Bayes’ Theorem is a statistical technique that calculates a final posttest probability based on an initial pretest probability and the results of a test of a given discriminating power. Thomas Bayes (1701-1761) first suggested this method, and Pierre-Simon Laplace published it in its modern form in 1812. It has generated quite a bit of controversy from frequentists (who work from a null hypothesis rather than an initial posited probability), but this technique has become much more popular in modern times.

Among many other applications, this is a common technique used in evidence-based medicine, in which statistical methods are used to analyze the results of diagnostic tests. For example, a diagnostic test might have a sensitivity of 98%, or in other words, the test will return a positive result 98% of the time for a person having that disease. It might have a specificity of 95%, which means it will return a negative result 95% of the time for a person who does not have the disease. For a disease that has a prevalence (a pretest probability) of 1% in the general population, Bayes’ Theorem provides, say, the probability of a person having the disease with a positive test result. Pretty darn likely, right? Well, it turns out it’s about 16% because the false positives from the 99% who do not have the disease overwhelm the true positives from the 1% who have it. This is the basis for recent recommendations to stop PSA screening in men, for example, as expensive and counterproductive. But all this is described in detail in the article. Enjoy!

]]>This project does not in any way affect the content here—essays will continue to be written as usual on lost arts in the mathematical sciences, including nomography. This is simply an outlet to provide an option for nomograms in poster form.

]]>Part I of this essay provided background information that demonstrates the difficulty of the problem of mental extraction of 13th roots and the efforts of calculators to master it. But can it be possible for us to extract 13th roots of 100-digit numbers without devoting portions of our life to it? With a basic talent in mental arithmetic and some study, it can certainly be done, even if not in record time. We present a new method that involves no logarithms, no antilogarithms and no factoring, one that works with 13th powers that end in 1, 3, 7 or 9 (the cases attempted by record holders). The memorization consists of one table and a few formulas. As always, a printer-friendly PDF version of Parts I and II is linked at the end of this essay.

]]>Mental calculators of note (so-called “lightning calculators”) developed areas of expertise in performing calculations that seem astonishing, even unbelievable, to the rest of us. One such specialty is calculating the 8-digit root of a 13th power of 100 digits. Achieving record times historically required massive memorization and calculating speed, racing through a procedure that remains a mystery to most people. Part I of this essay provides a historical overview of the extraction of 13th roots, including the methods used by a few mental calculators, methods that largely rely on a mix of intensive mental calculation and large-scale rote memorization. It demonstrates the creativity and drive of these marvelous people.

In Part II of this essay we will propose a new method for 13th roots like those posed to lightning calculators that is relatively easy to learn, one that makes this feat feasible for those of us with basic mental math capabilities and a desire to do something amazing. As always, a printer-friendly PDF version of Parts I and II is linked at the end of this essay.

]]>In 1982 H.A. Evesham produced his doctoral thesis, a review of the important discoveries in nomography. It is often cited in other works, but unless you know someone who knows someone, it is very difficult to find—I have never been able to locate a copy. However, just recently Mr. Evesham’s thesis has been professionally typeset and released as a book (click here or here) by Docent Press under the aegis of Scott Guthery. If you have an interest in the theoretical aspects of nomography beyond the basic construction techniques of most books and of my earlier essays, you will appreciate this book as much as I do. Mr. Evesham does a wonderful job of weaving mathematical discoveries in nomography from many contributors into a readable but scholarly work.

]]>NOTE: If you choose to have the calendar professionally printed by Lulu.com rather than on your own printer, through December 31 you can enter the code REMARKABLEYEAR305 at checkout for 20% off the price of $17.60. The Lulu site is also posting daily coupon codes in December here.

]]>The article, a significantly revised version of my blog essay, has now been published in the *UMAP Journal*, and per the standard agreement I can post the PDF of the article here for anyone to download. More information and a link to the article are below.

There are two formats available: two-sided 8-1/2″ x 11″ sheets of paper printed in landscape mode that can be connected at their edge as shown in the photo on the left, and two-sided 11″ x 17″ sheets of paper printed in portrait mode with two pages per side that can be folded as a group and stapled in the middle. Either of these could be printed to fit on A4 or other sizes, I’m sure. White paper can be used, but the color scheme is really designed for a light beige or ivory paper and it looks so much more professional when it’s printed on paper of some color (gray might work). The stapled format requires no other binding. As you can see from the photo on the left, I printed the first (non-stapled) format and took the printed sheets (24 lb. Southworth ivory linen paper from OfficeMax) to a local office shop (Kinko’s FedEx) and had them add clear plastic sheets to the front and back and install a spiral wire (a 60-second job that costs $5). Drilling a hole in the center along the top to hang it completes the calendar. Using 3 rings through punched holes along the top may be a cheaper option.

Continue below to see thumbnail images and the download instructions.

]]>Are you intrigued by nomograms but have no idea how to go about drawing them? PyNomo is an amazing, free software package for drawing precision nomograms. The output is in vector form in a PDF or EPS file, so it can be printed in any size and still retain its sharpness. PyNomo directly supports 9 basic types of nomograms based simply on the format of the equation, so for these types there is no need to convert the equation to the standard nomographic determinant or use geometric relations. But it also supports compound nomograms as well as more complicated equations that have been cast into general determinant form, so it can produce output for any equation that can be plotted as a nomogram.

When I started writing an essay on using PyNomo my plans were to show three examples of nomograms. But I had so much fun making really cool nomograms that the essay turned out to be more of a user’s manual, with examples of all the supported types and descriptions of the many parameters you can use to customize your nomograms. Leif Roschier, the author of the software, spent a great deal of time reviewing draft versions of the essay and making software updates for new features that were rolled into it, so the essay is comprehensive in scope and quite complete in details and practical advice. PyNomo is clearly my choice for drawing nomograms going forward, and I think you will find it as uniquely wonderful as I have.

The essay is too long and the example nomograms too detailed to be rendered in HTML here. The PDF version of the essay (Version 1.1) can be found** here**. The PyNomo website, which also contains many examples, is found** here**.

[NEW - December 21, 2011]: Detailed instructions on downloading and installing all required software applications onto a Windows XP or Windows 7 PC can be found **here**.

——————-

Updated October 19, 2009, to Version 1.1 for the new features of PyNomo Release 0.2.2:

- Automatic spacing of tick marks along scales—more tick marks where space is available and less where it’s crowded.
- Drawing of sample isopleths between specified values on scales.
- Printing of only significant digits of scale values by default, producing a cleaner-looking nomogram overall.

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.

]]>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.

]]>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. For convenience the method is described in both English and Chinese.

]]>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.

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.

]]>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.

]]>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.

]]>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!

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 each entry.

Comments on the posts are appreciated! A forum has also been added for discussing anything related to lost art in the mathematical sciences at http://www.myreckonings.com/forum. Also, feel free to use the Contact link 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)