distinguished into parts. No progress in ascending higher can be expected, till a regular habit of demonstration is thus acquired; it is much to be feared, that he who has declined the trouble of tracing the connexion between the proposition already quoted and those that are more simple, will not be very expert in tracing its connexion with those that are more complex; and that, as he has not been careful in laying the foundation, he will never be successful in raising the superstructure.

COLLEGE OF EDINBURGH,

Dec. 1, 1813.

ELEMENTS

OF

GEOMETRY.

BOOK I.

DEFINITIONS.

I.

"A Point is that which has position, but not magnitude." (See Notes.)

II.

A line is length without breadth. "COROLLARY. The extremities of a line are points; and the inter"sections of one line with another are also points.

III.

"If two lines are such that they cannot coincide in any two points, "witnout coinciding altogether, each of them is called a straight "line."

"COR. Hence two straight lines cannot inclose a space. Neither can "two straight lines have a common segment; that is, they cannot "coincide in part, without coinciding altogether."

IV.

A superficies is that which has only length and breadth.

"COR. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines."

V.

A Plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

VI.

A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

The definitions marked with inverted commas are different from those of Euclid.

C

N. B. When several angles are at one point B, any one of themt 'is expressed by three letters, of which the letter that is at the ver'tex of the angle, that is, at the point in which the straight lines that 'contain the angle meet one another, is put between the other two 'letters, and one of these two is somewhere upon one of those straight 'lines, and the other upon the other line: Thus the angle which 'is contained by the straight lines, AB, CB, is named the angle 'ABC, or CBA; that which is contained by AB, BD, is named the 'angle ABD, or DBA; and that which is contained by BD, CB, is 'called the angle DBC, or CBD; but, if there be only one angle at 'a point, it may be expressed by a letter placed at that point; as the 'angle at E."

VII.

An obtuse angle is that which is greater than a right angle.

IX.

An acute angle is that which is less than a right angle.

X.

A figure is that which is enclosed by one or more boundaries.-The word area denotes the quantity of space contained in a figure, without any reference to the nature of the line or lines which bound it.

XI.

A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one

another.

XII.

And this point is called the centre of the circle.

XIII.

A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

XIV.

A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

XV.

Rectilineal figures are those which are contained by straight lines.

XVI.

Trilateral figures, or triangles, by three straight lines.

XVII.

Quadrilateral, by four straight lines.

XVIII.

Multilateral figures, or polygons, by more than four straight lines.

XIX.

Of three sided figures, an equilateral triangle is that which has three equal sides.

XX.

An isosceles triangle is that which has only two sides equal.

AAD

XXI.

A scalene triangle, is that which has three unequal sides.
XXII.

A right angled triangle is that which has a right angle.

XXIII.

An obtuse angled triangle, is that which has an obtuse angle.

XXIV.

An acute angled triangle, is that which has three acute angles.

XXV.

Of four sided figures, a square is that which has all its sides equal, and all its angles right angles.

XXVI

An oblong, is that which has all its angles right angles, but has not all its sides equal.

XXVII

A rhombus, is that which has all its sides equal, but its angles are not right angles.

ᄆᄆ

XXVIII.

A Rhomboid, is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.

XXIX.

All other four sided figures besides these, are called Trapeziums.

XXX.

Parallel straight lines, are such as are in the same plane, and which being produced ever so far both ways, do not meet.