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.
I’ve been fascinated by astrolabes for a very long time, roughly 20 years. It was this avocation that led to my interest in sundials and, because they share museum space, my interest in clocks. When I lived in Rockford, Illinois, I would haunt the Time Museum, an institution that produced the most beautiful book on astrolabes. Adler Planetarium in nearby Chicago has one of the best astrolabe collections in the entire world, producing another beautiful book solely on Western astrolabes and a gorgeous book on antique scientific instruments in general. None of these provide the mathematical details of astrolabe design beyond a description of stereographic projection, and indeed this kind of detailed information is rarely found. The Astrolabe, a new book by James E. Morrison, is an absolutely unique and wonderful book on the mathematics needed to create accurate, beautiful designs of astrolabes, quadrants and other related instruments. I can’t recommend it enough to those who share the interests of this blog.
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.
Nomography, truly a forgotten art, is the graphical representation of mathematical relationships or laws (the Greek word for law is nomos). These graphs are variously called nomograms (the term used here), nomographs, alignment charts, and abacs. This area of practical and theoretical mathematics was invented in 1880 by Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision.
Along with the mathematics involved, a great deal of ingenuity went into the design of these nomograms to increase their utility as well as their precision. Many books were written on nomography and then driven out of print with the spread of computers and calculators, and it can be difficult to find these books today even in libraries. Every once in a while a nomogram appears in a modern setting, and it seems odd and strangely old-fashioned—the multi-faceted Smith Chart for transmission line calculations is still sometimes observed in the wild. The theory of nomograms “draws on every aspect of analytic, descriptive, and projective geometries, the several fields of algebra, and other mathematical fields” [Douglass].
This essay is an overview of how nomograms work and how they are constructed from scratch. Part I of this essay is concerned with straight-scale designs, Part II additionally addresses nomograms having one or more curved scales, and Part III describes how nomograms can be transformed into different shapes, the status of nomograms today, and the nomographic references I consulted.
An operational calculus converts derivatives and integrals to operators that act on functions, and by doing so ordinary and partial linear differential equations can be reduced to purely algebraic equations that are much easier to solve. There have been a number of operator methods created as far back as Leibniz, and some operators such as the Dirac delta function created controversy at the time among mathematicians, but no one wielded operators with as much flair and abandon over the objections of mathematicians as Oliver Heaviside, the reclusive physicist and pioneer of electromagnetic theory.
Pendulums are the defining feature of pendulum clocks, of course, but today they don’t elicit much thought. Most modern “pendulum” clocks simply drive the pendulum to provide a historical look, but a great deal of ingenuity originally went into their design in order to produce highly accurate clocks. This essay explores horologic design efforts that were so important at one time—not gearwork, winding mechanisms, crutches or escapements (which may appear as later essays), but the surprising inventiveness found in the “simple” pendulum itself.
It is commonly known that Galileo (1564-1642) discovered that a swinging weight exhibits isochronism, purportedly by noticing that chandeliers in the Pisa cathedral had identical periods despite the amplitudes of their swings. The advantage here is that the driving force for the pendulum, which is difficult to regulate, could vary without affecting its period. Galileo was a medical student in Pisa at the time and began using it to check patients’ pulse rates.
Galileo later established that the period of a pendulum varies as the square root of its length and is independent of the material of the pendulum bob (the mass at the end). One thing that surprised me when I encountered it is that the escapement preceded the pendulum—the verge escapement was used with hanging weights and possibly water clocks from at least the 14th century and probably much earlier. The pendulum provided a means of regulating such an escapement, and in fact Galileo invented the pin-wheel escapement to use in a pendulum clock he designed but never built. But it took the work of others to design pendulums for truly accurate clocks, and here we consider the contributions of three of these: Christiaan Huygens, George Graham and John Harrison.
Have you ever had to calculate the positions of astronomical objects? Orbital calculations relative to an observer on the Earth require derivations and time-consuming solutions of spherical trigonometric equations. And yet, these kinds of calculations were accomplished in the days prior to the advent of calculators or computers!
For example, to find the zenith angle (angle to overhead) and azimuth (angle from North) of the sun at any day and time of the year for any location on Earth, the laws of spherical trigonometry produce the formulas below. Here the solar declination δ is a function of the solar longitude λ and ecliptic angle ε as shown in the figure to the left.
These calculations can be automated today—but did I mention that these solutions were found before electronic calculators?
… or slide rules, or logarithms?
… or trigonometric formulas?
… or even algebra??
In fact, Vitruvius (ca. 50) and Ptolemy (ca. 150) provided mathematical and instrumental means of calculating the sun’s position for any hour, day, and observer location by the use of geometric constructions called analemmas (only indirectly related to the figure-8 analemma on globes). An important application of analemmas was the design of accurate horizontal and vertical direct and declining sundials for any observer location. These analemmas are awe-inspiring even today, and as the study of “Descriptive Geometry” has disappeared from our schools they can strike us as mysterious and wondrous inventions!
This journal attempts to capture my occasional encounters with the technically elegant but nearly forgotten in the mathematical sciences—artistically creative works that strike me as particularly brilliant. These can be small, clever things (say, an algorithm for calculating roots), or they can be ingenious technical inventions of more general application, basically anything that makes me think ‘Wow, that’s neat!’ Think of pendulum clock escapements; of beautiful precision sundials, astrolabes and other antique scientific instruments; of music theory and instrument design; of early, desperate attempts to calculate logarithms and trigonometric values; of stereo photography and linkage mechanisms; of difference engines, trinary arithmetic and slide rules; of old map projections and vacuum tube op-amps.
Posts here are brief or not-so-brief essays of unusual things of this nature that I read or hear about, supplemented with references and some amount of research I typically do on these topics. Any longer papers that emerge (particularly on mental calculation and antique scientific instruments) will be placed in my main website area http://www.myreckonings.com. To avoid printing difficulties with this wide format, there will be a link to a PDF version at the end of most essays.
Feel free to use the Contact tab along the top to send me general comments or any ideas (or text!) for new topics.
Ron Doerfler
(The figure above is from Oronce Fine’s Second Book of Solar Horology, translated with interpretation by Peter Drinkwater)
Dead Reckonings 