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

The 19. Theoreme. The 21. Proposition. Euery solide angle is comprehended vnder playne angles lesse then fower right angles.

SVppose that A be a solide angle contayned vnder these superficiall angles BAC, DAC and DAB. Then I say that the angles BAC, DAC and DAB are lesse then fower right angles.* 1.1 Take in euery one of these right lines ACAB and

AD a poynt at all aduentures and let the same be B, C, D. And draw these right lines BC, CD and DB.* 1.2 And forasmuch as the angle B is a solide angle, for it is contayned vnder three superficiall angles, that is, vnder CBA, ABD and CBD, therefore (by the 20. of the eleuenth) two of them which two so euer be taken are greater then the third. Wherefore the angles CBA and ABD are greater

[illustration]

then the angle CBD: and by the same rea∣son the angles BCA and ACD are grea∣ter then the angle BCD•• and moreouer the angles CDA and ADB are greater then the angle CDB. Wherefore these sixe angles CBA, ABD, BCA, ACD, CDA, and ADB are greater thē these thre angles, namely, CBD, BCD, & CDB. But the three angles CBD, BDC, and BCD are equall to two right angles. Wherefore the sixe angles CBA, ABD, BCA, ACD, CDA, and ADB are greater thē two right an∣gles. And forasmuch as in euery one of these triangles ABC, and ABD and ACD three angles are equall two right angles (by the 32. of the first). Wherefore the nine angles of the thre triangles, that is, the angles CBA, ACB, BAC, ACD, DAC, CDA, ADB, DBA and BAD are equall to sixe right angles. Of which angles the sixe angles ABC, BCA, ACD, CDA, ADB and DBA are greater then two right angles. Wherefore the angles remayning, namely, the angles BAC, CAD and DAB which contayne the solide angle are lesse then sower right angles. Wherefore euery solide angle is comprehended vnder playne angles lesse then fower right angles: which was required to be proued.

If ye will more fully see this demonstration compare it with the figure which I put for the better sight of the demonstration of the proposition next going before. Onely here is not required the draught of the line AE.

Although this demonstration of Euclide be here put for solide angles contayned vnder three super∣ficiall angles, yet after the like maner may you proceede if the solide angle be contayned vnder superfi∣ciall angles how many so euer. As for example if it be contayned vnder fower superficiall angles, if ye follow the former construction, the base will be a quadrangled figure, whose fower angles are equall to fower right angles: but the 8. angles at the bases of the 4. triangles set vpon this quadrangled figure may by the 20. proposition of this booke be proued to be greater then those 4. angles of the quadrangled fi∣gure: As we sawe by the discourse of the former demonstration. Wherefore those 8. angles are greater then fower right angles: but the 12. angles of those fower triangles are equall to 8. right angles. Where∣fore the fower angles remayning at the toppe which make the solide angle are lesse then fower right angles. And obseruing this course ye may proceede infinitely.

Notes

• * 1.1

  Construction
• * 1.2

  Demonstra∣••ion.