Science begets science. In a letter to fellow natural philosopher Robert Hooke in 1676, Isaac Newton famously decreed that his own achievements were merely a matter of "standing on the shoulders of giants".
The more we know about something, the more we can study it, whether it's particles firing in a Swiss bunker, as with Geneva's Large Hadron Collider, or Newton's fabled fleshy fruit toppling from a tree.
Scientists have been examining their own careers for centuries, but only relatively recently as a separate field of research. This intellectual analysis, called "scientometrics", emerged in the 1960s, and is essentially the "science of science". It posits questions such as, "How is productivity changing?" or "How many researchers do we need?" and now, "Are scientific discoveries getting more difficult?"
This latter poser arrived in May, courtesy of Samuel Arbesman, the Harvard postdoctoral fellow, journalist and evangelist for academe, writing in the journal Scientometrics. His conclusions? That Newton wasn't being modest: his experiments really were a breeze.
Newton and his peers studied springs and apples; now we need supercomputer networks to check out broadly the same things. In an objective sense, says Arbesman, science is getting harder. "Today, if you want to make a discovery in physics, it helps to be part of a 10,000-member team that runs a multi-billion dollar atom smasher," he says. "It takes increasingly more money, more effort, and more people to find out more things."
So what is scientometrics, and what else can it tell us? In simple terms, scientometrics is an "information science" that uses statistical techniques to put scientists under the microscope. Interested in how productive a certain university is? Proceed as follows: work out how many scientists it has, deduce how productive each of those brain-boxes is (why not rack up how often their research gets published?), add up all this output, and you have a reasonable means of quantifying a university's performance.
Much of modern scientometrics is based on the work of London-born information scientist Derek Price. He is best known for his 1963 book Little Science, Big Science, which made the distinction between the cottage-industry-sized experiments of the immediate post-war years and huge, international projects with budgets of billions of dollars. He argued that modern science had migrated from the former to the latter.
You can use scientometric techniques to analyse the reach of Price's own work. He wrote or edited 14 books and around 240 papers before his death in 1983. By 1987, his work had been cited in at least 2,200 articles, putting him in the top 1 per cent of cited authors at that time.
"Once of the first quantities to be studied in the field of scientometrics was the number of scientists over time," says Arbesman.
"The first PhDs in the United States were granted by Yale University in 1861. Since then, the number of scientists in the US and throughout the world has increased rapidly, even exponentially in some cases, and the rate of growth has actually been faster than the growth of the general population."
Ambitious questions can be built on sturdy foundations. What are the effects of war on scientific discovery? What does cheap air travel do for the accessibility of remote research destinations? And these analyses can feed into public policy. Last year, the US government committed an increase of $21bn (£13.5bn) for science funding. France has recently announced a €35bn boost. Our own Government, meanwhile, has a comprehensive spending review due in October. Scientometricists say their techniques will help decide what the coalition can slash and burn.
"Our methods are widely used," explains scientometricist par excellence Eugene Garfield, who set up the US Institute for Scientific Information in 1960. "But as is always the case when you're dealing with probabilities and statistics that are employed by policymakers, they can be used for good and for bad purposes."
On a basic level, Arbesman's paper is a good scientometrics case study. First, he scoured the web for three fields of research that had data easily to hand, then he set about analysing the data to answer his posited question: "How difficult has it been to discover things throughout history?" There were three fields of discovery that matched his needs and provided comprehensive data: the study of mammalian species, minor planets (such as asteroids), and the chemical elements. He assumed that the size of something equates to how easy it was to unearth (the smaller a creature or interstellar rock, the harder it is to spot – although larger elements are often the most difficult to synthesise). He then plotted this "ease of discovery" against time.
And what did he discover? "The difficulty levels don't drop by the same amount every year," Arbesman told The Boston Globe last month. "It declines by the same fraction each year. Think about Zeno's Paradox, where the runner keeps on getting halfway closer to the finish line of the race, and thus never quite makes it to the end."
Like many scientometric studies, the ramifications of this could be profound. Governments may need to spend exponentially larger amounts of money to continue research in particular fields. Recent reports marvelled at the 230,000 ocean species catalogued by scientists; in fact, global biodiversity is still an area that is largely terra incognita, and needs even more funds to maintain current rates of discovery.
Of course, Arbesman's findings cannot be extended everywhere. In some areas, at certain times, discovery becomes easier. At the turn of the 20th century, the formulation of quantum mechanics revolutionised how we understood elementary entities – the world's smallest things – and prompted a surge in research possibilities. Equally, advances in technology have sent genome sequencing into overdrive. "We are continually devising more clever methods of investigating the world," says Arbesman.
"Interdisciplinary studies between fields induce clever questions. We're not in the same position as we were at the end of the 19th century, when people genuinely believed we'd reached the limits of scientific endeavour."
While his findings are captivating, Arbesman is the first to acknowledge that his work is just a sneak peek at a diverse discipline. As with his ever-decreasing curve, with enough effort, scientometrics possesses a potentially infinite further number of possibilities. "Just as science grows exponentially more difficult in some cases, affordable technology can also proceed along a similar curve, and sometimes make science a lot easier," he concludes.
"An exponential increase in computer processing power means that problems once considered hard, like visualising fractals, proving certain mathematical theorems, or simulating entire populations, can now be done quite easily. And sometimes, discoveries can be done by being clever and more innovative, without much money."
Under the white coats: what scientometrics reveals
Still confused? A 2001 text by Dutch academic Loet Leydesdorff, 'The Challenge of Sciento- metrics: The Development, Measurement, and Self-Organisation of Scientific Communications', attempts to clarify the science of science.
The author explains how a mathematical consideration of communications can help to improve the study of technology and society.
In 2007, scientometricists Ulf Sandström and Martin Hällsten assessed persistent nepotism in the funding of medical research projects in Sweden. They found that a prior relationship with someone involved in the peer-review process increased an applicant's success by 15 per cent, but also that female grant applicants were treated 10 per cent better than their male colleagues.
In 2005, two scientists at Shanghai Jiao Tong University in China, Ying Cheng and Nian Cai Liu, compiled a scientometric list of the top 500 universities worldwide, assessing institutions on their proficiency in six academic areas. There was an articulate spluttering from Oxford University when it was placed a full five places below Cambridge, with Harvard coming out on top.
In a 2006 paper in the journal 'Scientometrics', three scientists took their native Chile as a case-study to look at the culture of scientific innovation in developing countries. They found that the importance of science within such societies was not well understood – putting such developing countries at a disadvantage to nations such as the US.
American scientists Henry Small, Ann Kushmerick and Doug Benson investigated the motives behind scientific research. Around 80 per cent of scientists surveyed said they believed their work had a significant positive impact on society, with some saying that scientific advancement itself was a benefit. The rest presumably were more motivated by curiosity than philanthropy.