November 23, 2015 at 20:27Claude Perrault's Rabdological Abacus
French scientist and architect Claude Perrault, brother of the famous storyteller Charles Perrault, invented the original counting device - the rhabdological abacus. The summing machine became a valuable addition to the XVII – XVIII centuries, a period rich in inventions of calculating devices.
Claude Perrault (1613-1688)
Perrot gave his invention a name in the spirit of ancient times, when a small board on which numbers were written was called an abacus. And since the machine performed arithmetic operations with the help of small sticks with numbers, it was considered rhabdological.
roman abacus
For the first time, the device was mentioned in the book "Collection of a large number of machines of his own composition" (1700) by Claude Perrault, published posthumously. The collection contained descriptions of almost all the inventions of the author (“pendulum clock”, “machine for lifting weights”, “machine for increasing the effect of firearms”, etc.). Rabdological abacus was listed at number ten.
It was a compact and simple computing machine, noted in history due to a significant difference from other invented devices in its field. Instead of the usual gears of Pascal, in the rhabdological abacus, gear racks (kremalery) were used.
The calculating machine was in the form of a plate with a finger thick, the height reached about 30 cm, and the width was 14 cm. Two windows were cut on the front side of the device, where the results were displayed. In the upper window, the result of subtraction was shown, and in the lower - addition. The multiplication table was engraved at the bottom of the front side.
In addition, seven grooves were cut on the front side, along which scales with divisions 1, 2, 3, 4, 5, 6, 7, 8, 9 were placed. Rulers were in the grooves. Using a pin with a pointed tip, they moved up and down to the base of the device.
Deep risks divided each line into 26 parts. The pin point was inserted into the risks, which allowed the rulers to be easily and precisely necessary. In the upper eleven divisions of the line was an increasing sequence (0,1,2,3,4,5,6,7,8,9,0) for subtraction. In the lower eleven divisions, a sequence was placed that decreased (0.9.8.7.6.5.4.3.2.1.0). Accordingly, it was used for addition. The numbers of these sequences were shown in the result boxes. In the lower window, they were the desired value of addition, and in the upper window, the desired value of subtraction.
The sequences plotted on the rulers were separated from each other by four empty divisions.
In total, the device used seven rulers, which were separated from each other by thin plates. In the lines, the sequence in ascending order was observed. The far right symbolized the category of units, the next line represented the category of tens, then hundreds, and so on up to the category of millions.
Each plate separating the rulers from each other had a hole. It was used to transfer overflows from the low order to the senior one and was located at the base of the ruler, pushed to the very top of the device. The size of the hole reached three divisions of the line.
Each ruler near the base of the right side had 11 teeth - one per division. On the other side of the ruler (if you count from the bottom - under 11 and 12 division) was a spring-loaded hook. Cogs and hooks were used to transfer overflows from the low order to the high order.
The transfer from the least significant to the highest was as follows. When the low-level bar was in the upper part of the device (that is, zeros were in the windows of the result of addition and subtraction), the spring-loaded hook was hidden in the body of the ruler, resting against the plate located to the left of it. As the ruler moved down, the spring-loaded hook approached the hole in the plate. When the number 7 belonging to the movable ruler appeared in the window of the addition result, the spring-loaded hook began to extend into the hole of the bar, and at the time when the number 9 appeared in the window of the result, it entered into engagement with the teeth of the senior discharge ruler. In the future, the movement of the ruler of the lower order moved the ruler of the senior discharge. As a result, when the digit 0 of the low order line appeared in the window of the addition results,
Example of the addition operation 127 + 65
Step 1. First, all the bits of the rhabdological abacus are set to 0. To do this, with the help of a pin, the rulers move to the highest position.
Step 2. The pin is placed in the groove at the risk of the low-order ruler, which is opposite the figure 7. The ruler moves until the pin rests against the lower end of the groove. In this case, the input number 7 will be displayed in the window of the addition result in the least significant digit. An important point is that in the window of the subtraction result in the least significant digit, the number needed to supplement the seven to ten, that is, number 3, will be displayed.
Step 3. A similar operation is extended with the discharge of ten , only in this case the groove is set opposite the number 2.
Step 4. For the discharge of hundreds, the groove is set opposite to the number 1. Next, the groove moves down to the stop, that is, by one division. The number 127 will be displayed in the window of the addition result.
Step 5. The next step is to enter the second term. The pin is installed in the groove at the risk of the low-order ruler, which is opposite the number 5, the ruler is shifted down to the stop. In this case, the pin will stop opposite the number 2, since the ruler will abut against the lower wall of the device before the pin reaches the lower end of the groove. At the same time, a dozen discharge line will drop one division down due to the work of the overflow transfer mechanism. In the result window, the number 130 will be displayed.
Step 6. In order to get the correct digit in the category of units - 2, it is necessary, without removing the pin from the slot, move the ruler up until the pin rests against the end of the groove. Thus, the number 132. will be displayed in the addition result window.
Step 7. The final stage - the discharge of tens of the second term is introduced. To do this, the pin is installed in the groove at the risk of the tens discharge line, opposite the number 6, the line moves down to the stop. This completes the calculation, and the desired value is displayed in the window of the addition result: 192.
Example of the subtraction operation 68-23
Step 1. All the digits of the rhabdological abacus are set to 0. To do this, use the pin, the rulers move to the highest position.
Step 2. Enter the least significant bit of the decremented one. The low-order ruler moves so that the figure 8 is displayed in the window of the subtraction result in the low-order position. To achieve this, you need to put the pin in the groove at risk, opposite the number 2, and slide the ruler down until the pin rests against the end groove.
Step 3. Next, enter the rank of ten to be reduced, for which the corresponding ruler moves so that in the window of the result of subtraction in the second position the number 6 is displayed. To do this, the pin is placed in the risk slot located opposite the number 4, and the ruler moves down until until the pin rests against the end of the groove. As a result, the window of the result of the subtraction will display the number 68.
Step 4. The deductible is introduced as well as the term in addition. To enter the low-order digit of the subtracted one, the pin is installed in the groove at the risk of the low-order line, opposite the number 3. The line moves down until the pin rests against the end of the groove. Next, the senior digit of the subtracted one is introduced, for which the pin is installed in the groove at the risk of the second left of the ruler, opposite the number 2. The ruler moves down until the pin rests against the end of the groove. The difference between the two numbers was found, the desired result is displayed in the window of the subtraction result: 45.
When subtracting with the help of the rhabdological abacus, we used the method of complementing up to ten, similar to the method used in Pascalin.
As an example, the solution to the equation: Y = 68-23 = 45. Using the complement method, the number 68 is represented as the difference between the numbers 100 and 32 (68 = 10-32). As a result, the equation reduces to the following form: Y = 68-23 = 100-32-23 = 100- (32 + 23) = 27. Such a transformation replaces the subtraction by addition and subtraction of the result of addition from 100, which is the transformation opposite to complement. Therefore, it remains to solve the problem of automatic addition to ten, for which two lines of numbers are plotted on all rulers, and on the lid of the rhabdological abacus there are two result output windows arranged so that the sums of two numbers displayed in the windows and located one under the other always equal ten.
In his design, Perrault used elements that were first used by the Scottish mathematician John Napier in a simple multiplier device. It was intended for multiplication, division, extraction of the square root and was described in detail by Nepher in the last publication during his lifetime (Rabdologia seu Numerationis, 1617).
sticks of Napier
The description of the rhabdological abacus shows its difference from the computers existing at that time. And, importantly, the device was quite easy to use. However, the rabdological abacus did not receive due recognition. Perhaps due to the unreliability of the device of the spring-loaded hook (it was used not only for transfer, but also for fixing the rods in the positions corresponding to the numbers entered), which during continuous operation often failed. But if the realized invention did not become widely used in practice, then Perrot’s ideas subsequently found application in other simple and reliable calculating devices (“Kummer’s calculator”, “Hans Zabelny comptator”, etc.).
The Kummer Reader (1846)
PS We carry out an action especially for readers of Habr. Details post here .
Claude Perrault (1613-1688)
Perrot gave his invention a name in the spirit of ancient times, when a small board on which numbers were written was called an abacus. And since the machine performed arithmetic operations with the help of small sticks with numbers, it was considered rhabdological.
roman abacus
For the first time, the device was mentioned in the book "Collection of a large number of machines of his own composition" (1700) by Claude Perrault, published posthumously. The collection contained descriptions of almost all the inventions of the author (“pendulum clock”, “machine for lifting weights”, “machine for increasing the effect of firearms”, etc.). Rabdological abacus was listed at number ten.
It was a compact and simple computing machine, noted in history due to a significant difference from other invented devices in its field. Instead of the usual gears of Pascal, in the rhabdological abacus, gear racks (kremalery) were used.
The calculating machine was in the form of a plate with a finger thick, the height reached about 30 cm, and the width was 14 cm. Two windows were cut on the front side of the device, where the results were displayed. In the upper window, the result of subtraction was shown, and in the lower - addition. The multiplication table was engraved at the bottom of the front side.
In addition, seven grooves were cut on the front side, along which scales with divisions 1, 2, 3, 4, 5, 6, 7, 8, 9 were placed. Rulers were in the grooves. Using a pin with a pointed tip, they moved up and down to the base of the device.
Deep risks divided each line into 26 parts. The pin point was inserted into the risks, which allowed the rulers to be easily and precisely necessary. In the upper eleven divisions of the line was an increasing sequence (0,1,2,3,4,5,6,7,8,9,0) for subtraction. In the lower eleven divisions, a sequence was placed that decreased (0.9.8.7.6.5.4.3.2.1.0). Accordingly, it was used for addition. The numbers of these sequences were shown in the result boxes. In the lower window, they were the desired value of addition, and in the upper window, the desired value of subtraction.
The sequences plotted on the rulers were separated from each other by four empty divisions.
In total, the device used seven rulers, which were separated from each other by thin plates. In the lines, the sequence in ascending order was observed. The far right symbolized the category of units, the next line represented the category of tens, then hundreds, and so on up to the category of millions.
Each plate separating the rulers from each other had a hole. It was used to transfer overflows from the low order to the senior one and was located at the base of the ruler, pushed to the very top of the device. The size of the hole reached three divisions of the line.
Each ruler near the base of the right side had 11 teeth - one per division. On the other side of the ruler (if you count from the bottom - under 11 and 12 division) was a spring-loaded hook. Cogs and hooks were used to transfer overflows from the low order to the high order.
The transfer from the least significant to the highest was as follows. When the low-level bar was in the upper part of the device (that is, zeros were in the windows of the result of addition and subtraction), the spring-loaded hook was hidden in the body of the ruler, resting against the plate located to the left of it. As the ruler moved down, the spring-loaded hook approached the hole in the plate. When the number 7 belonging to the movable ruler appeared in the window of the addition result, the spring-loaded hook began to extend into the hole of the bar, and at the time when the number 9 appeared in the window of the result, it entered into engagement with the teeth of the senior discharge ruler. In the future, the movement of the ruler of the lower order moved the ruler of the senior discharge. As a result, when the digit 0 of the low order line appeared in the window of the addition results,
Example of the addition operation 127 + 65
Step 1. First, all the bits of the rhabdological abacus are set to 0. To do this, with the help of a pin, the rulers move to the highest position.
Step 2. The pin is placed in the groove at the risk of the low-order ruler, which is opposite the figure 7. The ruler moves until the pin rests against the lower end of the groove. In this case, the input number 7 will be displayed in the window of the addition result in the least significant digit. An important point is that in the window of the subtraction result in the least significant digit, the number needed to supplement the seven to ten, that is, number 3, will be displayed.
Step 3. A similar operation is extended with the discharge of ten , only in this case the groove is set opposite the number 2.
Step 4. For the discharge of hundreds, the groove is set opposite to the number 1. Next, the groove moves down to the stop, that is, by one division. The number 127 will be displayed in the window of the addition result.
Step 5. The next step is to enter the second term. The pin is installed in the groove at the risk of the low-order ruler, which is opposite the number 5, the ruler is shifted down to the stop. In this case, the pin will stop opposite the number 2, since the ruler will abut against the lower wall of the device before the pin reaches the lower end of the groove. At the same time, a dozen discharge line will drop one division down due to the work of the overflow transfer mechanism. In the result window, the number 130 will be displayed.
Step 6. In order to get the correct digit in the category of units - 2, it is necessary, without removing the pin from the slot, move the ruler up until the pin rests against the end of the groove. Thus, the number 132. will be displayed in the addition result window.
Step 7. The final stage - the discharge of tens of the second term is introduced. To do this, the pin is installed in the groove at the risk of the tens discharge line, opposite the number 6, the line moves down to the stop. This completes the calculation, and the desired value is displayed in the window of the addition result: 192.
Example of the subtraction operation 68-23
Step 1. All the digits of the rhabdological abacus are set to 0. To do this, use the pin, the rulers move to the highest position.
Step 2. Enter the least significant bit of the decremented one. The low-order ruler moves so that the figure 8 is displayed in the window of the subtraction result in the low-order position. To achieve this, you need to put the pin in the groove at risk, opposite the number 2, and slide the ruler down until the pin rests against the end groove.
Step 3. Next, enter the rank of ten to be reduced, for which the corresponding ruler moves so that in the window of the result of subtraction in the second position the number 6 is displayed. To do this, the pin is placed in the risk slot located opposite the number 4, and the ruler moves down until until the pin rests against the end of the groove. As a result, the window of the result of the subtraction will display the number 68.
Step 4. The deductible is introduced as well as the term in addition. To enter the low-order digit of the subtracted one, the pin is installed in the groove at the risk of the low-order line, opposite the number 3. The line moves down until the pin rests against the end of the groove. Next, the senior digit of the subtracted one is introduced, for which the pin is installed in the groove at the risk of the second left of the ruler, opposite the number 2. The ruler moves down until the pin rests against the end of the groove. The difference between the two numbers was found, the desired result is displayed in the window of the subtraction result: 45.
When subtracting with the help of the rhabdological abacus, we used the method of complementing up to ten, similar to the method used in Pascalin.
As an example, the solution to the equation: Y = 68-23 = 45. Using the complement method, the number 68 is represented as the difference between the numbers 100 and 32 (68 = 10-32). As a result, the equation reduces to the following form: Y = 68-23 = 100-32-23 = 100- (32 + 23) = 27. Such a transformation replaces the subtraction by addition and subtraction of the result of addition from 100, which is the transformation opposite to complement. Therefore, it remains to solve the problem of automatic addition to ten, for which two lines of numbers are plotted on all rulers, and on the lid of the rhabdological abacus there are two result output windows arranged so that the sums of two numbers displayed in the windows and located one under the other always equal ten.
In his design, Perrault used elements that were first used by the Scottish mathematician John Napier in a simple multiplier device. It was intended for multiplication, division, extraction of the square root and was described in detail by Nepher in the last publication during his lifetime (Rabdologia seu Numerationis, 1617).
sticks of Napier
The description of the rhabdological abacus shows its difference from the computers existing at that time. And, importantly, the device was quite easy to use. However, the rabdological abacus did not receive due recognition. Perhaps due to the unreliability of the device of the spring-loaded hook (it was used not only for transfer, but also for fixing the rods in the positions corresponding to the numbers entered), which during continuous operation often failed. But if the realized invention did not become widely used in practice, then Perrot’s ideas subsequently found application in other simple and reliable calculating devices (“Kummer’s calculator”, “Hans Zabelny comptator”, etc.).
The Kummer Reader (1846)
PS We carry out an action especially for readers of Habr. Details post here .