Note 10 — NEW WAYS OF MEASURING THE SUN’S DISTANCE.

It may be briefly sketched, perhaps, in a note. The force necessary to draw the earth inwards in such sort as to make her follow her actual course is proportional to (i) the square of her velocity directly, and (ii) her distance from the sun inversely. If we increase our estimate of the earth’s distance from the sun, we, in the same degree, increase our estimate of her orbital velocity. The square of this velocity then increases as the square of the estimated distance; and therefore, the estimated force sunwards is increased as the square of the distance on account of (i), and diminished as the distance on account of (ii), and is, therefore, on the whole, increased as the distance. That is, we now regard the sun’s action as greater at this greater distance, and in the same degree that the distance is greater; whereas, if it had been what we before supposed it, it would be less at the greater distance as the square of the distance (attraction varying inversely as the square of the distance). Being greater as the distance, instead of less as the square of the distance, it follows that our estimate of the sun’s absolute force is now greater as the cube of the distance. Similarly, if we had diminished our estimate of the sun’s distance, we should have diminished our estimate of his absolute power (or mass) as the cube of the distance. But our estimate of the sun’s volume is also proportional to the cube of his estimated distance. Hence our estimate of his mass varies as our estimate of his volume; or, our estimate of his mean density is constant.