Produced by Tim Madden and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) [Transcriber's Notes All apparent printer's errors retained. Variation in punctuation are as in the original, but missing full stops at end of paragraphs have been supplied. There are inconsistencies in the use of italics, spacing of words and use of full stop after 'AXIOME', abbreviations etc. All are retained to match text. There is a great variation in spelling including multiple spellings of the same word, all spelling has been retained to match text. There are several instances of obviously missing letters or inverted n & u. These have been changed or obvious letters replaced, with the changes surrounded by {}. All instances are detailed at the end of the text. It should also be noted that in the original text there is a missing line at the end of page 24 in original text. There are a number of instances in the original text where 'that' is immediately followed by a second 'that' in the sentence. These could be potential printer's errors or, since several of them make sense, part of the author's style. They have been left in the text as they appear in the original text. The original text has many sidenotes, some are true sidenotes, introductions to paragraphs etc, some acting as footnotes with some marked in original text with *. These have been dealt with in three ways with the footnotes placed after their relevant paragraph and sidenotes place before their relevant paragraph. 1) Footnotes marked with capital letter. These were sidenotes in original text marked with * in the original text and thus acting like footnotes. 2) Footnotes marked with number. These were sidenotes in original text that were unmarked but acting like normal footnotes. The anchor in the text was placed at the most suitable relevant place in comparison with the placement of the sidenote text in the margin, but still should be considered only an approximate placement. 3) Sidenotes placed at start of the relevant paragraph. Some sidenotes were considered not to be relevant as footnotes, introductions to paragraphs etc, and were left as sidenotes before their relevant paragraph.] * * * * * A DISCOURSE _PRESENTED_ TO THE MOST SERENE Don Cosimo II. GREAT DUKE _OF_ TUSCANY, CONCERNING The _NATATION_ of BODIES Vpon, And _SUBMERSION_ In, THE WATER. By GALILEUS GALILEI: Philosopher and Mathematician unto His most Serene Highnesse.