OT for a friend;
Adopting the known results above referred to, for proportions between lines having direction in a single plane (though varying a little the known manner of speaking on the subject), it may be said that, in the horizontal plane, ``West is to South as South is to East,'' and generally as any direction is to one less advanced than itself in azimuth by ninety degrees. Let it be now assumed, as an extension of this view, that in some analogous sense there exists a fourth proportional to the three rectangular directions, West, South, and Up; and let this be called, provisionally, Forward, by contrast to the opposite direction, Backward, which must be assumed to be (in the same general sense) a fourth proportional to the directions of West, South and Down. We shall then have, inversely, Forward to Up as South to West, and therefore, as West to North: if we admit, as it seems natural and almost necessary to do, that (for directions, as for lengths) the inverses of equal ratios are equal; and that ratios equal to the same ratio are equal to each other. But again, Up is to South as South to Down, and also as North to Up: and we can scarcely avoid admitting, or defining, that (in the present comparison of directions) ratios similarly compounded of equal ratios are to be considered as being themselves equal ratios. Compounding, therefore, on the one hand, the ratios of Forward to Up, and of Up to South; and on the other hand the respectively equal (or similar) ratios of West to North, and of North to Up, we are conducted to admit that Forward is to South as West to Up. By a reasoning exactly similar, we find that Forward is to West as Up to South; and generally that if X, Y, Z denote any three rectangular directions such that A:X::Y:Z, A here denoting what we have expressed by the word Forward, then also A:Y::Z:X (and of course, for the same reason A:Z::X:Y); so that the three directions may be all changed together by advancing them in a ternary cycle, according to the formula just written, without disturbing the proportionality assumed. But also, by the principle respecting proportions of directions in one plane, we may cause any two of the three rectangular directions XYZ to revolve together round the third, as round an axis, without altering their ratio to each other. And by combining these two principles, it is not difficult to see that because Forward has been supposed to be to Up as South to West, therefore the same (as yet unknown) direction ``Forward'' must be supposed to be to any direction X whatever, as any direction Y, perpendicular to X, is to that third direction Z which is perpendicular to both X and Y, and which is obtained from Y by a right-handed (and not by a left-handed) rotation, through a right angle, round X; in the same manner as (and because) the direction West was so chosen as to be to the right of South, with reference to Up as an axis of rotation. Conversely we must suppose that if any three rectangular directions, , be arranged, as to order of rotation, in the manner just now stated, then Z:Y::X:A; or in other words, we must admit, if we reason in this way at all, that the direction called already Forward, will be the fourth proportional to . And if we vary the order, so as to have Z to the left, and not to the right of Y, with reference to X, then will the fourth proportional to become the direction which we have lately called Backward, as being the opposite to that named Forward.
Again, since Forward is to Up as South to West, that is in a ratio compounded of the ratios of South to East and of East to West, or in one compounded of the ratios of West to South, and of any direction to its own opposite; or, finally, in a ratio compounded of the ratios of Up to Forward and of Forward to Backward, that is, in the ratio of Up to Backward, we see that the third proportional to the directions Forward and Up is in the direction Backward: and by an exactly similar reasoning, with the help of the conclusions recently obtained, we see that if X be ANY direction in tridimensional space, then A:X::X:B; B here denoting, for shortness, the direction which has been above called Backward. |
