Postcards from South America

Herewith some random scenes...

Jorge Luis Borges and I had a meeting today at Cafe Tortoni in Buenos Aires. He waxed on, but I was sincere, in the etymological sense. Work it out.

Frutilla is a little-known monster which emerges at night from the Tigre Delta to hunt down vegans. As the vegans have learned to fight back, dealing it a deadly blow with a steak through the heart, Frutilla now has its infective viral particles smuggled onto breakfast tables in jars that we must not open. At least, that is what an old Indio lady told me for a small fee.
















A cayman in the river above the Iguazu falls.

A nearby catfish.

Butterflies dito. I don't think the cayman or caiman got fat on THEM.

A coatimundi, a raccoon relative, same area. People are warned not to feed thrm.






Me near the falls.

Chris ditto. It looks as though she is in front of a backdrop because I used flash-fill.

The falls themselves...
















The Parque das Aves. We could not find the Parque das Avenots

More sunsets and stuff later...

Triangular numbers

Background

When I can't sleep, I do calculations in my head: always have done, always will.  I might try to get the cube root of 17, or the successive powers of three, or other number patterns, like sets of three primes in arithmetic progression.

This story has its beginnings with a boring plane trip when I was playing with the triangular numbers, 1, 3, 6, 10, 15, 21 . . . which are generated by 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, 1+2+3+4+5+6...

These are called triangular numbers because you can make them up into neat triangles like this:
                        *
              *        **                                                  
      *      **      ***
*    **    ***    ****
1    3     6         10  and so on                                                                                                                     
There are several really cute things about these numbers that were known, way back in the days of Diophantus, an Ancient Greek who liked playing with really big numbers. Every perfect square is the sum of two consecutive triangular numbers, as you can see if you plot the square numbers as asterisks or dots like the diagram above. With a bit of effort, you can use a similar method to prove that every odd perfect square is of the form 8T + 1, where T is a triangular number.

The original problem

It occurred to me that the 8th term in the series of the numbers in the series was 36, which is a perfect square, and then I wondered if there are any other numbers in the series which are perfect squares, and if so, how you could generate them. It was late at night, I mistakenly calculated the value of T(15) as 121 (it is actually 120), and so it looked worth trying, but I went to sleep before I found any more.

Next day, I realised my error, and since I suspected that there would not be many numbers fitting the pattern, I created a spreadsheet that would generate them, knowing that term number T in the series is given by

T(n) = n * (n + 1) / 2, which means that term 11, for example, is 11 * 12/2 = 66

The Spreadsheet

Diophantus would have killed for even a simple spreadsheet program. Here is my spreadsheet solution to test for numbers that fit:

A1 is the starting number of the terms to be explored (in the first run, that will be 1).

A2 =A1 + 1

B1 =(A1*A1+A1)/2

C1 =IF(SQRT(B1)=INT(SQRT(B1)),1,"")

B2 and C2 are filled down from B1 and C1, which has similar formulas. Column A is copied down from A2.

I then fill down columns A, B and C to row 50000, and insert the value 1 in cell A1 to test the first fifty thousand terms. To get the next 50,000 terms, I type 50001 in cell A1, and so one.

A "hit" is indicated by the value 1 appearing in column C in the same line as the hit.

I found it a nuisance searching for the hits, so I created 20 cells in column D to help me find them:

D1 =sum(c1:c2500)

D2 =sum(c2501:c5000)

and so on to D20 -- this told me roughly where the hits were.

Then to eyeball for ANY hits in a block (they get fewer and fewer) I created E1 =sum(d1:d20), then all I had to do was insert my starting values 50001, 100001, 150001 etc in A1 and trawl through the terms, zeroing in on a specific 2500 rows to search for the hit when one was indicated in cell E1 and column D. Once I learned to calculate the approximate value, I could jump almost to the correct row, but that calculation is jumping ahead of the story.

The Results

What I had now was a curious pattern, where the terms that were perfect squares went like this:

The Mysterious Constant

About this time, I started to see some patterns. I found that these term numbers appear to be in a geometric progression with the ratios of two successive term numbers converging. Leaving out the first couple of terms of the converging series, I found these ratios between the key numbers:

5.87755102040816, 5.83680555555556, 5.82986317668055, 5.82867346938776, 5.82846938954150, 5.82843437620146, 5.82842836889813, 5.82842733820888, 5.82842716137060, 5.82842713102994, 5.82842712582431 and 5.82842712536459

So I wrote to an email list with  a few number-players in it, drawing attention to the curious patterns that were emerging, and asking for help. I ended by noting that the ratios of the square roots of the successive terms that are perfect squares rapidly converge and stabilise on the value 5.828427124746190, which I thought might give a hint about the value being converged upon above!

It was not long after that Ben Morphett told me that my mystery number was explicable:

"Your number 5.828427124746190 is equal to (3 + 2.sqrt(2)), and is the solution to the quadratic x^2 - 6x + 1 = 0"

I still don't know why, but I suspect that this relationship is just a coincidence, because somebody else managed to explain a lot of it for me in a different way — but that may just prove that I haven't looked hard enough.

But before we get to that, though, I commented to the list that the square roots of the perfect squares I had generated appeared relatively uninteresting, but Bruce Harris saw otherwise. My factorisation missed the point, he suggested:

" . . . if instead you factorise them into just two factors, allowing the factors themselves to be composite numbers, an obvious pattern emerges:

1 = 1 * 1

6 = 2 * 3

35 = 5 * 7

204 = 12 * 17

1189 = 29 * 41

6930 = 70 * 99

In each line, the first of the two factors is the sum of the two factors in the previous line, and the second is this number plus the first of those two factors again. Thus in the fourth line, 29 = 17 + 12, 41 = 29 + 12; in the fifth line, 70 = 41 + 29, 99 = 70 + 29."

He is right in this. Here is a table which extends this — the rest you can do yourself.

So what is the story with that strange factor? Tony Morton explained it to us in full, but I will reveal less than all of his solution, to leave you room to do some discovering yourself. Before I go to that, a few points:

Some hints

A famous mathematician called Gauss once was given, as a young boy, the task of summing the numbers from 1 to 100. He realised that if he added the first and the last terms in the sum, he got a total of 101, and the same if he added the second and second-last, or the third and the third-last: in all, there would be 100/2 instances of 101, making the final result he needed 5050. In fact he had multiplied (n+1) by n/2. Remember that.

    288 may be 2*12*12, but it is also 2*9*16, while 9800 can be 2*70*70 OR 2*49*100. Remember that.

    Every triangular number can be expressed either in the form n/2 * (n+1) OR n * (n+1)/2. Remember that.

    Geometrical proofs are possible for many problems with triangular numbers. Remember that.

You are now equipped to construct a proof which explains why the patterns arise, but you may not be able to explain that factor of 5.828427124746190 from this. Then again, maybe I have missed something :-)

The Morton Solution

So here is part of Tony Morton's solution, enough to get you thinking, and which does explain that factor. I have changed the notation to make it consistent with what has already appeared here.

If we want to know the nth triangular number - the sum of the numbers 1 through n — then we can use the formula T(n) = n (n + 1) / 2.

But suppose we are given a triangular number and want to know which one it is, to calculate n given T. The above formula can be written as n^2 + n - 2T = 0 which we can solve to get

n = (sqrt(8T + 1) - 1) / 2.

So for example, given T = 36, I can apply this formula to deduce that it is the 8th triangular number. So far so good. But if we look at this formula more closely we see it implies that if T is any triangular number, then 8T + 1 is always a perfect square. More explicitly, we can rearrange to get

8T + 1 = (2n + 1)^2.

Now, we are interested specifically in those triangular numbers T which are also perfect squares themselves. If we set T = m^2 for some (undetermined) integer m, then

8m^2 + 1 = (2n + 1)^2.

There is accordingly a one-to-one correspondence between the triangular numbers that are squares and pairs of positive integers (n,m) that satisfy the above equation. That is, given any square triangular number we can find n and m, and given n and m solving the equation we can find the corresponding square triangular number. So the problem of finding square triangular numbers is equivalent to that of solving the above equation.

Now, if n is a positive integer then so is 2n + 1, and if m is a positive integer then so is 2m. So let x = 2n + 1 and y = 2m to obtain

2y^2 + 1 = x^2

or x^2 - 2y^2 = 1.

Any solution in positive integers (x,y) to this equation gives us a solution in positive integers (n,m) to the earlier equation provided x is odd and y is even. It turns out, however, that every solution (x, y) satisfies this property. To see this, note that x^2 has to be odd because it is the sum of an even number (2y^2) and an odd number (1). Since x^2 is odd, x must be odd. But the square of any odd number leaves a remainder 1 when divided by 4, because (2k + 1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1. In mathematical jargon, x^2 is 'congruent to 1 mod 4'. If y were also odd, then y^2 would be congruent to 1 mod 4, and x^2 = 2y^2 + 1 would be congruent to 3 mod 4; a contradiction. Thus x is odd and y is even.

The equation x^2 - 2y^2 = 1 is known as Pell's equation. We thus see that every solution to Pell's equation gives a square triangular number, and conversely every square triangular number is obtained from a solution to Pell's equation. Given (x,y) we can calculate the triangular number T and its index n as

T = (y / 2)^2, n = (x - 1) / 2.

How to solve Pell's equation? Well, it's fairly easy to find the smallest solution: x = 3, y = 2. This corresponds to the trivial case n = 1, T = 1, which is obviously a square. So we'd like to find some larger solutions. It turns out that we can generate larger solutions from the smallest solution by the following trick. Write

1 = x^2 - 2y^2 = (x + y sqrt(2)) (x - y sqrt(2))

and raise this to an arbitrary power k to obtain

1^k = 1 = (x + y sqrt(2))^k (x - y sqrt(2))^k.

Now, given integers x and y, (x + y sqrt(2))^k can be written in the form A + B sqrt(2), where A and B are integers, and similarly (x - y sqrt(2))^k = A - B sqrt(2) by symmetry. We therefore have

1 = (A + B sqrt(2)) (A - B sqrt(2)) = A^2 - 2B^2.

So the numbers A and B, obtained from the kth power of x + y sqrt(2), provide a new solution to Pell's equation. Setting k = 2, 3, 4, 5 and so on through all the positive integers thus generates an infinite number of solutions to Pell's equation, hence an infinite number of square triangular numbers.

Thus, using the smallest solution x = 3, y = 2 we get (using Q to represent sqrt(2))

(3 + 2Q)^2 = 17 + 12Q x = 17, y = 12, n = 8, T = 36

(3 + 2Q)^3 = 99 + 70Q x = 99, y = 70, n = 49, T = 1225

(3 + 2Q)^4 = 577 + 408Q x = 577, y = 408, n = 288, T = 41616

and so on. It turns out that the powers of 3 + 2 sqrt(2) generate _all_ the solutions to Pell's equation in this way. Because, if we have any solution (A,B), we can always find C and D such that

A + B sqrt(2) = (3 + 2 sqrt(2)) (C + D sqrt(2))

and C, D are both positive, with C smaller than A. (Proof left to the interested reader.) If C is larger than 3 we can repeat this process with (C,D), and continue until we arrive at the smallest solution (3,2) (which we must do because C is getting smaller at every step). We thus obtain A + B sqrt(2) as a product of factors 3 + 2 sqrt(2).

***********************

We will leave Tony's proof at that point, but you are well on the way to the full explanation.

Possible enquiries

What is left for the honest enquirer, the interested reader, aside from testing the effects described, or seeking a new proof? Quite a lot, actually!

For a start, you have a method that may be new: using a spreadsheet to identify the interesting cases, and you have a couple of incomplete solutions.

Wondering what else might be explored led me to look at what David Wells had to say on pentagonal numbers in his "Penguin Dictionary of Curious and Interesting Numbers".

As I suspected, my discovery was not new to mathematics: on page 93, Wells notes that "Some numbers are simultaneously square and triangular . . . they are found by using a fact already mentioned . . . the Pell equation: 8x^2 + 1 = y^2" He goes on to give a general formula:

1/32 ((17 + 12*SQRT(2))^n + (17 - 12*SQRT(2))^n - 2)

and he adds that if T(n) is a perfect square, so is T(4n(n+1)) and that there are 40 palindromic numbers below 10^7: 1, 3, 6, 55, 66, 171, 595, 666 and 3003 among them. T(2662) = 3544453, and T(1111) and T(111*111) are also palindromes.

For every triangular number T(n), there is an infinite number of other triangular numbers T(m), such that T(n)*T(m) is a perfect square.

Also (T(n+1))^2 - (T(n))^2 = (n+1)^3, so the sum of the first n cubes is the square of the nth triangular number.

And (T(n))^2 = T(n) + T(n+1)*T(– 1)

And 2 * T(n) * T(– 1) = T(n^2-1)

The sum of the reciprocals of the triangular numbers converges on 2

The triangular numbers 16 and 21 have triangular numbers as both their difference and their sum, and there are others.

The triangular number 6 is said to be the only example under 660 digits whose square is also a triangular number (that sounds like a challenge to find the next one!)

And by the way, 1189 = (204 * 6) - 35

Pentagonal numbers

Pentagonal numbers are generated like this:

The numbers in the series are 1, 5, 12, 22, 35, 51, 70 . . . and there is a general formula which generates the whole set. What is it?

(Interestingly, the first factor in the general formula for triangular, square, pentagonal, hexagonal, heptagonal, octagonal numbers and so on is always 1/2n, and the second factor shows a fascinating pattern, which I leave you to discover when you work out the other formulae, but here are the first few numbers in a few of the polygonal series.)

Polygonal numbers
Polygonal numbers are generated by drawing similar patterns but with more sides, as seen in the figure above. If you draw them correctly, this is what you will get:

    Aside from the trivial cases of 0 and 1, there is another number under 500 which is both triangular and pentagonal, and there is another one under 50,000: are there any others? Can a computer be set the task of looking for these?

    There are three numbers under 2 billion which are at once triangular, pentagonal and hexagonal.

    There is at least one number under 1000 which is both square and heptagonal: are there any others?

    The 49th triangular, square (obviously!) and heptagonal numbers are all perfect squares: is this a record? (Excluding the trivial answers 0 and 1, of course, I have found nothing greater than three so far, but I have found a case where the triangular, square and octagonal numbers are all perfect squares. Maybe I should have tried nonagonal numbers . . .)

    There is an obvious link between triangular numbers and hexagonal numbers. Do any other sets of polygonal numbers have a relationship like this?

    The triangular numbers appear in Pascal's triangle. Do any of the other sets appear there? Why or why not?

    Think up some others yourself by deducing the formula for a set.

Postcard from Buenos Aires

We are ensconced in the grandeur of a truly grand hotel, and outside, the streets are full of honking cars, and the odd bang. It must be either politics or the religion of the area, which is futbol.  I think the red-and-whites (Rio Plata) won.

Some things remain the same, all over the world.

Like the inefficiencies of bureaucracies.  Lan Chile had us all lined up at gate 17 in Santiago, and USians, Canadians and Australians had to have a piece of paper. A woman demanded that everybody report to gate 13 with their bits of paper, and everybody went to the gate 17 desk to expostulate.

We just walked through, showed the ticket person our bit of paper, and got on the plane.  It worked!

Later, I watched a slack-jawed official run all the luggage through an X-ray machine "for security" who then wandered off, ignoring the screen as the things went through.  Today, I saw the sign above right, and decided that this was the department in charge of making mistakes, but being a bureaucracy, it probably makes them very slowly.

Or maybe it's the set for a Spanish-language remake of Fawlty Towers, where Manuel is a comic Englishman who speaks bizarre Spanish, and don't mention the war relates of the Malvinas War?

Maybe I got it all wrong, because we saw what the tiger meant when we went out today to look out at part of the optimistically named Rio de la Plata (River of Silver), which had no silver, but did get hundreds of islands, mad of mud from the Parana River, which some Year 6 teacher thought we should be able to place on a map. I recall the Parana, but preferred the Araguaya, which had a brilliant vowels to consonants ratio, but no matter: I cannot draw a map of either of them today.

The Parana-mud delta is hundreds of islands in a shape that looks a bit like a tiger's head, which may explain why it is called the Tigre Delta, or it may be the jaguars that used to live there.  Now it's full of real estate developers, moving in on the holiday shacks there.

We took our binoculars for the birds, but aside from pigeons, we saw only a bunch of all-black cormorants, against the sun, so no shot.

Back in town we saw some grand figs that were tricky to shoot, so these were the best I could do.  They may be Australian: I have seen a lot of river redgums (Eucalyptus maculata) and a few Tasmanian bluegums (Eucalyptus globulus) today, as well as a number of she-oaks (Casuarina or Australian pine to Americans, Allocasuarina to pedants).

Buenos Aires means "good air", but presumably the gums were planted to reduce the chances of malaria.  I think I have blogged about that before.  Yes: you can read about it at this link, and also at this link.

Anyhow, here are the figs, or at least, their roots, which are the best part:

The strange case of Bacterium termo

I came across this case by accident, while looking for something completely different for the book I am working on.

It shows how desperate people were in the late 19th century to find a way of curbing consumption or as it was then being called, tuberculosis.

Another example of the desperation: the following week, Science mentioned another treatment being tried:

Another method of treatment which is now attracting the attention of physicians is by injecting remedies directly into the lungs by means of the hypodermic syringe, the needle of which is passed through the wall of the chest, the effort being made to apply the medicinal agent as nearly as possible to the affected portion of the lungs. Some very encouraging cases are reported, in some of which the improvement has been so great as almost to justify one in speaking of them as cures. Carbolized iodine appears to have produced the best results, causing the complete cessation of cough and expectoration, and the further progress of the disease.

— News report, Science, 19 November 1886, 447.

I knew that one had never progressed very far, but I wondered if there might be some link to Coley's work. Back-tracking, I found a report in the British Medical Journal in October, 1886:

At a recent meeting of the Odessa Medical Society (Proceedings of the Odessa Medical Society, 1886, No. 6, pp. 1-12), Dr. Filipovitch, of the Odessa Town Hospital, made a very instructive communication on six cases of advanced pulmonary pthisis, which had been treated by him after the bacterio-therapeutic method, recommended by Professor Arnaldo Cantani … Having obtained, by fractional cultivation, pure culture of the bacterium termo in meat broth, the author took 5 cubic centimetres of the bacterial fluid diluted them with 10 cubic centimetres of boiled water (37° C.), aromatised the mixture with one or two drops of tincture of peppermint (to disguise an offensive odour), and made the patient inhale the whole by means of Richardson's spray-producer.

— British Medical Journal 2 October 1886, 641–2.

This went on to report that the method had failed. Then in December, a doctor named A. Primrose Wells (what were his parents thinking of?) reported that he also had tried the method, Where others had reported failure, he said:

I think it: only fair to state what has been my experience in five cases subjected to this treatment. I can scarcely say that failure has been the result of my efforts, for in two of the five cases marked improvement resulted, which continues up to the present time.

— A. Primrose Wells, Five Cases Of Pulmonary Phthisis Treated By The "Bacterium Termo" Spray, British Medical Journal 18 December 1886, 1211–1212.

He had obtained a pure culture of "bacterium termo" in a meat infusion, but as a biologist, I could not place this beastie, but I read on. He described his cases and their responses in detail, but the issue was going to be identifying "bacterium termo". I tried to suppress my sniggers about Primrose spraying rotten meat at people, but I'm only human.  Anyhow, I ploughed on.

Sadly, we will never know what it looks like. The name was used over four decades in the 1800s to indicate "…almost any motile rod found abundantly in decaying organic matter", according to H. J. Conn, writing in 1919. According to Conn, even the purest culture probably had at least two species present, and a study in 1950 concluded that the species would never have a type specimen, and should not be recognised.

Still, if anybody is looking for a good basis for a medical fraud, if you can get a patient to accept having a putrid mass of unidentified soil bacteria squirted into their lungs, you will know that the patient is terminally (a word I used advisedly) gullible — and it's a fair bet that they won't be coming around asking for a refund!

---

Thoughts on books and writing

I have a large database of quotes that I mine for epigraphs and stuff, and they are sorted by subject.

Writer with a deadline as seen
by a publisher?

Here is a small selection.  There will be others to follow: look for the tag Treasure chest, or search on the key word Curtiosities.

Of making many books there is no end.

— Holy Bible, Ecclesiastes, 12:12.

Look for knowledge not in books but in things themselves.

— William Gilbert (1540-1603), De Magnete [All About Magnets], 1600.

Books must follow sciences, and not sciences books.

— Francis Bacon (1561 - 1626), Proposition Touching Amendment of Laws.

A fishmonger near the British Museum once discovered that parchment or limp vellum, though defaced by ancient ink or paint, was better than oiled paper for wrapping fish. Before the authorities caught up with him, numerous rare manuscripts had found their way into London kitchens, and from thence to the trash bin.

— Lloyd A. Brown, The Story of Maps, Dover edition, 1979, p. 6.

Whitehead caught the unhistorical spirit of the scientific community when he wrote, 'A science that hesitates to forget its founders is lost.' Yet he was not quite right, for the sciences, like other professional enterprises, do need their heroes and do preserve their names. Fortunately, instead of forgetting these heroes, scientists have been able to forget or revise their works.

— Thomas Kuhn, The Structure of Scientific Revolutions, 138-139.

Contrast this situation with that in at least the contemporary natural sciences. In these fields the student relies mainly on textbooks until, in his third or fourth year of graduate work, he begins his own research. Many science curricula do not ask even graduate students to read in works not written specially for students.

— Thomas Kuhn, The Structure of Scientific Revolutions, second edition, 1970, p. 165

Another damned, thick square book! Always scribble, scribble, scribble! Eh! Mr. Gibbon?

— William Henry, Duke of Gloucester (1743-1805) (attributed).

Persons attempting to find a motive in this narrative will be prosecuted; persons attempting to find a moral in it will be banished; persons attempting to find a plot in it will be shot.

— Mark Twain (Samuel Langhorne Clemens) (1835 - 1910), 'Notice' at the start of Huckleberry Finn, 1884.

You know in England we read their works, but seldom or never take notice of authors. We think them sufficiently paid if their books sell, and of course leave them in their colleges and obscurity, by which means we are not troubled with their vanity and impertinence.

— Sir Robert Walpole (1676 - 1745), to the philosopher, David Hume.

Some books are only cursorily to be tasted of: namely first, voluminous books, the task of a man's life to read them over; secondly, auxiliary books, only to be repaired to on occasions; thirdly, such as are mere pieces of formality, so that if you look on them you look through them, and he that peeps through the casement of the index, sees as much as if he were in the house.

— Thomas Fuller (1608 - 1661), Worthies of England.

If I were to pray for a taste which would stand me in stead under every variety of circumstances, and be a source of happiness and cheerfulness to me through life, and a shield against its ills, however things might go amiss, and the world frown on me, it would be a taste for reading.

— Sir John Herschel (1792 - 1871) 'Address to the Subscribers of the Windsor and Eton Public Libraries', Charles Mackay (ed.), A Thousand and One Gems of English Prose (n.d.), p. 73.

Publisher as seen by a writer?

King David and King Solomon
 Led merry, merry lives,
With many, many lady friends
 And many, many wives;

But when old age crept over them,
 With many, many qualms,
King Solomon wrote the Proverbs
 And King David wrote the Psalms.

— James Ball Naylor

My desire is . . . that mine adversary had written a book.

— Holy Bible, Job, 31:35.

Lily: 'We looked at the books about crystals but they are so dreadful.'

— John Ruskin (1819 - 1900) The Ethics of the Dust, Ten Lectures to Little Housewives on the Elements of Crystallisation, 1866.

If you happen to have an Elzevir classic in your pocket, neither show it nor mention it.

— Philip Dormer Stanhope, 4th Earl of Chesterfield (1694 - 1773), Letters from a Celebrated Nobleman to his Heir.

A classic is something that everybody wants to have read and nobody wants to read.

— Mark Twain (Samuel Langhorne Clemens) (1835 - 1910), Mark Twain's Speeches, 1910.

Thou should'st be living at this hour,
Milton, and enjoying power.
England hath need of thee and not
Of Leavis and of Eliot.

— Heathcote William Garrod

To such a person my hope has been that my treatise would prove of the very greatest assistance. Still, such people may be expected to be quite few in number, while, as for the others, this book will be as superfluous to them as a tale told to an ass.

— Galen, On the natural faculties III, 10

I was in a Printing-House in Hell, and saw the method in which knowledge is transmitted from generation to generation.

— William Blake (1757 - 1827).

A visit to Old Man's Hat

The route we followed. The buildings on the left are part of the Q Station.

Today, I combined a number of things that I enjoy best: scrub-bashing in difficult terrain, social history, biology and geology.

This post is to help others who may want to go there: there are notes on finding the site at the end.

Sydney's Quarantine Station operated for about 130 years, and a lot of people, having suffered sea sickness, fear, discomfort and maybe disease, all the way to Australia, saw no more of our land than this station, some six miles/10km from Sydney itself, in a secluded place on the shores of Sydney Harbour, not far from my home.

You can relate this to the map above,
and note the wall running down-slope.

People arrived by ship, back then, and if there was disease on the ship, they were held at a special facility on North Head near the North Head Sanctuary, where I work as a volunteer. Many of them went up to a headland and carved their names, their initials, or other forms of memorial, in the flat sandstone.


Today, three of we gardeners went bush to look at these stone engravings on the headland. My thanks to my companions, Sharyn and Christabel.

By the way, all of the pictures here have been smallified, but if you click on them, you can embiggen them. 

The wall.

To start, we walked down alongside this wall.  This used to be the boundary that the passengers had to stay inside. 

You can see the same wall in the map and route pictures above, and it starts about 50 metres west of the bus stop that is across the road from the North Fort (Bella Vista) Cafe. If you can't work out where that is, you probably should not go there.

The route up to Old Man's Hat

The area is called Old Man's Hat: perhaps you can see why? This shows roughly the route we followed: this is a slightly energetic climb, and the bush is reputed to be full of ticks.  There is supposedly no Lyme Disease in Australia, but we seem to be getting a few cases of something similar, so ticks are no joke.

South Head and Sydney

 
A small view of a bit of the bush on North Head, which I am standing on, and South Head, opposite. These make up "the heads", the only entrance to Sydney Harbour.


The city of Sydney (aka the CBD or Central Business District) is on the horizon, over to the right.

The cliff near the gully

This is Triassic sandstone, a deposit known as the Hawkesbury sandstone, about 200 million years old, give or take a bit.

Some of the beds show cross bedding, and a few of them are rather shale-like.

More of that later.

The gully and a shale-like bed

The white bed you can see on the right was very fine-grained. At this point we had walked down the wall, along cliff-tops, down a few interesting drops, and we had started to scramble up the other side.

Note again that if you are thinking of trying this, you need a bit of a head for heights.

Heading up the slope: an easy part.

Here, we had already crossed the gully. 

Incidentally, in May, which we regard as well into autumn, we counted 17 species of plant in bloom.

Finding stuff

 
And soon we were finding stuff.


The area on top was reasonably open, if only because a lot of it was bare rock.

Looking down Sydney Harbour

And enjoying views. Here, from South Head to Sydney CBD.

The bush here is technically heath.


An engraving.  The "inmates" could make their way up here, where they were still inside the Quarantine Station, and they must have either had tools, or been lent them. Some of the engravings are no more than initials, but others are works of art. There are, by the way, no Aboriginal rock engravings in the area, that I know of.


Another engraving.





But this one is different. In World War I, Japan was in the war on the allied side. In 1916, some Japanese sailors were detained here, it seems.

They appear to have been the crew of 'Nikkon Maru'.

More engravings,

Current bedding.

I also caught this view of cross-bedding in the cliff. Cross-bedding or current-bedding happens when a current pushes sand along so it flows down a sloping face, making a series of beds that are laid down on about a 30-degree angle.

Ribbon markers.

One important thing to remember is that people are studying and marking the engravings with pink ribbons like these. One, in Cyrillic script, has yet to be tagged. We found that when we headed out across-country to the Q-station.

Tick warning sign.

We made it out, but going in that way would be a nightmare. because you simply would not know where you were going. We were warned to avoid the area because of ticks,


We came out through the Quarantine Station, where they didn't want us following one of the tracks. Curiously, the coin for scale is an American one.

Small ant hill, about 80 mm across.

And I had this find as we headed back to our cars: an ant hill where the ants, under the surface, see a different sand to what we see on top. With engravings or anything else, a lot of the story is hidden until you dig for it.

Me near the wall.

Now here are three shots that Sharyn took:

Christabel and I, most of the way down.

Me, climbing up the other side.

How to get there.

You can drive in and park. Be aware that there is a fee to park in the National Parks car park, but as you are driving along the road to North Head, turn in left at the North Fort entrance near the cafe, and parking is free.

You can also walk up from Manly Wharf (about 2 km) or you can catch the 161 bus from across the road from the wharf.

The wall that you walk down beside is about 50 metres back from (west of) the North Fort entrance, and on the other side of the road.

You need good shoes, the ability to climb and scramble, a bit of food, some water, sunscreen, a hat and some sort of anti-insect spray to discourage ticks.  There is no safe swimming there.

Aside from that, you are on your own. If you can't work out how to get there from this, you probably shouldn't even think of going in there.  If you do go in there, be ready at any point to give up and go back.  Look at the climbs with a view to going back the same way, as the route out that we followed was very overgrown and needed a trained eye. It is very slippery and wet, after rain.

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In July 2017, Brian Concannon made a comment (see below) about the rockfall there and asked if I had looked into it. I had, but I had failed to post it here. See The Great North Head Calamity, for an updated form of a piece I wrote for use elsewhere.

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This blog covers quite a few different things, so I tag each post. I also write books, and you can find me by Googling my name.  Or, if you want to buy my books, go here!

The growth of science

Science often advances when somebody comes up with a new method, but sometimes that method is a new machine. When my wife and I were courting, as it was still called in them there days, we spent a certain amount of time in swamps, attaching cobalt chloride paper to the leaves of assorted Banksias, so she could see how long they took to change colour. I was no use on the colour changes, being fairly "colour blind", but at least I could do the hack work and provide support.

Our daughter, of whom we are excessively proud, combines being a provider of grandchildren with being a post-doc who uses whizz-bang machines. In particular, she uses a laser to measure something similar to what my wife assessed with coloured bits of paper.

That led us, while driving to yum cha this morning, to reflect on how things have changed. I recalled that in 1988, the Irish PM (I think it was Bertie Ahern, but don't hold me to that) came to Australia and presented us with a file with all the convicts who left Ireland, plus their details, in it.

There was a problem: at 40 megabytes, the file was too large to load on any computer that we had, so we needed to set it aside until technology caught up. A while back, I went out to talk to students at Birrong Girls' High about the year 1859 from a particular viewpoint, and I took a PowerPoint file of artworks from the period that was almost 40 megabytes.

I carried it on a USB thumbstick, along with a huge range of other files that I might have needed, which used up almost none of the 8 gigabytes that it offered. The thumbstick cost me about one ten-millionth of what that much memory would have cost me in 1981: if you factor in inflation, you can make that one fifty-millionth!

We lose track of the way things are changing. As Robyn Williams observed, history is something we only perceive in hindsight. All the same, we keep increasing our expectations. Now, when somebody dies, it is a matter for fury and demands of "why the doctors did not do more?"

In the end, I suspect that we will run out of new machines, new meters, new metrics, and I wonder how society will react to that. Just as economists never saw the possibility that the economy might stop growing, we are unprepared for science to stop growing. I wonder if we should start to brace ourselves?

Maybe I should have called this "the end of science"?

Perhaps, but I would hate to be accused of tabloidism.