All which he concludes 1, With an Intimation of his sense concerning two other Comets, pretended to have been lately seen, _One_ at _Rome_, about the {303} _Girdle of Andromeda_, in the Months of _February_ and _March_, 1664. the other in _Germany_ in _Capricorne_, about _Saturne_ in the head of _Sagittary_, during the Months of _September_ and _October_, 1665. 2ly, With an Advertisement of what he has done in that important Work for the Advancement of _Astronomy_, the due _Restitution of the Fixt Stars, vid._ That he has almost finish't it; himself alone, without trusting to any other mans labour, that was not directed by him. The _Second_ Part of this Book (the _Mantissa_ to the _Prodromus Cometicus_) endeavours to justifie the Authors Observations touching the former Comet, excepted against by M. _Auzout_, in several particulars; as 1. That it had not pass'd to the _First_, but _Second_ Star in _Aries_, and had mov'd in quite another Line, than He had described. 2. That its _proper motion_ about the end of _January_ and the beginning of _February_, 1665. had not been rightly assigned. 3. That the _Bignesse_ of its _Diameter_ had not been truly delivered; Nor 4. The _Faces_ of its _Head_ in due manner represented. To all which the Author endeavors to answer: 1. By delivering all his Observations of that Comet, thereby to shew, what care and diligence he had used, _particularly_ to make out, how great its _Diurnal motion_ had been; in what proportion, and how far, it decreased, and where and in what degree it increased again: Which being, as he conceives, duly and exactly deduced, and demonstrated, he esteems it afterwards to be easie for every one, versed in these matters, certainly to collect and to judge, what way the Comet, after it became invisible to the naked Eye, and could be no longer observed with _Sextants_ and _Quadrants_, had taken, and what Line it had described. 2, By subjecting all those Observations, with great diligence and labour, to a rigid _Calculus_, thereby to obtain, for every day, the _Longitudes_, _Latitudes_, _Right Ascensions, Declinations_, _Proper motion_, _Angle_ of the _Ecliptick_ and the _AEquator_, and the _Nodes_ of that Comet; for the construction of an _Ephemerides_ of its whole Motion. From all which he pretends to prove, that he has not erred in his Observation of _February_ 18, nor been prepossest by any _Hypothesis_, nor deluded by any _Fixt Star_, as M. _Auzout_ thinketh; but that near the _First Star_ of _Aries_ there _then_ appear'd a _Phaenomenon_, most like to that Comet, that was seen some dayes before, if compared with the Observations make thereof _Febr._ 12, 13, 14. Though he will not hitherto positively determine, whether that _Phaenomenon_, which appear'd to him _February_ 18. was {304} indeed that very Comet, which he saw with his naked Eye, and observed with his Geometrical Instruments, the said 12, 13, and 14. dayes of _February_; or whether it was another, and whether he had