Produced by Adrian Mastronardi, Hazel Batey and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) _With the respectful compliments of the author._ Address, Mr. R. BALLARD, Malvern, ENGLAND. Illustration: Plumb Line over Pyramids THE SOLUTION OF THE PYRAMID PROBLEM OR, PYRAMID DISCOVERIES. WITH A NEW THEORY AS TO THEIR ANCIENT USE. BY ROBERT BALLARD, M. INST. C.E., ENGLAND; M. AMER, SOC. C.E. CHIEF ENGINEER OF THE CENTRAL AND NORTHERN RAILWAY DIVISION OF THE COLONY OF QUEENSLAND, AUSTRALIA. NEW YORK: JOHN WILEY & SONS. 1882. Copyright, 1882, By JOHN WILEY & SONS. PRESS OF J. J. LITTLE & CO., NOS. 10 TO 20 ASTOR PLACE, NEW YORK. NOTE. In preparing this work for publication I have received valuable help from the following friends in Queensland:-- E. A. Delisser, L.S. and C.E., Bogantungan, who assisted me in my calculations, and furnished many useful suggestions. J. Brunton Stephens, Brisbane, who persuaded me to publish my theory, and who also undertook the work of correction for the press. J. A. Clarke, Artist, Brisbane, who contributed to the Illustrations. Lyne Brown, Emerald,--(photographs). F. Rothery, Emerald,--(models). and--A. W. Voysey, Emerald,--(maps and diagrams). CONTENTS. PAGE §1. The Ground Plan of the Gïzeh Group 9 Plan Ratios connected into Natural Numbers 13 §2. The Original Cubit Measure of the Gïzeh Group 15 §3. The Exact Measure of the Bases of the Pyramids 19 §4. The Slopes, Ratios and Angles of the Three Principal Pyramids of the Gïzeh Group 20 §5. The Exact Dimensions of the Pyramids 23 §6. Geometrical Peculiarities of the Pyramids 28 §6A. The Casing Stones of the Pyramids 32 §7. Peculiarities of the Triangles 3, 4, 5 and 20, 21, 29 35 §8. General Observations 38 §9. The Pyramids of Egypt the Theodolites of the Egyptian Land Surveyors 41 §10. How the Pyramids were made use of 44 §11. Description of the ancient Portable Survey Instrument 51 Table to explain Figure 60 64 §12. Primary Triangles and their Satellites;--or the Ancient System of Right-angled Trigonometry unfolded by a Study of the Plan of the Pyramids of Gïzeh 64 Table of some "Primary Triangles" and their Satellites 74