Euclid
DEFINITIONS
The Point
1i. A point is that which has position but no dimensions. A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions—that is, it has neither length, breadth, nor thickness.
The Line
2ii. A line is length without breadth. A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness.
3iii. The intersections of lines and their extremities are points.
4iv. A line which lies evenly between its extreme points is called a straight or right line, such as AB. If a point move without changing its direction it will describe a right line. The direction in which a point moves in called its “sense.” If the moving point continually changes its direction it will describe a curve; hence it follows that only one right line can be drawn between two points. The following Illustration is due to Professor Henrici: “If we suspend a weight by a string, the string becomes stretched, and we say it is straight, by which we mean to express that it has assumed a peculiar definite shape. If we mentally abstract from this string all thickness, we obtain the notion of the simplest of all lines, which we call a straight line.”
The Plane
5v. A surface is that which has length and breadth.
A surface is space of two dimensions. It has no thickness, for if it had any, however small, it would be space of three dimensions.
6vi. When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane. A plane is perfectly flat and even, like the surface of still water, or of a smooth floor.
Figures
7vii. Any combination of points, of lines, or of points and lines in a plane, is called a plane figure. If a figure be formed of points only it is called a stigmatic figure; and if of right lines only, a rectilineal figure.
8viii. Points which lie on the same right line are called collinear points. A figure formed of collinear points is called a row of points.
Question
Which statement BEST describes the organization of the ideas in the text?
Responses
A The concepts are shown from the general to the specific size.The concepts are shown from the general to the specific size.
B The ideas are presented in their order of importance.The ideas are presented in their order of importance.
C The elements described are ordered from the smallest to the largest.The elements described are ordered from the smallest to the largest.
D The ideas are given in the correct chronological sequence.
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The correct answer is C: The elements described are ordered from the smallest to the largest.
In the excerpt, Euclid starts with the most fundamental concept, the point (which has no dimensions), and then progresses to a line (which has one dimension), followed by a surface or plane (which has two dimensions), and concludes with plane figures (combinations of points and lines). This organization reflects a progression from the simplest and smallest concept to more complex and larger entities.
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