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allude to it. Let it then be sufficient to say, that to this proposition we are indebted for a solution of the relations which not only the sides, but also the angles of a triangle, bear to each other.
Snowball's "Plane and Spherical Trigonometry" contains all that is necessary upon this subject, but a knowledge of Algebra as well as of Euclid is requisite as a preliminary.
CHAPTER X X X V.
LINES CAPABLE OF PASSING THROUGH ANY FOUR POINTS.
To a person who has never studied Geometry, it may possibly seem easier to draw and discuss a line that can be made to pass through many points than a line which is more strictly confined. But the student will, by this time, have perceived that so far is this from the truth that each fresh point of limitation adds immensely to the difficulty of the inquiry. Thus, the simplest form is the straight line, which is definite in every respect as soon as two
points in it are fixed and which admits of no variety of shape whatever. Next to this comes the circle requiring three points to make it definite and capable of infinite variety of curvature as the length of its radius varies (1.) These two kinds of loci we have already considered, but hitherto have only alluded to the classes of lines which can be made to pass through more than three points. But these latter are too important not to receive some attention, though, at the same time, they are too difficult to receive more than description, even that being very imperfect. This, indeed, will be at once apparent when it is perceived that of this class of line there are no less than three distinct forms, each being infinitely various. These classes are respectively called the Ellipse, Hyperbola, and Parabola. The mode of constructing the last of these has been already shown (ch. xxii.) and the only difference between the construction of the three figures is that,
In an Ellipse,
LP is always greater than PS in the same proportion (2.)
In an Hyperbola,
LP is always less than PS in the same proportion (3.)
(The figures are drawn opposite, but we are not sufficiently advanced to be able to trace them.)
These three curves have many curious properties, a few of which we shall mention here:
In the first place, they are all capable of being obtained from a cone by cutting it in various directions, and hence their usual name of "Conic Sections."
In the second place, the Ellipse and Hyperbola are the orbits of the planets, the earth included, and of the comets respectively, as indeed they are of all bodies subject to the same force as that which pervades the solar system. A Parabola, again, is the common path of a projectile upon which two forces act at the same time-one being the force of impulsion, the other that of gravitation. Thus :
Another property of this latter curve has been described in ch. xxii. as the principle involved in the burning glass. One very
similar is found in an ellipse,
which we shall describe.
In an Ellipse there are not one only but two foci (S and H), i.e. two points by means of which the curve may be drawn in the manner
above described. Let RPT be at tangent to the curve at P and SP, and PH be drawn then it can be shown that the angle RPS = the angle TPH. Anything, therefore, of a perfectly elastic nature, such as sound travelling from S to any point of the curve, would be reflected to H (6.) Now, it is the property of sound as of light to disperse itself in all directions. In general, therefore, when a man speaks, the sound is scattered, and consequently cannot be heard at any great distance. But if a person stands in the focus of any ellipse, however large, all he utters will strike against some point of the curve, and consequently being reflected to the other focus, will be distinctly heard there, however low he may speak. The Whispering Galleries at St. Paul's Cathedral and the Polytechnic are made upon this principle.
DESCARTES' SYSTEM OF RECTANGULAR CO-ORDINATES.
THE complications which increase thus rapidly as the number of points which determine the curve increases have even now outgrown the simple machinery which we have hitherto employed. And it is evident that unless some method can be found whereby the various properties of curves may be expressed in an easier and more general manner, it is hopeless to attempt to analyse more difficult loci. For a solution of this difficulty we are indebted to the celebrated Descartes, whose beautiful invention both opens out all the higher forms of lines, and renders the expression of their
properties so simple that in any given locus they can all be read at a single glance. His principle, we can only indicate here. To appreciate it fully would require a knowledge of algebra, but the geometrical part of it we are in a position to understand. Let there be a fixed point O, called the Origin, and through it two straight S
lines of unlimited length, OR, OS, at
right angles to each other, called the
axes (of x and y respectively.) Then
the position of any point P (1) will be fixed if either
1. The line OP and the angle POR be both definite (2) or,—
2. If a straight line, PM, be drawn to OR, parallel to OS, and OM and PM be both of definite length (3.)
In 1, OP and the angle POR are called the polar co-ordinates of the point P (O being called the pole.)
In 2, OM and MP are called the rectangular co-ordinates of the point P (O being the origin.)
The method of determining the position of any place on the Earth's surface by the ordinary lines of Latitude and Longitude is a simple example of Rectangular Co-ordinates, Greenwich being the Origin.
Now, let the point move from P in a S straight line to Q, and let ON and NQ be the co-ordinates of Q (4.)
Then, ON and NQ will be of different lengths from OM and MP. And the co