Abraham Zacuto, as mentioned before, had done some fine work that allowed ships to find out how far South and North they were, and in what direction they were heading, without reference to the Pole Star. However what it did not do was allow sailors to work out how far West or East they were. In fact, by 1490 people still didn't know how big Europe and Asia were, which explains why Columbus thought he was in Asia - he wasn't stupid, but he was using a map that over-estimated the size of Asia.
The problem of working out how far West you've travelled is known as the "Longitude Problem", and it was the biggest challenge of navigation right up until the early 20th Century. Among other things, if you don't quite know how far west you are then you might run into some rocks that you know are there, but didn't realise that you were right on top of them until too late - this happened several times.
Thankfully there's a solution. In modern times the world is divided into timezones and this directly relates to the problem because what timezone you are in - specifically, how many more hours it takes for the Sun to get to its highest point in the sky - tells you where you are in the world relative to Greenwich. So, all you need to do to know how far west of Greenwich you are would be to know what time it is in London when it's mid-day local time. Unfortunately, even by 1750 there was no good way of doing this.
Two possible solutions came out. In the first one you use a telescope to find the moon and some other astronomical object, such as the star Regulus in the constellation Leo, measure the relative altitudes of both of these, and find the apparent angle between them. Now, you correct for parallax error due to the differences between where you are and where the Greenwich Observatory is, and also correct for atmospheric effects that broaden the apparent widths of both objects, and subtract the Moon's semidiameter. This gives a value that can be made accurate to one half of a minute of arc. This value can then be compared which a table that tells you what time of day it is in Greenwich when those angles occur, and because in fact the angle is the same (or as near as) for all observers at the same exact moment in time, knowing what time it was when you made the observations allows you to deduce the time in Greenwich and thus to work out the time difference, and so the distance, between you and London...zzz...
Alternatively, you could use a watch that was set in Greenwich and keeps time accurately. Then you look to see when it's mid-day where you are, and read the time of the watch you've got. That's much easier.
Actually, that "fall-asleep" method I described first (known as the method of Lunar Distances) was the one most people used for over a hundred years. The reason is that all watches made before the discovery that quartz could be used to keep time (guess what I'll be writing about later...) were mechanical and broke down at sea, because of winds, salt water, tides, etc.. So the first person to build a watch to overcome these problems would revolutionise navigation, or at least save the all the hassle of the first method.
And there is such a man. John Harrison, the self-taught watch- and clock-maker, designed and built 5 timepieces capable of keeping time at sea. Each of these is a marvel of engineering, and the last ones, known as H4 and H5, were good enough to be as accurate as the Lunar Distance method, if not more accurate, and by 1850 when prices had come down were in use in most ships that sailed on the high seas, improving both naval safety and navigational ability by a huge amount. James Cook's 2nd and 3rd voyages to Australia made use of (a copy of) one of these clocks.
Some of the technology used in Harrison's clocks are still in use today, including the "grasshopper" escapement that reduces the need for oiling clocks; the "bimetallic strip" that doesn't expand in warm weather - making pendulum clocks accurate whatever the weather; and the "roller bearing", that reduces friction in many mechanical applications.
All of this, and more - Harrison also worked on the mathematics of musical tuning.