The Way To Geometry. Petrus Ramus

of the ancient Geometers, as appeareth out of Simplicius, in his commentaries upon the 8. chapter of Aristotle's iij. booke of Heaven: which kinde of figures Aristotle in the same place deemeth to bee onely ordinate, and yet not all of that kind. But only three among the Plaines, to witt a Triangle, a Quadrate, and a Sexangle: amongst Solids, two; the Pyramis, and the Cube. But if the filling of a place bee judged by right angles, 4. in a Plaine, and 8. in a Solid, the Oblong of plaines, and the Octahedrum of Solids shall (as shall appeare in their places) fill a place; And yet is not this Geometrie of Aristotle accurate enough. But right angles doe determine this sentence, and so doth Euclide out of the angles demonstrate, That there are onely five ordinate solids; And so doth Potamon the Geometer, as Simplicus testifieth, demonstrate the question of figures filling a place. Lastly, if figures, by laying of their corners together, doe make in a Plaine 4. right angles, or in a Solid 8. they doe fill a place.

      Of this probleme the ancient geometers have written, as we heard even now: And of the latter writers, Regiomontanus is said to have written accurately; And of this argument Maucolycus hath promised a treatise, neither of which as yet it hath beene our good hap to see.

      Neither of these are figures of this nature, as in their due places shall be proved and demonstrated.

      28. A round figure is that, all whose raies are equall.

      Such in plaines shall the Circle be, in Solids the Globe or Spheare. Now this figure, the Round, I meane, of all Isoperimeters is the greatest, as appeared before at the 15. e. For which cause Plato, in his Timæus or his Dialogue of the World said; That this figure is of all other the greatest. And therefore God, saith he, did make the world of a sphearicall forme, that within his compasse it might the better containe all things: And Aristotle, in his Mechanicall problems, saith; That this figure is the beginning, principle, and cause of all miracles. But those miracles shall in their time God willing, be manifested and showne.

      Rotundum, a Roundle, let it be here used for Rotunda figura, a round figure. And in deede Thomas Finkius or Finche, as we would call him, a learned Dane, sequestring this argument from the rest of the body of Geometry, hath intituled that his worke De Geometria rotundi, Of the Geometry of the Round or roundle.

      29. The diameters of a roundle are cut in two by equall raies.

      The reason is, because the halfes of the diameters, are the raies. Or because the diameter is nothing else but a doubled ray: Therefore if thou shalt cut off from the diameter so much, as is the radius or ray, it followeth that so much shall still remaine, as thou hast cutte of, to witt one ray, which is the other halfe of the diameter. Sn.

      And here observe, That Bisecare, doth here, and in other places following, signifie to cutte a thing into two equall parts or portions; And so Bisegmentum, to be one such portion; And Bisectio, such a like cutting or division.

      30. Rounds of equall diameters are equall. Out of the 1. d. iij.

      Circles and Spheares are equall, which have equall diameters. For the raies, which doe measure the space betweene the Center and Perimeter, are equall, of which, being doubled, the Diameter doth consist. Sn.

      The fifth Booke, of Ramus his Geometry, which is of Lines and Angles in a plaine Surface

      1. A lineate is either a Surface or a Body.

      Lineatum, (or Lineamentum) a magnitude made of lines, as was defined at 1. e. iij. is here divided into two kindes: which is easily conceived out of the said definition there, in which a line is excluded, and a Surface & a body are comprehended. And from hence arose the division of the arte Metriall into Geometry, of a surface, and Stereometry, of a body, after which maner Plato in his vij. booke of his Common-wealth, and Aristotle in the 7. chapter of the first booke of his Posteriorums, doe distinguish betweene Geometry and Stereometry: And yet the name of Geometry is used to signifie the whole arte of measuring in generall.

      2. A Surface is a lineate only broade. 5. d j.

      As here aeio. and uysr. The definition of a Surface doth comprehend the distance or dimension of a line, to witt Length: But it addeth another distance, that is Breadth. Therefore a Surface is defined by some, as Proclus saith, to be a magnitude of two dimensions. But two doe not so specially and so properly define it. Therefore a Surface is better defined, to bee a magnitude onely long and broad. Such, saith Apollonius, are the shadowes upon the earth, which doe farre and wide cover the ground and champion fields, and doe not enter into the earth, nor have any manner of thicknesse at all.

      Epiphania, the Greeke word, which importeth onely the outter appearance of a thing, is here more significant, because of a Magnitude there is nothing visible or to bee seene, but the surface.

      3. The bound of a surface is a line. 6. d j.

      The matter in Plaines is manifest. For a three cornered surface is bounded with 3. lines: A foure cornered surface, with foure lines, and so forth: A Circle is bounded with one line. But in a Sphearicall surface the matter is not so plaine: For it being whole, seemeth not to be bounded with a line. Yet if the manner of making of a Sphearicall surface, by the conversiō or turning about of a semiperiphery, the beginning of it, as also the end, shalbe a line, to wit a semiperiphery: And as a point doth not only actu, or indeede bound and end a line: But is potentia, or in power, the middest of it: So also a line boundeth a Surface actu, and an innumerable company of lines may be taken or supposed to be throughout the whole surface. A Surface therefore is made by the motion of a line, as a Line was made by the motion of a point.

      4. A Surface is either Plaine or Bowed.

      The difference of a Surface, doth answer to the difference of a Line, in straightnesse and obliquity or crookednesse.

      Obliquum, oblique, there signified crooked; Not right or straight: Here, uneven or bowed, either upward or downeward. Sn.

      5. A plaine surface is a surface, which lyeth equally betweene his bounds, out of the 7. d j.

      As here thou seest in aeio. That therefore a Right line doth looke two contrary waies, a Plaine surface doth looke all about every way, that a plaine surface should, of all surfaces within the same bounds, be the shortest: And that the middest thereof should hinder the sight of the extreames. Lastly, it is equall to the dimension betweene the lines: It may also by one right line every way applyed be tryed, as Proclus at this place doth intimate.

      Planum, a Plaine, is taken and used for a plaine surface: as before Rotundum, a Round, was used for a round figure.

      Therefore,

6. From a point unto a point we may, in a plaine surface, draw a right line, 1 and 2. post. j.

      Three things are from the former ground begg'd: The first is of a Right line. A right line and a periphery were in the ij. booke defined: But the fabricke or making of them both, is here said to bee properly in a plaine.

      The fabricke or construction of a right line is the 1. petition. And justly is it required that it may bee done onely upon a plaine: For in any other surface it were in vaine to aske it. For neither may wee possibly in a sphericall betweene two points draw a right line: Neither may wee possibly in a Conicall and Cylindraceall betweene any two points assigned draw a right line. For from the toppe unto the base that in these is only possible: And then is it the bounde of the plaine which cutteth the Cone and Cylinder. Therefore, as I said, of a right plaine it may onely justly bee demanded: That from any point assigned, unto any point assigned, a right line may be drawne, as here from a unto e.

      Now the