of, equal between themselves, each of these equal Angles is a Right one ; and that Right Line, wbich 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. A Term (or Bound) is that which is the Extreme of any Thing. XIV. A Figure is tbai which is contained under one or more Terms XV. A Circle is a plain Figure, contained under one Line, called the Circumference ; to whicb all Right Lines, drawn from a certain Point within the Figure, are equal. XVI. And that Point is called the Centre of the Circle. XVII. A Diameter of a Circle is a Right Line drawn ibrough the Centre, and terminated on both Sides by the Circumference, and divides the Circle into two equal Paris. XVIII. A Semicircle is a Figure contained under : a Diameter, and that part of the Circumfe rence of a Circle cut off by that Diameter. XIX. A Segment of a Circle is a Figure contain ed under a Right Line, and Part of tbe Circumference of the Circle (which is cut off by ibat Right Line.] XX. Right-lined Figures are such as are con tained under Right Lines. XXI. Three-fided Figures are such as are con tained under three Lines, XXII. Four sided Figures are such as are con tained under four Lines. XXIII. Many fided Figures are those that are contained under more iban four Right Lines. 1 XXIV. Of three-sided Figures, that is an Equi lateral Triangle, which bath three equal Sides. XXV. That an Isoceles, or Equicrural one, which hath only two Sides equal. XXVI. And a Scalene one, is that which bath ibree unequal Sides. XXVII. Also of three fided Figures, that is a Right-angled Triangle, which hath a Right Angle. XXVIII. That an Obtufe-angled one, which bath an Obtufe Angle. XXIX. And that an Acute-angled one, which baib tbree Acute Angles. XXX. Of four-sided Figures, that is a Square, whose four Sides are equal, and its Angles all Right ones. XXXI. Tbat an Oblong, or ReEtangle, which is longer than broad; but its opposite fides are equal, and all its Angles Right ones. XXXII. Thai a Rbombus, which haib 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 fame Plane, which, if infinitely produc'd both Ways, would never meet. POSTULAT E S. 1. RANT that a Right Line may be drawn from any one point to another. II. That a finite Right Line may be continued di really forwards. III. And that a Circle may be described about any Centre with any Distance. AXIOMS. B 2 A X I 0 MS. 1. THING $ equal to one and the fame 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 unequal. Tbing, are equal between tbemselves. Tbing, are equal between themselves. equal lo one another. Jelves. Right Lines, makes the inward Angles on the where the Angles are less than Rigbt ones. any one of them is express'd by three Letters, of PRO PROPOSITION I. PROBLEM. To describe an Equilateral Triangle upon a given finite Right Line. ET A B be the given finite Right Line, upon Triangle. About the Centre A, with the Distance A B, defcribe the Circle BCD*; and about the Centre B, **Poß. 3. with the same Distance B A, describe the Circle A CE*; and from the Point C, where the iwo Circles cut each other, draw the Right Lines CA, CBt. + Poß. 1. Then because A is the Centre of the Circle DBC, A C shall be equal to A B I. And because B is the t Def. 15. Centre of the Circle C A E, B C shall be equal to BA: But C A hath been proved to be equal to A B; therefore beth C A and C B are each equal to A B. But Things 'equal to one and the same Thing, are equal between themselves *, and consequently A C is * Ax. 1, equal to CB; therefore the three Sides C A, A B, B C, are equal between themselves. And so, the Triangle B A C is an Equilateral one, and is described upon the given finite Right Line A B; which was to be done. PROPOSITION II. PROBLEM. At a given Point to put a Right Line equal to a Right Line given. Line BC; it is required to put a Right Line at Draw • Poft. 3. Poft. I. Draw the Right Line AC from the Point A to C *; + 1 of tbis. upon it describe the Equilateral Triangle DAC+; | Poft, 2. produce D A and DC directly forwards to E and GI; about the Centre C, with the Distance B C, describe Now because the Point C is the Centre of the Circle + Def. 15. BGH, B C will be equal to CG t; and because D is the Centre of the Circle KGL, the Whole DL that are equal to one and the same Thing, are equal * Ax, 1. to one another *; and therefore likewile A L is equal Whence the Right Line A L is put at the given | Ax. 3. PROPOSITION III. PROBLEM 2 of tbis. Two unequal Right Lines being given to cut off a Part from ihe greater, equal to the leffer. given, the greater whereof is A B; it is required Put * a Right Line A D at the Point A, equal to the Line C; and about the Centre A, with the Diftance AD, describe a Circle D E F +. Then because A is the Centre of the Circle DEF, A E is equal to AD; and so both A E and Care each equal to AD; therefore A E is likewise equal to C I. And so, there is cut off from A B the greater of two given Right Lines A B ana C, a Line A E equal to the leffer Line C; which was to be done. PRO. & Poft. 3. Ax. 1. |