PAGE Prop. 18. Problem : { To find a line which shall be a mean proportional between 39 40 40 41 lateral triangle, &c., &c. The square on the diagonal of a square has twice the area To describe a parallelogram areally equal to a given equi From a given point outside the circumference to draw a The square on half a line has one-fourth the area of the The square on a line has to the square on a second line greater or less than the first the duplicate ratio, &c., &c. . 29. Theorem:-The rectangle contained by the diagonal of a square, &c., &c. 30A. Theorem:-The area of the rect. contained by the diag. of a sq., &c., &c. 47 30в. Theorem:-The area of the rect. contained by the greater side, &c., &c. 48 ,, 31. Theorem :-The square on the diagonal of a rectangle, &c., &c. 32. Theorem :-If a straight line be unequally divided into two parts, &c., &c. 50 50 51 51 53 40. Theorem: {" 41. Prop. 39. Problem :-To find the centre of description of a given arc. وو If a triangle be described standing upon the circumference in Problem :-To describe within a regular octagon a parallelogram, &c., &c. PAGE 56 57 58 Problem :-To describe within a regular octagon two triangles, &c., &c. 59 43. Problem : To construct an isosceles triangle having each of the angles at the base treble the vertical angle ,, 45. Problem :-To inscribe a regular nonagon in a circle. Euclid's Postulates; and Pure Geometry Some Remarks on "Euclid's Elements ;" with Problems and Suggestions To cut a given straight line in extreme and mean ratio; as a lineal problem. Prop. 1. Theorem :- -If with the one side of an angle as a radius, &c., &c. وو 36 Corollary:-The relationship of the transverse arc to the primary. 3. Problem :-To divide a given angle into any required number of equal angles Second Method of Trisection, &c. Introductory Note 4. Problem:-To trisect a given angle:-Supplementary Demonstrations To divide a given angle into any required number of angles 104 7. Problem :-To describe an equilateral triangle on a given straight line 11. Problem :-' 108 A succession of Lunar Arcs applied to ascertain the Circle's Ratio 116 The precise nature of the Fallacy in the Erroneous Conclusion To describe in a square a rhombus, such that the areal spaces on 2. Problem-To describe a quadrantal sector equal to a given circle 118 120 121 122 Lunar Problems. Prop. 1. Problem : { 3. Problem :-- Problem: To describe the quadrantal section of a reg. polygon, equal in area to a given octagon 124 The Lune of Hippocrates doubled. 125 PAGE PART FOURTH -Duplication of the Cube. Prop. 1. Problem :-To find two mean proportionals between two given straight lines . 127 shall be twice that of a given cube. (To find the dimensions of a cube of which the solid contents shall 128 129 To find a cube which shall have the ratio to a given cube of three To find a cube which shall have the ratio to a given cube of one to three. 6. Problem - To find a cube equal in solid contents to a given sphere Problem :-To find a sphere equal in solid contents to a given cube Problem :-To find a sphere having twice the solid contents of a given sphere. 133 133 PART FIFTH.-Quadrature of the Circle. Rolling a Circle or Arc on a Straight Line-Definitions and Postulate. 135 The part by which an octantal arc is greater than its sine is 139 Three Modes of Demonstration-Direct Geometrical Proof, Trigo- Inductive Reasoning based on known facts Scholium :-The fundamental denary (common) basis of "Number" and PAGE 142 Quadrature Prop. 3. Problem : Geometrical Quadrature and Rectification of the Circle To describe a square lineally equal to a given circle; or vice versâ 143 To describe a square areally equal to a given circle; , 4. Problem :— or vice versa 144 Corollary;-Each of the four angular figures, &c., &c. 144 As an Appendix to Part Fifth, it is purposed to append "Demonstration of the Circle's Ratio," published in 1879, with the Plates belonging to it. |