A treatise on some new geometrical methods, Volume 11873 |
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Common terms and phrases
²+v² a²+b² asymptotic axes of coordinates axis chord circle circular sections coefficients cone confocal surfaces conic section conjugate polar consequently constant coordinate planes cos² cubic equation cubical parabola cycloid dF dF diameter distance eliminating ellipse ellipsoid envelope epicycloid equal equation becomes evolute expression fixed point focal foci formula given plane given straight line Hence hyperbola hypocycloid intersection limiting tangent locus logarithms meet minor directrices origin parabola parallel perpendiculars let fall plane drawn plane of XY point of contact polar focus polar plane polarizing sphere pole preceding equation projective coordinates projective equation reciprocal polar second order semiaxes semicubical parabola shown side sin² square Substituting these values surface of revolution tangent plane tangential coordinates tangential equation theorem triangle umbilical directrix plane vector vertex ΑΦ
Popular passages
Page xix - Here, whole branches of continental discovery are unstudied, and indeed almost unknown, even by name. It is in vain to conceal the melancholy truth. We are fast dropping behind. In mathematics we have long since drawn the rein, and given over a hopeless race. In chemistry the case is not much better.
Page xx - The discharges of 1902 and 1903 are good as far as they go, but they do not go far enough.
Page 349 - XL. We might pursue this subject very much further, but enough has been done to show the analogy which exists between the trigonometry of the circle and that of the parabola. As the calculus of angular magnitude has always been referred to the circle as its type, so the calculus of logarithms may in precisely the same way be referred to the parabola as its type. The obscurities which hitherto have hung over the geometrical theory of logarithms are, it is hoped, now removed. It is possible to represent...
Page 342 - The locus of the point T, the intersections of the tangents to the parabola with the perpendiculars from the focus, is a right line ; or in other words, while one end of a subtangent rests on the parabola, the other end rests on a right line. So in the circle ; while one end of the subtangent rests on the circle, the other end rests on a cardioide, whose diameter is equal to that of the circle, and whose cusp is at S. SPA is the cardioide. The length of the tangent VN to any point N is wi(sec 0+...
Page 342 - ... of an equilateral hyperbola might be expressed by real exponentials, whose exponents are sectors of the hyperbola; but the analogy, being illusory, never led to any useful results. And the analogy was illusory from this ; that it so happens the length and area of a circle are expressed by the same function, while the area of an equilateral hyperbola is a function of an arc of a parabola, as will be shown further on. The true analogue of the circle is the parabola.
Page xiv - Gravitation, are the following similar remarks : " The exercise of the mind in understanding a series of propositions, where the last conclusion is geometrically in close connexion with the first cause, is very different from that which it receives from putting in play the long train of machinery in a profound analytical process. The degrees of convictiOD in the two cases are very different.
Page 331 - If we now extend this inquiry, and ask what is the magnitude of the amplitude of the arc of the parabola which shall render the difference between this parabolic arc and its subtangent equal to n times the distance between the focus and the vertex, we shall have, as before, by the terms of the question, 'I !(/// . 6)—m sec 0 tan 0=nm. But, in general, n(me)_m gec fl t&j} e=m$sec()d0. hence we inu-t have «= JsecO J0 = log (sec 0 + tan 6), or secfl+ tan0=e".
Page 344 - Therefore as the hyperbolic area is equal to a constant multiplied into the corresponding arc of the parabola, the evaluation of the hyperbolic area depends on the properties of logarithms. It also follows, from what has been established in the preceding part of this paper, that hyperbolic areas may be multiplied and compared according to the laws which regulate parabolic, arcs. Let...
Page 293 - ... devise a curve that shall represent one condition of a theory, or one truth of many, is easy enough. Thus, if we had first obtained by pure analysis all the properties of the circle without any previous conception of its form, and then proceeded to find a geometrical figure which should satisfy all the conditions developed in the theory, we might hit upon several geometrical curves that would satisfy some of the established conditions, though not all. That all lines passing through a fixed point...
Page 295 - These views will suggest to us the reflection, how very small is the field of that vast region, the Integral Calculus, which has hitherto been cultivated or even explored ! When we find that the highest and most abstruse of known functions, not only circular functions and logarithms, but also elliptic integrals of the three orders, are exhausted,