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 21. Theoreme. The 24. Proposition. If a solide or body be contayned vnder* 1.1 sixe parallel playne superficieces, the opposite plaine superficieces of the same bo∣dy are equall and parallelogrammes.

SVppose that this solide body CDHG be contained vnder these 6. parallel plaine superficieces, namely, AC, GF, BG, CE, FB, and AE. Then I say that the opposite superficieces of the same body, are equal and parallelogrāmes, it is to wete, the two opposites AC and GF, and the two opposites BG

[illustration]

and CE, and the two opposites FB and AE to be equall, and al to be parallelogrammes.* 1.2 For forasmuch as two pa∣rallel plaine superficieces, that is, BG, and CE are deui∣ded by the plaine superficies AC, their common sections are (by the 16. of the eleuenth) parallels. Wherfore the line AB is a parallel to the line CD. Again•• forasmuch as two parallel plaine superficieces FB and AE are deui∣ded by the plaine superficies AC their common sections are by the same proposition, parallels. Wherfore the lyne AD is a parallel to the line BC. And it is also proued, that the line AB is a parallel to the line DC. Wherfore the superficies AC is a parallelogramme. In like sort also may we proue, that euery one of these superficices CE, GF, BG, FB, and AE are parallelo∣grammes. Draw a right line from the point A, to the point H, and an other from the point D to the point F.* 1.3 Aud forasmuch as the line AB is proued a parallel to the line CD, and the lyne BH to the line CF, therfore these two right lines AB and BH touching the one the other, are parallels to these two right lines DC and CF touching also the one the other, and not being in one and the selfe same plaine superficies. Wherfore (by the 10. of the eleuenth) they compre∣hend equall angles. Wherfore the angle ABH is equall to the angle DCF. And forasmuch as these two lines AB and BH are* 1.4 equall to these two lines DC and CF, and the angle ABH is proued equall to the angle DCF•• therfore (by the 4. of the first) the base AH is equall to the base DF, and the triangle ABH is e∣quall to the triangle DCF. And forasmuch as (by the 41. of the first) the parallelogramme BG is double to the triangle ABH, and the parallelo∣gramme CE is also double to the triangle DCF, therfore the parallelo∣gramme

BG is equall to the parallelogramme CE. In like sort

[illustration]

also may we proue that the parallelogramme AC is equall to the parallelogramme GF, and the parallelograme AE to the parallelogramme FB. If therfore a solide or body be contained vnder sixe parallel plaine superficieces, the opposite plaine super¦ficieces of the same body are equal & parallelogrammes which was required to be demonstrated.

I haue for the better helpe of young beginners, described here an other figure whose forme if it be described vpon pa∣sted paper with the letters placed in the same order that it is here, and then if ye cut finely these lines AG, DE and CF not through the paper, and folde it accordingly, compare it with the demonstration, and it will shake of all hardenes from it.

Notes

• * 1.1

  M. Dee (to auoide cauillation) addeth to Euclides proposition this worde sixe: whome I haue follow∣ed accordingly, and not Zamberts, in this.

  This kinde of body mencioned in the proposition is called a Paral∣lelipipedō according to the di••fini∣tion before geuen thereof.
• * 1.2

  Demonstra∣tion that the opposite sides are parallelo∣grammes.
• * 1.3

  Demonstratiō that the oppo∣site superficies are equall.
• * 1.4

  AB is equall to DC, because the superficies AC is proued a pa∣rallelogrāme, and by the same rea∣son, is BH equall to CF, because the superficies FB is proued a pa∣rallelogramme: therefore the 34. of the first is our proofe.