A Relic Reflects - Calculation, Logarithms, Slide Rules, Modeling

A Relic Reflects Calculation, Logarithms, Slide rules, Modeling by ©2006 James S Phillips

Introduction I hatched the notion of this little discourse at the end of the 2006 spring semester. In our department offices, we have on old classroom model of a Pickett demonstration slide rule (the predecessor to the TI84 with overhead projector screen). It has been in our department since I started my adjunct teaching career around 1994 and has become a bit of an oddity in this era. Most of my colleagues in the department know what it is, but have never actually used one for calculation. Occasional comments have been made about learning more about it. In addition, there has been some curiosity expressed by some of the engineering students about this “old fashioned” calculator. Having started my undergraduate education in 1971, I had some personal experience with this tool (as well as the CRC math tables) both in high school and college. My experience is not as extensive as you might think. In 1972, the first hand-held scientific calculator, the HP-35, was introduced. A year later, the HP-45, a more sophisticated model, came out. After observing classmates using these calculators and seeing what a time saver they were, I forsook the slide rule and purchased the HP-45. So my trusty old slide rule has lain dormant for some 35 years. I never had the heart to dispose of it, so it continued traveling with me. Occasionally, I would open its drawer, but never thought much about it. Now I have decided to reacquaint myself with my old friend, at least for a short time. (It is interesting to note that my HP-45 died in about 7 years and was relegated to the rubbish heap long ago. It didn’t seem to evoke much sentimentality for me.) As I began my research, I was surprised at what a passionate following the slide rule has. Several internet sites have plenty of information concerning usage instructions, types, models, history, and collector information. I was able to locate “newer” books as well as some old classroom manuals and histories. I came to realize that I really had nothing significant to add to the discussion of slide rule, that hasn’t already been discussed in minute detail elsewhere. This made me question what the purpose of this short opus might be. After some deliberation, I have chosen to broaden my topic a bit and discuss logarithms and their use in two page 1

specific applications. The first application will be the basic computation principles of logs and how they are implemented on a slide rule. Included in this discussion, I will discuss the trigonometric scales on the slide rule as well. The second application is mathematical modeling of data, primarily the use of logarithmic graph paper to derive exponential models and power curves using an extension of basic linear algebra. Both of these applications made the concept of logarithms and their use tangible to me as a student and were useful to me in my early years in industry. I think for the students of today, logarithms are much more abstract and their relevance not as clear. Because the act of calculation at this point in history requires only that the user input the numbers into a machine correctly and depress the appropriate operational key, there is very little “thought” required on the part of the user. The act of calculation has become too easy in many respects. Now, this is not going to be another lament by an aging luddite. Though, after 15 years in engineering research and now 12 years of part and full time teaching, I do have some observations I would like to make. The technology discussed here has been supplanted by something far better. However, some of the estimation skills that allowed an individual to maintain some intuitive feel for the results of a calculation have, for the most part, disappeared. My discussions do not contain a lot of rigor, but I hope you enjoy the read. A Brief History The old saw has it that “necessity is the mother of invention.” I would say that laziness (or should I say “effort reduction”) is the father of creativity. As proof, I would offer the following quotes. The first from John Napier, the Baron of Merchiston in Scotland, in 1616: “Seeing there is nothing (right well beloved Students in Mathematickes) that is so troublesome to Mathematicall practise, nor that doth more molest and hinder Calculators, then the Multiplications, Divisions, square and cubical Extractions of great numbers, which besides the tedious expence of time are for most part subject to many slippery errors. I began therefore to consider in my minde, by what certaine and ready Art I might remove those hindrences.”1

1

Napier, John; A description of the Admirable Table of Logarithmes - London 1616; A Facsimile of the Old English translation of Mirifici logarithorum canonis descriptio 1614; Amsterdam, Theatrum Orbis Terrarum Ltd.; New York, Da Capo Press, 1969. (spelling and punctuation as in Napier’s preface) page 2

How many of our students would believe that the inventor of logarithms, had it in his mind to make their calculational life easier? Or how about this quote almost 300 years later from Florian Cajori: “Of the machines for minimizing mental labor in computation, no device has been of greater general interest than the slide rule.”2 I would venture to say that now almost 100 years after Florian, some might comment that with our electronic calculators and computers, that “mental labor” (or thought) in computation has been all but eliminated. Just look at the confused clerk, next time you provide them $11.01 for a $5.46 purchase. Napier’s invention was first published in 1614, though there is evidence that he privately communicated his work to a Danish astronomer, Tycho Brahe, in 1594.3 Napier’s invention converted the operation of multiplication to addition and the operation of division to subtraction -- as long as you had a table of logarithms. Probably doesn’t seem like much of a step forward for the multiplication of single digit numbers ( 3× 6), but think about the multiplication, by hand, of two six digit numbers (123456 × 654321). His initial set of logs were based on a geometric definition rather than a specific base (though it can be related to our natural logarithms3 noted by €  10 7  7 € NAP = 10 ln ln in the equation  , where NAP is the Naperian logarithm and N is the number  N  of interest). Henry Briggs, a geometry professor at Gresham College in London and later at Oxford, € collaborated with Napier and is given credit for developing the base 10 logs or common logs we use today. In 1617, Briggs published tables of fourteen place logarithms for counting numbers 1 to 1000. In 1624, the sequel to this effort was published, fourteen place logarithms from 1 to 20,000 and from 90,000 to 100,000. And in 1628, a Dutch Bookseller, Adrian Vlacq, published a set of ten place logarithms for numbers 1 to 100,000 4,5 filling in the gaps of Briggs’ tables. As much of a help as this was, it was still cumbersome to carry the table of logarithms wherever you went. In 1620, Edmund Gunter, an astronomy professor also at Gresham College, developed a two foot long rule where the distances between the numbers were proportional to their logarithms. To multiply two numbers, the user would use dividers to step along the scale to add or subtract as necessary, much like addition and subtraction on a number line discussed in 2

Cajori, Florian; A history of the Logarithmic Slide Rule; The Engineering News Publishing Company, London, 1909 (the first sentence of the preface) 3 Thompson, J. E.; The Standard Manual of the Slide Rule, Its History, Principle and Operation, Second Edition; D. Van Nostrand Company, Inc.; Toronto, New York, London; 1952 4 Boyer, C.B., revised bu Uta C. Merzbach; A History of Mathematics, Second Edition; John Wiley and Sons, 1989. 5 Clason, C. B.; Delights of the Slide Rule; Thomas Y. Cromwell Company, New York; 1964. page 3

many arithmetic texts. This was an improvement over the tables in terms of ease of use, but with some loss of precision (difficult to read more than three to four decimal places on a ruler, certainly not the ten to fourteen of the published tables). The actual notion of two sliding logarithmic scales adjacent to one another is credited to William Oughtred somewhere between 1620 and 1625. 6,7 This innovation eliminated the dividers, which made the calculation process a bit more straightforward. Note here that this calculator was for multiplication, division, and exponentiation, later models could be used to determine values of plane and hyperbolic trigonometric functions and determine common and natural logarithms of numbers. Addition and subtraction were still done “the old fashioned” way -- paper and pencil. This “calculator” was used for about 350 years and, for better or worse, helped in the development of some of the most complex accomplishments of the twentieth century -- nuclear physics, the space program that landed man on the moon, computer/calculator design, as well as large civil engineering design projects such as the Hoover Dam and the Golden Gate Bridge. Some Basic Math As stated above, logarithms converted multiplication and division to the simpler operations of addition and subtraction, respectively. For multiplication, the sum of the logs is equal to the log of the product. Or shown symbolically, log( A ) + log(B ) = log (AB) . In division, the A difference of the logs is equal to the log of the quotient. Symbolically, log( A ) − log( B) = log  . B € For exponentiation, the product of the exponent and the log of a number is equal to the log of the n

number raised to the exponent. Symbolically, nlog( A ) = log(€A ) . 8 Using tables, the process of multiplying two numbers involved looking up the logarithms for each number and then adding the two. Once the sum is determined, a search through the table for € this sum to find the “antilog” (the inverse value) is performed. As an example look at 3× 6. The log of 3 is 0.47712; the log of 6 is 0.77815. The sum of these logs is 1.25527. The inverse value of this logarithm is 18. Not all that impressive with simple operations, but very useful with large € numbers. Division is similar, find the difference of the two numbers. 6 7 8

Stoll, C.; “When Slide Rules Ruled,” Scientific American, Volume 294, Number 5; May 2006. Hopp, P. M.; Slide Rules, Their History, Models, and Makers; Astragal Press, Mendham, New Jersey, 1999. Note these rules are true for all logarithmic bases. I will only consider common and natural logs here. page 4

Just for fun, lets look at an exponential example, 36 . The most direct calculation would be to multiply 3 by itself 6 times - 3× 3 × 3× 3 × 3× 3. For logarithms, we take the log of 3, 0.47712, and multiply it by 6 which is 2.86272. Now find the antilog of this number which is € 729. Fewer steps ... maybe. € Now, believe it or not, I have left some of the tedious details out in this brief description of using logarithms. Understanding how the tables were put together, what mantissas are (decimal fraction of the logarithm), what characteristics are (the whole number of the logarithm and the location of the decimal place), and how to interpolate between values in the table are non trivial items to learn, though quickly mastered with frequent use. The slide rule provided a mechanical way to do these calculations but there were some tricks the user had to learn. The slide rule only provides one cycle of logs-- or a single decade of numbers -- numbers from 1 to 10 say. This means that you have to devise a means by which to express numbers in different decades in the same way. That is done by using scientific notation. You express the number as a product of a number between 1 and 10, and indicate the location of the decimal point in the number as a power of 10. For example 1,234 would be 1.234 ×10 3 . As a first step to all the calculations numbers were expressed in scientific notation. Next, it was important to make an estimate of the answer so a “common sense” check of the answer was € available. Then the calculation was performed. Precision9 of the calculation should be mentioned at this point. Using log tables, especially the fourteen place variety of Briggs, would produce precise results. The slide rule, by it’s nature, is less precise. In general, there are three places of precision in the model I used in school. Compare this to the 10 digits a TI84 displays. Please realize that in application problems, where numbers were based on measurements, it was not uncommon to have underlying values in engineering calculations to have accuracies not radically different from the precision of the slide rule. Thus, this apparent lack of precision was not that much of a limitation. 10

9

Precision refers to the number of decimal places in a number. Accuracy refers to the “closeness” of a quantity to the “real” value. 10 It is interesting to note that in the eleventh (newest) edition of the statics text that I teach from, author R. C. Hibbeler makes the statement; “In engineering we generally round off final answers to three significant figures since the data for geometry, loads, and other measurements are often reported to this accuracy.” page 5

So How Do You Use It? My discussion will be limited to my “big ten inch” (apologies to Bull “Moose” Jackson) linear Post Versalog 1460 (See photo 1). There are several other slide rule varieties that can be explored in Hopp’s book7.

Note the C and D scales

Photo 1: Post Slide Rule

The basic scales on a slide rule are the C and D scales. These scales have one cycle (1 to 10) of logs etched on them. These are used for multiplication and division. To multiply, place either of the rightmost or leftmost 1 (the index) on the C scale over one of the multipliers on the D scale. Move the hairline to the other multiplier on the D scale and find the product below this number on the D scale. Photo 2 shows the multiplication of 3× 2 = 6. Note that Photo 2 also demonstrates division as well. To divide, place the divisor on the C scale over the dividend on the D scale and read the quotient under the index on the D scale ( 6 ÷ 2 = 3). € €

Index

6÷2= 3

€

3× 2 = 6

Photo 2: Multiplication of 3 and 2 Division of 6 and 2 € page 6

The above example covers the basic idea of how the slide rule is used, but what do you do for a more general problem such as 127 ÷ 43? The first step is to rewrite the problem in terms of scientific notation so you can keep track of the decimal point. Remember that you only have one decade of numbers represented on the slide rule, so you need to express these numbers in such a € way as to place them in the same decade -- scientific notation. So the problem would look like 1.27 × 102 . At this point, in an attempt to avoid errors, an estimate of the answer should be 4.3× 101 made --- 1.27 ÷ 4.3 ≈ 0.3 Using the laws of exponents, I know10 2 ÷ 101 = 10, so an estimate for my answer would be 0.3 ×10 = 3. Now, I go to the slide rule and do the calculation. 4.3 on the C

€

scale is placed above 1.27 on the D scale. I place my hairline on the index and read my answer on € € the D scale. The answer is approximately 2.96. The result is shown in Photo 3. €

4.3 on C Scale over 1.27 on D Scale

2.96 on D Scale below Index

Photo 3: Division of 127 by 43 Trigonometric angle values are etched on three scales of the slide (see Photo 2). These scales are the TT, SRT, and the Cos S scales. The Cos S scale provides the sine values for angles from 5.74° to 90° (sine values from 0.1 to 1 -- left to right) and cosine values from 84.26° to 0° (cosine values from 0.1 to 1 -- left to right) using the fact that cosθ = sin(90 − θ ) . Sine and cosine values are read on the C scale below the desired angle value. Tangent values for angles between € 5.71° and 84.29° ( tangent values from 0.1 to 10) are read using the TT scale. Angles from 5.71° € to 45° are read from left to right on the C scale below the desired angle value. Angles from 45°

€

to 84.29° are determined by inverting the value on the C scale or reading the value on the CI € €

€ € €

page 7

€

(inverted C scale on the back of this rule) at the desired angle value. Small angle values for tangent and sine are determined by using the SRT scale and the C scale. The small decimal numbers running from right to left (from 0.6°, 0.7°, etc.) on the SRT scale are the angles and the C scale provides the function values. This scale takes advantage of the fact that the for small values of θ , sinθ ≈ θ ≈ tanθ , when θ is measured in radians. Cosine values of angles from 89.4° to 84.26° are provided by using this scale in conjunction with the C scale from right to left. €

€ Additional scales on the model I have can be used to determine square roots, square and € cube numbers, invert numbers, determine multiples of π , determine mantissas of common logs, and find any power or any root of any number. This model is a fairly “basic” engineering model. Some models included scales for hyperbolic sines and tangents. € All-in-all the slide rule was (is) an effective and useful tool. And like the graphing calculator of the current age, there was a lot of capability that the normal practitioner did not need to use and plenty of short cuts for the aficionado. In my opinion, the fact that the user had to take a more active role in the calculation than an electronic calculator requires is an advantage. The user had to keep track of the decimal place; have an intuitive feel for the numbers that went into the calculation; and an estimate of the answer. Of course, the lack of precision is its major problem. Modeling This section is still related to the general subject of logarithms with the addition of the lost art of graphing data values by hand. In my career, I graphed a lot of experimental data. The main purpose of this effort was to try to determine a predictive mathematical equation that described the phenomena observed in the field. The basic approach of the effort was to present the data in a graphical format on which the data was best described as linear. There are three basic graph formats: graph paper with arithmetic or linear scales; paper with an arithmetic horizontal scale and a logarithmic vertical scale -- semilog; and paper that has logarithmic scales in both horizontal and vertical directions -log-log.

page 8

First let’s review a little basic algebra. If I have a graph of points on arithmetic paper, and they appear to be trending in a linear fashion, I could develop an equation for these data by calculating a slope of the line and then use the slope-intercept form to determine the complete equation (Figure 1). The calculations on Figure 1 use the ordered pairs shown on the graph.

(0,5)

5

slope:

4

3

€ 2

y 2 − y1 0 − 5 5 = =− x 2 − x1 6 − 0 6

y = mx + b equation: 5 y =− x+5 6

1

(6,0)

0 0

1

2

3

4

5

€

6

Figure 1: Linear Equation on Arithmetic Scales

page 9

Now suppose you graph some data on arithmetic paper that is non linear (in Figure 2a)11. If this same data is graphed in semilog space, it approximates a line (Figure 2b). Using the basic concept of the slope-intercept form of a linear equation, the linear equation in this space is represented by the equation ln y = mx + lnb .12 Using the properties of logarithms this can be converted to the more familiar looking exponential model y = be mx . The calculations, using ordered pairs shown are provided in Figure 2. € € (0,30)

100

30

20 10 10

(60,3)

0 0

10

20

30

40

50

60

1 0

a) Arithmetic Graph Figure 2: Modeling Exponential Data

10 20 30 40 50 60

b) Semilog Graph ln y 2 − ln y1 ln 3− ln 30 −1 m= = = ≈ −0.03837 x 2 − x1 60 − 0 60 ln y = −0.03837 + ln b y = 30e−0.03837x

€

11

These examples were taken from: Hammond, R. H., et al; Engineering Graphics, Design-Analysis-Communication Second Edition; The Ronald Press Company; New York, NY; 1971. 12 The apparent slight of hand of using base 10 log paper and a base e equation is strictly legit -- trust me :). page 10

The final type of graph is data on a log-log graph. A graph that appears linear in log-log space can be written, using linear equation concepts, as log y = m log x + logb. Using the concepts of logs, this can be rewritten as y = bx m or a power curve. The graphs and calculations for this type of curve are shown in Figure 3.

€

€ 40

100

(9,33)

30

20

10

10

(0.1,2.5)

0 0

2

4

6

8

10

1 0.1

a) Arithmetic Graph log y 2 − log y1 log 33− log2.5 m= = = 0.5734 log x 2 − log x1 log9 − log 0.1 y 33 b = m = 0.5734 = 9.36 x 9 y = 9.36x 0.5734

€

1

10

b) Log-Log Graph

Figure 3: Modeling a Power Curve

This really impressed me as a student. I liked the fact that the concepts from basic linear algebra, something I had a good understanding of, could be used to derive these more complicated models with such ease. I also believe that the physical activity of graphing individual points on a graph allowed me to gain insight as to the actual behavior of the phenomena, especially the outliers, I was working to understand. Clearly, the graphing calculators of today provide a more precise model in a fraction of the time, but I’m not sure the underlying phenomena is as readily appreciated.

page 11

So What’s My Point? I started this little opus out by saying it was not another lament by an aging luddite. I hope that has been true. The improvements in technology have been a good thing. I didn’t think twice about leaving my slide rule for the HP-45. I am composing this on my laptop computer - not with paper and pencil - unthinkable when I started college in 1971 or even for my graduate project in 1980. I think Henry Petroski captured the motivation to advance technologically in his introduction to his book Success Through Failure; he writes: “Desire, not necessity, is the mother of invention. New things and the ideas for things come from our dissatisfaction with what there is and from the want of a satisfactory thing for doing what we want done. More precisely, the development of new artifacts and new technologies follows from the failure of existing ones to perform as promised or as well as can be hoped for or imagined. Frustration and disappointment associated with the use of a tool or the performance of a system puts a challenge on the table: Improve the thing.”13 This is what drove Napier to develop logarithms, Oughtred to develop the slide rule and HP to develop the “electronic” slide rules of the 1970’s. It continues to drive our development of calculational tools today. It is true that the slide rule is inferior to our calculators of today. Cliff Stoll in his Scientific American article about the slide rule states: “...the lack of precision a given, mathematicians worked to simplify complex problems. Because linear equations were friendlier to slide rules than more complex functions were, scientists struggled to linearize mathematical relations, often sweeping higher-order or less significant terms under the computational carpet. ... Engineers developed shortcuts and rules of thumb. At their best, these measures led to time savings, insight and understanding. On the downside, these approximations could hide mistakes and lead to gross errors.” 6 This statement is very true. On the other hand, we need to be aware that our current tools, especially complex computer 13

Petroski, Henry; Success Through Failure, The Paradox of Design; Princeton University Press, Princeton, NJ; 2006 page 12

models, may in fact isolate the designer, or researcher, from some basic understanding -- some common sense -- an intuitive “feel” for the problem. We need to avoid the dependance on the “black box.” Today’s students need to be reminded that a check of their sophisticated tools using first principles in a “back of the envelope” calculation can generate a great deal of insight and a quick “order of magnitude” check. They should never be satisfied with an answer until they examined their solution in this light. Henry Petroski in his book Remaking the World quotes an engineer named Mario Salvadori: “When my engineers come to me with millions of numbers on a high rise, I know there is one number that tells me a lot of things-how much the top of the building will sway in the wind. If the computer says seven inches and my formula which takes thirty seconds to do on the back of an envelope, says six or eight, I say fine. If my formula says two, I know the computer results are wrong.” 14 Students today need to be encouraged to develop and maintain some of the old fashioned skills from the past to help them become better designers, critical thinkers, and researchers.

14

Petroski, Henry; Remaking the World, Adventures in Engineering; Vintage Books; New York, NY; 1999 page 13