It took 200 years and the greatest minds on the planet, including Isaac Newton, to find a way to determine longitude at sea. The solution was to observe an event in local time and know the time of some other place, say London. The chronometer was invented that kept London time at sea, which was tricky, because clocks don't keep good time at sea. This solved the problem and let ships determine their longitude. It took hundreds of years.
My question is, why couldn't they just have noted the time of the port they departed from, say London, and then just recorded the number of hours that passed by way of hour glass? That way they would always know what London time was by taking the departure time and adding the number of hours.
Granted, it would have been tedious and you'd have to have chronomotists take shifts and such. But, considering before the chronometer's creation, ships had to sail all the way up to their correct latitude, and then just try their best to sail in a straight line, which could add weeks to their journey and significantly increase the chances of them dying. So, it seems worth the trouble in the absence of anything better.
Still, they didn't do this. This idea must have occurred to Galileo, Newton and Euler, so there is an important reason this wouldn't work, but I don't understand what it could be.
Please help.
To be accurate within 1 degree for a time-based measurement of longitude, the time needs to be known within 4 minutes. (1 degree corresponds to approximately 100km near the equator; this was the minimum accuracy required for the British 1714 Longitude Prize - and needed to be achieved after a voyage to the West Indies, approximately 6 weeks (Harrison's chronometer was 5 seconds slow on arrival in Jamaica).)
The sandglass/hourglass isn't accurate enough. While a good sandglass will be more accurate than a poor mechanical clock (an error of 15 minutes per day was small for the early mechanical clocks used at sea; 1 hour per day was common), it won't achieve 4 minute accuracy over 6 weeks. At sea, a sandglass is affected by the motion of the ship, changes in temperature, and (on long voyages) changes in local gravitational acceleration. On land and at sea, they are affected by irregularity of the flow of sand (or other grains), wear, and variation in the time taken to turn them over so that they can run again. It's possible (but not usual) for a land-based sandglass to be accurate to within 1 minute per day, but at sea, and error of about 15 minutes per day is common. After a few days, the error easily reaches an hour, at which point the error in position from determining longitude exceeds 1000km. For a single day, with an error of 15 minutes in time, the error in longitude is 3.75 degrees, or about 400km.
The sandglass was a common instrument for measuring time at sea (at least on European (and later, American) ships) from the 14th century to the 19th century. They were used for navigation, via measurement of speed. Position was then determined from the direction of travel (as determined by astronomical observations, or by a compass) and the speed giving the change in position over the day (this method of navigation is called dead reckoning). The typical error over 1 day would be about 10-15 nautical miles, or about 20km - note that this is much less than the error that would result from longitude measurement using a sandglass (about 400km per day).
For a short but useful discussion of the relative accuracy of longitude measurement with a marine chronometer, with a sandglass, and navigation accuracy using dead reckoning, see B. C. Rennie (2009) The dauphin map, Journal of Australian Studies, 2:3, 75-78, DOI: 10.1080/14443057809386783