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Updated: June 19, 2025
The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin.
This was Comte's contention. Others conceded that there might be such an area, but claimed that we can have no knowledge of it. Even the theologians, after their first shock, were disposed to concede that, concerning the magnitudes in which they were most interested, as for example, God and soul, we have no knowledge of the sort which the method of the physical sciences would give.
There are in the world valleys which are longer and a few which are deeper. There are valleys flanked by summits loftier than the palisades of the Kaibab. Still the Grand Canyon is the sublimest thing on earth. It is so not alone by virtue of its magnitudes, but by virtue of the whole its tout ensemble."
The distance is 3", p. 184°. A very pretty sight. We now pass to the extremity of the other arm of the cross, near which lies the beautiful little double 49, whose components are of magnitudes six and eight, distance 2.8", p. 50°. The colors are yellow and blue, conspicuous and finely contrasted.
The magnitudes are seven and seven and a half or eight, distance 1.2", p. 133°. This star is variously named Sigma 1126 and 31 Can. Min. Bode. Star No. 14 is a wide triple, magnitudes six, seven, and eight, distances 75, p. 65°, and 115", p. 154°. In the Head of Hydra we find Sigma 1245, a double of the sixth and seventh magnitudes, distance 10.5", p. 25°. The larger star shows a fine yellow.
Sigma 2166 consists of two stars of magnitudes six and seven and a half, distance 27", p. 280°. Sigma 2173 is a double of quite a different order. The magnitudes of its components are both six, the distance in 1899 0.98", p. 331°. It is evidently a binary in rapid motion, as the distance changed from about a quarter of a second in 1881 to more than a second in 1894.
For it is the peculiarity of linear extension that it alone allows its magnitudes to be placed in absolute juxtaposition, or, rather, in coincident position; it alone can test the equality of two magnitudes by observing whether they will coalesce, as two equal mathematical lines do, when placed between the same points; it alone can test equality by trying whether it will become identity.
The magnitudes in 12 are five and eight, distance 66", p. 168°. So also the wide double 17, magnitudes five and a half and six, distance 145", exhibits a tinge of lilac in the smaller component; the triple 35, magnitudes five, eight, and nine, distances 1", p. 77°, and 28.7", p. 124°, has four colors yellow, lilac, and blue, and the double 24, magnitudes five and six, distance 20", p. 270°, combines an orange with a lilac star, a very striking and beautiful contrast.
"It must be mighty dull bein' an officer." We must not forget the Great Cañon. The voyagers were amid magnitudes and sublimities of nature which oppressed as if they were powers and principalities of supernature.
We set to work then in the roaring, steaming valley with the vapour swirling about us, sometimes concealing us, sometimes half revealing us gigantic, again in the utterness of exposure showing us dwindled pigmies against the magnitudes about us. The labour was not difficult. By the time Darrow returned we had a pile of the saplings ready for his next direction.
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