The Slide Rule Watch

Now that’s a word that’s a blast from the past: The Slide Rule.

I was born in 1991 and when I was old enough to learn about numbers, the digital calculator has already replaced the slide rule – an analogue calculator that, as the name suggests, required the physical action of sliding markers around to do calculations such as multiplication, division, exponents and trigonometry. For many millenials like myself, the slide rule is like a cassette tape to Gen Z – most of us have no idea what it is and it really doesn’t matter because we have better technology now.

However, as a mathematician, I am can’t help but be in awe of such a device, no matter how obsolete it may be, for its brilliant and ingenious application of logarithms.

slide rule
My dad’s slide rule

How It Works

In this article, I’ll explain the basics of how it works, but first let’s talk about using two rulers for adding and subtracting:

To do a simple sum like 6 + 4, we place the origin of the yellow ruler on top of the 6 of the metal ruler, and read the answer off the metal ruler where the yellow ruler reads 4.

To do a subtraction of say 11 – 5, we place the yellow 5 above the metal ruler’s 11, and then read back at the origin of the yellow ruler to get an answer of 6.

This is pretty intuitive.

Logarithmic Scale

When it comes to the slide rule, the markers are not equally spaced apart, but rather they are spaced according to a logarithm scale of base 10. Which means the origin of the log scale starts at 1, not 0, since \(\log_{10}(1) = 0\), and these are followed by log 2, log 3, log 4, … log 9.

Now, using log laws notice the following:

\(\log_{10}(10) = 1\)
\(\log_{10}(20) = \log_{10}(10\times 2) = \log_{10}(10) + \log_{10}(2) = 1 + \log_{10}(2)\)
\(\log_{10}(30) = 1 + \log_{10}(3)\)
\(\ldots\)

\(\log_{10}(100) = 2\)
\(\log_{10}(200) = 2+\log_{10}(2)\)
\(\log_{10}(300) = 2+\log_{10}(3)\)
\(\ldots\)

This means every time we get to the next power of 10, the exponent of all the markers after them are increased by one as well. i.e. when you measure a \(2 + \log_{10}(5)\) marker on the ruler, it represents the number 500 and a \(3 + \log_{10}(5)\) marker represents 5000.

For more precision, the slide rule would usually exclude the numbers 1, 2, 3, through to 9 but instead mark out the logarithms of 10, 11, 12, …, 99 then move on to 100, 110, 120, 130, …, 990, 1000, 1100, 1200, so on. The more significant figures marked will mean more precision.

Multiplication and Division

Consider the multiplication \( 1.5 \times 70\).

Now consider the following:

\(\log_{10}(1.5 \times 70) = \log_{10}(1.5) \times \log_{10}(70) = -1 + \log_{10}(15) + \log_{10}(70)\).

Now the calculator will have markings for \(\log_{10}(15) and \log_{10}(70)\), and it also becomes an addition problem. We saw from my example with the yellow ruler and metal ruler that it is easy to calculate the addition and subtraction of two numbers when we have their markings.

Using the same method as that, we can find the answer for \(\log_{10}(15) + \log_{10}(70)\). All we have to do is keep track of the exponent which will tell us what power of 10 to multiply the reading by.

The Slide Rule Watch

During the second world war, increasing demand by pilots for a quick way to do calculations in the cockpit prompted the watchmaker company Breitling to meet this request. Willy Breitling asked mathematician Marcel Robert to design a watch bezel and dial to do logarithmic calculations such as those mentioned above. Thus, the Breitling Chronomat watch, which was later improved upon by the Breitling Navitimer watch, was created.

In the creation of this design, Marcel Robert would have had to apply his knowledge of radians, arc lengths and trigonometry to the logarithmic scale! I can see a lot of potential mathematics problems he would have had to solve to finally arrive at this solution.

A modern Breitling Navitimer watch

Unfortunately, I do not have the funds to purchase such an iconic piece myself, so I make do with the Seiko SSC-009 until one day, perhaps… For calculations, the only complaint that I have with the SSC-009 is the presence of parallax error from the bezel being slightly raised above the dial.

Seiko SSC-009

Calculations on the Slide Rule watch

Let’s have a look at what we can do with this.

Multiplication

\(1.5 \times 70 = 105\)

To find \(1.5 \times 70\), we position the bezel’s marking for 15, above the origin of the inner dial, 10. Remember the arc length between 10 and 15 represents a distance of \(\log_{10}(15)\). We then add a distance of \(\log_{10}(70)\), so from the inner dial we add an arc length between 10 and 70. Reading off the outer bezel, we get the reading 10.5. Remember we need to keep track of exponents, which requires us to have some sense with numbers, and we know the answer is therefore 105.

Division

\(74 \div 5.5 = 13.5\)

Consider this scenario: the exchange rate in Hong Kong is 5.5 HKD for 1 AUD. If I wanted to buy an item for 74 HKD, find the equivalent price in AUD.

For this problem, I position the 74 above the number 55, then read off the answer from above the inner dial’s 10. This works because \(\log_{10}(74 \div 5.5) = \log_{10}(74) – \log_{10}(5.5)\) and it becomes a matter of subtraction. Using the technique outlined in yellow/metal ruler scenario, I read off my answer from the origin, 10.

Hence, 74 HKD is about 13.5 AUD.

Unit Conversion

The great thing about unit conversions is that they’re usually just simple multiplication or divisions by a set constant. For example, converting miles to kilometres is just a multiplication by 1.609. On the logarithmic scale, therefore, the number of miles is always the same distance away from the number of kilometres.

70 km = 43.5 statute miles = 38 nautical miles = 23 000 feet

Positioning the arrow for km above 70, I can then read off the other distance markings for their equivalent conversions!

Final Remarks

My brother is training to become a commercial pilot. Cadets training to become pilots still need to know how to operate a circular slide rule in their examinations but realistically, they just use GPS and automatic instruments. However, when these instruments fail, they will need to default back to these methods without a digital calculator.

Furthermore, when a plane experiences turbulence, one cannot use a digital calculator – it could fall out of your hands and buttons are difficult to push. A slide rule on your wrist? That will always work despite electrical and physical interference!

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