# Age of Invention: Pennies for Your Thought

Welcome to my weekly newsletter, Age of Invention, on the causes of the British Industrial Revolution and the history of innovation. You can subscribe here:

**W**hen we think of labour-saving inventions, the kind of labour that springs to mind tends to be manual. We think of machines replacing the muscle of limbs and the dexterity of fingers, and we worry about their effects on unemployment and unrest. But there’s a subset of labour-saving inventions that rarely gets discussed. They might best be called thought-saving.

A few weeks ago I mentioned the introduction of mathematical techniques to navigation. Before the mid-sixteenth century in England, pilots very rarely even knew how to calculate their latitude, let alone their longitude. But over the course of just a few decades, England became one of the world leaders in navigational improvements. A handful of mathematicians saved pilots the trouble of calculation, by coming up with tables, instruments, diagrams, and rules of thumb. In the process, they improved navigation’s accuracy, and ushered in an age of English dominance of the high seas.

The historian Eric H. Ash gives a few great examples. In the 1590s, the explorer John Davis shared a way to calculate the time of high tide, without requiring multiplication. Likewise, William Bourne, a self-taught mathematician and gunner, in the 1560s provided an easy means of calculating the linear distance in one degree of longitude, at any given latitude. He provided a diagram — really an instrument, even if it wasn’t made of wood or brass — which with just a simple piece of string could be used to derive the answer without needing to understand cosines, or really any trigonometry.

The mathematicians did the same with maps, too (after all, aren’t all maps thought-saving?) The sixteenth-century cartographic innovations simplified the pilot’s ability to chart a route, for example by taking away all need to worry about the curvature of the earth. The famous 1560s map projection of Gerardus Mercator stretched the distance between the lines of latitude as they got closer to the poles, so that charting a course on such a map was a simple matter of drawing straight lines rather than complex trigonometry. The Mercator projection may well make Africa look smaller than Greenland — it’s actually almost fifteen times as large — but it made life significantly easier for mariners. For similar reasons, the mathematician John Dee designed a special chart — what he called the “paradoxall compass” — to aid the English explorers who in the 1550s went in search of a northeast and northwest passage to Asia. Conventional charts made navigating high latitudes confusing, as the north pole was a straight line — the map’s top border. Dee’s map made things easier by putting the pole at the centre, as a point, with the lines of latitude as concentric circles.

And then there were the instruments. It was one thing to use a cross-staff to find the altitude of some heavenly body. But when using the reading to calculate latitude, had the pilot accounted for the fact that their eye was higher than the surface of the earth? It sounds like a small error, but it could throw the latitude calculation off by a few miles. To correct for it, the mathematician Edward Wright drew up a simple table: no need to bother with calculation, just find the relevant row and column. Or what about the fact that the end of the cross-staff was not right at the centre of the observer’s eye? An even smaller error, but an error nonetheless. Thomas Harriot suggested placing the staff right in the corner of the eye, rather than on the cheekbone or the bridge of the nose, and having the remaining discrepancy written on the staff itself so that the pilot would remember to adjust for it. Notes like these, much like check-lists today, helped reduce the human error that was a risk in any calculation. Harriot even designed a flowchart, setting out precisely what kind of simple calculations to make, depending on roughly how far north or south the ship was.

What all of these thought-saving inventions had in common was that they involved a bit of up-front, sometimes complicated calculation by a handful of experts, which allowed for those after them to save on thinking things through for themselves. And what’s striking, I think, is how many of the more recent, world-changing innovations have been similarly thought-saving. The great-grandfather of computing, Charles Babbage, was first motivated to design a mechanical calculator — the difference engine — after getting bored of compiling navigational tables, reportedly exclaiming “I wish to God these calculations had been executed by steam!” The programmers or coders of today, or even just the resident office spreadsheet expert, who happens to know a few extra formulae to plug into a table, are also much like the applied mathematicians of the sixteenth century. They do the thinking, and even design and train machines to do the thinking, so that we don’t have to. All in all, thought-saving inventions shrank the globe, brought us to the moon, and now allow for near-telepathic communication with anyone on the planet. Who knows how many trillions of pennies they have saved us for our thoughts, and how many more are yet to be saved.

Until next time,

Anton