' 
 
THE ARMS AND CREST 
 
 UNITED STATES OF AMERICA, 
 
 AND THE 
 
 OBVERSE OF THE NATIONAL SEAL. 
 
 Ho! to the land shadowing with wings, which is beyond the rivers of Ethiopia: 
 that sendeth ambassadors by sea, even in ships of -whirling things upon the waters. 
 ISA. xviii. i. 
 
AN IMPORTANT QUESTION 
 
 IN 
 
 METROLOGY, 
 
 BASED UPON 
 
 RECENT AND ORIGINAL DISCOVERIES: 
 
 CHALLENGE TO "THE METRIC SYSTEM," 
 
 AND 
 
 AN EARNEST WORD WITH THE ENGLISH-SPEAKING PEOPLES 
 ON THEIR ANCIENT WEIGHTS AND MEASURES. 
 
 BY 
 
 CHARLES A. L. TOTTEN, M.A., 
 
 
 
 FIRST LIEUTENANT FOURTH UNITED-STATES ARTILLERY; PROFESSOR OF MILITARY SCIENCE 
 AND TACTICS AT "THE CATHEDRAL SCHOOL OF ST. PAUL;" LIFE-MEMBER OF INTER- 
 NATIONAL INSTITUTE FOR PRESERVING AND PERFECTING ANGLO-SAXON 
 WEIGHTS AND MEASURES; AUTHOR OF " STRATEGOS, THE 
 AMERICAN GAME OF WAR;" " OUR INHERITANCE 
 IN THE GREAT SEAL OF THE UNITED 
 STATES," ETC.. ETC. 
 
 'Remove not the ancient landmarks which thy fathers have set." 
 
 V;"; -i l\ ; ^ ??-> PROVERBS xxii. 
 
 NEW YORK : 
 JOHN WILEY & SONS, 
 
 15 ASTOR PLACE. 
 1884. 
 
COPYRIGHT, 1884, 
 BY LIEUT. C. A. L. TOTTEN, 
 
 FOURTH ARTILLERY. 
 
 JFranfclfn 
 
 RAND, AVERY, AND COMPANY, 
 BOSTON. 
 
TO THE 
 
 ENGLISH-SPEAKING PEOPLES 
 
 OF THE EARTH, WHO ALREADY POSSESS 
 
 THE WORLD'S GATES OF COMMERCE; WHO RAISE ITS 
 
 FOOD-SUPPLY, LOCK UP ITS SURPLUS, AND CONDUCT ITS TRADE; 
 
 WHO OWN ITS MINES, COIN ITS MONEY, AND CONTROL ITS INDUSTRY; WHO 
 
 INVENT ITS MEANS OF PROGRESS, CULTIVATE ITS INTELLECT, AND 
 
 ELEVATE ITS RELIGION; WHO TONE ITS MORALS, 
 
 LIBERATE ITS INHABITANTS, AND STEM 
 
 ITS TIDE OF ERRORS, 
 THIS BRIEF SURVEY OF A TOPIC PREGNANT 
 
 WITH WEIGHTY IMPORT 
 
 IS MOST RESPECTFULLY DEDICATED 
 
 BY THE AUTHOR. 
 
 226551 
 
"/ swear unto the Soul of Isis> that I have not 
 altered the Sacred Cubit of my fathers." 
 
 THE OATH OF THE EGYPTIAN DEAD (MONUMENTS OF EGYPT). 
 
METROLOGY. 
 
 " But thou shalt have a perfect and just WEIGHT, a 
 perfect and just MEASURE shalt thou have : that thy days 
 may be lengthened in the land which the LORD thy God 
 giveth thee." DEUTERONOMY xxv. 15. 
 
"O cos /AoAiora iravrwv ycco/xcrpet. 
 
 " He created Wisdom, and saw her, and numbered her, 
 and poured her out upon all his works." ECCLESIASTICUS i. 9. 
 
 "Thou hast ordered all things in measure and number 
 and weight.^ WISDOM xi. 20. 
 
PREFACE. 
 
 " And of Joseph he said, Blessed of the LORD be his land, for the precious things of 
 heaven, for the dew, and for the deep that coucheth beneath, 
 
 And for the precious fruits brought forth by the sun, and for the precious things put 
 forth by the moon, 
 
 And for the chief things of the ancient mountains, and for the precious things of the 
 lasting hills, 
 
 And for the precious things of the earth and fulness thereof, and for the good will of him 
 that dwelt in the bush : let the blessing come upon the head of Joseph, and upon the top of 
 the head of him that was separated from his brethren. 
 
 His glory is like the firstling of his bullock, and his horns are like the horns of unicorns : 
 with them he shall push the people together to the ends of the earth : and they are the ten 
 thousands of Ephraim, and they are the thousands of Manasseh." DEUT. xxxiii., 13-17. 
 
 THE race to whom these pages are dedicated is by blood, by letter, 
 and by the spirit, literally " called in Isaac's name." As Saxons, they 
 are the lineal descendants of the " Saka-i-Sunnia," or " Sons of Saac ; " 
 while, by the particular branch through which they derive their lofty 
 genealogy, they are the posterity of Joseph, "the beloved son," "to 
 whom pertained the birthright." 
 
 The Egyptian extraction of his two sons, Ephraim and Manasseh, 
 bequeaths to them, together with all the blessings "of him who was 
 separated from his brethren," an inheritance none the less royal and 
 significant in the mysterious land of their mother, "Asenath, the daughter 
 of Potiphera, Prince of On." Though, like another and a greater Son of 
 Jacob, it is true that they were, in early days, " called out of Egypt," it 
 is none the less true, that the summons was simply given to them for the 
 purpose of extending the possibilities of their ultimate dominion. 
 
 In these latter days, when the ends of the earth seem literally to draw 
 nigh unto us, when every nation is so deeply impressed with the uncer- 
 tainties surrounding the long-contested solution of " the Eastern ques- 
 
x PREFACE. 
 
 tion," when an intense and ever-increasing expectancy has settled down 
 upon the whole human race, it may well be asked if it is not a little re- 
 markable to see the flags of these two brother nations united for the first 
 time, since their independence, in the streets of Alexandria? In 1882 the 
 bronzed tars of England and America, of " Brothers John and Jonathan," 
 the only nations called " brethren " upon the face of the earth, 
 landed together on the shores of the delta of that ancient stream upon 
 whose banks their fathers, also brothers, had lived as princes twice 
 eighteen centuries before. Shall we endeavor to convince ourselves, that, 
 in this act, there was no overruling exercise of that Will which weaves 
 the thread of destiny? or shall we cease to doubt, and yield to the con- 
 viction that there is indeed a power that giveth the dominion unto whom 
 it will? 
 
 In the same year, 1882, both England and America struck off com- 
 memorative medals, upon whose reverse faces the two most mysterious 
 emblems of Egypt, the Sphinx and Pyramid, were severally displayed as 
 central devices. In their inception these medals had not the remotest 
 connection. The one was the Egyptian war-medal of Great Britain : 
 the other was the centennial seal-medal of the United States. Neverthe- 
 less, a strenuous effort was made to induce the Queen to adopt the 
 Pyramid as the central emblem for the British war-medal. It had already 
 been used by America that very year ; and it was not likely that an occa- 
 sion for its simultaneous employment by the two nations would soon, if 
 ever again, occur. But Ephraim is not Manasseh, and so the idea of 
 the Sphinx was adhered to by our fraternal nation as for it the most 
 appropriate. Was Providence, which counts the hairs upon a human 
 head, also an unconcerned spectator then? And were these matters 
 really trivial things, and, after all, of no historic moment ? 
 
 " There's a divinity that shapes our ends, 
 Rough-hew them how we will." 
 
 And it was no accident that the greatest commercial city of each of 
 these two brother and Egyptian nations was at this same time graced 
 with one of the two obelisks, that, when their father Joseph married the 
 Princess Asenath, had stood, like Jachin and Boaz, in strength and 
 beauty on each side the portal of her father's temple. 
 
PREFACE. xi 
 
 Who, indeed, shall say that, in youthful sports around the entrance to 
 that noble shrine, their fathers did not choose, as children do to-day, and 
 even name as " Ephraim " and " Manasseh," each one, the self-same 
 pillar, which, in centuries then to come, the powers that overrule have 
 now brought by such natural means to the more modern homes of their 
 descendants, and have stationed at the very gates of all their greatness? 
 
 There is undoubtedly an inheritance in the land of Egypt for the 
 Anglo-Saxon race, and the day has dawned when it shall be given unto 
 those whose right it is. 
 
 The word IV (zr) is the Hebrew equivalent for the Egyptian hiero- 
 glyphic Ra, and this latter is one of the most significant names by which 
 the Great Pyramid of Gizeh was known. It signifies rock, or the rock. 
 
 In the overflow of the Nile, this monument seems to rise out of the very 
 water itself, and thus to be a unique symbol of the fabled land on which, 
 it has stood since long before the days of Joseph. The Hebrew word 
 for water is D"D (maim) ; and if we read these two words, water and 
 rock, combined, we have D"iyo (mizraim), or the word employed for 
 Egypt itself throughout the Hebrew text. It literally signifies the rock 
 out of the water. Thus, as Mr. J. Ralston Skinner shows, the very name 
 of this land contains a symbol of its importance in a picture which may 
 be represented by sketching a river of water with a rock pyramid rising 
 therefrom upon its bank. The word "iy being put upon the pyramid, 
 above the water, and the one. D'~o, so divided, being written beneath the 
 water, a part upon either side of the pyramid. The two read, thus com- 
 bined, from right to left, as in the Hebrew, give us as one word the 
 famous name of Egypt, the cabalistic symbol of the earth itself saved 
 
xii PREFACE. 
 
 from the greater flood, and, even at an earlier date, drawn from the 
 maternal waters of the very womb of chaos. 
 
 In the following pages I am going to request the descendants of these 
 two branches of Joseph's family to look with me " unto the -i whence 
 they are hewn," and listen to some of the momentous truths with which 
 its metrologically proportioned blocks reply, in cosmic ratios, to the 
 grand dimensions of the earth on which they live. 
 
 That Jehovah has said, " Out of D"TOO have I called my son," is a 
 fact significant of the essential importance which this land has always 
 had, and ever will have, in the divine economy of the Scriptures, rightly 
 understood. From the earliest record of the dawn of time, when first 
 "the Spirit of God moved upon the face of the waters " (DTD), unto its 
 close, as described in Revelation, when, we are told, there shall be no 
 more sea, the whole Bible is harmonious in its employment of these 
 
 roots ; and whether it refers to the actual creation of the earth, the macro- 
 cosm, as in the first chapter of Genesis ; or of the Pyramid, the mesocosm, 
 or intermediate type thereof, as in the thirty-eighth chapter of Job ; or to 
 the lightening up of the darkness which covers the face of the deep 
 submerging the unregenerated soul of man, the microcosm, as it does 
 throughout its pages, the same fundamental cabalistic play upon 
 these mystic words occurs. Truly, this rising from the waters, this idea 
 of baptism, has a meaning as far-reaching as the deep from which it 
 springs. It is not in vain that creation and that the soul of man, covered 
 with primeval waters, " cry out of the deep unto Him who hath formed 
 the heavens and earth and all that therein is ; " and we may rest assured, 
 that in the beautiful proportions of that nv, thus raised from out the 
 
PREFACE. xiii 
 
 midst of the D'D, at the centre and the border of this fabled land of 
 mystery, D'lVO, we may learn a lesson whose teachings to us, who in 
 former days were, in Ephraim and Manasseh, thus just as truly lifted out 
 of the Egyptian deep, will perhaps be of most momentous import. 
 
 In our treatment of this subject, we shall first examine into facts which 
 lie near home, and see in how much, or in how little, the Anglo-Saxon 
 race is actually in possession of the blessings promised unto Ephraim 
 and Manasseh, as the sons of Joseph. We shall then as our readers to 
 accompany us through some studies of this remarkable monument, and 
 by the way shall gather not a little from the God-designed metrology of 
 Israel, of highly scientific import. 
 
 The hierarchy of science, so called, has long since agreed to disregard 
 as totally unreliable every structure a single stone of whose foundation 
 has been quarried from the eternal word of God. Any appeal to the 
 Scriptures, no matter how modest, is so thoroughly at variance with the 
 modern methods of philosophers, that the book of an author who has 
 searched the Scriptures for guidance towards the eternal truth of things, 
 is condemned unread, stamped with the seal of disapproval unopened, 
 and burned relentlessly without a hearing, lest the people, having read 
 it, should pronounce it true, and learn to disregard their would-be 
 teachers. 
 
 But it is not to be expected that a book so full of stumbling-blocks, 
 offensive to their theories of evolution, as is the Bible, should find any 
 favor, or receive the least toleration, at the hands of modern scholars. 
 From the dust they love to feel that all around has risen, and without a 
 God ; and into the silt of a disintegrated, dead, and formless universe, 
 they trust at length their worn-out essences will sink into eternal sleep. 
 Modern science, disguise it as we may, is thus at heart not merely far 
 at sea upon the waves of doubt, but is essentially an atheistic school, 
 that has no God, and which has long since closed its doors against 
 the written Word. 
 
 From this school, therefore, the present volume does not expect a 
 single meed of praise. But here I am content. Were it to be other- 
 wise, were recognition granted to the thoughts advanced, by our self- 
 styled scientific teachers, I should feel that the light in which these 
 pages had been written was but an ignis fatuus from the swamp of things 
 
xiv PREFACE. 
 
 that soon must pass away. I write, however, for the people, whose con- 
 cern alone this matter is, and who still read the Word with simple trust ; 
 and, though my subject is of the highest scientific nature, I doubt not 
 they will find it clear, as truth is ever found to be, and not so intricate 
 but that it will be filled with what to every mind is deeply interesting, 
 and also find it to be well within the scope of even moderate education 
 and capacity. 
 
 As a matter of fact, the Bible deals at great length with this very subject 
 of Metrology. Around it the whole of the Hebrew polity harmoniously 
 arranges itself. And very naturally too ; for a just and perfect life was 
 all it aimed at inculcating ; and the very measure of fair dealing, of justice, 
 and of truth, is centred in, and squared and righted at, an honest and an 
 accurate standard, too sacred to be ever lengthened or diminished by 
 any possibility of double dealing. . 
 
 As the study of Metrology inevitably leads us to the study of the Great 
 Pyramid of Egypt, so, too, it leads all dwellers in the land of Manasseh 
 to look with greater interest on the arms and crest and seal chosen for 
 the nation's blazonry by ancestors who wrought more wisely than they 
 knew. The United States of America has been a nation marked out by 
 special manifestations of Divine Providence from their very beginning 
 until now. 
 
 It was in their earliest struggles that they looked towards this Western 
 wilderness ; and, behold, the glory of the Lord appeared in the cloud, 
 and led them to their favored habitation. By faith, like Abraham, their 
 ancestor, when called upon to go out into a place which they should 
 afterwards receive for an inheritance, they obeyed, and went out. not 
 knowing whither they went, and dwelt in their land of promise as in a 
 strange country. But the clouds of the Almighty were about their habi- 
 tation, so that the sun smote them not by day, neither the moon by 
 night. It is, therefore, in their crest they fittingly commemorate how by 
 faith there sprang from even one, and him as good as dead, so many as 
 the stars of the sky in multitude. It is, therefore, in their motto they 
 repeat this reference, and intensify the idea of union by that beautiful 
 allusion to the universal brotherhood of all mankind, who in Christ, as 
 Saint Paul truly says, are " many in one" 
 
 This was, indeed, the nation that fled into the wilderness borne upon 
 
PREFACE. xv 
 
 eagle's wings, the Goddess of Liberty, clothed with the sun, bathed in 
 the cloud-reflected colors of her flag, and crowned with the stars that 
 marked the union of her States, and pointed out their lofty origin. And 
 her eagle guardian was the Lord of hosts himself; for as an eagle stirreth 
 up her nest, fluttereth over her young, spreadeth abroad her wings, taketh 
 them, beareth them on her wings, so the Lord alone did lead his people 
 towards the land of freedom, and there was no strange god with him. 
 
 Truly, then, may all the nations of the earth exclaim, " Who is like 
 unto thee, O people saved by the Lord, who is the shield of thy help ? 
 thine enemies shall be subdued unto thee, and thou shalt tread upon 
 their high places. In peace thou shalt be like unto thy father Joseph, 
 a branch planted by the rivers of water that bringeth forth his fruit in his 
 season : thy leaves also shall not wither, and behold whatsoever thou 
 doest it shall prosper." 
 
 Then may Jeshurun, the wise people, fittingly reply, " There is, indeed, 
 none like unto the God of Manasseh, who rideth upon the heavens in 
 my help, and in his excellency on the sky. The eternal God is my 
 refuge, and underneath are the everlasting arms." 
 
 Upon the reverse of our national seal, the references to our birthright 
 as descendants of Joseph and Manasseh, and thus of Egyptian origin, 
 are even still more pointed. The leading motto, " ANNUIT COEPTIS," 
 " He has prospered our beginnings," is a direct use of an expression 
 so often reiterated in the Bible -story of Joseph, that he has become the 
 very type of " a prosperous man." 
 
 In the capstone we have again, not only the emblem of that Divine 
 Providence which crowned our efforts as a struggling people, but of the 
 Saviour of his-people, in whom alone our building, fitly joined together, 
 groweth upward into that perfect union of the human and divine. The 
 building, a pyramid unfinished, an emblem of stability, of perfect 
 measure, just weight, and of eternal truth, and harmony with nature, 
 man, and God, is eminently the Egyptian emblem of Manasseh. The 
 date upon its bas is his year of maturity, " MDCCLXXVL," and 
 marks the dawn of another golden age, as the motto below expressly 
 indicates : 
 
 "NOVUS ORDO SECLORUM." 
 
xvi PREFACE. 
 
 This motto is an intentionally altered quotation from Virgil's Fourth 
 Eclogue, and was borrowed in turn by Virgil from the mystic Sibylline 
 records. The text opens as follows : 
 
 "Ultima Cumaei venit jam carminis aetas; 
 Magnus ab integro sadorum nascitur ordo. 
 Jam redit et Virgo, redeunt Saturnia regna ; 
 Jam nova progenies ccelo demittitur alto ; 
 Tu modo nascenti puero, quo ferrea primum 
 Desmet ac toto surget gens aurea mundo, 
 Casta, fave, Lucina," etc. 
 
 Translation. 
 
 " The last age of Cumean song now comes ; * 
 Nevus ordo sedorum* a mighty order of ages is born anew. 
 Both the prophetic virgin 3 and the Saturnian kingdoms* now return; 
 Now a new progeny 5 is let down from the lofty heavens ; 6 
 Favor, chaste Lucina, the boy 7 soon to be born, 
 In whom the iron age 8 shall come to end, 
 And the golden one arise again in the whole earth," etc. 
 
 Words would be exhausted in any attempt to do justice to the thoughts 
 that find birth in the contemplation of the Air"- '^n era. Unique with 
 the rest of the symbolism upon the long-concealed face of our Great 
 Seal, this motto comprehends in itselt the whole of the Virgillic Sibylline 
 fragment just translated. It is unique in its reference to the birth and 
 genius of American institutions, institutions that cannot pass away, and 
 whose full development no hand can stay from reaching the goal ot their 
 most perfect realization. The new order of things has been let down 
 from heaven, not again to be withdrawn from earth. But this motto is 
 also most beautifully in harmony with the entire reverse of the seal itself ; 
 and it conceals a hidden reference to the Great Pyramid above, the 
 legacy of an earlier Golden Age to ours. 
 
 1 The seventh, or last, or perfect, cycle. 2 The sabbatic age of rest, the age of freedom. 
 3 The Goddess of Liberty. 4 The new republic. s A progeny among nations, diverse 
 from all predecessors a government " of the people, by the people, and for the people ! " 
 a nation of independent States, and yet the union of a multitude of individuals. b " The many 
 and signal interpositions of Providence in our behalf." 7 "Young America." 8 Old-World 
 ideas and bondage. 9 One founded upon individual liberty, freedom, and progress. "All 
 men are born free and equal." 
 
PREFACE. xvii 
 
 In this, " the last age of Cumean song," it is our task to rebuild the 
 monument of just weights and perfect measures. In the day of liberty, 
 now fully dawned, the recognized equality of all, demands, as the founda- 
 tion of society, perfect justice in the dealings of man with man ; and it is 
 only in the rediscovery of the secrets of true pyramidal construction that 
 the new order of the ages can be founded in stability. 
 
 When at length, therefore, we Americans, as the children of Ma- 
 nasseh, have fully come to read our title clear to this inheritance, so 
 grand and so far-reaching, how pointedly will the blessing of the great an- 
 cestor who adopted us, for we were half Egyptian, and made us 
 equal to his own, be named and numbered on us ! 
 
 "He" (Manasseh, said Jacob when he blessed our fathers) "also 
 shall be a great people" Thus he, whose name was changed to Israel, 
 made us greater than his own ; since from them all he took the birthright, 
 and conferred it upon the two adopted sons of Joseph ! upon Ephraim 
 and upon Manasseh, upon England and America; that is, upon the 
 Anglo-Saxon race. 
 
CONTENTS. 
 
 PREFACE. 
 
 PAGE 
 
 The Race addressed ix 
 
 England and America x 
 
 The Cabalistic Origin of Metrology xi 
 
 Science essentially Atheistic xiii 
 
 The Significance of our Heraldry . " .-''..' xiv 
 
 Pyramid Metrology Manasseh's Inheritance xvii 
 
 INTRODUCTION. 
 
 Importance of the Subject . . . . . . . , I 
 
 The Powers of Congress ". '. '. .';''. . . . 2 
 
 Garfield and the International Institute *. ' 3 
 
 The Earth-commensuric Inch ' .- 1 . - . . . . . 4 
 
 The Pyramid as a Factor in this Question '. . > . . . 4 
 
 PART I. 
 WEIGHTS AND MEASURES. 
 
 Why Anglo-Saxon Metrology should not be abandoned . : . . ., 6 
 
 The Metric System versus the Anglo-Saxon . . . . r ....*, . 7 
 
 The English-speaking Nations, and the Commerce of the Earth . . .. ; 8 
 
 The Balance-Sheet of the World 9 
 
 Remarkable Facts, and the Future of Anglo-Saxondom .... 10 
 
 Our Duty certainly to preserve and perfect Our Own System 16 
 
 Destroy not the Ancient Landmarks 17 
 
 PART II. 
 LINEAR MEASURE. 
 
 Our Linear Unit, the Inch, grandly Earth-commensuric 18 
 
 Determination of Polar Axis 21 
 
 An Additional Standard Decimal and International Table 23 
 
 The Possible System absolutely Perfect 24 
 
 Our Present Tables may be rectified and preserved 26 
 
 No Change of Names necessary ......... 26 
 
 The Duty of the Anglo-Saxon Race 27 
 
 xix 
 
XX CONTENTS. 
 
 PART III. 
 CAPACITY MEASURE. 
 
 PAGE 
 
 The Coffer in the Great Pyramid ........ 29 
 
 The Caldron of the Anglo-Saxons . . 29 
 
 The Ark, or Laver, of the Hebrews . .. ; 30 
 
 The Three Standards Identical . . . . *^ . . . . ' . 32 
 
 The Pyramid a Metric Monument . . 32 
 
 Professor Smyth's Table . . . . Y 32 
 
 Noticeable Correspondencies - .-, .--- . 33 
 
 A Proposed Standard Decimal and International Table ..... 34 
 
 Further Considerations . . .- . , : 35 
 
 Table continued downward . . 37 
 
 Table continued upward . . . . . ... 38 
 
 Comparison of Results . . . -- ....... . . 38 
 
 Roots of the System . . . . . ^-n - '.-'' '-' : '-. . . . 40 
 
 Comparative Table . . . * . =-. '- - . . - 42 
 
 What have we to do with the Metric System? 43 
 
 PART IV. 
 WEIGHT MEASURE. 
 
 Relation of Capacity to Weight ''..> . . . . . . 44 
 
 The Standard Chamber at Gizeh . . . .--.'. ... 45 
 
 Present Anglo-Saxon Weight Measures . - ' *'- . . . -45 
 
 Their Lack of Harmony . . . ... ... . 46 
 
 Pyramid Weight Measure 47 
 
 Its Ultimate and Grain rectifies Anglo-Saxon Measure . 47 
 The Use of a Greatest Common Divisor . ... . . .48 
 
 Origin of British Metrology 49 
 
 The Saxon Tables as Unified 51 
 
 Proposed Decimal and International Table 52 
 
 Constants of System . . . . . 54 
 
 Peculiar Subdivision of the Ounce 54 
 
 Possibilities of Complete Resolution 55 
 
 The Last Drop in the Coffer 55 
 
 The System continued downward 56 
 
 The Cubic Inch in Grains and Ultimates ...... 56 
 
 Mean Density of the Earth . 57 
 
 Its Use in connecting Weight and Capacity 58 
 
 The System continued upward . . . . 60 
 
 Value of Cubic Inch in Grains and Ultimates a Most Important Number . 61 
 
 Wonderful Significance of this Cubic Inch . 61 
 
 Can the Metric System accomplish Such Feats ? 64 
 
 Position of Mr. Proctor and President Barnard ...... 64 
 
 Manufacture of Weights 65 
 
 The Teachings of the Great Pyramid 67 
 
 Ancient Metrology of Israel Earth-commensuric 71 
 
 Wonderful Cubical Properties involved -.. . 74 
 
CONTENTS. xxi 
 
 The Challenge to the Metric System . . 75 
 
 The French System examined 77 
 
 Its only Feature of Value not its Own 78 
 
 Duty of the Few who use the Metric System 79 
 
 Duty of the Many who use the Saxon System 79 
 
 PART V. 
 RECAPITULATION. 
 
 Principles upon which the Rectified System is based Si 
 
 Its Primary Unit . '- . . , , , -.- . ... . . 84 
 
 Standard Physical Conditions . . . .'-, f . . ': . .... 85 
 
 Special Standards . . . . .,'... 86 
 
 Units and Primaries . . . . . ,- ,,-' -, - ... , 87 
 
 Constants 88 
 
 PART VI. 
 AEROSTATICS. 
 
 The Atmosphere in its Relation to the Rectified System 90 
 
 Its Weight 90 
 
 Its Specific Gravity 90 
 
 Its Mean Barometric Height 90 
 
 The Dew-Point .'.--.. 91 
 
 Moisture ............. 91 
 
 Rate of Expansion _.... 92 
 
 Actual Height of Atmosphere . 92 
 
 PART VII. 
 CONCLUSION. 
 
 Metrology the Universal Science 95 
 
 There is no Exception to the Law of Number 95 
 
 Man's Love of Exact Numbers 96 
 
 No Such Thing as " Mere Coincidence " 97 
 
 How Nature replies to us in Terms of the New Metrology .... 98 
 
 The Standard Cube 98 
 
 The Unit Cube 99 
 
 Solomon's Wisdom ... TOO 
 
 The Babylonian pivision of the Circle ........ 100 
 
 Its Chronological Employment by Daniel Appropriate .... 104 
 
 Daniel's Chronological Scheme 106 
 
 These Facts are Older than Solomon . . 107 
 
 The Great Pyramid 107 
 
 The Standard, or King's Chamber 108 
 
 The Sanctum Sanctorum of All Science . . m 
 
XX11 
 
 CONTENTS. 
 
 The Sacred Cubit 
 
 APPENDICES. 
 APPENDIX A. 
 
 PAGE 
 
 "3 
 
 Rational Circular Subdivision 
 
 APPENDIX B. 
 
 203 
 
 APPENDIX C. 
 
 Metrological Notes and Queries . 
 
 Note i. The Origin of the Grain .* 
 Note 2. Units and Constants . - 
 Note 3. The Dimensions of the Coffer 
 Note 4. Postscript . . *." . 
 
 213 
 215 
 
 222 
 
 22 5 
 230 
 
ILLUSTRATIONS. 
 
 PLATES. 
 
 PAGE 
 
 The Arms and Crest of the United States . . . . . FRONTISPIECE. 
 
 The Cabalistic Origin of "Egypt "-"V ". xi 
 
 The Pyramid from an actual Photograph ........ xii 
 
 The Reverse to the Great Seal of the United States ^^^-~- " B--^ .68 
 
 FIGURES. 
 
 FIG. i. Direct Vertical Section of Great Pyramid 119 
 
 FIG. 2. Diagonal Vertical Section of Great Pyramid 119 
 
 FIG. 3. Equality of Boundaries 119 
 
 FIG. 4. 7r-angle of Casing-Stones ,~ 119 
 
 FIG. 5. Equality of Areas No. I 119 
 
 FIG. 6. Equality of Areas No. 2 119 
 
 FIGS. 7, 8, 9. Equality of Areas No. 3 120 
 
 FIG. 10. Equation of Boundaries . 120 
 
 FIG. ii. Pyramid Diagram, Ideal 122 
 
 FIG. 12. Pyramid Diagram, Workman's . . . . ... 123 
 
 FIG. 13. Temple Diagram . . 129 
 
 FIG. 14. Pyramid and Temple harmonized ........ 134 
 
 FIG. 15. The Ante-Chamber verified 139 
 
 FIG. 1 6. The Pyramid Diagram and the " Boss " . . . . . .143 
 
 FIG. 17. Workman's Rule for "squaring the Circle" 148 
 
 FIG. 18. The Double Diagram 152 
 
 FIG. 19. The Standard Bar 157 
 
 FIG. 20. The f-part Division 160 
 
 FIG. 21. The Cubit, Square, and Circle 163 
 
 FIG. 22. The Cubit and Primary Circular Functions . . . . .167 
 
 FIG. 23. The Cubit and Pythagorean Triangle ...... 169 
 
 FIG. 24. The Point, Line, and Cubit 171 
 
 FIG. 25. The Cubit and Angle 171 
 
 FIG. 26. The Cubit and Equilateral Triangle 172- 
 
 FIG. 27. The Cubit and Square 173 
 
 FIG. 28. The Cubit and Pentalpha 173 
 
 FIG. 29. The Cubit and Hexagon 174 
 
 FIG. 30. The Cubit and Octagon 174 
 
 xxiii 
 
xxiv ILL USTRA TIONS. 
 
 Fro. 31. The Cubit and Decagon 175 
 
 FIG. 32. The Cubit and Duodecagon . . . . . . . -175 
 
 FIG. 33. The Cubit and 15-Sided Polygon 176 
 
 FIG. 34. The Cubit and i6-Sided Polygon. . . . . . . . 176 
 
 FIG. 35. The Cubit and Heptagon 177 
 
 FIG. 36. The Cubit and Nonagon 178 
 
 FIG. 37. The Cubit and Hendecagon 178 
 
 FIG. 38. The Cubit and 13-Sided Polygon . . . . . . .179 
 
 FIG. 39. The Cubit and 'i4-Sided Polygon . . .^ I T . . 179 
 
 FIG. 40. The Cubit and Rational Polygons . . . . . . . . 180 
 
 FIG. 41. The Cubit and Irrational Polygons . '. . . ' ' .. . 180 
 
 FIG. 42. The Cubit and Trigonometry . ; . . . . . . 181 
 
 FIG. 43. The Cubit and Number in the Abstract 182 
 
 FIG. 44. The Pyramid, Pentalpha, and Canon of Proportions . . . 187 
 
 FIG. 45. The Pyramid Diagram and Human Figure . . . ., . 188 
 
 FIG. 46. The Pyramid Diagram and Human Figure 189 
 
 FIG. 47. The Universal Problem 196 
 
 FIG. 48. The 240 Division of the Circle . . . . . . . .208 
 
METROLOGY. 
 
 INTRODUCTION. 
 
 " The powers not delegated to the United States by the Constitution, nor prohibited 
 by it to the States, are reserved to the States respectively, or TO THE PEOPLE." Art. X. 
 Amendments to the Constitution of the United States of America. 
 
 METROLOGY is the science of round numbers, just measures, 
 and of perfect weights. In it all other sciences find common 
 ground. Its scope is co-extensive with the farthest reach of 
 every special subject of investigation. There is no other sub- 
 ject that is worthy of sharing with it the holiest place in the 
 temple of universal civilization. It is the very headstone in 
 which the whole building fitly joined together shall find at last 
 its own ideal fully realized. Upon the mount of intellect, and 
 at the very dawn of time, its model was revealed to man ; and 
 ever since his aim has been to shape his mental architecture 
 by it. 
 
 Metrology has been the question of the ages. It is the 
 question of the present, and forever it will be a question of 
 most vital import to the human race. 
 
 To a brief consideration of this pregnant subject, I invite the 
 earnest attention of every statesman, and of every thoughtful 
 citizen. To such of them as still revere the old traditions of 
 our race, the review that we shall give cannot but prove both 
 interesting and important ; while to those who, willing to give 
 up their birthright, are now so assiduously and insidiously 
 advocating the repression of our hereditary weights and meas- 
 ures, and the compulsory adoption of a foreign system with 
 
2 METROLOGY. 
 
 which our practice and our history have naught in common, 
 the gauntlet is thrown down. 
 
 We challenge them to an open and a free discussion. Let 
 them produce their case. It is the PEOPLE'S QUESTION, this 
 one of WEIGHTS AND MEASURES ; and as a right inherent, and 
 one never ceded in the CONSTITUTION, they alone can change the 
 " times and seasons" 
 
 Our representatives have no more right to force the metric 
 system upon us than they have to make our babies beg for 
 bread in foreign idioms. It will be objected by some, that 5 
 Sec. viii., Constitution of the United States, conveys to Congress 
 the direct power to "fix the standard of weights and meas- 
 ures." This is the exact wording of the document, but it by 
 no means implies all that the advocates of change would have 
 us believe. A standard is one thing ; the unit quite another : 
 thus Congress has already and legitimately made the yard the 
 national standard of linear measure ; but as to the unity that is 
 fixed. The inch has never been disturbed, nor does the right 
 inherent exist in Congress to disturb it. It was never con- 
 templated that our representatives would sweep away our units, 
 any more than that they would attack the roots of our lan- 
 guage. Moreover, the very context of this paragraph convinces 
 us that the province of Congress was merely to regulate and 
 define the standard in such exact terms, that, as in the case of 
 the value of foreign coins, etc., future legal complications as to 
 values in the interchange of commodities might be avoided. 
 Within the limits of the Anglo-Saxon system the power of 
 Congress is clearly confined ; and the minute it passes to new 
 Units, or beyond the borders of our native metrology, expressed 
 and handed down from the remotest times in Anglo-Saxon 
 terms, and linked to Anglo-Saxon history, at that minute it 
 trespasses upon rights undoubtedly reserved unto the people. 
 If the French system of metrology is to become Anglo-Saxon, 
 it must become 3o by the popular vote of those whose right 
 alone it is to assume the inconvenience and the great responsi- 
 bility. 
 
 Let those, then, who are so continuously knocking at the 
 doors of Congress come rather to the forum ; let them mount 
 the rostrum in the market-place, and show their wares to those 
 
INTRODUCTION. 3 
 
 who buy and sell. It is there alone the question can be set 
 tied. A law made elsewhere upon such a subject can be 
 enacted but in vain. 
 
 In the mean time we offer to the people the following brief 
 review of our native Anglo-Saxon metrology, and claim that it 
 is the most truly earth-commensuric system now in use among 
 men. It needs but a slight rectification to make it absolutely 
 perfect. 
 
 This " rectified system " is our special theme. As an asso- 
 ciate of the International Institute for Preserving and Per- 
 fecting (Anglo-Saxon) Weights and Measures, these pages 
 originated in an address intended primarily for them. But 
 from the important character of the discoveries to which our 
 researches have led us, we are now induced to offer their 
 results to the Institute in a more public manner, and to invite 
 all others who are interested to an open meeting and a free 
 discussion. 
 
 Human law is absolutely powerless to enforce, as the UNIT of 
 metrology, that which is not harmonious to nature. Our ven- 
 erated and beloved GARFIELD/ in a speech at Boston in 1878, 
 regarding the standard of weights and measures, said, 
 
 " I challenge the intelligence of any man who hears me, to think of such 
 a thing as a measure of length which has no length in itself. No : by laws 
 HIGHER than human legislation, length, depth, and height were created ; 
 men can only name and declare a definite length as the standard." 
 
 We believe that our ancestors have bequeathed to us a sys- 
 tem based upon these HIGHER laws. By the attrition of full 
 four thousand years, our Anglo-Saxon system has lost so little 
 of its ancient truth, that we believe its present possessors 
 direct descendants of a mighty race may return to the ancient 
 PERFECT standards without any inconvenience, and without alter- 
 ing a single name familiar to our children and our history. 
 
 To a survey of these facts we therefore invite that race, 
 which hitherto has not removed its ancient landmarks, and ask 
 
 1 Garfield was the first elected president of this Institute. He took a deep interest 
 in the cause. In his letter of declension, Nov. 29, 1879, he says he thinks he can serve the 
 cause more effectually, and without indelicacy, as an independent judge, when Congress, of 
 which he is a part, shall be memorialized in relation to measures the Institute may bring 
 before it. 
 
4 METROLOGY. 
 
 that every Anglo-Saxon give them all the consideration that 
 they justly merit. The subject is too important, the time too 
 critical, the interests involved are too far-reaching, and the 
 labors of our national adversaries are too incessant, to admit of 
 any loss of time. We bring this matter to the bar of the 
 people themselves ; and to that bar we bid our adversaries 
 come, and bring this challenge with them. 
 
 In the following pages, the present Anglo-Saxon linear inch 
 being taken as unity (i.), an "earth-commensuric," or ''pyra- 
 mid-linear" inch will be expressed by i.ooi,. an infinitesimal 
 correction yet to be determined by astronomers. In the text, 
 it is sometimes referred to as a rectified inch, and at others as 
 a pyramid inch. Those familiar with the pyramid literature of 
 the present day will of course understand these distinctions 
 without the foregoing explanation. The controversy now going 
 on among scientific men, relative to the true import of the Great 
 Pyramid, is waxing hotter every day, and at the same time 
 is becoming more dignified. Even the opponents of what is 
 opprobriously termed the " religious theory " by Mr. Proctor, 
 are becoming more and more convinced that the mystery 
 of this ancient mountain of the Nile is not yet solved, and 
 that the secret of the greatest of the world's seven wonders, 
 the only one yet standing on the earth, has not been 
 watched without a purpose for so many eras by the silent 
 Sphinx. To both sides of this controversy the following pages 
 will have much to say. And to those without the arena, the 
 merely interested laity, the whole topic, developed now upon 
 most interesting and important lines, assumes the proportions of 
 a world-wide question. No subject that has ever engaged the 
 attention of the intellect has elicited the interest that now sur- 
 rounds the solution of the problem of the Great Pyramid. It 
 has held the attention of man in every age, and to-day retains 
 it more intent than ever. If any fair-minded, candid spirit 
 once enters upon the subject, it will never cease to be ab- 
 sorbing ; and the grand truths of nature that are now unfolding 
 beneath the general scrutiny focused on it, will be seen to rise 
 pyramidally far above the loftiest subjects that have ever held 
 the mind entranced. Nor should the moderately informed turn 
 hopelessly aside, for fear the theme demands more erudition 
 
INTRODUCTION. 5 
 
 than is generally possessed. The subject is as simple as the 
 figure of the monument itself ; while at the same time, in its 
 scope, it is so mighty that the loftiest genius may find its meas- 
 ure far beneath its dizzy summit. Its story may be made a 
 nursery-tale, as easily as it becomes a midnight vision in the 
 study of the great philosopher. The only answer, then, to him 
 who asks, "Can any good come out of Nazareth?" is that 
 made years ago, and still as full of meaning, " Come and 
 see." 
 
I. 
 
 WEIGHTS AND MEASURES. 
 
 "Just measure and a perfect weight, 
 Called by their ancient names." 
 
 "The Philistines be upon thee, Samson." JUDGES xvi. 9, 14, 20. 
 
 Why Anglo-Saxon Metrology should not be abandoned. The Metric System ver- 
 sus the Anglo-Saxon. The English-speaking Nations, and the Commerce of 
 the Earth. The Balance-sheet of the World. Remarkable Facts, and the 
 Future of Anglo-Saxondom. Our Duty certainly to preserve and perfect our 
 own System. Destroy not the Ancient Landmarks. 
 
 HARDLY a year passes that either openly, or in covert ways, 
 the National Legislature at Washington is not asked to pass 
 laws or resolutions looking towards the ultimate adoption of 
 the metric system of France. 
 
 Fortunately, however, Congress is slow to move in such mat- 
 ters, and has hitherto shown a disposition to regard this subject 
 rather as a national one, and therefore as one for " the people," 
 as such, to settle at the polls. 
 
 The design of bills such as that of the late Alexander H. 
 Stephens, is to make compulsory the use of the new French 
 device in place of our present system of weights and measures, 
 regardless of our great pecuniary loss, and the lamentable evils 
 it would necessarily entail upon our people. Thanks to the 
 efforts of the Institute, to which this paper is more particularly 
 addressed, such bills have hitherto come short of their con- 
 templated ends. It is a source of regret, however, that this 
 foreign system has received at the hands of the authorities at 
 Washington even a permissive sanction. There was no need 
 of any such act of recognition. In a free country the metric 
 system has equal rights with any other, and should stand upon 
 its own merits. It should certainly have received the tacit 
 
WEIGHTS AND MEASURES. ^ 
 
 adherence of the majority, before any such public step or act 
 bringing it into notoriety was advisable. The advocates of 
 this system, however, are too deeply pledged to the ends they 
 have in view not to take advantage of the natural apathy of a 
 people merged in business and industry of every nature ; and 
 so, without our knowledge even, we find them strenuously at 
 work in every direction, introducing it into all our schools and 
 colleges, and here and there, without even the show of legal 
 right, forcing it into the very acts of the government itself. 
 
 " Every man of genuine practical experience realizes the 
 absurdity and impracticability of substituting this inconvenient 
 French metric system for our own hereditary system of weights 
 and measures, which has been in use from time immemorial, 
 and has woven itself into all our history. 
 
 " Now the great danger lies in the encouragement given to 
 the persistent advocates of the metric system by those who 
 are indifferent, unadvised, or ignorant of its demerits in com- 
 parison with the merits of the system to which it is opposed." 
 
 The arguments of the metric philosophers are mostly specious 
 ones, and will not bear a deep research into the constituted 
 nature of things. Its decimal feature is the only one that 
 demands serious consideration ; and this is but a borrowed one, 
 by no means French, and one of doubtful importance to 
 man, who habitually quarters and halves, and in all matters of 
 dimension and metrology seems to comprehend common frac- 
 tional ideas more instinctively than he does those of tens. 
 Indeed, the very continuity of the French system itself has been 
 broken by the introduction and constant use of halves and 
 quarters throughout its series. Such natural subdivisions have 
 been found as absolutely necessary as the half and quarter dol- 
 lar is in our own decimal system of currency. 
 
 In a country constituted like ours, the subject of so radical a 
 change as would be involved in an entire removal of our ancient 
 metrology is certainly one for the popular ballot only to decide, 
 and one with which, when its true import is made manifest, 
 even politicians (to say nothing of statesmen) will be found loath 
 to identify themselves. 
 
 The time has therefore come when it is necessary that the 
 people should be informed what this danger means. It is a 
 
METROLOGY. 
 
 11 Battle of the Standards." It is Anglo-Saxon metrology 
 the Metric System. The one or the other must become obsolete, 
 and that right speedily ; for we are now fully within the thresh- 
 old of an age that demands the recognition of universal ideas 
 and standards over the whole surface of the globe. The easy 
 introduction of standard time into this country, the unanimity 
 with which the meridian of Greenwich has been accepted as 
 the standard meridian for geographical and commercial pur- 
 poses, are significant. The next step is naturally one adoptive 
 of an universal system of weights and measures. Some twenty 
 nations have already recognized that of France; and in name, 
 at least, the metric system is already international. 
 
 But it is not so in fact. It is in the Anglo-Saxon system that 
 the actual business of the world is even now transacted. This 
 is a startling statement, for it puts an entirely new phase upon 
 the question in controversy. It is a novel and interesting 
 stand-point, from which, as Anglo-Saxons, we can afford to 
 review the whole topic before finally committing ourselves to an 
 almost irrevocable decision. 
 
 From such a stand-point the subject can be dealt with as it 
 were ad Jwminem ; and from it we will be surprised at the over- 
 whelming facts which go to establish beyond a doubt that the 
 Anglo-Saxon system is, in reality, the de facto " International 
 system " of the earth to-day, and that the irresistible march of 
 events has already given it, and will continue to maintain it in, 
 the ascendency, until all rival systems, particularly the metric, 
 shall have passed quietly into oblivion. 
 
 If any one is at all doubtful of the future of Anglo-Saxondom, 
 or disposed to question the statistical FACT that already, to-day, 
 this race stands head and shoulders above all other peoples of 
 the earth, and is moving onwards, in every line of progress, at 
 a constantly more accelerated rate than they, we refer them to 
 a volume which should be in the hands of every English-speak- 
 ing statesman, "The Balance-sheet of the World," by M. G. 
 Mulhall, F.S.S., London. 
 
 As generic, and essentially preliminary to our subject, we 
 propose here to briefly review some of the startling facts pre- 
 sented in this work. By such an examination it can and will 
 be conclusively shown that Anglo-Saxon metrology is by no 
 
WEIGHTS AND MEASURES. 9 
 
 means the incongruous failure its adversaries have endeavored 
 to demonstrate, and to make it clear, that, if without any par- 
 ticular attention it has already accomplished so much, it will 
 certainly accomplish all metrologists can wish for when once it 
 shall have been unified and rectified. 
 
 In a paper upon "Weights and Measures," written in favor of 
 the metric system, and delivered before the American Society 
 of Civil Engineers, Mr. Frederick Brooks, C. E., laid great 
 stress upon the fact, that, of the imports into the United States 
 (which, in the year ending June 30, 1879, amounted to $445,- 
 777,777), only twenty-eight per cent was produced in Great 
 Britain and countries using the Anglo-Saxon weights and 
 measures ; while more than half (fifty-nine per cent) was pro- 
 duced in countries, that, within the last hundred years, have 
 adopted a common international system of metrology. The 
 remaining thirteen per cent was produced in countries which 
 use various other systems of weights and measures. These 
 facts were all exhibited in detail by Mr. Brooks, in an accom- 
 panying diagram ; and from them, as of primary importance, it 
 seems as though we were expected to draw the conclusion that 
 Anglo-Saxons, at any rate those of the United States, should 
 also unhesitatingly come into the International Congress, and 
 speedily adopt the metric system. 
 
 At first glance this inference seems to be of some weight, 
 but it is only so upon its surface. Indeed, the reverse consid- 
 eration of the subject entirely overturns it. The argument is 
 drawn entirely from our imports, which are sold to us, and 
 therefore from things in which naturally the purchaser is enough 
 concerned to look after his own interests, and learn the metric 
 system, if needs be, to better guard them. But importing is 
 by no means the business of this country. It certainly did not 
 monopolize our wealth in 1879, and still less does it do so to-day. 
 
 The combined industry of the country in 1879 was repre- 
 sented by a money (gold) value of some two thousand million 
 pounds sterling, or at least ten thousand million dollars, of 
 which the import business barely represented twenty-two per 
 cent. 
 
 Shall the seventy-eight per cent involved in other industry 
 adopt "new times and seasons," and change its manners and 
 
10 METROLOGY. 
 
 its customs simply to accommodate the business of so small a 
 minority ? Surely one must have a strange idea of American 
 institutions, to found an argument upon such a basis, and a 
 stranger one of American perspicuity, if it is presumed that an 
 audience will draw such conclusions even from such data, to 
 say nothing of those which result from a more general consider- 
 ation of the whole subject. 
 
 Now, the imports into the United States are generally luxu- 
 ries and high-priced articles, or only raw material. They are 
 mostly the surplus from arts and trades of long standing in 
 foreign countries, and from them in particular is derived the 
 wealth of the countries whence we procure them. In times of 
 patriotic war, our people have given ample proof of their willing- 
 ness to sacrifice them all. 
 
 But there is another and far more important side of the sub- 
 ject, that of our Exports, with which foreign countries are 
 now so deeply concerned. 
 
 In the very year selected by Mr. Brooks for this discussion, 
 our exports were in value $736,634,834, or in excess of our im- 
 ports by $290,000,000. This excess alone is more than half of 
 the total imports for that year. Moreover, if there be any value 
 in the implied argument of Mr. Brooks, it is, of course, turned 
 back upon his own conclusions from this opposite stand-point. 
 Of our exports, $426,000,000, or about sixty-six per cent, went 
 to the Empire of Great Britain alone (an empire using Anglo- 
 Saxon metrology) ; and this amount, let it be noticed, was practi- 
 cally equal to the bulk of our whole import business. 
 
 With our exports it is not as with our imports. While we 
 can easily get along without the latter, the world itself cannot 
 live without the former. We hold the food-surplus of the 
 earth ; and the bulk of our exports is in breadstuff s, provisions, 
 meat, etc. ; that is, in staple articles. Our exports in food, in 
 1879, were thirty-two times our imports ($10,000,000) therein. 
 
 While carefully examining this matter, it is intended that the 
 data used shall be drawn largely from the London book already 
 referred to. Speaking of the United States, the author re- 
 marks, "Ten years ago the balance of trade was against the 
 country, but now the exports are thirty-one per cent over the 
 imports." This is still more true in 1883. "The Americans 
 
WEIGHTS AND MEASURES. II 
 
 now make one-fifth the iron, and one-quarter of the steel, of the 
 world. . . . The United States raise one-half the gold, and one- 
 half the silver, of the world's supply. . . . Taking in globo all 
 the mining-industries of the world, the United States repre- 
 sent thirty-six, Great Britain thirty-three, and all other nations 
 thirty-one, per cent of the total." Thus Anglo-Saxondom rep- 
 resents sixty-nine per cent of the mining-industry of the earth. 
 
 "The sailing-vessels of the world now trade mostly to the 
 United States." This being a fact, it follows, that, no matter 
 what weights and measures they use at home, they use the 
 Anglo-Saxon ones in our own markets. " But in comparison 
 with commerce, the Americans use three times as much money 
 as the English, and nearly twice as much as all Europe." 
 Moreover, in the past ten years the United States has coined 
 one-fourth of the gold, and one-sixth of the silver, used by all 
 the earth. " The net income of the United States per inhabit- 
 ant is double the European average." 
 
 America is a peaceful country : its aegis holds the olive- 
 branch of tranquillity within its hand of friendship. Our mili- 
 tary expenditure is the least of all nations, and is less than 
 one-fourth of the general European average. 
 
 Our national debt has been reduced twenty-two per cent in 
 ten years ; in fact, the ratio of debt per inhabitant has fallen 
 forty-two per cent, that of interest fifty-four per cent, in ten 
 years. " Population has increased thirty-one per cent since 
 1870 (i.e., to 1880), being the largest number gained in any dec- 
 ade of the Union." The increase of births over deaths " is three 
 times the average European increase, and double that of England 
 or Germany." It is even superior to that of Australia, a newer 
 country. " Every thing seems to promise, that twenty years 
 hence, at the close of the nineteenth century, the United States 
 will have between ninety and a hundred million inhabitants." 
 
 "Food supply is so abundant that the grain-crop is eighteen 
 and a half per cent over consumption, and the meat thirty-six 
 per cent in excess. The United States produced thirty per 
 cent of the grain, and thirty per cent of the meat, of the world." 
 
 "It appears, that, in spite of the population increasing 1,250,- 
 ooo yearly, the supply of grain is growing faster, and of meat 
 as fast. So that the exportation to Europe is likely to go on 
 
12 METROLOGY. 
 
 rising for many years to come." In spite of the home con- 
 sumption of meat being 120 pounds per inhabitant, equal to 
 2,740,000 tons, a surplus of 1,076,000 tons is annually left, 
 one-half of which is exported to the over-populated and hungry 
 foreign nations. " The Americans are apparently the best fed 
 of all nations." They are likewise the most comfortably 
 clothed in cottons, woollens, and linen. 
 
 Concerning our railroads, telegraphs, and internal systems 
 of river communication, it is needless here to speak : they have 
 no compeers on the globe. 
 
 Now, in view of all this wealth, and the constantly increasing 
 importance of this country as the market of the whole world, 
 how ridiculous is any appeal to our imports as an argument in 
 favor of changing our hereditary weights and measures, and 
 abandoning, to our inconceivable discomfort, our own traditions 
 for those of other nations, or for traditions which are no tradi- 
 tions, as those of France are not a century old ! No : rather 
 let all other nations who buy, who are glad to buy, and who 
 must buy in our markets double, ay, quadruple, what we buy 
 abroad, return to their own "ancient landmarks;" and, in the 
 mean time, let us only strive to perfect the heirloom we have 
 hitherto so well preserved. 
 
 Thus far this argument has only been answered from a single 
 stand-point, that of our own nation, and it should be borne 
 in mind that the statements have been drawn from the candid 
 pages of a foreign book. There is a grander view to take of 
 this topic. It is from the stand-point of all Anglo-Saxondom 
 compared with the rest of the world at large. The world has 
 increased in population in the decade from 1870 to 1880 about 
 9! per cent. In the mean time, Great Britain increased io, the 
 United States 31, Australia 56^, Canada 14^, and South Africa 
 (Anglo Saxon) 73 \ per cent. No other nation, save Belgium, 
 has increased over 11 per cent, and France but 1.67 per cent. 
 Even Turkey (2.01 per cent) has increased at a higher rate than 
 the mother of the metric system. At such a rate of increase 
 all other nations must in time be smothered out, and France 
 among the very first, before the Anglo-Saxon race. 
 
 But, further, the tabulated statement of the port-entries of all 
 nations for the ten years under consideration shows, that, while 
 
WEIGHTS AND MEASURES. 13 
 
 the tons burden of the world were 50,000,000, the United 
 Kingdom, British Colonies, and the United States contributed 
 28,000,000, or more than half. 
 
 One-half of the whole industry of the world is already in 
 Anglo-Saxon hands. In millions sterling the increase for the 
 ten years was, for all the earth, 1,866; while it was for Great 
 Britain 337, United States 525, Australia 57, Canada 28, and 
 South Africa 14; making a total of 961 millions of pounds 
 sterling increase. To this increase can also fairly be added that 
 of South American industry, 24 millions, almost all of which is 
 represented by British capital. In 1880 the industry of the 
 earth, expressed in millions of pounds sterling, was 2,024 f r 
 Great Britain, and 2,004 f r the United States. These two 
 nations headed the list, being followed by France at 1,325, by 
 Germany at 1,269, an d by other nations at a greater distance. 
 By industry we mean commerce, manufactures, mining, agricul- 
 ture, carrying-trade, and banking. 
 
 The increased consumption of cotton, wool, flax, jute, etc., in 
 the decade has been 1,666,000,000, of which .922,000,000 
 much more than half has been in Anglo-Saxondom. The 
 increase for the world in manufacturing has been .558,000,000, 
 of which 324,000,000 has been among those using pints, and 
 pounds, and inches. That for all Europe (non-Anglo-Saxon) 
 was but 212,000,000. 
 
 Out of 1 18,000,000 tons increase in production of coal, Anglo- 
 Saxon weights have measured 78,000,000. Out of 7,233 in- 
 crease in thousand tons of iron, they have measured 5,250; and 
 of steel, out of 3,068, they have measured 2,255 5 i- e -> m every 
 case far more than half of all the earth. Anglo-Saxondom pro- 
 duces and measures out by the pound and yard more than fifty 
 per cent of all the wool, and the United States alone seventy- 
 five per cent of all the cotton, raised upon the earth ; and other 
 nations are glad to purchase all these things in pounds and 
 yards. In general terms, the study of the commerce of the 
 world for the past twenty years (1861 to 1880) shows, that, out 
 of 40,000,000,000 (giving the value of the exports and imports 
 in round numbers), Anglo-Saxon metrology has measured and 
 re-measured far more than half. 
 
 The value of the shipping of the earth has increased 
 
14 METROLOGY. 
 
 ,40,000,000 in the decade, 26,000,000 of which was in Anglo- 
 Saxon bottoms ; that of all the metric nations put together was 
 only 13,000,000, the remaining 1,000,000 being scattering. 
 TJius two-thirds of the carrying-trade is already Anglo-Saxon, and 
 but one-third " metric." Which, therefore, it may well be asked, 
 is the de facto international system ? Does it not rather appear 
 that France, leading the Opposition in "the Napoleonic day," 
 when all her interests are known to have clashed with those of 
 England, strove for the mastery in commerce by this politic 
 though vain attempt at banding subjected Europe in a new 
 metrology? And does not the irresistible march of industry 
 prove that her dying system is international only in its self- 
 assumed, high-sounding name ? 
 
 Again : in 1879 the " tonnage" on sea of the earth was 18,- 
 000,000, 10,000,000 of which was Anglo-Saxon; the "carrying 
 power" on sea was 34,000,000, 21,500,000 of which was 
 Anglo-Saxon. For the United States alone the carrying- 
 power at home and on sea was 9,000,000 tons, and the tonnage 
 4,500,000. 
 
 There were 882,000,000 passengers carried upon Anglo-Saxon 
 railways in 1879, against 1,497,000,000 for the whole world, 
 and against 603,000,000 for the whole of Europe, " the Con- 
 tinent," only partly "metric" after all, since the great Rus- 
 sian Empire still remains without the metric union. 
 
 There were in 1879 in England 26,000 miles of active tele- 
 graph-wire, in the United States 119,000, and in the British 
 colonies 59,000 ; that is, for all Anglo-Saxondom 203,000, 
 against 250,000 for the partly metric Continent, and against but 
 303,000 for all the rest of the world. But upon these wires, as 
 an element going to show the magnitude of their thrift, busi- 
 ness, and enterprise, the Anglo-Saxons sent twice as many 
 messages per inhabitant as the Continental nations. England 
 sent 77, the United States 62, and the Continent but 30 mes- 
 sages, per hundred inhabitants. 
 
 Since 1870 (and to 1880) the mines of the earth have pro- 
 duced 360,000,000, of which 215,800,000 were from Anglo- 
 Saxon mines. And Anglo-Saxon mints have coined in the same 
 decade 224,420,000, out of 526,781,000 coined over all the 
 earth. Furthermore, out of 905,000,000 in coined specie 
 
WEIGHTS AND MEASURES. 15 
 
 current in the decade ending 1880, .524,000,000 were used in 
 Anglo-Saxon import business, against 367,000,000 on the 
 European Continent, and 514,000,000 in export business, 
 against 339,000,000 upon the Continent. 
 
 In accumulated wealth in 1880, Great Britain and the United 
 States led all the earth, followed next by France and Germany, 
 and far behind by all other nations taken individually. 
 
 The accumulated wealth of the whole European Continent 
 was 28,000,000,000, that of the Anglo-Saxon nations 18,- 
 000,000,000, while that of the world was only 47,000,000,000. 
 Considered from another stand-point, the world had 113 per 
 inhabitant; Europe, including Great Britain, m; Great 
 Britain alone, 260 (more than any other nation except Hol- 
 land (283), and more than double that of the world) ; the 
 United States, 158; Australia, 172; and Canada, 148. 
 Thus the Anglo-Saxons, as individuals, are worth per inhabit- 
 ant some 184; the world average being 113, and that of 
 the whole continent of Europe being but 91. 
 
 But a consideration of the public debt of nations is even more 
 significant. In the decade 1870 to 1880 the debt of all Europe 
 increased 52,000,000, and that of the world 44,000,000. 
 But three nations of the earth effected any reduction of their 
 national debts. These were the United States, by 86,000,- 
 ooo ; Great Britain, by 24,000,000 ; and Denmark (a former 
 "resting-place" of the Anglo-Saxon), by 3,000,000. The 
 public debt of the world in 1880 was 5,207,000,000, that of 
 the continent of Europe alone being 4,513,000,000; while 
 that of all Anglo-Saxondom together was but 1,276,000,000. 
 Now, it is also noticeable in this connection, that, while the 
 debt of Anglo-Saxondom is almost entirely held in native hands, 
 that of the rest of the world is in foreign hands, and that far 
 more than one-half of it is actually held by Anglo-Saxons. This 
 race, in fact, has bonds and mortgages on all the world. 
 
 In earnings .the United States lead all other nations, their 
 earnings for 1880 being 1,406,000,000. They were followed 
 by Great Britain and her colonies at 1,381,000,000, and far 
 behind by France at 927,000,000, by Germany 851,000,- 
 ooo, and by other nations at continually lower figures. The 
 earnings of the Continent were 3,797,000,000 ; of the world, 
 
16 METROLOGY. 
 
 ;6, 773,000,000 ; those of Anglo-Saxondom being 2,787,000,- 
 ooo, or more than two-thirds that of the Continental nations, 
 and far more than one-third of all the earth. 
 
 Furthermore, taxation has declined in Anglo-Saxondom alone. 
 In Great Britain it has declined in its ratio to income, while in 
 the United States it has done so, not only notably in this rela- 
 tion, but also in the absolute. In every other nation taxation 
 has increased both positively and relatively. 
 
 As to the food-supply of all nations, Europe in 1880 had a 
 deficit of 380,000,000 bushels of grain, while the United States 
 alone had a surplus of 370,000,000 bushels. In tons of meat, 
 Europe had a deficit of 853,000, while Australia alone had a 
 surplus of 838,000, the United States of 1,076,000, and Canada 
 of 170,000. Of the grain-surplus of the world (22,000,000 
 bushels), in 1880 1 7,000,000 were held by Anglo-Saxons ; and 
 of the 2,144,000 tons of meat, then surplus, 1,931,000 were 
 also owned by Americans, Australians, and Canadians. The 
 balance was held in South America and Algeria, and almost 
 entirely controlled, as in fact is almost all So'uth-American in- 
 dustry, by English capital. 1 
 
 But to what purpose shall we here continue this interesting 
 review of man's affairs ? The world is already Anglo-Saxon ; 
 and, in the face of such figures as we have just reviewed, the 
 claims of those who continue to urge us to adopt the metric 
 system are merely vain words, and uttered to no purpose. 
 
 The figures we have quoted are facts, hard facts, and 
 from their very nature they are international facts. They show 
 that the metric system is only international in name, and that 
 the truly international system of metrology is, in fact, our own 
 Anglo-Saxon one. To abandon it to-day, when all the world is 
 really using it in buying food and raiment at our ports, would 
 be to introduce more confusion into human affairs than the 
 earth has seen since the days of Nimrod. In changing Anglo- 
 Saxon weights and measures for the metric system, we would 
 
 1 For further sources whence to draw even more significant deductions as to the signs of 
 the present and the future of the Anglo-Saxon race, the reader is referred to The Balance- 
 sheet of the World, The Progress of the World, The English in South America, Hand 
 book of the River Plate, Hand-book to Brazil, etc., all by Mitchel G. Mulhall, F.S.S., 
 London, 1881. 
 
WEIGHTS AND MEASURES. I/ 
 
 not only disastrously and to no purpose disturb our own affairs, 
 but inconvenience those of all mankind. What the English- 
 speaking races, therefore, need to do is, not to adopt an alien 
 system, but to perfect their own. 
 
 " DESTROY NOT THE ANCIENT LANDMARKS." 
 
 Rather let us strive to unify the system, and to rectify it 
 back to its original and grandly earth-commensuric proportions ; 
 then may we transmit it proudly to posterity, an heirloom still 
 more valuable than when it was intrusted by our sires to us. 
 Let us, therefore, continue our investigation of this de facto 
 international system of metrology, that of the Anglo-Saxon 
 world, and determine how far it is from being perfect, from 
 being actually earth-commensuric, and thus in how much it 
 must be improved and unified and rectified, in order to make 
 it more than ever, and for all future time, a blessing to ourselves 
 and to our fellow-men. 
 
II. * 
 
 LINEAR MEASURE. 
 
 " So long as the human mind remains mathematical, it will prefer a diameter to a circum- 
 ference." HERSCHEL. 
 
 Our Linear Unit, the Inch, grandly Earth-commensuric. An Additional Standard 
 Table proposed for Decimal and International Purposes. The Possible System 
 absolutely perfect. Our Present Tables may be rectified, and retained so long 
 as useful. No Change of Names necessary. 
 
 IF all human life were at this moment to be blotted out, and 
 every book and instrument destroyed ; and if this building only 
 in which we sit to-night were left standing upon the surface of 
 the earth ; and if a body of scientific explorers, coming from 
 some other planet, were to be landed here, do you think they 
 could rediscover the unit of length used in its construction ? 
 Undoubtedly they could. 
 
 No matter how different their own linear unit and standard 
 might be, they would find, by its successive applications to the 
 various parts around them, certain repetitions, doublings, divis- 
 ions, etc., and could not but at last arrive at the conviction 
 that an inch had been the unit, and a measure of twelve such 
 inches the standard, of its builders. They would not of course 
 call it a "foot," nor know the unit as an " inch ; " but they would 
 not fail to conclude that these two lengths were the common 
 divisors of all the measures they could make. They would 
 soon be positive in their conviction, and assert it as a fact. 
 Nay, further : if they were equal to ourselves in science and 
 astronomy, they might ask themselves why we had employed 
 the smaller one, the inch, as a unit ; and perhaps in their 
 natural study of the earth itself, and in the light of intellect 
 (which must be similar throughout the universe), they would 
 
 18 
 
LINEAR MEASURE. 19 
 
 at last discover why, in the fact that an "inch" was exactly 
 commensurable with the grandest dimension of the earth it- 
 self. 
 
 This is just such another investigation, as, during the past 
 fifty years, has been brought to bear upon the ancient pyramid ; 
 and we are now as sure that it was constructed with a unit 
 inch, and by a standard cubit of twenty-five such inches, as if 
 its stones should cry aloudf and tell us its dimensions. We 
 also know that this inch is the most perfectly earth-adapted 
 unit of which we can conceive ; and all this information, and 
 whole volumes of astronomic and other metrological formulas, 
 have in the mean time also been discovered, as due solely to the 
 architectural study of this ancient monument. 
 
 How beautiful a conception is that of subdividing the polar 
 radius to obtain a standard ! The axis of the earth is its short- 
 est diameter ; about it the habitations of man are all conven- 
 iently disposed. It is a line so grandly fixed, that, through 
 all the cycles of time, it knows no shadow of a change in length 
 or in its general direction. About it day and night unerringly 
 return. It sweeps the sky in a majestic cycle, that but once in 
 25,827 years returns into itself, and forms a dial upon which 
 the duration of races, and the eras of geology, may well be meas- 
 ured. One-half of its length, the polar radius, is the very unit 
 which astronomers employ in measuring distances within the 
 solar system. 
 
 This terrestrial polar radius forms in turn the nobler unit of 
 a still greater radius, that of the solar system itself, with 
 which ambitious man essays even to pass beyond, to stars and 
 systems floating upon the very horizon of siderial immensity. 
 Who, then, can fail to see the grandeur of such a unit of me- 
 trology, or fail to catch the beauty of a system of such universal 
 application ? 
 
 Without at present reviewing the pyramid system of linear 
 measure, as heretofore tabulated by Professor Piazzi Smyth, let 
 us proceed directly to the consideration of our Anglo-Saxon 
 linear measures, and try and determine how they may be recti- 
 fied to earth-commensuric utility, duly prepared for decimal 
 use ; and, while being thus perfected, be at the same time 
 jealously preserved in all those essentials of name, familiar 
 
20 METROLOGY. 
 
 rationale, and common-fractional sequence, which tradition, time 
 without beginning, has made so sacred to us as a race. 
 
 Our Institute has been frequently accused of giving up too 
 much time to pyramid studies, and of leaving out of consid- 
 eration the more important subject of the improvement of our 
 tables. I therefore propose to consider the subject of Saxon 
 length-measure simply and solely from the stand-point of earth- 
 commensurability, and fitness for future international use. 
 
 If some of us have already found the ancient monument of 
 Egypt to teach the same great lessons, and if the world at 
 length shall read the " stone book " in the same bright light, 
 then, as pyramid-students, we shall of course rejoice ; but, as an 
 institute, we are incorporated to preserve and perfect the Anglo- 
 Saxon system as our prominent object. 
 
 It is of little consequence, then, where we find the truth, so 
 long as what we bring up is the priceless pearl itself, and par- 
 ticularly so long as we can prove its value from its own inher- 
 ent radiance. 
 
 Without further circumlocution, therefore, I shall ask your 
 attention to what, it seems to me, offers a practical means of 
 accomplishing the ends for which, as a society, we exist. 
 
 Years ago the great Herschel advocated the measurement 
 of the polar radius of the earth, the adoption of its one 10- 
 millionth as the standard international measure, and that a 
 twenty-fifth (^5) thereof be taken as the primal unit of all meas- 
 ure whatsoever. 
 
 Time has shown the wisdom of his proposition. And now 
 at length, in our day, we find the very president of the metric 
 bureau in this country President Barnard of Columbia Col- 
 lege regretting that the radius was not taken as the basis of 
 the French metric system ; and publicly declaring, that, were 
 the metric system to be formed anew, it would undoubtedly be 
 founded on this very line, so sure and fixed. Not only, there- 
 fore, impressed with the singular fitness of such a measurement 
 for standard and unit purposes, but profiting by fair experience, 
 and by the candid admissions of those who have already given 
 up their own traditions, let us cling to ours, and endeavor to 
 right them at this, the noblest dimension of the earth on which 
 we live. 
 
LINEAR MEASURE. 21 
 
 For the unit of linear measure I therefore join in proposing 
 that this Institute resolve upon and adopt the exact five-hun- 
 dred millionth of the terrestrial axis ; and that this unit be 
 named after its present wonderful Anglo-Saxon approximation, 
 and called "one inch." 
 
 The value of the unit will not differ from that of the present 
 Anglo-Saxon unit inch (British or American) by more than .001. 
 Of this we are already certain from our present knowledge of 
 earth size and shape. 
 
 In order to obtain this measure exactly, and for all future 
 time ; to provide for its preservation and accurate practical 
 renewal at any time ; and to establish it beyond danger of loss 
 or variation, I also propose that the Institute shall offer a 
 prize for the best essay upon the following theme : 
 
 "What is the simplest and most certain method of practi- 
 cally obtaining and preserving the measurement of the polar 
 axis of the earth ? " 
 
 The method proposed should be based upon the universal 
 principles of sound astronomy, be convenient of application, 
 avoid national prejudice, be accompanied by a full and complete 
 mathematical demonstration of its certain accuracy, and be in- 
 sured, by several checks, against the internal mechanical errors 
 of calculation. 
 
 The inch is the grand central unit of the entire Anglo-Saxon 
 system ; and its numerical and earth-commensuric value must 
 be rigidly determined before we can reasonably attempt to per- 
 fect any other part of the wonderful inheritance we have re- 
 ceived from earlier ages. 
 
 It should be clearly manifest that we must needs have actu- 
 ally in our possession the measured length of this unit before 
 we can determine the numerical nature and earth-commensuric 
 relations of any others ; as, for instance, those of capacity, 
 weight, area, volume, etc. 
 
 We may, however, in the mean time, profitably study and 
 examine the internal arrangement of the various subordinate 
 branches of the Anglo-Saxon system of metrology, and deter- 
 mine what ought to be, and what shall be, the future and inter- 
 national arrangement of the whole system. 
 
 It is an entirely feasible proposition to determine the rationale, 
 
22 
 
 METROLOGY. 
 
 of a system before we have obtained (or fixed by resolution) and 
 adopted the exact numerical length of its desired units ; thus, 
 for instance, the foot, the yard, the 24" gauge, long measure, 
 in fact, should be carefully maintained and preserved. It is of 
 manifest utility, of traditional value, and will find employment 
 to the end of time. In the mean while it will probably be 
 necessary to adopt some decimal system founded upon the 
 inch, for purposes of rapid calculation and international use. 
 
 Without reference, therefore, to the absolute length of the 
 unit inch, save that it shall be, when determined, accurately 
 one 5OO-millionth of the polar axis (and until so determined 
 shall be held at its present statute value), we may agree before- 
 hand that the rectified long measure shall be as heretofore, and 
 as follows : 
 
 RECTIFIED LONG MEASURE (STATUTE). 
 
 I inch = the central "unit" = one 5oo-millionth polar axis ; may be subdivided, as at 
 present, either decimally or common-fractionally, to suit necessities. 
 
 j-i inch 
 
 
 
 point. 
 
 
 -^ inch 
 
 = 
 
 hairbreadth. 
 
 
 ^2 inch 
 
 = 
 
 line. 
 
 
 -g inch 
 
 ss 
 
 barleycorn. 
 
 
 3 inches 
 
 = 
 
 palm. 
 
 
 4 inches 
 
 = 
 
 hand. 
 
 
 5 inches 
 
 = 
 
 span. 
 
 
 10 inches 
 
 = 
 
 double span. 
 
 
 12 inches 
 
 = 
 
 foot. 
 
 2 feet 
 
 = 
 
 gauge (24-inch). 
 
 3 feet 
 
 = 
 
 yard = 36 inches. 
 
 6 feet 
 
 = 
 
 fathom = 72 inches. 
 
 5^ yards 
 
 = 
 
 rod = :6j feet = 198 inches. 
 
 40 rods 
 
 =: 
 
 statute furlong = etc. 
 
 8 furlongs 
 
 = 
 
 statute or land mile = etc. 
 
 3 miles 
 
 = 
 
 statute or land league. 
 
 Let a "cable's length," as heretofore, equal 120 fathoms or 
 720 feet; and let " Gunter's chain," "Superficial measure," 
 "Cubical measure," etc., remain upon our books for those who 
 desire to use them. So long as they are valuable, and subserve 
 a purpose, they will be used : when they cease to be so, they 
 will naturally fall into disuse. 
 
 For decimal purposes I propose the following system, based 
 upon pyramidal studies, and upon the traditional lore of our 
 
LINEAR MEASURE. 23 
 
 race ; and to distinguish it from the present statute or long 
 measure, I shall denominate it " Standard Linear Measure." 
 When I come to the subject of capacity and weight measures, 
 as rectified by certain principles which I have discovered to 
 exist in the constitution of all nature, the parallelism of the 
 following table will be more clearly manifest. Its facility for 
 mathematical use is of course already apparent. 
 
 STANDARD LINEAR MEASURE (A NEW TABLE). 
 
 Inch, span, cubit, etc., all to be subdivided decimally, or common-fractionally, to suit 
 
 convenience. 
 
 5 inches = i span = 5 inches. 
 
 2 spans = i decimal foot = 10 inches. 
 
 5 spans = I cubit 1 = 25 inches, 
 
 j 2^ decimal feet = i cubit 
 
 ( 4 cubits = i linear standard = 100 inches. 
 
 ,o cubits 5'i cubits ='" l = Ipole . 3 
 
 ( 4 ells = i pole ) 
 
 ( 2- poles = i chain ) 
 
 10 poles { . . > = i acre-side. 
 
 ( 4 chains = i acre-side > 
 
 ( 2i acre-sides = i furlong ") 
 
 10 acre-sides < f . > i metron. 
 
 ( 4 furlongs = i metron \ 
 
 ( 2\ metrons = i mile 4 ) 
 
 10 metrons < > = i league. 
 
 ( 4 miles = i league > 
 
 1,000 leagues = i linear axis or polar radius. 
 
 Commencing with the cubit, which is one lo-millionth of the 
 polar radius, the foregoing table is strictly decimal. It is based 
 upon the unit inch, by and through which it connects with all 
 other Anglo-Saxon linear measures founded thereon. It culmi- 
 nates in the polar radius, by and through which it reaches up- 
 wards into all solar and sidereal distances. But the system is 
 doubly decimal, for running through it we find the following 
 subordinate one : 
 
 SUBORDINATE STANDARD LINEAR MEASURE. 
 2^ cubits = 62^ inches = one ell. 
 
 < 4 ells = i rod ) 
 
 io ells \ 4 , . . [ = i chain. 
 
 \ 2\ rods = i chain > 
 
 1 25" rectified = 25.025 i English inches; for analogues see International Standard for 
 May, 1883, p. 79. 
 
 2 62^" ; for analogues, see International Standard for May, 1883, p. 88. 
 
 3 i pole = 250"; for analogues to which, see tables published in International Standard 
 for May, 1883, and compiled by Jacob M. Clark, C.E. 
 
 * i standard mile = 2,500 cubits ; for analogues, see International Standard for May, 1883, 
 p. 89. 
 
24 METROLOGY. 
 
 . . (4 chains = 
 10 chains < , 
 
 ( 2^ acre-sides = 
 
 10 furlongs \ , 
 
 (2^ metrons 
 
 4 miles 
 
 acre-side 7 
 
 > = I furlong, 
 furlong > 
 
 metron 
 
 , = i mile, 
 mile 
 
 league. 
 
 semi-axis or polar radius. 
 
 1,000 leagues = 
 
 At the first glance, the beauty of the above double decimal 
 arrangement is apparent, and at the second, will be perceived 
 the convenience afforded for general calculation by quarters, 
 halves, fifths, etc. 
 
 These common-fractional subdivisions have been found to be 
 so necessary in practice, that even the " metric system " has had 
 to violate its decimal continuity by the admission and interpo- 
 lation of halves, quarters, etc. 
 
 Running through the standard tables, it will be seen that 
 specific names are for the most part given to these common- 
 f ractional parts. Thus : 
 
 i ell = i rod. 
 i chain = acre-side. 
 i furlong = ^ metron. 
 i mile = \ league. 
 
 The above values make, in fact, a convenient subordinate 
 system by themselves. Thus : 
 
 4 ells = i rod. 
 
 10 rods = 4 chains = i acre-side. 
 
 10 acre-sides = 4 furlongs = i metron. 
 10 metrons = 4 miles = i league. 
 
 Furthermore, the halves extend systematically throughout 
 the table, as follows : 
 
 5 cubits = rod. 
 5 ells = chain. 
 
 . 
 
 5 rods = \ acre-side. 
 
 5 chains = \ furlong. 
 5 acre-sides = metron. 
 5. furlongs = \ mile. 
 5 metrons = \ league. 
 
 The above values likewise form harmonious sequences of 
 practical, common, and decimal utility. There are two of them 
 which examination will show are alternates. Thus the first 
 one, starting with the half-rod, is as follows : 
 
LINEAR MEASURE. 2$ 
 
 5 cubits or 10 half-cubits = rod. 
 
 10 half-rods = ^ acre-side. 
 
 10 half-acre-sides = rnetron. 
 10 half-metrons = league. 
 
 For the alternate we have, 
 
 5 ells or 10 half-ells = ^ chain. 
 10 half-chains = ? furlong. 
 10 half-furlongs = mile. 
 
 Now, it is manifest that the decimal subdivision of the unit 
 inch can, as at present, be carried downward to the very limits 
 of numerical expression and microscopic appreciation. So, 
 too, beginning at the earth-commensuric inch, we may, for such 
 special purposes as shall be found convenient, run decimally 
 upward into the very largest measures of extension. For 
 instance : 
 
 i" = i unit (to be decimally subdivided). 
 
 10 inches = i link. 
 
 10 links = i linear standard. 
 
 10 linear standards = i measuring line. 1 
 
 Now, we need not tabulate farther than the measuring-line 
 of 1,000." Nevertheless, it must be manifest, that, in such a 
 system as we are here developing, the series runs upward 
 harmoniously alongside of, and through the entire system of, 
 rectified linear measure. 
 
 Thus, merely for sake of illustration, the above tables may 
 be thrown into rhythm with the whole system, as follows : 
 
 10 inches 
 
 10 links 
 
 10 linear standards 
 
 i inch 
 
 2^ inches = 
 4 measures = 
 2\ links = 
 4 cubits 
 2\ 1 in. stand. = 
 4 rods = 
 
 unit. 
 
 measure 
 
 link 
 
 cubit ") 
 
 r link or decimal foot. 
 
 , , > 1 linear standard, 
 linear standard > 
 
 rod 
 
 measuring line 
 
 I measuring line. 
 
 Furthermore, from an inspection it will be seen that two and 
 a half measuring lines equal one acre-side ; or, in other words, 
 
 25 linear standards = i acre-side. 
 250 linear standards i metron. 
 2,500 linear standards = i league. 
 2,500,000 linear standards = i polar radius. 
 
 1 2^ "measuring-lines " = 2,500"=: i side of a square acre; hence 10 "measuring-lines" 
 form the entire perimeter of a square acre. 
 
26 METROLOGY. 
 
 It is here worthy of note, in passing, that there are the same 
 number (2,500) of linear standards (100") in a league, as there 
 are of cubits (25") in a standard mile ; and, as we shall see later, 
 of rectified avoirdupois pounds in a standard ton, etc. 
 
 Again : based upon the unit inch, and familiar, not only to 
 pyramid students, but to all Anglo-Saxon readers of the Scrip- 
 tures, we can derive from these rectified measures the ancient 
 sacred measures of the Jews. Thus : 
 
 ECCLESIASTICAL LINEAR MEASURE. 
 
 i" = i unit (subdivided decimally). 
 
 5" = i span. 
 
 5 spans = i cubit. 
 
 6 cubits = i reed. 1 
 
 It is the consummate perfection of this system that makes it 
 of such universal application. From the great diversity of its 
 applications, it may seem at first glance somewhat complicated ; 
 but the most cursory examination into the principles upon which 
 it is based will show that it is just the reverse. We have here 
 presented the system in such a way as to show its full scope, 
 and in so doing may have given the idea of complication ; 
 nevertheless, the system itself consists simply of the standard 
 table just given. All of the others, including our well-known 
 linear or long measure, Gunter's measure, superficial measure, 
 cubic measure, etc., are merely subordinate branches of the 
 radical one, based upon the unit, and for special purposes. 
 
 Now, is there any need for these subordinate tables ? Un- 
 doubtedly there is at present. These subordinate systems 
 have sprung up from time to time with the growth of special 
 branches of trade, art, and science. They are not confusing 
 
 1 Mr. Jacob M. Clark, C.E., has shown that the linear measures of the Hebrews (Ezekiel's) 
 were undoubtedly founded upon a radial unit rather than upon a circumferential one. (See 
 Transactions of American Society of Civil Engineers, December, 1882; also International 
 Standard, 188^-84.) It is noticeable in this connection, that Ezekiel's ecclesiastical reed of six 
 cubits ( 1 50") consecutively chords the circle, i.e., by means of the hexagon, while it avoids 
 the " impossible problem " of measuring it. Ezekiel thus " vindicated geometry, and pro- 
 claimed the divorce of direct measure from circular, and itinerary." This divorce is absolute, 
 from the very nature of things : their forced conjunction is miscegenation, and illegal. The 
 sooner, therefore, we accept the geometry of Ezekiel, and put asunder things which can 
 never have been interiorly conjoined, the sooner we shall accept the facts about us, and 
 square our practice with the things that are. 
 
LINEAR MEASURE. 27 
 
 to tJiose who use them ; they are exactly suited to their wants, 
 having been the direct outgrowth of those wants. They have 
 arisen from the necessity of having various standards. The 
 confusion of the past in Anglo-Saxon weights and measures 
 has resulted, not so much from the use of numerous standards, 
 as from the loss of unity in the units, by means of which they 
 should have been, and probably were originally, all linked 
 together. 
 
 When once more we have rectified our unit inch back to its 
 earth reference, the standard system may be legalized ; and with 
 it all the wants of an international, earth-commensuric, and not 
 only thoroughly decimal, but particularly common-fractional 
 system, will at once be realized. 
 
 In the mean time the full use of the subordinate systems, 
 rectified at the same time by the adoption of the true unit, can 
 be fully enjoyed by all who are now familiar with them, or may 
 hereafter find them convenient for any special purposes. 
 
 It is our NATIONAL duty to perfect our own unit, and to: 
 found thereon a system that shall be standard. But no nation 
 constituted like the branches of the Anglo-Saxon race will ever 
 so forget the grand principles of individual freedom, as to make 
 any system compulsory upon its members. Nor, may we firmly 
 trust, will the Anglo-Saxon race ever so forget its own far- 
 reaching records and traditions, as to exchange a birthright, 
 whereby it compasses the earth with commerce, for a mess, of 
 metric pottage. 
 
 Such, then, is the Anglo-Saxon system of linear measure, a 
 system so beautiful and perfect, that, whether it were monu- 
 mentalized or not at Gizeh, would be worthy of the deepest 
 consideration of the English-speaking race. 
 
 This race stands in a peculiar relation to the rest of men. 
 Its future is secure : that of no other peopled. It has not yet 
 adopted the French system. In both Great Britain and Amer- 
 ica that system has hitherto been only legally tolerated, and 
 every effort of its votaries to make it compulsory upon us has; 
 thus far most signally failed. Nearly all of the Continental 
 nations of Europe have in the mean while accepted the metric 
 system, and have made it compulsory upon their people. We 
 alone, therefore, among the more enlightened peoples of the 
 
28 METROLOGY. 
 
 day, stand out against it, at present content with our own 
 traditions, and not ready for a change. 
 
 But for all Europe to accept' the metre does not insure its 
 ultimately universal acceptance at the hands of the rest of the 
 world ; far from it, if our vote is against it. The world is not 
 to be European, but Anglo-Saxon. 
 
 Leaving the pyramid entirely out of the question, we utterly 
 fail to see ivhy after finding that our own traditional unit of 
 linear measure has been proved to be more ancient than his- 
 tory, of marked personal reference, already in perfect rhythm 
 with the laws of human thought, earth-commensuric, ay, and 
 commensuric with the very universe itself we should entertain, 
 even for a moment, the thought of adopting one which we know 
 to be just the reverse, and dictated by a people whose customs 
 and manners have so little in common with our own. 
 
 We see no reason why a people already outnumbering the 
 inhabitants of all Europe, and who, over the whole face of the 
 globe (where other nations have successively failed to maintain 
 themselves in competition with us), rule as many more mil- 
 lions ; a people who actually own the capital, control the indus- 
 try, and, as a statistical fact, lock up the surplus of the whole 
 earth ; whose commerce and practical science has, in a century 
 and a half, changed its face and future, should do aught but 
 take such steps as shall be necessary to PRESERVE and perfect 
 the system of their own wise ancestors* 
 
 1 See Appendix A. 
 
III. 
 
 CAPACITY MEASURE. 
 
 " He layeth up the depth in storehouses." Ps. xxxiii. 7. 
 "He weigheth the -waters by MEASURE." JOB xxviii. 25. 
 
 The Coffer in the Great Pyramid. The Caldron of the Anglo-Saxons. The 
 Ark, or Laver, of the Hebrews. The Three Standards Identical. Our Na- 
 tive Measures over Four Thousand Years Old. Pyramid Measures according 
 to Professor Smyth. Noticeable Correspondences. A Proposed Standard 
 Decimal-Table for International Use. Further Considerations. The Table 
 continued Downward And Upward. The Resulting System. Comparison 
 of Results. Roots of the System. Comparative Table. What have we to 
 do with the Metric System in View of Such Great Possibilities? New Light, 
 and a General Survey of the Rectified Measure. 
 
 WERE the command "Measure the Capacity of this Coffer" 
 engraved upon its sides, in the symbols of a universal language, 
 the mysterious granite box in the king's chamber of the Great 
 Pyramid could speak no plainer than it now does to the wise 
 of all the earth. It is a most peculiar geometrical solid. It 
 was so exquisitely constructed that its interior capacity 71,- 
 250 pyramid cubic inches was one-half the volume of its 
 exterior dimensions, and the volume of its thick bottom half 
 that of its sides. Measure it as you will, and employ any unit 
 you desire, still this same relation, so suggestive of capacity, 
 must result. The beauty of this accomplishment will be appar- 
 ent to all who have ever studied the most renowned mathemati- 
 cal problem of the ancients, that of "duplicating the cube." 
 
 In fact, the coffer is a most wonderful measure of capacity. 
 It is a perfect standard : it contains exactly four original Brit- 
 ish " quarters," or one measured ton. It is not a little strange 
 that the "quarter" is now the largest grain-measure upon the 
 statutes of Great Britain. The natural inference is, that, at 
 some time or other, far back into our history, it was a quarter 
 
 29 
 
30 METROLOGY. 
 
 of something else, four times greater. Though that something 
 has now dropped out of Anglo-Saxon grain-measure, it must 
 have once held a prominent place therein. The modern British 
 quarter contains 17,745.536 British cubic inches ; and four such 
 quarters would contain 71,000(70,982) British cubic inches, 
 a quantity still remarkably close to this ancient pyramid-coffer 
 of 71,250 cubic inches. But this cubic volume is also equally 
 close to another and even more significant ancient measure, 
 and to one from which we doubt not we have inherited the 
 caldron itself : we refer to the ark of the covenant, held in 
 such sacred veneration by the people of Israel. Of the ab- 
 solute contents of the ark, or laver, of the Hebrews, we are 
 not informed ; though its capacity may be calculated to a very 
 near degree of accuracy. The Bible gives its outside meas- 
 ures as 2\ cubits long, i cubits broad, and i| cubits high. 
 We say outside measures, because the dimensions of the crown 
 of gold which covered it are given in the very same figures ; 
 and, had not these been outside measures, the "mercy seat," or 
 cover, would have fallen down inside of the ark. Its outside 
 measures were (in terms of the sacred cubit of 25"), therefore, 
 62.5" X 37.5" X 37.5". Now we are also left in ignorance as 
 to the thickness of the wood o'ut of which it was constructed. 
 Philo-Israel, in discussing this matter, suggests if inches 
 " stuff " as suitable for a well-proportioned box of this capacity. 
 These dimensions would give an inside content of 71,282 cubic 
 inches. Gunner Thomas Henry, R.A., in charge of the officers' 
 workshops at the Royal-Artillery Institution, Woolwich, sug- 
 gests as rather more suitable working-dimensions a 2-inch bot- 
 tom, and if -inch sides; which latter material would afford an 
 interior content of 71,213 cubic inches capacity. The mean of 
 these two, undoubtedly very close approximations, is 71,248 cubic 
 inches ; or, in all human probability, we have in the ark a box 
 of the same volume as the coffer of the Great Pyramid and 
 the caldron of our forefathers. We may, however, continue the 
 argument a step farther ; for we can also examine the molten 
 sea of Solomon, another and greater measure of capacity, and 
 one also made in terms of the sacred cubit. We shall adhere 
 to the description of it given in I Kings vii. 23-26, "that in 
 Chronicles being fragmentary." From this account we find that 
 
i 
 
 j 
 
 CAP A CITY MEASURE. 3 1 
 
 it was 10 cubits from brim to brim, 30 cubits in circumference, 
 and 5 cubits high. It was round all about, a handbreadth thick, 
 and contained 2,000 baths. We assume nothing in claiming that 
 it was hemispherical in figure; because, 1st, It was round all 
 about ; 2d, Its height was half its diameter ; 3d, Josephus ex- 
 pressly says it was hemispherical. There is also no doubt that 
 the 10 cubits diameter is an outside measure from "brim to 
 brim." Its vertical dimension of 5 cubits being so expressly 
 stated as height, not depth, must also be taken as outside 
 dimension. Its 30 cubits of circumference, however, must as 
 manifestly be taken as inside dimension; because, ist, We 
 should otherwise have two conflicting measures of the same 
 thing ; 2d, The circle of 30 cubits circumference at the inner 
 lip would leave a margin between the inner and outer edge 
 of the brim of 5^ inches, which, with half an inch allowance 
 for the extra carved work on the brim, may be fairly taken to 
 express the handbreadth thickness of the brazen sea itself. 
 Calculating the volume of such a hemisphere, we find it to be 
 3,562,070 cubic inches, or, within a minute quantity, 50 times 
 the capacity of the ancient ark of the covenant. The exact 
 size of a fiftieth part of the above-calculated volume is 71,241 
 cubic inches, or well within the margin required in a casting 
 of such immense size. Moreover, the quantity 430 cubic 
 inches by which such a hemisphere actually fell short of 
 the desired amount, 3,562,500 cubic inches, could easily have 
 been compensated for in the ornamental work about the brim. 
 
 But the molten sea = 2,000 baths. 
 It also = 50 arks. 
 
 Therefore the ark = 40 baths. 
 But the laver also = 40 baths. 
 Therefore the laver = the ark of the covenant. 
 
 Thus we have taken a very valuable step towards arriving at 
 the sacred measures of Israel ; since we now know that 40 
 baths = i laver, and 50 lavers = i molten sea. The exact di- 
 mensions of these two measures of Israel serve as checks upon 
 each other ; and from them we are not only able easily to fill in 
 the whole biblical series, but to give their capacity in cubic 
 inches, and more fully realize the possible intermediate chan- 
 nel through which these same dimensions may have come down 
 
METROLOGY. 
 
 to us from the still more ancient monument of Gizeh. It would 
 thus appear, that the ark was the capacity standard of Israel, 
 and held the same place in the biblical system that the coffer 
 did in that of still earlier days, and that the caldron of our 
 own forefathers did, and does in Anglo-Saxon metrology to-day. 
 As just remarked, the quarter is a grain-measure; and this 
 brings us to one of the best explanations of the meaning itself 
 of the word "pyramid." It is a Greek word, and was first used 
 by Herodotus in describing the wonderful monument which he 
 had visited in his travels. Herodotus probably Hellenized the 
 meaning, if not also the pronunciation, of the word by which 
 the monument was designated. All sorts of significations 
 have been given to it ; but there is a very simple one, derived 
 without any contortions whatever, from the Greek word pyros 
 ("wheat" or "bread"), and metron ("a measure"), pyros 
 metron, or pyro-mct> or pyramid ("the bread-measure"), the 
 standard International Metric Monument. The pyramid is then 
 a metric monument, and the mysterious coffer its standard meas- 
 .ure of the staff of earthly, and the symbol of eternal, life. 
 
 Without further discussion, I shall proceed directly to the 
 table of capacity measure, which pyramid students have dis- 
 covered there. It is a decimal system, superior to any now in 
 use, and grandly Anglo-Saxon. In its scope, it can satisfy the 
 requirements of the minutest scientific analysis, or rise without 
 hiatus to the measure of the universe itself. Professor Smyth 
 arranges the subdivisions of this ancient coffer as follows : 
 
 PYRAMID CAPACITY MEASURE. 
 
 Division or 
 number of each 
 denomination 
 contained in the 
 whole coffer. 
 
 Intermediate 
 division. 
 
 Capacity of each 
 denomination in 
 pyramid cubic 
 inches. 
 
 Equivalent 
 weight in pyra- 
 mid pounds of 
 water. 
 
 Name now proposed to be 
 given each successive 
 portion. 
 
 I 
 
 
 
 71,250 
 
 2,500 
 
 coffer. 
 
 4 
 
 4 
 
 17,812.5 
 
 625 
 
 quarter. 
 
 10 
 
 2-5 
 
 7J25 
 
 250 
 
 sack. 
 
 2 5 
 
 2-5 
 
 2,850 
 
 100 
 
 bushel. 
 
 250 
 
 10 
 
 28 
 
 IO 
 
 gallon. 
 
 2,500 
 
 IO 
 
 28-5 
 
 I 
 
 pint. 
 
 25,000 
 250,000 
 
 10 
 
 IO 
 
 2.8\ 
 0.285 
 
 O.I 
 0.0 1 
 
 wine-glass, or fluid oz. 
 teaspoon, or fluid dr. 
 
 2,500,000 
 
 10 
 
 0.0285 
 
 O.OOI 
 
 ten drops. 
 
 25,000,000 
 
 IO 
 
 0.00285 
 
 O.OOOI 
 
 drop. 
 
CAPACITY MEASURE. 
 
 33 
 
 He then subjoins the following table, to illustrate how closely 
 our present Anglo-Saxon measures of capacity still correspond 
 with their ancient" source and standard. 
 
 PYRAMID AND BRITISH CAPACITY MEASURES COMPARED 
 THROUGH THE TEMPORARY MEDIUM OF ENGLISH CUBIC 
 INCHES, APPROXIMATELY. 
 
 PYRAMID. 
 
 Coffer 71,463.750 
 
 Quarter 17,865.938 
 
 Sack 7J46.375 
 
 Bushel 2,858.550 
 
 Gallon 285.855 
 
 Pint 28.585 
 
 Ounce, in wine-glass . . . 2.858 
 
 Dram, or teaspoon. . . . .286 
 
 Drop .003 
 
 ENGLISH. 
 
 4 quarters 70,982.144 
 
 i quarter 17,745-536 
 
 i sack 6,654.576 
 
 bushel 2,218.192 
 
 gallon 277.274 
 
 P^t 34.659 
 
 ounce, apothecaries' . . J -733 
 
 dram .217 
 
 drop .004 
 
 Professor Smyth then gives a series of tables to show that 
 these measures correspond now with the pyramid in quarter, 
 sack, and bushel, for many parts of the world. 
 
 But there are many other correspondences between these 
 ancient measures and our own well-known Anglo-Saxon ones, 
 which do not appear upon the surface of this table. For in- 
 stance, in the United States the pint of liquid measure equals 
 28.875 cubic inches, and is based upon the old British wine- 
 gallon. The present British pint weighs a pound and a quarter. 
 An ancient Anglo-Saxon pound, however, was an exact pint, 
 and contained 27.73 =h cubic inches, one-sixteenth of which, or 
 1.736 cubic inches, apparently accounts for the apothecaries' 
 ounce as above (1.733) given. Now, by inspection it appears 
 probable that the apothecary ounce of 1.733 cubic inches was 
 originally derived by taking the one lO-thousandth of the 
 quarter, a supposition which becomes irresistible when we 
 go to pyramid originals. Thus the pyramid "quarter" of 17,- 
 865.938 Britishtcubic inches + 10,000 = 1.78659 British cubic 
 inches ; and if we multiply by 16 (the subdivision for avoirdu- 
 pois pounds) we obtain 28.585 British cubic inches ; i.e., the 
 pyramid avoirdupois pound ; while the same quantity multiplied 
 by 12 gives us 21.438 British cubic inches, or what was un- 
 doubtedly the original troy and apothecaries' pound (now only 
 ! 733 X 12 = 20.796 British cubic inches). 
 
34 METROLOGY. 
 
 From the foregoing considerations, we may rest assured, that, 
 while the coffer is the grand standard of both pyramid and 
 Anglo-Saxon capacity measures, the measured ounce, or the one 
 lO-thousandth of its quarter, and the gill, or one lO-thousandth 
 of the coffer itself, are the units by means of which its several 
 kinds of pints and measured pounds have been derived and are 
 connected, and are still to be preserved and protected. 
 
 This is the same view we have already taken of the inch in 
 linear measure ; namely, that it is the unit whence, for the 
 purpose of various trades, the foot, the yard, and the ell were 
 all obtained. 1 
 
 I shall now invite attention to the following rearrangement 
 of the pyramid-capacity table, into which I have introduced a 
 number of other well-known English measures, and by means 
 of which the double decimal nature of the system becomes 
 instantly apparent. It will be noticed, however, that this table 
 differs from that of Professor Smyth, in that it brings into the 
 system several measures which he overlooked, while it leaves 
 intact the values of most all of those which he did employ. In 
 one or two cases, however, for obvious reasons, I have also re- 
 arranged some of his proposed designations. 
 
 STANDARD TABLE OF PYRAMID CAPACITY MEASURES, FROM 
 THE OUNCE UP TO THE COFFER. 
 
 10 meas. ounces . J < meas " oz ' = ' ^ } I = i stand. Ib. meas. 
 
 ( 2$ meas. gills i standard Ib. meas. ) 
 
 10 standard Ib. meas. \ 4 stand lbs ' = ' 1 uart - } = i pottle. 2 
 
 2i Quarts = Dottle. ) 
 
 
 ( 2^ quarts 
 
 pottle. 
 
 ) 
 
 10 pottles .... 
 
 [ 4 pottles = 
 ( 2$ sacks = 
 
 sack, 
 bushel. 
 
 > = i bushel. 
 
 10 bushels .... 
 
 ( 4 bushels = 
 ( 2\ coombs = 
 
 coomb. 2 
 quarter. 
 
 > = i quarter. 
 
 
 
 
 
 
 
 
 
 1 The word " pound " is derived through pondus (" a weight "), from the Latin verb pen- 
 dere (" to weigh "), and may be of any convenient number of ounces, depending upon the 
 various requirements of art and trade. It simply means a -weight (and pint an equivalent 
 measure}, and by every class of workmen was originally applied as a designation to that 
 particular measure or weight (i.e., number of units) which they handled most frequently. 
 Thus, in United-States liquid measure we have the pint of 28.875 cubic inches, and in dry 
 measure of 33.600 cubic inches, and, finally, in British imperial measure (both liquid and 
 dry), one of 34.659 cubic inches. It almost seems impossible that such diverse measures 
 can have had a common unit, nevertheless I doubt not we shall be able to establish it as 
 an incontrovertible fact before we close the topic. 
 
 2 Saxon terms at present in use. See British imperial measure. 
 
CAPACITY MEASURE. 
 
 35 
 
 An examination of the foregoing table will show that it is 
 strictly decimal, while it is at the same time conveniently 
 arranged for the purposes of every-day life, which demand 
 quarters, halves, eighths, etc., even more emphatically than 
 they do the employment of tens and hundreds. 
 
 It will be noticed, however, that there is a subordinate system 
 of tens running through this table, by means of which even the 
 subdivisions themselves are also decimally correlated. Thus 
 from the gill, of four ounces, upward, we have the following 
 arrangement : 
 
 4 ounces = i gill. 
 
 10 gills =i quart. 
 
 10 quarts = I sack. 
 
 10 sacks =i coomb. 
 
 10 coombs = i coffer. 
 
 I shall now ask your attention to the cubical capacity (in 
 inches) of the various measures above given, with a view to 
 determining how nearly they are related to our present Anglo- 
 Saxon measures. 
 
 PYRAMID CAPACITY MEASURES COMPARED THROUGH THE 
 TEMPORARY MEDIUM OF ENGLISH CUBIC INCHES. 
 
 (PART i. 
 
 OUNCE TO QUARTER.) 
 
 Measured 
 Ounces. 
 
 Subdivisions. 
 
 British Cubic Inches. 
 
 I 
 
 10 
 
 100 
 
 1,000 
 10,000 
 
 i measured ounce 
 10 measured ounces 
 10 measured pounds 
 10 pottles 
 10 bushels 
 
 = i unit of capacity. 
 = measured pound. 
 = pottle. 
 = bushel. 
 = quarter. 
 
 1.7865938 
 17-865938 
 178.65938 
 1,786.5938 
 17,865.938 
 
 (PART n. 
 
 GILL TO COFFER) 
 
 Gills. 
 
 Subdivisions. 
 
 British Cubic Inches. 
 
 i 
 
 10 
 
 IOO 
 
 1,000 
 
 10,000 
 
 I gill 
 10 gills 
 10 quarts 
 10 sacks 
 10 coombs 
 
 = 4 units of capacity. 
 = i quart. 
 = i sack. 
 = i coomb. 
 = i coffer. 
 
 7.1463750 
 71.463750 
 714.63750 
 7,146.3750 
 71,463.750 
 
 The above table presents the radical or "standard" pyramid 
 
36 METROLOGY. 
 
 system under the temporary medium of our present cubic inches. 
 With its primary and secondary units, the measured ounce 
 and gill, we are now particularly concerned ; since from them 
 it can be clearly established that avoirdupois, apothecary, troy, 
 liquid, dry, and even the British imperial measures, must have 
 been originally derived. Thus sixteen of these pyramid unit 
 ounces equal 28.5855 English cubic inches, and form the basis 
 for the capacity of the common or avoirdupois pound. Traut- 
 wine says, "The standard of the avoirdupois pound, which is 
 the one in common use, is the weight of 27.7015 cubic inches 
 of pure distilled water, at its maximum density, or about 39.2 
 Fahrenheit, in latitude London, sea-level barometer 30"." 
 This same amount of water at the mean temperature of the 
 earth (i.e., pyramid standard) would gain about .016 in bulk 
 (i.e., .443 rb British cubic inches), and become at least 28.14 i 
 cubic inches. 
 
 Turning now to United-States liquid measure, we find that 
 its pint (the old "liquid pound") contains 28.875 cubic inches; 
 which is proportionately about as much in excess of sixteen of 
 our pyramid units as the avoirdupois pound is short thereof. 
 The pint of the British imperial (liquid and dry) measure is one 
 and a fourth pounds (avoirdupois) of water, or almost equal to 
 twenty of our unit ounces, allowing for mean temperature, and 
 loss of traditions. Now it is manifest that these, the true bases 
 of avoirdupois, liquid, and imperial measures, once established, 
 their whole tables follow as a matter of course, and connect 
 with the radical system at Gizeh, through the real unit 1.7865 + 
 English cubic inches. 
 
 In troy and apothecaries' weights the pounds are the same, 
 and, by measure, are each closely equal to twelve of our unit 
 ounces, or to (12 X 1.78659 ) = 21.4390 =t English cubic 
 inches, mean temperature, pressure, etc. 
 
 From an examination of the gill as a cubic measure of capa- 
 city, similar relations will be found to exist. Thus Trautwine 
 puts the modern gill at 7.21875 cubic inches, which is very close 
 upon that of our table (7.1463). Allowing for increase due to 
 temperature, little, if any, difference would exist. Four of these 
 gills give him the pint of liquid measure, 28.875 cubic inches, 
 and connect directly with the capacity of the avoirdupois pound, 
 
CA PA CITY ME A SURE. 3 7 
 
 I 
 
 etc. The minute discrepancies which may be detected in 
 these comparisons are nothing more than should have been 
 expected from our loss of traditions, and the remoteness of the 
 time when they were last compared with their grand standards. 
 Moreover, a large proportion of these discrepancies also vanish 
 upon equalizing the temperature. The most astonishing con- 
 sideration, however, is that they should ever have agreed so 
 closely. 
 
 In the table of pyramid measures of capacity which we have 
 just considered, we chose the limits, from the measured ounce 
 and the gill, to the quarter and coffer respectively, as those 
 comprising the requirements of the more ordinary phases of 
 human life. The pyramid system, however, extends harmoni- 
 ously both above and below these limits, so as not only to sat- 
 isfy the minuter requirements of scientific analysis, but equally 
 to comprehend those of the gigantic commerce of the dawning 
 international future. 
 
 Thus, the decimal subdivisions of the ounce and gill may be 
 continued almost indefinitely downward, as follows : 
 
 TABLE OF PYRAMID CAPACITY MEASURES, EXTENDED 
 
 DOWNWARD. 
 
 10 millirninims 
 
 ( 4 milliminims = i atom. 
 
 = i centiminim. 
 
 
 ( 2\ atoms 
 
 centimmim. 
 
 
 10 centiminims 
 
 j 4 centiminims = 
 
 drop. . j 
 
 = i deciminim. 
 
 
 ( 2% drops = 
 
 deciminim. ! 
 
 
 10 deciminims . . 
 
 ( 4 deciminims = 
 
 dose. j 
 
 = i minim. 
 
 
 \ 2\ doses ~ 
 
 minim. j 
 
 
 10 minims . . . 
 
 i 4 minims = 
 
 tablespoon. 1 " r | 
 
 = i measured ounce. 
 
 
 ( 2% tablespoons = 
 
 measured ounce, j 
 
 
 4 measured ounces 
 
 
 = i Kill. 
 
 The subordinate system of the above table is as follows : 
 
 10 atoms = i drop. 
 
 10 drops = i dose. 
 
 10 doses = i tablespoon. 
 
 10 tablespoons i gill. 
 
 1 Usually 8 to 10 teaspoons, or 4 to 6 dessert-spoons, or 2 to 3 tablespoons of large 
 capacity, may be reckoned at one ounce ; or at ratio of 9 : 5 : 2\ per ounce. 
 
METROLOGY. 
 
 On extending the system upward from the quarter and coffer, 
 some such series as the following must necessarily result. : 
 
 TABLE OF PYRAMID CAPACITY MEASURES, EXTENDED 
 
 UPWARD. 
 
 10 coffers . . 
 10 car-loads 
 10 barges . . 
 10 elevators . 
 10 districts . 
 
 The above likewise involves the following subordinate 
 one : 
 
 ( 2-J coffers 
 ( 4 bins 
 ( 2\ car-loads 
 i 4 lighters 
 ( 2| barges 
 ( 4 ship-loads 
 ( 2\ elevators 
 1 4 wharves 
 ( 2\ districts 
 ( 4 sections 
 
 - 
 
 i bin. 
 i car-load. 
 i lighter. 
 i barge. 
 i ship-load, 
 i elevator, 
 i wharf. 
 i district. 
 i section, 
 i ark. 
 
 ! 
 ! 
 
 f 
 
 i 
 
 = i " standard," or car-load. 
 = i barge. 
 
 = I district. 
 = i ark, or harbor. 
 
 10 bins . . = 
 10 lighters = 
 10 ship-loads = 
 10 wharves = 
 4 sections = 
 
 lighter. 
 
 ship-load. 
 
 wharf. 
 
 section. 
 
 ark, or harbor. 
 
 The pyramid system is, in fact, as unique from the stand-point 
 of capacity measure as we have already found it to be from 
 that of linear measure. Though continuous throughout, it may 
 be naturally subdivided into six separate tables, severally com- 
 prised between the following limits : 
 
 TABLE I. Milliminim to measured 
 
 ounce. 
 TABLE II. Atom to gill. 
 
 For scientific use in arts, profes- 
 sions, and trades. 
 
 TABLE III. Measured ounce to quarter. ) 
 
 TABLE IV. - Gill to coffer. \ For ordmar y use of ever y da y life - 
 
 TABLE V. Quarter to section. 
 TABLE VI. Coffer to ark. 
 
 For commercial 
 use. 
 
 and international 
 
 Each table is decimally subdivided, and is intimately related 
 to its alternates by the simplest common fractions (I, 4, J, 2, |, 
 25, etc.) ; and reductions from one measure to another may be 
 accomplished in the fewest possible figures, and without endless 
 fractions. 
 
CAPACITY MEASURE, 
 
 39 
 
 Tables I. and II. are concerned particularly with processes of 
 the more exact sciences and minute analysis. Tables III. and 
 IV. comprise the measures of familiar, every-day life ; while the 
 last two, V. and VL, are capable of satisfying the demands of 
 the most gigantic commercial enterprise that shall arise in the 
 day of universal peace and national intercourse. 
 
 Let us now view the central portion of this system that 
 comprised between the measured ounce and coffer in the 
 light of its own integrity, as expressed in pyramid cubic inches. 
 
 PYRAMID CAPACITY MEASURES COMPARED THROUGH 
 THE MEDIUM OF PYRAMID CUBIC INCHES. 
 
 (PART i. 
 
 OUNCE TO QUARTER.) 
 
 Measured 
 Ounces. 
 
 Subdivisions. 
 
 Pyramid Cubic Inches. 
 
 I 
 IO 
 100 
 
 1,000 
 10,000 
 
 i measured ounce. 
 10 measured ounces 
 10 measured pounds 
 10 pottles 
 10 bushels 
 
 = i measured pound. 
 = i pottle. 
 = I bushel. 
 = i quarter. 
 
 1.78125 (exact). 
 
 17.8125 
 178.125 
 I,78L25 
 17,812.5 
 
 (PART n. 
 
 GILL TO COFFER.) 
 
 Gill. 
 
 Subdivisions. 
 
 Pyramid Cubic Inches. 
 
 I 
 
 IO 
 IOO 
 
 1,000 
 
 10,000 
 
 I gill. 
 
 10 gills 
 10 quarts 
 10 sacks 
 10 coombs 
 
 = i quart. 1 " 
 = i sack. 
 i coomb. 
 = i coffer. 
 
 7.1250 (exact). 
 71.250 
 712.50 
 7,125.0 
 71,250 
 
 The foregoing is not only the radical table of the entire 
 system, but from it may be renewed the various special ones, 
 such as avoirdupois, liquid, etc., now in use. For instance, 
 sixteen of the above unit measured ounces give us exactly 28.5 
 pyramid cubic inches as the capacity (in pure water at mean 
 earth temperature, pressure, and hygroscopic state, 50 pyramid) 
 of an avoirdupois pound, from which the whole avoirdupois 
 sequence may be duly determined. Now, the rationale of taking 
 sixteen of these unit ounces to form the capacity of the avoir- 
 dupois pound becomes at once apparent upon reviewing the 
 
 1 Four such quarts = 285. cubic inches, the present milk-gallon of New Hampshire. 
 
40 METROLOGY. 
 
 United-States liquid measure, where four gills are taken for a 
 pint. Taking, therefore, four pyramid gills of 7.125 pyramid 
 cubic inches each, we have one pyramid pint 28.5 pyramid 
 cubic inches, the same capacity as the pyramid avoirdupois 
 pound, in pure water. Hence, as four ounces make a gill, and 
 four gills a pint, sixteen measured ounces make both a pint 
 and a measured avoirdupois pound. And avoirdupois becomes 
 at once related to liquid measure, and the two, through the 
 radical ounce (1.78125), to the whole pyramid system. In the 
 British imperial system, since its pint is taken to be \\ avoir- 
 dupois pounds, a pyramid gill of 7.125 inches, added to the 
 pyramid avoirdupois pound (28.5), gives its capacity. Thus, 
 28.54- 7- 12 5 35-625 pyramid cubic inches = one rectified 
 British imperial pint ; in terms of which, if the imperial sys- 
 tem be renewed and perfected, it will become at once, not only 
 earth-commensuric, but directly convertible, through the me- 
 dium of the unit pyramid measured ounce (1.78125), into the 
 terms of any other rectified Anglo-Saxon system. Again : 
 since the basis (Trautwine) of United-States dry measure "is 
 the old British Winchester struck bushel" (given as 2150.42 
 English cubic inches), by connecting it with the pyramid sys- 
 tem we shall render all of its elements duly convertible into 
 any other Anglo-Saxon measures. Now, inspection of the 
 pyramid system shows that the original basis of this old 
 "struck bushel" must have consisted of 350 gills, with the 
 "heap," usually one-seventh, scraped off; i.e., 350 gills struck 
 to 3 sacks, 30 quarts, or 300 gills; i.e., of 2,137.5 pyramid 
 cubic inches. The use of this as the basis of subdivision will 
 consequently perfect the system, and render it convertible in 
 simple terms and whole numbers. 
 
 Finally : the pounds of troy and apothecary measures consist 
 of twelve ounces each ; i.e., pyramid standard ounces. Their 
 true capacity of pure water is, therefore, 1.78125 X 12 = 21.375 
 pyramid cubic inches ; or, in other words, it is exactly one- 
 hundredth of a "struck bushel." 
 
 Thus, through the medium of the radical pyramid, " measured 
 ounce," and "gill," every one of the Anglo-Saxon tables of 
 capacity and weight (i.e., measure and capacity of the water- 
 equivalent) becomes directly related to all the others. 
 
CAPACITY MEASURE. 41 
 
 To recapitulate, the roots of the various systems as rectified 
 are as follows : 
 
 Pyr. cubic inches. 
 
 " unit " pyramid measured ounce = 1.78125 
 
 standard pyramid gill = 7.125 
 
 pyramid avoirdupois pound 16 pyramid ounces 28.5 
 
 pyramid pint (liquid, U.S.) = 4 pyramid gills = 28.5 
 
 pyramid struck bushel (dry, U.S.) = 300 pyramid gills = 2,137.5 
 
 British imperial pint (dry and liq.) = 20 pyramid ounces = 35.625 
 
 troy or apothecary pound = 12 pyramid oz. or pyramid gills = 21.375 
 
 These values are all based upon the contents of the coffer 
 in pyramid cubic inches, being 71,250. Proportional values, of 
 course, result when we take the contents of this standard in 
 British cubic inches. 
 
 In prder now to set forth what will be the nature of the con- 
 nection between the several systems of special Anglo-Saxon 
 measures and the standard system, that of the Great Pyramid, 
 when the former shall have the units duly perfected thereby, 
 we subjoin the following table. By reading it across, there will 
 be seen at a glance the equivalent of any designated measure 
 in terms of the rectified unit of each particular system. Thus 
 the coffer itself, of 71,250 pyramid cubic inches capacity, is 
 seen to be equal to 4,000 standard (or 10 oz.) pounds, 2,500 
 avoirdupois (or 16 oz.) pounds, or liquid-measure pints, 3,333$ 
 troy or apothecary (12 oz.) pounds, 2,133$ United-States dry 
 measure (or 18.75) pounds, and 2,000 British imperial (20 oz.) 
 pints, in which, throughout, the ounce considered is the stand- 
 ard "unit" (or one lo-thousandth of the quarter taken for 
 capacity at 17,812.5 pyramid cubic inches). 
 
METROLOGY. 
 
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 to to to to to to to to to to to to to to to to to to tooo to o o o o o o o 
 
 to to totototototototo to to to to to to tOOO fOO OOOOOOOO 
 
 S^*iz jo spunod 
 
 3333 J^g 00 * 
 
 ,S3UBD3l{10dB pUE Xoaj 
 
 s^SSS^S* "^ " 
 
 ajEA.n 3 
 
 g^rooo to 
 
 'S3DUHO 
 
 ,.jiun M 91 jo spunod 
 ao 'juid aad saqDui 
 
 vn N 
 
 oiqno pituEaXd S'gz jo 
 siuid pmbij *g '^Q ao 
 'spunod siodnpjiOAB 
 ui XjiOEdBO juajBAinbg 
 
 IPP:^" 
 
 saouno tl jiun ,, 01 jo 
 (qj aad ''oja 'aanssajd 
 
 ^w OOOOOOOO 
 
 saqoui oiqno piuiBjXd 
 Ssig'^i 'aajBM jo) 
 sqj piuiBjXd pJEpUBjs 
 ui AipBdBO juajBAinbg 
 
 oooooooo^ 
 
 8 II 2 * " * 
 
 
 iD C4 w^ i^NOO N-t^r-.^OOOO 
 
 
 1 8 88 1 J rTS,^ H^H.?^^^" 
 
 
 ^,i?^5 i?^E? R -^" 
 
 
 ^2^K.J?^ J 
 
 
 
 suoi 
 
 u^^tnx^ ..nmi^^i^ ^ ^ ^ ^ 
 
 
 
 
 oooo ooooo oooo 
 
 
 
 'suoiuod 
 
 o .1 S 
 
 
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 j i3 > a, Sc5 ? ^i-S^r^-^i-c o^.Soj.g c/,5 ^ 
 
 <(/)J5?wOT<S3u3 u C^U ffl t/3 ON C^OH O O HiSQQQcj<S 
 
CAPACITY MEASURE. 43 
 
 Not the least remarkable feature in the foregoing table is the 
 wholeness in which the numerical exchange of values comes 
 out. The passage from one system to another is thus a process 
 of the very simplest character. 
 
 From the foregoing consideration of the capacity measures 
 now in use throughout the Anglo-Saxon world, it is evident 
 that what the English-speaking people need, in order to obtain 
 a correct and earth commensuric system, is not by any means 
 an abolishment of its present measures. All that is necessary is 
 that the units of the various systems shall be severally based 
 back once more upon what is now clearly shown to have been their 
 original common UNIT. 
 
 This once accomplished, the various trades may continue in 
 the unmolested employment of those particular multiples of 
 measure and capacity which have been the result of such long 
 (at least four thousand years) experience. No terms will be 
 lost, no confusion inaugurated, by a wholesale change, as it were, 
 of " times and seasons ; " and the small corrections introduced 
 will be actually well within the margin now allowed in daily 
 trade and commerce ; while for purposes of mutual comparison 
 and intercourse, the most desirable ends will thus be realized. 
 
 What, then, indeed, have Anglo-Saxons to do with the met- 
 ric system in the face of such possibilities as are now before 
 them, affecting the absolute perfection of their own ? Why, 
 having preserved our "traditions" so long, or, rather, having 
 had them preserved for us so wonderfully, should we now dis- 
 card them, just as their beauties are being fully discovered ? 
 What has the metric system to offer in exchange for what we 
 now have, and for what we may have by perfecting our own ? 
 Its decimal system is absolutely its only claim ; and this is 
 entirely cancelled against the doubly decimal arrangement of 
 what the Anglo-Saxon inherits in the Great Pyramid, and in 
 the earth on wJiicJi he lives. Eliminate its decimal arrangement 
 from the metric., system, and nothing whatever remains but an 
 erroneous basis, a miscalculated standard, a confusion of un- 
 known terms, and a set of unscientific atmospheric conditions, 
 all of which are absolutely hostile to the every-day and practi- 
 cal requirements of man. 
 
IV. 
 
 WEIGHT MEASURE. 
 
 Let Him who hath weighed the mountains in scales, weigh me in balances of justice. 
 JOB xxxi. 6; ISA. xl. 12. 
 
 Relation of Capacity to Weight. The Standard Chamber at Gizeh. Present 
 Anglo-Saxon Weight-Measures. Their Lack of Harmony. Pyramid Weight- 
 Measure. Its Ultimate and Grain used to unify the Saxon Measures. The 
 Use of a Greatest Common Divisor. The Origin of British Metrology. The 
 Saxon Tables as Unified and Rectified. A Proposed Decimal Standard Table 
 of Weights for International Purposes. The Constants of the System. The 
 Peculiar Subdivision of the Ounce. Wonderful Possibilities of Complete 
 Resolution. The Last Drop in the Coffer, the Last Atom of the Caldron. The 
 Cubic Inch in Grains and Ultimates. Mean Density of the Earth. Its Use 
 in connecting Weight and Capacity. Its Value in Grains and Ultimates a Most 
 Important Number. Can the Metric System accomplish Such Feats? Manu- 
 facture of Weights out of Standard Material, Lead, Iron, etc. The Ancient 
 Metrology of Israel Part of this Earth-Commensuric and Rectified Saxon System. 
 The Wonderful Cubical Properties involved. The Challenge to the Metric 
 System. The French System examined. Its Only Feature of Value, the Deci- 
 mal, is not its Own. The Duty of the Few who use the Metric System. The 
 Duty of the Many who use the Saxon. 
 
 INTIMATELY connected with the subject of capacity measure 
 is that of weight. They are indeed so necessarily correlative, 
 that we have already found it impossible to compare the sev- 
 eral units and systems of capacity measure, without partially 
 anticipating our present topic. 
 
 Now, the most natural method of obtaining a unit of weight 
 is to fill one of capacity with some convenient substance, and 
 assume the weight of the result as the quantity sought. It is 
 universally customary to employ water at some standard tem- 
 perature and pressure for this purpose. In fact, water is 
 employed as the standard of comparison for all weight what- 
 soever, since it is taken as the unit of specific gravity itself. 
 
 44 
 
WEIGHT MEASURE. 45 
 
 If the coffer, therefore, which we have already found to be 
 so remarkable a capacity measure, be now filled with pure water, 
 it will be instantly converted into a standard of weight ; and 
 in this, its new employment, we shall find it capable of afford- 
 ing us even more astonishment than ever. 
 
 The temperature and barometric pressure in the king's cham- 
 ber we already know to be those most convenient for all ordi- 
 nary human avocations. They are the mean of all the earth, 
 more easily maintained, and less frequently varied from, than 
 any other standard points that could possibly have been agreed 
 upon. Now, 71,250 pyramid cubic inches of water, under the 
 circumstances presented in the king's chamber, weigh exactly 
 2,500 rectified avoirdupois pounds, or liquid-measure pints. This 
 fact recalls most vividly the ancient Anglo-Saxon rhyme, 
 
 " A pint's a pound, 
 The world around." 
 
 If Anglo-Saxon weights and measures, duly corrected by the 
 pyramid system, are to survive in the " battle of the standards," 
 now being so fiercely waged, then the above distich is deeply 
 prophetic. Let us proceed, therefore, to examine the three 
 systems now in common use, troy, apothecary, and avoirdu- 
 pois weights, in the light afforded by the pyramid, and see if 
 they are really worthy cf our support in the struggle now in 
 progress. 
 
 In Trautwine's " Engineer's Pocket-book " we learn, that, for 
 the United States and Great Britain, these weights are as fol- 
 lows : 
 
 Troy. i pound = 12 ounces = 240 pennyweights = 5,760 grains. 
 
 Apothecary. i pound = 12 ounces = 96 drams = 288 scruples = 5,760 grains. 
 Avoirdupois. i pound = 16 ounces = 256 drams = 7,000 grains. 
 
 We also read in this manual, that at present " the troy ounce 
 is greater than the avoirdupois, but the troy pound is less than 
 the avoirdupois ; while in troy and apothecary weights, the grain, 
 ounce, and pound are the same." "An avoirdupois pound = 
 1.21528 troy pounds, and an avoirdupois ounce = .91 1458 of a 
 troy ounce. Thus it is manifest that the grain is the only meas- 
 ure common to all of them ; and this as a divisor is now of little 
 value, save for the temporary purpose of mutual reduction. 
 
46 METROLOGY. 
 
 Its present value was arbitrarily chosen by Parliament at the 
 close of the eighteenth century, in a moment of too hasty le- 
 gislation, when 24 artificial grains (i.e., "so called, but of no 
 known variety of plant employed for breadstuff") were put for 
 what had formerly been the number, 32, that went to make up 
 a pennyweight, troy. We say the present grain is of little 
 value save as a common divisor for the temporary purpose of 
 reduction. This is because it is based upon the erroneous idea, 
 that the ounces or units in the three systems, as well as the 
 pounds, are not the same. Now, we have already shown, while 
 discussing the capacities of these various systems, that the 
 standard unit of capacity, 1.78125 pyramid cubic inches, at 
 mean temperature and pressure, was undoubtedly the original 
 basis of these measures, and that it is the one by which their 
 rectification should now be effected. This once accomplished, 
 it is manifest that the value of the present artificial grain as a 
 common divisor will instantly disappear. But its retention, 
 even at present, involves the perpetuation of an inconvenience. 
 For example, while the table of troy weight begins " 24 grains 
 make a pennyweight," and that of apothecary weight com- 
 mences "20 grains make a scruple" (i.e., both even and con- 
 venient numbers), the table of avoirdupois weight (the very one 
 of all others that should be even, as it is the most universally 
 employed) opens with the awkward statement that 27.34375 
 grains make a dram. 
 
 The systems are .not harmonious; and it is on account of 
 such evident flaws as the above, that the now mutilated Anglo- 
 Saxon weights and measures have so many opponents. Let, 
 however, these systems once be rectified back to their ancient 
 standard, and no system that can be conceived of will be able 
 to maintain itself against them. 
 
 In the following table, commencing with the ounce-weight 
 and ending with the ton, the same system has, of course, been 
 followed as in the capacity measures. Designations have sim- 
 ply been altered to suit the subject (weight) under considera- 
 tion. It may also be noticed here, that, as in the case of the 
 capacity measures, this same system can be extended both 
 upwards and downwards to correspond therewith. We shall, 
 however, at present content ourselves with the consideration 
 
WEIGHT MEASURE. 47 
 
 of this table alone, and of the various Anglo-Saxon systems 
 thence derived through the medium of its unit ounce. 
 
 STANDARD TABLE OF PYRAMID WEIGHT-MEASURE. 
 (FROM OUNCE TO TON.) 
 
 gill weight. ) =IstanApoundwt . 
 
 stand, pound wt. \ 
 
 10 ounces (weight) I* oz. (weight) = 
 
 I 2| gill weights - 
 
 10 pounds (weight) J 4 . (weight) = 
 
 io stone J 
 
 10 hundred weight \ 4 cwt> 
 ( 2\ weys 
 
 quart weights 
 4 stone. = 
 
 2\ sack weights = 
 
 quart weight. 
 
 stone. 
 
 = i stone. 
 
 sack weight. ) = I hundred weight, 
 cwt., or quintal. ) 
 
 wey> i = i thousand wt. 
 
 thousand weight. ) 
 
 4 thousand weight = i ton = I coffer. 
 
 The subordinate decimal system is as follows : 
 
 4 ounces = i gill weight. 
 
 10 gill weights = i quart " 
 10 quart weights = i sack u 
 10 sack weights = i wey. 
 10 weys = i ton. 
 
 In the above tables the ounce is the weight of a unit of 
 capacity, 1.78125 pyramid cubic inches filled with pure water 
 at mean earth temperature (50 pyramid), pressure (50 pyra- 
 mid), and hygroscopic state (50 pyramid) ; and the gill-weight 
 that of four such units, or 7.125 pyramid cubic inches. 
 
 We have already tabulated under the head of capacity meas- 
 ures the comparative values, in terms of the weight of their 
 own rectified units, of the troy, avoirdupois, and apothecary 
 subdivisions, which correspond to the foregoing table. 
 
 It is now necessary to determine the proper number of grains 
 originally contained in the "pound-weights" of these systems 
 as now rectified ; since, as before mentioned, by the very process 
 of rectification, the numbers of arbitrary grains contained in 
 these pounds will lose all their present significance. 
 
 From various considerations, ramifying out into too many 
 subordinate subjects to be considered here, it seems probable 
 that the number of real ultimates originally contained in the 
 ancient coffer of 71,250 pyramid cubic inches capacity, was 25,- 
 000,000 = (5,ooo) 2 (25 x 2 X ioo) 2 = (50 X ioo). 2 Let us there- 
 fore apply this number, and see if it will harmonize the several 
 
48 METROLOGY. 
 
 systems, and assist us in determining their greatest common 
 divisor. Let x = 25,000,000 ultimates. Then there having 
 been found to be 4,000 standard (i.e., lo-oz.) pounds in the cof- 
 fer, we have 
 
 i pyramid standard pound weight = - = 6,250 " ultimates." 
 
 And for similar reasons 
 
 i pyramid troy or apothecary pound weight - j- = 7,500 " ultimates." 
 
 i pyramid avoirdupois pound weight = - = 10,000 " 
 
 Furthermore, dividing 71,250 pyramid cubic inches by 25,- 
 000,000, we learn that one such standard ultimate contains 
 exactly .00285 pyramid cubic inches of pure water ; i.e., the 
 unity material at mean temperature, pressure, hygroscopic state, 
 etc., or is ^^Vo ^ a $ ^ water The above numbers of 
 standard ultimates per pound give us for each of the several 
 systems the following numbers : 
 
 In troy weight 3 l ' z S = 3 1 ! (exact) ultimates i pennyweight. 
 
 In apothecary weight 26.04166 = 26-^ " " = i scruple. 
 
 In avoirdupois weight 39.0625 = 39^ " " = i dram. 
 
 At the first inspection, the above figures appear to be some- 
 what objectionable, as everyone of them involves a fraction; 
 while of those now in use only one (the avoirdupois) does so. 
 Such was at first my own impression after calculating them. 
 Though all of them are exact, they involve decimal terminations 
 which are the very things we wish to avoid in a (to be) popular 
 table of measures. Closer study, however, revealed to me their 
 hidden beauty. The greatest common divisor of these three 
 quantities is 1.302083 ultimates ; the cubic capacity of which 
 is .0037109375 (exact) pyramid cubic inches of water, or, in other 
 words, exactly ygV 6 ^ an avoirdupois gill of water. 
 
 I shajl now retain the present name of this common divisor, 
 and designate the quotients thus obtained as grains (i.e., YgVfr 
 part of an avoirdupoi-s gill). Dividing each of the above nu'm- 
 bers by it (i.e., by 1.30208$ ultimates), we have, 
 
 For troy weight 24 grains = i pennyweight (as before). 
 
 For apothecary weight 20 " = i scruple (as before). 
 For avoirdupois weight 30 " = i dram (as now rectified}. 
 
WEIGHT MEASURE. 49 
 
 Hence we have rediscovered the rationale of the traditional 
 subdivision of the pennyweight into twenty-fourths, and of the 
 scruple into twentieths, and, in rectifying the systems by com- 
 parison with their pyramid standards, have been enabled to 
 relegate into oblivion the awkward number of 27.34375 arti- 
 ficial grains with which the avoirdupois table now opens, and 
 substitute therefor the simple number 30. 
 
 These numbers (24, 20, and 30) result from the very nature 
 of things, and are manifestly of "great importance in conven- 
 iently continuing the final subdivisions down through halves, 
 quarters, thirds, etc., to the very terminals demanded by the 
 arts, trades, and professions specially concerned with the use 
 of each several subdivision of the general pyramid, earth-com- 
 mensuric, or Anglo-Saxon systems. It is thus manifest, that 
 what we now call a grain in our present apothecary and avoir- 
 dupois weights is not the real and original ultimate of weight- 
 measure ; but it is the result of dividing the terminal number 
 of the several tables expressed in. original ultimates by their 
 greatest common divisor, and that its introduction into the 
 systems was, and is, the means of clearing them of fractions. 
 
 Troy, avoirdupois, and apothecary weights pass back to- 
 gether into the most remote Anglo-Saxon times. From a prioii 
 reasoning, it is clear, that, in their origin (as we have now actu- 
 ally demonstrated), they must have had a common unit. It 
 would be a most difficult undertaking to trace their actual 
 history, to tell when they severally broke off from the parent 
 stock, why they did so, and which of them is the oldest system. 
 We could probably trace the ounce back to the most remote 
 times. Like the inch, its origin is shrouded in the deepest 
 mystery, a mystery all the more tantalizing because it turns 
 out to be so consummately perfect a unit. Let us put the ori- 
 gin of the troy system, for the sake of argument, back to the 
 time of thq Norman Conquest. Now, which is the more prob- 
 able, that the jewellers and goldbeaters actually originated a 
 whole system, unit, standard, and all, or that they merely took 
 the unit of their ancestors, and arranged its subdivisions and 
 multiples to suit the demands of their own trade; i.e., that they 
 adopted a new standard, and retained the old unit ? Manifestly 
 the latter. Otherwise there would have been no means of com- 
 
50 METROLOGY. 
 
 parison : otherwise an ounce of gold would not have been intel- 
 ligible to their purchasers, who lived upon ounces of bread. It 
 was absolutely necessary, that somewhere in each of the ele- 
 mentary systems, as they successively originated, there should 
 have been a connecting link with what had gone before. It is 
 one thing to adopt a standard ; as, for instance, to say to-day 
 that hereafter twenty-five inches shall be called a cubit (sacred 
 or ecclesiastical), and that all church-architecture shall be based 
 upon it : it is quite another to make an entirely new unit also ; 
 as, for instance, the metric system. In the one case, we pre- 
 serve our traditions, and work intelligently among our neigh- 
 bors : in the other, we ruin all things with a deluge. To adopt 
 the second's pendulum, or some other constant thing, as a stand- 
 ard capable of perpetual renewal, and to ordain that it shall be 
 interpreted in inches, a well-known previous unit, with a view of 
 squaring and renewing feet, yards, and ells forever at a constant 
 value, is an act of wisdom ; but to break away from every tradi- 
 tion, and make all things new by curving them against an arbi- 
 trary ay, even an imaginary, impossible, and absolutely inde- 
 terminate thing, is an act of folly, worthy only of the dreadful 
 "age of revolution." The ratio exists, and can be found, by 
 which the second's pendulum, at any latitude, is a constant func- 
 tion of the polar axis ; and as, cceteris pat ibtis, the pendulum 
 for a constant latitude must be a constant length, we can by 
 its means say what is the cubit or the one lo-millionth of the 
 polar radius. This alone is a standard fit for all times and 
 places, and is eternal in its permanence ; and a twenty-fifth of 
 such a standard is the Anglo-Saxon inch and unit. 
 
 But our discovery throws a flood of light in another direction. 
 Since we have now established that the true numbers of the 
 real ultimates actually contained in the various Anglo-Saxon 
 and pyramid pounds are much greater than those now in use, 
 the question instantly arises, How are we to account for the 
 numbers 4,800, 5,760, and 7,680, which have come down to us 
 traditionally as part and parcel of the Anglo-Saxon systems, 
 and have so long been used as grains ? The answer is a 
 very simple one. These numbers express the values of the 
 several Anglo-Saxon pounds (standard, troy and apothecary, 
 and avoirdupois), in terms of their greatest common divisor 
 
WEIGHT MEASURE. 51 
 
 (.00285 X 1. 30208 J =.0037109375 exact cubic inches of water 
 at standard circumstances, = \^^ of an avoirdupois gill = -^\-$ 
 of a unit ounce). 
 
 Will any one maintain that such facts as these, whereby all 
 the parts of the whole existing Anglo-Saxon capacity and weight 
 measures are not only connected with the pyramid, but are 
 made to harmonize so exactly with each other, are " mere coin- 
 cidences," and that they exist simply because they have been 
 looked for ? The class of facts to which these discoveries 
 belong does not exist unless intentionally created by a scientific 
 use of the intellect, and of the fact of such intention we may 
 be as certain as of the correlative one that it requires an intel- 
 lect to comprehend them. 
 
 The several tables of Anglo-Saxon weights which we have 
 thus rectified at the pyramid will, therefore, stand as follows : 
 
 RECTIFIED TROY WEIGHT. 
 
 24 grains = i pennyweight = 31^ ultimates. 
 
 20 pennyweights = i ounce = 480 grains = 625 ultimates. 
 
 12 ounces = i pound = 240 dwt. = 5?6o grains = 7,500 ults. 
 
 Upon its surface the above table, save in the use of the addi- 
 tional ultimate, does not appear to differ from troy weight as 
 given in any book of tables. This is because we preserve all 
 the terms and the form of the old system. This is one of the 
 chief merits in the value of our discovery, that, while by this 
 system the weights now in use become absolutely perfected, no 
 confusion can result in their adoption. 
 
 RECTIFIED APOTHECARY WEIGHT. 
 
 20 grains i scruple = 26^ ultimates. 
 
 3 scruples = i dram = 60 grains = 78-^4- ultimates. 
 
 8 drams = i ounce = 24 scruples 480 grains = 625 ultimates. 
 12 ounces = i pound =96 drams =288 scruples = 5,760 grains = 7, 500 ults. 
 
 Similar remarks are in order relative to the above table. 
 
 RECTIFIED AVOIRDUPOIS WEIGHT. 
 
 30 grains i dram = 39^ ultimates. 
 
 16 drams = i ounce = 480 grs. = 625 ultimates. 
 
 16 ounces = i pound 256 drs. = 7,680 grs. = 10,000 ultimates. 
 
 28 pounds = i quarter = 448 oz. = 7,168 drs. = 215,040 grs. = 280,000 ults. 
 
52 METROLOGY. 
 
 4 quarters = I cwt. =112 Ibs. = 1,792 oz. = 28,672 drs. = 860,160 grains 
 
 = 1,120,000 ultimates. 
 
 20 cwt. = I ton = 80 qrs. = 2,240 Ibs. = 35,840 oz. = 573,440 drams 
 
 = 17,203,200 grains = 22,400,000 ultimates. 
 
 In the avoirdupois table the present form and skeleton are 
 preserved, save that the effect of our discovery is to rectify the 
 form in one particular ; namely, 30 of our new grains equal 
 I dram, while in the present statute-tables 27.34375 of the old 
 Anglo-Saxon grains go to form the dram. The system thus 
 clears the most important or commercial weight of its awk- 
 ward fractional commencement, besides making its ounce, 
 grain, and ultimate of the constant or unit value of all the 
 other tables. 
 
 To complete the series of tables, we shall now add the fol- 
 lowing one of intermediate values derived from the standard 
 pyramid pound of 10 unit ounces, and which we shall desig- 
 nate as : 
 
 STANDARD WEIGHT MEASURE (RECTIFIED). 
 
 12 grains = i scruple = 15$ ultimates. 1 
 4 scruples = i dram, or minim = 48 grains = 62.5 ultimates. 
 
 ( 4 drams = i table or large spoon ) i ounce = 40 scruples 
 \ 2\ large or tablespoons = i ounce C 480 grains = 625 ults. 
 ( 4 ounces = i gill ) i pound = 100 drams = 400 scruples = 4,800 
 I 2\ gills = i pound \ grains = 6,250 ultimates. 
 < (4 pounds = i quart wt.) = 400 drams ) i stone = 100 ounces 
 i (2j quart wt. - i stone) = 10 Ibs., etc. J = 1,000 drams. 
 
 i (4 stone i sack wt.) = 40 Ibs. = 400 \ = i quintal = 100 Ibs. 
 10 stone ounces = 4,000 drams. 1,000 oz. = 10,000 
 
 ( (2^ sack wt. = i quintal) = 100 Ibs., etc. ) drams. 
 r (4 quintals = i wey) 40 stone = 400 } i thous. wt. = 100 st. 
 10 quintals Ibs. 4,000 oz. 40,000 drams. = i ,000 Ibs. = 10,000 
 
 C (2^ wey = i thous. wt.) = 100 stone, etc. ) oz. = 100,000 drams. 
 (4 thous. wt. = i ton) = 40 quintals = 1 = i bin = 100 quintals 
 400 st. = 4,000 Ib. = 40,000 oz., etc. I = 1,000 stone = 
 >us. wt. -\ _ _ ~ r 10,000 Ibs = 100,000 
 
 loo quintals, etc. ounces, etc. 
 
 4 bins = i standard = 10 tons = 40 thous. wt. = 100 weys, etc. 
 
 1 The " ultimate " in these tables is of special advantage in calculation. Having played, 
 however, its most important part as a factor in the process of rectification, it may, in fact, be 
 entirely left out of the tables, and employed only as a means of subordinate intercourse, as it 
 were, from one to the other. But as the system develops into its widening possibilities, this 
 subdivision from its direct connection with the decimal notation will prove of great con- 
 venience. 
 
WEIGHT MEASURE. 53 
 
 In the above table, which is a new one, but in which, for 
 facility of introduction, old and well-known Anglo-Saxon terms 
 are employed, the most noticeable feature is its decimal con- 
 tinuity. Thus, commencing at the standard dram, this table 
 may be read decimally as follows : 
 
 10 drams = i ounce. 
 
 10 ounces = i pound. 
 
 10 pounds = i stone. 
 
 10 stone = i quintal. 
 
 10 quintals = i thousand weight. 
 10 thousand weight = i bin. 
 
 4 bins = i standard. 
 
 Or, since it is doubly decimal, we have the following alternate 
 system pervading it : 
 
 10 spoonfuls = 
 
 gill. 
 
 10 gills = 
 
 quart. 
 
 10 quarts = 
 
 sack. 
 
 10 sacks = 
 
 wey. 
 
 10 weys = 
 
 ton. 
 
 10 tons = 
 
 standard. 
 
 Moreover, a careful study of the table of standard weight 
 measure rectified will show the facility with which any and 
 all of its various terms may be halved, quartered, and mul- 
 tifariously subdivided into common fractional parts. Thus, 
 i pound equals \ of a quart ; 5 ounces equal J a pound ; \ of 
 a sack equals I stone, etc. The completeness with which this 
 common fractional subdivision, so necessary in the daily use 
 even of a decimal system, may be effected, will be best under- 
 stood from the following : Since there are 4,800 grains in a 
 standard pound, and since 4,800 = 2X2X2X2X2X2 
 X5X5X3XI, it follows that the pound may be divided 
 into i's, fs,^'s, 's, fs, J's, T ys, fts, ^'s, T Vs, ^*> sY 8 ' 
 
 sV 8 ' *V S -sV 8 ' *V S > lY s > sV 8 ' sV 8 > sY s > A' 8 '. $V S oV 8 ' Tib' 8 ' 
 T2Tl' s 2il)' s > ete -> ar| d hence that every term in the table may 
 likewise be so subdivided without fractional remainders, etc. ; 
 hence, if a decimal table of weights and measures be actually 
 necessary at the present state of science and commerce, we 
 have one in the system now discovered whose introduction will 
 involve no change of terms in Anglo-Saxondom, whose unit (the 
 
54 
 
 METROLOGY. 
 
 ounce) is that of troy, apothecary, and avoirdupois weights, and 
 one, too, in which the decimal system is not only double, but 
 which offers extraordinary facilities for the still more neces- 
 sary employment of common fractions, etc. 
 
 /;/ all of the foregoing tables, the ultimate, grain, ounce, 
 and gill are CONSTANT, or have a "standard" value : the other 
 subdivisions are the particulars of special arts, yet all of them 
 are perfect and direct parts of the " unit ounce weight." The 
 constant values are as follows : 
 
 ANCIENT PYRAMID, ORIGINAL ANGLO-SAXON, AND 
 RECTIFIED. 
 
 .00285 (exact) pyramid cubic inches of pure water at mean 
 earth temperature, pressure, and hygroscopic state (pyra- 
 
 ounce gill coffer 
 
 mid 50) = - -s = 
 
 625 2,500 25,000,000 
 
 1.3020833, etc. (i.e. Vf? or ifi!) grains = .0037109375 (exact) 
 pyramid cubic inches pure water, pyramid 50, temperature, 
 
 ounce gill coffer 
 
 barometer, and hvgros. = ; = -* = 
 
 480 1,920 19,200,000 
 
 480 grains = 625 ultimates = 1.78125 (exact) pyramid cubic 
 inches of pure water at 50 pyramid, temperature, barome- 
 
 gill quarter coffer 
 
 ter, and hvgroscope = ==*- = . 
 
 4 10,000 40,000 
 
 1,920 grains = 2,500 ultimates = 7.125 (exact) pyramid cubic 
 inches of pure water at 50 pyramid, thermometer, barome- 
 ter, and hygroscope _ quarter _ coffer 
 2,500 ~ 10,000 
 
 " ultimate " = 
 
 "grain" = 
 
 ounce = 
 
 'gill" 
 
 It is noticeable, that, in every one of these tables, the stand- 
 ard ounce is ultimately divided into 480 grains. Thus, in 
 troy, we have 24 X 20 480; in apothecary, 20 X 3 X 8 = 
 480; in avoirdupois, 30 X 16 = 480; and, in the proposed 
 standard, 12 X 4 X 10 = 480. By means of these arrange- 
 ments, every fractional part of this unit ounce that trade, art, 
 or science can demand is thus specifically named. Thus, to 
 mention but a few, the half-ounce = 8 drams avoirdupois = 
 4 drams apothecary = 10 pennyweights troy = 5 drams stand- 
 ard ; \ ounce = 8 scruples apothecary ; -J ounce = 10 scruples 
 standard = 4 drams avoirdupois = 2 drams apothecary = 5 
 pennyweights troy ; -J ounce = 4 pennyweights troy ; \ ounce 
 = 4 scruples apothecary ; \ ounce = i dram apothecary ; 
 ^ ounce = 2 pennyweights troy ; ^ ounce = 2 scruples 
 
WEIGHT MEASURE. 55 
 
 apothecary ; ^ ounce i dram avoirdupois ; ^ ounce i 
 pennyweight troy ; ^ ounce i scruple apothecary ; -fa ounce 
 = | dram avoirdupois ; -fa ounce = f scruple apothecary ; ^ 
 ounce -J pennyweight troy ; ^ ounce = | scruple apothe- 
 cary, etc., by suitable combinations. 
 
 All of the above tables are strictly Anglo-Saxon, and they 
 are as strictly of Great-Pyramid origin. For all practical pur- 
 poses, the subdivision of the coffer into 19,200,000 parts fur- 
 nishes us with an elementary weight sufficiently small ; and it 
 is upon this fractional subdivision that our ancestors mani- 
 festly agreed when they established those subdivisions, or ali- 
 quot numbers, for the various pounds, 4,800, 5,760, and 
 7,680, which we have come to designate as "grains." This, 
 however, exceeds the true pyramidal ultimate of weight, as 
 we have shown, in the proportion of 250 : 192 = 125 : 96 ; so 
 that the 19,200,000 grains, into which the coffer is thus re- 
 solved, are actually equal to 25,000,000 real ultimates of .00285 
 pyramid cubic inches capacity of pure water each. Such an 
 ultimate is equal in weight to 1 f centidrams of water (at 
 .00178125 pyramid cubic inches per centidram), as given in the 
 "Table of Pyramid Measure, extended Downward." (See p. 56.) 
 
 Both of these numbers, 19,200,000 and 25,000,000, expres- 
 sive respectively of the number of grains and ultimates in the 
 standard coffer, are remarkable for the facility with which they 
 may be successively factored. This is a property of manifest 
 and primary utility for any radical that shall be assumed as the 
 basis of minute capacity or weight measures. Thus, eleven suc- 
 cessive divisions of the number 19,200,000 by the constant fac- 
 tor 2 gives us the quantity 9,375, which by six more successive 
 divisions by the factor 5 reduces to unity itself, i.e., to one 
 grain ; that is, 2X2X2X2X2X2X2X2X2X2X2X 
 5X5X5X5X5X5X1 = 19,200,000. Or, viewed from 
 another stand-point, it is successively divisible by 10 seven 
 times, the remainder in this case being 1.92 (exact) grains : 
 10 X 10 X 10 X 10 X 10 X 10 X 10 X 1.92 =19,200,000. 
 
 Now, 1.92 = |f|, and in ultimates = fgf X ^ = 2.50 ulti- 
 mates, the cubic capacity of which is but .007125 (exact) pyra- 
 mid cubic inches ; i.e., the drop. The subdivision can be thus 
 continued down to what is literally the last drop in the coffer. 
 
56 METROLOGY. 
 
 Again, taking the number 25,000,000, and treating it in a similar 
 way, we find, that, in spite of its complex relations to 19,200,000 
 (i.e., 125:96), it can be six times successively divided by 2, 
 yielding the number 390,625 grains, which, by eight more 
 successive divisions by 5, likewise reduces to unity. Thus, 
 2X2X2X2X2X2X5X5X5X5X5X5X5X5X1 
 = 25,000,000. Or, factoring by 10 six times, we obtain the 
 minute quantity of 25 ultimates ; after which we may still factor 
 once or twice more by 5. Thus, 10 X 10 X 10 X 10 X 10 X 10 
 X 5 X 5 X i = 25,000,000 ; leaving respectively either 5 or but 
 i, and this one absolutely the last ultimate of all its fellows. 
 , Now, as in the case of capacity measures, so, with even more 
 appearance of necessity, we may continue the standard table of 
 weight downward from the unit ounce and gill as follows : 
 
 STANDARD WEIGHT MEASURE, EXTENDED DOWNWARD. 
 
 lomillidrams H millidrams = i atom. 1 = l centidram . 
 
 ( 2$ atoms = i centidram. 
 
 locentidrams J* centidrams - i drop = , deddranh 
 
 ( 2\ drops = i decidram. j 
 
 10 decidrams \ * decidrams = * dose ' ) = , standard dram. 
 
 ( 2\ doses i dram. 
 
 10 stand, drams j 4 stand, drams - i large spoon. 
 
 ( 2$ large spoons = i ounce. 
 4 ounces = i gill. 
 
 It will be noticed that there runs through this table, as in 
 the case of all the other standard ones, a subordinate decimal 
 system of weights, which is as follows : 
 
 10 atoms 
 
 10 drops = 
 
 10 doses = 
 
 10 large spoons = 
 
 4 ounces = 
 
 drop. 
 
 dose. 
 
 large spoon. 
 
 ounce. 
 
 gill. 
 
 The exact number of grains in a cubic inch of water (pyra- 
 mid 50, temperature and pressure and hygroscope) is expressed 
 decimally by the repetend, 269.47368421052631578947368 + 
 etc. ; and the number of ultimates in such a cubic inch by 
 another repetend, as follows : 350.877192982456140350 + etc. 
 
 The above numbers (though capable of exact use in calcula- 
 tion, on account of their repeating character) look formidable 
 
WEIGHT MEASURE. 57 
 
 But for all practical purposes they can safely be employed, as 
 decimals, in the following forms : grains in I cubic inch = 
 269.4737 ( ), and ultimates in cubic inch 350.8772 ( ). We 
 shall return, however, to these two repetends in due time, when 
 it will be shown that the beauty of the whole system concen- 
 trates in them, and when their consummate simplicity for the pur- 
 poses of accurate subdivisions into practical commercial weights 
 of all descriptions for USE UPON THE SCALES will be revealed. 
 
 In the mean while let us return to the coffer, which is now, 
 and in this connection, to have for us an even higher signifi- 
 cation than any upon which we have yet touched. 
 
 The best modern estimate of the mean density of the earth 
 is about 5.7 times the weight of water. To determine this 
 accurately has been one of the greatest problems of modern 
 times. It has enlisted the ability of the greatest scientific 
 men. Newton shrewdly guessed, without experiment, that it 
 would be found between five and six times that of water. 1 
 Maskelyne followed with a series of unsatisfactory experi- 
 ments making it 4.8 water, with an addition whose amount he 
 could not determine. Capt. Ross Clark, R.E., determined upon 
 5.316. Sir George B. Airy, Dr. Whewell, and Richard Sheep- 
 shanks, working in the mines at Cornwall, met with failure in 
 1855. Later, however, Sir G. B. Airy, in a mine near New- 
 castle, was somewhat more successful, but arrived at "the un- 
 expectedly large result of 6.565." Rev. John Mitchell next 
 proposed a new mode of determination, which was later carried 
 out by Cavendish with 5.45 as a result. Professor Reich of 
 Saxony repeated the experiments of Cavendish, and deduced 
 5.44. Finally Francis Baily, after a most protracted series of 
 investigations, determined this mean density to be 5.675 dr 
 .0038. Even yet, however, modern science is not satisfied and 
 certain, though it is well convinced that the quantity sought 
 for is around the estimate made by Newton, 5.5, with a pos- 
 sible addition of from .1 to .3, in the positive direction, from 
 the indications of the more careful experiments of Baily. 
 
 The Great Pyramid, however, pronounces emphatically for 
 
 the figures we first quoted, or for a mean density of the earth of 
 
 t, 
 
 1 It was Newton, too, who also first made the remarkably close approximation to the 
 length of the " sacred cubit," 24.88 British inches. 
 
58 METROLOGY. 
 
 just 5.7 times that of water /;/ the king's chamber ; i.e., at mean 
 temperature 50 pyramid, barometer 50 pyramid, and hygro- 
 scope 50. 
 
 I shall not at present attempt to adduce the various inci- 
 dental references found in the pyramid, and particularly in the 
 neighborhood of the granite leaf and coffer, to this number 5.7, 
 but, assuming it to be the quantity sought and intended to be 
 employed in pyramid metrology, shall proceed at once to 
 examine into the circumstances that will attend its introduc- 
 tion as a factor into our investigations. 
 
 We have already noticed, that by filling the coffer with water 
 at 50 pyramid, barometer, thermometer, and hygroscope, it is 
 converted from a capacity measure into one of weight. Such 
 a cofferful will, of course, be 71,250 cubic inches of water. 
 The question now arises, What will be its equivalent at the 
 mean density of the earth ? Dividing 71,250 by 5.7, to obtain 
 an answer to our question, we obtain for the result the notice- 
 ably neat number 12,500 cubic inches. This, therefore, is the 
 bulk of the earth's mean density material which the coffer indi- 
 cates, and which will at all events exactly balance a cofferful of 
 water under standard circumstances. This is a point that should 
 be clearly appreciated before we proceed ; namely, that, be the 
 mean density of the earth 5.7 exactly or not, still, and neverthe- 
 less, 12,500 cubic inches of material thus (5.7) dense will ex- 
 actly balance 71,250 cubic inches, or a theoretical cofferful of 
 the unity (i.) material water in the Standard Observatory at 
 Gizeh as well as elsewhere all over the earth amid the habita- 
 tions of man, duly preserved at mean earth temperature, press- 
 ure, and hygroscopic state. 
 
 This fact established, we are now ready to examine our 
 number 12,500 at length. It is first noticeable, that this num- 
 ber is exactly one-tenth the cube of 50, a fact which connects 
 it in a remarkable way with the standard chamber, already filled 
 with architectural, barometric, hygroscopic, thermic, mathe- 
 matical, geometric, and other references to this marked num- 
 ber, 50, in which the coffer was preserved at Gizeh. 
 
 Now since, as we have already seen, a coffer contains 2,500 
 pyramid avoirdupois pounds, each such pound is represented by 
 5 cubic inches of the mean density material; i.e., 12,5004- 2,500 
 
WEIGHT MEASURE. 59 
 
 = 5": hence it follows, that 50 such cubic inches will exactly 
 balance 10 rectified avoirdupois or commercial pounds. To give 
 a still better idea of this, it may be stated, that, if we could com- 
 press 285 cubic inches of water (10 pyramid avoirdupois pounds) 
 into a space of 50 cubic inches, it would be at the mean density 
 of the earth, or, at least, at standard pyramid density. For 
 similar reasons, a standard or lo-ounce pound = 3^ cubic 
 inches of this material, and a rectified troy or apothecary 
 pound 3 1 cubic inches, etc. Now, the fact that 12,500 is -^ 
 of a cube of 50" of this standard material, leads to the idea that 
 the cube of 50 itself t or 10 coffers of water, is the actual pyra- 
 mid standard for weight. 1 Such a cube (50" X 50" X 50"), mean 
 density, would contain exactly 25,000 commercial pounds, the 
 cubit (25 X 1,000 = 25 X io 3 ) character of which is noticeable. 
 By referring now to our table of "Standard Weight Measure ex- 
 tended Upward," it will likewise be seen that this quantity has 
 been designated as a standard, and is an average car-load, and 
 that 10,000 such cubes io 4 X (50 mean density) 3 = an ark 
 avoirdupois, or as many pyramid pounds avoirdupois as there 
 are inches in the polar radius of the earth. 
 
 As in the case of capacity measures by means of this stand- 
 ard cube, the rectified system of weight measures may be ex- 
 tended upward, so as to comprehend the most extravagant 
 demands that can be made upon it by a world at peace, and 
 engaged in universal intercourse and international commerce. 
 A scheme for such extended use is given below ; and since a 
 standard equals ten tons, or an average car-load, we have, 
 
 1 !t is also noticeable here, since 50" is equal to 2 X 25", or to hvo cubits, that the cube 
 of 50", or the standard cube, is itself, when expressed in cubits, peculiarly tied, as it were, to 
 the octenary system of numeration. Thus (so") 3 = (2 cubits) 3 = 8 cubic cubits. It is thus 
 not only geometrically a cube, but arithmetically so, and this in a most important triple way ; 
 for taken in terms of the unit cube [125,000 = (5o") 3 ], or in terms of the cubit cube [(25" X a) 3 
 = (2 cubits) 3 8 cubic cubits], or in its own integrity as a standard, it is equally expressive 
 of this fundamental metrologic idea of cubicity so necessary in weight and capacity consider- 
 ations, while at the same time it has an inherent connection with all linear measure through 
 the inch, the cubit, and the axial reference. For important discussions upon the peculiar 
 advantages of an octenary basis for metrology over a decimal one, and of the direct relation 
 of the former to duodecimal arithmetic, see following articles : " Unification of Moneys, 
 Weights, and Measures, 1 ' International Review. " Extracts " therefrom, by Alfred B. Taylor, 
 International Standard," July, 1883. "The Metric System in our Workshops," by Coleman 
 Sellers, Journal of Franklin Institute, June, 1874. " Report by S. F. Gates," International 
 Standard, March, 1883, etc. 
 
60 METROLOGY. 
 
 STANDARD WEIGHT MEASURE, EXTENDED UPWARD. 
 
 10 tons 
 
 ( 2\ tons 
 
 i bin. 
 
 ) _ 
 
 i standard, or car-Joad. 
 
 
 ( 4 bins = 
 
 standard. 
 
 ) 
 
 
 10 car-loads 
 
 ( 2\ carloads = 
 I 4 lighters = 
 
 lighter, 
 barge. 
 
 l = 
 
 i barge. 
 
 10 barges 
 
 ( 2\ barges = 
 ( 4 ship-loads = 
 
 ship-load, 
 elevator. 
 
 l = 
 
 i elevator. 
 
 10 elevators 
 
 ( 2-J elevators = 
 
 wharf. 
 
 |- 
 
 i district. 
 
 
 ( 4 wharves = 
 
 district 
 
 ) 
 
 
 10 districts 
 
 ( z\ districts = 
 
 section. 
 
 \ = 
 
 i ark, or harbor. 
 
 
 ( 4 sections = 
 
 ark. 
 
 ) 
 
 
 In the above table, there runs the following subordinate 
 decimal system : 
 
 2-J tons = 
 
 bin. 
 
 10 bins = 
 10 lighters = 
 10 ship-loads 
 IO wharves = 
 
 lighter, 
 ship-load, 
 wharf, 
 section. 
 
 4 sections 
 
 ark, or ha 
 
 Now, it has been already shown how from the standard cube 
 we may descend (unifying as we go the grandest system of 
 weights the Anglo-Saxon that the world now possesses) to 
 the very ultimates of troy, apothecary, avoirdupois, and stand- 
 ard weights. Let me, however, add, before dropping this 
 explanation, that it is evident, if we consider the ultimate 
 (=: .00285 exact cubic inch pure water at mean temperature 
 and pressure) a small enough subdivision to denominate here 
 by a special name, nevertheless, shall science find it necessary 
 to decimate the ultimate, grain, the ounce, or gill, it can do so 
 to the very limit of numerical capacity by simply removing the 
 decimal point one place farther to the left at every division by 
 IO. In this case, if a name be necessary, let us denominate 
 (following our now well-known monetary phraseology of dimes, 
 cents, and mills) .000285 = -^ of an ultimate a " dimultimate," 
 .0000285 = yjfl- of an ultimate a " centultimate," and . 00000285 
 1 oVo f an ultimate a " milultimate," etc. In a similar way 
 we can have dimegrains, centigrains, and millegrains, and dime- 
 ounces, centiounces, and milleounces, etc. 
 
 One can hardly imagine even that the remotest science will 
 ever be able to appreciate so small a quantity as a millionth of an 
 
WEIGHT MEASURE. 6 1 
 
 ultimate. Nevertheless, as mathematics may write and employ 
 in calculation a millionth even of a millionth (.CXXXDOOOOOOOI) of 
 an ultimate (.00285), this system admits of it in an exact num- 
 ber, = .00000000000000285, since the figure 5 in 285 is always 
 final. 
 
 Let us now return to a consideration of the two repetends 
 previously determined, as decimally expressive of the capacity 
 of a cubic inch of water in grains and ultimates. These num- 
 bers are, 
 
 i cubic inch of water = 269.47368421052631578947368 -f etc., grains. 
 " " " " ' = 350.877192982456140350 + etc., ultimates. 
 
 Through the medium of the mean density material, these 
 formidable numbers (nevertheless, perfectly exact even in their 
 present condition) assume the form, not only of the most beauti- 
 ful simplicity, but also of the most astonishing utility. Thus, 
 
 12,500 cubic inches mean density = 19,200,000 grains, by weight ; and 
 12,500 " " " " = 25,000,000 ultimates, by weight. 
 
 A cubic inch of mean density, therefore, contains exactly 
 1,536 grains 2,000 ultimates \ (pyramid) avoirdupois pound 
 = T 4 5 (pyramid) troy and apothecary pounds = ^ (pyramid) 
 standard pounds, or = 3 J unit ounces ; i.e., pyramid and rec- 
 tified avoirdupois, troy, apothecary, or standard. In this same 
 connection we should notice that 10 times this coffer equiva- 
 lent of mean density material, or 125,000 cubic inches mean 
 density, or (50 inches mean density) 3 = 250,000,000 ultimates, 
 or again the semi-axial number, and = 192,000,000 grains. 
 
 Let us now imagine ourselves in possession of a cubic inch 
 of the mean density material, and examine the nature of its 
 characteristic facilities as to numerical subdivision for the pur- 
 pose of weight-making, or manufacture. The two numbers to 
 be considered are 1,536 and 2,000; these, as above, being 
 respectively the numbers of grains and ultimates in the cubic 
 inch. Each of these numbers is capable of astonishingly com- 
 plete resolution by factors. Thus, 
 
 1,536 = 2X2X2X2X2X2X2X2X2X3X1. 
 
 2,000 =2X2X2X2X5X5X5X1. 
 
 By inspection of the above remarkable series of factors, 
 occurring so appropriately in the very place where they are most 
 
62 METROLOGY. 
 
 needed, to wit, as components (in the known terms of grains 
 and ultimates of all our Anglo-Saxon weight measures) of a 
 convenient and definite cubical portion, i.e., of ONE UNIT INCH, 
 of mean earth, or standard material, it is manifest that every 
 conceivable fractional portion that may be demanded by art, 
 science, or commerce can be realized, and this without the 
 shadow of an error. The numbers 1,536 and 2,000, which thus 
 express the value of the unit cube in terms of grains and ulti- 
 mates, are in many other ways equally notable. Thus, 1,536, 
 the more important of the two, as it subdivides the cube into 
 the familiar grains, is three times the cube of eigJit (3(8) 3 = 1,536), 
 a fact of vast importance in weight manufacture when we 
 remember the many desirable properties of 8 as a radix of 
 metrological notation. Nor should it be overlooked here, that 
 the number 8 is also itself a perfect cube. Thus, as more com- 
 pletely analyzed, 1,536 = 3 times, the cube of t'cvo cubed ; or = 
 3((2) 3 ) 3 . Thus, too, from still another stand-point, the unit cube 
 is made up of 192 elementary cubes of 2 on an edge, or of 8 
 grains capacity, the number 192 being itself resolvable by 
 2X2X2X2X2X2X3X1, or into halves, thirds, fourths, sixths, 
 etc. Now, all of these- cubic and even parts of the unit and 
 elementary cubes will be found, upon inspection, to bear direct 
 and important practical relation to the whole series of Anglo- 
 Saxon weight measures as rectified, the advantage of which 
 cannot fail to be seen by the scientific metrologist. We have 
 already seen that the system grandly culminates in a still more 
 important and standard cube ; so that, from the elementary, 
 through the unit, and up to the standard cube, and beyond, this 
 unique system is not only upon its surface both decimal and 
 common, but, when rightly understood, is even still more perfect 
 in its complete compass of all the properties of the number 8. 
 
 Moreover, the number 1,536 is exactly divisible by every one 
 of the digits except 7, and yields in this latter case a conven- 
 ient repetend. In the possession of this latter property this 
 number, therefore, occupies the same vantage-ground as the 
 number 7,200, concerning the desirability of which for employ- 
 ment as the number of grains in the avoirdupois pound so 
 much argument has hitherto been advanced. In this connec- 
 tion we may also note that the whole series of numbers express- 
 
WEIGHT MEASURE. 63 
 
 ing the grains and ultimates in the several pounds as here 
 rectified, possess exactly this same property. Thus, 4,800, 5,760, 
 and 7,680, the number of grains per pound as rectified, and 
 6,250, 7,500, and 10,000, the corresponding number of ulti- 
 mates, all yield to exact subdivision by the several digits, and, 
 even in the case of subdivision by 7, yield convenient repe- 
 tends. Similar properties are possessed by the number 2,000 
 (i.e., the ultimates in a cubic inch), which is equal to 2(io) 3 , 
 and is divisible by the digits into commensurable parts. Hence 
 the series of numbers here advocated as the only natural ones 
 for a system of universal metrology, possess, besides those 
 enjoyed by the number 7,200, other and far more important 
 ones, which the latter and its connected numbers do not and 
 can not enjoy. Finally, the factoring of the numbers belonging 
 to the rectified series shows other important practical beauties, 
 such as the following: 4,800 (2) 4 X 3 X (io) 2 ; 5,760 = 10(24)* 
 = 2X3X5(2) 3 ; 7,680 = 3X5((2) 3 ) 3 ; 6,250 = io(25) 2 = io((5) 2 ) 2 
 = 2(5)5; 7,500 = 3X 25 X (io) 2 = 3(5)* X (2) 2 ; 10,000 = (io) = 
 (2)(5)+, etc. 
 
 In these numbers, in fact, as aliquot parts of this little unit 
 cube, the grand metrological system of the Great Pyramid of 
 Gizeh culminates. Crushed, as it were, into a toy, that would 
 hardly fill an infant's hand, we have here for our contemplation 
 the whole symbolism of this mysterious monument. 
 
 The unit cube ! Of what is it not significant f Its edge, the 
 linear unit, is the i-25th of the sacred cubit, which in turn 
 is the one lo-millionth of the polar-radius. By it the cube is 
 directly related to all Anglo-Saxon, and eternal, linear measures 
 whatsoever. By it also it is directly related to all chronology, 
 to every motion of the earth. By it the pyramid itself the 
 mesocosm, as it were is built anew. Yes, literally built anew ;; 
 for 20,612 grains-!- 1,536 = .r = 13.16158203125 (exact), and 
 6,561 grains -f- 1,536 =. y = 4.271484375 (exact), and 20,612:- 
 6,561 : : x : y or as *r : i by the PARKER MODULUS! This is; 
 the hardest blow of all that the pyramid can give to modern 
 science ; for it is easier for a camel to go through the " needle's 
 eye " than for the modern scholar to admit the circle could 
 have been squared to such a degree of accuracy at this early 
 day, or that the value of the ?r was ever better than a mechani- 
 
64 METROLOGY. 
 
 cal approximation, until the intellect of our vaunted days dis- 
 covered the calculus. 1 
 
 By the square of its edge, i.e., by one of its six small faces, 
 the unit cube is related to all area ; and by its six unfolded is 
 revealed to our astonished eyes the very symbol of salvation 
 unto all who dwell upon the surface of the earth. 
 
 By the volume of the cube, capacity itself is meted out. The 
 i-7i,25Oth of the mysterious coffer, it measures food and drink, 
 and is correlated through art and science with every thing on 
 earth. Finally by its weight it consummates itself. The 
 i-i25,oat)th of the perfect or standard cube, it is the earth itself 
 in miniature, and leaves within the range of all metrology no 
 unit lacking. 
 
 This is not rhapsody, nor poetry, nor the scientific, nor 
 unscientific, use of the imagination : it is solid fact, and all 
 concentred in a cubic inch, the standard mean of every thing 
 man needs for measure. 
 
 Into this narrow compass we have thus at last compressed 
 the whole significance of the Great Pyramid ! and, as it grew 
 the less, its wonder only grew the more ! Truly, then, can all 
 men say, " This is indeed that wonder in the midst of Egypt, 
 and that altar on its border," of which the prophet spake. It 
 makes no compromise whatever with false science or its advo- 
 cates. Without error itself, it stands as an enduring emblem 
 of confusion unto all who undertake to balance its eternal 
 TRUTH with "vanity and words." 
 
 "Who/* then, "is this that darkeneth counsel with words 
 
 x It is not my purpose, here, to enter into any such forbidden ground as the domain of 
 this much vexed question. Parkers modulus, be it true or false, is by no means a weak 
 attempt at its solution ; and his ratio, 20,612 : 6,561 =3.141594 + etc., arrived at by an entirely 
 independent course of reasoning, is, at any rate, of an eminently practical character. It 
 exceeds that of Playfair and Legendre by a little more than unity in the 6th decimal place, 
 or is about i-3,ooo,oooth greater. Modern science, therefore, which is fully satisfied with the 
 ratio TT = 3.1416, for all practical purposes, can certainly find no fault, for similar pur- 
 poses, with Mr. Parker's very simple common-fractional form ^V^A which gives us a value 
 of it true, to within unity, two decimal places beyond! 
 
 My only object in alluding to this subject here at all, is to point out the fact of this 
 astonishing n ratio being present in the value of the unit cube. Since it is here, it must be 
 likewise present in that of the corresponding unit sphere of 1,536 grains weight, whose linear 
 dimensions, surface, solidity, etc., will therefore necessarily reflect similar numerical beauties 
 throughout the region of what may be designated as spherical metrology ; or weights, meas- 
 ures, and volumes of spherical forms. 
 
WEIGHT MEASURE. 65 
 
 without knowledge ? " With Charles Latimer " I ask Mr. Proc- 
 tor and President Barnard to cease their flippancy, and answer 
 the following and the foregoing." 
 
 I challenge any astronomer or "metric philosopher" to refute 
 the arguments contained therein. These Anglo-Saxon facts 
 are true, or they are false. There is but one eternal truth, 
 one justice among men, one perfect weight. If the secret of 
 the pyramid, Israelitish, Anglo-Saxon, international, and earth- 
 commensuric metrology, be not comprised within this little 
 cubic inch of the earth's mean density material, then let it be 
 disproved ; but, if it be so, it is then a solemn fact to all who 
 dwell upon the earth. 
 
 Why, then, we may ask, with Eliphaz the Temanite, " Should 
 a wise man utter vain knowledge, and fill his belly with the east 
 wind?" Why "Should he reason with, unprofitable talk? or 
 with speeches wherewith he can do no good?" It is said that 
 "there is such divinity doth hedge a king," and, again, that an 
 undevout astronomer is "mad." Where is the divinity that 
 hedges an undevout astronomer? It is time that this hiding 
 and belittling of the grandest facts should cease ; and Mr. Proc- 
 tor should be asked, "Where wert THOU when the foundations 
 of the pyramid were laid ? " 
 
 But let us return to the consideration of this wonderful cubic 
 inch. Five of them make a pyramid avoirdupois . pound, and 
 a sixteenth thereof gives us the unit ounce. In other words, a 
 stick of the mean density material five inches long, and one 
 square inch in cross-section, assists us in passing from the unit 
 cubic inch to the unit ounce weight, by means of a careful 
 subdivision into i6ths. 
 
 But of what shall we manufacture this stick or span-length 
 of the mean density material, if we are to employ these facts 
 for practical purposes ? An alloy of aluminum (specific gravity 
 2.6 =F) and copper (specific gravity 8.8 rb) may be made to 
 answer, since the proportions of the alloy may be so adjusted 
 as to easily bring a cubic inch of it to a density of 5.7 times 
 that of water. Thus, as a basis of experiment, we have $ (8.8 
 
 There are also many other alloys by means of which we can 
 realize this standard density. But an examination of a table of 
 
66 METROLOGY. 
 
 specific gravities will reveal to us that one of our best-known 
 substances the one, in fact, whose Latin name is significant 
 of weight itself, i.e., LEAD (or plumbum) has a mean density 
 (11.4) exactly double that of the mean density material! 
 Thus : [(water = i) X (5-7 mean density material)] X 2 = 
 11.4 lead. 
 
 Of course, in the manufacture of weights, this double density 
 is an advantage ; since it lessens by one-half the size of every 
 weight. Perhaps, therefore, no better material can be obtained 
 than this one, offered, as it were, by Nature herself. It is cheap 
 as well as heavy, much heavier than iron (7. 125) ; and, though 
 not quite so hard, a trace of antimony will strengthen it ; and a 
 duly proportioned standard alloy (11.4), with lead as the basis, 
 will satisfy all our requirements. 
 
 Taking, however, pure lead (11.4), as a substance within easy 
 reach of every workman, and as one indicated by Nature her- 
 self, it will be noticed, that, in being twice as heavy as the 
 mean or standard density material, its number of aliquot parts, 
 i.e., grains and ultimates per cubic inch, become (1,536 X 2) 
 = 3,072, and (2,000 X 2) = 4,000 respectively. 1 In other 
 words, while the cubic capacity of the weight standard, (5o") 3 
 125,000 cubic inches, may be allowed to remain the same, we 
 will, if it be now filled with lead, have doubled its already ex- 
 haustive powers of aliquot subdivision. 
 
 The standard 5O-inch cube of pure lead (density 11.4) weighs 
 exactly 50,000 pyramid avoirdupois pounds, one-tenth of which 
 = 5,000 pounds = (2 X 50 X 50) will balance a cofferful thereof. 
 
 For the finer weights used in troy and apothecary measures, 
 heavier, and particularly harder, metals may be employed. So, 
 too, the standard national sets of weights may be made of still 
 better material. For these purposes the metals gold (19.26), 
 platinum (21.5), and iridium hammered (24), may be alloyed so 
 as to produce some convenient multiple of the mean density 
 material, as, for instance, 3(5.7) =17.1 and 4(5.7) = 22.8, etc. 
 
 1 Of course the actual cubic volume of lead necessary to balance a cofferful of water is 
 (12,500) = 6,250 cubic inches lead, each cubic inch of which weighs two of those of the 5.7 
 material. It is also noticeable in this connection, that there are the same number of cubic 
 inches of lead (at 11.4) in a ton or cofferweight, as there are grains in the lo-ounce pound of 
 pyramid standard weight; i.e., 6,250: and similar numeric repetitions will be noticed 
 running all through this beautiful system. 
 
WEIGHT MEASURE. 6/ 
 
 Such an employment of specific gravity, to wit, for the pur- 
 pose of obtaining exact cubical volumes, which shall, in turn, 
 be related to each other in the same numerical sequences as are 
 the terms of our tables of metrology themselves, and such an 
 employment, too, as shall result in equal facility for binary, 
 octenary, or decimal usage, is the concentrated " Wisdom of the 
 Great Pyramid." It is the lesson written in its wonderful pro- 
 portions by the mighty intellect that planned it for the modern 
 scholar. Its scheme is universal ; and not until the science of 
 this latter day shall willingly repair to Gizeh, and study in its 
 grateful shades, and square and right itself by its hoary cosmic 
 truths, can modern science found itself upon the everlasting rock 
 of stability, and rear itself a pyramid eternal as the earth itself. 
 
 These cosmic truths flash out and round about this mystic 
 monument from every point of view. For instance, it is built 
 of limestone on a limestone hill. Why, may we ask ? May it 
 not be because of the following significant facts ? The mean 
 specific gravity of marble or limestone is put down by Traut- 
 wine as 2.65 to 2.85 times that of water. The latter is a 
 familiar number to those who have followed us carefully in 
 our discoveries. It is one-half of 5.7, the mean density of the 
 earth. Furthermore, it is, as an abstract number, one-tenth of 
 28.5, the number which expresses the cubic capacity of a recti- 
 fied avoirdupois pound in water. Hence 10 cubic inches of 
 2.85 limestone or marble is a commercial pound. Moreover, 
 since there are 1,728 cubic inches in a cubic foot, a cubic 
 foot of such limestone will weigh exactly one-tenth as many, 
 or 172.8 avoirdupois, pounds. Surely such facts as these are 
 more than remarkable : they are calculated to actually stagger 
 modern science in their endless suggestions of practical em- 
 ployment in the arts and trades and intellectual pursuits of 
 man. 1 A cubic foot of the mean density material weighs 
 
 1 I do not wish to be understood as claiming here that the Great Pyramid is actually 
 built out of limestone of this particular (2.85) specific gravity, for such does not appear to be 
 the fact. I simply wish to note, in passing, what may, perhaps, be at least an indirect 
 reference to the subject, and to call attention to the possibility of such facts being not only of 
 practical value, but here and there implied in the pyramid's significance. The specific gravi- 
 ties of some of our best and most useful building materials, granite, limestone, marble, basalt, 
 porphyry, slate, etc., all arrange themselves around this particular degree ; and future study 
 of that all-important topic may result in discoveries of great value to commerce and science. 
 
THE REVERSE 
 
 TO THE 
 
 GREAT SEAL OF THE UNITED STATES 
 OF AMERICA. 
 
 " And the rain descended, and the floods came, and the winds blew, and beat upon that 
 house ; and it fell not : for it was founded upon a rock." MATT. vii. 25. 
 
 68 
 
WEIGHT MEASURE. 69 
 
 twice as much as one of 2.85 material, or 345.6 pounds. 
 Hence one of water weighs in avoirdupois pounds 345.6 
 5.7 60^1 (exact) pounds, or decimally =: 60.631526 -}-, etc., 
 pounds, or for " practical purposes " = 6o avoirdupois pounds. 
 From this fact we learn that a cubic foot of any substance 
 whatsoever (say specific gravity x] must contain 6o-}| x avoir- 
 dupois pounds, or 96|-| x standard pounds, etc. Thus, through 
 the unit density, water (i), the mean density of our planet (5.7), 
 that of certain grades of the well-known building material, 
 limestone (2.85), and other geometric, numerical, specific grav- 
 ity, and similar references, too numerous to mention, does the 
 study of this building reach upward from our planet, and 
 beyond to others, and from the grander numbers of the 
 entire solar system outward farther yet unto the very bound- 
 aries of the universe itself, in search of loftier truths. 
 
 There was a day when so-called astronomers arrogantly 
 looked down upon a certain class of numerical laws which the 
 laity or non-schoolmen had found to exist among planetary dis- 
 tances, rotations, revolutions, etc. That day has passed away ; 
 and we now find an opinion constantly growing stronger among 
 the leading scholars of this branch of science, that there must 
 and does actually exist some universal law, by whose complete 
 resolution into its appropriate terms and functions, an equa- 
 tion, as it were, of the solar system will result. Theoretically 
 such an equation certainly does exist ; and from the many simple 
 relations already known to exist between densities, diameters, 
 rates, and distances, we can be assured that it will be ultimately 
 found to be by no means a complex one. The first step towards 
 a true universal astronomy is the thorough understanding of all 
 the "elements" of our own orb. It is the science of METROL- 
 OGY that teaches this, a science, that, in view of the outlook 
 afforded from so lofty an elevation as the Great Pyramid, we 
 have already seen fit to define as " The Science of Round Num- 
 bers, Just Measures, and of Perfect Weights." Once awakened 
 to the certain possibilities of such a scheme as the one the 
 pyramid monumentalizes, man can, with ever deepening enthu- 
 siasm, look into Nature and her beauteous laws, and doubt not 
 that the victory will be accorded to his intellect assisted from on 
 high. We doubt not that our opinions here expressed will meet 
 
70 METROLOGY. 
 
 with bitter criticism. We challenge it indeed, for it will lead 
 us to still higher and yet more impregnable positions. There 
 are those who would presume to criticise the proportions of the 
 throne of God, as rashly as they do the measures of his foot- 
 stool. 
 
 But let us return to a consideration of the facilities afforded 
 by this system for the practical manufacture of just weights 
 and measures. 
 
 In all of the cases we have noticed above, the real object was, 
 to obtain accurate sets of weights for comparison rather than 
 for actual use. For this latter purpose, iron, as heretofore, will 
 undoubtedly continue to be the most suitable. This is on 
 account of its great hardness, and the consequent permanence 
 of its stamped values. 
 
 Now, Nature comes to our assistance at this point in quite as 
 marked a way as she did with the material lead. Trautwine 
 puts the specific gravity of cast-iron at from 6.9 to 7.4, and tab- 
 ulates the average at 7.15. This quantity does not appear to 
 have any direct or convenient relation to 5.7, or to that of our 
 mean density material, but it does have a remarkably close one 
 to the cubic capacity of the coffer itself ; i.e., to 71,250 cubic inches. 
 In fact, it is perfectly within the powers of easy art to obtain 
 an iron whose specific gravity shall always be exactly 7. 125 
 times that of water, and of which, therefore, 10,000 cubic 
 inches, = 10 X (io") 3 or to ten cubes of ten inches on an edge, 
 will exactly balance the coffer. A cubic inch of such iron will 
 weigh exactly one-fourth of a pyramid avoirdupois pound, and 
 will therefore contain 1,920 aliquot grains and 2,500 ulti- 
 mates respectively. Hence four such cubic inches weigh a 
 pyramid avoirdupois pound ; three of them, a pyramid troy or 
 apothecary pound ; and two and one-half of them, a pyramid 
 standard pound. A ten-inch cube of such iron will weigh 
 exactly 250 pyramid or rectified Anglo-Saxon pounds ; and as in 
 a standard cube (5o") 3 there are 125 ( = (5)3) such ten-inch 
 cubes, the standard cube of iron at 7.125 will weigh 31,250 
 pyramid avoirdupois pounds. 
 
 Now, the slightest investigation into the properties of the 
 numbers involved in this rectified Anglo-Saxon system of Me- 
 trology, develops beauties in every direction ; and their very 
 
WEIGHT MEASURE. 71 
 
 existence establishes the supreme dignity of the treasure-house 
 thus opened. Let us take, for instance, this last number, 
 31,250, pyramid or rectified Anglo-Saxon avoirdupois pounds, 
 expressive of the weight of a standard cube (or 125,000 cubic 
 inches) of iron. 
 
 The number itself is io(5) 5 . Now, the fluid measures of an- 
 cient Israel (of divine origin, let it be remarked) were as 
 follows : 
 
 i omer 178.125 (exact) cu. in. waterymean temp., press., and hygro. 
 
 loomers = I bath = 1,781.25 " " " " " " " 
 
 10 baths i homer = 17,812.5 " " " " " " " 
 
 4 homers = i laver . = 71,250 " " " " " " " 
 
 50 lavers = i sea = 3,562,500 " " " " " " " 
 
 50 seas =178,125,000 " " 4< " " " 
 
 Dividing now the cubic capacity of each of the above meas- 
 ures by 5.7, to reduce it from water to that of the mean density 
 material, we obtain the several familiar numbers 31.25, 312.5, 
 3,125, 12,500, 625,000, and 31,250,000 cubic inches of mean 
 density material. These numbers are respectively equal to T ^, 
 &, 5 5 > 4(5) 5 , io,ooo(5)V 
 
 But the dry measure of Israel corresponded exactly with its 
 fluid measure. Thus the unit of the former was the gomer of 
 178.125 cubic inches capacity of pure water at mean earth tem- 
 perature and pressure and hygroscopic state, which, for testing- 
 purposes, would have been balanced by exactly 31.25 cubic 
 inches of mean density material. This dry measure was as 
 follows : 
 
 i gomer = 178.125 cu. in. water = 31*25 m ' n dens, material, to bal. 
 
 iogomers = i ephah = 1,781.25 " " = 312.5 " " " 
 
 ioephahs= icor =17,812.5 " = 3,125 
 
 4 cors = i ark =71,250 " " =12,500 ** " " 
 
 The dry measure proper terminated at this point ; but there 
 are the very best of mathematical reasons for believing that the 
 
 1 I simply wish to note here, that the volume of the " Holy of holies " was 125,000,000 
 cubic inches, and, if filled with water, would have been balanced by an amount of mean den- 
 sity material equal to 125,000,000 -j- 5.7 = 21,929,824.5614035087719, etc., a repeten'cl which 
 seems to have no earthly connection with the sequences of figures, 178,125, 71,250, 31,250, 
 which we have noticed in the above tabulation. That it docs, however, we shall see lat_r en. 
 
72 METROLOGY. 
 
 actual contents of the great ark of Noah were based upon a 
 value equal to 100,000 arks of the covenant, each of which 
 held 71,250 cubic inches of water, or, as we have just seen, 
 would have each been balanced by 12,500 cubic inches of mean 
 density material. 1 
 
 Now, there can be no doubt but that the whole religion of 
 Israel was one concentred about the grand principle of the per- 
 fect equality of men in their dealings each with each. "A 
 just and perfect measure shalt thou have" and " With what meas- 
 ure thou metest, it shall be measured uuto thee again" were the 
 underlying tenets of the whole Hebrew polity. 
 
 These are facts, and not fancies. The whole furniture of the 
 temple, and of the earlier tabernacle, was constructed upon a 
 consummate system of weights and measures, and finally cul- 
 minated in a chamber called the " Holy of holies," whose spe- 
 cially peculiar construction has ever been a vast mystery. In 
 this sacred chamber was the ark, an imperishable standard 
 of weight and capacity,' and in the latter were carefully 
 stored a pot, or omer, of manna, the staple and symbol of the 
 dual life of man, the rod of Aaron (a cubit in length), and 
 the decalogue, or measure and standard of the perfect life ; 
 while outside and beside it were other measures of equal im- 
 port. Now, in all this careful arrangement, " by number, 
 weight, and measure," there is design, of course, and in that 
 design a system which our investigations clearly prove to be 
 the embodiment of earth-commensuric metrology. 
 
 I shall now ask you to notice its wonderful culmination in 
 the sacred cube called the Holy of holies. The dimensions of 
 this cubkal chamber were 20 cubits X 20 cubits X 20 cubits, 
 or, as the sacred cubit was 25", were 500" X 500" X 500". 
 
 125,000,000 cubic inches was therefore its volume. Now, 
 at the very first glance, this number is a familiar one. It is 
 1,000 standard cubes, 1,000 itself being also a perfect cube = 
 (10)'. . 
 
 In our examination of the fluid measures of Israel we have 
 just seen that the "sea" of 50 lavers contained 3,562,500 
 cubic^inches of pure water at mean temperature and pressure, 
 
 1 I refer those who desire to examine these reasons for themselves, to the Great Pyra- 
 mid of Egypt, by Philo Israel, W. H. Guest, 20 Warren Lane, London, England. 
 
WEIGHT MEASURE. 73 
 
 or exactly 2,000,000 unit ounces (i.e., rectified Anglo-Saxon and 
 pyramid ounces at 1.78125 cubic inches of water per ounce), and 
 that 50 seas contained 178,125,000 such cubic inches, or 100,- 
 000,000 such unit ounces. We have also noted that the dry 
 measure proper terminated with ark of the covenant, or laver 
 volume of 71,250 cubic inches of water. Considered as a part 
 of this intricate system, the volume of the cubic Holy of holies, 
 125,000,000 cubic inches capacity of pure water, seems, as we 
 have before noted, to stand alone. But, by its very isolation, 
 it is a marked number. 
 
 A more careful scrutiny shows that it is equal to 1,000 
 lavers, or "arks of the covenant," divided by the standard 
 density, 5.7, a fact which serves to bring this latter number 
 (5.7) intensely to the front, and one which is still further 
 marked when we note that the ratio between the ark of Noah 
 ( 7,125,000,000 cubic inches, or 100,000 arks of the cove- 
 nant) and the "sacred cube," or Holy of holies, is ten times 
 this number, or as 57 : i. 
 
 Thus we find continually repeated in the metrology of the 
 Great Pyramid, in that of ancient Israel, and finally in that 
 of the modern Anglo-Saxon, as now rectified, these important 
 sequences of figures, 17,812.5, 71,250, 31,250, 125,000, 5.7, with 
 whose unique fitness to the constitution of nature as a fact, 
 we have actually forced and demonstrated our acquaintance. 
 The first one is the cubic contents of a rectified quarter, the 
 io,ooo dth of which gives us the unit ounce ; the second is, 
 the coffer or ton, a io,ooo dth of which gives us the unit gill ; the 
 third is a standard cube, weight of iron in true avoirdupois, 
 pounds ; the fourth, the standard cube volume of mean density 
 material in earth-comme'nsuric cubic inches ; and the last, the 
 specific gravity of the earth itself, or the standard density by 
 means of which, at mean temperature, pressure, etc. (50), they 
 are all so intimately correlated into a universal system of. 
 metrology. 
 
 Moreover, their relation, in the abstract, to the number 50 
 a number so peculiar to the king's chamber in the pyramid, to< 
 the Holy of holies in the temple, and to the standard cube 
 in the rectified Anglo-Saxon system and to 25, the cubit 
 number, is by no means to be slighted 
 
74 METROLOGY. 
 
 i 
 But it is very difficult to convey any adequate idea of the 
 
 consummate numerical beauty of this system of cubes, so inti- 
 mately related to the Anglo-Saxon system of metrology as 
 rectified, or of its vastly practical character. Thus, the grain 
 being the element, 8 grains (2) 3 grains, gives us an elementary 
 or primary cube, which is ^^ of the cubic inch. The next cube 
 we meet, and which may be called the secondary cube, is that 
 of (8) 3 = 512 grains. The unit cube of 1,536 grains consists 
 of three such secondary cubes, and the standard cube of (50 ") 3 
 125,000 unit cubes. Again, 
 
 i grain = .00065104166666, etc. = ToVu cu m f mean density material. 
 
 i elementary cube = .005208333, etc. = -r^ " " " " " 
 
 i secondary cube = .33333333, etc. = " " " 
 
 i unit cube = i = i " " " = ( i") 3 
 
 , , , ( 125,000 ) t ( 50") ;J 
 
 i standard cube = I R ri , . ri , .. \ = 125,000 = 1. rn 
 
 cubic cubits ( - M 8cu.) 3 
 
 ( 125,000,000 i i (500")* 
 
 isacredcube = j : , 000X8 cub. cubits ^ I25 ' O C ' = i io'x8cu. 
 
 Now, it is manifest, that, in terms of the grain, the primary 
 and secondary quantities of mean density material are perfect 
 cubes, and that the most important or unit, standard, and sacred 
 quantities, are likewise so in terms of cubic inches ; the latter 
 fceing so in cubic cubits also. Hence, it is also evident, that, in 
 one form or the other of their legitimate expression (i.e., either 
 in grains, in cubic inches, or cubits), all of them may be put 
 into cubic form ; and hence that all of their derivatives, and 
 their infinite combinations and multiples, may likewise be con- 
 sidered in a cubical character for the purposes of art, science, 
 or trade, whenever such a consideration shall be desirable to 
 facilitate calculation or practical wants. In fact, the field opened 
 by these newly discovered elementary principles in metrology is 
 broad enough to promise an everlasting harvest to intellectual 
 man. 
 
 In terms of rectified Anglo-Saxon weights and measures, 
 these Israelitish ones were as follows : i laver i tun, or 
 4 hogsheads ; i ark = i ton, or 2,500 avoirdupois pounds ; i 
 homer I cor = i hogshead, or 62| gallons, at 285 cubic 
 inches pure water at mean temperature and pressure ; i bath 
 = i ephah = 6| gallons ; and i omer i gomer = 2-|- quarts. 
 
WEIGHT MEASURE. 75 
 
 They are thus important elements in the grand and earth- 
 commensuric system, and this is exactly what we might have, 
 expected ; for if the system we have discovered be, as it un- 
 doubtedly is, the universal and only scientific one, and if the 
 one of ancient Israel was God-given, the latter could not but 
 have been also a part of the true system. 
 
 Let us now return for a monrent to the standard cube of 
 iron, whose weight, as we have seen, was 31,250 avoirdupois 
 pounds. It consists of 125 subordinate cubes, each 10" on an 
 edge, and weighing respectively 250 pounds. Arranging these 
 so as to form a continuous block of iron (io") 2 in cross-section, 
 the block will be 1,250" long ; and 2 2 - thereof, or 100 inches, 
 will weigh 2,500 avoirdupois pounds, and balance a coffer or 
 avoirdupois ton of pure water under standard circumstances, 
 etc., for all avoirdupois weights; 25" will balance the " thou- 
 sand " pounds of standard weight, etc. In other words, a i" 
 slab cut therefrom in cross-section will contain (i" X 10" X 
 10") 100 cubic inches, and therefore Weigh 25 avoirdupois 
 pounds = 33& pounds troy and apothecary = 40 pounds stand- 
 ard weight, etc. So, too, all superior and inferior weights, 
 in each of the several Anglo-Saxon, Israelitish, and Pyramid 
 systems, or, in general terms, in the earth-commensuric sys- 
 tem, will find accurate cubical expression therein. 
 
 But we need not go into this subject at greater length. 
 Enough has surely now been adduced to establish beyond cavil 
 the superiority, at every point, of the Anglo-Saxon measures, 
 rectified at the pyramid and by the Bible, over the metric 
 system. 
 
 I challenge the advocates of the metric system to bring for- 
 ward their case, and to show that it can be put upon the same, 
 or upon a corresponding, basis, as that on which I have demon- 
 strated that the Anglo-Saxon one properly belongs. 
 
 I challenge them to produce any decimal part of their metre 
 by means of which they can reach upward to astronomic dis- 
 tances, and downward into the ordinary affairs of men who 
 halve and quarter by instinct. 
 
 I challenge them to show that the square of such a part shall 
 be any natural superficial unit whatsoever, or that its cube shajl 
 relate itself to the volumes of nature or of the solar system 
 
76 METROLOGY. 
 
 And finally, I then challenge them to show that such a cube 
 when filled with water, a unity material to which men most 
 refer ; or with cereals, which they measure most ; or the earth 
 density material, as the manifest standard of weight ; or with 
 lead and iron, out of which man naturally makes his commer- 
 cial weights ; or with marble, as an enduring material for build- 
 ing ; or with other things of similar importance to man, shall 
 in all these cases, or in any of them, possess even coincidences 
 that shall have any systematic character whatever, and be 
 related to the natural constitution of things, and therefore to 
 each other, in grand round numbers. In other words, I chal- 
 lenge them to show that the metric system is the greatest com- 
 mon divisor, or any common divisor, or a least multiple, or any 
 simple multiple, of the eternal constitution of things as exist- 
 ing around about us under measure, capacity, and weight. 
 
 I maintain, that from the very nature of measure, capacity, 
 weight, etc., they are, and must be, directly related to each 
 other, and to the earth itself, in some one system that shall 
 be expressive, at its lowest terms, of all the elements of earth- 
 commensurability. There cannot be two such systems, and 
 there can be but one unit to the true system. This one true 
 system, I maintain, is based upon the ic-millionth of the polar 
 radius as the standard of length, upon its 25th as the unit of 
 length, upon the square of this as the unit of area, upon its 
 cube as the unit of volume, upon that volume filled with mean 
 density material (5.7) as the unit cube, and upon that volume 
 filled with the unity material as a basis of weight and capacity 
 measures (1.78125 cubic inches = I ounce) in which unity shall 
 be pure water (i) at mean earth temperature, pressure, and 
 hygroscopic state. I c4aim, that by the aliquot subdivision of 
 this unit cube of mean density material into 1,536 aliquots 
 called "grains," and into 2,000 others called "ultimates" (from 
 whence the unit ounces of capacity and of weight result), the 
 entire Anglo-Saxon system becomes, not only harmonious, but 
 rectified to nature as constituted in things men weigh and 
 measure ; that it realizes the pyramid metrology in its perfect 
 scheme, and is throughout completely earth-commensuric. 
 
 I maintain that this, the true system, was not evolved by 
 man, save as he found it already existing, like any other law in 
 
WEIGHT MEASURE. 77 
 
 nature. I maintain, moreover, that man per se cannot create a 
 new one upon arbitrary principles, and even hope to have it sur- 
 vive, except by the exercise of constant tyranny. I point to the 
 experience of four thousand historic years of men constituted 
 like ourselves for carrying on the affairs of ordinary daily life, and 
 say that the universal testimony has been in favor of such sys- 
 tems of metrology as were primarily based upon aliquot parts 
 not decimal ; that decimal subdivisions, while important for cer- 
 tain branches of science as to the handling of particular units, 
 etc., are not so for men at large ; that, nevertheless, the natural 
 system in its entirety is the most truly decimal system that can 
 be imagined, and that to it all Anglo-Saxon measures are 
 related. 
 
 I point to the experience of the past century, and claim that 
 the French metric system itself, in this particular, is gradually 
 changing its nature. It has not a single subdivision that men 
 who are forced to use it do not halve and quarter. Look, too, 
 at our own neat system of money. Though purely decimal, we 
 halve and quarter far more familiarly than we decimate. Take 
 away its dime, and what remains of its decimal character ? 
 
 In another century, if it be in existence still, I predict that the 
 metric system will have been forced to adopt common-fractional 
 aliquots as at first a subordinate part of their metrology ; that 
 they will next have to name them ; and that, if left to the people 
 (as in modern times it must be) and to time, this system, from 
 the nature of eternal things, will square and right itself with- 
 out assistance by the earth-commensuric cubic inch. 
 
 What, then, could be more grandly earth-commensuric than 
 the strange coffer in which the inch, the pint, and the ounce, 
 length, capacity, and weight, temperature, pressure, hygro- 
 scopic state, and density, bread itself and water, iron, lead, 
 and all things else, are thus all joined in a most intimately 
 harmonious reference, not only to the earth on which we live, 
 but to our solar system and the universe itself ? 
 
 The French metric system actually blasphemes the science 
 of " Universal Metrology " in the light of such a system. It 
 was enunciated in the era and the hot-bed of an awful atheism. 
 Its falsely calculated metre was based upon the erroneous as- 
 sumption, that all meridian quadrants of the earth were of equal 
 
7.8 METROLOGY. 
 
 length. We now know that probably no two of them are equal. 
 It violated their first modern geometric principle in ever taking 
 a curve for the standard of straight linear measure. It violated 
 the very resolutions of its own Academy of Paris in so doing ; 
 for that academy has resolved, abiding by it yet, that a curved 
 line cannot be squared or rectified, and that an infinite differ- 
 ence exists between the element of a diameter (straight) and 
 that of a meridian (curved). If this be so, then it is ridiculous 
 to base a scientific system upon what cannot be righted; the 
 affairs of daily life being squared rather than curved. 
 
 The French system never dreamed of earth-commensuric 
 capacity and weight standards. This is so because th^ere can 
 be no reason given why any particular cube filled with water 
 shall be as (i.e., //AT/ as) earth-commensuric by weight as it is by 
 volume, and by length and area as it is by both. The cubic 
 inch (5.7) alone does all these things, and is alone the proper 
 unit. Nor is it scientific to establish as a standard pressure, 
 that which nature everywhere abhors (the vacuum), and for 
 temperature that point (maximum density) around about which 
 its very standard (water) most rapidly alters its condition. For 
 the very reason (now so well approved) that astronomers go to 
 the North star when off the meridian, and moving slowly in 
 maximum elongation, and avoid the star when on the meridian-, 
 and in rapid motion, for the determination of the cardinal lines, 
 so, in due regard to the eternal fitness of things, philosophy 
 should, and science and common Anglo-Saxon sense do, veto the 
 standard temperature on which the metric system is based. 
 
 The only advantage the French metric system possesses is 
 its decimal notation. But the decimal system did not originate 
 in France ; it is older than tradition, and is personal to the 
 human being ; it is as readily applied to any other system ; it is 
 more ancient than the pyramid itself, and is built into it from 
 the foundation up. 
 
 In our public schools this false, unscientific metric monstros- 
 ity is taught, to the utter confusion of all true metrological 
 ideas. It is full time that we should banish it. It is not yet 
 a century old. In its arrogant beginnings, it even sought " to 
 change times and seasons." It made weeks of ten days, and 
 contemplated years of ten months, and numerous other follies^ 
 
WEIGHT MEASURE. 79 
 
 which, briefly bolstered up by the guillotine, have each and 
 all died violent deaths. 
 
 So long as the moon continues to lunate sidereally in periods 
 of 27 days, 7 hours, 43', 3", 47'", 20"", so long will men divide 
 their time by weeks; and if tyranny divides time decimally, 
 which is contrary to the nature of things, the moon herself will 
 square it by the perfect cycle into weeks of seven days. 
 
 What is now known as the metric system is but the shadow 
 of its former dimensions. Let us not allow it to live upon our 
 soil to celebrate its first centennial. Let not an English-speak- 
 ing, English-thinking, English-working, hard-handed Anglo- 
 Saxon people forget their own traditions any longer. But let 
 them listen to the silent teachings of that great monument 
 upon the ancient Nile, from whence, mysteriously indeed, but 
 nevertheless somehow, and in due time to be made clear, their 
 own grand system is derived. Even the permissive use of the 
 metric system is a blot upon our statute-books. There is no 
 need of such a resolution. If men want to use an evil system, 
 they will do it anyway. It is wrong, however, to give such a 
 system any encouragement whatever, certainly short-sighted to 
 recognize it at the nation's capital. But it is far worse to suffer 
 it to be introduced into governmental acts and records. It is 
 making, however, no headway among the people. For this we 
 thank Almighty God. Out of 60,000,000 now among us, about 
 500,000 (!) and I question even this as a very large over- 
 estimate scientific men have learned and use it, because 
 their foreign books are written in it. The rest of our millions 
 buy and sell their food and raiment in the good old Anglo- 
 Saxon way. 
 
 Let the scientific read and use it : it is as much a part of 
 their right and duty as to read and use the foreign languages 
 themselves. They are our servants, and should translate these 
 books into Anglo-Saxon weights, as well as Anglo-Saxon words. 
 Shall the millions learn the metric system because a few thou- 
 sands find their foreign theories written in metric hieroglyphics ? 
 As well might they demand that the Anglo-Saxon tongue itself 
 should be declared dead and inarticulate, and literally bring 
 about us Babel's own confusion, till the nation, stricken dumb, 
 shall learn another speech. 
 
80 METROLOGY. 
 
 The day of an international and universally earth-commen- 
 suric metrology is fast approaching. It was foreseen of old, 
 from out the very shadow of that "sign and wonder" in the 
 land of Ham ; and thither to the future gate of universal com- 
 merce, man will surely yet return a willing pupil. 
 
 Why, then, shall we not rather hasten it by setting our own 
 mansion in order, so that against the records of our own great 
 generation may be also written, " At this time men rebuilt the 
 'Wonder of the World,' and universal progress dawned upon 
 the earth " ? 
 
V. 
 
 RECAPITULATION. 
 
 " Let us hear the conclusion of the matter." 
 
 Principles upon which the Rectified System is based. Its Primary Unit. Stand- 
 ard Physical Conditions. Special Standards. Units and Primaries. Con- 
 stants. 
 
 (1) THERE must be some one system of metrology which best 
 suits the whole constitution of all nature. There cannot be two 
 "best " systems, nor can there be two equally good ones ; for 
 there can be but one least common multiple, or but one great- 
 est common divisor, of any series of numbers, circumstances, 
 or things. 
 
 (2) Such a system, when found, will be ipso facto the univer- 
 sal system, and therefore will be the only one fit to be interna- 
 tional as such, in a day of general commerce, civilization, and 
 freedom. Whatever that system be, it will be recognized in- 
 stinctively by man as the Metrological Law of Nature ; and, 
 until it is discovered, man's metrology cannot but be a mere 
 temporary expedient. 
 
 (3) To such a system, there can be but one primary unit. 
 
 (4) The aliquot subdivisions of this primary unit must not 
 only suit the requirements of man's every-day life, and respond 
 to his experience, but must square and cube themselves with 
 geometrical magnitude, harmonize with the elementary constitu- 
 tion of things as expressed in specific gravity, and comprehend 
 to the fullest degree the properties of number in the abstract. 
 
 (5) The system, therefore, must be both decimal and common- 
 fractional, be tangent to nature m its elementary dimensions, 
 its elements of capacity, and of weight, and, from all physical 
 stand-points whence metrology is viewed, must rest upon natural 
 
 standards. 
 
 li 
 
82 METROLOGY. 
 
 (6) The standard circumstances of such a system should be 
 the mean (each for its kind) of all the earth. Because (i) It 
 is the most natural. (2) The mean is most easily preserved. 
 (3) It is as a fact more habitually, used and preserved. (4) Man 
 in perfect health and in full development, as the highest expo- 
 nent of nature, lives, moves, and has his being, best at the 
 mean of all terrestrial circumstances. (5) Extremes are always 
 to be avoided, particularly in founding the bases of observation. 
 (6) Variable circumstances change less in quantity, and less in 
 rapidity (i.e., degree), around the mean, 
 
 (7) To be able to survive in the future day of universal 
 science and commerce, a day very near at hand, an inter- 
 national system of metrology must satisfy all of the foregoing 
 requisites. For man, as intellectually constituted, will not retain 
 an unsound system ; and any system that is merely tolerated, 
 and which has apparent and inherent defects, must certainly go 
 under. 
 
 (8) We are at the dawn of this new era. The Anglo-Saxon 
 race transacts the business of the earth, and does it in its own- 
 time-honored system of metrology. The French metric system 1 
 has apparent and inherent defects, and is used only by the 
 decided minority of the civilized world, and by a minority des- 
 tined to be outstripped in the struggle for existence. These 
 defects are radical to the metric system; and, as such, they 
 doom that system. It is, therefore, useless to expend time, 
 thought, or money upon it. The Anglo-Saxon. system, however, 
 is already known to all the world, even to the uncivilized 
 thereof : it is already so nearly earth-commensuric, as to need 
 but a slight rectification to enable it to embody nature's own 
 law of metrology. This can and has been demonstrated. It 
 therefore behooves the Anglo-Saxon race to " preserve the 
 ancient landmarks," to square them and to right them, and to 
 found them upon an everlasting basis ; to set their own house 
 in order, and to transmit to posterity a system, already interna- 
 tional, and one that hereafter, to eternity, shall need no change. 
 No amount of rectification can right the metric system, how- 
 ever ; for it is founded upon unsound principles, and the princi- 
 ples themselves must be abandoned. It is not only non-lineal 
 at the outset, but makes no pretensions to take advantage of 
 
RECA PITULA TION. 83 
 
 any of the properties of numbers other than the decimal ones, 
 nor of specific gravity as regulating standard volumes of stand- 
 ard substances at standard circumstances. Its rationale, in fact, 
 is not harmonious to nature in such a way as to express any 
 grand law of metrology as a part of the constitution of things ; 
 and, failing here, it fails in toto, and must be abandoned. 
 
 (9) Let us now briefly reiterate the principles upon which 
 we have worked in the foregoing system, with a view of fulfil- 
 ling all the necessary requisites above enumerated. We have 
 taken a cube of fifty linear units on an edge, and filled it with 
 a material whose mean density is 5.7 at mean temperature and 
 pressure. Then, since it contains 125,000 cubic units (inches) 
 of such material, it must balance 712,500 cubic inches (units) 
 of pure water under similar circumstances ; and we have de- 
 clared that such a weight shall consist of 192,000,000 grains, or 
 250,000,000 ultimates. Now, if the linear unit shall vary so as 
 to make our present authorized inch either + or to any 
 small degree, it shall not alter the arithmetical fact that we will 
 still use fifty of such new units as the edge of our standard 
 cube. We shall maintain that specific gravity 5.7 is the proper 
 standard density in terms of pure water at whatever may be 
 mean temperature, pressure, and hygroscopic state of the earth 
 (I now believe them to be 68 Fah., and 30" barometer, and \ 
 saturation), and we shall have the same number of grains and 
 ultimates in the standard and unit cubes as we have above set 
 forth. Having thus fixed our standard cube, and taken as our 
 unit cube one of 1,536 such grains, or one 125-thousandth of the 
 standard, we then establish the fact, that as our ultimate will 
 occupy .00285 cubic inch of pure water at standard temperature 
 and pressure, and as our ounce shall contain 625 ultimates, 
 therefore our unit ounce weight shall balance 1.78125 cubic 
 inches of pure water at standard temperature and pressure, 
 or .3125 cubic inches of 5.7 material under the same circum- 
 stances ; and by this unit ounce weight we intend to rectify all 
 Anglo-Saxon weights whatsoever. We also declare that the 
 space occupied by this unit ounce weight, when in pure water 
 at standard conditions, to wit, 1.78125 cubic inches, shall 
 be, and is, the unit ounce of measure or capacity. 
 
 By and with these weights and capacities, and their multi- 
 
84 METROLOGY. 
 
 pies, ranging upward and downward over the whole field of 
 weight and capacity, we propose to rectify all present Anglo- 
 Saxon measures, and to use the same in the manufacture of 
 weights and measures. 1 
 
 THE PRIMARY UNIT. 
 This is ONE CUBIC INCH OF MATERIAL, 5.7 TIMES AS DENSE 
 
 AS WATER. 
 
 Its edge is the linear unit, or one inch (i"). Its face is 
 the superficial unit, or one square inch (i") 2 . Its volume is the 
 cubical unit, -or one cubic inch (i") 3 > and measures all geomet- 
 rical volumes. 1.78125 : this cubical unit is the capacity ounce, 
 or unit of dry and liquid measure ; and a measured ounce of 
 pure water at standard (i.e., mean) circumstances is the unit of 
 weight, commonly called an ounce weight. Thus, the primary 
 unit has a volume of |f- that of the unit of capacity, and a 
 weight of 3^ ounces. The sum of its linear edges is 12' ', or 
 one foot ; and its total surface is 6 square inches = ^ a square 
 foot. -^ .3125 of the primary unit is an ounce weight ; and 
 || the volume thereof i.e., of the primary is the unit or 
 ounce of capacity measure, etc. The aliquot parts of this 
 primary unit are 1,536 grains, or 2,000 ultimates, the cubical 
 and square properties of which numbers relate them to all geo- 
 metrical volumes and surfaces ; while, by their roots (8 and 10), 
 the whole Anglo-Saxon system based thereon is made to par- 
 take of all the advantages of both the decimal and octenary sys- 
 tems of notation. Thus, 1536 = 3(512), and 2,000 = 2(1,000), 
 or 1,536 = 3(8)2, in which 8 itself is also a perfect cube, and 
 2,000 = 2(io) 3 . From the cubical, even, and triple nature of 
 the first may be realized all the desirable aliquots in common- 
 fractional form ; and from the even and cubical form, and the 
 decimal root of the latter, all the necessary decimal fractions of 
 
 1 Should, however, a standard density of more or less than 5.7 (exact) be desirable, as, 
 for instance, one of -f-{$ = 3T^?oT = 5-7 2 9 i etc - tnen tne relations of the system 
 to the standard cube of 50'' on an edge will result in proportional figures. Thus, 
 125,000 X |f$ will equal 716,187 cubic inches of pure water at standard circumstances, and 
 all the figures throughout the system would correspondingly vary. For obvious reasons, how- 
 ever, I am of the opinion that the standard density (5.7) without further fractional termina- 
 tion, and because of its simple relation to the cubic volume 712,500, through the cube of 
 50" = 125,000", is the proper one to employ. 
 
RECAPITULATION. 85 
 
 weight, etc. Again, 1,536 = (i6) 2 6 = 2(2 X 8) 2 3, and 2,000 = 
 5(2o) 2 = 5 (10 X 2) 2 , factors which exhibit respectively the 
 even, square, circular, octenary, and triple properties of the 
 one, and the quintuple, square, circular, decimal, and even prop- 
 erties of the other. All of the above properties of the primary 
 unit connect it with number, length, surface, volume, capacity, 
 and weight, not only as elements of geometrical form, as such, 
 but as Anglo-Saxon elements, and as elements that are univer- 
 sal, since they are terrestrially commensuric, the inch (linear) 
 being the one 5oo-millionth of the polar axis ; and the density, 
 5.7, being the mean of earth density, etc. 
 
 STANDARD PHYSICAL CONDITIONS. 
 
 For international and earth-commensuric metrology, the 
 standard physical condition for every variable of nature (such 
 as temperature, atmospheric pressure, hygroscopic state, etc.) 
 concerned therein, shall be the mean of all the earth for that 
 variable. Expressed in degrees, upon the respective scales 
 for each of the several variables, this mean shall be indicated 
 by 50; and when any particular variable as, for instance, 
 temperature is being considered alone, all the others such 
 as atmospheric pressure, hygroscopic state, etc. shall be 
 maintained at their mean, 50, each upon its proper scale. 
 Thus, when testing and stamping the weights, measures, etc., 
 of earth-commensuric metrology, the international observatory 
 shall be at the mean temperature, pressure, hygroscopic state, 
 etc., of the whole inhabited earth, or shall be at 50, from which 
 they will vary less, and around which they will be handled most 
 by man, from day to day throughout the year, and here and 
 there upon the earth. These standard or mean physical con- 
 ditions are now believed to be as follows : 
 
 Standard or Mean Temperature equals -f- 68 Fahrenheit = 
 -f- 1 6 Reaumur = + 20 Centigrade, or Celsius, or, upon the 
 international scale, = + 50, on which same scale water freezes 
 at o, and boils at 250. Standard temperature is thus one-fifth 
 the way from freezing to boiling point of water. 1 
 
 1 Very beautiful and accurate thermometers of this graduation may now be obtained of 
 J. S. F. Huddleston, 242 Washington Street, Boston, Mass. They are made expressly for 
 the Institute. 
 
85 METROLOGY. 
 
 Standard or Mean Atmospheric Pressure is that indicated by 
 30 axial inches of barometric mercury (specific gravity = 13.557), 
 other physical conditions being standard. This height, at the 
 sea-level on the standard parallel (30 north latitude), corre- 
 sponds to 30.03 present British inches of mercury (at specific 
 gravity 13.598) 4 C., and on the international scale ir marked 
 by 50; 48 thereon corresponding to 28".8, and 52 to 3i".2; 
 o = o", and corresponds to "no atmosphere." A substance as 
 light as cork (specific gravity .24; i.e., 15 pounds per cubic 
 foot) would stand 1,694". 5 8 high in the barometric tube, or upon 
 such a scale balance the atmosphere at 2, 824. 375 ; and one as 
 heavy as hammered iridium (taken at specific gravity 24) would 
 stand thereon at 28. 24375 = 1 6". 9458.' 
 
 Standard or Mean Hygroscopic State of tJie Atmosphere is 50, 
 on a scale where o marks "no moisture;" and 250 indicates 
 saturation, other circumstances being standard. The hygro- 
 scope stands thus at one-fifth the way from dry to wet, and 
 indicates a dew-point at i8.O55 ~t~ etc., on the international 
 thermometer (Fah. 68, dew-point 45). This standard hygro- 
 scopic state is marked upon the hygrometer by an amount of 
 aqueous vapor in the air equal to T Jg its volume, or equal to 
 4". 3 cubic inches of water per column of air one square inch 
 at base, and equal in height to the atmosphere. Its pressure 
 = -5*5 that of the whole atmosphere. 1 
 
 Standard or Mean Density is taken as 5.7 time's that of water 
 at other standard circumstances. This density being indicated 
 by 50, water will be indicated by 8|f-, hammered iridium 
 (specific gravity 24) by 2io|^, hydrogen (specific gravity 
 .0000895 db) by . 00078 5087 + etc., and a substance whose 
 density is 28.5 by 250, etc., thereon. 
 
 SPECIAL STANDARDS. 
 
 Under the foregoing standard circumstances (i.e., the me- 
 trological chamber of observation being brought to 50 upon 
 each of the international scales), the following are the standards 
 in particular : 
 
 1 Arrangements are now being made with instrument-makers of standing for the manu- 
 facture of barometers and hygroscopes graduated upon this system, and will be placed with 
 the Institute for sale. 
 
RECAPITULATION. 87 
 
 Standard of length is one lO-millionth of the semi-polar axis, 
 is called a cubit, and equals 25 earth-commensuric inches = 25 
 present Anglo-Saxon inches -f- - 001 (). z 
 
 Standard of volume is that of a cube whose edge is two 
 cubits ; i.e., = (5o") 3 (125,000 cubic inches) for geometrical 
 purposes. 
 
 Standard of weight is a standard volume of mean density 
 material ; i.e., it 'is the weight of a cube of material whose 
 density is 5.7 times that of water, and whose edge is 50". 
 This cube weighs 400,000 ounces = 5o(2o) 3 ounce weights : 
 hence it is equal to 50 times a cube which weighs 8,000 
 ounces. Such a cube as the latter, if its volume were 8,000 
 cubic inches, would have to be of a density equal exactly to 
 1,781.25, water being 1,000. This is related to all Anglo-Saxon 
 weights. For practical cases of comparison this standard cube 
 of 125,000 cubic inches, mean density material, may be divided 
 into 1,000 cubes of 5 inches on an edge. Each such cube of 
 comparison will thus contain 125 cubic inches, mean density 
 material, and weigh exactly 400 ounces = 25 avoirdupois pounds, 
 33^ troy and apothecaries' pounds, 40 standard or lo-ounce 
 pounds, and be of handy size for comparative metrological 
 purposes. 
 
 Standard of capacity is 71,250 earth-commensuric cubic 
 inches, or one ton. It is the volume occupied by 40,000 ounces 
 of pure water under standard circumstances. Numerically it is 
 expressed, ^ (50")' 5-7 = tV ( I2 5ooo) 5-7 = t\r (712,500) = 
 71,250 cubic inches pure water, etc. All dry and liquid measures 
 of the rectified Anglo-Saxon system run up to and through this 
 standard. 
 
 UNITS AND PRIMARIES OF SYSTEM. 
 
 Primary temperature. Freezing-point of water = o on in- 
 ternational scale (= 32 Fah., etc.). 
 
 Primary barometric point. Vacuum = o on international 
 scale (= o" of mercury, etc.). 
 
 1 The standard cubit, of 25 inches, graduated in various ways to accomplish the several 
 geometrical problems described in Appendix A, are soon to be manufactured by Darling, 
 Brown, Sharpe, of Providence, R.I., and will be within reach of such as desire to obtain 
 
 them. 
 
88 METROLOGY. 
 
 Primary hygroscopic point. Dry air, or no moisture, = o 
 on international scale. 
 
 Primary density point. That of vacuum = specific gravity 
 o.o = o on international scale. 
 
 Unit density that of water, pure, air at the international 
 standard (50) points, as to temperature, pressure, and hygro- 
 scopic state, and on the international scale of density corre- 
 sponds to 8|f 
 
 The unit of length, or lineal unit, = i" (inch) = one 500- 
 millionth polar axis = I Anglo-Saxon inch + .001 (db). 
 
 The unit of area, or square unit, = (i") 2 a square on one 
 unit inch linear. 
 
 The unit of volume, or cubical unit, = (i")3 = a cube on one 
 linear inch. 
 
 The unit of capacity measure (dry and liquid), or the capacity 
 ounce, = 1.78125 (exact) units of volume space occupied by 
 one ounce weight of pure water under standard circumstances, 
 or that occupied by 5.7 ounce weights of mean density material. 
 
 The unit of weight, or ounce weight, weight of 1.78125 
 unit volumes of pure water at standard circumstances, or 
 weight of -fg a unit volume of the mean density material = 
 1,536 grains = 2,000 ultimates. 
 
 CONSTANTS OF SYSTEM. 
 
 The term standard =: 50, on every scale, = mean of all ter- 
 restrial or physical circumstances concerned in metrology; i.e., 
 = perfect health as to temperature (68 Fah.) = 5.7 specific 
 gravity as to density, 30" as to barometer, | as to saturation, etc. 
 
 Unit inch = one 5OO-millionth of polar axis of earth = 
 i Anglo-Saxon (A.D. 1883) inch, + .001" to rectify it. 
 
 Unit square = (i") 2 . 
 
 Unit volume = (i") 3 . 
 
 Unit cube = (i") 3 > mean density material. 
 
 Unit ounce weight = i/ 2 V ( = 1.78125, exact) cubic inches, 
 or unit volumes of water, standard ; = ^ (= .3125) cubic inches, 
 or unit volumes of mean density material, standard ; 480 
 grains =625 ultimates. 
 
 Ounce of capacity = 1.78125 (exact) unit volumes, standard, 
 
RECAPITULATION. 89 
 
 = space occupied by one unit ounce weight of water, standard, 
 or by 5.7 unit ounce weights of mean density material, stand- 
 ard. 
 
 i grain = yJg-g of unit cube = *- ultimates = .0x337109375 
 (exact) cubic inches of pure water, standard. 
 
 i ultimate = ^-fa of unit cube = .00285 (exact) cubic inches 
 pure water, standard. 
 
 The foregoing being premised, it is proposed, that, in addition 
 to the several systems of Anglo-Saxon weights and measures 
 now in use (and which shall all be rectified thereby), the several 
 tables heretofore designated as " standard " shall be made legal 
 for decimal and other purposes, but that no system shall ever 
 be made compulsory. This having been accomplished, and all 
 the tables being united by a common inch, ultimate, grain, 
 ounce, and gill, and by the process of rectification being made 
 strictly accurate, scientific, and earth-commensuric, either sys- 
 tem of subdivision may be employed, according to the demands 
 of special arts, sciences, and trades, and the several systems 
 shall be left to work out their destiny according to their several 
 inherent merits. 
 
 In order to introduce the standard or decimal systems here 
 proposed, and which are founded respectively upon 10 units of 
 each of the several rectified Anglo-Saxon tables now in use, it 
 will by no means be necessary to adopt the entire scheme of 
 the new tables as given in this volume. All that is actually 
 essential is, that 10 inches shall be declared and understood to 
 mean a standard or decimal foot ; 10 ounces a decimal pound ; 
 and 10 capacity ounces (= 17.8125 cubic inches) a decimal 
 pint, etc. Other multiples of the above, such as by io s , ioo ds , 
 i,ooo ds , etc., may, or may not, be especially mentioned. But, be 
 they mentioned or not, they will, nevertheless, all be logical and 
 potential parts of the system, and as such be within reach of 
 those who may desire to employ them, and be intelligible to 
 - those who read them. The whole point of this discussion 
 is to be summed up in the statement, that it is perfectly prac- 
 ticable for us to introduce the decimal system into our own 
 weights and measures, and (if we rectify at the same time our 
 units so as to comprehend cosmic principles) thus found a sys- 
 tem of metrology that can never more be shaken or disturbed. 
 
VI. 
 
 AEROSTATICS. 
 
 " The firmament showeth his handy work." Ps. xix. i. 
 
 The Atmosphere in its Relation to the Rectified System. Its Weight. Specific 
 Gravity. Mean Barometric Height. Dew-Point. Moisture. Actual Height 
 of Atmosphere. 
 
 IT is in the atmosphere that " man lives and moves, and has 
 his being." The business of his life is in weighing and meas- 
 uring the things about him ; and, to do so accurately, a thorough 
 knowledge of "the condition and changes of the atmosphere, and 
 of their corresponding effects upon metrology, is absolutely 
 necessary. 
 
 Viewed from the standard observatory of this new system, 
 we will briefly enumerate a few of the more important facts 
 and data which are concerned directly in this subject. 
 
 A cubic foot of atmospheric air at the surface of the earth 
 (mean sea-level) and at standard physical conditions, i.e., 50 
 upon each of the international scales (68 Fah., 30" mercury at 
 13.557 density, J saturation, etc.), weighs 570.2809536+, etc., 
 grains. 
 
 Its specific gravity, compared with water, pure, and at stand- 
 ard circumstances, is .0012246970-!-, etc. 
 
 A column of air at the circumstances which would exactly 
 balance an equivalent (as to cross-section) column of the at- 
 mosphere itself would be 332,100 inches high, or 5.2414828 
 miles. This, in other words, would be the height of a baro- 
 metric column of air if it could be maintained throughout its 
 height at standard circumstances. It corresponds to a baro- 
 metric height of water, at standard conditions, equal to 33.8925 
 feet, or to 406.710 inches Taking all of these barometric col- 
 go 
 
AEROSTATICS. 91 
 
 umns as of equivalent cross-section, and that cross-section one 
 square inch, the pressure indicated would be 14.27052631+* 
 etc., avoirdupois pounds (i.e., rectified pounds at 28.5 cubic 
 inches, pure water, standard circumstances, per such pound). 
 Expressed in troy or apothecary pounds, this pressure is 19.- 
 02736841 +, etc., pounds ; and in standard or lo-ounce pounds, 
 22.832842096-}-, etc., pounds. Expressed in grains, this press- 
 ure is 109,600 ; and in ounces, the most convenient and numeri- 
 cally satisfactory form for use, it is 228 i ounces, almost exactly. 
 At 50 standard thermometer, hygroscope, and barometer, 
 the standard dew-point is at i8.O5555 -f , etc., temperature. 
 This indicates that the total barometric force of the moisture 
 (aqueous vapor, or steam) of the air is about -%, or, more ac- 
 curately, gVy^Vs !' f tne wn l e atmospheric pressure. This 
 moisture exists in the form of atmospheric steam (specific grav- 
 ity .00086483342), and occupies about 89,792 times the volume 
 of water which would produce* it. Condensed to water, the total 
 amount of aqueous vapor in a standard column the height of 
 the atmosphere, and a square inch in cross-section, amounts to 
 3.698546 cubic inches. There are .003000 grains thereof per 
 cubic inch of mean atmospheric air, or about -j-J-^ (more accu- 
 rately, Yi$oy 9) ^e weight thereof. In general terms, 1 10 times 
 the actual weight of the moisture present in standard air is 
 equal to 95 times the pressure which it exerts in the form of 
 steam. Thus, weight when condensed : pressure in vapor : : 
 
 The maximum (250 hygroscope) capacity of air at otherwise 
 standard circumstances for moisture is .015 grains per cubic 
 foot, 1,875 grains per standard cube, etc. The mean hygro- 
 scopic state is .003 per cubic inch, 5.184 per cubic foot, and 
 375 per standard cube. 
 
 A cubic inch of atmospheric air at standard circumstances 
 weighs .3300237 grains, .003 grains of which we have already 
 seen to be of aqueous vapor, leaving .3270237 grains to be 
 divided among the other constituents thereof. Putting the 
 weight of air, as such, into a simpler and more beautiful form, 
 a standard cube thereof weighs 5i rectified troy pounds (accu- 
 rately this is 5.500395 pounds troy). 
 
 The relation between air at mean, and water thereat, is 
 
92 METROLOGY. 
 
 expressed by their specific gravities, .001224697 : i.oooo, or 
 816.520331 -f-. That is, air is 816.52, etc., times lighter than 
 water, the material of unit (i) density, and 46,540.64 times 
 lighter than the standard density material (specific gravity 5.7 
 times water). 
 
 13.4670462, etc., cubic feet of air weigh a pound avoirdupois 
 (rectified). 
 
 The rate of expansion of air from 50 to 250 standard ther- 
 mometer is uniform, and equal to .0015034 of its bulk at 50 
 for every standard degree of heat. As the temperature de- 
 scends below 50 to the o point, air decreases in volume uni- 
 formly, and at the same rate, from its standard bulk at 50. 
 Above 250 and to 1,000 it increases in bulk at the uniform 
 rate of .00144 of its volume per every degree of heat. 
 
 Air at 50 temperature, when completely saturated with 
 moisture, will hold .015 grains of atmospheric steam per cubic 
 inch. This corresponds to 18.492730 cubic inches of water 
 per mean atmospheric column (i.e., i inch in cross-section, and 
 3 3 2, i oo feet high). 
 
 The actual height of the atmosphere has been variously esti- 
 mated. It has been generally estimated at 53.3 miles. This, 
 however, is now regarded as far too small. Of course, the at- 
 mosphere has a limit, and that, relatively to the radius of the 
 earth, a very narrow one. Its mean height that is, the whole 
 atmosphere reduced to a standard condition such as it is at sea- 
 level when all the international scales are at 50 is, as we have 
 seen, 5.2413 -f- miles, which is about Y -|^ the mean radius of the 
 earth. Now, Monsieur Liais found, by experiments on the. twi- 
 light arc at Rio Janeiro, that the height of the atmosphere was 
 probably in the neighborhood of 212 miles. The indications 
 are, that about -fa the radius, or 220 miles, certainly includes 
 its extreme height. 
 
 When the altitude of the atmosphere is taken in arithmetical 
 progression, its rarity is found to be in geometrical progression. 
 Thus, 5.24132 -f-> etc., miles being the mean height at standard 
 circumstances for sea-level conditions, at twice that height, or 
 10.48262 miles, the air would be 4 times as rare, at 20.96 miles 
 1 6 times as rare, etc. This will best be seen from the follow- 
 ing table : 
 
A EROSTA TICS. 93 
 
 TABLE OF ATMOSPHERIC HEIGHT AND RARITY. 
 
 If 5.24132 miles high, rarity = I (i.e., unity). 
 
 " 10.48264 " " " = i times as rare. 
 
 " 20.96528 " " " = iV " " " 
 
 " 3I-44792 " " " = ifc " " " 
 
 " 4i-935 6 " " " - T&B " " " 
 
 " 52.41320 " " " = TOW " " " 
 
 62.89584 " " " 
 
 73.37S48 " " " 
 
 " 83.86112 " 
 
 " 94.34376 " " " 
 
 " 104.82648 " " 
 
 That is, at 20 times the mean height, or 104.82648 miles, the 
 rarity of the atmosphere would be expressed by a fraction whose 
 denominator was (2) 20 = 1,048,576, a number which is practi- 
 cally eq-ual to the height in miles 104.8264 multiplied by 10,000. 
 The specific gravity of air at this degree of rarity would, under 
 other standard circumstances, be .00000000116795+, etc. 
 Furthermore, at double this height, or 209.65296 miles, the 
 rarity would be ( |><o i -4- 1,099,577,163,776; and at 220 
 miles, which we have just considered to be practically the 
 absolute limit of the atmosphere, the rarity would be just short 
 of ( J- )4 of air in the standard observatory, and at standard cir- 
 cumstances. Numerically this may be put as follows : at 42 
 times the height (5.241, etc., miles) of a standard atmosphere, 
 the actual rarity of air (free to occupy that space by the atmos- 
 pheric law above noted) would be expressed by ( ^,>. 42 times 
 5.241, etc., miles = 220.1356 miles, and (2) 42 = (4) 21 =4,388,308, 
 655,104, say practically, for the rarity 4,400,000,000,000, or 
 equal to a trifle less than twice the height in miles multiplied 
 by (ioo) 5 , i.e., extreme rarity = s^onTTTo) 5 - 
 
 Now, the atmosphere is a sea of air, whose fluctuations at a 
 surface of such extreme rarity must produce waves of enormous 
 size, and waves that (from the nature of gases, earth motion, 
 and the earth-surface atmospheric commotion) can never be at 
 rest. Taking these figures, therefore, as representing, as closely 
 as possible, the actual mean height of the fluctuating atmos- 
 pheric surface of the earth, the specific gravity of air at this 
 surface will be represented by the standard specific gravity of 
 air (.001224697) divided by 2 (2 20) I oo 5 == .000000000000002- 
 
94 METROLOGY. 
 
 78340; i.e., one quadrillion times as light as a density 2.78, or 
 about one quadrillion times lighter than that of alabaster, 
 chrysolite, aluminium, and chalk. 
 
 At this limit the atmosphere may certainly be regarded as 
 being in equilibrium with the ethereal medium of space, than 
 which nothing rarer probably exists in nature. 
 
VII. 
 
 CONCLUSION. 
 
 " Felix qui potuit cognoscere causas." 
 
 Metrology the Universal Science. There is no Exception to the Law of Number 
 Man's Love of Exact Numbers. No Such Thing as " Mere Coincidence." 
 How Nature replies to us in Terms of the New Metrology. The Standard 
 Cube. Solomon's Wisdom. But these Facts are Older than Solomon or 
 Moses. The Great Pyramid. The Standard, or King's Chamber. The 
 Sanctum Sanctorum of all Science. 
 
 IT is the highest pleasure of the finite mind, to be allowed 
 to discover the ways of the Infinite in the ordering of the 
 universe : 
 
 " Felix qui potuit cognoscere causas." 
 
 When man has thus discovered facts which correlate them- 
 selves to the constitution of all nature, he calls them science ; 
 and it is only when they do thus satisfy the demands of eternal 
 fitness, that they may be classed with "science properly so 
 called." 
 
 Metrology is the universal science ; because He who created 
 Wisdom, and saw her, and poured her upon all His works, hath, 
 "ordered all things in measure, and number, and weight." ' 
 
 Thus, the system upon which creation rests is metrology 
 itself. Nor is the attempt to discover this science a hopeless 
 effort. Man has already made vast strides in the right direc- 
 tion ; and the Anglo-Saxon system of to-day, from whence- 
 soever it was originally derived, has been shown to be but: 
 slightly from the truth. In botany, man has learned that alii 
 vegetable-life, from the cell to the leaf, is but a numerical 
 arrangement of cells ; and from the leaf to the tree is likewise 
 
 1 Wisdom xi. 30 ; Eccles. i. 9. 
 
 95 
 
96 METROLOGY. 
 
 simply one of leaves. In crystallization, it is a law of numbers, 
 and a rigid one, that governs. On it (number) the whole 
 of chemistry is founded ; and so, too, with every branch of 
 science. These laws the special ones of each branch of sci- 
 ence are all simple: there are no complex general laws. 
 The more general the law, the simpler its numerical expression. 
 Just as force is correlated to all other kinds of force, so, too, 
 all forms of matter are undoubtedly likewise correlated ; and 
 whether it be in planetary forms, or those of art and trade, they 
 are ruled by simple numbers. 
 
 Throughout his practical life, man displays a universal ten- 
 dency to employ "round numbers," as his factors of use, in 
 dealing with physical things. This tendency is rooted, not 
 merely in a desire to shorten his calculations, but still deeper, 
 in an innate conviction that the beauty and symmetry existent 
 in the constitution of all nature consists in its simplicity, and 
 particularly in the simplicity of its numerical expression. Hith- 
 erto it has been the inaccuracy of our determinations, in the 
 field of universal science, that has introduced so many irregular 
 and awkward quantities into our tables of physical data ; and, 
 until this inaccuracy has been overcome, we cannot feel that any 
 branch of modern science is upon a firm and eternal basis. 
 
 But progress is slow. The prejudice of long use, and the 
 natural conservatism of, philosophers, conspire together against 
 every change, as such, no matter how great the promised benefit 
 therefrom may be. Nevertheless, the stronger tendency to sim- 
 plicity, and the deeply rooted conviction that it exists, if it 
 can but be discovered, govern practical life, and, when the 
 true secret of numerical arrangement is revealed, will over- 
 come all opposition, and round, perfect, and beautify our scien- 
 tific data by its own exalted standard. 
 
 Among some of the most familiar examples of man's defer- 
 ence to round numbers, we' shall mention but few. Thus, 
 under the present system of Anglo-Saxon metrology, though 
 14.72 -f- pounds per square inch is the tabulated pressure of 
 the normal atmosphere, it is habitually taken at 15 pounds. 
 The present Anglo-Saxon ounce is not earth-commensurable ; 
 and hence neither it, nor its multiples, will necessarily greatest 
 commonly divide, nor least commonly multiply, the forces of 
 
I 
 
 CONCLUSION. 97 
 
 nature. Man, therefore, has to be inaccurate in his practice, 
 for the sake of convenience, and in some of his most important 
 mechanical works, as those concerned with steam, call 14.72 -f-> 
 etc., arbitrary pounds, 15. Take, moreover, the weight of a 
 cubic foot of water. This is habitually regarded as 62 J pounds; 
 though it is, so accurately as known, by the old system of me- 
 trology, 62.379, etc., pounds per cubic foot. Then, there is 
 the number TT = 3.141592, etc., which we habitually take at 
 3.1416, and so on throughout all science. 
 
 Now, it cannot be that Nature herself is thus inaccurate, nor 
 that she is similarly irregular in her ratios, factors, and moduli. 
 And it must be, moreover, that, in a realm where "all things 
 are weighed and measured in number," there is a universal 
 law of number harmoniously running through the things so 
 weighed and numbered, and that this law may, in due time, 
 perhaps, be discovered by man, who is himself its highest 
 exponent. 
 
 Now and then some faint glimmering of this numeric sym- 
 metry appears, but it is merely regarded as a coincidence. 
 Instead of being studied with a view to following up the clew, 
 and reaching on to the universal system itself, these coinci- 
 dences are hardly ever tabulated, or even noticed, in what are 
 considered to be strictly scientific works ; although, whenever 
 known, they are universally employed by the practical man. 
 Astronomy is full of them, but onty because they are more 
 apparent in the wider sweep of its grand quantities and cycles. 
 As an actual fact, however, astronomy is no more occupied by 
 coincidence, as such, than any other branch of science. They 
 all teem with it ; and, to their deeper students, it crops out at 
 every turn. Coincidence is a law indeed, it is the grand 
 numeric law of nature. It is based upon certain general 
 underlying principles of number, weight, and measure, univer- 
 sally pervading things as they are ; and there is no such thing 
 as mere coincidence, or as accident, in its commonly oppro- 
 brious sense. 
 
 Let us now briefly consider Nature from the stand-point of 
 the new metrology, and ask her some of the same questions 
 which she so uncertainly answers in terms of any of the arbi- 
 trary systems hitherto in vogue. 
 
98 METROLOGY. 
 
 She gives us, as the pressure of the atmosphere, 228^ ounces, 
 an accurate as well as satisfactory and round number. This 
 can be put into any kind of pounds, or used in grains at 109,600, 
 or in cubic inches of water at 406.710, or of mercury at 30 ", or 
 in air at standard condition, as 332,100 cubic inches, etc. She 
 gives us, for the weight of a cubic foot of water at standard cir- 
 cumstance, 970 5 6 y ounces exactly, or 97 ^ standard (ro-ounce) 
 pounds. It is even being ably urged, by those who are skilled 
 in the logic and philosophy of geometry, that she gives us for 
 the true circummetric ratio the common fraction ^sVi^ anc * 
 pronounces Against the old value of TT. She gives us, as the 
 mean height of the atmosphere in miles, 4/144 = 5.2414828, 
 etc. (long measure rectified). She gives us, as the number of 
 cubic inches in a vertical square-inch column of this mean 
 atmosphere, 63,360 X -v/144 > i n which 63,360 is the number of 
 standard inches in a rectified mile (long measure). She gives 
 us, as the weight of a standard cube (50 ") 3 of mean atmos- 
 pheric air, 5j troy pounds, rectified, accurately (5.500395+, 
 etc.). 
 
 But let us examine this system still more closely, and, to do 
 so, consider the weight of (5o") 3 , or 125,000 cubic inches of 
 standard atmospheric air. Since its specific gravity is .001 22469, 
 etc., or since one cubic inch weighs .3300237 grains, it fol- 
 lows that 125,000 (5o") 3 cubic inches will weigh 41,252.9625 
 such grains. Now, at the first glance, this is an ordinary 
 series of figures ; but, upon examination, it proves to be a most 
 extraordinary one. Upon the Babylonish system of circular 
 division, it is one-fifth of the radius expressed in terms of 
 seconds of arc. Thus, in 180 there are 648,000 seconds, 
 648,000 -4- TT= 206,264.8125, etc.; and of this latter num- 
 ber = 41,252.9625, etc. If, upon the other hand, we take 
 the subdivision of the circle into 240, a division for which the 
 Great Pyramid pronounces in no uncertain terms as the only 
 correct and rational one, and as one which is closely in keep- 
 ing with the requirements of general metrology, the number 
 41,252.9625, etc., is even more expressive, and may be written 
 thus: 
 
 240 3 3 (2) 4 (^) 4 2 6 3 4 s 2 io 2 6 4 
 41,252.9625, etc., =-^- io 2 = v ' u/ io 2 = ^ 5- = 
 
 5 7T , 7T 7T 7T 
 
CONCLUSION. 99 
 
 Now, all this is called coincidence in common parlance ; and so 
 it might have been had our grain been an arbitrary thing, such 
 as the present unrectified Anglo-Saxon grain, or the metric 
 gram. But the grain we are using is not arbitrary. It is 
 directly related to the grandest linear dimension of the earth, 
 to its mean density, to its cubical units of capacity and weight, 
 and to the specific gravities of all substances as correlated in 
 water as unity, mean density material as standard, lead as our 
 synonyme for weight, iron as our practical weight material, air 
 as the element in which we live and move, and to the properties 
 of geometric form, and the elementary principles of numerical 
 expression. Of course the cube, on 50" as an edge, is but one 
 out of an infinity of cubes that might have been selected ; and air 
 is but one of a myriad of substances. Nevertheless, it is a cube 
 already chosen from other and general earth-commensuric and 
 numerical principles, as our standard one, and certainly, in the 
 light of such remarkable relations as those with which we have 
 found it pregnant, well worthy of its place. 
 
 From the converse of the foregoing discovery, it follows, that, 
 so nearly as we can grasp it (and, from the standpoint which we 
 occupy, we believe it to be so absolutely), the weight in grains 
 of standard air per cubic inch is* expressed by the term | 
 
 / 648 \ i /6V 
 
 = (-) grains. 
 
 \I,0007r/ 
 
 This is the discovery of a fact in nature. It is true, of 
 course, only for that particular value of the grain which we have 
 found reason to adopt ; but this is a value which rests upon 
 other and entirely independent earth-commensuric considera- 
 tions. 
 
 Now, the fact that an atmospheric cube of i terrestrial inch 
 on an edge, contains, at standard, or mean earth conditions, a 
 number of earth-commensuric grains expressed by the remarka- 
 
 bly circular and circummetric function, - - -- , is 
 
 6 I,OOOTT TT (5)5 
 
 of the utmost importance to metrologists. It is a common 
 term, as it were, to a vast number of independent series of 
 physical data, a point in which all of these various sequences 
 coincide and harmonize, or through which they pass in progress 
 from o to infinity. 
 
100 METROLOGY. 
 
 To man, so devoted a lover of the beautiful, a fact like this is. 
 very impressive ; and it seems to have governed the whole 
 metrological system of the ancient Hebrews, a system, be it 
 remembered, to which a most lofty origin has been universally 
 accorded. The fact may, indeed, have been connected with the 
 very origin of the grain itself. At any rate, it was too impor- 
 tant a common tangential point of all of Nature's physical data, 
 and of her laws of numerical expression, to be overlooked ; and 
 so it was given the very highest prominence at Jerusalem. 
 
 We have already seen sufficient reason for regarding the 
 most sacred precinct of Solomon's temple as a metrological 
 observatory. Whatever were its special religious uses, it was 
 also a national chamber of weights and measures, and one 
 whose equable atmosphere was weighed and measured and 
 
 numbered in terms of this very expression, - - i5A_. 
 
 The Holy of holies was a perfect cube, each of whose edges 
 was 20 sacred cubits, or 500 inches long. Hence this sacred 
 cube the ancient God-designed and Israelitish metrological 
 chamber contained a volume of 125,000,000 cubic inches 
 of atmosphere, or by weight, at standard circumstances, (soo) 3 
 
 |! = 5003 I f J^L \ = 41,252,962, etc., grains. 
 
 5 \I,OOO7r/ 
 
 This, on the Babylonian division of the circle, is numerically 
 equal to 200 times the length of radius in seconds of arc, or to 
 1,000 times as much as we have found in a standard cube. 
 Again, this expression for the weight of the atmosphere of the 
 Holy of holies in grains, is -J the aggregate length in seconds of 
 arc, of the radii upon 10,000 circles; or, even more concisely, 
 it is equal to the number of seconds in 100 circumferences 
 (36,000 = 1 29,600,000") divided by TT. 
 
 Now, it seems to be the conviction of a very large class of 
 American pyramid students, that the number 1,296,000 = 360 
 X 60 X 60, and its various derivatives, 648,000, 324,000, etc., 
 necessarily refer to the division of the circle now in universal 
 use among mathematicians, and that it can have no reference 
 to any other division. This is entirely an error, and has arisen 
 from the fact that they also persist in looking upon the Great 
 Pyramid as a monument erected to perpetuate direct measures, 
 
CONCLUSION. 101 
 
 rather than to preserve cosmic ratios by means of certain stand- 
 ard lengths. This error has been repeatedly pointed out in 
 "The International Standard," once by the author (vol. i., No. 3, 
 July, 1883, p. 208, lines 30, 31), then by Jacob M. Clark, C. E. 
 (vol. i., No. 5, November, 1883, p. 332, lines 3-6), and finally by 
 Mr. J. Ralston Skinner (vol. i., No. 6, p. 519, lines 46-48). In- 
 deed, as Mr. Skinner implies, the idea seems to have been 
 grasped by Professor Smyth long ago, though certainly not 
 clearly enunciated by him. Misled by the intentional conceal- 
 ment of these ratios here and there in inches (and many of the 
 most beautiful of them are undoubtedly so expressed but in 
 earth-commensuric or pyramid inches), and entirely ignoring their 
 frequent repetitions in the sacred cubit of 25 such inches, and 
 believing these inches to be our present Anglo-Saxon unit, 
 this school has built its entire theory upon the division of the 
 circle into 360. But there are insuperable objections to this 
 division of the circle. It can form no part of the geometric 
 scheme of the Great Pyramid, if that edifice is indeed an em- 
 bodiment of accurate metrologic principles. 
 
 Once establish the premise that the pyramid is an exponent 
 of "just weights and perfect measures," and that its erection 
 was supervised by an architect so inspired as to have been inca- 
 pable of erring as to the proper number of aliquots in a circle, 
 and it immediately follows, that the 360 X 60' X 60" division 
 could not have been the basis, since it is irrational. No means 
 is known to geometry of obtaining a 36oth of a circle, or of an 
 angle ; it involves trisection : and we shall " square the cir- 
 cle " so soon as we accomplish it. We can only obtain such a 
 division by the process of infinite approximation in practice, 
 and in theory have long ago given it up. One of the simplest 
 modes of establishing the theoretic impossibility of obtaining 
 the ^Q of a circle is as follows : 5 = 5^, 6 = g 1 ^, 6 5 = 
 
 J eV A lil<T MIT- 
 Now, if we can construct the -fa and the ^ of a circle, we can 
 
 by comparison obtain the -%%-Q. But while by two bisections 
 from the ^ (which is itself obtained from comparison of the 
 ^ and JQ, both rational) we can obtain the ^ , we cannot simi- 
 larly obtain the -fa. This fractional part of the circle depends 
 upon the ^, and is the J thereof, or might be obtained from the 
 
102 METROLOGY. 
 
 \ by three bisections if we could only obtain the \. But the 
 -i is irrational: hence the -%\- is equally so. This is usually 
 lost sight of because the number 360, being divisible by 9, i.e., 
 giving a quotient 40, the habitual rule of the text-books is to 
 lay off 40 (see Trautwine and others), and its chord is that of 
 the ninth. Who cannot see, however, that this is merely beg- 
 ging the question ? for to use a circle only approximately sub- 
 divided into 36oths by the instrument-maker, as a means of 
 geometrically (!) constructing an accurate Qth, is arguing liter- 
 ally "in a circle." 
 
 Geometry having therefore, no positive means of so subdi- 
 viding a circle, we must abandon the method in the interests of 
 true metrology, as in time we shall have to abandon all other 
 methods which we have derived from Babylon. 
 
 The methods of Babylon are so pregnant with the numeral 
 6, that, in all her " times and seasons," her measures and her 
 weights, her name and number, have from time immemorial 
 been branded far and near as 666. It is the prevalence of this 
 numeral 6, trailing, like the serpent it represents, over all of our 
 inheritance, that makes our modern metrology a science now 
 so thoroughly "at sixes and sevens" with itself, 6 + 7 = 13 
 has always been . regarded as a number of rebellion, and not 
 until we have thoroughly purged our science of the dominant 
 factor 6 can we hope to make any progress towards the num- 
 bers of perfection. There certainly can be nothing in common 
 between the Tower and system of Babel and the Pyramid and 
 earth-commensuric system of Gizeh. These monuments were 
 the very exponents of two diametrically opposite classes of ideas. 
 The one was marked for endurance, the other was instability 
 itself. Harmony pervaded the one : confusion presided at the 
 other. Promethean success crowned the capstone of the one : 
 the other was the Epimethean experiment of a man whose very 
 name has since become the synonyme for rebel. The one was 
 the wise man's building, literally "founded on a rock;" the. 
 other that of Folly herself, built upon the sand, and out of slime 
 and clay. 
 
 So diverse, in fact, was the genius that governed the me- 
 trology of Babylon from that of the Hyksos builders of the 
 pyramid, men, let it be remembered, who " came out of 
 
CONCLUSION. 103 
 
 Chaldaea" in every sense of the word, 1 that, just so sure as 
 it was undoubtedly Babylonian to subdivide the circle into 360, 
 so we may be certain that it forms no part of the true system 
 of metrology as monumentalized at Gizeh. 
 
 But the number 1,296 and its derivatives are by no means 
 dependent upon the 360 circle and its sexagesimal subdivision : 
 they come as directly from 12, 24, 240, etc. Thus, 1,296 is the 
 number of inches in 108 feet, or it is the number of hours in 
 54 days. Let us examine its reference, by way of further argu- 
 ment, to the rational division of the circle into 240. For those 
 who are not familiar with the overwhelming arguments in favor 
 of this beautiful subdivision, Appendix B has been written by 
 Jacob M. Clark, civil engineer of the New-York Central Rail- 
 road, to whose erudition we are in modern times solely indebted 
 for the discovery. 
 
 Now, at the first glance it is manifest that any ratio, depend- 
 ing upon the number 360, is equally related to the number 240, 
 since both are multiples of the common factor 120. However, 
 to proceed with the discussion, let us examine the table pro- 
 posed by, and founded upon the discoveries of, Mr. Clark. 
 
 RATIONAL CIRCULAR MEASURE. 
 
 Circle. Sectors. Deg. Min. Sec. Thirds. Fourths. Fifths. 
 
 D iv v.etc. 
 
 i 24 = 240 2,400 = 24,000 = 240,000 = 2,400,000 = 24,000,000 = etc. 
 
 i 10 100 = 1,000= 10,000= 100,000= 1,000,000 = etc. 
 
 i= 10= 100= 1,000= 10,000= 100,000= etc. 
 
 i= 10 = 100 = 1,000 = 10,000= etc. 
 
 i = 10 = 100 = 1,000 = etc. 
 
 i = 10 = ico = etc. 
 
 i = 10 = etc. 
 
 i = etc. 
 
 Now, by examining the number under discussion, 129,600,000, 
 m the light of the foregoing table, it will be seen to be far more 
 expressive than when viewed under the light of the sexagesimal 
 system. 
 
 1 " Ur," the Chaldsean name for Great, was a designation of the Tower of Babel (see 
 Dr. Redfield in "International Standard" for March, 1884): so that most significantly we 
 are informed that Abraham, the father of the faithful, came out from " Ur of the Chaldees ; " 
 i.e., from out the very shadow of their falsely conceived and uncosmically proportioned 
 monument and system. 
 
104 METROLOGY. 
 
 In the rational circummetric division this number is a con- 
 centrated symbol of the pyramidal system, whose potency, as 
 we have seen, consists in its use of the number 5. 
 
 129,600,000 = the number of fifths (v) in 5 and | circles. 
 
 In another form, which intensifies the significance while it sim- 
 plifies the expression, we have 
 
 129,600,000 the number of fifths (v) in *g- circles (= f more 
 than 2 f- circles). r : f . < 
 
 That is, it is the number of fifths in five complete circles about 
 the pentagon, plus the arc subtended by the side of that alU 
 potent emblem, the pentalpha. The rhythm of this relation can 
 finally be made even more apparent, as follows : 1 29,600,000 v is 
 the number contained in the continuous arc which circum- 
 scribes the five sides of the pentagon five times, and one-fifth 
 of the pentalpha once. 
 
 But there are other references centring in and about this 
 number which make it still more significant in its connection 
 with the atmospheric value of the Holy of holies. 
 
 While geometrically an irrational divisor of the circle, the 
 number 360 has, from the days of Nimrod, been employed as a 
 familiar divisor of time, or the year. This division is peculiarly 
 a Babylonish one ; and its employment by Daniel, who wrote in 
 Babylon, concerning the duration of "the times of the Gen- 
 tiles," under the type of Babylon as the "head of gold," is very 
 pointed. 
 
 The Hebrew meaning of this great prophet's name, Dan-i-El, 
 is "Judge of God." The Hebrew significance of ancient Baby- 
 lon, or Bab-El, is "Gate of God." It was in the gates that 
 judgments were rendered in Oriental nations. The Book of 
 Daniel thus acquires a new significance as the judgments of 
 God, enunciated by his representative judge, and issued from 
 the very gate of judgment. Thus Daniel, wise beyond mortal 
 standards, and called to pass such momentous judgments upon 
 the Gentile world, was pleased to typify its chronological dura- 
 tion under familiar Babylonish terms. If we mistake not, there 
 is also a pointed irony in this employment. His system is 
 thus at perfect unity with itself. He uses the king as a type. 
 
CONCLUSION. 105 
 
 and great Babylon as an archetype, and with reference to each, 
 as well as to the grander empire typified, throughout most 
 fittingly employs the sexagesimal notation as its standard of 
 duration. 
 
 Nebuchadnezzar, the antetype, is insane a week of years, or 
 7 years of 360 days each. So Babylon, the archetype put for 
 the whole Gentile world, was, according to Daniel, to be blind 
 in its own devices 7 "times." A "time" being a year of such 
 years, gives us a period of 360 years ; and the 7 times to pass 
 over the Gentiles is 7 X 360 = 2,520 years. This latter num- 
 ber is doubly significant ; for not only does it have the chrono- 
 logical import above noted, but it is one of the most marked 
 numbers in the entire decimal system, being the least common 
 multiple of all its digits, and likewise astronomically a syn- 
 chronologic cycle of the Earth, Moon, and Sun. 
 
 Furthermore, the number 1,260 years (Daniel's 42 months of 
 30 years each), the half of this prophetic week, is less than 
 1,296, the root of the number we are studying, by 36 = 6 X 6, 
 a most expressive Babylonish number ; while 1,290 years (another 
 of Daniel's most important numbers) is less than it by the root 
 itself, 6. 
 
 In this connection let us refer to the number 1,296, which, as 
 we noted above, was the number of hours in 54 common days 
 of 24 hours each. Now, on calculating what part of the Baby- 
 lonish year of 360 common days this period of 54 days may 
 be, we are surprised to find that it comes out as the intensely 
 Babylonish repetend, 6.6666 + etc., forever. 
 
 Again, as a "time " was 360 years, or the square of 360 days 
 (i.e., 129,600 days), and as a week of times* was 7 X 360 = 2,520 
 years, so a month of times was 360 X 30 = 10,800 years ; and 
 a "prophetic year," that is, one of 12 such months, was 10,800 
 X 12 = 129,600 years, or the square of 360 years. This num- 
 ber may be arrived at directly by noting, that, as a day upon. 
 Daniel's stupendous week of Gentile times was 360 years, so, 
 360 such days would be a prophetic year, or 129,600 common 
 years. What, then, may we ask, is the implied duration of the 
 millennium upon such a scale ? To this we may reply, that as a 
 prophetic year is thus 360 days, each of 360 years' duration, and 
 as we are assured that with the Almighty " 1,000 years are as 
 
106 METROLOGY. 
 
 i day, and i day as 1,000 years" (note the duplication of 1,000), 
 so a millennium may be taken as at least implying 360 days X 
 1,000, each of 360 years X 1,000, or 36o 2 X i,ooo 2 common 
 years, or as equal to 1 29,600,000,000 years. 
 
 The following table illustrates these deductions : 
 
 Thus, a prophetic day = 360 days. 
 
 2 
 
 a time = 360 years, = 360 days. 
 
 2 
 
 ( = 360 years. 
 
 a prophetic year I _ 4 
 
 ( = 360 days. 
 
 .2 
 
 a millennium = 360 X 1,000 . 
 
 In all of these numbers, of such grandly chronologic im- 
 port, the number 1,296 is intensely present; but it is rather 
 through the relation noted above in the repetend 6.6666 +, etc., 
 X 54 at 24 each, be the periods how large soever, than from 
 any significance of the 360 division of the circle as geometri- 
 cally correct and rational, that we must view this marked 
 number and its significant use at Gizeh, Babylon, and Jerusa- 
 lem, the three great metrological centres of former days. 
 
 It has nothing upon earth to do with the irrational division 
 of the circle into 360 at 60' at 60", etc., but is beautifully in 
 harmony with the whole pyramid and temple systems of me- 
 trology and symbology when viewed through the correctly 
 geometrical division into 240 at 10' at 10", etc. 
 
 So, too, it utters significant harmonies in the chronological 
 scheme of 360 days of 24 hours each. 
 
 Viewed in the light of such facts, the formula giving the 
 weight of the atmosphere in Solomon's Holy of holies is indeed 
 a wonderful emblem of the "breath of lives" which the Almighty 
 One breathed into the soul of the temple of clay, which, at the 
 beginning, he created in his own image. 
 
 In this connection, and as directly related to the atmospheric 
 value of this sacred chamber, I wish to call attention to the 
 sequences of numbers 648,000, 1,296,000, 20,626,481 -f-, etc. ; 
 41,252,962 +, etc. ; 141,592 +, etc. ; 125,000,000, etc. They 
 are all intimately and equally related to the cubes of 500", of 
 50", and of 5", the decimal point alone moving according to the 
 cube considered. 
 
CONCLUSION. 107 
 
 Now, each of these cubes, whether selected by the ancients 
 on account of their notable numeric properties and specific- 
 gravity relations, or not, certainly received the most marked 
 deference at the hands of their metrologists ; and, from what we 
 have now discovered concerning them, they are still worthy of all 
 the regard and prominence we can ourselves bestow upon them. 
 
 These are facts, and not fancies, and are valuable facts, both 
 for the scientific and the practical man. 
 
 But their recognition seems to have been far older than the 
 eras of Moses or Solomon. We have already seen that the 
 metrology of a golden age earlier still that of the Great Pyra- 
 mid was centred in and around the cube of 50" of standard 
 material at mean earth condition as to the physical circum- 
 stances of the atmosphere. Now, a mere glance into the king's 
 chamber the sanctum sanctorum of this edifice shows us 
 that it was also consummately constructed to perpetuate these 
 same ratios. Its very relation to the cube of 50" would have 
 been sufficient to establish its tangency to the circular and cir- 
 cummetric ratios we found in the Holy of holies, were they not 
 already there even far more boldly than they were at Solomon's 
 Temple. 
 
 They occur in this remarkable room in an entirely different 
 way, not at all concealed, but built into its sides and walls and 
 volumes in a way too wonderful, in the light of all else we have 
 discovered there, not to be a clear demonstration of intention 
 and design completely realized. Thus, 
 
 648 
 Its breadth = 206.264, etc., inches. 
 
 7T 
 1,296 
 
 Its length = = 412.529, etc., inches. 
 
 648 TT 1,296 
 Its side-wall height = - - = 235.470, etc., inches. 
 
 Its interior height of chamber = 230.328, etc., inches. 
 Height to which the granite floor rises up into the side-walls 
 = 5.1^.15, etc., inches. 
 
 Total height of lower course of wall-stories = 47.008909394-. 
 Clear height of lower course of wall above floor == 41.86730. 
 Average height of the 4 upper wall-courses = 47. 1 153. 
 
lOo METROLOGY. 
 
 It is in the first place noticeable, that not only are these the 
 identical ratios, but that the total perimeter, (412.52, etc., -}- 
 235.47, etc.) X 2, of one of the side-walls of this chamber, equals 
 1,296 exactly, or one i-thousandth of a circumference of 360 
 where each second is one linear inch. In the rational 240 cir- 
 cle, 1,296 is exactly 5| circles where each degree is an inch. 
 
 From calculations based upon these dimensions, we further 
 learn that the volume contained by the extreme dimensions of 
 the chamber is 20,036,252.7789, etc., cubic inches. The cubic 
 space occupied by the granite filling which constitutes the floor 
 of the chamber, is 437,500 cubic inches. The cubic contents 
 of the lower portion of the chamber " marked off " above the 
 floor by the first horizontal course of wall-stones = 71,250 X 50 
 = 3,562,500 cubic inches. Hence the part " filled in" below 
 the floor, added to that "marked off" above the floor, =437,500 
 + 3*562,500 = 4,000,000 cubic inches exact. 
 
 Now, in the light of the earth-commensuric, or rectified, 
 Anglo-Saxon system of metrology, which we have been exam- 
 ining, this number is of marked import. 
 
 Numerically, 4,000,000" 4(1,000,000") = 4(1 oo") 3 . (ioo") 3 
 = 8 cubes of 50": hence 4(ioo") 3 = 32(5o") 3 = 4(2)3 (50")'. 
 Had it not been desirable, therefore, to build the chamber sim- 
 ply upon the circular and circummetric function - , its dimen- 
 sions could have been 400 long by 200 wide, and its lower tier 
 have been 50" high, the first 5.46875 inches being filled in, and 
 leaving 200 X 400 X 44.53125 = 3,562,500 cubic inches = 50 
 X 71,250" marked off above the floor. 
 
 However, as the chamber now stands, it realizes both of these 
 ends ; and the number, 4,000,000 cubic inches, thus with so 
 much the more difficulty so accurately set apart, demands a 
 moment's consideration at our hands. 
 
 With reference to water as the material of unit density, this 
 volume of 4,000,000 cubic inches has, as we have already seen, 
 been particularly subdivided into two parts; to wit, 3,562,500 
 = 71,250 cubic inches X 50, and 437,500 = 71,250 cubic fnches 
 X 6- 5 8 T . The latter volume seems to be thus set aside as though 
 specially to enable the grandly even quantity of 50 coffers of 
 water to be indicated. In the inter-relations, however, of the 
 
CONCLUSION. 109 
 
 several volumes thus marked off, there is a remarkable promi- 
 nence given to the number 57. For instance, 4,000,000 cubic 
 inches = 56^- = -3-jjjp- cofferfuls ; 437,500 cubic inches 6 5 ^- 
 =. 3 5 5 f - cofferfuls, and 3,562,500 = 50 cofferfuls. Now, this 
 latter volume is 8| times = ^ times the volume filled with 
 granite ; or, in other words, the latter volume is ^y of the 
 5O-coffer space marked off above the floor. It naturally now 
 occurs to us to inquire what this volume, the -fa of a coffer, 
 may be. 71,250 -f- 57 = 1,250 = io(5) 3 cubic inches. It is, 
 therefore, in volume and amount equal to 10 times the smaller 
 of the several cubes, whose relations we have found to be so 
 important. Again, the whole volume, 4,000,000 cubic inches, 
 is seen upon examination to be just * = || of a coffer short 
 of being exactly 57 coffers in capacity, a quantity of water 
 which would be balanced by the very notable quantity, 10 
 coffers of mean or standard (5.7) density material. We have 
 already seen that 10 coffers of water are equal to a standard 
 cube, or to (5o") 3 of mean density material. These, for the 10 
 coffers of mean density material here indicated, are equal to 5.7 
 times as much, or to 5.7 (5o") 3 of that material whose density is 
 standard, or 5.7 times that of water. 
 
 References more pointed to the coffer, the cube of 50", 
 the standard density, 5.7, etc., with which this chamber thus 
 almost cries aloud, can hardly be conceived. But let us exam- 
 ine the marked-off portions of the chamber with reference to 
 some of the other important metrological substances of practi- 
 cal life. 
 
 If we consider that the entire 4,000,000 cubic inches marked 
 off is a volume of the mean density material, it will equal, in 
 terms of the rectified Anglo-Saxon system, the following quan- 
 tities : 
 
 4,000,000 cu. in. mean density material = 8,000,000,000 ultimates = (2,ooo) 3 . 
 
 = 6,144,000,000 grains = 4 (1,536) (ioo) 3 . 
 = 12,800,000 ounces. 
 = 1,280,000 pounds standard. 
 = 1,024,000 pounds troy and apothecary. 
 800,000 pounds avoirdupois. 
 
 Considering the volume to be one of lead, the values through- 
 out will be just double the above. 
 
1 10 METROLOGY, 
 
 Regarding the whole 4,000,000 cubic inches as filled with 
 iron, the practical weight material of every-day life, the 
 numerical relations to the rectified system become very notice- 
 able. Thus, 
 
 4,000,000 cubic inches iron = 1,000,000 pounds avoirdupois = (ioo) 3 pounds. 
 
 = 1 >333>333i pounds troy and apothecary, 
 
 = 1,600,000 pounds standard. 
 
 = 16,000,000 ounces. 
 = 7,680,000,000 grains I = 7,68o (ioo) 3 grains. 
 
 5 S (^536) (ioo) 3 grains. 
 
 = 10,000,000,000 ults. i = I0 ' 000 X < IOO > 3 ultimates - 
 
 ) = 5 X 2,000 X (ioo) 3 ultimates, 
 
 With reference to the atmosphere, at standard physical con- 
 dition, the consideration of this marked-off portion is likewise 
 fraught with interest. A cubic inch of standard atmos- 
 
 120 
 
 5 : 
 
 10(640)648 
 4,000,000 cubic inches will weigh grams, in terms 
 
 of which expression all the derivative quantities may be written. 
 Thus, in ounces, since there are 480 grains in an ounce,, there 
 
 10(640)648 . 
 are ~ in the marked off 4,000,000 cubic inches space ; 
 
 and in 50 coffers of air there are - ounces, etc. 
 
 7T 
 
 With but one other reference to the numerical beauties of 
 this rectified system of metrology, we shall close an otherwise 
 endless topic. 
 
 Since in a cubic inch of water there are, at standard circum- 
 stances, grains, and since in an equal volume of air there 
 
 pheric air, we have already seen, weighs - - grains : hence 
 
 are 
 
 I V ~ ) g rams > an d since the specific gravity of water 
 
 \I,OOO T/ 
 
 is unity, it follows that the specific gravity of standard atmos- 
 pheric air must be exactly expressed by | T^QTT ~' C7 = 
 
 4 ' 617 =.001224+, etc. 
 
 1, 200, 000 
 
 From the foregoing brief survey of some few of the numeri- 
 
CONCLUSION. Ill 
 
 cal beauties of the system, and their direct earth reference, 
 both as to general and particular specific gravity, etc., it seems 
 to be an inevitable conclusion, that in thus unlocking the stand- 
 ard observatories of Egypt and Jerusalem, and building one for 
 the world to-day, we have discovered the key by means of 
 which we may reasonably also hope to penetrate into the sanc- 
 tum sanctorum of every human science ; for to have entered 
 into the holiest place of the universal science is to be at once 
 within those of all its various branches. 
 
APPENDIX A 
 
 "THE SACRED CUBIT.' 
 
"And Aaron cast down his rod before Pharaoh, and 
 before his servants, and it became a serpent. Then Pha- 
 raoh also called the wise men and the sorcerers : now the 
 magicians of Egypt, they also did in like manner with their 
 enchantments. For they cast down every man his rod, and 
 they became serpents : but Aaron V rod swallowed up their 
 " EXODUS vii. 10-12. 
 
"THE SACRED CUBIT." 
 
 " Merpov uv0p<j7rov 6 iortv ayyeAou." Rev. xxi. 17. 
 
 Ix the study of the Great Pyramid, the clearest intellect may 
 find its measure of capacity, and at the end fall short of com- 
 prehending all the secrets it contains. It is a beautiful and 
 fascinating study ; and, the deeper one investigates it, the more 
 convinced will he become of the master masonry that planned 
 and realized its grand proportions. Its stones cry out in silence 
 far more eloquent than words, its lines and angles speak in lan- 
 guage plainer than a written character, against the man who is 
 weak enough to find in such a structure simply accident and 
 not design. If such there be whose inner heart is not amazed 
 with admiration at such a building, raised, remember, long 
 before the dawn of science, and confounding even modern sci- 
 ence, then let him know, that, in the chambers of his soul, 
 there can be no lines that square and harmonize in pyramidic 
 proportions with the heavens and earth. 
 
 NOTE. I am induced, at the earnest solicitation of friends, to whom I have shown my 
 diagrams, to publish the following article as an appendix to this discussion. The matter 
 treated belongs naturally to the subject of " linear measure," as based upon the unit earth- 
 commensurable INCH, and upon twenty-five such inches for a grand standard. This is the 
 " sacred cubit ; " and its discussion brings into linear measure as much of the new and impor- 
 tant as those who shall have followed us in capacity and weight measure will have seen the 
 latter, in their relations to this cubit, to possess. But, in thus publishing the following in its 
 present shape, I must apologize for its colloquial style. It was written as the third of a 
 series of lectures upon my pyramid labors, and asked for by the officers stationed with me at 
 the Presidio of San Francisco in 1880. It was also intended to deliver the same series before 
 the " Mechanics' Instkute " in San Francisco. A sudden campaign, and a subsequent change 
 of station to the East, interfered ; and the MS. has remained in its present form ever 
 since. There is now no time for recasting the matter into a style corresponding with the 
 body of the discussion ; so, with an apology, it is appended, in the trust that the interesting 
 considerations it reviews will lend as much to the all-important subdivision of linear measure 
 as the subject merits. It throws such a glare upon the " sacred cubit," in its mathematical 
 and geometric relations to the requirements of a perfect linear standard for modern metrology, 
 that we cannot but accord to its lofty and ancient origin a wisdom truly superhuman. 
 
 "5 
 
Il6 METROLOGY. 
 
 One of the greatest difficulties experienced by the student of 
 the pyramid, is that of systematizing the information that has 
 already been acquired. He experiences this difficulty at the 
 outset, and it increases as he penetrates into its higher and more 
 interior mysteries. The perpetual recurrence of such important 
 sequences of numbers as those which express the " circummetric 
 ratio" and the "solar ratio," 3.14159, etc., 365.242242, etc., and 
 a multitude of dependent series, plunges one into a sea of con- 
 fusion. No sooner is the intellect focussed upon any part of this 
 mysterious structure, than from the most diverse lengths, angles, 
 shapes, and groupings, the same familiar metric rhythm flashes 
 forth. Now in cubits, then in inches ; here in circles, there in 
 squares ; sometimes in area, at others in volume, often where 
 looked for, and more frequently where least expected, they 
 are ever present. The mind cannot resist the inevitable con- 
 clusion, that a most consummate plan regulates the whole 
 design, and links each part to every other and the whole. Seek- 
 ing for the thread to guide him through a labyrinth of scientific 
 architecture, so worthy of the fabled land it long has shadowed, 
 the bewildered mind a thousand times is led astray. At last 
 the idea seems to be within his grasp. A vision of the system 
 shines athwart his gaze, and for a moment the IDEAL pyramid 
 is before him. It is but the mirage ; the vision fades ; the 
 monument is yet beneath his horizon. Sand, Sahara, and 
 infinity still surround him. 
 
 But he does discover, here and there, a green oasis ; this he 
 names, an island, merely, in the trackless desert : for his map is 
 necessarily incoherent, from its lack of cardinal bearings. 
 
 Such, for years, was my own experience in this engrossing 
 study. The mirage that .lures the student on is but a picture, 
 mirrored on the moister atmospheres which bathe these grateful 
 spots of green. But the existence of the image proves the 
 reality of the object, beyond, indeed, our plane of vision, but, 
 nevertheless, real and tangible. 
 
 I have already alluded to one of the skeleton systems, which 
 I called "the pyramid method," and which enabled us to con- 
 nect the coffer with the king's chamber, which contains it, 
 and both with the monument itself. But there the analogy 
 seemed to cease ; though it does, however, take a most interest- 
 
"THE SACRED CUBIT." 1 17 
 
 ing step beyond. Let us re-examine it, briefly, and in a new 
 form. 
 
 2irR is the modern symbol for the circumference of any circle 
 in which TT is the circummetric ratio, and R the radius of the 
 circle under consideration at the time. 
 
 Now, a new, and perhaps the most interesting, form in which 
 we can apply the pyramid method, is as follows. You will be 
 kind enough to notice that I use the form 2ir into R through- 
 out this application. 
 
 (1) In the pyramid, as a whole, 2ir into //"(its height) equals 
 the perimeter of its base, or equals 4!., since it is a square base, 
 and all four sides are the same. (See Figs. 3 and 10.) 
 
 (2) In the king's chamber, 2ir into B (its breadth) equals the 
 perimeter of its side; i.e., equals 2(L+H). You will also 
 remember that the length of this chamber is just double its 
 breadth. We have, therefore, the same principles, though with 
 a marked variation in the shape of the geometrical figure to 
 which it is applied. 
 
 (3) In the coffer, 2?r into H (its height) is again equal to the 
 perimeter of the base ; i.e., equals 2(L -f- B). But, in regard to 
 this latter figure, you will also remember that none of the 
 dimensions are apparently in any other way related to each 
 other : hence we have the repetition of the principles, with 
 absolute dissimilarity of the. ruling dimensions, introduced as a 
 geometric feature. 
 
 Let us now repair to the ante-chamber. Upon the granite leaf 
 that stretches across this narrow room is "the boss," the little, 
 raised, and rather more than semi-circular, protuberance which 
 has been described as "the only ornament " in the pyramid. 
 We shall study this very closely later on. At present I shall 
 only allude to the principles under discussion, and which we 
 have just applied to the dimensions of the pyramid, king's 
 chamber, and coffer. This principle re-occurs here in its most 
 ideal simplicity. The boss is 2-n-R itself ; for 27r, into its own 
 radius, is equal to the indicated circumference of its base. But 
 you will say, that, in all the other examples, I have introduced 
 the element of height in one or the other term of the equation. 
 Why is it not done here, to keep up the analogy ? I answer, it 
 is there a silent factor, since it is " unity " itself. The height 
 
Il8 METROLOGY. 
 
 of the boss is just one inch, or "unity:" it therefore disap- 
 pears. And so this factor disappeared in each of* the other 
 three cases ; for, in all those cases, 2ir, by itself, is the circum- 
 ference of the "unit radius :" and the applied formula in every 
 case is as follows : 
 
 The unit circumference (271-) X the unit (i) radius X a special 
 radius (height or breadth), yields the other dimensions in 
 perimeter. 
 
 But the "pyramid idea" is not to be unfolded by this prin- 
 ciple alone. Its help enables us to articulate only a few bones 
 of the mammoth skeleton before us. Let us, therefore, proceed 
 in another direction. 
 
 We will return to John Taylor's first discovery, that the cir- 
 cumference described with the height is equal to the perimeter 
 of the square base of the pyramid. In studying this and its 
 kindred facts, at the opening of our last lecture, we were 
 enabled to discover a magistral line, 10303.30 inches in length, 
 which, squared and circled, formed a connecting line between 
 four separate diagrams. 
 
 (1) The square of this line gave us the circle of the height 
 of the pyramid. (See Fig. 6.) 
 
 (2) The circumference of such a circle gave us the perimeter 
 of its base. (See Fig. 3.) 
 
 (3) The circle of this line gave, us the area of the base of 
 the pyramid. (See Fig. 5.) 
 
 (4) The perimeter of this base gave us the circumference of 
 the height. (See Fig. 3.) 
 
 (5, and finally.) The square of half this line is equal to the 
 meridian section of the pyramid, and also to the area of the 
 circle whose diameter is the height of the pyramid. (See Figs. 
 7 to 9.) 
 
 What an intricate confusion ! Here are four figures, involv- 
 ing squares and circles, radii and sides, areas, perimeters, 
 circumferences, and other intimate equalities ; and yet their 
 simultaneous comprehension is well-nigh unthinkable. It puz- 
 zled me for two years. I could not resist returning to it 
 again and again ; but, so soon as one set of relations was fixed 
 in the mind, the passage to the next was a process of oblitera- 
 tion. 
 
"THE SACKED CUBIT: 
 
 9131- 05 P. X. or 
 365-242 S.<7. 
 
 DIRECT VERTICAL SECTION OF 
 
 GREAT PYRAMID 
 
 Fig. 1. 
 
 12913 -26 P. /. or 
 516-530&C 
 
 DIAGONAL VERTICAL SECTION OF 
 GREAT PYRAMID 
 
 Fig. 2. 
 
 EQUALITY OF BOUNDARIES 
 
 
 9131 -05 P. I. 
 
 Great Pyramid's tquare base, 
 and circle with radiu8Pyr'a Vert'l height 
 
 Fig. 3. 
 
 If ANGLES OF CASING STONES OF 
 
 GREAT PYRAMID; 
 Jia affected by its external slope 
 and horizontal "masonry courses* 
 1T -3: 14159 26535 + frc; 
 toy. -49714. 93726.-h<c. 
 
 Fig. 4. 
 
 EQUALITY OF AREAS No. 1, 
 
 EQUALITY OF AREAS No.2. 
 
 V 9131 -05 P. J. , 
 
 Area of .square bass of Great Pyramid 
 
 area of a Circle whose diameter is given 
 
 ~ 100 in the Ante-chamber. 
 
 P. I.' PYRAMID INCHES. 
 
 Fig. 6. 
 
 Area of circle with G.Pyr's height for radius 
 Area of square whose length of side is given 
 -f-100 in the Ante-chamber. 
 S. Cf SACRED CUBIT. 
 Fig. 0. 
 
120 
 
 METROLOGY. 
 
 EQUALITY OF AREAS No. 3. 
 
 4563-52. 
 
 9131- 05 P. 1. 
 Direct Vertical Sect Ion ofGr.Pyr'd 
 
 Circle with Diameter 
 Vert'l Height of G.Pyr'd 
 
 Square with side 
 computed by Tf 
 
 Fig. 7. 
 
 Fig. 8. 
 
 Fig. 9. 
 
 11626-02 Ante-chamber length x ,100 Sun's distance from the earth, 
 in terms of tte"breadth of the Earth" from Pole toj>ole+ 
 
 365-242-H&C.- 
 ~Base-side length of Qr.Pui*d 
 in Sacred Cubits. 
 
 EQUATION OF BOUNDARIES AND AREAS, 
 CIRCLES AND SQUARES,INCHES INSIDE AND SACRED CUBITS 
 OUTSIDE GREAT PYRAMID 
 
 Fig. 10. 
 
11 THE SACRED CUBIT? 121 
 
 At last the solution occurred to me in the form of a single 
 geometric figure, which I shall term, par excellence, the pyramid 
 diagram. Its discovery was prolific with attendant ones, some 
 of which I hope to review this evening. 
 
 THE IDEAL DIAGRAM. 
 
 The pyramid diagram may be put under two forms, the one 
 absolute, and the other approximate. I shall call the former the 
 architect's ideal, the latter a workman's diagram. 
 
 Let us proceed to the examination of the first, or ideal 
 geometric diagram. 
 
 Let ABC represent the meridian section of the Great Pyra- 
 mid. 
 
 Complete the square ABFG. It is equal to the square base 
 of the pyramid. 
 
 With the vertex C as a centre, and a radius CD equal to the 
 height of the pyramid, describe the circle DIH. 
 
 By John Taylor's discovery, its circumference is equal to the 
 perimeter of the square ABFG base of the pyramid. 
 
 By means of the modern circummetric ratio (3.141592, etc.), 
 determine a square JMLK y whose area is equal to the circle of 
 the height. 
 
 Inscribe in this square another circle EOPQ. 
 
 Then will the area of this new circle be equal to that of the 
 square base of the pyramid itself. 
 
 Finally, upon CD as a diameter, construct another circle : its 
 area will be equal to that of the meridian section of the pyra- 
 mid, and to one quarter of the square JMLK. . 
 
 This most remarkable sequence of squares and circles, 
 depending upon the triangular dimensions of the pyramid's 
 meridian section, not only unites into one, easily comprehended, 
 diagram, all the confusing relations we have just noticed, but, 
 as we shall see later on, is a key to one of the most universal 
 geometric solutions that has ever been discovered. 
 
THE PYRAMID DIAGRAM. 
 
 (ARCHITECT'S IDEAL.) 
 
 Fig. 11. 
 

 
 
 
 G 
 
 \ 
 
 M 
 
 
 \ 
 
 
 123 
 
124 METROLOGY. 
 
 THE WORKMAN'S DIAGRAM. 
 
 Let us turn now to the approximate or workman's diagram. 
 This is a geometric construction without the use of rr. 1 
 
 As before, ABC is the meridian section of the pyramid, and 
 ABED a square equal to its base. 
 
 Join C and N t the vertex of the pyramid, and the centre of 
 the square, and upon it as a vertical diameter construct the 
 circle NJCM. Then will the area of this circle be equal to that 
 of the square base of the pyramid. 
 
 Upon its horizontal diameter JM, construct the square 
 JKLM. The area of this square will equal that of the -circle of 
 the height of the pyramid. 
 
 With CG y this height, as a radius, and C as a centre, describe 
 this circle of the height : then, as we already know, its circum- 
 ference equals the perimeter of the pyramid's base. Upon CG 
 as a diameter construct a circle. Its area, as before, will be 
 equal to that of the meridian section of the pyramid, and also 
 to a quarter of JM* . 
 
 1 Mr. Charles Latimer has shown, in vol. i. of the International Standard (p. 29), that 
 the identical triangle which forms the key to the architecture of the Great Pyramid, may be 
 developed from the circummetric relations of the square whose side is 81 X 100 = 9* X io 2 . 
 The diagram he gives comprises all the dimensions of the skeleton. In discussing this dia- 
 gram upon p. 245 of the same volume, I have myself called attention to the fact, that had 
 Mr. Latimer continued his series of squares, circles, and triangles downward a degree far- 
 ther, he would have arrived at the very ones employed by Mr. Parker in his interesting essay 
 upon the Quadrature of the Circle (John Wiley & Sons, publishers). Mr. Parker's method 
 of arriving at the relation of circles to squares, is an entirely original one, and is based upon 
 a rigid discussion of pure geometric form, and of abstract number. As a result of this 
 discussion, he arrives at the remarkable fraction ^$^, as expressive of the natural 
 circummetric ratio. In his deductions, Mr. Parker does not, in the remotest degree, 
 touch upon, or employ, any of the methods made use of by modern geometers in dedu- 
 cing the TT value. He arrives at his ratio by a direct examination of the " opposite duplicate 
 ratio " [y^3 ' ((3) 2 ) 2 ] f equal equilateral triangles and circles to their mutually standard 
 unit square. It is noticeable that the Parker modulus is having considerable attention 
 turned to it in higher scientific circles to-day, after having been received with a cold shoulder 
 some score of years. Now, as Mr. Latimer has shown the relations of the pyramid triangle 
 to the square of 81 units, and as I have shown the relations of the Latimer diagram to the 
 Parker modulus, the relations of the latter to the pyramid triangle are logically established. 
 For all practical purposes, therefore, the " workman " may employ the Parker modulus in 
 constructing pyramid triangles. Whether the latter be absolutely true, as claimed by Mr. 
 Parker, or merely a remarkably close approximation, it is, at any rate, accurate beyond any 
 possible degree of detection, even for so stupendous a structure as the Great Pyramid cf 
 Cizeh. Hence, representing by 77 the height of a pyramid triangle, and by S its base, the 
 workman may employ tTie formula 10306^= 65616", to determine the height required for 
 any given ba^e j or, conversely, the base for any given height. 
 
"THE SACRED CUBIT? 125 
 
 In order to demonstrate more clearly to your minds the sim- 
 plicity with which this geometric figure clears away the 
 inevitable confusion surrounding the four separate diagrams, 
 by means of which these relations have hitherto been expressed, 
 I have represented the workman's diagram again, and indicated 
 its more important lines by their familiar numerical values in 
 inches. 1 
 
 We have a very important use to make of the pyramid dia- 
 gram this evening. I hope to show that it is the mystic problem, 
 upon the pyramid's masonic trestle-board, which fits as beauti- 
 fully into the interior chambers and entire furniture of the 
 edifice, as we have now seen that it certainly does into the build- 
 ing as a whole. 
 
 There are, however, one or two reflections which I desire to 
 make while this diagram is fresh in our minds. 
 
 The first is this : the pyramid diagram establishes beyond a 
 doubt the possibility of squaring the circle geometrically. 
 Mathematicians have for ages been contented with approxima- 
 tions, and have acquiesced in the interdiction placed upon this 
 problem by the learned. The French Academy and the Royal 
 Society of Great Britain, within our own century, have passed 
 resolutions not to entertain any demonstration looking towards 
 this greatly desired solution. Convinced that the demonstra- 
 tions of Playfair and Legendre are correct, and that TT has no 
 simpler expression than its infinite decimal form, they repudiate 
 every effort to the contrary, and refuse even to listen to those 
 students who are yet unconvinced. The result is, that he who 
 labors at this, problem is regarded as either ignorant or of un- 
 balanced mind ; and the great problem of squaring the circle is 
 thus in utter disrepute. For one, I am not convinced : indeed, 
 I am satisfied to the contrary. I will not make the rash asser- 
 tion here, that I can solve this problem ; that is not our theme 
 to-night : but I will say, and I think the pyramid problem fully 
 establishes it as a fact, that such geometric relations certainly 
 do exist between squares and circles, in nature, that there must 
 be, and is, a geometric means of passing accurately to and fro 
 from either to the other in all their parts. The study of the 
 
 1 Diagram omicted. For dimensions, see Figs, i to 10) etc., which may be written on cor- 
 responding lines of Fig. 12, and the rest calculated. 
 
126 METROLOGY. 
 
 Great Pyramid from the mathematical stand-point convinces 
 the student, that no problem was paramount to this one in the 
 mind of its architect. It is unreasonable to expect the work of 
 even the wisest of men to be void of error ; man is not even a 
 demi-god ; and his work must fall short of his theory, even if the 
 latter be correct ; hence we must expect, even in this great 
 metric monument, only enough of realization to suggest, at 
 length, to some lucky and painstaking student, the ideal aimed 
 at by its mortal builders. 
 
 The second consideration is this : that he who discovers a 
 geometric method of constructing a triangle similar to that of 
 the theoretical meridian section of this Great Pyramid, will have 
 squared the circle. For given a triangle geometrically con- 
 structed, whose base shall be 9131 +, etc., and whose height 
 shall be 5813 -{-, etc., to any degree of accuracy that "we may be 
 able to measure them, and the pyramid diagram follows. The 
 circle will then be squared and verified ; for all the squares and 
 circles in this intimate series, based upon that triangle, are 
 related by equalities. 
 
 Now, this triangle does exist : it is built into and symbolized 
 throughout the monument. Theoretically, we can renew it to 
 any desired degree of accuracy by using the modern value of TT. 
 
 The great question, therefore, is, How did the shepherd kings 
 obtain this triangle ? By approximations ? by a TT value ? by any 
 numerical expression ? or by a strictly geometric construction ? 
 My own studies of this intricate subject lead me to the conclu- 
 sion, that the architect of the Great Pyramid had a simple 
 geometrical construction ; that he could and did square the 
 circle. But it necessarily follows, that he also had a perfect 
 numerical expression for his work ; for it is absurd to suppose 
 that correct geometry is inexpressible in numbers. Not neces- 
 sarily in decimal numbers, but in some form or other of 
 numerical expression, this circummetric ratio can certainly be 
 written. The circummetric ratio is no more difficult of expres- 
 sion than the ratio of the perimeter to the diagonal of a square : 
 both are incommensurable, equally so and similarly so. In 
 
 r 4 
 the square whose side is one, this lattei ratio is 4 : V2 = -^ 
 
 = 2.82 -{-, etc., forever. Nevertheless, - is a perfect arith- 
 
 V2 
 
"THE SACRED CUBIT." I2/ 
 
 metical expression for this incommensurable decimal ; and its 
 geometric relation a diagonal within a square is shown on 
 every draughtsman's trestle-board. Number is the soul of form 
 or geometry, and every mathematic line must have its "per- 
 fect " counterpart in number. 
 
 The shepherds, you remember, were a mysterious people. 
 They appeared suddenly in Egypt : they came with a purpose, 
 
 to build the pyramid, at a spot elected from the whole surface 
 of the earth by its unique and solitary fitness. Their work com- 
 pleted, they departed as suddenly as they had come. We know 
 that they next settled in Palestine : we know that their de- 
 scendants thereafter built no pyramids. Hiram, king of Tyre, 
 
 and Hiram Abiff, were descendants of these shepherd kings. 
 A later race of shepherds, the Israelites themselves, were also 
 related to the shepherd kings ; and they in turn followed in the 
 footsteps of their more ancient predecessors, whom, the Bible 
 tells us, the Lord, in former days, had also led up out of Caphtor, 
 or of Egypt. Solomon, Hiram, king of Tyre, and Hiram 
 Abiff, were the master-masons of the only other symbolic build- 
 ing that these peoples were connected with ; and, though its 
 style was so utterly dissimilar, yet it is reasonable to suppose that 
 perhaps they both belonged to the same school of architecture. 
 
 Although enough has been already adduced in the body of 
 the work to establish this connection upon a positive basis, we 
 shall allude, in passing, to the following : 
 
 THE PYRAMID AND SOLOMON'S TEMPLE. 
 
 The temple of Solomon is destroyed, but its measures and 
 dimensions are sacredly preserved. Freemasonry is familiar 
 with them ; and the great book of symbols, the Bible, carefully 
 records them. 
 
 Some years ago (1878) Major H. A. Tracy, R.A., in a pub- 
 lished article entitled, " Was the Architect of the Great Pyramid 
 inspired ? " called special attention to the temple of Solomon, 
 and pointed out some remarkable mathematical relations, 
 between its dimensions, which stamped it as undoubtedly pyra- 
 midic. Taking its interior dimensions, after its finished casing 
 of fine cedar, he studied particularly the diagonals of the vari- 
 ous grand divisions of the temple. Now, every builder knows 
 
128 METROLOGY. 
 
 that diagonals are of primary importance in verifying the per- 
 fection of rectilinear work. Without going into detail, I will 
 say, that, among other peculiar relations, Major Tracy discov- 
 ered that 
 
 The diagonal of the floor of the porch = ioy/5 cubits ; i.e., sacred cubits. 
 
 The diagonal of the floor of the holy place = ioy/2o cubits ; i.e., sacred cubits. 
 The diagonal of the side-walls of the holy place = 10^25 cubits ; i.e., sacred cubits. 
 The diagonal of the floor of the Holy of holies = iooy5o inches, etc. 
 
 The latter expression, looy^b, is x x 9 to thousandths of 
 accuracy. These numeric values are all intimately connected 
 with the pyramid system of construction. To mention, in pass- 
 ing, but one, it may be stated that 5C/VJ m".8o3398 -f- is 
 the height of the western wainscot in the ante-chamber, a room, 
 that, as we shall see later on, is filled with similar intimacies. 
 They are also peculiarly suggestive of Mr. Simpson's treatment 
 of the diagonals in the king's chamber of the Great Pyramid, 
 and which were alluded to in our last lecture. In both of these 
 buildings, these concealed and rhythmic values yield only to a 
 sacred cubit of 25 earth-commensuric inches, or to the single 
 " unit inch " itself. 
 
 While studying these relations of Major Tracy, pencil in 
 hand, in 1878, and long before my discovery of the pyramid 
 diagram, I noticed that the proportions between the main parts 
 of the temple of Solomon could be expressed in a diagram that 
 involved the relations of squares and circles of the very simplest 
 order. The relations between the squares and circles in the 
 pyramid diagram are, upon the other hand, of the most com- 
 plex order. The two sets of relations are, in fact, as diverse 
 from each other as possible. This will be more manifest when 
 the temple diagram shall have been explained. 
 
 The discovery of the temple diagram, in fact, preceded that of 
 the pyramid by at least four years, and, in reality, led to the 
 latter. Let us examine it. 
 
 Draw the line AE equal to 1,000", and upon it, as a diameter, 
 construct the circle AKCMEPGR. Its circumference will be 
 equal to 3,141.59+", etc., and its area 785>397-5+ square 
 inches, etc., both of which series of numbers are of the utmost 
 importance to mathematicians. 
 
THE SACRED CUBIT? 
 
 I2 9 
 
 THE TEMPLE DIAGRAM. 
 
 (SOLOMON'S.) 
 
 Fig. 13. 
 
130 METROLOGY. 
 
 Upon the same line AE, as a diagonal, construct a square 
 ACEG. It will be inscribed in the primary circle, and its side 
 be equal to 707. 1067", etc. This, too, is a fundamental multi- 
 plier in geometrical calculations. 
 
 Inscribe in this square another circle, SBUDWFYH, and in 
 this latter circle inscribe, in turn, another square, SUWY. 
 
 Finally, circumscribe about the whole figure the square 
 IJKLMNOPQR. 
 
 Draw diameters to the original circle, AKCMEPGR, passing 
 through the successive 45 points, and commencing with the 
 primary line AE, and complete the figure as in the drawing. 
 
 It is the temple diagram. Its circles and squares are related 
 to each other by the simplest of geometrical constructions, that 
 of inscription and circumscription. Proceeding outwards and 
 inwards from the square ACEG, whose diagonal AE is 1,000", 
 we have a circle and a square circumscribed, and a circle and a 
 square inscribed. The successive squares and circles are re- 
 lated to each other, as to areas, by doubles, and the alternate 
 ones by quadruples ; and the various parts of each are related 
 by a most interesting and intricate series of equalities, etc. 
 
 So, too, the perimeters and circumferences are related by 
 notable proportions and multiples of the " circummetric ratio." 
 To mention but one circumferential reference, in addition to 
 those already noted, the perimeter of the original square is 
 9OO7T to within .99^-" of accuracy. This is a general principle, 
 and is true of all circles ; though I have never before seen it 
 stated or used. 
 
 The error is so small, that, with no attempt at correction, it 
 amounts to but 22^ inches in a mile of circumference. In the 
 whole circumference of the earth, it is only 8.8 miles, or about 
 ifa of the diameter. In other words, to bring the matter down 
 to a plane of comparison with ordinary handwork, the circum- 
 ference of a lo-inch circle measured by this method would err 
 by only .oil of an inch. This is very small when we remember 
 that our finest standard measures are only graduated to hun- 
 dredths, and that a thousandth of an inch is absolutely invisible 
 in the sense of indistinguishable to the naked eye. 
 
 I am digressing for the moment ; but my excuse is this, 
 that we have here, thus indicated in the temple diagram, one 
 
r 
 "THE SACRED CUBIT." 131 
 
 
 
 of the closest working approximations to the "square of the 
 circle" that I am aware of. I shall call it Solomon's "temple- 
 rule;" and, without occupying your time with the steps by 
 which I have succeeded in putting it into a beautifully simple 
 , and most accurate form, it may be stated as follows : 
 
 In any square, 1,000 perimeters diminished by a correction of 
 one diagonal is equal to 900 circumferences of the circum- 
 scribed circle. You will notice the round numbers which enter 
 into this statement, and also that the diameter of the circle is, 
 of course, the diagonal of the square. Now, the error in this 
 formula, 
 
 looo P D = gooC, 
 
 is only .00000709. In the whole 25,000 mile circumference of 
 the earth, it amounts to only .175 of a mile (or to about 900 
 feet), which is far closer than modern science is yet certain of 
 that distance. 
 
 By the temple-rule, the value of TT may also be put under a 
 very simple form, 
 
 2000^2 i 
 
 7T 
 
 9OO 
 
 whose decimal value is 3.1415856, or practically 3.14159, closer 
 by unity in the place beyond, than the modern working-rule of 
 
 "THE TEMPLE-RULE." 
 1000 perimeters of any square i diam. = 900 circumferences inscribed. 
 
 To verify this formula, let us apply it to the square whose 
 diagonal is 1,000. Since the diagonal of every square is equal 
 to a side into the square root of two, we have the following : 
 
 .-. J> = 2S*, 
 
 or 
 
132 METROLOGY. 
 
 Substituting for D 2 its value, 1,000,000, we have 
 
 S = V 500000 = 707.10678 + , etc., 
 /. P = 4$ = 2828.42712 +, etc. 
 
 Hence, substituting, in the temple formula, the above values 
 of P and D) we have, as follows : 
 
 1000 P D = 900 C, 
 or 
 
 2828427.12 1000 = 
 or 
 
 2827427.12 = 
 
 CV. = 3141.585+, say 3141.59. 
 
 Now, it is manifest that no use, in the above calculation, has 
 been made of the modern value -* ; and yet it is equally clear 
 that the result is correct to the hundredths, and this is a circle 
 whose diameter itself is 1,000. 
 
 The modern ?r value for the circumference is 2irR ; or as R in 
 this case is 500, 
 
 C = 1000 TT = 3.141592 +, etc., X 1000 = 3141.592 +, etc. 
 
 The formula is, of course, equally true for any other value of 
 D) and, if it is desirable, may be developed to almost any degree 
 of accuracy ; but, as a working-formula, its simplicity, both for 
 memorizing and application, is as noticeable as its extreme 
 accuracy. 
 
 But I have not yet explained why I have termed this geo- 
 metrical figure the temple diagram. There are several reasons. 
 The principal one is this : in a figure so constructed upon a 
 base-line of 1,000 sacred inches (or twenty-fifths of the sacred 
 cubit), the area of FPQRHGF equals that of the "porch" of 
 Solomon's temple (125,000 square inches) ; the area of the 
 original square AC EG equals that of the holy place (500,000 
 square inches) ; and the area of the interior square SUWY 
 equals that of the Holy of holies. It is to be noticed here, in 
 addition, that this interior square, or Holy of holies, is inscribed 
 in a nest of circles and squares whose original radius, 500", was 
 the side-length of the Holy of holies itself. Thus having set 
 
"THE SACRED CUBIT? 133 
 
 out with a given line as a radius, we have here arrived at the 
 same line employed as the side of an inscribed square. From 
 the Masonic stand-point the tesselated pavement of Solomon's 
 temple is of great symbolic import ; and an examination of the 
 temple diagram develops the fact that its whole groundwork 
 is a simple checker-board 250 inches, or 10 cubits, on a side, or 
 each square of which contains an area of 100 square cubits. 
 Nor is the beautiful border, of smaller squares and rectangles, 
 to be overlooked in the mathematical relations of this ancient 
 and still wonderfully symbolic pavement. 
 
 But we must draw this examination of the temple to a close. 
 It must certainly now be clearly manifest, that these two 
 remarkable buildings belong to the same school of architecture. 
 While constructively they are entirely dissimilar, they are, 
 nevertheless, intimately related in spirit. It was to the temple 
 diagram that I resorted in my search for the pyramid triangle. 
 I reasoned that it must be hidden there somewhere. Study soon 
 demonstrated that the two diagrams were indeed most strangely 
 related, as though, in fact, they were an involute and evolute. 
 Drawing the working-diagrams to the same scale, I found many 
 points of coincidence, and at last proceeded as follows : 
 
 Producing the sides YS and Yl ot the Holy of holies, they 
 intersected those of the outer square in the points J and N; 
 and, joining the points G,J, and N t I obtained an approximate 
 workman's "pyramid triangle." 
 
 I shall not take you through the maze of calculations that 
 followed, but, to establish the connection thus discovered be- 
 tween the diagrams and buildings, merely ask you to examine 
 the combined figure, in which I have superfixed the pyramid 
 diagram upon that of the temple. This is merely constructed 
 upon the " workman's " or approximate plan, but the numerous 
 coincidences throughout the diagrams cannot fail to establish 
 their intimate connection. The harmony between the two 
 diagrams is such that every geometrical line and important 
 point of the one is directed, as it were, or located, by those of 
 the other. Thus, in our combined diagram, we have marked 
 with a o the commonly important points of both figures, and by 
 a * the seven points of special coincidence. If built upon the rr 
 relations, the coincidences of these diagrams are approximations 
 
134 
 
 METROLOGY. 
 
 THE PYRAMID AND TEMPLE DIAGRAMS HARMONIZED. 
 
 (APPROXIMATE.) 
 
 Fig. 14. 
 
"THE SACRED CUBIT? 135 
 
 of the closest possible character, approximations than which 
 none closer can be realized in the whole range of geometry. 
 If, however, the pyramid be constructed upon the " extreme and 
 mean ratio," as Ballard opines, the relations will possibly come 
 out still closer, 1 nor do I doubt but that between the architect's 
 ideal diagram and that of the temple, there may yet be found 
 an exact connection. 
 
 THE DIAGRAM IN THE KING'S CHAMBER. 
 
 Let us now retrace our steps through the great monument of 
 Egypt, and, retaining in our hands its complicated diagram as 
 a gauge, remeasure its dimensions. 
 
 If it be a sacred shrine, let us obey the prophetic mandate of 
 the angel to the holy St. John, and "arise and measure the 
 temple and its altar." 
 
 If perchance we have indeed discovered a key to the mystic 
 combinations that have for so many centuries defied admission, 
 we may then prepare ourselves for some surprising revelations 
 of symmetry and design. 
 
 Let us repair immediately to the best known, or king's, 
 chamber. I have already alluded to the peculiar adjustment of 
 its floor to the lower tier of its wall-blocks, whereby the con- 
 tents of this portion of the room is made to equal 50 times that 
 of the coffer, and whereby the ratio of the ark to the brazen 
 sea in Solomon's temple is exactly realized. This adjustment 
 is accomplished by raising, as it were, the floor of the chamber 
 up into the room some 5 inches. It would never have been 
 discovered but for the partial destruction of this floor in one 
 corner. Here the treasure-seekers of former years have re- 
 moved some of the floor-blocks, and thus revealed the fact that 
 the side-walls continue downward a few inches beyond the 
 general level of the floor. This adjustment has given rise to 
 what are known as the first, second, and mean heights of the 
 king's chamber. The first height is that of the room itself, or, 
 in pyramid inches, is about 230"; the second height is- that of 
 the side-walls, laid bare, or about 235"; and the mean of the 
 two is 232.52". 
 
 1 I am indebted to Mr. Jacob M. Clark of Elizabeth, N.J., for this idea, and shall grate- 
 fully leave it for his own labors to establish or contradict. 
 
136 METROLOGY. 
 
 Using, now, the mean height (232.52") of the chamber as 
 the altitude of a pyramid triangle, and drawing the pyramid 
 diagram to the resulting scale, we find that its magistral line 
 is 412. 13", or the length of the chamber. The base of the 
 triangle is the year number, or 365. 24"; and the radius of the 
 circle, whose area is 365. 24 2 (or the base of our new pyramid), 
 is 206.06", or the width of the chamber. Moreover, the radius 
 of the circle whose area is that of the pyramid triangle, is 
 1 1 6. 26", etc. In fact, the diagram, thus applied, reproduces all 
 the dimensions of the chamber. It must also be manifest, 
 that, with these dimensions, a model pyramid may be con- 
 structed. 
 
 Viewed in the light of this diagram, the king's chamber is 
 indeed an image of the pyramid, and a beautifully proportioned 
 image, too ! For further examination shows that the two simi- 
 lar diagrams (that of the pyramid itself, and that of the king's 
 chamber) are related to each other by the cubit number 25 as a 
 ratio. In other words, line for line, dividing every dimension 
 of the original pyramid diagram by 25, we obtain the corre- 
 sponding one in that of the king's chamber. 
 
 There are many other interesting relations between these 
 two diagrams, that, for want of time, I must pass over. 
 
 Let us now construct a pyramid triangle whose height shall 
 be the width of the king's chamber, and complete the result- 
 ing pyramid diagram. 1 We shall thus obtain one in which 
 the year number is now the magistral line. The perimeter 
 of the base of the resulting model pyramid is made up of the 
 two other important dimensions of the king's chamber, each 
 one being doubled. These are the length (412. 1318") of 
 the chamber, and this time its remaining or second height 
 
 (235.2431"). 
 
 But since the magistral line of this new diagram and the 
 pyramid base of the former one are the same (i.e., the year 
 number), the two diagrams have a mutual square whose side is 
 365. 242", and may thus be made continuous ^ or to form, as it 
 were, a double diagram. The two halves of this double dia- 
 gram are thus united by a circle and a square, the diameter 
 
 1 We just constructed the diagram upon its height. 
 
"THE SACRED CUBIT." 137 
 
 and side of which are respectively equal to the all-important 
 number, 365. 242. 1 
 
 The upper half of this double diagram refers, then, to the 
 king's chamber by its mean height ; the lower half to the same 
 chamber by its second height ; and each, or rather both, to the 
 primary pyramid diagram itself, as thus doubled by the cubit 
 number 25. Surely this is the perfection of design ! 
 
 But, moreover, the pyramid diagram fits as beautifully into 
 the dimensions of the coffer as it does into those of the king's 
 chamber, in which it is found. So, too, the dimensions of the 
 ante-chamber verify by the same consummate rule. 
 
 But, even more wonderful to relate, if we divide the dimen- 
 sions of the upper half of the double diagram of the king's 
 chamber by 4, and those of the lower half by 5, the ante-cham- 
 ber diagram results in the one case, and that of the coffer in 
 the other. 
 
 The accuracy of these results involves inches to their tens of 
 thousandths. They satisfy the various dimensions of the monu- 
 ment to the last reliable figures of modern measure, and they 
 advance beyond the possibility of stone-cutting and architecture 
 into the regions of pure mathematics of the highest order. To 
 obtain these results, and to verify their intimate relations to the 
 pyramid and to each other, months of the very closest mathe- 
 matical application have been necessary. But such work repays 
 the student. They convince him of a design whose scope 
 is simply perfect. Every application of the pyramid diagram 
 unlocks some new mystery in this wondrous edifice ; and each 
 new discovery suggests a chain of others, that runs out almost 
 hopelessly beyond the power of man to master in a lifetime. 
 
 THE ARCHITECTURE OF THE ANTE-CHAMBER. 
 
 I shall now illustrate how this diagram may be employed to 
 verify the architectural work of such a noble metric monument, 
 and how, perhaps, it may be yet employed by future generations 
 to repair and reconstruct the pyramid thus theoretically re- 
 newed. 
 
 We will confine our attention to the ante-chamber. 
 
 1 See Fig. 18, and its discussion, for a representation of what is meant by a double dia- 
 gram ; the square S and circle C having the magistral line 365.242 for their respective side 
 and diameter. 
 
138 METROLOGY. 
 
 The world to-day stands at the threshold to the ante-chamber 
 of modern civilization. As the Great Pyramid is a masterly 
 Masonic emblem of human experience, so, in the momentous 
 initiation of the race into the mysteries of more perfect life, 
 man is just leaving the great step at the southern end of the 
 Grand Gallery, a yard high, and a yard + a cubit wide, and 
 stoops at the narrow passage of Masonic trial he must yet 
 undergo ere he may emerge into the millennium symbolized 
 beyond. The whole world is upon the tiptoe of expectation ! 
 We are nearing an awful crisis ! The struggle coming, even 
 now at hand, is to be one between ignorance and knowledge, 
 wealth abused and poverty made desperate ; between the power 
 of evil and the strength of right. Men run to and fro, and 
 knowledge is increased. The times are rich in signs ; and, even 
 as man stoops at the narrow passage-way he yet must traverse, 
 the dread roar of the Cerberus of .Nihilism greets his ear, at 
 once from unenlightened Russia and enlightened England. 
 
 But the right must ultimately prevail. Convinced of this, let 
 us precede him into the symbolic ante-chamber, "-and fear no 
 evil." We may already read there, for our encouragement, 
 some of its many teachings. 
 
 The figure we shall now examine is a cross-section of the 
 ante-chamber. It is made by a vertical plane passing down east 
 and west just in front of the granite leaf that stretches across 
 this apartment. Upon this plane, you will notice, I have pro- 
 jected the leaf, or standard bar as it is sometimes designated. 
 This bar consists of two pieces of granite. The lower one is 
 carefully squared and finished, as is the upper block, save upon 
 its top surface, which was left by the architect in a purposely 
 irregular condition, as indicated in the figure. These blocks 
 are held in place by two grooves, cut downward from the tops 
 of the two wainscots represented in cross-section. The bottom 
 of the lower block rests upon the bottom of these grooves ; 
 while that of the upper one is flushed down upon the one below, 
 so as to form, as it were, a continuous piece of masonry. No 
 cement is used : the separateness of the blocks is thus clearly 
 defined. It is upon the upper block that the boss, already 
 described as the only ornament in the pyramid, occurs. This 
 boss is noticeably displaced from the centre of its block, both 
 
"THE SACRED CUBIT." 
 
 139 
 
 THE PYRAMID DIAGRAM AND A CROSS-SECTION OF THE ANTE-CHAMBER. 
 
 (ITS ARCHITECTURE VERIFIED.) 
 
 Fig. 16. 
 
140 METROLOGY. 
 
 as to vertical and horizontal arrangement. You will perceive 
 that the western, or right-hand, wainscot is somewhat higher 
 than the eastern, which is 103. 03" from top to floor : the height 
 of this western wainscot is 111.803". The thickness of the 
 wainscots is one foot each, and the maximum width of the 
 chamber (over them) is 65.2". The width between the wain- 
 scots, or under the granite leaf, is 41. 2". The length of the 
 chamber is exactly 116.26", 103.03" of which is floored with 
 granite, the remaining 13.23" being of limestone. The above 
 dimensions are all in " pyramid inches." 
 
 But such an enumeration of dimensions conveys very little 
 information. I shall give you a far better idea of this chamber 
 by stating that it has circummetric or TT proportions. We have 
 already noticed that the pyramid diagram satisfies these dimen- 
 sions at one-fourth of those of the king's chamber; or, since 
 that is one twenty-fifth of the pyramid itself, we shall explain 
 ourselves still better by calling attention to the fact, that in the 
 diagram, as applied to this chamber, every line comes out 
 exactly one-hundredth of the similar line in the primary pyra- 
 mid diagram. 
 
 To illustrate more clearly the accuracy with which this dia- 
 gram fits the ante-chamber, I have located it in mathematical 
 lines upon the cross-section. You will notice that it is inverted. 
 The vertex of the pyramid triangle is at the centre of the floor : 
 its base accurately divides the lower block of the granite leaf. 
 The centre of the square base of the model pyramid is at the 
 height of the eastern wainscot, while the ceiling of the chamber 
 is limited by that side of the square which lies opposite to the 
 base of the triangle. The horizontal diameter of the large cir- 
 cle, which limits the floor of the chamber, is also exactly equal 
 to the whole north and*south length of this floor, 116.26"; while 
 its granite length of 103.03" is equal to the height of the east- 
 ern wainscot, and to the magistral line of the diagram. More- 
 over, as though to intensify the coincidence of this diagram with 
 the marked architectural and geometric features of the cross- 
 section, this magistral line of 103.03" exactly fo'sects the equal 
 height of the eastern wainscot. The difference, 13.23", between 
 the length of the whole (i 16.26") of the floor of the ante-cham- 
 ber, and that part (103.03") which is of granite, is also shown in 
 
"THE SACRED CUBIT." 141 
 
 the diagram to be the difference of the diameters of the two 
 circles, which form such important elements in its sequence. 
 
 A longitudinal section (i.e., north and south) of this chamber 
 results in just as beautiful an application of the pyramid diagram. 
 
 It is manifest that such a diagram may be used to verify the 
 finish of such a chamber. The process is geometric ; it is 
 architectural ; it is more, it is Masonic. It is one of the grand- 
 est diagrams upon the Freemason's trestle-board ! It affords 
 a theoretic standard by means of which we may renew this 
 building, or erect another as free from error as instruments can 
 work, and measures test the accuracy of our efforts. It is be- 
 yond conception that even boasted modern skill could better 
 the solid granite facts thus built into this remarkable room. In 
 spite of fifty centuries of age, its noble architecture is still 
 square to the rule, and true to the plummet ; and every mystic 
 line within it literally squares a circle ! 
 
 THE BOSS AND THE DIAGRAM. 
 
 Let us now examine the boss upon the granite leaf. 
 
 To the boss upon the granite leaf we have already frequently 
 alluded. It is a small raised ornament upon its upper block. 
 It is not located at the centre of this block, but seems to be 
 purposely displaced therefrom a short distance to the west and 
 downward. This displacement amounts to exactly one inch in 
 westing. It is, in fact, equal to one of the grand earth-com- 
 mensuric standard units of the polar axis. 
 
 As though to intensify this 1 reference to the unit inch, the 
 height, or relief, of the boss is also an exact inch. Plaster-of- 
 Paris casts of this feature of the ante-chamber have frequently 
 been made, and taken to England, where they have been studied 
 with some remarkable results. 
 
 The following are some of the more noticeable discoveries 
 that have thus far been made with reference to it. 
 
 Its cubical contents, or volume, is one pint ; and, as this pint 
 weighs an even pound when filled with water, we are forced to 
 regard the boss as an epitome of unit measures. The inch, the 
 pint, the pound, the three working-units of length, capacity, 
 and weight, are thus united in this wonderful little projection. 
 
 Now, the shape of the boss is equally remarkable. Its base 
 
142 METROLOGY. 
 
 is a circular arc of rather more compass than a semicircle. It 
 is limited at its lower edge by a horizontal chord which is ex- 
 actly 5 inches long, or a span ; i.e., one-fifth of a cubit. Nor is 
 this all. 
 
 The radius of the circle thus limited by the chord of 5 inches 
 is 2.820-f inches. Mathematically this is, perhaps, the most 
 wonderful dimension of the boss ; for a circle whose radius is 
 2.820+ inches, has the same area as a square whose side is 5 
 inches ! Thus the boss is at once both a practical example and 
 symbol of the circle squared. 
 
 No more beautiful geometric representation of this celebrated 
 problem could be conceived of, than to cut a circle by that par- 
 ticular chord which is the side of its equal square. This sym- 
 bol appears to have been, not only purely original with the 
 architect of the Great Pyramid, but to have been monopolized 
 by the boss from that day to ours, the one of its discovery. 
 
 Let us now apply the pyramid diagram to this boss. 
 
 Let its projecting base of 5 inches, or a span, be the base of 
 our model pyramid and of its vertical triangular section : with 
 the radius of the boss describe the equal area circle, and it is 
 manifest that the whole diagram thence results in its most 
 absolute simplicity, and that it involves the height, length of 
 circumference, radius, chord, and diameter, or breadth of the 
 boss, in their actual mathematical positions. The boss, there- 
 fore, may be as beautifully renewed by this mysterious diagram 
 as we have already shown that the ante-chamber may be. 
 
 While studying the position of the boss, Mr. St. John Vin- 
 cent Day, an English civil engineer, discovered that the 
 distance from its centre to the end of the leaf itself, in its well- 
 cut groove of the eastern wainscot, was exactly one sacred 
 cubit, or 25 inches. Thus, as plainly as a stone can speak, this 
 granite leaf tells us that 5 inches make a span, and 5 spans 
 make a cubit. 1 
 
 In 1875 another practical engineer, Mr. Waynman Dixon, 
 writing to Professor Piazzi Smyth concerning the granite leaf, 
 makes use of the following language : " The more I see of this 
 remarkable stone, or leaf, the more I am convinced that the 
 
 1 See Fig. 15. 
 
THE SACRED CUBIT: 
 
 143 
 
 
 \ 
 
 
 THE PYRAMID DIAGRAM AND "THE BOSS. 
 
 (ITS IDEAL SHAPB RESTORED.) 
 
 Fig. 10. 
 
144 METROLOGY. 
 
 upper irregular part is in its original condition, and not broken 
 away by specimen-mongers or Arabs." 
 
 Now, what could have been the intention of the architect in 
 leaving this upper surface so irregular that its height cannot be 
 accurately measured ? 
 
 This question was answered by the Rev. C. W. Hickson in 
 1877, with a rigid mathematical demonstration. The length 
 and width of the leaf can be accurately obtained, because they 
 are clear and square cut. They are respectively 41.2 and 15.7 
 inches. The only way, however, that we can arrive at the 
 height of the leaf, is by taking a mean : this is the tVue method, 
 by modern rules, of mensuration. Now, the maximum height 
 is 5 1.3 -(-, and the minimum height is 45.8 +> the mean of which 
 two is 48.55 +, possibly inclining to 48.57. Multiplying these 
 three dimensions together, to obtain the volume of the leaf, its 
 cubical contents comes out in the wonderful sequence of figures 
 known as the circummetric ratio. It is 31415.9+, etc., cubic 
 inches, or 10,000 times TT cubic inches. This is a most stupen- 
 dous repetition -of the TT references that we have already found 
 to exist in the shape and dimensions of the boss itself. 
 
 The contents of the lower block of the granite leaf can be 
 most accurately determined, because of its perfectly rectangular 
 finish. This is found to be an even fourth of the contents of 
 the coffer; or since the latter is a "caldron," or four English 
 "quarters," the contents of the lower block of the leaf is one 
 such English, Anglo-Saxon, or, rather, pyramid, "quarter." 
 
 Who can fail to be astonished at such remarkable discoveries ? 
 And who can doubt the importance of a system of metrology, 
 for the preservation of which such infinite pains seem to have 
 been taken. 
 
 But we are by no means through with the boss upon the granite 
 leaf. I shall now ask you to follow me in a rapid glance over 
 some other of my own discoveries in this neighborhood. 
 
 It will probably have been noticed, that, in all the representa- 
 tions you have seen of the " pyramid diagram," there has been 
 one line to which, as yet, no reference has been made. It is. 
 the line XY. This is the right line joining the two points in 
 which the circle of equal area to the pyramid base intersects 
 the square which represents that base. Its length is, of course, 
 
"THE SACRED CUBIT." 145 
 
 equal to the side of the square, or to the base of the pyramid 
 triangle. 
 
 Now, this line is one of the most important ones in the entire 
 diagram ; though I have not the slightest doubt but that its im- 
 portance, and its relation to the diagram, have entirely escaped 
 the notice of every one present. It certainly escaped my own 
 for many months. So completely did the importance of this 
 line escape me, that I do not believe I was even led to construct 
 it until, in my study of the diagram, I had followed its applica- 
 tions down throughout the pyramid to the very boss upon its 
 granite leaf, and found in it its actual ideal. When I finally 
 came to apply this diagram to the boss, I was forced to draw 
 this line, because it was one of its most expressive features ; 
 namely, its chord. In such a diagram, complicated by its very 
 nature, the utmost simplicity consistent with the ends in view 
 was, of course, requisite ; and, had it not been for the necessary 
 introduction of this new line ^YFwhen we arrived at the boss, 
 it would probably have been undrawn to this day. 
 
 But even such a forced construction did not at once point 
 out its supreme importance. Its discovery was entirely due to 
 an accident, but to a most natural one, and to one that is beau- 
 tifully in keeping with those that have led to the discovery of 
 almost every secret that has been wrenched from the Great 
 Pyramid. 
 
 THE WORKMAN'S RULE FOR SQUARING THE CIRCLE. 
 
 I had no sooner completed the figure showing the application 
 of the " pyramid diagram " to the boss, than the fact of the five- 
 inch length of its upper chord suggested to me the idea of 
 dividing the chord off into its five several parts or inches. 
 This I did, and continued the vertical lines of division down- 
 ward across the square. It then most naturally occurred to 
 me, as a sequence, to redivide it by a set of horizontal lines, so 
 as to represent each of the 25 square inches which enter into 
 the square of the chord itself. 
 
 While constructing this 25-part (or cubit) division, I noticed 
 with surprise, that, so close as I could draw the figure at its 
 own actual scale, the first horizontal division fell coincidently 
 with the chord of the boss itself. It at once struck me that this 
 
146 METROLOGY. 
 
 fact afforded a practical method of squaring" the circle. I shall 
 henceforth denominate this method the " Workman's Pyramid 
 Rule." 
 
 As shown in the "Workman's Diagram," you will notice that 
 the circle of equal area to the pyramid base, passes not only 
 through the vertex of the pyramid triangle, but also through 
 the centre of the square base. Now, the new discovery re- 
 vealed the additional fact, that it also practically passed through 
 two other points, X and F, which lay an inch, or one-fifth, 
 downward upon each of the vertical sides of this square. 
 
 Now, but one circle can be passed through three given 
 points, X, F, and N; and since in the diagram this circle is the 
 equal-area circle, it follows, that to the same degree, in every 
 square, an equal-area circle may be passed through its centre 
 and two other points situated one-fifth of the way down two of 
 its opposite sides. 
 
 Any line can be divided into five equal parts by a most 
 simple geometric construction. No one can study the boss 
 upon the granite leaf without having its one-fifth (or, in its case, 
 " an inch ") division sooner or later suggested to his mind. 
 The " Workman's Diagram " shows that this fraction (one-fifth) 
 is not only the lowest term of a most useful approximation, 
 but one whose error cannot be appreciated by the ordinary 
 workman. 
 
 But this error, slight as it is without any correction, can be 
 made to vanish in a most simple and practical manner, to a 
 point that is far beyond the power of recovery by means of the 
 best micrometer yet constructed. 
 
 This is of enough importance to explain at greater length. 
 
 The radius of the boss, calculated by the best modern ?r value, 
 so that its circle shall have the same area as the square of its 
 five-inch chord, comes out 2.820947-}-, etc., inches. 
 
 If, however, this radius be geometrically constructed by means 
 of the "pyramid rule" just given, it will measure 2.833333, etc -> 
 inches, ad infinitum. The difference between these two values 
 is but .012(385) inches. 
 
 This difference is actually less than the width of the mathe- 
 matical line that an architect would have to make in drawing a 
 working-plan of the boss, let him execute it with the utmost care. 
 
" THE SACRED CUBIT? 147 
 
 Until mathematical instruments have been so perfected that 
 distinct working diagrams can be drawn in lines twice as fine 
 as jet-black hair, we need feel no concern at errors of construc- 
 tion so minute. 
 
 The standard mathematical breadth of a human hair is one 
 forty-eighth of an inch. This is about the width of the least 
 practicable line of construction that could be advantageously 
 employed in making a suitable diagram of the boss at its own 
 actual scale. And yet such a line one that had only a hair's- 
 breadth would completely cover all the error introduced by 
 the proposed construction. Its inner edge would fall about as 
 much within the true circle as its outer edge (drawn coincident 
 with the line of construction) would fall beyond it. The centre 
 of a hair-line would, in fact, just about indicate the actual posi- 
 tion of the true circle. 
 
 The absolute width of the belt of error that surrounds this 
 construction is about -^ of a unit (the inch in this case) ; i.e., 
 it is i -^ (Q) 2 . If, therefore, the circumference, to be con- 
 structed by the "pyramid rule," be made -fa of the unit in 
 width, its inner edge, just at the vanishing-point, will indicate 
 the absolute position of the required line to a degree of accu- 
 racy whose error the microscope could hardly grasp (g-gViiy)- 
 
 To illustrate better the practical application of this workman's 
 "pyramid rule" for squaring the circle, I shall take the largest 
 diagram that I can conveniently handle. The cloth before you 
 is 90 inches square, and is short by about 30 inches of what 
 I need to complete the whole pyramid diagram. (Fig. 17 is 
 a reproduction at -^ of the diagram that accompanied this 
 lecture.) 
 
 Let AB represent the side of a square 10 times as long as 
 the chord of the boss. It is 50 inches in length, or two cubits. 
 This square contains 2,500 square inches. We wish to obtain 
 an equivalent circle ; that is, one whose area shall contain 2,500 
 circular inches. 
 
 By the pyramid rule, divide the side CD into 5 equal or 
 "unit" parts; lay off DX and CY each equal to one such 
 " unit ; " through XE and Y pass as fine a circumferential line 
 as possible, and consider it the outer edge of the bordering. 
 Make the width of the mathematical line that is to represent 
 
148 
 
 METROLOGY. 
 
 (38765432. 10 
 
 THE WORKMAN'S RULE FOR "SQUARING THE CIRCLE." 
 Fig. 17, 
 
"THE SACRED CUBIT." 149 
 
 this circumference, or border, equal to unity, 4- 9 2 ; i.e., equal 
 to Cc' t or, in this case, equal to 10" '-f- 81 .12345", etc - : tnen 
 will the inner edge of the border, thus constructed, limit the 
 circular area required, and that area be correct to within less 
 than .of, a square inch, or equal to 3-5 j^ of this desired 
 area. One could buy gold and precious stones by such a rule 
 without fear of appreciable loss. 
 
 The radius R', passing the points XY and E, and limiting 
 the outer edge of the border, is 28.33, etc - \ tne radius R',, t cor- 
 rected by -, and limiting inner edge of the border, is 28.209877 ; 
 
 thus J?', depending upon ir values, is 28. 209479.' 
 
 But, furthermore, since CDH is manifestly a pyramid triangle, 
 all of the properties and relations of the pyramid diagram result 
 from one construction. Thus, 
 
 as just demonstrated ; 
 
 2irHG = perimeter AB CD 4 CD 
 
 TrtfG 2 = (2FY) 2 
 and 
 
 HDC= FY 2 = ir(\Gffy, etc. 
 
 This is a practical method of constructing equivalent squares 
 and circles, either as to area or perimeters, the equal of which 
 has hitherto been unknown to modern mathematicians, and the 
 discovery of which certainly enhances the interest centred in 
 the boss upon the granite leaf or " standard bar." 
 
 But, before finally turning from this subject, I wish to call 
 attention to another phase of the method I have given. Its 
 discovery was even more purely accidental than that by which 
 we have been able to locate the position of the equal-area cir- 
 cle. It depends upon the same method, and is most beautifully 
 in accord therewith. Before completing the diagram before 
 you, it seemed desirable to place upon it, though merely for 
 purposes of comparison, the circle of equal perimeter to the given 
 square. The cioth, being too limited to place this circle in its 
 
 1 In Fig. 17, R', should 28.20987, and 1^ 28.20947, the values having become mis- 
 placed on the engraving. 
 
150 METROLOGY. 
 
 proper place, as shown in the regular pyramid diagrams, I was 
 forced to draw it (with, of course, the height of the pyramid 
 triangle as a radius, but) with the centre of its base as its centre 
 instead of the vertex of the triangle, as in the most natural 
 method. Upon describing this circle, I was again surprised to 
 see that it also passed through two noticeable and correspond- 
 ing points upon the sides of the given square. These points, 
 you will perceive, are situated as far below those which mark 
 the intersection of the equal-area circle with the given square, 
 as the latter are below the angles of the square. 
 
 In other words, the circle of equal circumference practically 
 passes through two points situated one-fifth or a unit distance 
 below those which locate the circle of equal area. 
 
 Thus the same principle holds good. Comparison with the 
 true TT value shows that the error in this latter construction is 
 also of no moment, but that, when desirable, it may be made to 
 vanish out of microscopic sight by applying the several indicated 
 corrections. 
 
 Thus, by simply deducting the primary correction of ^ 
 
 = of unity, the radius becomes 31.83044 +, which is only 
 
 .ooo54 // too small. This radius gives a value for the circum- 
 ference of 199.9965", or one that is short of the true one 
 (200") by but .0035", or .00035 of unity; or, finally, one that 
 is short of the whole circumference sought by but -sy^y, or 
 about e^-Joo" thereof. The simplicity of this graphic method of 
 practically squaring the circle, its entire originality with the 
 Great Pyramid, and its direct and logical deduction from its 
 rediscovered and most important diagram, are certainly matters 
 of deep interest to all students now interested in pyramid 
 metrology. 
 
 UNITY REFERENCES OF THE BOSS. 
 
 Before turning from our special mathematical consideration 
 of the boss upon the standard bar, I must call your attention 
 to a most remarkable addition to the unity references already 
 noted as centring in and around it. We have found in it the 
 inch, the pint, and the pound : we have also seen how beauti- 
 fully it satisfies what has been called "the pyramid method." 
 
"THE SACRED CUBIT." 151 
 
 In the present lecture we have likewise seen in it an ideal 
 realization of the "pyramid diagram," and such a suggestive 
 one as to afford us a practical solution a workman's solu- 
 tion of the problem of problems. 
 
 If the pyramid itself be a common multiple of earth-com- 
 mensuric references, then especially so is its safely guarded 
 boss. From every stand-point it is wonderful 
 
 It was while studying the mutual relations of its inch height, 
 its five-inch chord, and its special radius, that I noticed its indi- 
 cated circumference was none the less worthy of attention, and 
 that it could be factored into a most astonishing series of circu- 
 lar unit-references. Thus, 
 
 = 17-7245 +> etc -> 
 
 = 2. (3.14159, etc.) (2.82094, etc.), 
 
 = 2. ( 3 .i4i,etc.)(. 5 64i)(5)(i=^), 
 
 = (6.283, etc.) (.5641) (#= r)(.i59i5)(3-i4i5> etc.)io, 
 
 = 2 (6.283, etc.) (.5641) (H= OCiSQ^) (3-i4i5> etc -)5- 
 
 Or, translating these apparently unmeaning figures into words, 
 we have, as follows : 
 
 2irR 17.7245 -f, etc. = the circumference of the boss= 2 (circ. rad - x ) 
 (rad. area - I )(boss hei s ht - I )(rad. circ -- I )(area rad -- 1 ) chord of the 
 boss. 
 
 That is, twice the product of five factors that are intimately 
 related to unity, by five units or the chord of the boss itself, is 
 equal to the circumference of the boss. 
 
 GENERAL FORMULAS AND THE PYRAMID DIAGRAM. 
 
 The same set of relations which I have lately pointed out in 
 the "International Standard" for September, 1883, as existing 
 between the lines and dimensions of the " Latimer diagram," as 
 I there completed it by the addition of the circle whose diameter 
 is 8 1, are found to run through the pyramid diagram as here 
 given. Thus, in the double diagram, let S, be first square, C, 
 the first circle, 5 the second square, C the second circle, S 1 the 
 third square, and C the third circle. Then these relations are 
 as follows : 
 
152 
 
 METROLOGY. 
 
 THE DOUBLE DIAGRAM. 
 
 (KING'S CHAMBER, ETC.) 
 
 Fig. 18. 
 
"THE SACRED CUBIT." 153 
 
 I. The diameter of the first circle is the side of the second 
 square, that of the second circle the side of the third square, 
 and generally the diameter of any circle (say, the nth) is equal 
 to the side of the next (i.e., the n + first) square. 
 
 II. The perimeter of the first square equals the circumfer- 
 ence of the second circle, that of the second square equals the 
 circumference of the third circle, and so on ; so that, generally, 
 the perimeter of the nth square equals the circumference of the 
 ;/ -f- first circle. 
 
 III. The area of the first square equals that of the first circle,, 
 that of the ;/th square equals that of the ;/th circle. 
 
 IV. A pyramid triangle may be based upon the diameter of 
 any circle, as, for instance, that of the first, and have for its 
 height the radius of the second therefrom. Thus, the diameter 
 of the first circle ("/) and the radius of the third circle (C) give 
 us a pyramid triangle, and the diameter of the nth, and the 
 radius of the ;/ + second, another. 
 
 From the foregoing rules, it follows, that since to the diame- 
 ter of the first circle, there can be given any numerical value 
 whatever from o to infinity, such a sequence of squares and 
 circles may be made to represent the nucleus of any desirable 
 series of pyramid triangles ; and hence, that there is a series 
 which corresponds to the base side of the Great Pyramid as the 
 side of a second square and the diameter of a first circle. 
 
 Now, it is a fact that the height of this pyramid, so far as we 
 can now measure and estimate, be it in the terms of any linear 
 measure whatsoever, corresponds to the radius of a third circle 
 of such a series, while its base corresponds to the side of the 
 connected second square : hence we are justified in believing 
 that the Great Pyramid of Gizeh was built as an intentional 
 exponent of such a geometrical series, rather than that it is a 
 gigantic accident. To this conclusion we are forced without 
 reference to any knowledge of the absolute unit of measure 
 actually employed by its builders. 
 
 From a study, however, of these relations, as recast into what: 
 we have here termed, par excellence, the " pyramid diagram," 
 and from a further study of the wonderful mathematical con- 
 nections based upon the 25-inch, or sacred, cubit, we are forced 
 to the conclusion that this particular measure, based upon the 
 
154 METROLOGY. 
 
 earth-commensuric unit inch, and so extremely close to our own 
 Anglo-Saxon unit, is the true metron of its builders. 
 
 Before leaving this subject, therefore, let us so generalize the 
 relations we have discovered, as to determine, in the terms of 
 some common quantity, the whole series of inter-related values. 
 
 Denote the square base of the pyramid by S, the pyramid 
 triangle by T, the equal-perimeter circle by C, the square whose 
 area is equivalent to this latter by S', the equal-area circle by 
 C', and the circle on the altitude of T by C", and designate, 
 by corresponding notation, the elementary parts thereof. Let, 
 also, C, represent the circle upon the side of 5 as a diameter, 
 C the circle upon R' as a diameter, T, the pyramid triangle of 
 equal area to C,,, and 5, the square whose area is equal to that 
 of C, y etc. 
 
 These circles, squares, and triangles form the sequence known 
 as the double pyramid diagram. Commencing at S, they run 
 upward through the links of the upper half of the diagram so 
 far as C" . Commencing, also, at the same point 5, they run 
 downward through those of the lower half of the same diagram. 
 The square 5 and its equivalent circle C' are common to each 
 half of the double diagram. 
 
 The preservation of this system of notation will greatly facili- 
 tate the ready designation of any particular square, circle, or 
 triangle in this confusing series. We shall now enumerate the 
 formulas governing the functions of these several mutually 
 related figures, commencing with those of S. 
 
 S. Denote the side of S by 2x. Then 
 S= 2x; area S = ^x 2 ; perim. S= 8x; diag. 5= x.2^2. 
 
 C. 
 
 S'. s' = / P' = / area'= ; diag.' = -- 
 
 V^TT V^ V 77 " 
 
 2X AX 
 
 C. R' = -=; D' = 7=; 2*1? 
 
 yTT y 7T 
 
THE SACRED CUBITS 155 
 
 C". R"= ; &' = ^; 27rR" = 4 x; *(%')* = 4^. 
 
 7T 7T 
 
 C. ^?/ = JP/ /?/ = 2#/ 2-rrR, = 2irx; 7r(/?,) 2 = irx 2 . 
 C,,. R,, = -; D,, = ~; 27rR,, = 
 
 \7T \7T 
 
 T,. Base = #V^/ alt. = ^ ; P f = 
 
 V*- 
 
 ,&. 6 1 / = x\7r; P, = 4x\7r; area/ = irx 2 ; diag., = x\27r. 
 
 An application of the foregoing formulas to any value of x 
 will result in the determination of the corresponding numerical 
 values of the entire series resulting therefrom. 
 
 Thus, if such a series of values be constructed upon the 
 earth-commensuric unit inch as the side of S,, there will result 
 what may be properly termed the "unit diagram." Now, if we 
 redraw the diagram at a scale five times greater, the " span or 
 boss diagram " will result. And, if this latter series of values be 
 again multiplied by five, there will result the "cubit diagram." 
 
 The "pyramid diagram" itself is determined from the "cubit 
 diagram " by multiplying all of its lines by the year number 
 365. 242 rb. This latter diagram, multiplied by 10,000,000, gives 
 us, so to speak, the "terrestrial diagram;" and undoubtedly, 
 as the series runs upward, we may pass through "solar" and 
 "sidereal" diagrams in our progress towards the ultimate and 
 "universal diagram," whose terms of infinite scope span all 
 the borders of creation. Descending again, however, from 
 the numerical values of the "pyramid diagram," we find, 
 that, taking one twenty-fifth thereof (the cubit number), we 
 are landed at a diagram which locates the dimensions of the 
 king's chamber ; and still further, that one-fourth thereof, or 
 one-hundredth of the "pyramid diagram," yields us that of the 
 ante-chamber. In a similar manner,' one-fifth of the king's 
 chamber gives the "coffer diagram," etc. It seems inevitable 
 that the queen's chamber, the subterranean chamber, and the 
 yet to be discovered "royal arch-chamber" of the monument, 
 will yield similar diagrams, with multipliers (perhaps 52.17-}-, 
 the year of weeks ; 7 cubits = 175", * values, etc.) just as ex- 
 pressive, and of just as important -value to metrology. 
 
156 METROLOGY. 
 
 From another stand-point, since 116.26 is the ratio of TT : Y = 
 3.14159+, etc. :$6$.242, etc., and since this is the governing 
 dimension of the ante-chamber, this latter may have its diagram 
 founded directly upon ?r values themselves. Thus, the " pyra- 
 mid diagram" in cubits, divided by TT, gives us the "ante- 
 chamber diagram," in inches, etc. The very passages individu- 
 ally, and as a scheme collectively, will undoubtedly be likewise 
 found to yield themselves to such a diagram : indeed, I have 
 already determined some of them ; though space forbids their 
 enumeration here. 
 
 THE CUBIT, THE TWENTY-FOUR INCH GAUGE, AND 
 
 THE YARD. 
 
 The sacred cubit of the Jews was set aside for special pur- 
 poses. With it they proportioned the tabernacle and the 
 temple. It had peculiar and important uses in the mystic rites 
 of the sanctuary. 
 
 Whether, however, this sacred cubit was ever put into the 
 hands of the common workman has long been a subject of con- 
 troversy. It is probable that the laborer, at his ordinary task, 
 did not use this sacred length : indeed, we know that the Israel- 
 ites employed, in their every-day affairs, a cubit of far different 
 length. This is sometimes spoken of as "the cubit of a man ;" 
 it was a span shorter than the sacred cubit, and was closely 
 assimilated, if not exactly equal, to the cubits of Egypt and the 
 surrounding nations. 
 
 There were, in fact, several such every-day cubits in use 
 among the Jews ; and, included in this number, was one in par- 
 ticular, that is by far the most important. This cubit is the 
 origin of the modern "foot," which is inherited by almost every 
 northern European nation from their ancestors. It was differ- 
 ent from every Egyptian cubit, and is still the true workman's 
 measure. 
 
 It is best known in our day as the two-foot rule, or the cele- 
 brated " 24-inch gauge " that has been so carefully handed down 
 from pyramid and temple time-s by the Freemasons, and is now 
 to-day in the hands of every operative mason in the world. 
 
 It is of that special length which is still found to be most 
 convenient by masons and carpenters, and it is as beautifully 
 
"THE SACRED CUBIT: 
 
 157 
 
 O 
 
158 METROLOGY. 
 
 indicated upon the granite leaf of the Great Pyramid as is the 
 sacred cubit itself. 
 
 In obtaining the latter, we measured from the centre of the 
 mystic boss, and penetrated into the groove of the wainscot to 
 the very eastern end of the leaf. The sacred cubit was, in fact, 
 hidden from ordinary sight. The 24-inch guage, however, is 
 exposed to the full view of all who will measure the leaf, and 
 live by its masonic teachings. 
 
 No sooner was I convinced of the rightful claims of this 
 granite leaf to another and more expressive designation, that of 
 the " standard bar," than I went deliberately to work to find 
 upon it this 24-inch, or workman's, gauge. 
 
 The very simplest investigation of the dimensions exposed to 
 view, showed me, that, from the western extremity of the chord 
 of the boss to the face of the eastern wainscot, was exactly two 
 feet, or the length of the masonic standard. 
 
 This workman's gauge is thus most conveniently located upon 
 the standard bar for measure, for scientific subdivision, and for 
 comparison with the wonderful sacred cubit that rests above, 
 and upon it. 
 
 But another Anglo-Saxon measure of note is now brought 
 prominently to our view : it is the yard itself. The thickness 
 of the eastern wainscot is just 12 inches, or the "common 
 foot : " hence from the wall through the wainscot, and to the 
 western extremity of the lower chord of the boss, is i' -}- 2! = 3' 
 = 36" the Anglo-Saxon yard. 
 
 From its remarkable commensurability with the great magis- 
 tral line of the whole earth, the polar axis, the sacred cubit 
 lays direct claims to precedence over all other standards of 
 linear measure. We have already alluded to this quite particu- 
 larly, and have, throughout our investigations of the pyramid, 
 seen how clearly the sacred cubit and its " unit inch " were 
 ruling elements in the combinations that conceal its mysteries. 
 The earth-commensuric "inch" must be the integral part of 
 every standard of linear measure raised against the monument 
 of Egypt, if we wish to discover there aught besides its symme- 
 try. To the " metre " it is silent, and to the centimetre a 
 hundred times as dumb. 
 
 By means, however, of the earth-commensurability of the 
 
"THE SACRED CUBIT." 159 
 
 inch and sacred cubit, and of their axial reference, we can pass 
 understandingly beyond our world, and measure the radius of its 
 orbit, and the various other distances of the solar system, in 
 terms of the polar radius. Indeed, mounting a step higher, the 
 astronomer estimates the distance from our system, or its 
 centre, the sun, to the fixed stars, and uses for this purpose 
 the radius of the earth's orbit. 
 
 This most natural growth of standard out of standard may go 
 on forever throughout the vast and illimitable regions of space, 
 without ever losing its earth reference. The axis, the cubit, 
 the inch, accompany us in each step outwards, go we how far so 
 ever ; and all our information can be harmonized into a vast sys- 
 tem of linear measure, that, commencing among the little inches 
 of ordinary life, at last span out the whole creation with scale 
 indeed stupendous, but never quite beyond our comprehension. 
 
 From what we have seen to-night of the pyramid diagram, 
 and its perpetual harmonious repetition throughout the building, 
 who, then, can doubt but that the more gigantic harmonies of 
 the universe are just as beautifully balanced, and that after we 
 have expressed the harmonies of our own solar system into a 
 pertect geometric figure, or solar diagram, so to speak, we may 
 thereafter, by the careful measure of the greater radii beyond 
 us, magistral lines of other, but still similar, diagrams, 
 learn also all about the universe that lies within the reach of 
 our invention and discovery ? 
 
 But, over and above its eart/i-commensuric claims, the sacred 
 cubit possesses other properties that mark it out as unique 
 among all the standards of linear measure known to man. 
 
 As our time is drawing to a close, I can but barely enumer- 
 ate a few of the more wonderful facts concerning the cubit as 
 a standard, and which I have been fortunate enough to discover 
 in my pyramid studies. 
 
 THE FIVE-PART DIVISION OF THE CUBIT. 
 
 From its susceptibility of a five-part division (i.e., into five 
 spans), the cubit takes to itself all of the circular properties we 
 have just noticed in the boss* A " pyramid diagram," con- 
 structed upon a chord that is five spans, instead of five inches, 
 
i6o 
 
 METROLOGY. 
 
" THE SACRED CUBIT? 161 
 
 in length, will, of course, possess equivalent properties, and 
 show forth the same absolute ratios. This is the first point of 
 peculiar interest to which I ask your attention. I shall recur 
 to it more particularly later on. 
 
 Let us now separate one of these end spans from the other 
 four by a marked line of division. The cubit is thus divided 
 into two very important parts ; the one containing twenty unit 
 inches, and the other five of them. 
 
 Now, it is one of the noticeable features of this division, that 
 each such part contains a whole number of inches, to wit, five 
 and twenty ; and, furthermore, that each part may, in turn, be 
 similarly ^subdivided into five smaller parts, and that this 
 redivision may still be effected without the rupture of the unit. 
 
 By this process the "span," or smaller part of the cubit, is 
 reduced to its ultimate unit, the inch. 
 
 In this respect the sacred cubit differs essentially from 
 the foot, the 24-inch gauge, the yard, etc. None of these 
 standards can be subdivided into fifths without destroying* the 
 integrity of some of their earth-commensuric units ; and, unless 
 the standard be divided into fifths, it cannot be employed as the 
 basis of circummetric scales. Take, for instance, the yard : like 
 the cubit, it numbers a perfect square 36" = 6 X 6". Now, if 
 the side of a square be 6 inches long, its area will be 36 inches, 
 and its perimeter 4 X 6, or 24 inches. But 24 + 6 30" ; so 
 that there remains an unused surplus of six inches, by which 
 the yard is dissimilar in construction to the cubit. 
 
 The French metre must be entirely prohibited from entering 
 into such a comparison. As we have already shown, it is not 
 truly earth-commensuric, is erroneous in principle and calcula- 
 tion, and certainly is not a scientific standard for international 
 intercourse. Moreover, it is, in reality, the unit itself of the 
 metric system ; and the method does not propose to divide a 
 unit, but simply to obtain the latter by the subdivision, into 
 fifths, of the Standard. 
 
 The metre is decimal in its construction : so, too, may any 
 unit be. Of course the division by fifths may proceed through 
 any decimal system : but, if we admit into the comparison a 
 standard whose essence is this decimal system, then all stand- 
 ards whatsoever are admitted through the same door ; for they 
 can all be made thus decimal. 
 
1 62 METROLOGY. 
 
 Moreover, to illustrate this important characteristic of the 
 cubit even more plainly, suppose it also to be decimally divided 
 into 100 equal parts. A fifth of these parts, or -ffa of the 
 whole cubit, is still an even number of inches; i.e., 5 inches: 
 and one-fifth of this, or of -^Q, is also an even number of inches, 
 or the unit itself. 
 
 With the metre, however, it is- different, even in its decimal 
 relation. Its length is 100 centimetres. This is a perfect 
 square : but its fifth is 20 centimetres, the square of which 
 (400) does not give the value of the metre ; though the perimeter 
 of this square (80) equals the cut-off portion. 
 
 The cubit is, in fact, as purely decimal as any standard can 
 be, but with the overwhelming advantage over all others, that 
 every simple fractional part thereof, whether decimal or com- 
 mon, is truly earth-commensuric. It has not only a direct ratio 
 to the axial radius of the earth, but through it to the universe 
 beyond. 
 
 At the first glance it seems as though the 45-inch ell of cloth 
 measure likewise possesses this 5-part subdivision without the 
 rupture of the unit. The ell is a yard and a quarter, or five- 
 quarters of a yard, long, and is thus by construction divisible 
 into 5 equal parts. Each of these parts is 9 inches long. 
 But the process so well begun stops here. The " quarter" is 
 not the unit of the ell : its unit is a " nail," 2 j inches in length. 
 Now, one-fifth of 9 inches is not 2 j inches, but i| inches ; and 
 thus the unit inch is ruptured fruitlessly. Moreover, had the 
 unit " nail " itself resulted, it would have been only a broken 
 number of inches, and not suitable for earth-commensuric pur- 
 poses. The rule, in fact, universally fails except in the 25-inch 
 cubit. Thus the cubit is unique in its five-part division, re- 
 peated until its own unit the earth-commensuric inch 
 results. 
 
 THE CUBIT, THE SQUARE, AND THE CIRCLE. 
 
 Let us now see into what some of these properties develop 
 when thus united in a single standard of linear measure. 
 
 From the stand-point to which we have attained, it should be 
 clear without any further investigation whatever, that, as the 
 cubit has been shown to be so important a line in the pyramid 
 
" THE SACKED CUBIT: 
 
 163 
 
164 METROLOGY. 
 
 diagram, it must possess some remarkable powers for measuring 
 squares and circles, and be especially fitted for subdivision into 
 scales adapted to such measurements ; and so it is. 
 
 By drawing a line GH across the cubit, such as shall separate 
 its right-hand " span " of 5 inches from the remaining 20 inches, 
 a most noteworthy dividing-line is obtained. 
 
 The number 5, the central one of the nine digits, was pecul- 
 iarly hateful to the Egyptians. Sir Gardner Wilkinson says, 
 that, to this day, it is regarded as "the evil number" by the 
 inhabitants of that country. So superstitious are they of this 
 unlucky number, that it is marked by a zero (o) on their 
 watches ! 
 
 It was, however, a most sacred number to the shepherd 
 kings. These ancient wise men symbolized its importance over 
 and over again, throughout the 5-sided, 5-angled, and wonder- 
 fully 5 -proportioned, monument they left in Egypt. So, too, the 
 number 5 was of the greatest sacred import to their later kins- 
 people, the Hebrews. 
 
 The whole Jewish polity religious and civil was em- 
 bodied in the five books of Moses. This book of books was 
 kept separate from all other sacred writings, and was even 
 known by the significant name of the " Pentateuch." When 
 Israel left the land of bondage, they marched out five abreast ; 
 and, as though to intensify still more their independence of 
 Egyptian superstitions and influences, the Bible tells us they 
 went out "with high hands," themselves symbols of five, raised 
 in the very faces of their former taskmasters. 
 
 Now, the cubit number 25 is the perfect square of this 
 sacred number, 5. The dividing line GH we have just drawn 
 through the cubit, cuts off to the right 5 inches, or the side, 
 and to the left 20 inches, or the perimeter, of this very square ; 
 while its area is indicated by the number of inches in the whole 
 cubit. These two parts of the cubit, through their equivalents 
 in the pyramid diagram, are respectively equal to 2-rrR and to 
 
 JEVjr. 
 
 In these expressions R is the radius of a circle whose circum- 
 ference, and R' the radius of another whose area, are respectively 
 those of the special square that we have been constructing. 
 The length of a cubit is thus 2-n-R -f R f \lir 2$", in which 
 = S = 5", or the chord of a boss. 
 
"THE SACRED CUBIT." 165 
 
 Let now the left-hand division of the cubit be subdivided 
 into a scale that shall contain as many units and decimal 
 parts of a unit as there are inches in the radius R, or that 
 of circumferences. It will thus at once become a scale of 
 circumferences. 
 
 So, likewise, if the right-hand part be similarly subdivided 
 with respect to R', or the radius of area, a scale of circular areas 
 will result. 
 
 Thus new light is shed upon the nature of this ancient and 
 honored standard of, not only rectilinear, but of curvilinear, 
 measure. From our present stand-point, the cubit is a rectified 
 pyramid diagram. It is of the primary degree ; since its dimen- 
 sions are those of the boss upon the granite leaf, which we have 
 already seen to be thus primary in all of its relations. 
 
 Here, then, we have a standard of linear measure, capable of 
 being subdivided, without v/astage, into two such concordant 
 parts that every square and circle in the intimate series of pri- 
 mary lines, unlocked by the pyramid diagram, has thereon its 
 appropriate place and scale. 
 
 By means of these scales, having given any other square or 
 circle whatsoever, the workman, by the simple measurement of 
 a side or radius, may instantly obtain all the other square and 
 circular equivalents found in the diagram itself, and to that 
 degree of accuracy with which the modern scale-maker shall 
 have graduated its subdivisions. 
 
 For instance, let him have a circle of any radius, say 7|". 
 
 To find the side of a square with an equivalent area, he uses 
 the cubit as follows : 
 
 (1) Ride. Go to the scale AB, that of the areal radius, 
 and, with a pair of dividers, take off the length of its unit AC. 
 Apply this unit as many (7^) times to the cubit as there are 
 inches in the radius : the number of inches thus covered will be 
 the length of the side of the equal-area square (13. 29"-!-). 
 
 Again : to find the side (i 1.78") of a square whose perimeter 
 shall have the same length as the circumference of this given 
 circle. 
 
 (2) Rule. Go to the scale DE, that of the circumference 
 radius, and, with a pair of dividers, take off the length of its 
 unit DF. Apply this to the cubit as many times as there are 
 
1 66 METROLOGY. 
 
 inches in the radius of the given circle : the length thus cov- 
 ered will be the perimeter of the square required, whose side is, 
 of course, one-fourth cf it. 
 
 Conversely, having given any square, as, for instance, the one 
 whose side is J\ inches, required its circular equivalents. 
 
 First, The circle of equal area. 
 
 (3) Rule. Measure the length of its side in terms of the 
 scale AB. The result will be the length of the equivalent areal 
 radius in inches (4.23"+). 
 
 Second, The circle of equal circumference. 
 
 (4) Rule. Measure the perimeter of the square in terms of 
 the scale DE, or measure one side thereof by this scale, and 
 multiply by four : in either case the result will be the length of 
 the circumferential radius in inches (4.774"+). 
 
 THE PRIMARY CIRCULAR FUNCTIONS AND THE CUBIT. 
 
 One of the most important circles known to practical geome- 
 ters is the one whose radius is 5, or whose diameter is 10. 
 This circle is so important, that you will find the lengths of its 
 principal parts tabulated in every book of mathematical rules, 
 data, and working formulas. For instance, I have here Has- 
 well's "Pocket-book for Engineers and Mechanics." Upon 
 p. 251 we find tabulated some twenty-five of these principal 
 parts. These values are particularly important because they 
 are the unit values, as it were, of the lengths they repre- 
 sent. Knowing them for a circle whose diameter is 10, the 
 radical of the whole decimal system, we can, of course, readily 
 determine them for circles whose radii are of any value what- 
 soever. 
 
 In the course of my pyramid investigations, I have discovered 
 that this circle bears a most intimate and characteristic relation 
 to the sacred cubit, and one which it will be well to note in 
 connection with that branch of our subject which we have just 
 been investigating. 
 
 Upon the sacred cubit as a diameter, construct a semicircle. 
 
 At the division of the cubit into its 5 and 20 inch parts, 
 a division that we have already found to be a most nota- 
 ble one for many other reasons, erect a perpendicular, and 
 
THE SACRED CUBIT: 
 
 I6 7 
 
 produce it until it intersects the semi-circumference just 
 drawn. 
 
 This perpendicular is, of course, a mean proportional between 
 2T?R and R'y-rr ; that is, between the two main divisions of the 
 cubit. Its length is, therefore, 10 inches, or the diameter of 
 the circle of unit values, to which we have just referred. 
 
 Let this circle now be constructed upon its perpendicular 
 
 THE CUBIT AND THE PRIMARY CIRCULAR FUNCTIONS. 
 Fig. 22. 
 
 diameter CD. Its unique location for the direct comparison of 
 its elements with the other functions of the cubit will be appar- 
 ent. Each one of the lines represented in the figure is the abso- 
 lute length of a circular function. They are all, in fact, unit 
 functions and decimal unit functions, let me reiterate of 
 the most important character. Their growth from the primary 
 standard of all linear measure the sacred cubit itself is 
 thus most beautifully illustrated. Nothing could be more sci- 
 entific than such an arrangement of units as that now before 
 
1 68 METROLOGY. 
 
 you. They are functions, then, of the sacred cubit as truly as 
 they are those of the radical circle, whose diameter (10) is the 
 base of all modern or decinral arithmetic. Thus, the line DE 
 is the side of an inscribed square ; that is, it is the unit of such 
 inscribed squares : and the side of a square inscribed in any 
 other circle, as, for instance, the one whose diameter is 47-1%", 
 is 4-7i 8 2 times the line DE. In a similar way, HK is a unit 
 function of the cubit, and the side of a square circumscribed 
 about the primary or lO-inch diameter circle, and Z^Fthat of an 
 inscribed octagon, etc. Did the French metrologists, even 
 in their wildest flights of imagination, ever dream of such 
 results ? What, indeed, can result from the metre but confusion ? 
 While, on the other hand, what limits dare we place to the 
 benefits that may result to science and mankind, when 
 in the hands of workingmen we place a standard such as 
 this ? 
 
 THE CUBIT AND SCALES. 
 
 By dividing each "unit" inch of the sacred cubit into quar- 
 ters, the whole cubit becomes scaled off into 100 equal parts : 
 thus, a quarter inch is the one-hundredth of a cubit. 
 
 So, likewise, if each quarter inch be redivided into tenths, or, 
 in other words, if the unit inches be divided into fortieths, the 
 cubit becomes a scale of 1,000 equal parts, 
 
 Again, by a division of each "unit " inch into one-hundredths, 
 the cubit becomes a scale of ^^ ; or, if into one-thousandths, 
 a scale of ^^. 
 
 Finally, by the use of micrometers and millimetres (for the 
 deeper purposes of more advanced science), it is manifest that 
 this subdivision may be almost indefinitely continued : thus, by 
 a simple quartering of the last subdivisions, the cubit at once 
 becomes a scale of 100,000. 
 
 The fitness of the cubit for decimal scales in general is thus 
 apparent. It is, however, more particularly to be noticed, 
 in this connection, that the various resulting subdivisions of 
 such scales are all well known and ordinary fractions of the unit 
 inch. 
 
"THE SACRED CUBIT? 
 
 169 
 
 THE CUBIT AND THE PYTHAGOREAN TRIANGLE. 
 
 Another point of interest connected with this ancient and 
 most sacred metron is its relation to the celebrated " 3-4-5 tri- 
 angle " of Pythagoras, a triangle whose claims to holding 
 possession of the very foundation of geometry, so far from being 
 shaken, are surer now than ever. My attention was called to 
 this fact by Mr. Jacob M. Clark, C.E., of Elizabeth, N.J., who, 
 
 Q F 
 
 THE CUBIT AND THE PYTHAGOREAN TRIANGLE 
 
 Fig. 23. 
 
 upon studying over my diagrams with me, pointed out this and 
 numerous other equally remarkable relations. 
 
 Let ABCD be a sacred cubit of 25 inches in length, and a 
 span, equal to 5 inches, in width : subdivide it into its square and 
 elementary spans, as in the illustration. Upon AD construct a 
 semicircle, and with A as a centre, and a radius AE 4 spans, 
 describe the arc EH: from B as a centre, and a radius BL = 3 
 spans, describe another arc LI, intersecting the former upon 
 the semicircle at the point O. Join AO and BO: then will 
 BOA form the "3-4-5 triangle," which has the cubit for its 
 hypothenuse. It is through this triangle that the cubit is 
 
I/O METROLOGY. 
 
 directly related to the common foot of 12 inches, and also to 
 the Anglo-Saxon yard of three such feet ; for tlie vertical OP of 
 tJiis triangle is tJie " common foot." The decimal foot (10 inches) 
 is, as we have already seen, similarly related to the cub.it at its 
 marked division E. I shall leave it to Mr. Clark to point out, 
 in due time, the relations of the sacred cubit to the 5-pointed 
 star, or " Pentalpha," a star which, like the "3-4-5 triangle," 
 comprehends not only the fundamental principles of geometry, 
 but is, par excellence, the EMBLEM of the Great Pyramid. In his 
 articles upon " Metric Analogues," in the " International Stand- 
 ard," he has already shown, that, upon the relations of these two 
 figures, the Pentalpha and the triangle of Pythagoras, the only 
 logical subdivision of the circle into degrees depends. He 
 there shows that the proper subdivision into 240 "exhausts 
 the resources, and satisfies all the demands, of strict geometry." 
 The exquisite beauty of such a subdivision for use in geography, 
 astronomy, and chronology cannot be too highly extolled ; and 
 the step towards the unification of time lately made in this 
 country will certainly lead in the future to the adoption of the 
 subdivision of the circle proposed by Mr. Clark. 1 
 
 Besides the foregoing connections, the sacred cubit is like- 
 wise related to many other important branches of mathematics, 
 such as to the fundamental functions of trigonometry, to many 
 special geometrical principles, and to number in the abstract, to 
 which, for want of space, we can only afford the briefest refer- 
 ence. 
 
 THE CUBIT AND GEOMETRY. 
 
 In the practical geometry of the arts, the sacred cubit is also 
 of the utmost value to the workman. We can but glance, how- 
 ever, over some of its more salient points ; since to exhaust 
 them all, would be to write a treatise upon the Science of 
 Geometry itself. Some of the more noticeable ones are as fol- 
 lows ; and, in examining them, it will be observed that they 
 universally depend upon the five-part division, which we have 
 already found to be the basis of so many other important inves- 
 tigations and relations. In the diagrams which we shall now 
 
 1 See Appendix B. 
 
" THE SACRED CUBIT? 
 
 171 
 
 hastily scan, the cubit ABCD is divided into only 5 spans : its 
 width is also taken at a span. 
 
 Now, if upon the 2O-inch or 4-span segment, as a diameter, 
 we describe a circle, it will be that one whose radius is the 
 decimal foot, or 10 inches. 
 This circle is as important 
 as that one whose diame- 
 ter itself is the basis (10) 
 of the decimal system 
 of numeration. An ex- 
 amination of this circle 
 in its relations to the 
 cubit reveals the follow- 
 ing facts : 
 
 (1) The point falling 
 upon the top of the cubit, 
 and between the second 
 
 and third spans, gives us our first geometrical idea, a point: 
 from it, within the radius of our conceptions, we are enabled to 
 proceed outwards to the remotest regions of geometry. 
 
 (2) Our next geometrical conception is that of the line: and, 
 from the diagram, we learn that the shortest line which may be 
 
 drawn in the circle is that 
 from its point G to N ; i.e., 
 the radius ten. This 
 line separates the second 
 and third spans without 
 breaking their units, and 
 
 THE POINT. LINE, AND CUBIT. 
 Fig. 24. 
 
 is also half the 
 
 segment of the cubit. 
 
 also 
 
 longest 
 
 longer 
 
 We 
 
 the 
 
 THE CUBIT AND ANGLE. 
 Fig. 25. 
 
 perceive that 
 
 line which may 
 be drawn in the circle, is 
 the one AE, likewise pass- 
 ing through the point G t 
 and constituting its double radius, diameter, or the full length 
 of the cut-off portion of the cubit. 
 
 (3) Two lines in their geometrical signification indicate 
 angle ; and, from the cubit, we learn that BAD = 90 = the 
 
172 
 
 METROLOGY. 
 
 standard angle of geometry, may be constructed by the circle 
 with three important sets of lines. Thus, AE, the greatest 
 line in the circle, and the infinitesimal element of the circle at 
 A t which indicates the direction of its infinite tangent, gives us 
 the radical angle of the cubit at A. The lines AJ and EJ, the 
 two greatest lines which may be inscribed in this circle, also 
 give us this standard angle at the point J upon the division line 
 KG of the cubit produced to its intersection with the circle. 
 So, too, at G, the production of the diagonals GM and GM", 
 gives us the radii themselves forming the angle of the perfect 
 square at the centre G. 
 
 (4) The employment of more than two lines gives us the 
 means of confining space, or of indicating form. In this par- 
 ticular, the cubit is the most wonderful assistance ever discov- 
 ered. Its ability to comprehend the various regular polygons 
 depends upon its five-part division. This, of course, can be 
 shared by any other scale, since all may be subdivided into 
 fifths ; but no other scale than one formed upon twenty-five' 
 units, or a due multiple thereof, can be so divided without rup- 
 turing the units, nor have as four of its five parts a double 
 radius equal to twenty units. The cubit is, therefore, unique 
 in the full realization of the constructions which are to follow. 
 
 THE CUBIT AND THE EQUILATERAL. 
 Fig. 26. 
 
 The primary regular form composed of right lines, is the 
 equilateral triangle. The side of this triangle is indicated by 
 that portion HI of the base of the cubit which is cut off by the 
 circle on its larger segment. 
 
" THE SACRED CUBITS 
 
 173 
 
 (5) The next of the regular polygons is the square, or tetra- 
 gon, whose side is the diagonal AM produced to its intersection 
 
 THE CUBIT AND THE SQUARE. 
 Fig. 27. 
 
 with the circle. It is likewise to be observed, that another 
 important line of the cubit, i.e., AE, is the diagonal of this 
 square. 
 
 THE CUBIT AND THE PENTALPHA. 
 Fig. 28. 
 
 (6) To form the pentagon with the cubit, take K as a centre, 
 and the diagonal KA as a radius ; describe the arc AL ; then 
 will the chord AL be the side of a pentagon. 
 
174 
 
 METROLOGY. 
 
 THE CUBIT AND THE HEXAGON. 
 Fig. 29. 
 
 (7) The side of a hexagon is, of course, the radius or dis- 
 tance AG = 10", or to half the cut-off portion, AE, of the cubit. 
 
 THE CUBIT AND THE OCTAGON. 
 Fig. 30. 
 
 (8) To obtain the octagon, produce the diagonal GM to M'; 
 join this with the point A, or produce the cubit division GK t 
 till it also intersects the circle ; then will NM also be the side 
 of the required polygon. 
 
THE SACRED CUBIT." 
 
 175 
 
 THE CUBIT AND THE DECAGON. 
 Fig. 31. 
 
 (9) The side of the decagon is the distance from L (already 
 determined in section 6) back to the cubit, or LG. 
 
 THE CUBIT AND THE DUODECAGON. 
 Fig. 32. 
 
 (10) The side of the duodecagon is the chord width AH of 
 the cubit, determined by joining the points A and H t in which 
 the circle intersects it. 
 
METROLOGY. 
 
 (n) To obtain the 1 5-sided polygon, join the points D and 
 B, i.e., draw the full diagonal of the cubit ; then connect the 
 point Z, in which this line intersects the circle, with the point 
 
 THE CUBIT AND THE 15-SIDED POLYGON. 
 . Fig 83. 
 
 /, in which the latter intersects the base DC of the cubit : it is 
 the side of the polygon required. Or, from the point J lay off 
 the chord JG' equal to the radius AG, and from G r lay back the 
 chord G'Z' equal to GL the side of the decagon: then 
 be the chord of one-fifteenth of the circumference (J 
 or the side of the required polygon. 
 
 THE CUBIT AND THE 16-SIDED POLYGON. 
 Fig. 34. 
 
 (12) To determine the i6-sided polygon, produce the diagonal 
 GM of the cubit to M' where it intersects the circle ; draw M'E 
 
"THE SACRED CUBIT." 
 
 177 
 
 intersecting the base of the cubit at R ; through G draw GR y 
 and produce it to S ; likewise produce GK to N. Join NS, and 
 it is the side of the required polygon. 
 
 The foregoing comprise all those elementary polygons that 
 have hitherto been classified as rational, since they may be 
 obtained by a direct geometrical process. Those left out of 
 the series, the heptagon or 7-sided, the nonagon or Q-sided, the 
 hendecagon or ii-sided, the 13-sided and the 14-sided polygons, 
 are termed irrational, since they have not hitherto been so 
 obtained. An examination of the peculiar properties and 
 power of the sacred cubit reveals the fact, however, that not 
 only may the rational polygons and their corresponding stars 
 be determined directly from it, as we have shown above, but 
 also, that, by its all-potent means, a series of most remarkable 
 working-approximations to the irrational ones may be arrived 
 at. Thus, 
 
 THE CUBIT AND THE HEPTAGON. 
 Fig. 35. 
 
 (13) To obtain the heptagon, employ HK as a chord (i.e., half 
 of that portion of the base of the cubit cut off by the circle), 
 and it will be the side of the required polygon true to thou- 
 sandths, or to the third place of decimals. 
 
METROLOGY. 
 
 THE CUBIT AND THE NONAGON. 
 Fig. 36. 
 
 (14) To obtain the nonagon, join 7 and (7, and produce the 
 oblique line, so obtained, until it intersects the circle at O. 
 Join O and H, and it is the chord required, likewise true to the 
 decimal .00205. 
 
 THE CUBIT AND THE HENDECAGON. 
 Fig. 37. 
 
 . (15) To obtain the hendecagon, draw the diagonal from K to 
 A ; and one-half of it, PA, is the closest approximation known 
 to the chord required. 
 
"THE SACRED CUBIT? 
 
 179 
 
 (16) For the 1 3-sided polygon, draw the diagonal from C to 
 (7, and through P and P' produce the diagonal PP' to T, its 
 
 THE CUBIT AND THE 13-SIDED POLYGON. 
 Fig. 38. 
 
 intersection with the circle : then will QT be the side required 
 to its nearest geometrical construction. 
 
 THE CUBIT AND THE 14-SIDED POLYGON. 
 Fig. 39. 
 
 (17) For the 14-sided polygon, lay off the distance HK* four 
 times as a cho^d, commencing at/, and terminating at/'. Pro- 
 duce GK to N. Join NJ r : it is the chord required. Or (see 
 Fig. 41), from //"as a centre, and with HK as a radius, describe 
 the arc KY ; produce FE to U, making EU equal to EB ; 
 through the point U thus determined, and V, the middle point 
 of GK, draw the line UV, which intersects the base of the 
 
i8o 
 
 METROLOGY. 
 
 THE CUBIT AND RATIONAL POLYGONS. 
 Fig. 40. 
 
 cubit at W. Draw GW, and produce it till it intersects the 
 circle at X. Join XY: it is the chord required. 
 
 The foregoing constructions, dependent upon the five-part 
 division of the cubit, as related to the circle whose radius is 
 
 10, struck upon its longer 
 segment AE as a diame- 
 ter, are represented col- 
 lectively in the two follow- 
 ing diagrams. 
 
 Fig. 40 gives us the con- 
 struction of the several 
 rational polygons as fol- 
 lows : ABCD is the cubit ; 
 E, P' y G, and O' its points 
 of division into spans or 
 fifths'; JINH the circle 
 upon its larger segment 
 A E as a diameter. To 
 recapitulate, therefore, HI is the side of the equilateral triangle, 
 AJ or AN thai of the square or tetragon, AL that of the penta- 
 gon, AG of the hexagon, NM' of the octagon, GL that of the 
 decagon, AH that of the 
 duodecagon, JZ' or IZ 
 that of the 1 5-sided, and 
 NS that of the 1 6-sided, 
 polygon. 
 
 In Fig. 41, constructed 
 and lettered to correspond 
 to the foregoing, and to 
 all the preceding several 
 diagrams under this head- 
 ing, HK is the side of 
 the heptagon, HO that 
 of the nonagon, AP that 
 of the hendecagon, QT 
 that of the 1 3-sided, and XY that of the 14-sided, polygon. 
 From these primary polygons, by proper bisections, the 
 whole series of regular geometric forms which result from 
 them may be duly constructed, and throughout the process 
 
 THE CUBIT AND IRRATIONAL POLYGONS. 
 Fig. 41. 
 
THE SACRED CUBIT? 
 
 iSl 
 
 the power of the cubit itself, as a means towards the end, be 
 constantly felt. 
 
 THE CUBIT AND TRIGONOMETRY. 
 
 The cubit is likewise directly connected with trigonometry 
 through the circle described upon its longer segment (20") as a 
 diameter. This, as above, is the circle whose radius is numeri- 
 cally the decimal root 10. To illustrate this, let, as before, 
 ABCD be a cubit, and EF its extreme five-part division ; upon 
 the longer segment AE as a diameter construct a circle, and 
 let the radical functions of trigonometry be drawn therein ; 
 then will their individual 
 relations to the cubit be 
 severally apparent, as also 
 the convenience of the po- 
 sition of the cubit itself for 
 measuring them thereon, 
 either in radius or in inches, 
 the unit of the cubit it- 
 self. For instance, El be- 
 ing equal to radius, or to 
 10, the angle HGE is an 
 octant, or is 30 on a scale 
 of 240 to the whole circle ; 
 i.e., 45 to the circle of 360, 
 
 which is its present subdivision as derived from Babylon. Its 
 sine is ///equal to its cosine HK, the measure of which latter 
 is GJ upon the cubit. Now, it is manifest, that to whatever 
 degree of minuteness and accuracy our cubit itself is subdi- 
 vided, be it to twenty-fifths, or to millionths, to that same degree 
 of accuracy, without tables or formulas, may the practical work- 
 man obtain the numerical values of the circular functions he 
 may need by a direct appeal to the cubit itself. Thus, let it be 
 supposed that a workman desires to obtain the angle whose 
 cosine is \R> or 5": he will produce the cubit division U'Z to N; 
 then will NM " 5", or \R, and be the cosine of the angle NGB 
 required. In a similar way, the cubit subdivision UZ' produced 
 to its intersection N f with the circle, gives us the angle N'GB, 
 whose cosine is \R or 5", etc., ranging throughout the 
 
 THE CUBIT AND TRIGONOMETRY. 
 Fig. 42. 
 
1 82 
 
 METROLOGY. 
 
 whole scope of this branch of mathematics. Should the radius 
 of some circle be any thing else than 10, the decimal root, the 
 cubit still affords us the direct means of obtaining its trigo- 
 nometrical angles and functions ; since, for instance, one whose 
 radius is *j\ inches has for its cosine \R> a length equal 4o 
 .725 that of the 10" radius : in other words, a new circle cen- 
 tred at G y and with a radius 7.25", will limit all the lines and 
 functions at their new points, and determine the same angles.- 
 
 THE CUBIT AND ABSTRACT NUMBER, 
 Fig. 43. 
 
 THE CUBIT AND NUMBER IN THE ABSTRACT. 
 
 There is another illustration, which shows still further the 
 connection of the cubit, as a radical metron, with all mathe- 
 matics. In the whole range 
 of number in the abstract, as 
 applied to the expression of 
 the areas and circumferences 
 of circles, there is but one 
 circle whose numerical ex- 
 pressions for area and circum- 
 ference are identical : this is 
 the circle whose radius is 2. 
 The area irR 2 of such a circle 
 is 7r(2) 2 = 4:r. Its circumfer- 
 ence, 2irR 27r(2), IS also ^TT. 
 
 Thus the same sequence of figures, to infinity, expresses a 
 number of square units in the one, and of linear units in the 
 other; to wit, 4^= 12.56636-!-. Now, an examination of the 
 larger segment of the cubit, after making the important or five- 
 part division, reveals the fact, that this segment is 4 in terms of 
 the span, and hence that the circle upon it as a diameter, and 
 whose utility for polygonal constructions we have already dis- 
 cussed, is, in effect, that one whose area is equal to its circum- 
 ference in terms of the cubit fifths or spans. Fig. 43 illustrates 
 the properties of this circle. Its area = its circumference = 
 respectively 12.56636 + square and linear spans. Hence the 
 arc AJ = 3.141592 + spans ; the semi-circumference AJE = 
 3.1415.92 +, etc -> decimal feet; and the full circumference 
 3.141592+; etc., times the larger segment of the cubit, or is 
 
"THE SACRED CUBIT." 183 
 
 equal to 15.707960 -f-, etc., palms of 4" each, etc. So, too, the 
 sector AGJA = 3.14159 + square spans; the semicircle = 
 3.141592 -f- decimal square feet, etc. 
 
 Another remarkable numerical property of this circle is the 
 fact that the area of the lune AX'EJA, being equal to that of 
 the triangle AEN, is exactly 100", or 4 square spans (= (io") 2 ) ; 
 hence the semilune EX'JE = 2 square spans, or 50" ; or, since 
 the quadrant EGJ, as already shown, is equal to 3.1415924-) 
 etc., square spans, the semicircular segment EGX' is equal in 
 area to 1.31415924-) etc., such units; and the circular sector 
 NX'EN= 3.141592 square spans; that is, the octant of the 
 circle whose centre is N, and whose radius is NX' t is equal to 
 the quadrant of the one whose centre is G t and whose radius is 
 2 spans. Now, the radius NE = 2^2 spans is a remarkably 
 close working-approximation to that of the circle whose area is 
 a square cubit ; it being 2.8284 +, etc. (say, 2.83), spans, against 
 the required TT value, 2.8209 (say, 2.821) spans, values which 
 differ by only .009 of a span. This construction thus affords 
 the practical workman another wonderfully close means of 
 approximating to the square of a circle, and one that is inti- 
 mately related to that depending upon the five-part division 
 which we have already studied. 
 
 It is worthy of note, too, in this connection, to observe that 
 these two working-approximations to the square of the circle 
 the one from 5 with a radius SC, and the other from N with a 
 radius NE intersect each other at Q, upon the cubit diago- 
 nal GF produced. 
 
 PERSONAL REFERENCES OF THE CUBIT. 
 
 As it is <?<7r///-commensuric, so, too, the cubit is also pecul- 
 iarly commensuric with the human frame. This is most 
 fortunate for the poor man, and for emergencies. 
 
 Its "unit,"- the "pouce," or " polgida " as it is sometimes 
 called in other countries, that is, the " thumb," or inch, is the 
 thumb-breadth of an able-bodied man ; while the entire stand- 
 ard, as its name (cubit) implies, is such a man's arm-length. 
 
 Surely no one is so poor as not to have a fair average of these 
 measures always at his hand and side, wherewith to govern both 
 
1 84 METROLOGY. 
 
 his life and work. In the true canon of human proportions 
 founded upon the ?r relations of the pyramid diagram, and 
 drawn at its actual scale, we doubt not that the inch and cubit 
 will be found to be the unit and standard in terms of which all 
 of the principal parts will be commensurable. 
 
 THE TRUE CANON OF HUMAN PROPORTIONS. 
 
 We have already had occasion to coin the word mcsocosm, 
 as one specially applicable to the Great Pyramid in its interme- 
 diate position to man and to the universe : it is at once an 
 image of the microcosm and the macrocosm. 
 
 We need not here review the discoveries of others which go 
 to show the intimate cosmic references of the pyramid to the 
 earth itself, and to the universe, but, taking it for granted that 
 all are more or less familiar with these references, shall confine 
 ourselves to an examination of the proportions of the human 
 figure as viewed in the light of the pyramid diagram. 
 
 In his elaborate work upon the ancient and modern systems 
 of proportion, the sculptor, W. W. Story, is led to propose, as 
 the basis of a new canon, a circle circumscribing an equilateral 
 triangle and a square. From the relations between the right 
 lines in these several figures, he determines those between the 
 various parts of the human frame. The primary rule in this 
 canon is thus enunciated by Mr. Story : " The entire height 
 of the figure is 3|- diameters, or 7 radii ; 4 times the base of 
 the triangle, or 5 times the side of the square," and he finds 
 that this diagram "will satisfy all the scientific and practical 
 conditions demanded of such a canon." " It gives not only the 
 lengths of the body, but also the breadths of the body viewed 
 in front, and the depths as viewed in profile." Mr. Story then 
 enumerates and explains the various measures of the body 
 founded upon this canon. 
 
 At the close of his work, however, we notice the following 
 very important admission, and one which cannot but be fatal 
 to the strict accuracy of his rule of proportion. " In the fore- 
 going pages, 4 times the side of the triangle, 7 times the radius, 
 and 5 times the side of the square, are sometimes treated as 
 equal. This, however, in a rigid geometric sense, is not true. 
 
"THE SACRED CUBIT." 185 
 
 Four times the side of the triangle is a fraction less than 7 
 radii, and 5 times the .side of the square is a fraction greater 
 than 7 radii. The equations 
 
 however, though only approximative, are not only sufficiently 
 exact for all practical purposes (the error in each case being 
 greater than T ^, but less than T -| 7 of the radius), but also are 
 approximations of the most scientific kind ; the true equations 
 being 
 
 and J and J being convergents to v^3 and ^2 respectively." 
 
 Now, as a necessary matter of fact, the human form is not 
 made and proportioned upon any mere system of approxima- 
 tions, but upon some diagram of the most rigidly correct order. 
 For all the practical purposes of the ordinary workingman, Mr. 
 Story's canon will undoubtedly answer ; but his own admissions 
 show us, that, for the ideal, we must look higher, and actually 
 include these very TT proportions, if we hope to attain the goal. 
 But to introduce these is to admit the whole pyramid system 
 through the same gate ; for Mr. Story's canon would then be- 
 come but a different form of the pyramid diagram, a special 
 case. 
 
 Now, the celebrated canon of Polycletus was the recognized 
 standard of the ancients ; and it is universally admitted that it 
 came, at least indirectly, from Egypt. " It is to Egypt," says 
 Story, "that the formal and rigid divisions of all the early 
 Greek and Etruscan statues plainly point ; " for "the Egyptians 
 possessed a canon of exact and geometric character." 
 
 "The Greeks," says Diodortis, "had no knowledge of this 
 kind of art ; though the use of it was perfectly known to the 
 Egyptians. For the Egyptians did not measure with the eye 
 the composition of their statues, but with measures ; so that 
 out of many blocks, worked according to an exact and fixed 
 measure, they brought the statue to its perfection : and what 
 
186 METROLOGY. 
 
 may be called truly wonderful is, that various artists in differ- 
 ent places should, upon a certain and established measure, com- 
 pose a single statue in twenty, and sometimes forty, pieces, 
 which would all exactly agree." Every thing in this statement 
 of Diodorus points to a Promethean system, and eliminates 
 from it all that is Epimethean and haphazard. It savors of the 
 true masonic method, so intensified at Gizeh and Jerusalem, 
 where the work grew towards its completion without the sound 
 of hammer or of iron tool. 
 
 Such a method must have been founded upon the strictest 
 principles of geometry ; and, if the canon of Polycletus was a 
 perfect one, it, too, must have come up out of Egypt with circum- 
 metric proportions. "The canon of Polycletus has, however, 
 been lost, and with it the Doryp1wros> which was its practical 
 embodiment. The only vestige of it that we possess, is to' be 
 found in the pages of Vitruvius, who, writing upon architecture, 
 gives us, incidentally, a confused account of some of its princi- 
 pal features." 
 
 One of the rules preserved by Vitruvius is as follows : " So 
 also the navel is the natural centre of the body ; far if a man 
 be placed supine, with his hands and feet spread, and the navel 
 be taken as the centre of a circle, the circumference would 
 touch the extremities of the fingers and feet." But this state- 
 ment of Vitruvius is evidently in error ; for the pubis, and not 
 the navel, is the natural and actual centre of the human figure. 
 Mr. Story himself doubts the accuracy with which Vitruvius 
 enunciates the principles of Polycletus ; for he says, ," It is 
 more than doubtful whether he really understood the canon ; as 
 he was not a sculptor, and would have had little practical need 
 to study it." 
 
 Now, as a matter of fact, let us examine the human figure 
 under true pyramidal proportions, and see what this principle of 
 Polycletus actually was. Ballard has shown that the Pent'alpha 
 is the emblem of the Great Pyramid. Inscribe it, therefore, 
 together with an erect human figure, in a circle. The centre 
 of the star and circle will be found at the pubis, and not at the 
 navel ; and, if the arms and legs be spread out easily, the whole 
 figure will have its five extremities found to coincide with the 
 several corresponding points of the star. The latter is thus not 
 
"THE SACKED CUBIT? 
 
 I8 7 
 
 only discovered to be an accurate and mathematical emblem of 
 the five-sided and five-pointed ?r pyramid, but a reason is seen 
 for the tradition, that, from time immemorial, has made it the 
 mystic symbol of the human form divine. 
 
 In the figure, it will also be noticed that the natural division 
 into quarters is borne out ; the centre of the breast being mid- 
 way between the pubis and the crown of the head, and the base 
 of the patella midway from the pubis to the sole of the foot. 
 
 THE PENTALPHA, THE PYRAMID. AND THE CANON OF PROPORTION. 
 Fig. 44. 
 
 In order to develop these facts more clearly, it will be noticed 
 that an inverted profile of the human form is put upon the 
 same diagram. 
 
 Vitruvius alsp gives another rule of the Polycletian canon, 
 and in rather more accurate language. He says, "As the 
 scheme of the circle is found in the body, so also is the scheme 
 of the square ; for if the measure be taken from the soles of 
 the feet to the top of the skull, and applied to the hands out- 
 stretched on either side, it will be found that the width and 
 height are equal, so as to form a perfect square." 
 
i88 
 
 METROLOGY. 
 
 Now, this statement verifies upon the figure which we have 
 given, but not upon the one described from the navel as a centre. 
 For if the circle of our own diagram be enclosed in a circum- 
 scribing square, and the arms be outstretched horizontally, the 
 tips of the fingers will just come to the vertical sides of the 
 square. Thus the two, and only, rules of the celebrated canon of 
 Polycletus which have been preserved to us are actually verified 
 
 THE HUMAN FIGURE AND THE PYRAMID DIAGRAM. 
 Fig. 45. 
 
 by the most primary application of the true pyramid proportions. 
 Let us now apply the full pyramid diagram to the human 
 form. Taking the centre, as before, at the pubis, and standing 
 the figure upon the lower side of the pyramid's square base, let 
 the crown of its head come to the vertex of the capstone. It 
 will be at once noticed that a host of important and mathemat- 
 ical coincidences result. Thus, to mention but a few, the line 
 XY the intersection of the circle of equal area with the base 
 of the pyramid is that which divides the figure in twain ; the 
 small of tjie body, or waist, is on the line BC, which is the upper 
 
"THE SACRED CUBIT." 
 
 189 
 
 side of the square base ; and all the intersections of the various 
 lines of the diagram mark important points, from which the 
 measures of a true canon may be fully realized. 4 
 
 In Fig. 46 we give a side view of the human form against the 
 same diagram, and, in addition, have introduced the general 
 directions of the several known passage-ways of the pyramid, to 
 show that they locate respectively the womb, the heart, and 
 the lungs. We cannot here examine these diagrams at greater 
 
 THE PYRAMID DIAGRAM AND THE HUMAN FORM. 
 Fig. 46. 
 
 length. That they comprehend the true principles of the lost 
 canon, however, we are fully persuaded ; and we shall leave it 
 for others to develop its elementary proportions. 
 
 It is manifest, therefore, that if a perfectly proportioned 
 human figure be drawn at a height of 10", and upon its front 
 and profile views the pyramid diagram be laid, it will enable us 
 to reproduce, at any scale, the true proportions of the God-like 
 image ; and there can be little doubt but that the same wisdom 
 which comprehended the grandeur of the macrocosm in the 
 
IQO METROLOGY. 
 
 wonderful proportions of the Great Pyramid, failed not to grasp 
 the lines of beauty of the microcosm. 
 
 The cubit is but another form of the mathematical relations 
 of the boss upon the granite leaf, as exemplified by the pyramid 
 diagram ; and thus the cubit itself comprehends all of the 
 relations, so intimate and far-reaching, which we have hastily 
 reviewed. 
 
 Possessed of such a standard, then, and initiated into its 
 higher uses, the ordinary " workman" is rendered practically 
 independent of ?r values and mathematical tables. With it, he 
 can readily construct at their true dimensions any of the simpler 
 " units " that he is in the habit of employing ; while in it alone 
 the advanced student, the architect, the sculptor, the astrono- 
 mer, and the metrologist, can find the very ideal of linear 
 measures. 
 
 Such relations as we have noted will be sought for in vain at 
 the bar of every other standard of linear measure. We defy 
 the advocates of the "metric system" to bring forth such facts 
 as these with the wand by which they wish to make us "change 
 our times and seasons." The "sacred cubit," like the "rod 
 which Aaron " flung upon the sands of Egypt centuries ago, 
 shall literally swallow up the serpent of "science falsely so 
 called." In the bright future of universal metrology, the false, 
 unscientific standard of an age which was the very nightmare 
 of injustice and infidelity, cannot exist. The lying magician 
 may indeed perform a wonder, but his wand shall be devoured. 
 
 The cubit is unique among measures. It cannot be improved 
 upon. It is the least common multiple of all that we can desire 
 in a linear standard, while its unit inch is the greatest common 
 divisor of things man has to measure. 
 
 THE UNIVERSAL PROBLEM. 
 
 In the winter of 1881, while generally investigating the 
 mathematics of our subject, I had occasion to propose quite an 
 interesting problem in geometry to several of the army-officers 
 serving at the Presidio. This problem was, to construct an 
 equilateral triangle whose area should be equal to that of a 
 given square. 
 
"THE SACRED CUBIT? 191 
 
 It was necessary to obtain a simple solution of this question 
 as a step in certain pyramid investigations which I then had on 
 hand, and several very ingenious ones were soon proposed. 
 
 Of course, we shall not have time to examine these several 
 solutions : they are foreign to our immediate subject. I shall 
 ask your attention, however, to one of them, and to this one, 
 not only because it touches so closely upon that subject, but 
 more particularly because I discovered it with the assistance of 
 the already wonderful " pyramid diagram." 
 
 The principles involved in this solution are so general, that 
 their application gives rise to the most universal geometric 
 method you will probably ever yet have heard enunciated. On 
 this account, I shall call it "The Universal Problem." 
 
 We shall probably arrive at a better understanding of this 
 problem, and of its special application to the case in point, by a 
 short preliminary course of reasoning. 
 
 A thorough comprehension of the principles involved in the 
 now familiar pyramid diagram, has already shown us that such 
 a direct relation certainly exists between circles and squares as 
 to prove that their mutual quadrature and rectification is an 
 undoubted possibility. 
 
 These principles may be translated into the following geo- 
 metric enunciation or theorem : If the perimeter of any square 
 be put into the form of a circle, and another square be con- 
 structed whose area shall be equal to that of this circle, then 
 will the area of a second circle, whose diameter is equal to the 
 side of this second square, be equal to the area of the original 
 square. 
 
 But if these relations which we have discovered be true of 
 circles and squares, they must, of course, be true of circles, and 
 all other regular rectilinear figures whatsoever ; since all such 
 figures can be recast into equal squares, and vice versa. 
 
 Investigation not only fully established this proposition, but 
 afforded an analytical demonstration of its accuracy. 
 
192 METROLOGY. 
 
 REDUCTION OF CIRCULAR AREAS TO ANY OTHER " REGULAR" 
 
 FORM. 
 
 General Demonstration. 
 
 Let T?R 2 = the area, and 2-n-R the circumference, of any circle. 
 Denote by P any regular polygon of n sides, the total perimeter 
 of which shall equal 2-rrR. 
 
 Let 5 denote one of these sides, and let b = its apothegm ; 
 
 then area of P = 2vR - = irRb. 
 
 2 
 
 Let there now be another circle, whose area ^(R'y the area 
 of P irRb ; then R' = vG% 
 
 Circumscribe about this latter circle a new polygon P' of n 
 sides ; denote its area by A'. Then, since the polygons P and 
 P' are similar, we shall have vRb : A' : '. b 2 : (V^) 2 ; i.e., (the 
 apothegm of P') 2 . 
 
 Therefore, A'b 2 =irR 2 b 2 , or A' -rrR 2 \ i.e., the last polygon 
 P' of n sides will have the same area as the original circle. 
 
 Q.E.D. 
 
 EXAMPLE. Special application of the above, in the case of a 
 circular area put into the equilateral triangular form : 
 
 irR 2 = area of given circle, and 2-rrR its circumference. 
 
 P = an equilateral triangle whose perimeter 2-n-R. 
 
 S, then, = |TT^ ; altitude = ^R^3 ; apothegm = altitude 
 
 Area P = -|(base X alt.) = 
 
 Let, now, second circle be discussed : 
 
 R'.'. V^TT^N/S = JV^^V^. But, since P' is to be circum- 
 scribed about this circle, the altitude of P' will equal 3 times 
 'R', or =: VTT^VVS, and its apothegm equals R r itself; hence 
 
 or 
 hence 
 
"THE SACRED CUBITS 193 
 
 and 
 
 27 
 or, reducing, 
 
 ;4' = TrR 2 . Q.E.D. 
 
 Further investigation showed that the principle applied to all 
 regular polygons whatsoever, and also afforded an algebraic 
 demonstration of this fact. 
 
 I shall now try and enunciate the more general proposition, 
 and show its application, by solving thereby the problem pro- 
 posed last winter. 
 
 The general theorem is as follows : If a regular polygon (P r ) 
 of any number (;/) of sides be so constructed that its perimeter 
 shall be equal to that of a given regular polygon (P) of ;/ sides, 
 and a third polygon (P") be constructed similar to P, but of 
 the same area as P r t then, the area of a fourth polygon (P'") t 
 similar to P', and circumscribed with P" about the same circle 
 (i.e., of equal apothegm), will be equal to that of the original 
 polygon P. 
 
 Let P be a given regular polygon of n sides. Let a equal the 
 length of one of its sides, and b its apothegm. Then will its 
 
 area be equal to - , and its perimeter equal to an. 
 
 Let P' of ;/ sides represent another regular polygon of equal 
 perimeter (an) to P y and let its apothegm be denoted by b'. 
 Then 
 
 S'= a side of /"=-*, 
 n 
 
 and 
 
 A' = area of P' = an . 
 
 2 
 
 Construct now a third regular polygon P" of n sides (i.e., 
 
 similar to P\ but whose area = A" = - is the same as that 
 
 2 
 
 of P r . Then, since P and P" are similar, we have, 
 
194 METROLOGY. 
 
 or 
 
 2 2 
 
 therefore 
 
 b" = ^fbb'. 
 
 And since J (perimeter of P")Jbb f = area A" = ^- y we have 
 
 Perimeter of / y/ = 
 and 
 
 Let now P'" y a polygon of ;/ sides, and similar to P', be con- 
 structed, but with an apothegm equal to that of P" ; i.e., equal 
 to b" = V^. (P" and P"' can thus be circumscribed about the 
 same circle.) Now, since P' and P'" are similar, we have, 
 
 Area P r : area P" : : (apothegm P') 2 : (apothegm P'") 2 , 
 or 
 
 that is, 
 therefore 
 
 A' _ 
 
 Q.E.D. 
 
 EXAMPLE. Take a square area equal to a 2 , and let it be 
 required to pass it, by the above method, into the form of an 
 equilateral triangle. 
 
 Given square. S = a, A = a 2 , perimeter =r 4^. 
 
 First triangle. Perimeter =4#, .*. S' = |-#, altitude ^l^y^, 
 
 _ 
 
 Second square. Area = f <?V3 5' v/area = |<7W3 ; hence 
 the radius of its inscribed circle = %S = JtfV/vJ 
 
 Second and final triangle. Since this triangle is to be cir- 
 cumscribed about the same circle as the second square, its 
 
"THE SACRED CUBIT." 195 
 
 apothegm equals the radius of the latter ; i.e., = J^V. The 
 altitude of a regular triangle equals 3 times its apothegm, 
 equals rtV^3 ; and since the altitude of an equilateral triangle 
 
 X -7- a side thereof, we have 
 
 /. the area of this triangle = |(alt. x base) = ^f 0^/3 X 77==) = <* 2 - 
 
 Q.E.D. 
 
 Application of the above to the square whose side is 10 = a. 
 The area of the required triangle is thus ioo = a 2 . Applying 
 the formulas of the final triangle abov.e, we have 
 
 = 20 x ( ' 7598 + ) = I5 ' I96o+> etc " 
 
 Alt. = 0VV3 10(1.3160740 +, etc.) = 13.160740 + , etc. 
 Area /. = %(S" X alt.) = ^(15.1960 x 13.160740 +, etc.) 
 
 = 99.99 +, etc., = 100. Q.E.D. 
 
 And now for its application. The question proposed was, 
 to construct an equilateral triangle of equal area to a given 
 square. 
 
 "THE UNIVERSAL PROBLEM," SPECIAL CASE. 
 
 Let ABCD be a given square whose side is a, and whose area 
 is ~a ; then, by the universal problem, will the equilateral tri- 
 angle LMN, whose side is -,==, have the same area. 
 
 .Following the enunciation just given, the perimeter of the 
 triangle EFG was made equal to 4^. Each side is thus |/z in 
 length. The area of this equilateral triangle was calculated 
 and found to be equal to I^VJ. Now, the square root of this 
 quantity, or fflVV3> i g > f course, the side of a square, HIJK, of 
 equal area. One-half (J^Vv/I) f tn ^ s side is the radius PO 
 of the inscribed circle. Three times this radius, or dVf, is the 
 
196 
 
 METROLOGY. 
 
 altitude of a circumscribed equilateral triangle whose area is 
 consequently a 2 , or that of the given and original square. 
 
 But the principle we have thus employed is even more gen- 
 eral yet. We have thus far applied it only to regular figures :. 
 it applies to irregular figures equally as well. It is a universal 
 
 principle, by means of which 
 a polygon of any form, regu- 
 lar or irregular, may be con- 
 structed, such that its area 
 shall be equal to that of any 
 other given (regular or irregu- 
 lar) polygon. 
 
 The " universal problem " 
 may then be enunciated as 
 follows : 
 
 If any polygon P' (let it 
 be regular or irregular) of 
 any number, ;/, of sides, be 
 j constructed so that its PC- 
 S' rimeter shall be equal to that 
 of a given polygon P (regular 
 or irregular), and of any num- 
 ber, ;/, of sides, and a third 
 polygon P" be constructed 
 similar to P, but of the same 
 area as P f f then the area of 
 a fourth polygon P'" , similar 
 to P' t and similarly circum- 
 stanced to the circle about 
 which P" is drawn (i.e., of 
 equal apothegms), will be 
 equal to the area of the origi- 
 
 I! 
 
 nal polygon P. 
 
 Let P be any given polygon (regular or irregular), and let n 
 equal the number of elementary triangular segments into which 
 it may be divided. Denote by a and 5 respectively the mean 
 lengths of the altitudes and bases of such segments. Call ;/5 
 
 a 
 
 its " ruling perimeter," then will ;/5 - equal its area. 
 
"THE SACRED CUBIT? 197 
 
 Let it be required to put this area nS - into the form of any 
 
 other polygon P' (regular or irregular) of ri elementary triangu- 
 lar segments. 
 
 By the principles of similar geometric figures, construct P' so 
 that its ruling perimeter n' S' shall be equal to nS, and deter- 
 mine all its elements. Let the resulting mean altitude of its 
 
 triangular segments be a' ; then will its area = f 5V ] = 5;/ . 
 Construct now a third polygon P" y similar to the original one 
 (P), but of area equal to that of P' \ i.e., = Sn-. 
 
 Then, since P and P" are similar, we have 
 
 
 
 a a' 2 
 
 2 ' ' 2 
 
 hence 
 
 / . > a , & 
 
 2 2 
 
 and 
 
 "\ 2 _ a *$ nat _ r 
 
 or 
 therefore, 
 
 { Sn J -5- Vtf#' = 6"' = the ruling perimeter, etc. 
 
 Thus all the parts of P" may be constructed. 
 
 Finally, construct P'" similar to P r (i.e., of the required 
 form), but whose mean apothegm shall equal that of P"\ i.e., 
 a m a"\l~aa f . By similarity of polygons determine all its 
 parts. 
 
 Then 
 
 Area P' : area P"' : : (apothegm P')* : (apothegm P'")* ; 
 
 Sna' 
 
 : area />"'::(<) ^(VW) 2 * 
 
 or 
 
 , , . -,,, x Sna' 
 (a Warea^r ) = 
 
198 METROLOGY. 
 
 Therefore, 
 
 2 (a') 2 2 
 
 which equals the originally given area of P. Q.E.D. 
 
 Hence the universal problem is demonstrated. 
 
 The pyramid diagram (i.e., the "architect's ideal" form 
 thereof) is a geometric representation of that special case of 
 the universal problem which concerns circles and squares. 
 
 Its theorem may be enunciated as follows : If a circle be 
 described whose circumference is equal to the perimeter of a 
 given square, and the area of such circle be put into the form 
 of a second square, then the area of a second circle inscribed in 
 this second square will be equal to that of the original square. 
 
 Proof (algebraic}. Let 2x = the side of any square, R the 
 radius of the equal-perimeter circle, and R' that of the equal- 
 area circle. 
 
 Then %x = perimeter of given square, ^x 2 =. its area, and 
 X2\[2 = its diagonal, etc. 
 
 By construction, it is required that 2-n-R = %x ; and it is to be 
 proved, that, by following the method laid down in the enuncia- 
 tion, Tr(R'Y shall equal ^x 2 . 
 
 Since 2irR = 8x, fi = ~. 
 
 7T 
 
 Area of first circle /. = TrR 2 = TT! ) = 
 
 \7r/ TT 
 
 . /i6x 2 AX i 
 
 Hence the side of an equivalent square = v = -^ = 4x 
 
 0r V/TT 
 
 Now, the radius, R', of a circle that shall exactly inscribe the 
 
 square whose side is 4*-L, is, of course, equal to -| such side; 
 
 VTT 
 
 i.e., = 2^JL- But if 
 
 = 2X 7-, 
 
 vv 
 
 then 
 
 Trf 2JC-i J = TT =4**. Q.E.D. 
 
"THE SACRED CUBIT." 199 
 
 Before turning from the algebraic consideration of the pyra- 
 mid diagram, it will be as well to demonstrate the equality of 
 area which exists between the pyramid triangle, the circle upon 
 its altitude, and the square upon the radius (R f ) of the circle of 
 equal area to the pyramid square. 
 
 (1) We have already seen that R = to altitude of pyramid 
 
 triangle = , and that its base = 2x ; hence the area of this 
 
 7T 
 
 triangle =' J(alt.)(base) = J/^V 2 *) = ~. 
 
 (2) But if R (the altitude of this triangle) is equal to i, 
 ^R = is the radius of a circle upon it as a diameter. The 
 area of this circle = v[ ) 1-U 
 
 \ 7T/ r 
 
 (3) Since, also, R', the radius of the equal-area circle, is 
 
 ^ y 
 
 -y=, its square (i.e., a square constructed upon it as a side) 
 
 VTT 
 
 / "? l'\ ^ A f 2 
 
 = (-7-1=: -t-. But these three areas are all equal. Q.E.D. 
 
 WTT/ T 
 
 The circumference of the circle on the altitude of this tri- 
 angle as a diameter = 4^-. Other interesting relations in this 
 same series of figures (see diagram of " boss ") are as fol- 
 lows : 
 
 Denote the distance GP by y. Then 
 
 y = /T - (/? - R'} = 2^ - R 
 
 Denote by # ^ = ^ the shortest distance from centre (N) of 
 square ABDE to the circumference (P) of circle of equal area. 
 
 R 
 
 x-y = z = - 
 
200 METROLOGY. 
 
 In the case of the boss, these several values are numerically as 
 follows : 
 
 2* = 5; 
 
 R = 3.18309886182322430602500 -f- ; 
 TrR = 10. 2irR = 20 ; 
 7^ = 31.830988, etc.; 
 2^ = ^=6.3661977 + ; 
 
 R i 
 
 - = = .15915494, etc.; 
 
 ^= 2.82094791775 +, etc., = -jL. 
 
 VTT 
 
 2 = 25 ; 
 
 2 = 7-957747I5 + ; 27r^'=Z/7r = 17.7245385091 + ; 
 =jy= 5.64189583550 + ; (D'Y = 31.830988618232 + ; 
 
 ry rx 
 
 - = .564189583550 + = radius of circle whose area = i = . 
 
 Area of A = ^*= 7-957747 + ; y= 2.45879697350 +, etc.; 
 z = x y = .04120302650, etc., = distance PN. 
 
 There are converse and collateral problems to the above 
 which I cannot even touch upon : in fact, I must close the 
 study of the whole topic. The half has not been told. Though 
 I have devoted two hundred pages to this interesting sub- 
 ject, I could as easily devote two thousand ; but, should I do 
 so, even then the topic would have been barely touched upon : 
 we would have the same unsatisfactory feeling at its close, and 
 be oppressed with a sense almost akin to that of mental suffo- 
 cation. Personally, I am dissatisfied almost despairingly so 
 at what is now known of this greatest wonder of the world. 
 Into its mysterious shadows I see, vanishing, threads that ap- 
 parently lead through the labyrinths of every branch of human 
 knowledge. 
 
 The pyramid is indeed a mystery. The Promethean idea so 
 thoroughly pervades its plan and realization, that, by it, high 
 science, accurate history, and perfect art, are blended in most 
 exquisite harmony. 
 
 In it, Art is taught by the beauty, grandeur, endurance, and 
 perfect fitness of the monument itself ; Science by its universal 
 
" THE SACRED CUBIT? 2OI 
 
 earth reference and commensurability ; History by its symbolic 
 architecture ; Religion by the story and purposes of its erection, 
 and by the marvellous prophecies and teachings built in stony 
 parables into its everlasting walls ; Astronomy by master point- 
 ings to the stars and heavenly cycles ; Astrology by the true 
 readings of the constellations, and their order of precession ; 
 Chronology by time references, tubes, and passages, and an in- 
 dex that marks unerringly the initial year of the annus magnus ; 
 Natural GeograpJiy by its unique location at the gate of universal 
 commerce and the centre of the inhabited earth ; Physical Geog- 
 raphy by its thorough knowledge and appreciation of the bear- 
 ings of temperature, pressure, and climate upon normal human 
 life ; Agriculture by its very name, and by the placing in its 
 safest chamber of a just measure of the staff of earthly and the 
 symbol of eternal life ; Mineralogy and Geology by the wisdom 
 displayed in the selection of a material that should not only 
 endure, but secure, a certain constant specific heat and gravity ; 
 Metrology by the irresistible deductions from facts built thus in 
 the form of all its elementary units ; and, finally, Man himself 
 by its practical co-ordination of all the wants of his best develop- 
 ment as a civilized and social being. 
 
APPENDIX B. 
 
 RATIONAL CIRCULAR SUBDIVISION. 
 
"And round about the throne were four and twenty 
 seats : and upon the seats I saw four and twenty elders 
 sitting." REVELATION iv. 4. 
 
 "And under the brim of it round about there were 
 knops compassing it, ten in a cubit, compassing the sea 
 round about." i KINGS vii. 24. 
 
RATIONAL CIRCULAR SUBDIVISION. 
 
 " On this subject he (our ancient brother mason, Pythagoras) drew out many problems 
 and theorems ; and, among the most distinguished, he erected this, which, in the joy of his 
 heart, he called Eureka, in the Grecian language signifying, / have found //.... It 
 teaches Masons ..." TRESTLE-BOARD. 
 
 THE following paper upon the " Division of the Circle," writ- 
 ten by Mr. Jacob M. Clark, C.E., at the request of the author, 1 
 is appended because of its importance. Universal metrology 
 cannot avoid the discussion of the circle ; and it is a matter of 
 supreme importance to the practical man, what mode of treat- 
 ment this primary geometrical form shall receive. Babylon has 
 long advocated, and from her we have accepted, the division 
 into 360 parts of 60 degrees each. Metric France attempted 
 to make the subdivision into 400 parts, and thence decimally 
 downward, but failed. The Great Pyramid advocates, as the 
 truly rational division, one into 240 parts, and thence decimally 
 or otherwise downward. The Tatter will undoubtedly prevail,, 
 and the day of its adoption may now be regarded as close at 
 hand; for its notably neat relation to the standard-time division 
 of the watch, and to the requirements of geography and astron- 
 omy, insure its claims. 
 
 1 NEW YORK, Dec. 22, 1883. 
 MY DEAR LIEUT. TOTTEN, 
 
 I have written out pretty fully my argument on the Division of the Circle. If in any 
 respect it will be of u,se to you, I shall be pleased. I know it may be found defective in 
 logical arrangement ; for, you know, I have to write a sentence at one time, and another at 
 another, under different surroundings. You may, therefore, find it " candid, discursive,. and; 
 didactic." 
 
 I am fully aware that habit, antiquity of usage, prejudices of education, politico-economK 
 cal influence, and the like, will be generally against the suggestion. They are the same 
 " thrones, principalities, and powers," that are always invoked to bolster up that old serpent, 
 the Devil, and original sin. . . . 
 
 Truly yours, JACOB M. CLARK. 
 
 205 
 
205 METROLOGY. 
 
 THE DIVISION OF THE CIRCLE. 
 
 From the nature of the problem, the argument must be in 
 part tentative and inductive, rather than synthetic or analytical. 
 
 Hitherto, in the treatment of the circle for primary arcs, 
 geometers seem to have stopped at the -^. Below this they 
 have for many centuries divided the circle by 24 for the diurnal 
 cycle, and for general purposes, somewhat arbitrarily, by 36, 
 with continued sexagesimal subdivision in each case. 
 
 In strict geometry, so far as known, the only general method 
 of subdividing circular arcs is by simple bisection. There are, 
 however, a limited number of commensurable arcs which can be 
 found by other methods. 
 
 Taking radius, or some aliquot part of it, as a function, we 
 easily obtain the |, |, -J, \, J, J, and Jg. 
 
 By means of extreme and mean ratio applied to radius, we 
 have, by a very simple process, the \ and T ^. 
 
 It is by comparison of the \ with the -^ .that we obtain 
 
 the TV 
 
 Now, while comparison is not a general method of subdivision 
 in circular arcs, for the purpose of obtaining a common measure, 
 it is susceptible of being carried somewhat farther. To under- 
 stand this more fully, let us apply the general method of bisec- 
 tion to the arcs already found. The result is the following 
 fourfold series : 
 
 A. i, i, i, i 
 
 It is observable in the above group, that the smallest com- 
 mensurable arcs which have commensurable functions are the 
 J- and the ^ Compare them, and their greatest common 
 divisor is the ^V From the distinctive peculiarity expressed, 
 it may be regarded as the major unit of circular measure, unless 
 reasons can be afterwards found for displacing it from that 
 position. We already see that the ^ can be obtained without 
 bisection Let us see how far we can carry this principle of 
 comparison without resorting to bisection at all. 
 
RATIONAL CIRCULAR SUBDIVISION. 207 
 
 By comparing the \ with the \ we obtain the ^. 
 
 By comparing the -fa with the ^ we obtain the -$. 
 
 By comparing the \ with the T ^ we obtain the ^. 
 
 By comparing the -fa with the -fg 1 
 Or by comparing the T X Q with the -fa we obtain the 
 Or by comparing the ^ with the -fa 
 
 By comparing the ^V w ^h tne A 
 Or by comparing the -fa with the ^ 
 Or by comparing the ^ with the A 
 Or by comparing the ^ with the ^ 
 
 It is apparent that we can proceed no farther without resort- 
 ing to bisection somewhere. And it is evident, too, that the T | 7 
 is the greatest common divisor of all the arcs in the first four 
 columns, i.e., of all the commensurable arcs which have com- 
 mensurable functions, and of those which are obtainable by 
 means of extreme and mean ratio, and of those which result 
 from comparison of these arcs among themselves, taken together. 
 
 Now, if we bisect all the arcs in the fourth column, it is ap- 
 parent that the ^\-$ is the greatest common divisor, not only of 
 all those in the fifth column, but of all in the other four. And 
 the same arc, the ^ J 7 , is also obtainable by bisecting some other 
 members of the fourth column, and then instituting again the 
 principle of comparison. That is, 
 
 iV ~~ Ar = sV 
 
 A - A 
 
 Or sV ~~ T?(T ! 
 
 Or A- A ~ mm 
 Or T V - A 
 
 So that, after all, it is not strictly a matter of indifference 
 where we resort to bisection, if, after having proceeded as far 
 as the T ;]-Q, we desire to secure the most elementary result ; i.e., 
 to exclude unnecessary repetition of methods. We see that we 
 need not resort to bisection directly more than once in order to 
 secure the desired result, so far as it can appear in the fifth 
 column. 
 
 And, in this search for the purest division, it has seemed to 
 me a postulate, that all kno'wn methods should be exhausted, 
 
208 
 
 METROLOGY. 
 
 but that, in order to secure the greatest pure common divisor 
 of the results, no method should be repeated, unless it be that 
 of making a final comparison, after all the methods shall have 
 been once applied at the most primitive or fundamental point. 
 Otherwise the result would be vitiated by unequal weight 
 among the methods. 
 
 Accordingly we are led to examine whether there is not some 
 one of these arcs in the fourth column which has some peculi- 
 arity not possessed by the others, and which connects with all 
 the methods, and which will therefore indicate precisely where 
 
 the method of bisection ought, 
 firstly and finally, to be applied. 
 Now, the octant is precisely such 
 an arc. It is the largest arc in the 
 fourth column. Moreover, we have 
 just seen, that, by bisecting it, we 
 can obtain by comparison all the 
 other members of the fifth column, 
 and among the rest the 210 m 
 several ways. 
 
 It is also the only known com- 
 mensurable arc with a commensur- 
 able function, which is at the same 
 time the sum of two incommensur- 
 able arcs, each of which has a com- 
 mensurable function. Its tangent 
 is equal to radius ; and it is composed of two incommensurable 
 arcs, whose tangents are respectively equal to J and \ radius. 
 This latter property, well known to analysts, may, without 
 resorting to the rules of trigonometry, be shown by a strictly 
 geometrical test, which, while it reveals the wonderful power of 
 the Pythagorean form of the right-angle triangle, 3-4-5, at the 
 same time discloses a direct connection of the octant with all 
 the methods, such as cannot be found in any other known arc. 
 
 In the figure, let the radius CA of the main circle equal 10. 
 Bisect this radius in c, and with Cc Ac = 5 as a radius de- 
 scribe the minor circle CbAd. Upon opposite sides of this 
 minor diameter apply the 3-4-5 triangle in reverted positions, the 
 acute angles resting in c. The complementary angles will be in 
 
 THE 240 DIVISION 
 Fig. 48. 
 
RATIONAL CIRCULAR SUBDIVISION. 209 
 
 the minor circumference at b and d respectively, and the angle 
 bed will be a right angle. Draw the minor chords bA and dA ; 
 also Cb and Cd, produced indefinitely, so as to intersect the 
 greater circle in B and D respectively. From D draw a tangent, 
 intersecting CB produced in 7! Draw also through A a tan- 
 gent, intersecting Cb ajid Cd produced in E and F. The arc 
 BAD is an octant, whose tangent DTis known to be equal the 
 radius ; and, by simple triangles, the tangents AE and AF oi its 
 component arcs are equal respectively to CA and CA. The 
 angle ECA is the one by means of which we arrived at extreme 
 and mean ratio, and obtained the chords of the \ and ^. 
 
 The tangent of the octant BCD is equal to the chord of the J. 
 Half of it, the tangent of BCA t is equal to the cosine of the J 
 and the sine of the -^ ; taken as the height of a segment, it 
 gives the chord of the ^. If the height of a triangle equals the 
 radius, it is the factor by which we multiply the base to find the 
 area, and, indeed, the circumference to find the area of a circle. 
 One-third of it, the tangent of A CD, is the factor by which we 
 multiply the area of the base of a cone or pyramid (the*height 
 being equal to the radius), or the surface of a sphere, to obtain 
 the solidity. 
 
 Thus far these properties of the octant have been examined 
 in the light of simple geometry. It may now be added, that 
 analysts have found (see Calculus), that, by a simple but slowly 
 converging series founded on the tangent, we can compute the 
 length of the arc, and so of the circumference ; and that two 
 analogous but rapidly converging series founded on the tangents 
 of its component parts, furnish the best and shortest known 
 means of calculating the length of the circumference to any 
 required degree of exactness. 
 
 The octant, then, though not a unit of circular measure in the 
 strictest and most exhaustive sense, is a dominant arc. Being 
 thus related to all the other arcs, and to the methods by which 
 they are obtained, and being itself founded on the perfect appli- 
 cation of the square to the circle, it furnishes the final and 
 crucial test. And, from the foregoing considerations, it is 
 claimed, that as a matter of pure geometry, and in every as- 
 pect of mathematical propriety, we should bisect the octant in 
 preference to any other arc. 
 
210 METROLOGY. 
 
 Bisect it (see figure) in G' . BG, being equal to DG, is then 
 the -j 1 ^ Compare this with the y 1 ^, which, as we have seen, 
 exhausts, on the principle of non-repetition, all the methods ex- 
 cept bisection, and we obtain the 2 | 7 , the greatest common 
 divisor of all the geometric arcs down to that limit. 
 
 And we see also, that this division by 24 emphasizes the 
 geometry of the round bodies in a manner which no other 
 grand division of the circle can possibly do. From the cylin- 
 der on the diameter down to the cone erected upon radius, all 
 the relations are expressed in terms of integer hour-arcs. 
 
 We have, then, 
 
 First, As a grand unit of circular measure, the ^, or hour- 
 arc, the greatest common divisor of all tJie commensurable arcs 
 which have commensurable functions. 
 
 Second, As to the method of subdivision, we have, for the 
 geometric degree, the greatest common divisor, so to speak, of 
 methods and commensurable functions. 
 
 And, between the two, the modulus is the product of the only 
 two geometric divisors which subsist ; i.e., 2 X 5 = 10. And, 
 between those limits, that modulus, and that alone, covers all 
 the geometric arcs. Below the 2 -J^, so far as we know, there is 
 but one general method, bisection. And for finer divisions we 
 must resort to some empirical method, unless we choose to 
 continue the binary arithmetic indefinitely. But the general 
 modulus of arithmetic being 10, it is difficult to divine a reason 
 why the decimal method should not prevail indefinitely down- 
 ward from the ^\-$. 
 
 Scholium. This decimalization accords with the general 
 practice of the ancients, so far as we can judge from their itin- 
 eraries. Having fixed the major unit, they subdivided decimally. 
 The Egyptians so subdivided from the \, the Persians from the 
 J, the Syrians from the -^g, and the ancient Hebrews from the ^. 
 Eratosthenes seems to have undertaken it from the -^, and 
 Posidonius from the %. The French have latterly attempted 
 the same thing from the quadrant. 
 
 We may here notice that the number 3, as a circular divisor, 
 disappears below the fa. It is peculiar to circular measure, but 
 only above that limit. Below that, it can only be introduced 
 empirically. There is no reason why it should be introduced into 
 
RATIONAL CIRCULAR SUBDIVISION. 211 
 
 the general modulus of arithmetic at all. And the prime 
 division of the circle by 36, or by 18, is as ungeometrical as that 
 by 25. And the sexagesimal principle of subdivision is doubly 
 so. The other grand divisions spoken of are more rational, but 
 imperfect because not exhaustive. 
 
 Precisely when the present division of the circle into 360, 
 and the sexagesimal principle of subdivision, were invented, or 
 whether the two were coeval, is not certainly known. They 
 are traceable to Babylon, and generally assigned to the first, or 
 Nimrodic, monarchy. The whole structure, like its cousin the 
 duodecimal arithmetic, is the work of mathematical cranks, who 
 have attached undue significance to the multiples and powers of 
 the number 3. The systems just referred to, the Hebrew in 
 its decimalization below the grand unit (which was double of the 
 one in this system), and the others throughout, are strong 
 historical protests against it. They have failed, from political 
 causes in the main, but partly through imperfections in the 
 grand division. An exception in the latter respect appears in 
 the conclusion of Posidonius (about 90 B.C.), who, whether by 
 accident or design, seems to have struck the true geometric 
 division. The combined effect of causes, however, which have 
 contributed to the prevalence of the present system, belong to 
 the historian rather than to the mathematician. 
 
 But, in view of the present wide-spread discussion on the sub- 
 ject of simplifying the correlation of our own weights and meas- 
 ures, the question is put to thinkers plainly, and with some 
 confidence, whether, in this matter of circular division, the 
 gradual displacement system by one founded on the strictest 
 demands of simple geometry, will not, in the end, as a matter 
 of general economy, by simplifying relations, and reducing the 
 aggregate labor of operators and calculators in all departments 
 of applied mathematics, be worth a thousand times more than 
 
 its cost. 
 
 JACOB M. CLARK. 
 NEW YORK, Dec. 22, 1883. 
 
APPENDIX C. 
 
 METROLOGICAL NOTES AND QUERIES. 
 
"Who hath measured the waters in the hollow of his 
 hand, and meted out heaven with the span, and compre- 
 hended the dust of the earth in a measure, and weighed 
 the mountains in scales, and the hills in a balance?" 
 
 ISAIAH xl. 12. 
 
METROLOGICAL NOTES AND QUERIES. 
 
 " The sands and their numbers I know, 
 And the measure I know of the ocean." 
 
 DELPHIC ORACLE. 
 
 [NOTE I.] 
 THE ORIGIN OF THE GRAIN. 
 
 HASWELL places the average weight of a bushel of wheat at 
 60 avoirdupois pounds. Hence, as there were 32 bushels in the 
 ancient coffer (and Anglo-Saxon chaldron), its weight of wheat 
 was 1,920 avoirdupois pounds. It will be noticed that 10,000 
 times this number, 19,200,000, is the weight of our rectified 
 coffer in "grains." But the involved relation is even more 
 remarkable when we count the kernels of wheat thus weighed. 
 Now, there are, upon an average, 17,812,500 d= kernels of "ripe 
 wheat taken from the middle of the ear" in a cofferful. 1 
 
 As the capacity, therefore, of the coffer, is 71,250 cubic 
 inches, each cubic inch of space will hold exactly 250 kernels 
 of ripe wheat. It will be noticed that this quantity is remark- 
 ably close to the number of weight grains (252.6937) now taken 
 as the value of a cubic inch of pure water at its maximum den- 
 sity (39.8 Fah., 30" bar., Haswell) ; so that we cannot refrain 
 from conjecturing that this fact may have had at least a remote 
 bearing upon the original selection of the particular number of 
 elementary Anglo-Saxon weights in a cubic inch of the natural 
 weight material, water. It is, therefore, probable, that at stand- 
 ard circumstances (ther. 68 Fah., bar. 30", and hygro. \ wet) 
 the average number of wheat-kernels which could be contained 
 in the volume occupied by a pyramid cubic inch of water would 
 not materially differ from 250, and the division of this amount 
 
 1 Haswell estimates that a bushel contains 556,290 kernels ; hence, even at this estimate, 
 32 such bushels would contain 17,801,280. 
 
 215 
 
2l6 METROLOGY. 
 
 of water into the same number of elements would thus arithmo- 
 graphically realize the actual bread capacity (i.e., in wheat-ker- 
 nels) of the same cubic volume. 
 
 To many, of course, the unlooked-for character of these re- 
 sults, and their surprising fitness, stamp them at once with 
 such an air of mystery, that they will be disposed to reject them 
 as suspicious, and as merely the products of so-called " trick- 
 arithmetic." But to others they will bear upon their face a 
 stamp of too much genuineness to be set aside at the mere 
 objection of the doubter, who ignores the very records of the 
 subject under discussion. We know, for instance, that it was 
 in just such a way that our own, not so very remote, forefathers 
 actually arrived at their three-barley-corn inch. So, too, by the 
 entirely independent regulations of ancient China, an inch was 
 by statute made equal to the average breadth of 10 millet seeds, 
 and that of 100 millet seeds, or 10 inches, measured and estab- 
 lished the standard foot. The very next step in such a system 
 must have been to obtain the average number of such elements 
 in a superficial area, and the next the number in a given and 
 handy cubic volume. Such steps would have been, in fact, the 
 only legitimate and logical ones in the passage from linear 
 measure to that of area, and thence to that of simple capacity. 
 Sq likewise, in passing from capacity to weight measures, it 
 would have been, and was, most natural to observe the arith- 
 metical relation for some time, even after another substance 
 water had been found to be the most fitting basis for the 
 latter. In some such way alone could the numerical sequence 
 and rationale have been preserved ; and that it was so preserved, 
 there is ample testimony at hand in the history of metrology. 
 
 Grain measure was the end and object, as its name implies, of 
 the pyramid (pyros-metron) ; and the founding of an elementary 
 weight upon the arithmetical value of the volumetric capacity 
 for wheat-kernels would, in view of the beauty of the resulting 
 subdivision, have been neither unwise nor impracticable. But, 
 be this as it may, the fact of their remarkably close approxima- 
 tion remains, and, as we have seen farther back, gains credence 
 from entirely independent sources. 
 
 The objection, that our results are only those of "trick-arith- 
 metic," is of no moment; for such an objection holds 6 ; ually 
 
METROLOGICAL NOTES AND QUERIES. 217 
 
 good against any system. It would sweep away our whole deci- 
 mal system, and brand as unworthy any employment of the 
 properties of the number 9 in mathematical work. The term, 
 "trick-arithmetic," is, in fact, short-sighted, and has been in- 
 vented by some very poor mathematician to cover his ignorance 
 of the properties of numbers ; for it does not exist in the sense 
 intended. 
 
 But, before closing this subject, let us examine its numerical 
 bearings a little further. Since there are 1.78125 cubic inches 
 of volume in the ounce of capacity, and since each such cubic 
 inch holds on an average 250 kernels of ripe wheat, the number 
 of such kernels in an ounce of capacity is 445.3125 : 10,000 
 times this is therefore the number contained in a "quarter." 
 There being 16 ounces in the pint, it will contain 7,125 wheat- 
 kernels, a number which is one-tenth the cubic capacity of the 
 chaldron itself in inches. Now, one-half of 445.3125 is 222.65625, 
 and represents the number of wheat-kernels in half a capacity 
 ounce. The latter number, however, is one-tenth the volume in 
 dubic inches of the thirty-second of the chaldron, which is the 
 bushel; that is, the bushel contained 2,226.5625 cubic inches, 
 and 556,640.625 kernels of wheat. But as the bushel weighs 
 only 60 pounds, or the coffer of wheat weighs but 1,920 pounds, 
 it would have been balanced by 1,920 X 28.5 cubic inches of 
 water 54,720 cubic inches. Hence, by weight, a bushel of 
 wheat was balanced by 1,710 cubic inches of water. The ques- 
 tion now arises, whether, at any time in the past history of Saxon 
 metrology, such a method of measuring a bushel of wheat 
 namely, by balancing it against what was very nearly a cubic 
 foot of water (^1710 = 1 1.96") could have obtained? The 
 answer is, that it would have been a most natural method, and 
 one measuring capacity by weight still very common in the 
 commerce of cereals, etc. The next question which presents 
 itself, is, whether the measure intended to hold the water might 
 not, either through the ignorance or cupidity of an illiterate age, 
 have come at length to be employed as the actual measure of 
 the wheat itself ? Such a misappreciation of the original system 
 may easily have come about, and its assumption may perhaps 
 shed some light upon the origin of the very short bushel of the 
 fifteenth century. 
 
218 METROLOGY. 
 
 The system demands that a thirty-second of a coffer, or the 
 capacity of 2,226.5625 cubic inches of wheat, shall be balanced 
 by 1,710 cubic inches of water, and that each volume shall weigh 
 exactly 60 pounds. On the other hand, a thirty-second of the 
 water weight of the coffer would give 78-^ pounds, while 1,710 
 cubic inches of wheat would have weighed but 46^5 pounds. 
 
 It is a noticeable fact, that, in the reign of Henry VII. (A.D. 
 1496), one of the most archaic periods in the history of 
 British metrology, two just such bushels were in actual use. 
 The one a bushel which the people at length refused to toler- 
 ate was of but 1,792 cubic inches capacity, and yielded a coffer 
 of only 57,344 cubic inches. The other was a wonderfully close 
 approximation to the ancient standard, and held 2,224 cubic 
 inches, giving a coffer of 71,168 cubic inches, or one only 
 gig short of the true standard. In the very next reign, the 
 gallon was fixed at 282 cubic inches (very close to the rectified 
 value, 285), the bushel at 2,256, and the chaldron therefore at 
 72,192 cubic inches. 
 
 Now, in all of these approximations and returns to values 
 which are pointed out as probably related to those of the 
 ancient pyramid standards, there is certainly too much system 
 to be attributed solely to blind chance and accident. President 
 Barnard considers that it "argues lunacy" in the man who 
 makes the statement that the pyramid coffer has been the 
 standard from which the corn measures of Great Britain have 
 been derived. It strikes us, rather, that he who persistently 
 resents the natural conclusion that these things are related, and 
 must have been somehow related, no matter how remotely, in 
 the minds of the originators of the Anglo-Saxon metrology, 
 violates every rule of logical ratiocination. Of course, these 
 measures have varied more or less during the forty centuries 
 that have intervened since they first "came up out of Egypt." 
 We do not presume to maintain that they have been miraculously 
 preserved always exact, any more than in the face of facts we 
 presume to maintain that the English inch of to-day is abso- 
 lutely equal to the American inch, and that both are, or either 
 of them is, at their ancient cosmic value. Nevertheless, it may 
 be fairly maintained, that these measures have not varied in the 
 long-run "materially." The proportions of an ordinary door 
 
METROLOGICAL NOTES AND QUERIES. 219 
 
 are probably about the same the world over, and have probably 
 been maintained at a constant average ever since the human 
 structure attained its normal height. So, too, ever since wheat 
 became the staff of life, and so long as it shall continue to be a 
 staple of national and international commerce, the ratio of its 
 bulk and weight, its specific gravity, in fact, has maintained, and 
 will continue to maintain, its bushel around that handy size 
 which experience finds not only to facilitate its ready manipu- 
 lation, but which the human mind recognizes as a direct refer- 
 ence to the necessary "tabular numbers " of general metrology 
 which we have inherited. But there are entirely independent 
 considerations which go to establish most conclusively a direct 
 connection, through blood and race inheritance, between the 
 measures of northern Europe and those of ancient Israel and 
 Egypt. Unfortunately, for lack of space, we cannot touch upon 
 these at all in this place ; and probably, if we could, they would 
 be classed by Mr. Proctor, President Barnard, and their follow- 
 ers, in the catalogue of "the vagaries of lunatics." Our object 
 is now, more particularly, to establish the eminent superiority of 
 present Anglo-Saxon metrology over all other competitors, and 
 to show how little rectification will be required to render it a 
 system whose prestige can never thereafter be shaken. 
 
 The objection may be raised at this point, that 250 is not 
 the number arrived at in our investigations as that expressive, 
 of the grains of weight in a cubic inch of water, and whereby it 
 is advised that our present system of metrology be rectified.. 
 The latter number is 15360-^57 = 269.47+, or is 19.47 + 
 units too great. But an examination will show that this objec- 
 tion is itself without weight. The 250 grains so closely approxi- 
 mated to by the grain weight now in Anglo-Saxon use, were 
 originally grains of capacity, not of water weight, and indicate 
 the average number of ripe wheat-kernels from the middle of 
 the ear that a cubic inch of volume will actually hold. We thus 
 perceive that there is an entirely different stand-point from 
 which to consider this question, and one far more appropriate ; 
 namely, from that of weight itself. Let us therefore examine 
 the subject from this new stand-point. 
 
 A cofferful of wheat-kernels will weigh less than one filled 
 with water in the ratios of their respective specific gravities. 
 
220 METROLOGY. 
 
 Now, as a cofferful of wheat contained 32 bushels of 60 pounds 
 each, it weighed 1,920 pounds, or balanced 1920 X 28.5 = 54/20 
 cubic inches of water. But a cofferful of water contained 71,250 
 cubic inches. Hence the specific gravities of wheat -kernels and 
 water are as 
 
 54720 : 71250, or as 192 : 250 ! 
 
 From the foregoing it follows, that T -^ f tne weight of an 
 even cofferful of wheat-kernels will exactly balance ^ f a co ^ 
 ferful of water. This is the weight of 285 cubic inches of water. 
 Now, if a ioo,oooth of this volume be taken as a terminal 
 quantity, so far as weight is concerned, it will be equivalent to 
 .00285 cubic inches of water at standard circumstances, a value 
 already arrived at without reference to any kind of cereal as the 
 true "ultimate" of our rectified system. 
 
 But since 192 times the water-filled coffer is equivalent in 
 weight to 250 times that of the same space filled with wheat- 
 kernels, one of the water parts will weigh \\\ i. 30208 J parts 
 of the terminal (.00285), just determined ; i.e., will be equal to 
 .0037109375 (exact) cubic iaches, or give us the actual value of 
 our rectified weight-gram ! And the standard weight of a kernel 
 of ripe wheat taken from the middle of the ear will therefore be 
 .003072 (exact) cubic inches of water, or 1.0778947357 "ulti- 
 mates," or .82782315520 of a "grain." 
 
 It will take 322.265625 (exact) such kernels to balance a cubic 
 inch of water ; and, finally, the specific gravity of such a stand- 
 ard kernel will be exactly .768 = 1536-!- 2. What the modern 
 determination for this latter quantity the specific gravity of 
 ripe wheat in the kernel actually is, I have no means of 
 knowing, as none of the tables within my reach give it ; so I 
 leave it for future determination, and as a sort. of crucial test of 
 some of these deductions. 
 
 It is of no consequence to me in what particular method these 
 facts were arrived at in ancient Egypt. They are matters of 
 ratio and number, and could have been determined by treating, 
 as we have done, any equivalent volumes of wheat, water, and 
 5.7 density material ; i.e., by comparison and by extended series 
 of actual experiments as to weight and number of kernels, or by 
 a study of number in the abstract, cubic volume, and mean den- 
 
METROLOGICAL NOTES AND QUERIES. 221 
 
 sity. The facts themselves are not manufactured, but are of 
 cosmic import, and the question a scientific one as to the actual 
 ratios running through nature. 
 
 President Barnard ridicules the idea of man having, at that 
 early day, been able to attain by scientific process and discovery 
 to any such accuracy of knowledge as the pyramid theory de- 
 mands. In this position we are disposed in part .to agree with 
 him. From the stand-point occupied by the closely corporated 
 modern school of science, there is no other view to be taken. 
 But the dilemma is only intensified by President Barnard's 
 claim, for mere ridicule and denial will not sweep away such an 
 array of data as are built into the Great Pyramid. Whether 
 determined or not by self-education, and understood or not in 
 the land that the monument overshadowed, they were none the 
 less actually there, if only to be locked up and sealed down in 
 everlasting masonry. And this, after all, is the whole gist of 
 the mystery of this sign and wonder of the land of Ham, that 
 such knowledge should have been possessed, and not under- 
 stood, unless we accept at once the religious theory of the 
 monument, and admit that it may have been, and probably was, 
 made after a model shown in vision, as were the tabernacle, the 
 temple, and the still earlier ark of Noah. 
 
 As a last resort, we may imagine some objector claiming 
 that the whole of this system is the mere result of " arguing in 
 a circle." Such an objection, however, though easily made, is a 
 very powerless one, when we consider the great antiquity of the 
 system, its actual use in Anglo-Saxondom from time immemorial, 
 the \videly spread traces of it all over the earth, its eminently 
 practical character, its scientific beauty, its cosmic, geometric, 
 and numerical features, and its endless possibilities. So that, 
 even were the objection a valid one, we would be tempted to 
 amend and enlarge it by calling it an argument about the sur- 
 face of a sphere, and congratulate ourselves that we had found 
 so firm a footing. 
 
 To recapitulate, therefore, the facts we have been consider- 
 ing, we subtend the following table : 
 
 250. = No. of kernels of wheat in i cu. in. volume. 
 
 322.265625 = No. of kernels of wheat to balance i cu. in. of water. 
 269.47368-}- = No. of weight "grains" to balance i cu. in. of water. 
 
222 METROLOGY. 
 
 350.87719+ = No. of weight " ultimates" to balance i cu. in. of water. 
 
 371.09375 = No. of cu. in. of water to balance 100,000 weight grains. 
 
 307.2 = No. of cu. in. of water to balance 100,000 kernels wheat. 
 
 285. = No. of cu. in. of water to balance 100,000 weight ult. 
 
 400. = No. of cu. in. occupied by 100,000 kernels wheat. 
 
 7125. = No. of wheat-kernels in a pint measure. 
 
 9277.34375 = No. of wheat-kernels in an avoirdupois pound. 
 
 .768 = standard specific gravity of ripe wheat. 
 
 [NOTE II.] 
 UNITS AND CONSTANTS. 
 
 In due time science will undoubtedly be forced to establish 
 a uniform system of logically correlated units. Chaos itself 
 was order compared with the confusion of terms, ideas, and 
 definitions upon which the present scientific schools base their 
 investigations. Each branch of scholars seems to have evolved 
 its own units with closed doors, and to have jealously excluded 
 all recognition of the fundamental principles of others. Metrol- 
 ogy, however, is the universal bond which unites knowledge 
 upon all subjects ; and, until its true prerogatives are duly 
 appreciated, the deductions of scholars cannot be more than a 
 mere patchwork of disconnected results. Harmony of funda- 
 mental principles is the substratum underlying all the various 
 forms of Nature's manifestation, but discord is now written 
 across the tablet upon which the modern schools have rudely 
 scribbled their uncertain " constants." 
 
 The result is, that in order for one school to "tie on " to the 
 facts of another, often involves the translation of an almost in- 
 expressible function. Could liberal representatives from the 
 colleges of every branch of knowledge but unite in a convention 
 upon " Terms and Units," the elevated site might soon be cleared 
 away upon which to erect a monument as stable as that of an- 
 cient Egypt ; but until there is some such unity of purpose, 
 governed by the logic of harmony, we cannot expect our fabric 
 to be more than one of clay and slime upon a sandy plain. 
 
 Now, it seems to the writer that no definition of a unit should 
 involve terms, which, in the several lines of knowledge involved, 
 are not also units; i.e., of the same "degree." For instance, 
 
METROLOGICAL NOTES AND QUERIES. 223 
 
 pounds and feet are, in their order, of an entirely different de- 
 gree from seconds of time. The latter is a real unit. The 
 former are collections of units, and therefore, logically, can no 
 more come together in the definition of a compound unit than 
 bushels and dollars can be logically compared with each other 
 on the same sides of a proportion. A minute in time corre- 
 sponds to the foot of distance, and the pound of weight, just as 
 inches, ounces, and seconds correspond ; and until " unity," as a 
 fundamental idea, means numerically I, we can have no proper 
 starting-point for universal metrology. 
 
 We repeat, therefore, what seems to be a self-evident demand 
 of the logical processes of the mind, that the definition of a com- 
 pound unit should accept no terms which are not respectively 
 the well-known units in their several and individual lines of 
 study. In the investigations of science conducted upon such 
 foundations, there could be no misconceptions ; and, wherever 
 the wheels of one branch come into contact with those of 
 another, we should have a true tangential contact, and not a 
 mutual interference, caused by the cutting of the one into the 
 other. 
 
 For instance, let us try and define a compound unit, that of 
 "the RATE of '&ork" in a logical manner. This is a proper sub- 
 ject of measurement, and must be measured by units if we aim 
 at fundamental facts, and if we wish our definition to be directly 
 available, as unity, in all of the branches involved. In general 
 terms, therefore, we should define the "unit of the rate of work" 
 as the amount of force necessary to raise a unit of weight to the 
 height of one linear unit in one unit of time. If we substitute 
 in the above the actual elementary units, and remain logical, we 
 cannot avoid defining it as, the amount of force necessary to 
 raise one ounce, one inch, in one second. If we confuse the de- 
 grees of the things involved, our result has at once an entirely 
 different meaning. Thus the common definition, "the quantity 
 of force necessary to raise one pound one foot high in one sec- 
 ond," is in reality a standard of the rate of work, and not its unit ; 
 since the resistance overcome, and the distance passed over, are 
 the common standards of weight and length, and not at all their 
 units. In terms of the true units, this criticised definition is 
 therefore as follows : The quantity of force necessary to raise 
 
224 METROLOGY. 
 
 sixteen units of weight to a height of twelve units of distance, in 
 one unit of time and the objections to its fundamental charac- 
 ter, as a "unit," become instantly apparent. It will be replied 
 by some, that, after all, this is merely a captious misappreciation 
 of terms, and that the pound and foot are truly scientific units. 
 If this is so, then a minute instead of a second should be taken 
 ,as the unit of time ; for it is of the same degree. But, so far as 
 time is concerned in the working of human mechanisms, the 
 minute would be utterly impracticable. The velocity of even 
 the longest ranged projectile could not be appreciated upon 
 such a basis, and the legitimate unit of time a second is 
 naturally retained from sheer necessity. 
 
 And here it should be premised, that, as the structure of a 
 series of units upon such principles will involve no ideas save 
 those of a primary character, no further reductions can ever 
 after be necessary, or even possible ; while, from them, standards 
 of any degree of magnitude and of any degree of complexity 
 that special circumstances may seem to warrant, can instantly 
 be formed. Thus, for instance, suppose the rate of work of a 
 machine is found to be 691,200 units ; then, from the very nature 
 of our unit, we may at once say, that the rate of the machine is 
 also 6,912 standard foot-pounds, or 3,600 common foot-pounds, or 
 that its rate is if common foot-tons, or that it will accomplish 
 1 08 foot -tons per minute, or 6,480 per hour, or do the work of 
 4|f f horses per day, etc. 
 
 We do not propose to attempt here the definition of all the 
 units of science upon the basis discussed. It clearly involves 
 the united interest and study of liberal specialists to accomplish 
 such a task. But leaving out of present consideration those of 
 electricity, magnetism, sound, light, and of many other branches 
 whose constants are even yet very little understood, we will 
 venture to define a few of the more common units from this 
 stand-point : 
 
 Unit of velocity equals a uniform motion equal to one linear 
 unit per unit of time (i second). 
 
 Unit of force equals that force, which, uniformly accelerating a 
 unit of weight for a unit of time, would produce a unit of velocity. 
 
 Unit of work equals the amount of force necessary to raise 
 the unit of weight through the height of one linear unit. 
 
METROLOGICAL NOTES AND QUERIES. 22$ 
 
 Unit of rate of work equals the amount of force necessary to 
 perform a unit of work uniformly in a unit of time. 
 
 Unit of temperature equals the amount of heat necessary to 
 -raise the thermometric column one standard degree at standard 
 circumstances. 
 
 Unit of heat equals the quantity of heat necessary to raise a 
 unit weight of water through a unit of temperature. 
 
 Unit of circumference equals one degree of a circle. 1 
 
 Unit of angular motion equals a uniform motion about a centre 
 equal to one unit of circumference per unit of time. 
 
 [NOTE III.] 
 
 THE DIMENSIONS OF THE COFFER. 
 
 The labors of Mr. W. M. Flinders Petrie, detailed at length 
 in a handsome octavo of 250 pages, comprise the latest work 
 that has been carried on at Gizeh with a view of obtaining 
 both the present measures of the Great Pyramid, and some idea 
 of the rationale upon which its proportions were laid out. 
 
 The book, unfortunately, is written in a most pointed animus 
 against the metrological, or so-called religious, theory of John 
 Taylor and Professor Smyth, and, as such, has been gladly pub- 
 lished almost entirely at the expense of the British Association. 
 So far, however, from accomplishing its ends, the volume has 
 been proved by many able reviews to establish beyond perad- 
 venture the soundness of John Taylor's original propositions. 
 
 We do not intend to review this work ourselves ; since we take 
 it for granted that all who are disposed to study the significance 
 of the proportions of the Great Pyramid fairly, and upon their 
 own merits, and who have Mr. Petrie's book at hand, will also 
 post themselves upon the able replies which the deductions con- 
 tained in this volume have already received at the hands of 
 writers in "The International Standard" (see volumes for No- 
 vember, 1883, January and March, 1884). Mr. Petrie's measure- 
 ments themselves, however, are generally excellent, and will 
 always occupy a dominant position in future studies of the monu- 
 ment. Compared one by one with those of Professor Smyth, they 
 
 1 See Appendix B. . 
 
226 METROLOGY. 
 
 will be found to completely vindicate the entire honesty of the 
 latter, and to establish more firmly than ever the footing of 
 the metrological school. Indeed, it is not a little strange, that 
 arriving, by an extended series of measurements, at almost the 
 identical figures of Professor Smyth, and profiting by all the study 
 of their proportions and ratios which had gone before him, Mr. 
 Petrie should have fallen into such serious error as to his deduc- 
 tions, and have allowed his prejudices to so seriously warp his 
 better judgment. Elated at his misconceived measurement of 
 the side of the fundamental square base of the pyramid, Mr. 
 Petrie considers he has ruined at the foundation the whole 
 theory of Professor Smyth, and, believing that he has estab- 
 lished the tombic theory, ironically implies that all the loftier 
 ideas of the John Taylor school must be buried in the sarcopha- 
 gus. So far, however, from coming with Mr. Petrie, and his too 
 easily persuaded American friend, to a gloomy funeral, the modern 
 builders of the pyramid theory may truly rejoice, and shout for 
 joy, at the fitting way in which they now may crown the edifice, 
 and cry, Grace unto its capstone ! Berwick, Smyth, and Wood 
 have completely vindicated, from Mr. Petrie's own labors, the 
 365.242 cubit length of the horizontal base side at the level of 
 the lowest socket. Little, therefore, remains to be accomplished 
 in this direction. So far, however, from weakening in other 
 directions the firm conviction of all earnest students of this 
 monument, that it was built harmoniously from block to block, 
 and from base line to vertex, upon some consummate mathemati- 
 cal and cosmic scheme, Mr. Petrie's whole work, and many new 
 ratios and proportions yet to be studied, and first discovered by 
 himself, lead us to even firmer conviction that the monument is 
 yet to assume proportions far beyond the most sanguine antici- 
 pations of those who look upon it as "a sign and wonder to the 
 Lord of hosts." The particular dimensions of the elevated 
 base line that is, of the square marked out by the cutting of 
 the monument through its own pavement level, and which Mr. 
 Petrie makes to be around 9,068.8 inches have noticeable TT 
 relations to the coffer far within, and to certain standard specific- 
 gravity teachings of great import. So, too, the peculiar propor- 
 tions found by Mr. Petrie to exist between the areas and eleva- 
 tions of every horizontal course of masonry throughout the 
 
METROLOGICAL NOTES AND QUERIES. 22J 
 
 monument, and the area of its meridian section, so as to bring 
 out twenty-fifths (the all-important cubit number) thereof, each 
 marked by specially thick courses, is too stupendous a piece 
 of engineering, and too beautifully realized, not to have been 
 based upon some grand fact which the architect felt that no 
 undertaking would be too difficult to employ for monumen- 
 talizing. 
 
 But all of these, and many other similar studies, we leave for 
 others to undertake, and shall ask the attention of our readers 
 to a short study of the coffer dimensions, as viewed from Mr. 
 Petrie's late measurements. 
 
 In "Our Inheritance in the Great Pyramid," Professor Smyth 
 gives a table, in which he enumerates the actual measurements 
 of the coffer, made on the spot by no less than twenty-five dif- 
 ferent observers, who, at different times during three full centu- 
 ries, had preceded him. It is a remarkable fact, that they all 
 differ, and some of them very materially. But three sets of 
 these measurements are of any intrinsic scientific value, those 
 of Professor Greaves (1638), Jomard and Napoleon's French 
 expedition (1799), an ^ of Col. Vyse (1837). But, strange to 
 say, even in these sets, several of the individual dimensions are 
 so conclusively erratic, and at variance with the general average, 
 as to warrant their summary rejection. 
 
 The measurements of Professor Smyth and Mr. Petrie are 
 undoubtedly the most reliable ones that have yet been made, 
 and perhaps are now the best that we can ever hope for from 
 direct measurement, owing to the injured condition of this inter- 
 esting relic, which is rapidly being mutilated past measurement. 
 Centuries of pilgrims and professional iconoclasts cannot ham- 
 mer at even the hardest granite without wearing it away at last. 
 Thanks, however, to the Promethean proportions of the pyramid 
 itself, and the carefully conceived harmonies of ratios running 
 through all of its details, we can rest assured, that, from indi- 
 rect measurements, checks upon our work gathered from all 
 parts of the pyramid, we may yet arrive at the absolute theo- 
 retical size, shape, and proportions of this standard measure of 
 capacity. 
 
 In the following table we subjoin the three several measures 
 of Greaves, Jomard, and Vyse, considered as the most reliable 
 
228 
 
 METROLOGY. 
 
 by Professor Smyth, together with his own and those of Mr. 
 Petrie. We have put the erratic dimensions in smaller type, as 
 a most cursory examination will show they are of very little 
 value. 
 
 COFFER MEASUREMENTS. 
 
 Designation. 
 
 GREAVES. 
 1638. 
 
 JOMARD. 
 I 799 . 
 
 VYSE. 
 1847. 
 
 SMYTH. 
 1865. 
 
 PETRIE. 
 I 880-2. 
 
 Average. 
 1884. 
 
 Inside. . . . 
 
 \ Length . 
 Breadth . 
 [ Depth . . 
 
 77-856 
 26.6l6 
 34-320 
 
 77.836 
 26.694 
 
 37.285 
 
 78.0 
 26.5 
 
 34-5 
 
 77-93 
 26.73 
 
 34-34 
 
 78.06 
 26.81 
 34-42 
 
 77.936 
 26.669 
 
 34-395 
 
 Outside . . . 
 
 Length . 
 Width . . 
 Height . 
 
 87-5 
 39-75 
 39-75 
 
 90.592 
 
 39-45 
 44-765 
 
 90.5 
 
 39-Q 
 41.0 
 
 89.71 
 38.65 
 41.27 
 
 89.62 
 38.50 
 41.31 
 
 89.66 
 
 38-57 
 41.29 
 
 Thickness . 
 
 ' Sides 
 Bottom . 
 
 
 
 
 5-99 
 6.92 
 
 5.82 
 6.89 
 
 5.905 
 6.905 
 
 Greaves 
 
 Vyse . 
 
 Smyth , 
 Petrie 
 
 From the several dimensions thus tabulated, we are enabled 
 to calculate the corresponding volumes, as follows : 
 
 Brit. cu. in. 
 
 Inside measurement . . . 71,118 
 
 Inside measurement . . . 71,312 
 
 ! Inside measurement . . . 71,533 
 
 Outside measurement . . . 143,154 
 
 j Inside measurement . . . 72,034 
 
 I Outside measurement . . . 142,535 
 
 Total 571,686 
 
 Mean = J = 71,460 
 
 Again, taking the dimensions for the inside from the column 
 marked average, we obtain 
 
 Br. cu. in. 
 
 A volume of 71,489 
 
 And for the outside volume one of ...... 142,830 
 
 One-half of the latter gives us . . . . . . . . 71*415 
 
 The mean of this, and the average inside, is equal to 71,452 
 The mean of the volumes from Smyth's and Petrie's 
 
 measures, taken alone, is . . . . .... 71,542 
 
METROLOGICAL NOTES AND QUERIES. 229 
 
 Now, the absolute volume represented by 71,250 pyramid 
 inches, each of (i.ooi db British inches) 3 , is 
 
 ( 71 250) ( i. ooi )3 = 71463.963821250 British cu. in., 
 
 a quantity remarkably close to the mean of the whole table, as 
 given above, and also close enough to those which we obtained 
 from the column of average dimensions involving those of 
 Greaves, Jomard. and Vyse, the latter of whom is generally 
 accurate only as to whole numbers. 
 
 BT. cu. in. 
 
 The mean of Smyth's measures, taken alone, is . . 71,529 
 While that of Petrie's is 71*516 
 
 Nor should it be overlooked in this connection, that Petrie 
 himself allows that a possible error of from 60 to 100 i British 
 cubic inches may affect his results as to the coffer's volume ; so 
 that 71,516 71,464 = 52 is well within the limits of his lowest 
 estimate as to error. 
 
 Viewing the whole subject broadly, then, and convinced that 
 there are other reasons, of an entirely independent nature, which 
 lend their weight in favor of the special volume of 71,250 pyra- 
 mid cubic inches, or 71,463.9-}- British cubic inches, we cannot 
 but feel, that, in the present dilapidated condition of the coffer, 
 our modern measures are as accurate as they ever will be made, 
 and that we must depend upon the checks furnished elsewhere 
 throughout the monument for correcting them. In view, too, 
 of such and kindred facts, with all of which Mr. Petrie and 
 other opponents of the metrological theory show themselves to 
 be perfectly familiar, it is difficult to resist the effort to try and 
 harmonize the details of the monument, not only with them- 
 selves, but with those grand cosmic truths to which they 
 certainly approximate. And it is astonishing that scientific 
 gentlemen, who, in other directions, strain themselves so desper- 
 ately in their ceaseless search for "missing links" of a character 
 absolutely degrading to intellectual man, will so allow their pre- 
 judices to close the eyes and ears of their understanding against 
 the reception of images that perhaps may prove their origin to 
 have rather been divine. 
 
230 METROLOGY. 
 
 [NOTE IV.] 
 POSTSCRIPT. 
 
 The present volume does not pretend to be an answer to, or 
 concern itself at all with, the style of arguments (?) advanced 
 by Mr. Proctor and President Barnard. To ridicule is not to 
 argue : to call design accident and coincidence, or else to deny 
 its very identity as an intended realization of a means towards 
 an end, merely because the preconceived opinions or teachings 
 ot so-called science are opposed to .it, is illogical and illiberal. 
 Unless we are certain that modern science already stands upon 
 the broad basis of absolute knowledge, it is ridiculous to shut 
 out independent study, or to flatter ourselves that a monument 
 such as that at Gizeh is not a harmony of cosmic truths. If 
 the Great Pyramid, as we are now beginning to understand it, 
 is, though full of sympathy with the diverse laws of nature and 
 her ratios and proportions, still merely a tomb and a haphazard 
 freak of chance, then, of all things upon earth, it is still, and 
 ever will be, the most truly wonderful, and more than ever 
 worthy of its place in history. 
 
 But there is a limit where the possibility of chance must end, 
 and at which the convictions of even the most unwilling mind 
 will be forced to recognize design. This limit has certainly 
 been reached in matters relating to the Great Pyramid ; and it 
 is rather now the duty of our scholars to devote their attention 
 to its study and to the unravelling of the message its originators 
 intended it should convey to coming generations, than that any 
 longer they should persist in belittling such a topic. 
 
 The metrologic scheme discussed in the foregoing pages has 
 undoubtedly connected not only the teachings of the pyramid 
 in one harmonious mosaic, but has joined them on to the study 
 of nature, and of man himself. It has been entirely due to 
 the lessons taught the writer by this monument, that any such 
 scheme was even rendered conceivable ; so that he is satisfied 
 to leave it for those into whose hands it may come, to elaborate, 
 until in some one science, which may be called metrology in- 
 deed, posterity may look from Nature up to Nature's God, and, 
 glancing back again upon their neighbors, understand more 
 fully how they are created in the image of their Maker. 
 
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