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 yourself, at your Lodgings, where I was also present) by Mr. Pett, one of His Majesty's 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, straight 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 straight 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 troublesome (as indeed it would be) first to form such Solids, and then cut them by Plains in such Positions as he desired; he had (for avoiding that trouble) ingeniously contrived this Expedient. He caused divers Board's, 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 glued together in a parallel Position; those of the latter sort, he caused so to be glued together, as that their sharp edges met in one common Angle. And having thus form several Solids, of Board's thus glued together, he then caused them to be wrought into such a form as that before described: Which being done, he then caused the Glue to be dissolved in warm Water, whereby the several Board's, falling asunder, did exhibit, in their several faces, the respective Sections of those Solids. And such were those he showed us; which being put together, made up such Solids; and taken asunder, showed the several Sections of them. I do not intend at all to disparage the ingenuity of that Contrivance, which was indeed very handsome, and neatly performed, but do withal suppose, that it would not be unpleasing to yourself, 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, complete the Rectangled Triangle CQA. Supposing then from every Point of the Quadrantal Arch DQ, to their respective Points in the straight Line BASILIUS, in Plains parallel to the Triangle CQA, the straight Lines Sa to be drawn, completing 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 straight 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 straight 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 straight Line; and therefore, the Hypothenuse of a Rightangled 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. Fig. 1. 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. Fig. 16. 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. the Curve Line dσq will be an Ellipse: For (supposing this Plain to be cut in ρσ by RSa, any of those Triangles parallel to CQA;) then is, As AC to A, or are to aρ; So CQ to Eq, and RS to ρσ. And consequently (the Ordinates' ρσ being proportional to RS the Ordinates' of a Circle) Edq will be the Quadrant of an Ellipse, as CDQ is of a Circle. 14. The Quantities I thus design in Species. Fig. 1. 15. These Ellipses (if cut off by Plains set at equal distances) projected on the Quadrant CDQ (to which they are parallel) will appear as in the second Projection, Fig. 17. which is thus drawn: Having drawn a Quadrant CDQ equal to that in the Solid, let CQ be divided into any number of equal parts at the Points q, and every of the Ordinates' RS (parallel thereunto) at the Points σ; through which, if we draw the Ellipses Dσq, these in the Plain will represent the like Ellipses dσq in the Solid. 16. If the Solid be supposed to be continued downward below its Quadrantal Base CDQ, the parallel Sections will yet be Ellipses: But with this difference; CD, which is now half the longest Diameter, will then be half the shortest Diameter of the Ellipse, (such as are those in Fig. 17. beyond the Circular Quadrant DQ:) And if the Solid be continued upward, after a decussation in AB, the like Ellipses will occur in the opposite Solid as in this. 17. The Quantities in this Projection I thus design in Species. Fig. 17. 18. In Numbers thus; (putting R = 1, A = 2.) EQ. ρσ. ρσ. ρσ. 0.25 0. 242+ 0.217 − 0. 165+ 0.5 0. 484+ 0. 433+ 0.331 − 0.75 0. 726+ 0.650 − 0. 496+ 1. 0. 968+ 0. 866+ 0. 661+ 1.25 1. 210+ 1.083 − 0.827 − 1.5 1. 452+ 1. 299+ 0. 992+ 1.75 1. 694+ 1.516 − 1. 157+ 2. 1. 936+ 1. 732+ 1.323 − 19 Thirdly; If it be cut by cΣσq, a Plain parallel to the Rectangle CDBA, Fig. 2. the Curve Line will have this property: Drawing the Triangles as in the Scheme, it is, As SIR (the Ordinate from any Point S in the Arch ΣQ) to σρ, or Σ● (the Ordinate from Σ, where the Plain cΣq cuts the Quadrantal Arch): So is are or AC (the whole height), to aρ or ασ (the distance of the Point σ from the Plain Aaα parallel to the Quadrant CDQ). Because are, aρσ, are like Triangles. 20. The Quantities I thus design in Species. Fig. 2. 21. These Curve Lines (if made by Plains at equal distances) projected on the Rectangle CDBA (to which they are parallel), will appear as in the third Projection, Fig. 18. which is thus made: Having drawn a Rectangle CDBA (like and equal to that in the Solid), and the Quadrant CDQ; divide CQ into any number of equal parts at the Points c, and draw the Sins or Ordinates' cΣ, with the Co-sines ΣP: Then supposing from the several Points R in the Line ●D, the Lines Rσa parallel to CA, (cutting the Quadrant QD at S, and AB at a:) And therein, As RS to PΣ; So AC (= are) to aσ: The Curve Lines qσP in this Plain, represent their Respectives qσS in the Solids. Where note, That as the Lines Sσσσ in the former Projection, so are the Lines aσσσ in this cut into equal parts. 22. As the Solid may be continued downwards at pleasure, beyond its Quadrantal Base CDQ; so may these Curve Lines qσP, in like manner, be so continued infinitely: And they will then be Assymptotes, each to other; and to the straight Line BD so continued. And if the Solid be continued upward after a decussation in AB, the same Plains will cut off in the opposite Solid opposite Sections like to these. 23. The Quantities in this Projection I thus design in Species. Fig. 18. 24. In Numbers thus; (putting R = 1, A = 2.) A q. a σ. a σ. a σ. a σ. I. 0.5 0. 516+ 0. 577+ 0.756 − Infin. II. 1. 1.033 − 1.155 − 1.512 − Infin. III. 1.5 1. 549+ 1. 732+ 2.268 − Infin. IV. 2. 2.064 − 2.310 − 3.024 − Infin. 25. Fourthly; If it be cut by a Plain CQD, Fig. 3. perpendicular to the Rectangle CDBA, and passing by the Centre C and any Point d in the Side DB, the Curve Line will have this property:. Cutting this Plain in ρ σ (by any of the Triangles RSa, parallel to CQA); then is, As AC or are, to a ρ; So RS, to ρ σ. The length of a ρ being first found in this manner: As CD to CR, or Cd to C ρ; So is dD to ρ R: Which subducted from the whole height, (or added thereunto, if we suppose ρ to be taken in the continuation of dC beyond C) gives the length of a ρ. 26. The Quantities of this Section I thus design in Species. Fig. 3, 4. 27. But if it be cut by a Plain CQb, Fig. 4. which passing by CQ, cuts any Point b (in the Side AB) before it come at d in the Side DB produced; the Curve Line b σ Q will have this property: Cutting this Plain, as before, in ρ σ, by any of the Triangles RSa parallel to CQA; then is, As bA to ba; or, As bC to b ρ; or, As AC (or are) to a ρ: So RS to ρ σ. 28. In Species, thus: Fig. 4. 29. Or else, continuing Cb till it cut DB (continued) in d, the Proportions will be as before, at § 25, 26. 30. But with this difference; That the Curve Q σ bd will cut its Axis at b, and meet with it again at d, (the part bd being on the other side of the Axis, and of the Plain ABDC, in the opposite Solid.) And accordingly (R ρ being in this case greater than are) the Quantities a ρ, and ρ σ, will be Negative Quantities; a ρ falling beyond the Vertex AB, which was supposed short of it; and ρ σ below the Plain ABDC, which was supposed above it. 31. In both these cases (whether Cd cut or cut not the Vertex AB) the Lines dC continued (answering to a suitable continuation of the Solid) will again meet with their Axes continued at δ (as far beyond C, as d is on this side it). But the Ordinates' in this continuation will be greater than those of dC; because from CD upward the Solid grows thinner, but thicker from CD downward. And accordingly, a ρ, which between AB and CD is less than are, (and above AB, a Negative Quantity;) the same below CD becomes greater than are; (dC cutting▪ DC at C:) For there it is a ρ = are − R ρ; here it is a ρ = are + R ρ. 32. These Curves, in both cases, (when Cd cuts or cuts not the Line AB) supposing the Side DB divided into equal parts by the Lines Cd, (if projected on one and the same Plain) will appear as in the fourth Projection, Fig. 19 Where the Axes dC being continued to δ, are then (to avoid confusion in the Figure) removed from their proper place in the Plain ABDC, and set off in the same straight Line AC continued; and the Ordinates' ρ σ applied to them in that Position, in such proportion to RS, as a ρ is to are or AC: And moreover, they are so distributed, some on the one side, some on the other side of AD, to prevent the confusion which might arise in the Figure, if so many Curves should all intersect one another in the same Point Δ, beside another intersection afterwards. 33. The Quantities in this Projection, I thus design in Species. Fig. 19 34. In Numbers (putting R = 1, A = 2,) the Semi-axes to the six Curves described (whereof the first is the circumference of a Circle) are these: I. CD II. Cd. III. Cd. IV. Cd. V. Cd. VI Cd. 1. 1.11803 1.41421 1.80278 2.23607 2.69258 And the Ordinates', supposing the Semi-axes divided into four parts, are these: I. RS. II. ρ σ. III. ρ σ. IV. ρ σ. V. ρ σ. VI ρ σ. 0. 0. 0. 0. 0. 0. 0. 661+ 0. 537+ 0. 413+ 0. 289+ 0. 165+ 0. 041+ 0. 866+ 0.758 − 0.650 − 0. 541+ 0. 433+ 0.325 − 0. 968+ 0.908 − 0.847 − 0.787 − 0. 726+ 0.666 − 1. 1. 1. 1. 1. 1. 0. 968+ 1.029 − 1.089 − 1.150 − 1. 210+ 1.271 − 0. 866+ 0. 974+ 1.083 − 1. 194+ 1. 299+ 1. 407+ 0. 661+ 0. 785+ 0. 909+ 1. 033+ 1.158 − 1.282 − 0. 0. 0. 0. 0. 0. Or if (for a more accurate describing of the Curves) the Semi-axes be divided into 16 equal parts (and the whole Axes into 32), the Ordinates' thereunto appertaining are these. I. RS. II. ρ σ. III. ρ σ. IV. ρ σ. V. ρ σ. VI ρ σ. 0. 0. 0. 0. 0. 0. 0.3481 0.2665 0.1849 0.1033 0.0217 − 0.0598 0.4841 0.3782 0.2723 0.1664 0.0605 − 0.0454 0.5830 0.4645 0.3461 0.2277 0.1093 − 0.0091 0.6614 0.5374 0.4134 0.2894 0.1654 +0.0413 0.7262 0.6014 0.4765 0.3517 0.2269 0.1021 0.7806 0.6586 0.5367 0.4147 0.2921 0.1702 0.8268 0.7105 0.5943 0.4780 0.3617 0.2454 0.8660 0.7578 0.6495 0.5413 0.4330 0.3248 0.8992 0.8009 0.7025 0.6042 0.5058 0.4074 0.9270 0.8401 0.7532 0.6663 0.5794 0.4925 0.9499 0.8757 0.8015 0.7273 0.6531 0.5789 0.9682 0.9077 0.8472 0.7867 0.7262 0.6657 0.9823 0.9362 0.8902 0.8441 0.7981 0.7520 0.9922 0.9612 0.9301 0.8991 0.8681 0.8371 0.9980 0.9824 0.9668 0.9513 0.9357 0.9201 1. 1. 1. 1. 1. 1. 0.9980 1.0136 1.0292 1.0448 1.0604 1.0760 0.9922 1.0232 1.0542 1.0852 1.1162 1.1472 0.9823 1.0283 1.0743 1.1204 1.1664 1.2125 0.9682 1.0288 1.0893 1.1498 1.2103 1.2708 0.9499 1.0241 1.0983 1.1726 1.2468 1.3210 0.9270 1.0139 1.1008 1.1877 1.2746 1.3616 0.8992 0.9976 1.0959 1.1943 1.2926 1.3910 0.8660 0.9743 1.0825 1.1908 1.2990 1.4073 0.8268 0.9431 1.0593 1.1756 1.2919 1.4081 0.7806 0.9026 1.0246 1.1465 1.2679 1.3899 0.7262 0.8510 0.9758 1.1006 1.2254 1.3502 0.6614 0.7854 0.9095 1.0335 1.1575 1.2815 0.5830 0.7014 0.8198 0.9382 1.0566 1.1750 0.4841 0.5900 0.6959 0.8018 0.9077 1.0136 0.3481 0.4297 0.5112 0.5928 0.6744 0.7560 0. 0. 0. 0. 0. 0. 35. Fifthly; If it be cut by a Plain Dσ qc, (Fig. 5.) passing through D, and perpendicular to the Rectangle ABDELLA, cutting AC in any Point c, and AQ in q; the Curve Line D σ q will have this property: Cutting this Plain by any of the Triangles RSa in ρσ, it will be, As AC, or are, to RS; So aρ to a σ. The length of a ρ being first found thus: As DC to DR, or Dc to Dρ; So is Cc to Rρ: Which subducted from are, leaves a ρ = are − Rρ; and here DR = DC − CR: But if Dc be supposed to be continued beyond c, and consequently R fall beyond c, then is DR = DC + CR. 36. The Quantities of this Section I thus design in Species. Fig. 5, 6. 37. But if this Plain (passing by D) cut any Point b in the Line AB, before it come at c in the continuation of CA, (Fig. 6.) the Curve ●●●e will have this property: Cutting this Plain (as before) in ρ σ, by any of the Triangles RSa parallel to CQA, then is, As bB to ba; or, As bD to b ρ; or, As BD (or are) to a ρ: So is RS to ρ σ. 38. In Species, thus: Fig. 6. 39 Or else, continuing Db till it cut CA (continued) in c, the Proportions will be as before, at § 35, 36. 40. But with this difference; That the Curve Dσ bq will cut its Axis at b, (the part bq being on the other side of the Axis, and of the Plain ABDC, in the opposite Solid.) And accordingly (Rρ being in this case greater than are) the Quantities aρ, and ρσ, will be Negative Quantities; aρ falling beyond the Vertex AB, (which was supposed short of it) and ρσ below the Plain ABDC, which was supposed above it. 41. In both these cases (whether Dc cut or cut not the Vertex AB) the Curve Lines Dσc continued (answering to a suitable continuation of the Solid) will again meet with their Axes (continued) at δ (as far beyond c, as D is on this side it). And if, in the mean time, the Axis D δ cut the Vertex BASILIUS, or its continuation▪ beyond A, the Curve will, in the same point, cut its Axis, and (passing thenceforth on the other side) meet with it again at δ. 42. These Curves, in both cases, (whether Dc cut or cut not the Line BASILIUS, or its continuation) supposing the Line CA divided into equal parts by the Lines Dc, (projected on the same Plain) will appear as in the fifth Projection, Fig. 20. Where the Lines Dc being continued to δ, are (to avoid confusion in the Figure) removed from their proper place in the Plain ABDC, and all set off in the same straight Line CA continued; and the Lines ρσ are applied to them as Ordinates' (in this Position) in such proportion to RS, as aρ is to are. 43. The Quantities in this Projection I thus design in Species. Fig. 20. 44. In Numbers (putting R = 1, A = 2,) the Semi-axes of the six Curves described, Dc, are of the same length with Cd, § 34. And the Ordinates', supposing the Semi-axes divided into four equal parts, are these: I. RS. II. ρσ. III. ρσ. IV. ρσ. V. ρσ. VI ρσ. 0. 0. 0. 0. 0. 0. 0.661 0.620 0.579 0.537 0.496 0.455 0.866 0.758 0.650 0.541 0.433 0.325 0.968 0.787 0.605 0.424 0.242 0.061 1. 0.75 0.5 0.25 0. − 0.25 0.968 0.666 0.363 0. 6●0 − 0.212 − 0.545 0.866 0.541 0.217 − 0.108 − 0.433 − 0.758 0.661 0.372 0.083 − 0.207 − 0.496 − 0.785 0. 0. 0. 0. 0. 0. Or (for a more accurate describing the Curves) dividing the Semiaxe into 16 parts (and the whole Axis into 32), the Ordinates' will be these. I. RS. II. ρσ. III. ρσ. IV. ρσ. V. ρσ. VI ρσ. 0. 0. 0. 0. 0. 0. 0.3481 0.3426 0.3372 0.3318 0.3263 0.3209 0.4841 0.4690 0.4539 0.4387 0.4236 0.4085 0.5830 0.5556 0.5283 0.5010 0.4736 0.4461 0.6614 0. 6●01 0.5787 0.5374 0.4960 0.4547 0.7262 0.6696 0.6130 0.5564 0.4998 0.4452 0.7806 0.7075 0.6343 0.5611 0.4880 0.4148 0.8268 0.7364 0.6459 0.5555 0.4651 0.3747 0.8660 0.7578 0.6495 0.5413 0.4330 0.3248 0.8992 0.7728 0.6463 0.5199 0.3934 0.2670 0.9270 0.7822 0.6373 0.4925 0.3476 0.2028 0.9499 0.7866 0.6234 0.4601 0.2969 0.1336 0.9682 0.7867 0.6052 0.4236 0.2421 0.0605 0.9823 0.7827 0.5832 0.3837 0.1842 − 0.0153 0.9922 0.7750 0.5581 0.3411 0.1240 − 0.0930 0.9980 0.7641 0.5302 0.2963 0.0624 − 0.1715 1. 0.75 0▪ 5 0.25 0. − 0.25 0.9980 0.7329 0.4678 0.2027 − 0.0624 − 0.3275 0.9922 0.7131 0.4341 0.1550 − 0.1240 − 0.4031 0.9823 0.6906 0.3990 0.1074 − 0.1842 − 0.4758 0.9682 0.6657 0.3631 0.0605 − 0.2421 − 0.5446 0.9499 0.6382 0.3265 0.0148 − 0.2969 − 0.6085 0.9270 0.6084 0.2897 − 0.0290 − 0.3476 − 0.6663 0.8992 0.5761 0.2529 − 0.0713 − 0.3934 − 0.7166 0.8660 0.5413 0.2165 − 0.1083 − 0.4330 − 0.7578 0.8268 0.5038 0.1809 − 0.1421 − 0.4651 − 0.7880 0.7806 0.4635 0.1463 − 0.1708 − 0.4880 − 0.8051 0.7262 0.4197 0.1132 − 0. ●933 − 0.4998 − 0.8043 0.6614 0.3721 0.0827 − 0. 2●●7 − 0. 4●60 − 0.7854 0.5830 0.3188 0.0546 − 0.2095 − 0.4736 − 0.7378 0.4841 0.2572 0.0303 − 0.1967 − 0.4236 − 0.6505 0.3481 0.1795 0.0109 − 0.1577 − 0.3263 − 0.4949 0. 0. 0. − 0. − 0. − 0. 45. Sixthly; If it be cut by a Plain Bqc, passing through B (Fig. 7.) perpendicular to the Rectangle ABDC, and cutting the Side AC in c, (and any of the Triangles RSa in ρσ) the Curve Line will have this property: As AC, or are, to aρ; So is RS, to ρσ. The length of aρ being first found thus: As BASILIUS to Basilius, or Bc to Bρ; So is Ac to aρ. 46. In Species, thus: Fig. 7. 47. But if this Plain (passing by B) cut the Line DC in any Point P, before it come at c in the continuation of AC, the Curve Line will have this property: Cutting the same Plain (as before) by any of the Triangles RSa parallel to CQA, it will be, As DP to DR; or, As BP to Bρ; or, are (= AC) to aρ; So RS to ρσ. 48. In Species, thus: Fig. 7. 49. Or else, continuing DP till it cut AC (continued) in c, the Proportions will be as before at § 45, 46. 50. But in this Section, the Curve doth not cut its Axis at P (as in the two Sections last mentioned at b), but continues on the same side of it, till it meet again (if it be continued) at β. And (in case of such continuation) instead of DR = R − c, it will be DR = R+c, where the Point R is beyond the Line AC: And, in like manner, if after Bc have cut AC in c, it cut the continuation of AB in P, then, instead of DP = R − C, it will be DP = R+C. 51. In both cases (whether Bc cut AC above or below the Point C) the Curve Lines Bσc continued (answering to a suitable continuation of the Solid) will again meet with their Axes (continued) at●β, as far beyond c as B is on this side it; but continues all on the same side of its Axis, without cutting it in the way, as in the two last mentioned Sections. 52. In both cases (whether Bc cut or cut not DC) the Curve Lines Bσcβ transferred to one and the same Plain, (supposing the Line AC divided into equal parts at the Points c) will appear as in the sixth Projection, Fig. 21. where the Lines Bc are continued to β, and then (to avoid confusion in the Figure) removed from their proper place in the Plain ABDC, and all set off in the same straight Line AC (continued), and the Lines ρσ applied to them as Ordinates' (in this Position) in such proportion to RS, as aρ to are. 53. The Quantities in this Projection, I thus design in Species. Fig. 21. 54. In Numbers (putting R = 1, A = 2,) the Semi-axes of the five Curves described, Bc, are of the same length with Cd, and Dc, § 34, 44. save that the first of those (which is the circumference of a Circle) is here omitted; (instead of which, in this case, we should have a straight Line, coincident with its Axe BASILIUS.) I. Bc. II. Bc. III. Bc. IV. Bc. V. Bc. 1.11803 1.41421 1.80278 2.23607 2.69258 And the Ordinates', supposing the Semi-axis divided into four parts, are these: I. ρσ. II. ρσ. III. ρσ. IV. ρσ. V. ρσ. 0. 0. 0. 0. 0. 0.041 0.083 0.124 0.165 0.207 0.108 0.217 0.325 0.433 0.541 0.182 0.363 0.545 0.726 0.908 0.25 0.5 0.75 1. 1.25 0.303 0.605 0.908 1.210 1.513 0.325 0.650 0.974 1.299 1.624 0.289 0.579 0.868 1.157 1.447 0. 0. 0. 0. 0. Or (for a more accurate describing the Curve) dividing the Semiaxe into 16 parts (and the whole Axe into 32), the Ordinates' will be these: I. ρσ. II. ρσ. III. ρσ. IV. ρσ. V. ρσ. 0. 0. 0. 0. 0. 0.0054 0.0109 0.0163 0.0218 0.0272 0.0151 0.0303 0.0454 0.0605 0.0756 0.0273 0.0547 0.0820 0.1093 0.1366 0.0414 0.0827 0.1241 0.1654 0.2068 0.0566 0.1132 0.1698 0.2264 0.2830 0.0732 0.1463 0.2195 0.2927 0.3658 0.0904 0.1809 0.2713 0.3617 0.4521 0.1083 0.2165 0.3248 0.4330 0.5413 0.1265 0.2529 0.3794 0.5058 0.6323 0.1448 0.2897 0.4345 0.5794 0.7242 0.1633 0.3265 0.4898 0.6531 0.8163 0.1815 0.3631 0.5446 0.7262 0.9077 0.1995 0.3990 0.5986 0.7981 0.9976 0.2170 0.4341 0.6511 0.8681 1.0852 0.2339 0.4678 0.7018 0.9357 1.1696 0.25 0.5 0.75 1. 1.25 0.2651 0.5302 0.7953 1.0604 1.3255 0.2790 0.5581 0.8371 1.1162 1.3952 0.2916 0.5832 0.8748 1.1664 1.4581 0.3026 0.6052 0.9077 1.2103 1.5129 0.3117 0.6234 0.9351 1.2468 1.5585 0.3187 0.6373 0.9560 1.2747 1.5933 0. 323● 0.6463 0.9695 1.2926 1.6158 0.3248 0.6495 0.9743 1.2990 1.6238 0.3230 0.6459 0.9689 1.2919 1.6148 0.3171 0.6343 0.9514 1.2686 1.5857 0.3065 0.6130 0.9195 1.2260 1.5325 0.2894 0.5787 0.8681 1.1575 1.4468 0.2641 0.5283 0.7925 1.0566 1.3208 0.2269 0.4539 0.6808 0.9077 1.1347 0.1686 0.3372 0.5058 0.6744 0.8430 0. 0. 0. 0. 0. 55. Seventhly; If it be cut by a Plain A σ d, passing through A (Fig. 8.) perpendicular to the Rectangle ABDC, and cutting the Side BD in d, (and any of the Triangles RSa in ρσ) the Curve Line d σ A will have this property: As AC, or are, to a ρ; So is RS, to ρσ. And a ρ is thus found: As AB to Aa, or AD to A ρ; So is Bd to a ρ. 56. In Species, thus: Fig. 8. 57 But if this Plain (passing by A) cut the Line CD in any Point P before it come at d in the continuation of BD, (Fig. 9) the Curve Line will have this property: Cutting the same Plain (as before) by any of the Triangles RSa parallel to CQA, it will be, As CP to CR; or, As AP to A ρ; or, are (= AC) to a ρ; So RS to ρσ. 58. In Species, thus: Fig. 9 59 Or else, continuing AP till it cut BD (continued) in d, the Proportions will be as before at § 55, 56. 60. In this Section, the Curve Line d σ A cuts not its Axis at P (as in the fourth and fifth Section at b), but continues on the same side of it (above the Plain ABDC) till it meet with it at A; but (supposing the Solid to be farther continued in the opposite Position) cuts it at A, and thenceforth continues on the other side of it (below the Plain ABDC continued) till it meet again at δ. And (in case of such continuation) instead of CR = + c, we shall have CR = − c; (because now R falls on the contrary side of C, in the continuation of DC:) And consequently the Ordinates' beyond A (being on the contrary side) to be interpreted Negatively (with the sign—) as those on this side, Affirmatively, with the sign +. 61. In both cases (whether Add cut or cut not the Line CD as at P; that is, whether the Point d fall above or below D;) the Curve Lines dσA continued (answering to a suitable continuation of the Solid) cutting their Axis at A, will again meet with it (continued) at δ, as far beyond A as d is on this side of it: And the Ordinates' beyond A will be just the same as on this side, but with contrary signs + −. 62. And the Curve Lines dσAδ transferred to one and the same Plain, (supposing the Line BD divided into equal parts at the Points d) will appear as in the seventh Projection, Fig. 22. Where the Lines damn are continued to δ, and then (to avoid confusion in the Figure) removed from their proper place in the Plain ABDC, and all set off in the straight Line AC (continued), and the Lines δσ applied to them as Ordinates' (in this Position) in such proportion to RS, as a ρ is to are. 63. The Quantities in this Projection I thus design in Species. Fig. 22. 64. In Numbers (putting R = 1, A = 2,) the Semi-axes of the five Curves described, Ad, are of the same length with BC, § 54. I. Ad. II. Ad. III. Ad. IV. Ad. V. Ad. 1.11803 1.41421 1.80278 1.23607 2.69258 And the Ordinates', supposing the Semi-axis divided into four equal parts, are these: I. ρσ. II. ρσ. III. ρσ. IV. ρσ. V. ρσ. 0. 0. 0. 0. 0. 0.124 0.248 0.372 0.496 0.620 0.108 0.217 0.325 0.433 0.523 0.061 0.121 0.182 0.242 0.303 ±0. ±0. ±0. ±0. ±0. 303 − 0.061 − 0.121 − 0.182 − 0.242 − 0.523 − 0.108 − 0.217 − 0.325 − 0.433 − 0.620 − 0.124 − 0.248 − 0.372 − 0.496 − 0. − 0. − 0. − 0. Or (for a more accurate describing the Curve) dividing the Semi-axis into 16 parts (and the whole Axis into 32), the Ordinates' will be these: I. ρσ. II. ρσ. III. ρσ. IV. ρσ. V. ρσ. 0. 0. 0. 0. 0. 0.0816 0.1632 0.2447 0.3263 0.4079 0.1059 0.2118 0.3177 0.4236 0.5295 0.1184 0.2368 0.3552 0.4737 0.5921 0.1240 0.2480 0.3720 0.4960 0.6200 0.1250 0.2499 0.3749 0.4998 0.6248 0.1220 0.2440 0.3660 0.4880 0.6099 0.1163 0.2325 0.3488 0.4651 0.5814 0.1083 0.2165 0.3248 0.4330 0.5413 0.0984 0.1967 0.2951 0.3934 0.4918 0.0869 0.1738 0.2607 0.3476 0.4345 0.0742 0.1484 0.2226 0.2969 0.3711 0.0605 0.1210 0.1815 0.2421 0.3026 0.0460 0.0921 0.1383 0.1842 0.2302 0.0310 0.0620 0.0930 0.1240 0.1550 0.0156 0.0312 0.0468 0.0624 0.0780 ±0. ±0. ±0. ±0. ±0. − 0.0156 − 0.0312 − 0.0468 − 0.0624 − 0.0780 − 0.0310 − 0.0620 − 0.0930 − 0.1240 − 0.1550 − 0.0460 − 0.0921 − 0.1383 − 0.1842 − 0.2302 − 0.0605 − 0.1210 − 0.1815 − 0.2421 − 0.3026 − 0.0742 − 0.1484 − 0.2226 − 0.2969 − 0.3711 − 0.0869 − 0.1738 − 0.2607 − 0.3476 − 0.4345 − 0.0984 − 0.1967 − 0.2951 − 0.3934 − 0.4918 − 0.1083 − 0.2165 − 0.3248 − 0.4330 − 0.5413 − 0.1163 − 0.2325 − 0.3488 − 0.4651 − 0.5814 − 0.1220 − 0.2440 − 0.3660 − 0.4880 − 0.6099 − 0.1250 − 0.2499 − 0.3749 − 0.4998 − 0.6248 − 0.1240 − 0.2480 − 0.3720 − 0.4960 − 0.6200 − 0.1184 − 0.2368 − 0.3552 − 0.4737 − 0.5921 − 0.1059 − 0.2118 − 0.3177 − 0.4236 − 0.5295 − 0.0816 − 0.1632 − 0.2447 − 0.3263 − 0.4079 − 0. − 0. − 0. − 0. − 0. 65. There are many other Sections which may be made of the same Solid; but these being all that were proposed to be considered, I shall stay here. 66. But these four last mentioned (and divers others, though somewhat different from them) do all fall under one General, as so many Particulars of it: For the better consideration of which, I shall complete the Body (or at least the half of it) which is here but Quadrantal; and imagine it farther to be continued downward (below its Circular Base) so far as shall be necessary; and continued upward (after an intersection in the Line AB) in like manner, as opposite Cones are wont to be considered. 67. Supposing then (Fig. 10.) on the Centre C, and Diameter ΔD, a Circle described ΔSQD; and CQ perpendicular to the Diameter ΔCD, dividing the Semicircle ΔQD into two Quadrants; and (at Right Angles to the Plain of the Circle) a Rectangle ΔDBβ, divided into two equal parts by the straight Line CA; (and therefore, joining QA, the Triangle CQA will be at Right Angles to both the Plains:) And from every Point S in the Perimeter of the Circle, to the respective Points a in the Line Bβ (in Plains parallel to ACQ) the straight Lines Sa to be drawn, completing on either side of the Rectangle a Curve Superficies ΔSDBβ: These with the Circle contain a Solid, which I call a Cono-Cuneus, made up of four such quadrantal Solids as are above described at § 1. which Solid (and it's Opposite, made by a decussation in the Line Bβ) we suppose to be continued as far as is necessary. 68 If this Solid be cut by a Plain at Right-Angles, to the Rectangle ΔDBβ, the Section of that Plain, with this Rectangle, will be either parallel to BD, (and then the Section will be a Rightangled Triangle, as in our first Case, § 6.) or parallel to DΔ; (and then the Section will be an Ellipse, as in our second Case, § 13.) or at lest will obliquely cut the two opposite Sides βΔ, BD, (produced, if need be) in δ, d; which Line δd I call, the Diameter of the Curve Line, made by the Section of the Solid. 69. And, under this last case, fall the four last of these beforementioned, (the fourth, fifth, sixth, and seventh) as appears by the Scheme: Where Cd, CB, Cb, answer the fourth Case; Δb, Δc, ΔA, answer the fifth Case; Bc, BC, Br, answer the sixth case; and Aδ, AΔ, Are, answer the seventh Case. But the Curves answering these Diameters I have omitted, to avoid confusion in the Figure. 70. Now a Point being assigned (Fig. 11.) in any of the Diameters δd, the Ordinate, or perpendicular height of the Curve over that Point, is thus found Geometrically: By the Point assigned ρ, suppose aρR drawn parallel to BD, cutting βB in a, and ΔD in R; on which, suppose a Perpendicular Plain erected, cutting the Semicircle ΔSD in RS, and the Solid in a RSa Rightlined Triangle; wherein ρσ being drawn parallel to RS, will be Perpendicular to the Plain of the Rectangle, as is the Line RS: And therefore σ is that Point of the Curve-Super●icies which is over the Point ρ, (through which therefore the Curve-Line passeth, whose Axis pasteth through ρ:) And therefore ρσ is the Ordinate to that Point of the Axis or Diameter δd. 71. Therefore the Point ρ being given, a and R be known also; and consequently, RS the Ordinate or Right-Sine belonging to the Point R, in the Semicircle ΔSD: Then, because a RS, aρσ, are like Triangles, As are is to aρ, So is RS to ρσ. 72. Having thus found as many of the Points σ as shall be thought necessary, a Curve-Line regularly drawn by them is that Curve to which δd is the Axis. 73. The Arithmetical Calculation may be thus performed: Supposing ΔD to be divided into any number of equal parts at the Points R, the Lines Ra (produced if need be) will divide δd into the same number of equal parts at the Points ρ: And if, from d, be drawn de parallel to Δδ, cutting Δβ in e, and ρε from the Points ρ, the Parallels ρε will cut δε into the same number of equal parts. 74. Supposing then (for the more convenient calculation) ΔD = D, ΔR = d, δe = Ω, δε = ω, (and therefore δd = cradic;: D2+Ω 2, and δπ = √: d2+ω 2 βΔ = A (the Altitude of βB above the Base), and δβ = α (the Altitude of βB the Vertex of the Solid, above δ the lower Vertex of the Curve-Line); then is eβ = α − Ω, εβ = α − ω, the Altitudes of βB above e, ε, or d, ρ; (and therefore, when Ω, ω, happen to be greater than α, the Quantities α − Ω, α − ω, are Negatives; and consequently, e, ε, above the Vertex βB, in the opposite Solid) and RD = D − d; and therefore RS = √: D 2 − d2. (a mean Proportional between ΔR, RD; that is, between d, D − d.) 75. Then, because a RS, aρσ, are like Triangles, (whether ρ be higher or lower than βB;) As are, or βΔ, is to aρ, or βε; So is RS to ρσ: That is, As A to α − ω; So , to √: dD − d 2 Which is therefore the length of ρσ. 76. If therefore, in a Plain, a Line equal to δd, be divided at ρ into so many Parts as is supposed; and on every of the Points ρ, wherein it is so divided, be erected Perpendiculars equal to the Lines ρσ thus found respectively (observing still, that the Negative Quantities are to be applied on the contrary part of the Line thus drawn); the Curve-Line drawn by the Points σ in the Plain, agrees with that in the Solid made by the Section thereof: And may be therefore described without an actual cutting of the Solid, and may be fitted to any proportion of the Height to the Base of the Solid; and in whatsoever position the Diameter δd be supposed to cut the Rectangle ΔDBβ in that Solid. 77. This Calculation ●itted to the Circular Base ΔSD, may with the same case be applied to a Parabola, Hyperbola, or other Curve-Line whatever, whose Axis is ΔD, and Vertex Δ; if instead of √: dD − d 2, (which is here the Ordinate in the Circle) we put the Ordinate of that other Curve-Line: As , if a Parabola; , if an Hyperbola; (supposing D the Transverse Diameter, and L the Latus Rectum); and the like in other Curves. 78. And for this reason, I chose to design the Point R by its distance from Δ, rather than from C, forward and backward; and ρ by its distance from δ, rather than from c; which otherwise might as conveniently be done. 79. But if we should so design it, and put c = RC (and consequently ; and α = Ac, the Altitude at c; and ω the difference of Altitudes at ρ from that at c; then is a ρ = α − ω for the Points ρ in cd above the Point c, but a ρ = α + ω for those below it in c δ: And accordingly . 80. But if we put α = Bd, (the Altitude of B above d, the higher Vertex of the Curve, which therefore will be a Negative Quantity when B falls below d) and ω, the difference of Altitude from that at d, the process will be the same as before, save that then, instead of α − ω, we must put α + ω = α ρ (and consequently And the like if d were the Vertex of a Parabola, or Hyperbola, or other Curve, whose Axis d δ slopeth downward. 81. If it be desired rather to find Instrumentally, than by Calculation, the several Ordinates' ρ σ, to any Diameter Δ D, in any such Solid, and in any Position assigned; it may be very easily performed in this manner. 82. First; Let any straight Line, at pleasure, LM (Fig. 12.) be divided at the Points ρ, into any number of unequal parts, as a Line of Ordinates' at equal distances in the Quadrant of a Circle; (in like manner, as the Line CQ is divided at the Points S in the first Projection, Fig. 16.) and on the other side of M, let λ M be so divided also into the same number of Parts; and on the several Points ρ, M, erect Perpendiculars, continued both ways as far as shall be needful. Which general Construction is applicable to any case at pleasure; and being once drawn, may successively be applied to many. 83. Then (supposing, in the Solid proposed, Fig. 11. δ E, parallel to Δ D, cutting AC in E; and Eq, parallel to CQ, cutting AQ in q;) set off, in the Perpendicular at M, Fig. 12. a Line ME equal to that Eq in the Solid, and draw the straight Lines E λ, EL; which Lines will cut off, in the other Parallels, the Lines ρ S, equal to the Ordinates' of that Ellipse in the Solid, (Fig. 11.) whose Axis is δ E; (●uch as are R σ in the second Projection, Fig. 17.) 84. In like manner, (supposing, in the Solid, Fig. 11. de parallel to D Δ, cutting CA, or the continuation thereof, in η; and η q, parallel to CQ, cutting QA, or the continuation thereof, in q;) set off in the Perpendicular at M, (Fig. 12, 13.) a Line M η equal to η q; (on the same side of λ ML with ME, or on the contrary, according as d and δ are on the same, or opposite Sides B β;) and draw the straight Lines η λ, η L; which Lines will cut off, in the other Parallels, the Lines ρ s, equal to the Ordinates' of that Ellipse in the Solid, whose Axis is η d. 85. Then (dividing E η into as many equal parts at the Points ω, as are the unequal parts in λ L) from every of the Points ω, draw straight Lines to λ or L respectively; which Lines will cut off, in the respective Parallels, ρ S, (the first in the first, the second in the second, etc. numbering the Points ω from E, and the Parallels from λ;) the Lines ρ σ, equal to the desired Ordinates' of the Curve proposed. 86. Last; Drawing a straight Line δ d (Fig. 14, 15.) equal to that in the Solid (the Diameter of the Curve proposed); and dividing it in the Points ρ into as many equal parts, as are the unequal parts in λ L; and to each Point of Division, applying at Right Angles the Lines ρ σ, equal to those upon the Line λ L (on the same or contrary sides of δ d, as those are of λ L); and, by the Points σ, drawing the Curve-Line which they direct: This Curve-Line is the same with that which is made by the Section of the Solid proposed, by a Plain on the Line δ d at Right-Angles to the Rectangle Δ DB β. 87. The same may be performed by one of the Triangles λ ME, Fig. 13. reckoning the Parallels therein twice over, (the Ordinates' in each Quadrant being the same) and dividing E η into as many parts as before. 88 If the Base Δ SD be an Ellipse, this process will be the same as in a Circle; but if it be a Parabola, Hyperbola, or other the like Curve, the Line λ L, which is now divided as a Line of Ordinates' at equal distances in a Circle, must then be divided as such a Line of Ordinates' of that Parabola, Hyperbola, or other Curve, whose Axis is Δ D: And then the rest of the Operation pursued with very little alteration. 89. In the whole Progress, I have still supposed the Parallelogram ABDC to be Rectangular, and the Quadrant CDQ at Right-Angles with that Plain, and the Triangles ACQ, a RS, at Right-Angles to both of them (and consequently, the Body to be Erect, not Scalene); and the Plain cutting this Body, to be also at Right-Angles with that Parallelogram. But in case any of what we suppose to be Rectangular, should be Oblique, the Sections will be somewhat different from these described, in like manner, as the Sections of Scalene Cones, or the Oblique Sections of Erect Cones, differ from the Right Sections of Right Cones. But of these cases, I intent not here to discourse farther, contenting myself with the Perpendicular Sections of these Erect Solids. FINIS. Fig. X. Fig. XI. Fig. XII. Fig. XIII. Fig. XIV. Fig. XV. Fig. XVI. Project. 1.a a Fig. XVII. Project. 2. Fig. XVIII. Project. 3. Fig. XIX. Project. 4.a a Fig. XX. Project. 5.a a Fig. XXI. Project. 6. Fig. XXII. Project. 7.a a