The Way To Geometry. Petrus Ramus

instrument for the drawing of a right plaine is called Amussis, & by Petolemey, in the 2. chapter of his first booke of his Musicke, Regula, a Rular, such as heere thou seest.

      And from a point unto a point is this justly demanded to be done, not unto points; For neither doe all points fall in a right line: But many doe fall out to be in a crooked line. And in a Spheare, a Cone & Cylinder, a Ruler may be applyed, but it must be a sphearicall, Conicall, or Cylindraceall. But by the example of a right line doth Vitellio, 2 p j. demaund that betweene two lines a surface may be extended: And so may it seeme in the Elements, of many figures both plaine and solids, by Euclide to be demanded; That a figure may be described, at the 7. and 8. e ij. Item that a figure may be made vp, at the 8. 14. 16. 23. 28. p. vj. which are of Plaines. Item at the 25. 31. 33. 34. 36. 49. p. xj. which are of Solids. Yet notwithstanding a plaine surface, and a plaine body doe measure their rectitude by a right line, so that jus postulandi, this right of begging to have a thing granted may seeme primarily to bee in a right plaine line.

      Now the Continuation of a right line is nothing else, but the drawing out farther of a line now drawne, and that from a point unto a point, as we may continue the right line ae. unto i. wherefore the first and second Petitions of Euclide do agree in one.

      And

7. To set at a point assigned a Right line equall to another right line given: And from a greater, to cut off a part equall to a lesser. 2. and 3. p j.

      As let the Right line given be ae. And to i. a point assigned, grant that io. equall to the same ae. may bee set. Item, in the second example, let ae. bee greater then io. And let there be cut off from the same ae. by applying of a rular made equall to io. the lesser, portion au. as here. For if any man shall thinke that this ought only to be don in the minde, hee also, as it were, beares a ruler in his minde, that he may doe it by the helpe of the ruler. Neither is the fabricke in deede, or making of one right equal to another: And the cutting off from greater Right line, a portion equall to a lesser, any whit harder, then it was, having a point and a distance given, to describe a circle: Then having a Triangle, Parallelogramme, and semicircle given, to describe or make a Cone, Cylinder, and spheare, all which notwithstanding Euclide did account as principles.

      Therefore,

      8. One right line, or two cutting one another, are in the same plaine, out of the 1. and 2. p xj.

      One Right line may bee the common section of two plaines: yet all or the whole in the same plaine is one: And all the whole is in the same other: And so the whole is the same plaine. Two Right lines cutting one another, may bee in two plaines cutting one of another; But then a plaine may be drawne by them: Therefore both of them shall be in the same plaine. And this plaine is geometrically to be conceived: Because the same plaine is not alwaies made the ground whereupon one oblique line, or two cutting one another are drawne, when a periphery is in a sphearicall: Neither may all peripheries cutting one another be possibly in one plaine.

      And

      9. With a right line given to describe a peripherie.

      This fabricke or construction is taken out of the 3. Petition which is thus. Having a center and a distance given to describe, make, or draw a circle. But here the terme or end of a circle is onely sought, which is better drawne out of the definition of a periphery, at the 10. e ij. And in a plaine onely may that conversion or turning about of a right line bee made: Not in a sphearicall, not in a Conicall, not in a Cylindraceall, except it be in top, where notwithstanding a periphery may bee described. Therefore before (to witt at the said 10. e ij.) was taught the generall fabricke or making of a Periphery: Here we are informed how to discribe a Plaine periphery, as here.

      Now as the Rular was the instrument invented and used for the drawing of a right line: so also may the same Rular, used after another manner, be the instrument to describe or draw a periphery withall. And indeed such is that instrument used by the Coopers (and other like artists) for the rounding of their bottomes of their tubs, heads of barrells and otherlike vessells: But the Compasses, whether straight shanked or bow-legg'd, such as here thou seest, it skilleth not, are for al purposes and practises, in this case the best and readiest. And in deed the Compasses, of all geometricall instruments, are the most excellent, and by whose help famous Geometers have taught: That all the problems of geometry may bee wrought and performed: And there is a booke extant, set out by John Baptist, an Italian, teaching, How by one opening of the Compasses all the problems of Euclide may be resolved: And Jeronymus Cardanus, a famous Mathematician, in the 15. booke of his Subtilties, writeth, that there was by the helpe of the Compasses a demonstration of all things demonstrated by Euclide, found out by him and one Ferrarius.

      Talus, the nephew of Dædalus by his sister, is said in the viij. booke of Ovids Metamorphosis, to have beene the inventour of this instrument: For there he thus writeth of him and this matter:—Et ex uno duo ferrea brachia nodo: Iunxit, ut æquali spatio distantibus ipsis: Altera pars staret, pars altera duceret orbem.

      Therfore

10. The raies of the same, or of an equall periphery, are equall.

      The reason is, because the same right line is every where converted or turned about. But here by the Ray of the periphery, must bee understood the Ray the figure contained within the periphery.

11. If two equall peripheries, from the ends of equall shankes of an assigned rectilineall angle, doe meete before it, a right line drawne from the meeting of them unto the toppe or point of the angle, shall cut it into two equall parts. 9. p j.

      Hitherto we have spoken of plaine lines: Their affection followeth, and first in the Bisection or dividing of an Angle into two equall parts.

      Let the right lined Angle to bee divided into two equall parts bee eai. whose equall shankes let them be ae. and ai. (or if they be unequall, let them be made equall, by the 7 e.) Then two equall peripheries from the ends e and i. meet before the Angle in o. Lastly, draw a line from o. unto a. I say the angle given is divided into two equall parts. For by drawing the right lines oe. and oi. the angles oae. and oai. equicrurall, by the grant, and by their common side ao. are equall in base eo. and io. by the 10 e (Because they are the raies of equall peripheries.) Therefore by the 7. e iij. the angles oae. and oai. are equall: And therefore the Angle eai. is equally divided into two parts.

      12. If two equall peripheries from the ends of a right line given, doe meete on each side of the same, a right line drawne from those meetings, shall divide the right line given into two equall parts. 10. p j.

      Let the right line given bee ae. And let two equall peripheries from the ends a. and e. meete in i. and o. Then from those meetings let the right line io. be drawne. I say, That ae. is divided into two equall parts, by the said line thus drawne. For by drawing the raies of the equall peripheries ia. and ie. the said io. doth cut the angle aie. into two equall parts, by the 11. e. Therefore the angles aiu. and uie. being equall and equicrurall (seeing the shankes are the raies of equall peripheries, by the grant.) have equall bases au. and ue. by the 7. e iij. Wherefore