As you may have noticed, the history of graphical computing (nomograms and the like) has become one of the major themes of this blog. I did not foresee this, as I knew virtually nothing about the subject before I started researching my first essays on nomography a couple of years ago. This topic is still one of my main pursuits, and I’m as astonished by what I find now as I was back then. To capture a bit of this spirit, I’ve created a free 2010 calendar titled The Age of Graphical Computing that is available for downloading and printing. The fun thing is that you can test the examples right on the calendar to show that they work!
William Thomson called them “beautiful and ingenious geometrical constructions,” and in variance to their rather humdrum name dygograms are certainly charming to the eye. But these geometric constructions can conveniently generate and then calculate the magnetic deviation of a ship compass at a location.
With our electronic calculators and computers, we take for granted the effortless arithmetic and trigonometric calculations that so vexed our ancestors. Pre-calculated tables for roots and circular functions, generated through hard work, were often used to create tables of magnetic deviations for specific ships and locations. To reduce the chance of misreading these tables, a few types of graphical diagrams, not just dygograms, were invented to provide fast and accurate readings of magnetic deviation. These graphical calculators are the focus of this part of the essay.
The Scottish mathematician and lawyer Archibald Smith first published in 1843 his equations for the magnetic deviation of a ship, or in other words, the error in the ship’s compasses from permanent and induced magnetic fields in the iron of the ship itself. This effect had been noticed in mostly wooden ships for centuries, and broad attempts to minimize it were implemented. But the advent of ships with iron hulls and steam engines in the early 1800s created a real crisis. A mathematical formulation of the deviation for all compass courses and locations at sea was needed in order to understand and compensate for it, and Smith became the preeminent expert in this sphere of activity. With Capt. Frederick J. Evans he extended his mathematical treatment to detailed procedures for measuring the magnetic parameters for a ship, and he also invented graphical methods for quickly calculating the magnetic deviation for any ship’s course once these parameters were found, constructions called dynamo-gonio-grams (force-angle diagrams), or dygograms for short.
Today, radio navigational systems such as LORAN and GPS, and inertial navigation systems with ring and fiber-optic gyros, gyrocompasses and the like have reduced the use of a ship’s compass to worst-case scenarios. But this triumph of mathematics and physics over the mysteries of magnetic deviation, entered into at a time when magnetic forces were barely understood and set against the backdrop of hundreds of shipwrecks and thousands of lost lives, is an enriching chapter in the history of science. Part I of this essay presents a brief sketch of the problem and the analysis and solutions that were developed to overcome it. Part II sets out with a discussion of Smith’s graphical methods of computing the magnetic deviation and concludes with a list of the references cited in the essay.
Mental calculators of yesteryear were usually described in magazines, newspapers and books in ways that can be startling in our more cynical age. But even today newspaper articles, documentaries and television features on modern lightning calculators appear almost regularly, often with a “hook” such as diminished capabilities in other areas (the “Einstein” effect). Surely there must be some reports that try to be objective, but I haven’t found them. At best they are naively written by people with little mathematical background; at worst they use considerable license (deception, really, if only by omission) to present a better story. This part of the essay is not directly related to the historical art of mental calculation itself, but I think it serves as a cautionary tale in evaluating articles on it.
The types of calculations performed by lightning calculators were historically quite limited, notable mainly for the size of the numbers and the speed at which they were manipulated. But remember that the questioner had to verify every calculation by hand, making higher powers and roots (particularly inexact roots) much less feasible. The dawn of calculators and computers propelled some of these tasks into hitherto uncharted territories such as 13th or 23rd roots, deep roots of inexact powers, and so forth, much of it supported by more sophisticated mathematics. Here we will review the methods of calculation used in the past, many of them not commonly known, as well as other techniques that are relatively new.
Individuals with preternatural abilities to calculate arithmetic results without pen, paper or other instruments, and to do so at astonishing speed, are the stuff of mathematical and psychological lore. These “lightning calculators” were sometimes of limited mental ability, sometimes illiterate but of average intelligence, and sometimes exceptionally bright, this despite the popular notion of the idiot savant. The techniques used by these people are not generally well known. In fact, despite claims by educators that acquiring a mental facility with arithmetic operations is essential to a student’s mathematics education, I see little in the textbooks other than simple estimations based on rounding values, surely the most basic and least interesting mental task. The field of mental calculation may not be a lost art per se, but in this digital age it most certainly is a neglected one.
Part I of this essay attempts to take a fresh look at both historical and modern lightning calculators. Part II describes classic and modern methods of mental calculation. And finally, Part III demonstrates as a cautionary tale the shallow and deceptive nature of most media coverage of lightning calculators, an important consideration in analyzing reports on them.
by Liunian Li 李留念 and Ron Doerfler
Designing a nomogram for an equation containing more than three variables is difficult. The most common nomogram of this sort implements pivot points, requiring the user to create a series of isopleths to arrive at the solution. In this guest essay, Liunian Li describes the ingenious design of a nomogram that requires just a single isopleth to solve a 4-variable equation.
In Part III of my essay on The Art of Nomography, I mentioned the use of Weierstrass’ Elliptic Functions to create a nomogram composed of three variable scales overlaid onto a single curve. In particular, Epstein describes using this family of functions to create a nomogram for the equation u + v + w = 0, adding that the formula can be generalized for functions of these variables. This topic generated some interest, and it certainly is interesting to me, so I’ve explored it in more detail by designing a single-curve nomogram based on functions of u, v and w. This essay describes the procedure I followed to create a “fish” nomogram (found here) manifesting the formula for the oxygen consumption of rainbow trout as a function of weight and water temperature—a modest attempt to blend art with artifice.
In addition to providing sophisticated nomograms, the use of determinants as described in the previous Part II offers one other huge advantage. Often the scaling factors of variables have to be manipulated to get a nomogram that uses all the available area and yet stretches portions of the curves that are most in need of accuracy; alternatively, there may be a need to bring distant points (even at infinity) into a compact nomogram. This can be done by morphing the nomogram with any transformation that maps points into points and lines into lines. It is also intriguing to consider the aesthetics of such transformations, creating eye-catching nomograms as an artistic process.
This final part of the essay reviews the types of transformations that can be performed on a nomogram, and it concludes by considering the roles of nomograms in the modern world and providing references for further information.
The previous Part I of this essay described the construction of straight-line nomograms using simple geometric relationships. Beyond this, a brief knowledge of determinants offers a powerful way of designing very elegant and sophisticated nomograms. A few basics of determinants are presented here that require no previous knowledge of them, and their use in the construction of straight line nomograms is demonstrated. Then we will see how these determinants can be manipulated to create extraordinary nomograms.