On 1 August 1785 Jean-François de Galaup, comte de la Pérouse, left France with two ships at his command on an epic voyage of exploration across the Pacific. It was the greatest such voyage yet undertaken under the French flag and alongside the seamen on board were a range of experts, among them astronomers, engineers and naturalists. The immediate catalyst for the voyage was France’s recent loss of its territories in North America to the British. But the political imperatives behind the great undertaking were even more nakedly ambitious than that.

An 1817 painting by Nicolas-André Monsiau – itself far from modest in scale – shows La Pérouse receiving his instructions from Louis XVI. In the painting, Louis is pointing to a map of the ocean as if to indicate a possible trans-oceanic route for the ships. It that sounds loosely figurative, it really isn’t: Louis issued La Pérouse with detailed instructions, page after page of them, listing the various scientific and commercial objectives he was expected to fulfil. They are in the painting, in fact; a naval minister behind Louis’ shoulder is clutching a sheaf of them. And the king – a devotee of geography and engineering, whose private rooms were filled with scientific papers, mechanical apparatus and laboratory equipment – would do more than issue orders. He followed the ships’ progress religiously.

Monsiau’s painting dominates one room of a gallery in the Science Museum’s superbly thoughtful and provocative new exhibition Versailles: Science and Splendour. An adjoining wall features three vast maps on which Louis himself plotted the reported movements of the ships against the planned route. He did that for three years. But then, in 1788, the king’s marks stop. Both ships were lost with all hands in a storm off Vanikoro in the Solomon Islands. The truth did not emerge until the 1820s and Louis himself wouldn’t live long enough to learn it. Five years later he went to the guillotine still uncertain of their fate.

Those three great maps and the assiduity with which Louis XVI followed the voyage – fuelled, one must think, by a mixture of ambition, confidence, and hope – are rather poignant now. The catastrophe of the French Revolution is just over the horizon, but the king’s mind is on other – to his way of thinking, larger – things. The exhibition covers the reigns of three kings, Louis XVI and his two immediate predecessors Louis XIV and Louis XV. It is hard to blame him for his complacency: the latter two occupied the throne for 72 and 59 years respectively. Together, despite the abrupt end, the three reigned from 1643 until 1793. Bourbon authority must have felt serenely inevitable, a straight and open road into a future that, however filled with unknowns, was supremely mappable, and susceptible to order. It’s not a coincidence that man’s conquest of the air begins with the Montgolfier brothers under the auspices of Louis XVI.

All three kings, the exhibition makes clear, saw in the nascent sciences of the Enlightenment the political possibilities of new ways of describing, ordering and therefore controlling the world. Across geometry, astronomy, military engineering, medicine, chemistry, cartography, garden design, and the study of the natural world, knowledge was power, and order was beauty. Reason itself was a thrilling new aesthetic for a court in which science and splendour were not in opposition, but instead were alike expressions of imperial monarchic greatness.

One of the first things you see here are two globes – one terrestrial, one celestial – by Matthaus Greuter, made in 1632. North America, on the former, is almost entirely dominated by the territory of Nova Francia. However fanciful its coasts and borders look to us now, that was part of the point: borders were fluid and changing, but to define the limits of something was to make it your own. Beside the two globes is a 1679 engraving of a map of the moon made by the Italian astronomer Jean-Dominique Cassini at the Paris Observatory. It took Cassini eight years and it is realised in such extraordinary detail and clarity that it would remain unsurpassed for two hundred years. Mapping, as activity and metaphor, is everywhere.

Near Cassini’s map are two large golden astronomical instruments, a planetarium and an eclipsarium, designed by Ole Rømer and built by Isaac Thuret, Louis XIV’s clock maker. Copies of them were gifted to the emperor of China and the king of Siam in the 1680s, and the information panel describes them as ‘objects of royal propaganda’, which is undoubtedly true. But an engraving on the other side of the same gallery picks up the story and fleshes it out with more nuance. It shows six Jesuit mathematicians – chosen by France’s recently established Royal Academy of Sciences – observing a lunar eclipse at the observatory of the Siamese king in 1684, before continuing their mission to China and the observatory in Beijing. Scientific endeavour, and knowledge more broadly, is thus situated as an exploration of power among elites worldwide, intellectual territory just as precious as land itself, and just as rich in resources to be exploited.

Versailles: Science and Splendour is based on an exhibition, Sciences et curiosités à la cour de Versailles, which ran at Versailles itself in 2010-11. Both exhibitions share the objective of re-evaluating the apparent extravagance and luxury of the three king’s courts – or at least contextualising such pomp and decadence, if that’s what it was, with reference to the flowering of scientific enquiry and intellectual endeavour that characterised all three reigns. The visitor is challenged, therefore, both to consider how deeply embedded the rational, scientific underpinnings of modernity are in the imperial European hierarchies of wealth, privilege, and power; and, contrariwise, how it was that the seemingly reactionary dynasties and institutions of old Europe were at the forefront of intellectual and technological innovations that still shape our lives today. On both scores it is unsettling, as complexity is and should be.

There is certainly no escaping a sense of the uncompromising, untrammelled power of the Bourbon court as embodied by Versailles itself. The palace began as a hunting lodge, but for over a century, from 1682 until 1789 and the French Revolution, it was the seat of the French monarchy. The transformation took not just science and industry but immense labour. Versailles itself had no natural water supply sufficient to feed its eight square kilometres of garden, and that’s before you consider the more than 2,000 fountainheads fed by over 45 kilometres of pipework around the palace. Alexandre Dufour’s plans for the pipework for just one fountain are displayed here; they are as precisely and elegantly beautiful as you might expect. A map shows the network of vast reservoirs built for the palaces by the astronomer Jean Picard, who ensured that the whole site was levelled and built hydraulic projects that supplied Versailles with more water than reached the whole of Paris.

But it wasn’t deemed enough. In response to Louis XIV’s demands, the Marly machine was developed, which pushed hundreds of thousands of gallons of water some 162 metres uphill from the Seine, through two reservoirs and along a seven-kilometre aqueduct. The machine took 1,800 men to build it and comprised, among other things, 14 huge paddle wheels and some 260 hydraulic pumps. It was said to be deafening in operation. It was one of the highlights of the tours that Louis XIV designed to show off the palace’s water features: three ambassadors, on a reciprocal visit from the king of Siam in the autumn of 1686, were astonished by its infernal noise and power, as they were no doubt intended to be. There are two images of the Marly machine here, one a technical engraving, the other an oil painting from the 1720s by Pierre-Denis Martin in which it does indeed look somewhat monstrous, like an industrial-scale torture device.

Some might find the masculine energy of some of this off-putting, but the exhibition is at pains to emphasise that, while women were largely excluded from the worlds of science and intellectual exploration, they nevertheless engaged with it wholeheartedly in any way they could. It offers, for instance, the manuscript of Gabrielle-Émilie du Châtelet’s 1749 translation of Newton’s Principia mathematica. But it also offers one of the most visceral items in a show that very much centres intellectual order. Concerned to maintain France’s population strength, Louis XV gave a midwife named Madame du Coudray a royal warrant to train others in the healthy delivery of babies. Over the course of 25 years she trained some 5,000 women, and one of her teaching aids is here: a stuffed-cloth mannequin of a seven-month-old fetus huddled and vulnerable in an opened womb. It’s a powerful reminder of human frailty amidst all this wealth and power.

It’s instructive to compare this exhibition with The Great Mughals, recently opened at the V&A. Both explore the imperial culture of two partly contemporaneous courts through the reigns of three sequential rulers. It may simply be a reflection of the different curatorial strategies of the two museums, but where at the V&A the emphasis is on an imperial culture trading more in soft power, with its affectations of global authority seemingly more of a rhetorical flourish than a programme of government, the Bourbon court, on this reading, seems unashamed in its desire to order not just its own dominions but the whole of creation. And if there isn’t much here that immediately underwrites the splendour of the exhibition’s title, one item alone triumphantly bears it out: the massive Clock of the Creation of the World, designed by Claude-Siméon Passemant, is a breathtaking achievement in both mechanical and artistic technique showing a golden ray from a golden sun picking out midday on a rotating terrestrial globe, itself bursting gloriously from bronze rocks and waves. In the middle of the sun is a clock showing the hour, the day, and the month. It was created in the 1750s as gift for the nawab of the Indian state of Golconda as part of a projected alliance against regional Mughal hegemony: two imperial projects, two aesthetic approaches to validating that power, abutting one another in a single object.

Among all the beautiful order, is there an ugly moral disproportion? If you want to see it, certainly. The information panel for a world map charting French colonial and scientific activity c.1700 references slavery, of course. One might also point to how Louis XIV publicised and encouraged developments in surgery by allowing himself to be operated on by his surgeon Charles François Félix for an anal fistula. Admirable in itself, perhaps, but Félix had practised the technique – experimented, you might say – on 75 patients among the poor and impoverished, not all of who had survived. Food shortages in France, made worse by outdated agricultural methods, were endemic, despite scholarly discussions of the problem at Versailles. An engraving shows the young Louis XVI, while still dauphin, ploughing a field. It was no more than a gesture towards an urgent problem that had no obvious place in the long royal horizon.

Indeed, many of the activities memorialised here feel like powerful metaphors for the utilitarian way the intellectual community viewed life and matter around or – perhaps more accurately – beneath it: landscaping, engineering, classification, and so on. Then there is dissection, which features heavily in the section about understanding nature. There is an engraving of a dissected chameleon, for example, and an anatomical painting of a splayed, prone horse, which is all the more coldly resonant for the cultural associations of social nobility and prestige that have always attended on horses.

But in an exhibition replete with metaphors for the processes of both imperial power and science let’s end with another timepiece. If the Clock of the Creation of the World is almost monumental, this one is small: a watch that might be held in the hand. It was commissioned for Marie-Antoinette in 1783 from Abraham-Louis Breguet, who was given an unlimited budget. It has 823 parts, many of them in gold or sapphire. With no apparent financial or time constraints, Breguet kept adding new features, new complications, as he developed them. His work might have been a synecdoche for the court’s indifference to anything that lay outside its world view. How long did it take him to create something that articulated so meticulously the exquisite order and perfection through which the kings of Versailles sought understand reality? Too long. Marie-Antoinette’s hand would never hold it. Breguet’s work was interrupted by the French Revolution and it would be 40 years before his watch was ready for delivery to a long-dead queen of a long-dead court, which had foundered on social disorders its maps had no place for, and been surprised to find that a world of elegant and ravishing design could yet run out of time.

Versailles: Science and Splendour is at the Science Museum, London, until 21 April.