to the abacus. This simple mechanism of beads and rods, refined in civilisations around the world over thousands of years, allows users to count into the billions on a handheld device. It can be used to add and subtract, to multiply and divide, and to find square and cubic roots.

Other tools allow for even more complex calculations. The Antikythera mechanism, an ancient Greek analogue computer, is thought to have allowed users to track the movements of the heavens over decades. Researchers studying the machine have used both physical examination of the device, and models of the mathematics available to its designers, to develop ideas on how it may have been put together and used.

All of these devices have some common characteristics; they’re designed with practical applications in mind and they take some of the mental labour of working with numbers away, without added complexity that introduces new mental labour.

The faintest ink is more powerful than the strongest memory

In the thousand or so years after the fall of Persia and of the Western Roman empire, society became exponentially more complicated. New tools allowed for more complex thinking, and more complex thinking allowed for more advanced tools. Paper, developed in China and used thousands of years later by Brunel, allowed mathematicians to develop and share advanced ways of working with numbers.

Vellum was rare and expensive. A book written on vellum would require the soft skin from the bellies of many herds of animals. Scribes had to plan what they would write, before they set to work. It was also rough, making the act of writing laborious, while writing on paper feels frictionless, you can quickly jot down as many ideas as you like and share them around the world. As Newton wrote to Hooke, throwing shade at his diminutive rival, ideas could be developed by thinkers ‘standing on the shoulders of giants’.

Precise calculation allowed for the development of more reliable metallurgy. Which, along with his paper notebooks and the calculus he performed in them, allowed Brunel to design structures like the Clifton Suspension Bridge.

The first computers

But Brunel was working at the limits of that tool for thinking. And moving beyond Brunel requires us to enter the mind of Charles Babbage, the father of the computer.

“He was actually a contemporary of Charles Babbage. They lived and worked in England, Charles Babbage was a mechanical engineer who came up with the concept of the Analytical Engine,” says Khan. “It is a mechanical computer based upon gears, a mechanical device for doing computations. And the Analytical Engine actually had all of the basic concepts that we still use in modern computers, such as storage, arithmetic processing units, for loops, and so on. He never got to build the Analytical Engine. But what he came up with was the framework for modern electronic computers.”

As Babbage was developing his ideas for a computational device, his peers were working on ways that device could be used. One such was Ada Lovelace, who showed how the device could be used for something other than pure calculation, writing one of the first algorithms to be performed by the Analytical Engine. If Babbage was the father of computer hardware, Lovelace was the mother of computer software.

The modern era

But it was much later, a century later that these ideas came fully to fruition. When Alan Turing arrived on the scene, the work Babbage began with computation and the work Brunel was doing with engineering calculations, finally collided. And in the 1940s and 1950s the first electronic computers began to be developed. Even then, the proves took decades.

“That was when IBM started developing their first electronic computers. Back then, people used to write programs using punch cards. Punch cards evolved into assembly language; assembly language evolved into high level programming languages like Fortran. And that eventually led to the development of numerical libraries, some of which are still being used today, high level computer applications for doing engineering, computation, and modern application software,” says Khan.

Well into the 1970s, the sort of complex calculations used by academic mathematicians required room-sized machines. Generations of their more practical number-working peers, like engineers, would still rely on pen, paper, and slide rule to get their work done. A key step in the widespread use of these new tools came in the late 1970s, when two professors at the University of Waterloo developed Maple.

“Back then computer algebra tools ran on mainframe computers. So these big, unwieldy machines that sat in a large office. But back in the late 70s, early 80s, these new microcomputers started hitting the streets and they were cheaper and more widely available. So Keith Geddes and Gaston Gonnet just wanted to make computer algebra tools available to more people.”

Geddes and Gonnet were frustrated for themselves, and for their students. Accessing the mainframe needed to perform calculations needed booking and connecting over a 56k line. Maple, running on the desk scale microcomputers of the day, would let students and researchers access similar tools whenever they needed. It would reduce the friction of performing complex calculations, allowing these academic users to focus on developing new ideas, rather than being bogged down in the legwork of calculation, or worse, waiting for a slot on the mainframe to be available.

Maple takes the stage

Laurent Bernardin interned with the company one summer in the 1990s before returning to university in Zurich to complete his PhD. He returned, after completing his research, as a developer, and is now the company’s CEO.

Bernardin says, “[The] two professors got together realising that there was a need in math education for a tool to help with calculations and setting out. [They created] that tool. And Maple was born quickly, was adopted across universities around the globe. And, today, that’s a really important part of our user base of our market.”

Engineers of all kinds, architects, product designers, production managers working with complex supply chains use these tools in a way that suits their work.

Khan says, “Maple became commercialised in 1985. That was when Waterloo Maple was first formed to distribute this new computer algebra tool. It started being made available at more universities, companies started buying it. The interface went through several iterations and became more usable. There were interactive plots, you can overwrite documentation. And that’s eventually evolved into the spectrum of software tools that Maple provides today.”

When the original versions of these programs came out, there was no Windows, no mice, no virtual desktop, or folders. Just a command line. And for the original users of Maple, as for many academics and mathematicians today, that was just fine.

“That’s all it was. So it was a lot more utilitarian,” says Khan. “And it’s funny enough, we still have many people who like using Maple that way, there’s still a mode in the modern software tool, which imitates the design metaphor of early DOS prompts, for example. So you just see lines of code, you execute it, you see the results.

But as we’ve seen, tools evolve to suit users. Sumerian farmers, priests, and tax collectors could work quite well, just counting on their fingers. While the civil servants and generals of the Chinese and Roman empires needed abacuses to work with much higher numbers and much more precision.

Sasanian architects could design a graceful arch with just a chain and two posts. While Brunel needed the work of earlier scholars, and masses of paper, to build his bridge. Today’s tools for thinking must also be adapted to users’ ways of working.

“Different tools have different design intents,” says Khan. “Some tools are designed for programmers such as code development environments, like Visual Studio. Some environments are aimed at mathematicians, people who need precise control over the mathematical structure of their equations, and in some environments are designed for engineers who simply want to throw down a few equations on a virtual whiteboard and manipulate them and get results.”

That insight has driven the development of MapleFlow. It’s designed to take the power of the Maple computational engine, and present it in a two-dimensional environment, like that of Brunel’s notebooks, or a whiteboard.

Margaret Hinchcliffe, Senior GUI Developer for Maplesoft says, “Our design goal was really to make the calculations quick to create, quick to work with, intuitive to work with. We’re kind of working in comparison to our other product, Maple, which is More like a Word document or something, with content created from the top to the bottom.

With maple flow, we wanted to be able to just throw content into the document at any location and not have to worry too much about the alignment of things or kind of working from top to bottom, being able to kind of jump around in your document. And that’s allowed by having this kind of 2d grid arrangement of the whiteboard.

“The Maple product is intended more for the academic market. And MapleFlow is intended more for an engineering market that’s looking, you know, to work quickly, get something that they can show people in a quick amount of time.”

MapleFlow is by no means the first tool for working with numbers in a graphical environment. Many of us will remember tools like Lotus 1-2-3, or will use modern alternatives like Microsoft Excel or Google Sheets. A lot of organisations essentially run all their business processes through spreadsheets like this. But there are distinct flaws to this approach, one is that it doesn’t allow you to show your calculations. Another is that these complex environments, developed over decades, attempt to cater for every potential use. And instead of being a tool for thinking, become an obstacle to thinking, constantly interrupting the user by making them perform the menial mental labour of working within a sprawl of different ‘improvements’.

“Excel is a great tool for capturing data and working with data tables. It’s not a great tool for engineering,” says Bernardin. “So where Maple differs is that Maple allows you to, just like a paper notebook, capture your calculations, capture the process of your thinking, from start to end, allows you to capture your calculations in equation form, equations that look the same, and just as accessible and just as visible and readable as when you’re writing down on paper.

“But of course, you have the computational power at your disposal. So you don’t have to do all the calculations by hand, you get support, to make sure that you get to the calculations faster. And you get support to make sure that you don’t make mistakes. And when you compare it to what you do in Excel in particular, in Excel, you also enter formulas, but those formulas, they’re kind of cryptic, right there behind the hidden and they are like a one-dimensional string of operations that is very hard to decipher and really decide whether you got it right or not.”

So, like we were all told at school: show your working. Whether in a raw DOS-like environment, or in MapleFlow. Maple users can show every step in their argument, seeing for themselves—and showing their colleagues and customers—how they reached their conclusions. But it also seeks to make it much easier to focus on constructing that argument.

Leveraging human nature

At that same time in the 1970s that computing power was moving from mainframes to microcomputers, psychologists were investigating how we can think efficiently. One was Mihály Csíkszentmihályi, who coined the term ‘flow state’. This is the idea that, by controlling our environment, freeing ourselves from distraction, and splitting tasks into discrete units, which each deliver their own reward, we can access a state of intense focus, not aware of anything else beyond the task. It’s something we can experience when cooking a meal.

Imagine making a soffritto, the flavour base of dishes across the Mediterranean and Latin culinary universe. You collect your ingredients: carrots, celery, onions, perhaps peppers, or garlic, or herbs. You clear and prepare your workspace, then start slicing. Each precise slice delivers a little hit of satisfaction, without being fully conscious of your end goal, or of time passing, you soon find yourself with the basis for a delicious ragu or arroz con pollo. That approach is now almost as widespread as the microcomputer or PC. It’s used in sports, in writing and in software development.

And its evil twin—hyperfocus—is used to design digital products that keep us glued to a screen, delivering subscription revenues or user metrics for advertisers – the infinite scrolling of social networks, or the carefully designed skill trees and crafting mechanics of role playing games.

“I actually have an addictive personality,” says Khan. “I get addicted to things really easily. It might be salted caramel popcorn, or it might be the World of Warcraft. And I remember when I started playing, I became addicted and I played nothing but the World of Warcraft, for the next month or so. I was even forgetting to eat or sleep.

“World of Warcraft is fantastic because it’s designed for people like me, so there’s constant stimulation. It looks pretty. You can do complex things, but the game mechanics are very, very simple. Now those concepts are actually reproduced in many other software tools. So something I become addicted to recently, and I’m not proud of this is TikTok. I like scrolling through the videos, seeing new things all the time, there’s constant stimulation.”

The challenge Khan and his colleagues set themselves was to turn these addictive qualities to good, to take the tiny dopamine hits you get from seeing a new video of a cute kitten trying to wear a melon peel as a hat, or of a guy blissfully skateboarding to work, or of gradually collecting the materials to craft a shiny new virtual axe. And to use that satisfaction to deliver useful work, to achieve flow state.

“Game mechanics, the way that you interact with Tik Tok is very, very simple,” says Khan. “I think we can take some of the lessons from tools like World of Warcraft, from social media, and actually translate those into calculation tools. The key words for me are that you have to make the game mechanics of calculation software simple. So there have to be a few simple rules, you have to learn some basic grammar before you can use the software. But based upon those simple rules, you have to be able to do complex things, solve real problems, you have to remove the cognitive overhead from using engineering calculation software.”

Hinchcliffe, Maple’s GUI Developer, adds, “I think the challenge we’ve come up with the most is just making the typical tasks that people want to do the most, and make those the most immediate. So really focusing on you know, how many keystrokes do they need to do this task? Or, how many mouse clicks do they need to do? if you want to type text, you can just start typing text, or if you want to start doing math, or creating plots, or dropping images to fill out your document. Those should just be immediately available. We have customers from aerospace, trying to land rovers on the moon on Mars, we have customers in automotive trying to design new car engines, designing automated factories for producing these cars, we have customers in manufacturing, we have customers in consumer products.”

But even clicking and dragging with a mouse is too distracting for Khan. He and Hinchcliffe, and the rest of the MapleSoft team, want to get even closer to the experience of jotting ideas down on a whiteboard. But doing that will require even better tools.

“There’s a certain tactile feedback to paper that you can’t get from a spreadsheets or a programming tool,” says Khan. “And I want to reproduce that design metaphor on the computer. So we’ve taken the first step with having a virtual whiteboard. But MapleFlow still relies on keyboard and mouse input. The next step is making use of modern advances in stylus and electronic pen inputs, turning handwritten equations into something computers can understand is actually pretty difficult because there’s inherent ambiguity in mathematics. And you need to remove that ambiguity.

“So for example, if you take something like ‘f, open brackets, x, closed brackets’, what does that actually mean? Well, it depends on the context. It could mean F is multiplied by whatever’s in the bracket x, or it could mean a function application instead. There are many instances like that in mathematics where you need context and understanding to decipher. Additionally, equations are 2d. They go from side to side, and they go up and down. Whereas handwritten text only goes from side to side.

“Traditionally, we’ve needed humans to do that interpretation. And we’ve been able to do that quite well. It’d be interesting to see if we can take advantage of modern advances in deep learning and AI to imitate what humans are doing and interpreting handwritten mathematics.”

Partner: Maplesoft
Main image: The Taq Kasra, 1940 by Roald Dahl