The nature of the present work is such that perhaps few students will find interest in each part of it alike. The ends and the means appeal to separate classes: the antiquarian, whose are the ends, will look askance at the means, involving co-ordinates, probable errors, and arguments based on purely mechanical considerations; the surveyor and geodetist, whose are the means, will scarcely care for their application to such remote times; the practical man who may follow the instrumental details, may consider the discussion of historical problems to be outside his province; while only those familiar with mechanical work will fuUy realize the questions of workmanship and tools here explained. An investigation thus based on such different subjects is not only at a disadvantage in its reception, but also in its production. And if in one part or another, specialists may object to some result or suggestion, the plea must be the difficulty of making certain how much is known, and what is believed, on subjects so far apart and so much debated. The combination of two apparently distinct subjects, is often most fertile in results; and the mathematical and mechanical study of antiquities promises a full measure of success. It is sometimes said, or supposed, that it must be useless to apply accuracy to remains which are inaccurate; that fallacies are sure to result, and that tie products of such a method rather originate with the modem investigator than express the design of the ancient constructor. But when we look to other branches of historical inquiry, we see how the most refined methods of research are eagerly followed: how philology does not confine itself to the philological ideas of the ancient writers, but analyzes their speech so as to see facts of which they were wholly unconscious; how chemistry does not study the chemical ideas, but the chemical processes and products of the ancients; how anthropology examines the bodies and customs of men to whom such inquiries were completely foreign. Hence there is nothing unprecedented, and nothing impracticable, in applying mathematical methods in the study of mechanical remains of ancient times, since the object is to get behind the workers, and to see not only their work, but their mistakes, their XIV INTRODUCTION. amounts of error, the limits of their ideas; in fine, to skirt the borders of their knowledge and abilities, so as to find their range by means of using more comprehensive methods. Modem inquiry should never rest content with saying that anything was " exact;" but always show what error in fact or in work was tolerated by the ancient worker, and was considered by him as his allowable error. 2. The materials of the present volume have been selected from the results of two winters' work in Egypt. Many of the points that were examined, and some questions that occupied a considerable share of the time, have not been touched on here, as this account is limited to the buildings of the fourth dynasty at Gizeh, with such examples of later remains as were necessary for the discussion of the subject All the inscriptions copied were sent over to Dr. Birch, who has published some in full, and extracted what seemed of interest in others; Dr. Weidemann has also had some of them; and they do not need, therefore, further attention on my part. Papers on other subjects, including the Domestic Remains, Brickwork, Pottery, and travellers' graffiti, each of which were examined with special reference to their periods, are in course of publication by the Royal Archaeological Institute. The mechanical methods and tools employed by the Egyptians were discussed at the Anthropological Institute, and are more summarily noticed here.