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Published online by Cambridge University Press: 20 January 2009
The following paper was written last summer, and was submitted to Dr Mackay with a view to eliciting his opinion particularly on the curious passage referred to in § 3, and on the remarks contained in § 8. I was not aware of the intention of Mr T. L. Heath to follow up his excellent edition of Apollonius by an edition of Archimedes on similar lines, and when I saw the announcement of his Archimedes in the month of October, I at once concluded that the notes I had made would have been anticipated by him. Since reading his masterly work, however, I am disposed to think there is still sufficient interest in the notes I have written to justify me in laying them before the Society; I therefore submit them in their original form, although I should have omitted certain details had I been acquainted with Mr Heath's work before writing the paper.
page 3 note * The references are to Heiberg's edition.
page 4 note * The text of Heiberg in this passage differs considerably from that of Torelli, but it is hardly possible that the latter can be correct. The sentence (Torelli, p. 287, at foot) “Apa kaì o gλlvδpos K.T.λ” is a mere repetition of that preceding it, while the position of apa at the beginning of the sentence is at variance with Greek usage. The deletion of 
before apa is stated in Heiberg's note to be due to Commandine, and it is easy to understand the deletion, for in Torelli's text the inscribed figure is only compared with the whole of the circumscribed cylinder and not also with a part of it as in Heiberg's text. In all the texts there is a certain ambiguity as to the precise meaning of the phrases “all the lines” and “all the lines cut off between AB, BΔ, but lines 14–16 in Heiberg make the meaning quite clear, for there it is explicitly stated that the circumscribed cylinder diminished by one of its elementary cylinders is more than double of the inscribed figure while it is obviously exactly double. Heath's rendering of the proposition is, of course, quite accurate in its mathematics, but in the condensation of the original text the erroneous statement has apparently been overlooked. So far as I am aware, the slip has not been previously pointed out.
page 10 note * In the diagram of Archimedes (Opera I., p. 462),
page 12 note * As another illustration of the application of the inequality theorems, I had worked out the value of the area of a segment of a parabola from the figure used in the mechanical quadrature, but as the method I followed is identical with that given by Heath (p. cliv.), I omit my investigation.
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