EUCL I D's I ELEMENTS BOOK I. DEFINITIONS. 1. POINT, is that which hath no Parts, or Magnitude. А II. A Line is Length, without Breadth: III. The Ends (or Bounds) of a Line, are Points. IV. A Right Line, is that which lieth evenly between its Points. V. A Superficies, is that wbich bath only Length and Breadth: VI. The Bounds of a Superficies are Lires. VII. A Plain Superficies, is that which, lieth evenly be tween its Lines. VIII. A Plain Angle, is the Inclination of two Lines to one another in the same Plane, which touch each other, but do not both lie in the same Right Line. IX. If the Lines containing the Angle be Right ones, then the Angle is called a Right-lind Angle. B X. When T a one. X. When a RightLine, standing on another Right Line, makes Angles on either Side thereof, equal between themselves, each of these equal Angles is a Right one; and that Right Line which stands upon the other, is. called a Perpendicular to that whereon it stands. Right one. of any Thing. or more Terms. Line, called the Circumference; to which all Right through the Center, and terminated on both Sides by ameter, and that part of the Circumference of a Circle cut off by that Diameter. XIX. A Segment of a Circle, is a Figure contain d un der a Right Line, and Part of the Circumference of the Circle [which is cut off by that Right Line.] Right Lines. der three Lines. der four. XXIII. Many sided Figures, are those that are contain'd under more than four Right Lines. XXIV. Of three-sided Figures, that is an Equilateral Triangle, which hath three equal Sides. XXV. That an Isosceles, or Equicrural one, which hath only two Sides equal. XXVI. And a Scalene one, is that which hath three un ' equal Sides, XXVII. Also of Three-sided Figures, that is, a Rightangled Triangle, which bath a Right Angle. XXVIII. a XXVIII. That an Obtuse-angled one, which hath an Obtuse Angle. XXIX. And that an Acute-angled one, which hath three Acute Angles. XXX. Of Four-sided Figures, that is a Square, whose four Sides are equal, and its Angles all Right ones, XXXI. That an Oblong, or Rectangle, a Figure which is longer on one side than the other, which is Rightangled, but not equal fided. XXXII. That a Rhombus, which hath four equal Sides, but not Right Angles. XXXIII. That a Rhomboides, whose opposite Sides and Angles only are equal. XXXIV. All Quadrilateral Figures, besides these, are called Trapezia. XXXV. Parallels are such Right Lines in the same Plane, which if infinitely produc'd both Ways, would a never meet. POST U L A T E S. COM 1. RANT that a Right-Line may be drawn from any one point to another. may be tinued directly forwards. III. And that a Circle may be describ's about any Center, with any Distance. 1. HINGS equal to one and the same 17 Thing, are equal to one another. II. If to equal Things, are added equal Things, the Wholes will be equal. III. If from equal Things, equal Things be taken away, the Remainders will be equal. IV. If equal Things be added to unequal Things, the Wholes will be unequal. V. If equal Things be taken from unequal Things, the Remainders will be ungenal. VI. Things which are double to one and the same Tbing, are equal between themselves. VII. Things, which are' half one, and the same Thing, are equal between themselves. VIII, Things which mutually agree together, are equal to one another. IX. The Wbole is greater than its-Part .. X. Two Right Lines do not contain a Space. XI. All Right Angles are equal between themselves. XII. If a Right Line, falling upon two other Right Lines, makes the inward Angles on the same Side thereof, both together, less than two Right Angles, those two Right Lines; infinitely produc’d, will meet each other on that Side where the Angles, are less than Right ones. Note, When there are several Angles at one Point, any one of them is express’d by three Letters, of which that at the Vertex of the Angle is plac'd in the Middle. For Example ; In the Figure of Prop. XII. Lib. I. the Angle contain’d under the Right Lines AB, BC, is called the Angle ABC; and the Angle contain’d under the Right Lines AB, BE, is calựd the Angle A BE. PRO PROBLEM. Right Line. ET AB be the given finite Right Line, upon which it is required to describe an L Equilateral Triangle. About the Center A, with the Distance AB, describe the Circle BCD*;* 3 Polie and about the Center B, with the same Distance BA, describe the Circle ACE; and from the Point C, where the two Circles cut each other, draw the Right Lines CA, CBt. + Poft. Then because A is the Center of the Circle DBC, AC shall be equal to A Bt. And because B is the $ 15 Def. Center of the Circle CAE, BC shall be equal to BA: but CA hath been proved to be equal to AB; there- . fore both CA and CB are each equal to AB. But things equal to one and the same thing, are equal between themselves, and consequently CA is equal to CB; therefore the thțee Sides CA, A B, BC, are equal between themselves. And so the Triangle BAC is an Equilateral one, and is described upon the given finite Right Line AB; which was to be done. PROPOSITION II. PROBLEM. At a given Point, to put a Right Line equal to a Right Line given. to put a Right Line at the Point A, equal to the given Right Line BC. B 3 Draw |