The determination of the apparent distances and positions of such double stars as could be measured with micrometrical instruments and high magnifying powers, was suggested by Sir William Herschel more than forty years ago, and in his hands it led to a new department of physical astronomy, by the discovery of sidereal phenomena referrible to the agency of attractive forces; but the determination of the existence of annual parallax, the immediate object for which the inquiry was instituted, was soon lost sight of in the more extensive views of the construction of the universe, which gradually unfolded themselves. Nor has the investigation been resumed, although from the precision with which such observations can be made, it seems, in the opinion of the authors of this paper, likely to be the mode by which the existence or non-existence of sensible parallax will ultimately be determined. The results of Sir William Herschel’s observations, from 1779 to 1784, were published in the Philosophical Transactions from 1782 to 1785; and a re-examination, after a lapse of twenty years, was undertaken by him in 1801, -2, -3, and -4; and in the Transactions for 1802 and 1804, unexpected phenomena were communicated. Instances in which two stars were performing to each other the offices of sun and planet were proved to exist; and to more than one pair the period of rotation was, according to the observations of the authors of this paper, ascertained with considerable exactness. Immersions and emersions of stars behind each other had been witnessed; and real motions among some of them had been observed rapid enough to be detected in very short intervals of time.