An Elementary Treatise on Descriptive Geometry...

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J. Weale, 1851 - 137 páginas
 

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Página i - DESCRIPTIVE GEOMETRY, an Elementary Treatise on; with a Theory of Shadows and of Perspective, extracted from the French of G. MONGE. To which is added, a description of the Principles and Practice of Isometrical Projection ; the whole being intended as an introduction to the Application of Descriptive Geometry to various branches of the Arts.
Página 84 - AM is lengthened at each instant of the motion by the same quantity as the distance BM is diminished. The velocity of the describing point in the direction AM is therefore equal to the velocity in the direction M Q. If, then, equal straight lines be cut off from MB, and from AM produced, and the parallelogram MPEQ, be completed, the diagonal ME of this parallelogram will be the direction of the motion of the generating point at M, and consequently the tangent to the curve at this point. It is clearly...
Página 115 - The surfaces may be divided into two classes, according to the manner in which they receive and remit the light ; viz., polished surfaces and dull surfaces.
Página 134 - ... sinister distance, and its altitude. And it is manifest they need not be taken in this order, but in any other that may be more convenient to the artist, there being six ways in which this operation may be varied. If any point in the same isometrical plane with the point required to be found, is already represented in the picture, that point may be assumed as a new regulating point, and the point required found by taking two distances ; and if the new assumed regulating point is in the same isometrical...
Página 6 - ... projections of a straight line whose length is required. Draw through a" an indefinite straight line H e meeting bb" at the point e, and from e measure a distance, e H, in the direction e a", equal to ab ; then H b", the hypotheneuse of the right angled triangle H e b", will be the length required. Since the two planes of projection are at right angles to each other, the same result might have been obtained by making the construction on the horizontal plane. It appears then that the dimensions...
Página 82 - ... the great circle parallel to the vertical plane of projection ; and these projections, which will be circular arcs, described from the point a as center, and with arbitrary radii, will cut the two generating lines in points k, p. This being premised, each spherical surface will cut the first surface in the circumference of a circle, the plane of which will be perpendicular to the axis a a', the vertical projection of which. will be obtained by drawing the horizontal ko, and its horizontal projection...
Página 83 - ... vector can be determined by elementary trigonometry ; and in its application to many other curves, of high and low degree, a like simplicity characterizes this elegant process. Nevertheless, in his treatise on Descriptive Geometry, Mr. JF Heather makes this curious remark : " This method, which Roberval invented before Descartes had applied algebra to geometry, is implicitly comprehended in the processes of the differential calculus, on which account it is not noticed in elementary mathematics";...
Página 10 - ... so that its radius at each instant is equal to the distance between the two points in which the plane of the circle cuts the axis, and a curve given in space. The generating curve in this way changes at the same time both its form and position. These three examples are sufficient to make it evident that all curved surfaces may be generated by the movement of certain curved lines, and that there is not any surface whose form and position cannot be completely determined by an exact and complete...
Página 84 - FIG. 13. point. It is clearly seen from this, that in the ellipse, the tangent bisects the angle BPN formed by one of the focal distances and the production of the other," etc., etc. Mr. Heather elsewhere explicitly states that if the components in two directions are given, whatever their relative magnitudes, the method of Roberval consists in " completing the parallelogram and drawing the diagonal.
Página 84 - AP produced, and the parallelogram /WKWbe completed, the diagonal PO of this parallelogram will be the direction of the motion of the generating point at P, and consequently the tangent to the curve at this point. It is clearly seen from this, that in the ellipse the tangent bisects the angle BPN, formed by one of the focal distances and the production of the other, '

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