By 1785, you need to know two things to determine your position on the globe: your latitude and longitude. Once you know those, you can compare them to a chart, and hey voila, you get a position. (Charts in 1785 were far from perfect, and there were errors in measurement, but the navigational tools of the time could give you a pretty good idea of positioning.)
I want to emphasize that these measurements are useless without charts, which is often lost in discussions of early navigation; it does you no good to know how far north/south or east/west you are without knowing where landmarks are.
To find your latitude (north/south position), the ship's sailing master (a warrant officer) and other officers (often the captain and first lieutenant, generally the midshipmen as this was part of their training) would measure the altitude of the sun over the horizon at local noon. By 1785, this would be done using a sextant. You can find descriptions of the process here and here. People often assume that observation is your latitude, but it's not -- the Earth's axis is tilted, and as the season changes so does the height of the sun at local noon. The British Admiralty and other navies made tables showing this progression (technically, the "declination" of the sun). So you would then compare the observed height of the sun (its angle over the horizon) with your handy table, and the resulting numerical angle is your latitude.
Longitude is more tricky. The most straightforward way of determining longitude is comparing local time to time elsewhere on the globe (usually, Greenwich Mean Time) and figuring out the time difference; if it's 1 p.m. at Greenwich at your local noon, you're 15 degrees west of Greenwich. Once marine chronometers became widespread, the longitude problem was easier to solve; but, chronometers were only provided to British naval ships traveling in far distant waters starting in the 1790s, and did not become standard issue until the 1840s. (Captains or masters could buy chronometers, although they were horribly expensive -- 60 to 100 guineas new, plus 5 or 10 per year for cleaning/resetting, and ships needed three to correct for errors.)
So our theoretical 1785 captain had three options for dealing with the question of his longitude:
Dead reckoning -- that is, plotting the ship's speed and course over time, accounting for wind and currents and latitude measurements, to arrive at an approximate position for the ship;
Running down a line of latitude -- widely used before the "invention of the longitude" around 1760, this implied that you'd sail to an easily-found point of latitude and turn dead east or west, steering for a landmark. This could be very risky -- there's only something like 1.75 degrees of latitude between the Scillies and Ushant, the entrances to the English Channel, so you better be darn sure of your latitude to enter the Channel that way.
calculate your time, and thus position, based on "lunar distances" -- either finding the degrees between the moon and another celestial body, or by measuring the positions of the moons of Jupiter, to compare it to tables and find Greenwich mean time. I am way in over my head on the math on these, but Wiki has what I am told is a [perfectly cromulent summary](https://en.wikipedia.org/wiki/Lunar_distance_(navigation).
One thing you may have noticed reading those descriptions is that the observations (before the chronometer) necessarily take place at different times of the day, and only once a day or maybe twice for longitude. So the business is still very much one of trial and error.
Hope this helps. Keep in mind that on a British ship of this era, although the captain is legally responsible for the ship, the sailing master is responsible for navigation. Many captains would take an interest in navigation, and midshipmen, to become lieutenants, would have to pass an exam that could include navigation, but not all captains were outstanding navigators.