§ 9. Our assemblage of people would not be homogeneous unless its members were all equal and similar and in precisely similar attitudes, and were all looking the same way. You understand what a number of people seated or standing on a floor or plain and looking the same way means. But the expression ‘looking’ is not conveniently applicable to things that have no eyes, and we want a more comprehensive mode of expression. We have it in the words ‘orientation,’ ‘oriented,’ and (verb) ‘to orient,’ suggested by an extension of the idea involved in the word ‘orientation,’ first used to signify positions relatively to east and west of ancient Greek and Egyptian temples and Christian churches. But for the orientation of a house or temple we have only one angle, and that angle is called ‘azimuth’ (the name given to an angle in a horizontal plane). For orientation in three dimensions of space we must extend our ideas and consider position with reference to east and west and up and down. A man lying on his side with his head to the north and looking east, would not be similarly oriented to a man standing upright and looking east. To provide for the complete specification of how a body is oriented in space we must have in the body a plane of reference, and a line of reference in this plane, belonging to the body and moving with it. We must also have a fixed plane and a fixed line of reference in it, relatively to which the orientation of the moveable body is to be specified; as, for example, a horizontal plane and the east and west horizontal line in it. The position of a body is completely specified when the angle between the plane of reference belonging to it, and the fixed plane is given; and when the angles between the line of intersection of the two planes and the lines of reference in them are also given. Thus we see that three angles are necessary and sufficient to specify the orientation of a moveable body, and we see how the specification is conveniently given in terms of three angles. § 10. To illustrate this take a book lying on the table before you with its side next the title-page up, and its back to the north. I now lift the east edge (the top of the book), keeping the bottom edge north and south on the table till the book is inclined, let us say, 20° to the table. Next, without altering this angle of 20°, between the side of the book and the table, I turn the book round a vertical axis, through 45° till the bottom edge lies north-east and south-west. Lastly, keeping the book in the plane to which it has been thus brought, I turn it round in this plane through 35°. These three angles of 20°, 45°, and 35°, specify, with reference to the horizontal plane of the table and the east and west line in it, the orientation of the book in the position to which you have seen me bring it, and in which I hold it before you.