Chap. I. Angle be lefs than a Right, it is peculiarly call'd an acute Angle; if greater, an Obtufe. Thus, Fig. 4, ACB and BCD are both right Angles; DCE an Acute; ECA an Obtuse. ous Ar Def.18. Two Angles, which have one and the Contigu- fame common Leg, are call'd (*) contigles, what. guous Angles: As ACB and BCD, Fig. 4; for the Leg BC is common to both. Def.19. Two Angles, which touch one anoVertical ther only at their Vertices or Heads, are call'd vertical Angles. Thus, Fig. 33. ABC and EBD are vertical Angles; and likewife CBD and ABE. Angles, what. dicular Line, Def 20. The Legs of a right Angle are faid to 4 perpen- be (†) perpendicular one to the other. Thus, Fig. 4, BC is perpendicular to AC what. or CD. And whenever the Legs are perpendicular, the Angle contain❜d between them is a right Angle. Def.21. near Fi gures, what. Rectilinear Figures are thofe which are comprehended by right Lines. And fuch are all the Figures spoken of in Chap. 1, and 2, of this Treatife, except the Circle. (*) They are ftil'd in Latin, Anguli deinceps. (+) By a Perpendicular is denoted a Line, which inclines no more to one Side than another, but ftands exactly Upright, and fo has the fame Pofition, as a perpendicular (i. e. hanging) Thread, with a Plummet at its End, will reft in, if left to it felf. It is alfo call'd a normal Line, from the Norma or Square-rule, whofe Sides are made perpendicular one to the other. The The Lines, whereby Figures are com- Chap. I. prehended, are call'd their Sides. And rectilinear Figures have, either three, four, or more Sides. Def.22. or The Sides of Figures, Among trilateral (i. e. three-fided) Fi- what. gures, call'd in one Word Triangles, an AnequilaDef.23. equilateral Triangle is that, which has teral Triall its three Sides equal one to the other, angle, as Fig. 7. what. An Ifofceles, what. A Triangle, which has only two Sides Def.24. equal, is call'd an Ifofceles; as Fig. 8. A Triangle, which has none of its Def.25. Sides equal, is call'd a Scalenum; as 4 ScaleFig. 9. num,what A Triangle, which has () one right Def.26. Angle, is call'd a rectangular Triangle. A rectanAnd the Side oppofite to the right An- gular Trigle, is call'd the Hypotenuse or Subtenfe. HypoteThus, Fig. 10, A is the right Angle, and nuse, BC the Hypotenuse. angle, and what. blygoni A Triangle, which has an obtufe An- Def.27. gle, is thence call'd an (*) Amblygonium; An Amas Fig. 9, where the Angle A is obtuse. um,what. A Triangle, which has all three An- Def.28. gles acute, is call'd an (*) Oxygonium; An Oxyas Fig. 7, and 8. (That no Triangle can have more than one right Angle, is prov'd from Theorem 4. (**) Thefe Words denote in the Greek Tongue refpectively, that which has an obtufe or acute Angle. And it is to be noted, that as a Triangle can have no more than one right Angle, fo can it have no more than one Obtufe, for the like Reason to be learnt from Theorem 4. Among gonium, what. Chap. I. Among quadrilateral (i. e. four-fided) Figures, a quare is that, which has four 4 Square, equal Sides, and four right Angles; as Fig. 11. Def.29. what. Def.30. An (†) Oblong is that, which has four An Ob right Angles, but only the two opposite long, what Sides equal; as Fig. 12. Def.31. A Rhombus has four equal Sides, and A Rhom- four oblique Angles; as Fig. 13. bus,what. A Rhomboid has four Sides, and four Def.32. 4 Rhom- oblique Angles; whereof only the two boid, oppofite, both Sides and Angles are ewhat. qual; as Fig. 14. Def.33. Any other quadrilateral Figure, is call'd 4 Trape- a Trapezium; as Fig. 15; zium, what. Parallel Lines, Lines, which are all along equally diDef.34. ftant one from the other, are call'd Parallels. Thus, Fig. 16, the right Lines what. AB and CD are Parallels; as are alfo, Fig. 17, the curve Lines ABC and DEF. Def.35. A Parallelogram is any quadrilateral Figure, whofe oppofite Sides are parallel, and confequently ()_equal. And fuch are all quadrilateral Figures, but a Trapezium. A Paralle logram, what. Figures, (+) It is call'd by common Workmen, a Long-fquare; and by Mathematicians frequently a Rectangle from its four right Angles. The other quadrilateral Figure, that has alfo all right Angles, being specified by the Name of a Quadratum or Square. C) That the oppofite Sides of a Parallelogram are equal, is wont indeed to be made a Propofition, and to be demonftrated. tagon, Figures, which have more than four Chap. I. Sides, are in general call'd Polygons; and Def.36. are diftinguish'd one from another by A PolyNames denoting (in the Greek Language) gon, Pentheir respective Number of Angles, and Hexagon, confequently of Sides. Thus, a Penta- &c. what. gon, Hexagon, Heptagon, Octogon, &c. denotes refpectively a Figure of Five, Six, Seven, Eight, &c. Angles or Sides; fuch as are reprefented, Fig. 18, 19, 20, and 21. Any Polygon, that has all its Sides, Def.37. and confequently all its Angles, equal a regular one to another, is call'd a regular Poly- Polygon, gon. Such are the Pentagon, Hexagon, c. defcrib'd, Fig. 18, 19, 20, and 21. what. ral, and As it is obvious, that a Figure, confi- Def.38. der'd by it felf, is then faid to be equila- Equilateteral, when all its own Sides are equal; equianguand equiangular, when all its own An- lar Figles are equal; fo it is to be observ'd, gures, that two or more Figures, confider'd in respect one of the other, are then faid to be equilateral, when the feveral Sides in monftrated. But I have chofen rather to infert it into the very Definition of a Parallelogram, as being really included therein. For fince to be parallel, is no other than to be equi-diftant, it follows, that to fay, the two oppofite Sides of a Parallelogram are Parallel, is the fame in Effect as to fay, that the other two oppofite Sides are equal; it being the Equality of thefe, that makes the other two equidiftant, i. c. parallel. one, Chap. I. one, are refpectively equal to the feveral Sides in the other; and likewise are then faid to be equiangular, when the feveral Angles in one, are refpectively equal to the feveral Angles in the other ; although, each Figure, confider'd in it self, is neither equilateral nor equiangular. Thus, Fig. 22, and 23, the Triangles ABC and abc, are equilateral and (*) equiangular Figures; because the Side AB=ab, BC=bc, CA=ca, and the Angle A=a, Bb, C=c. Def.39. Similar what. Similar (or like) Figures are those, which have their Angles mutually equal Figures, each to each, and alfo the Sides about the equal Angles (†) proportional. Thus, the Triangles ABC and atc, (Fig. 24, and 25,) are fimilar Figures. For the Angle Aa, Bb, C=c; and also the Side AB: BC : : ab : bc; and BC: CA:: be: ca; and CA: AB::ca: ab. And it (*) It is to be noted, that although equilateral Triangles, are equiangular by Theorem 5; yet equiangular Triangles, are not always equilateral, (as Fig. 24, and 25.) but the Sides about the equal Angles, are always proportional by Theorem 9. (+) It is fuppos'd, that the young Gentleman proceeds in his mathematical Studies regularly; and therefore has gone through the young Gentleman's Arithmetick, and fo understands what Ratio or Proportion is, and the feveral Sorts of it, viz. Altern, Inverfe, &c. Proportion; as alfo what the Antecedent and Confequent of a Proportion is; they being explain'd in the Chapter concerning Proportion in the faid Arithmetical Treatife. |