Newtons Principia For The Common Reader (Pb 2012) Paperback – January 1, 2012
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Newton discovered his calculus within the context of the analytic geometry of Descartes and there is little doubt that his mechanics was originally formulated in this framework also. Maybe Newton wanted to learn Euclid afresh for himself, maybe he believed the ancient Greek geometers had superior knowledge, though we now know he surpassed them considerably but his marriage of the calculus with Euclidean geometry extended its range considerably and was a prelude to differential geometry if not its most basic form.
I'd like to give you a sample of Newton's creation where there is an interplay of geometry ideas and his calculus. For this I've chosen the first of Newton's "Superb" theorems which is in section 76 of chapter 15 in the Chandrasekhar text. If you imagine a particle placed anywhere within the interior space of a very thin spherical shell it will experience no net gravitational force from the shell or its mass. Draw a chord through the particle point which intersects the shell at two points on the inner side of the shell (inner and outer effectively coincide since
thin). Draw another line through the particle point which makes a small angle dw with the chord-the line is assumed to cut outside the sphere in both of its directions. Now imagine this chord/line construct to be rigid and spin it about the chord as axis. It sweeps out a cone or if you like, two cones with the particle as apex. These cones obviously form identical solid angles (vertical angles are equal). If we assume the particle is not equidistant with the ends of the chord, we consider spheres with the particle as center and these two chordal distances
for the lengths of the radii and we only consider the parts of the spheres which contain the solid angles or cones,i.e., the shorter cone is part of the smaller sphere and similar holds for the taller. These area caps at the ends of the cones partly extend outside the original sphere or shell. Next consider the original caps that we first cut from the shell when we spun about the chord. Draw radii from the shell center to the chordal endpoints. Note also the two radii just drawn and the chord form an isosceles triangle with base angles equal to the angle between the normal to the sphere at the chordal point and the extended chord line at each end. Now we apply a limiting procedure and we consider just one side as the other follows the same. Since we assumed at least within context that the chord is not a diameter the original cut cap is somewhat oblong with the shorter part
coinciding with the cut circle diameter. Though we replaced the cut circle with a circular cap of different curvature due to sphere radius, as dw gets smaller they begin to coincide and go flat near zero. As the oblong original cap shrinks with dw it flattens forming an incline with the cone-an ellipse with minor axis the diameter of the cut circle. A bit of calculus to find ellipse area shows that the ratio of the ellipse area to the cut circle area is the major axis divided by the circle diameter and this forms a right triangle as dw approaches zero. Note the normal to the shell at the chordal point is perpendicular to the major axis (ellipse) and the normal to the circular cap is in line with the chord and also perpendicular to the circle diameter, i.e., the angle between the major axis and circle diameter is the same as a base angle of the isosceles triangle recently noted. Equivalently the ratio of the area of the spherical circular cap having chord end as vertex to the area of the portion of the spherical shell intercepted by the same cone is the cosine of the aforementioned base angle. This will be seen as needed to justify the last formula in the proof. If you get the book you'll understand the need for this monster-pardon my awkward prose. The idea is the spherical area parts originally cut by spinning about the chord each have area or mass proportional to the square of their respective distances from the particle as the areas become small or cone is thin but force goes as the inverse square so the force cancels for every chord-force goes as mass divided by inverse square. Those original parts were oblong and we found area to be the circular cap area divided by that base angle cosine as the cone thinned. The circular cap areas go as the squares of those aforementioned respective distances. So all the R's cancel and the cosines do too -base angles of an isosceles triangle. G-G=0. In truth it's easier by integrals- see Wikipedia or Vol. 1 of the Feynman Lectures.
Later in the chapter Chandrasekhar notes that Newton used the method of images in a gravitational context and this was done 150 years before the credited discovery of Kelvin-Surprise Newton was first. No one noticed till Chandra did his study. In truth most physicists and engineers get predigested Newton or Newton lite and there's probably more that may be undiscovered here.
That little blurb that I did above-Newton leaves it to the reader to complete the steps as did Chandrasekhar. You may yet profit from Newton and his biggest fan Chandrasekhar.