{ "id": "f5f3845cfa19ba1b99b9a2424bd596a688215792", "text": "II. An Account by Mr. John Eames, F. R. S., of a Book entitled, A Mathematical Treatise, containing a System of Conic-Sections, with the Doctrine of Fluxions and Fluents, applied to various Subjects. By John Muller.\n\nThe ingenious Author of this Work, observing how much Time is necessarily spent, and Pains taken, in learning these valuable Parts of Mathematics, thought it would be very well worth his while to lessen both, which he hopes he has done considerably, in the following Treatise. He has divided it into three Parts, contained in so many Books.\n\nIn the first of these, he considers the Properties of the three Sections of a Cone, as well in, as out of the Cone. And to make this Part of the Work of more Service to the Reader, Mr. Muller has not only selected the most considerable Properties of these Curves, that are to be met with in other Writers, both Ancient and Modern; but has added several new ones, which, as he informs us, are inserted in their proper Places. And that such Gentlemen as are desirous to read Sir Isaac Newton's Principia, but are at a loss for want of a sufficient Acquaintance with Conic-Sections, may be the more obliged, he has taken particular Care to demonstrate such Properties as Sir Isaac presupposes his Reader to be acquainted withal. Accordingly, he has prefix'd a Table of such Propositions, informing him as well where they are to be met with\nin this Book, as in Sir Isaac Newton's Principia Mathematica.\n\nThe Proofs made use of in his Demonstrations, are sometimes Algebraical, at other times Geometrical, according as he finds the one to be plainer and shorter than the other.\n\nBook II.\n\nThe second Book treats of the direct Method of Fluxions. And here he hopes the first Principles of this Method are laid down, not only in a new, but very plain and concise manner. He proceeds to shew the Use of Fluxions in the Solution of the common Problems of finding the Maxima and Minima of Quantities, the Radii of the Evolution of Curves, and the Radii of Refraction and Reflection. Under the first of these Heads he tells us, particular Care has been taken to distinguish the Maximums from the Minimums, a thing which has not been taken Notice of so much as it ought to have been. And whereas some Mathematicians, having made use of what they call infinitely small Quantities, are forced to reject something out of the Equation, for finding the Fluxion of a Rectangle, whose Sides are varying Quantities, Mr. Muller uses only finite Quantities; and finds the Fluxion of such a Rectangle after a new manner, without rejecting any Quantity for its Smallness. He does the same in finding the Fluxion of a Power. And to avoid the Use of infinitely small Quantities, introduces a new Principle, viz. That a curve Line may be consider'd as generated by the Motion of a Point carried along by two Forces or Motions, one in a Direction always parallel to the Absciss, and the other in a Direction\nrection always parallel to the Ordinate. Hence he in-\nfers, that the Fluxion of the Ordinate is to the Fluxion\nof the Absciss, as the Ordinate is to the Subtangent\nof the Curve.\n\nHaving likewise proved from the first Supposition,\nthat if the describing Point, when arrived at any Place\ngiven, should continue to move onwards, with the\nVelocity it has there, it would proceed in a Right\nLine, which would touch the Curve in that Point;\nhe concludes that the Direction of the Force in that\nPlace is in the Tangent to the Curve: Consequent-\nly, the three Directions being known in each Place,\nthe Proportion between the Velocities of the urging\nForces will be likewise known. So that the Nature\nof the Curve being given, the Law observ'd by these\nVelocities may be found; and if the Law of the Ve-\nlocities be given, the Nature of the Curve may like-\nwise be given.\n\nBook III.\n\nIn the third and last Book, we have the inverse Me-\nthod of Fluxions, with its Application to the severa-\nProblems solvable by it; such as the superficial and so-\nlid Contents of Curvilineal Figures, the Rectification\nof Curve Lines, Centres of Gravity, Oscillation and\nPercussion. Here also Mr. Cotes's Tables of Fluents\nare explain'd and illustrated by Examples.\n\nHe finishes this Book with a great Variety of Pro-\nblems, that are of a Physico-Mathematical Nature, se-\nveral of which are new, and proposed to him by Mr.\nBelidor. Some, indeed, are not so, having been\nsolved by Messieurs Varignon and Parent; but then\nhe has solved them after a different, and, as he hopes,\na more agreeable Manner, the Construction being\nmore simple, and the Process much shorter.", "source": "olmocr", "added": "2026-01-12", "created": "2026-01-12", "metadata": { "Source-File": "/home/jic823/projects/def-jic823/royalsociety/pdfs/103895.pdf", "olmocr-version": "0.3.4", "pdf-total-pages": 4, "total-input-tokens": 6042, "total-output-tokens": 1177, "total-fallback-pages": 0 }, "attributes": { "pdf_page_numbers": [ [ 0, 0, 1 ], [ 0, 1396, 2 ], [ 1396, 2939, 3 ], [ 2939, 4579, 4 ] ], "primary_language": [ "en", "en", "en", "en" ], "is_rotation_valid": [ true, true, true, true ], "rotation_correction": [ 0, 0, 0, 0 ], "is_table": [ false, false, false, false ], "is_diagram": [ false, false, false, false ] }, "jstor_metadata": { "identifier": "jstor-103895", "title": "An Account by Mr. John Eames, F. R. S. of a Book Entituled, a Mathematical Treatise, Containing a System of Conic-Sections, with the Doctrine of Fluxions and Fluents, Applied to Various Subjects. By John Muller", "authors": "John Muller, John Eames", "year": 1737, "volume": "40", "journal": "Philosophical Transactions (1683-1775)", "page_count": 4, "jstor_url": "https://www.jstor.org/stable/103895" } }