The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

¶ A Theoreme. 4.

The superficies of the segment or protion of any sphere, is equall to the circle, whose semidiameter, is equall to that right line which is drawne from the toppe of that segment to the circumference of the circle, which is the base of that portion or segment.

As in the Sphere

[illustration]

A, a Segment being cut of by the circle, whose diameter is CE: & the same circle being the base of the Segment, whose top also is D: the croked superficies sphe∣ricall of the same Seg∣ment, is equall to a cir∣cle whose Semidiame∣ter is equall to the right line DC. As is the cir∣cle B.

This hath Archimedes demonstrated in this first booke of the Sphere and Cylinder, in his 40. and 41 propositions: and I remitte them thether, that will herein demonstratiuely be certified: I would wish all Mathematiciens, as well of verities easy, as of verities rare and obscure, to seeke the causes demon∣stratiue, the finall fruite thereof, is perfection in this art.

Note.

Besides all other vses and commodities, that are of the Croked superficieces of the Cone, Cylin∣der, and Sphere, so easely and certaynely, of vs to be dealt with all: this is not the least, that a notable Er∣ror, which among Sophisticall brablers, and vngeometricall Masters and Doctors, hath a long time bene vpholden: may most euidently, hereby be confuted, and vtterly rooted out of all mens fantasies for e∣uer. The Error is this, Curui, ad rectum, nulla est proportio, that is:* 1.1 Betwene croked and straight, is no pro∣portion•• This error, in line••, sup••••ficieces, and solides, may with more true demonstrations be ouer∣throwne, then the fauourers of that fond fa••tasie are able, with argument, either probable or Sophi∣sticall to make shew or pretence to the contrary. In lines, I omitte (as now) Archimedes two wayes, for the finding of the proportion of the circles circumference to a straight line. I meane, by the inscription and ci••cumscription of like poligonon figures, and that other, by spirall lines. And I omitte likewise (as now) in solides, of a parallelipipedon, equall to a Sphere, Cone, or Cylinder•• or any segment or sector of the sayd solides. And onely, here require you to consider in this twelfth booke, the wayes brought to your knowledge, how to the croked superficies of a cone and cylinder, and of a sphere,* 1.2 (the whole, any segment or sector thereof) a playne and straight superficies may be geuen equall:

Namely, a Circle to be geuen equall, to any of the sayd croked superficieces assigned, and geuen. And then farther by my Additions vpon the second proposition, you haue meanes to proceede in all proportions, that any man can in right lines geue•• or assigne. The••fore, Curui ad rectum, proportio omnimoda potest dari. One thing it is, to demonstrate, that betwene a croked line and a straight or a croked superficies and a playne or straight superficies, &c. there is proportion. And an other thing it is, to demonstrate a particular and speciall kinde of proportion, being betwene a croked superficies and a straight or playne superficies. For this al∣so confi••meth the first.

This short warning will cause you to auoyde the sayd error, and make you also hable to cure them, which are infected therewith.

Notes

• * 1.1

  A great error commonly maintained.
• * 1.2

  Betwene straight and croked all maner of propor∣tiō may be geuen.