SECTION III. THE AREAS OF PARALLELOGRAMS AND TRIANGLES.

PROPOSITIONS 35-48 .

56

72

NOTE ON THE METHOD OF LIMITS AS APPLIED TO TANGENCY

Theorems and Examples on Book III.

163

167

231

I. ON THE CENTRE AND CHORDS OF A CIRCLE

233

II.

ON THE TANGENT AND THE CONTACT OF CIRCLES.

The Common Tangent to Two Circles, Problems on

Tangency, Orthogonal Circles

235

IV. ON THE CIRCLE IN CONNECTION WITH RECTANGLES.

III. ON ANGLES IN SEGMENTS, AND ANGLES AT THE CEN-

TRES AND CIRCUMFERENCES OF CIRCLES.

The Orthocentre of a Triangle, and Properties of

the Pedal Triangle, Loci, Simson's Line

Further Problems on Tangency

240

251

The First Proposition of Book XII. will be found at p. 364,
worked out as an Example on Proposition 20 of Book VI.,
of which it is a development. Prop. 3 of Book XII. is
briefly treated as a Corollary to Prop. 1.

EUCLID'S ELEMENTS.

BOOK 1.

DEFINITIONS."

1. A point is that which has position, but no magnitude. 2. A line is that which has length without breadth.

3. The extremities of a line are points, and the intersection of two lines is a point.

4. A straight line is that which lies evenly between its extreme points.

Any portion cut off from a straight line is called a segment of it.

5. A surface (or superficies) is that which has length and breadth, but no thickness.

6. The boundaries of a surface are lines.

7. A plane surface is one in which any two points being 2. taken, the straight line between them lies wholly in that

surface.

A plane surface is frequently referred to simply as a plane.

NOTE. Euclid regards a point merely as a mark of position, and he therefore attaches to it no idea of size and shape.

Similarly he considers that the properties of a line arise only from its length and position, without reference to that minute breadth which every line must really have if actually drawn, even though the most perfect instruments are used.

The definition of a surface is to be understood in a similar way,