DEFINITIONS OF BOOK III.

DEF. I. 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. This point is called the centre of the circle. A radius of a circle is a straight line drawn from the centre to the circumference.

DEF. 2.

DEF. 3. 4 diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

DEF. 4. An arc is a part of a circumference. Two arcs which together make the whole circumference are said to be conjugate. The greater of the two is called the major conjugate arc and the smaller the minor conjugate arc.

DEF. 5. The conjugate angles formed at the centre of a circle by two radii are said to stand upon the conjugate arc opposite them intercepted by the radii, the major angle upon the major arc, and the minor angle upon the minor arc.

DEF. 6. A sector is a figure contained by an arc and the

radii drawn to its extremities.

The angle of the sector is the angle

at the centre which stands upon the arc of the sector.

DEF. 7. A chord of a circle is the straight line joining any

two points on the circumference.

DEF. 8. A secant is a straight line of unlimited length which meets the circumference of a circle in two points.

DEF. 9. A segment of a circle is the figure contained by a chord and either of the arcs into which the chord divides the circumference. The segments are called major or minor segments according as the arcs that bound them are major or minor arcs.

DEF. 10. The angle formed by any two chords drawn from a point on the circumference of a circle is called an angle at the circumference, and is said to stand upon the arc between its arms.

DEF. 11. An angle contained by two straight lines drawn from a point in the arc of a segment to the extremities of the chord is called an angle in the segment.

DEF. 12. If the angular points of a rectilineal figure be in the circumference of a circle, the figure is said to be inscribed in the circle, and the circle to be circumscribed about the figure.

DEF. 13. The straight line which, meeting the circumference of a circle in one point, does not meet it again, is said to touch, or to be a tangent to, the circle at the point. The point is called the point of contact.

DEF. 13.

Aliter. If a secant of a circle alters its position in such a manner that the two points of intersection continually approach, and ultimately coincide with one another, the secant in its limiting position is said to touch, or to be a tangent to, the circle. The point in which the two points of intersection ultimately coincide is called the point of contact.

DEF. 14. Two circles are said to touch externally at a point where they meet, if each lies outside the other; to intersect if a part of each lies inside, and the remaining part outside the other; and to touch internally at a point where they meet, if one of them lies inside the other.

DEF. 15. called the inscribed circle of the triangle.

The circle that touches the three sides of a triangle is

DEF. 16. A circle that touches one side of a triangle and the other two sides produced is called an escribed circle of the triangle. If all the sides of a rectilineal figure touch a circle lying within the figure, the circle is said to be inscribed in the figure, and the figure is said to be circumscribed about the circle.

DEF. 17.

BOOK IV.

FUNDAMENTAL PROPOSITIONS OF PROPORTION.

INTRODUCTORY REMARKS.

In this book the subjects of the propositions, in the first Sections of both Parts I. and II., on Ratio and Proportion, are magnitudes in general, and not merely geometrical magnitudes—such as lines, angles, areas, etc. Many of the propositions are familiar, under somewhat different forms, to those who have studied Arithmetic and Elementary Algebra; but, as the starting-point of these latter sciences is the notion of number and not of magnitude in general, it is important that the more general treatment of Ratio and Proportion, either in the exact and complete form of Part II., or in the equally exact but restricted, and thereby simplified, form of Part I., should be mastered before proceeding to their application to Geometry.

A Magnitude is anything of which a greater or a less can be predicated.

Thus, number, length, area, value, probability, time, speed, weight, etc., and so also hunger, love, courage, talent, wisdom, etc., are magnitudes, each different in kind from each of the others,