It’s not often you hear anyone say people had it easier in the old days, but there is one group for whom perhaps things have got a bit harder: transport pioneers.

In 1731, the first stagecoach from London to Birmingham took two and a half days. A decade later, simple improvements to the poor quality roads meant that journey time had been reduced by 12 hours.

Today, a train from London to Birmingham takes about 1hr20. In a decade, if the construction of High Speed 2 goes to plan, the most expensive and technologically advanced railway project in British history will have reduced that by… 30 minutes. You can’t help feeling a bit short-changed.

Italy last week rejected a plan to increase the speed of its high-speed Milan to Rome line from 300kmph (185 mph) to 350kmph (220 mph). This would have been a fairly big speed increase – but the trains would only reach Rome 10 minutes earlier, and the transport ministry decided it simply wasn’t worth the cost of the extra power needed and the damage to the tracks caused by faster trains.

This all comes down to a cruel twist of mathematics: a small improvement to a slow mode of transport saves far more time than a much bigger improvement to fast one. If you double your speed then, ignoring complications like acceleration and corners, you halve the journey length. But each doubling requires a bigger increase in speed, while each halving results in less time being shaved off.

Here’s a quick example. Let’s say you want to go 640 miles – that’s roughly the straight-line distance from Paris to Vienna, or from London to Edinburgh and back. How long will that take you?

On foot, your average walking speed without roads would probably be about 2-3 mph. Let’s assume 2 mph for now. Walking those 640 miles would take you 320 hours, or about 13 days (not taking into account time to sleep or rest).

With a fit horse, your walking speed could be a sustained 4 mph. That tiny 2mph increase doubled your speed, and halved your travel time, cutting the journey by an incredible 160 hours – almost a week.

With a few more horses, good roads and some regular coaching inns, a stagecoach could do 8 mph. That lets you cut another 80 hours off the journey.

Once railway technology arrives, we can double the speed again to 16 mph – roughly the limit on early lines like Stockton-Darlington or Manchester-Liverpool. At this speed, your journey takes 40 hours. The savings aren’t as great as they were before, but cutting nearly 2 days off the journey isn’t bad.

Double the speed to 32 mph, the journey now takes 20 hours. Double again to 64 mph, it takes 10. At 128 mph – just about the highest speed possible on the British train network outside High Speed 1 – those 640 miles can be covered in just 5 hours.

But now things start getting difficult.

To double the speed again means increasing by another 128 mph, to 256 mph. No regular train runs at this speed anywhere in the world, but there is one contender: the Shanghai maglev. This can reach speeds of 270 mph but at a huge cost. The line is incredibly expensive, and makes a loss every year. If you travelled those 640 miles at 256 mph, the journey would take around 2.5 hours. For all that effort, you’ve saved 150 minutes. It might still be worth it – but it’s a lot of investment for a small gain.

Finally, we can double it one more time to 512 mph. This is roughly the cruising speed of an Airbus A320. Flying all the way, those 640 miles are covered in 75 minutes. The time savings are getting very limited now.

Speed (in miles per hour) and time taken (in hours) to cover 640 miles. The introduction of the horse was probably more important than maglev. Sorry.

Take into account the time needed to accelerate up to top speed and stop at stations along the way, and all this explains why the benefits of exciting high-speed rail projects can be underwhelming. Most intercity transport in the western world is already quite fast: eking out a few more miles per hour makes little difference to journey times except on the longest journeys.

This doesn’t mean there’s never a case for high-speed rail – its convenience can attract air passengers, and the (relative) glamour of express trains can attract passengers who wouldn’t otherwise use railways. But saving time is not necessarily a major advantage.

There is an exception: very long journeys. Over thousands of miles those small time savings can still add up. France and Germany have highly successful intercity high speed lines, because their routes stretch right across the country and beyond into neighbouring countries. If the government ever extends High Speed 2 into Scotland, or manages to integrate cities like Manchester into the Eurostar network, rail passengers might get to enjoy some really good time savings.

For an extreme example, see China, where the 1,400 mile long Beijing-Guangzhou high-speed line more than halved journey times between the two cities. Given that the slower train took over 21 hours, shaving 13 hours off meant it was well worth it (even if many people prefer to fly). Just don’t expect anything quite so dramatic on our little island.