Leonhard Euler ~ Sakinorva Databank

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Leonhard Euler

Mathematician

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Leonhard Euler ~ Sakinorva Databank

Leonhard Euler

Mathematician

Science, Technology, and Engineering

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	nicotineseries IXFP 6w5, 649 EII-Fi 2025/07/23 (Wed) 06:18:55 #10203
	Excuse me, should I do this? First of all, I know how to write this in terms of defending Ti using but I don't know yet it is are definitely defend them or probably criticize you all that even vote him intp. Yes of course I have the reason and I had a results. I do some of my reading, researching and a conclusion in depth, unfortunately waste my time for doing this at least I do better analysis for this, the things is aren't that easy to understand that the info, the whole thing about him when I learn more, the more suspicious I am. I as an individual wouldn't had to judge anything and take them easy from what I see and revise again from what I know. So, here is it: First of all, if you want to q&a with me that's okay, I really open to any idea! Or ur opinion So, in here the summarization I only did this with 5 problem: 1.) The description of many website 2.) For the sake of being cynic and skeptical about anything (lol) about what you all wrote with this comment 3.) When I saw an info of him whenever the website is you know I take it as the truth but what is that makes me wanna question it and think or not make me judge that was him with even single glance that is him personally and that it is. 4.) I see some Te 5.) The more I dig into the info, the more curious I become ;) woow 1.) Isn't that he really deductive>inductive users? Euler: "The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." >>> Comparing into some scholar by G. L. Alexanderson's journal: Ars Expositionis: Euler as Writer and Teacher (cite: https://www.jstor.org/stable/2690366) open admission of failure to prove a conjecture for which he had convincing evidence. George Pólya sums it up when he writes (italics ours): "A master of inductive research in mathematics, he [Euler] made important discoveries (on infinite series, in the Theory of Numbers, and in other branches of mathematics) by induction, that is, by observation, daring guess, and shrewd verification. In this respect, however, Euler is not unique; other mathematicians, great and small, used induction extensively in their work. Yet Euler seems to me almost unique in one respect: he takes pains to present the relevant inductive evidence carefully, in detail, in good order. He presents it convincingly but honestly, as a genuine scientist should do. His presentation is " the candid exposition of the ideas that led him to those discoveries" and has a distinctive charm. Naturally enough, as any other author, he tries to impress his readers, but, as a really good author, he tries to impress his readers only by such things as have genuinely impressed himself". What sets Euler apart from other great masters who wrote mathematics, including textbooks,many of whom even wrote very clearly? As Pólya has noted, part of the answer probably lies in Euler's approach to mathematics and the kind of mathematics that he did so well. He used a great deal of induction in his work, for pattern recognition and discovery of formulas were his forte. Against Ti (±probability): Even among his contemporaries, Euler was known as someone whose writings were particularly clear and elegant. Nicholas Fuss, his assistant in St. Petersburg and a fellow citizen of Basel, wrote in his eulogy on the death of Euler in 1783 about the clarity of his ideas, the precision in their statement and the order in which they were arranged. The son of Nicholas Fuss, Paul-Heinrich Fuss, in his preface to the collection of letters he edited in 1843 wrote of Euler's: "remarkably simple and lucid exposition of his profound research and his skillful choice of examples" Against Te it's Fe?(still questioning): He nevertheless seems to have had remarkable success at teaching on those occasions when he took on students. Some of text that against Gauss (it maybe because their different behavior? I don't know: Unlike others-Gauss' name comes to mind-Euler did not attempt to hide the origins of his theorems, but, on the contrary, he went out of his way to motivate them by giving many example Pólya quotes Condorcet as saying that "He [Euler] preferred instructing his pupils to the little satisfaction of amazing them. He would have thought not to have done enough for science if he should have failed to add to the discoveries, with which he enriched science, the candid exposition of the ideas that led him to those discoveries" Euler admits to having trouble proving certain conjectures, but then proceeds to use unproved results, cautioning the reader that certain steps yet completely proved. In reading Euler we can see a great, creative mind at work." =>Ne (just kidding my guys) It is this Ti?: A full account of Euler's presentation, taken from and translated by Pólya, can be found in. Here we extract a few sections to demonstrate Euler's style. In discussing the function (n), which gives the sum of the divisors of n, he begins: Euler: "Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of the primes, which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order. I am myself certainly far from this goal, but I just happened I to discover an extremely strange law governing the sums of the divisors of the integers which, at the first glance, appear just as irregular as the sequence of the primes, and which, in a certain sense, comprise even the latter. This law, which I shall explain in a moment, is, in my opinion, so much more remarkable as it is of such a nature that we can be assured of its truth without giving it a perfect demonstration. Nevertheless, I shall present such evidence for it as might be regarded as almost equivalent to a rigorous demonstration". 2.) In regarded of this website: https://www.irishtimes.com/news/science/euler-a-mathematician-without-equal-and-an-overall-nice-guy-1.4455424#:~:text=He%20was%20described%20by%20his,not%20essential%20for%20mathematical%20prowess. Sum it up almost thanks to the original sp 4 but I'm sorry for this site are to easy to take grain of salt of it and didn't explain in detail about anything in him and just a description per-se and don't believe it will prove anything "The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways, the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit." “On the character of Euler his biographers agree. A talented child, he was open, cheerful, and sociable.” Yeah that is that it was aux Fe right, so say that he had high Fe?? On where what do you state it was. So, another side ones that maybe because of Fi because he cling, firmly and honest with what his nature personal belief? idk, I don't judge this either. But try to think it now, figure it out and imagine that just once sentence=Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs is probably sign of Fe right? But figured it out where the timing is. It was in the Enlightenment Era where himself are against their the freethinkers too, is incessantly stating "to think to yourself" or about the Monadism of Leibniz/Wolffian philosophy is based on the Enlightenment too. So because that the origin of the atheism on that time is tremendously many (examples is the French encyclopedia editors) or he's really... you know is a biblical literalist too. Then their (almost) disregard his Calvinist beliefs. I think to compare it with Rousseau (he same devout Calvinist) he was in the same way because they try to protect the whole truth about their Christians (Calvinists) so that they conform to what is supposed to be in the Bible of their religion. Just him want to defend his personal beliefs or his own self moral, right? In my opinion my guys! Don't take it too seriously, or don't take it to your heart, it's just a theory (lol)! The trouble is while I'm compare it with Kant he was there are few thoughts or ideas that are influenced by Euler himself. Well, he had little of his(Euler) idea and then he would sends letters to Euler's to propose the idea sometimes, doesn't it? And Kant himself could've been against the Wolffian philosophy itself (probably). The evidence: "Any counterarguments?" -Espresso Cookie 3.) A more detailed, fixed and explanation for Berx from me (my opinion,ok!): From a biographical article on Euler by Walter Gautschi [functional indications are my own]: "Euler's writings have the marks of a superb expositor. He always strove for utmost clarity and simplicity (almost I think it's Ti but it cited clarity and simplicity is probably because he didn't want to exaggerated his idea/theories so it gives his mind is still on single-minded? Te? Fi??) and , and he often revisited earlier work when he felt they were lacking in these qualities (still it's not sensation perspection Si

Editing post #10203 by nicotineseries Excuse me, should I do this? First of all, I know how to write this in terms of defending Ti using but I don't know yet it is are definitely defend them or probably criticize you all that even vote him intp. Yes of course I have the reason and I had a results. I do some of my reading, researching and a conclusion in depth, unfortunately waste my time for doing this at least I do better analysis for this, the things is aren't that easy to understand that the info, the whole thing about him when I learn more, the more suspicious I am. I as an individual wouldn't had to judge anything and take them easy from what I see and revise again from what I know. So, here is it: First of all, if you want to q&a with me that's okay, I really open to any idea! Or ur opinion So, in here the summarization I only did this with 5 problem: 1.) The description of many website 2.) For the sake of being cynic and skeptical about anything (lol) about what you all wrote with this comment 3.) When I saw an info of him whenever the website is you know I take it as the truth but what is that makes me wanna question it and think or not make me judge that was him with even single glance that is him personally and that it is. 4.) I see some Te 5.) The more I dig into the info, the more curious I become ;) woow 1.) Isn't that he really deductive>inductive users? Euler: "The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." >>> Comparing into some scholar by G. L. Alexanderson's journal: Ars Expositionis: Euler as Writer and Teacher (cite: https://www.jstor.org/stable/2690366) open admission of failure to prove a conjecture for which he had convincing evidence. George Pólya sums it up when he writes (italics ours): "A master of inductive research in mathematics, he [Euler] made important discoveries (on infinite series, in the Theory of Numbers, and in other branches of mathematics) by induction, that is, by observation, daring guess, and shrewd verification. In this respect, however, Euler is not unique; other mathematicians, great and small, used induction extensively in their work. Yet Euler seems to me almost unique in one respect: he takes pains to present the relevant inductive evidence carefully, in detail, in good order. He presents it convincingly but honestly, as a genuine scientist should do. His presentation is " the candid exposition of the ideas that led him to those discoveries" and has a distinctive charm. Naturally enough, as any other author, he tries to impress his readers, but, as a really good author, he tries to impress his readers only by such things as have genuinely impressed himself". What sets Euler apart from other great masters who wrote mathematics, including textbooks,many of whom even wrote very clearly? As Pólya has noted, part of the answer probably lies in Euler's approach to mathematics and the kind of mathematics that he did so well. He used a great deal of induction in his work, for pattern recognition and discovery of formulas were his forte. Against Ti (±probability): Even among his contemporaries, Euler was known as someone whose writings were particularly clear and elegant. Nicholas Fuss, his assistant in St. Petersburg and a fellow citizen of Basel, wrote in his eulogy on the death of Euler in 1783 about the clarity of his ideas, the precision in their statement and the order in which they were arranged. The son of Nicholas Fuss, Paul-Heinrich Fuss, in his preface to the collection of letters he edited in 1843 wrote of Euler's: "remarkably simple and lucid exposition of his profound research and his skillful choice of examples" Against Te it's Fe?(still questioning): He nevertheless seems to have had remarkable success at teaching on those occasions when he took on students. Some of text that against Gauss (it maybe because their different behavior? I don't know: Unlike others-Gauss' name comes to mind-Euler did not attempt to hide the origins of his theorems, but, on the contrary, he went out of his way to motivate them by giving many example Pólya quotes Condorcet as saying that "He [Euler] preferred instructing his pupils to the little satisfaction of amazing them. He would have thought not to have done enough for science if he should have failed to add to the discoveries, with which he enriched science, the candid exposition of the ideas that led him to those discoveries" Euler admits to having trouble proving certain conjectures, but then proceeds to use unproved results, cautioning the reader that certain steps yet completely proved. In reading Euler we can see a great, creative mind at work." =>Ne (just kidding my guys) It is this Ti?: A full account of Euler's presentation, taken from and translated by Pólya, can be found in. Here we extract a few sections to demonstrate Euler's style. In discussing the function (n), which gives the sum of the divisors of n, he begins: Euler: "Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of the primes, which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order. I am myself certainly far from this goal, but I just happened I to discover an extremely strange law governing the sums of the divisors of the integers which, at the first glance, appear just as irregular as the sequence of the primes, and which, in a certain sense, comprise even the latter. This law, which I shall explain in a moment, is, in my opinion, so much more remarkable as it is of such a nature that we can be assured of its truth without giving it a perfect demonstration. Nevertheless, I shall present such evidence for it as might be regarded as almost equivalent to a rigorous demonstration". 2.) In regarded of this website: https://www.irishtimes.com/news/science/euler-a-mathematician-without-equal-and-an-overall-nice-guy-1.4455424#:~:text=He%20was%20described%20by%20his,not%20essential%20for%20mathematical%20prowess. Sum it up almost thanks to the original sp 4 but I'm sorry for this site are to easy to take grain of salt of it and didn't explain in detail about anything in him and just a description per-se and don't believe it will prove anything "The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways, the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit." “On the character of Euler his biographers agree. A talented child, he was open, cheerful, and sociable.” Yeah that is that it was aux Fe right, so say that he had high Fe?? On where what do you state it was. So, another side ones that maybe because of Fi because he cling, firmly and honest with what his nature personal belief? idk, I don't judge this either. But try to think it now, figure it out and imagine that just once sentence=Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs is probably sign of Fe right? But figured it out where the timing is. It was in the Enlightenment Era where himself are against their the freethinkers too, is incessantly stating "to think to yourself" or about the Monadism of Leibniz/Wolffian philosophy is based on the Enlightenment too. So because that the origin of the atheism on that time is tremendously many (examples is the French encyclopedia editors) or he's really... you know is a biblical literalist too. Then their (almost) disregard his Calvinist beliefs. I think to compare it with Rousseau (he same devout Calvinist) he was in the same way because they try to protect the whole truth about their Christians (Calvinists) so that they conform to what is supposed to be in the Bible of their religion. Just him want to defend his personal beliefs or his own self moral, right? In my opinion my guys! Don't take it too seriously, or don't take it to your heart, it's just a theory (lol)! The trouble is while I'm compare it with Kant he was there are few thoughts or ideas that are influenced by Euler himself. Well, he had little of his(Euler) idea and then he would sends letters to Euler's to propose the idea sometimes, doesn't it? And Kant himself could've been against the Wolffian philosophy itself (probably). The evidence: "Any counterarguments?" -Espresso Cookie 3.) A more detailed, fixed and explanation for Berx from me (my opinion,ok!): From a biographical article on Euler by Walter Gautschi [functional indications are my own]: "Euler's writings have the marks of a superb expositor. He always strove for utmost clarity and simplicity (almost I think it's Ti but it cited clarity and simplicity is probably because he didn't want to exaggerated his idea/theories so it gives his mind is still on single-minded? Te? Fi??) and , and he often revisited earlier work when he felt they were lacking in these qualities (still it's not sensation perspection Si<Ti/Fi. Si is Pi not Judging function). Characteristically, he will proceed from very simple examples to ever more complicated ones (word "complicated" is had been debunked in my argument that his want the whole theories to be clear and simplicity) before eventually revealing the underlying theory in its full splendor (probably that Te or probably Ne, you can judge?). Yet, in his quest for discovery, he could be fearless, even reckless, but owing to his secure instinct, he rarely went astray when his argumentation became hasty (not explain any indicate cognitive function but probably Te but idk). He had an eye for what is essential and unifying (Si/Ni, in this context it's the Pi function working imo). In mechanics, Gleb Konstantinovich Mikhailov writes, 'Euler possessed a rare gift of systematizing and generalizing scientific ideas (Thinking, but idk systematizing and generalizing it's seem Je imo :/), which allowed him to present large parts of mechanics in a relatively definitive form (Thinking, maybe Pi function).' Euler was open and receptive to new ideas (idea≠intuition, probably Ne/Se). In the words of André Weil, '...what at first is striking about Euler is his extraordinary quickness in catching hold of any suggestion, wherever it came from (Te, ±probably)... There is not one of these suggestions which in Euler's hands has not become the point of departure of an impressive series of researches (Ne<Te, look at that it is about inductive reasoning?) ... Another thing, not less striking, is that Euler never abandons a research topic, once it has excited his curiosity (Ni/Si, but probably Fe because excited); on the contrary, he returns to it, relentlessly, in order to deepen and broaden it on each revisit (Si/Ni, Pi function). Even if all problems related to such a topic seem to be resolved, he never ceases until the end of his life to find proofs that are "more natural," "simpler," "more direct." (Fi/Ti)" So, my answer is not completely correct, all of it. And anything that bother me is his lack of practicality is another my questions about him. 4.) If anyone want to ask me "is he... well he must be using Fe, right?" Well done, right of course he was in any website are potrait to be in positive way that describe he is a talented child, he was open, cheerful, and sociable. But any people can do the same and just do things the same like when their are yes, because in the environment where he live, within ourselves being a person we must be able to adapt with where we live and undisturbed whatever cognitive function we use. It's just a trait and behavior, right not about cognitive function. We can know people who had mentally ill to use Fe function or it can be Fi. We all naturally a healthy person as it should be and had a positive traits too. As they can be a courteous people, and the function is still Te. And then pair them with Fi function and vice versa (Ti/Fe pair), some of people can be healthy person is good and unhealthy. Yeah, idk I think that is it but honestly I also have a lot of mistakes that I have written here. I'm so sorry. We as humans make lots of mistaken too, right?
Replying to post #10203 by nicotineseries

	nicotineseries IXFP 6w5, 649 EII-Fi 2025/07/23 (Wed) 06:17:15 #10202
	6.) Quotes about Euler: (cite: https://en.m.wikiquote.org/wiki/Leonhard_Euler?searchToken=4ksauftzw0wiksimssnoeuwmm) Frederick the Great, Letters of Voltaire and Frederick the Great (1927), translated by Richard Aldington, letter 221 from Frederick to Voltaire (25 November 1777): "Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!" Carl Friedrich Gauss, as quoted by Louise Grinstein, Sally I. Lipsey, Encyclopedia of Mathematics Education (2001) p. 235: "The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it." George Pólya, Induction and Analogy in Mathematics (1954) Vol. 1 Of Mathematics and Plausible Reasoning: "Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive". William Whewell, History of the Inductive Sciences (1859) Vol. 1, pp. 363-363: "As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler." 7.) Another eulogy of Euler in 1783 by Nicolas Fuss: Website: http://eulerarchive.maa.org/historica/fuss.html By Marquis de Condorcet: Website: http://eulerarchive.maa.org/historica/condorcet.html 8.)?: For another you that wanna searching or research or even know in deep about the man is now right now isn't that difficult there are many on the website, you can search if you have any intention to know it. 9.) So another problem when he with Voltaire. I confused, so summary on he was a target of Voltaire wit right? 10.) Wow you guys read his letter to German princesses? But natural philosophy are. Ow I have a pdf of it. https://drive.google.com/file/d/16sIGvRWlQp1YypXfKUBzVaGD1QzAzpnm/view?usp=drivesdk 11.) Guys I found there is something that really interesting! Wait, okay here is it. https://myhero.com/leonhard-euler-the-catalyst-behind-the-revolution-of-mathematics 12.) If you want some book of his biography that I recommend to reads for. I definitely have an pdf on it so you can read. https://drive.google.com/file/d/1dEnKzINjAFUMIzRZjujfl0PiRIXzf32A/view?usp=drive_link 13.) About his mistakes: https://drive.google.com/file/d/1YLGD2fUGLRn3VTeEijjLm7QHpXpKAzLu/view?usp=drivesdk 14.) A little bit out of context is this "Mathematicians are People, Too!" Is the book I recommend to read about: https://drive.google.com/file/d/1YMD6HTCr1YsPsKPL9XnUfoQKsb8pbYD9/view?usp=drivesdk

Editing post #10202 by nicotineseries 6.) Quotes about Euler: (cite: https://en.m.wikiquote.org/wiki/Leonhard_Euler?searchToken=4ksauftzw0wiksimssnoeuwmm) Frederick the Great, Letters of Voltaire and Frederick the Great (1927), translated by Richard Aldington, letter 221 from Frederick to Voltaire (25 November 1777): "Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!" Carl Friedrich Gauss, as quoted by Louise Grinstein, Sally I. Lipsey, Encyclopedia of Mathematics Education (2001) p. 235: "The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it." George Pólya, Induction and Analogy in Mathematics (1954) Vol. 1 Of Mathematics and Plausible Reasoning: "Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive". William Whewell, History of the Inductive Sciences (1859) Vol. 1, pp. 363-363: "As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler." 7.) Another eulogy of Euler in 1783 by Nicolas Fuss: Website: http://eulerarchive.maa.org/historica/fuss.html By Marquis de Condorcet: Website: http://eulerarchive.maa.org/historica/condorcet.html 8.)?: For another you that wanna searching or research or even know in deep about the man is now right now isn't that difficult there are many on the website, you can search if you have any intention to know it. 9.) So another problem when he with Voltaire. I confused, so summary on he was a target of Voltaire wit right? 10.) Wow you guys read his letter to German princesses? But natural philosophy are. Ow I have a pdf of it. https://drive.google.com/file/d/16sIGvRWlQp1YypXfKUBzVaGD1QzAzpnm/view?usp=drivesdk 11.) Guys I found there is something that really interesting! Wait, okay here is it. https://myhero.com/leonhard-euler-the-catalyst-behind-the-revolution-of-mathematics 12.) If you want some book of his biography that I recommend to reads for. I definitely have an pdf on it so you can read. https://drive.google.com/file/d/1dEnKzINjAFUMIzRZjujfl0PiRIXzf32A/view?usp=drive_link 13.) About his mistakes: https://drive.google.com/file/d/1YLGD2fUGLRn3VTeEijjLm7QHpXpKAzLu/view?usp=drivesdk 14.) A little bit out of context is this "Mathematicians are People, Too!" Is the book I recommend to read about: https://drive.google.com/file/d/1YMD6HTCr1YsPsKPL9XnUfoQKsb8pbYD9/view?usp=drivesdk
Replying to post #10202 by nicotineseries

	nicotineseries IXFP 6w5, 649 EII-Fi 2025/07/23 (Wed) 06:16:30 #10201
	5.) Presenting the explicit versions of his quotes from Berx 1.) Euler: "Since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear." =>introduction to De Curvis Elasticis, Additamentum I to his Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes 1744; translated on pg10-11, "Leonhard Euler's Elastic Curves", Oldfather et al 1933. "All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction." 2.) Euler: "The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." =>Opera Omnia, ser. 1, vol. 2, p. 459 Spcimen de usu observationum in mathesi pura, as quoted by George Pólya, Induction and Analogy in Mathematics Vol. 1, Mathematics and Plausible Reasoning (1954) "It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful."

Editing post #10201 by nicotineseries 5.) Presenting the explicit versions of his quotes from Berx 1.) Euler: "Since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear." =>introduction to De Curvis Elasticis, Additamentum I to his Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes 1744; translated on pg10-11, "Leonhard Euler's Elastic Curves", Oldfather et al 1933. "All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction." 2.) Euler: "The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." =>Opera Omnia, ser. 1, vol. 2, p. 459 Spcimen de usu observationum in mathesi pura, as quoted by George Pólya, Induction and Analogy in Mathematics Vol. 1, Mathematics and Plausible Reasoning (1954) "It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful."
Replying to post #10201 by nicotineseries

	nicotineseries IXFP 6w5, 649 EII-Fi 2025/07/23 (Wed) 06:14:30 #10200
	So when I read it carefully and I think it reservedly. Now I'll do an opinion, and my resume of all what I read about this person. Yet I know if all of my presentations about the comment that I write are wrong and not consistent with. So this is the list of my reasons: 1.) Thanks for @Berx and his knowledge for writing of his topology and @watcher. But really? If I know if it was true he has Ti/Fe axis function but why? Well I'm open to any argument for opposing me but I think that was he is not think of something about "harmony"? Pre-establised harmony? Euler turns away from the monistic philosophy in his time and from monadology, and he returns to the Cartesian dualism of body and soul, but he does not accept the Cartesian concept of body. He refuses also the opinion that animals are mere automatic machines, and he reproaches the Wolffians for considering human beings in the same way. In letter No. 81 Euler explains “There is a special place in the brain where all the nerves come to an end, and just there the soul is seated or there it feels all its impressions which are effected on it by the senses.” In letter No. 92 he says that an hour is not bound to a place. Then he continues: “Similarly I may say that my soul is neither in my head nor outside my head nor elsewhere, ... Therefore my soul does not exist at any place, but it operates at a certain place ...” (What did it mean to that?) In letter No. 83 he tries to ridicule pre-established harmony by the fiction of a connection between his soul and the body of a rhinoceros in Africa, but in letter No. 93 he argues for a stationary soul, and he thinks it is possible that immediately after death God could connect his soul with a body on the moon. (what else could you possibly want? Huh, Fe again?) Since Euler returns to dualism, to him those problems that provoked the monistic philosophy return, along with the system of pre-established harmony for either resolving them or avoiding them. He is well acquainted with these difficulties, and he refers to them in the letters again and again: If body and mind are based on two completely distinct substances, how could it be possible that the perceiving soul is able to “assume” something of the material world? Indeed, sensible perception needs the fulfilment of some conditions of the body, but nevertheless the picture on the retina is not yet the object of the seeing soul. The problem of the provenance of sensation, which is a hard problem in every theory of knowledge, especially in the Kantian theory, is treated in letter No. 82. Euler compares the soul with a man sitting in a dark room and seeing the things outside the room via a camera obscura. Similarly the soul is considering the ends of the nerves and receiving the impressions of the sense organs. “Though it is absolutely unknown to us what the similarity is between the impressions on the ends of nerves and the objects causing those impressions, these impressions are appropriate to deliver to us a very adequate idea of the objects.” Descartes had at least endeavoured to argue for the adequacy of that idea, but in Euler’s writings it is merely claimed. Obviously he does not advance to the Kantian question of whether there could be any similarity at all. For Euler it is certain in virtue of God’s omnipotence that a connection between body and mind can exist and, indeed, does exist. The second principal question in the theory of knowledge is “How are they connected?” This question was treated by the atomists of antiquity and it is still discussed by modern brain scientists. Euler’s answer is succinct: It is a great mystery (grand mystére)! (What do you think guys?) 3.) I don't know why you say about this person use Ti while you said "Ti doesn't merely seek a formula that 'works'. It seeks the most elegant version of the formula, i.e. the one that expresses the 'most truth' using the least amount of notation." Did he seems mention that? In 1767 Euler was sixty years old and losing his eyesight. He dictated the book to a servant, a tailor's apprentice, so one might imagine that under such circumstances his account of the rules for the multiplication of roots contained simple mistakes or oversights, in what was otherwise a masterpiece (Ti?). However, neither old age nor blindness slowed Euler's productivity or dulled his sharpness of mind, as is well known. Besides, Euler was increasingly acknowledged as the person who strikingly solved the vexing puzzle of taking logarithms of negative and complex numbers, as the latter eventually became known. So it seems stunning that he would have been confused about elementary multiplication. A passage in his Algebra seems even to state that √-1 x √-4 = 2, ridiculous though it seems. Some historians, Florian Cajori for one [4, p. 127], have suggested that perhaps such errors stemmed merely from typographical printing miscues for which Euler can not be held accountable. Others say that it involved a systematic confusion. Actually, neither is the case. Surprisingly, Euler committed no mistake on the matter. The solution to this puzzle is found buried in history under layers of ambiguous expressions, notations, and changing conventions on the definitions of basic operations. Stranger still, the historical analysis reveals defects and arbitrariness in the approach to this subject that became incorporated into elementary algebra as we know it. (It's it Ti?). As you want to know more again you can check this: https://drive.google.com/file/d/1YLGD2fUGLRn3VTeEijjLm7QHpXpKAzLu/view?usp=drivesdk 4.) Actually, if you read that article that I write in my past comment. In my opinion, He is use his principle to refine and perfecting a few encompassing ideas rather than generate more or more. He is not a someone who seek impartial and universalized foundational principles and he want his formula to proving the other problem that someone probably know are missing. Like did he making a new formula than improving the Newton's second law or a rigid bodies? Or a pioneer of new possibilities did he do that? I know he was being impartial but it was not the way about principle, it was all about his manners, his ethic/moral that he had in his daily life. And I know he is too a pioneer of pure mathematics but did he seek for understand of it's own sake? For the play of being intelligent? He had do saying like "Oh" "my pencil is more intelligent than I am". I think that what his want to be mathemicians is all about his interest and he did that with passion for changing the past principle that not yet completed or repaired by someone. Think that about Fermat's number of theory, is one of his dedicated things to solved the problems. Or Konigsberg bridges? 5.) I highly recommend or totally agree you should read what I said before in my past comment. In there I almost give all of the resources that I collected from many articles, scholars, or even scientific book about his life in generals. Please please I begging you to not close your eyes and your mind about what I wrote in it. Thanks to all of you ♡

Editing post #10200 by nicotineseries So when I read it carefully and I think it reservedly. Now I'll do an opinion, and my resume of all what I read about this person. Yet I know if all of my presentations about the comment that I write are wrong and not consistent with. So this is the list of my reasons: 1.) Thanks for @Berx and his knowledge for writing of his topology and @watcher. But really? If I know if it was true he has Ti/Fe axis function but why? Well I'm open to any argument for opposing me but I think that was he is not think of something about "harmony"? Pre-establised harmony? Euler turns away from the monistic philosophy in his time and from monadology, and he returns to the Cartesian dualism of body and soul, but he does not accept the Cartesian concept of body. He refuses also the opinion that animals are mere automatic machines, and he reproaches the Wolffians for considering human beings in the same way. In letter No. 81 Euler explains “There is a special place in the brain where all the nerves come to an end, and just there the soul is seated or there it feels all its impressions which are effected on it by the senses.” In letter No. 92 he says that an hour is not bound to a place. Then he continues: “Similarly I may say that my soul is neither in my head nor outside my head nor elsewhere, ... Therefore my soul does not exist at any place, but it operates at a certain place ...” (What did it mean to that?) In letter No. 83 he tries to ridicule pre-established harmony by the fiction of a connection between his soul and the body of a rhinoceros in Africa, but in letter No. 93 he argues for a stationary soul, and he thinks it is possible that immediately after death God could connect his soul with a body on the moon. (what else could you possibly want? Huh, Fe again?) Since Euler returns to dualism, to him those problems that provoked the monistic philosophy return, along with the system of pre-established harmony for either resolving them or avoiding them. He is well acquainted with these difficulties, and he refers to them in the letters again and again: If body and mind are based on two completely distinct substances, how could it be possible that the perceiving soul is able to “assume” something of the material world? Indeed, sensible perception needs the fulfilment of some conditions of the body, but nevertheless the picture on the retina is not yet the object of the seeing soul. The problem of the provenance of sensation, which is a hard problem in every theory of knowledge, especially in the Kantian theory, is treated in letter No. 82. Euler compares the soul with a man sitting in a dark room and seeing the things outside the room via a camera obscura. Similarly the soul is considering the ends of the nerves and receiving the impressions of the sense organs. “Though it is absolutely unknown to us what the similarity is between the impressions on the ends of nerves and the objects causing those impressions, these impressions are appropriate to deliver to us a very adequate idea of the objects.” Descartes had at least endeavoured to argue for the adequacy of that idea, but in Euler’s writings it is merely claimed. Obviously he does not advance to the Kantian question of whether there could be any similarity at all. For Euler it is certain in virtue of God’s omnipotence that a connection between body and mind can exist and, indeed, does exist. The second principal question in the theory of knowledge is “How are they connected?” This question was treated by the atomists of antiquity and it is still discussed by modern brain scientists. Euler’s answer is succinct: It is a great mystery (grand mystére)! (What do you think guys?) 3.) I don't know why you say about this person use Ti while you said "Ti doesn't merely seek a formula that 'works'. It seeks the most elegant version of the formula, i.e. the one that expresses the 'most truth' using the least amount of notation." Did he seems mention that? In 1767 Euler was sixty years old and losing his eyesight. He dictated the book to a servant, a tailor's apprentice, so one might imagine that under such circumstances his account of the rules for the multiplication of roots contained simple mistakes or oversights, in what was otherwise a masterpiece (Ti?). However, neither old age nor blindness slowed Euler's productivity or dulled his sharpness of mind, as is well known. Besides, Euler was increasingly acknowledged as the person who strikingly solved the vexing puzzle of taking logarithms of negative and complex numbers, as the latter eventually became known. So it seems stunning that he would have been confused about elementary multiplication. A passage in his Algebra seems even to state that √-1 x √-4 = 2, ridiculous though it seems. Some historians, Florian Cajori for one [4, p. 127], have suggested that perhaps such errors stemmed merely from typographical printing miscues for which Euler can not be held accountable. Others say that it involved a systematic confusion. Actually, neither is the case. Surprisingly, Euler committed no mistake on the matter. The solution to this puzzle is found buried in history under layers of ambiguous expressions, notations, and changing conventions on the definitions of basic operations. Stranger still, the historical analysis reveals defects and arbitrariness in the approach to this subject that became incorporated into elementary algebra as we know it. (It's it Ti?). As you want to know more again you can check this: https://drive.google.com/file/d/1YLGD2fUGLRn3VTeEijjLm7QHpXpKAzLu/view?usp=drivesdk 4.) Actually, if you read that article that I write in my past comment. In my opinion, He is use his principle to refine and perfecting a few encompassing ideas rather than generate more or more. He is not a someone who seek impartial and universalized foundational principles and he want his formula to proving the other problem that someone probably know are missing. Like did he making a new formula than improving the Newton's second law or a rigid bodies? Or a pioneer of new possibilities did he do that? I know he was being impartial but it was not the way about principle, it was all about his manners, his ethic/moral that he had in his daily life. And I know he is too a pioneer of pure mathematics but did he seek for understand of it's own sake? For the play of being intelligent? He had do saying like "Oh" "my pencil is more intelligent than I am". I think that what his want to be mathemicians is all about his interest and he did that with passion for changing the past principle that not yet completed or repaired by someone. Think that about Fermat's number of theory, is one of his dedicated things to solved the problems. Or Konigsberg bridges? 5.) I highly recommend or totally agree you should read what I said before in my past comment. In there I almost give all of the resources that I collected from many articles, scholars, or even scientific book about his life in generals. Please please I begging you to not close your eyes and your mind about what I wrote in it. Thanks to all of you ♡
Replying to post #10200 by nicotineseries

date	username	vote
25/12/31 22:44	nicotineseries	N
23/08/21 05:48	jamelspy	INTJ
23/08/21 05:39	salim	INTJ
19/03/03 16:04	tch	INTJ
19/03/03 02:05	LadyX	INTJ

date	username	vote
23/08/21 05:48	jamelspy	INTJ
23/08/21 05:39	salim	INTJ
20/01/21 17:56	tch	INTP

date	username	vote
19/03/03 02:05	LadyX	1w9