Length of Martian year

The Danish astronomer Tycho Brahe made surprisingly accurate calculations of the position of Mars 20 years before the telescope was invented! In 1576, Brahe set up an observatory in Hven, an island near Copenhagen where he studied the stars for 20 years.
Tycho Brahe, Astronomiae instauratae mechanica, 1598
With a combination of his naked eyesight and large instruments, Brahe calculated the position of Mars over time to within four arcminutes. If you examine this data you will find that Mars returns to the same position in its orbit every 687 days. This timespan - a Martian year - is almost two Earth years.
The current known value of Mars’ orbital period: 686.98 Earth solar days

Length of Martian day

How would we go about determining the length of a full day on Mars?
Here is a hint: we need to be able to resolve surface features on Mars. That is, by seeing a surface feature at a given time and measuring how long it takes for the feature to appear in the same place again, we can measure how fast Mars rotates on its axis (a measure of a “day”).  The ability to resolve surface features wasn’t possible until the invention of telescopes.
Giuseppe Bertini, Painting of Galeleo Gelilei, 1585
Although Galileo observed Mars with a telescope in 1609, Mars still appeared to be too tiny to resolve features due to the magnification power, which was around 20 times that of the human eye. He simply described it as a “spherical body illuminated by the Sun.”  To him, Mars would have appeared to be about the same size as a pea held at a distance of 2.4 meters (8 feet). He wasn't able to resolve any surface features at that scale.
Advancements in telescope design resolved these problems by extending the lengths of telescopes to 15 - 20 feet by the middle of the 17th century.
Astroscopia Compendiari, compound telescope without a tube, 1684
The Dutch astronomer Christiaan Huygens (1629 - 1695) drew Mars using an advanced telescope of his own design. It was 23 feet long and magnified objects about a hundred times. With his telescope, Mars would have looked about as big as the Moon does with a naked eye, though with far less detail:
Artist rendition of Mars' details as seen through Huygen's telescope. Image: Brit Cruise
On November 1659, Huygens saw a prominent and irregularly shaped dark area on the surface of Mars. Over the course of the evening, he made note of observable changes in the position of that dark area, recognizable to modern observers as a visibly darker area on Mars known as Syrtis Major.
Image: International Planetary Cartography Database, 1659
Huygens noticed that the spot returned to the same position at the same time the next day, and calculated that Mars has a 24-hour period of rotation. In 1666, Gian Domenico Cassini refined this estimate, measuring Mars’s rotational period as 24 hours 40 minutes, which is very close to the actual value known today:
Length of Martian solar day: 24h 39m 35.2s
Today, a Martian day is called a “sol.”