Antiquated "Odometer"

Introduction: The Odometer

The term 'odometer' originates from ancient Greek [hodómetron, from hodós (“path” or “gateway”) and métron (“measure”)] and refers to a device that measures the distance traveled by a vehicle, whether on land or at sea. While most modern odometers are electronic or electromechanical, their ancient counterparts were purely mechanical. Although designs have evolved over time, the fundamental purpose of odometers has remained unchanged – to measure distance. These devices have long been used in a variety of vehicles, from modern automobiles and bicycles to horse-drawn carts of the past.

Possibly the first evidence for the use of an odometer can be found in the works of the ancient Greek Strabo (64 – 63 BC-24 AD) and the ancient Roman Pliny (23 – 79 AD).

Both authors list the distances of routes traveled by Alexander the Great (356 – 323 BC) as by his bematists Diognetus and Baeton: bematists were trained to measure distance while counting their steps but the high accuracy of the bematists’ measurements rather indicates the use of a mechanical device. In fact from the nine surviving bematists’ measurements in Pliny’s Naturalis Historia eight show a deviation of less than 5% from the actual distance, three of them being within 1%: the overall accuracy of the measurements implies that the bematists already must have used a sophisticated device for measuring distances, although there is no direct mention of such a device.

The first descriptions of this instrument have reached us through the manuscripts of the Roman architect and military engineer Marcus Vitruvius Pollio (80 – 15 BC).

Around 30 – 15 BC he wrote the famous “De architectura” dedicated to his patron, the emperor Caesar Augustus (Vitruvius, 1990): it is a ten books treatise meant as a guide for realizing projects such as buildings, military camps, facilities, instruments and machines. The odometer described by Vitruvius will be deeply explained in the next section.

As a matter of fact, the odometer has been studied by many scholars throughout history: Hero of Alexandria (10 – 70 BC) described an odometer set in motion by the rotation of a cart’s wheel and Leon Battista Alberti in the mid-fifteenth century studied this instrument in the treatise “Ludi rerum mathematicarum”. Between 1500 and 1504 Leonardo da Vinci (1452-1519) too was interested in the design of the odometer and drew few sketches on folio 1 of the Atlantic Codex preserved today in the Veneranda Biblioteca Ambrosiana in Milan.

In the following centuries prototypes of odometers have been built by many other scientists and inventors, such as Blaise Pascal (1623-1662) who used the same concept for the calculating machine called “Pascaline”, Thomas Savery (1650-1715) who perfectioned an odometer for ships, Benjamin Franklin (1706- 1790) who created a simple odometer to help measure the mileage of the routes when he decided to analyze the best routes for delivering the mail and many others.

In the end, it must be noticed that the odometer was also independently invented in ancient China (Needham, 1965) possibly by the prolific inventor and early scientist Zhang Heng (78 AD – 139 AD) or by the later Ma Jun (active during 220 – 265 AD); despite such associations, there is evidence to suggest that the invention of the odometer was a gradual process in Han Dynasty China that centered around the huang men court people (i.e. eunuchs, palace officials, attendants and familiars, actors, acrobats, etc.) that would follow the musical procession of the royal “drum-chariot”.

Vitruvius’ Odometer

Vitruvius describes two versions of the odometer in the Chapter 9 of book X of “De Architectura”: one for measuring marine distances and the other for land travels, which is the one that will be commented here.

Before turning to Vitruvius’ odometer, it is useful to briefly recall the most important measurement units used in ancient Rome:

    the “foot”, corresponding to 296.4 mm;

    the “double step”, which was 5 feet long or 1482 mm;

    the “mile”, corresponding to 1,000 double steps (the word mile derives from the Latin expression “millia passuum”), which was therefore equal to 1482 m.

Note: Odometer described by Vitruvius was 4 Roman feet in diameter ( circumference = 12.566 Roman feet), therefore 400 turns would be equal distance of 5025.4 Roman feet = 1489.8 m = 1 Roman mile). The 4 feet diameter of the wheel mentioned by Vitruvius, must have had slight reduction in circumference when we account for the “teeth”, because the Roman mile was 1,482 m.

Figure 3. Reconstruction of Vitruvius’ odometer according to Schofield (2016) (a 180 teeth gear is represented for sake of readability).

Vitruvius’ odometer, Figure 3, is driven by one of the two rear wheels of a four-wheeled carriage, called “raeda”. The wheels have a diameter of 4 feet (about 1.22 meters) which roughly corresponds to a circumference of 12.566 feet (3.83 m) ; therefore, it requires 400 turns of the wheels to measure the distance of 1 mile.

Figure 4. Sketch of Vitruvius’ odometer (a 180 teeth gear is represented for sake of readability).

With reference to Figure 4, Vitruvius reduces the 400 turns of rotation of the wheels to the single turn of a shaft using the coupling between a single tooth integral with one of the rear wheels (1) and a gear wheel with 400 teeth placed on a vertical plane (2). The latter in turn has only one tooth on its internal face: such tooth meshes with a third gear wheel (3), laid on a horizontal plane. With each complete rotation of the gear wheel (2), the tooth integral with it advances the wheel (3) by one position. So, every 400 rotations of the cartwheels, this has travelled 5000 feet, that is one mile, the 400 tooth wheel (2) has made a complete rotation and the upper disk (3) has rotated by one position.

On the wheel (3) in correspondence to each tooth there is a hole that contains a pebble while the lower face of the wheel is in contact with a container that has a single hole: in this way at every mile travelled a pebble falls into the container, allowing to measure the number of miles travelled during the day.

Figure 3 presents a CAD reconstruction of the Vitruvius’ odometer realized by the authors according to the hypotheses of Schofield (2016), but it must be said that different alternatives are also possible, especially as regards the orientation and positioning of the 400 teeth vertical disk (2), about which Vitruvius does not provide further information. Figure 5, for example, shows the reconstruction of Sleeswyk (1981).

Figure 5. Reconstruction of Vitruvius’ odometer according to Sleeswyk (1981) (a 100 teeth gear is represented for sake of readability).

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DE ARCHITECTURA by  MARCUS VITRUVIUS POLLIO

BOOK X CHAPTER 9

1. Let us now consider an invention by no means useless and delivered to us by the ancients as of ingenuity, by means of which, when on a journey by land or sea, one may ascertain the distance travelled. It is as follows. The wheels of the chariot must be four feet diameter; so that, marking a certain point thereon, whence it begins its revolution on the ground, when it has completed that revolution, it will have gone on the road over a space equal to twelve feet and a half. [ Editor’s Note: 4 feet x Pi = 15.566 feet circumference ]

2. This being adjusted on the inner side of the nave of the wheel, let a drum-wheel be securely fixed, having one small tooth projecting beyond the face of its circumference; and in the body of the chariot let a small box be fastened, with a drum-wheel placed to revolve perpendicularly, and fastened to an axle. The latter wheel is to be equally divided, on its edge, into four hundred teeth, corresponding with the teeth of the lower drum-wheel: besides the above the upper drum-wheel has on its side one tooth projecting out before the others.

3. Above, in another enclosure, is a third horizontal wheel toothed similarly, and so that the teeth correspond with that tooth which is fixed to the side of the second wheel. In the third wheel just described are as many holes as are equal to the number of miles in an usual day’s journey. It does not, however, signify, if they be more or less. In all the holes let small balls be placed, and in the box or lining let a hole be made, having a channel, through which each ball may fall into the box of the chariot, and the brazen vessel placed under it.

4. Thus, as the wheel proceeds, it acts on the first drum-wheel, the tooth of which, in every revolution, striking the tooth of the upper wheel, causes it to move on; so that when the lower wheel as revolved four hundred times, the upper wheel has revolved only once, and its tooth, which is on the side, will have acted on only one tooth of the horizontal wheel. Now as in four hundred revolutions of the lower wheel, the upper wheel will only have turned round once, the length of the journey will be five thousand feet, or one thousand paces. Thus, by the dropping of the balls, and of the noise they make, we know every mile passed over; and each day one may ascertain, by the number of balls collected in the bottom, the number of miles in the day’s journey.

5. In navigation, with very little change in the machinery, the same thing may be done. An axis is fixed across the vessel, whose ends project beyond the sides, to which are attached wheels four feet diameter, with paddles to them touching the water. That part of the axis within the vessel has a wheel with a single tooth standing out beyond its face; at which place a box is fixed with a wheel inside it having four hundred teeth, equal and correspondent to the tooth of the first wheel fixed on the axis. On the side of this, also, projecting from its face, is another tooth.

6. Above, in another box, is enclosed another horizontal wheel, also toothed, to correspond with the tooth that is fastened to the side of the vertical wheel, and which, in every revolution, working in the teeth of the horizontal wheel, and striking one each time, causes it to turn round. In this horizontal wheel holes are made, wherein the round balls are placed; and in the box of the wheel is a hole with a channel to it, through which the ball descending without obstruction, falls into the brazen vase, and makes it ring.

7. Thus, when the vessel is on its way, whether impelled by oars or by the wind, the paddles of the wheels, driving back the water which comes against them with violence, cause the wheels to revolve, whereby the axle is also turned round, and consequently with it the drum-wheel, whose tooth, in every revolution, acts on the tooth in the second wheel, and produces moderate revolutions thereof. Wherefore, when the wheels are carried round by the paddles four hundred times, the horizontal wheel will only have made one revolution, by the striking of that tooth on the side of the vertical wheel, and thus, in the turning caused by the horizontal wheel every time it brings a ball to the hole it falls through the channel. In this way, by sound and number, the number of miles navigated will be ascertained. It appears to me, that I completed the description in such a manner that it will be easy to comprehend the structure of the machine, which will afford both utility and amusement in times of peace and safety.

Translation Source

Sleeswyk (1981) suggests that the odometer could have been invented by Archimedes instead and just reported by Vitruvius in the De Architectura. In fact the concept of leaving a pebble drop to set the passing of time is indisputably due to Archimedes who used it in several inventions, first of all the famous water clock. Moreover, Archimedes is the inventor of the worm screw that we saw is fundamental to odometer functionality. In the end, Sleeswyk brings also historic arguments to his thesis: when Archimedes was a promising young scientist, the Romans experienced the maximum expansion efforts and needed new routes to be built. Archimedes was probably a relative of Hieron II, the tyrant of Syracuse in Sicily and in the end allied with Romans during the first Punic War: in this period the Romans built about 750 kilometers of new routes, that needed the pose of milestones during the way, and Archimedes could well have helped them measuring the distances.

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Conclusion

The study of the writings of Vitruvius and Leonardo tells us how the liaisons between the two scholars were strong despite the 1500 years that separate them. Leonardo was fascinated by the work of Vitruvius and has been looking for a copy of “De Architectura” for many years.

The exhibition in Fano (July 12-Oct. 13, 2019) has shown how much Vitruvius influenced Leonardo’s work. As for the odometer, Leonardo tried to draw the original concept by Vitruvius in the central drawing of folio 1r-b of the Codex Atlanticus (Figure 6) but probably gave up understanding it was practically impossible to realize. In fact the teeth of gear wheels in past years had the shape of equilateral triangles as shown for example in the Antikythera gearwheels and a 1:400 gear ratio causes a too large diameter or too small teeth, as would happen todays too with current design criteria. Leonardo overcame the problem in his design (see the left of Figure 6) by using a worm gear and a single tooth meshing with a peg wheel.

It is strange, however, that a practical and experienced engineer as Vitruvius had not taken into consideration this aspect of the design. Sleeswyk suggests that Vitruvius never built the odometer but simply described an invention dated back to Archimedes of Syracuse, who lived and cooperated with the Romans when they built some 750 kilometers of roads. In this case, the sentence “drum equipped with a single tooth protruding from its circular surface” would refer to a worm, that Archimedes well knew. In this way the odometer would connect three giants of the past: Archimedes, Vitruvius and Leonardo.

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