Dead Reckonings has a new home at http://www.deadreckonings.com. I moved my blog to improve site performance and to maintain the latest version of WordPress. This will also for a time reduce the tedium of deleting spam comments that every blog owner experiences. (Believe it or not, the WordPress software has caught well over half a million spam comments to my blog overall, and I’ve manually deleted many that got through).

This present site will be retained for some time to support legacy bookmarks and links, but commenting has been turned off here. Please have a look at the new site and let me know what you think! I will be posting new material again on the new site very soon (!).

I’m very proud of the feedback and support from readers of my blog, a fantastic group of very knowledgeable people. I have learned a lot of marvelous things from you since I started this blog back in 2007. I know that no one here is aware of the amount of interaction I have had with people through this blog, the friendships that have developed, and the extensive collaborations on nomogram designs that have occurred almost continuously. This has made the process of moving the blog so worthwhile. I look forward to continuing the conversation.

Ron

]]>This modern application may seem at odds with the theme of my blog, but in fact it was Maurice d’Ocagne who coined the term *parallel coordinates* in regard to his parallel scale nomograms. A hundred years later Alfred Inselberg extended this idea of parallel scales as a way of visually analyzing multi-dimensional data. Inselberg and others make use of some of d’Ocagne’s work on point-line duality to characterize functional relationships between variables as structures and envelopes of the lines drawn between their axes.

In a parallel coordinate plot the axes of values for the variables lie parallel to each other, typically running vertically with a linear mapping from the minimum to maximum value of each variable. Each instance of measurement of the variables (i.e., each row of the input CSV data file) is represented by a segmented line, or polyline, that passes through the corresponding value on each variable axis. Correlated groups of lines can be color brushed, and this color brushing propagates across any other 2D or 3D plots generated from the data. The parallel coordinate plot shown here does not display axes or labels, but this larger version does display axes labels (best viewed by downloading and opening it rather than in your browser).

These highly visual displays reveal correlations, patterns, trends and anomalies in multivariate systems, and these in turn can significantly aid in the diagnostic analysis of sensitivities and error sources in a system. I have been an advocate of nomograms as system models, and this aspect of parallel coordinate plots is what drew me to study them and ultimately to create Sliver. This linkage will be more complete when I report later on the use of Sliver to test an application of nomograms to image processing.

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Using a cardioid to represent a heart is not new, but I re-created the complex construction on the front from an old doctoral dissertation of 1900 by Raymond Clare Archibald at Kaiser-Wilhelms -Universität Strassburg. I plotted the curves in vector form in Mathematica and added the lines and text in Adobe Illustrator. I stylized the cardioid in the center of the figure by plotting several versions of it where I added a small parabolic term to the equation so for each curve the width of the cardioid in the middle varied. The straight lines are gray, actually, so the figure will be less cluttered when printed than when you view it normally in the PDF (you can zoom into the PDF to see this).

The image below provides a low-resolution view of the outside and inside of the card before folding down the middle, although the gray background is not included in the actual PDF. As you can see from the inset picture at the top of this essay, the curves and lines on the front side are much sharper then they appear here. The link downloads the vector-resolution PDF with rotated images that you would print in double-sided Portrait mode to line up the outer and inner images correctly for a side-opening card. Of course, it looks best on photo paper or glossy color cardstock, and perhaps at a bit smaller size than 8.5″x11″. It should scale appropriately for A4 or other paper sizes. You will probably want to write the event on the front and write something on the inside.

There is no indication on the card of its origin. If you ever want to know where you downloaded it from, you can find my contact information in the PDF file under the File—>Properties menu option of Adobe Acrobat. And if you would like something different printed inside or on the front, just write me using the *Contact Me* link in the lower right side panel of this blog and I’ll be happy to customize it for you.

— Ron

** Download PDF of Valentine’s Day Card**

P.S. I stumbled across a nice image for a music-themed Valentine’s Day card: The sheet music of Baude Cordier (c. 1380-1440) titled *Belle, Bonne, Sage* (*Beautiful, Good, Wise*). Have a look! According to Wikipedia, the red notes in this love song indicate rhythmic variations.

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]]>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!

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]]>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.

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]]>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.

The procedure as written here seems more complicated than it is in use. In practice insignificant digits are dropped as they occur, and hundreds digits are dropped when finding a result mod 100. Performing this calculation with pen and paper and no memorization is perhaps the best way to become comfortable with the process.

**The Last Four Digits of the Root**

First, we find the last 4 digits of the root. We find these from the last 4 digits of the power, which we denote as *dcba* where *a* is the final digit.

1. We have already seen in Part I the rule for finding the last digit of the root:

The last digit of the root is the same as that of the power.

2. The second-to-last digit of the root, as evident in the table of 2-digit endings in Part I, is

7b mod 10 for a = 1 or 7

7(b-2) mod 10 for a = 3 or 9

3. The next pair of digits of the root (the 3rd and 4th from the end) are found from the formulas below. The mod 100 operation at the end is the remainder when divided by 100, so it is the last two digits if the result is positive, or 100 minus the last two digits if the result is negative. In fact, the formulas are much simpler than they appear because only the last two digits are kept in each term. The ceiling function rounds the result up to the next integer, and the floor function rounds the result down to the previous integer.

70d – 26b

^{2}+ b(20c + 8 ) – ceiling(b/3) mod 100 for a = 170d + 17(c+1) + 32b

^{2}+ b(40c + 42) – floor(b/3) mod 100 for a = 3For a = 7, we find 10000-

dcba, then find the last four digits of the root from this new value that ends in 3, then subtract the answer from 10000For a = 9, we find 10000-

dcba, then find the last four digits of the root from this new value that ends in 1, then subtract the answer from 10000

A fast way to subtract *dcba *from 10000 is to subtract each digit from 9 and add 1 to the result.

Now we have the last four digits of the root and we can proceed to find the first four digits.

**The First Four Digits of the Root**

It remains to find the first 4 digits of the root from the initial 5 digits of the power. We can do this by memorizing a short table of root vs. power starting digits, and then applying an offset.

The table to memorize is shown on the right. These values represent a minimum distribution that provides 4-digit accuracy over the interval of 100-digit powers. The values are actually accurate to 5 digits rather the 4 digits, so in fact the next digit is 0 for each value. Therefore, we have values for 5-digit accuracy that only require memorizing 4-digit numbers. Also, the values of n are convenient multipliers to use in our formula.The steps to find the first 4 digits of the root are:

1. Find S as the first 5 (rounded) digits of the power divided by 10, so the 5th digit follows the decimal point. Find the values of P in the table on either side of S. If S is less than 1/3 the distance between these values of P, choose the R and P from the lower entry row, otherwise choose R and P from the higher row.

2. Find the difference D below to the 4th decimal place. Then calculate the correction to 3 decimal places and add it to the value of R from the table:

D = (S – P)

/ 10000correction = nD – 6(nD)

^{2}/R

Generally only the first term is required, but if D is larger the second term may provide an additional small correction that can be taken to a digit or two. Note that R ranges from 42 to 49 so we can replace 6/R with 1/7 or 1/8 depending on which end of the range R lies on.

3. Then merge the first 4 digits with the last 4 digits to get the most reasonable last digit of the first four digits. For example, assume the first four digits were 2222. If the last four were 3333 the answer would be a simple merge to 22223333, but if the last four were 7777 the last digit 2 in 2222 would be adjusted down to 1 to get 22217777 because 1.7 is closer to 2.0 than 2.7. If the first digit of the last four digits is 5, we would use the 5th digit we calculated for the first digits to decide what is the closest merge.

**Example Problems**

The exercises presented below are intended to show the method in use with published problems faced by mental calculators. As mentioned earlier, achieving the record times listed here would require a great deal more memorization and practice.

**Example 1: Alexis Lemaire Record 13th Root Problem (13.55 seconds)**

The first example finds the 13th root of the following number:

29288115834875201060553567352783652122196502020937

13928425510086152669633464222587770308279739304053

The last digits of the power are 4053, so the last digit of the root is 3, and 7(5-2) mod 10 = 1 so the last two digits of the root are 13. Then we work left-to-right from the formula below, ignoring any hundreds digits as we go:

70(4) + 17(0+1) + 32(5)

^{2}+ 5(40(0) + 42) – floor(5/3) mod 100 = 06so the last 4 digits of the root are 0613.

The first 5 digits of the power (rounded) are 29288 which we divide by 10 to get 2928.8, and the closest match in the table is 2869 so our first approximation is 44.73. Now D = (2928.8 – 2869) / 10000 = 0.0060 and the first correction nD = 12(0.0060) or 0.072 which we add to 44.73 to get a closer value of 44.802. The second term (0.0060

^{2}) / 7 is too small to make a difference. Therefore we merge 44.80 and 0613, and we end up with 44800613.

**Example 2: Gert Mittring’s Record 13th Root Problem (11.8 seconds)**

This example finds the 13th root of a number ending in 1:

70664373816742861022340088302401573757042331707026

32731269721516000395709065419973141914549389684111

The last digits of the power are 4111, so the last digit of the root is 1, and 7b mod 10 = 7 so the last two digits of the root are 71. Then we work left-to-right from the formula below, ignoring any hundreds digits as we go:

70(4) – 23(1) + 26(1)

^{2}+ (1)(20(1) + 8 ) – ceiling (1/3) mod 100 = 10so the last 4 digits of the root are 1071.

The first 5 digits of the power (rounded) are 70664 which we divide by 10 to get 7066.4, and the closest match in the table is 7398 so our first approximation is 48.11. Now D = (7066.4 – 7398) / 10000 = -0.03316 and the first correction nD = 5(-0.03316) or -0.166 which we add to 48.11 to get a closer value of 47.944. Now (-0.166

^{2}) / 8 is very small, about 0.003, and subtracting this gets 47.941. Therefore we merge 4794 and 1071, and we end up with 47941071.

**Example 3: Record 13th Root Problem by Wim Klein (1 minute 28 seconds) and Gert Mittring (39 seconds)**

Here is an example for a power ending in 7:

88008443440489299575219015772236417859411720052615

65487280650870412023307854274990144578442271602817

The last digit is a 7, so we replace the final digits 2817 with 10000 – 2817 = 7183. The last digit of this root is 3, and 7(8-2) mod 10 = 2 so the last two digits of the root are 23. Then we work left-to-right from the formula below for 7183, ignoring any hundreds digits as we go:

70(7) + 17(1+1) + 32(8)

^{2}+ 8(40(1) + 42) – floor(8/3) mod 100 = 26so the last 4 digits of the root for 7183 are 2623. Then we find 10000 – 2623 = 7377 as the last four digits of our original power ending in 2817.

The first 5 digits of the power (rounded) are 88008 which we divide by 10 to get 8800.8, and the closest match in the table is 9437 so our first approximation is 49.02. Now D = (8800.8 – 9437) / 10000 = -0.0636 and the first correction nD = 4(-0.0636) or -0.254 which we add to 49.02 to get a closer value of 48.766. The second term (-0.25

^{2}) / 8 = 0.008 so we subtract that to get 48.758. Therefore we merge 48.758 and 7377, and the closest 8-digit match is 48757377.

**Additional Examples**

As an additional exercise, you may want to try this record by Alexis Lemaire (3.625 seconds):

38934589793526802773496632556519305532657006082154

49817188566054427172046103952232604799107453543533

The answer is given at the very end of this essay.

Another exercise is this problem of Gert Mittring (11.8 seconds):

34288725041442601391808603643426837852427296517260

61936285121642529526002848517356482932010681285881

This answer can be found in the earlier photograph of Gert Mittring.

Additional examples can be created by installing Python. Then download this Python script. Double-clicking on that filename will launch a window that provides 13th powers ending in 1, 3, 7 and 9 along with their roots.

**Conquering the 13th Root**

Despite the perception of a general decline in mathematical ability, there are a number of outstanding mental calculators today who use advanced algorithms and memory techniques to eclipse their forerunners. The 13th root problem is clear evidence of this advancement. The early attempts by lightning calculators to find 13th roots of 100-digit numbers were beyond ordinary comprehension at the time, and the practitioners were considered marvels of nature. Today this problem is performed an order of magnitude faster and sometimes, with enough attempts, in a matter of seconds.

However, these feats require an astonishing dedication and number knowledge that still places these lightning calculators in a category beyond the understanding of the vast majority of people. We have attempted in this essay to present typical approaches taken by lightning calculators to this problem, both past and present, and we have described a new method that allows the mathematically-inclined person—with a reasonable degree of time and effort—to extract 13th roots of 100-digit numbers, and to do so at speeds that were once considered unthinkable.

**Appendix: Derivation of the Doerfler-Forster Method**

The final digit of the root is always the same as the final digit of the power since the power is of the form 4k+1. The formula for the second-to-last digit is apparent from the table of 2-digit endings presented above. The formulas for the two digits prior to these were found by manually searching for patterns in 4-digit tables of 13th roots, a task too mind-numbing to relate. Lemaire described this in an earlier excerpt quoted in this paper as “working out patterns.” The formulas were then verified over all the 4-digit endings by a software program.

The Newton-Raphson iterative approximation for solving the equation a^{p} = N is given by

or

For an initial estimate R of N^{1/13}, we can improve this estimate with the correction from one iteration:

Defining n = R / (13R^{13}) = 1 / (13R^{12}) and D = N – R^{13}, the correction we add to the initial estimate is

Rather than attempt a second iteration of the Newton-Raphson method, we can simply add another correction called the Chebyshev term [Doerfler 1993].

The total correction is the sum of correction_{1} and correction_{2}:

A table was created in software of all numbers n to 4 decimal places between 4×10^{-22} and 32×10^{-22}. A column was added for R from the equation n = 1/(13R^{12}), or R = (13n)^{-1/12}, and another column was added for P, where P = R^{13}×10^{-18}. This table was manually searched for rows in which n (without the exponent) was an integer to high accuracy, while R and P rounded to 4 digits with 5-digit accuracy. These were candidates for rows in the memorized table, where the powers of 10 are incorporated into the method rather than the table. Finally, the values were tested until there was a sufficient distribution of table rows to provide accuracy over the range of possible roots R to 3 decimal places in the correction formula. The result is a table that provides sufficient accuracy of results while minimizing the task of memorization.

**References**

Doerfler, R. W. 1993. *Dead Reckoning: Calculating Without Instruments*, Gulf Publishing, Houston.

Hope, J. A. 1985. *Unravelling the Mysteries of Expert Mental Calculation*, Educational Studies in Mathematics, 16:355-374. Available from JSTOR at http://www.jstor.org/stable/3482445

Lemaire, A. *13th Root: The Official Integer Root*, http://www.13throot.com/index2.html

Lemaire, A. and Rousseaux, F. 2009. *Hypercalculia for the mind emulation*, AI and Soc, 24:191-196. Available from SpringerLink at http://www.springerlink.com/content/c3v31q6k0r4854n3/

Mittring, G. 2004. *In 11,6 Sekunden die 13. Wurzel ziehen*, Spiegel Online, http://www.spiegel.de/wissenschaft/mensch/0,1518,333380,00.html

Rekord-Klub SAXONIA, *Mental Calculation Records: Extracting Roots*, http://www.recordholders.org/en/list/roots.html

Smith, S. B. 1983. *The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present*, Columbia University Press, New York.

**Answer to Additional Example #1:**

38934589793526802773496632556519305532657006082154

49817188566054427172046103952232604799107453543533

has a 13th root of 45792573.

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**<<< Go to Part I of this essay**

**Printer-friendly PDF file of The 13th Root of a 100-Digit Number (Parts I-II)**

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.

**Why 13th Roots?**

It is generally known that the difficulty of mentally solving for integer roots depends on the number of digits in the root rather than the number of digits in the power. So why did the 13th root of a 100-digit number become the standard?

First, the appeal of a prime power must be acknowledged. Square root extraction is another popular category in itself. Composite roots such as the 4th root or 12th root can be calculated as a sequence of roots of their factors (two square roots to get a 4th root, followed by a cube root to get a 12th root). In fact, for a given number of digits in the root, even-numbered roots are more difficult because the final digit of an odd root can be found from the final digit of the power. It turns out that orders of powers that are one more than a multiple of 4 (such as the 13th power) have a root with the same final digit, while orders of powers that are one less than a multiple of 4 (such as a cube) have a root with a unique final digit relative to the final digit of the power. So here we have the first clue: the final digit of a 13th root will be the same as that of the power.

Second, 100 is an impressive round number of digits, and this produces a 13th root consisting of 8 digits. This number of digits proves to be non-trivial while not beyond the capabilities of the best mental calculators. If there were only 3 digits in the root of an odd power, the problem is easy. The final digit is found from the rules just described. The first digit can be inferred from memorized or estimated ranges of the powers. The properties of modular arithmetic can reveal the middle digit; here the root and power are replaced with remainders after division by an integer while still retaining the 13th power relationship, and we can deduce the missing digit.

For example, consider the 3-digit root of an odd power such as 24137569. Since 200^{3} = 8000000 and 300^{3}=27000000, we know the root has a 2 as the first digit. Since 3 is one less than a multiple of 4, the final digit of the root will not necessarily equal that of the power, but there will be a unique mapping—in this case 9^{3} ends in 9 so the last digit is in fact 9 and we are left with 2b9, where b is unknown. The remainder when a number N is divided by 11 (or N mod 11) can be found by subtracting the odd-place digits from the even place digits and repeating, adding 11’s until the result lies between 0 and 10. Here 24137569 mod 11 = (9+5+3+4) – (6+7+1+2) = 5. Now (2b9)^{3} mod 11 must equal 5 as well. So [(9+2) – b]^{3} mod 11 = (11 – b)^{3} mod 11 = 5. As noted above, for cube roots there is a unique mapping between n and n^{3} mod 11 and this mapping would be memorized by the mental calculator. Here we can see that 3^{3} mod 11 = 5 since 27 = 22+5, so 11–b = 3 or b = 8. So the cube root of 24137569 is 289.

For the 13th root of a 100-digit number, the first digit is always 4, and we know the last digit is the same as the power. But the 13th root has 8 digits and 7992563 possibilities so there is much more ambiguity, even when the performer has memorized long tables of 2-digit or 3-digit sets of beginning and ending digits. For this reason, the *Guinness Book of World Records* created the category of 13th root extraction of 100-digit numbers, recording in the eleventh edition of 1972 a time of 23 minutes by Herbert B. de Grote of Mexico.

**Historical Methods**

In the years since de Grote’s initial record, great efforts have been taken to solve 13th roots, and as a result the times required for it have steadily decreased. Here we will discuss three of the major players in this field: Wim Klein, Gert Mittring and Alexis Lemaire.

**Wim Klein**

Wim Klein of the Netherlands, a lightning calculator who worked at CERN, bested de Grote with a time of 5 minutes and then proceeded to lower his time even further. He eventually attained a record time in 1981 of 1 minute 28 seconds to calculate the 13th root of

88008443440489299575219015772236417859411720052615

65487280650870412023307854274990144578442271602817

He found the answer to be 48757377.

How did he calculate these roots? Klein used logarithms to find the first five digits, and then used his knowledge of 13th root endings and modular arithmetic to deduce the last three digits. To find the logarithms he would factor the initial digits and add up those 5-digit logarithms from memory, interpolating between values as needed for offsets. Then he would divide by 13 and use the reverse process to find the antilogarithm, the number whose logarithm would be that value. This would be the initial five digits of the root. It is far easier said than done.

Smith [1983] discussed with Klein his method for finding the 13th root of

14762420839370760705665953772022217870318956930659

27236796230563061507768203333609354957218480390144

Here is Smith’s account of Klein’s procedure:

The first five digits of the root are fixed through the use of logarithms. Klein has memorized to five places the logs of the integers up to 150; this, coupled with his ability to factor large numbers, allows him to approximate the log of the first five digits of the power, which is usually sufficient to determine the first five digits of the root, though, as he says “the fifth digit is a bit chancy.”

Klein began by factoring 1,476 into 36 times 41 and taking the (decimal) log of each: log 36 = 1.55630 and log 41 = 1.61278; adding the mantissas yields 0.16908, but this is, of course, too little. Through various interpolations Klein estimated the mantissa of the log of 147,624 as 0.16925 (it is more nearly 0.16916).

Klein now had an approximation of the log of the 100-digit number above – 99.16925. This must be divided by 13 to obtain the log of the 13th root. Since 99=13 × 7 with a remainder of 8, to obtain the mantissa of the antilog of the 13th root he divided 8.16925 by 13, which is approximately 0.62840. He estimated the antilog to be about halfway between 4.2 and 4.3 and decided to try 4.25. The result was exact, so the first five digits of the root should be 42500, as indeed they are.

It is now necessary to determine the last three digits of the root. This he does from an examination of the last three digits of the power. In the case of odd powers, these uniquely determine the last three digits of the root, but in the case of even roots, like this one, this method yields four possibilities; in the case of 144 they are 014, 264, 514, and 764. (The choices always differ by 250.) To select the correct one Klein divides the original number by 13 and retains the remainder. In the case of 13th roots, the root remainder and the power remainder must be the same. The power remainder is 7; only 764 as the final three digits of the root will yield 7 as the remainder. Thus the 13th root is determined to be 42,500,764.

As we will see, there are multiple endings possible when the 13th power ends in 2, 4, 6, or 8, so these are not going to be record attempts. In fact, the above account appears to be unique; other accounts of 13th roots are limited to odd final digits, and the method described in Part II of this paper is also limited to odd final digits.

Here is Klein’s account for another 13th power, also from Smith:

75185285487713563581947553291145079861723813162341

53935861550997297991815299022662358976308065985831

The first five digits of the power are 75185, which is nearly 7519, and 7519 is 73 times 103. The mantissa of the log of 73 is 0.86332 and that of 103 is 0.01284. Their sum is 0.87616. Dividing 8.87616 by 13 yields 0.68278. This falls between the mantissas of the logs of 48 and 49, but is much closer to 48. Since 481 is 13 (mantissa 0.11394) times 37 (mantissa 0.56820), the mantissa of its log will be 0.68214; close, but still a bit low; 4,816 can be factored into 16 (mantissa 0.20412) times 7 (mantissa 0.84510) times 43 (mantissa 0.63347). This gives a mantissa of 0.68269. Then 4,818 factors into 66 (mantissa 0.81954) times 73 (mantissa 0.86332), which yields a mantissa of 0.68286. Thus, in the interpolation we want 9/17 of 20 which is about 10 1/2. The first five digits of the root should be 48170 (48160 + 10). This, in fact, is correct.

When Klein actually did the calculation he made a minor error (he was looking for the antilog of 0.68277 instead of 0.68278) and first took 48169 for the first five digits of the root. In this case, however, since the root is odd, the last three digits are uniquely fixed—since the power ends in 831, the root must end in 311. Upon dividing the power by 13 Klein got a remainder of 7. But dividing 48,169,311 by 13 gives a remainder of 8. To make these two remainders come into line he changed his solution to 48,170,311, which is correct.

Hope [1985] remarks that “acquaintances of Klein report that during these complex mental calculation tasks, Klein mutters constantly in Dutch while calculating, and a good part of his muttering consists of swearing.”

Klein also worked with higher-order roots. In 1976, he found the 73rd root of a 500-digit number in 2 minutes 43 seconds, a feat recorded in the two photographs of Klein in this essay. In 1983 he found the 73rd root of a 505-digit number in 1 minute 43 seconds. A 73rd root duplicates the final digit of the power just as the 13th root does, since they are both of order 4k+1. No doubt Klein used the same procedure described above but divided the logarithm by 73 rather than 13. In this case there are only 7 digits with 273696 possibilities, a factor of nearly 30 fewer than the 13th root of a 100-digit number. As we have seen, the mass of digits in the middle of the power mean nothing to the mental calculator, only the starting and ending digits. Klein talked freely of how he would sometimes write down intermediate results, particularly in front of audiences [Smith 1983].

**Gert Mittring**

These examples of Klein demonstrate a great deal of effort by an extremely talented and dedicated mental calculator, far beyond reasonable expectations for the rest of us. But modern calculators have eclipsed Klein, claiming records of mere seconds for particular attempts. One such calculator is Gert Mittring of Germany. On August 25, 2011, Mittring won his eighth consecutive title in the Mental Calculations World Championship, evidence of formidable talent. In 2004 Mittring (shown here with a different record attempt) achieved a record of 11.8 seconds for the 13th root of

70664373816742861022340088302401573757042331707026

32731269721516000395709065419973141914549389684111

Writing in *Spiegel Online* [Mittring 2004], Mittring describes one strategy for finding this particular root, as translated below:

There are different strategies in the present problem of finding the solution. I will try to explain a variation that I’ve used. As you like, you may devise alternative strategies. It is desirable in any case to have a “memory-economic” variant.

The determination of the 8-digit (integer) 13th root of a 100-digit number is done in three major steps:

- The estimate of the logarithm.
- The division by 13 (the root exponent). The result is the logarithm of the solution.
- The conversion of the logarithm to its antilogarithm.
It is sufficient to look at the first six digits of the 100-digit number. Rounded, it starts with 706644 × 10

^{94}. How can the logarithm be estimated effectively? I only know the logarithms of the primes up to 100 (2, 3, 5…, 97). That amounts to 25 logarithm values to 7 decimal places each, so the memory effort is equivalent to 15 phone numbers.

:The first big stepA first estimate is the following simple application of the rules of logarithms by factoring the number 706644 × 10

^{94}into prime factors and then summing them up (where log 2 + log 5 = 1):log (706644 × 10

^{94})

~ 94 + 2 log 2 + 5 log 3 + (6 log 3 + 2 log 5 + log 29) / 2

= 95 + 2 + 8 log 3 + log 29 /2 ~ 99.849199… (the exact value would be 99.849200…)

:The second step is simpleThrough my experience with multiplication I immediately know that 7.68 × 13 = 99.84. Right away I get the additional decimal places 07077 (rounded up, and knowing very well that the estimate for the original logarithm is a lower bound).

:The third stepIn the last step I have to find the antilogarithm of 7.6807077. A rough linear approximation over

log 48 = 4 log 2 + log 3 and log 47

gives me the first digits of the solution: 4794. It is apparent that

log 4794 × 10

^{4}= 4 + log 2 + log 3 + log 17 + log 47 (where log 47 is already known above)~ 7.6806980

The difference now is only 97 × 10

^{-7}. Obviously, 97 times ‘a bit over 11 units’ still needs to be added. The estimated solution is then ‘a little over’ 47941067.As a check, one can analyze the end digits of the original number. Because the 100-digit number ends in 11, a rule tells me that the solution needs to end in 71. Therefore, everything speaks in favor of 47941071, which I then spoke out and which was indeed the right answer.

Let’s study Mittring’s third step for finding the antilogarithm of 7.6807077. For N = 10^{7.6807077}, he uses his knowledge of logarithms to find N such that log N = 7.6807077:

- Mittring would know that log 47 = 1.6721. The original number minus 6 yields 1.6807077 which is greater than log 47 by 0.0086. So he checks log 48, and in this case he factors it into primes (2
^{4}× 3) and finds log 48 as 4 log 2 + log 3 = 1.6812, which is a little high by 0.0005. In practice, Mittring has no doubt memorized the table of logarithms on the right that spans the range of the initial two digits of the 13th root. - The “rough linear approximation” is a linear interpolation between log 47 and log 48. The difference between the values of log 48 and log 47 is 0.0091 per the table, and log 48 is 0.0005 too high, so 0.0005/0.0091 = 0.06 is subtracted from 48 to yield 47.94. Adding the 6 that was subtracted from the original number of 7.6807077 gives 6 + log 47.94 or log 10
^{6}+ log 47.94 or log 47940000. Now 4794000 becomes our initial estimate of the root. - Now Mittring finds the exact value of log 47940000 in order to find its difference from the exact value of 7.6807077. From his innate knowledge of numbers, he factors 47940000 into 10
^{4}×2×3×17×47, so log 47940000 = 4 + log 2 + log 3 + log 17 + log 47, which he adds up from memorized values of logarithms to get log 47940000 = 7.6806980. - Now log 47940000 is 0.0000097 lower than 7.6807077. How much would we increase 47940000 if we increase its log by 97? In other words, for n=47940000 what is Δn for a given increase of Δ(log n)? Mittring at least implicitly uses the linear approximation
- The final two digits of a 13th root of a power ending in 1, 3, 7, or 9 are easily found from the final two digits of the power (the other endings of 13th powers do not produce unique endings of their roots). The table on the right shows the two digit endings of 13th roots for any two-digit endings of the powers. For any final two digits of the power shown in the body of the table, the corresponding digits of the root are found at the edges. The last two digits of the power are 11, so the root ends in 71. The estimate of 47941067 is therefore modified to 47941071.

Δn ≈ 2.3nΔ(log n)

= 2.3 × 4.794e7 × 97e-7 ≈ 11 × 97 = 106

and so 47941067 is our next estimate. The “bit over 11 units” that is multiplied by 97 in Mittring’s explanation is evidently from approximating 2.3 × 4.8.

Did Gert Mittring do all this in 11.8 seconds? He described it as simply one strategy he has used, a variant that is “memory-economic,” but it requires significant knowledge of factors and logarithmic values. The details of the presentation of the problem and the timing parameters of the test are not known, but it seems inconceivable that this was the method used to achieve this record time. Perhaps this attempt was achieved through a different, “memory-intensive” method. In the next section we will discuss another mental calculator who bested Mittring’s time and refers to these as simply tests of memory, albeit of prodigious memory.

**Alexis Lemaire**

Less than a month after Gert Mittring’s 2004 record of 11.8 seconds, Alexis Lemaire of France is reported to have extracted the 13th root of the number below in 3.625 seconds. As presumably the case with Mittring, this time included the time to write the answer of 45792573:

38934589793526802773496632556519305532657006082154

49817188566054427172046103952232604799107453543533

What are we to make of this feat? On Lemaire’s website [Lemaire], he relates how he moved on extracting 13th roots of 200-digit numbers because the former “can now only be a record of memorization.”

[The 3.625 record time] means the 13th root of a 100-digit number is an immediate calculation (1 second), and the recordholder will be the one with the fastest time for writing, not with the fastest time for calculating. Every left part and right part of the 13th root of a 100-digit number can be memorized: there are only 3 and 4 digits to be memorized. Therefore the 13th root of a 100-digit number can be only a record of memorization, whereas the 13th root of a 200-digit number is a true task of mental calculation.

Lemaire here downplays the difficulty of memorizing all the starting and ending combinations; this is a stunning achievement of memorization. Referring to this record time and the number of possible answers, Lemaire in another paper [Lemaire 2009] states

Nearly 8 million combinations have been learned beforehand (consciousness of the future) through a

generalization axiom which compresses these numbers into softer rules by working out patterns… Furthermore, this reverse artificial intelligence uses fuzzy sets to compute faster when dealing with the central most difficult part of the calculation; the fuzzier the computation, the faster it is but also the less accurate. We use this point to break records set after a great number of attempts.

The complete set does not need to be memorized; a “fuzzy” but fast attempt is made and if the answer is incorrect the next number is attempted. A competitor may also be asked a familiar number by chance. It is important to remember that mental calculators make many, many attempts at record times. Lemaire remarks on his website that it took 742 attempts in 2005 to beat his previous record in the 200-digit realm. This is not a criticism—athletes do the same—but simply an aspect of the competition.

**The Current State of 13th Root Extraction**

Wim Klein once said, “What is the use of extracting the 13th root of 100 digits? ‘Must be a bloody idiot,’ you say. No. It puts you in the *Guinness Book*, of course” [Smith, 1983]. But in fact *Guinness World Records* dropped the category of 13th roots of 100-digit roots because the result is so dependent on the particular value of the power. And with the memorization capabilities witnessed today (most visible in the memorization of π), the field does not have the intellectual depth it once enjoyed.

Today a set of 13th roots is required for the world record as maintained by a separate organization, *Rekord-Klub SAXONIA*, and purposely includes even-numbered final digits of the powers that allow multiple combinations of final digits [Rekord-Klub SAXONIA]. It appears that no one has attempted this, and in fact it is likely that only Wim Klein, who passed away in 1986, could do it.

Part II of this essay describes a new way of extracting 13th roots of powers ending in 1, 3, 7 or 9. This method requires practice, but it allows someone to master 13th roots with a reasonable amount of number sense and memorization.

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**>>> Go to Part II of this essay**

**Printer-friendly PDF file of The 13th Root of a 100-Digit Number (Parts I-II)**

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.

This is a book about the theory of nomography—there are mathematical derivations and sketches of the corresponding scale curves, but few finished nomograms. The mathematics presented here are tools that can be extremely useful in designing your own nomograms, and in fact the nomograms presented in this review were made by me using Leif Roschier’s PyNomo software. (If you haven’t read anything about nomograms yet, please see this essay or this article.)

Two major threads in the history of nomography are **existence criteria** and **anamorphosis**. Existence criteria can tell you if the equation you are interested in can be represented as a nomogram (i.e., can be expressed as a determinant equation in standard nomographic form). It provides guidance in algebraically converting your equation to one of the canonical forms that have matching nomographic forms. This process generally involves derivatives, so some basic knowledge of how to take a derivative or find an integral (or simply a book of tables of these) is needed to really test your equation. Anamorphosis is the process of manipulating your equation into different forms of nomograms, say from three parallel scales to one curve containing two scales and one line, or even a single curve along which all three scales lie. It is more than simple geometric transformations or projections.

A brief overview of the history of nomography is given in a later paper by Evesham; the thesis, of course, goes into much greater mathematical detail. Evesham first describes the work of Leon Lalanne, who in 1843 adjusted the x and y axis scaling on a Cartesian plot to morph the curves for the equation xy = k into straight lines. In the process he created the first log-log plot in history and ultimately produced his “Universal Calculator” graphical computer seen on the cover of Evesham’s book. I have been meaning to write an essay on this calculator for some time. Lalanne is the one who termed this process anamorphosis.

The road from Lalanne to d’Ocagne’s invention of nomography strikes me as a strange one, a slow trip until 1867, and then an explosion of ideas in the 1880s. The event in 1867 was Paul de Saint-Robert’s presentation of his test to determine if an equation can be represented by two fixed scales and a sliding scale (like a special slide rule). This is equivalent to the three parallel scales of d’Ocagne’s later nomogram for simple addition of three functions. In other words, we are interested in whether an equation in the form F(x,y,z) = 0 can be rewritten in the form Z(z) = X(x) + Y(y). The Saint-Robert criterion says this is possible if

where the partial derivative ∂F/∂x is the derivative of F with respect to x assuming all other variables are constants, and ∂F/∂y is the derivative of F with respect to y assuming all other variables are constants. To find ∂^{2} ln R / ∂x∂y, we find the partial derivative of ln R with respect to x and then we find the partial derivative of the result with respect to y.

Let’s see if this works. Say we want to create a nomogram for the equation z = ax + by + cxy + d, or F(x,y,z) = –z + ax + by + cxy + d = 0. I created a nomogram last summer for this equation for an upcoming blog essay on nomograms to model 2-, 3- and 4-factor linear equations in Design of Experiments (DOE) methodology; this equation is the general form for a weighted, linear 2-factor equation. Can we draw it as three parallel scales in x, y and z? Here,

So yes, we find that we can create a simple addition nomogram of three parallel scales as Z(z) = X(x) + Y(y). Saint-Robert also provides a way of finding the functions X(x), Y(y) and Z(z) to plot on the scales:

which in the logarithmic form is the correct addition form for the parallel-scale nomogram, where the functions on the z, x and y scales are given by the corresponding terms Z(z), X(x) and Y(y). This nomogram is shown below for the specific equation z = 5x + 10y + 2xy. The sample isopleth (index line) demonstrates the graphical calculation of z=150 when x=2 and y=10. *Click on this figure and all figures in this essay for full-resolution versions.*

So you might say, well, I could have figured out that z = ax + by + cxy + d can be written as z – d – ab/c = (cx + b) (y + a/c) without going through all this. In my case, I didn’t realize it was generally possible when I was creating the nomogram, and it took some painful transformations of the nomogram I originally created to arrive at this parallel-scale form. Only then did I think to algebraically derive it. Doh!

Saint-Robert’s criterion is also useful for more complicated equations for which it is difficult to determine if a simple addition form exists. For example, an equation of the form

can be cast as

In fact, the variables x, y and z can be replaced here by any function f_{1}(x), f_{2}(y) and f_{3}(z) and this formulation holds. I’ve written out the complete derivation for this example using Saint-Robert’s criterion that you can find here.

Evesham also describes two different criteria by Massau and Lecornu for expressing F(x,y,z) = 0 in the form Z_{1}(z)X(x) + Z_{2}(z)Y(y) = 1, which is equivalent to the general form Z_{1}(z)X(x) + Z_{2}(z)Y(y) + Z_{3}(z) = 0 after division by Z_{3}(z).

Evesham continues his presentation by discussing the important existence theorems of Grönwall, Kellogg and Džems-Levi. Ultimately he points to Warmus’ publication *Nomographic Functions* in 1959 as the first comprehensive treatment of existence criteria for functions of three variables. Warmus uses exhausting algebraic techniques based on classifications of equations to either find the determinant elements or show that it is not possible. Thank goodness it’s written in English despite being published in *Rozpravy Matematyczne*. I have this little book, but it’s one that I have no plans to read through until I really need to.

An overview of d’Ocagne’s *Traitè de Nomographie* of 1899 (found at Google Books and as modern facsimiles 1 and 2 on Amazon) is provided. This is followed by a detailed look at J. Clark’s nomographic discoveries. I found this particularly exciting, as d’Ocagne, Soreau and others include Clark’s nomograms in their books and they discuss his methods of anamorphosis, but I had never found any book authored by Clark. In fact, I learned that Clark instead authored in 1907-1908 a series of articles on nomography in a French mechanical engineering journal, which to my surprise I subsequently found digitized at Google Books (see pages 321-335 and 575-585 of here and pages 238-263 and 451-472 of here).

Let’s take a look at some of Clark’s ideas discussed in Evesham’s book. We’ll use an equation that Clark used, the addition formula for the tangent function:

As Soreau points out, this can actually be drawn as a basic parallel scale addition nomogram. In one approach that I can only call inspired, you first create a nomogram for (*a*+*b*) = *a* + *b*, which for a parallel-scale nomogram consists of identical linear scales for *a* and *b* and a linear scale for (*a*+*b*) with half the spacing of the other scales and located exactly between them. Then you just plot the tangent of each value on the other side of the scales, and (*voila!*) you have an addition nomogram for the tangent function. A straightedge (*isopleth*) placed across the values on the tan *a* and tan *b* scales will cross the corresponding value on the tan (*a*+*b*) scale.

Clark looked at what nomographic forms are possible with given orders of equations. One of his results is that an equation in the form f_{1}f_{2}A_{3} + (f_{1 }+ f_{2})B_{3} + C_{3} = 0 can be represented by the following general determinant equation:

where A_{3}, B_{3} and C_{3} are functions of the third variable. If you actually multiply out the determinant you will get the equation (f_{2 }– f_{1})[f_{1}f_{2}A_{3} + (f_{1} + f_{2})B_{3} + C_{3}] = 0 which satisfies the original equation but consists of an extra factor (f_{2 }– f_{1}). More on that later!

With the substitution f_{1} = tan *a*, f_{2} = tan *b* and f_{3} = tan(*a*+*b*), the tangent addition formula can be written as

f_{3} = (f_{1} + f_{2}) / (1 – f_{1}f_{2}) or f_{1}f_{2}f_{3} + f_{1} + f_{2} – f_{3} = 0

If we match this to Clark’s form above, we find that A_{3} = f_{3}, B_{3} = 1 and C_{3} = -f_{3} so the determinant equation is of the form

And dividing the third row by f_{3} we arrive at the standard nomographic form:

When a nomogram is in this standard form, the first two columns are the x and y functions for the scales for each variable, so you can see that the scale for tan *a* lies on a parabola (since x = tan *a* and y = tan^{2}*a* means that y = x^{2}), and the tan *b* scale lies on exactly the same curve. The tan(*a*+*b*) scale lies on the line y = -1. A straightedge (isopleth) is placed across the value of tan *a* and the value of tan *b* on the parabola, and tan(*a+b*) will be the value crossed on the linear scale.

Clark calls this a *conical *nomogram, and it’s very interesting that two of the three scales lie on the same curve, although if f_{1} and f_{2} are not the same function the scale labels will be different (and therefore usually printed on opposite sides of the curve). Evesham provides examples for the equation f_{1} + f_{2} = f_{3} and f_{1}f_{2} = f_{3} expressed in Clark’s form.

Evesham also mentions that a homographic transformation can always transform this parabola into another conic such as a circle. A circle is far easier to draw by hand, and I think it provides a very striking nomogram. Evesham does not explain how this is done. I had in fact converted a hyperbolic scale to a circle previously in my nomography essays but it was a multi-step process, so I went to Clark’s original articles on Google Books and found his method for doing this transformation (on page 262 here).

For Clark’s general determinant equation given above, a circle will result if the elements of the determinant are replaced with the functions below:

(Note that Clark uses columns 2 and 3 in his articles for the x- and y-elements, but I’ve moved them to the more standard columns 1 and 2 in the determinant above.)

We can choose the angle α, and in fact this positions the opening of the parabola on the circle. We can’t choose α = 0 here because sin α = 0 and so the y-elements are 0. If we choose α= 45° we get the nomogram below, where the opening is at this 45° angle:

The circular scale is compressed asymmetrically, but if we get the figure below if we choose α = 89.99° (we can’t choose 90° or the row 3, column 3 element is 0 since A_{3} = -C_{3} and we can’t divide that row by any number to get the determinant in standard form):

The thing that Clark did that is most astonishing to me is to find a way to generalize his equation form and provide a means of getting all three scales to lie along a single curve! Clark deduced that the extraneous factor (f_{1} – f_{2}) in the determinant for the conical determinant is responsible for aligning two of the three scales (f_{1} and f_{2} here but we can use any two of the variables) on a common curve, so he searched for a factor that would align all three scales. For an equation in the general form

f_{1}f_{2}f_{3} + A(f_{1}f_{2} + f_{2}f_{3} + f_{1}f_{3}) + B(f_{1 }+ f_{2} + f_{3}) + C = 0 (1)

he found that the extraneous factor should be (f_{1} – f_{2})(f_{2 }– f_{3})(f_{3} – f_{1}), and this leads to the determinant equation:

You can see from the x- and y-elements of each row that the three scales lie on exactly the same curve, although if the functions are not identical the scale divisions will be different.

So let’s see, we can rewrite the tangent addition formula as f_{1}f_{2}f_{3} – (f_{1} + f_{2} + f_{3}) = 0 for f_{1} = tan *a*, f_{2} = tan *b* and f_{3} = -tan(*a*+*b*). Matching terms with the general form above, we find that A = 0, B = -1 and C = 0, so the determinant equation is:

Now we can add the first column to the third, divide by the second column, and shift the first column to the third column to get this into standard nomographic form:

For f_{1} = tan *a*, the scale curve is given as x = tan *a*, y = tan *a* / (tan^{2}*(a)* + 1) and the same for f_{2} = tan *b* and f_{3} = tan(*a*+*b*) and we have this remarkable nomogram below for tangent addition, which Clark termed the *acnodal* form:

In fact, the general equation form breaks into three distinct nomographic curve types that cannot be transformed into each other, represented by the following forms:

f_{1}f_{2}f_{3} = 1 (crunodal form)

1/f_{1} + 1/f_{2} + 1/f_{3} = 0 (cuspidal form)

f_{1}f_{2}f_{3} – (f_{1} + f_{2} + f_{3}) = 0 (acnodal form)

We have just seen the general acnodal form in our tangent addition nomogram. You may have seen the crunodal form below in my other essays or nomography calendar. This curve is also known as the *folium of Descartes*. The equation f_{1}f_{2}f_{3} = 1 corresponds to A=0, B=0 and C=-1 in Clark’s general equation (1) given above:

The cuspidal form is used for the harmonic relation, which can represent, for example, the focal length of two lenses in series or the equivalent resistance of two resistors in parallel: 1/f_{3} = 1/f_{1} + 1/f_{2}. This can be written as 1/f_{1} + 1/f_{2} + 1/(-f_{3}) = 0 to get it into our form above. It’s not very practical for graphical calculation as you can see from the example below of 1/950 + 1/700 = 1/403, but it’s a very interesting nomogram nonetheless. The equation 1/f_{1} + 1/f_{2} + 1/f_{3} corresponds to A=1, B=0 and C=-f_{1}f_{2}f_{3} in Clark’s general equation (1) given above, because you can rewrite 1/f_{1} + 1/f_{2} + 1/f_{3} as (f_{1}f_{2}f_{3})^{-1} (f_{1}f_{2 }+ f_{2}f_{3 }+ f_{1}f_{3}). With work you can find this determinant equation and its corresponding nomogram:

Actually, it’s possible to find a single curve nomogram for f_{1} + f_{2} + f_{3} = 0 as well, as shown below. Another way of looking at this is that the 3 real roots of a cubic equation ax^{3}+bx^{2}+cx+d = 0 sum to –b/a, so a plot of y = x^{3} marked with its x-values provides a single scale nomogram for addition:

Clark called all of these *cubic *forms of nomograms because the common curve in each case is given by a third-degree equation.

Evesham concludes his thesis with a discussion of Russian advances in nomography from the 1950s, including the use of oriented transparencies of scales laid across other printed scales as treated by Khovanskii. This type of nomogram can incorporate more variables in a single alignment. He refers to earlier work by Margoulis, and I have Margoulis’ book, *Les Abaques à transparent orienté ou tournant*, from 1931 that includes extremely interesting nomograms for aeronautics (such as attack angles for helicopters!), and which uses oriented transparencies to solve very complicated equations of several variables.

H.A. Evesham’s thesis weaves together many different threads in the history of nomographic theory. If the presentation above intrigues you, then you will find real treasures in this book. The bibliography in the book is valuable in itself for searching out more details. I enjoyed reading the book immensely. Mr. Evesham does a great service in making these advancements available to those of us today with an interest in the field of graphical calculation. Again, the book can be found here or here.

(I have no financial interest in the book or with Docent Press.)

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]]>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.

A different field of mental calculation is treated each month in this calendar, from 2-digit multiplication to square roots to factoring to logarithms and more. The upper half of each calendar page describes methods that can be used to solve problems of that month, and the lower half offers opportunities to test yourself and practice these techniques daily using the actual dates shown in the calendar. Of course, you should feel free to try all the dates at once, or bounce around between months, or do whatever you want. After all, this is all for fun!

Every month includes a legend that describes how the dates are used in the calculations. In nearly all cases the answers are provided right in the calendar itself, printed in number boxes connecting the dates involved in the calculation but small enough that they are not visible from a distance. Every day starting in February also poses a day-date calculation. In all, there are nearly 4000 exercises in mental calculation embedded in this calendar!

The calendar is available for downloading and printing at the link below. The format is two-sided 8-1/2″ x 11″ sheets of paper printed in landscape mode that can be connected at their edge. This could be printed to fit on A4 or other sizes, I’m sure. I recommend printing it on colored paper; although it looks fine on white paper, 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. In the past I’ve had a local office shop (Kinko’s FedEx) 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.

Alternatively, and in response to comments on last year’s calendar, it is available for ordering from Lulu.com if you prefer. The downloadable PDF file below has all the resolution of the Lulu.com version—the only difference between them is that the Lulu version has a light beige/ivory background that I did not include in the downloaded PDF to save printer ink. The photo above is of the Lulu.com version, which is beautifully printed. The calendar is listed on the site as a coil-bound book that you turn sideways to use (it’s not in the calendar category), and you will have to punch or drill a hole in it to hang it on a hanger.

Small images of the front and back and each page of the calendar are shown below. These provide only a rough idea of the content, but if you think you might be interested, the complete PDF file is just over 10MB, so it is easily downloaded and viewed at full resolution. The last page is flipped upside down in order that the back of the calendar has the same orientation as the front cover.

The PDF of the calendar for personal printing is found here.

The calendar on Lulu.com is found here.

Hope you like it. Have a happy 2011!

Ron

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**The Principles of the Hexagonal Chart**

The *L’Abaque Triomphe*, as mentioned earlier, is an example of a hexagonal chart as invented by Lallemand in 1885. This design is sometimes referred to as an early example of a nomogram, but it only fits this description in its broadest sense. Hexagonal charts can be used to plot a function *z* in terms of *x* and *y* without resorting to multiple plots or a difficult-to-read 3D plot. Primarily utilized by Lallemand and Provost, hexagonal charts appear to have lived a short life as the hexagon overlay yielded to the simple straightedge used for nomograms.

But hexagonal charts have their own charm, and in some ways they excel over nomograms. For example, today a hexagonal chart can be printed on isometric drawing paper (which has light blue lines at 0°, ±60° and sometimes 90°) or even hexagonal grid paper. Then by using the lines as visual guides there is no need to use an overlay at all! Nomograms, on the other hand, do require the use of a straightedge because the lines occur at whatever angle connects the relevant points on the scales.

The other advantages of hexagonal charts will become clearer as we look at the mathematics behind their design.

Consider the figure shown here [after Runge 1912]. Angle *AOC* is an arbitrary angle, but *OB* is a bisector of that angle. We mark points *a* and *c* such that *Oa* = *Oc*. Then we draw perpendiculars *ab* and *cb* that meet at a point on *OB* by the symmetry of the diagram. Then *Oa* = *Oc* = *Ob* cos *AOB*, or *Oa* + *Oc* = 2 *Ob* cos *AOB*. Now move *b* out to *b′* on the perpendicular *b-b′*. Since *b-b′* makes the same angles with *OA* and *OC*, *aa′* = *cc′*, and *Oa′* + *Oc′* again is equal to 2 *Ob* cos *AOB*. In this way we can see that we have an addition chart where two perpendiculars from offsets *a* and *c* along *OA* and *OC* intersect along *OB* with an offset *b* of (*a* + *b*) / (2 cos *AOB*).

We therefore provide scales for *a* and *c* that are linear and have the same scaling factors. Then if angle *AOC* = 90°, the scale for their sum *b* will need to have a scaling factor of 1 / (2 cos 45°) = 1/√2. If *AOC* = 120°, the scale for their sum *b* is the same as the scales for *a* and *c*, since cos 60° =1/2, and this is the hexagonal chart. This property is sometimes expressed as this: the algebraic sum of the projection of a segment of a line on two axes having an angle of 120° between them is equal to the projection of the same segment onto the internal bisector of the angle between these axes. The equation then reduces to

f(b) = g(a) + h(c)

For any chosen angle *AOC*, an overlay is needed that provides arms perpendicular to the three scales. This is shown below for *AOC* = 120° (the hexagonal overlay) and for *AOC* = 90°, with corresponding examples shown below each of these [Werkmeister 1923]. (It’s a bit hard to make out at this resolution, but the first example shows 3 x 2 = 6 and the second example shows 3 x 5 = 15.)

If the logarithms of *a*, *b* and *c* are plotted along their axes, we have a multiplication rather than an addition chart, and this allows very complicated formulas to be plotted as hexagonal charts. For example, consider the formula

P = (S + 0.64)^{0.58}(0.74V)

Here we take logarithms of each side to express this as a simple sum.

log P = 0.58 log (S + 0.64) + log 0.74 + log V

Then we plot 0.58 log (S + 0.64) along the *a* scale, log V along the *b* scale, and log P – log 0.74 along the *c* scale.

Now you can see that it doesn’t matter if the scales are moved apart as long as they are shifted along an axis perpendicular to their original scale (so they are moved perpendicular to the arm of the hexagon, or in other words, parallel to their current orientation). This makes it easy to create multiple scale triplets for different ranges of the variables by placing these scales parallel to each other. With matching titles or colors, the user reads values off the appropriate triplet for the range of interest, as shown in the figure on the right.

Below is an example of the use of multiple scales offset at convenient distances here and there along the 120° axes of the hexagon [d’Ocagne 1891/1899/1908/1921, Soreau 1921]. The chart is for simple multiplication of numbers, although with different labeling it could be the multiplication of functions of variables. Plotting the logarithms of the values converts the product to a simple addition. The middle scales AA, AB and BB provide the product when using scales A and A, A and B, and B and B, respectively. As examples, the intersecting dashed pair of lines at the top connect 3 on A and 3 on B, and the third vertical leg of the hexagon will terminate at 9 on the scale AB, and so forth for the other combinations shown.

Hexagonal charts can be easily extended to support the addition of four or more functions of variables. One way to do this is to group pairs of functions into grids, which we will see in later examples. This provides the value of an unknown variable with one placement of the hexagonal overlay, but is limited to three grids or six variables. Lallemand implicitly used grids when he embedded the magnetic variation effects into his maps on the chart. For simple sums such as f(a) = f(b) + f(c) + f(d), a different approach is to place the overlay to find k = f(b) + f(c) on an intermediate scale k, then slide the overlay along the arm crossing k until one of the other arms crosses f(d) on its scale, with the third arm showing the final value f(a) on that scale. This concept provides solutions for an indefinite number of summed functions, while the freedom to shift scales parallel to themselves means that the chart can be made quite compact. Later we will see an example of a tree network of hexagonal charts.

**The Fate of Hexagonal Charts**

Lallemand had barely publicized his highly useful invention of the hexagonal chart, originally in an internal publication of his directorate, when Maurice d’Ocagne announced his discovery of another radical means of graphical calculation, also an abaque but using what he called “parallel coordinates.” Today we call these constructions *alignment charts*, *nomograms* or *nomographs*.

A nomogram for the equation we considered earlier is shown in the figure on the left. A straightedge is used to cross values of S, V and P that satisfy the equation. Zooming in a bit, I get P = 1.955 for S = 2.1 and V = 1.48. The actual value of P is 1.965. Notice that there are no grid lines and no interpolating between gridlines required, a feature only possible in a hexagonal chart by using a transparent hexagon overlay. Nomograms are also indifferent to affine transformations (such as linear stretching during the printing process) and projections, which hexagonal charts are not. A survey of the field of nomography, including the derivation of this particular nomogram, can be found in several essays on this blog.

However, hexagonal charts were extensively treated even in books on nomography, as seen in the *References* section at the end of this essay, and work was done to create and use them in the late 1800s and early 1900s. Below are some hexagonal charts from sources contemporary to that time. **Click on an image if you’d like to see a higher resolution version**. A printable hexagonal overlay is found in the Appendix of the PDF file if you are interested in exercising the charts.

The figure below on the left is a hexagonal chart for the average error in the sighting of a level goniometer, per the equation printed along the top [Soreau 1902]. The figure on the right is a hexagonal chart for the force on the land for a vertical wall support, per the equation printed along the top of that chart [d’Ocagne 1891/1899/1921, Soreau 1902/1921].

On the left below, a hexagonal chart (plotted as logarithms to convert to addition) for the Law of Sines and the corresponding tangent equation is shown, where

α_{1} sin α_{2} = α_{4} sin α_{3}

α_{1} tan α_{2} = α_{4} tan α_{3}

The tangent scale is split and shifted along its axis to avoid crossing the Longueurs (“Lengths”) scale for α and to maximize space on the page. A remarkable aspect of this chart is that the third scale, which would be horizontal, is missing! Here we do not need to know α_{1} sin α_{2} (or in this case log α_{1} + log sin α_{2}), which would correspond to a mark along the vertical arm on the horizontal scale. Rather, we position the overlay for one set of values of α_{1} and α_{2} and simply slide the overlay vertically (per the arrows) to find all other values that provide the same ratio. [d’Ocagne 1899/1921].

On the right below is a hexagonal abaque by Lallemand for altitude H and latitude λ corrections η_{1} and η_{2}, respectively, based on difference in heights d [Lallemand 1889, d’Ocagne 1899/1921, Soreau 1902/1921], where

η_{1} = -0.0019 d H

η_{2} = -26 d cos 2 λ

On the left below is a hexagonal chart for compound interest [d’Ocagne 1891/1899/1921, Schilling 1900]

A = a (1 + r)^{n}

where:

a = Capital place

r = Taux

n = Temps

A = Capital produit

On the right is a hexagonal chart for six variables (three grids) for calculating the effects of refraction. For information on the equation, see the PDF file. [Lallemand 1889, d’Ocagne 1899/1921, Soreau 1902/1921]:

Below are two hexagonal charts for excavations and embankments, important calculations in d’Ocagne’s line of work [d’Ocagne 1899].

It is possible, and sometimes done even today, to compute the sum of multiple functions by chaining hexagonal charts in tree arrangements. Below is an example of such a chart [Haskell 1919].

**Triangular Coordinate Systems**

The ability to shift the scales also led to other graphical computers called *triangular coordinate systems*, and these continue in some form or other to the present day.

Consider Lallemand’s *L’Abaque Triomphe*. Since only the offsets from the green axes are used to align the hexagonal overlay, we can shift, say, the conical scale vertically (perpendicular to its horizontal scaling, or along the green line representing the hexagonal arm through its origin) without affecting the resulting computation. And as you can see below, if the conical scale is shifted vertically by the amount shown, the scales of the three variables can be represented along the sides of an equilateral triangle.

When three variable scales are located on the sides of an equilateral triangle, a triangular coordinate system is obtained. This is something I never encountered in school, i.e., another way of plotting a function *z* of two variables *x* and *y* without a 3D plot or a family of curves. As we have seen, a great number of complicated functions can be represented in this way, and with isometric or hexagonal ruled paper they are as easy to read as a 2D Cartesian plot.

Let’s return to our example of a nomogram using the same ranges as before:

P = (S + 0.64)^{0.58}(0.74V)

Again we take logarithms of each side to express this as a simple sum.

log P = 0.58 log (S + 0.64) + log (0.74) + log V

Below are shown two graphical calculators for this function. In the left figure, 0.58 log (S + 0.64) is plotted horizontally with a range of 1.0 to 3.5, and log V is plotted at an angle of 120° with a range of 1.0 to 2.0. Then log P – log (0.74) is plotted at 60° with a range of 1.0 to 3.4. The light blue isometric (60°) lines aid in the computation. For example, when S = 1.2 on the horizontal scale and V=1.9 on the 120° scale, the perpendiculars point to a value of P = 2.0 on the 60° scale. The actual value of P is 2.00. You have to estimate the path of perpendiculars for values that lie between the blue grid lines, but this is much easier than it first appears. And of course we can still use a hexagonal overlay for greatest precision.

In the second figure, the scales are shifted parallel to their original positions to form the triangular shape, with the 120° scale for V truncated to 1.7 so it doesn’t extend past the long 60° scale for P. However, this was for aesthetic reasons only, and it would work perfectly fine if any line extended past any other line; the only effect would be that the perpendiculars would lie mostly outside the triangle.

Let’s see, we no longer have the value V = 1.9 because of its shortened scale. Let’s try values between the grid lines, say, S = 2.1 on the horizontal scale and V = 1.48 on the 60° scale. Then I estimate P = 1.97 on the 120° scale (using the corner of a sheet of paper at the intersection of the estimated perpendiculars helps). The actual value of P is 1.965. Enlarging the figures and drawing a finer grid would provide greater accuracy. See the higher resolution versions of the left figure and the right figure.

Often in practice the sides of the triangle are moved outward and drawn longer than the scales (so the scales lie only on the portions of the sides) so that all the perpendiculars, or at least the perpendiculars for the ranges of interest, lie inside the triangle. The example on the right is from Otto [1963].

Fasal [1968] provides a derivation of the general case of an acute triangular coordinate system of three unique angles. This general equation is

g(y) = k_{1}f(x) + k_{2}h(z)

where k_{1} = cos β / cos α and k_{2} = sin (α+β) / cos α when f(x), g(y) and h(z) have identical scaling. It can be seen from the figure that the three angles of the triangle are α+β, 90°- α and 90°- β. Fasal provides a graphical way of choosing these internal angles of the triangular coordinate system for given ranges of the variables in order to minimize the overhang of the scales beyond the vertices of the triangle. Aiken [1937] also discusses these relations in a very readable paper.

Shown here are two different graphical representations for the friction head H in feet per 1000 feet of water of water flowing in a pipe of diameter d with a velocity of V feet per second [Hewes 1923]:

H = 0.38 V^{1.86} / d^{1.25}

The first figure shows this function plotted in on hexagonal axes, where the solution can be found with using the grid or with a traditional hexagonal overlay. The second figure shows the same function plotted in a triangular coordinate system. Click on them to see higher resolution versions.

The figure below shows a triangular coordinate system where the triangle has been trimmed to save space, which therefore requires that the scales be chopped into two parts apiece [Lacmann 1923]. The equation represented by this chart is

v = 43.1 d ^{0.62} J^{0.55}

Now it’s also possible to map the scales of a hexagonal chart onto the sides of an equilateral triangle as shown in the figure on the right, where the three scales (or in this case, three grids) are mapped onto the sides of the triangle [Lacmann 1923]. Note that here the guidelines are parallel to the sides rather than perpendicular to them. A hexagonal overlay with its center inside the triangle can align its arms to the three sides of the triangle. Based on the workings of a hexagonal overlay, you can see that for a triangle with equally-spaced scales along each edge, the line segments from any node in the middle of the triangle will terminate at three scale values that add to a fixed value. This construction is called a *trilinear chart* or *trilinear diagram*.

Below is a three-variable equation constructed as a trilinear chart. The fixed sum can be readily seen as the sum at any vertex of the triangle, such as the (100+0) sum at the lower corners. The upper vertex is the same except that v^{2}/2g is plotted so you would have to perform the inverse to find 100 along the right scale at the vertex. This is a triangular coordinate system for solving Bernoulli’s Equation for incompressible flow [Lacmann 1923]. Click on it to see a higher resolution version.

z + p/γ + v^{2}/2g = 100

Trilinear charts are actually in use today, generally for finding the result of mixing three components (such as gases, chemical compounds, soil, color, etc.) that add to 100% of a quantity. These usually have percents of each component listed on the three scales along the sides of the triangle, as shown in the figure on the left below [Newski 1955].

By itself there is not much information here, but it is possible to add a fourth variable as contour lines within the triangle, as in the rightmost figure above. For example, variations in the percentages of metals in an alloy result in different hardness, brittleness, etc. The user can select a mixture whose node lies on the contour line for the desired attribute. This is equivalent to having curves in the middle of a hexagonal chart for the center point of the overlay as seen in the figure to the right [Soreau 1921].

J. Williard Gibbs is credited with the first use of trilinear coordinates (for thermodynamics) in 1873 [Howarth 1996]. In 1881 Robert Thurston published a paper using trilinear coordinates to express the properties of copper-zinc-tin alloys using contours [Aiken 1937]. Therefore, it seems likely that these constructions preceded hexagonal charts, and perhaps Lallemand drew inspiration from them. In any event, trilinear graphs exist today for the same types of applications. The Piper trilinear diagram, for example, is used to plot measured values of concentrations of major ions in samples of ground water onto an array of such triangles.

The trinary diagram on the left below was published by S. F. Taylor in 1897 to graph the critical curve (the solid line) and tie-lines (the broken lines) between five conjugate pairs of compositions (I-VI) of a benzene-water-alcohol system. The crosses represent experimental data [Howarth 1996].

The figure on the right below shows an equilateral triangular coordinate system with truncated corners to characterize vacuum tubes. The logarithmic scale values that would have been along the removed sections are simply wrapped around the edges of the truncated sections [Aiken 1937]. Click on it to see a higher resolution version.

**A Whisper of the Past**

Hexagonal charts share an interesting legacy with their triangular cousins. Today we don’t see hexagonal charts at all, partly due to the need for a hexagonal overlay. Triangular coordinate systems are also rare with the exception of the easily plotted and readily-understood trilinear diagram. Graphical constructions based on intersection of lines and curves on a Cartesian x-y coordinate system (*intercept charts* or *lattice charts*) are sometimes encountered, as in phase diagrams of thermodynamics. Hexagonal charts filled a relatively brief need for a graphical calculator prior to the invention of nomograms, which in turn fell victim to the development of calculators and computers. The study of these developments, so treasured at the time and so overlooked today, evokes a real appreciation, at least in me, of the ingenuity and pressing human efforts that were made in the past. I can sense a real purpose, and sometimes a surprising turn of imagination, in the writings of these individuals.

**References**

*NOTE : The references with Google Books links are fully viewable and downloadable within the United States, but not necessarily in other countries due to variations in copyright laws.*

Aiken, Howard. 1937. *Trilinear Coordinates*. Journal of Applied Physics, 8:470-472. Available for download with IEEE Xplore access such as in university libraries.

d’Ocagne, Maurice. 1908. *Calcul Graphique et Nomographie*. Paris: Gauthier-Villars. Available for download.

d’Ocagne, Maurice. 1891. *Nomographie; les Calculs usuels effectués au moyen des abaque*. Paris: Gauthier-Villars. Available for download.

d’Ocagne, Maurice. 1899. *Traité de nomographie*. Paris: Gauthier-Villars. Available for download.

d’Ocagne, Maurice. 1921. *Traité de nomographie. 2nd Ed*. Paris: Gauthier-Villars.

Fasal, J. H. 1968. *Nomography*. New York: Ungar.

French, Thomas E. and Vierck, Charles J. 1958. *Graphic Science; Engineering Drawing, Descriptive Geometry, Graphical Solutions*. New York: McGraw-Hill.

Haskell, Allen C. 1919. *How to Make and Use Graphic Charts*. New York: Codex. Available for download.

Hewes, Laurence J. and Seward, Herbert L. 1923. *The Design of Diagrams For Engineering Formulas And the Theory Of Nomography*. New York, McGraw-Hill. Available for download.

Howarth, Richard J. 1996. Sources for a History of the Ternary Diagram. The British Journal for the History of Science, Vol. 29, No. 3. Available for download with JSTOR access.

Lacmann, Otto. 1923. *Die Herstellung gezeichneter Rechentafeln: Ein Lehrbuch der Nomographie*. Berlin: Verlag von Julius Springer.

Lallemand, Charles. 1885. *Les abaques hexagonaux: Nouvelle méthode générale de calcul graphique, avec de nombreux exemples d’application*. Paris: Ministère des travaux publics, Comité du nivellement général de la France.

Lallemand, Charles. 1889. Nivellement de Haute Précision, in Lever des Plans et Nivellement. Paris: Librairie Polytechnique. Available for download.

Lipka, Joseph. 1918. *Graphical and Mechanical Computation*. New York : John Wiley & Sons. Available (in a 1921 volume) for download.

Newski, B. A. 1955. *Prakitum der Nomogramm-Konstructionen*. Berlin: Akademie-Verlag.

Otto, Edward. 1963. *Nomography*. New York: Macmillan.

Peddle, John B. 1919. *The Construction of Graphical Charts*. New York: McGraw-Hill. Available for download.

Runge, Carl. 1912. *Graphical Methods*. New York : Columbia University Press. Available for download.

Schilling, Friedrich. 1900. *Ǖber die Nomographie von M. d’Ocagne*. Leipzig: Druck und Verlag von B. G. Teubner. Available for download.

Shirazi, Mostafa, et. al. 2001. *Particle-Size Distributions: Comparing Texture Systems, Adding Rock, and Predicting Soil Properties*. Soil Sci. Soc. Am. J. 65:300–310.

Soreau, Rodolphe. 1902. *Contribution á la théorie et aux applications de la nomographie*. Paris: Ch. Bèranger. Available for download.

Soreau, Rodolphe. 1921. *Nomographie ou Traité des Abaques*, Paris: Chiron.

Werkmeister, P. 1923. *Das Entwerfen von graphischen Rechentafeln (Nomographie)*. Berlin Verlag von Julius Springer.

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

**<<< Go to Part I of this essay**

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As an engineer Lallemand (1857-1928) [1, 2] created a number of ingenious devices to assist in determining altitudes, water and tides in France involving water levels, mercury baths, air bubbles, and other gauge techniques. Slow changes in these measurements led him to theoretical investigations of lunar tides in the Earth’s crust. He also created the modified polyconic form of map projection. Maurice d’Ocagne was his deputy from 1891 to 1901; his indebtedness to Lallemand is evident in his detailed treatment of hexagonal charts and his brief description of *L’Abaque Triomphe* in his masterpiece *Traité de Nomographie* and other works on nomography and its foundations [d’Ocagne 1891/1889/1921, Soreau 1902/1921].

**What is Magnetic Deviation?**

A well-constructed compass on a ship will fail to point to true (geographic) north due of two factors:

Magnetic variation(ormagnetic declination): the angle between magnetic north and true north based on local direction of the Earth’s magnetic field, and

Magnetic deviation: the angle between the ship compass needle and magnetic north due to iron within the ship itself.

**Magnetic variation** has been mapped over most of the world since the year 1700, although it changes over time due to drifting of the magnetic poles of the Earth. The compass correction for magnetic variation can be made based on published magnetic variation tables.

**Magnetic deviation** arises from the magnetic effects of both hard and soft iron in the ship. *Hard iron* possesses permanent magnetism as well as semi-permanent magnetism imprinted by the Earth’s magnetic field under the pounding of the iron during the ship’s construction, or from traveling long distances in the same direction under the influence of this field. Collisions, lightning strikes and time will cause significant changes in this magnetism. External fields such as the Earth’s magnetic field induce magnetism in *soft iron* in the ship on a near real-time basis, an effect that varies with location as the Earth’s magnetic field varies in strength and direction. The combination of magnetic fields from the iron of a particular ship produces a magnetic field that affects the accuracy of compasses onboard that ship, sometimes dramatically. A detailed account of the origins and history of magnetic deviation can be found in another essay of mine.

**The Equations of Magnetic Deviation**

Lallemand’s *L’Abaque Triomphe* is shown below (a high-resolution version is also available). It provides a graphical means (an **abaque**) for calculating the magnetic deviation of the ship *Le Triomphe* for a given compass course and location on Earth using equations developed by Archibald Smith in 1843. The magnetic deviation essay mentioned above provides the background and analysis of these equations (where the mathematical derivation is given by a hyperlink in the online version of the essay and in the Appendix of the PDF version hyperlinked at the end of the webpage).

The magnetic deviation equations use both non-bold and bold variables A, B, C, D and E, as well as measured magnetic parameters of the ship. Here the angle ξ′ is the compass course, or the angle from north indicated by the compass needle, and δ is the magnetic deviation, or the angle correction to be applied to the compass course to counteract the effects of magnetic deviation.

where at a given location of the ship,

and A, D, E, λ, c, P, f and Q are parameters deduced for a particular ship. These formulas assume a magnetic deviation of less than about 20° in order that B and C can be expressed as simple arcsine functions, and so a certain amount of correction for magnetic deviation may be needed in the binnacle holding the compass. Also, the *heeling* of the ship, i.e., the leaning of the ship due to wind as well as transient rolling and pitching of the ship, is not taken into account in these equations.

Now the equations for magnetic deviation are provided along the top of Lallemand’s chart, along with the measured values of the ship magnetic parameters. The coefficients in bold in the equations are represented on the chart in their more traditional German Blackletter (Fraktur) font. Also, the term “ctg θ” on the chart should be understood as “c tan(θ)” and “ftg θ” should be understood as “f tan(θ)”.

You can see that there is a mistake in the printed formulas on the chart—the terms in the inner parentheses in B and C should be divided, not multiplied. With D given as 6°45′, the value of ½ sin D is relatively small (about 0.06), and the error has a quite small effect on the overall result. It is not clear whether these incorrect formulas were used in designing the chart or whether they are due to an error on the part of the letterer or printer. As I will discuss a bit later, I performed quite a few tests of the accuracy of this chart based on a model of the Earth’s magnetic field at that time; from those tests it appears that the chart design itself was based on the incorrect formulas, but the differences in the results are small and the inherent inaccuracies in the chart and model make the distinction difficult.

**Using Lallemand’s Chart**

Directions for the use of the *L’Abaque Triomphe* are provided along the bottom of the chart, and there is even an example in dashed lines worked out on the chart itself.

Let’s follow the dashed line example marked on the chart highlighted in the figure below. The ship *Le Triomphe* is located at latitude 42°N and longitude 20°W and has a compass heading (or compass course) of 41.5° (read clockwise from North).

Step 1: The navigator locates the lat/long point on the map along the left side, moves from this position horizontally to the radial line pointing to the 41.5° course along the top, and marks this point.

Step 2: The same lat/long position is found in the upper map on the right side and followed along the guide lines to the line pointing again to the 41.5° course along the edge, marking that point.

Step 3: A transparent or translucent overlay about the size of the paper and marked with a hexagon as shown in lower right of the figure is aligned square to the page with two of its radial arms crossing the two marked points from Steps 1 and 2. The Appendix of this essay contains a printable hexagonal overlay for use with the charts in this essay.

Step 4: The course correction is read from intersection of the next hexagon arm and the deviation scale (11.8°).

The compass course has this 11.8° deviation easterly from North, so the compass course has to be adjusted to obtain a true course of 41.5°. It surprises me that the correction for magnetic variation is not included in the result, as we will see that it was used in the calculation of the magnetic deviation.

If the compass course were southerly (90° to 270°), step 2 would be performed based on the lower map on the right side rather than the upper map.

**The Accuracy of Lallemand’s Chart**

So how accurate is it? The U.S. Geological Survey has modeled the magnetic variation around the world over the last few centuries. The figures below show the horizontal component and inclination (dip) of the Earth’s magnetic field in 1884, the year prior to the creation of the abaque. One microTesla is equivalent to 1/100 Gauss, so for example the horizontal intensity of the magnetic variation in Paris in 1884 was 19µT or 0.19G.

We can insert values from these figures at different locations on Earth into Lallemand’s equations and compare the result to that obtained graphically from the abaque. It is important to note that the prime meridian (0° longitude) is located at Paris in Lallemand’s chart; the French did not accept Greenwich as the prime meridian until 1911.

Also, there is no indication of the units of the “magnetic force” used in Lallemand’s chart, and any units could be used since the constants would scale any units appropriately; unfortunately, there are no units listed with these constants. Initially I presumed that H would be magnetic flux density in units of Gauss, since Maxwell and Thomson extended the cgs system of units with such electromagnetic units in 1874, but these units do not produce consistent results in the abaque. I later bought a copy of the *Admiralty Manual for the Deviations of the Compass* from 1893, in which Archibald Smith and F.J. Evans lay out the rationale for the equations used in Lallemand’s chart, and discovered that they normalized H to 1.0 at its value at Greenwich. We can assume that Lallemand normalized H to 1.0 at Paris instead (the difference is not large), so a horizontal intensity H from the 1884 USGS figure has to be multiplied by 1.0/0.19 = 5.26 before using that value in Lallemand’s formulas.

The results of my tests for various locations and courses are found below. The first row compares the computed value with the graphical value at the canonical location of 42°N 20°W. The rest of the rows are for different locations and/or different compass courses. The top spreadsheet compares the graphical results with computations based on the formulas listed on the chart, while the second spreadsheet uses the mathematically correct formulas for B and C.

A lower average absolute error over the tested locations and compass courses is found in the top spreadsheet, suggesting that the chart was drawn using the incorrect formulas found on the chart, although the uncertainties in the graphical readings make this less than certain. In any event, the small difference between the two formulas is apparent.

The results are not bad at all given that we are estimating values off a model, certainly much, much better than not correcting for magnetic deviation at all. In addition, once you start taking measurements off the chart, you begin to notice that the abaque is a bit sketchy at places (look closely at the spacing of the vertical longitude lines in the map along the left side) and was most likely a proof-of-principle graphic.

**How Does Lallemand’s Chart Work?**

So how does it all work? Hexagonal charts in general are the subject of the next section of this essay, but at this point we state the conclusion: the hexagon arms point to three scales oriented 120° to each other and the value (offset) of the magnetic deviation scale δ is the sum of the values (offsets) of the other two scales. In the figure below the green lines cut the three scales at their zero points, and since these lines nearly intersect at the same point, then within some small error the hexagon will connect the zero values on the three scales (0+0=0). The values Y_{1}, Y_{2} and Y_{3} are the values (offsets) of the scales for the example of 42°N 20°W. With a ruler you can verify that Y_{3} =Y_{1} + Y_{2} in length except for a small error (~1mm at the scale of the full-page version shown earlier) due to the inaccuracy in the chart as manifested by the inexact intersection of the green lines.

Let’s look at the construction of the three scales. Each represents an offset from an axis, and one of the advantages of this type of chart is that it doesn’t make any difference where along this axis this offset occurs. So in the first scale the offset Y_{1} can occur anywhere along the vertical green x-axis of the scale, and this is true for Y_{2} on the second scale. This allows the scales to be shifted anywhere along the green axes for optimum placement of the scales, and in fact it allows the Deviation (δ) scale to be tucked in the narrow space between the leftmost map and the central cone.

Now the leftmost map is drawn in such a way that the value of B for any latitude and longitude position provides a vertical offset B from the axis passing through the center of the cone. Extending this offset to the right provides a Y_{1} value for the first term in the formula for magnetic deviation:

Y_{1} = B sin ξ′

All of the other terms are combined into an offset generated from the maps on the right side of the chart :

Y_{2} = A + C cos ξ′ + D cos 2ξ′ + E cos 2ξ′

To demonstrate the construction of the twisted cylindrical plot on the right side of the chart, I’ve plotted a graph here that shows Y_{2} as a function of ξ′ for values of

- Latitude = 42°
- Longitude = 20° West (-20°)
- Inclination θ = +70° downward
- Horizontal H = 16 microTesla = .16 Gauss
**B**= (1/.84)[0.106*tan θ + (-.033)/H)] = 0.101- B = arcsin[B[1 + (sin(6.75°) / 2)]] = 6.15°
**C**= (1/.84)[-.013*tan θ + (-.020)/H)] = 0.106- C = arcsin[C[1 – (sin(6.75°) / 2)]] = 5.73°

Rotating this plot counterclockwise by 30° yields a y-axis that is 60° clockwise from the vertical axis of the chart, lying along the next arm of the hexagon. Note that the angles shown in this plot vary from 0° to 90° and 270° to 360°, which corresponds to compass courses ξ′ in northern half of the compass rose. The range ξ′ = 90° to 270° cross in the opposite direction, which is why there is a separate map used for compass courses ξ′ in the southern half of the compass rose. The offsets for the various latitude and longitude locations, using H and θ for the local magnetic variation, provides the curved lines on the lower and upper maps. This is where the enormous manual effort by Lallemand to create this chart is most apparent.

In the end the third arm of the hexagon overlay provides the sum of these two offsets, or

Y_{3} = Y_{1} + Y_{2} = A + B sin ξ′ + C cos ξ′ + D cos 2ξ′ + E cos 2ξ′

which is the required equation for magnetic deviation.

Lallemands *L’Abaque Triomphe* is a uniquely interesting graphic because of the sophistication inherent in this first published hexagonal chart. No other chart of this type exists to my knowledge, although the use of hexagonal charts continued for some time until nomograms finally displaced them for good. The principles and history of hexagonal charts and their relatives, triangular coordinate systems, are the subject of Part II of this essay.

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**>>> Go to Part II of this essay**

]]>

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.

A blog article is quite a different animal from a journal article, but fortunately the *On Jargon* feature of the journal is intended for expository articles. In writing it I expanded the topics of the original essay while abbreviating or eliminating other areas. For example, grid nomograms are now included in the article along with a method for creating the initial determinant of an equation without guesswork. In addition, 21 new nomograms were created for the article using PyNomo software—some will look familiar from my PyNomo essay and calendar. But the original blog essay proceeds at a more relaxed pace, and it includes more of my thoughts on today’s status of nomograms at the end. For this reason I certainly recommend my original essay as well, particularly as a first introduction to nomography.

So if you’d like to read the article, here’s the link:

The journal has a small 6″x9″ page format, so you may find the text to be a bit large when it’s printed. On the other hand, the figures are vector-drawn and will be shown larger than in the journal (in fact, you can zoom and print them on large paper for use). You can select a smaller paper size when printing if you like the smaller format.

I’d like to thank another friend and nomography advocate, Leif Roschier, for writing the PyNomo software I used to create the nomograms. I can’t imagine drawing them in any other way.

=========

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Copyright ©2010 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP. Request permissions from COMAP, Inc., 175 Middlesex Turnpike, Suite 3B, Bedford, MA 01730, USA, 1–800–772–6627 = (800) 77–COMAP or (617) 862–7878; (617) 863–1202 (fax); or info@comap.com.

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]]>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.

Small images of the front and back and each page of the calendar are shown at the bottom of this page. These provide only a rough idea of the content, but if you think you might be interested, the complete PDF file is just over 5MB, so it is easily downloaded and viewed at full resolution (for viewing you will want to download one of the first two PDF files listed below in which all pages but the back cover are right-side up).

**Flip-Up Landscape Printing with Binding**: The following two PDF files are designed for two-sided printing in landscape mode on 8-1/2″ x 11″ paper by default (choose the “flip up” mode of two-sided printing). Some sort of binding is required, as in the spiral binding shown in the photograph at the top. Note that the last page in the PDF file is upside down in order that the back of the calendar has the same orientation as the front cover.

The PDF of the calendar with major U.S. holidays can be downloaded

here.The PDF of the calendar with no holidays can be downloaded

here.

**Double Portrait Printing with Stapling**: The following two PDF files are designed for two-sided printing in portrait mode with two pages per side on 11″ x 17″ paper by default (do **not **choose the “flip up” mode of two-sided printing). Note that the pages are in a strange order and are often upside down in the PDF file to provide the correct order and orientation when the packet of papers is folded and stapled in the middle.

The PDF of the stapled calendar with major U.S. holidays can be downloaded

here.The PDF of the stapled calendar with no holidays can be downloaded

here.

In all cases the moon phases (shown as empty or filled circles) are referenced to U.S. Eastern Standard Time, or GMT-5, so the date could be off by a day depending on where you live.

Have a happy 2010!

Ron

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

]]>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.

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

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.

**Computing the Magnetic Deviation**

There were a few graphical methods of looking up the magnetic deviation on a ship at sea. A common type of chart, or deviation card, is shown in the figure on the right. Here the sailor would follow his compass heading on the inner circle and arrive at a course on the outer circle that is corrected for the magnetic deviation of the ship at the location for which the card was constructed.

A more typical chart invented in 1851 is called a **Napier’s diagram** after James Napier (1821-1879). The simple nature of this chart belies its advantages in obtaining a fast and reliable correction. To create it, vertical linear scales of compass courses are drawn (two for good resolution), along with with dotted and solid lines at 60° angles from the scales as shown in the figure (originally these were at 90° and 45°).

Then the magnetic deviation is plotted opposite the scale, left (for west) or right (for east) such that the deviation for a given course is located on the solid line running 60° from the scale value. The semicircular and quadrantal components are also individually plotted as dashed curves in the ones I have seen. To read the correction, you would find the compass heading on the scale, proceed up the dotted line to the solid curve, then back to the scale along the solid line (or parallel to them for intermediate scale values). Then since the triangles are all equilateral, you’ve directly found the corrected compass heading without doing any arithmetic. Simple in design, robustly effective in use!

Napier’s diagrams could be directly plotted from data obtained as a ship was swung, and much was made of the then-new method of least squares in drawing the curves. It can be seen here that the solid curve is the sum of the plotted semicircular curve, the quadrantal curve, and the constant deviation A when significant. Only the semicircular deviation and the new total need to be re-plotted for other locations.

As ship design evolved from simple iron plating to iron hulls with larger engines, the inaccuracy of the inexact coefficients A, B, C, D and E became noticeable. On the other hand, the equation expressing the magnetic deviation in terms of the exact coefficients **A**, **B**, **C**, **D** and **E** was difficult to compute, despite a set of mathematical tables and rules specifically prepared for this purpose by Smith. In an attempt to deal with this, Smith invented the geometric constructions he called **dygograms** to provide a graphical calculation of the magnetic deviation for any compass course using the exact rather than inexact coefficients.

**Dygograms**

The *Admiralty Manual for the Deviations of the Compass*, by Smith and Evans,* *describes in detail two main types of dygograms, called Dygogram I and Dygogram II. We will briefly investigate each of these.

To create a dygogram for a given ship, it is assumed that the exact coefficients **A**, **B**, **C**, **D** and **E** are known. However, Smith does derive inverse series for extracting these exact coefficients from the inexact coefficients A, B, C, D and E that can be obtained by harmonic analysis from measurements taken as the ship is swung.

**Dygogram I**

Equation (2) in Part I of this essay expressed the magnetic deviation δ in terms of the inexact coefficients as reproduced below:

We see that the magnetic deviation consists of a constant value, semicircular terms with a period of 360°, and quadrantal terms with a period of 180°, each constant in magnitude. So as the ship swings completely around, the semicircular contribution make one revolution while the quadrantal contribution makes two revolutions. For the Dygogram I type, Smith modeled the net deviation as one point revolving on a circle at a certain angular speed, with a second point revolving on an outer circle centered on the first point but at *half* that speed. With correct scaling, the inner circle represents the contribution of the quandrantal terms while the outer circle represents the contribution of the semicircular components. An offset of A between the inner circle’s center and the origin completes the model.

This is an epicyclic motion similar to the Ptolemaic epicycles that modeled positions of the outer planets, except that Ptolomy had the outer circles rotating faster than the inner ones. This can also be modeled as a point on a circle as the circle rolls around the circumference of another circle, where here the effective radius of the inner circle is increased by the radius of the outer circle. This is akin to a penny rolling around another penny, in which Abraham Lincoln rotates twice for every revolution of the outer coin. Smith originally assigned the faster quadrantal rotation to the outer circle, and to me this is the instinctive way to do it, but by the 3rd edition of the Admiralty Manual he reversed these to take advantage of simplifications in construction proposed by Lieut. Colongue of the Russian Imperial Navy. It’s also a fact that the quadrantal force does not depend on the ship’s location, so he could affix this as the inner circle and then redraw only the outer curve for various locales.

The overall curve that results from a point rotating on a circle that is revolving around another point rotating on a circle is called The Limaçon of Pascal after its discoverer Étienne Pascal, father of Blase Pascal. It was named (from the Latin limax for snail) by Gilles-Personne Roberval in 1650 in his use of it to draw tangents as a means of differentiation. It has the general parametric form

x = b cos θ + a cos 2θ

y = b sin θ + a sin 2θ

or in polar coordinates,

r = b + a cos θ

For a particular ship and location, the limaçon can be a circle if a = 0, it can approach but not touch the inner circle (as a dimpled circle) if a < b < 2a, it can just touch the inner circle (as a cusped curve, or a cardioid) if a = b, or it can loop within the inner circle and exit it if a > b. It’s a popular curve. As Smith describes it for p = b and q = a/2,

It is at once a conchoid of the circle, an epitrochoid, and a Cartesian oval. When p = q it gives, as was shown by Pascal, a solution of the problem of the trisection of the circle; when p = 2q it is a caustic by reflexion of the circle.

A Type I dygogram in high resolution from Lyons’ book can be seen by clicking **here**. We will be using miniature versions of it as we go. The dygogram calculates the magnetic deviation δ from the following equation in terms of the exact coefficients and the magnetic course ξ:

Now when the magnetic course is due north, we have ξ = 0° and the formula reduces to:

Referring to the dygogram components in red in the figure to the right, we place a point *O* at the bottom of the page and at a convenient distance above it we place a point *P*. This distance is defined as the length of 1 in the dygogram and represents the mean directive force to north λH. Then using this as a unit length we move right a distance **A** to plot the point *A*, right again a distance **E** to plot the point *E*, up a distance **D** to plot the point *D*, up again a distance **B** to plot the point *B*, and right a distance **C** to plot the point *C*. From the figure you can see that Equation (4) for tan δ with ξ = 0° holds for this diagram, and therefore if magnetic north lies along the vertical line from *O*, that the angle from it to *C* (which is marked NORTH here) is the magnetic deviation when the ship is pointing north. So we mark this point *C* as *N* for the north magnetic course ξ = 0° and for clarity draw a ship pointing north at this point.

Now consider the blue markings we’ve added to our dygogram figure below on the left. For a magnetic course of due south, ξ = 180°, and from Equation (3) the only effect is that **B** and **C**, the distances that define the outer curve, change sign. So we extend (the math term is *produce*) *ND* a distance equal to *ND* in an opposite direction from point *D* and we get the appropriate point *S* (SOUTH) on the curve that will provide at *O* the magnetic deviation for a magnetic course of due south, that is, the angle between the vertical line *OX* and the line *OS*.

Now from the order in which we have placed these coefficients horizontally and vertically on the diagram you can see that the radius of the inner circle *AD* equals the magnitude of the sum of the quadrantal terms, or (**D**^{2} + **E**^{2})^{1/2}. The distance *DC* equals the magnitude of the sum of the quadrantal terms, or (**B**^{2} + **C**^{2})^{1/2}. The distance **A** simply produces a constant offset angle.

So let the magnetic course ξ proceed clockwise around from the north point *N*. If we rotate the constant radius *AD* by 2ξ about the point *A* while we rotate the constant distance *DC* by ξ around *D*, then the horizontal distance from the vertical line *OX* of the point on the curve will equal **A** + **B** sin ξ + **C** cos ξ + **D** sin 2ξ + **E** cos 2ξ while the vertical distance above O will equal 1 + **B** cos ξ + **C** sin ξ + **D** cos 2ξ + **E** sin 2ξ. Therefore, the tangent of the angle at *O* will be the ratio of these, and by Equation (3) the angle at *O* between the vertical line and the line to the point on the curve is the magnetic deviation δ for that magnetic course.

A limaçon is the curve that results from these required relationships. We mark a point *Q* where the north-south line *NS* intersects the inner circle. As the angle ξ rotates about *Q*, the segment *DC* will rotate at the same rate but the angle *DAE* will rotate at twice the angle. The point *Q* is called the pole of the dygogram and ξ is measured clockwise about it from *QN*. You can see on the figure that east-northeast is marked at a position 67.5° clockwise from *N*.

We construct the outer curve by laying a straightedge along *NS* and marking the positions of *N*, *D* and *S* on it, where *D* will be the midpoint of *NS*. We always need *DC*=*DN* to be a constant length from *D*, so we place the straightedge at various locations with *D* lying on the inner circle and passing through *Q*. At each location we mark points on the paper corresponding to *N* and *S* on the straightedge and we connect them to construct the curve. Then with a protractor centered at *Q* we mark various cardinal and intercardinal points of the compass on the outer curve and the dygogram is complete. The outer curve can indeed loop within the inner circle in some cases, as seen in the dygogram at the top of this essay which is also reproduced on the right. In practice most of the construction lines are removed in the final dygogram. From this graphical calculator we can now easily find the magnetic deviation for any magnetic course of the ship.

But again we would like to have the deviation in terms of the compass course ξ′ that we are reading on the ship. We know that δ = ξ – ξ′, so we can calculate from the dygogram the deviation for a given magnetic course and find the compass course as ξ — δ, so we have the compass course for that magnetic course, but finding the deviation from a known compass course would be a trial and error process. Smith recommends laying out a Napier’s diagram for all the deviations, plotted offset along the solid lines from the compass course scale, and then using that diagram for any desired compass course. Smith also provides a couple of constructions to approximate the magnetic course for a desired compass course. But the best construction he describes is again due to Lieut. Colongue.

To demonstrate how we can find the magnetic deviation from the compass course, let’s first work backwards from the sample dygogram we’ve been using. In the figure on the right, for a north magnetic course of ξ = 0° we can read a magnetic deviation δ = 18° on the protractor (the angle between the vertical line *OX* and the north position of the ship in the upper right). Since δ = ξ – ξ′ we find that ξ′ = -18°. Now what we are going to do is to start with a compass course ξ′ = -18° and see if we end up calculating the same deviation δ = 18°.

The method is to first draw the (red) line from *Q* at the compass course angle ξ′ from the (green) line *QN* and mark the point where it intersects the vertical line *OX*. Then with dividers we draw the arc of a circle that passes through three points: *O*, *Q* and this intersection point. It will intersect the dygogram curve at the magnetic course ξ, and we proceed as before to find the deviation by the angle between *OX* and a line from *O* to that point. As you can see, it works out perfectly here, as that intersection corresponds to the north magnetic course that gave us δ = 18°. Using dividers to find the arc is analogous to trial and error, I suppose, but it’s a lot easier than doing math by trial and error, and this is one big advantage of graphical calculators in general.

Again, let’s verify it for the *ENE* location marked on the dygogram on the left. The magnetic course for this intercardinal point is 67.5° around *Q* from the north point *N*. We read from the protractor at the bottom that the deviation for this magnetic course is 36°, so we start with ξ′ = 67.5° — 36° = 31.5°. We draw a (red) line from *Q* at this angle from the (green) line *QN* and mark its intersection with *OX*. We draw the arc connecting *O*, *Q* and this point, and it indeed intersects the dygogram curve at the *ENE* point where ξ = 67.5°, from which we can read the deviation. So these two examples demonstrate that we can find the deviation for any compass course on the dygogram with a few extra steps.

Once the dygogram is constructed for one location, Smith provides simple procedures for plotting the dygogram at any other location from as few as two measurements of deviation vs. magnetic course at that place. In the process the values of the location-specific coefficients **B** and **C** are also found. Smith suggests a mechanical way of tracing the curve with a roller revolving about a roller in the same way as a coin revolves around the coin, but there are other ways to custom-draw a limaçon; I came across this mechanism in a book on linkages.

**Dygogram II**

Smith created an intermediate type of dygogram that consisted of an ellipse and a circle in lieu of the limaçon, but we are going to proceed to his ultimate form consisting of just two circles. I have only seen this discussed in the *Admiralty Manual* and in Smith’s obituary by Thomson, and only Thomson describes the case where **A** and **E** are non-zero. It is a clever transformation.

Smith relates that it occurred to him to turn the paper with the same velocity that the ship turns, i.e., at the half-speed rotation of the outer generating circle, while tracing the limaçon in a same direction. Then the fixed point *Q* traces an inner circle, and the limaçon also ends up tracing another, outer circle. Here’s my guess as to how he might have come to that. Let’s zoom in on the innermost orbits in the figure of the Ptolemaic system we saw earlier. Here we see, from an Earth-centered point of view, the Sun revolving in a circular orbit around the Earth and Mars circling in what appears to be a perfect dygogram! And it very nearly is—it certainly is an epicycloid. If Mars were to have an orbital period of 2 Earth-years instead of 1.88 Earth-years it would be a dygogram, because the Sun point is modeled as revolving about the Earth with a period of 1 year and to match observations Mars must be modeled as revolving about the Sun point with a period of 2 years. Now as we know, Copernicus demonstrated that by changing the reference frame to a Sun-centered system (by fixing the paper on the Sun as the orbits trace) we find that the orbits of Earth and Mars end up as simple nested circles. We have an analogous situation here.

To construct this type of dygogram, a circle of radius 1 of some unit of length is drawn with a center *O*. Then using this unit length we move up from *O* a distance **B** and right a distance **C** and mark this point *o*. Then we move up a distance **D** and right a distance **E**. We draw a circle with its center at *o* that passes through this last point. This small circle is marked *n*, *e*, *s* and *w* and degrees are marked on it in a clockwise direction. The large circle is marked *S*, *E*, *N* and *W* but the position of these is rotated by **A**. Degrees are marked on it counterclockwise from north. Again Smith provides methods of plotting such a dygogram for different locations from a few observations. The finished dygogram appears as in the example below on the right.

Once the dygogram is created, Smith provides the following succinct procedure for reading the magnetic deviation for a given magnetic course:

Let ξ be the given magnetic course. Take

R, a point on the circumference of the large circle, andron the circumference of the small circle, such thatNOR=nor= ξ, and joinOR,Rr. ThenORris the deviation which is +, or easterly, ifris to the right ofOlooking fromRtoO. If the large circle is graduated we may measure the angleORrby producingRO,Rr, to intersect the circle inJandj. The arcJjwill then be twice the required angle.

But, again, on the ship we only know the compass course ξ′ we are reading from our compass. To use this we can place a straightedge that intersects the center *O* of the large circle and the value of ξ′ on the outer circle. Then we move it parallel to itself (there are linkages for parallel rulers) until it intersects the two circles at the same marked angle. These are the points *R* and *r* with values equal to the magnetic course ξ for this compass course, and we proceed from here as before to find the magnetic deviation. Again we see the power of a graphical calculator to naturally close in on a solution that would otherwise be a tedious trial-and-error arithmetic calculation.

Smith describes drawing a large set of lines between points on the two circles that have corresponding marked angles to provide a convenient, overall visual layout. Also, if the large circle is considered to be an upside down compass card, we can glue the (upside-down) small circle onto the compass card itself. In those days the needle was attached to the compass card, so the card turns with the needle and the compass course is the reading of the card in the forward direction the ship is facing. When the ship is at sea, we can find which drawn line between the two circles is fore-and-aft (which will be parallel to the compass course shown on the fore edge of the rotated card), and this will cut the two circles in points corresponding to the magnetic course. Or better yet, we can steer a magnetic course by turning the ship until the line connecting the desired magnetic course values on the two circles is fore-and-aft. Smith termed this a **steering dygogram**. It seems absolutely brilliant to me, but the lack of ready literature on it suggests it was never really taken up.

These graphical calculators—the Napier’s diagram and the various incarnations of the dygogram—are convenient devices for obtaining the magnetic deviation of a particular ship at a particular place. But a ship at sea would have to carry a set of charts like these for various locales, one more variable that could lead to disastrous errors. However, in 1885 a French engineer named Charles Lallemand created a uniquely designed and somewhat famous graphic, a hexagonal chart of his own invention, to calculate the magnetic deviation of the ship *Le Triomphe* no matter where it was located. The design and workings of this chart is the subject of another essay of mine, *Lallemand’s L’Abaque Triomphe, Hexagonal Charts, and Triangular Coordinate Systems* (under construction).

**A Brief “Dygression”**

In an article in 1907 A.G. Greenhill used a dygogram to model the reaction force on the axle of a pendulum. And just now as I write the final paragraphs of this essay, I look at the steering dygogram and it reminds me of sun compasses used as very early navigational aids. And that reminds me of portable sundials, which sparked my interest in nomography and graphical calculators in the first place. And *that* reminds me that the Equation of Time correction (the difference between the mean clock time and the actual solar time that must be accounted for in any good sundial) is composed of a sinusoidal term with a period of a year and a sinusoidal term with half that period!

Why this didn’t occur to me before is a mystery. During the few months I’ve been researching this essay, including reading through all the references and buying the only copy of Smith’s *Admiralty Manual* I could find in this hemisphere, I’ve also been plotting figure-8 analemma curves to represent the Equation of Time (EOT) correction on a sundial I’m designing for my house.

The non-circularity (or eccentricity) of the Earth’s orbit is one component of the EOT. The other component is the angle between the equator and the plane of the orbit (or the ecliptic tilt). Whitman provides a relatively simple but accurate formula for the EOT for a given day:

where Δt is the correction in minutes and (t — t_{0}) is the number of mean (clock time) solar days since the Earth’s perihelion (or closest approach to the sun).

The perihelion date varies with the year depending mostly on leap year differences, and it ranges from Jan. 3 to Jan. 5, so we can let t_{0} = 4. The EOT is typically averaged over the four-year leap day cycle anyway. Then let α = 2π/365.24 and we have

We can twice apply the identity

and find that

which is in the form of the magnetic deviation equation for the inexact coefficients A = 0, B = -7.637, C = 0.526, D = -9.134 and E = -2.920.

So a dygogram can model the EOT if the 365.24 days of the year are spaced out equally through the 360 degrees (about *Q* in Dygogram I and along the outer circle in Dygogram II). Now in an equatorial sundial the hour lines are also equally spaced around the circumference of a circle. So it might be possible to adapt the steering dygogram to provide the EOT correction right on the sundial. I will have to think about that.

**An Exquisite Endeavor**

The centuries it took to untangle the mysteries of magnetic deviation represent an enormous, sustained effort by scientists, mathematicians, ship captains and crews. Many people provided the mathematical and scientific tools and data needed to analyze the problem, and in fact this essay has focused only on contributors in the West. Captain Flinders, Edmund Halley, George Airy, William Scoresby, Archibald Smith and F. J. Evans sparked major advances in this area before gyrocompasses, ring gyros, and now LORAN and GPS diminished the importance of the compass. But it was so important then—a friend of his remarked that Smith was “penetrated by the conviction of the usefulness of his work.” Through his *ténacité passionnée* Smith produced a mathematical framework that defined magnetic deviation in a new and practical way, an achievement so beneficial that by the time he died, as Thomson relates,

From every ship in Her Majesty’s Navy, in whatever part of the world, a table of observed deviations of the compass, at least once a year is sent to the Admiralty, and is therefore subjected to [Smith’s] harmonic analysis.

From my point of view, the heroic efforts made by these men to overcome the deadly consequences of magnetic deviation comprise a very heartening thread of history and an inspiring illustration of the role of the mathematical sciences in advancing our civilization.

**REFERENCES**

The sources linked to Google Books are all fully viewable in the U.S., but not necessarily in other countries. If you find you are unable to read particular pages that you are interested in, please contact me and I will try to provide an excerpt for personal research under Fair Use provisions.

**A Practical Manual of the Compass, A Short Treatise on the Errors of the Magnetic Compass, with the Methods Employed in the U.S. Navy for Compensating the Deviations and a Description of Service Instruments, Including the Gyro-Compass**, The United States Naval Institute (1921), pp. 1-114. A collection of papers on the subject of magnetic deviation, measurement and corrections. It can be found here .

Barber, G.W, and Arrott, A.S. *History and Magnetics of Compass Adjusting*, IEEE Transactions on Magnetics, Vol. 24, No. 6, November, 1988. A refreshingly short and highly readable history of magnetic deviation.

Evans, F.J. **Elementary Manual for the Deviations of the Compass in Iron Ships, 3rd Edition**. London: J. D Potter (1875). As the title page says, this is “arranged in a series of questions and answers, intended for the use of Seamen, Adjusters of Compasses, and Navigation Schools, and as an introduction and companion to the Admiralty Manual for the Deviations of the Compass.” Worth reading to understand the basis of the deviation formulas, but oddly enough given his co-authorship of the Admiralty Manual, Evans does not treat dygograms at all. It can be found here.

Gibson, Lt. John. *The Dygogram; Its Construction, Description and Use* in **Proceedings of the United States Naval Institute**. United States Naval Institute (1984) pp. 555-571. A mathematically detailed article on the Type I dygogram, including a unique analysis of its validity by treating each magnetic component in isolation. It can be found here.

Grattan-Guinness, Ivor. **The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel**. Princeton University Press (2000). This is the source for the assertion that the first known appearance of the phrase *harmonic analysis* is in Thomson’s obituary of Archibald Smith, as cited by Jeff Miller at his website on *Earliest Known Uses of Some of the Words of Mathematics* (see *harmonic analysis* here). Thomson does say here that it is “commonly called” harmonic analysis, however.

Gray, Andrew. **A Treatise on Magnetism and Electricity, Vol. I-II. Maps, Tables, Diagrams**. New York: Macmillan (1898), Chap. IV, pp 85-100. Provides a derivation of Smith’s formulas for magnetic deviation, including the heeling error, and the construction of a Type I dygogram. It can be found here.

Greenhill, A.G. *The Dygogram of Axle Reaction of a Pendulum*, in **Proceedings of the Edinburgh Mathematical Society, Vol. XXVI, 1907-1908**, pp. 21-29 in second half of the volume. An interesting use of the dygogram to model other dynamic systems. Unfortunately, Figures 5 and 6 referenced in the article, which appear at the end of the volume, are only half-shown due to poor attention paid during scanning. It can be found here.

Gurney, Alan. **Compass: A Story of Exploration and Innovation**. New York: Norton (2005). A very entertaining, popular and surprisingly informative read on the history of the compass, including the effects and corrections due to magnetic deviation. Highly recommended!

**Handbook of Magnetic Compass Adjustment**. National Geospatial-Intelligence Agency (2004). A succinct and very clear summary of magnetic deviation, its effects on compasses, and sequenced steps to adjust compensators. It excludes the mathematical details of the derivations of the formulas and all graphical constructions.

Love, J.J. *Review of Earth’s Magnetism in the Age of Sail (A.R.T. Jonkers, 2003).* (2004). An interesting review and analysis of the history presented in the book (which I have not read). It can be found here.

Lyons, Timothy A. **A Treatise on Electromagnetic Phenomena, and on the Compass and Its Deviations Aboard Ship: Mathematical, Theoretical, and Practical, Vol. II**. J. Wiley & Sons (1903). An incredibly thorough account of the whole subject of compasses and magnetic deviation, including Type I dygograms and various means of correction. It can be found here.

Maor, Eli. **Trigonometric Delights**. Princeton University Press (1998). Chapter 7 of this book deals with epicycloids and hypocycloids. This is the source of the figure here of Ptolemy’s planetary scheme. The full book is available online here.

Muir, William C. P., **A Treatise on Navigation and Nautical Astronomy: Including the Theory of Compass Deviations, Prepared for Use as a Text-book at the U. S. Naval Academy**. United States Naval Institute (1906), pp. 55-203. A very good, comprehensive overview of the derivation of the formulas for magnetic deviation and Type I dygograms. It can be found here.

Smith, Archibald. *Mr. Archibald Smith’s Introduction to Dr. Scoresby’s Journal of the Royal Charter*, in **The Magnetism of Ships and the Deviations of the Compass, Comprising the Three Reports of the Liverpool Compass Commission**, Navy Department, Bureau of Navigation in Washington (1869). This volume if chock-full of detailed reports on magnetic deviation, but perhaps the one of most interest is the one on pp. 287-316 by Archibald Smith, in which he briefly describes the derivation of his formulas and refers to the long dispute between Airy and the then-deceased Scoresby over compass correction techniques, where Airy’s correction in fact did not work south of the (magnetic) equator. Smith attempts to be diplomatic, assigning most of the disagreement to a misunderstanding but allowing that “This mode of correction Dr. Scoresby, in common with many or most of those who examined the question, among whom I may rank myself, considered to be not only erroneous in principle but dangerous in practice.” I believe this is Smith’s introduction to Scoresby’s book that caused an immediate printed response by Airy that I read about. It can be found here.

Smith, Archibald. *On the Mathematical Formulae Employed in the Computation, Reduction, and Discussion of the Deviations of the Compass, with Some Practical Deductions Therefrom as to the Mode of Construction of Iron-Built Vessels*, in **Transactions of the Institution of Naval Architects**. Royal Institution of Naval Architects (1862), pp. 65-73. A relatively brief overview of the subject by Archibald Smith. It can be found here .

Thomson, William. *Obituary Notices of Fellows Deceased* in **Proceedings of the Royal Society of London, Vol. XXII, from December 1, 1873 to June 18, 1874**, pp. i-xxiv. A very nice, oft-referenced summary of the accomplishments of Archibald Smith in regard to magnetic deviation, including his derivations and dygogram construction, by his friend William Thomson (later Lord Kelvin). It can be found here. Note that this obituary is located nearly halfway through this two-volume book, not at the beginning as you would expect from the page numbers.

Whitman, Alan M. *A Simple Expression for the Equation of Time*. A relatively simple but accurate formula for a the Equation of Time is provided here, expressible as the sum of sinusoidal terms with the period of a year and the period of half a year in the same relationship as magnetic deviation. It can be found here.

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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.

**The Sources of Compass Error**

Assuming they are constructed well, compasses on ships fail to point to true (geographic) north due to two factors:

**Magnetic variation**(or magnetic declination), the angle between magnetic north and true north due to the local direction of the Earth’s magnetic field, and**Magnetic deviation**, the angle between the compass needle and magnetic north due to the presence of iron within the ship itself.

The algebraic sum of the magnetic variation and the magnetic deviation is known as the compass error. It is a very important thing to know.

**Magnetic Variation**

Magnetic variation has been known from voyages since the early 1400s at least. Certainly Columbus was distressed as he crossed the Atlantic to find that magnetic north and true north (from celestial sightings) drifted significantly, and in fact by 1542 it was known that an **agonic line**, where no difference between the two exists, runs through the Atlantic.

In 1581 Robert Norman published his conclusion, based on his records of magnetic variation, that the “point of respect” of a compass lies within the Earth rather than, say, a mountain of lodestone in the North as many supposed. This led to Dr. William Gilbert’s construction of spherical lodestones to model the Earth and his proposal that it acts as a giant dipole magnet.

We now know that the locations of the Earth’s magnetic poles are not coincident with the geographic poles—not even close, really—and they are always wandering around. Even then, the Earth’s magnetic field is not a simple dipole, and geological masses can also affect the local magnetic field. Henry Gellibrand discovered in 1635 that there are also *secular *variations that change in time: slower ones due to changes in the Earth’s magnetic field, and more sudden and temporary ones due to sunspot activity and magnetic storms in the ionosphere.

These magnetic variations are important, particularly on long ocean voyages. The mapping of these values led at one time to proposals to use on-board measurements of magnetic variation to determine the longitude of the ship; with sightings taken for the latitude this would provide a ship’s location anywhere in the world. And knowing your location at sea was paramount. When the famous Longitude Prize was announced in England in 1714 (triggered by the loss of 2000 sailors and four ships of the Royal Navy off the Scilly Islands in 1707), the three main contenders for it were measurements of lunar parallax, clock time, and magnetic variation.

The first ocean voyages dedicated specifically to scientific research were those of Edmund Halley. In 1698 he commanded the ship *Paramour*, traveling the South Atlantic measuring the latitude, magnetic declination and longitude when possible. (The longitude was found by observing eclipses of the moons of Jupiter to retroactively determine universal time when compared against these events as recorded in England.) Based on this and a subsequent voyage in 1699, Halley published in 1701 the world map below displaying the known magnetic variations (in *A new and correct sea chart of the whole world showing the variations of the compass as they were found in the year M.D.CC.*, by Edmund Halley, found here), information that was later ignored to his peril by Admiral Shovell, the commander of the fleet in the 1707 disaster. The bold line in this map that emerges from the southeastern U.S. and veers southward across the two halves and past West Africa is labeled the The *Line of No Variation*, the agonic line of his time. His “Curve Lines” of equal magnetic variation (today called **isogones**) were first used by Christovao Bruno in the 1620s, but this is the earliest surviving example of such a map. It doesn’t seem much of a stretch to me to imagine that this might have influenced Faraday in his concept of magnetic field lines 150 years later. Captain James Cook used a copy of Halley’s map on his voyages around the world.

As Halley noted, these maps must be regularly updated, and ships studiously logged the differences between their compass heading and true north obtained from astronomical sightings. A map of the world’s magnetic variation in 1800 is shown to the right (from here using US Geological Survey models). You can see that the bold agonic line had moved in that span of 100 years. The variation off the coast of England, as another example, increased to -30° from less than -10°. Ultimately Halley modeled the Earth’s magnetic field as four magnetic poles, two in the crust and two in an interior rotating ball, none with symmetry. It’s a shame that Halley only infrequently measured the dip, or inclination, of the Earth’s magnetic field, thinking it not nearly as important as the horizontal force.

**Magnetic Deviation**

There is an additional effect on the compass needle that took much longer to appreciate and even longer to understand. This *magnetic deviation* is due to the iron in a ship, and even the small amount of iron in wooden ships had an impact, although it was often masked by shoddy compass construction. The first notice in print of this effect was by Joao de Castro of Portugal in 1538, in which he identified “the proximity of artillery pieces, anchors and other iron” as the source. As better compass designs appeared, a difference in compass readings with their placement on the same ship became more apparent. Captains John Smith and James Cook warned about iron nails in the compass box or iron in steerage, and on Cook’s second circumnavigation William Wales found that changes in the ship’s course changed their measurements of magnetic variation by as much as 7°.

Captain Matthew Flinders (1774-1815) spent years in the very early 1800s on voyages to investigate these effects, discovering that in addition to the horizontal magnetic field of the Earth, the inclination (or dip) of the field contributes to the magnetic deviation as well, or in other words, that both the vertical and horizontal components of the Earth’s magnetic field affect a compass. He eventually discovered that an iron bar placed vertically near the compass helped overcome the magnetic deviation, and this **Flinder’s bar** is still used today in ships’ binnacles. Dr. William Scoresby later isolated the *soft iron* in the ship as being magnetized by the Earth’s magnetic field and thereby affecting the compass.

The effects became much more pronounced after 1822 when construction of ships with iron hulls and steam engines commenced. Here it was discovered that *hard iron*, which becomes magnetic when pounded as during the construction of a ship, turns the ship into a permanent, multi-pole magnet. And this magnet changes slowly under the pounding of waves or vibrations from engines, or suddenly through collisions. George Airy, Royal Astronomer of Greenwich, initiated the procedure of swinging a ship to measure its deviation, and then to correct it with permanent magnets and chains of soft iron in the binnacles, but his corrective actions did not work worldwide, particularly south of the magnetic equator where the dip reversed sign or under the changing conditions mentioned above. Public disputes occurred between Airy and Scoresby over magnetic compensation methods for ships [see Smith, 1869], with a third front opening from those such as F. J. Evans who preferred to simply subtract tabulated values of magnetic deviation at ship locations.

**The Mathematical Description of Magnetic Deviation**

Beginning in 1843, Archibald Smith (1813-1872), a warm man “behind a reserve which is perhaps incident to engrossing thought” [Thomson], derived and developed his set of equations for magnetic deviation. They are expressed in terms of the ship’s magnetic or compass course, the horizontal component and dip of the Earth’s magnetic field at a given location, and magnetic parameters unique to a given ship. For narrative flow the essence of the derivation can be found in a separate document **here** (it is the Appendix in the PDF version of this essay). The two principal results are

and

where the ship’s head is pointing at an angle ξ from magnetic north at a given location (the *magnetic course*), and at an angle ξ′ as shown on the compass (the *compass course*). The magnetic deviation δ of the compass needle from magnetic north due to the ship is then ξ – ξ’. The first equation above provides the deviation using exact coefficients **A**, **B**, **C**, **D** and **E** but is expressed in terms of the non-observable magnetic course, while the second equation is expressed in terms of the observable compass course but uses inexact coefficients A, B, C, D and E. Therefore, the second (inexact) equation is most useful on-ship, while the first (exact) one is more useful in characterizing the ship. The relationships of the terms in the second equation to those in the first are:

where at a given location of the ship,

and A, D, E, λ, c, P, f* *and Q* *are parameters deduced for a particular ship.

This set of equations takes into account the magnetic effects of both hard and soft iron in the ship. Hard iron possesses permanent magnetism, while external magnetic fields such as the Earth’s induce magnetism in soft iron. In essence magnetic deviation has the following components:

- A constant term A due to any misalignment of compass north to the ship’s keel line and to asymmetrical arrangements of soft iron horizontally in the ship.
**Semicircular**forces (those with a period equal to 360°):- Fore and aft components of the permanent magnetic field due to hard iron and the induced magnetism in asymmetrical vertical iron forward or aft of the compass–the B sin(ξ′)term.
- Athwartship component of the permanent magnetic field due to hard iron and the induced magnetism in asymmetrical vertical iron port or starboard of the compass–the C cos(ξ′) term.

**Quadrantal**forces (those with a period equal to 180°):- Induced magnetism in symmetrical arrangements of horizontal soft iron–the D sin(2ξ′) term.
- Induced magnetism in asymmetrical arrangements of horizontal soft iron–the E cos(2ξ′) term.

The terms P/H in **B** and Q/H in **C*** *are due to the permanent magnetism of the ship. The magnetic forces P and Q are not dependent on H, but the reciprocal of H appears because the countering force tending to keep the needle on magnetic north is given by H.

Even though the quadrantal terms D sin(2ξ′) and E cos(2ξ′) are dependent on the induced magnetism in the soft iron, it turns out that they are not dependent on location. This is because the quadrantal force is proportional to H, and the force that keeps the needle on magnetic north is H, so as H changes the two forces vary in the same proportion and the net deflection is constant. The semicircular terms B sin (ξ′) and C cos (ξ′) do depend on the Earth’s magnetic field, including its dip, at each ship location.

It may surprise you that the magnetic dip affects the compass—it certainly surprised me. After all, it would seem that only the horizontal component would have an effect on a horizontal compass. But in fact the dip θ is used with the horizontal component H to derive the vertical component H tan(θ) which does have an effect on the induced horizontal magnetic field of the ship. The vertical component of the Earth’s magnetic field bends in the presence of soft iron as shown in the figure to the left. Another way to look at this is that the net field is the superposition (the vector sum) of the vertical magnetic field and the dipole magnet induced in the iron. This creates an induced horizontal component due to the vertical field, and this will affect the compass needle. This is negated somewhat by a vertical Flinder’s bar with its upper end at the level of the needle.

It’s interesting to note that all this was well-known prior to Maxwell’s unification of electromagnetic theory—in fact, he refers to it in his famous treatise. And the basic theory is unchanged from that time; the composite graph shown earlier of the components of magnetic deviation is from the *Handbook of Magnetic Compass Adjustment* issued by the National Geospatial-Intelligence Agency in 2004.

**Other Sources of Magnetic Deviation**

There is another effect not taken into account in this chart, the distortion due to *heeling* of the ship, i.e., the leaning of the ship from wind as well as the transient rolling of the ship. Smith derived equations for all that as well in his manual [Gray presents a nice summary of it all]. Heeling produces a maximum deviation when the ship is heading north or south, and no deviation when heading east or west (although the needle will have less directive force to north in this case). The complementary pitching action of the ship, being more transient that heeling, does not produce a significant difference in deviation on average. Another transient effect found in practice, the *Gaussin error* (not Gaussian error), is a time lag in magnetic change with heading change of about 2 minutes due to opposing magnetic fields in the soft iron created by eddy currents (by Lenz’s Law). Of greater concern is the *retentive error*, or the tendency to retain residual, subpermanent magnetism in the hard iron that is accumulated as the ship maintains a set course for a long time (say, several days) while being hammered by waves, an effect that can last from hours to more than a day after a heading change. This certainly required some experience and a good bit of art to reckon in the past.

Add to this the changing effects on magnetic deviation from variable quantities of ammunition on board, varying turns on cable reels, attached boats and nearby ships, personal effects such as watches and belt buckles, stowing of the anchor chain, lightning strikes, the heating of smoke stacks and exhaust pipes, and so on. Newer aluminum boats, for example, don’t provide magnetic shielding of sources below deck, and Barber relates that in one new aluminum cutter the compass deviation obediently tracked the generator speed. With all these effects it’s not surprising that at one time a magnetic compass was often placed high on the mizzenmast for a sailor to climb to take readings, a very effective solution in calm seas but a problematic one when a bobbing ship magnified the needle bounce and swing! (The construction of a compass that would minimize needle swing due to the motion of a ship was also a long-running debate, with the version by William Thomson—later Lord Kelvin—in his popular commercial binnacle eventually losing out to liquid-filled models.)

The direction of permanent magnetism of hard iron is related to the direction that the ship was facing when it was built; the compass needle will be attracted to the part of the ship that was south of it during construction. Smith held that an iron ship should be built with its head in a north-south direction, and preferably south. The effect is due to the alignment of magnetic domains in the iron with the external magnetic field of the Earth while being worked and pounded. In fact, Gilbert had created magnets by hammering iron rods laid in a north-south direction as part of his demonstration that the Earth acts as a mostly dipole magnet. But this initial permanent magnetism doesn’t last, and in some cases over half of a ship’s original permanent magnetism is lost in the course of its first year of use. And while the permanent magnetism of a ship is fairly constant after that point, any collision or repair of the ship will alter that permanent magnetism, requiring a new set of measurements and corrections to be applied.

**Ascertaining Ship Parameters**

The measurement of the magnetic deviation parameters for each compass location on a given ship was a tedious job that prompted many proposals for the best method. In general the dip θ at the measurement location was found on shore with a dipping needle. The horizontal component of the Earth’s magnetic field (H) was also found at that location, normalized to 1.0 for a standard location (Greenwich, for Smith). Then the combined horizontal magnetic force of the Earth and ship (H’) was found at each compass location on the ship itself. To do this, the needle of a precision compass was manually rotated to one side and released, and the time for a set number of vibrations was recorded, both on board (T′) and on shore (T). Then H′ / H = T^{2} / T′^{2}. Measurements were obtained as the ship was *swung* to different *rhumbs* of the compass rose, 8 directions for inexact values and 32 for more exact results, executed slowly and in opposite directions to reduce the Gaussin error. The difference between the compass course and the known magnetic heading, or the deviation δ, was also recorded at each rhumb. With varying levels of complexity, the required constants were extracted. For example, the *mean directive force* to magnetic north λH is found as the average value of H′ cos δ.

For a sub-optimal suite of only 8 measured deviations at 45° intervals (i.e, *N*, *NE*, *E*, *SE*, *S*, *SW*, *W* and *NW*), we can find the inexact coefficients from these rules:

A is the mean of the algebraic sum of all the deviations

B is the mean of deviation at *E* and the negative of the deviation at *W*

C is the mean of the deviation at *N* and the negative of the deviation at *S*

D is the mean of the deviations at *NE* and *SW* and the negatives of the deviations at *SE* and *NW*

E is the mean of the deviations at *N* and *S* and the negatives of the deviations at *E* and *W*

For any more than 8 rhumbs the derivation of the parameters is very complicated and involves solving a complex system of equations developed by Archibald Smith and others. Smith’s equation in A, B, C, D and E is actually a truncation at 2ξ’ of a Fourier series expansion in sines and cosines of multiples of the course ξ′, so these values are the corresponding Fourier coefficients—in fact, the first use of the phrase *harmonic analysis* is found in William Thomson’s obituary of Archibald Smith [Grattan-Guinness]. For more detailed information on deriving the ship’s parameters in this way, or for deriving the exact coefficients **A**, **B**, **C**, **D** and **E**, please see Smith’s *Admiralty Manual for the Deviations of the Compass* or *The Magnetism of Ships and the Deviations of the Compass, Comprising the Three Reports of the Liverpool Compass Commission*.

**Compensating for the Magnetic Deviation**

The magnetic deviation of a ship is typically corrected, even today, by components located in the binnacle holding the compass. Permanent magnets are positioned to compensate for the permanent magnetism of the ship. A vertical soft iron bar (the Flinders bar) is also located near the compass to counter the effects of the vertical component of the Earth’s magnetic field. These correct for the semicircular forces. Soft iron spheres on a rotating base serve to correct for the quadrantal forces, but their positions have to adjusted for the magnetic latitude.

The spheres also help negate magnetic deviations from heeling of the ship. There are also adjustable permanent magnets included to overcome these heeling effects. A permanent magnet mounted vertically directly beneath the compass does not have any effect when the ship is upright, but will correct for heeling error as the compass needle dips a bit in the lean. These permanent magnets also need to be adjusted with latitude.

Finally, there are current-carrying coils in the binnacle that are energized to counter the effects when the ship activates its degaussing coils to elude mines that trigger on the magnetic fields of passing ships.

Occasionally the net effect of magnetic deviation on an uncompensated compass completely negates the magnetic effect from the Earth, and the compass has no preferential direction at all, or only a weak one that makes observations uncertain. For this reason compensation is usually preferable to simply adding a correction in degrees from a table or diagram. I might add that Thomson once said that the chances were 50-50 that the navigator would get the sign wrong in calculating a compass correction [Barber]. Also, the equations given earlier assume a magnetic deviation of less than about 20° in order that B and C can be expressed as simple arcsine functions, and so a certain amount of ship correction is generally needed in an iron ship to ensure this.

For centuries, long compass needles (say, up to 15 inches) were thought better for higher magnetic strength (true enough) and better stability in rough seas (not true at all). But it happens that there are sextantal deviation terms in 3ξ′ for these long compass needles due to their response to the permanent magnet compensators, and octantal terms in 4ξ′ due to the interaction of these needles with their magnetic images in soft iron compensators. Smith early on had developed a rule for compass card needles, that when there are two needles they should be placed with their ends on the card at 30° to each side of the ends of the north-south diameter of the compass (the lubber line), and when there are four needles they should be placed with their ends at 15° and 45° from the ends of the diameter (30° apart). In this way there are equal moments of inertia parallel and perpendicular to the compass axis, which eliminates wobbling of the card But twenty years later, in analyzing ship data exhibiting higher-order effects, Evans and Smith discovered that small needles arranged in just the way he had prescribed exhibited less sextantal and octantal deviation than one short needle—and exactly zero mathematically! It was “a happy coincidence” according to Smith, and this justified moving the compensating permanent magnets and soft iron correctors much nearer this type of compass for more accurate elimination of the semicircular and quadrantal deviations [see Lyons for proofs].

Because of the calculations required for tabulating the magnetic deviation for all magnetic or compass courses for a location at sea are time-consuming and error-prone, much work was done to create graphical ways of plotting the values for all courses from a few measurements taken at the location. The result could also be used by navigators to read the magnetic deviation quickly and easily while at sea. These graphical calculators are the subject of Part II of this essay.

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**Hans the Clever Horse**

Let’s take a quick look at a historical example of media misrepresentation, in this case an unintentional one. In the late 1800s and early 1900s the horse shown in the pictures here (Clever Hans) was thought to have the ability to perform arithmetic as well as other reasoning tasks expressed by tapping a hoof a certain number of times. The New York Times wrote a feature on the horse in 1904 (BERLIN’S WONDERFUL HORSE; He Can Do Almost Everything but Talk—How He Was Taught) that brought enough attention to the matter that the German board of education created a team of experts to investigate the situation. Following extensive testing, the New York Times reported (correctly) that the committee had found no evidence of trickery and concluded the horse was exhibiting genuine skills. Only later did the psychologist Oskar Pfungst conduct enough blind tests to determine that the horse was reacting to unconscious cues by the questioners. (For a really good read on this, see **here**.)

So you can’t always believe what you see or read, and more generally it takes a critical look (and maybe some cynicism) to separate the chaff from the wheat. And when the story is worth retelling, and particularly when the calculator is a savant, it’s often difficult to be objective about the subject.

**Innocent Sources of Hyperbole**

Often information for newspaper articles is taken from promotional material or verbal descriptions by biased acquaintances or naïve observers, and of course there is always the temptation to embellish the truth a bit. There are accounts I’ve read of confederates in the audience asking questions or seeding problems with numbers having special properties that make, say, multiplication or division with another genuinely produced number much easier. This isn’t unlikely if you think of the “showman” type of lightning calculators, ones who mix these demonstrations with mentalism or magic (Arthur Benjamin, however, is a true lightning calculator as well as a magician). I can also say that the few times I have seen a mental calculator in action, the audience cannot distinguish between presentations of pure mental calculation and simple, standard ways of completing a magic square, for example, that I think of as dross. I also think it’s fair to say that incorrect answers are seldom reported, particularly if the calculator corrects the error.

Sometimes the problems posed by honest people turn out to be simple for the calculator given their extensive practice. For example, Smith records a number of questions that involve the number of seconds in some number of years; hours in some months, days and hours; cubic yards in some cubic miles; and so on for various simple multiples of common unit conversions known by any calculator of the time.

It also happens that the calculator gets lucky in a problem selection or in an answer, and then it’s one for the record books. Although I’m not a lightning calculator by any stretch, I’ve certainly benefited from a lucky guess. There was a particularly complicated calculation in my physics class once, involving many terms with powers in the numerator and denominator. As I was wont to do while students reached for their calculators, I wrote down what I thought the first few digits were, which was actually a stretch for me given the problem, and then just wrote two more digits randomly. When the first student read out the answer from the calculator only the last digit was off, and only by 1. There was absolute silence in that classroom as I turned and changed that last digit, and of course I never said a word about it.

People sometimes stumble by providing a problem they think is difficult, such as choosing odd numbers or even prime numbers, without realizing that the particular numbers offer a convenient shortcut or, more likely, that the calculator has already memorized the results for these numbers. We will see later that in a documentary on Tammet the researchers decided to ask him a high power of a 2-digit number. Were they going to pick an even number, or maybe one in which the digits were the same? Not likely—primes seem ideal, and there are limited numbers of them. In fact the limits of the calculator display and the powers they selected limited the number to less than 40, so how many likely numbers are there? (he was asked for 37^{4}, 27^{7} and 31^{6} in the documentary, and it is true that 27 is not prime). I can’t claim that Tammet knew these, but he may have at least known some intermediate powers of these that might have helped (and Tammet has prodigious powers of memory for numbers). This is fine and fair game for any lightning calculator in my opinion. Klein, for example, knew a wealth of number facts, such as “the first 32 powers of 2, the first 20 powers of 3, and so on.” In fact, in referring to Dase’s calculational efforts, Gauss wrote

One must distinguish two things here; an extraordinary memory for numbers and true calculating ability. These are, in fact, two completely separate quantities that may be connected, but are not always.

And it’s easy to read through an account and unthinkingly accept the writer’s assumptions. When I was young I read an account in which the interviewer wrote down a 20-digit number on a napkin and presented it to a memory expert for 15 or 30 seconds, after which the person could read it backwards and forwards. I realized that I could certainly do that, and a lot of people can, say by mnemonics or by grouping it into five 4-digit numbers. In tests by Binet, Diamandi was able to memorize on average 11 digits in 3 sec, 16 digits in 5 sec, and 17 digits in 6 sec, although Binet indicates a significant error rate. Is this so tough? After all, 3 seconds is really a longer span of time than you might think. At Eberstark’s request, Smith tested him by reading aloud single digits at a tempo specified by the calculator, about 1.75 sec between digits, for 20 digits, (Eberstark at the end extended this to 40 digits). Is this hard? There are those who do in fact perform amazing feats of quick memorization (in tests Salo Finkelstein repeated a 20-digit and 25-digit number after exposure for 1 sec apiece, a 33-digit number exposed for 2 sec, and 39 digits exposed for 4 sec), but the lesson here is to be critical when reading articles or watching programs.

As a final example, in a 2005 performance in the second Arthur Benjamin video listed earlier and found **here**, four audience members were brought on stage at the start to verify his answers on calculators. Benjamin did a variety of calculations, most of them correctly, but it’s interesting that despite his turning to them to request verification, two of his five answers in squaring 3-digit numbers were incorrect. But none of the four challenged his answers, and to be honest, I wouldn’t have had enough confidence that I entered the digits correctly to have held up a show like that either. So don’t trust observers *or* judges.

**Conscious Bias in Reporting**

Sometimes the writer or director purposely skews the reporting in ways that are probably conscious but not that serious—sins of omission and that sort of thing in a light piece. For those with an interest in the subject beyond casual reading, it’s important to notice these nuances.

Smith’s book is rife with contemporary accounts that use phrases such as “in an instant” or “in a flash” or “in the blink of an eye” and so forth. And the details of the task are seldom presented, even in structured tests by researchers, a fact that absolutely amazes me. Did the calculator repeat the problem back to the questioner? Did the questioner write the question down in front of the calculator, did the calculator have the problem in view during the test, and was timing (if there was any) stopped when the first digit of the answer was being written or the last, or when the calculator said “Done” or when the last digit was recited? And we have seen that there are particular types of problems (e.g., the number of seconds in a given number of years) that benefit hugely from memorized facts. Smith also points out instances in which the set of test questions by researchers all shared the same shortcut property—why is that? How many questions were asked in all? Were just the correct, speedy ones reported? One rarely if ever has these facts in an account of a lightning calculator.

This continues today, of course. Let’s look at a common example of subtly slanted reporting. You might think I’m seeing bias where there is none, but when you read enough of these accounts you begin to see patterns. Alexis Lemaire is an extremely talented mental calculator with a specialty of extracting 13^{th} roots of 200-digit numbers, a feat that I don’t believe is attempted by others. So my comments here are only on the reporting and in no way reflect on Mr. Lemaire or his abilities.

So here’s a typical news report of a record time set by Lemaire at the Oxford Museum of the History of Science, dated July 30, 2007, by BBC News and found **here**.

The task is to find the 13th root of 85,877,066,894,718,045,602,549,144,850,158,599,202,771,247,748,960,878,023,151,

390,314,284,284,465,842,798,373,290,242,826,571,823,153,045,030,300,932,591,615,

405,929,429,773,640,895,967,991,430,381,763,526,613,357,308,674,592,650,724,521,

841,103,664,923,661,204,223.

The answer’s 2396232838850303. Multiply that by itself 13 times and you get the above. Even with a calculator you wouldn’t beat Alexis Lemaire doing the calculation in his head.

Another article from 2005 on such a record, shown in the figure above, can be found **here**.

Now let’s look at this article of another such record from any of the seven reports I found very quickly online via Google. For direct comparison with the first one above I’ll choose BBC News again, dated December 11, 2007, and found **here**.

The fastest human calculator has broken his own mental arithmetic world record.

Alexis Lemaire used brain power alone to work out the answer to the 13

^{th}root of a random 200-digit number in 70.2 seconds at London’s Science Museum.The 27-year-old student correctly calculated an answer of 2,407,899,893,032,210, beating his record of 72.4 seconds, set in 2004.

The so-called ‘mathlete’ used a computer package to randomly generate a number before typing in the answer.

So here we are given a little more information on how the test was performed. But do you see the real difference here? The randomly generated number is not reported, just the root. Why is that—generally the huge number is much more impressive to present than the root. Well, let’s see what that randomly generated number was:

91474397281474512894803677416201430283564210503432385339561327276933454229

60930464647192509451811477101625889659290744142634989755650414557096020392

5503679105245199142338806082494254050610000000000000

It doesn’t look so random. In fact, it wasn’t the power that was randomly generated (after all, what are the odds that a randomly generated number would be a 13^{th} power?), but rather the root was randomly generated and the power calculated from that value. Which is fine, but it seems apparent to me, at least, that they would have reported the power if it didn’t end in thirteen zeros. The reader might immediately intuit that last digit of the root is 0, so it detracts slightly from the effect and makes them consider that it was a lucky break. In fact the last digit is always identical in a number and its 13^{th} power so it’s always trivial to find it, but this now makes the second-to-last digit trivial (the digit 1). I see a little bias on the part of the reporter, and I saw this in every report of this event I could find.

Let’s take a news report of another record-breaking event from November 16, 2007, found **here**.

27-year-old Alexis Lemaire from France has set a new world record by mentally calculating the 13th root of a 200-digit number in 72.4 seconds. He correctly identified the answer as 2,397,207,667,966,701. The previous record was 77 seconds.

No 13^{th} power listed here either. And I believe the reason is that this power is

86332348800352843610126990022313468510477370930755992152681390347795323097

51168717005763648080727141383324712170576311110855841562345802001852561285

2897226196105357173387251523920946707380414694987101

With the power and root in view it doesn’t take too long to figure out that any power ending in 01 would have roots ending in 01, so this was again a case in which the last two digits are found instantly. This may not have been a lot of help to Lemaire, as he probably knows all two digit endings of 13^{th} roots, but again this is all about the reporting. Also, these are situations where it’s easy for us to see the advantages of a particular number, whereas lightning calculators have a wealth of stored number facts that can make certain problems much easier in a less apparent way, and it only takes one lucky number to break a record.

**Deconstructing the BrainMan Documentary**

Let’s take a look at a documentary that in my opinion inadvertently reveals itself as duplicitous. This may seem a bit tedious, but I’m kind of proud of the mathematical detective work I did in uncovering this.

Daniel Tammet currently holds the European/British record for memorizing pi (22,514 digits in all!) as listed here, he has appeared on *60 Minutes *and David Letterman’s show, and he has written a recent autobiography titled *Born on a Blue Day: Inside the Extraordinary Mind of an Autistic Savant*. He has some talent with mental calculation as well, and he was featured in a popular 2004 documentary called *BrainMan *(titled *The Boy With The Incredible Brain* in the UK). The documentary won a Royal Television Society award in December, 2005, and was nominated for a BAFTA in 2006.

Tammet experiences synaesthesia, the ability to see or experience numbers as shapes, colors and textures. Here’s a typical excerpt from an article on him:

Tammet is calculating 377 multiplied by 795. Actually, he isn’t “calculating”: there is nothing conscious about what he is doing. He arrives at the answer instantly. Since his epileptic fit, he has been able to see numbers as shapes, colours and textures. The number two, for instance, is a motion, and five is a clap of thunder. “When I multiply numbers together, I see two shapes. The image starts to change and evolve, and a third shape emerges. That’s the answer. It’s mental imagery. It’s like maths without having to think.

Now I was asked awhile back about Tammet’s solution of 13/97 in the *BrainMan* documentary. I had not seen it, but I replied that division by a two-digit number like 97 is not difficult (130/97=1 remainder 33, 330/97=3 remainder 30+3(3)=39, 390/97=4 remainder 2, etc., so we get .134… and so on—lightning calculators can fly through this). However, we saw earlier here than the reciprocal of 97 consists of repeating groups of 96 digits. I thought it likely that either half the repeating group or all the repeating group of 1/97 was memorized, because changing the numerator to any 2-digit number less than 97 simply cycles the starting position of the repeating group to another location. In fact, Aitken had remarked on the commonly proposed problem of this reciprocal in his **talk**:

Here the remark was made that memory and calculation were sometimes almost indistinguishable to the calculator. This was illustrated by the recitation of the 96 digits of the recurring period of the decimal for 1/97, checked by Dr. Taylor. Probably because 97 was the largest prime number less than 100, this particular example had been frequently proposed.

Actually, I suspect division by 97 is often asked because it takes a whole 96 digits before the digits start repeating.

Later that day it occurred to me that there might be a way to detect whether that problem was solved by Tammet through a memorized repeating group. The reciprocal of 97 is

1/97 =

.010309278350515463917525773195876288659793814432|

98969072164948453608244226804123711340206185567|

010309278350515463917525773195876288659793814432|

98969072164948453608244226804123711340206185567|

01030927…

and also

13/97 =

.134020618556701030927835051546391752577319587628|

865979381443298969072164948453608247422680412371|

134020618556701030927835051546391752577319587628|

865979381443298969072164948 453608247422680412371|

13402062…

I’ve added vertical bars at each half of the 96-digit repeating group. We can see that each decimal expansion starts repeating every 96 digits. In addition, each digit in one half of the repeating group is the difference from 9 of the corresponding digit in the other half of the repeating group as mentioned earlier in the Fast Division section of this essay. So to produce 1/97 we can just memorize the first 48 digits of the repeating group, and then repeat that it but subtract each digit from 9. Again, Aitken had done that in anticipation of being asked for it in performances.

You can see that 13/97 has simply cycled the repeating group to the position near the end that starts with 134. The starting point for a given numerator is not predictable, but we can just divide 13 by 97 to a few digits (.134) to find the start, or we can multiply 13 by the first few digits of the repeating group for 1/97 (.0103) to find 134 as the starting point. If only the first half of the repeating group of 1/97 is memorized, we would see if 134 is in that group, and if it isn’t there (like now) we look for the 9’s complement 865 in that group, which is found near the end. So we start at that point, listing the 9’s complement of each digit as we cycle around the start of that half-group, and when we reach 865 again we repeat the steps but don’t take the 9’s complement. So we can always get away with memorizing just 48 digits to divide any number by 97. If the numerator is greater than 97, it’s just a whole number and a fraction with a numerator less than 97, so we end up being able to divide any number at all by 97 this way. (Technically speaking, we only need to memorize 47 digits because the repeating group for division by any number ending in 7 ends in 7, but then we’d have to remember that fact.).

Well, I thought that given the different starting position of 13/97, if Tammet were using a memorized half or full repeating group of 1/97, a verbal hesitation might be detectable at the end of this group as he “resets” his memorized or mnemonic digits to the start of this group. The 1/97 repeating group ends with the digits 567 and then cycles back to the beginning to 010… This 567 sequence occurs in the 11th to 13th position in 13/97. If Tammet hesitated between stating the 7 in the 13th position and the 0 in the 14th position in reciting 13/97, it would be evidence that the memorized group (or half-group) of 1/97 was being utilized.

So after all of this I finally watched the documentary online. You can find the complete UK version of the documentary in 5 parts on YouTube. The first part is found **here**. (Google Video has the U.S. version in a single video, but it does not include the scene with the whiteboard that I will refer to below.)

The 13/97 calculation is not far into the documentary (at 1:49 in Part 1), so it’s quite interesting to watch all of it up to that point to see how the shapes/crystallization process is described. He actually does the 13/97 calculation twice back-to-back, it turns out.

We don’t hear the problem posed to Tammet in the video, so we don’t know the delay between when the question was asked and the response was begun. Tammet slowly recites the digits of the answer all the way up to 5..6..7 and then—he comes to a total halt! He has lost his place, the questioner mentions that he is carrying on, he says he’s carrying on, then he says to tell him to stop or … and the interviewer stops him. I about fell off my chair. The interviewer asks him how many places he can do it to, and Tammet replies, “A hundred—nearly a hundred.” To me this reveals that he has done division by 97 before, and as we know if you can do 96 digits you can repeat them as long as you want to.

And this brings me to something very misleading—that the interviewer implies here that division gets harder as you get further into the solution. This is absolutely false—in division it’s just as easy to get the 1,000th digit as it is to get the 2nd digit. It is not like, say, a square root.

And I’m in luck—Tammet is going to repeat his answer after the interviewer retrieves a computer to get more digits. And here I will say that anyone who digs up an old 8-digit calculator to go test a savant on his calculating abilities, especially on division, can only be setting up a dramatic scene in a fluff piece. And another thing: the interviewer tells Tammet to wait, they are going to go find a computer, then come back, boot it up, sync the camera to the vertical sync of the video card, run the calculator application, and ask him the very same problem to see how many digits he can do?? And what would someone like me be doing the whole time they’re fiddling around? I’d be furiously calculating more digits in my head, that’s what.

But Tammet starts reciting again at about the same rate as the video scans the digits on the computer screen. He gets to the 5..6..7… and to my astonishment at that exact moment (and I mean *exactly* at the instant the next digit would be uttered) a voiceover is spliced into the video, saying two superfluous things—that every digit is correct (which we would know if the voiceover wasn’t there) and that he will eventually exceed the 32 digits of the computer (which we find out when we get there). Also, at this same point a video cut is shown of his hands making movements on the table. Then the audio and video return to him reciting about a dozen places later. If you replay the video and you continue reciting the correct digits during the voiceover, you find that Tammet would have had to have sped up significantly to have been at that point when they return to his recitation, although to be fair he does speed up at the very end of it all.

So it strongly appears to me that the documentary covered up for him on what I think must have been some sort of difficulty at the point I predicted. Not a big deal, maybe, but the producers went to a lot of trouble to deceive us on this, and that’s makes me question the validity of the whole enterprise.

In Tammet’s book on page 4 he says he “calculates” divisions like 13/97 by seeing spirals rotating in loops that seem to warp and curve, and in fact if you go to nearly the end of Part 4 of the documentary (from 9:41 to 9:44 **here**) and look carefully when Tammet is tracing such a spiral on the whiteboard (see the frame capture here), you’ll find it is being drawn right below “1/97”. So this divisor pops up again, lending credence to the theory that it was half- or fully-memorized.

Finally, let’s look at the three integer powers that are asked of him. Very early in Part 1 (at 0:58 in **here**) the narrator says that Tammet was asked to find 37^{4}. We don’t see it asked, we just see the interviewer punching 37 x 37 x 37 x 37 extremely slowly into the calculator, followed by a continuous pan to Tammet, who looks up and recites the answer. So the question was asked at some unknown time prior to the entry into the calculator.

Later in the Part 4 of the documentary (at 0:40 in **here**) Tammet is asked to find 27^{7} and 31^{6} in two apparently unplanned, poorly executed tests by two neuroscientists (are there no x^y keys on these calculators??). We’ll never know how long it took to find the results because the documentary has so many cuts injected there that our sense of time is destroyed while the background music gives a false sense of continuity. We do see that they don’t start a small timer until 4 seconds after one of the problems is given. Again, the producers of the documentary mislead the audience by compressing the timescale. And for those who still might have thought the documentary to be unbiased, a voiceover appears during the latter calculation to blame whatever delays there were (what were they?) on jet lag.

So to summarize all this, in the process of trying to analyze Tammet’s method I found strong evidence that the *BrainMan *documentary in several ways actively misled the viewers. And of course this all has to do with the producers of the documentary themselves, not Daniel Tammet. And that’s why you have to be critical of these sorts of things.

Now let’s consider the first part of the documentary listed earlier on Rüdiger Gamm found **here**. Very early into it, just after 0:40 sec, Gamm announces to an auditorium that he will attempt to divide the prime number 109 into a 2-digit number provided by an audience member. He will attempt to go 100 digits after the decimal place. After receiving a number of 93, Gamm repeats the problem “93/109” and focuses on the problem for a total of 11 sec. Then he starts reciting the digits, very soon accelerating and reciting the digits as fast as he can say them.

Every alarm in your head should be going off about now. Is the number Gamm chose (109) one of those primes whose reciprocal has the maximum possible repeating group (108 digits)? Did he recite only 100 digits so the repetition after 108 digits wouldn’t be noticed? Yes, and in my opinion, yes. Here’s the reciprocal of 109 with vertical bars separating halves of the repeating group as in our earlier example for 97:

1/109 =

.009174311926605504587155963302752293577981651376146788|

990825688073394495412844036697247706422018348623853211|

009174311926605504587155963302752293577981651376146788|

990825688073394495412844036697247706422018348623853211|

0091743119…

So the repeating group does have 108 digits, and as always in this case, the digits in the second half are the 9-complements of the corresponding ones in the first half. So let’s take the submitted numerator 93 and mentally divide it by 109 to three digits. We can divide 93 by 110 instead and adjust for the offset in each step by adding the previous digit, as presented earlier in the *Fast Division* section for division by a number ending in 9:

93/11 = 8 remainder 5

(50+8)/11 = 5 remainder 3

(30+5)/11 = 3 remainder 2

and we locate 853; it’s in the second half of the group near the end. If the repeating group for 109 is memorized (or even half of it as described earlier since the other half is the 9’s-complement), it’s child’s play to recite the digits.

Now I don’t know how Gamm actually performed this feat. If you practice just a bit with the adjusted division process you can develop a kind of rhythmic cadence as you go:

* 93 ***8 ***58* **5 ***35 * **3** *23 ***2 ***12* **1** *11* **0 ***10* **0** *100 ***9** *19 * **1** *81* **7 **…

This is remarkably easy if you try it without reading it. Stating the bolded digits out loud really helps to append them to the remainder of the next division. Gamm does seem to develop a sort of cadence in the video, and he is a phenomenal calculator, so it’s likely that he is just amazingly fast at this. In any event, to judge the performance it’s important to realize that an adjusted division technique exists, and it’s also worth noting that with some memorization you too could walk into an auditorium and perform as well as Gamm on this.

**The Appeal of the Mental Calculator**

The study of lightning calculators of the past is a fascinating one for me from a mathematical aspect more than a psychological one. We’ve seen years of articles by educators bemoaning the dependence of students on calculators, but I see little in school textbooks on mental math other than simple estimation. And yet when I have presented basic methods of mental calculation to classes (elementary and college), I’ve met with incredible interest. Certainly the *BrainMan* documentary is a very popular one. But these types of presentation generally ascribe abilities in these areas to mysterious machinations in the minds of remote geniuses, which makes for a good story but can be discouraging. In fact, these individuals through talent and training acquired a knack for racing headlong through calculations that are not mysterious at all once the methods are taught.

And they are not being taught. Mental calculation can be a highly creative and satisfying endeavor offering a variety of interesting strategies, more than I have presented here and many more than most people realize. It is a skill that engages both children and adults, and one that naturally leads to a real familiarity with the properties and relationships of numbers. It provides a useful and fun approach for developing a number sense and generating a true appreciation for the elegance of elementary mathematics. It should not be a neglected art.

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

**<<< Back to Part II of this essay**

**Printer-friendly PDF file of Lightning Calculators Parts I-III.**

The traditional demonstrations of lightning calculators fall into the following categories:

- Fast addition of numbers (not that common, actually)
- Fast multiplication of multi-digit numbers (very common, and with more success than Superman demonstrates)
- Fast division (uncommon)
- Factoring of large numbers or finding them to be prime (common)
- Extraction of roots of perfect powers (very common)
- Extraction of roots of numbers that are not perfect powers (rare)
- Raising numbers to various powers (common)
- Finding logarithms of numbers (uncommon)
- Finding one or more sums of four squares that add to a given number (occasional)
- Calendar calculations (exceedingly common)
- Compound interest (isolated)

In addition, there are more modern methods that can be used, particularly for approximating logarithms, exponentials and trigonometric functions, that have been constructed for those interested in these types of problems.

The main techniques will be highlighted in sections below devoted to each of these tasks. It is important to realize that lightning calculators were highly individual in how they approached these tasks, and most calculators have such a vast knowledge of number facts that answers were often obtained immediately from memory or following only slight adjustment. As one example, Klein learned through experience the multiplication table through 100×100 and used it to great advantage doing cross-multiplication in 2-digit by 2-digit chunks. He also knew squares of integers up to 1000, cubes up to 100, and roughly all primes below 10,000. He also knew logarithms base 10 to 5 digits for integers up to 150.

Sometimes calculators used a mnemonic scheme, often of their own design, to aid in remembering these number facts. Mnemonics is the association of digits with images or letters in a sentence. Arthur Benjamin presents in his book, *Secrets of Mental Math*, the mnemonic scheme he uses to remember intermediate values during long mental calculations, based on a phonic method a few hundred years old. I ran across a chapter from a 1910 book that uses this same scheme to encode the cubes of all 2-digit numbers, and on a lark I modernized its quaint phrases and extended its scope to provide squares as well, and I wrote it all up in a paper found **here**.

But the more involved calculations also involve algebraic methods deduced by the performer through familiarity with the processes or, increasingly today, by consciously applying mathematical relations, number theory and numerical approximations. Some of the methods described below receive greater attention in Smith’s book, while others are described in greater detail in other references. An excellent source for the world records in various categories of memorization and mental calculation can be found **here**.

I might as well mention here that there is a movement to assign discovery of quite a few of these algebraic techniques to an ancient system of Vedic Mathematics rediscovered between 1911 and 1918 from the Sanskrit texts known as the Vedas by Sri Bharati Krsna Tirthaji (1884-1960) and expressed as sixteen Sutras. See **here** for an overview of these beliefs. For a detailed presentation of these Sutras as well as outright criticism of the supposed origin of them and their overall effectiveness as an educational tool, see **here**. In my opinion, and I know this is not a popular one in some circles, systems such as this (and including resurgent schools for teaching the abacus and soroban in China, Japan and elsewhere) divert students’ time in much the same way as the “New Math” introduced in U.S. schools in the 1950’s and 1960’s.

**Fast Addition**

It might seem that rapid addition would be a common demonstration for lightning calculators, but Smith notes that lightning calculators, driven by their interest in numbers, typically found addition and subtraction too dry for study. Inaudi and Bidder would add several multi-digit numbers, and there were a few more who specialized in just this task, but their methods were necessarily straightforward. The common theme seems to be grouping numbers into groups of digits to add separately, minding any carries or borrows as needed. I remember reading somewhere of a tip for adding a column of 3-digit numbers such as a grocery bill that proved a surprisingly helpful technique. In such a case it’s easier to add the tens and ones digits as groups and then add the hundreds digits at the end. So say you are presented with a column of numbers such as

245

814

152

81

696

317

——

Adding them as single digits can be slow and adding them as 3-digit numbers can be confusing, so we might add 45+14=59, add 52=111, add 81=192, add 96=288 (where 96=100-4), add 17=305. Then add 2+8+1+6+3=20 and with the carry of 3 we have 2305 as the sum. Now this might be a lot slower for you than just adding the columns in individual digits, but a practiced calculator can add 2-digit numbers in a flash (or 3-digit numbers and so on), so with some development this can be a faster alternative.

**Fast Multiplication**

I haven’t heard of any lightning calculator who didn’t or doesn’t perform multiplications of multi-digit numbers. Many of them had such an intimate knowledge of factors and multiples built up after years of practice that often such a problem could be re-arranged into a known one plus some correction such as an additional factor. Some common products produce numbers that are easy to multiply by another number, so knowing such convenient products can be a real help. For example,

67 x 3 = 201 23 x 13 = 299 19 x 21 = 399

17 x 47 = 799 89 x 9 = 801 53 x 17 = 901

37 x 27 = 999 7 x 11 x 13 = 1001 23 x 29 x 3 = 2001

31 x 43 x 3 = 3999 and so forth…

The two most common techniques used by lightning calculators for mental multiplication are adding *partial products *and performing *cross-multiplication *on the digits.

Partial products are the combinations of the individual digit multiplications, and they are added up from left to right to find the product:

46 x 58 = 40×50 +40×8 + 6×50 + 6×8

= 2000 + 320 + 300 + 48

= 2668

The terms are added as they are calculated, so when 40×8 is calculated, it is added to 2000 to get 2320, then 6×50 is added to get 2620, and finally 6×8 is added to yield 2668. There is only one running total to remember.

Cross-multiplication does not involve these large sums. The digits of the product are found one at a time, but the procedure has the disadvantage that the digits are produced from right to left, so they must be remembered and reversed to recite the answer verbally. Typically the digits are written as they are obtained from right to left. In this method the combinations of single-digit products that contribute to each digit of the result are added, including carries. For example,

46 x 58:

6×8 = 48, or **8** with a carry of 4

4×8 + 6×5 + 4 = 66 , or **6** with a carry of 6

4×5 + 6 = **26**

Answer: 2668

Both of these methods have the advantages that they can produce results very quickly with practice, they scale up very well with larger multipliers, and they don’t require any multiplications beyond one-digit by one-digit (Klein used 2-digit by 2-digit cross-multiplication). They are simple and very practical methods.

There are various ways to simplify multiplication based on the properties and relationships of the numbers involved. We might notice in a problem that one of the multipliers is quite near a very round number, say, a multiple of 10 or 25. We can multiply by that round number instead and adjust for the difference at the end. For example,

29 x 34 = 30×34 − 34

To find 30×34 here, we would multiply from left to right: 30×30 + 30×4. Now if a multiplier exceeds a multiple of 10 by the amount of the multiple, we can use the multiple of 10 and add 1/10 of that result. If a multiplier lies below the multiple of 10, we subtract 1/10 of the result. Multiples of 11 and 9 have these properties.

33 x 62: Find 30×62 = 1860, then 1860 + 186 = 2046

36 x 62: Find 40×62 = 2480, then 2480 − 248 = 2232

We would not subtract 248 directly in the last example, but rather subtract 250 and add 2, a slightly different view of subtraction that makes a large practical difference.

We can also look at a number as a collection of convenient groupings. For example, we can multiply 124726132 by 5 by first halving each even grouping in the first number and then appending zero:

12 4 72 6 132 x 5 = 6 2 36 3 066 0 or 8 32 6 31 x 5 = 4 16 3 15 5

Multiplying a number by 15 can be done by multiplying by 10 and adding half the result. We can think of adding a zero, and then adding half of each even grouping to itself, working left to right and keeping the same number of digits in the grouping as it started with. If a grouping ends up with an additional digit, the upper digit is added to the grouping to the left. The presentation below makes the calculation look more difficult than it actually is—the result is generated smoothly from left to right, with perhaps a correction for a carry from the next grouping, as with the carry of 1 from the (72+36) grouping below to the group on its left:

12 4 72 6 132 x 15 = (12+6) (4+2) (72+36) (6+3) (132 + 66) 0

= 18 7 08 9 198 0

Multiplication by 25, or 100/4, can be thought of as appending two zeros and dividing by 4. Multiplying by 50 can be done as 100/2, 75 as 300/4, 125 as 1000/8, and so forth.

These are reasonable and readily understood concepts that involve looking at the whole number rather than individual digits. This is a mental shift that is subtle but critical in developing a number sense. Methods like these are also more general than they seem at first, because if they *almost* apply, we can use them on nearby numbers and then apply a correction at the end.

These methods all involve thinking about the properties of numbers, so they appeal to me as methods for somewhat specific circumstances. However, there is a type of method that is useful in a very wide variety of multiplications. When the multipliers are a distance c and d from a round number, their product can be represented by the product of the round number and the sum of the round number and the two differences, with the product of the two differences added at the end as a small correction. There does not seem to be a consistent name for this method in the literature; I call it the *Anchor Method*:

(a+c)(a+d) = a(a+c+d) + cd Anchor Method

This is much easier to use than it might appear, as we will see, and a knack for it is easily developed with a small amount of practice. The concept can be taught to children. I visualize “anchoring” one multiplier at the round number, and then literally stringing out the differences from the original numbers from this anchor to find the other multiplier. It will turn out that the original multipliers move outward, their product will be less than the original, so the correction at the end needs to be added, and if they move inward, the correction is subtracted. This corresponds to the intuitive (and correct) concept that a square has the greatest area for a given sum of side lengths; the rectangle produced by shifting length on a square from one side to another side will have a smaller product of the two sides because (x+n)(x−n) = x^{2} − n^{2} is always less than x^{2}.Below are three representative problems and a visualization of each solution. (The numbers are shown on vertical number lines because I “see” number lines as vertical rather than horizontal. I remember having difficulty learning the number line concept in grade school, and I believe it was due to think a vertical layout would be much more intuitive to children (and me) who think numbers go up as they get higher.)

12 x 13 = 10 x 15 + 2 x 3 = 156

18 x 16 = 20 x 14 + 2 x 4 = 288

18 x 24 = 20 x 22 − 2 x 4 = 432

An anchor of 100 is very common, say, 84^{2} = 100×68 + 16^{2}. With 100 as the anchor, we can find 68 as the last digits of 84 doubled rather than by finding the difference between 100 and 84 and subtracting this from 84.

If the numbers to multiply are far apart, though, we can end up with a large correction term *cd*. There are a few strategies to bring the multipliers nearer to each other:

1. Subtract one number from a very round number (or add it to a very round number) to bring it closer to the other number:

23 x 67 = 23(100−33) = 2300 − 23×33 = 2300 − (20×36 + 3×13)

2. Divide or multiply one number by a low integer and add a correction:

23 x 67 = 23x33x2 + 23 = 2(20×36 + 3×13) + 23

3. Break one number into two convenient parts:

23 x 67 = 23(50+17) = 2300/2 + 23×17 = 1150 + 20^{2} − 3^{2}

In the end we can use our creativity and experience to manipulate the calculation as we wish.

One of the most powerful tools in mental calculation is converting the multiplication of two different numbers into the square of the average minus the square of the distance to the average. This is shown by the *Midpoint Method*, an algebraic identity:

(a+c)(a−c) = a^{2} − c^{2 }Midpoint Method

where *a* is the average of the two numbers, (a+c) is one of the numbers, and (a−c) is the other number. This is algebraically equivalent to the Anchor Method formula if d = -c, or in other words when the anchor is midway between the two multipliers. The choice of the anchor as the midpoint or some other number depends on the problem and on personal preferences, but there is no doubt that using the midpoint is a very common technique. For example,

28×32 = 30^{2} − 2^{2}

52×78 = 65^{2} − 13^{2}

or, considering the first problem in this section,

46 x 58 = 52^{2} − 6^{2}

Less convenient multipliers can be manipulated in a number of ways to use this technique. We might have the case where there is no midpoint of the two multipliers—here we can adjust one of the multipliers by 1, do the calculation, and then provide a correction to account for the original adjustment, as for 28×33 = 28×32 + 28 = 30^{2} − 2^{2} + 28, but in this particular case it may be easier to use the Anchor Method from the last section: 28×33 = 30×31 − 2×3.

To calculate squares we might use the Midpoint Method *in reverse*. We can split a square into the product of two numbers equidistant from the original number, and add the square of that distance, again one scenario of the Anchor Method. For example, let’s continue with one of our examples from earlier:

52×78 = 65^{2} − 13^{2}

Now we find 65^{2} by spreading 65 in both directions by an equal amount and adding the square of that amount. Here a good spread is by 5, yielding 65^{2} = 60×70 + 25 = 4225. Similarly, 13^{2} = 10×16 + 9 = 169. So we can turn a general multiplication into a square plus a small correction, and we can turn that square into an even simpler multiplication and one more small correction if needed. Again, I find it helpful to remember that the average squared will always be larger than the spread numbers multiplied, so when collapsing two multipliers to a square you *subtract *the correction, and when spreading a square to the product of two numbers you *add *the correction. These transformations become automatic and very fast after a bit of practice.

Many of you may recognize in the example of 65^{2} the trick for squaring numbers ending in 5: multiply the number left of the units digit by that number plus one, and then append 25, as in 6×7 | 25 = 4225. Now we can see why that works.

The Midpoint Method described earlier applies to larger numbers, e.g., 244×376 = 310^{2} − 66^{2}. But 310^{2} is really just a square of a two-digit number followed by two zeros—what if we had ended up with a three-digit square here? Again we split the square into two numbers equidistant from the original number, adding the square of that distance. To illustrate, 244×382 = 313^{2} − 69^{2} = [300×326 + 13^{2}] − 69^{2}, and we end up with a simple calculation if we know the two-digit squares.

And there are indeed a variety of other techniques for finding squares. Most of these involve expressing the number to be squared as the sum of two other numbers that are more easily squared, using the *Binomial Expansion for Squares*:

(a+b)^{2} = a^{2} + 2ab + b^{2 }Binomial Expansion for Squares

To illustrate,

34^{2} = (30+4)^{2} = 30^{2} + 2x30x4 + 4^{2} = 1156

69^{2} = (70−1)^{2} = 70^{2} − 2x70x1 + 1^{2} = 4761

313^{2} = (300+13)^{2} = 300^{2} + 2x300x13 + 13^{2} = 90000 + 7800 + 169 = 97969

In another application of the binomial expansion, one of the most intriguing and useful techniques easily finds the square of a number near 50. Here we add the difference from 50 to 25, multiply by 100, and add the difference squared. If the number is within 10 of 50, we can add the difference to 25 and simply append the distance squared rather than adding it. Let’s use the vertical bar “|” to separate two-digit groups. Note that if we end up with a 3-digit result in a grouping, its most significant digit would be added to the group to its left. In this notation,

(50+a)^{2} = (25 + a) | a^{2}

so,

52^{2} = (25+2) | 2^{2} = 2704

44^{2} = (25−6) | 6^{2} = 1936

38^{2} = (25−12) | 12^{2} = 13 | 144 = 1444

This is a simpler way of thinking of the binomial expansion (50+a)^{2} = 2500 + 100a + a^{2}.

We can also use the fact that multiples of 25 are fairly round numbers. We can square numbers near 25 using the expansion (25+a)^{2} = 625 + 50a + a^{2}, as 27^{2} = 625 + 100 + 4 = 729. The relation (75+a)^{2} = 5625 + 150a + a^{2} can be used to find, say, 78^{2} = 5625 + 450 + 9 = 6084. We can reformat these into our notation, noting that a .5 in a group is converted to a 50 in the group to the right of it:

(25+a)^{2} = (6 + a/2) | (25 + a^{2})

27^{2} = (6+1) | (25 + 2^{2}) = 729

(75+a)^{2} = (56 + a + a/2) | (25 + a^{2})

78^{2} = (56+3+1.5) | (25 + 3^{2}) = 60.5 | 34 = 6084

Alternatively, we can re-arrange the binomial expansion of two-digit squares ending in 9, 8, or 7 in another interesting way:

(10a+9)^{2} = 100a(a+1) + **8**0(a+1) + **1**

(10a+8)^{2} = 100a(a+1) + **6**0(a+1) + **4**

(10a+7)^{2} = 100a(a+1) + **4**0(a+1) + **9**

where the digits in bold comprise the square of the units digit. So 79^{2} = 5600 + 640 + 1 = 6241, 87^{2} = 7200 + 360 + 9 = 7569, and so on.

If a neighbor of the number has a square that is known or easily calculated, we can use this convenient square and adjust for the difference. Since (a+1)^{2} = a^{2} + a + (a+1), we can find 31^{2} = 30^{2} + 30 + 31 = 961. Similarly, 29^{2} = 30^{2} − 30 − 29 = 841. For other neighboring numbers we can find the square of the convenient number, then add or subtract the original number, the final number, and twice each number in between, so 32^{2} = 30^{2} + 30 + 2×31 + 32 = 1024, a square that we recognize from powers of 2. Ultimately we will find that the field is quite crowded for squaring numbers less than 100, and in a surprising development we eventually start looking to three-digit numbers for more interesting challenges.

Three-digit numbers can be treated like two-digit numbers in all these methods if we treat the leftmost two digits as a single digit, as in using the technique for squaring numbers ending in 5 to find 235^{2} = 23×24 | 25 = 55225. We can also alter some of the methods slightly for three-digit calculations. The square of a number near 500 can be found by adding the difference from 500 to 250 and appending the difference squared as a three-digit group delineated by a comma:

(500+a)^{2} = (250 + a) , a^{2}

so,

513^{2} = 263,169

492^{2} = 242,064

Multiplying larger numbers extends these rules further with a corresponding increase in difficultly. A more recent method of multiplying two 4-digit numbers is discussed in the Newer Methods section of this essay. I might add that an excellent, free training program for practicing multiplications up to 4×4 can be found **here**.

** ****Fast Division**

Division was not a common task except in the limited context of factoring a number, which is not really division in the truest sense. When this was done, as in the case of decimalizing a fraction, it was often done by reversing known multiplications or by taking advantage of properties of division by small integers (which might be factors of the actual divisor).

There are some properties of reciprocals 1/t that help in finding their decimal expansions, and of course a calculation of s/t might first calculate 1/t and then multiply the answer by s. For a denominator t with prime factors of 2 and 5 only, the number of decimal places in its decimal expansion will equal the highest power of 2 or 5, so any s/16 will terminate after the fourth decimal place since 16 = 24 x 5. If t is a prime number, the decimal expansion of 1/t will consist of some zeros followed by repeated groups of digits. The length of this group will be (t-1) or a factor of (t-1), the first type occurring for t = 7, 17, 19, 23, 29, 47, 59, 61, 97, … So 1/7 will have a 6-digit repeating group and in fact 1/7 = 0.142857142857142857… When t is one of these special primes, the corresponding digits in the two halves of the group will add to 9, so here if we find 1/7 to three places (0.142) we immediately know the next three digits (857) and we now have the whole repeating group. A numerator here other than 7 that is less than 7 simply rotates the digits of the repeating group, maintaining this relationship. A numerator greater than 7 will consist of some digits to the left of the decimal point, followed by the repeating groups based on the remainder. Also, the repeating group of 1/t for a prime t with a units digit of 1 will have a last digit of 9 and vice-versa, otherwise the last digit of the repeating group will be the same as the last digit of t. So for 1/7 we know immediately that the last digit of the repeating group will be 7, so we take the reciprocal to 2 places (0.14), then we know the next digit is (9-7) or 2, then we complete the entire group as 142857.

There are many such relationships that make such divisions faster and simpler. People will generally request division by prime numbers anyway, so it’s possible to memorize some of these repeating groups (or half of each group). Some questioners are aware that 1/97 has 96 digits in the repeating group and ask for that reciprocal. Aitken remarked that this was sometimes asked and then rattled off the answer, and we will see a critique in the *Media* part of this essay of such a question posed in a modern documentary.

Aitken used these properties of reciprocals to decimalize fractions, but he also would use straight division but with a simpler divisor, making corrections as needed in each step. For example, when a divisor ended in 9, such as 1/59, he would divide instead by 60 as described in his talk:

6)1.016949152 = .0169491525…

where the adjustment for the simpler divisor amounts to adding the previously obtained digit to the next digit in the dividend (here always 0).

If there is situation that involves dividing by, say, a four-digit value, we can try to reduce the denominator to an integer of one or two digits at most, as short division by numbers of this size are not too difficult. First, we convert the denominator to an integer by shifting its decimal point and shifting the decimal point in the numerator by the same amount. For example, 4.657/.07 = 465.7/7 = 66.53 to four digits. Then we look to simplify the fraction by dividing the numerator and denominator by low common factors. For example, .2420/7.2 = 2.420/72 = .605/18 = .0336 to four digits. We could have twice divided through by 2, but the last two digits of both numbers are divisible by 4, so the entire numbers are divisible by 4. The division by 18 can be done directly (I would count up by 18’s here, so for 60 we have 18–>36–>54 gives 3 remainder 6, then for 65 we know 54 again gives 3 remainder 11, then for 110 we double 54 to give 6 remainder 2, etc.), or we can divide .605 by 2, then by 9. Division by 2 is easiest if the number is split into even number groups, so .605 is split into .(60)(50), so half of each even group gives .3025, and dividing this by 9 yields .0336 as before. In other words, we can divide the denominator by a convenient factor even when the numerator is not evenly divisible by it, e.g., 35/36 = 5.833/6 = .9722 .

We can also adjust the denominator a little bit to get it to a round number as long as we adjust the numerator by the same percentage. If we are solving 247/119, we see that the numerator is about twice the denominator, so if we adjust 119 up to 120, we need to adjust 247 by about 2, and we arrive at 249/120 = 24.9/12 = 2.0750 compared to the actual value of 2.0756… With experience, we might notice that 247 is twice 119 plus about 10%, so we could add 2.1 to 247 to get a more accurate 24.91/12 = 2.0758. If we have 91.5/353, we can adjust the denominator down to 350 and double the fraction to have a single-digit division, so 91.5/353 = 90.75/350 = 181.5/700 = 1.815/7 = .2593 , where we reasoned that decreasing 353 by 3 was roughly equal to decreasing 91.5 by ¾ . Our answer will be a bit high, since 91.5 is a bit more than 1/4 of 353, so we might subtract a tiny bit from our answer (which is in fact in excess by .0001). This shifting technique may not seem like much, but as a graduate teaching assistant I impressed more than one physics class by using it to mentally calculate answers to problems.

Finally, we can generalize an approximation that is valid for small b, that is

1 / (1+b) ≈ 1−b

to get

a / (c+b) ≈ (a/c) (1 – b/c)

a / (c-b) ≈ (a/c) (1 + b/c)

The error here is about .01 of a/c when b/c is 1/10, and about .0001 of a/c when b/c is 1/100 , low for both approximations. This is a nice alternative to shifting the denominator when the numerator is not a simple multiple or fraction of the denominator. For example, 27/61 ≈ (27/60)(1 – 1/60) . Here we can find 27/60 = 2.7/6 = .4500 , then subtract .4500/60 = .0450/6 = .0075 to get .4425 compared to the actual value of .4426 . Since 1/60 is 1/6 of 1/10 and the error follows a square law, we are low by about (.45/36)(.01), or .0001 , but this is for better calculators than me.

In short, long division should not be as intimidating as it might seem, particularly since we have flexibility in our accuracy. If the problem is difficult to rearrange, we settle for less accuracy; if it can be easily manipulated, we take what we are offered.

**Factoring and Primality Testing**

Factoring, I imagine, would have fired the imagination of a lightning calculator. Here every carefully preserved number fact, every trick in the book could be thrown at the problem in a wild attempt to unlock the puzzle in a highly-charged atmosphere of anticipation. If an answer were to emerge immediately, either through luck or a creative leap, the solution is recorded as an instance of true genius. The fact that methods of factoring are not commonly known, and that no closed form method of factoring exists in general, lends this feat an aura of mystery that high-order roots once had prior to calculators. In fact, determining whether a number has no factors (other than 1 and itself, of course)—or in other words declaring a number to be prime—is more difficult than finding factors of a compound number. This one category may be the true measure of the depth and brilliance of a mental calculator. Many were adept at it, including Klein and Aitken. At 8 years of age, Zerah Colburn could factor 6-digit numbers or declare them prime.

Trial and error was the most common method, but only after reducing the possible factors of the number to a minimum. This can be done by looking at the last few digits of the given number and having memorized products that end in those numbers. As trivial examples, an even last digit such as 0,2,4,6, or 8 is obviously divisible by 2, and a last digit of 0 or 5 is divisible by 5. If the last two digits of a number are divisible by 4, the number is divisible by 4, and if the last three digits are divisible by 8, the number is divisible by 8. There are a number of divisibility tests for small primes that are commonly known (see **here**). Beyond that, lightning calculators often knew all products of two numbers that would end in any two digits, and it’s a good bet that they knew a lot that ended in various 3-digit numbers. These would significantly limit the number of factors to verify by multiplication, which only have to be tested up to the square root of the given number, and unless the number turns out to be prime there is no need to test every one before a true factor is found.

The mathematician Fermat produced the first methodical method of finding factors of integers. Since a^{2} – b^{2} = (a+b)(a-b), then if two squares can be found whose difference equals the given number, two factors will have been found. As mentioned in the *Fast Multiplication *section, squaring numbers is generally easier than multiplying two numbers, and calculators could also memorize tables of last digits of differences of squares to limit these possibilities as well. To simply test whether a number of the form (4n+1) is prime, possible **sums **of two squares could be checked, as a prime of this form can only be expressed as a single such sum (Smith reports that Aitken and Klein used this fact).

Factoring is a fun diversion. I know at least two people who practice factoring car license plates or the last few digits of the odometer while driving. Not recommended.

**Integer Roots**

Producing high-order roots of perfect powers is extremely common, generally possessing all the drama of factoring or primality testing (and assuredly more) without the nuance or difficulty of the latter. It makes great press, though (see the later discussion on the media). I say “generally” because at the highest levels of this task, a distinction lost on the public, a calculator does have to stretch his/her capabilities in remarkable ways to find the answer. Klein was an expert on this, along with Dagbert and Marathe, but it’s safe to say that integer roots were asked of all lightning calculators, then and now.

As a good rule of thumb, the difficulty of extracting a root does **not** depend on the order of the root (unless it is an even root, which is rarely asked) but rather on **the number of digits in the answer**. This is critical in any evaluation of such a feat—remember this the next time you hear that someone extracted the cube root of a number near a billion, or the 13^{th} root of a 39-digit number or the 23^{rd} root of a 69-digit number, all of which have at most 3-digit answers.

It turns out that the last digit of a root of an order (4k+1), such as a 5^{th} root, a 9^{th} root, etc., is the same as the last digit of the power, so for example the 13^{th} root of 79,469,020,066,571,739,979,222,359,560,551,645,783 has a last digit of 3. The last digit of a root of an order (4k+3), such as a cube root, a 7^{th} root, etc., is different but unique compared to the last digit of the power, so with some memorization the digit-pairing is also known for these roots. Between these two rules we have one of the three digits of an exact, odd root.

Now the highest digit can be found by memorizing the ranges of powers for the various starting digits 0 through 9. In our example above, of the 13^{th} root of 79,469,020,066,571,739,979,222,359,560,551,645,783 we might have memorized the fact that the 13^{th} power of a 3-digit number starting with 8 ranges from 55×10^{36} to 250×10^{36}, and here we have 79×10^{36}, so we now immediately know the number is of the form 8n3, where n is the final digit to determine.

We can use remainders from divisibility tests to find the missing middle digit. For example, the remainder after a cube is divided by 11 (the 11-remainder) is uniquely paired with the remainder after its cube root is divided by 11, as

(0,0), (1,1), (2,8), (3,5), (4,9), (5,4), (6,7), (7,2), (8,6), (9,3), (10,10)

We can find the 11-remainder by subtracting the sum of the even-place digits of a number from the sum of the odd-place digits, then adding or subtracting multiples of 11 to find a number between 0 and 10. For example, if we have a cube 300763, the 11-remainder is (3+7 +0) – (6+0+3) = 1. Therefore the 11-remainder of the cube root is, from the pairing, also 1. We know from earlier than a cube and its root has a unique pairing of last digits, which we can also memorize as

(0,0), (1,1), (2,8), (3,7), (4,4), (5,5), (6,6), (7,3), (8,2), (9,9)

So the last digit of the cube root must be 7 since 300763 ends in 3, and since the cube is less than a billion it is a 2-digit number n7. The 11-remainder (7-n) must equal 1, so n = 6 and we find the cube root of 300763 to be 67.

The 9-remainder can be tried for fifth roots, as it produces (power, root) pairs of

(0,0), (0,3), (0,6), (1,1), (2,5), (4,7), (5,2), (7,4), (8,8)

The only ambiguity is when the 9-remainder of the fifth power is 0, and in this case the 11-remainder can then be used to distinguish them.

In our running example of the 13^{th} root of 79,469,020,066,571,739,979,222,359,560,551,645,783 the 13-remainder will be the same as the 13-remainder of the root. We could do short division by 13 working from left to right one digit at a time, or since 7x11x13=1001, we can divide out multiples of 1001 from the original number by subtracting each thousands group from the thousands group to its right:

79,469,020,066,571,739,979,222,359,560,551,645,783

390,020,066,571,739,979,222,359,560,551,645,783

(-370),066,571,739,979,222,359,560,551,645,783

436,571,739,979,222,359,560,551,645,783

135,739,979,222,359,560,551,645,783

604,979,222,359,560,551,645,783

375,222,359,560,551,645,783

(-153),359,560,551,645,783

512,560,551,645,783

48,551,645,783

503,645,783

142,783

641

Dividing 641 by 13 we arrive at a 13-remainder of 4. So 8n3 must leave 4 as a 13-remainder, and it doesn’t take long to find the middle digit to be 2 and we have found the 13^{th} root of a very large number indeed.

With some additional memorization of two-digit endings of powers it’s possible to get the last **two **digits for a given root (and calculators often specialize in certain orders of roots), and this is also possible to do for the first two digits. This provides the ability to find roots of greater numbers of digits. Klein also used logarithms he memorized to calculate the first five digits of the answer, which also increased his range—he is an example of someone who has raised the bar on these calculations to extremely impressive heights. Alexis Lemaire, a present-day lightning calculator is another—his specialty is finding the 13^{th} root of 200-digit numbers, which contains up to 16 digits.

I might add that I use the variable precision arithmetic (*vpa*) command in the MATLAB software package to generate arithmetic results to many digits. Here the command “vpa 823**13 200” will provide all digits of 823^{13} up to a maximum of 200 digits. **Octave** is a free open-source alternative to MATLAB that is designed to accept MATLAB commands.

**Non-Integer Roots**

Irrational roots, that is, decimal roots of number that are not perfect powers, is historically rare, although it is more popular today because it is easy to use calculators and computers to generate problems. Aitken was able to approximate square and cube roots using numerical approximation techniques he was aware of as a mathematician.

Aitken could find the square roots of non-squares to five significant digits in about 5 sec. From an initial approximation n (a decimal or fraction) of the square root of a number N, he used the Newton-Raphson method for iterating a function to find a correction as (N – n^{2}) / (2n). So for N=85, we can estimate the square root as 9 and find a better answer as 9 + (85 – 9^{2})/18 = 9.22 compared to the actual value of 9.219544… A closer initial value yields a much closer answer, so if we can do two-digit multiplications and divisions we can take an initial estimate of 9.2 to find a better answer of 9.2 + (85 – 9.2^{2})/18.4 = 9.219565…

Cube roots can be approximated by a similar mechanism—for a description (and for more examples of square roots) I heartily recommend reading Smith’s book or Aitken’s 1954 talk on mental calculation **here**.

If logarithms and anti-logarithms can be mentally calculated, this provides a different way of approximating roots, even higher-order roots. The 12^{th} root of N, for example, can be calculated by finding log N, dividing by 12, and then finding the antilogarithm, all at whatever accuracy the calculator can produce. Klein used his memorized logarithms and simple interpolation to do this.

**Powers of Integers (Involution)**

As we saw earlier, squares and cubes of numbers offer advantages to the calculator. In general it’s easier to use the binomial expansion of (a+b)^{n} for a round number “a” and a small correction “b” than to multiply the number (a+b) by itself n times.

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

etc.

When multiplying, Mondeux would factor problems if possible, and if this reduced the problem to powers such as squares and cubes, he would employ the binomial expansion. In 1952 Klein raised 87 to the 16^{th} power, which Smith assumes was most likely by successive squaring. Marathe is an expert on raising single digits to powers up to 20 (but how many different results is this, really?). Euler is reported to have mentally calculated the first six powers of all numbers less than 20 in one restless night.

**Logarithms**

Logarithms occur naturally in formulas, and as we have seen earlier they can be used to find roots of any order, even fractional roots, if there is a means of finding the antilogarithm as well.

We saw earlier than Klein had memorized the logarithms of the first 150 integers to 5 digits. Bidder had memorized those for the first 100 integers to 8 digits. By factoring a number and scaling the factors by multiples of 10 when needed, corresponding logarithms can be added to find the logarithm of the answer. For example, log 483 = log (3 x 7 x 23) = log 3 + log 7 + log 23 = 0.47712 + 0.84510 + 1.36173 = 2.68395. But what about log 487? Well, 487 = 483(1+ 4/483) ≈ 483(1 + 1/120) so log 487 ≈ log 483 + log(1 + 1/120). The well-known power series for the natural logarithm (denoted by ln) of a value 1+x for x ≤ 1 and x ≠ -1 is:

ln (1+n) = n – n^{2}/2 + n^{3}/3 – …

We can truncate this series very quickly if n is small. To find the common logarithm (to base 10) rather than the natural logarithm (to base e), we have to multiply ln(1+n) by log e = .4343. So

log(1 + 1/120) = .4343 ln (1 + 1/120) ≈ .43/120 = 0.00358

Adding this value to our earlier result log 483 = 2.68395 we arrive at log 487 ≈ 2.68753 which is actually correct to the last digit shown.

There are other methods as well. Bidder used the following relation to arrive at the last correction above:

log (1 + n) ≈ 10^{m} n log (1+10^{-m})

where m is chosen so that 10^{m} n lies between 1 and 10. Bidder memorized the values of log (1+10^{-m}):

log 1.01 = 0.00432…

log 1.001 = .000434…

log 1.0001 = .0000434…

etc.

where the digits approach log e = .4343… as m increases. The correction above was log (1 + 1/120) = log (1 + .00833), so m=3 will give 10^{3} x .00833 = 8.33 and multiplying this by log 1.001 = .000434 we arrive at .0036 if we simplify the multiplication to 2 places. This is quite near the correction we calculated by our last method.

Note that n can be positive or negative in these relations, so the relations are useful when it is easier to find the logarithm either above or below the desired number.

**Sum of Four Squares**

Every positive integer can be written as at least one sum of four squares, so this task was occasionally asked of lightning calculators, particularly those who specialized in it such as Ruckle, Finkelstein and Klein. As a typical case, Ruckle expressed 15663 as a sum of four squares in 8 sec, followed immediately by a second sum. The same was done for 18111 in 26.5 sec and 63.5 sec, and for 53116 in 51 sec immediately followed by a second sum.

Like factoring, a solution for reducing an integer to the sum of four squares cannot be expressed in closed form, and success relies in part on the experience and cleverness of the calculator. I have written a paper summarizing methods for such reductions (a subject not covered by my book, by the way) that can be found **here**.

In a different vein related to squares, given an integer c, Mondeux could find two squares a and b that have a difference of c. He apparently knew that if d = a – b, then b = (c – d^{2}) / (2d). Then it becomes a matter of finding d such that b is a positive integer, whence a = b + d. If c is odd then he could set d = 1 and then b = (c-a)/2 and a = b + 1.

**Calendar Calculations**

Calendar calculations are probably the most commonly performed feat of calculators, particularly aspiring calculators, but this happens to be my least favorite task. It usually involves finding the day of the week for any day in history, which has to take into account leap years and the Gregorian calendar change (which was adopted in various years by various countries, actually). Since this is an area I haven’t studied in detail, I will simply provide some good websites that describe calendar algorithms: **here**, **here**, **here**, **here**, and **here**.

An algorithm for mentally computing the phase of the moon with 2-day accuracy between 2000 and 2009 can be found **here**.

**Compound Interest**

Bidder mentally calculated simple interest on money at 10 years old and compound interest later in life as described in his 1856 talk found **here**. This interest is periodically compounded rather than continously compounded, which would require calculating exponentials. I am not aware of any other historical calculator who dealt with this area of mathematics.

**Newer Methods**

New methods for calendar calculation seem to appear now and then, and I presume these are being used by some. My book from 1993 also contains quite a few algorithms invented or adapted for mental calculations to high precision. In addition I have written quite a few papers on methods—the papers linked below reside on the Online Materials page of the section of my main website devoted to the book.

For example, in the *Logarithms *section above we saw a method of calculating the logarithm of a number based on a nearby round number N whose logarithm is much easier to find. However, there are various other approximation schemes for finding such a logarithm. The most generally useful one of these, I think, is the following relation for n small compared to N:

ln (N + n) ≈ ln N + 2n/(2N + n)

Compared to the problem in the earlier *Logarithms *section, the correction term to add to log 483 to find log 487 when using this new formula is .4343(8/970) = 0.00358, accurate as before to the last digit. If additional digits were taken for ln N and for this correction term, it would be found to be more accurate than the earlier result obtained by truncating the first term of the power series.

It’s also possible to extend the Newton-Raphson method to churn out digits of a square root one or two at a time indefinitely, or at least until the calculator has reached their limit of time or ability. It’s not necessary to read the book for this, as the method is provided in full in the papers found **here**, **here** and **here**.

Manny Sardina has produced approximation algorithms for integer and fractional roots of numbers based on continued fraction representations in a paper found **here**.

There are also various methods for calculating exponentials that were not used historically by mental calculators. The most promising one from the book is detailed in the paper **here**.

John McIntosh has discovered another method for exponentials that only requires knowing log 2 = .3010300 and log 3 = 0.477121. His presentation of this can be found **here**.

Algorithms for approximating trigonometric functions are also presented in the book. And in an odd grouping of functions sharing a similar approximation technique, I have written a paper that describes methods for mentally calculating the tangent, hyperbolic tangent, exponential and logarithmic functions to high accuracy, found **here**.

Finally, although lightning calculators historically could find products of two 4-digit problems or more (Zerah Colburn at 7 years old could multiply two 4-digit numbers), I wrote a paper on what I believe is an easier way to perform this task, one that is particularly useful when the difference between the first half and the second half of one of the numbers is small, found **here**.

It seems that some modern calculators have picked up on some of these methods, particularly the one for inexact square roots, which has now appeared in simplified form in a couple of other books.

This, then, represents a short summary of some of the methods that have been developed for mental calculation. Again it is important to realize that lightning calculators historically developed highly individualized ways of doing things, and many of those ways were fairly inefficient. But optimum efficiency was not necessarily critical, particularly considering the lack of objectivity among those reporting the exploits of these individuals. To grasp the true history of lightning calculators and their art it is important to recognize this media partiality, and this is the subject of the next part of this essay.

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**<<< Back to Part I of this essay** **>>> Go to Part III of this essay**

**Printer-friendly PDF file of Lightning Calculators Parts I-III.**

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.

The subject of lightning calculation has been an interest of mine for many years. Although I’m certainly not a lightning calculator, as a graduate teaching assistant in physics in the early 80’s I enjoyed mentally calculating the results of problems to quite high accuracy while the students were working their calculators, and I would typically end the semester with a class on such methods. In 1988 I started gathering material for a book on methods of high-precision mental calculation of arithmetic as well as elementary functions such as logarithms, exponentials and trigonometric functions (*Dead Reckoning: Calculating Without Instruments*, 1993). A few years ago I started my main website MyReckonings.com mainly to devote a portion of it to notes and errata for the book, as well as serve as a repository for papers I’ve written on topics of mental calculation. The website area devoted to the book can be found **here** and the page devoted to additional papers is found **here**, where some of the links in this essay are directed.

The history of lightning calculators, at least to 1983, is presented most comprehensively in Steven B. Smith’s book, *The Great Mental Calculators: The Psychology, Methods and Lives of Calculating Prodigies Past and Present*. This is a fascinating read and an honest attempt to analyze the capabilities and methods of a number of these individuals. Smith notes that isolation (at least mental isolation), generally in children, is a condition favoring the development of this ability, and it’s hard to argue with that. He describes a gamut of lightning calculators who run the spectrum of mental acuity.

So let’s see who we have in the book. Grouping people by their overall mental acuity is a dangerous sport prone to misinterpretation and error (for example, it is impossible for me to classify Thomas Fuller (1710-1790) because he was a victim of the slave trade in America). As a rough interpretation by me from Smith’s book, among those that seem to be of low intelligence (which represents a large range) are

- Jedidiah Buxton (1702-1792)
- Henri Mondeux (1826-1861)
- Jacques Inaudi (1867-1950)

Among those who seem of average intelligence are

- Zerah Colburn (1804-1839)
- Johann Martin Zacharias Dase (1824-1861)
- Pericles Diamandi (1868- )
- Arthur Griffith (1880-1911)
- Salo Finkelstein (1896/7-?)
- Maurice Dagbert (1913-?)

Those who seem to have exceptional intelligence include

- George Parker Bidder (1806-1878)
- Truman Henry Safford (1836-1901)
- Frank D. Mitchell
- Gottfried Ruckle (1879-1929)
- Wim Klein (1912-1986)
- Hans Eberstark (1929- )
- Shyam Marathe (1931- )
- Shakuntala Devi (1932- )
- Arthur Benjamin (1961-)

And those with ability in this area who left a permanent mark on mathematics and science certainly include

- Alexander Craig Aitken (1895-1967)
- John Wallis (1616-1703 )
- Andrè Ampere (1775-1836)
- Leonhard Euler (1707-1783)
- Karl Frederich Gauss (1777-1855)
- John von Neumann (1903-1957)

So in fact we see a predominance of ability in those with higher mental acuity, and it turns out that as we proceed to later names here we find that these abilities remained or were developed in adulthood. Savants are certainly more fascinating because of their lack of ability in other areas, but the talents of those without disabilities is certainly in contrast to the popular conception of lightning calculators.

Watch a child who is doing math homework—when they are calculating the answer they generally get quite physically agitated, tapping a pencil, shaking or hitting their heads, standing up and sitting down, talking, etc. It’s quite striking when you are looking for it, a strange association between mathematical reasoning and motor functions that makes you wonder if the standard, ultra-quiet testing environment in school is really ideal. Some (probably most) lightning calculators such as Inaudi, Colburn, Safford, and Benjamin, were or are quite agitated while performing. These are termed auditory calculators, but there are visual calculators as well (Diamondi, who had a “photographic” memory, Ruckle, Marathe, Dase, etc.) and those who don’t fit neatly into either category (Klein, Aitken). There are also those who experience synaesthesia, seeing colors when hearing or visualizing numbers (such as Daniel Tammet, who also visualizes “landscapes” and “spirals” of numbers. Smith even describes a “tactile” calculator.

Another topic of great interest and historical misunderstanding concerns the calculation process itself. It is often assumed that the results are spontaneously produced by an unconscious, mysterious and instantaneous process. Smith concludes that this is false, that the calculation proceeds through a sequence of operations that is conscious or semi-automatic, much like spoken language or touch typing. The brain scan figures that are shown later, in fact, show in red the areas of the brain used by the modern lightning calculator Rüdiger Gamm (in green and red) compared to several non-expert calculators (in green) as described in a paper **here**.

So how fast were/are these lightning calculators? The short answer is that the reported times, those that have any validity at all, are all over the map. Often those who report on times did not recognize attributes of specific problems that led to easy solution, or based their reports on second-hand or promotional material, or ignored delay tactics such as writing down or repeating the problem. Often the reports don’t indicate when timing began and stopped, and it often goes unreported whether the problem was in sight during the calculation and whether the answer was produced digit-by-digit or as a complete solution. Some times are considered beyond credibility or markedly inconsistent with the difficulties of various questions.

Smith attempts to sort through the available data on historical calculators, a seemingly frustrating enterprise. A decent set of tests were conducted by the noted psychologist Alfred Binet in 1894 and there was so much confusion on the best way to measure response times that a device that traced respiration on a revolving cylinder was settled on, but even Binet didn’t record whether the answers were written down or if the first or last digit of the answer was the initial trigger. Inaudi took an average of 2.0 seconds for 2-digit by 2-digit (2×2) multiplications, 6.4 seconds for 3×3 multiplications, 21.0 seconds for a 4×4 multiplication, and 40.0 seconds for a 6×6 multiplication. Diamandi did much worse but the problem was removed from view during the test and timing was definitely stopped after the last digit was written, so it is not a fair comparison. Ruckle and Finkelstein did worse than Inaudi in later identical tests. Klein was demonstrably fast, but in tests in 1953 he was allowed to view the problem and write down digits as he obtained them, a definite advantage for someone like him who used cross-multiplication. Nonetheless his time of 48 seconds to multiply two 9-digit numbers and 65-2/3 seconds to multiply two 10-digit numbers is very impressive. Klein also extracted integer roots of numbers, particularly 13th roots of 100-digit numbers, achieving a 1 min 28.8 sec time at one point.

As for the size of problems, Dase reportedly multiplied two 8×8 digit multiplications in 54 sec, a 20×20 digit multiplication in 6 min, a 40×40 digit multiplication in 40 min, and a 100×100 digit multiplication in 8 ¾ hours. Gauss later posed the question of who checked that last answer (adding that it was “a crazy waste of time”), and in fact lightning calculators often made errors, but of course those aren’t typically reported. Buxton, who could not read or write numbers, squared 725,958,238,096,074,907,868,531,656,993,638,851,106 in his head over the course of 2-1/2 months, an astonishing feat of concentration that is scarcely marred by an error of one digit in the answer (this was an attempt to square 2139 but based on a flawed value for 2138). Alexander Craig Aitken extracted non-integer roots among other calculations, merging his natural speed in arithmetic with mathematical approximation and iteration formulas. And John Wallis before him extracted the square root of 3×1040 to 21 digits during one sleepless night and of a 55-digit number to 27 places during another night. Dase is also reported to have extracted square roots of perfect squares of 100 digits and 60 digits, but with no times given.

The biographical details of these lightning calculators make interesting reading but are not the focus of this essay. There are a number of sites that provide the history of their lives or links to them, such as those listed found **here**, **here**, **here**, **here**, **here**, **here**, **here**, and especially Oleg Stepanov’s site that contains many historical articles on various mental calculators **here**.

There are modern-day lightning calculators, of course. Some of those listed above who have a good amount of history in this field (such as Arthur Benjamin and Shakuntala Devi) still perform in public. There are also relative newcomers that I am aware of through the Yahoo Mental Calculation Group, such as Rüdiger Gamm, Gert Mittring, Alexis Lemaire, Robert Fountain, George Lane, John van Koningsveld, Alberto Coto, Willem Bouman, Andy Robertshaw, Matthias Kesselschläger, Yusnier Viera Romero and Jorge Arturo Mendoza Huertas, but this is a highly Eurocentric view (other than the last two) because of the makeup of the Yahoo group and the participants in the Mental Calculation World Cup held in Europe. Unfortunately, I’m not familiar with the many other newer lightning calculators from around the world—Chan Hee Yi of Korea has been pointed out, and India certainly has a number of such talented individuals. (I nearly decided not to list any modern calculators to avoid slighting other people who certainly deserve to be listed—rest assured that all omissions are due to my limited knowledge in this area.)

Below are some links to good videos on lightning calculators in various languages:

- Wim Klein and Hans Eberstark at CERN (where Klein worked) from 1973 is found
**here**. - Wim Klein at CERN in 1959 is found
**here**. - A two-part documentary on Rüdiger Gamm can be found
**here**and**here**. - Videos of Arthur Benjamin are found
**here**and**here**. - A lecture by Gert Mittring on extracting cube roots is
**here**. - A video on Jan van Koningsveld is found
**here**. - A Dutch TV program on Willem Bouman is found
**here**.

The history of lightning calculators is interesting from a human standpoint, but it’s perhaps more intriguing because the methods they learned or developed are uniquely suited for fast mental calculation. These methods are different from the ones taught in school for pencil-and-paper solution, and therefore most people are quite surprised when they find out that other algorithms such as these exist. Techniques designed specifically for mental calculation are the subject of the second part of this essay.

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**>>> Go to Part II of this essay**

**Printer-friendly PDF file of Lightning Calculators Parts I-III.**

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.

We are interested in designing a nomogram for the following equation in m, l, k and θ:

where θ > 0 and m and l lie between 0 and 100. However, the general design here is valid for other ranges of these variables.

Below is the completed nomogram, including an isopleth for the solution l = 20, m = 40, k = 50 and θ = 25°. The derivation of the design follows this figure. A high-resolution version of the nomogram can be found here.

To create this nomogram, we first draw a 100×100 grid with the origin at (0,0). The m-scale lies along the left side and increases from bottom to top. The l-scale lies on the right side and increases from top to bottom. In terms of x and y, these scales can be described by the equations (a) and (b):

The slope of the line drawn between the l and m scales can be expressed in terms of either variable:

Substituting equations (a) and (b) into (c), we arrive at

We also have the following equation that must be satisfied for this isopleth:

By substitution, equations (d) and (e) can produce independent equations for l and m:

The next step is key. These equations must be valid for any values of l and m. Therefore, if we rewrite l and m as

then l and m are arbitrary only if A=0, B=0, C=0, and D=0. Setting A=B=0 in the first equation in (f) and solving the two resulting equations for x and y provides

The same set of equations for x and y is obtained when we set C=D=0 in the second equation in (f).

Now we let k = 0, 1, 2, 10, 50, 100, 150, … and plot k-curves for the variable θ. Then we let θ = 0°, 1°, 2°, 3°, 4°, … and plot θ-curves for the variable k. This forms the nomogram shown in the figure above, which provides a linear mapping of solutions to the original equation.

We can verify this result by substituting equations (a), (b) and (g) into the standard determinant form that describes our nomogram:

After substitution we arrive at

which is true from our original equation.

The method is equivalent to converting an equation into determinant form as

This method is generally suitable for 3, 4, 5, or 6-variable equations, but is complicated for equations of 5 or 6 variables.

四变量诺模图（李留念 和Ron Doerfler）

设计有三个以上变量方程的诺模图是很有难度的一件事。在同类问题中，大多数都借用轴点，还需要用户自己创建一组等值线才能求得最终结果。在此客户的这篇论文中，李留念概述了一种巧妙的诺模图设计过程，仅需要一条等值线就能求得四变量方程。为了方便大家，此方法用英语和汉语两种语言描述。

我们有意于给下面方程设计一诺模图，其含有变量m、l、k和θ：

此处θ>0，m和l在0到100之间。此设计一般也适用于这些变量的其他范围。

以下是完整的诺模图，且含有一条满足l = 20, m = 40, k = 50 和 θ = 25°时的一条等值线。得到次图的推导过程紧随其后。此诺模图的高分辨率版可以在此处找到。

为了创建此诺模图，我们首先画一100X100的方框，（0,0）为原点。m坐标轴在左侧，方向自下向上且均匀增加。l坐标轴在右侧，方向自上至下且均匀增加。用x和y坐标表示两坐标轴方程为（a）和（b）：

在l和m之间直线的斜率可同时用几个变量表示为（即直线方程）：

把(a)和(b)代入(c)，可以得到

同时我们也必须让下面方程满足等值线：

通过代换，由(d)式和(e)式可求得l和m，两式且不相关：

接下来这一步是关键。在l和m为任意值时这些方程都必须满足，因此，若把l和m再写为：

只有当A=0、 B=0、C=0和D=0都成立时，l和m才可为任意值（即l和m与（f）式中其他四个变量不相关）。令（f）式中第一个方程A=B=0，求解两方程得x和y

由（f）式中第二个方程C=D=0联立求解也可得到x和y同样的方程。

现在我们可以令k = 0, 1, 2, 10, 50, 100, 150……，θ作为自变量得一族k曲线。同样的，令θ = 0°, 1°, 2°, 3°, 4°……，k作为自变量可得到一族θ曲线。这样就形成了如上所示的诺模图，他提供了一种对原方程进行线性变换的解法。

我们可以验证此结果，把求得的(a)、(b)和(g)代入到标准行列式中：

代入后求得：

和原方程相符。

这种方法相当于把方程式变换为行列式，就像

一样。

这种方法广泛适用于3、4、5和6变量方程，但是对于5和6变量方程又过于复杂。

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]]>

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.

To review, Epstein displays two examples of nomograms based on these elliptic functions as shown below. A line (isopleth) crossing any three points anywhere on the entire curve produces three label values that add to zero. When two of the values are identical, the isopleth is drawn tangent to the curve at that point. In the first figure, the isopleth must cross two numbers on one scale and a third number on the other overlaid scale (such as u = +0.2524, v = +0.3842 and w= -0.6366). In the second figure the isopleth generally crosses both curves, although it could just cross the rightmost curve three times along the shallow curve if the scale were stretched enough to distinguish the crossing points.

The values on the x and y axes are not used in reading the nomogram but rather relate to the plot of the Weierstrass’ Elliptic Function providing the outline of the nomogram:

The first plot above is for g_{2} = 1200 and g_{3} = -12000 (with one real root around x = -21), while the second is for g_{2} = 1200 and g_{3} = -4000 (with three real roots at about x = -19, 3.5 and 15.5). Somewhere between these two functions (at g_{2} = 1200 and g_{3} = 8000) the two curves just touch, representing a double real root. These plots were presented as a demonstration of the level of some of the mathematics under development for nomograms at the time.

Looking at this in more detail now, the scale values for all points on these curves are calculated from

Like most things elliptic in mathematics, this integral cannot be solved directly but must be numerically computed to the accuracy desired. Epstein’s book from 1958 discusses methods of approximating the integral in different ranges; today we have computer programs that can easily perform numerical integration. In practice the positive-value scale above the x-axis in the first figure above is initially calculated from this integral. Now the difference in any two lower scale values is the negative of that for the upper scale, so the positive-value scale that wraps below the x-axis is calculated at each x by adding the difference between the value at y=0 and the value at x on the upper scale to the value at y=0. For example, the value +.7803 on the lower scale is found by adding (0.6751 – 0.5699) to 0.6751, and in this way the entire positive-value scale can be labeled. The negative-value scale halves are simply labeled with negative values of their opposite positive-scale halves.

Epstein provides a proof that these scales on this family of curves produce a nomogram in which three crossed values add to zero.

Now assume we have the more general formula f(r) + g(s) + h(t) = 0. We can then assign scale values for r, s and t such that f(r), g(s) and h(t) result in the original scale values found above, and we can use the same curve. Notice in the first figure above that when an isopleth intersects the upper curve at two or three points, the negative-value is added to the two positive-values. So we can map, say, f(r) to the negative-values and g(s) and h(t) to the positive-values. However, when an isopleth intersects the lower curve in two or three points, two negative-values are added to a positive-value, so we can continue to map, say, f(r) and g(s) to the same negative- and positive-values as they wrap continuously around through the lower curve, but h(s) has to change its mapping to negative-values. This took me a while to realize, and it means that h(s) scale values will be discontinuous when crossing the x-axis. We will see how I handled that.

Let me say a word about the software tools we will use in creating the fish nomogram. I hasten to add that nomograms can be calculated and plotted by hand or in Excel or PowerPoint or Word or AutoCAD or Visio or whatever—I use tools that I’m familiar with from other applications and these will probably be different tools than you would use. The only real software need for this project is a math program to do numerical integration to find the default scale values for a given set of g_{2} and g_{3}, which only needs to be done once. However, one of my aims was to create a general set of tools that allow me to quickly create new nomograms based on elliptic functions.

So here are my specific tools. All software listings will be linked in this essay. The MATLAB program is used to perform the numerical integration and to provide plots for solving a system of equations for the desired scale ranges. This is an expensive program, but GNU Octave is a free, open-source program for various platforms that is mostly compatible with MATLAB scripts. The mathematics typesetting engine LaTeX is used to output the PDF of the actual nomogram. I use the free MiKTeX distribution of LaTeX, and the free, matching editor and user interface TeXnicCenter. I also use the powerful Ti*k*Z drawing package for LaTeX, which also provides plotting capability for PDF output once the free gnuplot program is installed. LaTeX has limited programming capability, so I use a separate program to calculate label values and coordinates for the scales and store them into text files that are read in by the LaTeX program. This program is written in the free, open-source scripting language Tcl, for which I use the free ActiveTcl download. However, Tcl and its companion TK are installed by default on most Unix systems and on Mac OSX.

To begin, we need to find values of g_{2} and g_{3} for the elliptic function that provide the best overall fish outline—I settled on g_{2}=1200 and g_{3}=-8500. (I used the plot command in LaTeX that you will see in the later listing, but any plotting program will do.) We can calculate the positive-values on the upper curve for each x-value by running the script below in MATLAB.

g2 = 1200.0;

g3 = -8500.0;

h=@(x) (1.0 ./ sqrt(4*x.^3-g2*x-g3));

for xvalue = -40:1:40

% Output x

xvalue

% Do numerical integration and output result

result = quad(h,xvalue,10000000000)

end

And, yes, I did have to take the integration all the way out to x=10,000,000,000 to get the answer to four decimal places. But this was completed instantly in MATLAB on my PC, indicating that the program does an intelligent distribution of integration intervals. This script prints out the values to the screen, but I also added some script lines to print the output values to a file as well for convenience—this longer version can be found here. The presence of a non-zero imaginary component in any result indicates that x lies off the end of the curve and there is no y-value to plot.

The wrap-around part of the positive-value scale as well as the negative-value scale and its wrap-around to the upper curve are found by simple manipulations of these numbers as described earlier. Below is the nomogram for **u + v + w = 0** for g_{2}=1200 and g_{3}=-8500. Click here for a full-resolution version.

Looks like a fish to me. But we want to develop this nomogram for a more general equation than **u + v + w = 0**. I found a fish-related equation developed by Liao that provides the oxygen consumption rate O_{2} in weight units of dissolved oxygen (DO) per 100 weight units of fish per day as a function of water temperature T in °F and average individual fish weight W in lbs.:

O_{2} = KT^{n }W^{m}

where the rate constant K and the slopes m and n for rainbow trout are given by:

for T>50: K = 3.05×10^{-4}, n = 1.855, m = -0.138

for T<50: K = 1.90×10^{-6}, n = 3.130, m = -0.138

Apparently this is important in the design and calibration of aerators for rainbow trout “racetracks” at trout farms.

Since there is a different equation for T<50 than for T>50, we can take advantage of that to skirt the difficulty of the discontinuity of one of the positive-value scales as it wraps around from the upper curve to the bottom curve as mentioned earlier. We will implement the equation for T>50 on the upper curve, with T using a positive-scale, W using a negative-scale, and O_{2} using a positive-scale and wrapping around the lower curve continuing as a positive-scale. So for T>50 the isopleth has to cross two or three points above the x-axis since there are no W or T scales on the lower curve. This allows the third scale O_{2} to wrap around and maintain positive values. The equation for T<50 will be implemented on the lower curve, with T′ and W′ using negative-value scales and O′_{2} using a positive-value scale and wrapping around to the upper curve continuing with the positive-value scale.

Using logarithms the oxygen consumption equation can be cast in the correct format for the nomogram:

n log T + m log W + (log K – log O_{2}) = 0

The term log K could be paired with any term, actually.

So we need to place labels T, W and O_{2} on the plot above such that each corresponding term in this equation is equal to the u, v and w values shown in the plot for **u + v + w = 0**. Setting each term in the equation above to a corresponding variable u, v or w, we have

u = n log T or label “T” = e ^{u/n}

w = m log W or label “W” = e ^{w/m}

v = log K – log O2 or label “O_{2}” = e ^{log K – v} = K e ^{–v}

Consider the upper curve only for the moment. Let’s assign T to the positive-value scale, W to the negative-value scale, and O2 to the positive-value scale that wraps around to the lower curve. The range of the u scale is 0.2142 to 0.8686 for the upper curve. The range for the w scale is -0.8686 to -1.5230 (at x=24), and the range of the v scale is 0.2142 to 1.5230 if we let this scale wrap all the way around to x = 24. With these limits, the above equations for T>50 produce ranges of 4.7217<W<62083.3 and 1.1224<T<2.2839, wildly different from what we want for the nomogram.

But there are at least two ways to change the range of our T and W scales without changing the basic equation **u + v + w = 0**. First, we can multiply u, v and w by the same constant A. Second, we can add offsets u_{0} and w_{0} to u and w if we subtract (u_{0} + w_{0}) from v.

According to Wikipedia, the world record rainbow trout was a 43.6 pounder caught from the shore at Lake Diefenbaker, Saskatchewan in June, 2007. If we desire ranges of 50<T<100 and 0.5<W<50, say, then we want from the above equations

e ^{A(0.2142 + u0) / 1.855} = 50

e ^{A(0.8686 + u0) / 1.855} = 100

e ^{A(-.0.8686+ w0) / (-0.138)} = 0.5

e ^{A(-1.5230 + w0) / (-0.138)} = 50

This is an over-determined system comprising 4 equations with only 3 variables to define: A, u_{0} and w_{0}. There’s no guarantee that even 4 variables would allow this system of equations to be solved, so unless one of the equations happens to be a duplicate of another there’s no way we will be able to exactly solve this system. The best we can do is to find values of A, u_{0} and w_{0} that will approximately provide the ranges we want. The way to do this is to move all the terms to left sides of the first two equations and plot them against each other (setting y=A and x=u_{0}) to see where they intersect, thereby solving for A and u_{0} for these first two equations. Then we do the same for the second pair of equations, solving for A and w_{0} for these equations. Then we cross our fingers and hope that the values of A are about the same so we can pick one that roughly satisfies all 4 equations.

A MATLAB script to plot the two sets of equations on a single grid is shown below, followed by the resulting plot (after some interactive editing to add grids, colors and a legend).

hold on; % To overlay multiple plots

syms x y;

eq1 = ‘exp(y*(.2142+x)/1.855)-50’;

eq2 = ‘exp(y*(.8686+x)/1.855)-100’;

ezplot(eq1,[-10,10,-6,6]), ezplot(eq2,[-10,10,-6,6]);

syms x y;

eq1 = ‘exp(y*(-.8686+x)/-.138)-0.5’;

eq2 = ‘exp(y*(-1.523+x)/-.138)-50’;

ezplot(eq1,[-10,10,-6,6]), ezplot(eq2,[-10,10,-6,6]);

Looking at the intersections of the two pairs of plots, the values of A (the y-value) for them are remarkably close. So we pick the intersection most sensitive to differences (that for the second pair of equations) and we find approximately that A = 0.9725 and w_{0} = 0.9725 by interactively zooming into the plot. (The fact that these two values are identical is coincidental—I did these plots for several different ranges and they were always different). For this value of A the average value of u_{0} (the x-value) for the first two equations is about 7.65.

The only rigid constraint on our range is the minimum value of 50 for T, so we need to adjust u_{0} to achieve this. Substituting A = 0.9725 into the first of the 4 equations above we find that u_{0} = 7.2478 provides a value of 50 on the right side. This means that v_{0} = -(7.2478 + 0.9725) = -8.2203, and we end up with the following scale equations for the upper curve:

T = e ^{0.9725 (u + 7.2478) / 1.855} using positive-scale values of u

W = e ^{0.9725 (w + 0.9725) / (-0.138)} using negative-scale values of w

O_{2} = 3.05×10^{-4} e ^{0.9725 (v – 8.2203)} using positive-scale values of v

which gives ranges of 50<T<70.46 and 0.4809<W<48.40, which is acceptable. And we can now list these values in place of the original u, v and w values on our upper fish outline and we have a nomogram for the T>50 equation!

We repeat all this for the lower curve for the equation for T<50. We assign T′ to the negative-value scale on this lower curve, W′ to the negative-value scale as well, and O′_{2} to the positive-value scale that wraps around to the upper curve. With ranges of 32<T′<50 and 0.5<W′<50, say,

e ^{A(-0.8686 + u0) / 3.130} = 32

e ^{A(-0.2142 + u0) / 3.130} = 50

e ^{A(-.0.8686+ w0) / (-0.138)} = 0.5

e ^{A(-0.2142 + w0) / (-0.138)} = 50

We choose A = 0.9725 as before (although we have the freedom to choose a different A for the lower curve), with w_{0} = 0.335 and u_{0} ≈ 12.3 from the MATLAB plots. To have a maximum T of 50, u_{0} = 12.805, so v_{0} = -13.14 and we arrive at the following scale equations for the lower curve (with primes on the variables to distinguish this set of scales):

T′ = e ^{0.9725(u + 12.805) / 3.130} using negative-scale values of u

W′ = e ^{0.9725(w + 0.335) / (-0.138)} using negative-scale values of w

O′_{2} = 1.90×10^{-6} e ^{0.9725 (v – 13.14)} using positive-scale values of v

which gives ranges of 40.8<T′<50 and 0.4269<W′<42.96, which we’ll accept as well. And now we can list these values in place of the original u, v and w values on the lower fish outline for the T′<50 equation.

A Tcl script is used to perform the calculations of T, W and O_{2} and store the values along with their scaled x and y coordinates and angles. The script reads in a set of prepared files that must be placed in the directory of the Tcl script. These files contain the unscaled values of u, v and w for **u + v + w = 0** when g_{2}=1200 and g_{3}=-8500 as calculated by MATLAB and shown in the black-and-white plot above (click on the filenames to see their contents):

- positive_values_uppercurve_g2_1200_g3_-8500.txt — (x,labelvalue) pairs
- negative_values_uppercurve_g2_1200_g3_-8500.txt — (x,labelvalue) pairs
- positive_values_lowercurve_g2_1200_g3_-8500.txt — (x,labelvalue) pairs
- negative_values_lowercurve_g2_1200_g3_-8500.txt — (x,labelvalue) pairs

The following files are output to this same directory by the Tcl script for reading by the LaTeX files when placing the tickmarks and labels on the elliptic function plot. Separate files for each portion of the curve are created to ease LaTeX code.

- uppertickmarks.txt — (xcoord,ycoord,angle)
- T_upper.txt — (xcoord,ycoord,labelvalue,angle)
- W_upper.txt — (xcoord,ycoord,labelvalue,angle)
- OX_upper_upper.txt — (xcoord,ycoord,labelvalue,angle)
- OX_upper_lower.txt — (xcoord,ycoord,labelvalue,angle)
- T_lower.txt — (xcoord,ycoord,labelvalue,angle)
- W_lower.txt — (xcoord,ycoord,labelvalue,angle)
- OX_lower_lower.txt — (xcoord,ycoord,labelvalue,angle)
- OX_lower_upper.txt — (xcoord,ycoord,labelvalue,angle)

The Tcl script can be found here (remove the .txt suffix to run it). It performs the required calculations, formats the labels as 4-digit strings with a decimal point, scales the x and y coordinates for the final plot, and calculates the angles for tickmarks and labels from the slopes between neighboring points.

These output files are then copied to the directory of the LaTeX script, which plots the curve and reads label values and coordinates from the files, placing the tickmarks and labels at the required positions and angles on the curve. The script also creates the fish features such as the eye and fins, and it routes labels around the fins and adds lines within the fins to relate these labels with their tickmarks. Finally, it adds the title, legend and scale names. This LaTeX script can be found here (remove the .txt suffix to run it).

The final nomogram in full resolution can be downloaded from **HERE**, but it is also appended to the PDF file of this essay. A reduced, lower quality version is shown below for discussion purposes. The scale colors are meant to suggest a rainbow trout, but I wanted long scales on the tail so I replaced the rounded tail of the trout with a V-shaped tail. So you should consider this a simple fish illustration.

Here the scales on the nomogram marked O_{2}, T and W apply to T>=50 (as you can see since the T scale starts at 50.00 and goes up), where the blue O_{2} scale also wraps around the bottom of the fish. The scales O’_{2}, T’ and W’ apply to T<50. (I manually changed the T label on the lower curve from 50.00 to 49.99 to avoid confusion on which T scale to use.) All the scales reside along the main fish body, and the lines through the fins are just pointers from the outer scales to their tickmarks. An isopleth crossing any three points on the primed set of scales or three points on the unprimed set of scales will provide three values that solve the oxygen consumption equation.

One of the intriguing features of nomograms is that they can solve for any variable in the equation even when there is no explicit solution for that variable. In this nomogram, any isopleth that does not cross the center line will encompass 6 possible combinations of T, W and O_{2} locations (O_{2}TW, O_{2}WT, WO_{2}T, TO_{2}W, TWO_{2}, WTO_{2}). An isopleth crossing the center line will have only 2 possible combinations since O_{2} or O’_{2} must lie by itself at one end of the isopleths (TWO_{2}, WTO_{2}). All combinations in our ranges are available this way except for isopleths near the mouth at an angle where the isopleth misses the end of the tail. The ranges could have been mapped to stop at the top of the head rather than the mouth if this were really an issue. Compare this with the situation in the first figure from Epstein, in which there are many possible isopleths that intersect the curve in only two points.

Some possible isopleths are shown in the figure above. A significant amount of testing demonstrates that the nomogram works great, which is still a bit surprising to me. As an example, consider the isopleth in this figure whose right end touches the upper edge of the tail closest to the tip. This line provides oxygen consumption values for the two variable combinations listed in the upper table on the right.

As another example, the isopleth that crosses three points on the upper curve provides the six variable combinations shown in the lower table on the right.

One thing that was impressed upon me during this exercise is the amount of work needed to create a nomogram that is effective and simple to use, regardless of the shape! In retrospect I wish I had expended my efforts on an equation more useful or more interesting to me. I could have rotated the plot 90° clockwise, drawn rocket fins rather than fish fins, and nomographed an equation related to rocketry, astronomy or space science. But my objective was to learn how to create nomograms based on Weierstrass’ Elliptic Functions, which has happened, and software components now exist to quickly create other nomograms of this sort. It’s also true that I have never before seen a nomogram in the shape of anything specific, although it seems to me that even traditional nomograms could be incorporated in some way into drawings of objects they represent.

**References**

Epstein, L. Ivan. **Nomography**. New York: Interscience Publishers (1958). An advanced book that treats determinants throughout as well as projective transformations. It should not be the first book or two to read on nomography.

Soderberg, Richard W. **Flowing Water Fish Culture**. Liao’s 1971 oxygen consumption equation of the fish nomogram is presented on page 51, along with parameters for rainbow trout and salmon. This page is among the viewable snippets for this book on Google Books here. If this link does not work directly, you can search for the book on the Google Books site by title or author.

**Elliptic Curve**, Wikipedia. Found here, this is an excellent article on elliptic curves, including the fact that they form an abelian group under the “+” group operation, which is fundamentally the basis of the discussion here. The figures and description of this additive property of points on the curve will look very familiar to you. Elliptic curves are also used in new cryptographic and integer factorization methods, so they are very interesting creatures indeed!

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

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In 1996 I discovered James Morrison’s excellent website, still the best one on astrolabes, and ordered both versions of *The Personal Astrolabe*, a precision astrolabe on laminated card-stock customized for my latitude and longitude. Its 50-page booklet is excellent in explaining the basics of astrolabes and their use. I’ve used one of these astrolabes on camping trips to identify stars and tell time, and from that day to this, through 3 jobs, the other one has been hanging on my office wall. I’ve assembled and used two instructional astrolabe kits, one put out by the National Maritime Museum in Greenwich and another lesser one. I’ve devoured papers on dating astrolabes from variations in star locations due to the Earth’s precession, on detecting forged astrolabes, on Islamic astrolabe decorations, and so forth. So I like astrolabes.

And in the same year of 1996 Mr. Morrison sent me a draft copy of his translation of Henri Michel’s *Traité de l’Astrolabe*, the only text on the mathematical details of astrolabe design I had encountered other than the short appendix of Harold Saunders’ book, *All the Astrolabes*. But the constructions in Michel’s book were geometric; the only equations were provided in two chapters inserted by Morrison. In an accompanying letter Morrison mentioned that he was working on a book—and now, suddenly, 11-1/2 years later, here it is. And it is a stunning, 437-page, large-format book containing 265 detailed drawings from PostScript files created by the author, presenting the design and mathematics of astrolabes, quadrants and related instruments. In addition to the technical details, the book is infused with the history of these instruments, describing their global spread from Greek mathematical and astronomical thought through their maturation in the Islamic world followed by their re-introduction into Europe by the Moors in Spain, as well as presenting the origins and inventors of specific instrument types. A clear overview of astronomical basics is included for those with little background in this area. Program listings are provided in BASIC and C to calculate typical design parameters, and there is even an introduction to generating PostScript output for astrolabe graphics (I will be using LaTeX and the Ti*k*Z vector drawing package, thank you). I am thrilled by this book—it eclipses any other book I’ve seen on how to design astrolabes. It also provides specific examples on using these instruments—for example, this is the only book that I know of that describes how to use a linear astrolabe in sufficient detail for me to understand it.

You can find more information and a sample chapter from the book to download here (you may recognize Gunter’s quadrant in this chapter from Part III of my nomography essay). Morrison writes that he is from a family of grammarians, and it shows in the care and clarity of the prose in this book. If you want to see photographs of astrolabes you will want a different book; if you are interested in astrolabe design you will love this book.

To be clear, I have absolutely no financial connection with James Morrison beyond my purchases in 1996 and the Christmas gift of this new book from my brother. I also have no connection to Amazon or to the Adler Planetarium.

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

]]>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.

There are several standard transformations of the Cartesian coordinate system that preserve the functionality of nomograms. After they are presented we will walk through a sequence of these to see their effects on the shape and scales of a nomogram.

**Translation**

We can translate the nomogram laterally, which is equivalent to translating the x-y axes to new x’-y’ axes. We can add *c* to all determinant elements in the x column and *d* to all determinant elements in the y column to shift the axes left by *c* and down by *d* (or in other words shift the nomogram right by *c* and up by *d*).

x_{n}‘ = x_{n} + c

y_{n}‘ = y_{n} + d

**Rotation**

We can rotate the nomogram about the origin of the axes by an angle θ (positive for counter-clockwise rotation) by replacing each determinant element x_{n} in the x column and the each determinant element y_{n} in the y column with

x_{n}‘ = x_{n} sin θ + y_{n} cos θ

y_{n}‘ = x_{n} cos θ – y_{n} sin θ

**Stretch**

We can stretch a nomogram in the x direction by multiplying each determinant element in the x column by a constant, and likewise for the y direction.

x_{n}‘ = cx_{n}

y_{n}‘ = dy_{n}

**Shear**

Shear is a slewing of perpendicular axes to oblique axes or vice-versa. This is perhaps best understood by referring to this figure showing a shear from one set of axes x-y to another set x’-y’ in which the x’ axis is canted at an angle θ to the x axis but the y’ axis aligns with the y axis. For this case,

x_{n}‘ = x_{n} cos θ

y_{n}‘ = y_{n} + x_{n} sin θ

Shear can be used to convert a traditional Z chart with a slanting middle line to one with a perpendicular middle line and vice-versa. It could have been used to convert the determinant for the earlier nomogram of equivalent friction radius to plot it relative to the oblique x’-y’ axes rather than to the perpendicular x-y axes.

**Projection**

Referring to the first figure below, a projection uses a point P (called the *center of perspectivity*) to project rays through points of a nomogram in the x-y plane to map them onto the z-y plane (also called the x’-y’ plane), foreshortening or magnifying lines in varying amounts in the x’ and y’ directions. It is also possible for P to lie above the x-y plane, where rays from points on the nomogram pass through P to the x’-y’ plane as shown in the second figure. (This can be used to convert a nomogram in the shape of a trapezoid to a rectangular one, changing scale resolutions to maximize the available space, but we will see an easier method later.) In either case, for a P location (x_{P},y_{P},z_{P}) and a nomogram x-element x_{n} and y-element y_{n},

x_{n}‘ = z_{P} x_{n} / (x_{n} – x_{P})

y_{n}‘ = (y_{P} x_{n} – x_{P} y_{n}) / (x_{n }– x_{P})

The line BD under P in the second figure never maps onto the x’-y’ plane because it is parallel to that plane—it is a “straight line at infinity” and will become important in our example below.

**A Transformation Example**

Epstein outlines a sequence of transformations to convert a nomogram for the equation **q ^{2 }– aq + b = 0** to a more convenient circular form. Unfortunately, he provides only the final nomogram, so I have taken up his challenge and traced through the details of each step while creating my own intermediate nomograms. These nomograms were created using the freely-available LaTeX typesetting engine and the free vector drawing package TiKz, which supports plotting parametric functions and has enough flexibility to draw and nicely label tick marks on the curves. Excel and MATLAB, for example, can plot parametric functions but do not appear to support labeled tick marks on the curves corresponding to the parametric variable. Chung uses Python code to create his nomograms (described here) and there is a very powerful Python program to plot nomograms here. An online tool to create custom, interactive parallel scale nomograms only can be found here. I find that the LaTeX code is quite simple and very flexible and is especially convenient for those of us who already use LaTeX to create technical articles. My LaTeX code that created the nomograms below can be found here.

A determinant representing the equation **q ^{2} – aq + b = 0** can be constructed (and verified it by multiplying it out) as

(Note: In all determinants in this essay the x elements are in the first column and the y elements in the second column, which follows most presentations but is reversed from Epstein’s.)

This nomogram is plotted in the figure to the right (the x-y axes and grid would be deleted from the final nomogram). The q-scale is found in practice by taking a range of q values and calculating x = q / (q+1) and y = q^{2} / (q-1) as a coordinates to plot, but if we eliminate q between the two parametric equations we arrive at x^{2} – xy + y = 0, demonstrating that the q-scale is in fact a hyperbola. A straightedge placed across any two values of a and b will intersect the q-scale in two points if there are two real solutions, one point if there is a double real root, and no points if there are no real roots. However, the layout of the q-scale is problematic, as the two halves stretch toward infinity very quickly and it is not possible to accurately locate q points for isopleths near the asymptotes of the hyperbola. So we will transform this nomogram into one in which the q-scale is finite. The labels for the tick marks will not be displayed on the following plots, but the tick mark spacing and colors will provide a guide for how the curves are re-mapped.

First we will rotate this nomogram clockwise by 45° (or θ = -45°) and stretch it in both dimensions by 2^{1/2 }for a reason that will become apparent in the next transformation. Since cos -45° = 2^{-1/2} and sin 45° = -2^{-1/2}, the earlier rotation formulas after the stretch become

x_{n}‘ = x_{n} + y_{n}

y_{n}‘ = –x_{n} + y_{n}

Performing this substitution for the x element and y element of each row of the determinant, we arrive at

which is plotted on the right.

We rotated the nomogram because we wanted a vertical line (say, x=1) that does not intersect the hyperbola. A projection transformation can convert a scale with two branches like this hyperbola into a single connected scale (an ellipse) if a straight line separating the two branches is projected to infinity, that is, if the line is parallel to the y-z axis (which x=1 is) and the projection point P is located directly above or below it in its z value (as the line BD in the earlier projection figure). Choosing P = (1,-1,1), the earlier projection formulas become

x_{n}‘ = x_{n} / (x_{n} – 1)

y_{n}‘ = (-x_{n} – y_{n}) / (x_{n} – 1)

and the determinant becomes

and the ellipse magically appears in the plot on the right.

Now let’s shear the nomogram so the green b-scale lies on the y-axis while keeping the red a-scale parallel to the x-axis. The shear formulas are slightly different as we are shearing to the y-axis, and for a b-scale slope of –½ they reduce to:

x_{n}‘ = x_{n} + y_{n} / 2

y_{n}‘ = y_{n}

and the determinant becomes

which is plotted on the right.

Now we’ll translate the nomogram upward by 2 to place the intersection point on the origin.

x_{n}‘ = x_{n}

y_{n}‘ = y_{n} + 2

This is plotted on the right.

And finally we shrink the nomogram in the y direction by a factor of 2 to get a circular scale for q.

x_{n}‘ = x_{n}

y_{n}‘ = y_{n} / 2

yielding

The figure on the right is the plot of my final determinant, and by changing the scales of the axes we arrive at Epstein’s figure below.

The entire range of q from -∞ to +∞ is now represented in a finite area, and certainly the range less than 1.5, which veered to infinity in our original nomogram, is nicely accessible. The larger numbers are not as accessible, but the ranges can be skewed to spread out any range by multiplying the original equation by a constant. We could have stopped at any of the nomograms containing an ellipse, but it is easier to draft the circle. An elliptical nomogram for the quadratic equation ax^{2} + bx + c = 0 is shown here for comparison.

It’s interesting to play around with a straightedge on the circular nomogram we derived above to see that it works. In particular, an isopleth through an a-value and b-value will just touch the q-circle if the discriminant from the quadratic formula is 0 (for the equation Ax^{2} +Bx + C = 0, the discriminant is the value B^{2}-4AC whose square root is taken in the quadratic formula, or a^{2}-4b here). When the discriminant is less than zero the isopleth misses the q-circle, denoting no real roots, and when it is greater than zero it crosses two real roots on the q-circle.

And in fact if you eliminate the a-scale, then the b-scale represents the product of two numbers on the q-circle. This is because if we have two solutions q_{1 }and q_{2} of **q ^{2} – aq + b = 0**, then the equation can be written as (

The transformations we have discussed can also be represented as matrices. Transformations are performed by matrix multiplication of the transformation matrix and the nomogram determinant. Two or more transformations can be combined by multiplying their transformation matrices. It often happens after such a matrix multiplication that the nomogram determinant needs to be manipulated again into the standard nomographic form. For example, the transformation matrices for rotation and projection are

It is possible to use matrix multiplication to map a trapezoidal shape (such as the boundaries of a nomogram that does not occupy a full rectangle) into a rectangular shape. This would increase the accuracy of the scales that can be expanded to fill the sheet of paper. Consider the following matrix multiplication:

By the rules of matrix multiplication and some manipulation of the result, each y’ and x’ in the resulting matrix can be represented as

x’ = (xk_{11} + yk_{21} + k_{31}) / (xk_{13} + yk_{23 }+ k_{33})

y’ = (xk_{12} + yk_{22} + k_{32}) / (xk_{13} + yk_{23 }+ k_{33})

Now if we want to remap an area such that the points (x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3}) and (x_{4},y_{4}) map to, say, the rectangle (0,0), (0,a), (b,0) and (b,a), we insert the final and initial x’s and y’s into the formulas above, giving us eight equations in nine unknown k’s. So we choose one k, solve for the other eight k’s and multiply the original nomogram determinant by the k matrix and convert it back to standard nomographic form, then replot the nomogram—a fun way to spend an afternoon.

There are non-projective transformations as well that can be used to create nomograms in which all three scales are overlaid onto one curve (although the third will have different tick marks). This is highly mathematical and involves things called Weierstrass’ Elliptic Functions, so Epstein is a resource if there is interest in the details. Epstein provides nomograms of this sort for the equation **u + v + w = 0** (which can be generalized to any equation of this form, including ones in logarithms). In the first figure below, the isopleth must cross two numbers on one scale and a third number on the other overlaid scale (such as u = +0.2524, v = +0.3842 and w= -0.6366). In the second figure the isopleth crosses two curves containing three scales. Ignore the numbers on the x-axis and y-axis—these relate to the function used to derive the nomograms. I’m showing these simply to demonstrate the advanced mathematics that was targeted at nomographic construction at one time. (Please see my later essay *A Zoomorphic Nomogram* for a detailed discussion on using these elliptic functions to create a nomogram.)

Transformations provide a creative way of morphing nomograms into designs that are most useful, such as increasing the accuracies of particular ranges of variables. They also allow us to create nomograms that are eye-catching and artistic.

**The Status of Nomograms**

Today the use of nomograms seems scattered at best. Chung’s interest in nomograms shown here grew from his desire to provide quick and easy calculations of hit strength, etc., in strategic wargaming. There are many simple nomograms that exist for doctors to quickly assess attributes and probabilities, such as here (or here if you can’t draw a line), here, here, and here. A Body Mass Index (BMI) nomogram is common (such as here) and is derived in this PowerPoint presentation on nomograms. There are also some engineering nomograms found here and there online (such as here, here, here, here, here, and there).

I first heard of nomograms in the context of sundials. As Vinck and Sawyer describe (see references), Samuel Foster published a treatise in 1638 titled “The Art of Dialling” that contained a dialing scale for the construction of horizontal, vertical and inclining sundials as shown in the figure from Vinck to the right. Foster’s construction scale is actually a circular nomogram, a tool discovered nearly 300 years earlier than its attributed discovery by J. Clark in 1905! Sawyer writes that Foster did write of the more general computing applications of his scale. Here an isopleth from one point on a perimeter scale through a point on the middle scale will cross their product on the other perimeter scale, and with suitable trigonometric scaling one can lay out hour lines on a wide variety of sundials. Certainly many sundial designs (nearly all) rely on graphical plots with the gnomon shadow or a weighted, hanging string serving as the isopleth. Card dials are particularly complicated because they map a 3-D geometry to a 2-D plane, as you can see on this webpage. Sawyer has designed a few dials that employ nomograms, two of which are shown below.

Masse uses the same projection transformation method as we did to create a sundial in which the gnomon tip shadow during the day traces a circle rather than a hyperbola on the face of the dial. My interest in nomography (including the transformation techniques) and my efforts to plot nomograms using the LaTeX typesetting engine are partly due to my intention to create new sundial designs based on nomograms.

The use of analog graphic calculators is actually much older—the quite complicated grids and curves on old astrolabes and quadrants, as shown below, effectively serve as nomograms. There are also curves on the backs of most astrolabes to convert equal hours to unequal hours using the alidade (the sight on the back) as an isopleth, and the *qibla* diagram on the back of Islamic astrolabes provides the direction to Mecca for any hour of any day by using the alidade when the astrolabe is aligned correctly.

And I think there could be more applications today. I was at a picture framing store while I was writing this essay to get a matte board cut as a frame for some calligraphy created by my son. The pricing was based on three variables (the window height, the window length and the border width) and possibly the matte board type. Everyone who came up to get a price for a certain configuration or a variety of configurations had to wait while the clerk wrote down the three parameters, punched them into some formula on a calculator, and referred to a chart to find the corresponding price in quarter-dollars. I was thinking the whole time that having photocopies of a nomogram laying around would let customers use a straightedge (and there are a lot of those in a framing store!) to figure the pricing out and quickly optimize their parameters without having to wait in line, and it would certainly be faster for the clerk.

But nomograms have their own intrinsic charm. As a calculating aid a nomogram can solve very complicated formulas with amazing ease. And as a curiosity a nomogram provides a satisfying, hands-on application of interesting mathematics in an engaging, creative activity.

**References**

* Update 10/17/2008*: Leif Roschier, author of the powerful Pynomo software for creating nomograms, has updated his site found here with beautiful nomograms he has created. Be sure to visit the

Douglass, Raymond D. and Adams, Douglas P. **Elements of Nomography**. New York: McGraw-Hill (1947). This book is as concerned about the drafting aspects of nomograms as with their (exclusively geometric) derivations. The printed forms it champions for calculating nomograms are more trouble than they are worth, in my opinion.

Epstein, L. Ivan. **Nomography**. New York: Interscience Publishers (1958). An advanced book that treats determinants throughout as well as projective transformations. It should not be the first book or two to read on nomography.

Hoelscher, Randolph P. *et al*. **Graphic Aids in Engineering Computation**. New York: McGraw-Hill (1952). The best all-around book on nomography of the ones listed here. Geometric derivations are given first and determinants introduced later in a very understandable presentation. It also contains a very interesting chapter on special slide rules created for solving particular engineering equations.

Johnson, Lee H. **Nomography and Empirical Equations**. New York: John Wiley and Sons (1952). A good, solid reference that is very readable. Uses geometric derivations throughout (no determinants) and treats all the common nomographic forms.

Levens, A.S. **Nomography**. New York: John Wiley and Sons (1937). A very readable book that is organized in a very convenient manner based on equation types. A very nice collection of nomogram examples of various types is presented in the Appendix. For nomograms created from geometric derivations only (and only the most sophisticated nomograms types really require solutions by determinants) this is an excellent reference book with many examples.

Masse, Yvon. *Central Projection Analemmatic Sundials*, The Compendium, Journal of the North American Sundial Society, Vol. 5, No. 1, Mar. 1998, pp.4-9.

Otto, Edward. **Nomography **(International Series of Monographs in Pure and Applied Mathematics).New York: Macmillan (1963). An advanced but (with some work) a readable book that uses determinants throughout to create nomograms as well as projective transformations to square them up or expand scales for greater accuracy. There is a long, interesting chapter on problems of theoretical nomography, such as determining what equation forms can be expressed as nomograms.

Sawyer, Fred. *Ptolemaic Coordinate Sundials*, The Compendium, Journal of the North American Sundial Society, Vol. 5, No. 3, Sep. 1998, pp.17-24. Describes three sundials incorporating nomograms.

Sawyer, Fred. *A Universal Nomographic Sundial*, Bulletin of the British Sundial Society, Oct. 1994. 94(3):14-17, also in **Sciatheric Notes**, a collection of sundial articles by Fred Sawyer from this publication. This collection and all the sundial-related articles referenced here can be found in the outstanding North American Sundial Society Repository CD orderable on the NASS site.

Vinck, Rene J. *Samuell Foster’s Circle*, The Compendium, Journal of the North American Sundial Society, Vol. 8, No. 3, Sep. 2001, pp.7-10. Also see Fred Sawyer’s *The Further Evolution of Samuel Foster’s Dialing Scales* that follows this article in the same issue of the journal.

**Blog Update (not yet in the current PDF file): **Google has digitized the entire book **Graphical and Mechanical Computation** (1918) by Joseph Lipka that can be read or downloaded here (link updated 1/27/08). Chapters 3-5 on nomography have example after example of creating nomograms from the geometrical formulas.

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A **matrix **consists of functions or values arranged in rows and columns, as shown within the brackets in the figure on the right. The subscript pair refers to the row and column of a matrix element. A **determinant** represents a particular operation on a matrix, and it is denoted by vertical bars on the sides of the matrix. The determinant this 3×3 matrix is given by

a_{11}a_{22}a_{33 }+ a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32 }– a_{13}a_{22}a_{31} – a_{11}a_{23}a_{32 }– a_{12}a_{21}a_{33}

But there are visual ways of deriving it. In the first figure the first two columns of the determinant are repeated to the right of the original, and then the products of all terms on diagonals from upper left to lower right are added and the products of all terms on diagonals from upper right to lower left are subtracted. A convenient mental shortcut is to find these diagonal products by “wrapping around” to get the three components of each term. Here the first product we add is the main diagonal a_{11}a_{22}a_{33}, then the second is a_{12}a_{23}a_{31 }where we follow a curve around after we pick up a_{12} and_{ }a_{23 }to pick up the a_{31}, then a_{13}a_{32}a_{21 }by starting at a_{13} and wrapping around to pick up the a_{32 }and a_{21}. We do the same thing right-to-left for the subtracted terms. This is much easier to visualize than to describe. Determinants of larger matrices are not considered here.

There are just a few rules about manipulating determinants that we need to know:

- If all the values in a row or column are multiplied by a number, the determinant value is multiplied by that number. Note that here we will always work with a determinant of 0, so we can multiply any row or column by any number without affecting the determinant.
- The sign of a determinant is changed when two adjacent rows or columns are interchanged.
- The determinant value is unchanged if every value in any row (or column) is multiplied by a number and added to the corresponding value in another row (or column).

That’s it. Now consider the general diagram to the right from Hoelscher showing three curvilinear scales and an isopleth. Similar triangle relations give

(y_{3}-y_{1})/(x_{3}-x_{1}) = (y_{2}-y_{1})/(x_{2}-x_{1}) = (y_{3}-y_{2})/(x_{3}-x_{2})

The first two parts of this can be rewritten as a cross product:

(x_{3}-x_{2}) / (y_{2}-y_{1}) = (x_{2}-x_{1}) / (y_{3}-y_{2})

and when this is expanded it is equal to the determinant equation

We get this result regardless of which pair we choose to use in the cross product. The x and y elements can be interpreted as the x and y values of f_{1}(u), f_{2}(v) and f_{3}(w) if we don’t mix variables between rows (the first row should only involve u, etc.) and if the determinant equation is equivalent to the original equation. This is the *standard nomographic form*. Here y is not expressed in terms of x as we normally have when we plot points at (x,y) coordinates, but rather x and y are expressed in terms of a third variable, that is, x_{1} and y_{1} are expressed in terms of a function of the variable u, x_{2} and y_{2} are expressed in terms of a function of the variable v, and x_{3} and y_{3} are expressed in terms of a function of the variable w. These are called *parametric equations*. One way to plot them is to algebraically eliminate the third

variable between x and y to find a formula for y in terms of x. Another way is to simply take values of the third variable over some range, calculate x and y for each value, and plot the points (x,y)—a more likely scenario when we have computing devices.

Let’s consider the equation **(u + 0.64) ^{0.58}(0.74v) = w** that we used earlier to create a parallel scale nomogram. We converted this with logarithms: to 0.58 log (u + 0.64) + log v = log w + log (0.74), or

0.58 log (u + 0.64) + log v – [log w + log (0.74)] = 0

We could have grouped log (0.74) with any term, but we’ll stay consistent with our earlier grouping. This is an equation of the general form **f _{1}(u) + f_{2}(v) – f_{3}(w) = 0**, so let’s find a determinant that produces this form. We want each row to contain only functions of one variable of u, v or w, so we’ll start with f

Now let’s set the lower left corner to f

Now we can place f

But the second determinant is simply the first one after the third column is subtracted from the second column and the third determinant is simply the second one after the third column is added to the second column. These are operations that will not change our determinant equation as described in our earlier list, so they are all equivalent.

Now we want the flexibility to scale our f_{1}(u) and f_{2}(v) by m_{1 }and m_{2} calculated for parallel scale charts as the desired height of the nomogram divided by the ranges of the functions. If we take the first determinant of the three possible ones shown, notice that we can introduce the scaling values without changing the determinant equation if we write it as

So now we have to convert this to the standard nomographic form having all ones in the last column (and continuing to isolate variables to unique rows). First we add the second row to the third row, where 1/m_{1 }+ 1/m_{2} = (m_{1}+m_{2})/m_{1}m_{2}:

Then we multiply the bottom row by m_{1}m_{2}/(m_{1}+m_{2}) and swap the first two columns so that the y column (the middle column) contains the functions:

and we have the determinant in standard nomographic form. The first column represents x values and the second column represents y values of the functions. The scaling factors of m_{1} and m_{2} result in a scaling factor m_{3} for the w-scale of m_{1}m_{2}/(m_{1}+m_{2}) as we found earlier from our geometric derivation. We had calculated m_{1} = 25.72 and m_{2}=19.93 before, giving m_{3}=11.23. This determinant also shows that we place the u-scale vertically at x=0 and the y-scale vertically at x=1, with the w-scale at x= m_{ }(m_{1}+m_{2}) = 0.5634, but in fact we can multiply the first column by 3 to get a scale of 3 inches, and in this case the w-scale lies vertically at x=1.69 inches, and so we end up with *exactly* the same nomograph we found in Part I using geometric methods.

This was a bit of work, but we have found a universal standard nomographic form for the equation **f _{1}(u) + f_{2}(v) – f_{3}(w) = 0** including scaling factors.

Let’s derive a Z chart for division using determinants. For **v** = **w/u**, we rearrange the equation so the right-hand term is 0, or **uv – w = 0**. One possible determinant we can construct is

which graphs to the nomogram on the right, a Z chart with a perpendicular middle line. A different determinant would result in a Z chart of the more familiar angled middle line. An interesting aspect of such a chart is that the u-scale and v-scale have different scaling factors despite the fact that they can be interchanged in the equation.

There is a definite knack to all of this, and at this point I’d like to recommend the webpages on nomography by Winchell D. Chung, Jr. at this site. His webpages are quite interesting to read—there are quite a few examples of nomograms, and the determinant approach is used throughout. In particular, he provides other examples of expressing an equation into determinant form here. He also gives a few examples of converting the determinant to the standard nomographic determinant form here, where examples 2 and 3 are from Hoelscher. Most importantly, for equations of several standard formats Chung also reproduces tables that map these equations *directly *to standard nomographic determinant forms here.

Determinants are most useful when one or more of the u, v and w scales is curved. The quadratic equation **w ^{2} + uw + v = 0** can be represented as the first equation below, and dividing the last row by w-1 we immediately arrive at the standard nomographic form shown in the second equation:

The u-scale runs linearly in the negative direction along the line y=1. The v-scale runs linearly in the positive direction with the same scale along the line y=0. The x and y values for the curve for w can be plotted for specific values of w (a parametric equation), or w can be eliminated to express the curve in x and y as

x/y = w

y = (x/y) / [(x/y) – 1] = x / (x – y)

resulting in the figure at the right in which the positive root w_{1} of the quadratic equation can be found on the curved scale (the other root is found as u – w_{1}).

Hoelscher presents the equation for the projectile trajectory **Y = X tan A – gX ^{2 }/ (2V_{0}^{2}cos^{2}A)** where A is the initial angle, V

Y = X – 0.0322X2 /V_{0}^{2}

One determinant for this is shown in the first equation below, which can be manipulated into the standard nomographic form shown in the second equation:

Hoelscher assumes -2000<Y<7000 ft and 800<V_{0}<4000 fps and a chart of 5 inches square, so after some more manipulations (including the swapping of the first two columns) we arrive at the final form:

This is shown as the curve for A=45º along with curves for other angles in this figure (a *grid nomogram* such as this can be used to handle an equation with more than 3 variables).

It’s possible to have two or three scale curves depending on how the determinant works out, and it is possible to have two or all three curves overlap exactly. The equation for the equivalent radius of the friction moment arm for a hollow cylindrical thrust bearing is **R = 2/3 [R _{1}^{3} – R_{2}^{3}] / [R_{1}^{2} – R_{2}^{2}]** or 3RR

Here the R

Otto provides an interesting alternate determinant for the equation **f _{1}(u) + f_{2}(v) + f_{3}(w) = f_{1}(u) f_{2}(v) f_{3}(w)**.

The nomogram for the particular equation of this type

Finally, Otto describes a very interesting determinant that can be created for the equation **f _{1}(u) f_{2}(v) f_{3}(w) = 1**.

For

Determinants provide a creative way of generating nomograms whose designs are varied and interesting. We can also apply transformations to the x and y elements of the determinants to morph nomograms into shapes that are most accurate for given ranges of the variables or that utilize the available space most efficiently (or are simply more pleasing to the eye). Transforming nomograms is the subject of Part III of this essay.

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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.

The simplest form of nomogram is a scale such as a Fahrenheit vs. Celsius scale seen on an analog thermometer or a conversion chart. Linear spacing can be replaced with logarithmic spacing to handle conversions involving powers. Slide rules also technically qualify as nomograms but are not considered here. A slide rule is designed to provide basic arithmetic operations so it can solve a wide variety of equations with a sequence of steps, while the traditional nomogram is designed to solve a specific equation in one step. It’s interesting to note that the nomogram has outlived the slide rule.

Most of the nomograms presented here are the classic forms consisting of three or more straight or curved scales, each representing a function of a single variable appearing in an equation. A straightedge, called an **index line** or **isopleth**, is placed across these scales at known values of these variables, and the value of an unknown variable is found at the point crossed on that scale. This provides an analog means of calculating the solution of an equation involving one unknown, and for finding one variable in terms of two others it is much easier than trying to read a 3-D surface plot. We will see later that it is sometimes possible to overlay scales so the number of scale lines can be reduced.

**The Geometry of Nomograms**

We can design nomograms composed of straight scales by analyzing their geometric properties, and a variety of interesting nomograms can be constructed from these derivations. Certainly these seem to be the most prevalent types of nomograms.

The figure on the right shows the basic parallel scale nomogram for calculating a value f_{3}(w) as the sum of two functions f_{1}(u) and f_{2}(v):

f_{1}(u) + f_{2}(v) = f_{3}(w)

Each function plotted on a vertical scale using a corresponding scaling factor (sometimes called a scale modulus) m1, m2 or m3 that provides a conveniently sized nomogram. The spacing of the lines is shown here as a and b. Now by similar triangles, [m_{1}f_{1}(u) – m_{3}f_{3}(w)] / a = [m_{3}f_{3}(w) – m_{2}f_{2}(v)] / b. This can be rearranged as:

m_{1}f_{1}(u) + (a/b) m_{2}f_{2}(v) = (1 + a/b) m_{3}f_{3}(w)

So to arrive at the original equation f_{1}(u) + f_{2}(v) = f_{3}(w), we have to cancel out all the terms involving m, a and b, which is accomplished by setting m_{1} = (a/b) m_{2} = (1 + a/b)m_{3}. The left half of this relationship provides the relative scaling of the two outer scales and the outer parts provide the scaling of the middle scale:

m_{1}/ m_{2} = a / b m_{3} = m_{1}m_{2 }/ (m_{1}+ m_{2})

Also note that the baseline does not have to be perpendicular to the scales for the similar triangle proportion to be valid. Now a = b for the case where the middle scale is located halfway between the outer scales, and in this case m_{1} = m_{2} and m_{3} = ½m_{1}. For a smaller range and greater accuracy of an outer scale, we can change its scale m and move the middle line away from it and toward the other outer scale. In fact, if the unknown scale w has a very small range it can be moved *outside* the two other scales to widen the scale. Additions to u, v or w simply shift the scale values up or down. Multipliers of u, v and w multiply the value when drawing the scales (they are not included in the values of m in the above calculations). Subtracting a value simply reverses the up/down direction of the scale, and if two values are negative their scales can simply be swapped. The example on the right shows a parallel-scale nomogram for the equation **(u–425) – 2(v–120) = w** designed for ranges 530<u<590 and 120<v<180.

So this looks like a lot of work to solve a simple linear equation. But in fact plotting logarithmic rather than linear scales expands the use of parallel scale nomograms to very complicated equations! The use of logarithms allows multiplications to be represented by additions and powers to be represented by multiplications according to the following rules:

log(cd) = log c + log d log c^{d} = d log c

So if we have an equation such as f_{1}(u) x f_{2}(v) = f_{3}(w), we can replace it with

log [f_{1}(u) x f_{2}(v)] = log f_{3}(w)

or

log f_{1}(u) + log f_{2}(v) = log f_{3}(w)

and we have converted the original equation into one without multiplication of variables. And note that there is actually no need to solve symbolically for the variable (we just plot these logs on the scales), a significant advantage when we come to more complicated equations.

Let’s create a nomogram for the engineering equation **(u + 0.64) ^{0.58}(0.74v) = w** as given in Douglass. We assume that the engineering ranges we are interested in are 1.0<u<3.5 and 1.0<v<2.0.

0.58 log (u + 0.64) + log (0.74v) = log w

0.58 log (u + 0.64) + log (0.74) + log v = log w

0.58 log (u + 0.64) + log v = log w – log (0.74)

We will directly plot the three components here as our u, v and w scales. To find the scaling factors we divide the final desired height of the u and v scales (say, 6 inches for both) by the ranges (maximum – minimum) of u and v:

m_{1} = 6 / [0.58 log (3.5 + 0.64) – 0.58 log (1 + 0.64) ] = 25.72

m_{2} = 6 / [log 2.0 – log 1.0] = 19.93

m_{3} = m_{1}m_{2} / (m_{1}+ m_{2}) = 11.23

Let’s set the width of the chart to 3 inches:

a / b = m_{1} / m_{2} = 1.29 so a = 1.29b

a + b = 3 so 1.29b + b = 3 yielding b = 1.31 inches and a = 1.69 inches

We draw the u-scale on the left marked off from u = 1.0 to u = 3.5. To do this we mark a baseline value of 1.0 and place tick marks spaced out as 25.72 [0.58 log (u + 0.64) – 0.58 log (1.0 – 0.64)] which will result in a 6 inch high line. Then 3 inches to the right of it we draw the v-scale with a baseline value of 1.0 and tick marks spaced out as 19.93 (log v – log 1). Finally, 1.69 inches to the right of the u-scale we draw the w-scale with a baseline of (1.0 + 0.64)^{0.58}(0.74)(1) = 0.98 and tick marks spaced out as 11.23 (log w – log 0.74). And we arrive at the nomogram on the right, where a straightedge connecting values of u and v crosses the middle scale at the correct solution for w, and in fact any two of the variables will generate the third. Flexibility in arranging terms of the equation into different scales provides a means of optimizing the ranges and accuracies of the nomogram. A larger scale and finer tick marks can produce a quite accurate parallel scale nomogram that is deceptively simple in appearance, and one that can be manufactured and re-used indefinitely for this engineering equation.

It is also possible to create a circular nomogram to solve a 3-variable equation. Details on doing this from geometrical derivations are given in Douglass.

**N or Z Charts**

A nomogram like that shown in the figure on the right is called an “N Chart” or more commonly a “Z Chart” because of its shape. The slanting middle scale joins the baseline values of the two outer scales (which are now plotted in opposition). The middle line can slant in either direction by flipping the diagram, and it can be just a partial section anchored at one end or floating in the middle if the entire scale isn’t needed in the problem, thus appearing, as Douglass puts it, “rather more spectacular” to the casual observer. A Z chart can be used to solve a 3-variable equation involving a division:

f_{3}(w) = f_{1}(u) / f_{2}(v)

By similar triangles, m_{1}f_{1}(u) / m_{2}f_{2}(v) = Z / [L – Z]. Substituting f_{3}(w) for f_{1}(u) / f_{2}(v) and rearranging terms yields the distance along Z for tick marks corresponding to f_{3}(w):

Z = L f_{3}(w) / [(m_{2}/m_{1}) + f_{3}(w)]

The f_{3}(w) scale does not have a uniform scaling factor m_{3} as before. We could have used a parallel scale chart with logarithmic scales to plot this division, but the Z chart performs this with linear scales for u and v and it was once a real chore to calculate logarithms. But further, the linear scales of the Z Chart are much more suitable for combining a division with an addition or subtraction than compound parallel scales with their logarithmic scales. And of course if the scale for the unknown variable is an outside one, we have a Z chart for multiplication.

An example of a Z chart is shown here for the equation **Q ^{2} = (8R+4) / (P-3)**. To create this, the desired height of the nomogram and the ranges of P and R provide their scaling factors m

**Proportional Charts**

The proportional chart solves an equation in four unknowns of the type

f_{1}(u) / f_{2}(v) = f_{3}(w) / f_{4}(t)

If we take our Z chart diagram and a second isopleth that intersects the Z line at the same point as the first, we have by similar triangles:

m_{1}f_{1}(u) / m_{2}f_{2}(v) = m_{3}f_{3}(w) / m_{4}f_{4}(t)

which matches our equation above if we choose the scaling of the outer scales such that

m_{1} / m_{2} = m_{3} / m_{4}

We then overlay two variables on each outer scale with this ratio of scaling factors, as shown in the nomogram to the right from Josephs for the approximate pitch of flange rivets in a plate girder, where p is the rivet pitch in inches, R is the rivet value in lbs, h is the effective depth of the girder in inches, and V is the total vertical shear in lbs: p = Rh/V.

Another type of proportional chart uses crossed lines within a boxed area, as shown below. Again, the scaling factors for the four variables are given by m_{1 }/ m_{2} = m_{3} / m_{4} where these are related as before to the u, v, w and t scales, respectively. (Actually, similar triangles still exist and the ratios still hold for any parallelogram, not just a rectangle.)

But there are other types of proportional charts as shown below. In the ones labeled Type 3 an isopleth is drawn between two scale variables, then moved parallel until it spans the third variable value and the fourth unknown variable. The flange rivet example done in this manner is shown here. In the Type 4 nomogram the second isopleth is drawn perpendicular rather than parallel to the first one; it’s actually easier to draw a perpendicular than a parallel line if you have a drafting square or even a rectangular sheet of paper.

**Concurrent Scale Charts**

The concurrent chart solves an equation of the type

1/f_{1}(u) + 1/f_{2}(v) = 1/f_{3}(w)

The effective resistance of two parallel resistors is given by this equation, and a concurrent scale nomogram for this is shown on the right.

The derivation is somewhat involved, but in the end the scaling factors m must meet the following conditions:

m_{1} = m_{2} = m_{3} / (2 cos A)

where A is the angle between the u-scale and the v-scale, and also the angle between the v-scale and the w-scale. The scaling factor m_{3} corresponds to the w-scale. The zeros of the scales must meet at the vertex. If the angle A is chosen to be 60°, then 2 cos A = 1 and the three scaling factors are identical, as is the case in this figure.

To solve the 4-variable equation 1/f_{1}(u) + 1/f_{2}(v) + 1/f_{4}(t) = 1/f_{3}(w), the equation is first re-arranged as 1/f_{1}(u) + 1/f_{2}(v) = 1/f_{3}(w) – 1/f_{4}(t). Then the two halves are set equal to an intermediate value f(q). A compound concurrent chart is then created in a similar way to other compound charts as shown in this figure (here A is chosen to be less than 60°).

**4-Variable Charts**

A 4-variable equation with one unknown can be represented as a combination of two separate charts of any type. The first step is to break the equation into two parts in three variables that are equal to one another. For f_{1}(u) + f_{2}(v) + f_{3}(w) = f_{4}(t) and t unknown, we can re-arrange the equation into f_{1}(u) + f_{2}(v) = f_{4}(t) – f_{3}(w) and create a new variable k to equal this sum. Then a *blank* scale for k is created such that a parallel scale nomogram for f_{1}(u) + f_{2}(v) = k marks a **pivot point** on the k-scale, then a second straightedge alignment from this point is used for a parallel-scale nomogram for f_{4}(t) – f_{3}(w) = k to find f_{4}(t). The scaling for u, v and w and the position chosen for the k-scale can be optimized to minimize errors at the pivot point for small errors in the straightedge alignment. The figure on the right shows a compound parallel scale nomogram. Below are examples from Levens of compound nomograms of Z charts and concurrent and proportional charts. A key often provides instructions on the use of a compound nomogram as shown in the first figure. Of course, this concept can be extended to equations with additional variables, where color coding would be helpful.

**Curved Scale Charts**

It is possible to geometrically derive relationships for nomograms that have one or more curved scales, but the design of these more complicated nomograms is so much easier using determinants. Designing nomograms with determinants is the subject of Part II of this essay.

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**>>> Go to Part II of this essay**

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The name of Oliver Heaviside (1850-1925) is not well-known to the general public today. However, it was Heaviside, for example, who developed Maxwell’s electromagnetic equations into the four vector calculus equations in two unknowns that we are familiar with today; Maxwell left them as 20 equations in 20 unknowns expressed as *quaternions*, a once-popular mathematical system currently experiencing a revival for fast coordinate transformations in video games. Heaviside also did important early work in long-distance telegraphy and telephony, introducing induction-loading of long cables to minimize distortion and patenting the coaxial cable. At one time the ionosphere was called the Heaviside layer after his suggestion (and that of Arthur Kennelly) that a layer of charged ions in the upper atmosphere (now just one layer of the ionosphere) would account for the puzzlingly long distances that radio waves traveled. But Heaviside was an iconoclast who saw little need to ingratiate himself with others or spend time justifying his methods to them. It took later mathematicians such as Carson and Bromwich to demonstrate that his operators are analogous to later, well-developed integral equations and contour integrals in the complex plane.

Actually, absence of rigor is less unusual historically than it might appear—much mathematical science has progressed on very shaky ground indeed, and often proofs of mathematical techniques lag by many years their application. The famous mathematician G. Hardy wrote

All physicists and a good many quite respectable mathematicians are contemptuous about proof.

and certainly mathematicians often discover things by intuition that require a great deal of time and labor to prove. Often mathematics progresses from the specific to the general, bottom to top. Carl Friedrich Gauss once wrote in his diary about one of his most important discoveries, which was based on a purely computational observation: “I have the result, but I do not yet know how to get it.” The mathematician J. Hadamard observed, “The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.”

Today “experimental mathematics”—using computers to search for number-theoretic results that can be generalized—is a hot field in mathematics. As David Berlinski writes,

The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen.

And even back in 1951, Kurt Godel wrote,

If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.

Heaviside was openly dismissive of attempts to provide rigor for his operator calculus. Here’s what he had to say regarding his bold generalization of an experimental result to a general one that we will discuss shortly:

Those who may prefer a more formal and logically-arranged treatment may seek it elsewhere, and find it if they can; or else go and do it themselves.

Perhaps his refusal to validate it was a good thing because his operator calculus was not generally rigorous, particularly when initial values are non-zero. Heaviside used a great deal of intuition to guide him in the process of applying his calculus. In the end Laplace transforms, easier to use with a more rigorous structure and incorporating the powerful tool of convolution, overtook the operational calculus of Heaviside, and his methods largely fell victim to history. But the ingenuity of Heaviside appeals to me (along with his sheer indifference to the complaints of those he left in the dust) and so this essay is a short appreciation of his work in this regard.

Let’s first get a flavor of how his operators worked. Heaviside took the basic equations for voltage v and current i for a discrete resistance R, capacitance C and inductance L and rewrote them using his operator p, which performed the derivative with respect to time on the function to the right of it (effectively d/dt), as in p·f(t). He also assumed that the inverse of p, or 1/p, is the operator performing the integral of a function, so that p · 1/p = 1/p · p = 1. This is not generally true because when the integral is taken after the derivative there is a constant term, but it is true for a function where f(0)=0. So we have

v = iR

i = Cpv from v = (1/C) ∫ i dt

v = Lpi from v = L di/dt

Then he separated the operators from their functions, giving them an independent existence, to form a general AC *impedance* defined as v/i that he called Z (a term we still use today):

Z = R

Z = 1 / (Cp)

Z = Lp

Heaviside generally treated problems in which a constant voltage is applied to a circuit at time t=0, in other words an impulse (or step function) such as might be encountered as transient signals on cables. In telegraphy these transient effects limit the signaling speed, while in telephony the transient effects limit the line length. But this step function also has the property that f(0) can be treated as 0, the condition for the commutivity of the inverse operator given above. Using the superposition property from this waveform also allowed him to analyze distortion at all signal frequencies. In fact, Josephs remarks that, explicit or not, it is always assumed in the differential equations for transient currents that the applied voltage is 0 for t<0. Heaviside wrote this step function as the bold symbol **1**. So if we assume we have a circuit with a resistance and inductance,

Z = R + Lp

Then for v = **1** here,

i = v/Z

= **1 **/ (R+Lp)

= (**1**/R) [1 / (1 + Lp/R)]

= (**1**/R) [R/(Lp) / (R/(Lp) + 1)]

= (**1**/R) [R/(Lp)] [1 / (1 + R/(Lp))]

= (**1**/R) [R/(Lp)] [1 – R/(Lp) + (R/(Lp))^{2} – (R/(Lp))^{3} + · · ·] by the Binomial Theorem

= (**1**/R) [(R/L) · 1/p – (R/L)^{2} · 1/p^{2} + (R/L)^{3} · 1/p^{3} – · · ·]

Now for the step function **1**, the “integral” 1/p · **1** = t, and in general 1/p^{n} · **1** = t^{n}/n! , so

i = 1/R [R/L · t – (R/L)^{2} · t^{2}/2! + (R/L)^{3} · t^{3}/3! – · · ·]

But the power series expansion of e^{-t} is

e^{-t} = 1 + t + t^{2}/2! + t^{3}/3! + · · ·

so i can be rewritten as:

i = 1/R [1 – e^{-(R/L)t}]

yielding the correct exponential rise of current through the circuit with a time constant of 1/(R/L) = L/R.

To solve the differential equation y” – y = 0 for t>0 and y(0)= 0 and y'(0) = 0, Heaviside would rewrite this as p^{2}y – y = **1**, or y = **1** / (p^{2} –1). But 1/(p^{2} – 1) can be expanded into the series (1/p^{2} + 1/p^{4} + 1/p^{6} + · · · ), and since we have 1/p^{n} · **1** = t^{n}/n! from above, then y = t^{2}/2! + t^{4}/4! + t^{6}/6! + · · · = ½(e^{t} – e^{-t}) – 1. You can see how easily one can get in the habit of dropping the **1** altogether and working solely with p’s once the knack is acquired.

As another example, Heaviside presented an operator function [p/(p+B)]^{1/2} acting again on a step function. Here he divides through by p to get (1 + B/p)^{-1/2} and again expands this into a power series, arriving at a solution in terms of modified Bessel functions. Can p’s be handled in these ways? Apparently they can, for Heaviside was able to provide correct solutions using his operator calculus.

Now in circuits with continuous, distributed impedances such as in telegraph lines, (particularly distortionless ones in which an infinite current appears instantaneously at t=0) Heaviside was faced with fractional powers of p such as p^{1/2}. Unfazed, he found a specific problem that had been solved using Fourier series methods (the solution of the diffusion equation) and applied it to a problem for which p^{1/2} appeared when written in his operator form (the current from a step voltage in an infinitely long cable). Heaviside equated the form of his solution to the Fourier solution to deduce p^{1/2} and he declared the result to be generally true! (Later, Heaviside presented a direct derivation based on the gamma function, but it is also derivable using Carson integrals and other methods).

He then arrived at his fractional powers of p:

p^{2} **1** = 1 / (πt)^{1/2}

p^{3/2} **1** = p(p^{1/2} **1**) = – t^{-3/2} / 2π^{1/2}

p^{-1/2} **1** = t^{1/2} / (1/2)!

p^{-3/2} **1** = 1/p (p^{-1/2}) **1**

and so forth.

Heaviside developed the *Heaviside Expansion Theorem* to convert Z into partial fractions to simplify his work. For i=1/Z and Z a polynomial in p, the roots of Z can be found and i expressed as a sum of terms consisting of constants divided by the simpler factors. A similar thing is done when using Laplace Transforms, but Heaviside developed his own method of calculating these constants, the *Heaviside Cover-Up Method* (See this webpage for a description).

Heaviside expressed the use of this theorem for a step function as

i = v/Z_{0} + v ∑ e^{pt} / [p(dZ/dp)]

So let’s look again at our first problem above of a series circuit of a resistance and an inductance, where Z = R + Lp and the steady state Z0 = R. Now dZ/dp = L and setting R+Lp = 0 implies a root p = – R/L. Then from the equation just above after setting the step value v to 1:

i = 1/R + e^{-(R/L)t} / [(–R/L) · L]

or

i = 1/R [1 – e^{-(R/L)t}]

which is the same solution we obtained earlier using power series.

Nahin describes Heaviside’s ingenious solutions when using his operators for the time-varying current in circuits with additional continuously-distributed parameters such as found in actual telegraph lines. He provides an example in which Heaviside adds a section of cable to the beginning of an infinite line, finds the current as a function of time for that configuration, and then “removes” the initial section to end up with the solution for the original cable. In fact, Heaviside used his operator calculus to design a transmission line with zero distortion (but with exponential attenuation over distance).

Now when Z is a polynomial of degree greater than 4, its roots are difficult or impossible to find directly. Also, the Expansion Theorem does not work for a Z with a root of zero or with any repeated roots, a situation not encountered in passive networks but one that can occur when an internal component such as an amplifier sources energy. In true Heaviside fashion we can in this case treat equal roots as unequal, solve for the transient current, and let the roots approach equality as a limit!

But Heaviside removed even these difficulties by instead expanding Z in inverse powers of p and then replacing p by t^{n}/n! exactly as we saw in our first example. I’ve seen this referred to as Heaviside’s *Extended Expansion Theorem*, and I really have to wonder whether punchier names for his methods would have preserved more interest in them. Heaviside, who coined some terms much disliked in his day but which have stuck (such as impedance, inductance, conductance, admittance and reluctance), awkwardly referred to “algebrising” a differential equation with his operators and “logarising” when taking a logarithm, and he christened his operator e^{-ph} the “Spotting function” because it isolates, or spots, a certain value of the function. Along with his “Cover-Up Method” these are not exactly memorable names.

Now as one more example of this method, consider a series circuit of R, L and C, where Q represents the charge in the circuit, so the current i = dQ/dt = p·Q:

Z = R + pL + 1/pC

i = **1** / [R + pL + 1/pC]

[R + pL + 1/pC] i = **1**

[p^{2}L + pR + 1/C] Q = **1**

Q = **1** / [p^{2}L + pR + 1/C]

= **1** { p^{2}L · [1 + (R/L)p^{-1} + (1/LC) p^{-2}]^{-1}}

Expanding the term on the right using the Binomial Theorem, we have

Q = (**1**/L) · {p^{-2} · [1 – [(R/L)p^{-1} + (1/LC)p^{-2}] + [(R/L)p^{-1} + (1/LC)p^{-2}]^{2} – · · · }

= (**1**/L) · {p^{-2} – (R/L)p^{-3} – [(1/LC) + R^{2}/L^{2}]p^{-4} + · · · }

and as before we replace p^{-n} · **1** by t^{n}/n! to arrive at a power series in t without having to find the roots of Z. These power series are sometimes not expressible in terms of elementary functions, but Heaviside dismissed this with the astute observation that calling the sum of a particular power series an exponential or trigonometric function didn’t simplify finding the original solution of it.

In 1893 Heaviside published the first of a three-part series describing his operator calculus in the *Proceedings of the Royal Society*. Later in the year the second part appeared, but this “was the last straw for mathematicians” [Nahin], and his third part was rejected, leading to a series of written attacks best enjoyed by reading Nahin’s book.

What killed the third part was Heaviside’s cavalier use of divergent series, dismissing their apparent tendencies to infinity while producing accurate results by manipulating them at whim, or at least this was how it appeared. In fact, Heaviside often produced two versions of his power series solution, a convergent one that was useful for small t but was too slow to converge for large t, and a divergent one that was useful for large t when it was taken to a small number of terms. Josephs provides an example of two such series for the Bessel function solution of a current entering a particular type of transmission line:

e^{-φt}I_{0}(φt) = 1 – (φt) + (1·3/(2!)^{2}) (φt)^{2} – (1·3·5/(3!)^{2}) (φt)^{3} + · · ·

e^{-φt}I_{0}(φt) = (2πφt)^{-1/2} [ 1 + (8φt)^{-1} + 1^{2}·3^{2}/(2!) · (8φt)^{-2} + 1^{3}·3^{3}·5^{3}/(3!) · (8φt)^{-3} + · · · ]

The first expression is convergent but is slow to converge for larger values of φt, while the second expression is a divergent series of an asymptotic type (a series in inverse powers of the argument). The plot below shows the effect of taking terms in each series for relatively large values of φt. The first expression is shown as convergent, but slowly, while the second expression converges quickly to approximately the final value of the convergent expression as the values of the terms decrease, then takes off and rises to much larger values. The larger the value of t, the less is the error at the point of the minimum term. The trick is to stop taking terms after the one with the smallest value, which Heaviside could do empirically.

In general, Heaviside found that if an operational equation is a series in integral and fractional powers of p, he could discard the terms with integral powers of p, express the fractional powers of p as p^{n} p^{1/2} or p^{n} (πt)^{-1/2}, and create a divergent series in p^{n} useful for large t. Josephs provides an example of this procedure in deriving a convergent and divergent series for the voltage at the terminals of a non-inductive cable when a 1-Volt battery is applied through a terminal resistance.

Heaviside used physical intuition to guide him in handling these series, and he was unparalleled in his electromagnetic intuition. And of course he went far beyond the short flavor of his operator calculus I’ve described here—Heaviside found solutions for very complicated electromagnetic problems whose solutions were intractable by any other method at the time.

But Heaviside also checked his steps with exhausting numerical calculations to make sure he didn’t make a misstep. I really appreciate this because I have often done that in deriving approximations to elementary functions, an interest of mine apparent on my MyReckonings.com website—I was really gratified to find a professional doing something like this. Heaviside also took his solution and verified that it satisfied the original equation, although of course this doesn’t validate the uniqueness of it. Because Heaviside’s work was results-oriented, he sometimes provided *ad hoc* arguments to support his derivations. He often suggested in rather abrupt prose that mathematicians should provide rigorous proofs for what he did. I see a bit of his humanity in a footnote in Volume 2 of his **Electromagnetic Theory** (a volume almost exclusively concerned with his operator methods):

It is rather disagreeable to have to be self-assertive and dogmatic (especially when one thinks of the always possible risk of error); but there may be times when it becomes a duty—e.g., when mathematical rigourists are obstructive.

And Heaviside certainly did exercise the option, often with a Victorian flair for the erudite put-down:

I think I have given sufficient information to enable any competent person to follow up the matter in more detail if it is thought to be desirable. It is obvious that the methods of the professedly rigorous mathematicians are sadly lacking in demonstrativeness as well as in comprehensiveness.

Mathematicians were not amused, but I am.

**References**

Heaviside, Oliver. **Electromagnetic Theory, Vols. 1-3**. New York: Cosimo Classics (2007). These three volumes, available at Amazon, are reprints of Heaviside’s books on electromagnetic theory. Volume 2 contains a modified version of the rejected Part III of his series on operators for the Proceedings of the Royal Society. The three volumes were also printed as one large book by Dover in 1950.

Heaviside, Oliver. *On Operators in Physical Mathematics, Part I.* Proceedings of the Royal Society, vol. 52, Feb., 1893, pp. 504-529. Online scans of this can be found at http://gallica.bnf.fr/ark:/12148/bpt6k56145p/f512.item and its following page selections.

Heaviside, Oliver. *On Operators in Physical Mathematics, Part II*. Proceedings of the Royal Society, vol. 52, Feb., 1893, pp. 504-529. Online scans of this can be found at http://gallica.bnf.fr/ark:/12148/bpt6k56147c/f112.item and its following page selections.

Josephs, **H.J. Heaviside’s Electric Circuit Theory, 2nd Ed**. Methuen’s Monographs on Physical Subjects, London: Methuen and New York: Wiley (1950). This tiny book (without the cover I measure it at 6-5/8″ x 4-1/8″ x ¼”) is packed with useful information on Heaviside’s operational calculus.

Lindell, I.V. *Heaviside Operational Rules Applicable to Electromagnetic Problems*. Progress in Electromagnetics Research, PIER 26 (2000), pp. 293-331, also found at http://ceta.mit.edu/PIER/pier26/11.9909172jp.Lindell.pdf. A comprehensive, highly mathematical collection of valid rules for Heaviside’s operator calculus along with their derivations.

Nahin, Paul J. **Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age**. Baltimore: Johns Hopkins University Press (1988). This is one of my favorite books of all time, a fascinating biography of Heaviside and his contemporaries and brimming with the controversies he engendered. It has a technical bent as well, as do all of Nahin’s excellent books.

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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.

It was Christiaan Huygens (1629-1695) who built the first pendulum clock as we know it on Christmas, 1656. His pendulum swung in a wide arc of about 30° and consisted of a metal ball suspended by silk threads. There are a few design aspects of pendulums that may appear obvious in retrospect but which were novel enough at the time. First, there is the matter of air and gear friction. To minimize these effects there must be sufficient mass to make frictional forces irrelevant, the rod of the pendulum should be thin, and the pendulum must be enclosed to avoid drafts. It is also true, unlike in Huygen’s pendulum, that the bob itself should be thin—later bobs were made to slice through the air, and this feature along with the requirement for significant mass results in the tapered lens-shaped disk that we see today on pendulums.

The second design feature we see in this clock design of Huygens is the concentration of mass at the bottom of the pendulum. While the mass itself does not contribute to the period of the pendulum (after frictional forces are overcome), the length of the pendulum certainly does, and this length is measured from the pivot point to the center of mass of the pendulum. The common “seconds” pendulum (which actually has a two-second period) is convenient for gearing ratios since most clock escapements move the seconds hands on each swing, and this pendulum has a length of a little over 39 inches. Huygens and Christopher Wren proposed this distance as a standard unit of length, but were unaware at that time of the geographic variation of this length (in 1793 the meter was standardized as one ten-millionth of the distance between the North Pole and equator based on curvature estimates from triangulations by Delambra and Mechain). Later measurements by Jean Richter of this geographic variation led Huygens to assign this variation to centripetal force from the Earth’s rotation, which is indeed latitude dependent. The “seconds” length requirement makes for non-portable pendulum clocks, particularly if the center of mass of the pendulum is not as low as possible (and also exacerbated if the pendulum has a wide swing to accommodate as in Huygens’ first clock).

Mahoney points out that Huygens’ invention of the pendulum clock contained an important feature: the independent suspension of the pendulum and the crutch linked the clock mechanism and the pendulum, but in a way that allowed separate, one-way adjustments. The driving force of the escapement could be adjusted without affecting the operation of the pendulum, and varying the characteristics of the bob (such as increasing the bob mass to overcome variations in crutch coupling or streamlining it to decrease air resistance) did not affect the operation of the escapement. This allowed a practical means of calibrating the individual components. Also, the silk threads of the pendulum rod in this design were extremely light and strong with little stretch and high resistance to rot, and they also minimized friction at the pivot point. They were ideal for Huygens.

But the silk threads offered a huge advantage in one other way—they allowed the effective length of the pendulum to vary so the pendulum is truly isochronic. It had been discovered by Marin Mersenne (1588-1648) that the period of the pendulum does indeed depend on the amplitude of its swing, with isochronism true enough only for small angles. Wider swings produce slightly longer periods in the same proportion as *sin x* varies from *x *as the angle *x* increases. This error is called the circular error or circular deviation. The wrapping of the silk threads through the “cheeks” seen in the clock diagram effectively shorten the pendulum length with its distance along its arc, constraining the pendulum to a **cycloidal** path and providing true isochronism. Huygens revolutionized the design of pendulums through such mathematical analysis of this and other characteristics of pendulums.

A cycloid is the curve defined by the path of a point on the edge of a circle as it rolls along a straight line, as shown in Huygens’ figure on the left. It is a **tautochrone**, a curve for which a frictionless particle sliding on it under gravity to its lowest point will take the same amount of time regardless of its starting position on the curve. By definition an isochronic pendulum needs to follow a tautochronic path.

But let’s back up a bit to see how Huygens found this curve. In deriving the relation for the period of a pendulum in 1659 equivalent to T = 2π(L/g)^{1/2}, Huygens found he had to make an approximation that was only negligible for small amplitudes of oscillation, one that defined a curved path of a cycloid with a vertical axis of half the pendulum length L. In one of those fortuitous circumstances that occur so frequently in history, he had studied precisely that curve for a mathematical challenge issued by Blaise Pascal in 1654. Following this lead Huygens found that a body falling from any point along the cycloid will reach the bottom in the same amount of time, and the ratio of this time to the time for free fall from rest along the axis of the cycloid is π : 2. In 1673 he published his masterpiece, the *Horologium Oscillatorium*, in which he directly proved that the cycloid was the needed curve for an isochronous pendulum.

To derive this, Huygens begins by presenting the Galilean properties of free-fall, i.e., that the distance fallen is proportional to both the time squared and the velocity squared. In addition, the distance fallen in a given time is equal to the distance traversed in the same time with a constant velocity half that of the velocity at the end of the fall. After proving 20 more propositions too detailed to present here, he arrives at the figure to the right. Here the arc ABC is a cycloid created by point A as the circle AVD rolls along the top line DC. As translated from the Latin by Blackwell, Huygens states and then proves that:

The time in which a body crosses [spans] the line MN, with the uniform velocity acquired after it has fallen through the arc BG of the cycloid, will be related [proportional] to the time in which it would cross the line OP, with half of the uniform velocity which it would acquire by falling through the whole tangent BI, as the tangent ST is related to the part QR of the axis.

After one more proposition in which he considers infinitesimal arcs of travel along the cycloid (we’ll touch on that later), Huygens arrives at the culmination of Part II of his work:

On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other; and these times are related to the time of a perpendicular fall through the whole axis of the cycloid with the same ratio by which the semicircumference of a circle is related to its diameter.

where the last clause provides the π : 2 ratio mentioned earlier.

So an isochronous pendulum must be constrained to move along a cycloidal path. Huygens needed to find the curve (the **evolute**) that would “unwind” to form this cycloid (the **evolution** or **evolvent**), and he created a new branch of mathematics, the theory of evolutes, to do it. The problem reduced to finding a curve such that

- Each leaf is tangent to the centerline.
- Each leaf is perpendicular to the cycloidal arc of the pendulum at its point of contact.
- The leaf length to the point of contact with the bob equals the pendulum length, so it must have an arc length of twice the diameter of the circle generating the cycloidal path of the pendulum, which is half the cycloid measured from its base to vertex.

This leads to the fact that the evolute must be a curve having the same base, height and length as the cycloidal path of the pendulum, and Huygens came to the startling realization that the evolute is a cycloid generated by the same circle as the cycloid derived for the pendulum path, or in other words, the cycloid is its own evolute! (In 1692 Jacob Bernoulli showed that a logarthmic spiral also is its own evolute.)

But for a practical pendulum Huygens further proposed that it is necessary to know its “center of oscillation,” and using an axiom equivalent to the conservation of energy he defined the center of gravity of a pendulum in terms of the modern concept of its moment of inertia. Taking the limits of infinitesimal points of mass, he calculated the centers of oscillation of many types of pendulums; for example, his spherical bob of radius *r* on a weightless string produced a center of oscillation 2r^{2}/5L below the center. This provides the analysis of the effect of sliding weights on pendulums to adjust for (or measure) geographical differences. He derives the practical technique of using the period of a known pendulum to find *g*, the acceleration due to gravity of free-falling bodies. Finally, Huygens describes the conical pendulum and produces theorems on centrifugal force equivalent to (but preceding) Newton’s F=mv^{2}/r. This is the only place where force appears, as his work is based on the concept of conservation of energy, still a popular approach to physical problems involving complicated motions.

The *Horologium Oscillatorium* was written in the style of geometric physics in which quantities are related by proportions demonstrated with geometric constructions, a method soon superseded by the analysis tools of mathematical physics. But Huygens used infinitesimal time intervals and distances and extrapolated them to limiting cases, prescient in his anticipation of the development of the calculus. Blackwell points out that while later physicists relied on the extensive foundations of calculus and mechanics to build arguments, Huygen’s work “may be enjoyed as a beautiful specimen of [his] explicit handling of physical concepts and argument.” It is a self-contained jewel with a brilliant clarity seen in great works of all fields.

As an example of the practical nature of Huygens, he provides in this work an alternative method of drawing a cycloid that is shown in the figure on the left. Equal arcs AC, CD, and so forth are drawn on a circle whose diameter is half the length of the pendulum. Now join these points with horizontal lines and construct LM as the arc length AF, divided into as many parts as points marked off on AF. Then on the horizontal lines mark GO and CN with lengths equal to one part of LM, then HQ and DP with lengths equal to two parts of LM, etc. The curves connecting these points are the required cheek curves for the given length of the pendulum. To construct LM from the arc length AF, XZ is drawn equal to the sum of the two chords of half of AF. Then overlay XY as the length of the chord of AF. Finally add ZΔ as 1/3 of YZ. Then if AF is 1/6 or less of the circumference of the circle, XΔ equals AF to 1 part in 6000. What strikes me about this construction is that it creates an *approximate* solution, something I wouldn’t expect from a geometric construction and surely an indication of how closely Huygens aligned his mathematics with the practical construction of mechanical clocks.

And maybe that’s what makes the *Horologium Oscillatorium* such a fascinating piece of work. Mahoney points out something I hadn’t noticed, that there are three layers of meaning in the diagrams and sketches of Huygens. In this work we see the overlay of physical shapes (the pendulum cord and cheeks) onto geometric constructions proving theorems about those shapes. In notes from 1659 in which Huygens first finds the cycloid as the isochronous curve, he also overlays an auxiliary curve, a parabola that describes the velocity of the bob as it moves along the cycloid. He created “a curve in physical space, the properties of whose normal and ordinate could be mapped by way of a mathematical curve so as to generate another mathematical curve congruent to a graph of velocity against distance” [Mahoney]. Later we will see that Huygens created a *mathematical relation* that defines isochronous systems, thereby lifting the mathematics out of the geometrical physics and anticipating analysis as the new physics.

In this work Huygens also studied conical pendulums (in which the bob swings in a horizontal circle rather than in a vertical plane) as well as compound pendulums. He also designed pendulums for use aboard ships, a rocking platform for which pendulum clocks were never successfully produced. For this environment Huygens in 1672 created a clock that utilized a triangular pendulum, that is, one that is suspended from two separated cords and thereby constrained to move in one plane only, theoretically eliminating most effects of the rocking of the ship. The clock itself was suspended first in a ball-and-socket mount and later on gimbal mounts in an attempt to eliminate rocking in the plane of pendulum motion.

Huygens later invented the ingenious tri-cordal pendulum, a ring suspended at three points by threads and made to oscillate around its center as shown in his sketches below. Radial placement of weights could be used to calibrate the pendulum. From his analysis of conical pendulums he discovered that this mechanism would be isochronous if any point of the ring moves along a parabola curved around the cylinder defined by the ring. To fine-tune the tri-cordal pendulum to this constraint he considered adding cheeks but eventually just went with longer threads.

After the publication of his Horologium Oscillatorium, Huygens found that in a cycloidal pendulum the force on the bob is proportional to the distance or angle from the neutral position, and he deduced that any mechanical system that met this constraint would be isochronous. He came up with a number of mechanisms of this type. In 1675 this led him to invent the horizontal balance spring as a clock oscillator, in which the force varies directly with angle in the same way that force varies directly with distance in ordinary springs, although Hooke did not publish his law on this until 1678. (There is some debate today on whether Hooke actually invented the balance spring.)

But the elasticity of the balance spring suffered under temperature variations, and in 1693 Harrison revisited notes he made in 1684 to create the “Perfect Marine Balance.” Here the spring balance was replaced with something like a physical pendulum, a balance bar on a knife edge swinging in a vertical plane and controlled by another weight suspended by a thread from cycloidal cheeks that are mounted on the balance shaft and oscillate with the balance. To get the returning force to vary directly with the angle, he experimented with a weighted chain arrangement and with a float partly submerged in oil or mercury.

Meanwhile, in 1670 the anchor escapement was invented, possibly (and certainly claimed) by Robert Hooke. Some authors attribute its discovery to Thomas Tompion (1639-1713), but a more correct attribution may be to William Clement (1643-1710) [Heldman]. The workings of this escapement are outside the scope of this essay, but its effect on pendulum design was significant because it was used to reduce the pendulum swing to 4-5°. (It is worth noting here that the verge escapement can have as small an angle of escape as desired by designing very long pallet arms and a large distance between the horizontal escape wheel and the pivot arbor for the pallets and crutch—there are provincial French pendulum clocks, mostly of the 19th century, with this arrangment [Heldman]. But certainly the anchor escapement triggered clock designs at the time that had small-angle swings.)

A reduced pendulum swing makes possible much longer pendulums for a given horizontal space. Clocks with 14ft. pendulums were built, for example, and Tompion produced a clock with a 13ft. pendulum hung above the movement. Longer periods are more directly geared to clock time, but the small swing provided by the anchor escapement also significantly reduced friction at the pivot point. And now that the swing was small, the silk cords that rolled over the cycloidal cheeks were replaced with a short strip of flat metal (a brass suspension spring) that simply flexed around the shorter arc of the cheeks, decreasing the friction even more. When a metal rod and bob were connected to the strip, the entire pendulum manifested a permanent, all-metal construction. It might be noted here that Huygens and others looked to long, slow pendulums for stability, but in fact more success in pendulum clocks was ultimately had with short, fast-moving pendulums.

The other major problem with pendulums was the change in length, and therefore the center of gravity, with temperature. Huygens never fully realized the effect of temperature on his clocks. On hot days a pendulum lengthens slightly and the clock slows, and the opposite happens on cold days. George Graham (1673/4-1751) attempted to devise a pendulum using the varying expansion rate of metals to remain isochronous over temperature ranges. In these designs the expansion in temperature of one metal is offset by the expansion of the other, designed so that the net length of the pendulum remains constant. Failing to arrive at a suitable design, Graham settled on mercury-compensated pendulums as his solution. Here the pendulum is designed to hold mercury in a glass cylinder in much the same way as a mercury thermometer. When the pendulum length increased with temperature, the mercury expanded as well, and vice-versa. When properly designed, the net center of gravity of the pendulum remained unchanged regardless of temperature variations. Ingenious! Graham also invented the deadbeat escapement that made for quite small pendulum arcs.

(As an interesting notion Matthys mentions that if a pendulum is not temperature compensated, one might support the bob at the bottom edge. In this way the upward expansion of bob partially compensates for downward expansion of pendulum rod.)

John Harrison (1693-1776) approached Graham in 1730 with his invention of the “gridiron pendulum.” Graham, a respected member of the Royal Society in London (and considered a humble, generous man according to all references I’ve seen), encouraged Harrison to pursue it, and it is the design seen in serious grandfather clocks (but faked in most commercial ones sold today). The design from a later clock is shown in the figure here. The expansion of brass rods in the pendulum compensates for the expansion of iron rods in order to keep the effective length the same over temperature. The key is to have a high ratio between the thermal coefficients of the two metals used in the gridiron. If the ratio is 2:1, two rods can be used to expand downward and one rod upward, and so forth for different ratios. Anything less requires more rods to achieve the ratio. The ratio for iron and brass is 1.7:1, so Harrison used three rods expanding downward and two upward, and included two of these sets to equalize the weight. Since one rod can be common to both sets, nine rods were needed.

Well, just about. Harrison was also the first to realize the effect of atmospheric density on the period of a pendulum. Colder temperatures produce higher air densities, which alter the buoyancy of the bob and therefore the restoring torque. Another factor, absolute humidity, affects the density and viscosity and thence the rate of energy loss, equilibrioum amplitude and period of a pendulum [Emmerson]. From experiments performed with evacuated bell jars, Harrison adjusted his gridiron pendulum to account for the effect of temperature-induced density changes as well as for thermal expansion! Harrison also first confronted the effect of air resistance after he invented the grasshopper escapement—the frictional losses were now so low in his clock that the pendulum swung wildly until he attached small vanes to the pendulum. Air resistance is important, as over 90% of the drive energy imparted to a pendulum is lost through air drag. (Actually, some amount of air resistance can provide stability to the amplitude of the pendulum swing.) Through this and many other innovations Harrison claimed a pendulum clock accuracy of 1 second per month, an achievement still very much envied (and challenged).

There are other aspects of pendulums that are not considered here. For example, two or more pendulums that are lightly coupled (such as in clocks sitting on the same mantelpiece) will synchronize their swings, but in the opposite direction. Huygens made the first observation of a coupled oscillator in just this way in 1665 while recovering from an illness. For unconstrained simple pendulums with the same natural period, such a loose coupling results in a modal phenomenon in which the total swinging motion moves back and forth between them. Highly coupled oscillators, such as a compound pendulum where one pendulum is hung from the bob of another, exhibit chaotic motion. Coulomb used a torsion pendulum in 1784 to quantify the electrostatic force, and Cavendish determined the density of the Earth in 1798 using a pendulum. Foucault also famously used a pendulum in 1851 to directly demonstrate the rotation of the Earth. But in the end my fascination lies with the creative, technical pursuits seen in the early designs of pendulum clocks.

**References**

Andrewes, W. J. H. (Ed.). **The Quest for Longitude: The Proceedings of the Longitude Symposium Harvard University, Cambridge, Massachusetts, November 4-6, 1993**. Cambridge: Collection of Historical Scientific Instruments, Harvard University (1996). Wow, is this a neat book, a lavishly illustrated collection of fascinating essays by experts in horology on the pursuit to determine the longitude of a person at sea, a huge historical problem. Essays that provided information for the present essay include *The Longitude Timekeeper of Christiaan Huygens*, by J.H. Leopold; *‘John Harrison, Clockmaker & Barrow; Near Barton upon Humber; Lincolnshire’: The Wooden Clocks, 1713-1730*, by Andrew L. King; and *The Scandalous Neglect of Harrison’s Regulator Science*, by Martin Burgess.

Emmerson, Alan. The papers in Horological Science by Mr. Emmerson presented here are a pedagogical treat, presenting clear mathematical explanations of pendulum physics such as the non-isochronous behavior of a rigid pendulum suspended between cheeks as mentioned in this essay.

Heldman, Alan W. *Personal Communications*. Mr. Heldman’s horological knowledge led to several corrections and improvements to this essay, which is much appreciated.

Huygens, Christiaan. **The Pendulum Clock or Geometrical Demonstration Concerning the Motion of Pendula as Applied to Clocks, Translated with Notes by Richard J. Blackwell**. Ames: Iowa State University Press (1986 translation of 1673 *Horologium Oscillatorium*). Surprisingly, this is the first English translation of Huygens’ book, and it’s a really interesting read. This is also the culminating scientific work presented as geometrical physics (i.e., using geometric constructions as derivations and proofs, with relations between quantities of different dimensions expressed as proportions rather than equations). Later works by others trended toward analytical approaches, particularly following the invention of calculus. Interestingly, Blackwell also notes that this book is based on an axiom equivalent to the conservation of energy rather than the concept of forces developed later by Newton. An actual scan of the 1673 book from which the manuscript figures of this essay were drawn is found at http://kinematic.library.cornell.edu:8190/kmoddl/toc_huygens1.html.

King, Henry C. **Geared to the Stars: The Evolution of Planetariums, Orreries, and Astronomical Clocks**. Toronto: University of Toronto Press (1978). An encyclopedic, out-of-print work on a niche subject that is quite an expensive volume to buy on the used market. I was able to borrow it from a local library.

Mahoney, Michael S. Various fascinating papers, most of which involve Huygens, can be found at http://www.princeton.edu/~mike/17thcent.html. In particular, details of Huygens’ original cycloidal derivations from 1659 can be found in *Christiaan Huygens: The Measurement of Time and Longitude at Sea*, and an interesting discussion of the physical and mathematical layers within Huygens’ drawings is presented in *Drawing Mechanics*.

Matthys, Robert J. **Accurate Clock Pendulums**. Oxford University Press (2004). Lots of practical advice can be found in this book.

[Please visit the new home for Dead Reckonings: http://www.deadreckonings.com]

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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!

Consider the diagram at the right of the motion of the sun on the celestial sphere. NCP is the North Celestial Pole, equivalent to the North Pole of the Earth about which the Earth spins. The sun lies directly over the equinoctial line (essentially the same as the equator) on the vernal and autumnal equinoxes (~March 21 and September 21). However, since the Earth’s axis is tilted by 23.5° relative to the plane of its orbit, the sun appears to move above and below the equator during the course of a year as shown by the circle in bold (the ecliptic), achieving its northerly peak at the summer solstice (~June 21) and tracing in the course of the day the circle labeled Y, and achieving its southerly peak at the winter solstice (~December 21) and tracing in the course of the day the circle labeled U. The fact that the sun is much larger than the Earth (so the rays are “parallel”) justifies displacing an observer located on the top of the sphere to the middle of the sphere, simplifying the analysis. During the course of the year the observer sees the sun rise above the horizon circle from an easterly direction and trace an arc to set below the horizon circle in a westerly direction, with the summer arc higher than the winter arc. The objective is to provide a set of angles that uniquely identify the sun’s position in the sky at any time; although the analemmas provided all such angles, the only two we will consider here are the zenith angle and azimuth mentioned earlier.

The shadow of the tip of a sundial will trace out an arc over the course of a day, and the path of this arc depends on the location and the day of the year. On the equinoxes, the shadow will trace a straight line (we are speaking here of latitudes between the Arctic and Antarctic Circles), while on other days the shadow will trace hyperbolas. Knowledge of the sun’s position at all times provided the Greeks with the ability to mark time by drawing these shadow traces on sundials. Note that the Greek sundials marked 12 divisions of daylight, regardless of the length of daylight for that day, so-called **unequal (or seasonal) hours**.

The information contained in the 3-D sphere above is neatly contained in the 2-D geometric construction below, the initial stage of the analemma described but not invented by Vitruvius. This figure is reproduced from Gibbs’ book (see References). The observer is a shadow-casting stick (or **gnomon**) BA rising vertically from the ground BR on which it will cast a shadow over the course of the day. The circle about A is the meridian circle, which contains the zenith of the observer and is oriented North-South (the longitude line of the observer). BC is the shadow length of the gnomon at noon (so the sun is due south) at an equinox, so line NC represents the equator and therefore drawn at an angle equal to the latitude. The axis of the Earth PQ is constructed perpendicular to the equator line.

So far everything lies in the meridian plane, the plane of your monitor. Now we will take the horizon circle, perpendicular to the meridian plane and moved parallel from the ground BR, and project it edge-wise into the meridian plane, resulting in EAI. Remember that this is actually a circle extending through your monitor screen, as this will become important.

Vitruvius marked off 1/15 of the meridian circle (or 24° rather than the more correct 23.5° ecliptic angle used by Ptolemy) on either side of the intersection of the equator with the meridian circle at F (at H and G) and drew the small corresponding circle centered at F. Then because H and G are located at the extreme angles of the ecliptic, KH (drawn parallel to the equator) is the projection into the meridian plane of the circle the sun takes in at the winter solstice and LG is the same for the summer solstice. We can now mark the points of two shadows of the gnomon *due south* *at local noon*: T in the summer from extending LAH, and R in the winter from extending KAG. LAH and KAG represent the ecliptic. Also, remembering that LG, KH and all other projections of the parallel circles of the sun’s path between these extremes, we can take, say, the summer sun circle and swing it about LG into the meridian plane, becoming the half-circle LYG. Y is found by constructing a perpendicular to LG from the point S where LG intersects the horizon EAI. Now since EAI and LG are projections of circles that are actually at right angles to your monitor, do you see that when LYG is swung back out of the monitor to its proper position the point S represents the point where the sun meets the horizon, i.e., sunset? Therefore, LY is the half-daylight duration and the fraction LY/LG represents the fraction of day to night for the summer solstice. The figure also shows the same construction for the winter solstice.

We can find the sun circles for other days between the two solstices. Again considering the same summer sun circle on LG swung around into the meridian plane, as shown in the figure on the right, we can mark off 6 equal arcs along LG to represent the 6 signs of the zodiac that the sun passes through in a half-year, and we can drop perpendiculars to the diameter LH to find the sun circles for the days of entries into those signs (and others days). The other 6 signs produce the same lines as the first 6 lines. Constructing lines parallel to the equator at these ecliptic longitudes provide noon shadow tips on the ground BR for those days. However, it turns out that dividing the half-circumference of the small circle into 6 parts and creating parallels to the equator produces the same sun circles, and this is such a useful trick that this small circle has its own name, the **menaeus**. Gibbs, Heilbron and Drecker provide proofs of this, but in fact it is easy to see. Referencing my figure below, horizontal lines are drawn from 12 equidistant locations on the menaeus on the left. The menaeus rotated edge-on is shown, then two edge-on and one facing ecliptic circles oriented at different angles relative to the menaeus but having the same diameter projection.

As you can see, angling the ecliptic spreads the sun positions proportionally, so in fact a menaeus can be drawn at any angle to the ecliptic as long as its diameter equals the projected length of the ecliptic. Lines parallel to the equator have to be drawn at the points on the ecliptic, though, so having the menaeus centered on the equator provides these parallel lines directly. Also, this location does not significantly interfere with the other lines of the analemma, allowing the menaeus to consist of a separate attachment to an instrument, permanently subdivided.

The menaeus is a trick that you may notice now and again in some geometric constructions. The menaeus also makes an appearance in the figure on the left demonstrating the construction of the elliptical trace on an analemmatic sundial. This type of sundial has a vertical gnomon but provides the time in modern equal hours, requiring the gnomon to be moved to a specific offset for a given day. Its construction is based on the analemma of Vitruvius, hence its name. The menaeus actually makes an appearance and functions in at least two other portable sundials that are based on the sun’s altitude and the day of the year. Click **here** to see the Capuchin card dial (which will be located on the left), as well as the Regiomontanus dial and its very interesting construction (from Drinkwater).

So how do we find the sun’s position for times other than noon on a given day? We had seen earlier that the line from the upper end of the sun circle for the day through the tip of the gnomon onto the horizon provides the noon shadow length on that day (due South) measured from the base of the gnomon. In fact, we can find the South component of the shadow length for *any* hour of the day by dropping a line from the hour point on the sun circle perpendicular to its axis and projecting this axis point through the gnomon tip to the baseline, as shown in the figure to the right. In this figure the arc from noon to sunset is divided into 6 equal lengths (giving seasonal hours), and the black dots represent the perpendicular projections onto the axis of the sun circle for the day (here the summer solstice). Blue lines are drawn from these points to the green baseline through the top of the gnomon, and the distance from the base of the gnomon to these intersections provides for every hour the South component of the shadow length for the given height of the gnomon. At sunset the sun is on the horizon, so that blue line is horizontal.

But we need to know more than just the South component of the shadow length from to plot the day curves. Look at the figure below with the ivory background (also displayed in my “Welcome” post), reproduced from Peter Drinkwater’s translation and interpretation of the 16th century works of Oronce Fine. The figure is flipped left-right from that above, and the menaeus is at the upper end, but you should recognize some things here (ignore the radiating lines from the center for the moment). See the menaeus, the projections of the sun circles for the equinoxes and solstices, and the winter solstice half-circle swung upward into the meridian plane?

Here the quarter-circle up from the horizon (*eac* here) is divided into 90°. The sun half-circle at the summer solstice that has been swung into the meridian plane is divided into 12 parts (one per hour), which makes sense because the axis of the circle is normally parallel to with the Earth’s axis about which the sun seems to rotate. At the points on the axis where the perpendiculars to the hour lines meet, lines parallel to the horizon are drawn to the degree scale, giving the sun’s altitude at that hour on that day. If you remember the sun’s half-circle as actually perpendicular out of the screen, you can visualize why this would be so—drawing the parallel lines is equivalent to rotating the meridian circle about the center to become a circle through East, West and the zenith points. Note that **equal hours** are measured here, while the Greeks and Romans would simply divide the daylight hours into 12 parts—since we are seeing only a half circle here, the arc between the “4/8” mark (equivalent to sunset point S in the earlier diagram) and the almost hidden “12” mark at the upper termination would be divided into 6 parts to mark the unequal hours as we did earlier.

You can see that drawing lines from these altitude degree readings through the center point provides shadow lengths along the ground at the base of the vertical gnomon *da* for every hour of the day. In this case, however, the author chose to provide readings on a **shadow square**, a two-dimensional scale that provides the equivalent of the tangent of the angle as the ratio of the height and vertical distances—for example, typically there are 12 units displayed in each dimension to provide simple fractions, so the tangent of the ray that is shown intersecting the shadow square 8 units from the bottom right is 12/8 and the tangent of the ray intersecting 6 units from the top is 6/12. The shadow square is generally seen under the alidade (the sight) on the back of astrolabes, which is useful because the tangent of a sighted angle to the top of a building, say, can be used to determine the height of the building by multiplying this tangent by the measured distance to the building.

But as I mentioned, extending these lines through the shadow square onto the baseline provides the absolute length of the gnomon shadow, measured from its base, for the day corresponding to the sun circle. So we have the situation where we know the South component of the gnomon shadow for any hour of the day (by drawing lines from the hour point on the sun circle perpendicular to its axis, then from this point through the gnomon tip to the baseline, shown here in blue), and we know the full shadow length (by extending the points on the altitude quarter-circle through the gnomon tip, shown here in red). We can construct the sundial layout, viewed from above, directly below the analemma. We first mark a point for the gnomon (here in green) below that of the analemma. Then for each hour we drop a line corresponding to the South component of the shadow (a blue line), and we drop a line corresponding to the full shadow length (a red line). We draw circles centered on the gnomon with radii equal to the full shadow lengths, and the intersections of these circles with the South component lines provide the locations of the tip of the gnomon shadow for every hour of that day. On the equinoxes these points form a straight East-West line a distance from the gnomon base equal to the extension of the equator line to the baseline; otherwise a hyperbola results. Drawing lines through shadow points for the same hour on different days (i.e., from different sun circles) provides the hour lines of the sundial—they are slightly curved, although all surviving Greek planar dials simply have straight lines between the points for the two solstices. Evans provides examples of constructing these layouts, including tips such as drawing the baseline higher than the base of the meridian circle in order to work with a larger meridian circle without having such long projected lengths on the baseline.

A century later comes Ptolemy, who finds the spherical model used by “the ancients” to be inelegant, as the equator in the first figure to the right has a coordinate system that depends on the geographic location of the observer. (By the way, all these angles shown in this figure were derivable by those ancients!) Ptolemy replaced the equatorial circle with a moveable circle that passes through the East and West points and the sun (the *hectemoros circle*), as shown in the rightmost figure, and in the process created a new analemma.

Ptolemy’s analemma (from his text *Analemma) *is shown below, again from Gibbs. Similar components to Vitruvius’ analemma include the meridian circle centered on E, the horizon AB, the half-circle HXK of the sun’s path swung into the meridian plane, and the daylight fraction HN/NK. The rest is a compression of Gibbs’ text:

The sun X is at the second hour point here, and O is found on HK by drawing a perpendicular from X. Y on the meridian circle is found such that OY equals XO, and ΛE is drawn perpendicular to EO. POR and SOC are drawn perpendicular to EB and EG, respectively. Q is marked on POR such that PQ=OX, and point F is marked on SOC such that SF=OX. Then the hectemoros angle is equal to ΛEY; the horarius angle is equal to GEC; the meridian angle is equal to BEW; the vertical angle is equal to GEΨ; the horizon angle is equal to GEΩ.

Drecker and Gibbs provide the derivations, and in fact all the angles are derivable geometrically from this analemma, such as the zenith distance and azimuth angle whose formulas from spherical trigonometry are given at the top of this post. Ptolemy then created a mechanism to rapidly create these analemmas for different latitudes and solar declinations. Elements that are unchanged, such as the meridian circle and parallels to the equator, were engraved or painted on metal, stone or wood along with scales for major latitudes and equinox hour lines. A wax layer was applied to the plate to allow date- and location-specific lines to be drawn, and by rotating the disk and using a right-angled straightedge, the solutions could be quickly found. Neugebauer conjectures that the Greek interest in conic sections originated in the study of sundials, and that their interest in trisecting an angle with compass and straightedge was related to dividing the analemma circles into 12 parts, i.e., two bisections and then a trisection within each quarter-circle. I assume he means dividing the daylight portions of the sun circles into 12 parts, as dividing a full circle (such as the menaeus) into 12 parts is easy because marking off a chord length equal to half the radius will span 60° of the circle.

Actually, the way that the modern trigonometric calculations were embedded in the geometric constructions is through the chords or half-chords that are drawn between points on the circles. As shown in the figure on the right, sin α = (chord 2α) / 2. In the sexagesimal number system of Ptolemy’s day, sin α = (chord 2α) / 120. From this figure it is also apparent that cos α = [chord (180° – 2α)] / 2. As an aside, to create a menaeus with a span +/-23.5°, I would propose marking off on the meridian circle 2/5 of the meridian radius, as it turns out that the chord formula for this span yields a menaeus diameter that is 0.3987 of the meridian radius.

As an example of solving trigonometric formulas using chords, it is possible to find the latitude φ of a location by the length of the longest day (at the summer solstice) through the formula tan φ = (-cos (M/2) / tan ε), where M is the length of the longest day converted to an angle at 15° per hour and ε is the obliquity of the ecliptic (23.5°). The day lengths would have been measured at the time with water clocks (**clepsydras**) that drained water to measure time. In the first Vitruvius analemma figure above, angle NAL=ε and angle MAS=φ, with the sun circle LYG representing the summer solstice. The entire figure can be drawn other than the horizon lines EAI and BR that specify the latitude, setting Y such that the length LY/LG represents the half-daylight fraction observed for that day. Then the horizon line at A is drawn along AS, completing the analemma and identifying the latitude without solving the trigonometric formula directly (see Wilson). Rather than an absolute length of the longest day, it is also possible to perform this construction with the ratio of the lengths of the longest day to the shortest day. This is more than a passing curiosity—the Greeks used the ratios of longest to shortest days to map the world. In fact, Neugebauer tells us that the ancient Babylonians accepted the round value 3:2 as this ratio, apparently without realizing the geographic variation, and this led the Greeks to calculate the **clima** (a zone roughly equivalent to latitude) of Babylon at what would be 35° rather than the correct 32.5°, “seriously affecting the shape of the eastern part of the ancient map of the world.”

I find all this intriguing from a creative as well as a technical aspect. To my knowledge, the only prevalent use of geometric constructions today is in graphical layouts of sundials. Cousins’ book on sundials is chock-full of these fascinating constructions, and the best part is that when you are finished with the design and have built the physical sundial, you can take your layout sheet, frame it, and mount it on the wall as a piece of art.

**References**

Calvert, J.B. *Vitruvius and the Analemma*, at http://mysite.du.edu/~etuttle/classics/analemma.htm. Discusses the analemma from the point of view of orthographic projection.

Cousins, Frank W. **Sundials: The Art and Science of Gnomonics**. New York: Pica Press, 1970. This book on sundials has a particular focus on using geometric constructions to understand and lay out all manner of dials.

Drecker, Joseph. **Theorie der Sonnenuhren**, Berlin, 1925. I have not seen this work, but Gibbs and Neugebauer both refer to this work for more derivations related to the analemma of Ptolemy.

Drinkwater, Peter I. **Oronce Fine’s Second Book of Solar Horology**. Warwickshire, England: Self-published, 1993. This is a unique booklet, a translation and interpretation of this work of Oronce Fine (1494-1555), Dialist to Francis I of France. The figure with the ivory background is reproduced from this work, although it is not clear how much of this figure is due to Fine and how much to Drinkwater.

Evans, James. **The History and Practice of Ancient Astronomy. **New York and Oxford: Oxford University Press, 1998. This is a fantastic book that I recommend to anyone with an interest in this topic (it can be found on Amazon.com here). It also includes, for example, templates and instructions for creating your own astrolabe and an equatorium for the position of Mars. Perhaps it’s a good thing I found this reference so late in preparing this essay—I learned more about the analemmas by working hard to figure them out, and I would not have located most of the other interesting references here.

Gibbs, Sharon L. **Greek and Roman Sundials. **New Haven and London: Yale University Press, 1976. This was my primary source for detailed information on the analemmas of Vitruvius and Ptolemy, as well as the source of a few of the diagrams.

Heilbron, J.L. **The Sun in the Church: Cathedrals as Solar Observatories.** Cambridge and London: Harvard University Press, 1999. A very interesting book that briefly describes the analemma of Vitruvius and gives a short proof of why the menaeus works in Appendix B. The figure of the sun’s motion on the celestial sphere is reproduced from this book. See also *A Mathematical Supplement to “The Sun in the Church: Cathedrals as Solar Observatories”* by Ng Yoke Leng and Helmer Aslaksen at http://www.math.nus.edu.sg/aslaksen/projects/heilbron/.

Neugebauer, O. **The Exact Sciences in Antiquity, 2nd Ed**. New York: Dover Publications, 1969. This book has a wider scope, but a brief discussion of Ptolemy’s analemma and its mechanization into an instrument appears in Appendix II, *On Greek Mathematics*. A more comprehensive journal article on the topics of this appendix can be found online at http://www.ams.org/bull/1948-54-00/S0002-9904-1948-09089-9/S0002-9904-1948-09089-9.pdf.

Neugebauer, O. **Astronomy and History: Selected Essays**. New York: Springer-Verlag, 1983. This book contains is a collection of papers by the author over many years. Some information overlaps that of the previous reference (and in fact it contains the journal article linked in the previous reference), but this book is my source on the mistaken calculation of Babylon’s location (pp. 64,240).

Wilson, Curtis. *Hipparchus and Spherical Trigonometry*, article #2 of http://www.dioi.org/vols/w71.pdf. Describes how the analemma incorporates the formula for the latitude based on the length of the longest day. (I’m not knowledgeable enough to recommend other articles of this journal, but they do make sensational reading as all-out academic battles.)

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