EUCLIDE's The whole FIFTEEN BOOKS compendioufly Demonftrated : 4 WITH ARCHIMEDES's Theorems of the Sphere and Cylinder Investiga ted By ISAAC BARROW, D.D. late Mafter To which is Annex'd, EUCLIDE's Data, and a brief The Whole revis'd with great Care, By THOMAS HASELDEN, Teacher Καθαρμοὶ ψυχῆς λογικῆς εἰσιν αἱ μαθηματικαί ἐπιςήμαι. LONDON: Printed for Daniel Midwinter and Aaron 5-20-46ADE 56 To the READER. to F you are defirous, Courteous Reader, to know what I have performed in this Edition of the Elements of Euclide, I shall here explain it you in short, according to the Nature of the Work. Ihave endeavour'd to attain two Ends chiefly; the first, to be very perspicuous, and at the same time fo very brief, that the Book may not swell to such a Bulk, as may be troublesome to carry about one, in both which I think I have fucceeded. Some of a brighter Genius, and endued with greater Skill, may have demonstrated most of these Propositions with more nicety, but perhaps none with more fuccinctness than I have; especially fince I alter'd nothing in the Number and Order of the Author's Propositions; nor prefum'd either to take the Liberty of rejecting, as less necessary, any of them, or of reducing some of the easier fort into the Rank of Axioms, as several have done; and among others, that most expert Geometrician A. Tacquetus C. (whom I the more willingly name, because I think it is but civil to acknowledge that I have imitated him in Some Points) after whose most accurate Edition I had no Thoughts of attempting any thing of this Nature, 'till I confider'd that this most learned Man thought fit to publish only Eight of Euclide's Books, which he took the pains to explain and embellish, having in a manner rejected and undervalued the other Seven, as less appertaining to the Elements of Geometry. But my Province was originally quite different, not that of writing the Elements of Geometry after what method foever I pleas'd, but of demonstrating, in as few Words as poffible I could, the whole Works of Euclide. A a2 to Four of the Books, viz. the Seventh, Eighth, Ninth, and Tenth, although they don't so nearly appertain to the Elements of plain and Solid Geometry, as the fix precedent and the two subsequent, yet none of the more skilful Geometricians can be so ignorant as not to know that they are very useful for Geometrical Matters, not only by reason of the mighty near affinity that is between Arithmetick and Geometry, but also for the Knowledge of both commensurable and imcommensurable Mag nitudes, so exceeding neceffary for the Doctrine of both plain and folid Figures. Now the noble Contemplation of the five regular Bodies that is contained in the three laft Books, cannot without great Injustice be pretermitted, fince that for the fake thereofour σοιχειωτής, being a Philosopher of the Platonic Sect, is faid to have compos'd this univerfal System of Elements; as Proclus lib. 2. witneffeth in these Words, "Ὅθεν δὴ κὴ ἡ συμπάσης σοιχειώσεως τέλο προεςήσαζο τὴν τις καλεμθύων πλαζωνικῶν χημάτων σύςασιν. Besides, I eafily perswaded my self to think, that it would not be unacceptable to any Lover of these Sciences to have in his Poffeffion the whole Euclidean Work, as it is commonly cited and celebrated by all Men: Wherefore I refolo'd to omit no Book or Proposition of those that are found in P. Herigonius's Edition, whose Steps I was oblig'd closely to follow, by reason I took a Resolution to make use of moft of the Schemes of the said Book, very well foreseeing that Time would not allow me to form new ones, though Sometimes I chose rather to do it. For the same Reason I was willing to use for the most part Euclide's own Demonstrations, having only express'd them in a more fuccinct Form, unless perhaps in the Second, Thirteenth, and very few in the Seventh, Eighth, and Ninth Bocks, in which it seem'd not worth my while to detriae in any Particular from him: Therefore I am not without |