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Updated: May 11, 2025
Nevertheless, toward evening the leaders took some wise strategic precautions; they protected the army, which rested in the rear on Sedan and the Meuse, by two battle fronts, one composed of the Seventh Corps, and extending from Floing to Givonne, the other composed of the Twelfth Corps, extending from Givonne to Bazeilles; a triangle of which the Meuse formed the hypothenuse.
The extreme southwestern corner of Germany is an irregular right-angle, formed by the course of the Rhine. Within this angle and an hypothenuse drawn from the Lake of Constance to Carlsruhe lies a wild mountain-region a lateral offshoot from the central chain which extends through Europe from west to east known to all readers of robber-romances as the Black Forest.
There, on the vast plains, immense geometrical figures were to be traced by means of luminous reflectors; amongst others, the square of the hypothenuse, vulgarly called the 'Ass's Bridge. 'Any intelligent being, said the mathematician, 'ought to understand the scientific destination of that figure.
Thus the perpendicular Osiris, or the active, male principle being represented by a line whose measurement is 3; and the base Isis, or the passive, female principle by a line whose measurement is 4; then their union, or the addition of the squares of these numbers, will produce a square whose root will be the hypothenuse, or a line whose measurement must be 5.
As for my uncle, he was beginning to storm against the horizontal road. He loved nothing better than a vertical path; but this way seemed indefinitely prolonged, and instead of sliding along the hypothenuse as we were now doing, he would willingly have dropped down the terrestrial radius.
"You can if you want to, but it is better not to," rejoined Holmes, simply, as though not observing the sneer, "because the ripple represents the outer lines of the angle of disturbance in the water; and as any one of the sides to an angle is greater than the perpendicular from the hypothenuse to the apex, you'd merely be going the long way.
In other words, the base and hypothenuse of the supposed triangle would be so long when compared with the perpendicular, that the two former may be regarded as nearly parallel. In this manner the horizon of the æronaut would appear to be on a level with the car.
The Generalizations which respectively affirm that all the angles of a triangle are equal to two right angles, or that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, rest upon an entirely different basis of proof from those upon which the Generalizations rest which respectively assert that water is composed of certain chemical constituents combined in certain proportions, or that the nerves are the instruments of sensation and of motion.
When in the first room which I visited, a slight, slim creature was had up before me to explain to me the properties of the hypothenuse, I fairly confess that, as regards education, I backed down, and that I resolved to confine my criticisms to manner, dress, and general behavior. In the next room I was more at my ease, finding that ancient Roman history was on the tapis.
It would not occur to a geometer to ask with trepidation what difference it would make to the Pythagorean proposition if the hypothenuse were said to be wise and good. Yet metaphysicians, confounding dialectic with physics and thereby corrupting both, will discuss for ever the difference it makes to substance whether you call it matter or God.
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