Jesus may not have died nailed to the cross because there is no evidence that the Romans crucified prisoners two thousand years ago, a scholar has claimed.
The legend of his execution is based on the traditions of the Christian church and artistic illustrations rather than antique texts, according to theologian Gunnar Samuelsson.
He claims the Bible has been misinterpreted as there are no explicit references the use of nails or to crucifixion - only that Jesus bore a "staurus" towards Calvary which is not necessarily a cross but can also mean a "pole".
Mr Samuelsson, who has written a 400-page thesis after studying the original texts, said: "The problem is descriptions of crucifixions are remarkably absent in the antique literature.
"The sources where you would expect to find support for the established understanding of the event really don't say anything."
The ancient Greek, Latin and Hebrew literature from Homer to the first century AD describe an arsenal of suspension punishments but none mention "crosses" or "crucifixion."
Mr Samuelsson, of Gothenburg University, said: "Consequently, the contemporary understanding of crucifixion as a punishment is severely challenged.
"And what's even more challenging is the same can be concluded about the accounts of the crucifixion of Jesus. The New Testament doesn't say as much as we'd like to believe."
Any evidence that Jesus was left to die after being nailed to a cross is strikingly sparse - both in the ancient pre-Christian and extra-Biblical literature as well as The Bible.
Mr Samuelsson, a committed Christian himself, admitted his claims are so close to the heart of his faith that it is easy to react emotionally instead of logically.
Mr Samuelsson said the actual execution texts do not describe how Christ was attached to the execution device.
He said: "This is the heart of the problem. The text of the passion narratives is not that exact and information loaded, as we Christians sometimes want it to be."
Mr Samuelsson said: "If you are looking for texts that depict the act of nailing persons to a cross you will not find any beside the Gospels."
A lot of contemporary literature all use the same vague terminology - including the Latin accounts.
Nor does the Latin word crux automatically refer to a cross while patibulum refer to the cross-beam. Both words are used in a wider sense that that.
Mr Samuelsson said: "That a man named Jesus existed in that part of the world and in that time is well-documented. He left a rather good foot-print in the literature of the time.
"I do believe that the mentioned man is the son of God. My suggestion is not that Christians should reject or doubt the biblical text.
"My suggestion is that we should read the text as it is, not as we think it is. We should read on the lines, not between the lines. The text of the Bible is sufficient. We do not need to add anything."
BNN June 27, 2010
Monday, 5 July 2010
Monday, 14 June 2010
The pope and a puzzling African king
By Alex Beam, Globe Columnist | August 4, 2005
It is traditional for newly elected popes to adopt a new coat of arms, displaying heraldic imagery symbolizing their heritage or their attitudes toward the church. It has been widely noted, for instance, that a bishop's hat, or miter, sits atop Pope Benedict XVI's new coat of arms, and not a crown. That is viewed as an expression of the new pope's humility, that he comes not to reign over the church but to continue his work as one of God's servants.
Another prominent feature of the pope's new crest has also attracted attention: the picture of the ''African king" facing left on the coat of arms. For one thing, the portrait is practically a caricature of an African male, with exaggerated lips painted ruby red. ''It's not good," says Holy Cross professor of religious studies Matthew Schmalz, who has written about the crest for the Catholic magazine Commonweal.
There is little doubt why the image of the African king appears on the crest. It symbolizes the Pope's commitment to rallying Catholic worshipers in Africa, the fastest-growing province of the church. (In June, Benedict said he would summon a special synod of African bishops, the first since 1994.) ''For me, [the African king] is an expression of the universality of the Church," the then-Cardinal Ratzinger wrote in his 1998 book ''Milestones: Memoirs 1927-1977." He also wrote that he did not know where the African image, which appeared on his coat of arms when he was archbishop of Munich-Freising, came from.
He is not alone. No one really knows who the African king might be. ''It probably strikes people as being really odd," Schmalz says. ''They look at it and think, 'What is going on here?' "
Mario Valdes, a researcher for WGBH's ''Frontline," strongly believes the black king can be one of only two people: Balthazar, one of the three kings from the ''Orient," who brings myrrh to the newborn Jesus Christ; or Prester John, the mythological king of the ''three Indias," or, alternatively, Ethiopia. Prester John, who in some versions of his legend is white, was supposed to have ruled over a land where spiritual and temporal powers coexisted in perfect harmony. Pope Alexander III is said to have written a letter to Prester John in 1177, although the Catholic Encyclopedia casts doubt on this event.
But there are two other possible identities for the unknown Moor. He could be St. Maurice, a Roman commander from Africa whose Christian soldiers refused to sacrifice to the pagan gods after an important victory, and were themselves massacred. (Valdes discounts this, noting that Maurice is usually depicted wearing a centurion's helmet.) And there is a more grisly possibility. At the time of the Crusades, some Christian kings displayed a severed Moor's head on their flags or crests to symbolize victories over their Islamic enemies. It is conceivable that the king, known as the ''Moor of Freising," evolved from such an image, although the figure shown on Benedict's coat of arms is wearing a collar and has suffered no violence. ''It's certainly one of the possibilities," Schmalz says.
'Twas ever thus
In a recent outing, Berkshire Eagle columnist Howard Herman reviewed the first season of the Pittsfield Dukes, the New England Collegiate Baseball League team owned by former Red Sox general manager Dan Duquette. Herman noted that ''Subtract the on-field aspect of the 2005 NECBL season, and the Dukes were a roaring success." The Dukes finished last in their division, with a won-lost record of 11-31.
At the end of his column, Herman includes this quote from NECBL executive vice president Joel Cooney: ''[Duquette's] going through a learning curve in how to recruit players. [The record] is not a direct reflection on the organization. If you give him time, he will definitely turn it around."
Alex Beam is a Globe columnist. His e-dress is beam@globe.com
Wednesday, 12 May 2010
A lesson in displacement
When an ice cube melts in a glass of water, does the level of the water fall or rise?
Neither. Archimedes' principle states that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid. When the ice cube is floating, the buoyant force is equal to the weight of the ice cube, but since the buoyant force is equal to the weight of the water displaced by the submerged portion of the ice cube, the floating ice cube displaces its own weight in water. When the ice cube melts, it turns into its own weight in water, thereby leaving the overall water level unchanged.
Neither. Archimedes' principle states that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid. When the ice cube is floating, the buoyant force is equal to the weight of the ice cube, but since the buoyant force is equal to the weight of the water displaced by the submerged portion of the ice cube, the floating ice cube displaces its own weight in water. When the ice cube melts, it turns into its own weight in water, thereby leaving the overall water level unchanged.
Saturday, 6 March 2010
Swimming with Dolphins: Green or Not?
When is it not okay to swim with dolphins, and why?
Swim with the Dolphins (SWTD) programs at resorts and get-aways seems like so much fun, right? How neat it is to be able to get in the water and play with these smart, wonderful mammals! So often, it seems like a great, green activity - after all, you're hopping in a lagoon or well-cared-for tank and hanging out with healthy, loved animals. The United States alone has between 14 and 18 SWTD attractions, and the NMFS estimated that in 1990 over 40,000 people swam with captive dolphins in the US, with the number dramatically increasing over recent years.
But there's a few big problems with these programs that take the green shine right off the activity. If you're thinking of adding swimming with dolphins to your vacation itinerary, check out why you may want to reconsider the idea.
1. Harmful and Traumatic Round-Up Techniques
As shown in the documentary "The Cove," dolphins that end up in SWTD programs are often caught under the most horrifying of circumstances. In Japan, tens of thousands of dolphins are rounded up into a small cove. Trainers come to pick out which dolphins they want for various aquatic centers. The dolphins are then rounded up as the other members of their pod are slaughtered. Not all capture techniques are so abusive, but they are all always traumatic for the dolphins, often causing a fatal condition known as capture stress or capture myopathy. According to the World Society for the Protection of Animals, of those dolphins that survive the capture and are brought into captivity, 53% will die within their first 3 months in a tank. Captive breeding programs, on the other hand, avoid the trauma of capture, but often do little to ensure that capture of wild dolphins ends.
2. Unnatural Habitats and Living Situations
Wild dolphins swim upwards of 40 miles a day with their family pods, hunt for their own food, and spend only 20% of their time at the surface of the ocean. On the other hand, captive dolphins swim in circles, are fed fish already caught and killed for them, are no longer part of their family pod - and dolphins are extremely social animals - and spend around 80% of their time at the surface. The environment, no matter how clean, goes against a dolphin's basic instincts and is usually not nearly stimulating enough to keep the animals from suffering stress and boredom.
Additionally, the conditions of aquatic centers where SWTD programs exist are barely regulated in the US, and often not regulated in other countries. From the numbers known, every seven years, at least 50% of all captive dolphins die due to the violence of their capture, intestinal disease, chlorine poisoning and stress-related illness. Even Sea World reports an average of 3 dolphin deaths per year.
3. Ensuring Continued Capture of Wild Dolphins
While some aquatic centers have breeding programs and express the desire to promote dolphin conservation, the fact is dolphins are taken in large numbers from the wild to support SWTD programs. The practice is ultimately inhumane, which is why Brazil, Italy, and the U.K. have banned interactive marine-mammal programs. It may seem like a great green activity while on vacation, but in the larger scheme of ocean and animal health, SWTD programs aren't so eco-friendly.
What You Can Do!
The Humane Society of the United States suggests:
Write or visit SWTD attractions and express your concerns. There are currently almost 400 bottlenose dolphins in captivity in the United States, with many more in facilities around the world, many of whom are used in breeding programs. If these programs are as successful as captive facilities claim, then capture from the wild can and should be eliminated.
Consider bypassing hotels, resorts, and cruise lines that offer SWTD attractions to tourists. Let the facility know that while you would normally have booked your stay with them, you see that they support this un-green activity and are therefore booking elsewhere.
Contact your U.S. senators and representative and tell them to support an amendment to the Marine Mammal Protection Act prohibiting the capture of marine mammals from the wild for public display during the Act's next re-authorization.
By Jaymi Heimbuch
San Francisco, CA, USA Thu Aug 6, 2009
Swim with the Dolphins (SWTD) programs at resorts and get-aways seems like so much fun, right? How neat it is to be able to get in the water and play with these smart, wonderful mammals! So often, it seems like a great, green activity - after all, you're hopping in a lagoon or well-cared-for tank and hanging out with healthy, loved animals. The United States alone has between 14 and 18 SWTD attractions, and the NMFS estimated that in 1990 over 40,000 people swam with captive dolphins in the US, with the number dramatically increasing over recent years.
But there's a few big problems with these programs that take the green shine right off the activity. If you're thinking of adding swimming with dolphins to your vacation itinerary, check out why you may want to reconsider the idea.
1. Harmful and Traumatic Round-Up Techniques
As shown in the documentary "The Cove," dolphins that end up in SWTD programs are often caught under the most horrifying of circumstances. In Japan, tens of thousands of dolphins are rounded up into a small cove. Trainers come to pick out which dolphins they want for various aquatic centers. The dolphins are then rounded up as the other members of their pod are slaughtered. Not all capture techniques are so abusive, but they are all always traumatic for the dolphins, often causing a fatal condition known as capture stress or capture myopathy. According to the World Society for the Protection of Animals, of those dolphins that survive the capture and are brought into captivity, 53% will die within their first 3 months in a tank. Captive breeding programs, on the other hand, avoid the trauma of capture, but often do little to ensure that capture of wild dolphins ends.
2. Unnatural Habitats and Living Situations
Wild dolphins swim upwards of 40 miles a day with their family pods, hunt for their own food, and spend only 20% of their time at the surface of the ocean. On the other hand, captive dolphins swim in circles, are fed fish already caught and killed for them, are no longer part of their family pod - and dolphins are extremely social animals - and spend around 80% of their time at the surface. The environment, no matter how clean, goes against a dolphin's basic instincts and is usually not nearly stimulating enough to keep the animals from suffering stress and boredom.
Additionally, the conditions of aquatic centers where SWTD programs exist are barely regulated in the US, and often not regulated in other countries. From the numbers known, every seven years, at least 50% of all captive dolphins die due to the violence of their capture, intestinal disease, chlorine poisoning and stress-related illness. Even Sea World reports an average of 3 dolphin deaths per year.
3. Ensuring Continued Capture of Wild Dolphins
While some aquatic centers have breeding programs and express the desire to promote dolphin conservation, the fact is dolphins are taken in large numbers from the wild to support SWTD programs. The practice is ultimately inhumane, which is why Brazil, Italy, and the U.K. have banned interactive marine-mammal programs. It may seem like a great green activity while on vacation, but in the larger scheme of ocean and animal health, SWTD programs aren't so eco-friendly.
What You Can Do!
The Humane Society of the United States suggests:
Write or visit SWTD attractions and express your concerns. There are currently almost 400 bottlenose dolphins in captivity in the United States, with many more in facilities around the world, many of whom are used in breeding programs. If these programs are as successful as captive facilities claim, then capture from the wild can and should be eliminated.
Consider bypassing hotels, resorts, and cruise lines that offer SWTD attractions to tourists. Let the facility know that while you would normally have booked your stay with them, you see that they support this un-green activity and are therefore booking elsewhere.
Contact your U.S. senators and representative and tell them to support an amendment to the Marine Mammal Protection Act prohibiting the capture of marine mammals from the wild for public display during the Act's next re-authorization.
By Jaymi Heimbuch
San Francisco, CA, USA Thu Aug 6, 2009
Tuesday, 9 February 2010
‘Hypocritical’ EU gives €34.5m to fleets fishing tuna to extinction
The European Union has given out tens of millions of euros to subsidise the Mediterranean tuna fishing fleets despite warnings from scientists that overfishing is pushing the species close to extinction.
Between 2000 and 2008 a total of €34.5 million (£31.4 million) was given by the EU to support the fishing fleets, Joe Borg, the European Commissioner, revealed after a parliamentary question from a Spanish MEP.
“I am shocked at the scale of the subsidies given to the bluefin fleet,” said Raül Romeva i Rueda, who represents the European Green Party. “This shows clearly the hypocrisy of the EU, which insists on the need to conserve fish stocks while simultaneously encouraging the rapid expansion of a fleet that was already too large.”
Spain received more than half of the subsidy, with French and Italian fleets the next biggest beneficiaries. Cyprus, Malta and Greece were also given money.
Over the eight-year period, €23 million was given to fund the construction of new boats, including ultra-modern purse seiners that are able to land 100 tonnes in one haul. A further €10.5 million was given to modernise existing vessels, increasing their ability to track down and catch the tuna. Only €1 million was used to decommission vessels, but mainly for small-scale, local boats.
Overcapacity has been a problem in the world’s fishing fleet, with too many boats chasing too few fish. According to the European Commission, EU vessels are able to catch almost 21,900 tonnes of tuna a year, approaching twice the EU’s 2009 quota of 12,400 tonnes.
Stocks of the Eastern Atlantic bluefin tuna are at risk after more than half a century of overfishing.
The International Council for the Conservation of Atlantic Tuna was established in 1969 after concerns that the species was being fished unsustainably when the fish came to spawn in the Mediterranean.
Since 1955 populations have shrunk to a quarter of their former size, with the bulk of the reduction occurring since 2002. Between 2001 and the present, the average size of bluefin tuna has shrunk by half.
In October the organisation’s scientists found that the stock was below 15 per cent of its pre-exploitation levels, qualifying it for a ban on trade via the Convention on International Trade in Endangered Species (CITES). It is not known if the EU will support a proposal from Monaco to ban the trade when CITES meets in March next year.
“We’ve always suspected the amount of public subsidy was very high, but until now it’s been very hard to get a good picture,” said Sergi Tudela of WWF. “These figures are not even complete yet. This is just the funding from Brussels, and the figures do not include the national subsidies, which in many cases equal them. It’s a real scandal,” he said.
“The EU has now committed to reducing overcapacity, but we’re going to have to pay again for that. We’ve paid once to make these ships that have been used to make a few people rich. They’ve destroyed the bluefin — a common stock — and now they are going to ask for more money.”
The Times December 4, 2009
Between 2000 and 2008 a total of €34.5 million (£31.4 million) was given by the EU to support the fishing fleets, Joe Borg, the European Commissioner, revealed after a parliamentary question from a Spanish MEP.
“I am shocked at the scale of the subsidies given to the bluefin fleet,” said Raül Romeva i Rueda, who represents the European Green Party. “This shows clearly the hypocrisy of the EU, which insists on the need to conserve fish stocks while simultaneously encouraging the rapid expansion of a fleet that was already too large.”
Spain received more than half of the subsidy, with French and Italian fleets the next biggest beneficiaries. Cyprus, Malta and Greece were also given money.
Over the eight-year period, €23 million was given to fund the construction of new boats, including ultra-modern purse seiners that are able to land 100 tonnes in one haul. A further €10.5 million was given to modernise existing vessels, increasing their ability to track down and catch the tuna. Only €1 million was used to decommission vessels, but mainly for small-scale, local boats.
Overcapacity has been a problem in the world’s fishing fleet, with too many boats chasing too few fish. According to the European Commission, EU vessels are able to catch almost 21,900 tonnes of tuna a year, approaching twice the EU’s 2009 quota of 12,400 tonnes.
Stocks of the Eastern Atlantic bluefin tuna are at risk after more than half a century of overfishing.
The International Council for the Conservation of Atlantic Tuna was established in 1969 after concerns that the species was being fished unsustainably when the fish came to spawn in the Mediterranean.
Since 1955 populations have shrunk to a quarter of their former size, with the bulk of the reduction occurring since 2002. Between 2001 and the present, the average size of bluefin tuna has shrunk by half.
In October the organisation’s scientists found that the stock was below 15 per cent of its pre-exploitation levels, qualifying it for a ban on trade via the Convention on International Trade in Endangered Species (CITES). It is not known if the EU will support a proposal from Monaco to ban the trade when CITES meets in March next year.
“We’ve always suspected the amount of public subsidy was very high, but until now it’s been very hard to get a good picture,” said Sergi Tudela of WWF. “These figures are not even complete yet. This is just the funding from Brussels, and the figures do not include the national subsidies, which in many cases equal them. It’s a real scandal,” he said.
“The EU has now committed to reducing overcapacity, but we’re going to have to pay again for that. We’ve paid once to make these ships that have been used to make a few people rich. They’ve destroyed the bluefin — a common stock — and now they are going to ask for more money.”
The Times December 4, 2009
Tuesday, 26 January 2010
Know Your Champagne
New Year's Eve means champagne. And nothing loosens up a crowd (read: the ladies) faster than a little bubbly. Whether you'll be drinking it all night, or just for a private midnight toast for two, you'll need to know what, and how much, to get. And what to do once you get it.
Here are five things every guy should know about champagne:
1 - What is Champagne?
All champagne is sparkling wine. But not all sparkling wine is champagne. The difference? True champagne is produced in the Champagne region in northeastern France. If it's made in California from champagne grapes it cannot be called "champagne". It has to be called "sparkling wine."
When buying a sparkling wine look for the words "methode champenoise", meaning the wine has undergone a second fermentation in the bottle. This is what creates the bubbles. Lesser quality sparkling wines create bubbles by adding carbonation. Just like 7-Up. Enough said.
Most champagne is made as a "cuvee", meaning it's a blend of three kinds of grapes: two reds and a white, usually Chardonnay, Pinot Noir and Pinot Meunier.
If you see "Blanc de Blancs" on the label that means the champagne is made entirely of white grapes and will have a lighter, more elegant taste. "Blanc de Noirs" means it is a white wine made from red grapes, like Pinot Noir, making the wine more full bodied, and giving it a golden color. "Rose" is made by blending in a little red wine and gives the wine a pink color. (Don't equate this with cheap rose wines. Rose sparkling wines are not lesser in quality.)
2 - What Does "Brut" Mean?
The level of sugar in sparkling wine determines if it will be dry or sweet. "Brut" is very dry, and is considered the standard for fine champagne. The scale runs as follows:
Extra Brut (Too dry for most people)
Brut
Extra Dry (Which is not as dry as Brut, so keep that in mind)
Sec (this is where it begins to get sweet)
Demi-Sec (usually served with dessert and wedding cake)
Doux (could put you in a diabetic coma)
3 - How Much Should I Spend?
This is a matter of personal preference. Rappers sing the praises of Crystal, and wouldn't be caught by the paparazzi drinking anything less. But they can afford the $300+ price tag. You, being the middle class warrior you are, need to be more realistic. You can get a great bottle of California sparkling wine - like Mumm Cuvee Napa NV or Roederer Estate Brut NV - for about $20. And when you're buying five or six bottles for a party, that's a helluva lot easier to handle.
4 - How Much Should I Get?
Depends on how loose you want your guests. The average bottle serves six glasses. If your party is going all night, figure about a bottle per person. If you're not partying that hard, figure 2-3 glasses, or half a bottle, per person.
Guy at the liquor store trying to talk you into buying a magnum? And you don't know what a "magnum" is? Easy. It's a bottle equivalent to two regular bottles of champagne. If you really want to impress your guests, get a Jeroboam. That big boy is equivalent to four bottles. And women love a guy with a big bottle.
5 - How Do I Serve It?
One word: cold. Champagne is meant to be served between 43 and 48 degrees Fahrenheit. Which means you should put it in the fridge for 2-3 hours before the party. Don't have that kind of time? Never put champagne in the freezer. The contents are under pressure and the bottle tends to explode in freezers. And you don't want to pop prematurely. You can chill it in an ice bucket filled with half ice and half water for about a half hour. And if you sprang for the Crystal and don't want the label soaking off in the water before your guests can see how much you spent? Don't worry. Champagne labels are adhered with waterproof glue. The label isn't going anywhere.
Once the bottle is open, (and a detailed explanation of how to open a champagne bottle is coming next), pour into thin flutes about three quarters full. Never fill the glass to the top. And never use wide, shallow glasses. Flutes preserve the bubbles. And you know how she loves it when the bubbles tickle her nose.
Here are five things every guy should know about champagne:
1 - What is Champagne?
All champagne is sparkling wine. But not all sparkling wine is champagne. The difference? True champagne is produced in the Champagne region in northeastern France. If it's made in California from champagne grapes it cannot be called "champagne". It has to be called "sparkling wine."
When buying a sparkling wine look for the words "methode champenoise", meaning the wine has undergone a second fermentation in the bottle. This is what creates the bubbles. Lesser quality sparkling wines create bubbles by adding carbonation. Just like 7-Up. Enough said.
Most champagne is made as a "cuvee", meaning it's a blend of three kinds of grapes: two reds and a white, usually Chardonnay, Pinot Noir and Pinot Meunier.
If you see "Blanc de Blancs" on the label that means the champagne is made entirely of white grapes and will have a lighter, more elegant taste. "Blanc de Noirs" means it is a white wine made from red grapes, like Pinot Noir, making the wine more full bodied, and giving it a golden color. "Rose" is made by blending in a little red wine and gives the wine a pink color. (Don't equate this with cheap rose wines. Rose sparkling wines are not lesser in quality.)
2 - What Does "Brut" Mean?
The level of sugar in sparkling wine determines if it will be dry or sweet. "Brut" is very dry, and is considered the standard for fine champagne. The scale runs as follows:
Extra Brut (Too dry for most people)
Brut
Extra Dry (Which is not as dry as Brut, so keep that in mind)
Sec (this is where it begins to get sweet)
Demi-Sec (usually served with dessert and wedding cake)
Doux (could put you in a diabetic coma)
3 - How Much Should I Spend?
This is a matter of personal preference. Rappers sing the praises of Crystal, and wouldn't be caught by the paparazzi drinking anything less. But they can afford the $300+ price tag. You, being the middle class warrior you are, need to be more realistic. You can get a great bottle of California sparkling wine - like Mumm Cuvee Napa NV or Roederer Estate Brut NV - for about $20. And when you're buying five or six bottles for a party, that's a helluva lot easier to handle.
4 - How Much Should I Get?
Depends on how loose you want your guests. The average bottle serves six glasses. If your party is going all night, figure about a bottle per person. If you're not partying that hard, figure 2-3 glasses, or half a bottle, per person.
Guy at the liquor store trying to talk you into buying a magnum? And you don't know what a "magnum" is? Easy. It's a bottle equivalent to two regular bottles of champagne. If you really want to impress your guests, get a Jeroboam. That big boy is equivalent to four bottles. And women love a guy with a big bottle.
5 - How Do I Serve It?
One word: cold. Champagne is meant to be served between 43 and 48 degrees Fahrenheit. Which means you should put it in the fridge for 2-3 hours before the party. Don't have that kind of time? Never put champagne in the freezer. The contents are under pressure and the bottle tends to explode in freezers. And you don't want to pop prematurely. You can chill it in an ice bucket filled with half ice and half water for about a half hour. And if you sprang for the Crystal and don't want the label soaking off in the water before your guests can see how much you spent? Don't worry. Champagne labels are adhered with waterproof glue. The label isn't going anywhere.
Once the bottle is open, (and a detailed explanation of how to open a champagne bottle is coming next), pour into thin flutes about three quarters full. Never fill the glass to the top. And never use wide, shallow glasses. Flutes preserve the bubbles. And you know how she loves it when the bubbles tickle her nose.
Thursday, 14 January 2010
A History of Zero
One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.
The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)
Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.
One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.
The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.
If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.
We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.
Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.
Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.
The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-
... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.
What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.
In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.
We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.
Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
Subtraction is a little harder:-
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.
In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-
... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.
However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-
A number remains unchanged when divided by zero.
Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.
You can see a separate article about Mayan mathematics.
The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-
If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.
The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-
An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.
In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.
One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.
Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!
Article by: J J O'Connor and E F Robertson
The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)
Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.
One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.
The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.
If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.
We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.
Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.
Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.
The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-
... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.
What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.
In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.
We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.
Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
Subtraction is a little harder:-
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.
In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-
... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.
However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-
A number remains unchanged when divided by zero.
Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.
You can see a separate article about Mayan mathematics.
The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-
If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.
The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-
An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.
In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.
One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.
Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!
Article by: J J O'Connor and E F Robertson
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