Progress, as conceived by most people, consists in replacing older objects, techniques, or philosophies by newer, better, ones. Sometimes indeed the change is for the better, but sometimes it is just change for change—ever had an older device of some sort that was perfectly adequate for your usage, yet you still replaced it with a newer version with no net gain? Unfortunately, the same happens with ideas, especially with mathematics and computer science.
But there are lessons to be learnt from the past. I’m not talking about fables and cautionary tales; I’m talking about the huge body of science that was left behind, forgotten, superseded by modern techniques.
The past is fraught with surprises. A good example of this is the Antikythera mechanism (main fragment shown above). Built somewhere around 150 to 100 B.C, this magnificent piece of engineering—that would have to wait for the middle ages to be rivalled—was an analog computer of sorts: a device to predict the position of celestial objects; a mechanical planetarium.
When I first read of the Antikythera device, I thought it could only be a hoax as it was unexpectedly sophisticated. It was exquisitely crafted; sets of wheels within wheels made with astonishing precision; something of an seemingly impossible feat for this era. The first comparable devices started appearing in the late 14th century (the astronomical clock of Strasbourg, for example) and the next big jump occurred somewhere in the 16th or 17th century, when people like Pascal and Schickard started building small arithmetic computers.
This device is just extraordinary, the only thing of its kind. The design is beautiful, the astronomy is exactly right. The way the mechanics are designed just makes your jaw drop. Whoever has done this has done it extremely carefully…in terms of historic and scarcity value, I have to regard this mechanism as being more valuable than the Mona Lisa.
Pr Michael Edmunds of Cardiff University
Needless to say, that’s quite astonishing that people from Antiquity—despite known examples of surprising devices and buildings—had the knowledge and engineering skills to build an analog computer predicting the (apparent) position of celestial objects with great precision.
First lesson: previous generations were just as smart as we are.
While there’s a great deal of knowledge that was built since the Renaissance, or, more exactly, the scientific revolution, a good part of it is now inaccessible. Partly because it is written in obsolete languages such as Latin, old french or English, but even more so because the knowledge is presented in a way incomprehensible for the modern reader. If you read mathematical texts from before the mid 19th century, you will find comparatively little mathematical notation, and if any, most of it is ad hoc, a pet notation from the specific author; text filled with complicated periphrases to express simple mathematical concepts that today have simple and unambiguous notations . Reading Descartes in the original, for example, is really a less than cromulent experience if you really want to understand something; the same can be said for Newton. Fortunately, modern-language and commented editions of the classical texts exist.
Since the mid-19th century, or maybe at the beginning of the 20th, mathematical notation evolved significantly and started to standardize, making the global body of knowledge accessible to all. The evolution and standardization not only made exchange but also collection of mathematical knowledge possible.
The simplest form of such a collection is the compendium, where mathematical facts—almost always limited to formulae and tables—are piled up in broad categories, given as is, without much explanation. Compendia are the closest analog of a dictionary for mathematical knowledge. But, as all dictionaries, the knowledge doesn’t form a unified body, but merely a list of data, and whatever explanation it contains, if any, is, by definition as one might say, terse.
It is most interesting is look at compendia written before the 1950s, at a time where computer meant a person rather than a machine, you may discover surprising techniques that aren’t taught today and are now nowhere to be found.
One such treasure trove is the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, simply known as the Abramowitz and Stegun book. Published in 1964, with its 1050-something pages, it is maybe the last of the great compendia. Still in print today (; there are also PDF versions, but some have more than half of the pages missing) it is one of the rare must have any one should have handy on his (e-)bookshelf.
When we have a look at the Abramowitz and Stegun, what we first notice are the lengthy tables—now perfectly useless. But do not think that the extensive tables had all their entries computed exactly! No, some entries were computed exactly, but the rest were interpolated with an ad hoc method with known error bounds for the function being listed in the table. This means that some of the values were computed at length but that most values, values in between, were interpolated to sufficient precision.
Today, we would use the artillerie lourde to address the problem of interpolation, using very general but computationally intensive and not necessarily well adapted methods. One might think about splines, or, for higher-dimensional data, kriging. But if you look at the Abramowitz and Stegun, you will find many pages devoted to the problem of interpolation. What is very different from a modern chapter on the subject is the number of ad hoc and specialized methods one finds; each tailored to a specific class of function, sometimes even a single specific function, each with its error bounds. They even have methods for grids of whimsically placed points.
Second lesson: artillerie lourde is not always the best way to go; some people have addressed specific problems in the past, it may be worthwhile to see what has been done before.
Years ago, while a post-doc at McMaster University, I found a book (in a second-hand bookstore, of course) on how to use slide rules to perform all kinds of calculations. This also contained a lot of information; and I realized that I already used some of them instinctively. Needless to say, I didn’t replace my (OS-specific desktop) calculator for a slide rule (despite some still being manufactured!) but it gave me ideas I can apply once in a while.
So maybe the moral of all this is that while you should know about what’s currently state-of-the-art, you should also pay attention to what has been done before. Old books are a very good way of achieving this and the next time you think of not getting a used math book because it may be too old, think again.
 Florian Cajori — A History of Mathematical Notations: Two Volumes Bound as One — Dover, 1993