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Updated: May 29, 2025
This was when abstract geometrical reasoning commenced, and astronomical observations aiming at precision were recorded, compared, and discussed. Closely associated with it must have been the construction of the forms of logic.
Another practical demonstration which Thales was credited with making, and to which also his geometrical studies led him, was the measurement of any tall object, such as a pyramid or building or tree, by means of its shadow. The method, though simple enough, was ingenious. It consisted merely in observing the moment of the day when a perpendicular stick casts a shadow equal to its own length.
At 32° Fahrenheit, and far below that temperature, oxygen and hydrogen are elastic gaseous bodies, whose particles tend to rush away from one another with great force. Water, at the same temperature, is a strong though brittle solid, whose particles tend to cohere into definite geometrical shapes, and sometimes build up frosty imitations of the most complex forms of vegetable foliage.
They passed down the terrace steps into a garden ablaze with tulip beds in geometrical patterns; at the foot ran a yew hedge, and beyond it, in a side-walk, they came upon a scullion boy chasing a sulphur-yellow butterfly.
Is it a mere coincidence, or is it one of the rules of the religious architecture of the remote past? And which are the imitators the builders of the Egyptian pyramids, or the unknown architects of the under ground caves of India? In pyramids as well as in caves everything seems to be calculated with geometrical exactitude.
"I know of no greater shame than that which you now inflict upon me. Will you be so good as to question me? It is your duty." "You carry yourself very high, sir! We shall see presently whether this be a legitimate pride." "Proceed, sir; I wait for you." M. Monge then put to me a geometrical question, which I answered in such a way as to diminish his prejudices.
It is the perception of this unity in diversity, of this similarity of plan, for instance, in all tree-like forms, however diverse, the sprig of mignonette, the rose-bush, the fir, the cedar, the fan-shaped elm, the oval rock-maple, the columnar hickory, the dense and slender shaft of the poplar, which charms the eye of those who have never heard in what algebraic or arithmetical terms this unity may be defined, in what geometrical or architectural figures this diversity may be expressed.
For example, a complete event is never disclosed in sense-awareness, and thus the object which is the sum total of objects situated in an event as thus inter-related is a mere abstract concept. Again a right-angle is a perceived object which can be situated in many events; but, though rectangularity is posited by sense-awareness, the majority of geometrical relations are not so posited.
The rise of meta-geometrical methods and other branches of scientific speculation have led in recent years to a considerable amount of very interesting inquiry into the nature of our fundamental geometrical conceptions. Strange to say, a large body of respectable mathematicians have been found to favour the extraordinary view that our mathematical conceptions are derived from Sensation.
In spite of lessons at Paris and private lessons which he afterwards took at Valence, he was never a dancer: his bent was obviously for the exact sciences rather than the arts, for the geometrical rather than the rhythmical: he thought, as he moved, in straight lines, never in curves.
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