COMPENDIOUS ASTROSCOPY.

It remains for us to examine a few things a little more accurately, which might perhaps cause scruples for those not yet experienced. They will fear, first, that as the thread reaching to both lenses sags, its curve, although small, might nevertheless interfere with their parallel position at those great lengths of one hundred or two hundred feet. And indeed, if a heavier rope were needed, its curvature would cause no small harm, and no intensity of tension could overcome this disadvantage. But now, with the larger lens suspended and balanced as we have done, we guide it with the pull of only a very light silk thread; its weight does not exceed half a drachma original: "semidrachmam" — approx. 1.75 grams for fifty feet, and the same thread supports seven pounds before it breaks. Therefore, its bending does not hinder anything in this, nor in a much greater distance of the lenses, even if it is not pulled with more than moderate force, equivalent to two or three pounds; certainly, since geometric perfection is by no means required here, as is known to any experienced person.

Indeed, it is certain that in the same ratio as a rope is lighter than a rope, the force of tension is diminished, by which both approach a straight line equally. Thus, a cord fifty feet long and weighing an ounce would need a force of forty-eight pounds, whereas our thread, of equal length, will need no more than three pounds. And this is clearer by itself than to be proven by demonstration. For it is exactly the same as when sixteen half-drachma cords are pulled each by the weight of three pounds, and when they are pulled together forming an ounce-weight cord, it is tensioned by sixteen times three pounds combined.

But these matters, which pertain to the bending of the thread, can also be further weighed by reasons of geometry and experiments. Namely, the tensioned thread, because of that small bending, expresses a parabolic line so closely that it may be taken for true without error. The depth of this parabola, in a length of one hundred and fifty feet, we found to be about a quarter of a part of a foot; when the thread was tensioned parallel to the horizon, and only with a force of two and a half pounds. Let the parabola of the thread be abc, its depth db, drawn from the straight line adc. Furthermore, let the lines ae and cf be tangent to the parabola, to which ce and af meet, parallel to db. Therefore, to the observer from point a, along the line ae, it was noted that the space ce becomes one foot; from which it follows that db is the fourth part of a foot. But af is equal to ce. Thus, the thread cba pulls the lens placed at c in such a way that it is not directly opposite to the eye, which is at a, but to point f. So that the eye is at an interval of one foot from the true place: which in that distance of 150 feet cannot be harmful. For the angle of deflection cae or acf becomes only two-fifths of one degree; so that there is no need for the remedy, which we shall nevertheless give. But with distance gh taken as double the former, or three hundred feet, so that the curved thread is gbh, the measure of the cavity kb will indeed be quadruple the former db, but the angle of deflection will be only double, that is, one degree; as is easily seen, by drawing the tangent gl, which meets the perpendicular hl. For hl itself will be quadruple to kb or ce; but the distance gh to ac was double. Therefore, the angle of deflection hgl, of the previously found cae, can be considered double.

This aberration, however, of 48 minutes original: "scrupulorum 48" — archaic term for arcminutes is still of no importance, nor will it cause harm if neglected. Nevertheless, so that there remains no room for complaint, I will now show what kind of correction can be applied, and indeed of such a kind that, in one operation, it restores every other declination of the lens.

Therefore, once from the beginning, for the superior preparation of the large lens, let this which we shall say be added. Namely, with the lens balanced as we have instructed, and fixed at the height of the eye, let the thread attached to the tail be taken with the other hand, and brought to the eye...