In his fifteenth letter, he tells Hypatia that he was so unfortunate, or found himself so ill, that he wished to use a _hydroscopium_, and he requests that she would cause one to be constructed for him. “It is a cylindrical tube,” adds he, “of the size of a reed or pipe. A line is drawn upon it lengthwise, which is intersected by others, and these point out the weight of water. At the end of the tube is a cone, the base of which is joined to that of the tube, so that they have both only one base. This part of the instrument is called _baryllion_. If it be placed in water, it remains in a perpendicular direction, so that one can discover by it the weight of the fluid.” Petavius, who published the works of Synesius in the year 1640, acknowledges in his annotations, that this passage he did not understand. An old scholiast, he says, who had added some illegible words, seemed to think that it referred to a water-clock; but this he considers improbable, as a clepsydra was not immersed in water, but filled with it. He conjectures, therefore, that it may allude to some such instrument as that which Vitruvius calls _chorobates_. The latter however was employed for leveling; and it appears that Synesius, who complains of the bad state of his health, could not think of leveling. Besides, no part of the description in Vitruvius agrees with that which is given in so clear a manner by Synesius. Petau published his edition of the works of this philosopher in the time of Peter de Fermat, conseiller au parlement de Toulouse, a man of great learning, who was an excellent mathematician, and well-acquainted with antiquities and the works of the ancients. We have by the latter a commentary upon some obscure passages of Athenæus, annotations on the writings of Theon of Smyrna, and emendations from a manuscript to the Stratagemata of Polyænus, which may be found also in his Miscellanies. Mursinna, in his edition of the same author, has added them to the end of the preface. As Fermat was often consulted respecting difficult passages of the ancients, he could not be unacquainted with that in the new edition of Synesius. He drew up an explanation of it, and gave it to a friend who was then about to publish a French translation of Bened. Castelli’s book, Della Misura dell’Acque Correnti, and who caused it to be printed along with that work. Fermat died in the year 1665. After his death his son published some of his writings under the title of Varia Opera Mathematica[414]; and in this collection is inserted his short treatise on the _hydroscopium_, from which I have extracted the following explanation.