Those who are anxious for further information respecting the history of this ancient problem may be referred to the "Penny Cyclopædia," by Professor De Morgan, and the Rectification of the Circle, by Mr. Shanks.

() and tan.-1) each

239

The latter has computed tan." to 609 decimals, the base of Napier's logarithms to 137 decimals, the modulus of the common system of logarithms to 136 places of decimals, and the powers of 2 as far as 2721, commencing with 213, and proceeding 225, 237, 249, 261, &c.

I shall now conclude with the hope that what I have done may be useful to the teacher, the pupil, and the artisan, in facilitating the study and practice of a pleasing and important science, which has numerous applications in the ordinary course of practical experience.

ROBERT RAWSON.

PORTSMOUTH, May 1st, 1856.

EUCLID'S DEFINITIONS.

I.

A point is that which hath no parts, or which hath no magnitude.

II.

A line is length without breadth.

III.

The extremities of a line are points.

IV.

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

V.

A superficies is that which hath only length and breadth.

VI.

The extremities of a superficies are lines.

VII.

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

VIII.

"A plane angle is the inclination of two lines to one another in a plane, which meet together but are not in the same direction." IX.

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.

A

N.B.-"When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex 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, A B, C B, is named the angle A B C, or C BA; that which is contained by A B, D B, is named the angle A B D, or D B A; and that which is contained by D B, C B, is called the angle D B C, or C B D ; 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.”

X.

When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

XI.

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

XII.

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

XIII.

"A term or boundary is the extremity of any thing."

XIV.

A figure is that which is inclosed by one or more boundaries.

XV.

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.

XVI.

And this point is called the centre of the circle.

XVII.

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

XVIII.

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

XIX.

"A segment of a circle is the figure contained by a straight line, and the circumference it cuts off."

XX.

Rectilineal figures are those which are contained by straight

lines.

XXI.

Trilateral figures, or triangles, by three straight lines.

XXII.

Quadrilateral, by four straight lines.

XXIII.

Multilateral figures, or polygons, by more than four straight

lines.

XXIV.

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

XXV.

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

XXVI.

A scalene triangle is that which has three unequal sides.

XXVII.

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

XXVIII.

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

XXIX.

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

XXX.

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

XXXI.

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

XXXII.

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

XXXIII.

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.

XXXIV.

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

XXXV.

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

POSTULATES.
I.

Let it be granted, that a straight line may be drawn from any one point to any other point.

II.

That a terminated straight line may be produced to any length in a straight line.

III.

And that a circle may be described from any centre, at any distance from that centre.