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 THE UNIVERSITY 
 OF CALIFORNIA 
 
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 PRESENTED BY 
 
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 GEOMETKY IN MODERN LIFE. 
 
 TWO LECTURES AT ETON.
 
 iGEOMETRY IN MODERN LIFE^ 
 
 BEING THE SUBSTANCE OF TWO LECTURES 
 
 USEFUL GEOMETRY", 
 
 GIVEN BEFORE 
 
 C|re Sftterarg j&arietij at tott. 
 
 J. SCOTT RUSSELL, M.A., F.R.S., 
 
 ETC. ETC. ETC. 
 
 ETON: WILLIAMS AND SON. 
 
 LONDON : SIMPKIN, MARSHALL, AND CO. 
 
 1878.
 
 LONDON : 
 E. CLAY, SONS, AND TAYI.OP, 
 
 BREAD STREET HI I.I, E.C.
 
 "l8ot<? Be real ov povov ev tol<j Batp,oviovi teal 6eioi<; 
 irpdypaa't rav rw apidfiw <f)vaiv /ecu rc\v Bvvap.iv la^yov- 
 o~av, dXka ica\ ev to?? dvdpooTUKOi? epyoi? /ecu \6<yoi<; 
 nrdai iravrd, Kal Kara, ras Bap,iovpyla<; ra? Te)(yLKd<; 
 7ruaa<i ical Kara, rav fiovaiicdv. ^eOSo? Be ovOev Bexerai, 
 d tw dpiOpto <pvcri<i ovBe dppovla' ov yap olicelov avTol? 
 ivri' rd<i yap direipw Kal dvorjreo ical aXoyw <f>vaio<; to 
 tyevBo? Kal 6 (pOovos ivri 
 
 Philolaus, Frag. 
 
 Kal to pev Br) irpo tovtov irdvra TavT e^etv aXoycos Kal 
 dpeTpcos' ore B' eVe^etpetTo Koapelcrdai to irdv, irvp irp&Tov 
 Kal vBcop Kal <yr)v Kal depa, Xyyr) p>ev eypvTa avTcov aTTa, 
 iravTairacri p,r)v BioKeipeva gj<? irep et/co? e%eii> 7rav oTav 
 dirfj twos 6Je6?, ovtw Brj totc Tre<pVKOTa Tavra irpwTov 
 Btea^rj/jLarlaaTO e'iBeac t Kal dpiOpols. to Be fj Bwarov a? 
 KaWiGTa dpiaTa re ef ov-% ovtcos eyovTwv tov 6eov avrd 
 %vvLo~Tdvai, irapd irdvTa i)plv a<? del tovto \eyop,evov 
 i/Trap^ero). vvv cT ovv rrjv Bidra^iv avT&v eTn^zt-pyTeov 
 eKaaTWV Kal yeveaiv dr)6ei \6yu> 777)05 vpd<; BrjXovu. dWd 
 yap iire\ peTe%eT twv Kara iraiBevaiv 6B5)v, oY wv evBeiK- 
 vvoQai Ta Xeyopeva avdyKrj, ^vve^eade. 
 
 Plato, Timceus, p. 53, B.
 
 PREFACE. 
 
 The two separate views of Education in Science, 
 which are expressed in these pages, will I doubt not 
 be entirely approved by the experienced teachers to 
 whose pupils they are addressed. Geometry, as a 
 pure Science, gives logical training, is a discipline of 
 thought, is an instrument of human culture, and 
 has high educational value. But Geometry is equally 
 the development of a method pervading Nature ; its 
 mastery gives Man a power to govern Matter. The 
 training, which enables him to comprehend the 
 mechanism of the universe, enables him also to make 
 creations of his own in harmony with those greater 
 designs of which his own are but a small portion. 
 These two uses of geometric education, the one purely 
 gymnastic, the other practical and technic, may be so
 
 viii PREFACE. 
 
 combined that each shall aid and not impede the 
 other. The order, number, arid measure which per- 
 vade the universe can be easily brought within the 
 scope of elementary education, and so form the fit 
 preparation for scientific observation and experiment 
 in later life, by means of which the standards of 
 application of abstract truths to matter and events 
 in human life are determined and made familiar. 
 But the one learning cannot be too soon begun, nor 
 the other too long continued, and each is a material 
 aid to the other. 
 
 Whoever has had experience of the difficulty with 
 which entirely new thoughts are conveyed into the 
 youthful mind, will frankly agree with the author in 
 the absolute necessity of initiating new knowledge 
 in simple words and self-evident forms, and of 
 making meanings plain by repetition and emphasis, 
 in a degree which may seem to the initiated tedious 
 and useless. In this little work there is much, in 
 matter and in manner, open to this just criticism. 
 On the other hand it must not be forgotten that the 
 aim and purpose of preliminary education is "to lead 
 the mind on from simple beginnings to arduous 
 conclusions " ; and for this object and with these
 
 PREFACE. ix 
 
 views the first steps must before all be made secure 
 and unquestionable. In like manner it may be 
 found that while much at the beginning seems too 
 simple, much towards the end will inevitably be 
 too recondite and inaccessible. 
 
 The highest education of our time will be that 
 which combines the literature and philosophy of the 
 past with the science and arts of the present, and so 
 enables us to forecast, plan out, prepare, and execute 
 skilful, wise, and useful work in the future.
 
 CONTENTS. 
 
 I. 
 
 THE GREEK PHILOSOPHY AND GREEK GEOMETRY. 
 
 Antient Philosophy and Modern Science are not antagonistic in Use or 
 Nature, although often so represented in Argument Equally in both, 
 Order, Measure, and Number rule the Universe They applied 
 the Laws of Mind to Matter We seek the Laws of Mind through 
 Matter Page 1 
 
 II 
 
 MODERN SCIENCE AND MODERN GEOMETRY. 
 
 The Modern Sciences are Methods of questioning Matter for useful 
 Knowledge, and the Modern Geometry contains the Methods of using 
 matter for man's material work in daily life Why is Geometry little 
 known ? Page 8 
 
 III. 
 
 GEOMETRY ANTIENT AND MODERN. 
 
 Some great Antient Geometers and their works The greatest part of 
 their writings destroyed by fire in the Alexandrian Library Modem 
 Geometers have first tried to restore the lost Ancient Geometry Next 
 to extend it and apply it to Modern Use Geometry, a Study A 
 Science An Art A Tool of the other Sciences .... Page 14
 
 xii CONTENTS. 
 
 IIII. 
 
 USEFUL GEOMETRY. 
 
 Uses of Geometry In Mental Training In Science In Laws of Light 
 and Visible Form In Measuring and Mapping In Ordering, 
 Arranging, Planning, and Building Training for Trades and 
 Professions .... Page 22 
 
 V, 
 
 FIRST STEPS IN GEOMETRY. 
 
 Here There Yonder Hence Thence Hither Thither Starting 
 Point Stand Point Midway -Measure Distance . . Page 27 
 
 VI. 
 
 FIRST WAYS IN GEOMETRY. 
 
 Straight On Right Across Forwards and Backwards Right and 
 Left Upwards and Downwards North and South East and 
 West '. Page 35 
 
 VII. 
 
 LAND-MEASURING. 
 
 Base Line Starting Point Stand Point Stations Chain Measure 
 Off Sets Cross Measures Acres Records Plan Calculations 
 Map Page 44 
 
 VIII. 
 
 THE GEOMETRY OF THREE WAYS. 
 
 Three Stations Three Ways Three Distances " Trigonometry" On 
 the Square On Ihc Skew Page 56
 
 CONTENTS. xiii 
 
 Villi. 
 
 THREE POINT MEASURE. 
 
 Geometry of Navigation " Trigonometric Surveying " Geometer's Fore- 
 cast " Trigonometric Tables " " Trigonometric Diagrams " Sailors' 
 Trigonometry Page 65 
 
 X. 
 
 MEASURE AND NUMBER. 
 
 Measures make Numbers The Measure is One Another of the same 
 makes Two The Usefulness of the Number Two In Parting In 
 Cutting In Ways In Navigation In Matter Nature In Music 
 Number Three Number Five Even Numbers Odd Numbers 
 Groups of Numbers Square Numbers Uses of Numbers in Geometry 
 of Place, Order, and Shape, Ordering, Building, Enlarging Shapes 
 by Numbers Movements, in Space by Number .... Page 74 
 
 XI. 
 
 FAMILIES AND PEDIGREES OF NUMBERS. 
 
 Common Numbers Selected Numbers Choice Number Agreeing 
 Numbers Family Numbers Prime, Patriarchal, Primitive, or 
 Princely Numbers II, III, V, VII, XI Their Families . Page 106 
 
 Table of Degrees of the First Five Families Page 116 
 
 Pedigrees of the First Numbers \. . Page 123 
 
 Table of Sixty -four Family Pedigrees Pages 132, 133 
 
 XII. 
 
 SYMMETRY, HARMONY, MELODY. 
 
 Symmetry Harmony Melody Page 135 
 
 Note Page 141 
 
 Table of Melodic Sounds Page 154 
 
 Tables of their Numeric Pedigree ........ Pages 133, 156 
 
 Symmetry in Work Divine and Human Page 157 
 
 Examples of Symmetric Form growing out of the numbers 
 
 II and V in their Families Page 162
 
 xiv CONTENTS. 
 
 XIII. 
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 
 
 Light goes Straight We see Straight Light marks out lines for the 
 Geometer An Etonian Geometer in his yacht in Northern Seas finds 
 his place by. Geometry How to shew on paper the shapes of things 
 you see as they seem to eyesight Artists should be Geometers 
 A great Artist Geometer Page 164 
 
 XIIII. 
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 
 
 Forces and Motions in Worlds of Matter obey the laws and follow the 
 lines of the Geometer The Geometer's Diagrams and Forecasts on 
 pages 70 and 71 are also Diagrams and Measures of the Forces and 
 Motions of Matter in Nature Page 178 
 
 XV. 
 
 ENDING ENDLESS. 
 
 The Straight The Bent The Crooked The Curved The Infinite 
 The Infinitesimal Page 189 
 
 ERRATUM. 
 
 Page 43, line 3 from bottom, for " October,'' read " September."
 
 GEOMETRY IN MODERN LIFE.
 
 GEOMETKY 
 
 MODEEN LIFE 
 
 THE GREEK PHILOSOPHY AND GREEK 
 GEOMETRY. 
 
 I hope no Eton Man will think that because I 
 am invited here to speak in favour of Geometry, as 
 a Science of which much Practical Use may be made 
 in Modern Civilized Life, that therefore I undervalue 
 in my own mind or wish to depreciate in yours the 
 Studies of Antient Language, Literature, Politics and 
 Philosophy which form the Classical Education of 
 this famous School of Young Englishmen No ! 1 
 love Greek Literature, I delight in the Records of 
 Greek Thought I was early a disciple of the 
 School of Plato an early student of Science as 
 taught by Aristotle. If I have learned to think truly 
 and to reason wisely and to follow out principles 
 into acts of life and their due results in human
 
 2 GEOMETRY IN MODERN LIFE. 
 
 progress, I owe my initiation into right ways of 
 thought and right principles of procedure chiefly to 
 an early training in Greek Thought expressed in 
 the words of the Greek Schools of Philosophy of 
 some 2000 years ago. 
 
 I therefore love the Literature which is a main 
 subject of your thought and work at Eton aud 
 if I venture to make any suggestion to you, it 
 would be rather that you should carry out your 
 study of Greek Thought and Philosophy to its 
 fit developement, in its due application to our 
 new conditions of Modern Life, Modern Science, 
 and Modern Civilization ; and not at all that you 
 should abandon or curtail any portion of the Mental 
 Gymnastic which you are now going through as a 
 training for life. 
 
 Allow me to tell you that Modern Chemistry is 
 but a realization and a materialization of the Antient 
 Greek Thought about the Constitution of the 
 Material Universe. 
 
 Modern Chemistry is the creation of John Dalton 
 John Dalton discovered that all Matter is made 
 up of little Atoms that these atoms, though not 
 alive, have each a little nature of its own, and if 
 not a will, at least a way of its own These atoms 
 are not a confused mass as you might think them 
 when you look at a rude stone or a lump of clay 
 or a sand hill on the sea shore Every rude mass, 
 when you look close into it, is built up of exquisitely 
 made little particles, nicely shaped, neatly fitted
 
 GEOMETRY IN MODERN LIFE. 3 
 
 together, and all grouped about each other in regular 
 proportions, and although all alike in certain ways, 
 yet so invented contrived and shaped and propor- 
 tioned that out of simple elements you can form 
 endless beautiful combinations. 
 
 Now, of these pretty little particles, out of which 
 the whole Earth and Heaven, Sea and Air are 
 formed there are about 64 sorts or kinds. Some 
 of these sorts love each other and cling together 
 some hate each other and keep apart. But the 
 strangest thing of all is that each Family or Eace 
 of Atoms, out of these 64 families, has a name or 
 rather a number, which it takes by nature, and 
 it will take no other. By this it rules its conduct, 
 and if forced for a time to group with one it does 
 not like, will break away from it on the first chance 
 and take back to its old companions. 
 
 Now among these 64 families of elements which 
 make up our World and some of which inhabit the 
 Moon and the Stars also some love the number 3 
 some love 4 others love 5 or 6 and some take to 7. 
 
 LOVES OF THE ATOMS. 
 
 Among the Elements of the World 
 
 Oxygen loves the number. . 8 
 Nitrogen loves the number . 7 
 Carbon loves the number . . 6 
 
 Boron loves the number . . 11 
 
 B 2
 
 6 GEOMETRY IN MODERN LIFE. 
 
 also pervades the living mind of Man, made 
 Divine by its divine origin. - -Greek Philosophers 
 then believed in Divine Intention pervading 
 Creation ; in Divine Science Containing The 
 Mysteries of the Thoughts of that Divine Mind. 
 They believed that Humanity is but a Micro- 
 cosm of that Divine Mind They believed that 
 the Thoughts of God, carried out in Creation, are 
 expressed in it in a manner meant to be intelligible 
 to the human mind They believed that the Mind 
 of Man, through the Divine Nature planted within 
 him, has the germs of Divine Truth inherent in 
 his nature, and is able to detect in the Harmony 
 of Man's Nature with Matter Nature, the Harmony 
 of both with the Mind of God ! 
 
 Now if I am asked which was the Great Antient 
 Science by which the Ancient Philosophers expressed 
 their thoughts concerning the Divine Methods of 
 thinking out the Universe and working out the 
 Matter World I venture to say, that the Method 
 which led them to the Truths of Creation and Com- 
 prehension of God's Truth, was, the Antient Geo- 
 metry. Now do not understand that when I speak 
 of the Antient Geometry I am thinking of Euclid 
 only. The burning cf that glorious treasure house 
 of human and divine thought the Library of 
 Alexandria some 1,200 years ago was an incalcul- 
 able misfortune to modern scholars and to humanity. 
 Euclid's books were a part and but a small part 
 of Antient Geometry, and much of his work was
 
 GEOMETRY IN MODERN LIFE. 7 
 
 burnt and lost of the fragments saved no one can 
 esteem the value more than I do, as the spark of 
 fire that has rekindled Modern Science. I say then 
 that Euclid's books are to us a valued treasure of 
 Greek Philosophy but only a fragment of the Divine 
 Literature of Greek Philosophy a guide, through the 
 first steps of the exact thinker, groping his way to 
 the Governing Truths of the Universe of Matter 
 and its Divine Mind.
 
 GEOMETRY IN MODERN LIFE. 
 
 II. 
 
 MODERN SCIENCE AND MODERN 
 GEOMETRY. 
 
 Modern Life is distinguished from Antient Life 
 mainly by the Developement of Thought and 
 Thinking into Act and Work The Antients 
 thought and sang and philosophised, and it is our 
 delight to read their thoughts in their clear Logic 
 and chant their songs in their living poems They 
 created Deities out of their own heads and built 
 up an Ideal Universe out of their Creative Genius 
 We Moderns owe them much They have taught us 
 to think and reason they did more they taught 
 us to embody thought in Matter Their Sculpture 
 and their Architecture are embodiments of refined 
 thought cut with exquisite skill out of shapeless 
 masses of stone and marble. But Modern Life and 
 their Life have little in common beyond community 
 of thought and feeling The things and works of 
 our life and theirs are radically different We start 
 life in their grooves of thought and we straightway 
 diverge into modern ways of thinking and working.
 
 GEOMETRY IN MODERN LIFE. 9 
 
 Geometry is one of the links that binds us and 
 them together Euclid has survived as well as 
 Homer The Conic Sections of Apollonius and the 
 Helix of Archimedes, are the staple of modern 
 astronomy and modern mechanics Their thoughts 
 and their ways of thinking are indeed our heritage. 
 
 But all the uses we make of Thoughts and Thinking 
 are quite different from theirs They created Divinities 
 out of their own fancy We seek Divinity in deeds 
 not fancy "We seek a Deity, in a Power that creates 
 worlds and makes them carry themselves, with no 
 need of a fancied Atlas to carry them on his own 
 shoulders We seek a Deity in a Practical Ruler of 
 the Universe Whose Will gives changeless Law to 
 Matter and to Mind In Whom we live and move 
 and have our Being Who prepared us to live in this 
 World, and Who prepared the World for us to live 
 in. Who put in the World stored for us Stone to 
 build with, Coals to burn, Iron to melt, Brass to 
 mould, Steel to cut, Water to drink, Air to breathe 
 Who covered the Earth with green carpets, and 
 raised noble woods above to shelter us, and dressed 
 lovely flowers up for our joy ; and bade the trees 
 prepare fruits and strew them along our paths and 
 set us in a Paradise. We seek also a Divinity to 
 Whom we have some near affinity Who made us in 
 His own image, so that we too are Divine Who has 
 placed us here in this beautiful Earth to make it 
 more beautiful Who has given us all its treasures 
 that We might show our divine nature by doing
 
 10 MODERN SCIENCE 
 
 all the work in the world which He left undone in 
 order that we might do it He gave us gardens, 
 trees and flowers, that we may also make gardens, 
 trees and flowers, and make them more beautiful 
 than we found them and we have done so in 
 a measure by modern horticulture and make 
 lands more fertile and fruits more fruitful, which we 
 have done by modern agriculture. He gave us 
 horses to carry our burdens and oxen to draw our 
 ploughs, and left us to make horses much more 
 powerful, and ploughs much more helpful, which we 
 have done by making engines every one as powerful 
 as many horses, and each fleeter than the fleetest 
 racehorse and by setting other engines, instead of 
 oxen, to plough our fields, and dig our mines, and 
 drain our marshes, and spin our yarn, and weave 
 our clothes. Thus we now rejoice, in a nobler 
 Divinity than the Greeks, and we have a nobler 
 vocation than theirs, the vocation of Acts and 
 Deeds and Works which carry out the aims ot 
 the World Maker as far as made known to us little 
 men who try to become worthy of our divine 
 pedigree. 
 
 Thus then we have thoughts in common with the 
 great thinkers of old But we differ from them in 
 our acts We think that we were set in this world 
 for a purpose That we were destined to work out 
 part of a great plan That our part of that plan is 
 to make perfect all the work for which the materials 
 have been collected, arranged, and stored up for
 
 AND MODERN GEOMETRY. 11 
 
 us and that work Modern Science enables us to 
 fulfil, that work we are busy doing. 
 
 Now the Great Instrument the Divinity has used 
 in creating our world is Geometry. The Tool He 
 has given to us for our work is Geometry. This 
 Instrument we owe to the Antients who told us, 
 " God is the Great Geometer." The Use of this 
 Instrument they commended to us when they wrote 
 over the Schools of Greece, " Let no Ignoramus in 
 Geometry enter here." 
 
 D 
 
 MHAEIS AfEnMETPHTOZ EIZITI2. 
 
 D 
 
 If I were asked to define Modern Geometry I 
 should call it the " Science of Shape." Perhaps I 
 should add to its fulness by adding also Size as well 
 as Shape ; but we might consider Size as an Element 
 of Shape. 
 
 If I were asked to extend and explain my 
 Definition, I should say that Geometry Compre- 
 hended all Knowledge of Space and Place Distance 
 and Measure Direction and Way Shape and Size 
 Bulk and Extent ; I should further define Geometry
 
 12 MODERN SCIENCE 
 
 as an Art as well as a Science. The Antients called it 
 Earth Measuring But we Moderns use it for Sea 
 Measuring as well as Land Measuring, and for Sky 
 and Star Measuring equally. 
 
 But it is much more than this It comprehends 
 the whole of the Arts of Shape Design Shape 
 Drawing Shape Cutting Shape Building Shape 
 Creating. But Science and Skill in Shape Making 
 is really the Essence of most of the important Arts 
 of common life We take shapeless clay and shape 
 it into bricks we take shapeless piles of bricks and 
 shape them into shapely Queen Anne's palaces we 
 quarry masses of stone and our masons hew them 
 into columns, arches, palaces and churches we give 
 a block of marble to the sculptor and he gives it us 
 back a Venus de Medicis or an Apollo Belvidere we 
 give a shipwright great trees and he shapes them 
 into sea-going ships we give the goldsmith an 
 iu sot of cold and some stones and he gives back a 
 princely crown our potters and glassblowers give 
 shapes to marvellous dreams and produce triumphs of 
 Art in form and varied hue. All modern work is 
 but the transformation of some mass of Matter out of 
 one shape into a better shape. Geometry is of the 
 Essence of Modern Life If Geometry thus pervades 
 everything round about us were it not well that We 
 should all of us know something about it \ If most 
 Work done in the world of Matter is merely trans- 
 formation of Shape, should not each of us make 
 himself more useful to the world if he knew how to
 
 AND MODERN GEOMETRY. 13 
 
 transform things out of the shapes one doesn't want 
 into the shapes we need ? 
 
 The best answer to all this is given by the Antient 
 Greek " God is the Great Geometer." Let us be 
 little ones ! 
 
 But if Geometry be thus the One Universal 
 Science of the Nature of all the Material World 
 all round about us If it is necessary, in order to 
 explain to us the difference between Things Created 
 and Matter in Chaos before Creation If Creation was 
 merely the Giving Shape and Substance to Things 
 which were formerly without Form and Void it is 
 plain that the chief work of the Matter world 
 comes under the Science of Geometry ! 
 
 Why then is Geometry to the greatest number 
 of human beings The Unknown, both as Science 
 and Art ? 
 
 The answer to this is not difficult Geometry is 
 and was always an Exact Precise and Hard Study. 
 Geometry is hard to learn and hard to practice 
 Geometry requires of the learner, self-sacrifice. 
 Geometry requires of the worker, dexterity, exact- 
 ness, Forethought. Thus the nature of Geometry 
 makes it hard to get and hard to use. 
 
 I will now try to make it more easy for you to learn 
 and more convenient for you to use, by helping you 
 to avoid the crooked paths that confuse the learner, 
 and the clumsy ways that lead him to do bad work.
 
 14 GEOMETRY IN MODERN LIFE. 
 
 III. 
 
 GEOMETRY ANTIENT AND MODERN. 
 
 Geometry as we now have it, is almost wholly 
 the Antient Geometry. But the aims of Modern 
 Life and the uses to which we put the knowledge the 
 Antients have left us are new and manifold. 
 
 Geometry as a study is now twofold, 
 
 a Science and an Art. 
 
 As a Science we have little more to learn than 
 what the Greeks knew and taught. 
 
 As an Art, useful in modern life, we have done 
 much and have still more yet left to do. 
 
 Unluckily for us, we have only fragments of their 
 knowledge. The work of Modern Geometers has 
 been mainly to seek, gather, and join these fragments, 
 to fill the void spaces, to recover and restore their 
 lost Geometry to its original completeness. 
 
 That Geometry was the result of 1,000 years of 
 Antient work : and ours is the result of 300 of 
 modern research and restoration.
 
 GEOMETRY ANTIENT AND' MODERN. 15 
 
 Foremost among the great Antient Geometers 
 were : 
 
 OAAHZ. 
 
 560 546 B.C. 
 Milesian of Phoenician origin. 
 Founder of the Ionic School. 
 
 ANAZIMANAPOI. 
 
 547 B.C. 
 
 Milesian. Father of Geography. 
 
 First Map. Dissertation on Magnitude of Stars. 
 
 nYGATOPAZ. 
 
 539472 B.C. 
 
 Native of Samos. Migrated to Croton in Italy. 
 Founder of the Pythagoreans. 
 
 nAPMENIAHZ. 
 
 502455 B.C. 
 
 Native of Elea. Chief of the Eleatic School. 
 Author of Terrestrial Spheric Geometry.
 
 16 GEOMETRY IN MODERN LIFE. 
 
 nAATnisi. 
 
 429 347 B.C. 
 Athenian, born in iEgina. 
 Founder of the Academy. 
 
 APIZTOTEAHZ. 
 
 384323 B.C. 
 
 Native of Stageira in Chalcidice. 
 Founder of Peripatetic School. 
 
 EYKAEIAHI. 
 
 Of Alexandria, b.c. 323. 
 Founder of the Mathematical School of Alexandria. 
 
 APXIMHAHZ. 
 
 Of Syracuse, b.c. 287212. 
 Geometer Astronomer Engineer. 
 
 euprjKa. 
 
 i o 
 
 AnOAAHNIOZ 
 
 Of Perga. b.c. 222. 
 Conic Sections.
 
 GEOMETRY ANTIENT AND MODERN. 17 
 
 nTOAEMAIOZ. 
 
 Of Pelusium. a.d. 139. 
 
 Lived at Alexandria. 
 Geometer Geographer Astronomer Musician. 
 
 0EOAOZIOZ. 
 
 Of Tripoli, a.d. 96 (?). 
 Master of Spherical Geometry. 
 
 nPOKAOZ. 
 
 A Lycian. Born at Byzantium, 412 485 a.d. 
 
 Studied at Alexandria. 
 Taught at Athens. 
 Treatise on the Sphere. 
 
 ALEXANDRIAN LIBRARY DESTROYED. 
 
 By order of Khali f Omar. 
 a.d. 640. 
 
 The Library of Alexandria, said to have contained 400,000 volumes, 
 was divided into two parts. One in the Brucheium, the other in the 
 temple of Serapis. The Library in the Brucheium was destroyed 
 during the blockade of Julius Caesar, B.C. 47. The Serapeion was partly 
 destroyed by Christian fanatics in the fourth century, a.d. It was 
 finally destroyed by the Khalif Omar in 640, a.d.
 
 18 GEOMETRY IN MODERN LIFE. 
 
 The great Modern Geometers have been 
 
 GALILEO GALILEI. 
 
 1564, a.d. 1642. 
 
 JOHN KEPLER. 
 
 1571, a.d. 1630. 
 
 RENE DES CARTES. 
 
 1596, a.d. 1650. 
 
 ISAAC NEWTON. 
 
 1642, a.d. 1726. 
 
 G. W. DE LEIBNITZ. 
 
 1646, a.d. 1716. 
 
 BERNOUILLIS. 
 JAMES. JOHN. DANIEL. 
 
 1655, a.d. 1782. 
 
 LEONHARD EULER. 
 
 1707, a.d. 1783.
 
 GEOMETRY ANTIE NT AND MODERN". 19 
 
 During the present century some eminent geo- 
 meters have lived, whose works will form part of the 
 history of our century. I trust also that more 
 eminent geometers may still come forth with great 
 discoveries, some of which may set in order the 
 chaotic accumulations of material facts by special 
 records of minute observations. But it is not our 
 duty here to anticipate the great achievements which 
 may still be left for the young geometers of the 
 nineteenth century. 
 
 The great codes of Geometric Truth given us by 
 the Antients, and recovered for us by Modern 
 Philosophers, are one of the most important sources 
 of Modern Education. 
 
 Geometry is a Study. 
 
 It feeds Thought, with Changeless Truths. 
 It trains Minds to, Methods in Thought. 
 
 It disciplines character to 
 
 Precision in Fact, 
 Method in Eeason, 
 Modesty in Opinion. 
 
 Geometry is a Science, 
 
 Giving Universal Law to all Space, 
 
 and to All Matter Filling Space. 
 
 C 2
 
 20 GEOMETRY IN MODERN LIFE. 
 
 The essence of Geometry in Matter is 
 
 to give it Place, Order, and Symmetry 
 out of Chaos, Disorder, Confusion. 
 
 Geometry is an Art, 
 
 For practical use in daily life and common things. 
 
 It teaches, 
 
 Measuring, Shaping, 
 
 Planning, Moulding, 
 
 Drawing, Cutting Out, 
 
 Mapping, Building Up. 
 
 Geometry is a Tool of Science. 
 
 It helps us to measure truly, 
 
 Lands, Moon, 
 
 Seas, Stars, 
 
 Continents, Heavens. 
 
 The methods of Geometry 
 
 Explain, the Principles of the Universe. 
 Express, the Laws of Matter, 
 
 as laid down bv The Great Geometer.
 
 GEOMETRY ANTTENT AND MODERN. 21 
 
 The Study of Geometry teaching us, the 
 
 Nature of the Universe, 
 Methods of Construction, 
 Organization of the Matter World, 
 
 teaches us also, the Nature of the World in which 
 we have to do each his own work : and shews the 
 methods, by following which man may imitate God 
 in giving like perfection to his work.
 
 22 GEOMETRY IN MODERN LIFE. 
 
 IIII. 
 
 USEFUL GEOMETRY. 
 
 We now understand That Geometry is the gift 
 of the Greek Geometers to us, for use in our life 
 That it is the great instrument of Modern Science 
 That without it no man can do good work in the 
 world of Matter in which we each occupy a responsible 
 place. 
 
 I. Use of Geometry in Mental Training. 
 
 Teaching to 
 
 Think Exactly, 
 
 Word Thoughts Precisely, 
 
 Reason Truly. 
 
 Training to 
 
 Submit Opinion to Fact, 
 Bend Will to Rule, 
 Enforce Law, 
 Maintain Truth.
 
 USEFUL GEOMETRY. 23 
 
 II. Use of Geometry in Science. 
 
 Giving Exact Knowledge of 
 
 P1 aces, Shapes, 
 
 Distances, Sizes, 
 
 Directions, Extents, 
 
 Wa y s > Contents, 
 
 of all the things round about us, the world at large, 
 the Universe of Space, 
 
 under the divisions of 
 
 Surveying, Navigation, 
 
 Geography, Astronomy. 
 
 III. Use of Geometry. 
 
 In teaching Laws of Sight and Visible Forms- 
 Lines of Sight, 
 Lines of Light, 
 Images of Things as Seen, 
 Images of Great and Small, 
 Images of Far and Near. 
 Visible Forms and Tangible Forms.
 
 24 GEOMETRY IN MODERN LIFE. 
 
 IIII. Use of Geometry. 
 
 It teaches 
 
 Measuring and Mapping the Near and Afar- 
 
 Gardens, Seas, 
 
 Fields, Islands, 
 
 Lakes, Continents. 
 
 V. Use of Geometry. 
 
 It teaches the 
 
 Placing of Things Aright, 
 Setting Things in Order, 
 Shaping Things Aright, 
 Fitting Things to Each Other. 
 
 Forming truly the 
 
 Straight and the Bend, 
 Square and the Skew, 
 Upright and the Slope, 
 Flat and the Round. 
 
 It teaches 
 
 Building up the Great out of the Little, 
 Outtino- down the Great into the Many ;
 
 USEFUL GEOMETRY. 25 
 
 How the 
 
 Great grows into the Infinite, and the 
 
 Small dwindles down into the Infinitely Little. 
 
 VI. Use of Geometry. 
 
 It teaches 
 
 Geometric Laws of Force. 
 
 Geometry of Balance, 
 
 Geometry of Movement, 
 
 Persistence of Force in Line, 
 
 Persistence of Motion in Line, 
 
 Deviation of Line by Deflection of Force, 
 
 Compounding of Motions and Forces, 
 
 Stored Force, 
 
 Spent Force, 
 
 Growing Speed, 
 
 Steady Speed, 
 
 Waning Speed, 
 
 Paths of Projectiles, 
 
 Pendulums, 
 
 Planets.
 
 26 GEOMETRY IN MODERN LIFE. 
 
 VII. Use of Geometry. 
 
 Geometry gives useful training in the following 
 Professions, Occupations, and Trades. 
 
 Philosopher Master Professor Teacher. 
 Sailor Soldier Engineer Mason. 
 Architect Builder Plasterer Plumber. 
 Farmer Ploughman Hedger Ditcher. 
 Miner Smelter Mould er Fou n der. 
 Woodman Sawyer Carpenter Joiner. 
 Spinn er Warper Weaver Tailor. 
 Potter Glass-blower Glass-cutter Glazier. 
 Goldsmith Coppersmith Tinsmith Blacksmith. 
 Sculptor Carver Decorator Painter.
 
 FIRST STEPS IN GEOMETRY. 27 
 
 V. 
 
 FIRST STEPS IN GEOMETRY. 
 
 The first Thoughts in Geometry arise in Your 
 Mind and in Mine, when You and I standing 
 apart ask 
 
 How far asunder are We ? 
 
 The first Words of Geometry arise when we 
 want to tell each other what we think, about our 
 Places and our Distance 
 
 I stand Here You stand There. 
 
 I call the place where I stand " Here," 
 You call the place where I stand " There," 
 
 I call the place where You stand "There," 
 You call the place where You stand "Here." 
 
 Thus when we both mean the same thing we 
 two express our thought in opposite words.
 
 28 GEOMETRY IN MODERN LIFE. 
 
 The Words of Geometry have therefore different 
 meanings and even opposite, according to the place 
 of the Speaker or the place of the Listener at the 
 time the words are spoken. 
 
 Our first work as Geometers is to give clear 
 meaning to each of the simple words we use. 
 
 Here ! There ! ! Yonder ! ! ! 
 are the words I use for the Places we two speak of. 
 
 Here means My Place, 
 
 Where I stand, 
 My Stand -point ; 
 
 There means Your Place, 
 
 Where you stand, 
 Your Stand-point ; 
 
 Yonder means The other Place, 
 
 Where the other stands, 
 His Stand -point. 
 
 Geometers mark these Places thus : 
 
 My 1 You | Your 
 
 Stand I Q Q Stand- 
 
 point I Hero. There I point.
 
 FIRST STEPS IN GEOMETRY. 29 
 
 Geometric Ways. 
 Hence, Thence, Hither, Thither, 
 are the words for the Ways we think of. 
 
 We mark these Ways and show them thus : 
 
 I f Going Coming^ You 
 
 Q 1 ^_> Thither Hither < & \ Q 
 
 Here I Hence. Thence There. 
 
 Thus 
 When I go " Thither," you call it " Hither," 
 When You come "Hither" you call it "Thither." 
 
 Thus, once more, meaning the same ways, we 
 use opposite words, I calling that way "Hither" 
 which You standing opposite call "Thither." 
 
 Thus the meanings of Words about Place and 
 Way change with the Speaker or the Listener or their 
 place or attitude. 
 
 To make words about Place and Way clear and 
 exact, the Geometer first settles one place which 
 he calls the Starting-point, another which he calls 
 a Stand-point, and any or many others which he 
 calls Stations or Marks.
 
 30 GEOMETRY IN MODERN LIFE. 
 
 His first work is to learn or fix how far the Starting- 
 point and Stand-point are asunder from each. This 
 he sometimes calls his Base Line sometimes he calls 
 it his Standard (stand-hard) of di-stare (di = dwy 
 = two and so apart) or Distance (di-stantia) 
 sometimes it is a very long distance or a short one, 
 a Standard Mile or a Standard Yard or a Standard 
 Inch. 
 
 Having settled my Starting-point here and your 
 Stand-point there, let us take the first steps to get 
 accurate measure of Far and Near. 
 
 You standing There, 
 I standing Here, 
 
 Let us agree to Meet (Mete). 
 
 Starting Hence I take 1 1 steps your way, 
 Starting Thence you take 1 1 steps my way. 
 
 We Meet. 
 
 We call our Meeting Place " Midway." 
 
 "Midway" between Starting-point and Stand- 
 point is now a settled Station. 
 
 We call this Midway Station, the same distance 
 from your Place and mine. 
 
 If our steps agree our Midway is equidistant from 
 our two stations.
 
 FIRST STEPS IN GEOMETRY. 31 
 
 We must therefore agree our steps. I therefore 
 walk from Midway to your Station and find it 11 
 of my steps ; you walk from Midway to my Station ; 
 if you find it 1 1 of your steps : we are at one. 
 
 It is agreed that 
 Midway is 11 steps away from both stations. 
 
 Whence it comes that 
 Start-point and Stand-point are 22 steps asunder. 
 
 Start- _ Stand- 
 
 . O ... ii ...+... ii ... o . 
 
 point v -^ ^-^ point 
 
 x <-22-> x 
 
 An agreed Step is commonly called a Pace. 
 
 An agreed Standard among English Geometers is 
 called the Yard Measure. 
 
 An agreed Standard among English sailors is 
 called the Fathom Measure. 
 
 The Yard Measure measures half-way of the 
 Fathom ; " Two Yards " mean the same distance as 
 " One Fathom." 
 
 The Basis of all Geometry is therefore a 
 
 Settled Start-point, and 
 Settled Standard of Distance.
 
 32 GEOMETRY IN MODERN LIFE. 
 
 England has settled her Start-point, from which 
 she has measured England, Europe, the World, and 
 the Heavens. 
 
 England's Start -point lies in Greenwich Park. 
 There it is preserved in a national building, in care 
 of the Geometers of the nation. From this Start- 
 point in Greenwich Park all England is measured 
 all English maps are measured and made from this 
 Start-point Maps or Charts of all Seas round the 
 World are measured and made from it, and printed 
 for the use of all English ships and sailors. 
 
 Wherever an English sailor in an English ship has 
 to find his way across the sea, an English map shows 
 him how he should go showing him how far he now 
 is to the East or to the West of this Start-point 
 in Greenwich Park. On his map this Start-point 
 is marked Q and the other stations are marked 
 & c -> as they go Eastward from Greenwich 
 round the World. 1 
 
 England has also settled her Standard of Distance 
 as well as her Standard of Place One English Yard 
 has long been safely preserved by the English 
 nation as the Standard of all English Measure. 
 
 The English Yard is the Standard of Distance 
 
 o 
 
 on Land. 
 
 1 These marks show '' Meridians " and give " Longitudes " on Maps 
 and Charts.
 
 FIRST STEPS IN GEOMETRY. 33 
 
 The Fathom double the Standard Yard is the 
 Standard of Distance at Sea. 
 
 The English Standard Measure is as carefully 
 kept for the service of the English nation as the 
 English Start-point. 
 
 The necessity for the establishment of a national 
 Standard of Measure is very plain We cannot all 
 step alike we all want to measure truly, not to 
 take distance as mere guess work, or matter of 
 personal fancy. As men's strides differ in stretch, 
 the English nation found that they must take one 
 man's measure, instead of leaving his own measure to 
 each man's choice. So they took the measure of the 
 King's arm's-length, and agreed that should be 
 Standard English measure. Both his arms out- 
 stretched make the English Fathom, and each one 
 a Yard. That King was Henry I. But the French 
 took the length of Charlemagne's foot as their 
 Standard, calling it pied du-roi. It is also on record 
 that the girdle of the Saxon Kings kept at Winchester 
 was ordered by King Edgar to be the Standard 
 "gird" of English measure, and that it was ordered 
 by William the Norman not to be changed. The 
 English Standard was afterwards deposited in the 
 Crypt Chapel of Edward the Confessor in the 
 Cloisters of Westminster Abbey, and was carefully 
 preserved there till recently, when it was trans- 
 ferred into the keeping of the present " Warden of 
 the Standards."
 
 34 GEOMETRY IN MODERN LIFE. 
 
 It is instructive to find how man's measures of this 
 world and of other worlds grew out of his own person 
 and his own wants. A sailor measures a line or a 
 rope, by unwinding it, holding a length of it out- 
 stretched between his arms, and then shifting the line 
 length after length until he has measured it all by 
 arm's lengths. To tie a knot at each arm's length was 
 an easy way of marking a line, and at every 11 
 arm's lengths. Of these arms' lengths, or fathoms, 
 11 are the breadth of one acre, 110 being the length 
 of the acre ; 880 are an English land-mile, 1014 make 
 a sea mile, and 21, GOO sea-miles gird the globe. The 
 foot, once the " pied du roi," is in England somewhat 
 shorter than the foot of Charlemagne, being | of the 
 fathom, or ^ of the yard. Thus man takes himself as 
 the measure of all things round about him. Perhaps 
 also we Finite Beings can only comprehend the 
 Universe about us by the measure of our own little 
 bodies, little ways, and little minds !
 
 FIRST WAYS IN GEOMETRY. 35 
 
 VI. 
 FIRST WAYS IN GEOMETRY. 
 
 The StartiDg-point in our Geometry once settled, 
 our Measures agreed, our Distance taken in Number 
 and Measure we may now proceed to find our way 
 further on and otherwise. 
 
 You and I standing apart, I Here, You There ; 
 have paced towards each other 11 paces and have 
 met Midway. Let us now each turn aside. You go 
 to your Right, I go to my Right, or as a third person 
 would say, one go to the Right and number two go to 
 the Left. Let us each make 11 paces and take our 
 stand there turning round face to face. We now face 
 each other, 22 paces asunder. Let us next move 
 towards each other and meet Midway. Let us next 
 return, you to your Left, I to my Right, each to our 
 old place. 
 
 L> 2
 
 36 
 
 GEOMETRY IN MODERN LIFE. 
 
 Your 
 Second Station. 
 
 O 
 
 I 
 
 O 
 
 Here 
 
 Meeting 
 
 m> 
 
 Meeting 
 
 You 
 
 -O 
 
 There 
 
 O 
 
 My 
 
 Second Station.
 
 FIRST WAYS IN GEOMETRY. 37 
 
 What have we done ? We have both paced new 
 ways " Eight Across " our former ways. 
 
 Formerly, we paced Eight Forward and met Now, 
 we went Eight Across, returned and met. 
 
 This New Way is called the "Eight Across" to 
 mark its difference from the Eight Forward. 
 
 Sometimes also we say " Eight On " and " Square 
 Across," meaning the same difference of way. 
 
 But whatever words we chose, 
 
 Straight On, and 
 Eight Across 
 
 are the two first clearly distinct w T ays in Geometric 
 Thought and Work.
 
 38 
 
 GEOMETRY IN MODERN LIFE. 
 
 Your 
 Station. 
 
 O 
 
 on? 
 
 My 
 
 Starting 
 
 ^-v Returning 
 
 Point. 
 
 -< W 
 
 Returning 
 
 Your 
 Starting 
 
 -O 
 
 Point. 
 
 M 
 
 o 
 
 My 
 
 Station.
 
 FIRST WAYS IN GEOMETRY. 39 
 
 The most perfect means of pointing out these two 
 ways, and marking the nature of their difference to 
 the eye is to take a sheet of paper, with no fold in 
 it, then to double it, folding it flat, and opening it 
 out again, when you see a mark right through the 
 paper. Call that mark a "Line." Next, take the 
 two ends of that line and lay them on each other by 
 folding the paper over. Then press the doubled 
 paper quite flat and open it out again. You will now 
 see two marks going right through the paper and 
 going right across each other. 
 
 These marks in the flat sheet are two marks of 
 the difference of way called 
 
 Straight On, and 
 Right Across.
 
 40 
 
 GEOMETRY IN MODERN LIFE. 
 
 be 
 
 < m 
 
 Square Across. 
 
 To the Left. 
 
 01 
 
 Q 
 
 Square Across. 
 
 To the Right. 
 
 tc 
 
 -
 
 FIRST WAYS IN GEOMETRY. 41 
 
 This process of folded measures, may assist us 
 in thinking clearly out the matter of " Way and 
 Measure," as distinguished from our former process of 
 "Distance and Measure." 
 
 Our first folding of our flat sheet gave us one 
 marked track along the paper. We call it straight 
 along or right along, because it shews no turning aside 
 either way. The sheet of paper on either side seems 
 the same, shewing no difference between one side 
 of the line and the other side of the line. Both sides 
 in the overfold agree and are at one, there is no dif- 
 ference between the right side and left ; and the line 
 continues always right on one way without turning. 
 
 This therefore we call the Eight Forward or Straight 
 Onward way and line. 
 
 Our second folding of our flat sheet gave us a 
 second marked track across the former marked track. 
 We call it Straight Across or Right Across or Square 
 Across, because it shews no leaning to either side. Each 
 side covers the other side and so measures it exactly by 
 itself. The two are one they agree. Right Across 
 is the middle way between the two opposite ways. 
 
 Thus, then, when we call the one way Right Forward 
 we call the second or middle way Right Across Or 
 if we called the one way Right Backward, this same 
 middle way would still be right across the one way, 
 whether you called it forwards or backwards. When 
 we face each other, forwards from you is backwards 
 from me, when ways cross us both, Right Across to 
 you and Loft Across to mo are one.
 
 42 
 
 GEOMETRY IN MODERN LIFE. 
 
 
 
 j. 
 
 
 a 
 
 w 
 
 - 
 
 E 
 O 
 
 O 
 
 03 
 
 P4 
 
 fc 
 
 < 
 
 O 
 
 
 
 O 
 
 
 
 fz
 
 FIRST WAYS IN GEOMETRY. 43 
 
 Thus the language of Way becomes settled and 
 clear. And we next ask ourselves Can we have 
 a starting way, just as we had a starting point 1 
 Can we have a Standard Measure of Difference of 
 Way, just as we made a Standard Measure of Dif- 
 ference of Distance ? The North Pole Star the 
 Eising and Setting Sun, give us our Leading Ways 
 or Starting Directions for our sailing voyages, for our 
 land travels, and for our Geometric Surveys. 
 
 The Pole Star and the Mariner's Compass shew us 
 the one way of North and South. 
 
 The rising sun and the setting sun on the 
 21st March and 21st October, shew us the other 
 way of East and West. 
 
 And these two ways go Right Across.
 
 44 GEOMETRY IN MODERN LIFE. 
 
 VII. 
 
 LAND MEASURING. 
 
 The ways of the Land Measurer and of other 
 " Geometers/' are reckoned by the methods and 
 guided by the principles we have initiated in our 
 last two Sections. 
 
 Suppose I want to make the plan of a field ? I 
 first walk round it, and make a sketch on a page 
 of paper shewing the shape it seems to me ; one side 
 may be on the bends of a stream, the other on a 
 road with trees and cottages, that may give it an 
 irregular shape difficult to measure. But I get over 
 the difficulties by the use of the two kinds of line, 
 " Straight Forward " and " Right Across." 
 
 I take first my stand at a point whence 1 see right 
 through to the farthest off point of the field. I then 
 walk my own measured pace along this line, keeping 
 count as J go, and marking in figures any objects 
 I pass -a tree a stream a fence a path.
 
 LAND MEASURING. 
 
 45
 
 46 GEOMETRY IN MODERN LIFE. 
 
 This first measured straight way I may call my 
 Base Line. My Starting-point being at one end and 
 my Stand-point at the other. My measure of this 
 is, say 330 of my paces or yards. That I mark on 
 my plan. 
 
 Start Bush Oak Station 
 
 o -o o o 
 
 70 60 230 
 
 Base. 
 
 When I have reached my Stand-point at the far 
 end of the field and recorded my distance ; I take 
 that as my Base line. I next proceed to take 
 measures Right Across my base line. These measures 
 taken right across the base line, Land Surveyors 
 sometimes call "Offsets." I take my measures right 
 across as follows. Standing at my far station, I 
 return along my base line. Returning I measure 
 22 paces backwards. Here I stop. Turning to the 
 right, I walk Straight Across, to the boundary, pacing 
 as I go, and pacing as I return to the base line as 
 a check on my accurate pacing. I mark the 
 measure of this on my sketch, and this I measure 
 off exactly on my plan across the base line to the 
 right. Next I turn to the left of the base line ; I 
 walk straight to the boundary, pacing as I go, mark 
 my measure on the sketch, and check it by pacing 
 as I return. Thus I have measured my first pair 
 of offsets right and left at my first halting-point on 
 my way back along the base line.
 
 LAND MEASURING. 
 
 47
 
 48 GEOMETRY IN MODERN LIFE. 
 
 The Second process is like the first. I return along 
 the base line a second stage of 22 paces, stopping 
 at 44 yards from the far Stand-point. Here I pace 
 my second offset to the right and my second offset 
 to the left, and I check both and record them on 
 my sketch. 
 
 Thus I continue my work without change of 
 method, to a Third, Fourth . . . and Fifteenth 
 Stopping place, recording a pair of offsets at each. 
 I am now at the original Start-point. My work of 
 measure is done. My work of making a plan of the 
 work done still remains. I take a flat board, and a 
 flat sheet of paper stretched on it. I draw a Base 
 Line, I measure 330 equal parts on it ; I call each 
 part a pace, I mark every 22nd pace, and draw a 
 line right across at each mark, going off to Right 
 and Left, and I mark on each and measure on each 
 its recorded length. I place a mark on the end of 
 every offset at its place of measured length. Then 
 I see plainly the outline of the Field truly " marked 
 out " or " set off" by this line of marks. 
 
 Thus I have a Plan of my Field, recording its 
 measures. But J want to know more. I want to 
 know how many acres my measured field encloses 1 
 What is an acre ? An acre means the day's work of 
 a plough. What does a plough do in a day ? A 
 good plough covers with a day's work a stripe of land 
 22 yards wide and 220 yards long, which on water
 
 LAND MEASURING. 
 
 49
 
 50 GEOMETRY IN MODERN LIFE. 
 
 would be called 11 fathoms wide and 110 fathoms 
 long. Or if turned into feet it would be 66 feet wide 
 and 660 feet long. That is an Acre. 
 
 Now in measuring the field, we stopped every 
 22 yards and took an offset. We cut the land right 
 across by our offsets, into stripes, each 22 feet wide. 
 If then we had 15 stripes of 22 yards wide, and if 
 the average length of each stripe were 220 yards, 
 then our field was exactly 15 acres, and so our work 
 is done. 
 
 Now the simple result would be quite correct if the 
 offsets had been all alike long, as they would be in 
 a field everywhere the same breadth. But if they 
 varied in length, we have only to add all the lengths 
 together, and we should get the same result, as the 
 longer offsets would make good the defects of the 
 shorter offsets.
 
 LAND MEASURING. 51 
 
 Record of Measurements. 
 Base line 15 stations of 22 yards = 330. 
 
 Across 
 
 
 Start-point 
 
 Across. 
 
 Eight 
 
 
 o 
 
 
 Left 
 
 11 
 
 
 22 
 
 
 13 
 
 22 
 
 
 44 
 
 
 23 
 
 33 
 
 
 66 
 
 
 34 
 
 44 
 
 
 88 
 
 
 59 
 
 55 
 
 + 
 
 110 
 
 + 
 
 88 
 
 66 
 
 
 132 
 
 
 82 
 
 84 
 
 
 154 
 
 
 86 
 
 90 
 
 
 176 
 
 
 121 
 
 73 
 
 
 198 
 
 
 165 
 
 63 
 
 + 
 
 220 
 
 + 
 
 196 
 
 50 
 
 
 242 
 
 
 157 
 
 37 
 
 
 264 
 
 
 116 
 
 26 
 
 
 286 
 
 
 80 
 
 14 
 
 
 300 
 
 
 39 
 
 O 
 
 + 
 
 330 
 
 + 
 
 o 
 
 E 2
 
 52 GEOMETRY IN MODERN LIFE. 
 
 Calculation of Acres. 
 On the Right of the Base Line. 
 
 On the Left of the Base Line. 
 I | Completed Stripes. 
 
 11 
 
 13 
 
 24 
 
 
 I 
 
 22 
 
 23 
 
 45 
 
 
 II 
 
 33 
 
 34 
 
 67 
 
 
 III 
 
 44 
 
 59 
 
 103 
 
 
 mi 
 
 55 
 
 + 88 + 143 
 
 + 
 
 V 
 
 66 
 
 82 
 
 148 
 
 
 VI 
 
 84 
 
 86 
 
 170 
 
 
 VII 
 
 90 
 
 121 
 
 211 
 
 
 VIII 
 
 73 
 
 165 
 
 238 
 
 
 vim 
 
 63 
 
 + 196 + 259 
 
 + 
 
 X 
 
 50 
 
 157 
 
 207 
 
 
 XI 
 
 37 
 
 116 
 
 153 
 
 
 XII 
 
 26 
 
 80 
 
 106 
 
 
 XIII 
 
 14 
 
 39 
 
 53 
 
 
 XIII1 
 
 668 
 
 1259 
 
 1927 
 
 
 On the Right 668 yards length of Stripe 22 yards 
 wide. 
 
 On the Left 1259 yards. 
 
 On the whole 1927 yards length, or 8 acres 
 and | [8.759].
 
 LAND MEASURING. 53 
 
 Thus to measure a field, to tell how many acres 
 it covers, you merely cut it into stripes 22 yards 
 broad, and you measure the length of each of the 
 stripes, and you add these lengths all together, and 
 you see how many lengths there are of 220 yards 
 each, and that number of lengths is the number of 
 acres, which gives the following scale of acres. 
 
 22 yards Broad. SSL* MHe. Ten Miles. 
 
 Length of Stripes . 220 440 660 880 1760 17600. 
 
 Number of Acres . One Two Three Four Eight Eighty Acres. 
 
 Hence we see that a railway passing through land 
 and taking a breadth, for fences, works and way, of 
 22 yards, would require eight acres of land per mile 
 of railway. Where the works were small half that 
 would do. 
 
 This measure of acres of plough fields, and the 
 small measures of cloth, are done the same way. A 
 web of cloth may be one yard wide measured right 
 across its length, and may be 33 yards long. This we 
 call one piece of cloth. We also call it 33 square 
 yards of cloth. 
 
 A wall we should measure the same way. A 6 feet 
 wall, might measure two yards upright and one mile 
 long. That we should call twice 1760 yards, or 3520 
 square yards of wall. A window having panes of
 
 54 GEOMETRY IN MODERN LIFE. 
 
 glass, each 1 foot wide and 18 inches long, of 
 12 panes in number, would make one stripe of 
 glass, 18 feet long of 1 foot wide measured square 
 across. We should call that an extent of 18 square 
 feet. 
 
 In like manner a piece of silk ribbon 2 inches 
 wide and 33 inches long would be said to contain 
 66 square inches, and laid on a piece of cloth that 
 ribbon would be said to cover 66 square inches of 
 cloth. 
 
 In like manner a stream, canal, or river, might be 
 half a mile wide and 66 miles long : and that water 
 we might say covered an extent of 66 miles long, 
 half a mile wide or 33 miles long of 1 mile wide, or 
 33 square miles in extent. Thus, the thoughts of 
 " Distance " and " Measure/' " Straight Forward," 
 and " Right Across," correctly warded and properly 
 worked out, give all the knowledge necessary to 
 enable you to do the Geometry of Field and Land 
 and Flood Measuring. 
 
 All this simple Geometry of the two Elementary 
 ways Straight Forward and Right Across is good 
 for all measures, easily within reach. But when you 
 have to take larger and longer measures, than our own 
 paces can roach, where you have to measure impassable 
 streams, wide lakes, steep cliffs, high mountains, broad 
 sens and distant stars : you luive to seek the aid of
 
 LAND MEASURING. 55 
 
 instruments of direction and distance, which the 
 
 Practical Geometer has made, which enable your 
 
 sight to see afar and to measure afar, distances which 
 
 your own senses and your personal measures, cannot 
 
 compass without their aid. Such methods and in- 
 
 . . , . ( Topographic 
 struments are explained in < ~ 
 
 I Geometry.
 
 56 GEOMETRY IN MODERN LIFE. 
 
 VIII. 
 
 THE GEOMETRY OF THREE WAYS. 
 
 We have now seen how the Geometry of " The Two 
 ways," The Straight Forward and Right Across 
 enables us to take the shapes and draw the plans and 
 measure the acreage and tell the extent of Fields and 
 Walls, and Stuffs. 
 
 We have next to study a Third Way the intro- 
 duction of which leads us to another branch of 
 Science. 
 
 The Geometry of Three Ways. 
 
 When You and I met face to face I here, You 
 There : when we met midway and measured our 
 distance asunder, we called that the Way from Stand- 
 point to Stand-point. We called it the Right For- 
 ward Way and the Straight Forward Way. When 
 next we Two met Midway, and you turned to the 
 Left while I turned to the Right, and then turned 
 and faced each other, and then returning met each 
 other Midway ; We called that Second Way Right 
 Across or Square Across the former way.
 
 THE GEOMETRY OF THREE WAYS. 57 
 
 But we may now introduce a third person into our 
 group. We must give him his own Stand-point. 
 This third Stand-point introduces new relations of 
 Place and Way, as Elements of our Geometry. 
 
 I Here. 
 
 You There. 
 
 He Yonder. 
 
 My 
 
 Your 
 
 His 
 
 Stand-point. 
 
 Stand-point. 
 
 Stand-point 
 
 I. 
 
 II. 
 
 III. 
 
 Now the first question is, where shall we place 
 him ? shall we first give him a Stand-point ? and 
 next let him choose one ? Let us first give him one 
 of our Stand-points, say at the end of our way Right 
 Across. Then he stands where one of us stood, and 
 we three face the same Middle Point. He facing Right 
 Across us. 
 
 The Three Stations. 
 
 in 
 
 o 
 
 i ii
 
 58 
 
 GEOMETRY IN MODERN LIFE. 
 
 Suppose now we wish to meet. We have only 
 each to march as formerly towards the common 
 Middle-point and we meet and return each to his 
 own Station. 
 
 Next he may wish to come to me. For this our 
 two ways suffice. He has only to march to the Mid- 
 point, face to the right and he reaches my station, or 
 to face to the left and reach yours. This he does and 
 returns back the ways he came. 
 
 The Three Distances. 
 m 
 
 o 
 
 o 
 
 < m 1 m > 
 
 X 
 
 -< m 10 m > 
 
 ii 
 
 o 
 
 Thus the two ways Right Across suffice for the 
 communication between our three places, until a new 
 question occurs to him.
 
 THE GEOMETRY OF THREE WAYS. 59 
 
 May / not come 
 
 Straight from III to I ? 
 
 May / not come 
 
 Straight from III to II ? 
 
 Might you not come 
 
 Straight from I to me ? 
 
 Might not he come 
 
 Straight from II to me ? 
 
 The answer is " Certainly." 
 
 It is now plain, that between each couple of us 
 three there is now one Straight Forward way, which 
 makes it unnecessary for either of us to go round by 
 the Midway and then turn right across a second 
 way before reaching the other's Stand-point. 
 
 Between you li and me I, 
 
 there lies One Straight unbroken Way. 
 
 Between you II and him III, 
 
 there lies One Straight unbroken Way. 
 
 Between him III and me I, 
 
 there lies One Straight unbroken Way.
 
 60 GEOMETRY IN MODERN LIFE. 
 
 Thus these three places, give rise to three Straight 
 Ways. 
 
 The Three Ways. 
 
 hi 
 
 o 
 
 N* 'r 
 
 / X 
 
 [ I 
 
 The study of 
 
 These Three Places, and 
 These Three Ways 
 
 gives rise to what is called, 
 
 "Trigonometric Geometry." 
 
 Trigonometric Geometry is the basis of Navigation, 
 and Topography, and our National Trigonometric
 
 THE GEOMETRY OF THREE WAYS. 61 
 
 Surveys and Maps are conducted by its laws. It is 
 therefore most worthy of study. 
 
 The Geometry of the Three Ways, starts from the 
 same modes of thought and action, which gave us to 
 understand the Geometry of the Two Ways. 
 
 When we were only Two, You and I stood apart. 
 We marched and met Midway, measuring as we met. 
 We returned to our former station checking our way 
 as we went back. 
 
 Now that we are Three, let us do the same thing. 
 Let us two, I and II, repeat our measure. Let us 
 two, I and III, measure our distance asunder, meet 
 and return. You two, II and III, measure your 
 distance asunder, meet and return. We three have 
 now met and measured our Three Ways. 
 
 Let us now compare together, our original Two 
 Ways, with our new Three Ways. Our original Two 
 Ways made a complete system of communication 
 between the same Three Places as the Three Ways 
 now do. Thus the question arises. Which is the 
 better 1 The Three Ways or the Two ? What are the 
 uses of the one and the other system ? 
 
 These two systems of Geometric Thought and 
 Work are distinguished by the names the Square 
 and the Skew. We ask, shall we go on the Square 
 or go Askew ? Shall we work on the Square or on
 
 62 GEOMETRY IN MODERN LIFE. 
 
 the Skew ? Sometimes the system of the Square or 
 Working and Measuring Right Across, is called the 
 System of Rectangular Co-ordinates, and tle other 
 is called Oblique Triangulation. But the meaning 
 is much simpler than these words. 
 
 It means, shall we measure lengthways and right 
 across ? or shall we go on the slope and measure 
 askew, in short, shall we take " Square Measure or 
 Skew Measure " ? 
 
 Let us go back to our original plan. Let us place 
 III on his station at the same distance Midway as 
 ourselves. We meet and part the nearest way, all 
 three going together to the Mid-point. But a new- 
 question arises. How far apart are He and We 
 on the skew ? This is the great question of 
 Trigonometry. 
 
 It is the question of 
 
 The Three Places, 
 The Three Ways, 
 The Three Measures. 
 
 To answer these questions, w T e have to proceed 
 systematically and compare and measure the difference 
 between The Square and The Skew. For this end, 
 place I and II in line, 20 yards asunder ; place III 
 right across 10 yards from Mid-point. We are all 
 three 10 yards from Mid-point.
 
 THE GEOMETRY OF THREE WAYS. 
 
 Ill 
 
 63 
 
 O 
 
 V 
 
 t 
 
 4- 
 
 / 
 
 
 \ 
 
 O * & 10 b > 
 
 o 
 
 < S 10 ^ > o 
 
 i 
 
 
 II 
 
 The first Question of Trigonometry is, 
 
 How far are we two from him ? 
 
 To this question there are two answers. 
 
 The first is, 
 
 20 yards on the Square. 
 
 Second, 
 
 14 yards on the Skew. 
 
 The Second question is, 
 
 How far are we two from each other ? 
 
 Answer, 
 
 20 yards on the Square. 
 28 yards on the Skew. 
 
 There is yet a third answer, which is, 
 "Nobody can exactly tell!"
 
 64 GEOMETRY IN MODERN LIFE. 
 
 The reason of this particular answer is, that at 
 20 yards asunder between I and II, and 10 yards 
 off to III on the square from Mid-point, the way 
 from I to III is 14 and on to II is 14 more, making 
 28 in all. But these numbers in measure have a 
 slight inaccuracy, which we cannot conquer ; and 
 which makes us call them " discords " or " incom- 
 mensurables." But the inaccuracy is so slight as to 
 be of no practical value and easily corrected for 
 use, by adding the decimal fraction, viz. 1 4. 142136 
 
 The great object of the Geometry of the Three 
 Ways, is to tell in all cases, the exact measured 
 distance from station to station, in any three ways, 
 for every degree of Skew, and to compare it with the 
 distances on the Square or Eight Across. 
 
 These measures are treasured in books made at 
 great cost, at Government cost, for the Public Good, 
 and called 
 
 Trigonometric Tables. 
 
 By the use of these Tables, Travellers, Sailors, 
 Astronomers, and Geometers of all sorts are saved 
 a life-time of calculation and trouble, by having 
 placed at their disposal the invaluable results of 
 whole lives of many other able men, spent in 
 calculations for our good.
 
 THREE-POINT MEASURE. 65 
 
 Villi 
 
 THREE-POINT MEASURE. 
 
 Three-point Measure grows out of the Geometry 
 of Three Ways, and is one of the parts of Geometric 
 Science of most use in the World of Thought and 
 Work. It is commonly called Trigonometry or Trigo- 
 nometrical Calculation ; and is best known as the 
 Geometry of Navigation, the Geometry of the Globe, 
 and is employed in Britain for the Trigonometrical 
 Survey. It is the basis of our calculations of the 
 places and distances of the stars. 
 
 My Start-point, your Stand-point, and our Distance 
 Apart, are the basis of all further measures. This 
 we call our Base Line. By means of these we proceed 
 to settle the Place and Distance of all round about 
 us. For this purpose we send another to his station, 
 and ask him to measure as he goes, and mark his 
 
 F
 
 66 
 
 GEOMETRY IN MODERN LIFE. 
 
 Stand-point. Thus, we choose a Third Station, and 
 he measures a Second Line of Distance, and takes 
 his stand. 
 
 Thus we stand 
 
 o 
 
 Here 
 
 Yonder 
 III 
 
 O 
 
 II 
 
 o 
 
 There 
 
 I Here You There He Yonder. 
 
 Let us agree to make our Base Line 10, 100, or 
 1,000 inches long, on paper ; or feet long, on a floor ; 
 or yards long, on the field : or chains long, across 
 the country ; or miles long, in world measure
 
 THREE-POINT MEASURE. 67 
 
 Let us agree with him that he shall make his 
 Line Right Across from your Station of the same 
 measure. 
 
 Now comes the important question 
 
 Can we foretell this Third New Distance from him 
 to me, without measuring ? 
 
 Answer 
 
 We can ! 
 
 How ? 
 
 By Three-point Measure ! 
 
 By Antient Geometry ! 
 
 But why need we calculate, when we have simply 
 to measure it by walking over the ground 1 
 
 Because a lake, or sea, or chasm, or precipice, 
 or void, may come between and render your step- 
 ping or measuring inexact or impossible ! Or in war 
 that space might be occupied by the enemy, and you 
 want to take the measure and get round him without 
 his knowing it. Your 3 stations and 2 lines might 
 lie outside a fortress. 
 
 f 2
 
 68 GEOMETRY IN MODERN LIFE. 
 
 Returning now to our question, we ask, How far 
 is Station I from Station III ? 
 
 Answer given by Geometry. 
 
 If I to II is measured exactly 100 Chains, Base 
 Line. 
 
 If II to III is measured exactly 100 Chains, 
 Right Across. 
 
 Then the Third Distance, 
 
 from 1 to III, on the Skew, 
 
 Must Measure 141 Chains, and 42 Links. 
 
 And this the Geometer tells you without ever 
 having made the measure. If you test it by trying 
 you will find his estimate correct. 
 
 This Three-point Geometry is of the most simple 
 and useful character it implies very few simple 
 elements 
 
 Two Points in Measured Line. 
 
 A Third Point, in a Second Measured Line, 
 Right Across. 
 
 The Third Distance foretold without Measure.
 
 THREE-POINT MEASURE. 69 
 
 The Geometer's Forecast of the Third Distance. 
 
 Just as the common Almanac tells you the days 
 of the week, and year, and school- days, and holidays, 
 so the Geometer's guide book tells us the measures 
 of ways for our guidance. Out of the Geometer's 
 book of the Three Ways, I give you the following 
 table and diagrams
 
 GEOMETRY IN MODERN LIFE. 
 Geometer's Forecast. 
 
 Base 
 
 Line 
 
 Line Run 
 
 Skew Line 
 
 Measured. 
 
 Right Across. 
 
 Foretold. 
 
 Chains of 
 22 Yards. 
 
 Chains of 
 22 Yards. 
 
 Note The big Figur 
 are Chains, the Small 
 ones are Links. 
 
 100 Chains. 
 
 10 
 
 Chains. 
 
 100. 49 
 
 100 
 
 33 
 
 20 
 
 13 
 
 101. 98 
 
 100 
 
 33 
 
 30 
 
 33 
 
 104. 40 
 
 100 
 
 33 
 
 40 
 
 ,, 
 
 107. 70 
 
 100 
 
 33 
 
 50 
 
 33 
 
 111-80 
 
 100 
 
 35 
 
 60 
 
 33 
 
 H6. 
 
 100 
 
 33 
 
 70 
 
 33 
 
 1 22. 06 
 
 100 
 
 33 
 
 80 
 
 3 3 
 
 128.06 
 
 100 
 
 33 
 
 90 
 
 33 
 
 134. 53 
 
 100 
 
 33 
 
 100 
 
 3 1 
 
 141.40 
 
 100 
 
 35 
 
 110 
 
 33 
 
 148.oo 
 
 100 
 
 33 
 
 120 
 
 33 
 
 156.20 
 
 100 
 
 33 
 
 130 
 
 33 
 
 164. 01 
 
 100 
 
 33 
 
 140 
 
 > 
 
 1 72. 04 
 
 100 
 
 33 
 
 1 50 
 
 - 
 
 i 80. , 7 
 
 100 
 
 33 
 
 160 
 
 3 1 
 
 188. 67 
 
 100 
 
 3< 
 
 170 
 
 3 3 
 
 197. 23 
 
 100 
 
 3) 
 
 180 
 
 3 
 
 205. 91 
 
 100 
 
 33 
 
 1.00 
 
 3 3 
 
 1 1 
 
 - ' 4 -70 
 
 100 
 
 
 200 
 
 33 
 
 '-co
 
 THREE-POINT MEASURE. 
 
 71 
 
 Diagrams of Distances as Forecast by the 
 
 Table. 
 
 
 100 
 
 /.*' 
 
 /# 
 
 loo 
 
 
 IOC 
 
 If I 
 
 100 
 
 Forecast. 
 JOl'M 
 
 loo 
 
 
 100 
 
 ,'1
 
 72 GEOMETRY IN MODERN LIFE. 
 
 Let us now see how this forecast of figures helps 
 us in life. 
 
 At sea, under sail, the question arises 
 
 I want to go due North in the wind's eye. My 
 sailing craft has made good her course N.W. on the 
 starboard tack, 50 knots, and is now on the port tack, 
 N.E., doing other 50 knots. How much due Northing 
 shall I have made out of these 100 knots tacking? 
 
 Answer 70 knots to the good. 
 
 This answer is got from the line of figures in 
 the Table 
 
 100 100 141. 
 
 We halve each figure and get 
 
 50 50 70 70 
 
 On board a ship you propose to have a foremast 
 square up 100 feet high, and 50 feet abaft the stem. 
 How long must the forestay be ? exclusive of the 
 ends to make fast. 
 
 Answer from the Table Close on 112 feet. 
 
 And how long must the forestay of the bowsprit 
 be which runs out 50 feet further ? 
 
 Answer from the Table Close on 142 feet. 
 
 On land you may ask this question If I go round 
 the corner of the fence 100 paces, and then turn 
 square to the right 150 paces more, how much 
 should 1 save by avoiding the corner, and going 
 aslant through the field ? 
 
 Answer from the Table 
 
 180 feet across. 
 instead of 250 feet round the corner, 
 saving 70 feel of distance.
 
 THREE-POINT MEASURE. 
 
 73 

 
 74 GEOMETRY IN MODERN LIFE. 
 
 X. 
 
 MEASURE AND NUMBER. 
 
 We have already watched the first steps of the 
 Geometer as he leaves his Starting-point, and paces 
 his way to his intended first Station. Arrived there 
 he has measured his first distance. This distance 
 he may call one mile, or two half miles, or four 
 quarters of a mile, or eight eighths of a mile, or 
 eighty chains, or 8,000 links. All these numbers 
 equally represent the one distance from Start to 
 Station. Thus, One measured distance may have 
 Many representative numbers. In this case there 
 are 1, f, i, f, 80, 8,000; also 1760 and 5280 
 represent the same distance in yard and foot 
 measure. 
 
 Tims the invention of Standard Measure converts 
 Distances into Numbers. Numbers from that moment 
 become integral parts of Geometry parts that can 
 be kept and studied quite apart but which are
 
 MEASURE AND NUMBER. 75 
 
 closely allied, and which in modern life render great 
 practical aid, each to the other. 
 
 One of the most simple and convenient numbers 
 for use in Antient Geometry was the Number 2, and 
 its family. 
 
 The Number 10 now contests its supremacy in 
 what moderns have widely adopted as the Decimal 
 system. Each has its advantages and disadvantages. 
 
 TWO. 
 
 Two and its family have the advantage of pro- 
 ducing a great multitude of Number in Measure, 
 with great simplicity in thought and in work. The 
 consequence of this is that in antient measures we 
 find it generally used.
 
 76 GEOMETRY IN MODERN LIFE. 
 
 Example. 
 
 Take 
 
 i_ 
 
 A Standard Measure. 
 Cut that in Two. 
 
 Two. 
 Cut each of these in Two. 
 
 Four. 
 
 
 Cut each of these in Two. 
 
 
 Eight. 
 
 
 Cut each of these in Two. 
 
 
 Sixteen. 
 Cut each of these in Two. 
 
 Thirty-two. 
 
 Cut each of these in Two. 
 
 Sixty- four.
 
 MEASURE AND NUMBER. 
 
 77 
 
 Now "Cutting in Two" is one of the simplest 
 operatioDS. If your measure is a fine thread, or string, 
 or rope, you merely fold it so that one half exactly 
 fits over the other, and the folding point marks the 
 division of Two ; and this you repeat as you choose. 
 
 This process of cutting in Two may be carried 
 on as far as you choose, and it amounts to a division 
 into more than a thousand minute measures in the 
 tenth operation of cutting. 
 
 Thus- 
 
 First cutting makes 
 Second cutting 
 Third cutting 
 Fourth cutting 
 Fifth cutting 
 Sixth cutting 
 Seventh cutting- 
 Eighth cutting 
 Ninth cutting 
 Tenth cutting 
 
 2 
 
 4 
 
 8 
 
 16 
 
 32 
 
 64 
 
 128 
 
 256 
 
 512 
 
 1024 
 
 Therefore, if you want to divide any measure into 
 very minute parts, in the simplest, quickest, easiest 
 way, the young geometer has no easier way than 
 using the process of repeated division in Two.
 
 78 
 
 GEOMETRY IN MODERN LIFE. 
 
 The antient mariner made a similar use of the 
 Number Two and its family, in Points of the 
 Compass. 
 
 He first divided the sea all round about him into 
 Two Ways 
 
 East and West. 
 
 Next into Two more Ways 
 
 North and South.
 
 MEASURE AND NUMBER. 
 
 79 
 
 Next, these Four Ways were divided each in two 
 by Four new Ways. 
 
 These Eight Ways, divided in two, give other Eight 
 Ways, making Sixteen.
 
 80 
 
 GEOMETRY IN MODERN LIFE. 
 
 And each of the Sixteen, parted in two, showed 
 Thirty-Two Ways. 
 
 Thus the antient mariner settled his Thirty-Two 
 Ways by the same Numbers which gave him his 
 antient Measures of Distance. 
 
 While we moderns have abandoned the antient 
 measure of 2, 4, 8, 16, and 32, in Measure of 
 Distance, we still adhere to the antient division of 
 ways into 2, 4, 8, 16, and 32, according to the 
 Points of the Compass. 
 
 My own opinion also is that the antient Knot, 
 or Nautic Mile, used in sea measure, was the length
 
 MEASURE AND NUMBER. 81 
 
 of B-V of a degree of the earth's girth, which by 
 division into 2, 4, 8, &c, ten times over, became 
 1,024 fathoms. 
 
 This differs in the number of fathoms from the 
 modern Admiralty rule, which gives 1,014 English 
 fathoms of two yards as measure of the Sea Mile. 
 This would make a difference of ^ between the 
 length of the Antient and Modern fathom. 
 
 This example shows clearly the great value of well- 
 chosen numbers. The English Admiralty Fathom and 
 Mile of 1,014, when halved, gives 507. When 
 quartered it gives 2531, or a "broken number;" 
 an eighth gives 126f, another broken number; a 
 sixteenth gives 63f, a number still more broken ; and 
 so on, worse and worse. 
 
 Whereas, by adopting for Fathom Measure 1,024 
 to one Nautic Mile, that would still be as true as 
 now, to a man's arm's-length, and 1,024 would be 
 halved 10 times without a single broken number. 
 
 G
 
 82 GEOMETRY IK MODERN LIFE. 
 
 Halving a Theoretic Nautic Mile of 1,024 Fathoms. 
 
 Halvings. Fathoms. 
 
 Once gives 512 
 
 Twice 256 
 
 Thrice 128 
 
 Four times 64 
 
 Five 32 
 
 Six 16 
 
 Seven 8 
 
 Eight 4 
 
 Nine ........ 2 
 
 Ten 1 
 
 Thus Measure and Number, wisely chosen, go 
 together, and help each other. Well chosen Numbers 
 also give aid to clear Thought. These numbers 1 
 have given as growing out of " Two," are numbers 
 never to be forgotten by thoughtful men who desire 
 to understand nature, creation, and common life. 
 To show that matter and nature in creation "Feel 
 these Numbers," even if they do not understand 
 them, may be gathered from one simple illustration. 
 
 Sounds composed each of some of those numbers 
 would be all in harmony with each other, and each 
 would sound a perfect octave, to the one below and 
 to the one above it. A string stopped at parts whose 
 length fits these numbers, would give a series of
 
 MEASURE AND NUMBER. 83 
 
 octaves in j)erfect tune. A series of organ pipes, 
 whose lengths were measured to fit those numbers, 
 would sound a series of octaves all in har- 
 mony. 
 
 Thus in Nature Order, Measure, and Number rule. 
 As in antient philosophy, so in modern science, 
 Numbers enter into Geometry, in the First Steps 
 of that science. My Stand-point and my Start- 
 point are marked in Geometry by a mark about 
 each, and this mark we commonly call 
 
 Zero Q or Starting Point. 
 
 And next, when I have completed One Measure, 
 say a Yard, a Fathom, a Mile, or a Knot, I call that 
 One " Standard" "Unit" "Measure," and I mark 
 it thus 
 
 .Zero One 
 
 G Standard Length. ^. 
 
 Unit; ~~ ^ 
 
 Starting Measure. 
 Point. 
 
 In Numbers, in like manner, we have a starting 
 Thought and a starting Mark, which precedes Measure 
 or Number, and which means neither Measure nor 
 Number. It means Nothing Zero a Cypher 
 
 G 2
 
 84 GEOMETRY IN MODERN LIFE. 
 
 Ought or Nought. It is the starting place of 
 Number, and is marked thus 
 
 Zero Ought Nought. 
 
 In Arithmetic, as in Geometry, there must be a 
 Standard, or Unit, or Constant, or Measure, or One, 
 or Something, in order to give meaning to Numbers, 
 and to give meaning to marks of Numbers. This 
 Standard Mark is called 
 
 Unity 1, or I, or the Unit One. 
 
 A Starting Point we call Zero, and a Standard or 
 Unit we call One, for the foundation from which 
 spring together the sciences of measure and number. 
 
 It is most desirable both for the Student of 
 Geometry, and the Student of Arithmetic, that they 
 should cultivate early the study of numbers, the 
 nature, characteristics, classes, and special uses of each 
 number. Each number has a special nature, a 
 common nature, a pedigree, and a peculiarity. Clear 
 thought and exact work depend mainly on selection 
 of fit Numbers and risht use of each Number. 
 
 Whether is a number or not is a moot question. 
 In our Decimal Arithmetic it occupies the same place 
 as the other numbers, and fills a similar place, and
 
 MEASURE AND NUMBER. 85 
 
 sometimes means even much more than they do. 
 Still, it is best to say that 
 
 " Zero, 0, is no number," and is only the 
 " Starting Point of Number." 
 
 The same question arises about 1, or I. Is " One " 
 a number ? Some people say " Yes ! it is the First 
 Number, the First of the Numbers. It is No. I ! " 
 But others say, " No ! Two is the smallest Number." 
 One is One, and is not a Number. 
 
 Perhaps it is wisest to say 
 
 " One, I, is not itself Number, but is only 
 " The Standard or Measure of Number." 
 
 If we agree on this mode of speaking, we shall 
 make 2 our first Number. Alone is One. Two 
 together is a Pair, a Couple, a Brace, and may be 
 marked several ways 
 
 " Two, II. One more than One alone. 
 " One Pair, I Couple, 1 Brace, 2 Uuits." 
 
 Two alike therefore make a Couple or Pair, in 
 Geometry. In Geometry, if you pace a Chain to 
 meet me, and I pace a Chain to meet you, we call 
 our distance 2 Chains. But we have to make sure 
 beforehand, that our paces are alike. And if we
 
 86 GEOMETRY IN MODERN LIFE. 
 
 take a string and double it, and cut it in two, we 
 must take care they are alike before we call them 
 2 Halves. 
 
 Two therefore, is in exact ways of thought and 
 word, the First Number. It requires exact measure 
 before we admit that the word and mark Two, II, 2, 
 are correctly applied. It applies equally to an 
 increase of I to I making Two Units ; and to a 
 division of I into Two Halves, provided the two 
 parts are duplicates, or alike. 
 
 One then is our Standard both in Geometry and 
 Arithmetic, and One, and One Alike taken together, 
 give us our First Number in both Sciences, viz., 
 " Two alike without difference," II, 2. 
 
 THREE. 
 
 Three is the next Number. We call it a "Perfect 
 
 Number," and it is sometimes called the " Perfect 
 
 Number" because it combines the fewest elements 
 with closest ties. 
 
 Three has a Starting Point and a Stand Point, as 
 No. I has. Three has a eouple of its parts alike, as 
 No. II has. But Three has what the others have not, 
 it has a Centre and Two Sides, it has a middle body 
 and a pair of wings, it is composed out of One and 
 Two together, and therefore includes the qualities of 
 both !
 
 MEASUKE AND NUMBER 87 
 
 Three consists of I and II, of I and I and I, and 
 may be represented thus 
 
 One and One and One, 
 
 or 
 
 I and I and I, 
 
 or 
 
 I 
 I I 
 
 or 
 
 ill 
 
 or 
 
 Whether in geometry, in architecture, or in nature, 
 the Number III and its family lead to symmetry, 
 beauty, and perfection. 
 
 Number Four has little to distinguish it beyond 
 other numbers. It is Two Twos a Pair of Pairs 
 a Couple of Couples. It is merely Two repeated.
 
 88 GEOMETRY IN MODERN LIFE. 
 
 It is merely the same over again. Four of the same 
 together have little variety, and mere monotony or 
 repetition. 
 
 FIVE. 
 
 Number Five has variety with uniformity. First, 
 it has One Unit, and Two Couples, as its elements. 
 In Geometry it has One in a Central Place, and One 
 Pair on either hand. Next, it adds to the Number 
 Three, a repetition of Two. 
 
 Five consists of 
 
 I I II I 
 
 One and Two and other Two. 
 
 II I I I 
 
 One and a Pair of Twos. 
 
 Mill 
 
 One and Two and Two. 
 
 .ill. 
 
 i, Li 
 
 Two and Three together.
 
 MEASURE AND NUMBER. 89 
 
 Five consists of the following groups of place 
 
 ooooo 
 oo oo 
 
 o 
 
 
 oo 
 
 
 oo 
 
 
 o 
 
 o 
 
 
 
 
 o 
 
 o 
 
 oo oo 
 
 By the same process we find that Six is merely 
 a Pair of Threes repeated. That Seven is a Centre, 
 with Six surroundings, or a Central Unit, with Two 
 Threes. That Eight is merely a repetition of Couples, 
 and that Nine is only a repetition of Threes. 
 
 Thus we gradually come to see Numbers in dif- 
 ferent groups, each with a special character. Know-
 
 90 GEOMETRY IN MODERN LIFE. 
 
 ledge of the nature of Number is therefore part of 
 the Science of Place, Order, and Measure ; and I 
 have taken the Number Two as a first example of 
 the value of Numbers, and of the special value which 
 belongs to one number, and distinguishes it from 
 others. 
 
 The first question worth knowing about any number 
 is this 
 
 Is the number " Even or Odd ? " 
 
 "Even" means that it belongs to the Number 
 Two. 
 
 " Odd " means that it disagrees with the Number 
 Two. 
 
 We thus get the following Classes 
 
 Even Numbers. 
 2 4 6 8 10 12 14 16 18 20 
 
 Odd Numbers. 
 1 3 5 7 9 11 13 15 17 19 21 
 
 Looking at these we soon see that the " Even " 
 Numbers are each 
 
 "A Pair of the same." 
 
 Thus, 4 is a Pair of Twos, is a Couple of 
 Threes, 10 is a Pair of Fives.
 
 MEASURE AND NUMBER, 91 
 
 Even Numbers are all therefore compounds of a 
 couple of another Number. 
 
 The ' ' Odd " ones, on the contrary, cannot be 
 parted in Pairs, except by leaving out the Odd 
 One. 
 
 Thus 
 
 3 is a Pair of Ones, with One left out. 
 
 5 is a Pair of Twos, with One left out, 
 
 7 is a Pair of Threes, with One over. 
 
 11 is a Pair of Fives, with One more. 
 
 Perhaps the best form in which the fitness of 
 Number to place, order and shape may be shewn, 
 is by setting out Places in order. Take Geometer's 
 marks for places, and arrange them in Order and 
 Number 
 
 I. II. 
 
 III. IIII. 
 
 O GO 
 
 ooo oooo 
 
 v. 
 
 VI. 
 
 ooooo 
 
 oooooo 
 
 We soon notice that the Odd One is a middle 
 mark, and that the Even One has no one in the 
 centre.
 
 92 GEOMETRY IN MODERN LIFE. 
 
 Let us next analyse them by ordering in Couples 
 thus 
 
 o 
 
 
 
 
 
 o 
 
 
 
 o 
 
 oo 
 
 oo 
 
 oo 
 
 oo 
 
 oo 
 
 oo 
 
 
 
 
 
 o 
 
 
 o 
 
 
 oo 
 
 oo 
 
 oo 
 
 oo 
 
 
 oo 
 
 oo 
 
 oo 
 
 oo 
 
 
 oo 
 
 oo 
 
 oo 
 
 oo 
 
 
 oo 
 
 oo 
 
 The Odd Numbers are at once detected by the 
 eye from the exceptional place of the Odd One. 
 
 In this form the Odd One is offensive to the eye, 
 and occupies an exceptional place, and at first sight, 
 perhaps, an inferior place. 
 
 But when arranged in single lines the Odd One 
 may be set in the most distinguished place, and so 
 have given to it the greatest importance. 
 
 Thus, among Three, the Centre One is made most 
 important 
 
 o o
 
 MEASURE AND NUMBER. 93 
 
 Among Five the Centre One is dominant 
 
 oo oo 
 
 Among Seven the Central One is eminent - 
 
 ooo ooo 
 
 Among Eleven there may be Two dominant, and 
 One predominant 
 
 oo oo oo oo 
 
 The reason of this plainly is, that because Eleven 
 is made up of a Pair of Fives, each Five may have 
 its own dominant centre, and one remains over to 
 dominate the Pair of Side Centres. 
 
 The Odd Numbers thus admit of an order which 
 is symmetric about a Centre, and the Even Numbers 
 do not. 
 
 Thus we may range the Odd Numbers in the fol- 
 lowing orderly group
 
 94 GEOMETRY IN MODERN LIFE. 
 
 O 
 
 o o 
 
 o o 
 
 o o o 
 
 rf o o o o 
 
 o o o o o 
 
 s ooosooo 
 
 OO0OOOO0 
 O0OO09OO0 
 
 oooooooo 
 
 000 
 
 ooooooooo 
 
 O0OO000O0 
 OO0OOOO0 
 
 OOO0OOO 
 
 s o o o o o 
 
 O o o o 
 
 o o o 
 
 o o 
 
 o o 
 
 o 
 
 p 
 o 
 o 
 
 o 
 
 I I 
 
 P3 

 
 MEASURE AND NUMBER. 95 
 
 It is plain from this ordering of the Odd Numbers 
 round their Centres, that the Odd Numbers are 
 Aristocrats, and the Even Numbers are Democratic. 
 It will also become plain how the ordering of Atoms, 
 of Matter in Nature, may out of the same matter 
 give strange varieties of groups, according to the 
 wise selection of Number set in order. 
 
 Out of due ordering of Number in place grows 
 a " Third Order " of numbers, of a quite other nature 
 from Odd or Even numbers. 
 
 They are commonly called 
 
 " Square Numbers," 
 and they may be either Odd or Even. 
 
 Take the Geometer's marks for places, and let us 
 arrange them side by side, but agreeing to limit our- 
 selves to having only as many side by side as we 
 have in one row. 
 
 We get this Form. 
 
 
 
 
 OO OO 4 
 
 
 
 ooo s 
 
 OO OO 
 
 
 OO 2 
 
 ooo 
 
 OO OO 
 
 O 1 
 1 
 
 oo 
 
 4 
 
 ooo 
 
 y 
 
 OO OO 
 
 16
 
 90 GEOMETRY IN MODERN LIFE. 
 
 OOO OOO' 
 
 ooooo* OOO OOO 
 ooooo OOO OOO 
 
 00000 OOO OOO 
 OOOOO OOO OOO 
 OOOOO OOO OOO 
 
 25 36 
 
 OOOOOOO 7 
 OOO0OOO 
 
 ooooooo 
 
 
 OOO0OOO 
 
 oooo 
 
 OOO0OOO 
 
 49 
 
 OOOOOOOO' 
 
 ooooooo 
 oooooooo 
 oooooooo 
 oooooooo 
 oooooooo 
 oooooooo 
 oooooooo 
 
 64
 
 MEASURE AND NUMBER. 
 
 97 
 
 Thus each Number has a quality which fits it 
 for some special use, and unfits it for some other; 
 and each Number belongs to some family of like 
 Number : and when we know the family to which 
 each belongs, and its place in that family, we can 
 tell its usefulness in Geometry, the part it plays in 
 Nature, and its fitness or unfitness for the uses of 
 common life. 
 
 In Geometry. 
 
 If I take a simple element of three-cornered form, 
 and set it out in rows in Odd Numbers, it will grow 
 into larger figures of like form 
 
 Element 
 
 b, 
 
 One. 
 
 Addition 
 
 S\ 
 
 Three. 
 
 Sum 
 
 
 Four. 
 
 Whole 
 
 H
 
 98 GEOMETRY IN MODERN LIFE. 
 
 Addition f\f\K FlvE - 
 
 Sum 
 
 
 Nine, 
 
 Whole 
 
 Addition 
 
 Sum 
 
 
 Seven. 
 
 Sixteen. 
 
 Whole 
 
 Addition HSktSJ\ ^ INE - 
 
 Sum 
 
 
 ia 
 
 
 Twenty-Five. 
 
 Whole
 
 MEASURE AND NUMBER. 99 
 
 In these figures, the first row has 1 
 
 the second row has 3 
 
 the third row has 5 
 
 the fourth row has 7 
 
 the fifth row has 9. 
 
 The whole number of Elements in each is 
 1, 4, 9, 16, 25. 
 
 If I take simple elements of Square form, and group 
 them in like manner, by Odd numbers around the 
 first element, I shall have symmetric growth of 
 large out of little, all of like form. 
 
 ] Take one square add around one corner of j j 
 it three squares 
 
 Together 
 
 o 
 
 these four make 
 
 one square of four 
 
 | Take a square of 4' add around one corner 
 I of it five squares
 
 100 GEOMETRY IN MODERN LIFE. 
 
 Addition 
 
 Sum 
 
 Whole 
 
 
 
 
 
 
 
 
 
 h 
 
 
 
 
 
 9 
 
 Five. 
 
 Nine. 
 
 Addition 
 
 
 
 Seven. 
 
 Sum 
 
 
 
 
 1 1 1 
 
 
 Sixteen. 
 
 Whole
 
 MEASURE AND NUMBER. 
 
 q 
 
 u 
 
 Addition Li Nine. 
 
 101 
 
 Sum 
 
 tr 
 
 Twenty-Five. 
 
 Whole 
 
 Even if I next pass to Kound forms, which cannot 
 be built up by numbers of like elements, but which 
 grow by transformation from small to great, by 
 additions of parts unlike the whole, I still find 
 that the same laws of Odd Number rule the growth 
 and the same Square Numbers represent the resulting 
 extent. 
 
 When we go further into circles, the same reason 
 still holds true A Circle of One, with a belt of 3 
 rolled round, it becomes doubled in diameter and in 
 size four fold.
 
 102 GEOMETRY IN MODERN LIFE. 
 
 SEPARATE. 
 
 Circle 
 Bolt 
 
 Circle 
 Belt 
 
 Circle 
 
 Belt 
 
 Circle 
 
 ONE 
 THREE 
 
 FOUR 
 FIVE 
 
 NINE 
 
 SEVEN 
 
 Belt 
 
 NINE
 
 MEASURE AND NUMBER. 
 
 103 
 
 JOINED. 
 
 RESULT. 
 
 Circle 
 FOUR 
 
 Circle 
 
 NINE 
 
 Circle 
 SIXTEEN 
 
 Circle 
 
 TWENTY 
 
 FIVE
 
 104 GEOMETRY IN MODERN LIFE. 
 
 When we pass from Forms in Space to Laws of 
 
 Matter, we find that the same laws of Number 
 
 rule Moving Mass. The Apple falling from the 
 Tree falls thus : 
 
 In the 1st quarter second it falls 1 foot 
 
 2nd 3 feet 
 
 3rd 5 
 
 4th 7 
 
 5th 9. 
 
 Thus the speed of the falling apple grows gently 
 but evenly with the time of fall, according to the 
 Odd Numbers, and we can thus sum up the total 
 heights fallen at the end of each period of fall. 
 
 In One Quarter Second 
 
 1 foot is the whole fall. 
 
 In Two Quarters or Half a Second 
 
 1 + 3 feet gives a whole fall of 4 feet. 
 
 In Three Quarters 
 
 1 + 3 + 5 gives 9 feet fall. 
 
 In Four Quarters 
 
 1 + 3 + 5 + 7 gives 16 feet fall. 
 
 In One and a Quarter Second 
 
 1+3 + 5 + 7 + 9 gives 25 feet fall.
 
 MEASURE AND NUMBER. 105 
 
 And this law of Odd numbers for each quarter, 
 and of Square numbers for the whole time, is the 
 Law of Falling Bodies, equally for 
 
 Apples from the Tree, 
 Shot from the Gun, 
 Planets Round the Sun. 
 
 Such Laws of 
 
 Numbers, Order, Reason, 
 
 as 1, 3, 5, 7, 9, 11, 13 . . . and 1, 4, 9, 16, 25, 36, 49 
 
 pervading 
 
 Forms, Shapes, Quantities of Things, 
 
 ruling 
 
 Geometry, Men ; s Acts, 
 
 Earth's Phenomena, Orbits of Heaven, 
 
 lead to the thought, that 
 
 Number, Measure, 
 
 Order, Reason, 
 
 Pervade Equally, the 
 
 Thinking Mind, Nature of Matter, 
 
 Forms of Space, Force, 
 
 Shapes of Things, Motion, 
 
 throughout the Earth and the Heavens.
 
 10G GEOMETRY IN MODERN LIFE. 
 
 XI. 
 
 FAMILIES AND PEDIGREES OF NUMBERS. 
 
 Between Chaos and Creation, we find this striking 
 difference, that in one all things seem thrown 
 together without purpose, plan or method, while 
 in the other all things are set out systematically 
 in settled place, order and number. 
 
 Ordered Place means Shape ; 
 Ordered Number means Symmetry. 
 
 Together they give 
 
 Symmetric Form. 
 
 Place without Order means Disorder ; 
 Place without Number means Confusion. 
 
 Together they give shapeless crowd or 
 Chaos.
 
 FAMILIES AND PEDIGREES OF NUMBERS. 107 
 
 The first step towards Symmetric Form is giving 
 to each element, Settled Place, giving precise meaning 
 and measure to the thoughts and words 
 
 Here, There, Yonder. 
 Close, Wide, Afar, 
 Beside, Between, Beyond. 
 
 The next step towards Symmetric Form is ordering 
 each element into Clear and Exact Relation to each 
 other, as 
 
 In Line The Same Way. 
 
 In Lines Parting Ways. 
 
 In Lines Like Ways. 
 
 In Groups Round Centres. 
 
 Thus Order and Rule displace Disorder and 
 Confusion. 
 
 The third step towards Symmetric Creation is 
 the settlement of a Standard of Unity, or group, 
 or model or measure, out of which all things round 
 about shall grow in proportioned bond or relation, 
 measured by Number. This gives birth to the One, 
 or the Unity, or the Root, or the Base, or the 
 Parent or the ancestor of the rest who follow, 
 surround, or grow out of the Primitive One. 
 
 In our thoughts of things we have therefore Our 
 Starting Point or Place, which is not a thing nor
 
 108 GEOMETRY IN MODERN LIFE. 
 
 a measure, nor a number ; though it is the place 
 of one, and though it is often spoken of and thought 
 of as if were a number. 
 
 It is called 
 
 Ought, Nought, or Zero. 
 
 It is marked 
 
 o, 0> or o. 
 
 We shall call it in our Geometry, " The Starting 
 Point of Number." 
 
 The First Number in the common list of numbers 
 is commonly called 
 
 Number One. 
 
 But it is not a number in the strict sense of the 
 word, when we take " a number " to mean several 
 or some, or part of many. Nevertheless, we may 
 with due precaution call it the Beginning of Number, 
 the Root of Number, the Foundation of Number, 
 the Unit or Standard by the repetition of which we 
 get, " two," " several," " more " and " many." 
 
 Number Two 
 
 Is in reality the First Number, as of use in ex- 
 pressing the connection of one thing with another,
 
 FAMILIES AND PEDIGREES OF NUMBERS. 109 
 
 or of one thought with another thought, or of one 
 person with another. 
 
 "Two of the Same" 
 
 express what we may consider the earliest stage 
 of Genesis of Number. 
 
 "Two" and "Twin" and "Twain," 
 
 "A pair" "A Couple" "A Brace," 
 
 All express this primitive thought of number, 
 or radical beginning of measured thought- It may 
 be called 
 
 The Elementary Number, 
 
 and the study of it gives most knowledge most 
 exactly with least pains. 
 
 But the Number 
 
 Three 
 
 is " the typical " and to many the most " perfect " 
 of numbers. It is the first in the whole series of 
 odd numbers, and is only second in the series of 
 families of numbers. The families of numbers, are 
 the various series of numbers, which flow from or 
 grow out of, or are created by some single primitive 
 number, or parent number, or ancestor.
 
 110 GEOMETRY IN MODERN LIFE. 
 
 Two 
 
 Has a large family of numbers, all growing up 
 out of it as their root, or descending down from 
 it as their parent. Each of these is the twin family 
 of the parent pair. Family follows family multi- 
 plying by Two, and each family is known by its 
 order, and the number of its descent is called its 
 pedigree. The parent of each family, of pure and 
 unmixed number, is sometimes called the Prime 
 Number, which means the same thing as Princely 
 Pedigree. 
 
 The following are the most distinguished Prime 
 or Primitive or Princely Numbers 
 
 + . - 
 I, U, III, V, VII, XI, XIII. 
 
 It may seem strange at first sight that these 
 numbers should belong to higher rank than the 
 omitted numbers IIII, VI, VIII and Villi; but it 
 will soon appear that those belong to the second or 
 next lower degree of numbers.
 
 FAMILIES AND PEDIGREES OF NUMBERS. Ill 
 
 The Family of The Number Two 
 
 Has the following series of generations and 
 degrees 
 
 First Generation 
 
 II 
 
 Marked 2 ] 
 
 Second Generation II and II 
 
 Marked 2 2 
 
 Third Generation 
 
 ( II II 
 
 i ii ii 
 
 Marked 2 3 
 
 Fourth Generation 
 
 II II II II 
 II II II II 
 
 Marked 2 4 
 
 Fifth Generation 
 
 II II II II 
 II II II II 
 ii II II II 
 II II II II 
 
 Marked 2 5 . 
 
 Thus the numbers 2, 4, 8, 16, 32, are the first 
 members of the family, and are technically called 
 "the 1st, 2nd, 3rd, 4th, and 5th Degrees, from the 
 Koot II."
 
 112 GEOMETRY IN MODERN LIFE. 
 
 The Family of the Number Three 
 
 Has the following series of generations or 
 degrees 
 
 First Generation III Marked 3 1 
 
 III 
 
 Second Generation T rr ryr Marked 3 2 
 
 III 
 
 Third Generation IH IH Marked 3 3 
 
 III III 
 
 III III III III 
 
 9 
 
 9 9 
 Fourth Generation Marked 3 4 
 
 9 9 
 
 9 9 9 9 
 
 9 
 9 9 
 
 9 9 
 
 Fifth Generation 9 9 Marked 3 5 . 
 
 9 9 
 
 9 9 9 9 
 
 9 9 9 9 
 
 9 9 9 9 9 9 9 9 
 
 Thus the Numbers 3, 9, 27, 81, 243, are the first 
 members of the family and are technically called, 
 "the 1st, 2nd, 3rd, 4th and 5th Degrees from the 
 Root TIT."
 
 FAMILIES AND PEDIGREES OF NUMBERS. 113 
 
 The Family of the Number Five 
 Has the following Members and Degrees 
 
 First Generation V. Marked 5 1 
 
 V, 
 
 Second Generation v V V V Marked 5 2 (25) 
 
 25 25 
 Third Generation 25 Marked 5 3 (125) 
 
 25 25 
 
 125 125 
 
 Fourth Generation 125 Marked 5 4 (625) 
 
 125 125 
 
 625 625 
 Fifth Generation 625 Marked 5 5 (3125). 
 
 625 625 
 
 Thus the Numbers, 5, 25, 125, 625, 3125, are the 
 First Numbers of the Family, and are technically 
 called " the 1st, 2nd, 3rd, 4th and 5th Degrees from 
 the Root V."
 
 114 GEOMETRY IN MODERN LIFE. 
 
 The Family of the Number Seven 
 Has the following Members and Degrees 
 
 First Generation Marked 7 1 
 
 VII 
 
 Second Generation Marked 7 2 (49) 
 
 VII 
 
 VII VII 
 
 VII VII VII VII 
 
 Third Generation Marked 7 3 (343) 
 
 49 
 
 49 49 
 
 49 49 49 49 
 
 Fourth Generation Marked 7 4 (2401) 
 
 343 
 
 343 343 
 
 343 343 343 343 
 
 Fifth Generation Marked T (16807). 
 
 2401 
 
 2401 2401 
 
 2401 2401 2401 2401 
 
 Thus the Numbers 7, 49, 343, 2401, 16807, are 
 the First Numbers of the Family, and are technically 
 called "the 1st, 2nd, 3rd, 4th and 5th Degrees from 
 the Root VII."
 
 FAMILIES AND PEDIGREES OF NUMBERS. 115 
 
 The Family of the Number Eleven. 
 
 First Generation Marked ll 1 
 
 XI 
 
 Second Generation Marked ll 2 (121) 
 
 XI 
 
 XI XI 
 
 XI XI XI 
 
 XI XI XI XI XI 
 
 Third Generation Marked ll 3 (1331) 
 
 121 
 
 121 121 
 
 121 121 121 
 
 121 121 121 121 121 
 
 Fourth Generation Marked ll 4 (14641) 
 
 1331 
 
 1331 1331 
 
 1331 1331 1331 
 
 1331 1331 1331 1331 1331 
 
 Fifth Generation- Marked ll 5 (161051) 
 
 14641 
 
 14641 14641 
 
 14641 14641 14641 
 
 14641 14641 14641 14641 14641 
 
 i 2
 
 116 GEOMETRY IN MODERN LIFE. 
 
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 FAMILIES AND PEDIGREES OF NUMBERS. 117 
 
 When we have made ourselves familiar with 
 these Princely Families of Numbers, so as to single 
 them out readily from among crowds of common 
 numbers, we have the great advantage of knowing 
 all about them with perfect certainty of accuracy 
 and with least trouble in calculation. We can also 
 see beforehand the many consequences to which 
 combinations of them may lead. 
 
 Thus if you look at the family of II, and ask 
 " what number would be the result if I multiply 
 16 by 64 ? " the family table of II at once leads 
 you to the answer. You see there that 16 is the 
 4th Degree and that 64 is the 6th Degree the 
 4th Degree with the 6th Degree, make the 10th 
 Degree. The 10th Degree in the Table of II is 
 " 1024," and that is the answer to your question. 
 In like manner, the numbers 81 and 729, belong 
 to the family Table III. 81 is the 4th Degree, 
 729 is the 6th Degree combined they yield the 
 10th Degree whence we learn that "the product 
 of 81 by 729 yields" the number found in the 
 Table of III in the 10th Degree, we find there the 
 result already foreshown, " 59049." 
 
 But the reverse process of calculation is in 
 common Arithmetic much more tedious. If I 
 wished to know " how often I could take 625 out 
 of 15625 V I seek in the Table of Degrees, and 
 in the Family of V I find 625 in the 4th Degree
 
 118 GEOMETRY IN MODERN LIFE. 
 
 and 15265 in the 6th Degree. There are 2 Degrees 
 of difference. If I look to the 2nd Degree I find 
 25 is the 2nd Degree of the Family V. "25 
 times " is the answer to my question. 
 
 This relationship flows all through these Families 
 of Princely Numbers, so that a man who has 
 mastered them becomes nearly an intuitive omnis- 
 cient calculator. Such men spend their lives in 
 making calculations for us, and giving us the 
 note-books of their Life's Work. Napier, Babbage, 
 De Morgan, and many such men, have stored up 
 all their work in millions of figures and multitudes 
 of pages. What you have to learn at College, is 
 the skill to understand and use their work, so as 
 in after life to spare your labour by utilising 
 theirs. 
 
 To do work wisely by using only the right 
 numbers, you must know thoroughly the nature 
 of the numbers you put your trust in, for many 
 men have made the greatest blunders in Life's 
 Work by putting trust in the wrong number. I 
 have seen the Water Works of a great City ruined 
 by an Engineer who was not master of the right 
 figures taking the wronoj ones, and when the works 
 were ready (as he counted) the water came in and 
 the works broke down ! He had chosen the wrong 
 familv !
 
 FAMILIES AND PEDIGREES OE NUMBERS. 119 
 
 If you will look down the columns of the 
 Families of the Prime Numbers, you will see some 
 curious characteristics of each, which may be of 
 great use to you in life, and may often keep you 
 from being deceived. Look down the family of 
 III and ask yourself if you see any likeness, visible 
 to the eye, on the face of all the members of that 
 family ? because if you see any one number wanting 
 that mark, that number is a blunder, and is not 
 one of the family. 
 
 The mark seen on the face of the Family III is 
 this mark The figures each member of the family 
 consists of, all make "nine." The Second Degree 
 is 9 ; the Third Degree is 27 ; and these two figures 
 
 2 and 7 together make 9. The Fourth Degree is 
 81, and these two figures 8 and 1 together make 9. 
 The Fifth Degree is 243 and these three figures 2, 
 
 3 and 4, together make 9, and so on all the figures 
 in each member of this family when added together, 
 make nine, or two nines, or three nines or more nines, 
 and any others show error. 
 
 On the face of the Family V is this mark, that 
 the two last figures in each member are " two and 
 five." The Second Degree is 25 ; the Third Degree 
 is 125 ; the Fourth Degree is 625, and the Fifth 
 3125, all ending in 25. The Family VII end 
 successively in 9, 3, 1, 7, 9, 3, 1, 7, &c. &c. All 
 the Family of XI end in 1 . Thus the Third Degree
 
 120 GEOMETRY IN MODERN LIFE. 
 
 is 1331 and the Sixth Degree is 1771561, quite 
 unlike excepting this ending in 1. 
 
 Besides these characteristics of each Princely 
 Family of Numbers, there are a multitude of others 
 which grow up in our minds as we become more 
 intimate with each of them and more familiar with 
 the uses to which they may conduct us. 
 
 But the most important results of these Numbers 
 of High Pedigree flow from their powerful influence 
 over other Numbers, which mingle with the vast 
 crowd of Vulgar Numbers and as it were give 
 organisation of Rank and Degree to large classes of 
 Ordinary Numbers. 
 
 Each Family of Prime Numbers, may enter into 
 alliance with other Families of Prime Numbers, and 
 so give rise to New Families of Aristocratic Pedigree 
 but of Mixed Family. Thus the Family of (2 2 ) Two 
 in the 2nd Degree may unite with the Family of 
 (3 2 ) Three in the 2nd Degree and the product of 
 the two will be a mixed breed of Number (2 2 , 3 2 ) 
 or (2, 3) 2 and the resulting Number 36, becomes 
 a Number of great practical Value in all Matters 
 which concern equally Matters compounded of 
 Twos and Threes. We say therefore of the Number 
 3G, that it conies of the united Families Two and 
 Three in the Second Degree ; and hence we at once 
 know many of its qualities, and can predict many 
 of their consequences.
 
 FAMILIES AND PEDIGREES OF NUMBERS. 121 
 
 We may next go a step onward and in a following 
 generation of Two, we may introduce a family con- 
 nection with Five. Take the Family of Two in the 
 Third Degree (2 3 ) already connected with (3 2 ) in 
 family bond (36) and let a .new member be taken 
 from the Family Five. The product of the Family 
 2 3 and 3 2 when united with 5 1 , or the First Degree 
 of the Family Five, gives us the Number 360. 
 
 The great value of this Number has been well 
 known and much used during many centuries of 
 time in many nations. It represents nearly, but not 
 exactly, the number of days in a year. But it is 
 so useful that we all divide the Circle of the Globe, 
 and the Heavens, and our own Little Circles, into 
 360 Little Degrees, or small parts of a Circle. 
 
 Some people have tried to find a better Number 
 than 360, but they have failed. The Number 352 
 is a favourite number of mine ; and if I had lived 
 at the time wdien the Family 360 was first installed 
 into Governing Power over the territory of the 
 Circle, I might perhaps have fought a battle in 
 favour of 352 ; but it is too late ! 360 dominates 
 the Circle for all time ! 
 
 And there are good reasons in favour of 360. If 
 [ want to halve it, it yields an unbroken number, 
 180. If I want to quarter it, it gives me 90, and 
 if f want eight equal parts it gives me 45, being
 
 122 GEOMETRY IN MODERN LIFE. 
 
 three unbroken numbers. But if I asked it to go 
 further, it would break down, and a sixteenth part 
 of 360 would be the very vulgar fraction or broken 
 number 22^. 
 
 But in compensation for this defect, 360 allows 
 me to part it in three parts of 120 each, into six 
 parts of 60 each, into twelve parts of 30 each, and 
 into twenty-four parts of 15 each all free of broken 
 numbers or vulgar fractions. I can also divide a 
 circle into five parts of 72 each, or ten parts of 36 
 each, or twenty parts of 18 each, or forty parts of 9 
 each. All without breaking a number or producing 
 a vulgar fraction. 
 
 Now all this convenience, a skilled man can fore- 
 see and foretell, by mere inspection of the Pedigree 
 of the Number 360. Its Pedigree being (2 3 , 3 2 , 5 1 ) 
 he foretells that you can divide by 2 three times 
 over, by 3 twice over, by 5 once ; and by 2 with 3, 
 by 3 with 5, by 5 with 2. Thus orderly unbroken 
 groups of Division and Subdivision grow out of the 
 wise choice for Daily Universal Use, of a number 
 having a varied and mixed but Primary Pedigree 
 of the Princely Numbers, II, III, and V, in the 
 
 Degrees 
 
 [2 :! , 3 2 , 5 1 ]. 
 
 It thus becomes plain, how much knowledge grows 
 out of the Pedigree of Numbers and how much
 
 FAMILIES AND PEDIGREES OF NUMBERS. 123 
 
 use comes from the wise selection of fitting numbers 
 for each given purpose in life. It is therefore worth 
 while to study the Pedigree of all Common Numbers, 
 in order to judge which to choose and which to 
 reject. 
 
 The following 
 
 Table of Pedigrees of the First Sixty-Four 
 Ordinary Numbers 
 
 has therefore been prepared in order to assist in the 
 study of the characteristics of each and their con- 
 nections with each other. It may be worth notice 
 that the Ordinary Numbers in their usual succession 
 form the First Eow on the Left without Distinction. 
 Along the top are the Five Prime Numbers of 
 Highest Distinction, the Roots of the Largest 
 Families, and below each Head of the Family are 
 ranged the successive Members of the Family 
 marked each by his respective order or degree. 
 
 On a line with each member of a Family is the 
 title and Degree of any Second Family with which 
 he is allied and of a Third or a Fourth Family 
 with whom he is allied. 
 
 On the extreme Right of this Table will be found 
 a Row of Numbers of No Pedigree. They may 
 be called, if we like, Ancestors themselves, as they
 
 124 GEOMETRY IN MODERN LIFE. 
 
 do produce families " few and far between " : but 
 they do not come in any way from any of the 
 Princely Families, although they may form connections 
 with them farther down. We sometimes call them 
 " Prime," but not Princely Numbers. 
 
 Availing ourselves of the information in this 
 Table let us see what we can gather from it, about 
 the common Numbers with which we are familiar. 
 
 At the top of Ordinary Number stands 
 
 ? 
 
 Zero is no Number it is Nought. But it marks 
 a Place and is the Starting Point of All Number. 
 
 I? 
 
 is not a "Number" it is not "some" or "any" 
 or "many." But it is the Standard of Number 
 the Measure of Number ; it forms the Standard 
 Unit by the repetition of which Numbers are 
 made. 
 
 Two therefore is our First Number, 
 
 II. 
 One and another, of the same make Two.
 
 FAMILIES AND PEDIGREES OF NUMBERS. 125 
 
 Two therefore implies a " Pair," " Measured/' 
 " Alike." It is the first result of Measure, the 
 first step in Number it is the Ancestor of the 
 First Family of Number ; and its product is a 
 multitude of Even Numbers, into all of which the 
 Family of Two enters in a lower or higher degree. 
 
 All Numbers which have any alliance with Two, 
 will find a portion of their pedigree in the column 
 of the Table headed II. 
 
 III. 
 
 Number Three takes its place next to Two, at the 
 Head of the Family of 3 ; and all numbers allied 
 to it, find their representative in the column of the 
 Table headed III. 
 
 Four is a most important valuable Number and 
 in some ways is a perfect number but it is not the 
 Head of a Family. It is the second member of the 
 Family of Two and under the head of Two it takes 
 rank in the Second Degree as 2 2 . 
 
 V. 
 
 Five is the head of a Family next in rank after 
 Three. There are few r er members of the Family of
 
 126 GEOMETRY IN MODERN LIFE. 
 
 Five in the Table, than of Two or Three. But 
 there are many numbers of high value which have 
 their Pedigree in the Family of Five. Thus 10 is 
 formed by the union of 2 1 with 5 1 . Again, 20 or 
 a " Score " is formed by the union of 2 2 with 5 1 , and 
 100 is derived from the union of 2 2 with 5 2 . Five 
 is the Patriarch of most important Families in 
 modern calculation, although the Family of Two 
 had higher prestige in Olden Time. 
 
 <;. 
 
 Six has little distinction among numbers, except 
 as half-a-dozen. It is the simple union of Two in 
 the first degree with 3 in the same. 
 
 VIT. 
 
 Seven is a Prime Number, next in importance 
 after Five, and the head of a family of which 
 still fewer members will be found in the Table. 
 Nevertheless all its progeny have interesting and 
 valuable characteristics. 
 
 Is a distinguished and most valuable number, 
 and is the first which attains the distinction of 
 the Third Degree. It is the Third Degree of the 
 Family Two, 2''.
 
 FAMILIES AND PEDIGREES OF NUMBERS. 127 
 
 9. 
 
 Nine is a most distinguished and interesting 
 number. It has all the good qualities of the 
 parent number Three : it is 3 2 , and is sometimes 
 considered as having all the qualities of Three 
 intensified. Three threes is a sort of perfection in 
 the Second Degree. 
 
 10. 
 
 Ten is the Type or Standard or Model Number 
 of Modern Measure and Modern Calculation. The 
 whole French Nation have become slaves to its 
 rule, and we are fast following. 
 
 The temptation to give Number Ten the sole 
 absolute rule of all Measure and all Number is so 
 great, that few of modern Nations have been able 
 to resist it. Logically it seems perfect. In fact 
 it is a painful tyranny to which Englishmen have 
 a natural antipathy, and against the compulsory 
 and universal adoption of which in English Rule 
 Measure, Money, and Thought, we continue to rebel. 
 
 Perhaps the true wisdom in such cases is a 
 compromise. Let us take the Number Ten at 
 what it is good for ; but accept it thus far and no 
 farther. It is good for our intercourse with most
 
 128 GEOMETRY IN MODERN LIFE. 
 
 continental nations ; it is the basis of our own 
 common arithmetic. But our own measures of 
 Distance have Eleven as their unit ; our own 
 measures of Weight have Seven as their unit ; our 
 own measures of Value have Twelve and Twenty 
 as their units, combining the primitive numbers 
 Two, Three, and Five. Let us preserve our own 
 measures for our own ways of thought but let 
 us learn the Decimal System of Weights, Measures 
 and Money, just as we learn foreign tongues, for 
 convenience of foreign intercourse and foreign travel. 
 
 Ten therefore must hold its high place as the 
 Standard of Decimal Counting and Decimal Measure 
 although it is not a Prime Number but a combination 
 of Five with Two, or 2 1 with 5 1 . It results from 
 the union of Two First Families of Number in the 
 First Degree. 
 
 It is also useful to remember that 
 
 10 being 10 1 or Ten in the First Degree 
 
 100 is 10 2 or Ten in the Second Degree. 
 
 1000 is 10 3 or Ten in the Third Degree. 
 
 10000 is 10 4 or Ten in the Fourth Degree 
 
 1,000,000 is 10 6 or a million. 
 
 1,000,000,000 is 10 or a thousand millions. 
 
 1,000,000,000,000 10 12 or a million millions.
 
 FAMILIES AND PEDIGREES OF NUMBERS. 129 
 
 But it may also be useful to remember that 
 
 2 10 exceeds 1,000. 
 
 2 20 exceeds 1,000,000. 
 
 2 30 exceeds 1,000,000,000. 
 
 2 40 exceeds 1,000,000,000,000. 
 
 XL 
 
 Eleven, which follows Ten in our ordinary 
 numbers, is the parent of a large family of 
 interesting Numbers, by union with other Families. 
 It is marked by the repetition of the other numbers 
 on the face of the product. Thus 22, 33, 44, 55, 
 66, 77, 88, and 99, are merely the product of 
 Eleven united with 2, 3, 4, 5, &c. Eleven thus 
 leaves its distinguishing mark on all the early 
 members of its family connections. 
 
 12. 
 
 Twelve, though not a primitive number, is yet 
 one of the most useful and convenient of early 
 Numbers. It is the union in One Number, of the 
 early numbers 2, 3, 4, and 6. Twelve consists of- 
 a Pair of Sixes, of Three Fours, of Four Threes, 
 and of Six Couples. This makes it the most con- 
 venient in the handling of Common Numbers in 
 daily Merchandize. 
 
 K
 
 130 GEOMETRY IN MODERN LIFE. 
 
 XIII or 13. 
 
 Is a Transition Number, from the Highest Pedigree 
 to Numbers without Pedigree. The fault of Thirteen 
 is, that its family are few, and their relations to 
 other Numbers very rare. Compared with the 
 family connections of its next neighbour Twelve, 
 it really has little or no connection, and is little 
 and rarely seen in union with other families. It 
 may be seen two or three times in our Table in 
 connection with 2 and 3. 
 
 14. 
 
 Fourteen has no particular quality except as the 
 double of Seven or 2 with 7. 
 
 15. 
 
 Fifteen is the first result of the union of Five 
 with Three, or 3 1 with 5 1 . 
 
 16. 
 
 Sixteen is a distinguished, important and useful 
 number of High Pedigree. It is Two raised to the 
 Fourth Degree. It comprehends in its pedigree the 
 lower numbers of Eight, Four and Two or 2 1 , 2 2 , 2 3 . 
 Its practical convenience is that you can halve it,
 
 FAMILIES AND PEDIGREES OF NUMBERS. 131 
 
 then quarter it, then split each quarter in two, and 
 that again in two ; and do this without breaking 
 the number or splitting it up into fragments or 
 making vulgar fractions of it, which is so often 
 disagreeable. Sixteen is a most convenient number 
 and is found in the Family Table II and is 
 marked 2 4 . 
 
 17. 
 
 Is a Number of no Pedigree and very little use. 
 Only a few scattered numbers come of it. Still it 
 is a Prime Number. 
 
 Eighteen is a convenient and useful number. It 
 is half-way between 12 and 24, or between One 
 Dozen and Two Dozen, and it is the half of Three 
 Dozen. It is the outcome of Two with Three, or 
 2 1 with 3 2 and its family is intimately allied with 
 all theirs. 
 
 19. 
 
 Nineteen has no Pedigree, little connection, very 
 rare family and very little use, excepting to fill up 
 the void between the very useful numbers Eighteen 
 and Twenty. It is a Prime Number. 
 
 k 2
 
 132 
 
 GEOMETRY IN MODERN LIFE. 
 
 FAMILY PEDIGREE OF 
 THE FIRST SIXTY-FOUR ORDINARY NUMBERS. 
 
 O 
 
 H 
 
 H 
 
 U 
 o 
 
 W 
 
 H 
 i i 
 
 w 
 m 
 
 i i 
 P 
 
 O 
 
 W 
 
 H 
 
 THE FIVE FIRST PRINCELY NUMBERS OR PRIMES. 
 
 II III V VII XI 
 
 
 
 
 2 ... 2 1 
 
 3 
 
 4 ... 2 2 
 
 5 
 
 6 ... 2 1 ... 
 
 7 
 
 8 ... 2 3 
 
 9 3* 
 
 10 2i ... 
 
 11 II 1 
 
 12 ... 22 ... 
 
 13 13 
 
 14 ... 2^ ... 
 
 15 3 1 ... 5 1 
 
 16 2* 
 
 17 17 
 
 18 2i ... 
 
 19 19 
 
 20 2 2 ... 
 21 
 
 22 2 1 IP 
 
 23 23 
 
 24 ... 2 3 ... 3i 
 
 25 5' 
 
 26 ... 21 13 
 
 27 3' 
 
 28 ... 2 2 V 
 
 29 29 
 
 30 ... 21 ... 3 1 ... 51 
 
 31 31 
 
 32 V 
 
 a 
 5 
 
 GC 
 
 Q 
 O 
 
 o 
 
 Q 
 fcg 
 
 Q 
 
 f 
 
 o 
 w 
 
 7 T A<? Degrees of Descent from each.
 
 THE FIVE^FIRST PRINCELY NUMBERS. 
 
 133 
 
 CONTINUATION. 
 THE] FIVE FIRST PRINCELY NUMBERS OR PRIMES. 
 
 II III V VII XI 
 
 2 5 
 2 
 
 2 2 
 2 1 
 2 3 
 
 2 1 
 2 2 
 2 1 
 2 4 
 2 1 
 2 2 
 2 1 
 2 s 
 2 1 
 2 2 
 2 1 
 2 fi 
 
 3 1 
 
 3 2 
 
 3 1 
 
 3 1 
 
 3 2 
 
 3 1 
 
 . 3 1 
 
 3 s 
 
 3 1 
 
 3 1 ... 5 1 
 
 3 2 
 
 ll 1 
 
 7 1 
 
 7 1 
 
 ll 1 
 
 7 2 
 
 ll 1 
 
 V 
 
 7 1 
 
 <-17 
 
 37 
 <-19 
 <-13 
 
 41 
 
 43 
 
 <-23 
 47 
 
 <-17 
 
 <-13 
 
 53 
 
 <-19 
 
 <-29 
 
 59 
 
 61 
 <-31 
 
 3 
 w 
 
 
 Q 
 O 
 
 o 
 
 I I 
 Q 
 W 
 H 
 
 O 
 
 77te Degrees of Descent from each.
 
 134 GEOMETRY IN MODERN LIFE. 
 
 20. 
 
 Twenty is one of our most important English 
 Numbers. It is called in pure English " A Score." 
 It is (or was) a Standard Number in Reckoning 
 Flocks, and Keeping Tale in our Pastoral Counties. 
 Threescore and Ten, Fourscore and Fivescore, are 
 well-known numbers of record and keeping of 
 count. Our Pound of 20 Shillings and our Ton 
 of 20 Hundredweight are in accord with Old 
 English Custom of Counting by Scores. 
 
 For other Numbers beyond Twenty and up to 
 Sixty-four the preceding Table tells its own tale. 
 But Students of Number will do well, after master- 
 ing that, to make for themselves a much longer 
 Table of Pedigrees and to select from it their own 
 Pet Numbers, to aid them in their own Familiar 
 Calculations. 
 
 Note. 
 
 Careful study of the preceding Table is useful to 
 the reader of the following Section.
 
 SYMMETRY, HARMONY, AND MELODY. 135 
 
 XII. 
 
 SYMMETRY, HARMONY, AND MELODY. 
 
 Symmetric form is to the eye, what symmetric 
 sound is to the ear. Both contain thought, and 
 carry meaning, and yield pleasure. There is a like 
 difference between music and noise as between 
 creation and chaos. The creations of Handel and 
 the creations of Phidias convey the same divine 
 thoughts to our minds by like means, though by 
 very unlike ways. The charm equally of architecture 
 and of symphony come from the Symmetric Agree- 
 ment of the Elements, out of which both are com- 
 posed. The placing of parts in order to the eye, 
 is like the sending of sounds in time to the ear ; both 
 are pleasant or disagreeable according as they give 
 true measure or false measure, whole measure or 
 broken measure, agreeing measure or disagreeable, 
 concords or discords. 
 
 " Noise " means orderless or disorderly sound. 
 " Music " means measured or " tempered " sound.
 
 136 GEOMETRY IN MODERN LIFE. 
 
 "Tune" means sound to an agreed standard. That 
 standard we call the Fundamental Sound or key 
 note. " Harmony " means sounds agreeing with that 
 sound when given together. " Melody " means sounds 
 agreeing when given out each in measured succession 
 after the other. 
 
 Now the same numbers rule Symmetric Shape and 
 Symmetric Sound. The proportions that are agreeable 
 to the eye in columns and temples and statues, are 
 equally agreeable to the ear in songs and in anthems, 
 and in symphonies ; they express like thoughts in the 
 mind of the composer and give rise to like feelings 
 in the mind of seer and listener. Whoever therefore 
 would become a great architect, artist or creator of 
 human work, should first study how the harmonies 
 of number have been given to God's creations in 
 matter, and these he will find developed in the laws 
 of sound and the phenomena of visible beauty. 
 
 The laws of sound and the principles of symmetric 
 composition are exquisitely simple, to those who have 
 made a study of Symmetric number and Geometric 
 order. The first key to this knowledge is to be 
 found in the study of the Families of Numbers, and 
 in the knowledge of the Pedigree of each ordinary 
 number. These were probably among the mysteries 
 well known to the Greeks, and aided their genius 
 in their marvellous creations.
 
 SYMMETRY, HARMONY, AND MELODY. 137 
 
 I will now try to lead you through some of the 
 first steps towards a comprehension of those great 
 principles, of rhythmic sound and symmetric form. 
 
 Noise is caused by a shock or stroke or beat given 
 to some mass of matter. That shock tells on the 
 mass of air near the struck body. It sends this mass 
 of air into a heap near the struck body. This heap 
 forms an air wave. This air wave flies away from 
 the struck body carrying with it some of the force 
 given out in the shock. This wave is the carrier of 
 the shock through great distances in the air, it travels 
 with great speed. This "carrier wave" takes the 
 shock of the stroke from afar, to the ear of the 
 listener. It delivers it, as it got it, into the ear of 
 the listener, where there is a musical instrument 
 prepared to receive it. This mechanism in the ear 
 enables the listener to notice the sound, to note its 
 nature, to remember it, to measure it by other knowm 
 sounds, and in an educated ear, to tell exactly its 
 measure, its number, its agreement or discord and to 
 know its numerical relations of family. 
 
 The exact nature of this sound- carrying w T ave is 
 knowm. It goes straight forward, like a beam of 
 light. It is turned aside by an obstacle or thrown 
 backwards by an immovable obstacle, and it carries 
 its message to the ear of the listener at a pace of 
 near 12 miles in one minute (or 700 miles an 
 hour).
 
 * 
 
 138 GEOMETRY IN MODERN LIFE. 
 
 This speed of the sound-carrier wave, is note- 
 worthy. Because much of the nature of different 
 sounds is explained by its measures. 
 
 The standard speed of the carrier wave of 
 sound is 
 
 1024 feet per second. 
 
 This number 1024 is the Tenth Degree of the 
 
 Family of Two. This gives it high value in the 
 
 Theory of Sound. By looking into the table of the 
 
 Family II. you may see the many interesting 
 
 numbers to which it is related. The most interesting 
 
 (to us) of these ten relatives is, the Fifth Degree 
 
 of Two, or 
 
 2 5 or 32 
 
 and 32 times 32 make 1024. 
 
 Now as we have seen that 1024 is the standard 
 speed of the carrier wave, so it is most interesting to 
 know that its near relative 32 is the standard of 
 musical tone, or 32 feet long is the standard of length 
 of pipes in the organ, and the lengths of all the 
 others arc fixed in precise numerical agreement with 
 this one, otherwise the organ would be " Out of 
 Tune." 
 
 All sounds are carried to our ears by the carrier 
 wave, ;it like speed, discords and chords, screams and
 
 SYMMETRY, HARMONY, AND MELODY. 139 
 
 shock, and songs, all travel alike fast ; fortunately 
 that is so, otherwise, if some sounds travelled fast 
 and others travelled slow, say high notes faster and 
 low notes slower, then sounds would reach us in quite 
 different order from that in which they were de- 
 livered. All the high sounds of a tune. might reach 
 us first, and all the low sounds might follow later, 
 and so the whole structure of the song would be 
 shattered and distorted. Or in listening to the far off 
 speech of an orator, all the consonants might come to 
 us first, and the vowels follow later, which would be 
 inconvenient. Happily 1024 is the speed of all sounds, 
 high and low, hard and soft. 
 
 An organ pipe 32 feet long when its breath rushes 
 in at one end, gives a shock to the air in the whole 
 pipe, which reaches the mouth of the pipe one 32nd 
 part of a second of time after it enters. Thus 32 
 such shocks can pass along this 32 feet pipe in one 
 second of time. Thus this standard 32 feet organ 
 pipe sends out 32 shocks, by 32 carrier waves in each 
 second, and making together 1024 feet travel. 
 
 When we take a like organ pipe, only .16 feet 
 long, each gush of entering air gets out of the pipe in 
 Jt part of a second. Therefore 64 beats can get 
 along in each second of time. Hence 64 shocks are 
 brought to our ear in each second of time out of the 
 1 6 feet pipe, while only 32 reached us from the 32 
 feet pipe.
 
 140 GEOMETRY IN MODERN LIFE. 
 
 Now the sounds carried to us from the 32 feet pipe 
 and the sounds from the 16 feet pipe, have a close 
 agreement. They are quite different, one being high 
 (64) and the other low (32) ; nevertheless, they are so 
 like in sound that we feel them to be in perfect 
 agreement. When sounded together, they help each 
 other : when sounded after each other, they have 
 an agreeable diversity. 
 
 We therefore in music consider them as of the 
 same family. We also give them the same name. If 
 we call the one by the name C, we call the other by 
 the name CC, or we call the one C and the other 
 c. Now the one number 32 is related to the other 
 number 64 ; just as one sound is to the other or as 
 2 5 is to 2 6 . In technic terms, they are Octaves 
 C 1 and C 2 . 
 
 The 32 feet pipe and the 1 6 feet pipe sending out 
 32 sound waves and 64 sound waves, send to our ears 
 two tones, which taken together seem " one with 
 a difference " in " perfect harmony," and taken before 
 and after each other, we feel them to give an agree- 
 able variety. 
 
 It is convenient to call this standard sound and all 
 other sounds belonging to the Family of Two, by the 
 name C, and by writing it thus 
 
 a 7 10 
 
 C 2 or C 2 or C 2 or C 2
 
 SYMMETRY, HARMONY, AND MELODY. 141 
 
 we should always know how many beats per second 
 are meant. 
 
 The following is a list of the lengths of channel 
 traversed by the sound wave, the number of waves 
 sent out of each organ pipe, the musical name of 
 each sound given into the ear, the degree in the 
 family of II to which each number of waves or its 
 tone of music belongs. 
 
 Note. 
 
 It may be convenient here to note, that the 
 Relations of the Numbers quoted in this Section 
 are perfect and exact, but the measures or standard 
 units of actual phenomena are variable. A foot is a 
 different unit of length in England, France, Switzer- 
 land, and Germany. The speed of a sound wave is 
 greater when the barometer is high, and less when it 
 is low. The tone of an organ pipe is higher when 
 the land on which it stands is lower down near the 
 sea level, and it sends out lower sounds when placed 
 on a hill. Also a very violent sound like a cannon- 
 shot goes slightly faster than a gentle sound of the 
 human voice. Nations also differ in their choice of 
 the pitch of tone they call C. Thus the chosen 
 measure, standard, or unit, varies, but the numerical 
 relations are unchanged.
 
 142 GEOMETRY IN MODERN LIFE. 
 
 H3 
 
 O Q O 
 
 O 
 
 
 II 
 
 o 
 
 ^ 
 
 ft 
 
 CI CI <M <M CM CI 
 
 <M 
 
 -* 
 
 CO 
 
 CO 
 
 cq 
 
 -tf 
 
 CO 
 
 to 
 
 cq 
 
 >o 
 
 i 1 
 
 d 
 
 
 
 r i 
 
 <N 
 
 lO 
 
 o 
 
 
 4-J 
 
 -t^> 
 
 +-> 
 
 -M 
 
 +J 
 
 +- 
 
 fl wj 
 
 o 
 
 <D 
 
 a> 
 
 O 
 
 0> 
 
 o 
 
 Ci 
 
 O) 
 
 .&> 
 
 ,v 
 
 <D 
 
 t$> 
 
 o 
 
 5yD ^ 
 
 ^-1 
 
 <4-H 
 
 -* I 
 
 t+_ 
 
 M-l 
 
 t+=l 
 
 J^ V 
 
 
 
 
 
 
 
 DPh 
 
 CJ 
 CO 
 
 r 1 
 
 00 
 
 -* 
 
 <M 
 
 1 1 
 
 C<l <M 6q CI CI cq 
 
 O O O o 
 
 T3 
 
 1 1 O
 
 SYMMETRY, HARMONY, AND MELODY. 143 
 
 If we used the degrees of Two to express the 
 degrees of sound then 
 
 5 6 7 8 9 10 
 
 C 2 C 2 C 2 C 2 C 2 C 2 
 would express the exact nature of each sound. 
 
 "When we have studied the nature of the foundation 
 on which the structure of music rests, and understand 
 the basis of tone, or Bass, from which melody grows 
 upwards, we may next proceed to consider the suc- 
 cessive steps, of Measured Melody, by which the 
 endless variety of musical structure is organized and 
 built up. 
 
 5 
 
 C 2 being taken as our standard basis, and 
 
 6 
 
 C 2 being its nearest relative or higher octave 
 
 let us now seek out in the families of Number, 
 other harmonies of number, which may lead us to 
 harmonic melody. 
 
 5 
 
 C 2 gives 32 sound waves in a second out of a 32 
 feet organ pipe. 
 
 The family of III if united with the family of II 
 would by substituting a 3 for a 2, modify that family. 
 Let us see how 
 
 Instead of 2 5 which gives us 32 waves, let us 
 take 2 4 , 3 1 .
 
 144 GEOMETRY IN MODERN LIFE. 
 
 Now 2 4 is 1 G ) ,.-.,., 
 
 t n , n M6 combined with 3 makes 48. 
 and 3 1 is 3 ) 
 
 Forty Eight is therefore a number which is the 
 product of the union of III in the first degree, with 
 II in the fourth degree. If therefore the harmony of 
 family number lead to harmony of waves and sounds, 
 it follows that 
 
 If an organ pipe be made to send out 48 sound 
 waves in the same time as the other organ pipe sends 
 out 32 sound waves, the two sounds might be in 
 perfect harmony. 
 
 In fact it is so ; and 
 
 32 waves of sound, along with 48 waves of sound 
 
 in the same time, are in perfect accord, and make 
 a perfect harmony. 
 
 This harmony is called in music the " Perfect 
 Fifth." 
 
 In number it is called the harmony of 
 (2 4 with 3 1 ). 
 
 The manner in which this harmony grows out of 
 this union, is quite conceivable. Such waves keep 
 time with each other
 
 SYMMETRY, HARMONY, AND MELODY. 145 
 3 groups of 16 make 48 
 6 groups of 8 make 48 
 12 groups of 4 make 48 
 
 Thus we see that in every second of time groups 
 of 3 waves, 6 waves, 12 waves, will begin and end 
 at the same instant of time, and will so measure the 
 same intervals, and so coincide, agree and harmonise 
 in space, in time and in number. Happily our own 
 nature is in agreement with this wave nature, and 
 this number nature, and this agreement of sound is to 
 us " sweet." 
 
 The Perfect Chord of the "Fifth" in music, is 
 the Perfect Harmony of 2 4 and 3 1 in waves. The 
 Symmetric Union of Families II and III. 
 
 If we seek a new harmony and a new melodic 
 sound, we have only to go into the Family of V to 
 find it. 
 
 Into the Family of II let us introduce the First 
 Member of the Family of V, thus 
 
 2 3 is 8 -) 
 
 cl . e f- 8 combined with 5 yields 40. 
 5 is d j J 
 
 Forty is therefore a number growing from the Union 
 of V in the first degree with II in the second degree.
 
 146 GEOMETRY IN MODERN LIFE. 
 
 If therefore an organ pipe be made of the right 
 length to send out 40 sound waves in the same time 
 as the other sends out 32 the two sounds might 
 agree. 
 
 They do in fact agree ! 
 
 In fact 
 
 32 waves of sound, along with 40 waves of sound 
 are in harmony. 
 
 This Harmony is called in Music the " Perfect 
 Third." 
 
 In Number it is called the Harmony of 
 (2 3 with 5 1 ). 
 
 The growth of this harmony is quite conceivable 
 
 5 groups of 8 make 40 
 10 groups of 4 make 40 
 20 groups of 2 make 40 
 or, 
 
 8 groups of 5 make 40 
 
 4 groups of 10 make 40 
 
 2 groups of 20 make 40.
 
 SYMMETRY, HARMONY, AND MELODY. 147 
 
 These groups of 5 waves, 10 waves, and 20 waves, 
 making up exact measures of time and space and 
 sending out coinciding waves of sound, do in fact 
 give into our ear an agreeable tone which we 
 call 
 
 The Perfect "Third" in Music. 
 
 The Harmony of 2 3 and 5 1 in Waves. 
 
 The Symmetric Union of II and V in Family 
 Pedigree of Number. 
 
 We have now seen that the Number II in different 
 degrees produces Harmonic Tones, in the relation 
 called " Octaves." 
 
 That the Number III produces Harmonic Tones in 
 the relation of " Perfect Fifths." 
 
 That the Number V produces Harmonic Tones in 
 the relation of " Perfect Thirds." 
 
 But we have produced a still higher result. All 
 these taken together make up a complete conchord 
 with each other, called a " Common Chord." 
 
 A solid foundation is thus laid on which to build 
 up harmonic structures of noble and grand volume 
 of sound, which at once soothe the feelings and 
 elevate the mind of the listener ; and we have next 
 to see how this monotonous harmony can be varied, 
 relieved and made playful and interesting as well as 
 soothing and grand. This we must try to do by 
 
 l 2
 
 148 GEOMETRY IN MODERN LIFE. 
 
 contriving new combinations or new tones, which 
 shall be quite different from those already agreed on, 
 and yet in close agreement with them so as to give 
 variety without discord. It is by such intermediate 
 members of our musical family numbers, that we 
 shall lay the " stepping stones to melody." 
 
 We have already gotten four foundation tones, 
 wide asunder in number and nature but in perfect 
 harmony. These are 
 
 5 3 1 4 1 6 
 
 C 2 E 25 G 23 C 2 
 
 Three wide blanks remain between these settled 
 tones, and the melody of music requires that they be 
 filled with Stepping Stones, that may lead us from 
 each to each by gentler strides and easier steps. Let 
 us now look in our Table of Family Pedigree for 
 numbers of allied families to those already found. 
 
 Let us seek a tone to divide the interval between 
 C and E. Let us try it after the same fashion in 
 which we originally found the tone G to fill the 
 
 "5 6 
 
 interval between C 2 and C 2 . We then introduced 
 the family III into the family II ; let us now intro- 
 duce a second member of the family. Take 3 2 and 
 unite him with 2 2 . Their joint product will be 
 3 2 with 2 2 or 9 with 4, and 4 by 9 is 3G. 
 
 If now we call this new tone of 36 waves per 
 second D and place it mid way betwixt C and E, we 
 shall have the following relations:-
 
 SYMMETRY, HARMONY, AND MELODY. 149 
 
 c 
 
 D 
 
 E 
 
 IP 
 
 (IP IIP) 
 
 (IP V 1 ) 
 
 2 5 
 
 2 2 3 2 
 
 2 3 5 x 
 
 32 
 
 4 by 9 
 
 8 by 5 1 
 
 32 
 
 36 
 
 40 
 
 Thus we have got a tone called D betwixt C and 
 E which is a member of the Family of Two in a lower 
 degree, and of Three in a higher degree, and thus we 
 have the three tones following symmetrically as the 
 successive numbers, 8, 9, 10. Thus 
 
 Our Triplet now 
 
 consists of 
 
 
 C 
 
 D 
 
 E 
 
 32 
 
 36 
 
 40 
 
 waves 
 
 waves 
 
 waves 
 
 4x8 
 
 4x9 
 
 4 x 10 
 
 or the melodic succession of 
 
 
 8 
 
 9 
 
 10 
 
 It is very interesting to know that our ear and 
 our feelings and our minds accept these three groups 
 of 8, 9, and 10 waves in each second of time as 
 elements of 
 
 Perfect Melody. 
 
 5 
 
 Having filled up this void next C 2 , let us next 
 see if we can likewise fill up the greater void 
 
 next to C 2 .
 
 150 
 
 GEOMETRY IN MODERN LIFE. 
 
 4 1 6 
 
 Between G 2 3 and C 2 lies the difference of number 
 
 48 and 64. 
 
 G grows out of 2 4 3 1 or 16 x 3 
 
 48 
 
 Cis2 6 
 
 or 8 x 8 
 
 64 
 
 Now as we have already seen that melodic suc- 
 cession was formed above C 5 by a succession of waves 
 in groups of 8, 9, 10 ; we may try if we can inter- 
 pose a like succession between G and C 6 . 
 
 G is 8 groups of 6 .... 48 waves. 
 
 Let us try a sound and let us call it 
 
 A. formed of 9 groups of 6 waves each 
 
 56 waves. 
 
 B. formed of 10 groups of 6 waves each 
 
 60 waves. 
 
 Then we shall have the following triplet 
 
 G 
 
 A 
 
 B 
 
 48 
 
 54 
 
 60 
 
 waves 
 
 waves 
 
 waves 
 
 6x8 
 
 6x9 
 
 6 x 10 
 
 Or a second melodic 
 
 8 
 
 succession 
 9 
 
 of 
 
 10. 
 
 Fortunately for us Nature is in harmony with 
 these numbers. They also are melodic successions.
 
 SYMMETRY, HARMONY, AND MELODY. 151 
 
 Matter Nature and Human Nature are agreed in 
 accepting these numbers as melodious groups of 
 sound waves. 
 
 These Two Triads have now built up our elements 
 of Melody into the following form and proportion 
 
 II 
 
 c 
 
 D 
 
 E 
 
 32 
 
 36 
 
 40 
 
 waves 
 
 waves 
 
 waves 
 
 4x8 
 
 4x9 
 
 4 x 10 
 
 8 
 
 9 
 
 10 
 
 III 
 
 , 
 
 
 
 
 G 
 
 A 
 
 B 
 
 C 
 
 48 
 
 54 
 
 60 
 
 64 
 
 waves 
 
 waves 
 
 waves 
 
 waves 
 
 6x8 
 
 6x9 
 
 6 x 10 
 
 8x8 
 
 8 
 
 9 
 
 10 
 
 
 Between our first melodic triad and our second 
 melodic triad, there remains an awkward interval. It
 
 152 GEOMETRY IN MODERN LIFE. 
 
 is awkward, because it is too narrow to be spanned in 
 two steps as the others were, and it is too wide to be 
 left void. 
 
 There are two candidates for this opening a 
 member of the family VII and a member of the 
 family XL Now neither of these families is in intimate 
 relation with the others. If we take 7 2 it is 49 : now 
 48 is already our chosen number as G, and 49 would 
 merely be a new competitor for the place of G and 
 therefore would be in antagonism and would form a 
 screeching discord. 
 
 The other candidate for the place is one of the 
 family XI and the number ll 1 with 2 2 yields 44 
 which to the eye and the mind seems likely for the 
 purpose. 
 
 But no ! The ear rejects it. The ear prefers 
 7 to 11 ! Why, it is difficult to say. There are 
 circumstances in which the ear is pleased to accept 11 
 as a favoured candidate for place in melodious 
 succession, but for this place it is rejected. 
 
 The candidate accepted by the ear is in this case 
 the family VII. Six sevens make 42, and when 
 42 is placed before 48, it gives rise to an equal 
 succession among the following groups of Six. This 
 agreement seems to be that which has given the 
 preference to VII over XI.
 
 SYMMETRY, HARMONY, AND MELODY. 153 
 
 Calling this new step in melody 
 
 2 1 3 1 7 1 F 
 
 we get the following series of melodic number. 
 
 Ill 
 
 F G A B 
 
 42 48 54 60 
 
 waves waves waves waves 
 
 6 X 7 6x8 6x9 6x10 
 
 7 8 9 10 
 
 We can now place in order 
 
 The complete series of melodic steps.
 
 154 GEOMETRY IN MODERN LIFE. 
 
 Ph 
 
 El 
 
 i> 
 
 oo 
 
 00 CO 
 
 p <v 
 
 > > 
 
 
 GO 
 
 00 <4_, O 
 
 O T I 
 
 co 
 
 CO 
 
 o 
 
 X 
 
 cq 
 
 
 05 
 
 CO 
 
 
 X 
 
 & 
 
 -d 
 
 03 
 
 
 CO 
 
 
 go 
 
 X 
 
 CO 
 
 X 
 
 
 ** 
 
 X 
 
 CO 
 
 CD 
 CO 
 
 as 
 X 
 
 O oq 3 x 
 
 CO 
 
 CO 
 
 a. 
 
 a> 
 
 .a * 
 
 CC of 
 
 SP d > 
 
 o 
 
 rt 00 
 GO 
 
 ce 
 
 oo 
 
 cq _, d ,-h 
 
 o 
 
 Oco 
 
 C7i [V, 
 
 
 
 -*
 
 SYMMETRY, HARMONY, AND MELODY. 155 
 
 Now each of these groups is melodious with the 
 others, because each group of 8, 9, and 10 is in near 
 alliance with the next, and because 7, 8, 9, and 10 is 
 a like series of steps in Dumber and sound. And 
 these same notes have harmonic relations to each 
 other, by the union in the same measure of time, of 
 like numbers of groups ; and by the fact that when 
 sounded together the tones are harmonious in the 
 exact degree in which numbers belonging to one 
 family bring about agreement. 
 
 Thus we see by the eye that 
 
 E must harmonise with B 
 because their family numbers are 
 
 2 3 5 1 and 2 2 3 1 5 1 
 
 Groups of 5 waves being given out in one case in 
 pairs, and in the other in triples, per second, or in 
 perfect harmony. 
 
 In like manner we may foresee that 
 
 D must harmonise with A, 
 because their family numbers are 
 
 2 2 3 2 and 2 1 3 s
 
 156 
 
 GEOMETRY IN MODERN LIFE. 
 
 Groups of nine waves being given out in both cases, 
 in the one 4 groups fitting in with 6 groups in the 
 other, in each second, a perfect harmony of 3 and 2. 
 
 All these tonic relations of sound, are therefore, 
 the embodiment in matter, and the development into 
 symmetric wave motion, of the harmonic numbers 
 given in our table on page 133, from which the fol- 
 lowing: has been taken. 
 
 
 II 
 
 II 
 
 32 
 
 . 2 5 
 
 
 36 
 
 2 2 
 
 . 3 2 
 
 40 
 
 2 3 
 
 
 42 
 
 2 
 
 . 3 
 
 48 
 
 2 4 
 
 3 
 
 54 
 
 2 
 
 . 3 3 
 
 60 
 
 . 2 2 . 
 
 . 3 
 
 64 
 
 . 2 6 
 
 
 VII 
 
 XI 
 
 c 
 
 D 
 E 
 
 7 
 
 
 F 
 G 
 A 
 B 
 
 C 
 
 Thus I have tried to show you that numbers of 
 high family pedigree, when they go out into the 
 world of material phenomena, rule the ways in 
 which matter works, that they express and explain 
 the phenomena of agreement and disagreement in the 
 air waves that carry sound into our ears, that they 
 are felt by our nature and accepted by our minds, as 
 agreeable and disagreeable even when we are not 
 familiar with their nature ; and that education in the 
 family nature and individual pedigree of number 
 enables us to give a rational explanation of the
 
 SYMMETRY, HARMONY, AND MELODY. 157 
 
 phenomena of concord and discord, and may 
 materially aid us in gaining mastery over the 
 mysteries of melodious succession and harmonious 
 symphony. 
 
 But I should fail in rewarding you for the pains 
 of this study, if I did not point out to you the many 
 gains it gives to wider knowledge of modern science, 
 and higher skill in Art and Life. Harmonic and 
 melodious number rule many of the marvellous 
 w T orks of God, as well as the occupations of man. 
 God has told the atmosphere in which we live, to 
 carry far and wide, sweet sounds, but to refuse to 
 carry discordant noise ; or rather to favour the transit 
 of sweet sounds and to hinder that of harsh ones. 
 
 When you are far away from sounding instruments 
 and from the voice of song, you will notice that its 
 effect on your ear and mind is much sweeter, than 
 when you were near. Why is this ? 
 
 It is because it is a law of creation, that Sounds in 
 Harmony shall be carried by waves that assist each 
 other, while Discords are dropped out and left 
 behind. 
 
 It might be difficult to explain this discriminative 
 faculty of the carrier wave of sound, but it is so. 
 
 Not merely the carrier wave has this natural bias 
 against transporting disagreeable sound ; but Tlw
 
 158 GEOMETRY IN MODERN LIFE. 
 
 Matters which cause sounds, and send forth the sound 
 waves to us, have an innate antipathy to discords. 
 The Eolian Harp is a simple set of stretched strings, 
 left to itself in an air-current. The breath of the 
 wind blowing on the strings produces in the ear of 
 the listener wild melody and exquisite harmony : why 
 not discord ? We only know that each string, struck 
 by the wind, produces only the tones which belong to 
 the harmonic series, which we have selected from our 
 table of family pedigree. It refuses to sound any 
 other. All the numbers in our last column are left 
 out. Hence inanimate nature loves Order, Measure, 
 and Number as we do. Perhaps it is the mind of 
 God which moves both it and us. 
 
 If I were to settle the right proportions for an 
 ordinary drawing room, in which the music of song 
 was to be well heard and which should at the same 
 time be seemly to the eye, I should give to it the 
 proportions of harmonic sound. 
 
 Say 30 feet for the length. 
 Take 12 feet for the height. 
 And 18 feet for the width. 
 
 Thus we have, 
 
 Length 5 times 6. 
 Breadth 3 times 6. 
 Reiefht '1 times (>.
 
 SYMMETRY, HARMONY, AND MELODY. 159 
 
 Therefore the families 2, 3, and 5, the prime 
 numbers of perfect harmony, are all represented here, 
 and are united by the common bond of 6, which is 
 itself the outcome of 2 with 3. 
 
 The physical consequence of this will be that all 
 three dimensions of the room will agree in favouring 
 these harmonic sounds and in suppressing discords. 
 The sound waves will also travel three ways ; along 
 the room, across the room, and up and down. The 
 waves of sound in these three sets, will form groups 
 of six, of which groups, every 2, 3, 5, will exactly 
 time each with the other, and so each will help all, 
 and there will be no hindrance to continuance and 
 perfection of harmony. 
 
 A room of 29 feet, by 17 feet by 13 feet, would 
 be nearly the same size, and to the eye nearly the 
 same proportion, but to the air and the sound wave 
 and the voice of the singer and the ear of the listener, 
 would be impracticable and intolerable. Every room 
 has a key tone and a harmony or discord of its own, 
 and a speaker or singer is heard ill or well and gives 
 voice with ease or pain, according as his voice and 
 the room's dimensions agree or disagree. 
 
 o 
 
 In a beautiful work of the sculptor, like proportions 
 rule the human form. Harmonic Form has like 
 number with Harmonic Sound. " The works of God 
 are made in Order, Measure, and Number."
 
 160 
 
 GEOMETRY IN MODERN LIFE. 
 
 Take a well-proportioned man, say of six feet. 
 The following are his measures and numbers 
 
 Inches. 
 
 Height 72 
 
 Outstretched arms 72 
 
 Arms' length from centre of body . . 36 
 
 Length of hand 8 
 
 Length of face 8 
 
 Length of head and neck from collar 
 
 bone to crown 16 
 
 Breadth from shoulder joint to shoulder 
 
 joint 16 
 
 Length from elbow to finger point. . 18 
 Length from shoulder joint to hip joint 
 Length from hip joint to knee joint . 
 Length from knee joint to ankle joint. 
 
 Length of foot 10 
 
 Length from shoulder joint to elbow 
 
 joint 10 
 
 Height of ankle joint above heel . . 2 
 
 Yards- 
 2 
 
 2 
 1 
 
 Now all these numbers are our harmonic numbers 
 2, 3, and 5 and none other. 
 
 2 3 with 3 2 make 8x9 
 
 2' J with 3' make 4x0 
 
 2' with 3 2 make 2 x 9 
 
 2 1 2- x 2-
 
 SYMMETRY, HARMONY, AND MELODY. 161 
 
 2 1 with 5 1 makes 2x5 10 inches 
 
 2 3 makes 8 
 
 2 1 is 2 
 
 Hence all grow out of the Prime Families of 
 Number and Harmony 
 
 2, 3, 5. 
 
 The total height is 
 
 Head and neck 16 
 
 Body 18 
 
 Upper leg 18 
 
 Lower leg 18 
 
 Heel 2 
 
 72 
 
 In like manner, when we go into the vegetable 
 kingdom, we see everywhere developments of form 
 and number, symmetric and harmonious. 
 
 The following are two examples of symmetric form 
 growing from the Koots II and V in conformity with 
 the Family Degrees of these two prime numbers> 
 already given in pages 162 and 163. 
 
 M
 
 162 
 
 GEOMETRY IN MODERN LIFE. 
 
 Example of Symmetric Form growing from the 
 Number II. (See page 111.)
 
 SYMMETRY, HARMONY, AND MELODY. 163 
 
 ^* 
 
 Example of Symmetric Form growing from the 
 Number V. (See page 113.) 
 
 m 2
 
 164 GEOMETRY IN MODERN LIFE. 
 
 XIII. 
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 
 
 Shapes of things as they are in themselves, shapes 
 of things as they seem to my eye, shapes of things 
 as they are in my mind, may be and often are 
 strangely different. 
 
 One of the arts of life is to shape things out of 
 matter to shape lovely forms out of lumps of clay, to 
 hew marble columns out of limestone rocks, to shape 
 graceful ships out of forest trees, to rear comely 
 mansions out of common bricks, to carve groups of 
 men out of blocks of stone, to paint a lovely scene 
 of forest, mountain, lake, and stream, to draw on 
 paper an interesting object so as to show it exactly as 
 we saw it, to draw on paper the shape of some new 
 thing with precision, so that another shall see it in 
 the mind's eye exactly as it will be seen by them 
 after it has come to exist ; these are some of the 
 practical works of common life, and all these are 
 merely Geometry.
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 165 
 
 Light is one of the most useful instruments of 
 human life, and one of the most useful tools of the 
 Geometer. Light shows shape, Light shows distance, 
 Light throws shadow, Light is the food of life, and 
 is also the food of the eye and the mind. To the 
 Geometer, Light going straight from far to near, 
 marks out straight lines in space ; striking on masses 
 of matter, it marks out shape by shadow and shade ; 
 coming into his eye it tells him place and direction, 
 and enables him to measure size, tell distance, and 
 mark out shape. 
 
 Light therefore deserves the study of the young 
 Geometer. It shines out of a light-sending body in 
 straight lines. It strikes on dark bodies and makes 
 them shine by its own shine, and makes them seem 
 to shine by a light of their own. Shining light 
 makes other bodies throw their shine off from them- 
 selves also in straight lines, and these straight lines 
 of light entering our eyes make us see the bodies 
 which sent them to us ; and when we look at a near 
 flower or a distant star, the coloured light of the one 
 and the clear light of the other enable us to see 
 them along the line of light which came to us, in the 
 near distance or far away. Hence we see straight 
 because light comes to us straight from the form 
 seen. 
 
 It is by the Geometry of lines of Light that an 
 astronomer tells the place of a fixed star or of a
 
 166 GEOMETRY IN MODERN LIFE. 
 
 moving planet ; that he foretells the aspect of Saturn's 
 rings ; that he tells from Jupiter's satellites the hours 
 in the heavens, in the same way as we do the hours 
 of the clock from the place of its hands; that he 
 foresees the moment when the moon will enter the 
 shadow of the earth, which shadow will be cast off 
 by straight lines of light coming from the sun and 
 stopping on the earth, while others escape past the 
 earth and carry light all around. 
 
 On the earth's surface, it is the lines of light 
 coming from the lighthouse that enable the mariner 
 to keep clear of rocks and sands, and steer his ship 
 safe home to her destined port. It is by "taking 
 sights " of sun, and moon, and satellites and stars that 
 the sailor is able to tell his latitude and longitude and 
 know his exact place in the wide ocean, when there is 
 nought in sight but sky and sea, and so Geometry of 
 lines of Light and Sight becomes essential to the art 
 of modern navigation, and is part of a seaman's skill 
 and sailor's education. In like manner, on land, it is 
 by a series of optical instruments, called telescopes, 
 sights, theodolites, reflecting circles, sextants, quad- 
 rants, altitude and azimuth circles, and such like 
 geometric tools for measure by lines of light, that 
 the Geometer performs all the operations of place- 
 fixing, time-marking, and way-finding through the 
 heavens and over land and sea. 
 
 A single example will show how much a skilful
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 167 
 
 Geometer may learn out of a single sight of one 
 phenomenon of light in the heavens. 
 
 A skilled yachtsman, an Etonian, was very lately 
 making researches in matters of science in his own 
 yacht, which he navigates himself, and had gone far 
 to the north in somewhat hazy weather, and wanted 
 much to know his " whereabouts." 
 
 Luckily, near midnight the weather was clear ; and 
 just then and there our yachtsman saw the sun, not 
 in the act of setting, but in the act of just touching 
 the sea and then rising up again, as it does only in 
 summer " very far north." This single sight carefully 
 noted, told him his yacht's exact place on our round 
 world. The sight was easy and simple, the calcula- 
 tion somewhat long and complex, and I have great 
 pleasure in presenting it to you as a sample of one 
 use of an Etonian's Geometry.
 
 168 
 
 GEOMETRY IN MODERN LIFE. 
 
 OUR ETONIAN'S OBSERVATION 
 
 AND 
 
 CALCULATION OF HIS YACHT'S PLACE IN THE NORTH SEA. 
 
 1877, July 23, Midnight ; in Longitude 21 55' E. The observed Altitude 
 of the Sun's upper limb (the lower limb being obscured by cloud) when on 
 the Meridian below the Pole was 1 T 40". Index Error, Nil ; Height of 
 Observer's Eye, 10 ft. The Latitude was found as follows : 
 
 h. m. s. o ' 
 
 Astronomical apparent time at J July 23 J2 Q Q L(mg> 21 5g 
 
 Long, in time E 1 27 40 ,, 4 
 
 Astronomical apparent time at 
 Greenwich 
 
 23 
 
 10 32 20 
 
 Sun's Declination at apparent ) July 23 N. 20 1 19*5- 
 
 Noon at Greenwich \ 
 
 Change in 24 hours . 
 
 Greenwich Time 
 
 h. m. s. 
 Change in 10 32 20 . 
 Sun's Declination as above 
 
 Reduced Declination . 
 
 Sun's Codeclination or Polar distance 
 
 Observed Altitude 
 Dip 10 feet 
 
 Refraction .... 
 
 Parallax .... 
 
 Sun's Semidiameter . 
 
 True Altitude of Sun's Centre . 
 Sun's Codeclination or Polar distance 
 
 Latitude .... 
 
 24 19 48 467 
 
 90 
 
 70 4 11 N. 
 
 1 7 40 
 
 - 3 10 
 
 1 4 30 
 
 - 23 58 
 
 40 32 
 + 09 
 
 
 40 
 
 41 
 
 
 - 15 
 
 47 
 
 
 24 54 
 
 + 
 
 70 4 
 
 11 
 
 
 70 29 
 
 5 N 
 
 h. m. s. 
 1 27 40 
 
 12 32*8 
 
 Log. 
 
 2817 
 
 h. m. s. 
 
 
 
 10 32 20 
 
 Log. 
 
 3574 
 
 .-05 30-5 
 
 Log. 
 
 6391 
 
 20 1 19*5 
 
 
 
 
 19 55 49- 

 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 169 
 
 Thus we learn that lines of light are the Geometer's 
 guides in the use of his instruments and tools, for the 
 finding of places, the measuring of distances, the 
 drawing of shapes and ways on paper, and the laying 
 out of lines of railway, lines of road, boundaries of 
 estates, courses of ships, shores of seas, outlines of 
 lakes, and paths of planets. 
 
 How to show on flat paper the shapes of things 
 you see round about you, some far off, some near at 
 hand ; how to show them truly as they are, and also 
 as they seem to your sight; to make true plans, 
 drawings, outlines, shades, shadows, and pictures 
 with absolute truth and perfect exactness that is 
 a hard problem and requires a Master- Geometer. 
 
 And to a sound Geometer, the problem is de- 
 lightful and beautifully simple. 
 
 First, you have to fix your own stand-point, which 
 we call the place of the seer. 
 
 Second, you have to settle the way the thing lies 
 from yourself, or the way you must look to see what 
 you want ; that we call the line of sight. 
 
 Third you must know how far off each thing lies 
 from yourself. Next, how far it lies to the right or 
 the left from your line of sight.
 
 170 GEOMETRY IN MODERN LIFE. 
 
 Thus you learn, or settle the place of each thing 
 you mean to place in your plan or picture. This you 
 must do before you begin, and it is called plotting 
 out or preparing or placing your objects. 
 
 This placing each thing in its right place is mere 
 plain Geometry. 
 
 You Take your stand-point. 
 Settle your line of sight. 
 Mark off distances from your stand-point. 
 Mark off places to the right. 
 Mark off places to the left. 
 
 Having marked off places and distances as they 
 are in fact, you have next to mark them off as 
 they seem. 
 
 1 Now the fundamental principle about placing things 
 as they seem instead of as they are is this 
 
 Things on the right of the seer, seem at less and 
 less distance from the line of sight, or the mid-line, 
 in exact proportion as they are further off from the 
 seer. Thus a nearer thing seems more to the right 
 than a further off thing which is exactly the same 
 distance off on the right from mid-line. Therefore, 
 if a row of trees are all equi-distant from a mid-line, 
 and each at distances of 1, 2, 4, 8, 16, 32, chains
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 171 
 
 from me ; their distances from the mid-line will seem 
 to me 1, i, t, 1, rV of the true distance. 
 
 What happens in distance of place to the eye 
 happens also in seeing size. Each tree of the same size 
 at distances from me of 1, 2, 4, 8, and 16 chains, 
 will seem to me successively as of the diminishing sizes 
 1, h t, I, tV. Thus the arrangement of place, the 
 settling of distance, and the measure of size, are all 
 quite different to the eye from what they are in fact ; 
 and if a picture or a plan is to show truly to the eye 
 what the eye has seen or is to see, the Geometer and 
 the Artist must draw them on paper, not as they are 
 but as they should seem and Geometry alone can do 
 that true and exact. What I have said of place to 
 the right and of place to the left of the sight-seer is 
 equally true of place above his own level or below his 
 own level. Heights above his eye and his line of 
 sight grow less and less on the plan, and seem less 
 and less to the eye, as they are further and further 
 from the seer, and exactly in the same proportions 
 
 of 1 I, t, I, tf, heights, 
 to 1, 2, 4, 8, 16 distances. 
 
 And what is true of heights above the eye is equally 
 true of depths below the eye. 
 
 The right placing of all things to be seen in a plan 
 or picture is a mere carrying out of the same system
 
 172 GEOMETRY IN MODERN LIEE. 
 
 of Geometric stand-point, measured way off, and 
 distance square across, which we have already studied 
 among the First Steps of the Geometer ; and is 
 equally applied to right and left distance and to 
 distance up or down ; and this law of seeming diminu- 
 tion of size and distance, with increasing way off as 
 measured from the seer, is the addition wanted to 
 carry it into use. 
 
 One of the most interesting and one of the most 
 common uses of the Geometer's art is the laying 
 down on a sheet of paper, or on a flat board, or on 
 a sheet of canvas stretched out flat, the 
 
 Shape, Outline, Form, or Appearance, 
 of 
 
 The things he sees. 
 
 This art of showing things in shape as they seem 
 to our eyes, by lines and shades and lights and 
 shadows, is a modern art of great interest and every 
 day use. It is commonly called 
 
 " The Art of Projection." 
 " Descriptive Geometry." 
 " The Science of Perspective." 
 
 In plain English it is 
 
 How to show the Shapes you see.
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 173 
 
 How to show to another on a flat sheet, the shapes 
 of things you have seen, so truly that he shall see 
 them, or seem to see them, just as truly as you did 
 see them in reality. That is the high art of the 
 modern painter ; it is the humbler art of the modern 
 draughtsman and engraver ; it is an art which must 
 be founded upon a profound and thorough knowledge 
 of the laws of light and sight. 
 
 The first Geometric law of light is that light comes 
 straight from light-giving bodies or sources of light, 
 and enters into or strikes on all other bodies round 
 about them. 
 
 The second Geometric law is that light comes 
 straight from visible forms to the seer's eye. 
 
 Next we have to take into account 
 
 The shape of the body in reality. 
 
 Its attitude as we see it. 
 
 Its seeming shape in that attitude. 
 
 Its seeming size in distance. 
 
 Its exact place from line of sight. 
 
 Take one example of 
 
 Simplest shape, 
 Say A Common Square Box.
 
 174 GEOMETRY IN MODERN LIFE. 
 
 The Geometer's technical term for this simple form 
 is a "rectangular parallelopipedon," but the words 
 "square box " are equally plain. A common square 
 box, if set straight in line with the seer's eye, on the 
 same level, will seem a common square flat front, and 
 he will only see that face of the box, and no other. 
 
 Now the box has six faces 
 
 Top and Bottom. 
 
 Front and Back. 
 
 Two ends Right and Left. 
 
 Let us distinguish these six faces by colour : 
 
 Top, Yellow. 
 Front, Red. 
 Bottom, Blue. 
 Back, Black. 
 Right end, Orange. 
 Left end, Green. 
 
 When the box stands on our level straight in front, 
 we see a square Red shape only. 
 
 If raised higher, we see the Red front and a Blue 
 bottom joined on. 
 
 If set lower, we see the Red front and the Yellow 
 top together.
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 175 
 
 If set on the level to our right, we see the Red 
 front with a Green end on its left. 
 
 If set on our level to our left, we see the Red front 
 with an Orange end on its right. 
 
 Then follow four more attitudes : 
 
 The box may stand 
 
 Below us on the Right. 
 Below us on the Left. 
 Above us on the Right. 
 Above us on the Left. 
 
 In four cases we see four sights 
 
 1. Yellow, Red, and Green. 
 
 2. Yellow, Red, and Orange. 
 
 3. Red, Blue, and Green. 
 
 4. Red, Blue, and Orange. 
 
 In these four last not only are the faces changed in 
 colour, they are all changed in visible shape. They 
 were all square, now all seem skew. There is still 
 another series of semblances, of the same box shape, 
 according to attitude. Fancy the same box shape 
 suspended in the air ; by a string from one corner, it
 
 176 GEOMETRY IN MODERN LIFE. 
 
 would present to the eye a new series of pictures as 
 it was spun round the axis or string. 
 
 Now in that box you have only to fancy there is 
 contained a statue or a work of art of any kind, and 
 it is plain that in each of these positions and attitudes 
 of the box the statue within takes a like attitude. 
 And if you fancy the box made of glass, the statue 
 would be visible to the seer's eye, but each sight of 
 the statue would be a different picture. 
 
 This serves to show you the chief difficulty of the 
 painter, planner, and designer. He has 1st, to know 
 exactly the things as they are ; 2nd, to know exactly 
 how they will seem. 
 
 This art of showing shapes on the flat as they are 
 seen in fact, or as they are intended to seem when 
 seen, requires the exercise of many faculties and much 
 precise knowledge. But of all these faculties the 
 most essential is the art of the Geometer. 
 
 That the great old painters were great Geometers, 
 their life and works indisputably show. Eubens, 
 Michael Angelo, and Leonardo da Vinci, were accom- 
 plished Geometers. When I was investigating this 
 question in the Low Countries, I had a curious 
 illustration of the union of Geometric w^ith Artistic 
 craft. An ancient town was hefeditary owner of 
 extensive lands. The town corporation held a meeting
 
 SIGHT, LIGHT, SHAPE, AND SHADOW. 177 
 
 at which, a councillor proposed that an exact Geo- 
 metric Survey of the town-lands should be made. It 
 was thought " worth while/' and it was agreed that 
 the survey should be paid for at so much per acre- 
 The question who should be the Geometer, or lands 
 surveyor, was decided by one councillor assuring the 
 members that there was only one good Geometer in 
 the town, and he named their greatest painter, one 
 whose fame still stands high. The painter accepted 
 the order. He carefully measured all the lands of 
 the corporation, including streams, trees, fences, gates, 
 farm -buildings and dwelling-houses. I had the 
 great pleasure of examining his work, then some 
 hundred years old, and carefully preserved in the 
 town-hall. He not merely laid down his plans 
 exactly on parchment, but also gave a plan, painted 
 in oil, and representing all the lands, trees, and 
 buildings, in one single view, in exquisite Geometric 
 perspective, showing everything true as on a picture. 
 
 This Geometric projection of land-surveying is 
 really a beautiful oil painting, and I should never 
 have fancied it other than a mere landscape, had 
 I not found this curious history in the well-kept 
 record of proceedings of the old town corporation. 
 The truthfulness of all this painter's pictures was 
 quite explained to me by his mastery of the mystery 
 of representing visible form by exact Geometric 
 method.
 
 178 GEOMETRY IN MODERN LIFE. 
 
 XIV. 
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 
 
 The machinery of mechanical forces and motions 
 which constitute the main elements and powers 
 employed in the Engines, Engineering, Arts, Manu- 
 factures and Industries of Modern Life, are matters 
 mainly of modern civilization and scarcely at all of 
 tradition. Our great mechanical inventions have 
 come to us mainly from the great discoveries of the 
 laws of the universe of matter by Galileo, Huyghens, 
 Kepler, and Newton. The discoveries of modern 
 science and the development of modern invention 
 have gone hand in hand ; and have created the 
 present marvellous navigation of the ocean and the 
 travelling over continents and around seas ; which 
 give to us little human beings a power of going 
 everywhere in small portions of our short lives, as 
 well as being nearly omnipresent in mind and thought 
 by electric telegraph ; while our steam-engines are 
 doing the work of thousands of horses and millions 
 of men.
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 179 
 
 All this has grown out of the modern discoveries 
 of the laws of force, mass, and motion. Happily for 
 us, these modern laws were found to be at one with 
 the antient laws of Geometry. It is quite conceivable 
 that Geometry of void space might have had one set 
 of rules, and that the massive, moving matter of the 
 universe might have had new laws of its own, quite 
 inconsistent with the laws laid down by the mere 
 Geometer. All this is conceivable, and has been 
 conceived : and has led many men to spend their 
 lives in inventions and discoveries which seemed 
 mathematically perfect and which are materially 
 impossible. 
 
 Modern discoveries tested by experience have 
 taught us that masses of matter, still or moving, 
 starting from rest, stopping from motion, move as 
 accurately, stop as exactly, change their speeds and 
 alter their ways with as perfect precision as if they 
 were celestial beings moving to music of the spheres, 
 and mapping out a Geometer s Forecast of their ways 
 and distance. A planet, a cricket-ball, and a swing, 
 all describe Geometric paths in precise measure, time, 
 and distance, as foretold in the Geometer's Forecast. 
 
 Matter moving free, goes on straight and even. 
 
 That is the first law of Matter Nature. 
 
 Matter in motion changes not, it neither goes faster 
 nor slower ; it neither turns to right nor left nor changes 
 
 N 2
 
 180 GEOMETRY IN MODERN LIFE. 
 
 upwards nor down. Ever onward, changeless. That 
 is the law and nature of Dead Matter. 
 
 But hindered matter takes heed of hindrance, and 
 yields. Helped matter accepts help and turns it 
 to use. 
 
 One moving mass lends help to another moving 
 mass, directs its course, turns it back, or increases its 
 speed. 
 
 But all this it does by settled way and measured 
 number. In all its ways matter follows the settled 
 rules of the Antient Geometry. 
 
 Matter moving free goes on straight. 
 
 Matter moving goes in measured distance. 
 
 Matter moving free keeps true time. 
 
 Matter yields by measured way to force. 
 
 Matter turns aside by measured degree yielding to 
 measured force. 
 
 Matter goes faster yielding by measure to measured 
 force. 
 
 Matter goes slower yielding by measure to measured 
 force.
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 181 
 
 How force turns moving matter aside ? is an 
 interesting, instructive, useful question. 
 
 You row a boat across a lake, going straight. 
 
 You move the water out of your way, that takes 
 force. 
 
 But you next row across a stream already going 
 straight another way. 
 
 What happens % 
 
 Which way do you go ? Not the way you row ! 
 Where do you land ? Not where you direct your 
 boat ! 
 
 The law of the matter is this 
 
 Each force does its own work as if the other did 
 not exist ! If you row 4 miles an hour your way, 
 you do your 4 miles just the same in the stream as in 
 the still. 
 
 While you do your 4 miles, the stream does 
 its 3 miles in the hour. 
 
 The double result is this 
 
 1. You have gone your 4 miles with neither 
 more nor less effort than in the still.
 
 182 GEOMETRY IN MODERN LIFE. 
 
 2. The stream has taken you its 3 miles without 
 other effort than if you had lain still. 
 
 3. But the joint' result is 
 
 A new way. 
 
 On which third way 
 
 The boat and load have gone 5 miles ; 
 
 While you were going 4 and the stream 
 going 3. 
 
 The result then is 
 
 A new way and a new speed. 
 
 This resulting way and speed arising from boat 
 and stream combined is identical with the pure 
 Geometer's Forecast of Three Points and Three 
 Ways, as given in Tables and Diagrams on pages 
 70, 71. 
 
 The general law of combined motions, forces, 
 and masses is, that each does its own work its 
 own way, and that any two forces and motions do 
 make one third force and motion exactly as the 
 Three Ways of the Geometer foretell. 
 
 Thus taking the Diagrams on page 71, which are 
 here repeated, we find
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 183 
 
 Diagrams of Distances as Forecast by the 
 Table. 
 
 
 
 100 
 
 
 /(? 
 
 
 100 
 
 .J 
 
 Forecast. 
 IQI'M 
 
 loo 
 
 ^tyf--"' 
 
 100
 
 184 GEOMETRY IN MODERN LIFE. 
 
 Thus in the first figure, if 200 tons were a force 
 acting upright, 100 tons a second force acting horizon- 
 tally, the result of the two would be a strain of 
 223*60 tons on the slope. 
 
 If altered to the form of the next figure, there 
 would only be a strain of 180 tons on the upright, 
 and 205*91 on the slope. 
 
 If in the third figure, the length of two roof 
 rafters were represented by 100, and of the tie-beam 
 by the sloping side, we should conclude that each 
 strain of 100 lbs. on a rafter would produce a strain 
 of 141*41 lbs. on the tie. From the fourth figure 
 we may learn that if a ship were moved by tide along 
 100 fathoms, while moving by wind 150 fathoms as 
 shown, the result of the two would move that ship 
 180 fathoms along the third side, and this third side 
 will show the true course of the ship. 
 
 The two last figures show that a weight of 101*98 
 tons may be pulled up one slope by a force of 20 tons, 
 while a weight of 111*80 tons drawn up the other 
 slope will require a force of 50 tons. 
 
 It is with the heavenly bodies as it is with 
 earth masses. A comet may be rushing past the 
 earth at a thousand miles an hour. The earth 
 attracts the comet as it does the apple. It pulls it 
 towards the earth's centre. The Geometer knows
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 185 
 
 the comet's speed. He knows the speed the earth 
 can give to the comet in a different course. He 
 makes out his diagram of the three ways, and this 
 third way tells him the new way and the new 
 speed of the comet. Thus Geometry w T ith laws 
 of matter mark out a new and formerly unknown 
 w r ay of this comet through the distant heavens. 
 
 Thus out of two ways of force and speed grows 
 a third way which is the result of their joint action. 
 But in this third way our moving mass might have 
 a new influence brought on it. This new force 
 with its new w T ay the Geometer would treat just 
 as before and out of the third way combined with 
 a fourth he would get a fifth. 
 
 Therefore we now know that all calculation of 
 force, mass, way, and speed, is merely an application 
 of Geometry to the grand law revealed to us by 
 experience of nature ; that the laws of Way, 
 Measure, and Number have been laid upon the 
 matter universe so as to be quite intelligible to 
 those students who know the ways of the Great 
 Geometer. 
 
 The science of mass, motion, and force is an 
 application of the principles of the Geometer to 
 the laws of matter by measures of length, distance, 
 and direction ; which represent space, speed, and 
 way.
 
 186 GEOMETRY IN MODERN LIFE. 
 
 When two motions go one way, we add their 
 speeds, and double the distance done. 
 
 When two motions are contrary, we deduct the 
 less from the greater, the difference being the 
 resulting motion. 
 
 When two motions go different ways, each does 
 its own distance ; we take the common starting-point 
 and the two separate arriving-points of the two 
 independent motions, and we join all three. The 
 first two lines measure the two independent ways, 
 and the third line measures the third way resulting 
 from the two. 
 
 And this process of the three ways will not only 
 resolve two forces into a third, but will find a fourth 
 force growing out of three, and a fifth force growing 
 out of four ; and will truly represent and measure 
 any number of forces continually changing the result 
 of any number of forces, successive or simultaneous. 
 
 Thus Geometry makes plain to our minds the laws 
 of matter, motion, and speed. But this same Geo- 
 metric solution of the problems of moving bodies, by 
 a diagram of three ivays, gives us the solution of 
 a multitude of other questions which are not questions 
 of motion, but questions of stedfastness. We want a 
 locomotive engine and a train to move over a bridge, 
 but we do not want the bridge to move with them,
 
 GEOMETRY, MATTER, FORCE, AND MOTION. 187 
 
 and therefore the bridge- maker has to learn how to 
 make his bridge stand stedfast, and not to suffer itself 
 to be induced to participate in the motions of the train. 
 This requires skill in the art of securing rest, and 
 preventing motion. Sound Geometers do this, but 
 clever men who are not skilled, build bridges which 
 cannot resist temptation to move, and so are sent 
 into the abyss of chaos. The ancient Roman viaducts 
 still standing are noble examples of Geometric 
 skill, and it is interesting to know that they were 
 designed and built by Greek Geometers for Roman 
 masters. 
 
 The art of bridge building, the skill of setting 
 work stedfast, consists mainly in knowing how to 
 stop two forces by one, and how to oppose one 
 destructive force by two others, which do not seem 
 directly opposed to it. A train on a viaduct weighs 
 down the top of an arch, a prop direct from below 
 might keep the bridge from falling ; but that is not 
 the purpose of the bridge, which has to keep the 
 train in the air, by support taken from two distant 
 piers, one before and the other behind the train. 
 
 How to hold up the train, with no support beneath, 
 by stedfast support brought from two distant places, 
 is the problem to be met. We have here three 
 points. We have here three ways. The Geometer 
 draws three lines from point to point. He groups
 
 188 GEOMETRY IN MODERN LIFE. 
 
 them together, and measures their length. One length 
 he takes to measure the weight of the train say 
 for each ton of train an inch of length. He next 
 measures the two remaining sides of his diagram, 
 when he finds that every ton weight of train produces 
 two tons, or three tons, or one ton and a half of 
 strain on the material of the bridge. Experience 
 tells him if his bridge can stand this strain or not, 
 and if so, for how long? Modern science tells us 
 that trains take some of the life out of a bridge every 
 time they cross it with a heavy load. 
 
 The builder of a modern viaduct is either therefore 
 a good Geometer, or he is good for nothing, and his 
 bridges give way because he does not know the 
 Geometric method of preventing motion, because he 
 does not know that one force prevents two, and two 
 forces prevent one, according as three points, three 
 distances, and three ways, are ill chosen or well 
 and so his work is foolish or wise, or wasteful, useful, 
 dangerous or enduring. 
 
 All youths who wish to become distinguished 
 designers, builders, engineers, architects, or sculptors, 
 must first become good Geometers.
 
 ENDING ENDLESS. 189 
 
 XV. 
 
 ENDING ENDLESS. 
 
 What I have shown you of Geometry is only the 
 first flight of a few steps upwards towards that great 
 temple of knowledge, where you may find stored for 
 you by the antient Geometers, and completed by the 
 modern philosophers, the great system of divine 
 thought and human knowledge, which constitutes the 
 laws of space and matter, of shape and substance, of 
 force and motion ; and by means of which you may, 
 in after life, foresee distant events, provide for difficult 
 undertakings, calculate ways and means, foretell dis- 
 tances, courses, and time, and do all your work in 
 the material world in order, measure, and number. 
 
 But I have merely shown you the threshold of 
 that vast system of truth, revealed to us in the works 
 of the Great Geometer. I have only tried to assist 
 you in selecting a few first steps of solid ground, on 
 which you could firmly set foot, and so placed in line 
 that you should see clearly which way they lead you
 
 190 GEOMETRY IN MODERN LTFE. 
 
 through the difficulties of the unknown, towards 
 those great heights of thought and those wide ex- 
 panses of knowledge which lie beyond. It is not 
 for me to conduct you onward through this journey 
 of school study, and of life's work. For that I leave 
 you in the w r orthy hands of your able masters, with 
 whom I am proud, in this small degree, to co-operate 
 in directing your ways towards the acquisition of 
 knowledge, wisdom, and usefulness. 
 
 I have only a few concluding words to say, for 
 the purpose of giving you some idea of that Geometry 
 of which hitherto I have said nothing. Of the 
 Geometry of Place and Distance, and of Points and 
 Lines, I have tried to give you some precise thoughts 
 and exact expressions, and practical uses. But of 
 the whole theory of curve lines [ have scarcely said 
 one word, though I have said some things which are 
 meant to lead you that way, and prepare you for 
 these after studies. 
 
 Of Circles I have said nothing, except where 1 
 showed you how they grew by additions, following 
 the laws of Odd Numbers ; and in like manner, as 
 Squares and Triangles grow, though in different 
 fashion. But I must now tell you that when men 
 come to make long journeys inland, over wide con- 
 tinents, and when men make voyages across wide 
 oceans, those ways that we call " straight," are not 
 so. The straight ways of every day life become
 
 ENDING ENDLESS. 191 
 
 great circles, when we wander wide over the ocean, 
 and the Geometer who surveys the globe, and the 
 navigator who sails the ocean, have to be well aware 
 that a line which seems straight is no longer so, and 
 that what seems a straight course is only 
 
 Part of a Geeat Circle of the Globe. 
 
 When you go further on in Geometry you will find 
 that the able seaman has to master the science of 
 " Spherical Trigonometry," instead of Plane Trigo- 
 nometry, and that instead of finding the shortest 
 way in the world is to go straight, he finds that the 
 art of navigation teaches him great circle sailing as 
 the art of " slow, sure, and quick." 
 
 "Great circle sailing" round the world teaches 
 us many truths, and also requires of us much skill. 
 When we turn to the right or the left, as we must 
 often do, to avoid an island, a rock, or a shoal, or 
 when driven aside by stream or storm, we have to 
 find our way back into our circle again, and verify 
 our place and distance. This tests the adept's skill 
 in Spheric Geometry. Also, it teaches us much we 
 on land have small chance to notice. As we go on 
 sailing in our great circle round the earth, we find 
 the sea "rounding" under our ship. Every mile 
 of distance forward on one course, the sea rounds 
 downwards under our feet ! The measure of this 
 rounding is 8 inches.
 
 192 ' GEOMETRY IN MODERN LIFE. 
 
 Having found that 8 inches is the measure of the 
 earth's rounding, we may take this as our standard 
 unit of deviation from the straight, when we go 
 round the earth on a "great circle." And when we 
 have taken this measure of 8, and called it our 
 standard I or unit, we can find other larger devia- 
 tions by the nature of the circle and our old law of 
 Odd Numbers. Thus 
 
 The earth bends round downwards in the 
 
 Together. 
 
 First Mile I. Unit. 
 
 Second III. 4 
 
 Third V. 9 
 
 Fourth VII. 16 
 
 Fifth Villi. 25 
 
 Sixth XL 36 
 
 Summing up any of these we choose, we find that 
 in 3 miles we have nine measures of deviation, or 
 
 9 X units of 8 inches. 
 Making 9 X 8, or 72 inches, or 6 feet. 
 
 Whence we learn that in 3 miles over sea, on a 
 great circle, we go downwards 6 feet. 
 
 In like manner, by summing any other descents, 
 we find that in 4 miles we have gone down 16 units
 
 ENDING ENDLESS. 193 
 
 of 8 inches or 10 feet, in 5 miles 25 units, or lGf feet, 
 and so on. 
 
 These deviations from the straight on the circles 
 of the globe, lead us on to the knowledge of all 
 the characters of the circle. But the circle is not 
 the only curve in nature, although the most common 
 in our ways of working and thinking. Knowledge 
 of the characteristics of the circle leads us forward 
 to many other ways of 
 
 Deviation from the straight. 
 
 After you have learned the Geometry of great circle, 
 circle sailing, and the Geometry of the globe, com- 
 monly called Spherical Trigonometry, you will then 
 merely have been introduced to the first member of 
 
 The Great Family of Curve Lines. 
 
 " The crow Hies straight ! " 
 " The arrow goes straight ! " 
 
 These are common sayings, but when you go 
 beyond common opinion, and get a little nearer the 
 truth, you find that the arrow does not go straight, 
 that like the ship it drops downwards as it goes. 
 When you study the path of the arrow closely, you 
 find that it droops, so as to form a curve that " looks 
 like " a great circle, or a bit of one. 
 
 o
 
 194 GEOMETRY IN MODERN LIFE. 
 
 At first it seems as though the arrow followed 
 exactly the great circle of the earth, in the rate of 
 drooping by gravity as it goes, only with a different 
 unit, differing according to the different speeds with 
 which the arrow leaves the bow ; but with wider 
 experience we find it not so. They are like, but not 
 alike. 
 
 The arrow from the bow droops by a 
 
 New path of descent, 
 which forms 
 
 A new curve line, 
 
 commonly called 
 
 "The Parabola." 
 
 This parabolic path deviates from the straight path 
 and forms a new path, and leads to new principles of 
 Geometry, gives new modes of calculation, and teaches 
 us new methods of work. 
 
 The parabolic curve shows us the paths of 
 
 The arrow from^the bow. 
 The bullet from the pistol. 
 The bolt from the rifle. 
 The shell from the cannon. 
 The comet from far away.
 
 ENDING ENDLESS. 195 
 
 The next curve which we meet in nature, and of 
 which Geometry "tells the tale," is 
 
 The Elliptic Curve. 
 
 It is the path of a planet round the sun, of our 
 earth through space, of the moon round about us, 
 and of Jupiter's satellites round about him. 
 
 The Geometer of the heavens has therefore to 
 master the elliptic curves, and to make the elliptic 
 calculations belonging to the paths of each, and all 
 of the heavenly bodies, before he can foresee and 
 foretell for us all the facts that will appear in the 
 heavens, as they will appear to the eyes of any sailor 
 in any part of the ocean, for years to come. All 
 this he does for you and me, and stores up for us for 
 years before they are wanted, in a plain book, for 
 the guidance of all sailors along their great circles 
 round the globe, and this you buy in a common sea- 
 man's shop for a few shillings, and it is called the 
 Nautical Almanack ! 
 
 The next curve we have to study is called 
 
 The Hyperbola. 
 
 It is a wilder and stranger curve than the circle, 
 ellipse, or parabola, and was thought to be " the 
 prodigal son " of the family, from his eccentricity and
 
 196 GEOMETRY IN MODERN LIFE. 
 
 his endless wanderings from which it seemed as though 
 he would never return. At last the hyperbolic path 
 was found amenable to rule and order, and made of 
 use in the work of modern life. The Hyperbolic Curve 
 tells us the true theory of the rebellions among atoms 
 of water in steam boilers, and how they strive against 
 the unwise intrusion of undue heat into the wrong 
 place, or at the wrong time, and when the right 
 principles of boiler-making and of boiler-heating are not 
 carried out or wrongly timed ; then the atoms following 
 the nature of the Hyperbola, break all bounds, burst 
 their iron bonds, tear in pieces all about them who 
 would control or cross them, and disperse and dis- 
 appear into infinite space ! These are merely the 
 first few families of curve lines ; after which there 
 follow generations and families in multitudes of order 
 and degree ; deeply interesting in nature, and highly 
 useful in application ; like the Families of Numbers 
 of which you already know the degrees, the appli- 
 cations, and the use. 
 
 And now in leaving off I have to make an end 
 of a real pleasure : but I feel deeply that I have 
 reached a conclusion in which nothing is concluded. 
 
 Geometry has beginnings. 
 
 It has no bounds. It is endless, for its last steps 
 arc the knowledge of the endless and the boundless.
 
 ENDING ENDLESS. 197 
 
 The highest Geometry is 
 
 The Geometry of the infinite 
 and 
 
 The Geometry of the infinitesimal. 
 
 But even there the human geometer's mind is in 
 some degree permitted to compass divine thought, and 
 he finds at every step of the progress of man's 
 thought towards the analysis of the infinitely great 
 and towards the integration of the infinitesimal, that 
 his thought of limits and measures becomes clearer 
 and clearer as his view stretches wider and farther, 
 and also goes narrower and deeper. He finds further 
 that as his mind comprehends more and more the 
 marvels of infinite space, and penetrates more and 
 more into the mysteries of the infinitely minute : 
 by so much the more does he become able to 
 appreciate the marvels of the creations, of earth and 
 heaven, and the more does he find his own mind 
 filled with the thoughts that pervade all the works 
 of the Great Geometer. 
 
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