Leonardo da Vinci was an artist who became a scientist by asking what underlay the world he so brilliantly depicted. Behind the asymmetric smile of the “Mona Lisa” lay a complex interplay of facial muscles; in sketching a waterfall, Da Vinci became fascinated by fluid dynamics. (Stefan Klein’s thought-provoking book, Leonardo’s Legacy, provides a persuasive account of this process of discovery.) The exact extent of Da Vinci’s innovations are a matter of debate, but surely there is no dispute that he was one of western civilisation’s great geniuses.

Yet there is no escaping the fact that Da Vinci was able to achieve so much, so broadly, because so little was known. It was possible to make leaps forward in scientific understanding armed with little more than a keen eye and a vivid imagination.

Those times are long gone. Approximately 3,000 scientific articles are published per day – roughly one every 10 seconds of a working day. We can now expect that these papers will, each year, cite around five million previous publications. And the rate of production of scientific papers is quadrupling every generation. (All these estimates are based on data from the Institute for Scientific Information.) The percentage of human knowledge that one scientist can absorb is rapidly heading towards zero. This side of a new Dark Age, there will never be another Da Vinci.

Benjamin Jones, an economist at Northwestern University, has been monitoring these trends for some years. The typical science paper or patent, finds Jones, is now produced by a large team. The specialisation of each of that team’s members, to the extent that it can be deduced, is narrower than before. And the members of the team are also older. All of this follows quite plausibly from the drift away from the world of Da Vinci: as the sum of human knowledge swells, individual researchers must spend longer and longer acquiring ever smaller slivers of it, before they are in a position to make their own contributions.

This means that funding new ideas that matter is almost certainly getting more and more expensive. By itself that fact need not be too disturbing: we can afford to spend more on science, and there are more qualified scientists across the world than ever. But it also suggests that scientific and technological innovation is, more than ever before, an organisational problem – and an organisational problem to which we have probably devoted too little attention.

I think we need to do more to combine the best features of public-sector and private-sector innovation. Governments have the long-term perspective and the financial firepower to fund the really big projects, while small technology start-ups, funded by venture capitalists, have the collective ability to take risks and attempt a huge range of different approaches. It should not be beyond human ingenuity to devise government grants that are more pluralistic and take bigger risks.

But there is more going on here than funding. Benjamin Jones points out that scientific institutions and science policy have yet to catch up with changing trends in science. Peer review, patent evaluations and grants all tend to rely on the expertise of individuals – expertise that is less and less likely to be able to cover the necessary bases. Science has always struggled with cross-disciplinary work, but now that disciplines are becoming so narrow, cross-disciplinary work is unavoidable.

Perhaps the most important discovery in Jones’s work is the ageing of scientists. Scientists may have to wait longer before reaching the frontiers of scientific knowledge – but who will keep them funded and engaged as they make the trip? No wonder Wall Street and the City find it so easy to recruit disaffected young physicists.

Also published at ft.com.