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 4. Probleme. The 26. Proposition. Vpon a right lyne geuen, and at a point in it geuen, to make a solide angle equall to a solide angle geuen.

SVppose that the right line geuen be AB, and let the point in it geuen be A, and let the solide or bodily angle geuen be D being contained vnder these superfici∣all angles EDC, EDF and FDC. It is required vpon the right line AB, & at the point in it geuen A to make a solide angle equall to the solide angle D. Take in the line DF a point at all aduentures, and let the same be F.* 1.1 And (by the 11. of the eleuenth) frō the point F. Draw vnto the superficies wherin are the lines ED & DC a per∣pendicular line FG, and let it fall vpon the plaine superficies in the point G, & draw a right line from the point D to the point G. And (by the 23. of the first) vnto the line AB, and at the point A make vnto the angle EDC an equall angle BAL, and vnto the angle EDG put the angle BAK equall: and by the 2. of the first, put the line AK equall to the line DG, and (by the 12. of the eleuenth) from the point K raise vp vnto the plaine superficies BAL a perpendicular line KH, and put the line KH equall to the line GF, and draw a right lyne

from the point H to the point A. Now I say that the so••ide angle A contained vnder the su∣perficiall angles BAL, BAH, and HAL is equall to the solide angle D contained vnder the superficiall angles EDCEDF, and FD••. Le•• the the li••es AB and DE be put e∣quall, and draw these right lines. HB, BK, FE, and EG. And forasmuch as the line FG is erected perpendicularly to the ground superficies,* 1.2 therfore by the 2. definition of the eleuenth, the lin•• FG is also erected perpendicularly to all the right lines that are in the ground super∣ficies and touche it. Wherfore either of these angles FGD and FGE is a right angle, and by the same reason also either of the angles HKA and HKB is a right angle. And foras∣much as these two lines KA & AB are equall to these two lines GD & DE, the one to the other, and they containe equall

[illustration]

angles (by construction). Wher¦fore (by the 4. of the first) the base KB is equall to the base EG, and the line KH is equall to the line GF, and they cōprehēd right angles. Wherfore the line BH is equall to the line FE. Agayne, forasmuch as these two lines AK and KH are equal to these two lines DG and GF, and they containe right angles. Wherfore y^e base AH is (by the 4. of the first) equall to the base DF. And the line AB is equall to the line DE. Wherfore these two lines AB and AH are equall to these two lines FD and DE, and the base BH is equall to the base FE. Wherfore (by the 8. of the first) the angle BAH is equall to the angle EDF. And by the same reason also the angle HKL is equall to the angle FGC. Wherfore if we put these lines AL and DC equall, and draw these right lines KL, HL, GC, and FC: for∣asmuch as the whole angle BAL is equall to the whole angle EDC, of which the angle BAK is supposed to be equall to the angle EDG, therfore the angle remayning, namely, KAL is equall to the angle remayning GDC. And forasmuch as these two lines KA and AL are equall to these two lines GD and DC, and they containe equall angles, therefore by the 4. of the first, the base KL is equall to the base GC, and the line KH is equall to the line GF, wherfore thes•• ••wo lines LK and KH are equall to these two lines CG and GF, and they cō∣taine right angles. Wherfore the base HL is (by the 4. of the first) equal to the base FC. And forasmuch as these two lines HA and AL are equall to these two FD and DC, and the base HL is equall to the base FC, therfore (by the 8. of the first) the angle HAL is equall to the angle FDC, and by construction, the angle BAL is equall to the angle EDC. Wherefore vnto the right line geuen, and at the point in it geuen, namely, A, is made a solide angle equal to the solide angle geuen D: which was required to be done.

In thes•• two 〈…〉〈…〉

[illustration]

here put, you may in 〈◊〉〈◊〉 clearely concerne the ••••••••mer construction and d••••monstratiō, if ye erect pe••••pendicularly vnto the ground superficies the tri∣angles ALB and DCE, & eleuate the triangles ALH and DCF that the lynes

LA and CD of them may exactly agree with the line•• LA and CD of the ••riangles erec••ed•• For so or∣dering them, if ye compare the former construction and demonstration with them, they will be playn•• vnto you.

Although Euclides demōstration be touching solide angles which are contained only vnder three superficiall angles, that is, whose bases are triangles: yet by it may ye performe the Probleme touching solide angles contained vnder superficiall angles how many soeuer, that is, hauing to their bases any kinde of Poligonon figures. For euery Poligonon figure may by the 20. of the sixt, be resolued into like tringles. And so also shall the solide angle be deuided into so many solide angles as there be triangles in the base. Vnto euery one of which solide angles you may by this proposition describe 〈◊〉〈◊〉 equall solide angle vpon a line geuen, and at a point in it geuen. And so at the length the whole solide angle after this maner described shall be equall to the solide angle geuen.

Notes

• * 1.1

  Co••struc••ion.
• * 1.2

  Demonstra∣••ion.