PAGE

Prop. 18. Problem :

{

To find a line which shall be a mean proportional between
a greater and a lesser line of known magnitude
Problem :-To find a third proportional to two given straight lines.
Problem :-To three proportionals to find a fourth proportional, &c., &c.
Problem-To find a fourth proportional to three given straight lines

39

40

40

41

lateral triangle, &c., &c.

The square on the diagonal of a square has twice the area
of the lesser square

To describe a parallelogram areally equal to a given equi

From a given point outside the circumference to draw a
straight line which shall touch the circumference.

The square on half a line has one-fourth the area of the
square on the whole line
Within a given square to describe a square of which the
area equals half the area of the given square

The square on a line has to the square on a second line greater

or less than the first the duplicate ratio, &c., &c. .
The square on a line which is a mean proportional between two
other lines, &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.
Theorem :-If three lineal magnitudes be in continued proportion, &c., &c. .
Problem - Equiangular parallelograms having two of the sides, &c., &c.
Theorem :-The areas of rectilineal figures are proportional to the areas, &c., &c.
36. Problem :-To describe a parallelogram, a square, and a circle, &c., &c.
Similar trilateral figures having one, two, or all three of their

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
the half of a circle, &c., &c. .

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
Problem :-'
-To inscribe a regular heptagon in a circle .

,, 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.
Tentative Problems

Prop. 1. Theorem :- -If with the one side of an angle as a radius, &c., &c.
Scholium:-On the ultimate chord, &c. .

وو

36

Corollary:-The relationship of the transverse arc to the primary.
Scholium; and Sub. Problem:-To divide a given angle, &c.
Polysection by the First Method

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
equal each to each

104

7. Problem :-To describe an equilateral triangle on a given straight line

11. Problem :-'
-To inscribe in a circle a regular polygon of thirteen sides

108

A succession of Lunar Arcs applied to ascertain the Circle's Ratio 116
The result apparently confirms that of "Modern Methods"; namely
T= 3.14159...

The precise nature of the Fallacy in the Erroneous Conclusion
explained

To describe in a square a rhombus, such that the areal spaces on
both sides of the rhombus, &c., &c.

2. Problem-To describe a quadrantal sector equal to a given circle

118

120

121

122

Lunar Problems.

Prop. 1. Problem :

{

3. Problem :--
--To describe an octantal sector equal to a given circle

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
To find the dimensions of a cube of which the solid contents

shall be twice that of a given cube.

(To find the dimensions of a cube of which the solid contents shall
be one-half those of a given cube

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
Augmentation and Diminution of the Sphere.

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
Prop. 1. Theorem :-
an aliquot part of the are
Repetition of the Demonstration-Varied

The part by which an octantal arc is greater than its sine is
the tenth part of the arc length

139

Three Modes of Demonstration-Direct Geometrical Proof, Trigo-
nometrical Proof, Inductive Reasoning on direct Evidence.

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.