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 32. Theoreme. The 43. Proposition. In euery parallelograme, the supplementes of those parallelo∣grammes which are about the diameter, are equall the one to the other.

SVppose that ABCD be a parallelograme, and let the diameter ther∣of be AC: and about the diameter AC let these parallelogrames EH and GF consist: and let the supplementes be BK and KD. Then I say that the supplement BK is equall to the supplement KD. For forasmuch as ABCD is a parallelograme

[illustration]

and the diameter therof is AC, therfore (by the 34. proposition) the triangle ABC is equall to the triangle ADC. A∣gayne forasmuch as AEKH is a pa∣rallelograme, and the diameter therof is AK, therfore (by the same) the trian∣gle AEK is equall to the triangle AHK. And by the same reason also the tri∣angle KFC is equall to the triangle KGC. And forasmuch as the triangle AEK is equall to the triangle AHK, and the triangle KFC to the triangle NGC, therfore the triangles AEK and KGC are equall to the triangles AHK and KFC: and the whole triangle ABC is equall to the whole triangle AD

C: wherfore the residue, namely, the supplement BK is (by the 3. common sen∣tence) equall to the residue, namely, to the supplement KD. VVherefore in e∣uery parallelogramme, the supplementes of those parallelogrammes whiche are about the diameter, are equall the one to the other: whiche was required to be proued.

* 1.1Those parallelogrames are sayde to consist about a diameter which haue to their diameters part of the diameter of the whole and great parallelograme, as in the example of Euclide. And such parallelo∣grames

[illustration]

which haue not to their diameters part of the diameter of the greater parallelograme, are sayde not to consist about the diameter. For thē the diameter of the greater parallelograme cutteth the side of the lesse cōtayned wythin it. As in the parallelogramme AB, the diameter CD, cutteth the side EH of the parallelogramme CE. Wherefore the parallelogramme CE is not about one and the selfe same diameter with the parallelo∣gramme CD.

* 1.2Supplementes or Complementes are those figures which with the two pa∣rallelogrammes accomplish the whole parallelogramme. Although Pel••tarius for distinction sake putteth a difference betwene Supplementes and Comple∣mente••. The parallelogrammes about the diameter he calleth Complementes, the other ••wo figures he calleth Supplementes.

* 1.3This theoreme hath three cases onely. For the parallelogrammes which consist about the diameter, eyther touch the one the other in a point: or by a cer¦tayne parte of the diameter are seuered the one from the other: or els they cutte the one the other.* 1.4 For the first case is the example of Euclide before set. The se∣cond case is thus.

* 1.5Suppose that AB be a parallelograme••

[illustration]

whose diameter let be CD: and aboute the same diameter let these parallelogrammes CK and DL consist: and betwene thē let there be a certayne part of the diameter, namely, LK. Then I say that the two supplementes AGLKE & BFKLH are equall. For we may as before (by the 34. proposition) proue that the triangle ACD, is equall to the triangle BCD, and the triangle ECK to the triangle KCF, and also the triangle DGL to the tri∣angle DHL. Wherfore the residue, namely, the fiue sided figure AGLKE is equall to the residue, namely, to the fiue sided figure, BFKLH: that is, the one supplement to the other•• which was required to be proued.

* 1.6But now suppose AB to be a parallelogramme, and let the diameter thereof be CD: and let the one of the parallelogrammes about it be ECFL, and let the other be

DGKH, of which let the one cut the other.

[illustration]

Then I say that the supplementes EG and EH are equall. For forasmuch as the whole trian∣gle DGK is equal to the whole triangle DHK (by the 34. proposition), and part also of the one, namely, the triangle KLM is equall to part of the other, namely, to the triangle KLN (by the same), for LK is a parallelograme: therefore the residue, namely, the Trapesium DLNH is equall to the residue, namely, to the trapesiū DLMG: but the triangle ADC is equal to the triangle BCD, and in the pa∣rallelograme EF, the triangle FCL, is equall to the triangle ECL, and the trapesium DGML is (as it hath bene proued) equall to the trapesium DHNL. Wherefore the residue, namely, the quadrilater figure GF is equall to the residue, namely, to the quadrilater figure EH, that is, the one supplement to the other: which was required to be proued.

This is to be noted that in ••••h of those three cases it may so happen, that the parallelogrammes aboute the diameter shall not haue one angle common wyth the whole parallelogramme, as they haue in the former figures. But yet though they haue not, the selfe same demonstration wil serue, as it is playne to see in the figures here vnderneath put. For alwayes, if from thinges equall be taken away thinges equall, the residue shalbe equall.

[illustration]

This proposition P••litarius calleth Gnomicall, and misticall,* 1.7 for that of it (sayth he) spring infinite demonstrations, and vses in geometry. And he putteth the conuerse thereof after this manner.

If a parallelogramme be deuided into two equall supplementes,* 1.8 and into two complements what∣soeuer: the diameter of the two complementes shall be set directly, and make one diameter of the whole parallelogramme.

Here is to be noted as I before admonished that Pelitarius for distinction sake putteth a difference betwene supplementes and complementes, which diffe∣rence, for that I haue before declared, I shall not neede here to repete agayne.

Suppose that there be a parallelogramme ABCD, whose two equall supplements let be AEFG and FHDK, and let the two complementes thereof be GFCK and EBFH: whose diameters let be CF and FB. Then I say that CFB is one right line, and is the diameter of the whole parallelogramme ABCD: for if it be not, then is there an

other diameter of the whole parrallelogramme, which let

[illustration]

be CLB being drawen vnder the diameters CF and FB, and cutting the line GH in the in the point L. And (by the 31. proposition) by the point L, draw vnto the line AC a parallel line MLN. And so are there in the whole paral∣lelogramme ABCD two supplements AMGL and LHND, which by this proposition shalbe equall the one to the other. For that they are about the diameter CLB. But the supplement AEFG is (by supposition) equall to the supplement FHDK: and forasmuch as FHDK is greater then LHDN, AEFG also shalbe greater then AMGL, namely, the part greater then the whole: which is impossible. And by the same reasō may it be proued, that the diameter cannot be drawen aboue the diameters CF and FB. Wherefore CFB is one diameter of the whole parallelogramme ABCD: which was required to be proued.

Notes

• * 1.1

  How parallelo∣grammes are sayde to consiste about a dia∣meter.
• * 1.2

  Supplements & Complementes.
• * 1.3

  Three cases in this Theoreme.
• * 1.4

  The first case.
• * 1.5

  The second case.
• * 1.6

  The third case.
• * 1.7

  This proposition called Gnomical and mistical.
• * 1.8

  The conuerse of this proposition.