What was the notion of limit that Newton used?Why is calculus missing from Newton's Principia?Did Michel Rolle say that the calculus is “a collection of ingenious fallacies”?Was English mathematics behind Europe by many years because of Newton's notation?Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity?When/How were the product and chain rules first proved?What is the modern understanding of the chronology of Newton's mathematical work?Who is Joshua King?When and who was the first mathematicians to prove rigorously that $sqrt[3]2$ was impossible number?Madhava and $pi$What is the correct statement of Cauchy’s erroneous theorem on continuity?What evidence is there that the Babylonians used the Babylonain method of estimating square roots?Did the Idea of Universal Gravitation predate Newton?How was the notion of the metacenter of a floating body discovered?
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What was the notion of limit that Newton used?
Why is calculus missing from Newton's Principia?Did Michel Rolle say that the calculus is “a collection of ingenious fallacies”?Was English mathematics behind Europe by many years because of Newton's notation?Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity?When/How were the product and chain rules first proved?What is the modern understanding of the chronology of Newton's mathematical work?Who is Joshua King?When and who was the first mathematicians to prove rigorously that $sqrt[3]2$ was impossible number?Madhava and $pi$What is the correct statement of Cauchy’s erroneous theorem on continuity?What evidence is there that the Babylonians used the Babylonain method of estimating square roots?Did the Idea of Universal Gravitation predate Newton?How was the notion of the metacenter of a floating body discovered?
$begingroup$
I have read that the notion of limit became rigorous two centuries after the discover of calculus
What Newton had in his mind regarding the notion of limit?
mathematics
edited 6 hours ago
KCd
1,927813
asked 9 hours ago
veronikaveronika
1434
$endgroup$
|
show 5 more comments
$begingroup$
I have read that the notion of limit became rigorous two centuries after the discover of calculus
What Newton had in his mind regarding the notion of limit?
mathematics
edited 6 hours ago
KCd
1,927813
asked 9 hours ago
veronikaveronika
1434
$endgroup$
$begingroup$
@MathWizard title has been changed.
$endgroup$
– KCd
6 hours ago
$begingroup$
He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him."
$endgroup$
– Conifold
6 hours ago
$begingroup$
@Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation.
$endgroup$
– terry-s
5 hours ago
$begingroup$
@terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example.
$endgroup$
– Conifold
5 hours ago
1
$begingroup$
@terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect.
$endgroup$
– Conifold
4 hours ago
|
show 5 more comments
$begingroup$
I have read that the notion of limit became rigorous two centuries after the discover of calculus
What Newton had in his mind regarding the notion of limit?
mathematics
edited 6 hours ago
KCd
1,927813
asked 9 hours ago
veronikaveronika
1434
$endgroup$
I have read that the notion of limit became rigorous two centuries after the discover of calculus
What Newton had in his mind regarding the notion of limit?
mathematics
mathematics
edited 6 hours ago
KCd
1,927813
asked 9 hours ago
veronikaveronika
1434
edited 6 hours ago
KCd
1,927813
asked 9 hours ago
veronikaveronika
1434
edited 6 hours ago
KCd
1,927813
edited 6 hours ago
KCd
1,927813
edited 6 hours ago
KCd
1,927813
1,927813
asked 9 hours ago
veronikaveronika
1434
asked 9 hours ago
veronikaveronika
1434
asked 9 hours ago
veronikaveronika
1434
1434
$begingroup$
@MathWizard title has been changed.
$endgroup$
– KCd
6 hours ago
$begingroup$
He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him."
$endgroup$
– Conifold
6 hours ago
$begingroup$
@Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation.
$endgroup$
– terry-s
5 hours ago
$begingroup$
@terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example.
$endgroup$
– Conifold
5 hours ago
1
$begingroup$
@terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect.
$endgroup$
– Conifold
4 hours ago
|
show 5 more comments
$begingroup$
@MathWizard title has been changed.
$endgroup$
– KCd
6 hours ago
$begingroup$
He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him."
$endgroup$
– Conifold
6 hours ago
$begingroup$
@Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation.
$endgroup$
– terry-s
5 hours ago
$begingroup$
@terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example.
$endgroup$
– Conifold
5 hours ago
1
$begingroup$
@terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect.
$endgroup$
– Conifold
4 hours ago
$begingroup$
@MathWizard title has been changed.
$endgroup$
– KCd
6 hours ago
$begingroup$
@MathWizard title has been changed.
$endgroup$
– KCd
6 hours ago
$begingroup$
He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him."
$endgroup$
– Conifold
6 hours ago
$begingroup$
He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him."
$endgroup$
– Conifold
6 hours ago
$begingroup$
@Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation.
$endgroup$
– terry-s
5 hours ago
$begingroup$
@Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation.
$endgroup$
– terry-s
5 hours ago
$begingroup$
@terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example.
$endgroup$
– Conifold
5 hours ago
$begingroup$
@terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example.
$endgroup$
– Conifold
5 hours ago
1
1
$begingroup$
@terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect.
$endgroup$
– Conifold
4 hours ago
$begingroup$
@terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect.
$endgroup$
– Conifold
4 hours ago
|
show 5 more comments
2 Answers
2
active
oldest
votes
$begingroup$
Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia, with a focus on parts of Book 1, section 1.
(When I return to my sources, I'm away from base right now, I will put in online references to the Principia in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in The Mathematical Principles of Natural Philosophy, vol.1 of 2, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.)
Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'. When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri. Newton clearly considered such methods as not well justified, hence his reliance on limits.
There is further material that contributes to an answer to the current question in Why is calculus missing from Newton's Principia? , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"? . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's Principia to provide the justification that did not appear to be available elsewhere. Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.
answered 8 hours ago
terry-sterry-s
2,233421
$endgroup$
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
|
show 2 more comments
$begingroup$
Newton did not have the explicit concept of limit. He just used the notion of infinitesimal, a quantity that sometimes is not zero (in operation) and sometimes is zero (in final result). For example, calculating the derivative of $x^2$ is like
$$
dotx=fracDelta yDelta x=frac(x+Delta x)^2-x^2Delta x=2x+Delta x=2x
$$
(Newton used $dotx$ for derivative and later Leibniz improved to $fracdydx$). In the last step, $Delta x=0$, but in $fracDelta yDelta x$, $Delta x$ can not be $0$ for $frac00$ makes no sense. This means that $Delta x$ (infinitesimal) sometimes is zero and sometimes not, a fact that Newton could not explain. Nor did Leibniz and others know the solution. However, this defect of infinitesimal has been ignored and the verity of Calculus has never been in doubt because the powerful method of Calculus through infinitesimal has solved so many and important problems mankind has even never dreamed before.
The rigorous explanation of infinitesimal through the notion of limit was not completed until two hundred years after Newton, through the work of Cauchy and Weierstrass. So it is an overstatement to say that Newton knew the notion of limit and rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is an overstatement again to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.
answered 9 hours ago
Math WizardMath Wizard
365111
$endgroup$
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
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2 Answers
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$begingroup$
Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia, with a focus on parts of Book 1, section 1.
(When I return to my sources, I'm away from base right now, I will put in online references to the Principia in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in The Mathematical Principles of Natural Philosophy, vol.1 of 2, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.)
Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'. When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri. Newton clearly considered such methods as not well justified, hence his reliance on limits.
There is further material that contributes to an answer to the current question in Why is calculus missing from Newton's Principia? , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"? . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's Principia to provide the justification that did not appear to be available elsewhere. Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.
answered 8 hours ago
terry-sterry-s
2,233421
$endgroup$
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
|
show 2 more comments
$begingroup$
Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia, with a focus on parts of Book 1, section 1.
(When I return to my sources, I'm away from base right now, I will put in online references to the Principia in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in The Mathematical Principles of Natural Philosophy, vol.1 of 2, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.)
Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'. When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri. Newton clearly considered such methods as not well justified, hence his reliance on limits.
There is further material that contributes to an answer to the current question in Why is calculus missing from Newton's Principia? , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"? . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's Principia to provide the justification that did not appear to be available elsewhere. Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.
answered 8 hours ago
terry-sterry-s
2,233421
$endgroup$
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
|
show 2 more comments
$begingroup$
Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia, with a focus on parts of Book 1, section 1.
(When I return to my sources, I'm away from base right now, I will put in online references to the Principia in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in The Mathematical Principles of Natural Philosophy, vol.1 of 2, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.)
Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'. When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri. Newton clearly considered such methods as not well justified, hence his reliance on limits.
There is further material that contributes to an answer to the current question in Why is calculus missing from Newton's Principia? , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"? . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's Principia to provide the justification that did not appear to be available elsewhere. Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.
answered 8 hours ago
terry-sterry-s
2,233421
$endgroup$
Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia, with a focus on parts of Book 1, section 1.
(When I return to my sources, I'm away from base right now, I will put in online references to the Principia in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in The Mathematical Principles of Natural Philosophy, vol.1 of 2, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.)
Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'. When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri. Newton clearly considered such methods as not well justified, hence his reliance on limits.
There is further material that contributes to an answer to the current question in Why is calculus missing from Newton's Principia? , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"? . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's Principia to provide the justification that did not appear to be available elsewhere. Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.
answered 8 hours ago
terry-sterry-s
2,233421
edited 8 hours ago
answered 8 hours ago
terry-sterry-s
2,233421
answered 8 hours ago
terry-sterry-s
2,233421
answered 8 hours ago
terry-sterry-s
2,233421
2,233421
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
|
show 2 more comments
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
"First and last ratios" is actually the idea of infinitesimal, a quantity sometimes is considered not zero and sometimes is zero. So it is an overstatement to say that Newton actually had a pretty explicit concept of limit.
$endgroup$
– Math Wizard
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
@math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion?
$endgroup$
– terry-s
7 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century
$endgroup$
– Math Wizard
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
@Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now?
$endgroup$
– terry-s
5 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
$begingroup$
Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar.
$endgroup$
– Math Wizard
4 hours ago
|
show 2 more comments
$begingroup$
Newton did not have the explicit concept of limit. He just used the notion of infinitesimal, a quantity that sometimes is not zero (in operation) and sometimes is zero (in final result). For example, calculating the derivative of $x^2$ is like
$$
dotx=fracDelta yDelta x=frac(x+Delta x)^2-x^2Delta x=2x+Delta x=2x
$$
(Newton used $dotx$ for derivative and later Leibniz improved to $fracdydx$). In the last step, $Delta x=0$, but in $fracDelta yDelta x$, $Delta x$ can not be $0$ for $frac00$ makes no sense. This means that $Delta x$ (infinitesimal) sometimes is zero and sometimes not, a fact that Newton could not explain. Nor did Leibniz and others know the solution. However, this defect of infinitesimal has been ignored and the verity of Calculus has never been in doubt because the powerful method of Calculus through infinitesimal has solved so many and important problems mankind has even never dreamed before.
The rigorous explanation of infinitesimal through the notion of limit was not completed until two hundred years after Newton, through the work of Cauchy and Weierstrass. So it is an overstatement to say that Newton knew the notion of limit and rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is an overstatement again to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.
answered 9 hours ago
Math WizardMath Wizard
365111
$endgroup$
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
add a comment |
$begingroup$
Newton did not have the explicit concept of limit. He just used the notion of infinitesimal, a quantity that sometimes is not zero (in operation) and sometimes is zero (in final result). For example, calculating the derivative of $x^2$ is like
$$
dotx=fracDelta yDelta x=frac(x+Delta x)^2-x^2Delta x=2x+Delta x=2x
$$
(Newton used $dotx$ for derivative and later Leibniz improved to $fracdydx$). In the last step, $Delta x=0$, but in $fracDelta yDelta x$, $Delta x$ can not be $0$ for $frac00$ makes no sense. This means that $Delta x$ (infinitesimal) sometimes is zero and sometimes not, a fact that Newton could not explain. Nor did Leibniz and others know the solution. However, this defect of infinitesimal has been ignored and the verity of Calculus has never been in doubt because the powerful method of Calculus through infinitesimal has solved so many and important problems mankind has even never dreamed before.
The rigorous explanation of infinitesimal through the notion of limit was not completed until two hundred years after Newton, through the work of Cauchy and Weierstrass. So it is an overstatement to say that Newton knew the notion of limit and rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is an overstatement again to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.
answered 9 hours ago
Math WizardMath Wizard
365111
$endgroup$
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
add a comment |
$begingroup$
Newton did not have the explicit concept of limit. He just used the notion of infinitesimal, a quantity that sometimes is not zero (in operation) and sometimes is zero (in final result). For example, calculating the derivative of $x^2$ is like
$$
dotx=fracDelta yDelta x=frac(x+Delta x)^2-x^2Delta x=2x+Delta x=2x
$$
(Newton used $dotx$ for derivative and later Leibniz improved to $fracdydx$). In the last step, $Delta x=0$, but in $fracDelta yDelta x$, $Delta x$ can not be $0$ for $frac00$ makes no sense. This means that $Delta x$ (infinitesimal) sometimes is zero and sometimes not, a fact that Newton could not explain. Nor did Leibniz and others know the solution. However, this defect of infinitesimal has been ignored and the verity of Calculus has never been in doubt because the powerful method of Calculus through infinitesimal has solved so many and important problems mankind has even never dreamed before.
The rigorous explanation of infinitesimal through the notion of limit was not completed until two hundred years after Newton, through the work of Cauchy and Weierstrass. So it is an overstatement to say that Newton knew the notion of limit and rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is an overstatement again to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.
answered 9 hours ago
Math WizardMath Wizard
365111
$endgroup$
Newton did not have the explicit concept of limit. He just used the notion of infinitesimal, a quantity that sometimes is not zero (in operation) and sometimes is zero (in final result). For example, calculating the derivative of $x^2$ is like
$$
dotx=fracDelta yDelta x=frac(x+Delta x)^2-x^2Delta x=2x+Delta x=2x
$$
(Newton used $dotx$ for derivative and later Leibniz improved to $fracdydx$). In the last step, $Delta x=0$, but in $fracDelta yDelta x$, $Delta x$ can not be $0$ for $frac00$ makes no sense. This means that $Delta x$ (infinitesimal) sometimes is zero and sometimes not, a fact that Newton could not explain. Nor did Leibniz and others know the solution. However, this defect of infinitesimal has been ignored and the verity of Calculus has never been in doubt because the powerful method of Calculus through infinitesimal has solved so many and important problems mankind has even never dreamed before.
The rigorous explanation of infinitesimal through the notion of limit was not completed until two hundred years after Newton, through the work of Cauchy and Weierstrass. So it is an overstatement to say that Newton knew the notion of limit and rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is an overstatement again to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.
answered 9 hours ago
Math WizardMath Wizard
365111
edited 3 hours ago
answered 9 hours ago
Math WizardMath Wizard
365111
answered 9 hours ago
Math WizardMath Wizard
365111
answered 9 hours ago
Math WizardMath Wizard
365111
365111
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
add a comment |
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing!
$endgroup$
– terry-s
4 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
$begingroup$
I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view.
$endgroup$
– Math Wizard
3 hours ago
add a comment |
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$begingroup$
@MathWizard title has been changed.
$endgroup$
– KCd
6 hours ago
$begingroup$
He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him."
$endgroup$
– Conifold
6 hours ago
$begingroup$
@Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation.
$endgroup$
– terry-s
5 hours ago
$begingroup$
@terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example.
$endgroup$
– Conifold
5 hours ago
1
$begingroup$
@terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect.
$endgroup$
– Conifold
4 hours ago