Was Ramanujan wrong, or wrongly reported?

 Most recreational mathematicians know the story of Godfrey Hardy’s taxi. In brief, Hardy called on his sick colleague, Srinivasa Ramanujan. In the course of making conversation, Hardy mentioned the number of his taxi-cab, his favourite form of transport. It had, said Hardy, a rather dull number, 1729. “No, Hardy! No, Hardy!” replied Ramanujan, “It is a very interesting number - it is the smallest number expressible as the sum of two cubes in two different ways.”

Ramanujan was referring here to the fact that 1729 is the sum of one cubed and twelve cubed, and also the sum of nine cubed and ten cubed. The two mathematicians then went on to discuss the fourth powers equivalent, but that has no part here. There is a solution, by the way, with 133 and 134 being the numbers on one side: the rest I leave to you, once you have my methodology, set out below. So Hardy is mainly remembered by mathematicians as the person who played straight man to Ramanujan.

There was more, as we shall see, but first, a small diversion: 1729 is one of a special group of numbers called Carmichael numbers, which are important in number theory. It is highly likely that Hardy was trying to find out if Ramanujan had discovered these numbers in his intuitive way, and got an answer from left field instead. As I am about to reveal, though, this was wrong, and given Ramanujan’s brilliance, it is far more likely that he was misquoted

It has been known for thirty years or so that there is an infinite number of Carmichael numbers, but is there an infinite number of them with factors in arithmetic progression? That description fits 1729 (7 x 13 x 19), but that may be just happenstance. On the other hand, I read recently that 91 is expressible as the sum of two cubes in two different ways: 91 = 3³ + 4³ = (-5)³ + 6³. 

At 0600 this morning, it was dark, I had fetched the newspaper, and was trying to remember the target number, and the cubes that composed it. Then I recalled that it was 91, which my mind had filed as interesting, because it is ¹/₁₉ of 1729, being 7x13.

That did it. I got up, fired up Excel, and set to work. But before I continue, what are Carmichael numbers? Mathematicians will understand when I note that there is insufficient space in the margin of the page to offer it in full…

OK, I won’t be mean to those interested but less familiar with the trivia. Pierre de Fermat (1601–1665) is  remembered today mainly for his “Last Theorem”, which took more than 300 years to prove. In the margin of his copy of Diophantus’ Arithmetica, Fermat wrote:

“To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.”

Now on with the spreadsheet and how I saved myself a lot of what we Australians call hard yakka. One way to solve knotty problems is to try all the possibilities and these are Diophantine solutions, named after the author of the book that Fermat scribbled his note in.

I once wrote in one of my books that Diophantus would have killed to get his hands on a computer and a spreadsheet program, and I meant it. I am still trying to find a way of using a spreadsheet to test Collatz' conjecture.

In cell A2, I entered the value -20, then I selected that column down to row 89, and used FILL – SERIES to integers down to 67. Next, in cell B2, I inserted this formula: =A2*A2*A2. This, of course, returns the value (-8000), being the cube of -20.

Next, I used COPY – DOWN, or CTRL-D, to fill column B with cubes. Then I was ready to laboriously typed in the first row: C2 (=B2+B3); D2 (=B2+B4); E2 (=B2+B5) and so on, all the way to column AQ. Then I could highlight rows 2 to 89 and columns C to AQ and fill those cells with COPY – DOWN, or CTRL-D.

As you can see, I now had more sums-of-two-cubes values than I could poke a stick at, and a few of my “hits” are marked with colour. I highlighted all of the values, copied them and did an unformatted paste into a Word file. This gave me tab delimited rows, so I had to get rid of the tabs. In Word, CTRL-h gives FIND AND REPLACE, and if you are smart-lazy like me, you either know, or need to know two codes to use. A tab marker is ^t, and a carriage return (end of paragraph) is ^p.

So in no time at all, I had 1845 values that could be sorted into numerical order and searched. After getting through less than a page, I muttered something that a passing kookaburra misheard as beggar this for a game of soldiers. The actual words are now lost to the mists of time, so we shall move on.

I highlighted the whole column (CTRL-A) and copied it (CTRL-C). Then back to the spreadsheet, open a new worksheet, click on A1 and paste (CTRL-V). Now I have all of my values in order, but no great desire to eyeball them, as I had had no breakfast, and no mug of tea, either. Time for smart-lazy again. In cell B2, I added this formula: =IF(A1=A2, "hit","").

 

As you can see, there was no need to scrutinise all the values, but look on the right, where there are some trebles. Now ignoring zero, which can be obtained in an infinite number of ways: x³ + (-x)³, where x is any integer, the first treble is well below Ramanujan’s 1729. You can get both 728 and -728 in three ways.

Here they are: 728 = (-10)³ + 12³ = 6³ + 8³ = (-1)³ + 9³

Numerology is a trap, a snare and a delusion, but the difference between 1729 and 728 is 1001 (7x11x13), while 91 is 7x13 and 1729 is, as noted above, 7, 13 and 19. You can see why people get drawn in, even if I don’t mention that only in base-13 notation is it true that 6x9=42!

He proves by algebra that Hamlet's grandson is Shakespeare's grandfather and that he himself is the ghost of his own father.
—James Joyce, Ulysses, 21.

I think I’ll stop there.

Collatz’ conjecture

 Number crunchers know that the word conjecture is always a warning that by the pricking of my sums, something evil this way comes. Conjectures are unsolved problems, and in fact, Paul Erdös, a noted Hungarian mathematician, was reported to have said of Collatz’ conjecture, “Mathematics may not be ready for such problems.” Others called it “dangerous” and “a quagmire”.

When it comes to mathematical challenges, the Four-colour map problem, Fermat’s last theorem and squaring the circle, are far too difficult to even consider on a bus, but the Collatz conjecture is nice and simple to play with. It was put forward by Lothar Collatz, who waited two years after receiving his doctorate, before offering this puzzle. Pro tip: always get your higher degree nailed to the wall before you make waves!

Choose any positive integer n to begin a series. For each following term, if the previous term is even, the next term is one half of the previous term. On the other hand, if the term is odd, multiply it by 3 and add 1. Collatz’ conjecture is that no matter what the value of n, the sequence will always reach 1. Here are five sample strings:

1, 4, 2, 1;

2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;

3, 10, 5, 16, 8, 4, 2, 1;

4, 2, 1;

5, 16, 8, 4, 2, 1.

The sequences generated are sometimes called the hailstone sequence or hailstone numbers, because the values usually go through multiple ascents and descents, like hailstones in a cloud.

If you are working through the numbers on your bus ride, can you see what the next number is that you need to test? From what you can see above, you can rule out 7, 8, 10, 11, 13, 16, 17 and lots more…

The Hungarian-born mathematician Paul Erdös (1913–1996), is considered to hold the world record for the number of papers he wrote in collaboration with other mathematicians. Erdös numbers are whimsical numbers given to mathematicians. Erdös himself has the Erdös number 0, and any person who has collaborated with Erdös on a paper has an Erdös number of 1, while a mathematician who has collaborated with a direct collaborator is given an Erdös number of 2, and so on.

Once in a thousand years

This is from my book for bright young people, Playwiths.

Consider the number of years between events described as “once in a thousand years”, such as floods. To the layperson, this immediately raises the question: how can the authorities access data, covering several thousand years? The answer is that they can’t, but they have what is usually referred to as the Poisson distribution to fall back on, and to understand that, we need to consider an old tale of Prussian cavalrymen who were kicked in the head by their horses.

Just in case you know any French, the Poisson distribution has nothing to do with handing out fishes. It was developed by (and named after) Siméon-Denis Poisson. It describes the probability of clusters in random events, given nothing more than the average occurrence of such events. (If you have no French, their word for fish is poisson, leading to dreadful puns about one man's meat being another man's poisson, but that is irrelevant.

This on the right is not irrelevant, but it is, instead, an elephant, which is a horse of a different colour, as we say in the writing trade. Now let's get back to the horses...

Poisson died in 1840, before the Prussians were kicked. Ladislaus Bortkiewicz published a book in 1898 in which he tried out the distribution of head kicks in each of the 14 corps of Prussian cavalry over a 20-year period, to see if it matched Poisson’s predictions.

Basically, the Poisson distribution works like this: given a sample average (or better, a population average), you can predict the probability of clusters of, say, breast cancer cases in a workplace, the number of calls to a call centre in a given minute, power failures on a grid, some types of traffic accident, the number of typographical errors on a page and the failure of light bulbs. And given some flood data for a few inundations, the Poisson distribution can predict about how often there would be a flood of a certain level.

Let us consider the Prussian data: there were several cases where a significant number of kicks had happened, and many more where no kicks had happened, so Bortkiewicz got hold of the data for 200 corps-years. In 109 cases, there were no injuries, but there were 65 instances of one injury, 22 cases of two, three cases of three head-kicks and one unfortunate corps, in one year, had four instances, a total of 122 cases. That meant the probability of a case in any given corps in any given year was about 6/10, or if you want precision, 0.61.

Bortkiewicz triumphantly showed that the known distribution was an almost perfect fit to the theoretical prediction. After that, people everywhere took up Poisson’s idea enthusiastically.

This story was popular, because most of the world liked the idea of Prussian cavalry being kicked in the head, but the main point was to say that there would be variation, and a high “score” did not necessarily imply carelessness or anything else. Ask anybody who has done some basic statistics, and they will all know about the Prussian head-kicks. It’s the example that is always mentioned.

What is less-mentioned is that you can calculate the flood height that, based on prior data, would happen once in a thousand years. This figure would be approximate, and the estimates would be refined after each flood, and they would be slightly invalidated if the risk is increasing rather than steady, but it’s better than nothing if you need a predictor.

I actually began looking into this issue, revisiting it after several decades, because somebody was questioning the science behind climate change and global warming, and as a throw-away line, poked fun at councils in Australia which have maps showing the limits of one-in-a-thousand-year floods. How, the idiot asked, could anybody know what has happened in the past?

Those who know my historical interests will not be surprised to learn that I point to 1859 as the year when scientists in unrelated disciplines began to be unable to understand one another. The public had started to feel lost around science a few years earlier, but after the 1860s, a great deal of science was either counter-intuitive or it relied on obscure methods. One way and another, science all got progressively more complicated.

Counter-intuitive science is in some ways the worst source of dissent and confusion, but if we know that mathematicians have a clever wrinkle that lets them estimate what a one-in-a-thousand-year flood would be like, we can accept that. The science that flies in the face of uninformed ‘common sense’, and the science that causes fears to arise, these are the sorts of science that cause trouble.

Even if the ancient Greeks knew that the world was a sphere, peasant minds were happy to say that the world they saw was clearly flat. In the same way, other equally simple and fearful peasant-quality minds attack the idea of evolution, misrepresenting what evolution is, even as they deny it. 

Climate is another case: the modern peasants who watch the weather on TV thinks they understand climate, but that, in fact, is a very different kettle of poissons.

A question of class

This is another selection from Mr Darwin's Incredible Shrinking World, my social history of science in the year 1859. Find out more here. We are looking at Britain as it was in 1859.

* * * * *

In London, May, June and July were once the months when Parliament met, and this determined ‘the season’ which ended on 12 August, the first day of grouse shooting. The well off, even those not involved in politics, came to London in the season for races at Ascot, operas, balls, parties, viewings of the Royal Academy and other social events.

Every member of society had obligations, but the obligations of some were less onerous than the demands society made of others. In an age before labour-saving devices though, the rich had a duty to hire labour.

In 1859, Isabella Mary Beeton, known today as “Mrs Beeton”, even listed how many servants a household should have, based on income. She had, that year, begun a series of 48-page monthly supplements to The Englishwoman's Domestic Magazine, her husband’s journal, and in 1861, these were released as a single volume, Mrs Beeton's Book of Household Management. We will visit this in chapter 7.

Army officers, having purchased their commissions, were obliged to lead, and sailors and soldiers were obliged to submit to flogging. Still, the writing was on the wall for the lash as a naval punishment after an incident at Plymouth in July, on board HMS Caesar (then in dock) when a sailor was flogged in front of civilian workers. The outraged watchers protested and argued with some of the officers. Flogging was not abolished in the British army and navy until 1881, but it effectively ended in 1859, thanks largely to a crusading doctor.

When Mary Ann Evans published Adam Bede in 1859, she wrote as George Eliot, and was unprepared for the attention it would bring her. Well known and admired in her own circle of intellectuals, she now found herself publicly identified with George Eliot, the clever ‘male’ novelist. By the time Middlemarch came out in 1871–72, she was well respected, and admired, in both names, for her social conscience.

In chapter 16 of Middlemarch, her fictional characters debate Wakley’s view that coroners need medical training, so as not to be bamboozled or misled by inadequate medical men. Unlike the characters in Eliot’s  book, Thomas Wakley was a real person. As a young doctor he cared about the reform of the medical profession, and political reformer William Cobbett suggested that he establish a medical journal, which he did in 1832, calling it The Lancet.

In 1835, Wakley was elected to Parliament, and his maiden speech attacked the conviction of the Tolpuddle Martyrs, a group of unionists transported to New South Wales in 1834 on trumped-up charges for daring to organise to defend themselves. He was an all-round decent human being of liberal outlook who opposed slavery, the Corn Laws, the 1834 Poor Law and the Newspaper Stamp Act.

Wakley deserves most of the credit for the creation of the Royal College of Surgeons in 1843 and also the General Council of Medical Education and Registration in 1858, but his public fame rests mainly on his work as a coroner. He not only held the beliefs mentioned in Middlemarch, he put them into action.

No observer of coronial inquests today can hear a coroner’s blunt demolition of an evasive witness without recalling what happened when Wakley confronted a workhouse master. The man complained that an exhumed pauper’s body, while it had undoubtedly been scalded to death, had not been properly identified as Thomas Austin, the subject of the inquest. Said the worthy coroner: ‘If this is not the body of the man who was killed in your vat, pray, Sir, how many paupers have you boiled?’

To get his reform campaign moving, Wakley needed to be elected as a coroner. He narrowly lost his first attempt in East Middlesex in 1830, but won in West Middlesex in 1839. From time to time, Wakley reported details of notable coronial hearings in The Lancet, and that brings us to his inquest into the death of Fred White, a young soldier of the Queen’s Own Hussars who died in 1846.

The true cause of death, a flogging of 150 lashes, was covered up, but with a jury’s support, Wakley ordered the body exhumed so the original post mortem could be assessed. The evidence of military cruelty was there for all to see, and an end to flogging in the army came a step closer. The practice finally ceased after a man called Davies was flogged almost to death at Woolwich in September 1859; Wakley’s pursuit of the White case had laid the foundations.

Wakley and Dickens met in 1841, and Dickens once served on a jury under him. The two undoubtedly influenced each other, but Wakley’s essential humanity is seen best in an instance where he may have cut the odd coronial corner.

Thomas Glover was a Civil Surgeon at Scutari during the Crimean War, and either during that time or on his return, became addicted to chloroform and opium, both then readily available to doctors. In April, 1859, he died as a result of an excessive dose of chloroform, and his colleague Thomas Wakley, acting as coroner, brought in a verdict of accidental death. It did no harm. But there was harm enough around.

The birth of the tourist

This is drawn from Mr Darwin's Incredible Shrinking World, a social history of science in the year 1859. Find out more here.

Travelling from London to Rome took 21 weary days in 1843. It took just two-and-a-half days by 1860, and tourism had become a mass commodity. Just as new technology brought fast travel to the masses, so Thomas Cook brought the masses to fast travel. He augmented the technology with group excursions, travellers’ cheques, hotel coupons and round-trip tickets, teaching first a nation, then the world, to obey timetables.

Cook began simply, arranging for a temperance group of 485 people to go from Leicester to Loughborough in 1841 at a shilling a head, with a brass band, speeches and food thrown in. He built up slowly, then left his job in 1845 to run tours full time. By 1848, he was taking parties to Scotland and the Lake District, then running ‘specials’ to see the Great Exhibition in 1851, and private trains to London to see the Duke of Wellington lying in state before his funeral in 1852.

Mr Cook’s train with 28 carriages of paying customers ran across Brunel’s magnificent Saltash Bridge between Devon and Cornwall, when it opened in 1859. Ten years later, he was offering a 105-day tour of Egypt and Palestine, and one of his 1872 brochures is reported to have inspired Jules Verne to write Around the World in Eighty Days. Verne seems to be one of the few writers of his time who was sensitive to how the world was shrinking.

Cook offered a world tour lasting a more relaxed 222 days but Verne’s Phileas Fogg used timetables to find a faster way. In 1859, it might have been possible for a New Zealander to reach London comfortably, via Suez and Marseilles in eighty days, but getting around the world in that time was still a bit of a challenge.

Some travellers had more pressing reasons to travel than ‘merely’ seeing the world, motives like dodging spouses or obligations. The classifieds in London’s The Times of 3 January reveal that people could advertise for missing friends in Australia; a New York enquiry agency offered to provide information about traders or to collect debts. In London, Charles Frederick Field, a former Chief Inspector of the Detective Police of the Metropolis, offered London and Continental Private Inquiries, and access to his New York agent.

Just before Christmas 1859, the New York Times reported that a burglar named Hod Annis had been taken back to Boston. He had been arrested in Philadelphia, and managed to escape the clutches of the law, but he was undone when he telegraphed his mistress to send him money, and the authorities intercepted his requests. His travels had been in vain.

Up until 1859, travel books were written either by intrepid explorers to recoup their costs, or as guides for intending emigrants. With steamships making foreign travel more available, the need for guidebooks grew—Murray’s Guide to Madras and Bombay Presidencies for 1859 was the beginning of a long line of them. The author was Edward Eastwick and the ‘Murray’ who gets more credit than the author was John Murray, the publisher of Charles Darwin’s The Origin of Species. Eastwick advised his readers that on arriving at Madras (Chennai), the trick was ‘to get into a palankeen and be carried to the club, if a bachelor; or if travelling with ladies, to some friend’s house. There are, indeed, hotels which may be repaired to as a dernier ressort’.

Isabella Bird Bishop, who became the first woman member of the Royal Geographic Society in 1892, published her The Englishwoman in America in 1856. She was publishing still in 1900 having started when her clergyman father sent her to America to research American Christianity for him. A relative of William Wilberforce and cousin to John Bird Sumner, Archbishop of Canterbury from 1848 to 1862, she was well connected enough to get an introduction to John Murray, who accepted her first work, and she never looked back.

After the death of her parents, Bishop sailed for Melbourne in 1872, then visited New Zealand and Hawaii before crossing America and reaching New York at the end of 1873. There were more travels, and she circumnavigated the world three times, a record that few humans could have matched in the nineteenth century.

And it was all down to steam transport. Steamships were also used to take an England cricket team to North America in 1859, the first international tour involving any team sport. England won every match.

Sometimes, travel failed to broaden the mind. An American visitor to Berne had returned claiming that everything in Berne smelled of cheese. ‘Cheese is the Bernese otto of roses’, he grizzled, playing on the words of the then common use of ‘attar of roses’, an oil distilled from rose petals, and still used in many perfumes. Every city had its smells, but the odour of horse dung was common to all of them.

Oersted's experiment

 This is from a book I am about to start pitching:

Hans Oersted is remembered in the name of the unit of magnetic field strength, the oersted. He was also the person who coined the term ‘electromagnetic’. With that sort of introduction, it should not be hard to work out that it was Oersted who first observed the magnetic effect of an electric current. All the same, Oersted was trained in metaphysics (a branch of philosophy), rather than in physics.

Nonetheless, in 1806, he became professor of physics and chemistry at Copenhagen. As well as being the first to prepare metallic aluminium, Oersted is remembered for his discovery of electromagnetism, which he made during a lecture. His discovery of the electromagnetic effect was immediately translated into several languages, though not entirely reliably. The 1826 English source I found for his work contained a contradiction which was not in the 1820 French version I happened to have to hand, so my quotation below is a mix of the two versions.

If he had written in Latin, and we had all been forced to learn Latin, this problem would not have arisen, but even by the 1820s, Latin was no longer universally understood.

The first experiments…were set on foot in the classes for electricity, galvanism and magnetism, which were held by me in the winter just past. By these experiments it seemed…that the magnetic needle was moved from its position by the help of the galvanic apparatus…when the galvanic circuit was closed, but not when open, as certain very celebrated physicists in vain attempted several years ago…

A modern re-enactment of Oersted’s experiment. My choice of the aluminium ruler was deliberate.

You can see the way this worked in the illustration above, but as this is simple enough for the reader to try, let me note that the entire apparatus is one compass, one AA cell, a length of insulated wire and some sticky tape, plus an aluminium ruler which is optional, but it would have pleased Oersted. I bared one end of the wire, taped it to one end of the dry cell (this was sloppy practice but good enough) and bared the other end.

I taped the dry cell to the ruler (or to the rule if you are a pedant), taped the compass to the ruler to stabilise it, and that was it. As you can see, a single dry cell was enough to bring about a noticeable swing. Incidentally, if you reverse the wire (and as a result, the current), the swing reverses, and the same reversal happens if the wire is under the compass.

One of the great continuing arguments in science relates to the need to justify research in advance, usually for the benefit of bean counters, weasels and other parasites, by showing what research is useful for. Even the most useless-looking piece of science can become useful, as Karl Pearson was to discover. Here, Lord Kelvin reflects upon Oersted’s researches:

Oersted would never have made his great discovery of the action of galvanic currents on magnets had he stopped in his researches to consider in what manner they could possibly be turned to practical account; and so we would not now be able to boast of the wonders done by the electric telegraphs. Indeed, no great law in Natural Philosophy has ever been discovered for its practical implications, but the instances are innumerable of investigations apparently quite useless in this narrow sense of the word which have led to the most valuable results.
—Lord Kelvin (1824–1907), 1846, quoted R. A. Gregory, Discovery (1916), 241.

Aside from Luigi Galvani and Alessandro Volta, the main players in the unravelling of Faraday’s electromagnetism include Georg Ohm, Hans Oersted, and James Clerk Maxwell, who brought us to the point where we could see light as an electromagnetic wave, much as Michael Faraday had expected, leading on to George FitzGerald, and then to Heinrich Hertz, Guglielmo Marconi and beyond.

There are also the users of electricity and magnetism, from Joseph Henry and Edward Davy, who both invented an electric relay, Charles Wheatstone, Alexander Graham Bell, and people like Joseph Swan, Thomas Edison and Nikola Tesla who made our modern uses of electricity possible.

There was far more to magnetism than compasses for navigation. After Oersted found that a variable current in a wire would make a compass needle deflect from its usual direction, André Marie Ampère (1775–1836), found that like currents attract, then he discovered the solenoid in 1826: this was a coil of insulated wire with a current passing through it, and it would be the basis of transformers, electric motors, relays and electromagnets. The most common and audible household use of the solenoid today is probably in the switching systems which commonly turn the water flow on and off in washing machines.

Ampère completed his work while believing incorrectly in two ‘magnetic fluids’, which he called a northern fluid, and a southern fluid. So long as he observed correctly, and so long as his theory allowed him to make sensible predictions to test, it mattered little. Then in 1831, Michael Faraday discovered electromagnetic induction, and soon after, invented the very first electric motors.

Following on from this in 1845, Faraday discovered what we now call the Faraday effect, where a magnetic field makes the plane of polarised light rotate. This later influenced James Clerk Maxwell to come up with the idea of electromagnetic radiation, which led to Hertz inventing radio. And it all came from one simple observation by Oersted, a lifetime earlier!

Science is like that…

And that, by the way, is my preferred name for the book.

Of timetables and tourism

I have been busy, cleaning up old books, so here's a bit of new stuff I have written for one of them. 

Despite what my publishers often assume, I am not really an historian. I do, however, write about historical matters, because like any good scientist, I am kin to Rikki-Tikki-Tavi, Kipling’s mongoose, who always had to “run and find out”. I want to know why things happened, and what made them happen as they did, and not differently. That is why I can argue that without the steam engine and the telegraph, Einstein may not have started the line of thought that led to special relativity.

It all began with a need to synchronise clocks, but nobody needed to do that until railways came in, and timetables were needed to make sure that up and down trains did not collide on the single tracks that were normal. (The single tracks between towns worked because train drivers would pull into sidings to let other trains pass. That is why they needed synchronised clocks and watches.)

Before long-distance railways, the only people who needed accurate times were those on ships who needed reliable time for navigation. The distance of a ship north or south of the equator, its latitude, was measured by taking sightings on certain stars, or on the sun, preferably at noon.

The harder task was finding your longitude, how far east or west you were. British sailors would set their chronometers by the standard time, as measured by the observatory at Greenwich. Using that as a basis, if the sun reached the highest point in the sky four minutes earlier than at Greenwich, you were one degree east of Greenwich, and if noon came four minutes later, you were one degree west of Greenwich.

Up until the 1850s, anybody with a telescope could set a ‘local noon’, but even in a small country like Britain, this system was no longer safe for the railways, and so Greenwich Mean Time was adopted in Britain.

An international convention in 1881 met to establish what we now know as the international date line as the prime meridian, but the convention had a majority of European members, and so the meridian through Greenwich became the standard for time-keeping, simply because it was closer to home for a majority of members. But if that meridian was the base line for timing, it still took more than 30 years for French and German geographers to accept the use of the Greenwich meridian as the base for all world mapping.

Einstein’s thinking about time arose from the needs of steam train drivers, but until the telegraph was available, synchronisation was hard. Without telegraphy, Einstein may not have pursued the deep problems of time and space.

Probably he would have got there, because trains and telegraphs were helpful precursors, rather than what scholars back then designated as sine qua non, a Latin tag meaning without which, nothing. There were many essential precursors like the invention of calculus, and probably a few enabling technologies.

In cities and towns, steam whistles, bells, flags and lights on towers indicated impending or recent arrivals and departures, summoning those to start walking who planned to meet (or be) passengers, and those seeking to send or collect goods or mail.

Timetables also fed tourism, the great leveller, that all too often pulls every culture down to the lowest common denominator, but which also opens the door to the acquisition of knowledge. Mass tourism corrupts, most of the time, causing hordes to rush from ticking Mona Lisa on their bucket list, to watching a “crocodile show” in Australia, or ogling the bare-breasted apsaras at Angkor Wat.

Angkor Wat apsaras.

Along the way, though, one learns to drink rauchbier, or to put lemon in weissbier, to gain a degree of cultural perspective. Tourism also eliminates weak vessels, the ones who want to play with grizzly bears or lions, so it has its good points.

The idea of the Grand Tour began in the late 1600s, but as late as 1843, travelling from London to Rome took 21 weary days, though in 1860, steam ships and trains got you there in just two and a half days. In those earlier and slower days, when people went on the Grand Tour, they tried to tick as many boxes as possible, and after Pompeii was rediscovered in 1748, it went on the list of marvels to see. Seeing things like that helped educate people more widely.

The Mona Lisa crowd. This was as close as I wanted to go.

Just as the new technology brought fast travel to the masses, so Thomas Cook brought the masses to fast travel. He augmented the technology with group excursions, travellers’ cheques, hotel coupons and round-trip tickets, teaching first a nation, then the world, to obey timetables.

Cook began simply, arranging for a temperance group of 485 people to go from Leicester to Loughborough in 1841 at a shilling a head, with a brass band, speeches and food thrown in. He built up slowly, then left his job in 1845 to run tours full-time. By 1848, he was taking parties to Scotland and the Lake District, then ran ‘specials’ to see the Great Exhibition in 1851, and private trains to London to see the Duke of Wellington lying in state before his funeral in 1852.

Mr Cook’s train with 28 carriages of paying customers ran across Brunel’s magnificent Saltash Bridge when it opened in 1859. Ten years later, he was offering a 105-day tour of Egypt and Palestine, and one of his 1872 brochures is supposed to have inspired Jules Verne to write Around the World in Eighty Days.

Verne seems to be one of the few writers of his time who was sensitive to how the world was shrinking. Cook offered a tour lasting a more relaxed 222 days but Phileas Fogg used timetables to find a faster way. In 1859, a New Zealander could reach London in comfort in 80 days, via Suez and Marseilles, but getting around the world in that time was still a bit of a challenge.

Some had more pressing reasons to travel, like dodging spouses or obligations. The classifieds in The Times on January 3 reveal that people could advertise for missing friends in Australia, while a New York enquiry agency offered to provide information about traders or to collect debts. In London, Charles Frederick Field, a former Chief Inspector of the Detective Police of the Metropolis, offered London and Continental Private Inquiries, and access to his New York agent.

Before about 1859, travel books were written either by intrepid explorers to recoup their costs, or as guides for intending emigrants. Now steamships made foreign travel more available, and Murray’s Guide to Madras and Bombay Presidencies for 1859 was the beginning of a long line of guides. The author was Edward Eastwick and the “Murray” who usually gets more credit was John Murray, the publisher of Charles Darwin’s Origin of Species.

Eastwick advised his readers that on arriving at Madras (now Chennai), the trick was “to get into a palankeen and be carried to the club, if a bachelor; or if travelling with ladies, to some friend’s house. There are, indeed, hotels which may be repaired to as a dernier ressort.”

Isabella Bird Bishop published her The Englishwoman in America in 1856. She became the first woman member of the Royal Geographic Society in 1892 and publishing still in 1900. She started when her clergyman father sent her to America to do research on American Christianity for him. A relative of William Wilberforce and cousin to John Bird Sumner, Archbishop of Canterbury from 1848 to 1862, she was well-connected enough to get an introduction to John Murray, who accepted her first work– and she never looked back.

After the death of her parents, she sailed for Melbourne in 1872, then visited New Zealand and Hawaii before crossing America and reaching New York at the end of 1873. There were more travels, and she circumnavigated the world three times, a record that few humans could have matched in the 19th century — and it was all down to steam transport. Steamships also carried an England cricket team to North America in 1859, the first international tour involving any team sport. England won every match.

Sometimes, travel failed to broaden the mind. An American visitor to Berne had returned claiming that everything in Berne smelled of cheese. “Cheese is the Bernese otto of roses”, he grizzled, using the then common version of ‘attar of roses’, an oil distilled from rose petals and still used in many perfumes. Every city had its smells, but the odour of horse dung was common to all of them.

But that's another story.

Swimming naked and other scandals

 This is taken from my LONG Australian history, You Missed a Bit, which is much cheaper as an e-book.

Today, swimming is among Australia’s most popular pastimes, but in the 19^th century, ‘bathing’ as it was called was severely restricted. When we read about our Victorian ancestors in neck-to-knee bathing costumes, we assume that was the norm, but originally, public bathing was banned between sunrise and sunset, simply because most people did away with “togs” altogether.

Miles Franklin reflects this practice, even at the end of the 19^th century. On a hot day, she and a number of girl visitors “…went for bogeys in a part of the river two miles distant…”

Aunt Helen…was the only one who bothered with a bathing-dress. The rest of us reefed off our clothing, in our hurry sending buttons in all directions, and plunged into the pleasant water. [i]

Confirming evidence comes from official announcements like the one below, dated 4 October 1810. The key term is found in variants and derivatives of the word ‘indecent’, which was the officials’ code word for ‘naked’. This command was signed by the Governor’s secretary, J. T. Campbell:

A very indecent and improper custom having lately prevailed of soldiers sailors and inhabitants of the town bathing themselves at all Hours of the Day at the Government Wharf, and also in the Dock-yard, His Excellency the Governor directs and commands that no person shall Bathe at either of those Places in future, at any Hour of the Day; and the Sentinels, posted at the Government Wharf and in the Dock-yard are to receive strict Orders to apprehend and confine any Person transgressing this Order. [ii]

On the other hand, it was fully accepted that people would go to a secluded place to bathe, as indicated by an 1817 warning in the Sydney Gazette, about dangerous places near Farm Cove and Cockle Bay (Darling Harbour), where “coral” [in reality tubeworms, Galeolaria] and oysters lined the rocks:

Some of the accidents have been very serious, and in one instance threatened the necessity of amputation. Parents will we trust benefit by this caution, and admonish their children against the unnecessary hazard, as there are retired situations much better adapted to the recreation of bathing. [iii]

In other words, the problem was not the act of bathing, or being naked: it was being seen naked that was the issue. Unobserved open-air nakedness was fine, all the best people did it. At the end of 1821, wealthy parents could enrol their sons at the Sydney Academy, Macquarie-street, conducted by Isaac Wood. The school-house was described as standing on a hill, next to the Domain, in a healthy spot with excellent views.

This House has been erected at a considerable expence, affording to the Pupils every accommodation, with commodious apartments, &c. &c. Its contiguity to the Domain (where the Pupils have a liberty of walking, and, in the season, bathing, &c. &c.) renders it one of the most delightful situations possible. [iv]

There, young Gentlemen, of all ages, would be “…received and educated in the several branches of useful and polite literature, according to their destinations in life.” That said, there was only one school-room, though each student had a separate bed, and would learn English, plain and ornamental Writing, Arithmetic, Latin, French, Drawing, Dancing, Music and Fencing.

Boarders paid £40 a year from the age of ten upwards, and below that, £30—and even that amount was a shepherd’s pay for a year, so these pupils were “upper crust”. Clearly, the habit of bathing in a suitable place was normal for all ages and classes.

Indeed, in 1824, somebody writing as “An Independent Whig” wrote to the liberal newspaper, the Australian, protesting at a sign which had appeared, a fortnight earlier. The sign read:

NOTICE.—All Persons found Bathing, at the Military Bathing-house, and which is fenced in, will be prosecuted with the utmost rigour of the law. N. B.—A sentry is now posted to guard the ground and fence and to secure all depredators. [v]

The writer was furious at this high-handed action by the officers of the garrison,

…forbidding any persons bathing at the military bathing-house, not only myself, but I presume the inhabitants generally, would be much obliged by being informed, through the medium of your Independent Paper, if the officers have any legal right to usurp to themselves exclusively, a public place, for their own private use. [vi]

Here, there was no suggestion of indecency. But as the population grew, secluded places became harder to find, and enclosures and bathing machines began to appear. The first was in Hobart in 1829:

The public will be pleased to learn that Mr. Easton has completed a bathing machine, and is well advanced with a second for the accommodation of the inhabitants of Hobart town. [vii]

And it wasn’t only gentlemen who used Easton’s facilities, given this, four weeks later:

We are happy to observe, that the laudable enterprise of Mr. EASTON, in having erected bathing machines, for the accommodation of invalids ordered sea-bathing, and those fond of that recreation, is likely to meet with encouragement—daily remarking several groups of respectable females visiting the same at Sullivan’s Cove. [viii]

In Sydney in 1833, things were apparently getting out of hand, and the Sydney Gazette praised the new orders about bathing:

Years ago we called attention to this. No gentleman could take his wife, sister, daughter or female acquaintance, to any of the public walks adjoining the harbour without keenly feeling the want of some regulation like that which is now in force, and which, we trust, will be strictly enforced. [ix]

Towards the end of the year, The Sydney Gazette and New South Wales Advertiser reported that men had been bathing openly at 1 pm on a Saturday on “the Domain side of the Cove”. [x] That was all very well, said the Sydney Monitor the next month, but where were people to be allowed to bathe? The relevant Act referred to Sydney Cove and Darling Harbour, but was Woolloomooloo Bay included in the list? At that time, swimmers wanting a bathing spot within three or four miles of the town either went through private property, or chose Woolloomooloo Bay.

As a prevention of the indecencies that occurred in Sydney Cove and Darling Harbour, the Police Act is considered an important benefit. But if Woolloomoollo Bay be to be included in the restriction, it becomes a public nuisance, which, however, might easily be rectified by a reservation to a particular spot, and which, when known as a bathing-place, would be avoided by females. [xi]

Perhaps the writer thought women would not wish to bathe, but across the continent, even vice-regal ladies did it. In Perth in 1836, it was announced that the Governor, Sir James Stirling and Lady Stirling and family, would all be visiting Fremantle to go sea bathing.

Captain Scott, the Harbour Master, has a bathing machine in readiness, and others no doubt will soon be prepared, when the number of visitors attracted to this fashionable resort increase. [xii]

The term ‘bathing machine’ was fairly elastic. When a hairdresser named Robinson established a bathing machine at Figtree Sands, Woolloomooloo Bay in 1838, it was soon referred to as “floating baths”. It was “…sixty feet in length, completely roofed in, and provided with dressing-rooms, and other conveniences…”. [xiii]

These were clearly swimming baths, an enclosure off the sea. By 1840, Robinson’s establishment was catering to both men and women, but never at the same time and place.

HOT AND COLD BATHS. Robinson’s Australian Floating Baths, adjoining the Domain and Woolloomoolloo Bay.—J. ROBINSON with great pleasure announces to the Ladies and Gentlemen of Sydney, and its vicinity, that his Baths have undergone a thorough repair and are now ready for use…The great benefits derived from such a delightful recreation as Bathing are too well known to need any comment from the proprietor of the Baths.—Terms Twelve Months £2, Six ditto £1 5s, Three ditto 15s., One ditto 7s., single Bath 1s., warm ditto ditto 2s. 6d. All subscriptions to be paid in advance. [xiv]

As an additional attraction, both baths offered Coffee, Lemonade, Soda Water, Biscuits and Cigars, and Mrs Robinson also had bathing dresses and oil skin caps for sale, though Mr Robinson offered no “bathing dresses” for sale.

In March 1845, the earliest recorded swimming race took place. The competitors were male, the races were around buoys in the harbour, and they started at 6:30 a.m. and 6:45 a.m., suggesting that clothing was not involved.

This noble and manly exercise is much practised by our colonial youth; but the merit of bringing together some of our best swimmers to compete for the honour of a prize, is due to Mr. Robinson, who has for some years catered for the public comfort by establishing his baths in Woolloomooloo Bay. [xv]

Over time, other cities became involved, and in 1854, Adelaide was planning two plunging baths, “…that for gentlemen measuring 88 feet by 40 feet, and that for ladies 48 feet 6 inches by 28 feet; besides numerous warm baths, dressing rooms, towel rooms, waiting-rooms, committee-room, engine-houses, residence for bath keeper, &c., &c.” [xvi]

In 1861, Elizabeth Ramsay-Laye was full of praise for the bathing facilities in Melbourne, something which she called one of the gems of the city.

No places could be better adapted for the purpose. Gentlemen have, at some distance from the shore, a bathing ship, from which they may jump into the water and swim a certain distance, buoys being placed to prevent the uninitiated from getting within reach of sharks. The ladies are equally cared for, and the arrangements made for them are most comfortable. [xvii]

Other Melbourne ladies swam from the beach, and a letter to The Argus signed “DECENCY” sounded a note of alarm which reveals in passing that while women wore “bathing dresses”, those swimming at Brighton, if they had no private dressing-box, got changed in the scrub.

A “well-known broker”, a married man, had been hiding in the bushes to peep at the ladies in a state of undress, creeping up to them on hands and knees. On one occasion, he ran off when he was detected, but he had been recognised.

The same ladies met the same wretch near the same place the following morning, and accused him of it, when he again made use of his legs to the best advantage. And I may add, that this fellow, in the garb of a gentleman, will be well ducked if he persist in it. [xviii]

Throughout the later 19^th century, there were more and more reports of “indecency” charges, but the times were changing. In 1890, Dr E. G. Leger Erson delivered a scientific paper in Melbourne on “School hygiene.”

This was mainly about making children aware of the need for good (diffused, not too bright nor too dim) lighting, sewerage, and ventilation, and it also touched on posture, but Dr Erson also wanted to see children taught to ride horses, shoot and swim.

While the writers at the Bendigo Advertiser thought the horse riding part unnecessary, and said shooting was dangerous, they agreed that all children should learn to swim, and offered helpful suggestions, like adapting the dams scattered around the goldfield:

These could be levelled, and the bottom covered with gravel, and a bit of hoarding would be sufficient to protect from the public gaze, while modesty could still further be prevented from being shocked by the boys wearing swimming pants. By this means a great benefit could be obtained at a minimum cost. [xx]

Note how a concern for ‘decency’ has been replaced by a desire for ‘modesty’. Two years later, in 1892, Port Fairy in Victoria proposed to make mixed bathing legal, and the Portland Guardian agreed with keeping up with the times, noting that “some estimable persons will regard it as a very horrible and immodest course of proceeding…” It would, said the paper be a very desirable one, and no more immodest than a family game at tennis or rounders.

Judging by the pictures in the American papers, bathing dresses may be very handsome and becoming costumes, and as to the modesty of the attire—well, honi soit &c. They are Diana’s robes, compared to some of the displays I have blushed to look upon in the theatres and public ball-rooms. [xxi]

Mixed bathing became more popular in the 1890s, and all Australian newspapers reported the Parliamentary debates in the other colonies. One thing quickly becomes clear: in New South Wales, Newcastle had taken the lead in matters of bathing, and they were keeping themselves informed, as a report from Melbourne in the Newcastle Morning Herald and Miners’ Advocate shows.

In spite of police prudery and the cost of “neck-to-knee” bathing costumes, the residents of Port Melbourne approved of mixed bathing in the open. Their Mayor, Dr Malcolmson, headed a deputation to the Harbour Trust…

The chairman of the trust said they could not interfere with the Police Offences Statute, but as a practical remedy suggested that the residents of Port Melbourne, following the example of other suburbs, should pack the Bench, and fine offenders 1d only instead of the customary 5s. [xxii]

There was debate in the NSW Legislative Council over bathing at Newcastle, and the decision was made to legislate for the whole colony. [xxiii] By August 1894, the battle had been fought, and won, with a Newcastle Borough Council by-law, allowing daylight bathing (but banning mixed bathing).

“Bathing at the Beach: It shall be lawful for all persons, whether male or female, to bathe in the sea at all times and at all hours of the day at those parts or portions of the sea beach within the municipality duly set apart and defined as male and female places. Provided such persons desirous of bathing, being male or female, shall be clothed or covered from the neck and shoulders to the knees with a suitable bathing dress or costume to prevent exposure or indecency; such clothing or covering shall be approved of by the council or other persons employed by the council as caretaker or caretakers. That portion or part of the beach set apart for females or young children shall not be encroached or trespassed upon by any male or males above the age of 7 years, nor shall the part or portion of the beach set apart for males be encroached or trespassed upon by any female or females. Any person infringing the provisions of this bylaw shall be liable on conviction to a penalty not exceeding £1 nor less than 1s. [xxiv]

By 1900, swimming was an Olympic sport, and one Australian swimmer went to the Olympic Games in Paris. While few Australians have heard of him today, F. C. V. (Frederick) Lane won a gold medal for the 200 metres freestyle in 2 minutes 25.25 and a second gold for winning a “200 metres obstacle race”.

This race involved climbing up and down a pole, scrambling over a row of boats and diving under another row of boats, as well as swimming 200 metres. He completed that race in a creditable 2 minutes 38.4 seconds. [xxv] Notably, the race was held in the Seine, where the swimmers had to beat not only the current but the sewage of Paris, which flowed into the river.

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[i] Miles Franklin, My Brilliant Career, ch 17.

[ii] The Sydney Gazette and New South Wales Advertiser, 13 October 1810, 1, http://trove.nla.gov.au/ndp/del/article/628098#pstart6682

[iii] The Sydney Gazette and New South Wales Advertiser, 5 April 1817, 2, http://trove.nla.gov.au/ndp/del/article/2177167

[iv] The Sydney Gazette and New South Wales Advertiser, 29 December 1821, 4, http://trove.nla.gov.au/ndp/del/article/2180705

[v] The Australian (Sydney), 11 November 1821, 4, http://trove.nla.gov.au/ndp/del/article/37072524

[vi] The Australian (Sydney), 25 November 1824, 4, http://trove.nla.gov.au/ndp/del/article/37071701

[vii] The Hobart Town Crier, 17 January 1829, 2, http://trove.nla.gov.au/ndp/del/article/4218444

[viii] Colonial Times (Hobart), 13 February 1829, 3, http://trove.nla.gov.au/ndp/del/article/8644113/666470

[ix] The Sydney Gazette and New South Wales Advertiser, 28 February 1833, 3, http://trove.nla.gov.au/ndp/del/article/2210938

[x] The Sydney Gazette and New South Wales Advertiser, 15 October 1833, 2, http://trove.nla.gov.au/ndp/del/article/2214382

[xi] The Sydney Monitor, 20 November 1833, 2, http://trove.nla.gov.au/ndp/del/article/32145054

[xii] The Perth Gazette and Western Australian Journal, 19 March, 1836, 670, http://trove.nla.gov.au/ndp/del/article/640502

[xiii] The Australian, 24 July, 1838, 2, http://trove.nla.gov.au/ndp/del/article/36857967

[xiv] The Sydney Herald, 13 October 1840, 1, http://trove.nla.gov.au/ndp/del/article/12866022

[xv] Sydney Morning Herald, 3 March 1845, 2, http://trove.nla.gov.au/ndp/del/article/12877795

[xvi] South Australian Register, 18 October 1854, 3, http://trove.nla.gov.au/ndp/del/article/49197848

[xvii] Elizabeth Ramsay-Laye, Social Life and Manners in Australia, 96 – 7.

[xviii] The Argus, 14 March, 1861, 7, http://trove.nla.gov.au/ndp/del/article/5698419

[xix] William Morrison Bell Other Countries, vol 1, 279, and E. E. Morris (ed.) Australia’s First Century, 75.

[xx] Bendigo Advertiser, 14 January 1890, 2, http://trove.nla.gov.au/ndp/del/article/88979480

[xxi] Portland Guardian (Victoria), 11 March 1892, 3, http://trove.nla.gov.au/ndp/del/article/65390320

[xxii] Newcastle Morning Herald and Miners’ Advocate, 19 March 1894, 6, http://trove.nla.gov.au/ndp/del/article/137177795

[xxiii] The Argus, 10 March 1894, 9, http://trove.nla.gov.au/ndp/del/article/8731262

[xxiv] Sydney Morning Herald, 24 August 1894, 4, http://trove.nla.gov.au/ndp/del/article/13965281

[xxv] Sydney Morning Herald, 15 August 1900, 7, http://trove.nla.gov.au/ndp/del/article/14330670