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A38722 The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...; Huict livres des Eléments d'Euclide rendus plus faciles. English Dechales, Claude-François Milliet, 1621-1678.; Euclid. Elements.; Williams, Reeve, fl. 1682-1703. 1685 (1685) Wing E3399; ESTC R10241 136,603 430

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inscribed by the little Rectangles through which the Circumference of the Circle passeth and all those Rectangles taken together are equal to the Rectangle AL. Imagine that the Semi-Circle is made to roul about the Diameter EB the Semi-Circle shall describe a hemi-sphere and the inscribed Rectangles will describe inscribed Cylinders in the semi-sphere and the Circumscribed will describe other Cylinders Demonstration The Circumscribed Cylinders surpass more the inscribed than doth the hemi-sphere surpass the same inscribed Cylinders seeing that they are comprehended within the Circumscribed Cylinders Now the Circumscribed surpass the inscribed by so much as is the Cylinder AL therefore the hemi-sphere shall surpass by less the inscribed Cylinders than doth the Cylinder made by the Rectangle AL. The Cylinder AL is less than the Cylinder MP for there is the same Ratio of a great Circle of the Sphere which serveth for Base to the Cylinder AL as of MP to R so then by the foregoing a Cylinder which hath for Base a great Circle of the Sphere and the Altitude R would be equal to the Cylinder MP Consequently the hemi-sphere which surpasseth the quantity D by the Cylinder MP and the inscribed Cylinders by a quantity less than AL surpasseth the inscribed Cylinders by less than the quantity D. Therefore the quantity D is less than the inscribed Cylinders What I have said of a hemi sphere may be said of a whole Sphere LEMMA II. LIke Cylinders inscribed in Two Spheres are in Triple Ratio of the Diameters of the Spheres Lemma Fig. II. If the two like Cylinders CD EF be inscribed in the Spheres A B they shall be in Triple Ratio of the Diameters LM NO Draw the Lines GD IF Demonstration The like Right Cylinders CD EF are like so then there is the same Ratio of HD to DR as of QF to FS as also the same Ratio of KD to KG as of PF to PI. And consequently the Triangles GDK IFP are like by the 6th of the 6th so there shall be the same Ratio of KD to PF as of GD to IF or of LM to ON Now the like Cylinders CD EF are in Triple Ratio of KD and PF the Diameters of their Bases by the 12th therefore the like Cylinders CD EF inscribed in the Spheres A and B are in triple Ratio of the Diameters of the Spheres PROPOSITION XVIII THEOREM SPheres are in triple Ratio of their Diameters The Spheres A and B are in Triple Ratio of their Diameters CD EF. For if they be not in Triple Ratio one of the Spheres as A shall be in a greater Ratio than Triple of that of CD to EF therefore a quantity G less than the Sphere A shall be in Triple Ratio of that of CD to EF and so one might according to the first Lemma inscribe in the Sphere A Cylinders of the same height greater than the quantity G. Let there be inscribed in the Sphere B as many like Cylinders as those of the Cylinder A. Demonstration The Cylinders of the Sphere A to those of the Sphere B shall be in Triple Ratio of that of CD to EF by the preceding Now the quantity G in respect of the Sphere B is in Triple Ratio of CD to EF there is then the same Ratio of the Cylinders of the Sphere A to the like Cylinders of the Sphere B. So then the Cylinders of A being greater than the quantity G the Cylinder B that is to say inscribed in the Sphere B would be greater than the Sphere B which is impossible Therefore the Spheres A and B are in Triple Ratio of that of their Diameters Coroll Spheres are in the same Ratio as are the Cubes of their Diameters seeing that the Cubes being like solids are in Triple Ratio of that of their Sides FINIS ERRATA PAge 14. Line 21. read AFD p. 23. l. 17. for DF r. EF. l. 16. r. DFE l. 27. r. FD. p. 24. l. 1 and 9. for FB r. FD. p. 26. l. 8. r. HI p. 52. l. 14. r. ACB p. 55. l. 22. r. ACF p. 71. l. 3. r. ACB p. 82. l. 19. r. ABC p. 117. l. 13. r. GFE p. 125. l. 9. r. AD. p. 178. l. 16. r. CFD p. 194. l. ult r. AF. p. 197. l. 2. r. CDA p. 218. l. 7. r. ⅓ and ⅓ p. 268. l. 24. r. CE. p. 281. l. 3. r. ECD p. 289. l. 4. r. are simular p. 294. l. 13. r. eight ¾ p. 309. l. 15. r. DBE Advertisement of Globes Books Maps c. made and sold by Philip Lea at the Sign of the Atlas and Hercules in the Poultry near Cheapside London 1. A New Size of Globes about 15 Inches Diameter made according to the more accurate observation and discoveries of our Modern Astronomers and Geographers and much different from all that ever were yet extant all the Southern Constellations according to Mr. Hally's observations in the Island of St. Helena and many of the Nothern Price Four Pounds 2. A size of Globes of about Ten Inches Diameter very much Corrected Price 50 s. 3. Concave Hemispheares Three Inches Diameter which serves as a Case for a Terrestial Globe and may be carried in the Pocket or fitted up in Frames Price 15 or 20 s. 4. Another Globe of about Four Inches Diameter fitted to move in Circular Lines of Brass for Demonstrating the Reason of Dyalling or being erected upon a small Pedestal and fet North and South will shew the hour of the day by its own shadow or by the help of a moving Meridian will shew the hour of the Day in all parts of the World c. 14. There is in the Press a Particular description of the general use of Quadrants for the easie resolving Astronomical Geometrical and Gnomonical Problems and finding the hour and Azimuth universally c. whereunto is added the use of the Nocturnal and equinoctial Dial. 15. A new Map of England Scotland and Ireland with the Roads and a delineation of the Genealogy of the Kings thereof from William the Conqueror to this present time with an Alphabetical Table for the ready finding of the places Price 18 d. 16. The Elements of Euclide explained and demonstrated after a new and most easie method with the uses of each Proposition in all the parts of the Mathematicks by Claude Francois Millet Dechales a Jesuite 17. New Maps of the World four quarters and of all the Countries and of all sizes made according to the latest discoveries extant may be had pasted upon Cloath and Coloured also Sea Plats Mathematical Projection Books and Instruments whatsoever are made and sold by Philip Lea. FINIS EUCLIDS ELEMENTS with the uses EUCLID London Printed for Phillip Lea Globe Maker at the Atlas and Hercules in Cheapside Near Friday Street There all Sorts of Globes Spheres Maps Sea-plats Mathematical Books and Instruments are Made and Sold Plate 2. Propositions and Uses of the first Book See Plate 3. Plate 1. Definitions of the first Book Propositions and Uses of the first Book See Plate 2. Plate 3. Propositions Uses of the first Book Definitions Propositions Uses of the Second Book Plate 4 Definitions of the third Book Propositions Propositions and Uses Plate 5. Definitions Propsitions of the fouth Book Plate 6. Definitions of the Sixth Book Propositions Uses Plate 7. Definitions of the Eleventh Book Propositions Plate 8. Proposi of the Twelfth Book