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

Definitions.

A rectiline figure is sayd to be inscribed in a rectiline figure,* 1.1 when euery one of the angles of the inscribed figure toucheth euery one of the sides of the figure wherin it is inscribed.

As the triangle ABC is inscribed in the triangle DEF, because that euery angle of the triangle inscribed, namely, the triangle ABC toucheth euery side of the triangle within which it is described, namely, of the triangle DEF. As the angle CAB toucheth

[illustration]

the side ED the angle ABC toucheth the side DF, and the angle ACB toucheth the side EF. So likewise the square ABCD is said to be inscribed within the square EFGH. for euery angle of it toucheth some one side of the other. So also the Pentagon or fiue angled figure ABCDE is inscribed within the Pentagon or fiue angled figure FGHIK•• As you see in the figure••.

Likewise a rectiline figure is said to be circumscribed about a rectiline figure,* 1.2 when euery one of the sides of the figure cir∣cumscribed, toucheth euery one of the angles of the figure a∣bout which it is circumscribed.

As in the former descriptions the triangle DEF is said to be circumscribed about the triangle ABC, for that euery side of the figure circumscribed, namely, of the trian∣gle DEF toucheth euery angle of the figure wherabout it is circumscribed. As the side DF of the triangle DEF circumscribed, toucheth the angle ABC of the triangle ABC about which it is circumscribed: and the side EF toucheth the angle BCA, and the side CD toucheth the angle CAB. Likewise vnderstand you of the square EFGH which is circumscribed about the square ABCD: for euery side of the one toucheth some one side of the other. Euē so by the same reason the Pentagon FGHIK is circumscribed a∣bout the Pentagon ABCDE, as you see in the figure on the other side. And thus may you of other ••ectiline figures consider.

* 1.3By these two definitions it is manifest, that the inscription and circumscription of rectiline figures here spoken of, pertayne to such rectiline figures onely, which haue equall sides and equall angles, which are commonly called regular. It is also to be noted that rectiline figures only of one kinde or forme can be inscribed or circumscribed the one within or about the other. As a triangle within or about a triangle: A square within or about a square: and so a Pentagon within or about a Pentagō, & likewise of others of one forme. But a triangle can not be inscribed or circumscribed within or aboute a square: nor a square within or about a Penta∣gon. And so of others of diuers kyndes. For euery playne rectiline figure hath so many angles as it hath sides. Wherfore the figure inscrided must haue so many an∣gles as the figure in which it is inscribed hath sides: and the angles of the one (as is sayd) must touche the sides of the other. And contrariwise in circumscription of figures, the sides of the figure circumscribed must touch the angles of the figure a∣bout which it is circumscribed.

* 1.4A rectiline figure is sayd to be inscribed in a circle, when eue∣ry one of the angles of the inscribed figure toucheth the cir∣cumference of the circle.

A circle by reason of his vniforme and regular distance which it hath from the centre to the circumference may easily touche all the angles of any regular recti∣line figure within it: and also all the sides of any figure without it. And therfore a∣ny regular rectiline figure may be inscribed within it, and also be circumscribed a∣bout it. And agayne a circle may be both inscribed within any regular rectiline fi∣gure, and also be circumscribed about it.

As the triangle ABC is inscribed in the circle ABC•• for that euery angle toucheth some one pointe of the circumference of the circle. As the angle CAB of the triangle ABC toucheth the point A of the circumference of the circle. And the angle ABC

of the triangle toucheth the pointe B of the circumfe∣rence

[illustration]

of the circle. And also the angle ACB of the tri∣••ngle 〈…〉〈…〉 the pointe •• of the circumference of the circle. In like manner the square ADEF is inscri∣bed in the same circle ABC: for that euery angle of the square inscribed, toucheth some one poynte of the circle in which it is inscribed. And so imagine you of rectilined figures.

A circle is sayd to be circumscribed about a rectiline figure,* 1.5 whē the cir∣cumference of the circle toucheth euery one of the angles of the figure about which it is circumscribed.

As in the former example of the third definition. The circle ADEF is circumscri∣bed about the triangle ABC, because the circumference of the circle which is circum∣scribed toucheth euery angle of the triangle about which it is circumscribed•• namely, the angles CAB, ABC, and BCA. Likewise the same circle ADEF is circumscribed about the square ADEF by the same definition as you may see.

A circle is sayd to be inscribed in a rectiline figure,* 1.6 when the circumference of the circle toucheth euery one of the sides of the figure within which it is inscribed.

As the circle ABCD is inscribed within the triangle

[illustration]

EFG, because the circumference of the circle toucheth euery side of the triangle in which it is inscribed•• namely the side EF in the point B, and the side GF in the pointe C, and the side GE in the point D. Likewise by the same reason the same circle is inscribed within the square HIKL. And so may you iudge of other rectiline figures.

A rectilined figure is said to be circum∣scribed about a circle,* 1.7 when euery one of the sides of the fi∣gure circumscribed toucheth the circumference of the circle.

As in the former figure of the fift definition, the triangle EFG is circumscribed a∣bout the circle ABCD, for that euery side of the same triangle beyng circumscribed toucheth the circumference of the circle, about which it is circumscribed. As the side EG of the triangle EFG toucheth the circumference of the circle in the point D: and the side EF toucheth it in the point B: and the side GF in the point C. Likewise also the square HIKL is circumscribed about the circle ABCD, for euery one of his sides toucheth the circumference of the circle, namely, in the pointes A, B, C, D. And thus consider of all other regular right lined figures (for of them onely are vnderstanded these definitions) to be circumscribed about a circle, or to be inscribed within a cir∣cle: or of a circle to be circumscribed or inscribed about or within any of them.

* 1.8A right lyne is sayd to be coapted or applied in a circle, when the extremes or endes therof, fall vppon the circumference of the circle.

As the line BC is sayd to be coapted or to be appli∣ed

[illustration]

to the circle ABC, for that both his extremes fall vpon the circumference of the circle in the pointes B and C. Likewise the line DE. This definition is very necessary, and is properly to be taken of any lyne ge∣uen to be coapted and applied into a circle, so ••hat it exceede not the diameter of the circle geuen.

Notes

• * 1.1

  First defini∣tion.
• * 1.2

  Second de∣finition.
• * 1.3

  The inscripti∣tion and cir∣cumscription of rectiline ••ig••res pertai∣••eth only to regular fi∣gures.
• * 1.4

  The third de∣finition.
• * 1.5

  The fourth definition.
• * 1.6

  The fift de∣finition.
• * 1.7

  The sixt de∣uition.
• * 1.8

  Seuenth de∣finition.