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Updated: June 22, 2025
The position and figure of any object are determined by determining the position of a sufficient number of points in it; and the position of any point may be determined by the magnitude of three rectangular co-ordinates, that is, of the perpendiculars drawn from the point to three planes at right angles to one another, arbitrarily selected.
And as the parallel rays are merely perpendiculars to the waves which fall on the concave surface, which waves are parallel to AD, it will be found that as they come successively to encounter the surface AB, they form on reflexion folded waves composed of two curves which originate from two opposite evolutions of the parts of the curve AFE. So, taking AD as an incident wave, when the part AG shall have met the surface AI, that is to say when the piece G shall have reached I, it will be the curves HF, FI, generated as evolutes of the curves FA, FE, both beginning at F, which together constitute the propagation of the part AG. And a little afterwards, when the part AK has met the surface AM, the piece K having come to M, then the curves LN, NM, will together constitute the propagation of that part.
Here the clash of rich primary colours, the perpendiculars which began with bronze girls' heads and ended with bronze girls' feet or animals' claws, the vast flat surfaces of furniture, the stiff curves of wood and a drapery, the morbid rage for solidity which would employ a candelabrum weighing five hundredweight to hold a single wax candle, produced a real and imposing effect of style; it was a style debased, a style which was shedding the last graces of French Empire in order soon to appeal to a Victoria determined to be utterly English and good; but it was a style.
It is hard to give an idea of the difficulties in climbing up from Bogni to the city, where the hardiest traveler feels vertigo in picking his way over a path often but a yard wide, with perpendiculars on either hand. Finally, after many strange feelings in your head and along your spinal marrow, you thank Heaven that you are safe in Kalaa.
He was nevertheless unwilling to adopt the opinion of Rheticus that the number six was sacred, maintaining that the "sacredness" of the number was of much more recent date than the creation of the worlds, and could not therefore account for it. He next tried an ingenious idea, comparing the perpendiculars from different points of a quadrant of a circle on a tangent at its extremity.
Looking about me I saw surprise, perplexity, doubt, wonder, and uncertainty in every countenance, if I did not find conviction. One fact embarrassed even me. Our friends the Horizontals were evidently quite as much at fault as our opponents the Perpendiculars, instead of being, as I had good reason to hope, in an ecstasy of pleasure on hearing their cause sustained by an authority so weighty.
Now if about the centres A, B, one describes the circles DK, EL, which represent the spreading of the waves which originate from these two points, and if one draws the straight line KL which touches these two circles, it is easy to see that this same line will be the common tangent to all the other circles drawn about the centres F, H, etc.; and that all the points of contact will fall within that part of this line which is comprised between the perpendiculars AK, BL. Then it will be the line KL which will terminate the movement of the particular waves originating from the points of the wave AB; and this movement will be stronger between the points KL, than anywhere else at the same instant, since an infinitude of circumferences concur to form this straight line; and consequently KL will be the propagation of the portion of wave AB, as has been said in explaining reflexion and ordinary refraction.
Here was matter for discussion, nor was I at all sanguine as to the result; but to be thus knocked on the head by a club, in the outset, was too much for the modesty of a maiden speech. I took my seat in confusion; and I plainly saw that the Perpendiculars, by their sneers, now expected to carry everything triumphantly their own way.
It would be quite possible to be first led to this proposition by actually drawing perpendiculars in many cases, and finding that they always met in a point; this experience might lead us to look for the general proof and find it. Such cases are common in the experience of every mathematician. The other point is more interesting, and of more philosophical importance.
Swing a branch of a heavy-leaved tree across the top of the wide window in Japanesque arrangement, put two men, two pipes, and two pegs in the foreground, the rising bubbles sparkling yellow in the level sunset rays, and the pipe's incense ascending in blue perpendiculars, and you have a suggestion of the perfect peace and entire absence of bustle which we associate with a certain Valley of Pong.
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