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The line A B must equal the line C D; somewhere in space must be found the centre or the focus of every curve; and every angle must subtend a certain arc, to be easily found on reference to the tables of the text-books.
These ideals will be a spiritual reality; and they will be expressed in nature in so far as nature supports that regulative interest. Many a perfect and eternal realm, merely potential in existence but definite in constitution, will thus subtend nature and be what a rational philosophy might call the ideal.
The terms or goals of thought have for their function to subtend long tracts of sensuous experience, to be ideal links between fact and fact, invisible wires behind the scenes, threads along which inference may run in making phenomena intelligible and controllable. An idea that should become an image would cease to be ideal; a principle that is to remain a principle can never become a fact.
The value of size becomes immediate only when we are at close quarters with the object; then the surfaces really subtend a large angle in the field of vision, and the sense of vastness establishes its standard, which can afterwards be applied to other objects by analogy and contrast.
It is self-evident that the hypocycloid must return into itself at the point of beginning, without crossing: each branch, then, must subtend an aliquot part of the circumference, and can be traced also by another and a smaller describing circle, whose diameter therefore must be an aliquot part of the diameter of the outer pitch-circle; and since this last must be equal to the sum of the diameters of the two describing circles, it follows that the radii of the pitch circles must be to each other in the ratio of two successive integers; and this is also the ratio of the number of pins to that of the epicycloidal branches.
The latter is the fact that the claims of spiritism lift man into an entirely new domain of the universe, remove him from the great field of the material with which he is physically affiliated and to which his senses are closely adapted, and place him in a region beyond the scope of the senses, a vast kingdom which is held to underlie or subtend the physical, and which the ordinary outlook of the scientist fails to perceive.
The same infallible guide cuts off mountain-tops and deeply snowed ridges, which to the unaided eye appear far lower than the point from which they are viewed; but which, from the quantity of snow on them, must be many thousand feet higher, and, from the angle they subtend in the instrument, must be at an immense distance.
How do we discover that, in an isosceles triangle, the sides which subtend the equal angles are equal? We do not go about collecting the opinions of individuals upon the subject, nor do we consult the records of other peoples, past or present. We do not measure a great number of triangles and arrive at our conclusion after a calculation of the probable error of our measurements.
The more obvious results and superficial harmonies perceived in those regions will receive names and physics will be arrested at natural history. Where these unexplained facts are mental it will not be hard to do more systematically what common sense has done already, and to attach them, as we attach love or patriotism, to the natural crises that subtend them.
What, therefore, must have been my amazement, on awakening from a brief and disturbed slumber, on the morning of this day, the seventeenth, at finding the surface beneath me so suddenly and wonderfully augmented in volume, as to subtend no less than thirty-nine degrees in apparent angular diameter! I was thunderstruck!
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