How To Draw A Straight Line
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Publicado: 7 de noviembre de 2012
How to Draw a Straight Line
Introduction:
Cornell University’s Reuleaux kinematic model collection includes many linkages; the most popular of these among mathematicians is the Peaucellier-Lipkin linkage S35. This article is a short introduction (not complete) to the history of the problem of how to change circular motion into straight-line motion and vice versa. Some mathematicians formulatedthis problem as: "How can you draw a straight line?" The Peaucellier-Lipkin linkage was the first precise solution to this problem.
When using a compass to draw a circle, we are not starting with a model of a circle; instead we are using a fundamental property of circles that the points on a circle are at a fixed distance from its center, which is Euclid's definition of a circle. Is there atool (serving the role of a compass) that will draw a straight line? If, in this case, we want to use Euclid's definition: "A straight line is a line which lies evenly with the points on itself" it will not be of much help. One can say, "We can use a straightedge for constructing a straight line!" Well, how do you know that your straightedge is straight? How can you check that something isstraight? What does "straight" mean? Think about it!
As we can see in some 13th-century drawings of a sawmill (at right), mechanisms for changing circular motion to straight-line motion were in use in the 13th-century and probably originated much earlier. In 1588 Agostino Ramelli published his book on machines where linkages were widely used. But, of course, there is a vast difference between thelinkages of Ramelli and those of James Watt (1736-1819), a pioneer of the improved steam engine and a highly gifted designer of mechanisms. Watt's partner, machine builder, Matthew Boulton, built engines in his shop "...with as great a difference of accuracy as there is between the blacksmith and the mathematical instrument maker." [Fergusson 1962]
It took Watt several years to design astraight-line linkage that would change straight-line motion into circular motion. He wrote to Boulton:
"I have got a glimpse of a method of causing the piston-rod to move up and down perpendicularly, by only fixing it to a piece of iron upon the beam, without chains, or perpendicular guides, or untowardly frictions, archheads, or other pieces of clumsiness…. I have only tried it in a slight model yet, socannot build upon it, though I think it a very probable thing to succeed, and one of the most ingenious simple pieces of mechanisms I have contrived…". [Fergusson 1962]
Years later Watt told his son: "Though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other mechanical invention I have ever made." [Fergusson 1962]
"Parallel motion" is a name Wattused for his linkage (see model S24), which was included in an extensive patent of 1784. Watt's linkage was a good solution to the practical problem. But this solution did not satisfy mathematicians who knew that it only traced an approximate straight line. An exact straight-line linkage in the plane was not known until 1864. In 1853 Pierre-Frederic Sarrus (1798-1861), a French professor ofmathematics at Strassbourg, devised an accordion-like spatial linkage that traced exact straight line but it still was not a solution of the planar problem.
There were several attempts to solve this problem before Peucellier. Other linkages in this Reuleaux model collection are connected with some of the names of 19th century mathematicians who tried to solve the problem of how to draw a precisestraight line. Reuleaux thought that these mechanisms were so important that he designed 39 straight line mechanisms for his collection, including those of Watt, Roberts, Evans, Chebyshev, Peuaucellier-Lipkin, Cartwright and some of his own design. See all models in the S-series.
The appearance in 1864 of Peaucellier's exact straight-line linkage went nearly unnoticed. Charles Nicolas Peaucellier...
Introduction:
Cornell University’s Reuleaux kinematic model collection includes many linkages; the most popular of these among mathematicians is the Peaucellier-Lipkin linkage S35. This article is a short introduction (not complete) to the history of the problem of how to change circular motion into straight-line motion and vice versa. Some mathematicians formulatedthis problem as: "How can you draw a straight line?" The Peaucellier-Lipkin linkage was the first precise solution to this problem.
When using a compass to draw a circle, we are not starting with a model of a circle; instead we are using a fundamental property of circles that the points on a circle are at a fixed distance from its center, which is Euclid's definition of a circle. Is there atool (serving the role of a compass) that will draw a straight line? If, in this case, we want to use Euclid's definition: "A straight line is a line which lies evenly with the points on itself" it will not be of much help. One can say, "We can use a straightedge for constructing a straight line!" Well, how do you know that your straightedge is straight? How can you check that something isstraight? What does "straight" mean? Think about it!
As we can see in some 13th-century drawings of a sawmill (at right), mechanisms for changing circular motion to straight-line motion were in use in the 13th-century and probably originated much earlier. In 1588 Agostino Ramelli published his book on machines where linkages were widely used. But, of course, there is a vast difference between thelinkages of Ramelli and those of James Watt (1736-1819), a pioneer of the improved steam engine and a highly gifted designer of mechanisms. Watt's partner, machine builder, Matthew Boulton, built engines in his shop "...with as great a difference of accuracy as there is between the blacksmith and the mathematical instrument maker." [Fergusson 1962]
It took Watt several years to design astraight-line linkage that would change straight-line motion into circular motion. He wrote to Boulton:
"I have got a glimpse of a method of causing the piston-rod to move up and down perpendicularly, by only fixing it to a piece of iron upon the beam, without chains, or perpendicular guides, or untowardly frictions, archheads, or other pieces of clumsiness…. I have only tried it in a slight model yet, socannot build upon it, though I think it a very probable thing to succeed, and one of the most ingenious simple pieces of mechanisms I have contrived…". [Fergusson 1962]
Years later Watt told his son: "Though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other mechanical invention I have ever made." [Fergusson 1962]
"Parallel motion" is a name Wattused for his linkage (see model S24), which was included in an extensive patent of 1784. Watt's linkage was a good solution to the practical problem. But this solution did not satisfy mathematicians who knew that it only traced an approximate straight line. An exact straight-line linkage in the plane was not known until 1864. In 1853 Pierre-Frederic Sarrus (1798-1861), a French professor ofmathematics at Strassbourg, devised an accordion-like spatial linkage that traced exact straight line but it still was not a solution of the planar problem.
There were several attempts to solve this problem before Peucellier. Other linkages in this Reuleaux model collection are connected with some of the names of 19th century mathematicians who tried to solve the problem of how to draw a precisestraight line. Reuleaux thought that these mechanisms were so important that he designed 39 straight line mechanisms for his collection, including those of Watt, Roberts, Evans, Chebyshev, Peuaucellier-Lipkin, Cartwright and some of his own design. See all models in the S-series.
The appearance in 1864 of Peaucellier's exact straight-line linkage went nearly unnoticed. Charles Nicolas Peaucellier...
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