4 Preliminaries Ch. 1
vice versa. But we must havesomeaxioms to get an approach under way.
1.3 Euclid’sThe Elements.........................
1.3.1
The Elementsinvolved the earliest surviving deductive system of reasoning, having
axioms or postulates and common notions, and proceeding by way of careful state-
ments of results and proofs. Up to c. 1800, geometry was regarded as the part of
mathematics which was best-founded logically. But its position was overstated, and
its foundations not completed until c. 1880-1900. Meanwhile the foundations of al-
gebra and calculus were properly laid in the 19th century. From c. 1800 on, some
editions used algebraic notation in places to help understanding.
1.3.2 Definitions...............................
The Greeks did not appreciate the need for primitive terms, andThe Elementsstarted
with an attempt to define a list of basic terms.
DEFINITIONS- APOINTis that which has no parts, or which has no magnitude.
- ALINEis length without breadth.
- TheEXTREMITIESof a line are points.
- ASTRAIGHT LINEis that which lies evenly between its extreme points.
- ASUPERFICIESis that which has only length and breadth.
- TheEXTREMITIESof a superficies are lines.
- APLANE SUPERFICIESis that in which any two points being taken, the
straight line between them lies wholly in that superficies. - APLANE ANGLEis the inclination of two lines to one another in a plane,
which meet together, but are not in the same direction. - APLANE RECTILINEAL ANGLEis the inclination of two
straight lines to one another, which meet together, but are not in the same
straight line. - When a straight line standing on another straight line, makes the adjacent an-
gles equal to one another, each of the angles is called aRIGHT ANGLE;and
the straight line which stands on the other is called aPERPENDICULARto
it.