ABSTRACTThe worksheets that presumably contained Newton's early development of the fundamental concepts in his Principia have been lost. A plausible reconstruction of this development is presented based on Newton's exchange of letters with Robert Hooke in 1679, with Edmund Halley in 1686, and on some clues in the diagram associated with Proposition 1 in Book 1 of the Principia that have been ignored in the past. A graphical construction associated with this proposition leads to a rapidly convergent method to obtain (...) orbits for central forces, which elucidates how Newton may have have been led to formulate some of his most fundamental propositions in the Principia. (shrink)
SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in 1920 studied the (...) early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend. (shrink)
In Proposition 10, Book 2 of the Principia, Newton applied his geometrical calculus and power series expansion to calculate motion in a resistive medium under the action of gravity. In the first edition of the Principia, however, he made an error in his treatment which lead to a faulty solution that was noticed by Johann Bernoulli and communicated to him while the second edition was already at the printer. This episode has been discussed in the past, and the source of (...) Newton’s initial error, which Bernoulli was unable to find, has been clarified by Lagrange and is reviewed here. But there are also problems in Newton’s corrected version in the second edition of the Principia that have been ignored in the past, which are discussed in detail here. (shrink)
In 1662 Christiaan Huygens carried out the famous Torricelli experiment to test the existence of atmospheric pressure by inserting the apparatus in the glass receiver of a vacuum pump, and evacuating the air inside it. He reported that when the air was exhausted, a column of water remained suspended in a 4-foot tube. This unexpected result was in stark contrast with earlier experiments of Boyle and Hooke that apparently had confirmed Torricelli’s explanation that such a water column was supported by (...) outside air pressure, and would fall when the air was removed. Huygen’s “anomalous suspension” led to the continuation of controversies in the seventeenth century about the nature of the vacuum that these experiments were expected to resolve. Surprisingly, the origin of Huygens’ unexpected result has remained a puzzle up to the present time. In this paper, I discuss the dynamics of such a column of water under the experimental conditions reported by Boyle and by Huygens, that turned out to be different, and present the results of a replication of their experiments with a modern vacuum pump. Contrary to the conventional explanations of these experiments, I demonstrate that in the Boyle–Hooke version of this experiment, the water column descends initially because it is forced down by the gas pressure due to air dissolved in the water which is released inside the Torricelli tube after the external pressure is sufficiently decreased. Huygens, however, first removed this trapped air before he carried out his experiment. In the absence of this internal gas pressure, the early rudimentary vacuum pumps were inadequate to decrease the air pressure sufficiently inside the receiver to demonstrate the descent of a Torricelli column of airless water 4-foot in height or less. (shrink)
The central claim that understanding quantum mechanics requires a conscious observer, which is made by B. Rosenblum and F. Kuttner in their book “Quantum Enigma: Physics encounters consciousness”, is shown to be based on various misunderstandings and distortions of the foundations of quantum mechanics.
In Book 1 of the Principia, Newton presented two different descriptions of orbital motion under the action of a central force. In Prop. 1, he described this motion as a limit of the action of a sequence of periodic force impulses, while in Prop. 6, he described it by the deviation from inertial motion due to a continuous force. From the start, however, the equivalence of these two descriptions has been the subject of controversies. Perhaps the earliest one was the (...) famous discussion from December 1704 to 1706 between Leibniz and the French mathematician Pierre Varignon. But confusion about this subject has remained up to the present time. Recently, Pourciau has rekindled these controversies in an article in this journal, by arguing that “Newton never tested the validity of the equivalency of his two descriptions because he does not see that his assumption could be questioned. And yet the validity of this unseen and untested equivalence assumption is crucial to Newton’s most basic conclusions concerning one-body motion” (Pourciau in Arch Hist Exact Sci 58:283–321, 2004, 295). But several revisions of Props. 1 and 6 that Newton made after the publication in 1687 of the first edition of the Principia reveal that he did become concerned to provide mathematical proof for the equivalence of his seemingly different descriptions of orbital motion in these two propositions. In this article, we present the evidence that in the second and third edition of the Principia, Newton gave valid demonstrations of this equivalence that are encapsulated in a novel diagram discussed in Sect. 4. (shrink)