What if you get lost somewhere on earth? Is it based on GPS? OK, but what if there’s no signal and no equipment? A map? Good idea, but what if you don’t see the landmark when you travel on the high seas? For thousands of years, many people have discovered the problem. The brave sailor then switched to the sun and the stars. For this reason, they need some geometry knowledge, especially Trigonology. Let’s say you’re on the high seas and want to calculate the latitude position. The sun and most stars change their position in the sky over time, but some stars seem to be in the same place all the time. Polaris, for example, also known as North Star, always seems to be directly above the Arctic. As it turns out, your latitude corresponds to the angle of the Polaris above the horizon. Consider a plane that includes the Arctic, your point x, and the earth’s center o. Strictly speaking, Polaris is not perpendicular to the upper x, as shown in the figure, but it is so far away from earth that the line of vision from X to Polaris is almost vertical, so we can assume that it does. The angle theta of “Polaris above the horizon” is the angle indicated in the figure. This is the angle that we form from the line t (which is our view towards the horizon) tangent to the line of view of Polaris (as l) and point x on earth. The latitude at point x is defined as the angle Φ between the line r from O to X and the plane containing the equator. In our two-dimensional image, the equatorial plane is only a horizontal line e through o. It intersects with the vertical line L at point L and tangent t at point t. As R is the radius of the circle and t is the tangent of the circle, we know that R and t form a right angle at point x, and because t and L form angle theta, we can know that the angle between L and R is 90 ° – theta. “< p > < p > consider triangles with O, x, and l as their vertices now.”. As we can see, the angle at point x is 90 ° – theta. Since l and E are perpendicular to each other, the angle at point L is 90 °. We know that the sum of the inner angle of the triangle is 180 °, so the angle Φ, the latitude of our position, is: [/p > < p > so, your latitude is defined by the Greek astronomer Hipparchus in this way, when the Arctic star was on the horizon years ago. He didn’t even know that the earth was round at the time, but the illustrations we showed here explain why the definition of hippachas was consistent with the modern definition. The southern hemisphere does not have Polaris in the same sense, but if you are in the southern hemisphere, you can find your latitude using a constellation called “South Cross” (marked on the Australian flag) and two stars called the “South pointer”. For thousands of years, navigators have used different devices to measure the angle of stars above the horizon. It includes beautiful astrolabes and sextants, which you can often see in antique shops and museums. We have seen how knowledge of geometry can help adventurous sailors determine latitude. For thousands of years, people have been using stars to do this. But longitude is another story. It was not until the 18th century that a reliable method of longitude measurement had been found – before that, countless people had died at sea, and the solution was ultimately provided by the clock. We can use the position of Polaris above the horizon to measure latitude because it changes as you move north or South (i.e. latitude is changing). This is not true for longitude: the position of the Polaris does not change longitude as you move east or west. You can see this in the two-dimensional image below. The points P and Q are located on different meridian lines, but the angles caused by the position of the Polaris are the same. When you move east or west, the change is the time of the day. Every 15 degrees east, local time moves one hour forward, and 15 degrees west, and you move back for an hour. So if you know local time and Greenwich time, you can use time difference to calculate longitude. Even without a clock, it’s easy to find local time: you just need to look at the position of the sun. But the only way to know Greenwich time is to carry a clock with you. It seems easy to do so today, but until recently, it was a huge problem. The existing clocks are too sensitive to be taken to the ship: swinging and rolling can make them inaccurate. Sailors were unable to determine that longitude would have disastrous consequences. One example is the Sili sea crash of 1707, of which four British ships sank near the siley islands, very close to their Portsmouth home port. As sailors were unable to determine their exact location and, under the influence of severe weather, the ships hit rocks, killing as many as 2000 people, and were one of the most serious maritime disasters in British history. Several countries, including the Netherlands and Spain, offered huge rewards to anyone who could solve the problem of longitude. In 1714, the UK followed this practice and provided a huge reward of up to 20000 pounds, so various solutions were proposed. Someone was holding an injured dog on board, and according to him, because of mysterious alchemy, the dog barked more than every noon in Greenwich, while others were more scientific. For example, comparing the location of the moon with other stars, and then looking at the detailed catalog of stars, you can determine the local time fairly accurately. However, the method is time-consuming and error prone. It was the working class from Lincolnshire, who won the prize by a joiner. Since 1730, John Harrison has been working on the clock at sea, and he has been working on the latest two clocks that are accurate enough to win the prize. However, the claim for a prize has become a long process because the Commission refuses to pay the full prize. He appealed to King George III, and then to parliament to correct it. “God testify, Harrison, I’ll see you get the right way!” It is said that King George said. Harrison finally received the rest of the money and the honors he deserved in 1773. Three years later, he died at the age of 83. In addition, the story of longitude is described in Dava Sobel’s longitude, and you are interested in reading it.