The Way To Geometry. Petrus Ramus

superficiall angle: [And such also are the angles ouy. and bcd.] so is the angle o. a solid angle, to witt comprehended of the three surfaces aoi. ioe. and aoe. Neither may a surface, of 2. dimensions, be bounded with one right line: Nor a body, of three dimensions, bee bounded with two, at lest beeing plaine surfaces.

      4. The shankes of an angle are the bounds compreding the angle.

      Scèle or Crura, the Shankes, Legges, H. are the bounds insisting or standing upon the base of the angle, which in the Isosceles only or Equicrurall triangle are so named of Euclide, otherwise he nameth them Latera, sides. So in the examples aforesaid, ea. and ei. are the shankes of the superficiary angle e; And so are the three surfaces aoi. ieo. and aeo. the shankes of the said angle o. Therefore the shankes making the angle are either Lines or Surfaces: And the lineates formed or made into Angles, are either Surfaces or Bodies.

      5. Angles homogeneall, are angles of the same kinde, both in respect of their shankes, as also in the maner of meeting of the same: [Heterogeneall, are those which differ one from another in one, or both these conditions.]

      Therefore this Homogenia, or similitude of angles is twofolde, the first is of shanks; the other is of the manner of meeting of the shankes: so rectilineall right angles, are angles homogeneall betweene themselves. But right-lined right angles, and oblique-lined right angles between themselves, are heterogenealls. So are neither all obtusangles compared to all obtusangles: Nor all acutangles, to all acutangles, homogenealls, except both these conditions doe concurre, to witt the similitude both of shanke and manner of meeting. Lunularis, a Lunular, or Moonlike corner angle is homogeneall to a Systroides and Pelecoides, Hatchet formelike, in shankes: For each of these are comprehended of peripheries: The Lunular of one convexe; the other concave; as iue. The Systroides of both convex, as iao. The Pelecoides of both concave, as eau. And yet a lunular, in respect of the meeting of the shankes is both to the Systroides and Pelecoides heterogeneall: And therefore it is absolutely heterogeneall to it.

6. Angels congruall in shankes are equall.

      This is drawne out of the 10. e j. For if twice two shanks doe agree, they are not foure, but two shankes, neither are they two equall angles, but one angle. And this is that which Proclus speaketh of, at the 4. p j. when hee saith, that a right lined angle is equall to a right lined angle, when one of the shankes of the one put upon one of the shankes of the other, the other two doe agree: when that other shanke fall without, the angle of the out-falling shanke is the greater: when it falleth within, it is lesser: For there is comprehendeth; here it is comprehended.

      Notwithstanding although congruall or agreeable angles be equall: yet are not congruity and equality reciprocall or convertible: For a Lunular may bee equall to a right lined right angle, as here thou seest: For the angles of equall semicircles ieo. and aeu. are equall, as application doth shew. The angle aeo. is common both to the right angle aei. and to the lunar aueo. Let therefore the equall angle aeo. bee added to both: the right angle aei. shall be equall to the Lunular aueo.

      The same Lunular also may bee equall to an obtusangle and Acutangle, as the same argument will demonstrate.

      Therefore,

7. If an angle being equicrurall to an other angle, be also equall to it in base, it is equall: And if an angle having equall shankes with another, bee equall to it in the angle, it is also equall to it in the base. è 8. & 4. p j.

      For such angles shall be congruall or agreeable in shanks, and also congruall in bases. Angulus isosceles, or Angulus æquicrurus, is a triangle having equall shankes unto another.

      8. And if an angle equall in base to another, be also equall to it in shankes, it is equall to it.

      For the congruency is the same: And yet if equall angles bee equall in base, they are not by and by equicrurall, as in the angles of the same section will appeare, as here. And so of two equalities, the first is reciprocall: The second is not. [And therefore is this Consectary, by the learned B. Salignacus, justly, according to the judgement of the worthy Rud. Snellius, here cancelled; or quite put out: For angles may be equall, although they bee unequall in shankes or in bases, as here, the angle a. is not greater then the angle o, although the angle o have both greater shankes and greater base then the angle a.]

      And

      9. If an angle equicrurall to another angle, be greater then it in base, it is greater: And if it be greater, it is greater in base: è 52 & 24. p j.

      As here thou seest; [The angles eai. and uoy. are equicrurall, that is their shankes are equall one to another; But the base ei is greater then the base uy: Therefore the angle eai, is greater then the angle uoy. And contrary wise, they being equicrurall, and the angle eai. being greater then the angle uoy. The base ei. must needes be greater then the base uy.]

      And

      10. If an angle equall in base, be lesse in the inner shankes, it is greater.

      Or as the learned Master T. Hood doth paraphrastically translate it. If being equall in the base, it bee lesser in the feete (the feete being conteined within the feete of the other angle) it is the greater angle. [That is, if one angle enscribed within another angle, be equall in base, the angle of the inscribed shall be greater then the angle of the circumscribed.]

      As here the angle aoi. within the angle aei. And the bases are equall, to witt one and the same; Therefore aoi. the inner angle is greater then aei. the outter angle. Inner is added of necessity: For otherwise there will, in the section or cutting one of another, appeare a manifest errour. All these consectaries are drawne out of that same axiome of congruity, to witt out of the 10. e j. as Proclus doth plainely affirme and teach: It seemeth saith hee, that the equalities of shankes and bases, doth cause the equality of the verticall angles. For neither, if the bases be equall, doth the equality of the shankes leave the same or equall angles: But if the base bee lesser, the angle decreaseth: If greater, it increaseth. Neither if the bases bee equall, and the shankes unequall, doth the angle remaine the same: But when they are made lesse, it is increased: when they are made greater, it is diminished: For the contrary falleth out to the angles and shankes of the angles. For if thou shalt imagine the shankes to be in the same base thrust downeward, thou makest them lesse, but their angle greater: but if thou do againe conceive them to be pul'd up higher, thou makest them greater, but their angle lesser. For looke how much more neere they come one to another, so much farther off is the toppe removed from the base: wherefore you may boldly affirme, that the same base and equall shankes, doe define the equality of Angels. This Poclus,

      Therefore,

      11. If unto the shankes of an angle given, homogeneall shankes, from a point assigned, bee made equall upon an equall base, they shall comprehend an angle equall to the angle given. è 23. p j. & 26. p xj.

      [This consectary teacheth how unto a point given, to make an angle equall to an Angle given. To the effecting and doing of each three things are required; First, that the shankes be homogeneall, that is in each place, either straight or crooked: Secondly, that the shankes bee made equall, that is of like or equall bignesse: Thirdly, that the bases be equall: which three conditions if they doe meete, it must needes be that both the angles shall bee equall: but if one of them be wanting, of necessity againe they must be unequall.]

      This shall hereafter be declared and made plaine by many