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

* 1.116 A cone is a solide or bodely figure which is made, when one of the sides of a rectangle triangle, namely, one of the sides which contayne the right angle, abiding fixed, the triangle is moued about, vntill it returne vnto the selfe same place from whence it began first to be moued. Now if the right line which abideth fixed be equall to the other side which is moued about and containeth the right angle: then the cone is a rectangle cone. But if it be lesse, then is it an obtuse angle cone. And if it be greater, thē is it an a cute∣angle cone,

This definition of a Cone is of the nature and condition that the definition of a Sphere was, for either is geuen by the motion of a superficies. There, as to the production of a Sphere was imagined a semicircle to moue round, from some one point till it returned to the same point againe: so here must ye imagine a rectangle triangle to moue about till it come againe to the place where it beganne. Let ABC be a rectangle triangle, hauing

[illustration]

the angle ABC a right angle, which let be contained vnder the lines AB and BC.

[illustration]

Now suppose the side AB, namely, one of the lines which cōtaine the right angle ABC to be fastened, and about it suppose the triangle ABC to be moued from some one poynt assigned till it re∣turne to the same agayne (as vppon the diameter in the definition of a Sphere ye imagined a se∣micircle to moue about): so shall the solide or body thus described be a perfect Cone. As you may imagine by this figure here set. And the forme of a Cone you may sufficiently conceaue by the figure set in the margent. There are of Cone three kindes, namely, a rectangle Cone, an obtuseangle Cone, and an acute angle Cone, all which were before in the former definitiō defined: Name∣ly, the first kinde after this maner.

If the right line which abideth fixed, be equall to the other side which moueth ro••••d about, and containeth the right angle, then the Cone is a rectangle Cone.

* 1.2As suppose in the former example, that the line AB which is fixed, and about which the triangle was moued, and after the motion yet remayneth, be equall to the line BC, which is the other line con∣tayning the right angle, which also is moued about together with the whole triangle•• then is the Cone described, as the Cone ADC in this example, a right angled Cone: so called for that the angle at the toppe of the Cone is a right angle. For forasmuch as the lines AB and BC of the triangle ABC are e∣quall, the angle BAC is equall to the angle BCA (by the 5. of the first). And eche of them is the halfe of the right angle ABC (by the 32. of the first). In like sort may it be shewed in the triangle ABD, that the angle ••DA is equall to the angle ••AD, and that eche of them is the halfe of a right angle. Wherefore the whole angle CAD, which is composed of the two halfe right angles, namely, DA•• and CA•• is a right angle. And so haue ye what is a right angled Cone.

But if it be lesse, then is it an obtuseangle Cone. As in this ex∣ample,

[illustration]

the line AB fixed is lesse then the line BC moued a∣bout. Wherefore the Cone described of the circumuoluti∣on of the triangle ABC about the line A••, is an obtusean∣gle Cone, for that the angle at the toppe DAC is greater then a right angle. Wherefore it is an obtuseangle. And therefore the Cone is called an obtuse angle Cone.

And if it be greater, then i•• i•• an acuteangle Cone. As in

[illustration]

this figure, the line AB fastened, is greater then the line BC moued about. Wherefore the Cone de∣scribed by the motion and turning of the triangle ABC about AB is an acuteangle Cone, hauing the angle at the toppe BAC an acute angle. Of whome the Cone is called an acuteangle Cone. For the ea∣sier sight & cōsideration of all these kindes of Cones, and also for the plainer demonstration of the varie∣ties of their angles in their toppes, I haue described them all three in one playne figure, of which the Cone ACB is a right angled Cone, hauyng his fix∣ed side CF equall to the line FB, and hys angle ACB a right angle: the Cone AEB is an obtuse angle Cone, and ADB an acuteangle Cone. By which figure ye may easily demonstrate (by the 21. of the first) that the angle ADB of the Cone ADB, whose fixed line DF is greater then the side FB, is lesse then the right angle ACB, and so is an acute angle. And also (by the same 21. of the first) ye shall with like facilitie perceaue how the angle AEB of the Cone AEB whose fix∣ed line EF is lesse then the side FB, is grea∣ter then the right angle ACB: and there∣fore is an obtuse angle.

This figure of a Cone is of Campane, of Vitellio, and of others which haue written in these latter times, called a round Pyramis,* 1.3 which is not so aptly. For a Pyramis, and a Cone, are farre distant, & of sundry natures. A Cone is a regular body produced of one circumuolution of a rectangle triangle, and limited and bordered with one onely round super∣ficies. But a Pyramis is terminated and bordered with diuers superficieces. Therefore can not a Cone by any iust reason beare the name of a Pyramis. This solide of many is called Turbo, which to our purpose may be Englished a Top or Ghyg: and moreouer, peculiarly Campane calleth a Cone the Py∣ramis of a round Columne, namely, of that Columne which is produced of the motion of a parallelo∣gramme (contained of the lines AB and BC) moued about, the line AB being fixed. Of which Co∣lumnes shall be shewed hereafter.

Notes

• * 1.1

  Seuententh diffinition.
• * 1.2

  First kinde of Cones.
• * 1.3

  A Cone called of Campane a ro••••de Pira∣mis.