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 23. Theoreme. The 28. Proposition. If a parallelipipedō be cutte by a plaine superficies drawne by the diagonall lines of those playne superficieces which are opposite: that solide is by this playne superficies cutte into two equall partes.

SVppose that the parallelipipedon

[illustration]

AB be cutte by the playne super∣ficies CDEF drawne by the dia∣gonal lines of y^e plaine superficieces which are opposite, namely, of the superficieces CF and DE. Then I say that the parallelipi∣pedon AB is cutte into two equall partes by the superficies CDEF. For forasmuch as (by the 34. of the first) the triangle CGF is equall to the triangle CBF,* 1.1 and the triangle ADE to the triangle DEH, and the parallelograme CA is equall to the parallelogramme BE, for they are opposite, and the parallelogramme GE is also equall to the parallelogramme CH, and the parallelogramme CE, is the common section by supposition: Wherfore the prisme contai∣ned vnder the two triangles CGF, and DAE, and vnder the three parallelogrammes GE, AC, and CE is (by the 8. definition of the eleuenth) equall to the prisme contayned vnder the two triangles CFB and DEH and vnder the three parallelogrammes CH, BE, and CE. For they are cōtayned vnder playne superficieces equall both in multitude and in magnitude. Wherefore the whole parallelipipedon AB is cutte into two equall partes by the playne superficies CDEF: which was required to be demonstrated.

A diagonall line is a right line which in angular figures is drawne from one angle and extended to his contrary angle as you see in the figure AB.

Describe for the better sight of this demonstration a figure vpon pasted

[illustration]

paper like vnto that which you described for the 24. proposition onely altering the letters: namely, in steade of the letter A put the letter F, and in steade of the letter H the letter C: moreouer in steade of the letter C put the letter H, and the letter E for the letter D, and the letter A for the letter E, and finally put the letter D for the letter F. And your figure thus ordered compare it with the de∣monstratiō, only imagining a superficies to passe through the body by the dia∣gonall lines CF and DE.

Notes

• * 1.1

  Demonstra∣tion.