Air bubbles with dripping taps

If you are tired of London

Jonathan's coffee house plaque
Jonathan’s Coffee House was the site of the first London Stock Exchange. A link between coffee and the financial markets can be made via the maths describing Brownian Motion.

London has a long history for tea and coffee drinkers. The first London coffee house opened in around 1652 in St Michael’s Alley*, followed quickly by a large number of coffee houses all over London. Isaac Newton, Edmond Halley and Abraham de Moivre were all known to drink coffee in the various establishments dotted around the city (de Moivre even charged a small fee for mathematical advice at Old Slaughters on St Martin’s Lane). Famous institutions such as the London Stock Exchange and even the RSPCA were first formed at London Coffee Houses. On the engineering front, Thomas Telford established the Institute of Civil Engineers at the Salopian Coffee House just off Trafalgar Square. Yet the story in this post comes not through the coffee houses but from historic tea drinkers and, in particular via a “prodigious drinker of tea”, Dr Samuel Johnson**.

The story is detailed in Boswell’s “Life of Samuel Johnson” and concerns a friend of Johnson: Mrs. Anna Williams (1706-83). Williams had gone blind in the 1740s and, in the later part of her life lived with Johnson in his various dwellings around Fleet Street. You can now visit the Samuel Johnson museum in one of those properties at Gough Square, just off Fleet St. Between 1759 and 1765 Williams was living a short distance from Johnson but Johnson regularly popped around for tea in the evening. It was on one such occasion that Boswell joined them. Williams would pour the tea for all the guests but Boswell wondered how she knew when to stop pouring given that she couldn’t see. Boswell thought it was possible that she had subtly dipped her finger into the tea to feel when it was about to flow over the edge of the cup. As he later found out, this was not true. Apparently she had “acquired such a niceness of touch as to know by the feeling on the outside of the cup how near it was to being full.”

The question then arises, how could she have known this just by touch?

a heat sensitive coffee mug
This coffee mug changes colour as it gets warm. The constellations slowly appear as the heat is transferred from the hot coffee inside to the outside of the mug.

A first solution would be that she was feeling the heat of the tea through the porcelain of the cup. Is this possible? It is certainly true that, once a cup is full, it feels warmer to the touch than an empty cup, but heat does not travel through a substance, such as a cup, instantaneously. Would Williams have had enough time between feeling that the cup had become warm to stopping pouring to prevent the cup from overflowing? A quick experiment with my colour changing cup (pictured) suggests that this may be unlikely. Even if she had the sensitivity in her fingertips to quickly discern a change in temperature, it took 20 seconds between boiling water being introduced to the cup before the top started to change colour. Although porcelain is thinner than my mug, and the change is probably quicker to feel than to see with the cup, it is likely that she would have carried on pouring the tea beyond the rim of the cup before realising that the cup was becoming full. Although it is possible that she could compensate for this, for example by touching the cup further towards the base, perhaps there are other ways that you can discern a cup is almost full without seeing it?

A second solution could be that her discernment occurred through a mix of touch and sound. There are a few ways in which audio signals will suggest that a cup is almost full. Coupled with the feeling of increasing warmth on the side of the cup, this may give the tea pourer increased confidence that the cup is comfortably full. One such clue is familiar to those of us who fill water bottles up from the tap, the change in pitch as the tea is poured. Similar to a Helmholtz resonator, a pourer would know when to stop as the musical note from the pouring tea changed. But the tea cup is open at the top so this effect is weak. Perhaps the sound clue instead comes from the sound that the poured tea makes as it impacts the surface of the cup of tea. When we hear a dripping tap, what we are actually hearing is the bursting of an air bubble just below the water surface. We would expect the sound of that to depend, for shallow containers such as the cup, on the depth of the tea. Consequently, if we poured the tea slowly and heard the drops of tea as they entered the cup, we may expect to gain an idea as to how full the cup is. As Boswell assumed a slight of hand (and of hygiene) on the part of Mrs Williams, it is probable that she did not pour the tea slowly enough for this to have been her primary route for knowing that the tea cup was nearly full. Perhaps Mrs Williams did not use sound after all, but relied on something else?

Bubble tea? There is a lot of physics to be found in tea as well as coffee. Take time out with your brew. What do you see, hear, feel?

There is one more clue that you could get about the relative fullness of a cup of tea using touch. If you very gently push against the cup with your finger, you can feel the resistance of the cup to movement. As the cup became heavier with tea, the resistance to its being pushed would increase (Newton’s second law). After pushing at the top of a few mugs with my finger, this seems to be possible. Unless the person watching the tea being poured was very observant, it is not clear that they would notice this. Can we gather anything of Boswell’s skills in observation? Throughout his book he describes people and social situations well and yet, he was unsure whether Williams had dipped her finger into the tea cup to know when it was full enough. I think we can probably gather that on this particular point, Boswell may well not have noticed had Williams been pushing gently at the tops of the cup to see how easily they moved. Although there is a risk of pushing the cup over, this does seem to be a very feasible way of filling a cup without using your sight.

There is however one last, maybe more boring possibility. Generally the speed at which liquid comes out of a tea pot is quite reproducible. Through the fact that she was pouring tea every evening for Johnson and his guests, it is quite possible that she knew how long to pour for each fixed angle of pour before the tea cup was properly filled. We’d still have to ask how she measured the time given that she wouldn’t have looked at her watch, but perhaps here we would have a clue from Galileo. He is said to have sung songs with a known rhythm in order to measure time (the use of the pendulum to measure time came later, partly as the result of Galileo’s work). The idea was that the rhythm is a surprisingly reproducible method for comparing time intervals. Maybe Mrs Williams sang to herself while pouring the tea for the guests.

What do you think? Perhaps you can think of another effect that could be used to determine when your tea cup is full, without any visual clues. Or maybe you just disagree with my deductions. Whatever your thoughts, do let me know in the comments below or on the various social media sites (FB, Twitter, Mastodon), I look forward to learning more and maybe, being able to pour my coffee with my eyes closed.

*Information on the London Coffee Houses can be found in the excellent “London Coffee Houses” by Bryant Lillywhite published in 1963

**The story of Mrs Williams, Dr Johnson, his tea drinking and the notes of Boswell is described in “Life of Samuel Johnson” by James Boswell