One way is by making a cup of tea.
Water boils at 100°C only at sea level. The temperature at which water boils varies based on air pressure. For every 1000 metres, the boiling point drops by about 5 degrees. Thus, with a kettle and a thermometer you can estimate altitude.
With great difficulty over the better part of a century.
The Great Trigonometrical Survey of India started from the ocean in 1802, and 100 feet at a time, took measurements and did a bunch of math, and worked there way across the sub-continent to the Himalayas, completing in 1871.
It was a great scientific achievement, lead during some of its more important years by George Everest, who received a knighthood for his efforts.
The basic technique is fairly simple. You start with two sticks at sea level, a decent distance apart, and measure their exact longitude and latitude. Then you put a third stick some distance away and inland, making a triangle. Based on the distance between the first two sticks and the angles they form with the third stick, you can compute the third stick's exact position and elevation.
Once all that is done, you repeat, planting a stick further inland and drawing a new triangle.
The Great Survey did this with better instruments, better technique, and on a greater scale than had ever been done before.
Also, math and triangulation. Trigonometry has been around a long long time.
See, with just one side of a triangle and the angle between it and another side, you can figure out the missing side. So if you make the triangle such that once side is easy to measure, and then you use a protractor and your vision to determine the angle, you can math the height.
A field of study called geodetics used instruments called Theodolites. Sometimes they were called diopters. The theodolites almost look like telescopes but with a lot of rulers on it so you can determine which direction the telescopes are pointing, up-and-down and side-to-side.
Using the measurements from the rulers on the theodolite, you can use math called Trigonometry to tell you how far away something else is.
In the case of Chomolungma (Mt Everest), the British started from the ocean in India and measured all the way across the country until they could see the mountain. After getting their measurements, they did the math and found out how tall they thought the mountain was.
Trigonometry. In high school we visited an amusement park. By being a known distance from the base of the top of the roller coaster hill (length and angle (90 degrees) of one side of the triangle) and then calculating the angle to the top of the structure using a sighting scope, we could calculate the height. [This pic](https://image.slidesharecdn.com/heightanddistances-120108094402-phpapp01/95/height-and-distances-12-728.jpg?cb=1326016394) sums it up.
They often didn't.
I can't find the citation easily, but in the late 1990s/early 2000s, a very large number of Australian mountains and waterfalls had their heights revised after someone realised that there were a stupidly large number of them that were listed as having a height of 305 metres. Some went up, most went down - a couple by more than half.
It turned out that the official height figures were often the estimates of the original explorer/surveyor and no-one had been arsed to actually measure them, especially because they weren't a suspiciously round number.
Of course, the reason that they weren't a round number was because during metricisation in the 1970s a height of 1,000 feet had been converted to 305 metres, but that was before heights were recorded on computers and easily checked against each other.
Angles and heights are very easy to measure using a sextant, some trigonometry, and a few known distances. If you have the angle and one side length of a triangle, you can determine the other unknown values of the triangle, including, in this case, the height, or distance at a perpendicular angle from the base to the uppermost vertex.
Along the same lines, I've always wondered how cartographers managed to draw the continents before airplanes. They weren't as accurate as in the last 100+ years, but they weren't too far off, either.
Apologies if this has already been said. With Trigonometry, you can estimate height by using the mountains shadow. Measure the length of the shadow from the base of the mountain to the tip of the shadow. Measure the angle of the sun to the tip of the shadow. With that information, you can determine the distance from the peak of the mountain, to the peak of the shadow. Now you can solve for the 3rd side of the triangle, ground level to peak of mountain.
I don't know about how they *actually* did it, but one thing that might work is to use the mountain's shadow. Find a place where you can see the mountain such that the sun will set directly behind it. Then check the exact time that the mountain's shadow hit you. It is easy to calculate your distance from the mountain (or at least easier), and the time of day gives you the angle that you made looking up at the mountain's peak. The rest is trig.
You could use the shadow of the mountain and use a proportion, but at that scale the curvature of the earth makes an impact.
You could use a barometer, which they had in the past. The density of air was known to be lesser the higher up you go. So the pressure read is related to your height on the mountain.
You could boil water as well.
Same way we do today - Trigonometry.
You select a [datum point](https://en.wikipedia.org/wiki/Geodetic_datum) and everything is measured relative to that.
Technology changes, we have satellites and laser theodolites that measure to greater levels of precision, but it's the same basic process that goes back to Pythagoras. Skilled surveyors mapped the world with [Theodolites](https://en.wikipedia.org/wiki/Theodolite) and Trig.
Oddly enough, the better question is how we accurately measure the height of a mountain NOW. Back in the day, surveyors used theodolites and other equipment to measure distances and elevations...but it's not just like measuring something with a ruler and calling it 8 inches. There's a very specific procedure for measuring something, moving a certain distance, measuring the distance you moved, measuring the height of the thing again, etc., and the net result is that you have a bunch of measurements with known error functions built in. Through an evaluation of the measurements and errors, you end up with not just the elevation of the mountain, but an error bar: i.e. 6,208.16 ft +/- 0.02 feet.
Currently, with GPS and photogrammetry, the accuracy is has a bias issue that is difficult to correct for. For example, if your satellite isn't exactly where you thought it was, it's going to always return a signal that's biased in one direction or another...and because there are only so many GPS satellites, the error bars are disproportionately large. There are differential GPS surveys - like you get the measurement at known point A and then hike your butt over to point B and check out the difference...but while it sounds great on paper, in reality it relies on a lot of assumptions and is prone to bias.
So the net result is that old timey theodolites + multiple measurements + math is more accurate than new fangled GPS.
Thales was one of the first mathematicians to estimate immeasurable heights. Around 2500BCE he was using notions of similarity and to estimate the height of Pyramids in Egypt.
I always understood it as the thing they teach you in algebra where they have a fixed hight next to something x feet tall, and measure their shadow.
I think its a subcategory of proportions.