It might have been the trying, frontier conditions of Burra that prompted Henry to look for “greener pastures”. Another possibility was the gold rushes in Victoria in the early and mid 1850s. He traveled on his own to Ballarat in Victoria. The family tradition goes that he was assisted by the aborigines; moving from “tribe to tribe” on his journey. Of the towns that he passed through in Victoria, he was particularly impressed with Avoca. With its Court, police station, gold wardens, churches, and schools it would have seemed a far more civilised place to live, earn an income and bring up a large family than Burra. He returned home to convince his family and friends to move to Avoca. He eventually took about 12 families with him to settle in Avoca.
The Google Map below shows Cornwall where the Trevithicks were born. Double click near the blue place indicator to zoom in. If you zoom enough you will see two blue place indicators, one for Sithney and the other for Helsotn. Click on each indicator for more information including pictures of Henry and Elizabeth in later life.
View Larger Map
The family migrated to Australia in the mid to late 1840s. There was a major emigration from Cornwall between 1946 and 1848 caused by potato blight and economic hardship. Henry’s family settled in Burra in South Australia. It is hardly surprising that this family of miners lived in Burra as it was one of the major mining centres in South Australia at the time. Copper had been discovered in 1845. The town began as a single company mining township that, by 1851, was a set of townships (company, private and government owned) collectively known as “The Burra”. The Burra mines supplied 89% of South Australia's and 5% of the world’s copper for 15 years, and the settlement has been credited (along with the mines at Kapunda) with saving the economy of the struggling new colony of South Australia.
From the beginning the township was
There are no houses, the swellings being excavated in the banks of the Burra creek.
The photograph shows some of these “homes”. Henry’s family initially lived in this type of primitive accommodation by the Burra creek.
The family had shifted to Avoca by 1854 as Henry Trevithick is mentioned in the Mount Alexander Mail(Castlemaine) of 13th October 1854. It seems that he was living in a “tent close to the police camp” in Avoca. My great-grandmother, Harriet, was born in there on 24th August 1857.
Henry made a living in the mines but quickly turned to other pursuits as well. He eventually owned two hotels, the Locomotive and the Commercial. The Locomotive was situated opposite the court building. A residential house now stands on the block. The Locomotive was probably not a large hotel, we cannot find a picture of it in the family records or in the Avoca records. It is described as “the house and premises known as the Locomotive Hotel” in a notice in the “Avoca Mail” of 1865. It seems that in that year Henry sold it to Joseph Henry Gawith. Henry also bought a fair amount of land in the town and owned shares in two of the large mines in the area. One of his money making ventures eventually led to a tragedy, in May 1859. Henry and a friend, Henry Knott, had a beer selling booth at the local race track.
The following is Henry’s testimony to the subsequent inquest:
I am a Publican residing at Avoca. Yesterday, I was driving the deceased home from the Race Course in my spring cart, he was not sober, the wheel went into a small rut in the road and the deceased fell out and pitched on his head. I did not fall out myself, the accident happened near the Park Hotel. I called out for assistance. Mr Holmes came to me and helped me to put the deceased in the cart. I then sent for a Doctor. The deceased was holding my arm as I drove along to keep himself steady in the cart. The deceased must have died instantly after he fell out, the hour was five o’clock.
Henry was probably “gilding the lily” as another witness claimed: “… I saw the deceased and last witness driving in a spring cart they were driving fast they were neither of them sober.”
It seems fairly clear from the Inquest that Henry was drunk and speeding in his cart. He was probably quite lucky not to be charged.
His friend, the deceased, Henry Knott had been another influential citizen who was also a member of the Avoca council. Not long before his death he gave some land to Avoca for a cemetery. His grave was one of the first in the new cemetery, and his wife had to pay 20 pounds for the grave site.
Henry made quite a success of his new life in Avoca. He was elected to the local council, and became quite well off financially. Not all of his financial ventures were necessarily successful. For instance, he owned a mine, for a while, and the family tradition is that he lost money in that venture.
On the other hand there is a report in the Avoca Mail, September 9th 1873, which states:Many of our readers will be pleased to hear of a slice of good luck which occurred to Mr Henry Trevithick in his claim on the Deep Lead, on Saturday last, when a welcome nugget of about 30 oz was unearthed.
Certainly there is a family story that at times he lit his pipe with 5 pound notes.
Henry was obviously a “rough and tumble” sort of man, but late in his life he turned to matters of religion. My mother has his bible in her care.
He died in 1887 as described by the Avoca Free Press:
Another of the links binding the past to the rising generation has been severed by the death of Mr Henry Trevithick, long and favourably known in former years as the landlord of the Locomotive Hotel, opposite the Court House. Deceased, who was in the 77th year of his age, has suffered for some considerable time from a general breaking up of the system. The funeral, which took place on Wednesday afternoon, was numerously attended.
Saturday, September 29, 2007
Family History: Henry and Elizabeth Trevithick
Tuesday, September 25, 2007
Equinoxes
Day and night are not exactly of equal length at the time of the March and September equinoxes. The dates on which day and night are each 12 hours occur a few days before and after the equinoxes. The specific dates of this occurrence are different for different latitudes.
On the day of an equinox, the geometric center of the Sun's disk crosses the equator, and this point is above the horizon for 12 hours everywhere on the Earth. However, the Sun is not simply a geometric point. Sunrise is defined as the instant when the leading edge of the Sun's disk becomes visible on the horizon, whereas sunset is the instant when the trailing edge of the disk disappears below the horizon. These are the moments of first and last direct sunlight. At these times the center of the disk is below the horizon. Furthermore, atmospheric refraction causes the Sun's disk to appear higher in the sky than it would if the Earth had no atmosphere. Thus, in the morning the upper edge of the disk is visible for several minutes before the geometric edge of the disk reaches the horizon. Similarly, in the evening the upper edge of the disk disappears several minutes after the geometric disk has passed below the horizon. The times of sunrise and sunset in almanacs are calculated for the normal atmospheric refraction of 34 minutes of arc and a semidiameter of 16 minutes of arc for the disk. Therefore, at the tabulated time the geometric center of the Sun is actually 50 minutes of arc below a regular and unobstructed horizon for an observer on the surface of the Earth in a level region.
For observers within a couple of degrees of the equator, the period from sunrise to sunset is always several minutes longer than the night. At higher latitudes in the northern hemisphere, the date of equal day and night occurs before the March equinox. Daytime continues to be longer than nighttime until after the September equinox. In the southern hemisphere, the dates of equal day and night occur before the September equinox and after the March equinox.
Using this site I calculated that at the latitude and longitude of Melbourne 20 September had days and nights of equal length. At the top of Cape York Peninsular, at 10 degrees south, equal day and night occurred on 12th September.
Monday, September 24, 2007
Satan
... His Pride
Had cast im out from heaven, with all his host
Of rebel Angels, by whose aid, aspiring
To set himself in glory above his peers,
He trusted to have equalled the Most High,
If he opposed, and, with ambitious aim
Against the throne and monarchy of God,
Raised impious war in Heaven and battle proud,
With vain attempt. He the Almighty Power
Hurled headlong flaming from the ethereal sky,
With hideous ruin and combustion, down
To bottomless perdition, there to dwell
In adamantine chains and penal fire,
Who durst defy the Omnipotent to arms.
Nine times the space that measures day and night
To mortal men, he, with his horrid crew,
Lay vanquished, rolling in the fiery gulf,
Confounded, though immortal. But his doom
Reserved him to more wrath; for now the thought
Both of lost happiness and lasting pain
Torments him: round he throws his baleful eyes,
That witnessed huge affliction and dismay,
Mixed with obdurate pride and steadfast hate.
At once, as far as Angels ken, he views
The dismal situation waste and wild.
A dungeon horrible, on all sides round,
As one great furnace flamed; yet from those flames
No light; but rather darkness visible
Served only to discover sights of woe,
Regions of sorrow, doleful shades, where peace
And rest can never dwell, hope never comes
That comes to all, but torture without end
Still urges, and a fiery deluge, fed
With ever burning sulphur unconsumed. ...
Powerful writing for believers and non believers alike!
My Last Duchess by Robert Browning
That's my last Duchess painted on the wall,
Looking as if she were alive. I call
That piece a wonder, now: Fra Pandolf's hands
Worked busily a day, and there she stands.
Will't please you sit and look at her? I said
"Fra Pandolf" by design, for never read
Strangers like you that pictured countenance,
The depth and passion of its earnest glance,
But to myself they turn (since none puts by
The curtain I have drawn for you, but I)
Ans seemed as they would ask me, if they durst,
How such a glance came there; so, not the first
Are you to turn and ask thus. Sir, 'twas not
Her husband's presence only, called that spot
Of joy into the Duchess' cheek; perhaps
Fra Pandolf chanced to say "her mantle laps
Over my Lady's wrist too much," or "Paint
Must never hope to reproduce the faing
Half-flush that dies along her throat"; such stuff
Was courtesy, she thought, and cause enohgh
For calling up that spot of joy. She had
A heart ... How shall I say? ... too soon made glad,
Too easily impressed; she liked whate'er
She looked on, and her looks went everywhere.
Sir, t'was all one! My favour at her breast,
The drooping of the daylight in the West,
The bough of cherries some officious fool
Broke in the orchard for her, the white mule
She rode with round the terrace - all and each
Would draw from her alike the approving speech,
Or blush, at least. She thanked men, - good; but thanked
Somehow ... I know not how ... as if she ranked
My gift of a nine hundred years old name
With anybody's gift. Who's stoop to blame
This sort of trifling? Even if you had skill
In speech - (which I hae not) - to make your will
Quite clear to such an one, and say "Just this
Or that in you disgusts me; here you miss,
Or there exceed the mark" - and if she let
Herself be lessoned so, nor plainly set
Her wits to yours, forsooth, and made excuse,
- E'en then would be some stooping, and I choose
Never to stoop. Oh, Sir, she smiled, not doubt,
Whene'er I passed her; but who passed without
Much the same smile? This grew; I gave commands;
Then all smiles stopped together. Thee she stands
As if alive. Will't please you rise? We'll meet
The company below then. I repeat
The Count your Master's known munificence
Is ample warrant that no just pretence
Of mine for dowry will be disallowed;
Though his fair daughter's self, as I avowed
At starting, is my object. nay', we'll go
Together down, Sir! Notice Neptune, tho',
Taming a sea-horse, thought a rarity,
Which Claus of Innsbruck cast in bronze for me.
It is amazing that a man of such cold blooded and casual violence could have such power over life and death. Pity the "fair daughter" of the Count who is to replace the Duke's former wife who had been murdered by his order - "I gave commands; then all smiles stopped together". She was killed because of her natural innocence and natural sexuality. It is quite possible that the transgressions of the former Duchess were only in the mind of the Duke. The new wife had better be warned very carefully about the sort of deferential behaviour required by her new husband.
What makes the Duke even more frightening is that although he has the instincts and morals of a gangster, he has the patina of a gentleman and is a connoisseur of fine art.
Book Review: Imagining Numbers by Barry Mazur
Imagining Numbers
(Particularly the square root of Minus Fifteen)
In his preface to the book Mazur states that the book is "written for people who have no training in mathematics ... but who may wish to experience an act of mathematical imagining and to consider how such an experience compares with the imaginative work involved in reading and understanding a phrase in a poem." A major aim of the book then is to draw analogies between the imaginative process of mathematics and poetry.
Mazur returns regularly to parts of the tone poem "Whatever It Is, Wherever You Are" by John Ashbery. In pondering the inventors of writing Ashbery writes:
To what purpose did they cross-hatch so effectively, so that the luminous durface that was underneath is transformed into another, also luminous but so shifting and so alive with suggestiveness that it is like quicksand, to take a step there would be to fall through the fragile net of uncertainties into the bog of certainty ...
I would like some of the "bog of certainty" in Mazur's description of the relationship between poetical and literary imagining and the mathematical variety. I have an interest in poetry and literature on the one hand and mathematics and science on the other, but if Mazur provides an explanation of the relationship between imagining in the humanities and the sciences then it is beyond my ken. This book can be seen as a contribution to breaking down the division between the two cultures described by C. P. Snow - the Humanities and the Sciences. Snow might have overstated his case but something of a division still exists. Even if the book does not finally succeed in drawing a convincing analogy between creativity in the sciences and creativity in the humanities, the erudition of Mazur in both literature and mathematics show that it is possible for a modern expert in the sciences to have and interest and skills in the humanities.
Consequently the rest of this post will concentrate on the mathematical ideas in the book. Mazur's description of the development of the idea of complex numbers by a range of mathematicians over three centuries is interesting and compelling.
A brief note on typology. Square foots will be mentioned frequently in this post. I will not use the conventional square root symbol - √ - as it will require too many graphics, instead I will represent square roots using the computer terminology, so that the square root of 4 will appear as: sqrt(4).
As Mazur describes (p39) the Italian mathematician Girolamo Cardono (1501-76) in his Ars Magna coming across an equation that contains (5 + sqrt(-15)) × (5 - sqrt(-15)) . There are three possible reactions to this piece of maths, but to make the discussion clearer I will use sqrt(-4).
A reaction from those who have little understanding of or interest in maths would be "what is all the fuss about? the answer is obviously -2" . Those who have a reasonably sophisticated understanding of maths would think 2i . Those who took some notice in their middle secondary school maths classes would be saying "that's impossible, there is no such thing as the square root of a negative number!" As i described in this post multiplying two negative numbers together will result in a positive one. If you multiply -2 by -2 the answer is +4, so there seems to be no solution to sqrt(-4). The problem for that point of view is that if you ignore the objection to square roots of negative numbers and just multiply out Cadano's expression you end up with the positive whole number 40, as demonstrated below:
(5 + sqrt(-15)) × (5 - sqrt(-15))
= 25 -5 × sqrt(-15) +5 × sqrt(-15) -(sqrt(-15)× (sqrt(-15)
-5 × sqrt(-15) +5 × sqrt(-15) cancel each other out os
= 25 -(-15)
= 25 +15
= 40
It is surprising that such an unusual expression evaluates to the simplest type of number - a positive integer. Although sqrt(-15) seems initially nonsensical if it is accepted without thinking too much about what it means it can produce acceptable results.
If we multiply the number line by -1 we rotate it by 180 degrees.
Writing sqrt(-1) is a little clumsy so mathematicians developed a notation for it" i . Sqrt(-4) can be written as sqrt(4) × sqrt(-1) which becomes 2i when sqrt(-1) is written as i . Numbers that contain i are called imaginary numbers but we can generalise the function by adding a real number to form a +bi. These are called complex numbers.
Previously we discussed multiplying by -1. What happens if we multiply a complex number by i? Lets see: i × (a + bi) = ai + b × i × i = ai - b (as i × i = -1). If you plot these numbers on a cartesian plane, with the real numbers on the X-axis and the imaginary ones on the Y-axis then it should be clear that multiplying by i is equivalent to rotating by 90 degrees. Mathematicians plot complex numbers on the complex plane with the horizontal axis for the real portion of the number and the vertical axis for the imaginary numbers. This makes sense if multiplying by i is equivalent to rotating by 90 degrees.
What happens if we multiply the complex number by i a second time. This is obviously another 90 degree rotation resulting in a cumulative 180 degree rotation - the same as multiplying by -1. That is multiplying by i2 produces the same result as multiplying by -1. This might not be particularly surprising as i2 = -1. We seem to have arrived at the same point from two different directions.
Many mathematicians use complex numbers without concerning themselves with the question of what imaginary numbers look like but for those or us who have less mathematical skill Mazur's description of i as a rotation of 90 degrees on the complex plane is satisfying.
This post just scratches the surface of the area of mathematics called Complex Analysis. Other reasons to believe in the reality of complex numbers is that complex maths is essential to a number of important areas of physics including Quantum Mechanics. Some time or other (when I learn something about it) I hope to further explain that claim.
Sunday, September 23, 2007
Weekend 21st to 23rd September
On Saturday 22nd we managed to sell my desk, one of about four items of furniture that we hope to sell before Settlement Day on November 30th.
On Sunday 23rd we attended the Storm Vs Paramatta at Telstra Dome. As described in the post it was an entertaining and exciting game.
Storm into another Grand Final
It might not be an advantage to have the week off in the finals, as yesterday Geelong only just beat Collingwood in an AFL Prelim Final after having the week off. Storm, who likewise had had a rest last week, started the game in a lethargic frame of mind. Paramatta, on the other hand, came to play.
Paramatta applied relentless pressure from the opening kick off. It was not a surprise when Eric Grothe ran a try across the
line in the eleventh minute. It was very wide so the conversion was difficult and was missed. From about the half way mark of the half Storm started to get their act together, and started to apply relentless pressure. At the 28 minute mark Steve Turner crossed over for a try which Cameron Smith did not convert. Four minutes later it was Cooper Cronk's turn to score close to the posts (see photo) and Cameron Smith converted easily. Storm continued to threaten to score more tries until half time but Paramatta held them out. With the score standing at 10 to 4 the predominantly Storm crowd enjoyed the half time break.Paramatta started the second half as well as the first with a try to Joel Reddy after only three minutes, which was converted by Krishan Inu. The score now was even at 10 all. On the score board it looked like a run away win for Storm from then on as they scored two penalty goals and two converted tries in the second half. It wasn't an easy win though, as Paramatta with the Grand Final at stake continued to battle for a try. They had two tries disallowed in close decisions by the Video Ref.
This tough fought win was a better preparation for next week's Grand Final than a run away win like Storm's first win of the Final Series against the Broncos.
Another success for Storm on the afternoon is that the game attracted a crowd of 33,427, the largest crowd at a Storm game in Melbourne.
Telstra Dome is a much more convenient venue to attend that Olympic park. Finding a park for the car is always a problem at the Graveyard but today we took the train to Southern Cross station, and walked the hundred metres or so to the ground.
Beleura House and Garden
The house has an intimate feel, it is not on the grand scale of Como or Rippon Lea. There are no grand rooms designed for the lavish entertaining of large numbers of people, rather the rooms are on a smaller scale appropriate for entertaining relatively small groups of friends and acquaintances. The decoration of the house gives some indication of the types of guests invited to Beleura. In one of the rooms there is a picture of Norman Lindsay, who was a family friend. In another room is a framed letter from Nelly Melba, recommending a New York interior designer. Another set of rooms were set aside for Melba's use when she visited.
Most walls are adorned with art - paintings and sculpture. The painting styles are varied with some fairly buttoned up 18th century pieces and in other rooms nudes. The ceilings of the entrance hall, main dining room and drawing room are covered by frescos, with liberal displays of flesh. John apparently complained to the painter of the excessive nudity in the frescos and the painter had his revenge, depicting John in his gardening clothes. The photograph at right displays - in wide angle - the main dining room. Part of the ceiling fresco can be seen.
John considered it wasteful to warm the whole house in the wintertime, as there were generally only two people living in it most of the time - John and his (male) housekeeper. So, he built two flats external to t
he house, one for himself and the other for the house keeper. He rarely left the property, as it contained all he required. He, or maybe his housekeeper, ordered by phone his requirements from Myer - he was a substantial shareholder.The house is set in 20 acres of beautiful gardens. The garden follows the same pattern as the house. There is only one large section,
the formal garden in front of the house, as illustrated at left. Most of the garden consists of relatively small "rooms" each following particular themes. The themed garden rooms include: the Italian Walk, the Hansel and Gretel garden, the Roman Ruin "Folly", the fruit and vegetable garden and the Japanese garden with a temple structure overlooking a lake, illustrated at right.The garden is undergoing constant renovation and new areas are being added so the House and Garden will repay regular future visits.
John Tallis died in 1996 and bequeathed the House and Garden to the people of Victoria. He set up a trust to assist with the maintenance of the house. As well the trust provides scholarships for young musicians.
Regular concerts are held in the Melba room in a building near the main house.
Saturday, September 22, 2007
Kiwi
What a fantastic concept! Sad and funny at the same time.
Theatre Review: The War of the Worlds
I had expected that it would be a theatrical experience in the form of a musical play, so I was surprised to see seating over most of the floor of the auditorium, and a stage at the end of the hall. It was immediately clear that most of the stage would be taken up with the musicians - orchestra and rock band - with little room for the actors. Consequently I had to recast my expectations. As the show progressed it was revealed as something of a hybrid - part rock concert, part musical theatre, part cinema.
A show with the music of Jeff Wayne's Musical Version of The War of the Worlds would always be interesting and the music played by The Black Smoke Band and backup orchestra was powerful and exciting. The problem for a show like this is to have the visuals live up to the quality of the audio. In the end even a huge screen will appear small from the back of a venue the size of the Rod Laver Arena. The visuals on the screen were interesting but not as spectacular as the music. The live singers were displayed on the screen with the prerecorded visuals. Although they didn't integrate well with the visuals it was an effective method of showing the singers in such a large auditorium.
The most interesting of the visual effects was projected on a screen on the side wall. One of the highlights of the original album was the narration by Richard Burton. If I had have thought of it I would have expected it to be a disembodied voice, but instead Burton's head appeared and started the narration. It seams that they used the same technique as that used to bring Gollum to life in the Lord of the Rings movies. An actor - Brian Mallon - lip-sinked Burton's original audio recording. That was used in the generation of the Burton video. It was interesting to see a young, unlined Burton saying "No one would have believed in th last years of the nineteenth century, that human affairs were being watched from the timeless worlds of space." This was a great achievement of computer animation. The producers claim that the image is holographic - ie 3D - but from our distance away it was impossible to see the 3D effect.
The special effects were not confined to the audio-visual sphere, though. Shortly after the Martian tripod fighting machines appeared in the story a 9 metre version descended to the stage from its hiding place amongst the lights and scaffolding above the stage. This was reasonably impressive.
The performers sang and acted well. I was particularly impressed with the range of Michael Falzon as the Artillery Man. Shannon Noll, as the crazy Parson Nathaniel, Rachel Beck, Justin Hayward, Chris Thompson and Anna-Marie Wayne also turned in fine performances.
Overall, once I had adjusted to they style of the performance, it was a pleasurable evening's entertainment.
One of the strong points of Wayne's original concept was to closely follow the original nineteenth century story. Most of the film versions have attempted to update the story with more realistic - modern - technology. These have generally been unsuccessful. Science fiction from an earlier era often has a jarring effect. We can accept their description of technologies that we have not yet developed - some of which are well into the future - but they often contain elements that show they are mired in an old and now outmoded paradigm. In the original book Wells described a powerful heat ray, which is clearly more powerful than any laser that has yet been invented, yet the tripods are very mechanical and the method of inter planetary travel is more like an artillery shell than a rocket. Wayne made the right choice in not attempting to update the story.
Wells's resolution to his story always seemed to me to be weak. It seems unlikely to me that a species that could span interplanetary space would overlook the effect of bacteria. Nasa was aware of this as an issue back in the 1960. Wells attempted to overcome that fundamental weakness in the Epilogue to the story when the Journalist considered the possibility of the Martians analysing their mistake and mounting a further invasion. Wayne includes this, but adds a sting with the "Houston Scene" at the end indicating that the Martians are back.
The video below is a promo for a DVD of the English production. It gives a good sense of the look and feel of the show.
Thursday, September 20, 2007
Negative times Negative equals Positive
Those with some familiarity with secondary school maths will be aware that a negative number multiplied by a negative number gives a positive number.
For example:
-2 X -3 = +6 , it does not equal -6.
I thought that rule was an axiom or assumption. An equivalent in standard English is the double negative. The statement "I don't want nothing" means that you want something because whatever you don't want it isn't nothing. Of cause in colloquial English and other versions of English it can still mean a negative. As well a double negative can mean negative in some other languages, for example Africans and Greek.
The Distributive Law
In mathematics though negative times negative equals positive turns out to be a provable law. The first issue to deal with is the Distributive Law of Mathematics, which can be written as : A x C + B x C = (A + B) x C . Stated in words the distributive law says: to multiply a given number C by the sum of two numbers is the same as multiplying C by each of those numbe
rs and and then adding the result. The Distributive Law links multiplication with addition and is one of the fundamental laws of maths.
Here is a demonstration of how the law works with positive whole numbers.
If A = 2, B = 3 and C = 4:
2 x 4 + 3 x 4 = (2 + 3) x 4
8 + 12 = 5 x 4
20 = 20
There are algebraic proofs of the Distributive Law, but they require considerable explanation. As a visual thinker I find the diagram at right compelling proof.
Let a and b be any two real numbers. Consider the number n defined by:
Applying the Distributive Law to Negative Numbers
Our task is to apply the distributive law to negative numbers.
When we do this two other rules will be resolved: positive x negative = ? and negative x negative = ?.
Positive x Negative = ?
Assume: positive x negative = positive ie -3 x 4 = +12 and -1 x 4 = 4
If A = 2, B = -3, C = 4
By the distributive law:
2 x 4 + -3 x 4 = (2 + -3) x 4
8 + 12 = -1 x 4
20 = 4
This result is clearly incorrect so our assumption (positive X negative = positive) is incorrect.
Assume: positive x negative = negative ie -3 x 4 = -12 and -1 x 4 = -4
By the distributive law:
2 x 4 + -3 x 4 = (2 + -3)x 4
8 + -12 = -1 X 4
-4 = -4
This is clearly correct so our assumption (positive X negative = negative) is correct.
Negative X Negative = ?
Assume: negative x negative = negative ie -3 x -4 = -12 and -1 x -4 = -4
If A = 2, B = -3, C = -4
By the distributive law:
2 x -4 + -3 x -4 = (2 + -3) x -4
-8 + -12 = -1 x -4
-20 = -4
This is clearly incorrect so our assumption (negative x negative = negative) is incorrect.
Assume: negative x negative = positive ie -3 x -4 = 12and -1 x -4 =4
If A = 2, B = -3, C = -4
By the distributive law:
2 x -4 + -3 x -4 = (2 + -3) x -4
-8 + 12 = -1 x -4
4 = 4
This is clearly correct so our assumption (negative x negative = positive) is true which is what we set out to do at the beginning of the post.
A More Concise Proof
The following proof is more shorter than that described above. The only extra mathematical rule that this contains is a number multiplied by zero is equal to zero, ie N x 0 = 0 .
An example is 5 x 0 = 0 which equals 0 + 0 + 0 + 0 + 0
which is of cause 0.
n = ab + (-a)(b) + (-a)(-b)
We can write
n = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a) (0)
= ab + 0
= ab
Also,
n = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 x b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b)
So we have,
n = ab
and
n = (-a)(-b)
Hence ab = (-a)(-b)
UPDATE: 28 / 01/ 2013
Here is an argument from the Khan Academy using the distributive law, demonstrating that a negative times a negative equals a positive:
Here are two more videos from the Khan Academy explaining negative times negative equals positive.
The first one explains why positive times negative equals negative:
The next video uses ideas from the previous video to demonstrate that negative times negative equals positive:
Monday, September 17, 2007
Weekend 14th to 17th September
That evening we watched the Collingwood Vs West Coast football match. West Coast are very formidable at Subiaco Oval but Collingwood were rated a chance to win for two reasons. They have been in very good form recently, beating Sydney convincingly last Weekend. West Coast have lost - to injury - their three supreme midfileders, Judd, Kerr and Cousins. West Coast were in front for most of the game with Collingwood seemingly to be just hanging on. Then late in the third quarter, Worsfold, the West Coast coach made what was probably the biggest clanger of his coaching career. Glass had dominated Rocca with the only goal that Rocca has scored being more of an accident than an intentional move. When Worsfold took off Glass for a rest with six minutes to go in the third quarter it might have seemed a safe move as West Coast was up by almost four goals. Without Glass making a fool of him, Rocca scored two goals and at Three Quarter time Collingwood were only down by four points. The last quarter was close with scores even at full time. Two five minute "halves" were then played. The first one was inconsequential, but in the second five minutes Collingwood showed its superior fitness and ran away to win by 19 points. This was one of the best games of football I have seen.
On Saturday morning I attended Windy Hill Fitness Center for training and Pilates class.
In the afternoon Margaret and I visited John's grave as it is his birthday. Then we went to Glenroy Salvation Army for the 50th anniversary celebrations. There are a number of dates when this celebration could be held. If held last year it would have celebrated the first meetings at Glenroy. This year celebrated the building of the hall. 2009 would have commemorated the appointment of the first officer - Neil Saunders - and 2013 would have been the 50th anniversary of Glenroy becoming
a corps. We sad an entertaining powerpoint presentation which included many photographs from the past. We caught up with many people from the corps' history and had a celebration meal.
In the evening we watched the Kangaroos beat Hawthorne and move into one of the Preliminary Finals next week.
On Sunday morning we attended a Celebratory service at Glenroy SA and another meal.
In the afternoon Margaret and I say The Bourne Ultimatum at Broadmeadows cinema. For my reaction to the film see this post.
Monday 17th was my birthday. In the morning we attended our monthly Probus Club meeting. The speaker outlined the changes to superannuation to start on 20th September. It was probably too late for anyone who needed to make a change to their super, but it was good for us to confirm that we have made the right choices (months ago) with our financial adviser.
In the evening we went out to the Flower Drum restaurant for a double celebration - my birthday and selling our house. The flower drum's service and food were up its usual excellent standard.
Movie Review: The Bourne Ultimatum
This is pure multiplex escapist entertainment, and as such is entertaining but not particularly memorable. If the mark of a great movie is to leave you thinking about the issues raised after you leave the the cinema, then this was not a great movie.
This is a very white hat / black hat story. The villains in the CIA who have set up the "Black Opps" assassination program which created Bourne are back without any shade of gray. The CIA operative who is trying to stop them is similarly unrelieved white. The program which equipped Bourne with such effective killing skills also seems to have given him other less likely skills, namely a fluency with language. Bourne speaks three languages fluently, four if you count English.
Bourne is shown solving many problems effectively, so that some unlikely achievements can be easily glossed over. He can open any locked door, and even enter a CIA controlled building effortlessly - though we are not shown how. He crosses national boundaries seamlessly unless he wants to be noticed. He can be blown up by a bomb or involved in a high speed car crash and walk away with a nary a scratch.
The cinematography is annoying. Virtually the whole film seems to have been photographed by a hand held camera. This works very well in the action scenes, increasing their realism and immediacy, but the camera bobbing around while two people sit quietly at a table holding a conversation is very annoying.
Irrespective of all of the criticisms, it is an entertaining multiplex thriller, if you don't think about it much.
Rating: 3 stars.
Wednesday, September 12, 2007
The Election
But is is certainly now clear that the Coalition will lose this election. It will probably be a substantial loss, though a bloodbath or a close loss are still possible.
I would quantify the levels as:
* Close loss - Coalition 48% Labor 52%
Seats
Labor 76, Liberal 72, Independent 2
* Substantial Loss - Coalition 45% Labor 55%
Seats
Labor 92, Liberal 56, Independent 2
* Bloodbath - Coalition 43% Labor 57%
Seats
Labor 105, Liberal 43, Independent 2
The bloodbath levels might seem an amazingly strong figure for Labor but they are less than the current four poll average figures which stand at: Labor 58.25% Coalition 41.75%
The seat estimates come from Anthony Green's election calculator.
One of the reasons that I have been holding off to make a final decision is that since the nadir of March the Coalition have been making a comeback. Between March and June the Coalition Two Party Preferred (TPP) vote increased by about 1% per month. An extrapolation of that trend, once it got underway indicated a close election in late November or Early December. Happily for those of us who hate Howard, the trend stalled in July and August and has turned back towards Labor during the first half of September. Currently (in mid-September) the TPP levels have returned to the levels of May.
The TPP trends that I have described came from monthly aggregates of all of the four polls, AC Nielsen, Galaxy, Newspoll and Morgan. This seems to be the best way of getting a handle on what is happening as the individual polls can be quite varied. For instance, an extrapolation of the Galaxy results in April, May and June predicted a Coalition win. Since then the Galaxy polls have move in Labor's favour and predict a Coalition TPP of 44% in mid November. The Nielsen, Newspoll and Morgan all predict a 45% Coalition TPP in mid November.
Of cause TPP does not tell the full story of an election. It is possible to for a party to win an election (ie pick up a majority of seats) even when it receives a TPP of less than 50%. Howard managed this in 1998. Although the Coalition got less votes than Labor nationally, they got them in the right places. Labor stacked up big majorities in its own seats, while the Coalition won seats by smaller margins. A similar pattern would result in Labor needing more than 50% of the vote to win. Happily this time around the Coalition are not getting support in the right places. The strongest swings to Labor, according to the polls, are in Coalition seats, then in marginals and the smallest in Labor seats.
Rudd has been a much more effective campaigner that many of us expected. He has very cleverly turned issues around on the Government. One of the Coalition's apparent strengths has been the economy, with very rosy macro figures. Rudd though framed the issue differently, focusing on how well off people subjectively feel. He struck a cord with this approach and it took some time for the Government to realise that it had been wrong footed. Eventually Howard started talking about how people felt economically, but soon after the Reserve Bank raised interest rates which gave credence to Rudd's line.
Howard has attempted to use his tried and true methods of smear and wedge to no avail. The personal attacks clearly irritated people and Rudd and Labor's popularity increased with each attack. Rudd has successfully avoided Howard's wedges, particularly on the Haneef afair, even thought the Government's behaviour at that time was totally reprehensible.
Howard probably expected to get a poll bump from "playing the statesman at APEC" but Rudd successfully gatecrashed that and showed that he could effectively mix it with the big boys.
The outbreak of leadership discontent in the Liberal Party will not help the next few polls, and if they continue to be bad for the Government (as I expect will be the case) it is possible that more leadership issues will appear. The cartoon opposite comments on Howard's announcement that he will retire sometime during the next term (if he wins the election) and then hand over the Prime Ministership to Costello. Joe Hockey talked about Howard being a Bradman. He of cause forgot that Bradman made a "duck" in his last test innings.
Howard might be forced to call an election, though possibly a long one as a circuit breaker. He seems amazingly confident that he can out campaign Rudd, even though Rudd has beaten him hands down all year. The coalition will of cause promise bucket loads of money in the election. It will be interesting to see how Rudd handles that. He could promise his own river of money or he could make the point that a desperate PM is ramping up inflation to save his own skin.
Monday, September 10, 2007
Two Irish Jokes
A surly English overseer is standing at the entrance to a construction site in London. It's a filthy wet day. He sees approaching him a shabby figure, with a clay pipe clenched in mouth and a battered raincoat, and scowlingly thinks, Another effing Mick on the scrounge. The Irishman shambles up to him and asks if there's any casual job going. "You don't look to me ," says the supervisor, "as if you know the difference between a girder and a joist." "I do too," says the Irishman indignantly, "The first of them wrote Faust and the second one wrote Ulysses."
This obviously revenges itself on centuries of English caricature of the Irish as stupid and sub-human. I suspect that the references to the weather, shabbiness, clay pipe and raincoat are Joycean but at the moment don't get the allusions. If anyone reading this understands the point of these, please let me know in the comments section.
After reading Hitchens piece on Joyce I decided to start reading some of his writing. I have started with his Portrait of the Artist as a Young Man as it describes the early life of Stephen Daedalus, who is one of the protagonists in Ulysses. Another reason why I haven't yet attempted Ulysses is that reading another preliminary book is probably required, The Odyssey.
Hitchens' joke reminded me of another Irish joke that I heard many years ago. It is a little risque so only click on the link below if you are not offended by a blue joke.
Men have always wondered about the function of the knob on the end of a man's penis. The English spent $100,000 on research to discover that it gave pleasure to the woman. The Scottish spent $500,000 and concluded that it gave pleasure to the man. The Irish spent $2.50 and discovered that it was to stop the hand slipping off.
I didn't realise it at the time I heard that joke, but now I know that it is very Joycean, though in a different sense to the first one. A fan once seized Joyce by the hand and exclaimed, "May I kiss the hand that wrote Ulysses?" Joyce responded, "No - it did lots of other things too."
Not Minor Premiers by Accident
Meanwhile at the MCG only a few hundred metres to the north, Geelong, the AFL minor premier, was handing out a similar thrashing to the Kangaroos. Geelong won by 106 points.
A 106 point win in the AFL is probably equivalent to a 40 point win in the NRL.
I kept tabs on both games by listening to the ABC broadcast of the AFL game while watching the NRL game. Often I had two earphones in my ears, one from the radio the other from NRL "sports ears" broadcasting the referee's comments.
Friday, September 7, 2007
Theatre Review: Who's Afraid of Virginia Woolfe
The play starred Garry McDonald as George, Wendy Hughes as Martha, Stephen Phillips as Nick and Alison Bell as Honey. The play is set in George and Martha's living room which is lined by book shelves, which are empty. George is an academic with empty book shelves, some sort of comment there.
The play opens with George and Martha entering the room at 2:00 am after a faculty party at the college where George works as a history teacher. Martha's father is founder and president of the college. They have both been drinking and keep drinking though out the show, but they are not badly effected by alcohol. Martha is discontented and describes their house as a dump. George is ready for bed, but Martha tells him that they will soon have visitors - a new teacher and his wife. She says the teacher is in the maths department but we later discover that he is a biologist. His name is Nick, although he is not called by name in the play, and his wife is Honey. When they arrive it is clear that Honey is badly effected by her drinking at the party.
As Shirley Galloway noted in a review of the play: " ...the audience watches George and Martha tear savagely at each other with the knives of hurled words, sharpened on pain and aimed to draw blood...". George and Martha's relentless verbal attacks on each other gradually reveal their story and the reasons for their unhappiness.
Possibly the greatest cause of their distress does not appear in the play - Martha's father. Martha's mother died when she was young and Martha is very attached to her father - something of a "Daddy's Girl". She wants to make her father proud of her. The play is set in the 1950s so Martha is not able to gain her father's approval by her own achievements, she must do this via her husband's successes. Unfortunately George is a failure in his career. During the war he was briefly Head of the History Department while all of the other faculty members were away fighting. Sadly for George none of his colleagues were killed and when they returned he lost his position as Department Head. Martha wants the unconditional love of her father and a loving relationship with George, but she can't have both.
George is also struggling to deal with issues from his childhood, though these are presented with more ambiguity. In Act 2 he tells Nick a story of a teenage boy who accidentally kills his mother and later his father. Later in the play we hear that this was the main theme of a novel that George had written. According to Martha, George claimed that the boy was himself. George clearly has feelings of guilt towards his parents, and the novel was probably an attempt at penance to expiate his guilt. This was thwarted by Martha's father who argued that it was not the type of book to be published by a member of his college and so he stopped publication.
Both couples are damaged by their childlessness. To compensate George and Martha invent a child, but they have an agreement not to mention him in front of other people. Martha breaks that rule and tells Nick that their son will be home tomorrow for his 21st birthday. Both Martha and George have invested heavily in this imaginary child, but George decides to announce in front of Nick and Honey that the child has been killed. The manner of the death harks back to the plot of the novel. He probably does this to wound Martha and in that aim succeeds magnificently. But there is also possibly a positive motivation. It might well be that "killing" the child will allow George and Martha to create something real and wholesome in their relationship. This possibility is suggested in the final scene between George and Martha after Nick and Honey leave.
Nick and Honey's relationship is less clearly fleshed out. They obviously have serious difficulties. They have known each other since childhood, but Nick feels that he was trapped into marriage by a phantom pregnancy of Honey's. This version of the play leaves out a scene that indicates that their childlessness is caused by Honey's fear of having a child. Possibly, like all of the other characters in the play she is still emotionally a child. Her abuse of alcohol might relate to this. At the end of the play George and Martha might have gained some self-knowledge and may be able to move on to a more positive relationship. It is clear that this is not possible yet for Honey and Nick.
The performances were all superb. Alison Bell as Honey was particularly interesting, see photograph at right. The character laughs one minute, cries another, and then runs to the bathroom to puke. She is not a loud drunk, much of the time she is quite "out of it". As Bell says of Honey, in an interview is Scenes, Spring 2007, the "trick is trying to work out how aware she is of what's going on and how much passes her by". She adds, "Honey isn't lucid ... I have to put myself in that foggy place and take that extra time to respond and react".
It was a three and a half hour tour de force, and very well worth seeing.
Tuesday, September 4, 2007
Book Review: Two Lives
I bought and read this book for two reasons.
The first is that Vikram Seth is a fine writer. I have read and enjoyed two of his other books: The Golden Gate a verse novel set in California and An Equal Music a book set in the world of classical music.
Two Lives is a biography of two of Seth's relatives. I am writing a Family History with my parents so my second reason for reading the book was to see how Seth approached the task of biography.
The two lives described in the book are those of Seth's great-uncle Shanti and his great-aunt Henny. These are a somewhat unlikely couple. They both lived more that half their lives in England. Shanti was an Indian and Henny was a German of Jewish extraction.
The story of how they steered their lives through the tempestuous 1930s and 1940s to finally marry is an interesting one. To read a brief description click on the link below. Shanti decided, reluctantly, to take up dentistry at the urging of his brother, Raj who said, "In our family we have an engineer, an accountant, a judge and a doctor but no dentist. Why don't you train for that?" Shanti already had a B.Sc. qualification but was not confident of obtaining a job, so he applied to Paris and Berlin Universities in dentistry, and was offered places in both. He tried out Paris but he didn't enjoy France. He quickly went on to England where he was uncertain about studying in Germany as he did not understand the language. He went to Berlin, in July 1931, all the same and after accommodation at a number of different locations boarded with a Jewish family called Caros. The Caros had three children, Lola, Heinz and Henny. Shanti and Henny became friends but the relationship did not develop any further as Henny was being by Hans Mahnert, whose father was Henny's boss at work. Shanti proved his talent with languages and dentistry as he succeeded in passing his course although it was in a foreign language. He had to learn German as well as dentistry. He passed his exams in April 1936, a few months before the infamous Berlin Olympics. Shanti's problems were not over though as his German qualifications were not recognised in England. He spent a difficult year in the beautiful city of Edinburgh sitting exams every two weeks and finally achieved his British dental qualification. World War II brought tribulations for all of those whose stories are related in the book. None worse than for Henny's mother and sister, Lola. Henny kept up an intermittent communication with them, which stopped after November 1942. The author found documents listing people who were transported on cattle trains to the concentration camps. He discovered that they were both transported in mid May 1943. There is of cause no record of their death, along with millions of other people, but it is unlikely that they survived in the camps for long. Henny did not discover the fate of her mother and sister until after the war. Shanti joined the British Army as a dentist and saw action in North Africa and Italy. He corresponded regularly and affectionately with Henny. At the battle of Monte Casino in Italy he had his right arm blown off. Henny was working in England sharing with the English the privations of wartime Britain. After the war Shanti and Henny were seen by their friends as a couple and in 1949 they were engaged and in 1951 they were married. Shanti managed to develop a successful dental practice even though he had only one arm. Their lives had been buffeted severely by the events of the first half of the 20th Century. They lived quietly as a married couple in the more relaxed and less violent second half of the century. The structure of the book Seth could have taken a similar tack writing as a disinterested and non-intrusive author, but he chose another approach. He started the book with his arrival as a 17 year old to stay with Shanti and Henny. He come to England to continue his education. The first section is a brief biography of the author with regular reference to Shanti and Henny.It finishes with Shanti and Henny's deaths and describes the process of Vikram's decision to write the book. Section 2 describes Shanti's life up to the end of the war. Section 3 is about Hunny. When the author came to write about Henny she was already dead. The conditions in Germany before, during and after the war, are described in correspondence between Henny and her friends, most of whom were in Germany. This is a fairly harrowing section of the book. The last two parts 4 and 5 describe the married life of Shanti and Henny. The book comes full circle in the end and the author appears explicitly again. He describes visiting some of the locations described earlier in the book finishing with the house where Shanti and Henny spent their married lives. One theme of Seth's book and our family history is the intertwining of private and public lives. Although my parents did not live in circumstances as dangerous as Seth's uncle and aunty, their lives were effected by the great events in the world, particularly depression and war.
How they met
The rise of the Nazis formed a backdrop to Shanti's time in Berlin and the Nazi government was the reason for Shanti's departure from Germany late that year. He had been offered an academic position following the completion of his dissertation. (Incidentally, Henny's sister Lola typed the dissertation and corrected infidelities in his German.) This appointment was overturned by the Ministry of Education. Shanti reluctantly returned to England, only with hindsight realising how lucky he was.
Meanwhile conditions in Germany went from bad to worse for the Jewish Caro family. Heinz managed to leave for South America in 1938. Henny, who knew some English, was sponsored by a relative of her boyfriend Hans. She arrived in England just a month before the war started. Her mother and sister stayed behind in Berlin.
As I read Seth's book, I was interested in how he structured it. In the family history that I am writing, I do not appear as the author. Although I am the author the sections on my father and mother are written in their voice. Dad's section is called Eric's Story and Mum's section, Edna's story. I will appear later in Stephen's story.