The first near-miss took place, appropriately enough, at the Carrefour de la Croix Rouge. Happily, I negotiated the high-speed, randomly swerving traffic at this Parisian crossroads without transferring any bright yellow paint to other vehicles or street furniture, and neither me nor my passenger needed the services of the Red Cross. Lesson One: when you are weaving through continental Europe's biggest city aboard an absurdly oversized yellow quad bike, even Parisian drivers will get out of the way.
A more useful Lesson One might have been some driving tuition from a qualified instructor. But my introduction to how to handle the mighty yellow beast was as perfunctory as a Parisian shrug: this is how you switch it on, this is the brake, don't touch this red button. Bon voyage.
As my confidence grew with each balletic swerve through the chaotic choreography of a Parisian rush hour, I realised this was like a theme-park ride through the great icons of the French capital - with the terrifying difference that you are in control. Or, in my case, not.
Pity my poor travelling companion, Alessia, who had to endure the staccato imprecision of a learner driver while perched on the back of what is basically a piece of overgrown agricultural machinery. Furthermore, as we lurched through a city that shimmers with grace, making too much noise and drawing unwarranted attention to ourselves, Alessia was required to map-read and point out some of the many hazards I had failed to spot while focusing on not pressing that red button. But she could, at least, see this magnificent city from a new perspective.
The adventure began at Place St-Sulpice, one of the intellectual nodes of the Left Bank. This quintessential Parisian square is at the heart of St-Germain-des-Prés. The west side is dominated by the elegant Mairie for the sixth arrondissement, the east side by the delicate 300-year-old façade of the St-Sulpice church. I ignored them both and headed straight for the dark underbelly of the place - specifically, the underground car park.
I must have visited a hundred car-rental offices during my non-vehicle-owning life. The transaction began in the traditional manner with lots of form-filling, but then veered away from convention: first, because of the price (£106), which is enough for a week's hatchback rental on the Costa del Sol but buys only two hours on a Parisian quad bike; next, because we emerged from the up-ramp aboard something that felt immorally inappropriate for the purposes of sightseeing in Paris. Indeed, the vehicle I christened Citron Grand seems inappropriate for almost any task other than crossing the difficult steppe terrain of northern Canada, the home of Messrs Bombardier - whose main business is making planes and trains, but which also manufactures the "Outlander Max XT Quad".
You may feel you know quad bikes. If you happen to be an ageing rock star with a place in the country and a steady stream of royalties that you have to spend on something, you may even have ridden on one. But trust me: you have never seen anything like this. The Bombardier quad is the Boeing 747 - no, the Airbus A380 - of the quad-bike world, towering above feeble automobiles such as the Citroën 2CV we nearly crushed while rotating around the Place de la Concorde. Like a Jumbo jet, it should require years of training before you are allowed anywhere near the controls of one. But thanks to a loophole in French law as wide as the Seine, anyone over 21 who has had a driving licence for at least two years is allowed to take to the wheel - well, handlebars.
Since I secretly hope you will try the thing yourself, let me offer some advice about handling this genetically modified Tonka toy. It has more in common with a motorbike than a car: the throttle is a twist-grip on the right-hand handlebar, while the brake is on the left. You enjoy (although that may not be the exact verb Alessia would use) a 360-degree view. And if you have a shred of sense, you wear a crash helmet.
Given the present unrest in Paris, wearing a crash helmet is a good plan for any tourist. Indeed, the closest shave we experienced - just outside the Louvre - was when we became unhappily tangled up in a convoy of riot police on their way to, er, calm things down at the Sorbonne. With an almost tangible wave of rebellion sweeping through the city, I momentarily worried about the quad bike company's assurance that my Big Lemon was "un véhicule révolutionnaire".
There is, though, much more to mobile sight-seeing in a fabulous city than a series of narrow escapes. You have probably seen Paris on foot, or by bus or taxi. But when you whizz in all your unconfined yellow glory along broad boulevards past exquisite 19th-century architecture, the traffic acquires a balletic aspect and the experience takes on a filmic quality. Just as at the movies, extravagant design is best viewed on a long, lingering pan - which is exactly what a quad bike provides. I counted at least 25 frames a second, as the city of light flickered past. Suddenly, you are an extra in Subway, The American Friend, Paris By Night ... or, since we have now survived our encounter with the CRS toughies and safely arrived at the Louvre, The Da Vinci Code. When the film of Dan Brown's book opens next month, I M Pei's Louvre pyramid will appear prominently. But the big screen cannot compare with sweeping in through the ancient stone portals and swooping around the circle at the centre of these sublime palaces, inadvertently sending an innocent party of Japanese tourists running for cover.
Riot police, striking students, urban unrest and now travel writers driving oversized quad bikes: you can see Paris rapidly sliding down the league table of favourite foreign cities. But in my estimation, it was getting better by the minute - which is about the time it takes to get from the Louvre to the Place de la Concorde aboard an 800cc, 62-horsepower quad bike. For the first time, I could see the ancient Egyptian obelisk for what it really is: an exclamation mark at the end of the Tuileries, preparing you for the grandest of boulevards.
You soon learn to swim in the constantly changing, complex currents of Parisian traffic. The quad bike has a trio of virtues: a tight turning circle, fierce acceleration and the ability to startle everyone from passers-by to truck drivers into acquiescence. When you are proceeding at the* *legal limit of 50km/h, the fact that everyone gets out of your way frees you to appreciate the regal scale and architectural eloquence of the Champs-Elysées.
"Paris is all yours!" chirrups the company that hires out quad bikes. Apparently, the clients so far have been almost all American. "You are the first Englishman," the helpful woman in the office said. Maybe that's because we're too mean: at a price approaching one pound per minute, too damn right that the city is mine, I concluded. And what a way to see it: when Baron Haussmann laid out the broad boulevards of Paris in the 19th century, he surely had the quad bike in mind. This is a vehicle that affords an omniscient perspective of the city planner's handiwork, while simultaneously claiming exemption from the normal rules of traffic. You can orbit the Etoile - the name for the grandiose roundabout surrounding the Arc de Triomphe - for as long as you, or your passenger wishes, or The Independent photographer demands.
You may be wondering what, exactly, Alessia was doing on the back of what was, for her, a big yellow taxi. I'm sure she was, too. (You can read her generous account of events above.) I needed to find a companion with whom to share the experience. "Be the freest twosome in town," the firm's publicity had urged. "Une balade à deux en toute securité" - "Two can sit on each quad - it's as safe as can be!" Assured of such total security, but with no friends prepared to join me on the quad bike, I took advantage of a weekly column in this travel section to ask for a volunteer to help with an unspecified mission in the French capital. Alessia was the first to respond. She is a student at Warwick University, which I recall attending for several years in the late 1970s. Unlike me, though, she is gifted in both French and geography, and guided us through the evening rush-hour back to Place St-Sulpice.
Some 135 minutes after stuttering out of the underground car park, I screeched down the ramp, switched off the Citron Grand and handed in the keys and helmets. By now we were a quarter-hour late, and I feared the Big Lemon - and presumably my credit card - were about to turn into soft fruits. But the rental woman was as forgiving as the city's motorists. Driving in Paris strikes me as an exercise in four-dimensional, high-speed chess, played according to a set of rules that are known only to the local drivers. But the vast majority of road users took evasive action long before I was obliged to (and, usually, long before I had seen them). By the end, my hearing was muffled because of the rasping growls of the engine; my legs were aching; and I was grinning inanely. Worth a pound a minute? Certainly, not least because you can immediately start dropping the experience into conversation.
"Where can you go quad biking in Paris?" a friend asked when, with meticulous casualness, I happened to mention my trip. I thought for a moment and then realised: anywhere you jolly well like. One outstanding issue, though: on a big yellow bike, what does that little red button do?
Simon Calder paid £79 for a London-Paris return (08705 186 186, www.eurostar.com) and €149 (£106) for a quad bike (00 33 1 40 68 03 03; parisquadvisit.com)
AN OFFER SHE COULDN'T REFUSE...
"At the age of 37, she realised she'd never drive through Paris, in a sports car, with the warm wind in her hair." I may yet end up like the woman who Marianne Faithfull sang about in "The Ballad of Lucy Jordan". But at least, at the age of 22, I can claim to have ridden through Paris, on a quad bike, on a bitingly cold day in March.
I am studying French at Warwick University, and spending this academic year at the Sorbonne (barricades permitting). But I happened to be at home for the weekend in London when Simon Calder's column asked for a participant for some undisclosed (but "legal and decent") activity in Paris. Three days and a Eurostar trip later, I found myself clambering on to the back of a large yellow quad bike in a central Paris car park. We set off on a whirlwind tour of the city's highlights with a seemingly fearless photographer hanging out of the car in front of us, taking pictures of us in all our quad-biking glory.
The afternoon was certainly not for those who wish to maintain an air of Parisian chic. But despite the helmet, the rather attractive Paris Quad Visit anorak and the paralysing cold, I did feel a bit like a film star, being photographed zooming around the Louvre. perched on the surprisingly comfortable back seat of a chauffeur-driven quad bike, while Parisians and tourists alike gaped.
It was a bit of a bumpy ride at first, but soon it became quite exciting, speeding around Paris at what felt a lot faster than our top speed of 50km/h. The quad was also effective for squeezing up the small pedestrianised roads that one normally wouldn't be able to - let alone be allowed to - navigate in a vehicle. We found that no one questioned the quad, although we might have got lucky as most of the police were off at the Sorbonne dealing with rioting students.
All in all, it was a fantastic way to explore Paris and weave in and out of those petites rues in record time. It would be worth braving the hills of Montmartre to discover what the quad really can do, but with all the posing we didn't have time. I'll have to try it again - but the next time, I insist on some warm wind in my hair. And I'll drive.