Actually  a system of taximeter was invented in ancient Rome, to calculate the taxi (chariots)  fare: 

An odometer for measuring distance was first described by Vitruvius around 27 and 23 BC, although the actual inventor may have been Archimedes of Syracuse (c. 287 BC – c. 212 BC) during the First Punic War. Hero of Alexandria (10 AD - 70 AD) describes a similar device in chapter 34 of his Dioptra. The machine was also used in the time of Roman EmperorCommodus (c. 192 AD), although after this point in time there seems to be a gap between its use in Roman times and that of the 15th century in Western Europe.[3] Some researchers have speculated that the device might have included technology similar to that of the GreekAntikythera mechanism.[4]

The odometer of Vitruvius was based on chariot wheels of 4 feet (1.2 m) diameter turning 400 times in one Roman mile (about 1400 m). For each revolution a pin on the axle engaged a 400 tooth cogwheel thus turning it one complete revolution per mile. This engaged another gear with holes along the circumference, where pebbles (calculus) were located, that were to drop one by one into a box. The distance traveled would thus be given simply by counting the number of pebbles.[4] Whether this instrument was ever built at the time is disputed. Leonardo da Vinci later tried to build it himself according to the description, but failed. However, in 1981 engineer Andre Sleeswyk built his own replica, replacing the square-toothed gear designs of da Vinci with the triangular, pointed teeth found in the Antikythera mechanism. With this modification, the Vitruvius odometer functioned perfectly.[4]
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see the video :

The Renault Taxi de la Marne (Marne Taxi) is an automobile manufactured between 1905 and 1910 by Renault and used as a taxicab. The name Taxi de la Marne was not used until the outbreak of World War I, when the fleet of Paris taxis was requisitioned by the French Army to transport troops from Paris to the First Battle of the Marne in early September 1914.

It was the first car produced after Marcel Renault's death in 1903, along with another four models.

A car-rental company in Paris ordered 1,500 cars in 1905 as a result of a new invention that automatically calculated how much the passenger had to pay. It was called a taximeter and had been invented in 1891. Soon the Taxi de la Marne was popular for the rest of the decade in Paris and also in London from 1907.
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Picture of Daimler 1898

Modern taxicabs
The Daimler Victoria was built in 1897 as the first gasoline-powered taxicab.
Electric battery-powered taxis became available at the end of the 19th century. In London, Walter C. Bersey designed a fleet of such cabs and introduced them to the streets of London in 1897. They were soon nicknamed 'Hummingbirds’ due to the idiosyncratic humming noise they made.[3] In the same year in New York City, the Samuel's Electric Carriage and Wagon Company began running 12 electric hansom cabs.[8] The company ran until 1898 with up to 62 cabs operating until it was reformed by its financiers to form the Electric Vehicle Company.

The modern taximeter was invented by German inventor Friedrich Wilhelm Gustav Bruhn.and the Daimler Victoria—the world's first meter-equipped (and gasoline-powered) taxicab—was built by Gottlieb Daimler in 1897 and began operating in Stuttgart in 1897. Gasoline-powered taxicabs began operating in Paris in 1899, in London in 1903, and in New York in 1907. The New York taxicabs were imported from France by Harry N. Allen who decided to paint his taxicabs yellow to maximise his vehicles' visibility.

Taxicabs proliferated around the world in the early 20th century. The first major innovation after the invention of the taximeter occurred in the late 1940s, when two-way radios first appeared in taxicabs. Radios enabled taxicabs and dispatch offices to communicate and serve customers more efficiently than previous methods, such as using callboxes. The next major innovation occurred in the 1980s, when computer assisted dispatching was first introduced.[citation needed]
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