In November 1660, the world was a mysterious place. There was no explanation for the rise and ebb of the tides. Air was a puzzling, invisible fluid with unexplained properties. There was no known way to measure the height of a mountain. Minerals were produced by "certain subterranean juices through veins of the earth".

A small group of men who began meeting at Gresham College that month and formed a society to promote experimental knowledge (the royal charter came in 1662; the first women fellows were elected in 1945) listened to strange reports from Iceland of smoking lakes and fire in the sea. They wondered why winter was colder than summer, and they speculated on the spontaneous generation of life in the absence of "certain seminal principles".

They did more than wonder: they experimented. They choked chickens, gagged fish, strangled dogs and dissected living cats. They transfused blood from a sheep to a human. They tried to imprison a spider inside a circle of powdered unicorn's horn. They also suffocated mice; but according to their first chronicler, they themselves breathed "a freer air" and conversed quietly "without being ingag'd in the passions, and madness of that dismal Age". These men lived in a world of plague, fire, war, public execution, witchcraft, alchemy, religious hatred, political ferment and precarious patronage: but they made it a rule to discuss neither God nor politics, nor news "other than what concern'd our business of Philosophy".

As well as collecting uncritical observations of monstrous births and listening to investigations into the supposed consequences of a tarantula's bite, they read a paper from a certain Mr Isaac Newton of Cambridge, which showed that white light was in fact made up of the colours of the rainbow. This was a landmark moment in science, but as James Gleick – the first of many impressive contributors to this substantial celebration of 350 years of the Royal Society – reminds us, we recognise landmarks after we have passed them. At the time, the society's own experimenter, Robert Hooke, dismissed Newton's hypothesis as wrong. This snub drove the sulking Newton back to his obsession with alchemy and scripture.

There is a solid case for regarding Hooke and Newton and their peers as the makers of the modern world, but it might not have seemed so at the time, even to members of this not so exclusive club. Past fellows have included Samuel Pepys, John Evelyn, Edward Gibbon and Lord Byron, and Bill Bryson appropriately enhances his celebratory mix with contemporaries better known for pen than pipette. Margaret Atwood traces the origin of that ever-popular B-movie figure, the mad scientist, back to book three of Gulliver's Travels, and points out that Swift can only have been lampooning the Royal Society when he invented the flying island of Laputa and the Grand Academy of Lagado. But she also identifies strands in the Swiftian mockery – the anal inflation of dogs, the extraction of sunbeams from cucumbers – that eerily prefigure later realities involving colonoscopy and cod liver oil tonics.

The science writer Margaret Wertheim uses Dante's Divine Comedy, with its journey through concentric spheres of increasing perfection, to illuminate the retreat from the Aristotelean cosmology that had been shared into the 17th century by papists, Puritans and the first professors of natural philosophy. Euclidian space in a few generations expanded to exclude celestial space. The telescope, and the Copernican principle that there was nothing special about the Earth or the solar system, began to banish God and his angels from any physical location in the heavens. Stark rationalism swiftly provoked its own enduring counter-revolution. "To the continuing horror of many champions of science," she warns, "belief in astral planes, psychic channelling, reincarnation and past lives seems to be growing stronger."

But out of untidy discovery, wonders emerged. The universe was an ever-open book, said Galileo, "but it cannot be understood unless you have first learned to understand the language and recognise the characters in which it is written. It is written in the language of mathematics." Out of the original "Colledge for the Promoting of Physico-Mathematicall Experimentall Learning" and the work of its correspondents, admirers and imitators, first in Europe and America, and later everywhere on the planet, came so many of the things we now recognise as modern science and technology. Bryson's contributors celebrate aeronautics and evolution; suspension bridges and systematic biology; X-ray crystallography and lightning conductors; Bayesian distribution and Bakelite; climate science and complexity theory.

This is a book of cerebral riches, heavy with history, to be consumed at leisure. It is also beautifully illustrated. All but one of its 22 contributors wrote specially for this anthology. Richard Holmes, fresh from his scientific history The Age of Wonder, provides new material on 18th-century balloon flights. Richard Dawkins sums up the significance of Darwin's achievement with renewed metaphorical force. The Natural History Museum palaeontologist Richard Fortey highlights the importance of collections; Steve Jones raises some of the puzzles of biodiversity; the physicist and science fiction author Gregory Benford contemplates the enigma of time.

Every now and then, the book begins to seem like a royal variety performance: well-known acts trip on to the stage, perform a much-loved routine and disappear, to be followed by something completely different yet equally familiar. But all contributors in their different ways also remind us that the show goes on. Do we see more clearly than Hooke and Newton did three and a half centuries ago? Oliver Morton argues that we may have traded one picture of the Earth for another, but our understanding of the globe remains incomplete; Ian Stewart reminds us that for all Galileo's astuteness, even scientists can be oblivious to the subtle mathematics that underpin their research; John Barrow considers the apparent simplicity of cosmological physics and points out that we do not observe the laws of nature, we see only the outcomes of those laws. "Outcomes are much more complicated than the laws that govern them."

The physicist and astrobiologist Paul Davies reminds us that even the keystone of the Copernican revolution – the assumption that there is nothing special about us – might be incompletely laid. Is the solar system typical? Perhaps, but supporting evidence began to emerge only 15 years ago, and carbon-based life exists on Earth but, as far as we know, nowhere else. Is there anything typical about our position in spacetime? Davies has his doubts: carbon, manufactured by burning stars, was not possible for the first five billion years, and may not be possible 100 billion years from now, although the universe could drag on, getting ever colder and darker, for another 10 billion billion empty years.

Gregory Benford makes the same point: "We seem to occupy an unusual niche in the long history of this universe." The novelist Maggie Gee takes global warming as a text for an entertaining sermon on fiction's love affair with apocalypse. The astronomer Martin Rees, president of the Royal Society, who in a 2003 book warned that we might already have begun Our Final Century, is sure that many mysteries remain. "Most of the questions still being addressed simply couldn't have been posed 50 years ago (or even 20): we can't conceive what problems will engage our successors."