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

* 1.1Like segmentes or sections of a circle are those, which haue equall angles, or in whom are equall angles.

* 1.2Here are set two definitions of like sections of

[illustration]

a circle. The one pertaineth to the angles whiche are set in the centre of the circle and receaue the circumferēce of the sayd sections:* 1.3 the other per∣taineth to the angle in the section, whiche as be∣fore was sayd is euer in the circumference. As if the angle BAC, beyng in the centre A and re∣ceaued of the circumference BLC be equall to the angle FEG beyng also in the centre E and receaued of the circumference FKG, then are the two sections BCL and FGK lyke by the first definition. By the same definition also are the other two sections like, name∣ly BCD, and FGH, for that the angle BAC is equall to the angle FEG.

* 1.4Also by the second definition if B

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AC beyng an angle placed in the cir∣cumference of the section BCA be e-angle EDF beyng an angle in the se∣ction EFD placed in the circumfe∣rence, there are the two sections BCA, and EFD lyke the one to the o∣ther. Likewise also if the angle BGC beyng in the section BCG be equall to the angle EHF beyng in the sectiō EHF the two sections BCG and EFH are lyke. And so is it of angles beyng equall in any poynt of the circumference.

Euclide defineth not equall Sections:* 1.5 for they may infinite wayes be described. For there may vppon vnequall right lynes be set equall Sections (but yet in vne∣quall circles) For from any circle beyng the greater, may be cut of a portion equall to a portion of an other circle beyng the lesse. But when the Sections are equall, and are set vpon equall right lynes, theyr circumferences also shalbe equall. And right lynes beyng deuided into two equall partes, perpendicular lynes drawen from the poyntes of the diuision to the cir∣cumferēces

[illustration]

shalbe equall. As if the two secti∣ons ABC and DEF, beyng set vppon equall ryght lynes AC & DF, be equall: then if ech of the two lynes AC & DF be deuided into two e∣quall partes in the poyntes G and H, & from the sayd poyntes be drawen to the circumferences two perpendicular lynes BG and EH, the sayd perpendicular lynes shalbe equall.

Notes

• * 1.1

  Tenth defini∣tion.
• * 1.2

  Two defini∣tions.
• * 1.3

  First.
• * 1.4

  Second.
• * 1.5

  Why Euclide defineth not equall Secti∣ons.