The Skeleton Proposition when filled up will appear as below, symbols and contractions being allowed.

PROP. 1.-PROB.

To describe an equilateral triangle on a given finite straight line. SOLUTION.-Psts. 3 and 1.-Pst. 3. A may be described from any centre at any distance from that centre.

Pst. 1. A line may be drawn from any one point to another. DEMONSTRATION.-Def. 15, and Ax. 1.-Def. 15. AO is a plane figure bounded by one continued line called its Oce, and having a certain point within it from which all st. lines drawn to the ce are equal. Ax. 1 Magnitudes which are equal to the same, are equal to each other.

USE AND APPLICATION.-This problem may be applied to the measurement of inaccessible lines, by drawing on wood or brass an equil. triangle, and using the instrument, by placing it at A, and along the line AB, so that C and B may be seen; then if it be carried along AB, until at B, C and A can be seen along the edges of the instrument, the side AB will have been traversed, and A B is equal to AC or BC. The AB, being measured, will equal the other distances, AC, or CB.

PART I.

GRADATIONS IN EUCLID.

ELEMENTS OF PLANE GEOMETRY.

BOOKS I. & II.

GRADATIONS IN EUCLID.

BOOK I.

DEFINITIONS.

1. A Point is that which has no parts, or which has no magnitude: it marks position.-THEON and PYTHAGORAS.

"A point is that of which there is no part."-EUCLID: or, "which cannot be parted or divided."-PROCLUS.

"A point is a monad having position."-PYTHAGORAS.

A mathema

tical point cannot be drawn; for a visible point is, in fact, a surface. 2. A Line is length without breadth; or it is extension in one direction.

A mathematical line cannot be drawn; for whatever is visible must have breadth.

A line is measured by the number of units, or monads, of length contained in it ;-as, 5 inches; 9 feet; 13 miles.

3. The extremities, or ends, of a line are points.

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

"A straight line is the shortest distance between two points."-ARCHIMEDES, adopted by LEGENDRE.

A straight line is that of which the extremity hides all the rest, the eye being placed in the continuation of the line."-PLATO. Plato's line was thus a visible, not a mathematical line.

5. A superficies, or surface, is that which has only length and breadth; it is extension in two directions.

6. The extremities, or boundaries, of a surface, are lines.