Building a Cathedral without Science or Mathematics: The Engineering Method Explained

Bill reveals the long-lost methods used by Medieval engineers to design stone cathedrals. Methods that required no science, mathematics, or literacy, yet which reveal the engineering method.

*Learn More: Companion Book*
Explore the ideas in this video series further with its companion book: _The Things We Make: The Unknown History of Invention from Cathedrals to Soda Cans_ (ISBN 978-1728215754)

*Other Videos in this Series*

*Video Summary*
0:00 Titles

0:07 Intro
In this first video of the series Bill notes that the engineering method is among the the oldest of human responses to fulfill human needs.

0:28 Göbekli Tepe
Bill illustrates this with a brief discussion of the ruins of Göbekli Tepe, a stone structure in southeastern Turkey built thousands of years before Stonehenge and the pyramids. A structure whose use and purpose remains mysterious. The age of Göbekli Tepe highlights that engineering existed long before science — at least what we mean by “science” today.

0:56 Precision of Göbekli Tepe
The precision of the placement of the stones in the ruin and their manner of preparation indicate that it is an engineered object.

2:39 Do Engineers Need Science?
This leads to the question “Do engineers need science to create.” To answer that question Bill considers the design of Sainte-Chapelle — a stunning thirteenth century stone building. He notes that it was designed and built — as were all medieval cathedrals — by mason who knew no science, or mathematics and who could not even read, nor did they have a measuring stick.

3:13 Gothic Cathedrals: Light
The medieval engineers strive to build cathedrals that could house large stained glass windows that could let in sunlight.

3:54 Pointed Arches
Medieval masons used pointed arches in their cathedrals to create high ceilings.

4:16 Pointed vs Circular Arches
The pointed arch allowed mason to build higher ceiling using less stone, as clearly seen in comparing Sainte-Chapelle to the Pantheon in Rome.

5:00 Width Circular Arch
The width of the circular arch grows proportional to its height.

5:29 Width Pointed Arch
The width of a pointed arch doesn’t expand because the pointed arch changes shape as it grows taller.

5:39 Proportional Rule
Bill explains that these early engineers used a “rule of thumb” inherited from antiquity — one used to build Roman buildings like the Pantheon — to size the supporting walls underneath the arches. The rule was simple: The width of the supporting wall should be between a fifth and a fourth of the arch’s span.

6:51 How Medieval Engineers Sized Walls
These early engineers could not perform the mathematics needed to implement this rule, so, as Bill demonstrates, they turned this into an action that required no calculation or measurement with a marked rule.

9:06 Why Arch Divided into Three Section
As Bill notes, this is just how the geometry works out. It is the same rule as used to size the supporting wall for a semi-circular arch.

9:52 Rule of Thumb
A rule of thumb is the heart of the engineering method: it allowed the masons to build without understand at a deep level the properties of stone or knowing mathematics.

10:29 The Engineering Method
This is defined as “Solving problems using rules of thumb that cause the best change in a poorly understood situation using available resources.” That’s a sharp contrast with the scientific method, because these rules of thumb are only guides that offers a high probability of success, but no guarantee.

10:40 Rule of Thumb Never Disproved
And, unlike a scientific theory, a rule of thumb is never, in a sense, disproved. That hundreds of cathedrals are still around today, standing for eight or nine hundred years is proof. Instead of being disproved, this rule of thumb for stone became outdated, not wrong, as iron and steel I-beams replaced stone.

11:10 Do Engineers Still Use Rules of Thumb?
Bill considers the question “is engineering based on rules of thumb antiquated in our scientific age?” He notes that that line of thought misunderstands the purpose of the engineering method, which is to solve practical problem before we have full scientific knowledge.

11:33 Next Video
Bill notes that in the next video he’ll explore how engineers work their way around that lack of scientific understanding, how they overcome uncertainty.

11:45 End Titles

[hullinstruments]

I'm so freaking hyped that you're making new content. I'm glad you are doing well.

Many of us owe you a lot

I'm in my mid-30s now....and when I started watching your stuff ... Back when you first started making videos...I had followed a path of strictly woodworking and being in business for myself dealing antique instruments. Always hated and avoided a lot of engineering things especially electronics and such.

But somehow fell backwards into systems engineering specializing in metrology equipment troubleshooting and repair. And your content was a huge part in making me see the world that way and find those things interesting. A huge part

[alexm.]

As a structural engineer my mind is absolutely blown! I have always wondered how they were able to create those stunning pieces of architecture without highly complex math. Never heared any explanation on this topic in university either. Thank you very much for this enlightening insight!

[scottb]

I trained as an associate electrical engineer and worked in the field for many years, but never had the engineering method explained to me. With so much emphasis nowadays on data and science it escapes people (even us engineers) that these rules of thumb, as explained so well by Bill here, are at the core of the built world of ours. I sometimes find myself immobilized trying to build something in adherence to strict scientific principles, when all that is needed is to keep the end in mind and create the best change using the available resources. Practicality over (but not excluding) theory is at the heart of engineering.

[michaelmuntean]

I am a practicing structural engineer, in continuous practice 40 years since I graduated from a prestigious university.
For years I’ve been interested in the design of arches, but nearly every reference or text I’ve seen used modern beam theory to explain how to design one. When these great structures were constructed, there was no ‘beam theory’, or the calculus used to find solutions to design problems using it.

Yet those medieval structures stand, for millennia in some cases.

Thank you for starting to describe the methods used by those very early masons and designers. It is very valuable, at least to this engineer.

[bradleygawthrop]

This is near to my heart, for many years I was a pipe organ builder, and methods like this were used to build those fantastically effective and sophisticated devices before we had any deep scientific understanding of the fluid dynamics and acoustic science and mechanical disciplines which underpin them. Centuries of refinement of the rules of thumb in organ building honed them to such degree that most of them have gone right on being used as the standards even as the scientific and analytical tools have became ubiquitous. Thanks so much for this!

[GeorgeCowsert]

The reason the Scientific Method is often pointed to to explain why the Engineering Method exists is because Science is literally just trying things and documenting results in the search of a pattern.

The Engineering Method, meanwhile, takes known information and pieces them together into the most optimal configuration.

One discovers stuff. The other makes that stuff useful.

Mathematics meanwhile meshes perfectly into making sure the science and engineering can be more precise.

[mmatt]

How the heck did that 12 minute video only take 2 minutes to watch?!? What a master at captivating an audience and explaining complex ideas so simply that even I can understand them! Bravo good sir.

[Audey]

Spent much of my youth studying math and science in college. I'm now an electrician. I didn't have a word for it until now, but adopting this Engineering Method was something of a hurdle for me at first. But now that I've ingrained it I'm faster, waste less, and have higher quality results. Sure, I can do the math to find the center of a room, but it's faster to simply snap chalk lines corner to corner (not to mention this naturally accounts for areas that are out of square). I can do all all kids of fancy trig and calculus, but it's always easier and less error prone to simply use rules of thumb or templates.

Great video! glad to see you're making stuff again.

[christinebeames]

Well well, you just enthralled a old granny with no knowledge of engineering, but with a curiosity about how they built the cathedrals, well past her bed time, I just subbed xx

[hellojudas]

“I never worked out why, geometrically, that works- because it doesn’t matter”

I love this a lot. You don’t have to understand why it works, you just have to know that it works. It’s the same reason you can be an electrician without being a quantum physicist

[DarthObscurity]

I look at the world differently after watching your videos. The can engineering is one of my favorites, with every curve and angle purposely decided on. Helped me see that a lot of the design choices for other areas like construction or even bike design aren't just about asthetics and how something as simple as the shape can help strengthen or improve something.

[willpugh]

I have been a fan subscribed and waiting for a video ever since the aluminum can episode, the way you teach has an almost asmr like aspect and its very thoughtful and full of information presented in a way thats not overwhelming kudos sir can’t wait

[idonnow]

This is what should have been shown on history channel instead of ancient aliens, an entire generation wouldn't have been fooled into ridiculous pseudoscientific nonsense

[peruibeloko]

I can easily see this series being used in classrooms in the future, just like those classic informational videos from the 90s, and this is honestly amazing. Glad you're back!

[plakor]

Not only that, the mediaeval cathedrals were/are acoustic marvels as well as visual masterpieces. Amazing stuff.

[jimurrata]

Grateful that youve decided to upload new content!
I realize that IRL takes priority but when i say "Thanks for coming back!" I speak for many. 😉

[UmbrehVR]

Thank you for posting such wonderful, informative and interesting content for so long! it's nice to see you posting after three years. Your videos started my interest in engineering and well... everything you've talked about! once again, thank you so much!

[CrossingThinIce]

I wonder about the process of developing one of these heuristics. Was it one person or dozens who were able to identify a possible solution to a problem and then refine that possible solution by successive approximation, until the result stopped improving. Did it happen quickly or over scores of years? What about the initial inspiration to try that type of solution to the problem. Fascinating. Thank you for this story.

[cosimo]

Wonderful. I always wanted to know how Gothic cathedrals were built. And clearly spoken with no background 'music' or sound effects which are the curse of most YT videos.

[asimo]

I wasn't sure if I could care about how arches were designed hundreds of years ago when your video began. Then your demo of how an arch was drawn without math sold it. Thank you Bill and welcome back!