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115-431: Pascal's calculator was especially successful in the design of its carry mechanism , which adds 1 to 9 on one dial, and carries 1 to the next dial when the first dial changes from 9 to 0. His innovation made each digit independent of the state of the others, enabling multiple carries to rapidly cascade from one digit to another regardless of the machine's capacity. Pascal was also the first to shrink and adapt for his purpose

230-421: A bevel gear , whose overall shape is like a slice ( frustum ) of a cone whose apex is the meeting point of the two axes. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called miter (US) or mitre (UK) gears. Independently of the angle between the axes, the larger of two unequal matching bevel gears may be internal or external, depending the desired relative sense of rotation. If

345-421: A bevel gear , whose overall shape is like a slice ( frustum ) of a cone whose apex is the meeting point of the two axes. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called miter (US) or mitre (UK) gears. Independently of the angle between the axes, the larger of two unequal matching bevel gears may be internal or external, depending the desired relative sense of rotation. If

460-487: A differential . Whereas a regular (nonhypoid) ring-and-pinion gear set is suitable for many applications, it is not ideal for vehicle drive trains because it generates more noise and vibration than a hypoid does. Bringing hypoid gears to market for mass-production applications was an engineering improvement of the 1920s. Lantern gear A gear or gearwheel is a rotating machine part typically used to transmit rotational motion and/or torque by means of

575-507: A lantern gear , used in turret clocks and water wheels . This innovation allowed the device to resist the strength of any operator input with very little added friction. Pascal designed the machine in 1642. After 50 prototypes , he presented the device to the public in 1645, dedicating it to Pierre Séguier , then chancellor of France . Pascal built around twenty more machines during the next decade, many of which improved on his original design. In 1649, King Louis XIV of France gave Pascal

690-465: A livre and 12 deniers to a sol . Length was measured in toises , pieds , pouces and lignes with 6 pieds to a toise , 12 pouces to a pied and 12 lignes to a pouce . Therefore, the pascaline needed wheels in base 6, 10, 12 and 20. Non-decimal wheels were always located before the decimal part. In an accounting machine (..10,10,20,12), the decimal part counted the number of livres (20 sols ), sols (12 deniers ) and deniers . In

805-587: A royal privilege (similar to a patent ), which provided the exclusive right to design and manufacture calculating machines in France. Nine Pascal calculators presently exist; most are on display in European museums. Many later calculators were either directly inspired by or shaped by the same historical influences that had led to Pascal's invention. Gottfried Leibniz invented his Leibniz wheels after 1671, after trying to add an automatic multiplication feature to

920-480: A tax commissioner , Pascal hoped to provide a shortcut to hours of number crunching performed by workers in professions such as mathematics, physics, astronomy, etc. But, because of the intricacies of the device, the relationship Pascal had with craftsmen, and the intellectual property laws he influenced, the production of the Pascaline was far more limited than he had envisioned. Only 20 Pascalines were produced over

1035-529: A transmission or "gearbox" containing a set of gears that can be meshed in multiple configurations. The gearbox lets the operator vary the torque that is applied to the wheels without changing the engine's speed. Gearboxes are used also in many other machines, such as lathes and conveyor belts . In all those cases, terms like "first gear", "high gear", and "reverse gear" refer to the overall torque ratios of different meshing configurations, rather than to specific physical gears. These terms may be applied even when

1150-529: A transmission or "gearbox" containing a set of gears that can be meshed in multiple configurations. The gearbox lets the operator vary the torque that is applied to the wheels without changing the engine's speed. Gearboxes are used also in many other machines, such as lathes and conveyor belts . In all those cases, terms like "first gear", "high gear", and "reverse gear" refer to the overall torque ratios of different meshing configurations, rather than to specific physical gears. These terms may be applied even when

1265-401: A 10,000-wheel machine, if one existed, the operator would have to set every wheel to its maximum and then add a 1 to the "unit" wheel. The carry would turn every input wheel one by one in a very rapid Domino effect fashion and all the display registers would be reset. The carry transmission has three phases: The Pascaline is a direct adding machine (it has no crank), so the value of a number

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1380-439: A carry right through the machine, is the most demanding task for a mechanical calculator and proves, before each operation, that the machine is fully functional. This is a testament to the quality of the Pascaline because none of the 18th century criticisms of the machine mentioned a problem with the carry mechanism and yet this feature was fully tested on all the machines, by their resets, all the time. Additions are performed with

1495-423: A decimal machine, the digits 0 through 9 are carved clockwise, with each digit positioned between two spokes so that the operator can directly inscribe its value in the window of complements by positioning his stylus in between them and turning the wheel clockwise all the way to the stop lever. The marks on two adjacent spokes flank the digit 0 inscribed on this wheel. On four of the known machines, above each wheel,

1610-625: A geared astrolabe was built in Isfahan showing the position of the moon in the zodiac and its phase , and the number of days since new moon. The worm gear was invented in the Indian subcontinent , for use in roller cotton gins , some time during the 13th–14th centuries. A complex astronomical clock, called the Astrarium , was built between 1348 and 1364 by Giovanni Dondi dell'Orologio . It had seven faces and 107 moving parts; it showed

1725-463: A geared astrolabe was built in Isfahan showing the position of the moon in the zodiac and its phase , and the number of days since new moon. The worm gear was invented in the Indian subcontinent , for use in roller cotton gins , some time during the 13th–14th centuries. A complex astronomical clock, called the Astrarium , was built between 1348 and 1364 by Giovanni Dondi dell'Orologio . It had seven faces and 107 moving parts; it showed

1840-419: A great variety of shapes and materials, and are used for many different functions and applications. Diameters may range from a few μm in micromachines , to a few mm in watches and toys to over 10 metres in some mining equipment. Other types of parts that are somewhat similar in shape and function to gears include the sprocket , which is meant to engage with a link chain instead of another gear, and

1955-419: A great variety of shapes and materials, and are used for many different functions and applications. Diameters may range from a few μm in micromachines , to a few mm in watches and toys to over 10 metres in some mining equipment. Other types of parts that are somewhat similar in shape and function to gears include the sprocket , which is meant to engage with a link chain instead of another gear, and

2070-526: A little monster appear, that lacks its principal limbs, the others being deformed, lacking any proportion.” Pascal operated his project with this hierarchy in mind: he invented and thought, while the artisans simply executed. He hid the theory from artisans, instead promoting that they should simply remember what to do, not necessarily why they should do it, i.e., until "practice has made the rules of theory so common that [the rules] have finally been reduced into art”. This stemmed from his lack of faith in not only

2185-517: A motor communicates motion' is from 1814; specifically of a vehicle (bicycle, automobile, etc.) by 1888. A cog is a tooth on a wheel. From Middle English cogge, from Old Norse (compare Norwegian kugg ('cog'), Swedish kugg , kugge ('cog, tooth')), from Proto-Germanic * kuggō (compare Dutch kogge (' cogboat '), German Kock ), from Proto-Indo-European * gugā ('hump, ball') (compare Lithuanian gugà ('pommel, hump, hill'), from PIE * gēw- ('to bend, arch'). First used c. 1300 in

2300-517: A motor communicates motion' is from 1814; specifically of a vehicle (bicycle, automobile, etc.) by 1888. A cog is a tooth on a wheel. From Middle English cogge, from Old Norse (compare Norwegian kugg ('cog'), Swedish kugg , kugge ('cog, tooth')), from Proto-Germanic * kuggō (compare Dutch kogge (' cogboat '), German Kock ), from Proto-Indo-European * gugā ('hump, ball') (compare Lithuanian gugà ('pommel, hump, hill'), from PIE * gēw- ('to bend, arch'). First used c. 1300 in

2415-482: A nobleman to do for a commoner at the time. Morland was able to recruit the best talent in Europe. His first craftsmen was the famous Peter Blondeau , who had already received protection and recognition from French statesman Richelieu for his contributions in producing coinage for England. Morland's other craftsmen were similarly accomplished: the third, Dutchman John Fromanteel , came a famous Dutch family who pioneered

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2530-702: A pointer on top of the chariot kept the direction of latter unchanged as the chariot turned. Another early surviving example of geared mechanism is a complex calendrical device showing the phase of the Moon, the day of the month and the places of the Sun and the Moon in the Zodiac was invented in the Byzantine empire in the early 6th century AD. Geared mechanical water clocks were built in China by 725 AD. Around 1221 AD,

2645-441: A pointer on top of the chariot kept the direction of latter unchanged as the chariot turned. Another early surviving example of geared mechanism is a complex calendrical device showing the phase of the Moon, the day of the month and the places of the Sun and the Moon in the Zodiac was invented in the Byzantine empire in the early 6th century AD. Geared mechanical water clocks were built in China by 725 AD. Around 1221 AD,

2760-521: A price; craftsmen were not able to legally experiment with Pascal's design, nor were they able to distribute his machine without his permission/guidance. Pascal lived in France during France's Ancien Régime . During his time, craftsmen in Europe increasingly organised into guilds , such as the English clockmakers who formed the Clockmakers guild in 1631, half-way through Pascal's efforts to create

2875-414: A series of teeth that engage with compatible teeth of another gear or other part. The teeth can be integral saliences or cavities machined on the part, or separate pegs inserted into it. In the latter case, the gear is usually called a cogwheel . A cog may be one of those pegs or the whole gear. Two or more meshing gears are called a gear train . The smaller member of a pair of meshing gears

2990-414: A series of teeth that engage with compatible teeth of another gear or other part. The teeth can be integral saliences or cavities machined on the part, or separate pegs inserted into it. In the latter case, the gear is usually called a cogwheel . A cog may be one of those pegs or the whole gear. Two or more meshing gears are called a gear train . The smaller member of a pair of meshing gears

3105-605: A series of wooden pegs or cogs around the rim of a wheel. The cogs were often made of maple wood. Wooden gears have been gradually replaced by ones made or metal, such as cast iron at first, then steel and aluminum . Steel is most commonly used because of its high strength-to-weight ratio and low cost. Aluminum is not as strong as steel for the same geometry, but is lighter and easier to machine. powder metallurgy may be used with alloys that cannot be easily cast or machined. Still, because of cost or other considerations, some early metal gears had wooden cogs, each tooth forming

3220-605: A series of wooden pegs or cogs around the rim of a wheel. The cogs were often made of maple wood. Wooden gears have been gradually replaced by ones made or metal, such as cast iron at first, then steel and aluminum . Steel is most commonly used because of its high strength-to-weight ratio and low cost. Aluminum is not as strong as steel for the same geometry, but is lighter and easier to machine. powder metallurgy may be used with alloys that cannot be easily cast or machined. Still, because of cost or other considerations, some early metal gears had wooden cogs, each tooth forming

3335-451: A similar lack of commercial success. Most of the machines that have survived the centuries are of the accounting type. Seven of them are in European museums, one belongs to the IBM corporation and one is in private hands. Pascal planned to distribute the Pascaline broadly in order to reduce the workload for people who needed to perform laborious arithmetic. Drawing inspiration from his father,

3450-538: A small quotient wheel is mounted on the display bar. These quotient wheels, which are set by the operator, have numbers from 1 to 10 inscribed clockwise on their peripheries (even above a non-decimal wheel). Quotient wheels seem to have been used during a division to memorize the number of times the divisor was subtracted at each given index. Pascal went through 50 prototypes before settling on his final design; we know that he started with some sort of calculating clock mechanism which apparently "works by springs and which has

3565-400: A surveyor's machine (..10,10,6,12,12), the decimal part counted the number of toises (6 pieds ), pieds (12 pouces ), pouces (12 lignes ) and lignes . Scientific machines just had decimal wheels. The decimal part of each machine is highlighted. The metric system was adopted in France on December 10, 1799, by which time Pascal's basic design had inspired other craftsmen, although with

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3680-473: A tax commissioner, and sought to produce a device which could reduce some of his workload. Pascal received a Royal Privilege in 1649 that granted him exclusive rights to make and sell calculating machines in France. By 1654 he had sold about twenty machines (only nine of those twenty machines are known to exist today), but the cost and complexity of the Pascaline was a barrier to further sales and production ceased in that year. By that time Pascal had moved on to

3795-400: A type of specialised 'through' mortise and tenon joint More recently engineering plastics and composite materials have been replacing metals in many applications, especially those with moderate speed and torque. They are not as strong as steel, but are cheaper, can be mass-manufactured by injection molding don't need lubrication. Plastic gears may even be intentionally designed to be

3910-400: A type of specialised 'through' mortise and tenon joint More recently engineering plastics and composite materials have been replacing metals in many applications, especially those with moderate speed and torque. They are not as strong as steel, but are cheaper, can be mass-manufactured by injection molding don't need lubrication. Plastic gears may even be intentionally designed to be

4025-402: A very simple design", was used "many times" and remained in "operating order". Nevertheless, "while always improving on it" he found reason to try to make the whole system more reliable and robust. Eventually he adopted a component of very large clocks, shrinking and adapting for his purpose the robust gears that can be found in a turret clock mechanism called a lantern gear , itself derived from

4140-454: A water wheel mechanism. This could easily handle the strength of an operator input. Pascal adapted a pawl and ratchet mechanism to his own turret wheel design; the pawl prevents the wheel from turning counterclockwise during an operator input, but it is also used to precisely position the display wheel and the carry mechanism for the next digit when it is pushed up and lands into its next position. Because of this mechanism, each number displayed

4255-762: Is ⁠ ( A − B ) {\displaystyle (A-B)} ⁠ . It feels like an addition since the only two differences in between an addition and a subtraction are the position of the display bar (direct versus complement) and the way the first number is entered (direct versus complement). The following table shows all the steps required to compute 54,321-12,345=41,976 Pascalines came in both decimal and non-decimal varieties, both of which can be viewed in museums today. They were designed for use by scientists, accountants and surveyors. The simplest Pascaline had five dials; later variants had up to ten dials. The contemporary French currency system used livres , sols and deniers with 20 sols to

4370-414: Is added to the accumulator as it is being dialed in. By moving a display bar, the operator can see either the number stored in the calculator or the complement of its value. Subtractions are performed like additions using some properties of 9's complement arithmetic. The 9's complement of any one-digit decimal number d is 9- d . So the 9's complement of 4 is 5 and the 9's complement of 9 is 0. Similarly,

4485-424: Is displayed and then mark the spoke under the stopping lever and the one to the right of it. Four of the known machines have inner wheels of complements, which were used to enter the first operand in a subtraction. They are mounted at the center of each spoked metal wheel and turn with it. The wheel displayed in the picture above has an inner wheel of complements, but the numbers written on it are barely visible. On

4600-506: Is independent of the other. When it is time to propagate a carry, the sautoir, under the sole influence of gravity, is thrown toward the next wheel without any contact between the wheels. During its free fall the sautoir behaves like an acrobat jumping from one trapeze to the next without the trapezes touching each other ("sautoir" comes from the French verb sauter , which means to jump). All the wheels (including gears and sautoir) have therefore

4715-425: Is not only acceptable but desirable. For basic analysis purposes, each gear can be idealized as a perfectly rigid body that, in normal operation, turns around a rotation axis that is fixed in space, without sliding along it. Thus, each point of the gear can move only along a circle that is perpendicular to its axis and centered on it. At any moment t , all points of the gear will be rotating around that axis with

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4830-425: Is not only acceptable but desirable. For basic analysis purposes, each gear can be idealized as a perfectly rigid body that, in normal operation, turns around a rotation axis that is fixed in space, without sliding along it. Thus, each point of the gear can move only along a circle that is perpendicular to its axis and centered on it. At any moment t , all points of the gear will be rotating around that axis with

4945-440: Is often called pinion . Most commonly, gears and gear trains can be used to trade torque for rotational speed between two axles or other rotating parts and/or to change the axis of rotation and/or to invert the sense of rotation. A gear may also be used to transmit linear force and/or linear motion to a rack , a straight bar with a row of compatible teeth. Gears are among the most common mechanical parts. They come in

5060-440: Is often called pinion . Most commonly, gears and gear trains can be used to trade torque for rotational speed between two axles or other rotating parts and/or to change the axis of rotation and/or to invert the sense of rotation. A gear may also be used to transmit linear force and/or linear motion to a rack , a straight bar with a row of compatible teeth. Gears are among the most common mechanical parts. They come in

5175-421: Is perfectly centered in the display window and each digit is precisely positioned for the next operation. This mechanism would be moved six times if the operator dialed a six on its associated input wheel. The sautoir is the centerpiece of the pascaline's carry mechanism. In his " Avis nécessaire... ", Pascal noted that a machine with 10,000 wheels would work as well as a machine with two wheels because each wheel

5290-512: Is produced by net shape molding. Molded gearing is usually powder metallurgy, plastic injection, or metal die casting. Gears produced by powder metallurgy often require a sintering step after they are removed from the mold. Cast gears require gear cutting or other machining to shape the teeth to the necessary precision. The most common form of gear cutting is hobbing , but gear shaping , milling , and broaching may be used instead. Metal gears intended for heavy duty operation, such as in

5405-512: Is produced by net shape molding. Molded gearing is usually powder metallurgy, plastic injection, or metal die casting. Gears produced by powder metallurgy often require a sintering step after they are removed from the mold. Cast gears require gear cutting or other machining to shape the teeth to the necessary precision. The most common form of gear cutting is hobbing , but gear shaping , milling , and broaching may be used instead. Metal gears intended for heavy duty operation, such as in

5520-637: Is reversed when one gear wheel drives another gear wheel. Philon of Byzantium was one of the first who used gears in water raising devices. Gears appear in works connected to Hero of Alexandria , in Roman Egypt circa AD 50, but can be traced back to the mechanics of the Library of Alexandria in 3rd-century BC Ptolemaic Egypt , and were greatly developed by the Greek polymath Archimedes (287–212 BC). The earliest surviving gears in Europe were found in

5635-472: Is reversed when one gear wheel drives another gear wheel. Philon of Byzantium was one of the first who used gears in water raising devices. Gears appear in works connected to Hero of Alexandria , in Roman Egypt circa AD 50, but can be traced back to the mechanics of the Library of Alexandria in 3rd-century BC Ptolemaic Egypt , and were greatly developed by the Greek polymath Archimedes (287–212 BC). The earliest surviving gears in Europe were found in

5750-462: The Antikythera mechanism an example of a very early and intricate geared device, designed to calculate astronomical positions of the sun, moon, and planets, and predict eclipses . Its time of construction is now estimated between 150 and 100 BC. The Chinese engineer Ma Jun (c. 200–265 AD) described a south-pointing chariot . A set of differential gears connected to the wheels and to

5865-403: The Antikythera mechanism an example of a very early and intricate geared device, designed to calculate astronomical positions of the sun, moon, and planets, and predict eclipses . Its time of construction is now estimated between 150 and 100 BC. The Chinese engineer Ma Jun (c. 200–265 AD) described a south-pointing chariot . A set of differential gears connected to the wheels and to

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5980-484: The timing pulley , meant to engage a timing belt . Most gears are round and have equal teeth, designed to operate as smoothly as possible; but there are several applications for non-circular gears , and the Geneva drive has an extremely uneven operation, by design. Gears can be seen as instances of the basic lever "machine". When a small gear drives a larger one, the mechanical advantage of this ideal lever causes

6095-432: The timing pulley , meant to engage a timing belt . Most gears are round and have equal teeth, designed to operate as smoothly as possible; but there are several applications for non-circular gears , and the Geneva drive has an extremely uneven operation, by design. Gears can be seen as instances of the basic lever "machine". When a small gear drives a larger one, the mechanical advantage of this ideal lever causes

6210-431: The transmissions of cars and trucks, the teeth are heat treated to make them hard and more wear resistant while leaving the core soft but tough . For large gears that are prone to warp, a quench press is used. Gears can be made by 3D printing ; however, this alternative is typically used only for prototypes or very limited production quantities, because of its high cost, low accuracy, and relatively low strength of

6325-431: The transmissions of cars and trucks, the teeth are heat treated to make them hard and more wear resistant while leaving the core soft but tough . For large gears that are prone to warp, a quench press is used. Gears can be made by 3D printing ; however, this alternative is typically used only for prototypes or very limited production quantities, because of its high cost, low accuracy, and relatively low strength of

6440-467: The 10 years following its creation. In 1649, King Louis XIV of France gave Pascal a royal privilege (a precursor to the patent ), which provided the exclusive right to design and manufacture calculating machines in France, allowing the Pascaline to be the first calculator sold by a distributor. Pascal feared that craftsmen would not be able to accurately reproduce his Pascaline, which would result in false copies that would ruin his reputation along with

6555-423: The 11's complement of 3 is 8. In a decimal machine with n dials the 9's complement of a number A is: and therefore the 9's complement of (A-B) is: In other words, the 9's complement of the difference of two numbers is equal to the sum of the 9's complement of the minuend added to the subtrahend. The same principle is valid and can be used with numbers composed of digits of various bases (base 6, 12, 20), like in

6670-801: The 17th century, had the progress for his machine halted due to his artisan selling the machine's parts for financial solvency. Pascal’s own conduct led to difficulty in recruiting artisans for his project. This was rooted by his belief that matters of the mind trumped those of the body. Pascal was not alone, as many natural philosophers of his time had a hylomorphic understanding of the inventing process: ideas precede materialisation, as form precedes matter. This naturally led to an emphasis on theoretical purity and an underappreciation for practical work. As Pascal described artisans: “[they] work through groping trial and error, that is, without certain measures and proportions regulated by art, produc[ing] nothing corresponding to what they had sought, or, what’s more, they make

6785-444: The Pascaline. In 1820, Thomas de Colmar designed his arithmometer , the first mechanical calculator strong enough and reliable enough to be used daily in an office environment. It is not clear whether he ever saw Leibniz's device, but he either re-invented it or utilized Leibniz's invention of the step drum. Blaise Pascal began to work on his calculator in 1642, when he was 18 years old. He had been assisting his father, who worked as

6900-571: The Thompson Manufacturing Company of Lancaster, New Hampshire still had a very active business in supplying tens of thousands of maple gear teeth per year, mostly for use in paper mills and grist mills , some dating back over 100 years. The most common techniques for gear manufacturing are dies , sand , and investment casting ; injection molding ; powder metallurgy ; blanking ; and gear cutting . As of 2014, an estimated 80% of all gearing produced worldwide

7015-486: The Thompson Manufacturing Company of Lancaster, New Hampshire still had a very active business in supplying tens of thousands of maple gear teeth per year, mostly for use in paper mills and grist mills , some dating back over 100 years. The most common techniques for gear manufacturing are dies , sand , and investment casting ; injection molding ; powder metallurgy ; blanking ; and gear cutting . As of 2014, an estimated 80% of all gearing produced worldwide

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7130-427: The artisanal work process, but in the artisans themselves: “artisans cannot regulate themselves to produce unified machines autonomously." In contrast, Samuel Morland , one of Pascal's contemporaries also working on creating a calculating machine, likely succeeded because of his ability to manage good relations with his craftsmen. Morland proudly attributed part of his invention to the artisans by name– an odd thing for

7245-450: The axes, each section of one gear will interact only with the corresponding section of the other gear. Thus the three-dimensional gear train can be understood as a stack of gears that are flat and infinitesimally thin — that is, essentially two-dimensional. In a crossed arrangement, the axes of rotation of the two gears are not parallel but cross at an arbitrary angle except zero or 180 degrees. For best operation, each wheel then must be

7360-450: The axes, each section of one gear will interact only with the corresponding section of the other gear. Thus the three-dimensional gear train can be understood as a stack of gears that are flat and infinitesimally thin — that is, essentially two-dimensional. In a crossed arrangement, the axes of rotation of the two gears are not parallel but cross at an arbitrary angle except zero or 180 degrees. For best operation, each wheel then must be

7475-438: The axis, meaning that it is congruent with itself when the gear rotates by 1/ N of a turn. If the gear is meant to transmit or receive torque with a definite sense only (clockwise or counterclockwise with respect to some reference viewpoint), the action surface consists of N separate patches, the tooth faces ; which have the same shape and are positioned in the same way relative to the axis, spaced 1/ N turn apart. If

7590-438: The axis, meaning that it is congruent with itself when the gear rotates by 1/ N of a turn. If the gear is meant to transmit or receive torque with a definite sense only (clockwise or counterclockwise with respect to some reference viewpoint), the action surface consists of N separate patches, the tooth faces ; which have the same shape and are positioned in the same way relative to the axis, spaced 1/ N turn apart. If

7705-476: The base of the wheel (6, 10, 12, 20), is displayed just above this digit. A horizontal bar hides either all the complement numbers when it is slid to the top, or all the direct numbers when it is slid toward the center of the machine. It thereby displays either the content of the accumulator or the complement of its value. Since the gears of the calculator rotated in only one direction, negative numbers could not be directly summed. To subtract one number from another,

7820-448: The best shape for each pitch surface is neither cylindrical nor conical but a portion of a hyperboloid of revolution. Such gears are called hypoid for short. Hypoid gears are most commonly found with shafts at 90 degrees. Contact between hypoid gear teeth may be even smoother and more gradual than with spiral bevel gear teeth, but also have a sliding action along the meshing teeth as it rotates and therefore usually require some of

7935-448: The best shape for each pitch surface is neither cylindrical nor conical but a portion of a hyperboloid of revolution. Such gears are called hypoid for short. Hypoid gears are most commonly found with shafts at 90 degrees. Contact between hypoid gear teeth may be even smoother and more gradual than with spiral bevel gear teeth, but also have a sliding action along the meshing teeth as it rotates and therefore usually require some of

8050-418: The bottom right of the stop lever. To add a 5, one must insert a stylus between the spokes that surround the number 5 and rotate the wheel clockwise all the way to the stop lever. The number displayed on the corresponding display register will be increased by 5 and, if a carry transfer takes place, the display register to the left of it will be increased by 1. To add 50, use the tens input wheel (second dial from

8165-455: The bottom was reached, similar to the way the rotary dial of a telephone is used. This displayed the number in the windows at the top of the calculator. Then, one simply redialed the second number to be added, causing the sum of both numbers to appear in the accumulator. Each dial is associated with a one-digit display window located directly above it, which displays the value of the accumulator for this position. The complement of this digit, in

8280-433: The calculator. This affected Pascal’s ability to recruit talent as guilds often reduced the exchange of ideas and trade; sometimes, craftsmen would withhold their labour altogether to rebel against the nobles. Thus Pascal was in a market that had a scarcity of skills and willing workers. Importantly, artisans were not as free as intellectuals to create the machine: Gottfried Leibniz , who built upon Pascal's calculator later in

8395-438: The corresponding cylinder to its maximum number, ready to be re-zeroed. To do so, the operator inserts the stylus in between these two spokes and turns the wheel all the way to the stopping lever. This works because each wheel is directly linked to its corresponding display cylinder (it automatically turns by one during a carry operation). To mark the spokes during manufacturing, one can move the cylinder so that its highest number

8510-526: The display bar moved closest to the edge of the machine, showing the direct value of the accumulator. After re-zeroing the machine, numbers are dialed in one after the other. The following table shows all the steps required to compute 12,345 + 56,789 = 69,134 Subtractions are performed with the display bar moved closest to the center of the machine showing the complement value of the accumulator. The accumulator contains ⁠ C P ( A ) {\displaystyle CP(A)} ⁠ during

8625-475: The first step and ⁠ C P ( A − B ) {\displaystyle CP(A-B)} ⁠ after adding B. In displaying that data in the complement window, the operator sees ⁠ C P ( C P ( A ) ) {\displaystyle CP(CP(A))} ⁠ which is A and then ⁠ C P ( C P ( A − B ) ) {\displaystyle CP(CP(A-B))} ⁠ which

8740-409: The method of nine's complement was used. The only two differences between an addition and a subtraction are the position of the display bar (direct versus complement) and the way the first number is entered (direct versus complement). For a 10-digit wheel (N), the fixed outside wheel is numbered from 0 to 9 (N-1). The numbers are inscribed in a decreasing manner clockwise going from the bottom left to

8855-434: The most common configuration, the axes of rotation of the two gears are parallel, and usually their sizes are such that they contact near a point between the two axes. In this configuration, the two gears turn in opposite senses. Occasionally the axes are parallel but one gear is nested inside the other. In this configuration, both gears turn in the same sense. If the two gears are cut by an imaginary plane perpendicular to

8970-434: The most common configuration, the axes of rotation of the two gears are parallel, and usually their sizes are such that they contact near a point between the two axes. In this configuration, the two gears turn in opposite senses. Occasionally the axes are parallel but one gear is nested inside the other. In this configuration, both gears turn in the same sense. If the two gears are cut by an imaginary plane perpendicular to

9085-522: The most efficient and compact way of transmitting torque between two non-parallel axes. On the other hand, gears are more expensive to manufacture, may require periodic lubrication, and may have greater mass and rotational inertia than the equivalent pulleys. More importantly, the distance between the axes of matched gears is limited and cannot be changed once they are manufactured. There are also applications where slippage under overload or transients (as occurs with belts, hydraulics, and friction wheels)

9200-522: The most efficient and compact way of transmitting torque between two non-parallel axes. On the other hand, gears are more expensive to manufacture, may require periodic lubrication, and may have greater mass and rotational inertia than the equivalent pulleys. More importantly, the distance between the axes of matched gears is limited and cannot be changed once they are manufactured. There are also applications where slippage under overload or transients (as occurs with belts, hydraulics, and friction wheels)

9315-445: The most viscous types of gear oil to avoid it being extruded from the mating tooth faces, the oil is normally designated HP (for hypoid) followed by a number denoting the viscosity. Also, the pinion can be designed with fewer teeth than a spiral bevel pinion, with the result that gear ratios of 60:1 and higher are feasible using a single set of hypoid gears. This style of gear is most common in motor vehicle drive trains, in concert with

9430-445: The most viscous types of gear oil to avoid it being extruded from the mating tooth faces, the oil is normally designated HP (for hypoid) followed by a number denoting the viscosity. Also, the pinion can be designed with fewer teeth than a spiral bevel pinion, with the result that gear ratios of 60:1 and higher are feasible using a single set of hypoid gears. This style of gear is most common in motor vehicle drive trains, in concert with

9545-517: The nymphs of the planthopper insect Issus coleoptratus . The word gear is probably from Old Norse gørvi (plural gørvar ) 'apparel, gear,' related to gøra , gørva 'to make, construct, build; set in order, prepare,' a common verb in Old Norse, "used in a wide range of situations from writing a book to dressing meat". In this context, the meaning of 'toothed wheel in machinery' first attested 1520s; specific mechanical sense of 'parts by which

9660-460: The nymphs of the planthopper insect Issus coleoptratus . The word gear is probably from Old Norse gørvi (plural gørvar ) 'apparel, gear,' related to gøra , gørva 'to make, construct, build; set in order, prepare,' a common verb in Old Norse, "used in a wide range of situations from writing a book to dressing meat". In this context, the meaning of 'toothed wheel in machinery' first attested 1520s; specific mechanical sense of 'parts by which

9775-438: The pendulum clock. In the end, Pascal succeeded in cementing his name as the sole creator of the Pascaline. The royal patent states that it was his invention exclusively. Besides being the first calculating machine made public during its time, the pascaline is also: Gears#Cage gear A gear or gearwheel is a rotating machine part typically used to transmit rotational motion and/or torque by means of

9890-481: The points p and q are moving along different circles; therefore, the contact cannot last more than one instant, and p will then either slide across the other face, or stop contacting it altogether. On the other hand, at any given moment there is at least one such pair of contact points; usually more than one, even a whole line or surface of contact. Actual gears deviate from this model in many ways: they are not perfectly rigid, their mounting does not ensure that

10005-481: The points p and q are moving along different circles; therefore, the contact cannot last more than one instant, and p will then either slide across the other face, or stop contacting it altogether. On the other hand, at any given moment there is at least one such pair of contact points; usually more than one, even a whole line or surface of contact. Actual gears deviate from this model in many ways: they are not perfectly rigid, their mounting does not ensure that

10120-460: The positions of the sun, the moon and the five planets then known, as well as religious feast days. The Salisbury Cathedral clock , built in 1386, it is the world's oldest still working geared mechanical clock. Differential gears were used by the British clock maker Joseph Williamson in 1720. However, the oldest functioning gears by far were created by Nature, and are seen in the hind legs of

10235-405: The positions of the sun, the moon and the five planets then known, as well as religious feast days. The Salisbury Cathedral clock , built in 1386, it is the world's oldest still working geared mechanical clock. Differential gears were used by the British clock maker Joseph Williamson in 1720. However, the oldest functioning gears by far were created by Nature, and are seen in the hind legs of

10350-451: The reputation of his machine. In 1645, in order to control the production of his invention, Pascal wrote to Monseigneur Le Chancelier (the chancellor of France, Pierre Séguier ) in his letter entitled "La Machine d’arithmétique. Lettre dédicatoire à Monseigneur le Chancelier". Pascal requested that no Pascaline be made without his permission. His ingenuity garnered the respect of King Louis XIV of France who granted his request, but it came at

10465-672: The resulting part. Besides gear trains, other alternative methods of transmitting torque between non-coaxial parts include link chains driven by sprockets, friction drives , belts and pulleys , hydraulic couplings , and timing belts . One major advantage of gears is that their rigid body and the snug interlocking of the teeth ensure precise tracking of the rotation across the gear train, limited only by backlash and other mechanical defects. For this reason they are favored in precision applications such as watches. Gear trains also can have fewer separate parts (only two) and have minimal power loss, minimal wear, and long life. Gears are also often

10580-672: The resulting part. Besides gear trains, other alternative methods of transmitting torque between non-coaxial parts include link chains driven by sprockets, friction drives , belts and pulleys , hydraulic couplings , and timing belts . One major advantage of gears is that their rigid body and the snug interlocking of the teeth ensure precise tracking of the rotation across the gear train, limited only by backlash and other mechanical defects. For this reason they are favored in precision applications such as watches. Gear trains also can have fewer separate parts (only two) and have minimal power loss, minimal wear, and long life. Gears are also often

10695-481: The right on a decimal machine), to add 500, use the hundreds input wheel, etc... On all the wheels of all the known machines, except for the machine tardive , two adjacent spokes are marked; these marks differ from machine to machine. On the wheel pictured on the right, they are drilled dots, on the surveying machine they are carved; some are just scratches or marks made with a bit of varnish, some were even marked with little pieces of paper. These marks are used to set

10810-425: The rotation axis will be perfectly fixed in space, the teeth may have slightly different shapes and spacing, the tooth faces are not perfectly smooth, and so on. Yet, these deviations from the ideal model can be ignored for a basic analysis of the operation of a gear set. One criterion for classifying gears is the relative position and direction of the axes or rotation of the gears that are to be meshed together. In

10925-425: The rotation axis will be perfectly fixed in space, the teeth may have slightly different shapes and spacing, the tooth faces are not perfectly smooth, and so on. Yet, these deviations from the ideal model can be ignored for a basic analysis of the operation of a gear set. One criterion for classifying gears is the relative position and direction of the axes or rotation of the gears that are to be meshed together. In

11040-456: The same angular speed ω ( t ), in the same sense. The speed need not be constant over time. The action surface of the gear consists of all points of its surface that, in normal operation, may contact the matching gear with positive pressure . All other parts of the surface are irrelevant (except that they cannot be crossed by any part of the matching gear). In a gear with N teeth, the working surface has N -fold rotational symmetry about

11155-456: The same angular speed ω ( t ), in the same sense. The speed need not be constant over time. The action surface of the gear consists of all points of its surface that, in normal operation, may contact the matching gear with positive pressure . All other parts of the surface are irrelevant (except that they cannot be crossed by any part of the matching gear). In a gear with N teeth, the working surface has N -fold rotational symmetry about

11270-415: The same size and weight independently of the capacity of the machine. Pascal used gravity to arm the sautoirs. One must turn the wheel five steps from 4 to 9 in order to fully arm a sautoir, but the carry transfer will move the next wheel only one step. Thus, much extra energy is accumulated during the arming of a sautoir. All the sautoirs are armed by either an operator input or a carry forward. To re-zero

11385-530: The sense of 'a wheel having teeth or cogs; late 14c., 'tooth on a wheel'; cog-wheel, early 15c. The gears of the Antikythera mechanism are made of bronze , and the earliest surviving Chinese gears are made of iron, These metals, as well as tin , have been generally used for clocks and similar mechanisms to this day. Historically, large gears, such as used in flour mills , were commonly made of wood rather than metal. They were cogwheels, made by inserting

11500-470: The sense of 'a wheel having teeth or cogs; late 14c., 'tooth on a wheel'; cog-wheel, early 15c. The gears of the Antikythera mechanism are made of bronze , and the earliest surviving Chinese gears are made of iron, These metals, as well as tin , have been generally used for clocks and similar mechanisms to this day. Historically, large gears, such as used in flour mills , were commonly made of wood rather than metal. They were cogwheels, made by inserting

11615-566: The study of religion and philosophy , which gave us both the Lettres provinciales and the Pensées . The tercentenary celebration of Pascal's invention of the mechanical calculator occurred during World War II when France was occupied by Germany and therefore the main celebration was held in London, England. Speeches given during the event highlighted Pascal's practical achievements when he

11730-464: The surveying or the accounting machines. This can also be extended to: This principle applied to the Pascaline: The machine has to be re-zeroed before each new operation. To reset his machine, the operator has to set all the wheels to their maximum, using the marks on two adjacent spokes , and then add 1 to the rightmost wheel. The method of re-zeroing that Pascal chose, which propagates

11845-490: The torque T to increase but the rotational speed ω to decrease. The opposite effect is obtained when a large gear drives a small one. The changes are proportional to the gear ratio r , the ratio of the tooth counts. namely, T 2 / T 1 = r = N 2 / N 1 , and ω 2 / ω 1 = 1/ r = N 1 / N 2 . Depending on the geometry of the pair, the sense of rotation may also be inverted (from clockwise to anti-clockwise , or vice-versa). Most vehicles have

11960-490: The torque T to increase but the rotational speed ω to decrease. The opposite effect is obtained when a large gear drives a small one. The changes are proportional to the gear ratio r , the ratio of the tooth counts. namely, T 2 / T 1 = r = N 2 / N 1 , and ω 2 / ω 1 = 1/ r = N 1 / N 2 . Depending on the geometry of the pair, the sense of rotation may also be inverted (from clockwise to anti-clockwise , or vice-versa). Most vehicles have

12075-416: The torque on each gear may have both senses, the action surface will have two sets of N tooth faces; each set will be effective only while the torque has one specific sense, and the two sets can be analyzed independently of the other. However, in this case the gear usually has also "flip over" symmetry, so that the two sets of tooth faces are congruent after the gear is flipped. This arrangement ensures that

12190-416: The torque on each gear may have both senses, the action surface will have two sets of N tooth faces; each set will be effective only while the torque has one specific sense, and the two sets can be analyzed independently of the other. However, in this case the gear usually has also "flip over" symmetry, so that the two sets of tooth faces are congruent after the gear is flipped. This arrangement ensures that

12305-416: The two gears are firmly locked together, at all times, with no backlash . During operation, each point p of each tooth face will at some moment contact a tooth face of the matching gear at some point q of one of its tooth faces. At that moment and at those points, the two faces must have the same perpendicular direction but opposite orientation. But since the two gears are rotating around different axes,

12420-416: The two gears are firmly locked together, at all times, with no backlash . During operation, each point p of each tooth face will at some moment contact a tooth face of the matching gear at some point q of one of its tooth faces. At that moment and at those points, the two faces must have the same perpendicular direction but opposite orientation. But since the two gears are rotating around different axes,

12535-533: The two gears are sliced by an imaginary sphere whose center is the point where the two axes cross, each section will remain on the surface of that sphere as the gear rotates, and the section of one gear will interact only with the corresponding section of the other gear. In this way, a pair of meshed 3D gears can be understood as a stack of nested infinitely thin cup-like gears. The gears in a matching pair are said to be skew if their axes of rotation are skew lines -- neither parallel nor intersecting. In this case,

12650-533: The two gears are sliced by an imaginary sphere whose center is the point where the two axes cross, each section will remain on the surface of that sphere as the gear rotates, and the section of one gear will interact only with the corresponding section of the other gear. In this way, a pair of meshed 3D gears can be understood as a stack of nested infinitely thin cup-like gears. The gears in a matching pair are said to be skew if their axes of rotation are skew lines -- neither parallel nor intersecting. In this case,

12765-528: The vehicle does not actually contain gears, as in a continuously variable transmission . The earliest surviving gears date from the 4th century BC in China (Zhan Guo times – Late East Zhou dynasty ), which have been preserved at the Luoyang Museum of Henan Province, China . In Europe, Aristotle mentions gears around 330 BC, as wheel drives in windlasses. He observed that the direction of rotation

12880-413: The vehicle does not actually contain gears, as in a continuously variable transmission . The earliest surviving gears date from the 4th century BC in China (Zhan Guo times – Late East Zhou dynasty ), which have been preserved at the Luoyang Museum of Henan Province, China . In Europe, Aristotle mentions gears around 330 BC, as wheel drives in windlasses. He observed that the direction of rotation

12995-467: The weakest part in a mechanism, so that in case of jamming they will fail first and thus avoid damage to more expensive parts. Such sacrificial gears may be a simpler alternative to other overload-protection devices such as clutches and torque- or current-limited motors. In spite of the advantages of metal and plastic, wood continued to be used for large gears until a couple of centuries ago, because of cost, weight, tradition, or other considerations. In 1967

13110-467: The weakest part in a mechanism, so that in case of jamming they will fail first and thus avoid damage to more expensive parts. Such sacrificial gears may be a simpler alternative to other overload-protection devices such as clutches and torque- or current-limited motors. In spite of the advantages of metal and plastic, wood continued to be used for large gears until a couple of centuries ago, because of cost, weight, tradition, or other considerations. In 1967

13225-411: Was already known in the field of pure mathematics, and his creative imagination, along with how ahead of their time both the machine and its inventor were. The calculator had spoked metal wheel dials, with the digit 0 through 9 displayed around the circumference of each wheel. To input a digit, the user placed a stylus in the corresponding space between the spokes and turned the dial until a metal stop at

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