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Sed Sc hoc amplius adnotare non pigebit, Superficies Cylindrorum duorumperforantium intra Sphseram, aequales esse Superficiei Sphaerae post perforatio-nem relictse, sive duplici Velo Florentino, hoc est duplo quadrato Diametri.

Atque hoc exinde patet quod Velum Florentinum aequale lit Figuris quatuorsinuum rectorum Quadrantis & Superficies perforans iisdem etiam iit aequalis,quoniam illis congruit si inflectio fiat ut supra.

Hoc tantum addam, Considerationem Figura Sinuum rectorum (cujus eti-am partes in Quadrata facile mutantur) sufficere ad Demonstrationem eo-rum omnium quae de aliis solidis Torno elaboratis vel Cylindro perforatis, eo-rumque Superficiobus ab Acuti firmo Geometra V. V. (Vineentio Vi viam nifallor) Dignifiimo Galilaei Discipulo proferuntur; dum Fabricam & Menfii-fam Testudinum docet. Spcciatim Superficies Testudinis Scaphoidis Romana:

[ Volta a Schifo alia Romana ] ex octo Figuris Sinuum rectorum Arcus Qua-drantalis constat, ac proinde Testudini Veuformi Florentinae aequalis est. Undepatet quomodo aequalibus quadratis superimponi possunt duae Testudines,quarum altera est undique clausit, altera quatuor Fenestris interrupta, utraqueQuadrati Baseos dupla.

VII. X. Dramng the ßreight Lines 'E A , and EB (cutting the Are AB in G.) The Quiirjtu;*and on A G, a Perpendicular E F, (vthich wiä eherefexe fitjs to rhe Center G Luntja| fcyAfr.bccause BiscBing A G at Right Angle s j) ehe Right-lined Trianglc AT E, is J.Perks, a httleequa! to A D E, ehe proposed Portion es ehe Lunula. Walks^n^?"

/>. 411.

The Demonstration is to this purpole; vi^. ADB being a Quadrantal k«- rx-Arc; the Aogle A G B will be three Hälves of a Right Angle ; (sind its con-junct Angle EGA, half a Right Angle- and that Angle (bfing externa! tothe Triangle AGE, is equal to the two opposite Intervals GEA -f EAG.

Whereof GEA (becausc an Angle in the Semicircle A E B) is a Right Angle,and therefore E A G is half a Right Angle, (as are also FEG, and F E A)and the three Triangles A F E, G F E, and GE A, each of them halt aSquare.

And AG to A E, as V % to r, ('proportional to the Reipective Radii of thetwo Circles). And the like Segments A D G, A E, in their reipective Circles(as the Squares of their reipective Radii) as i to 1. And therefore the Semi-(egment A F D, equal to the Segment A E. And confequently (one takingfrom the Triangle, as-much as the other adds toit) theportioii of the LunulaA DE, equal to the Triangle A F E. E. D.

If the Point E chance to be in K (the middle of the Arc A E B) therewillbe no Interlection at G (the points G, B being then eoincident, but withoutany disturbance to the Demonstration :) If it happen beyond it, to.ward ;B ;then G will be on the other side; and \Vhar is here sitid os EGB, must beaccommodated to E G A.

The Ground os the whole Procris is plairtlytbis; The Angle ACE, be-ing an Angle at the Center of the greater Circle, but at the Ctrcumfevence ofthe lesser, the Line CDE (as -it pasieth from C<A to C B) doth in the fameproportion, divide the Quadrantal Arc ADB, and the Semicivcutu* AE B:whcncc ali the rest doth naturally follow. .

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