(1) EUCLI D's ELEMENTS. BOOKI. DEFINITIONS. A POINT, is that which hath no Parts, IV. A Right Line, is that which lieth evenly between its Points. V. A Superficies, is that which hath only Length and Breadth. VI. The Bounds of a Superficies are Lines. VII. A Plain Superficies, is that which lieth evenly between its Lines. VIII. APlain 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 fame Right Line. IX. If the Lines containing the Angle be Right ones, then the Angle is c is called a Right-lin'd Angle. B X. When 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. XI. An Obtuse Angle, is that which is greater than a Right one. XII. An Acute Angle, is that which is less than a Right one. XIII. ATerm (or Bound) is that which is the Extreme of any Thing. XIV. A Figure, is that which is contained under one, or more Terms. XV. A Circle, is a plain Figure, contain'd under one Line, called the Circumference; to which all Right Lines, drawn from a certain Point within the Figure, are equal. XVI. And that Point is called the Center of the Circle. XVII. A Diameter of a Circle, is a Right Line drawn through the Center, and terminated on both Sides by the Circumference, and divides the Circle into two equal Parts. XVIII. A Semicircle, is a Figure contain'd under a Diameter, 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 under a Right Line, and Part of the Circumference of the Circle [which is cut off by that Right Line.] XX. Right-lin'dFigures, are such as are contain'd under Right Lines. XXI. Three-fided Figures, are such as are contained under three Lines. XXII. Four-fided Figures, are such as are contain'dunder four. XXIII. Many fided 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 unequal Sides. XXVII. Also of Three-sided Figures, that is, a Right angled Triangle, which hath a Right Angle. XXVIII. 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-fided 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 fide than the other, angled, but not equal fided. which is Right 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 never meet. POSTULATE S. I. RANT that a Right-Line may be drawn G from any one Point to another. II. That a finite Right Line may be continued directly forwards. III. And that a Circle may be describ'd about any Center, with any Distance. I. T ΑΧΙΟM S. HINGS equal to one and the same 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 unqeual. VI. Things which are double to one and the same Thing, are equal between themselves. VII. Things, which are half one and the same Thing, are equal between themselves. VIII. İhings which mutually agree together, are equal to one another.. IX. The Whole 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 call'd the Angle ABE. PRO PROPOSITION I. PROBLEM. To describe an Equilateral Triangle upon a given finite ET AB be the given finite Right Line, About the Center A, with the Di- 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, CB†. + Post Then because A is the Center of the Circle DBC, AC shall be equal to AB. 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; therefore 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 three Sides CA, AB, 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 LET the Point given be A, and the given Right Line A, equal to the given Right Line BC. B3 Draw |