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A67375 Cono-cuneus, or, The shipwright's circular wedge that is, a body resembling in part a conus, in part a cuneus, geometrically considered / by John Wallis ..., in a letter to the honourable Sir Robert Moray ... Wallis, John, 1616-1703. 1684 (1684) Wing W565; ESTC R34709 14,684 34

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CONO-CUNEUS OR THE SHIPWRIGHT'S CIRCULAR WEDGE THAT IS A Body resembling in part a CONUS in part a CUNEUS Geometrically considered By JOHN WALLIS D. D. Professor of Geometry in the University of Oxford and a Member of the Royal Society LONDON IN A LETTER TO THE HONOURABLE Sir ROBERT MORAY Knight LONDON Printed by John Playford for Richard Davis Bookseller in the University of OXFORD 1684. TO THE HONOURABLE Sir Robert Moray K t. SIR SInce I came home from London I have taken some time to consider of those Solids and Lines made by the Sections thereof proposed to Consideration to my Lord Brouncker and your self at your Lodgings where I was also present by Mr. Pett one of His Majesties Commissioners for the Navy and an excellent Shipwright The Bodies proposed to consideration were all of this form On a plain Base which was the Quadrant of a Circle like that of a Quadrantal Cone or Cylinder stood an erect Solid whose Altitude being arbitrary was there double to the Radius of that Quadrant and from every Point of its Perimeter streight Lines drawn to the Vertex met there not in a Point as is the Apex of a Cone nor in a parallel Quadrant as in a Quadrantal Cylinder but in a streight Line or sharp Edge like that of a Wedge or Cuneus On which consideration I thought fit to give it the name of Cono-Cuneus as having the Base of a Cone and the Vertex of a Cuneus By the various Sections of this Solid in several Positions he did rightly conceive that divers new Lines must arise in great variety different from those arising from the Section of a Cone Some of which he supposed might be of good use in the Building of Ships in order to which it was that he proposed them to Consideration Now because he judged it troublesom as indeed it would be first to form such Solids and then cut them by Plains in such Positions as he des●red he had for avoiding that trouble ingeniously contrived this Expedient He caused divers Boards of a good solid Wood to be exactly planed some of an equal thickness some meeting in a sharp edge those of the former he caused to be glewed together in a parallel Position those of the latter sort he caused so to be glewed together as that their sharp edges met in one common Angle And having thus formed several Solids of Boards thus glewed together he then caused them to be wrought into such a form as that before described Which being done he then caused the Glew to be dissolved in warm Water whereby the several Boards falling asunder did exhibit in their several faces the respective Sections of those Solids And such were those he shewed us which being put together made up such Solids and taken asunder shewed the several Sections of them I do not intend at all to disparage the ingenuity of that Contrivance which was indeed very handsom and neatly performed but do withall suppose that it would not be unpleasing to your self or him to see those Lines described in Plano which would arise by such Section of the Solid That therefore is the work of these Papers to represent the true nature of such Lines and the ways to draw them without the actual Section of a Solid Which I have the rather undertaken because this is a Solid which I do not know that any other have before considered And because this may be a Pattern according to which other Solids of like nature may be in like manner considered if there shall be occasion If beside these Sections which he hath already considered there be any other Sections of this or other the like Solids which he shall conceive useful to his purpose the same may in like manner be represented without the actual Section of such Solids by Lines thus described in a Plain But which of them may be most advantageous to his design I do not pretend to understand so well nor can with so much certainty affirm as that I am SIR Your very humble Servant Oxon Apr. 7. 1662. JOHN WALLIS CONO-CUNEUS OR THE SHIPWRIGHT'S CIRCULAR WEDGE The Sections of a CONO-CUNEUS 1. ON a Rectangle CDBA Fig. 1. erect at Right Angles the Quadrant of a Circle CQD and joining QA compleat the Rectangled Triangle CQA Supposing then from every Point of the Quadrantal Arch DQ to their respective Points in the streight Line BA in Plains parallel to the Triangle CQA the streight Lines Sa to be drawn compleating a Curve Superficies DSQAaB the Solid thus contained I call a Cono-Cuneus 2. It differs from a Quadrantal Cone in this only That what is here a streight Line AB is there a single Point all the Lines drawn from the Points S meeting there at the Point A. 3. It differs in this from a Wedge or Cuneus That what is here a Quadrant CQD is there a Rectangle 4. It differs in this from a Quadrantal Cylinder That what is here a streight Line AB is there a Quadrant equal and parallel to CQD 5. This Solid being cut by Plains in different Positions will produce in the Curve Surface DQAB great variety of Lines As for Example 6. First If it be cut by RSa a Plain parallel to the Triangle CQA the Line Sa is by construction a streight Line and therefore the Hypothenuse of a Right-angled Triangle SRa 7. And consequently this Cono-Cuneus is equal to half a Quadrantal Cylinder of the same Base and Altitude For every of the Triangles SRa in the Cono-Cuneus being half the respective Rectangle in the Cylinder the whole of That will be equal to the half of This. 8. The Quantities therein I thus design in Species 9. These Triangles if made by Plains set at equal distances projected on the Plain CQA to which they are parallel will appear as in the first Projection Fig. 16. which is thus drawn Having drawn a Triangle ACQ like and equal to that in the Solid and CQD the Quadrant of a Circle let CD be divided into any number of equal parts at the Points R from every of which the Ordinates RS being drawn take equal thereunto in the Line CQ the Lines Cs or Rs then joining As the Triangles sRA or sCA in this Plain represent the like Triangles SRA in the Solid 10. And if we suppose the Solid to be continued downward beyond its Quadrantal Base these Triangles must be so continued also And the like if we suppose it to be continued upward after a decussation in AB as in opposite Cones 11. The Quantities in this Projection I design thus in Species 12. In Numbers thus putting R = 1. A = 2. CR. Cs. As. 0. 1. 2 236+ 0.125 0 992+ 2.233 − 0.25 0 968+ 2 222+ 0.375 0.927 − 2 204+ 0.5 0 866+ 2 179+ 0.625 0.781 − 2.147 − 0.75 0 661+ 2 106+ 0.875 0 484+ 2.058 − 1. 0. 2. 13. Secondly If it be cut by Edq a Plain parallel to the Quadrantal Base CDQ Fig. 1.