Ellipse

“ A very near approach to the true form of a true ellipse may be drawn by the construction given in Figure 81, in which A A and B B are centre lines passing through the major and minor axis of the ellipse, of which a is the axis or centre, b c is the major axis, and a e half the minor axis. ”

“ It will be noted that, first, all the centres for the end curves fall on the line of the length or major axis, while all those for the sides fall on the line of width or the minor axis; and, second, that as the dimensions of the ellipses increase, the centres for the arcs fall nearer to the axis of the ellipse. Now in proportion as a greater number of arcs of circles are employed to form the figure, the nearer it will approach the form of a true ellipse; but in practice it is not usual to employ more than eight, while it is obvious that not less than four can be used. ”

“ If the ellipse, as a simply geometrical speculation, has had so much power in the education of the race, what are the intellectual relations of its beauty through its connection with astronomy? Who can estimate the influence which this oldest of physical sciences has had upon human destiny? Who can tell how much intellectual life and self-reliance, how much also of humility and reverential awe, how much adoration of Divine Wisdom, have been gained by man through his study of these heavenly diagrams, marked out by the sun and the moon, by the planets and the comets, upon the tablets of the sky? ”

“ If perturbations are ignored, a planet can be regarded as moving in an ellipse with the sun in one focus. The size and shape of the ellipse can be defined by the length of its axis and by the eccentricity; the plane in which the ellipse is situated is determined by the position of the line, called the line of nodes, in which it cuts a fixed plane, usually taken to be the ecliptic, and by the inclination of the two planes. ”

“ The earth does not describe a circle round the sun, but an ellipse, as it must by the laws of rational mechanics. Now, the earth occupies one of the centres of the ellipse, and consequently, at the time of its transfer, it is further from the sun, that is to say, at its apogee, and at another time nearer, that is to say, at its perigee. ”

“ Let a p = o q, and when the motion parallel to a c has vanished, the particle will be found at p; hence p must be the extremity of the major axis of the ellipse. In the next revolution, the extremity of the axis will advance a little more, and thus the ellipse moves round gradually. ”

“ If we hold it parallel to the face, with the whole breadth opposite to the eye, we see it as an entire circle. If we turn it a little sideways, it appears oval, or as an ellipse; and, as we continue to turn it more and more round, the ellipse grows narrower and narrower, until, when the edge is presented to the eye, we see nothing but a line. Now imagine the ring to be near a perpendicular wall, and the eye to be removed at such a distance from it, as not to distinguish any interval between the ring and the wall ”

“ Suppose then that a class, while becoming skilful in working problems, seems to have forgotten that the axes are no part of the curve itself. The teacher may wake it up by saying, "You don't imagine, surely, that axes form any necessary portion of a curve. An ellipse, for instance, is the path of a body moving round a focus of attraction. Suppose a planet were endowed with the power of leaving a track in the sky, the track would be an ellipse only, unencumbered with axes. ”

“ But this inability was a mere accident: the idea of an ellipse could have been acquired from the paths of the planets as effectually as from anything else, if the paths had not happened to be invisible. If the planet had left a visible track, and we had been so placed that we could see it at the proper angle, we might have abstracted our original idea of an ellipse from the planetary orbit. Indeed, every conception which can be made the instrument for connecting a set of facts, might have been originally evolved from those very facts. The conception is a conception of something ”

“ Is that pull to be always the same, or is it to be greater at some times than at other times? Newton showed that no ellipse other than a circle could be described, if the pull from the sun were always the same. ”

“ Let us return to the subject of Analytical Geometry.
The tangent to an ellipse is an imaginary straight line, representing the path which would be followed by the body tracing the ellipse, if its connection with the attracting focus were suddenly to cease. In its essence, the tangent is a sublime effort of the scientific imagination; it pictures the result of a sudden cessation of the action of gravity. In practice the tangent is a convenient line for indicating the curvature at any given point. ”

“ If the ellipse in Figure 30 be the orbit of a planet, S will be the place of the sun. The sun is at one focus of the ellipse described by every planet Every planet describes a different ellipse. The degree of flatness of every ellipse is different for every planet; the direction of the long diameter of the ellipse is different for every planet; there is every possible variety among them. ”

“ This ellipse we may call the directional ellipse of the bivector. In the case of a real vector, or of a vector having a real direction, it reduces to a straight line. In any other case, the angular direction from the imaginaiy to the real part of the bivector is to be regarded as positive in the ellipse, and the specification of the ellipse must be considered incomplete without the indication of this direction. ”

“ Thus, if a cone be cut by a plane at right angles to its axis we get a circle. If, instead of being perfectly at right angles, the plane subtends with the axis an angle of 89° 59′, we have an ellipse which no human eye, even when aided by an accurate pair of compasses can distinguish from a circle. Decreasing the angle minute by minute the ellipse becomes first perceptibly eccentric, then manifestly so, and by and by acquires so immensely elongated a form, as to bear no recognizable resemblance to a circle. By continuing this process the ellipse passes insensibly into a parabola ”

“ Then with centre O1 and radius OJ = OA1, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.
ELLIPSOID, a quadric surface whose sections are ellipses. Analytically, it has for its equation x2/a2 + y2/b2 + z2/c2 = 1, a, b, c being its axes; the name is also given to the solid contained by this surface (see Geometry: Analytical) . The solids and surfaces of revolution of the ellipse are sometimes termed ellipsoids, but it is advisable to use the name spheroid (q.v.) . ”

“ No one of them, however, continues to move in any one circle, but is perpetually passing from one to another, through an infinite number of circles, in the course of each revolution; for an ellipse, said he, is an oblique section of a cone, and in a cone, betwixt the two vortices of the ellipse there is an infinite number of circles, out of the infinitely small portions of which the elliptical line is compounded. ”

“ An elliptical field may be obtained, by giving the bounding line the shape of one quarter of an ellipse, and placing it in such a manner that the vertex of the conjugate axis falls upon one of the reflectors, and the vertex of the transverse axis upon the other. ”

“ A body is in stable equilibrium when its centre of gravity is at the lowest point: unstable when it is at the highest. This may be very simply illustrated by an ellipse, which I hold in my hand. The centre of gravity of this figure is at its centre. The ellipse, when resting on its side, is in a position of stable equilibrium and its centre of gravity is then clearly at its lowest point. But I can also balance the ellipse on its narrow end, though if I do so the smallest touch suffices to overturn it. ”

“ Even if the speed be correct the orbit will always be an ellipse if the direction of the initial throw be not perpendicular to the line joining the ball to the centre of the globe. We can make the ball describe a very long ellipse or an ellipse which differs but little from a circle. ”

“ In isometric perspective everything that is round in reality is drawn in the shape of an ellipse, that is a closed curve that is longer than it is wide as shown at A in Fig. 28; there are different shaped ellipses but [33] there is only one used for isometric drawing and this is always in the ratio of 11⁄4 to 2; that is if an ellipse is 2 inches long it will be 11⁄4 inches wide; an ellipse 4 inches long will be 21⁄2 inches wide and so on.
An easy, though rough way to draw an isometric ellipse is to make a line as long as the diameter of the disk or wheel you intend to represent ”

“ The importance of the ellipse to astronomers is greater than that of any other geometrical figure. In fact, all the planets, as they perform their long and unceasing journeys round the sun, move in ellipses ”

“ It is obvious that if we were to stick the two pins into the same place we should get a circle, so that a circle is a particular case of an ellipse, namely an ellipse which has zero eccentricity. All the planets move in ellipses which are very nearly circles, whereas the comets move in ellipses which are very far removed from circles. ”

“ A properly constructed elliptical figure is difficult,p. 105 principally, because two different sized curves are required, and the pen runs from one curve into the other. If the two curves meet at the wrong place, you may be sure you will have a distorted ellipse. ”

“ It is such a case as this which gives rise to the doctrine that the mind, in framing the descriptions, adds something of its own which it does not find in the facts.
Yet it is a fact surely, that the planet does describe an ellipse; and a fact which we could see, if we had adequate visual organs and a suitable position. Not having these advantages, but possessing the conception of an ellipse, or (to express the meaning in less technical language) knowing what an ellipse was, Kepler tried whether the observed places of the planet were consistent with such a path. He found they were so; ”

“ For instance, it is a contradiction to depict one body as constantly falling towards another, and as, at the same time, constantly flying away from it. The ellipse is a form of motion which, while allowing this contradiction to go on, at the same time reconciles it. ”

“ The equation of an ellipse expresses the universal conceivable relation between its co-ordinates, of which only the real values have geometrical significance. Similarly, the equations between the elements of phenomena express a universal, mathematically conceivable relation. Here, however, for many values only certain directions of change are physically admissible. As in the ellipse only certain values satisfying the equation are realised, so in the physical world only certain changes of value occur. Bodies are always accelerated towards the earth. ”

“ In cases where the eccentricity becomes very great, the beauty of the curve is [36] destroyed, and ultimately the ellipse is merged into one straight line. Most learned Professors, here we have a terrible warning of the awful result of too much eccentricity. ”

“ It is in vain, that mathematicians represent a plain surface as produced by the flowing of a right line. It will immediately be objected, that our idea of a surface is as independent of this method of forming a surface, as our idea of an ellipse is of that of a cone; that the idea of a right line is no more precise than that of a plain surface ”

“ In the course of our investigations, we can not fail to catch glimpses of wonderful changes in far away times; and can not help seeing what labor is involved in the solution of all questions relating to the same.13
The earth revolves around the sun in an orbit called an ellipse. This is not a fixed form, but slowly varies from year to year. It is now gradually becoming circular. It will, however, not become an exact circle. Astronomers assure us that, after a long lapse of time, it will commence to elongate as an ellipse again. ”

“ When is between and , (Fig. 44.) and the image is part of an ellipse, namely, all but that part, which we found in the first case.
If we suppose to go outside of the circle beyond , (Fig. 46.) we shall be led to the case of a convex mirror.
Our equation will still be
and the image will in all cases be a part of an ellipse, turning its convexity towards . ”

“ Thin cardboard is necessary. What does elliptical mean? A good size is 4 inches the long way and 21⁄2 inches across. All the girls know how to draw an ellipse. Cut the six pieces of silk 1⁄2 inch larger all around than the ellipses, and cover in same way as you did the round disks. If there are not enough scraps of one color, use a contrasting color for the lining. After the six pieces have been covered and joined together to make three ellipses, then the three are to be overhanded together very neatly, leaving open the third edge, which is the opening of the case where the spools, etc. ”

“ Even if we could observe the motions of a planet, of a perfect globular form, free from all perturbing457 or retarding forces, we could never prove that it moved in a perfect ellipse. To prove the elliptical form we should have to measure infinitely small angles, and infinitely small fractions of a second; we should have to perform impossibilities. ”