6. Thus we have four distinct kinds of things or ideas (i) Space, (ii) Surface, (iii) Line, (iv) Point. A solid is a portion of space; it is bounded by surface; any portion of its surface is bounded by lines; any portion of a line is bounded. by points. We may notice that when a point moves, its path is a line; when a line moves its path is a surface; when a surface moves its path is a solid. Any material thing occupies a solid; the thinnest soap-bubble film is a thin solid. An immense number of thin soap-bubble films laid one on the other would make up a solid containing soapy-water. The finest thread woven by a spider is a very fine solid rod; and with a sufficient number of these threads we could fill a solid of any size. The very smallest grain of dust is a solid; and by adding enough of them together we could make a solid of any size. But by adding surfaces together we can only get larger surfaces; by adding lines we can never fill a surface; any number of points can never make up a continuous line. Surface has no thickness; a line has only length; a point has no magnitude. Hence we see the meaning of Euclid's Definitions. 7. DEF. A point is that which has no magnitude; it has position only. 8. DEF. A line is that which has length without breadth. 9. DEF. A surface is that which has only length and breadth; it has no thickness. 10. In thinking of the shape of a solid, we are not to consider that any material thing necessarily occupies that solid. The science of geometry is the science of solids, surfaces, lines and points and it has nothing whatever to do with the properties of matter. We can think of a cube, or a square, without being able to make a perfect cube or a perfect square. The pictures which we draw of solids, surfaces and lines are pictures merely; they are aids to the proper understanding of the subject; just as any other pictures explain and illustrate the subject of a book. We can think of a surface, without thinking of it as the boundary of a particular solid. We may think of a line independently of a particular portion of surface. We may 11. DEF. A combination of surfaces, lines and points is called a geometrical figure. 12. It is assumed in Euclid that any figure is capable of being moved from one part of space to another without alteration of its size or shape. For example, the sides of a square are a series of lines having a particular size and shape. We can think of this series of lines as having different positions in space, the lines themselves still forming the sides of the same square. We can suppose for example that one of the corners of the square is at some particular point in space. We can suppose that the figure is turned in some particular way; and so on. 13. DEF. A representation of a figure on paper, or otherwise, is called a diagram. We cannot draw a line; but we may if we please agree that a certain mark shall represent a line pictorially; just as we may represent anything we please in a picture. For instance a map gives a representation of fields, roads and rivers; indicating lines by visible marks of ink; and it is understood that the thickness of the 'lines' indicates various circumstances, such as the boundary of a county or of a parish or of a field but it does not indicate thickness or width of itself. It is usual to use the word figure when speaking of a diagram in Euclid; meaning the figure of which the diagram is a picture. 14. A Theoretical Diagram is a diagram in which we suppose, for the purpose merely of argument, that we have actual theoretical lines and theoretical points. A Practical Diagram is the nearest approach we can make to a theoretical diagram by means of the lines drawn on paper, or in some other way. The diagrams in this book all represent, or are pictures of, theoretical diagrams; the 'lines' are, for the purposes of argument, supposed to be theoretical lines. Just as the lines drawn in a map or plan are supposed to indicate theoretical lines. And when one line is printed thicker than another, the thickness is for some purpose other than to suggest that a line can have any thickness. B A The above is a diagram representing a triangle, or three straight lines joining three points. In the above diagram two of the 'lines' are printed thicker than the others. This is not meant to indicate that those lines are of any thickness at all. The thickness is simply for pictorial effect. 15. It is customary in geometrical diagrams to refer to any particular point in the figure which it represents, by means of a letter placed in the diagram near the point. Thus the letter A indicates the corner of the triangle near which it stands. Euclid's Test of Equality. 16. Magnitudes which coincide, or which if superposed would coincide, are equal to each other. If a point is applied to another point the two do not make a double point; or, a larger point; for a point has no magnitude. They simply 'coincide' or become one point. If a line is applied to another line so as to 'coincide' they do not make a thicker line; for a line has no thickness; they simply become one line. Euclid supposes that we can apply one figure to another without alteration of shape or size and thus demonstrate their equality or inequality the one with the other. |