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ns 26 jan 02
Is this the one? We may have found the cell that will revolutionise medicine
IT MIGHT turn out to be the most important cell ever discovered. It's a stem cell found in adults that can turn into every single tissue in the body.
Until now, only stem cells from early embryos were thought to be able to do this. If the finding is confirmed, it will mean cells from your own body could one day be turned into all sorts of perfectly matched replacement tissues and even organs.
If so, there would be no need to resort to therapeutic cloning-cloning people to get matching stem cells from the resulting embryos. Nor would you have to genetically engineer embryonic stem cells (ESCS) to create a "one cell fits all" line that doesn't trigger immune rejection. The discovery of such versatile adult stem cells will also fan the debate about whether embryonic stem cell research is justified.
"The work is very exciting," says Ihor Lemischka of Princeton University. "They can differentiate into pretty much everything that an embryonic stem cell can differentiate into." The cells were found in the bone marrow of adults by Catherine Verfaillie at the University of Minnesota. Extraordinary claims require extraordinary proof, and though the team has so far published little, a patent application seen by New Scientist shows the team has carried out extensive experiments. These confirm that the cells - dubbed multipotent adult progenitor cells, or MAPCs-have the same potential as ESCs. "It's very dramatic, the kinds of observations [Verfaillie] is reporting,' says Irving Weissman of Stanford University. 'The findings, if reproducible, are remarkable." At least two other labs claim to have found similar cells in mice, and one biotech company, MorphoGen Pharmaceuticals of San Diego, says it has found them in skin and muscle as well as human bone marrow. But Verfaillie's team appears to be the first to carry out the key experiments needed to back up the claim that these adult stem cells are as versatile as ESCS.
Verfaillie extracted the MAPCs from the bone marrow of mice, rats and humans in a series of stages. Cells that don't carry certain surface markers, or don't grow under certain conditions, are gradually eliminated, leaving a population rich in MAPCs. Verfaillie says her lab has reliably isolated the cells from about 70 per cent of the 100 or so human volunteers who donated maerrow samples.
The cells seem to grow indefinitely in culture, like ESCS. Some cell lines have been growing for almost two years and have kept their characteristics, with no signs of ageing, she says. Given the right conditions,
MAPCs can turn into a myriad of tissue types: muscle, cartilage, bone, liver and different types of neurons and brain cells. Crucially, using a technique called retroviral marking, Verfaillie has show,n that the descendants of a single cell can turn into all these different cell types-a key experiment in proving that MAPCs are truly versatile.
Also, Verfaillie's group has done the tests that are perhaps the gold standard in assessing a cell's plasticity. She placed single MAPCs from humans and mice into very early mouse embryos, when they are just a ball of cells. Analyses of mice born after the experiment reveal that a single MAPC can contribute to all the body's tissues.
MAPCs have many of the properties of ESCs, but they are not identical. Unlike ESCs, for example, they do not seem to form cancerous masses if you inject them into adults. This would obviously be highly desirable if confirmed.
"The data looks very good, it's very hard to find any flaws," says Lemischka. But it still has to be independently confirmed by other groups, he adds.
Meanwhile, there are some fundamental questions that must be answered, experts say. One is whether MAPCS really form functioning cells. Stem cells that differentiate may express markers characteristic of many different cell types, says Freda Miller of McGill University. But simply detecting markers for, say, neural tissue doesn't prove that a stem cell really has become a working neuron.
Verfaillie's findings also raise questions about the nature of stem cells. Her team thinks that MAPCs are rare cells present in the bone marrow that can be fished out through a series of enriching steps. But others think the selection process actually creates th@ MAPCS. "I don't think there is la cell' that is lurking there that can do this. I think that Catherine has found a way to produce a cell that can behave this way," says Neil Theise of New York University Medical School. Sylvia Pag6n Westphat, Boston
Alchemy for beginners To turn base metals into gold, first boil your nuclei
THE atomic nucleus behaves so much like a drop of liquid, it can actually boil, say physicists who have measured the properties of nuclear "vapour" for the first time. Their discovery is helping to explain how heavy elements such as gold, lead and uranium are made inside supernovae.
Strong nuclear forces hold protons and neutrons together in nuclei in much the same way that electromagnetic forces bind the molecules in a droplet of water. In nuclear reactions the minuscule "droplet" can spin, bulge or split, but until now no one had found a way to discover whether it can boil. "You can't stick a thermometer in the nucleus," James Elliott at Lawrence Berkeley National Laboratory in California points out. Now Vic Viola of Indiana University in Bloomington and his colleagues have cracked the problem. At Brookhaven National Laboratory in New York, they accelerated particles called pions to 99.9 per cent of the speed of light and smashed them into gold nuclei. By looking at the size of the nuclear blobs that were thrown out, they were able to measure the nuclei's transition from liquid to vapour.
As the energy of the pions increased, so did the size of the blobs. But eventually they stopped getting bigger-showing that the additional energy was being used to change the state of the nuclei from liquid to gas. And when the researchers cranked up the energy even further, the chunks suddenly got smallel This indicates that all the nuclei have been vaporised, says Viola. 'If you vaporise a drop of water and look at the gas coming off, you see small clusters of just two or three molecules," he says.
When researchers at Lawrence Berkeley and Michigan State University used Viola's data to calculate the boiling points of different nuclei, they found they are typically billions of times greater than those of atoms, around 100 billion degrees kelvin. And when they measured the density of the nuclear vapour, its pressure was proportional to its temperature, just as in an ordinary gas.
This new understanding of nuclear matter is already helping other researchers to model supernovae-exploding stars that generate heavy elements such as gold from nuclei no heavier than iron. No one fully understands how the nuclei capture the extra neutrons to form heavy elements, but Chikako Ishizuka and colleagues at Hokkaido University in Sapporo, Japan, say that when nuclei boil, it's easier for them to incorporate extra neutrons. Then they can condense into heavy elements as they cool.
Viola says knowing how nuclei change from one state to another is crucial to understanding a range of processes. "Now we can describe nuclear physics under any conditions," he says. Eugenie Samuel More at: Physical Review Letters (vol 88, p 022701)
Keep young and beautiful Believe it or not there's already a drug that slows ageing
IT SEEMS like the ultimate elixir-a drug that extends your lifespan while maintaining your youthful health and vigour. What's more, in the US it's already approved for human use. There is just one snag: to reap the benefits, you have to be a fruit fly. Even so, the discovery promises to open up new leads in the quest to understand longevity and ageing, says Tom Kirkwood, an expert on ageing at the University of Newcastle upon @ne. 'What we are looking at here is a pretty fundamental mechanism."
This is the first time that simply feeding a drug to flies has made them live longer. And in a twist that contradicts theories on ageing, there seems to be no price to pay for this extra time. The flies are as healthy and fecund as their untreated peers.
Kyung-Tai Min and a team of researchers at the National Institutes of Health and the California Institute of Technology made the discovery by accident when they were testing a drug called 4-phenylbutyrate (PBA) on flies with neurodegenerative disease. They found that feeding the drug extended maximum lifespan of healthy flies by over 50 per cent, and their average lifespan by one-third. Intriguingly, higher doses of the drug were either toxic or less effective, hinting that you need to strike a delicate balance to maximise the repair mechanisms in cells.
Previous studies suggest that reduced fertility and semi-starvation can extend fruit flies' lifespan, so the team wondered whether PBA was mimicking these effects. But when they weighed the flies and counted their offspring, they found they were normal.
To test the flies' resistance to stress, they then starved them and fed them a chemical that generates free radicals. But far from having to pay for their longevity with a weaker constitution, the PBA flies survived better than the controls. 'We are going to test more, but so far, it seems they are perfect,' says Min.
Kirkwood, however, is sceptical. "The drug might be incur ring a cost that isn't visible yet,' he says. For instance, the pampered lab flies may have to eat more than usual to keep them going. 'They might not be so competitive in the wild."
PBA works its dramatic effects by blocking the activity of histone deacetylases, enzymes involved in switching genes on and off. Min found that 100 genes were switched on in response to PBA, including the one for superoxide dismutase, a protein well known for its anti ageing effects. About 50 others were switched off. Minis team are investigating these genes, and how PBA has its effect on the histone deacetylases.
So will we be popping anti-ageing pills any time soon? Kirk wood thinks not. "It's just not that easy to bring about changes which alter lifespan without deleterious effects,' he says.
Unlike flies, mammalian cells continue to divide and renew throughout life, so inducing large changes in gene expression may be risky. However, Min points out that PBA has been approved in the US for treating cystic fibrosis and sickle cell anaemia, and seems to have few side effects. He and his team will be testing the drug on mice very soon. Claire Ainsworth
More at: Proceedings of the Notional Academy of Sciences (vol
99, p 838)
Destroyer of worlds They bring death and destruction-and a fortune in gold
AN ASTEROID that wiped out huge swathes of life when it collided with Earth 360 million years ago may also have brought with it untapped mineral wealth.
Robert lasky of Western Australia's Department of Mineral and Petroleum Resources (DMPR) discovered the 120-kilometre wide Woodleigh impact crater accidentally as he was drilling for coal near Shark Bay, 650 kilometres north of Perth. When lasky and his colleagues drilled down into the crater, which is buried under up to 600 metres of rock and sand, they discovered a huge chunk of dense granite that had been dragged upwards as the ground rebounded when the asteroid hit.
Mineralogist Franco Pirajno of the DMPR has now analysed samples of the granite and was surprised to find that it contains valuable elements such as magnesium, copper, chromium and nickel, which is unusual in this kind of rock.
Instead of the asteroid being vaporised and spread through the air over a wide area, Pirajno concludes that many of its eompo-
nents must have been incorporated into the surrounding hot granite. The rock would have been crushed undeftpressures of around 100,000 atmospheres. "It must have created one hell of a bang," he says. Gold deposits may also have formed as streams of hot water ran through cracks in the rock.
But although the asteroid may have left behind a valuable legacy for mining companies, its impact at the time would have been devastating. Dating of the rock samples has revealed that the collision happened 360 million years ago-the time of a mass extinction towards the end of the Devonian period, when 85 per cent of all species were wiped out. Geologists have long speculated that an asteroid may have been responsible, but until now there was no known impact that could have triggered the catastrophe.
The Woodleigh asteroid certainly had enough destructive potential-it was about five kilometres wide, making it the fourth largest asteroid collision ever discovered on Earth, almost as big as the Mexican Chicxulub collision that's thought to have wiped out the dinosaurs 65 million years ago. The impact would have been felt globally, triggering earthquakes and volcanic activity, as well as creating a cloud of dust that spread around the world, blocking out sunlight.
Lasky believes more craters may lie hidden under Western Australia's red deserts and is keen to keep searching. But Pirajno says it is now up to the mining companies to investigate the economic potential of the Woodleigh site. Peter Hadfield, Sydney
Stay of execution Why one of the world's deadliest killers has won another reprieve
SMALLPOX is one of the biggest killers in history. But now we have it at our mercy, with the virus apparently surviving in only two labs, in the US and Russia. So why, instead of delivering the promised coup de grAce, did the World Health Organization last week vote to postpone its destruction?
The WHO says it is to allow research to continue. But as recently as last October, D. A. Henderson, who led the smallpox eradication campaign, argued that the last official stocks of virus should be destroyed. The live virus is useless for testing potential new drugs or vaccines, he wrote, as it only infects people, and people no longer get smallpox.
So what's changed? The answer, New Scientist has learned, is that Peter Jahrling's team at the US Army Medical Research Institute for Infectious Diseases at Fort Detrick, Maryland, has finally managed to infect a handful of cynomolgus macaques with smallpox. They do not yet have a perfect animal model for the disease. But the researchers hope they will soon be able to test new vaccines, drugs and diagnostic tools for smallpox.
Such progress helped the US persuade the WHO's executive council to vote to postpone destruction. Not that the US would have destroyed its stock even if the vote had gone the other way. This November, after the 11 September attacks and the anthrax scares, US Secretary of Health Tommy Thompson announced that the US would hang on to its stocks "until adequate medical tools are available to counter any future outbreak of this disease". Of course, if the official stocks really were all that's left of the virus, there would be no need for any defences once they were destroyed. After smallpox was officially eradicated in 1980, the WHO asked all countries to destroy their samples or send them to the official repositories in the US and Russia. It then planned to destroy these stocks. But in 1999, the US convinced WHO members to put off destruction until 2002, amid rising fears that other, clandestine stocks might be used as weapons, meaning live virus was needed for defensive research. "The evidence that anyone else has the virus is circumstantial," says Jonathan Tucker of the Monterey Institute of International Studies. Nevertheless, it is "highly likely", says Alan Zelicoff of Sandia National Laboratories in New Mexico. "It is not credible that all nations would give up their stocks at a UN request," he says. Antibodies in soldiers defecting ftom North Korea reveal recent vaccination against smallpox-probably, says Zelicoff, because North Korea has a smallpox weapon. Iraq is also implicated. And not all of the 100 tonnes of virus the Soviets weaponised per year in the 1980s may have been destroyed. While the existing vaccine is effective, says David Heymann, head of infectious diseases at the WHO, it is a live virus related to smallpox with rare but potentially fatal side effects. And it can make people with impaired immunity seriously ill. But you can only truly test a replacement for it against real smallpox virus. Enter the macaques. But there are problems. So far the animals only get sick after very large doses of virus, which makes testing vaccines difficult as enough bugs can defeat any vaccine. And health officials are looking for a drug that works after symptoms start-unlike cidofovir, the only known anti-smallpox drug. But most infected macaques die before developing full-blown symptoms like those in humans, making tests difficult. The system merely needs development, says Heymann. Whatever the fortunes of the team at USAMRIID, political pressures in the US are likely to ensure that this research will continue. Ironically, it will only be a success if it is never used. Debora MacKenzie
Daddy's girls A hint of your father tums a perfect stranger into an ideal mate
WOMEN seem to be drawn to men who smell like their father. In a test that involved women sniffing unwashed T-shirts, they appeared to unwittingly prefer the odour of men who have genes similar to their dad's.
This is no Freudian Oedipal complex. Instead, it appears to be a tactic in a poorly understood evolutionary game, where the prize is either greater resistance to disease among offspring, or an unconscious ability to spot distant relatives in a sea of strangers.
The genes in question form part of the major histocori3patibility complex, or MHC, which encode parts of the immune system. These genes are thought to be closely linked to others th,* dictate our natural odour.
These findings come as a surprise, because female mice are known to sniff out males with MHCs different from their own, preferring them to mates with a similar genetic make-up. And women were thought to do the same, according to a study in which women sniffed T-shirts that had been worn by men (New Scientist, 6 May 1995, p 19). But the new study paints a more complicated picture. Martha McClintock, Carole Ober and a team at the University of Chicago studied 49 women whose own and whose parents' MHC genes were known. The women sniffed T-shirts unrelated men had worn, as in the earlier study, but this time they had no idea what they were smelling. They were asked to say which odours they would prefer if they had to smell them all the time.
Surprisingly, the women preferred the odours of men who had some MHC genes, or alleles, in common with them. On average, the owner of a woman's most preferred odour shared 1.4 alleles with her, whereas the owner with the least appealing odour shared 0.6. What's more, the alleles that matched were those that the women had inherited from their fathers.
That goes against the prevailing theory that outbreeding-seeking mates who are genetically dissimilar to you-is always best. Going for a mate with different immune system genes from your own should ensure that your children have the biggest possible arsenal for attacking pathogens. Also, the rarer their MHC, the less likely it is that pathogens will have evolved to outsmart them.
But McClintock thinks that interpretation is too narrow. Limited inbreeding can work, as it may actually make sense to stick with combinations of genes that are proven to successfully fight disease. 'There's an intermediate number of matches that's probably optimal,' she says.
Wayne Potts of the University of Utah in Salt Lake City has a different explanation.
Although mice go for mates with different MHC genes, they prefer to share a nest with mice with a similar MHC genetic make-up, probably to ensure they are near their kin. Woman may be attracted to their father's odour for a similar reason-so they can home in-on relatives.
Potts says that Ober's own studies show women tend to marry MHC-dissimilar men (New Scientist, 10 February 2001, p 36). And marriage may be a more reliable indicator than old T-shirts, he points out. Alison Motiuk More at: Nature Genetics (DOI: 101038/ng8301
Bridge the world Now's your chance to find out how well connected you are
IF YOU get unexpected e-mail from an American university over the coming months, don't just assume it's junk. It could be from scientists investigating a fascinating social phenomenon.
According to urban folklore, everyone in the world knows everyone else via just a few intermediaries-an effect summed up by the phrase 'six degrees of separation". The number six emerged from an experiment performed in 1967 by the social psychologist Stanley Milgram, who sent packages to several hundred randomly selected people in America's Midwest, with the aim of getting them delivered to target people in Boston.
Each recipient was given some details about the target, such as their name and profession, and was asked to send the package to a personal acquaintance whom they believed was more likely to know the target personally.
Milgram discovered that on average the packages reached their targets after passing through astonishingly short chains, typically comprising just six people. In 1998, mathematicians Duncan Watts and Steven Strogatz at Cornell University showed that Milgram's finding can be explained by the 'small world effect", in which just a handful of people with very diverse friends can "short circuit" otherwise huge networks of acquaintances (New Scientist, 6 June 1998, p 7).
But attempts to replicate Milgram's findings have had mixed results-and in any case, the original experiment fell far short of proving that the "six degrees" effect holds true for the whole world. So a team at Columbia University is now using the Internet to attempt a global version.
Instead of a postal package, they are inviting people to use their network of acquaintances to get an e-mail message to targets spread across the world. According to Watts, who devised the experiment, e-mail is ideal for testing Milgram's claim as there are well over 100 million e-mail users worldwide. Only e-mails between genuine acquaintances will be deemed to complete a chain. People won't be allowed to short-circuit the sequence by just looking up the target's e-mail address.
Watts has set up a website giving full details about how to take part, and how to volunteer to act as a target. 'Ideally, we'd like to have, say, 100,000 people, each trying to reach around 20 targets," he says.
The team is keen to have as many people take part as possible, not least because they suspect people's mistrust of unsolicited e-mail might otherwise scupper their experiment. Early tests show that barely I in 4 e-mails are being passed on. With such a high rate of attrition, many thousands of people would have to take part to give much chance of even one chain of acquaintances reaching the target if Milgram's six degrees apply worldwide.
"Perhaps people can't be bothered to pass them on-or perhaps Milgram was just wrong," says Watts. "Either way, we need lots of people to take part so we can tell." Robert Matthews More at: hftp://smallwor[d.sociologycolumbia.edu/index.htm[
Men of God IT'S time to recant the orthodoxy that women are the more pious sex.
A new multi-faith StUdy of religious behaviour In Britain reveals that men are more likely than women to practise their religion-in every major faith bar Christianity.
That doesn't mean that women';s religious feelings or experiences ave weaker than men's, cautions psychologist Kate Loewervthal. The difference may reflect cultural influences. For instance, Muslims and Jews generally consider the religious role of men more important, and women are under less pressure to attend places of warship. In some cases they are actively discouraged-women are not supposed to enter a mosque ff they are menstrtating, for example. The traditional view that women are the more pious sex Is based on studies done almost exclusively on Christians, says Loewenthal. So her teem at Royal Holloway University of London studied 530 Christian, Hindu, Jewish and Muslim men and women about their religious pracuceThe team assessed activity such as prayer, study of religious texts and visits to a place of worship. "We found that in the non-Christian religions in Britain the men were more religious than the women,' says Loewenthal Daniel Batson, a prychologist at the University of Kansas, agrees that the findings could be due to cultural differences among the religions. But he cautions that British culture could also be having some unforeseen effect. "Europe is quite different in religion from the US," he says. "in Europe, religion is not seen as the bastion of culture that it once was." Betsy Mason More at: Personality and individual Differences (vol 32, p 133)
Power surge Renewable energy's latest contender is lurking by a coast near you
A TIDAL power station that taps the surging power of strong underwater currents will be tested for the first time this summer. In Britain, there are about 40 key locations around the coastlines where, in theory, there's enough energy in tidal streams to generate up to a quarter of the nation's electricity.
Wind creates waves on the sea surface, but as wind blows intermittently, wave power is quite unpredictable. But tides are regular, as they are caused by the gravitational pull of the Moon and Sun on large masses of water. If the local geography is right, ocean channels create fast-moving "tidal streams", where vast masses of rising or falling water are squeezed into a restricted space. But no one has proved that extracting energy from tidal streams is practical.
That could soon change. The British government is now looking for new energy sources to help cut carbon emissions. And this month, British offshore equipment company Engineering Business was given a fl.l million government grant to build a 1 SO-kilowatt prototype tidal power station. Dubbed Stingray, the machine should be installed between May and September on the seabed to the south-east of the Sound of Yell, off mainland Shetland. A pair of 15-metre-long hydroplanes, mounted on a stand, will oscillate with the tide to drive a hydraulic motor that generates electricity (see animation on the Web at www.engb. com/Pages/animation.htm).
Hydraulic pistons control the angle at which Stingray's hydroplanes face the tidal current to make the most of the onrushing water. Like an aircraft wing, their angle of attack-the angle at which it bites into the current-changes to create "lift", which pulls the hydroplane up and down. As they move, the hydroplanes yank on an attached arm that pumps high-pressure oil through a hydraulic motor, which turns an electric generator.
Although the design is still being finalised, the structure is expected to weigh 35 tonnes, rise to 20 metres above the seabed and work in currents of between 2 and 3 metres per second (4 to 6 knots). Most of it will be made of steel, though the hydroplanes could be cast in glassreinforced plastic.
Stingray may only work with the tide flowing in one direction, but its successors will swivel round or flip over four times a day so they can catch the tidal stream in both directions. Depending on the site, this should mean they generate power at least three-quarters of the time.
Engineering Business's managing director Tony Trapp told New Scientist he is "quietly confident" that Stingray will work, but stresses that its economics are less certain. He estimates that it will-generate electricity for between 4.7 and 12 pence per kilowatt-hour. Although this is more expensive than wind and nuclear power, it is comparable to wave power-and much more predictable.
Another contender in the tidal-stream game is an underwater windmill developed by Marine Current Turbines of London. A prototype that generates power using the circular motion of propeller-like turbines was originally due for installation off the coast of south-west England in 2000 (New Scientist, 20june 1998, p 38). But according to MCT's Peter Fraenkel, it was postponed because of delays in getting almost El million in government funding. Now he has the money, Fraenkel is planning to install a 300-kilowatt underwater windmill north-east of Lynmouth, Devon, in September. But both Fraenkel and Trapp insist they are not racing to be the first to generate tidal power. It is likely to be a multibillionpound industry, predicts Fraenkel. "There's room for both of us." Rob Edwards, Edinburgh
The Other Side of Zero
The opposite of infinity is a number so small that mathematicians almost missed it entirely. Good Job they didn't, says Ian Stewart
IT IS a number like no other. It is smaller than anything except zero, but it's not zero. It makes no logical sense, but it has endless uses. It's the infinitesimal, and it's back. Infinitesimally speaking, a circle is actually a polygon with infinitely many infinitesimal sides. A solid is in fact an infinite sandwich of infinitesimally thin slices. And velocity is an infinitesimal distance divided by an infinitesimal time. The whole world is made of these next-to-nothings.
Yet 19th-century purists found these tiny slices of nothing just too much to swallow, and they were banished from mathematics for more than a century. Only recently has the concept been restored to respectability, and it's back with a vengeance. Infinitesimals are now giving us insights in physics and simplifying our understanding of key areas of pure mathematics. But how on earth can such a peculiar notion make sense?
The idea of infinitesimals is an ancient one. Archimedes used them to calculate the volume of a sphere. He imagined cutting the sphere into infinitely many slices and hanging them on one arm of a balance. He rearranged them so that they exactly balanced a cone hanging on the other arm. Knowing how levers work, he related the volume of the sphere to that of the cone, and out popped his formula. It's a bizarre idea: if the slices are infinitely thin, how can they have any weight or volume? How can you add up a lot of zeros to get something substantial? Yet somehow it works.
The real arguments began after Newton devised his calculus, a way to calculate rates of change, such as velocities. He looked at the changing quantity (position, say) at two times separated by a very short interval, and calculated a formula for the average rate of change in that interval. Then to get the answer for a precise moment in time, rather than the average over a short interval, he shrank the interval in the formula to zero. That is, he did the calculation assuming it was non-zero, and then set it to zero. This vanishing quantity he called a 'fluxion" and, like Archimedes's slices, it worked,
But again the logic seems faulty. Over an interval of zero seconds, the position changes by zero, so the calculation becomes 0/0, and every mathematician knows that 0/0 can be anything you like.
This was too much for the bishop and philosopher George Berkeley, who in 1734 published a pamphlet called The Analyst, or a Discourse Addressed to an Infidel Mathematician ... (the title went on a bit). He sarcastically called Newton's fluxions "ghosts of departed quantities". His criticisms were spot on, but people kept on using calculus because it always gave sensible answers. It was more like magic than mathematics, but the spell worked.
Still, mathematicians in the 19th century were troubled. If a positive number is not zero yet is still smaller than any positive non-zero number, then it must be smaller than itself. That's impossible. So they found alternative ways to set up calculus, and called the resulting formalism "analysis" to emphasise its technically demanding foundations. Many people, especially school teachers and students of calculus, looked wistfully over their shoulders, because infinitesimals are much easier to handle than the rigours of analysis. But mathematicians were determined: infinitesimals could not possibly exist. That's because the basic ingredient of analysis is the concept of a "real" number. And in the technical mathematical sense of the word real, a real number is one that can be written as a decimal. Real numbers include all whole numbers and all fractions, together with more subtle numbers like 'A and 42. Crucially, no matter how small a real number is, there is always another number smaller than it-just divide 1 by some large enough number n. This property of reals is called the Archimedean axiom, and it leaves infinitesimals out in the cold, since it's impossible to imagine a real number that is like I divided by infinity. It would be a decimal with infinitely many zeros-in other words, zero itself. It was more than a century later, in the 1960s, before the logician Abraham Robinson came along and spotted a way round this argument. Why not simply accept that infinitesimals aren't, in the technical sense, real? At least six times in the history of mathematics, the meaning of the word number has been changed to accommodate new variants. Even fractions and negative numbers were once considered preposterous, but they proved indispensable and after a while no one batted an eyelid. Mathematicians could likewise accept and use infinitesimals, Robinson reasoned. There's no need to pretend that they are real numbers. So Robinson abandoned the Archimedean axiom and simply asked, what happens if we accept that there are numbers too small to be expressed as decimals? He worked out the arithmetic of such infinitesimals-how to do maths with them-and found that unlike Newton's paradoxical fluxions, they make perfect sense. You can add, subtract, multiply and divide infinitesimals, and even combine them with real numbers to produce "nonstandard" numbers. Adding an infinitesimal to a real number gives you a finite nonstandard number-"three and a vanishingly small bit", for example. Dividing a real number such as 1 by an infinitesimal produces infinity, as you might expect, but it is a specific infinity that makes perfect logical sense. What's more, an infinitesimal avoids the contradiction of having to be smaller than itself. lt only has to be smaller than all nonzero real numbers, and that's not a problem as they aren't real anyway. What Robinson ended up with was a new and consistent number structure that included all the usual real numbers and arithmetic operations, but supplemented them with the infinitesimals and infinite numbers. This new branch of maths, known as nonstandard analysis (NSA), cloaks each ordinary number with a cloud of nonstandard numbers, all closer to it than any other real number and differing from each other by an infinitesimal amount. It gives the real numbers an infinitesimal fuzz.
The upshot of Robinson's discovery is that approaches like that of Archimedes can be made rigorous. You really can find the volume of a sphere by adding together infinitely many infinitesimal slices. It's as though NSA gives you a knife sharp enough to cut it up that small and handle the pieces. This is close to our intuitive idea of why this method works. And if the test of a mathematical model is how easily it helps us understand the world, then infinitesimals pass with flying colours. In a way, it seems that the world really is made of infinitesimal pieces.
The infinitesimals and their other nonstandard companions are now coming into their own in many areas of mathematics and physics. Take the Boltzmann equation. This describes how a cloud of tiny particles, such as the molecules in a gas, changes in density as the particles move and collide. Physicists use computers to solve the equation, to predict gas flows in interstellar clouds, for example. The same equation can even be adapted to replace atoms with stars, and so predict the motions of stars in a galaxy.
But the equation held a dark secret. No one had proved that it wouldn't go haywire somewhere. That is, you couldn't be sure that for some new situation the equation wouldn't predict points of infinite density, or some other absurdity. And if you can't even rely on that much, you can't trust numerical approximations on a computer.
No one has yet found a "classical" proof that the equation always has neat solutions, because the huge numbers of particles that could be involved mean there are too many possible situations to cover. But in 1984, Lars Arkeryd decided to use infinitesimals instead. By making the molecules infinitesimally small, he could use the mathematics of NSA to do the proof once and for all, and he succeeded in proving that the equation is well behaved after all. Physicists can now trust their simulations. A case where NSA has actually given physicists new results is Brownian motionthe apparently random movement of small bits of dust or smoke particles caused by molecules of the surrounding fluid buffeting them. This kind of unpredictable behaviour also d6cribes the movement of prices in stock markets, the movement of data in computer networks and many other situations, so if you find a model for one you can apply it to all the related areas. In the 1920s, Norbert Wiener of MIT discovered how to calculate some of the characteristics of Brownian motion, by considering the average properties of the fluid as if it were a continuous substance-smoothing out the molecules into a continuous goo. But Wiener's approach was extremely technical. Worse, it only revealed the general properties of Brownian motion. There is nothing in his theory that corresponds to the trajectory of an actual bit of grit. It's like having a theory of the Solar System in which there are no planets and no orbits. In 1976, Robert Anderson discovered a much better and simpler approach to the problem using NSA. He started by modelling the whole problem rather like a game of 3D chess on a very large chessboard. The model chops space up into cubes, some of which contain a piece that represents a molecule. At each tick of a clock the pieces move randomly, either north, south, east, west, up or down. Because this system is discrete, the calculations can be done combinatorially-that is, by counting things. This avoids all the technicalities of Wiener's approach, and it's relatively easy to work out what's going on.
Of course, real molecules move continuously rather than on a lattice. But you can easily model this with infinitesimals: make your chessboard from infinitely many infinitesimal squares. The successive moves are separated by a fixed, but infinitesimal, interval of time. This way Anderson could work out real trajectories for single dust particles, or single shares on the stock market.
Much more recently, some of the oddest aspects of infinitesimals have been appearing in pure maths. In 2000, Viadimir Kanove of Moscow University and Michael Reeken of Wuppertal University in Germany published a nonstandard proof of the Jordan curve theorem. This theorem states that every joined-up or "closed" curve divides the plane into two distinct regions, an inside and an outside. It may sound obvious, but it's not. Curves can be very complicatedfor example, a spiral embellished with many smaller spirals and so on forever. Proving the Jordan theorem for such a curve is anything but straightforward.
It is straightforward, however, if the curve is a polygon. To tell if a point is inside or outside, simply draw a straight line away from the point far enough to get well away from the polygon. If the line crosses the polygon an odd number of times then your point is inside. If it crosses an even number of times-such as zero-then it is outside.
Kanove and Reeken have shown that you can approximate any closed curve by a single polygon with infinitely many infinitesimal sides. This differs from the original curve by an infinitesimal amount, but now you can use the same odd/even argument to prove the theorem. The twist is that you might cross the polygon an infinite number of times, but that's airight. With NSA you can tell if an infinite integer is odd or even: if you can make it by multiplying some other nonstandard infinite integer by two, then it's even. If you have to double another infinity and then add one, it's odd.
If all that is too abstruse for you, consider something more everyday: computer graphics. A monitor screen is composed of finitely many tiny rectangular pixels, but it can be modelled instead as a lattice with infinitely many infinitesimal pixels. jeanPierre [email protected] of Lois Pasteur University in Strasbourg has shown that this has many advantages, especially when you want to rotate an image through some angle.
In the ordinary case, when you rotate a finite lattice of points through some angle, they don't usually fit very neatly into the original grid of pixels you have on your screen, so working out the formula for how to do it is tricky. But if you instead work out how to do the rotation for an infinity of pixels, then you can ifistantly adapt that formula to rotate a finite number, instead of having to do the calculations from scratch each time. Other manipulations can also be simplified this way, and infinitesimals could one day be behind incredible special effects and computer games.
And perhaps more than mere games. juha Oikkonen of the University of Helsinki reckons that NSA will be able to answer questions about what a computer can in principle do. "In theoretical computer science, one studies extremely complicated finite situations," he says. The limits of computing power may be related to the passage between the infinite and the finite, and Oikkonen's guess is that infinitesimals may be the way to explore this borderline. But for now it is only a guess.
Admittedly, infinitesimals take a little getting used to. These slices of nothingness might seem like nonsense at first, but in mathematics you should never give up on a good idea just because it doesn't make sense, El