# Tag Info

22

The controversy was ostensibly over what gets to be the "true quantity of motion", momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first, a "skillful attack by Leibniz against an inadequate concept, $m|v|$, and its description of the world", it quickly deteriorated into a ...

17

Galileo followed a venerable tradition of distinguishing numbers, magnitudes of different kinds (lengths, times, areas, etc.) and ratios. This is somewhat analogous to the strictures of modern dimensional analysis used in physics, but even stricter, and ancient Greeks did not have dimensional constants to bridge the gaps. They did not even have enough ...

16

This usage of $\mathbf i$, $\mathbf j$, and $\mathbf k$ is not specific to physics. It is also used in mathematics, specifically when teaching linear algebra or multivariable calculus in $\mathbf R^3$ as well as group theory for the quaternion group $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$. The notation came from the use of the letters $i$, $j$, and $k$ in ...

13

See Heytesbury and the Physical Sciences and Nicole Oresme for detailed information about the so-called Oxford Calculators and their contribution (mainly) to mathematics. The issue is not so clearly assessed by modern historiography: There has been some discussion of the meaning of the work of Heytesbury and the other Calculators for the development of the ...

13

It was "solved" by Huygens in Horologium Oscillatorum (1673). The scare quotes are there because he never wrote down the equation, and even Newton's laws were not yet explicitly formulated. Huygens considered the motion of pendula, and for simple cases knew the "law of the conservation of living force" (mechanical energy), as Bernoullis later called it, see ...

12

The next to last sentence has all the reasons in a nutshell:"Buridan used the theory of impetus to give an accurate qualitative account of the motion of projectiles but he ultimately saw his theory as a correction to Aristotle". Buridan's account, as Aristotle's or Avicenna's before him, was qualitative, he never put it into equations, which would allow for ...

12

Such coordinates were called canonical because they are those in which equations of motion (or, of the hamiltonian flow of a function $H$) take the “canonical form” $$\frac{dq_i}{dt}=\frac{\partial H}{\partial p_i}, \qquad \frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i}$$ first written by Poisson (1809, pp. 272, 313), Lagrange (1810, p. 350), and Hamilton ...

11

By this standard why single out Galileo? Euclid "plagiarized" Elements, there isn't a single theorem in it that can be reliably attributed to him, and there are entire books that can be attributed to early Pythagoreans, Eudoxus or Theaetetus. In the whole 13 books he does not credit a single person by name. Descartes "plagiarized" analytic geometry, after ...

11

There was an opposition. The reasons were mainly philosophical. The main thing which was hard to accept was "action at a distance" through the void space. For example Huygens did not accept this. It contradicted Descartes theory which was prevalent at that time. So the question had too be decided by observations and experiments, as it always happens in ...

9

I have not read Buridan, but I am sure he was a philosopher, like Aristotle. The laws of nature are usually not named after philosophers. Philosophers can state all kinds of opinions, but this does not really add to the body of positive knowledge about nature. The difference between Newton and Buridan, is that Newton not just stated an opinion, but developed ...

9

Newtonian mechanics was not yet in place when Hooke published his De Potentia Restitutiva (On Restoring Force) in 1678, Newton's Principia only came out in 1687. Hooke inferred the law from experiments not only with springs but also with wood and a "body of air", concluding: "The power of any spring is in the same proportion with the tension thereof: ...

9

This is a very good question, I wish it got more attention. My answer will only be partial for I had difficulty finding early details on gyromagnetic effect and ratio. The concept comes up every time we have a rotating system of charged particles, because it has angular momentum and creates a magnetic dipole field, and it played an important historical role ...

9

If you are interested in descriptions of “everyday life” of human computers, here is an excerpt from Stan Ulam’s autobiography, Adventures of a Mathematician (University of California Press, 1991) concerning the years 1949–1952, when he was a part of the team working in Los Alamos on thermonuclear explosion. The account of the mathematical problems involved ...

9

At the time of Newton, the scientists could NOT detect any deviation of the Newton's laws from reality. As we know now, the only visible effect of this deviation in the Solar system is the anomalous precession of Mercury's perihelion. It was detected only in the middle of 19th century; it makes about 43 seconds per century, and it was not known at Newton's ...

9

Yes. Léon Foucault in 1851 published in the Comptes rendus a paper Démonstration physique du mouvement de rotation de la Terre au moyen du pendule detailing his experiment and the mathematical justification for it. (“Physical demonstration of the rotational movement of the Earth by means of the pendulum” is the English translation of that title.) He did not ...

8

Perhaps, the most insightful analysis (possibly to this day) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential ...

8

There are too separate issues here. The method of fluxions and fluents, Newton's version of calculus, is amply represented in Newton's extant papers, starting with 1669 On Analysis by Equations with an Infinite Number of Terms sent as a letter to John Collins, and disseminated by him to multiple correspondents, including Leibniz. The dotted shorthand was ...

8

The word originated with Aristotle, whose "energeia" and "entelecheia" can roughly be translated as enaction, that which makes matter move, and embodiment, that which makes matter take form, respectively. Over the middle ages the attention was focused on "impetus", roughly associated with mass times speed and usually taken to be the precursor of modern ...

8

(a) In answer, first a comment about proportionalities and equalities. Many 17th-century writers, including Huygens and Newton, commonly used proportions rather than equalities, especially to avoid doing what would now be calculation of ratios between physical quantities of unlike kind, such ratios being considered somehow illegitimate. This old tradition ...

7

The "classical" references for the sources and evolution of Newton's mechanics are : Alexandre Koyré, Richard S.Westfall and I.Bernard Cohen. Ivor Grattan-Guinness' comment is [in the Norton edition of the book] at page 259 refers to Richard S. Westfall, Force in Newton's physics: the science of dynamics in the seventeenth century (1971), Ch.8. I've no ...

7

There is a chapter "the history of the energy concept" of some 80 pages in Philip Mirowski's book More heat than light which is rather informative. The main point of the story appears to be that energy is something that is conserved, that is the concept really makes sense as an "invariant". The end of the chapter mentions various energetist and energetisms, ...

7

It is more accurate to say that Hamilton anticipated some of the ideas of mathematics and heuristics of quantum mechanics, that would later inspire Schrödinger to produce his formulation of wave mechanics. The reason he was able to anticipate those ideas is that the quantum wave-particle duality had a classical predecessor, the optico-mechanical analogy. ...

7

Goldstine, A History of Numerical Analysis from the 16th through the 19th Century (1977), describes Kepler's approach (p. 47), which may be found in Kepler's Epitome Astronomiae Copernicanae (1618), Ch. 4, Bk. V., pp. 665f. It is an iterative numerical algorithm Kepler called regula positionum. Goldstine describes the steps of an example, which begins on p....

7

Edited. The story is long and complicated. From the mathematical point of view, three statements are involved: A. Eigenvalues of a self-adjoint operator are real and simple, B. There exists an orthogonal basis consisting of eigenvectors, C. Every quadratic form can be reduced to principal axes. In 1715 Brook Taylor found that functions $$\sin\frac{\pi nx}... 7 Liouville shows (1838, pp. 347-349) that if x=\phi_t(a) is the "complete integral" of an ODE$$ \frac{dx}{dt}=P(t,x) \tag{*} $$on \mathbf R^n (meaning that a\in\mathbf R^n are n otherwise arbitrary parameters for the general solution), then the jacobian u=\det\bigl(\frac{\partial x_i}{\partial a_j}\bigr) satisfies (the identity you are after):$$ \...

7

Kinematics was distinguished from dynamics by the Merton school (a.k.a. Oxford calculators) of scholastics in 14th century, who worked out kinematics of uniformly accelerated motion. In particular, they formulated the mean speed theorem (a.k.a. Merton rule) (distance traveled is half the sum of the initial and final velocities, times the elapsed time), which ...

7

This probably refers to Galileo's "derivation" of Tartaglia's observation that cannon balls achieve maximal range when fired at 45°. Tartaglia's theory of projectile motion was wrong, he assumed that fired balls follow a line segment going up, then an arc of a circle to change direction, and finally fall vertically down, but the observation was ...

6

According to Truesdell [1954]: (p. xliii:) As far as I can ascertain, it is Euler [1750, p. 196] which contains the first general statement of “Newton’s equations”. (p. xlii:) The axioms which Euler asserts “include all principles of mechanics” are $$2M\frac{d^2x}{dt^2}=P,\qquad 2M\frac{d^2y}{dt^2}=Q,\qquad 2M\frac{d^2z}{dt^2}=R.$$ (...) Anyone who ...

6

Not much is really known about general "scientific environment". What we really have is only books and fragments of scientists and references on them in other works. It is commonly acknowledged that most writings did not survive. So we cannot have an adequate picture of the general scientific environment. The most developed sciences were mathematics and ...

6

This depends on which objects you have in mind and who you would call a scientist. There was a broad consensus in the ancient Greek natural philosophy that superlunar objects, like the stars and planets, were moving on their own steam, pardon, divine nature, and uniformly along circles at that. Early mathematical astronomers accepted this conviction to such ...

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