Pittsburgh Years






n my lifetime I have achieved some dexterity with three different pairs of sticks.  These are: drumsticks, chopsticks, and calculator-sticks (otherwise known as slip-sticks, but more properly as a slide rule).  Drumsticks are certainly the oldest in historical time; rhythm, their rationale, is perhaps the most underrated of the function of sticks; coordination and syncopation are surprisingly important elements of human society and drum sticks facilitate this most fundamental aspect of peoples’ learning.  As for chopsticks, anything that will help put food in the mouths of billions of people, and elegantly, without dirtying their hands, has obvious benefit on which I will not elaborate.  But for sheer cleverness it seems to me that it is hard to top the slip-stick, even though now, like the abacus, it is something of a museum piece, it was, in my working days, an indispensable tool. 

A Scotsman with the title of Baron John Napier seems, in the year 1614, to have realized that for every number, there is some exponent of 10 that, when exponentiated, will equal .  For the number 1000, that exponent would be 3, that is 10 to the third power equals 1000.  A natural reaction to this discovery is, I suppose: So what?  And one’s second reaction might be to ask oneself, Do Barons really have enough practical work to do?  Yet there is more here than meets the eye.  Quite a lot of science and mathematics have been built on this simple, non-obvious, notion.

Suppose one wishes to multiply 3 times 6.  This is trivially simple since most of us learned our multiplication tables up to at least 10 times 10 in grammar school, so long ago that by now we have quite forgotten just how it was that we learned it, but some small part of our memories remain devoted to these handy little functions.  I learned the algorithm for long division and remember it still, but as to that for square roots, a similar process, it has quite disappeared from my cerebrum since I never used it much, or at all, except to pass a quite useless test.  It seems to me quite remarkable how little utility this skill has in everyday life.  What can the teachers of the day have been thinking when they drilled into our heads this useless process?

In the Dominican Republic, where I have spent many winters, numeracy is limited; waitresses routinely use an electronic calculator to total the most trivial of quentas (checks, bills).  There are two reasons for this: the first is that many do not learn arithmetic very well in the first place, and the second is that the calculator can then simply be turned toward the customer, whose language may be completely unknowable, and the total will be displayed to him unequivocally in what seems to be the only truly universal language of our species: Arabic numbers which, curiously, were developed by the Hindus and merely spread around by the Arabs.  This tells us something of the utility of communication in comparison to invention.

But now imagine that one wishes to multiply 10,200 times 3.76 and then divide that product by 1.426.  Try doing that in your head.  Now we’re getting to the unintuitive utility of slip-sticks.  “Why,” you say, “just get out your calculator.” Yes, but unfortunately, while our species has wished to do such things for many thousands of years, it has only been in the last 40 or 50 that there has been such a thing as an electronic calculator.  Meanwhile, long before that time, the Empire State Building was built, along with the Brooklyn Bridge, the Hoover Dam, and innumerable other grand undertakings.  Generally speaking, it was done using the peculiar insight of Baron Napier.  And that is this:

If one wishes to multiply 1000 times 10,000—a trivial undertaking to be sure, but we’ll start simply—one can think of the problem in this way: the exponent of 10, which will yield 1000 when exponentiated, is 2, and that which will yield 10,000 is 3.  Now, oddly enough, if we add these two exponents we get 5.  If we now exponentiate ten to the 5th power, the sum of 2 and 3, we, as a result, get the number 1,000,000 which, as it turns out, is precisely the number that we are looking for when we wish to multiply 1000 times 10,000.  This is so simple that one ought to think about for a moment:  +  =  = 1,000,000.  Adding the exponents and then exponentiating to the same base (in this case 10), permits one to multiply.  But what about numbers that aren’t so trivially simple?  For example, what about the number 319 times the number 23?  Now, not so simple.  Can we do this?  As it turns out, we can.  But there is a problem: we need to know just what power of 10 yields 319, and what power of 10 yields 23.  This is not so easy, but it can be done.  We’ll start with 319:

Obviously  is too small, only 100, and  = 1000 is too big, so the answer lies somewhere in between.  Let’s split the difference and try .  Now the question becomes what, exactly, is 10 to the power of 2-1/2?  And here is just where we need square roots, the ones we forgot from grammar school.  Why do we need them?  Because , which is the equivalent of .  There’s that square root.  At this point, since I don’t remember how to do it by hand, and I haven’t the slightest interest in learning it, I’m going to cheat, much as I would certainly have done in grammar school had it been possible; I’m going to use a calculator.  This gives us 316, and a little change.  So we’re close; in fact, were close enough for most engineering purposes, for which approximation has been raised to an art form.

What has happened here is that we’ve gotten lucky.  It’s not always that easy.  But there are people, and I know one of them well who just seem to like numbers, and working them out.  It amuses them.  And a good thing for us too.  As it happens, when one gets into this in any depth at all, one realizes that these numbers come in patterns and there are people who love to work out these patterns, patterns of numbers which we now call logarithms.

Book of logarithms.jpg

In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals


The word itself is from the Greek, a combination constructed from logos + arithmos, which is to say proportion, or ratio, plus number, and the name was concocted by none other than the Baron himself.    

Once these notions had been worked out a number of people sat down and for amusement, and perhaps profit, worked out whole books of numbers—logarithms.  A monumental task certainly.  One of the first of these is pictured on the left.  To give an idea of the utility of this effort I should add that nearly 400 years later, after Henry did his thing I myself used tables of logarithms for trigonometry functions, another of their uses.

The subject of logarithms is in fact much more complex than the simple formulation that I have given it here, and if you’re interested you can follow the link.  I will say that it has been used in virtually every science from microscopy to astronomy, and just about everything in between.

But ordinarily, instead of parsing through tables of logarithms one can instead use the slide rule which is not much more than a table of logarithms printed on two sticks—to bring it down to its most simple explanation.  There are limitations as to the accuracy to which numbers can be read in this fashion simply because of how finely and accurately numbers can be positioned over each other, and of course the accuracy with which the slide rule is printed (actually embossed) in the first place.  And there are other limitations, but these, as it turns out, are quite easy to overcome.

The slip-stick


The C and the D scales (ignore the others) are used for multiplication.  Here, the number 1 on the C scale is positioned over the 2 on the D scale.  Then one can multiply 2 quite easily times any number on the C scale.  For example, notice that 2 on the C scale is directly above the number 4 on the D scale, and the 3, is directly over the number 6.  Intermediate values can also of course be read.



An answer reached using a slide rule is not precise: multiplying two times three on a slide rule gives exactly the same answer as multiplying 200 times 3 million; all you can tell from the slide rule is that it is 6…  something; it is up to the user to determine the “scale” of the answer, and this is done simply by estimation.

First it is necessary to develop a certain facility in exponentiation to help resolve the scale of the answer.  For a simple example: 200 = 2  102, while 3 million = 3 106, the exponents of ten in effect representing the number of zeros dropped implicitly.  So the answer here would be calculated as 2  3  108, or six followed by eight zeros, 600,000,000.  The two exponents of ten in the multiplicands, 2 and 3, are simply added together since adding exponents is equivalent to multiplication. 

When performing division one simply subtracts the exponents instead of adding them.  For example: 3 million divided by 200 can be expressed as:    = 15,000.

Fortunately, for the buildings and bridges in the world, the approximation inherent in a slide rule—since it can only be read to a close and not a certain accuracy—is insignificant when compared to the rough calculations already made when determining the loads and stresses, those real things that the individual numbers represent in the first place.  All engineering is approximation, the real world being what it is.  How does one estimate how many heavy trucks will be on a bridge at the same time, and how many cars? How many people will wander a department store together?  How many tables and shelves will be positioned on the floor of the department store?  And these are the easiest of loads to estimate.  How hard will the wind blow?  How much snow will fall on the roof?

But each engineer does not try to estimate these unknowns.  What has happened is that consortiums of engineers have gotten together, talked the issues through, and in some cases made tests, and then they produced a Manual of Standards to be used for these loads.  Some are based on the use to which the structure is to be put, others have to do with where, in the world, it is to be located; wind forces in Pittsburgh are not the same as those in the Caribbean; earthquakes in Tokyo are not ordinarily the same as those in Alabama.  And there are many other factors to be taken into account as well.  But the point is that approximations of these sorts far outweigh the minute inaccuracies that result from reading the slide rule.

Should you now be worried about crossing a bridge in your light little car, aside of a huge 16 wheeler?  The plain answer is no.  And the reason for this is that, in engineering, reasonable estimations are used routinely and then quite large safety factors are applied either to the loads themselves, or more likely to the ability of the structure to withstand them—or little of each.  For example, structural steel is well understood to yield at a certain rather closely known stress, so the permitted stress is cut just about in half thus adding “safety”.  The rarity with which large engineered structures fail, is testament to this judiciousness.  And I can assure you that when something substantial does collapse the reasons for its demise are pored over by rafts of engineers—not to mention lawyers—in an effort to see just what it was that caused the catastrophe.  And the standard safety factors, in use all over the world, are simply the summation of the knowledge gleaned from all of these investigations over centuries.

There is a sense in which the electronic calculator of today, or a computer, gives one a false sense of accuracy.  By providing an answer to perhaps eight digits after the decimal point one tends to assume an accuracy that simply doesn’t exist.  Since the numbers entered in the calculator in the first place, representing the forces to be resisted, were inescapably rather rough estimates, one should not be deceived as to the accuracy of the answer provided by this sort of equipment.