# When did architects learn how to calculate load on building members accurately?

For many years, large buildings with impressive spans between the building columns were erected without precisely calculating what loads the columns, walls, and buttresses would have to support. Load on a building is a complex calculation taking in account the weight of the roof and potential snow, the size and spacing of columns, how building members are braced, and in modern times wind loads and seismic forces.

I have read that the architects of Gothic cathedrals were not able to calculate with any accuracy the size and number of columns and flying buttresses they needed to support a given design. They worked from geometry, rules of thumb, and prior experience.

At what point in history did mathematics begin to make building design more predictable and foolproof? I suspect that at some point algebraic equations replaced geometry and that after the advent of calculus, the section modulus of various shapes began to be understood, but I cannot seem to find information as to where and when these things happened. A good answer would point out which mathematical concepts were first used rigorously in structural design and also what notable buildings were the result of this more formalized process.

• I'd be curious to know whether the ancient Greeks and/or Romans had any mathematical ideas about architecture that are worth mentioning. Having been to the Pantheon in Rome, I wonder whether they would have been able to build such a thing without serious calculations.
– Danu
Oct 31, 2014 at 7:55

Stevinus developed the basic equilibrium of forces rules in structures in the late 1500's. All the first editions of his works are available on google books and internet archive, but I highly recommend the technical summary with translation that is in: MINM

Or you can look at the first editions, most of them are quite short books.

One caution: the detailed general mathematical models of loads for material beam configurations are discovered only later by Euler, indeed, and those not so discovered were developed into their reliable and modern form in the 19th century only. There is no single source, but many different papers published in 19th century journals.

I highly recommend looking into the historical notes and references Clifford Truesdell included in his various technical books (and longer papers, 1950's-1970's) on classical mechanics. They have almost every historical reference (because Truesdell could and did read all the primary sources in the original languages).

It's not something that easily fits neatly in a post, requiring a review length treatment, but it exists completely in Truesdell (I'll have to check which of his books has the most concise listing and description of priority references in mechanics---look forward to an edit).

The Roman architect, author and civil engineer, Vitruvius wrote 10 books of architecture, De Architectura, the text is translated in the Project Gutenburg Ebook Vitruvius: The Ten Books on Architecture - while not necessarily being the first to apply mathematical principles to architecture, e.g Egyptians, Mayans etc.

However, the Romans certainly had a consistency of how their structures (aqueducts, amphitheaters etc) were built and have withstood disasters, modernisation and time itself. A pertinent quote in De Architectura is in the section The Fundamental Principals of Arctitecture include:

Architecture depends on Order (in Greek τἁξις), Arrangement (in Greek διἁθεσις), Eurythmy, Symmetry, Propriety, and Economy (in Greek οἱκονομἱα).

Some particular points are:

Order gives due measure to the members of a work considered separately, and symmetrical agreement to the proportions of the whole.

De Re Metallica was written by Georges Agricola and published in approximately 1550. It was written in Latin, and kept available in European churches. Builders (and readers in general) could physically access the book there. In addition, for those who were illiterate or could not read Latin, the priest or deacon would translate and read requested passage when needed.

Architects may have been able to accurately calculate building loads prior to 1550. After 1650, such matters became more standardized due to the existence and accessibility of De Re Metallica.

• Hmm, from leafing through the link to archive.org that you provided, it seems like that book is mostly about mining techniques and principles of geology. I do not see where it touches on the mathematics of structural design.
– Mike
Oct 31, 2014 at 10:39