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

An other conuerse of the same proposition.

If a parallelogramme be the double of a triangle, being both within the selfe same parallel lines:* 1.1 then are they vpon one and the selfe same base, or vpon equall bases. For if in that case their ba∣ses should be vnequal, then might straight way be proued, that the whole is e∣quall to his part: which is impossible.

A trapesium hauing two sides onely parallel lines,* 1.2 is eyther more then dou∣ble, or lesse then double to a triangle contayned within the selfe same parallel lines, and hauing one and the selfe same base with the trapesium, or table. ••ust the double it cannot be, for then it should be a parallelogramme. A trapesium ha∣uing two sides parallels hath of necessitie the one of them longer then the other: for if they were equall then should the other two sides enclosing them be also e∣quall (by the 33. proposition.) If the greater side of the trapesium be the base of the triangle, then shal the trapesium be lesse then the double of the triangle And if the lesse side of the trapesium be the base of the triangle then shall the trapesi∣um be greater then the triangle.

For suppose that ABCD be a trapesium,* 1.3 and let

[illustration]

two sides thereof, namely, AB and CD be parallel lines, and let the side AB be lesse then the side CD, & produce the side AB infinitlye on the side B to the point F. And let the triangle ECD haue one and the selfe same base with the trapesium, namely, the line CD. Then I say that the trapesium ABCD is lesse thē the double of the triangle ECD. For put the line AF equall to the line CD (by the 3. propositiō) and draw a line from D to F. Wherefore ACDF is a parallelo∣gramme (by the 33. proposition). Wherefore (by the 34 proposition) it is double to the triangle ECD. But the trapesium ABCD is a part of the parallelo∣gramme ACDF. Wherefore the trapesium ABCD is lesse then the double of the triangle ECD: which was requi••ed to be proued.

Agayne let the triangle haue to his base the side

[illustration]

AB.* 1.4 Then I say that the trapesium ABCD is grea∣ter then the double of the triangle AEB. For from the side CD cut of the line CF equall to the line AB (by the ••. proposition). And draw a line from B to F. Wherfore (by the 33. proposition) ABCF is a parallelogramme: and therefore is (by the 34. pro∣position) double to the triangle AEB. Where∣fore the trapesium ABCD is more then the dou∣ble of the triangle AEB•• which was required to be proued.

Notes

• * 1.1

  An other con∣uerse of the same propositiō.
• * 1.2

  Comparison of a triangle and a trapesium being vpon one & the selfe same base, and in the selfe same parallel lines.
• * 1.3

  When the grea∣ter side of the trapesium is the base of the tri∣angle.
• * 1.4

  When the lesse side is the base.