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S i< <^ -4 -i »^ U3 ;tr QB o i'l' Mz r" Is TO HIM THAT (ilVKTII HTIlKN'CiTlI Ti> TIIK AVKAK AND WISDOM TO THK Sl.MlM.K, TO WHOM BKI-«X(iS WHATKVKU MTIUT IS COXTAIXKI) IX IT, THIS BOOK Hl'Mlll.Y AXI) UKVKKKNTI.Y XSclrtcittrlr. CENTRIFUGAL FORCE AND GRAVITATION. > SOME OBSERVATIONS ON THE PURPOSE OP THE SERIES ; ^ SPECIFICATION OP CONTENTS, AMD INDEX. In regard to the title, the contents and the purpose, we are now in a position to give a more full and specific explanation than could be given in the introductory remarks at the commencement of this Series. The general character of the work is critical and judicial, and one of the forms in which its purpose may be exprcased is to discriminate and distinguish between Science and Unsound Science. The Preface (to Part First) asserts and undertakes to demonstrate that certain unsound and erroneous theories contaminate a large and important part of what is now supposed to be the orthodox teaching of Natural Science. Now, Unsound Science, although opposed to (Sound) Science, has this in common therewith, that it is system- ized and organized. In Unsound Science, one false statement is based upon some more primary false state- ment 5 and the whole scheme, in respect to its unsound- ness, is founded upon unreality, mystery and untruth, just as Science is necessarily based upon certainty and truth. We wish it to be particularly observed that we do not dispute or call in question the facts of observation n SPECIFICATION OF CONTENTS. .r--' , il m'':>: or experience, whether that experience bo the result of physicftl or ideal analysis; nor do we accuse or in the least degree suspect the observers and experjjuenters, acknowledged by science as such, of niisrcprest^ntation or intentional deception ; on the contrary, we have trustfulness and confidence in their observations and in the results of their experimental investigations as tiio^e ot men honourable and trustful as well as of natural ability (in many instances of great ability) specially trained and devoted to such investigations. For instance, although we have challenged and are disputing astronomical theories and doctrines wliicli were adopted and con- curred in by William Herschel, we do not question the correctness of his astronomical observaiions nor doubt the trutiifulness of the record which he gave to the public of those observations. In stating our reasons for con- cluding that the planets Uranus and Neptune do not belong to the solar-system, we have based the argument upon the careful observations of the satellites of the planet Uranus by that distinguished astronomer. (Part Third, page 15, et seq.) The specific cases on which we challenge the present teaching, the grounds upon which we dispute that teaching, and the arguments by which we support our protests, are now before the reader. It is true that these cases are amongst those which are looked upon as the most difficult, complex and abstruse of all the general subjects belonging to Natural Science ; but our argument is. . .that the difficulty and abstruseness arises, in a great measure from the fallacy and mystery which have become involved in the teach- ing on those cases, and does not necessarily belong to BFIOITIOATION OF CONTENTS. 5 the subjects themselves; that, moreover, the decisive issue is, in each case, as to the foundation of wliat is tauglit, as to whether the fundamental basis is real, reasonable and true, or unreal, deceptive and false ; and that, to determine the character of such fundamental basis, when brought separately and distinctly under consideration, does not require the formuhe of modern mathematics, nor any peculiar and arbitrary system of reasoning ; neither is the decision necessarily confined to a few highly trained and highly educated minds ; the essentially requisite qualification is that the mind, ordin- arily educated and possessed of ordinary intelligence, shall be dispassionate, unprejudiced and reasonable.* We will now briefly specify the most important of . those questions on which we have found occasion to condemn the present teaching. (1) On a hody revolving around a centre of gravitation. — Whereas it is now taught that the moving body re- volves in an arc (circle) compounded of the tangential motion and the centrifugal motion We have sliown that this teaching is erroneous, for, if it were true the velocity of the revolving body would continually increase . .And we aflirm, as the sound doctrine, that the revolv- ing body is restrained by the gravitating influence from deviating out of the arc (circle) of revolution. Fart Fibet, Skc. 1. • We are not to be understood as objecting, or entertaining any objection wimtever, to the strictest niatliematical investigation in any fornt that is sound and scientific ; but we insist that a knowledge of what are termed higher mathematics, and of n.odern niatliematical methods and processes, are unnecessary, fad we therefore en- tirely object to those who may not possesflS such technical knowledge but are quite well qualified to come to a sound reasonable decision on the several questions., being excluded from their rightful share and influence in determining the general conclusion. I i 6 8PECIPICATI0N OF C0NTBNT8. (2) Of the moori's influence on the tides. — Whereas it is now taugiit that the mr on draws the earth away from the water (and the water iway from the earth).. Wo allinn this to be impossible j for, if it were true, the earth and moon would soon come into collision. .And we hav«' shown that a reasonable and satisfactory explan- ation of the double tide may be found in the efl'ect of the moon's influence in shifting the earth's centre of gra- vity towards the side opposite to her, causing that part of the earth's surface to attract the water more power- fully, thereby occasioning a greater depth on tiiat side furthest from her ; whilst the direct influence of lunar gravitation attracts the water to the side nearest to the moon and causes a greater depth on that side also. Part Fikbt, Sec 12. (3) The penduhim as measuring variations in the inten- sity of gravitation at different parts of the earth'' s surface. — It is now taught that the squares of the times of oscil- lation vary in the inverse proportion of the force . . ..We have pointed out that such doctrine is not con- sistent with the law of accelerated motion under the influence of a continuously acting force such as gravita- tion ; because by that law the time occupied by a body in moving (falling) through a certain space is inversely proportional to the force simply. Wherefore, if the force of gravity be augmented, the times, and not the squares of the times, of oscillation of the pendulum will be dimin- ished in the same ratio. Fabt f insT, Seo. 14. ^:\ ■^' i^ 8PKCIFI0ATION OP CONTENTS. ' f (4) The inclined-axis of the earthy and the plane of the earth^s orbital revolution constantly coincident tvith the equatorial plane of the sun. which is now taught as according with the observed facts, and as a part of astro- nomical science; we have shown to be an untenable theory. . . .unsupported by the facts. And we affirm and have demonstrated by the observed facts, the actual relation to be the perpendicular position of the earth's axis relatively to the axis of the sun, and the obliquity of the earth's orbit of revolution relatively to the equatorial plane of the sun ; and which last we affirm to be the mean nodal plane of orbital revolution for each and every planet belonging to the solar-systfnii. Pabt 8k( nd. (5) Whereas it is now taught that the planets Uranus and Neptune belong to the Solar system as planetary members thereof, we have shewn that certain of the observed phenomena pertaining to those two planets do not accord with such hypothesis ; and we conclude, for the reasons set forth, that those two planets belong to a neighbouring stellar system, and therefore are not plane- tary subjects of our Sun. Pabt Third. (6) The Theory of Cometary Orbits. — Whereas it is now taught that the cometary body approaches the sun (in obedience to gravitation) from an enormous dis- tance, and that, after making part of a revolution j^iound the sun, in comparatively close proximity thereto, not- withstanding the exceeding intensity of the attractive force to which i,*; is then exposed, it suddenly departs in a right line without further obedience to the law of gravitation, until it has again attained an enormous dis- lli 8 SPEOIPIOATION OP CONTENTS. in tance from the sun, whence it returns. . .We have shown that this theory is quite inconsistent with the law of gravitation. pabt thibd. We have, moreover, for reasons set forth, objected generally and particularly to the eccentric orbit theory as not belonging to sound astronomical science ; and we have proposed, and affirmed as a positive doctrine the theory of the compound expanding orbit, which perfectly accords with the law of gravitation, satisfactorily ex- plains the facts of obsorvation, and is supported by the phenomena pertaining to cometaiy bodies. r&BT FOUBTH. Note. — We have also explained that the luminous train and light of the comet is caused by an accumulation of free (volumetric elec- tricity on that side of the cometary body furthest from the sun; which accumulation results from the inductive action of the sun's electricity. To shew that this explanation is satisfactorily supported, we have brought together a number of observed facts belonging to electricity and electrical phenomena. pabt Fifth. (7) The Corpuscular Emission Theory of Light.r—'We hvvve rejected this theory as untenable because it assumes that light, or the cause of light, consists in the reception of material particles emitted by the luminiferous body. Now if this be true, the recipient of the light, .i.e., the illuminated body, would necessarily increase in weight accoAJing to the quantity of light received, and which quantity would be proportionate to the time of its expo- sure to the rays of light. Experience has established that no such effect takes place, and has, therefore, demon- strated the supposition to be unfound^^d. Pabt Fifth. ti: SPECIFICATION OF CONTENTS. 9 It »; s re (8) The Undulatory Theory of Light. — ^We have shown this theory to be unsound and to be scientifically inad- missible. If the ether, with which the theory affirms that the universe is filled, be material (corporeal), it would necegoarily accumulate about the centres of gravitation ; and if it be not material (corporeal), no ground whatever is shown (even hypothetically) for the supposed undulations, for even the theory does not in- clude an assumption that an immaterial (mode) of force can of itself undulate. The theory is therefore either impossible or altogether unsupported. Again, the theory affinns, according to some teachers, that a definite por- tion of ether expands and contracts alternately ; accord- ing to other teachers that a number of particles of ether undulate or vibrate transversely, which is defined to mean that the particles move a certain distance with great rapidity in the one direction and then with equal rapidity in the reverse. In neither case is any cause of limitation in regard to the motion shown.. No cause whatever is assigned to restrain the portion of expanding ether, or the particles of ether in motion, from intermingling with the surrounding ether, in which their motions take place. If the particles commence to move rapidly in the one direction, no cause is assigned why they should not con- tinue to move in that direction. The so-called theory is therefore quite inadmissible and not even sufficiently complete to entitle it to con- sideration as an undemonstrated theory. We may remark, as an indepe" lent objection, that a wholly suppositious hypothetical fluid which is neither material nor immaterial, and is not analogous to anything known ; a fluid which has impossible properties attributed to it to suit the purposes of a theory, cannot be considered to belong to science. . ~ ; 4 10 SPIOIFIOATION OP CONTENTS. it (9) The Aberration-qf-Light Theory. — This theory we have shown to be whoKy suppositious, to be founded on a false analogy, and to be inconsistent with the known facts of astronomy. p^bt Third. (10) The Velocity-of-LigU Tlieory. — Whereas this theory is based upon the apparent variation in the time which elapses between the successive eclipses (of one) of Jupiter's satellites ; and was proposed expressly with reference to that variation, and assigned as the cause thereof ... we have shown that the perspective alterations in the visual angle arising from tlie increased and decreased distance of the earth from the planet, v/hich Roemer and his successors have neglected to take into consideration, does in itself naturally and satisfac- torily account for the observed apparent variation in the times of the eclipses ; the theoiy is, therefore, wholly superfluous. We have, moreover, shown that the theory is impossible and untenable, because iiTeconcileable with the known facts pertaining to light, belonging to astro- nomical science. p^^^ ^,^3 (11) We have stated as a generalization in physical science, that the natural world, as known to us, consists offeree and matter That matter, as directly cognized by us through our senses, (i. e., as naturally manifested to us by sensation) is compounded of force and matter That /orce is spiritual, and quite distinct from matter, upon which it acts, and by and through which its manifestations become known to us That light, heat, gravitation, motion, electricity, are various conditions of (free) force; and therefore all of them are spiritual. . SPECIFICATION OP CONTENTS. 11 . .That the idea of time is a recognition by the mind of successive manifestations of force on matter That time belongs, therefore, as, a necessity, to our cognition of the material (natural) world, but does not belong to force. And... that, therefore, the various conditions of force, such as light, heat, &c., have no velocity. . .i. e., that the intercommunication between separate bodies of free force (influence) does not involve the idea of time. i: Ili N»-,. «■ , ( , I i^^i^mmm mmm INDEX, AND SPECIFICATION OF CONTENTS, OF THE SERIES ENTITLED CENTRIFUGAL FORCE <& GRAVITATION. PART FIRST. The first general subject we have taken for particular ex- amination is that which gives the title to the whole series, viz., Centrifugal Force and Gravitation. Th's compound subject, which may be said to constitute the foundation of theoretical astronomy, occupies almost the whole of Part First, under the following divisions : — Page (1) Centrifugal Force 10 (2) Terrestrial Gravitation at the Distance of the Moon , 16 (3) Centripetal and Equilibriate Centrifugal Force of Revolution at the Centre of the Earth.... 18 (4) The Rotation of the Earth or other Planet, as modifying the influence of the Gravitating mass from within, at the Surface of the Sphere 21 (5) The Effect of Gravitation in sustaining tho Rotation of the Earth 26 (6) The Law of Gravitation and Centriftigal Force. (a) Linear Dynamical Equilibrium 30 (6) Angular Dynamical Equilibrium 33 (c) The Dynamical Conditions of Statical Equi- librium 38 (7) The Sun's Rotative mass and Gravitative influ- ence on the Surface of the Earth 40 (8) Planetary Systems 41 y . I i ' r! 14 (9) (10) (11) (12) (13) (14) (15) (16) (17) SPECIFICATION OF CONTENTS. Page The case of a Gravitating mass (a Planetary or Cometary Body) approaching or receding from the Sun ; and the law of Equable Areas 44 The Compounded Cer tripetal and Compounded Centrifugal Motion of the Planetary (or Comotary) Body in approaching and reced- ing from the Sun 55 Ellipticity of the Planetary Orbit 57 The influence of the Moon's Gravitation upon the Earth and the Theory of the Tides 58 The Form of the Earth and Terrestrial Gravi- tation 71 The Pendulum as a Measure of Terrestrial Gravitation... . 75 Terrestrial Gravitation compounded of the at- tractive force belonging to all and each of the parts of the Earth 78 Gravitation and the Atomic Theory 83 Habitability of the Moon 91 PART SECOND. The Subject of the Second Part is entitled, •' Theory of Orbital Eevolution. Demonstration of the Earth's Perpendicu- lar Axis. And, Oblique Plane of Terrestrial Orbit." Page (1) The Celestial Sphere and the relative position of the Solar System 9 (2) The Theory of the Earth s Perpendicular Axis. 14 The Compound AflSrmative Proposition 16 The Oblique Orbital Eevolution 18 (3) The Mechanical relation of the Planet's verti- tical motion to the gravitative force of the Sun 21 (4) The Ellipticity and Dynamical Elements of the Orbit 24 k. r r». '■^ V 8PE0IF10ATI0N OF CONTENTS. 15 Pago (5) The Theory of the Earth's Inclined-Axis 28 (6) The Two Theories contrasted 29 (m) Illustration of the Perpendicular-Axis Theory 32 (n) Illustration of Inclined-Axis Theory 33 (7) The Transits of Venus and Mercury 48 (8) Practical application of the Perpendicular- Axis Theory to the measurement of the Sun's distance 61 (9) Parallax of the Stars 64 (10) The general doctrine of the Compound Oblique Orbit 57 (o) The Transit of Venus illustrating the gen- eralization 58 (11) The Solar Spots and the Perpendicular-Axis Theory 69 (12) The Solar System and the physical arrange- ments of the Stellar Universe, 66 (6) The precession of the Equinoxes 67 (c) Relation of the vertical motion to the com- pound oblique orbit 69 (d) Vibration of the Earth's axis 71 (e) The Astronomical Record 74 Appendix. (13) Definitions belonging to the present ''octrine of the Celestial Sphere, &c 79 (14) The Diurnal Rotation of the Earth and the Polo of the Ecliptic 82 m I, * 4 i I I ! I P» 16 8PB0IFI0ATION OF CONTENTS. PART THIRD. The Third Part comprises in throe chapters two dis- tinct subjects, viz : — Chapter I. — The Solar System; and the general ARRANGEMENT OF THE SiDEREAL UNIVERSE. (1) The present state of Asti'onomical Science 9 (2) Stolhir Systems having their axis of revolution perpendicular to that of the Solar System... 13 (3) The planets Uranus and Neptune, and the question of a neighbouring Stellar system . . 16 (4) Masses of aggregated matter and their relation to the laws of the material universe 23 (5) The relative distances of the visiule stars 27 Chapter II. — The Present mixed doctrine of Par- allax AND Aberration. (6) Theory of Parallsvx 30 (7) Theory of AbeiTation 48 (8) A direct method of obtaining parallax of the distant stars 68 Chapter III. — The Aberration-of-Light Theory examined. (9) The conditions and I'equisitions of the theory.. 70 (10) Practical application of the theory 72 (11) The nature of Light as assumed by the theory (Immaterial Matter) 76 (12) Aberration a dynamical theory 77 M)t« a. The dynamical theory not in hai-mony with the laws of motion (13) Distrust in the gift of sight required by the Aberratior theory 80 Note b, A further dynamical test of the theory 82 Ifote c. The continuous radiation of light and heat into space. ..i.e., radiation to waste.. .re- quired by the theory 83 (14) Direct heliocentric methods of obtaining par- allax of the distant stars 86 ':■ I- ; .^ |ral R8E. 9 tion . 13 tho im.. 16 tion 23 27 Par- 30 .... 48 the .... 68 SORY joiy., 70 72 leory 76 ...... II fiony J tJie 80 eoiy 82 and ..re- 83 par- 86 SPECIFICATION OP CONTENTS. \*l PART FOURTH. Tho Fourth Part treats of Comets and examines tho Theory of Eccentric Cometary Orbits. Page Preface 5 Chap. 1 — Theory op Cometary Orbits. Introductory Observations 11 (1) Tho Compound Sideral Orbit 12 (2) Tho Compound Solar and Planetary Orbit 16 (3) Terrestrial and Solar Comets 20 (4) The Expanding Compound Orbit 20 Chap. II. — The Cometary Phenomena. (1) Theoretical Consideration of tho Phenomena. . . 23 (a) Physical condition of a Comet (2) Explanation of the Phenomena 27 (3) Natural Division of Comets into two classes 28 (4) Tho Theory of Eccentric Orbit.s, and tho Facts | g^ of Astronomy J (5) Biela's Comet 32 (6) Notices from the Record, of various Comets 36 (7) The Eccentric Oibit Theory 41 (a) Extravagant hypotheses of the theory 49 (8) The Comet's Luminous Train 51 (9) Concluding Eoraarks 53 Appendix. (1) Notice of Haliey's Comot 55 (2) Tho Comet of 1744 59 «; 18 I f, >: 'i -■:1 ': >. f f. SPBCIFIOATION OP CONTENTS. PART FIFTH. Theory of tho Nature of Light; The "' vo Theory of Sound ; The Nature of Force and MaiiifoHtutions of Force in the Phenomena of PhyHica! Science, FIT?ST DIVISION. The Nature op Lkiht. paoe (1) 'flu- Corpiiscular EiniHwion Theory 9 Objt'ctioiiH to tl»e Tlieory 10 (2) The Umhilatory Theory 14 Olijectioiis to the Tlieory 17 Science and Theoloijy 19 Science and Metaphysics 26 (3) Velocity oC Light Theory 31 ObjpctioiiH to the Theory 42 Examination of the Record Reference to Experience by Instances 49 Light. Suiiiii'nry ol' the Evidence. Conclusion by) inductive reasoning as to the nature of light . J (4) Generalization on the Nature of Force 61 Force and Matter 63 Allied Forces of Physical Science 65 (6) The relationship of Voliunetric Electricity and Light.. 66 The Luminous Train of the Comet ) Electrical Induction in its application to this > 79 case ] Evidence of the relationship from the general record 80 l; SECOND DIVISION. ' The Material Relations op Force. (1) The Wave Theory of Sound 93 Objections to the theory 102 Examples and Illustrations from the Rocord 115 The Nature of Sound 123 SPECIFICATION OF CONTKNTH. 19 (2) ManifpHtiitionfl of the variou.s lurniH (imwIoh) of force oil mutter. A . — Mii)j;netic and Theriiml eflectH of Klrctricity . . 127 B, — Mf«'"ciilnr (Voltaic) Klwitrinity. nnd tlic Mii-) j 1 oo lerial Kl.'nieiits of ciiinpounil matter ) C— Caloric-Force and Matter 139 D. — Magnetiwrn aiid Mr.f^neto-Electric Force . . , . 142 E. — Maj^netizatioii of Lijilit and Dia-MajrnetiHin,, 148 (3) The Material World and the Uiiiverfle 1.57 Conclusion IfiO APPKNDIX. (1) Her^clicl'M dcHcriptioii of the KclipHes iiinl Occulta-) j^, tioiiH o( Jupitcr'H HatcllifcH j (2) Tyiidull'n Observations on the Ktlier of the Undiila-i tory theory j '^' (3) Appendix to the Allied ManifcslutioiiH of Force on matter. a.— Magneto-Electric Force 173 6.— Volumetric Electricity 177 c— Physical and Chemical EU'ectxof Molecular j Electricity j fi.— Acoustic Force, Motion and Electricity l'J5 e.— Dynamical etlect.s of Electro-Magnetism 19d (4) Subjective applications of Electric Force to mechan-i ical purposes j The Coa! supply. The Pa.«t ami F-itnie 214 Hf lib' I' li"» : f.«: tm-. THEOLOGY and THE SCIENCE of GOVERNMENT. Being a Review of Macaulay's Teaching on that Division of Science, I m t t NATURAL PHILOSOPHY AND DIVINE REVELATION. THE SCIENCE of IDEAL THEOLOGY As Taught bv the Bible. POLITICAL ECONOMY. CENTRIFUGAL FORCE and GRAVITATION. In Six Parts. PARTS. I. — The Central Attractive Force and the Tangential Motion. 2. — ^Thc Doctrine of the Celestial Sphere, and the Theory of Orbital Revolution. 3. — ^The Solar System and the Theory of the Stellar Universe. The mixed doctrine of Parallax and Aberration. • 4. — Comets and Ccmetary Orbits. •• 5.— Theories of Light. The true nature of Light. The Wave-Theory of Sound, and Manifestations of the various forms of Force. 6. — Index and Specification of Contents. THE CIRCLE and STRAIGHT LINE. In Four Parts. PARTS. I. — ^The Geometrical Relationship Demonstrated. 2. — The Construction of the Circle. Quantitive Geometry. 3. — Elements of the Circle. Theory of Curvature. 4. — Analysis of Circle. Decimal Cyclometry. (KUKLOS.) JOBN HARRIS. John Lovsll, montbsal. A. Williams & Co., Boston. Trubner &, Co., London PROSPECTUS OF A WORK ON GENERAL SCIENCE. flON. I Br (KUKLOS,) John Harris. N. : volution. ixed doctrine of ' of Sound, and Co., [tONDON In tho concluding part of the troatiso called ' Centrifugal Force and Gravitation' we have specified tlio common purpose to which ita several parts are related. That treatise, however, is itself intended to form part of a work having a more general purpose. Belonging to this m iro comprehensive work four treatises are now published, namely, 'Centrifugal Force and Gravitation;' ' The Circle and Straight Line ;' * Theology and the Science of Govirnmcnt;' ' Ideal Theology as taught by the Bible.' The general purpose, which tho complete work is intended to serve, is demonstratiori of the true relationship of Theology to Science. This relationship may be stated in two ways, different, .yet in harmony with each other accordingly as tho terms ' Theology ' and 'Science ' are used in their universal and highest sense, or in the restricted sense of systemizcd knowledge belonging to humanity— «. e., of human Tlieology and human Science. Now in tho universal or divine sense, it should be immediately evident that Science (i. c, all science) is included in and belongs to Theology. In the restricted sense, we believe it to be more correct to define tho relationship by stating — that Theology belongs to Science as a part thereof; but (and to demonstrate this is the more general and comprehensive purpose of our work), it (the Science of Theology) is a part of Science absolutely essential to the well-being, to the progress, and, ultimately, to the existence of the whole. It may be pro- perly considered as the all-important centre of coherence and stability, having a relationship to the other divisions of knowledge comparable with that of the Sun to the various membern of the Solur system. If this dependence of tho other departments of Science upon Theology, for the healthy progress and well-being of the whole, be distinctly recognized and appreciated, the supreme importance of endeavoring to preserve that central Science fi*om the admixture of unreason _ able theories, and the contamination of unsound and false conclusions, will be manifest. Having thus stated the general purpose of the Series, we refer to the accompanying tabulated summary for the particular contents of each treatise, and we will, for the present, leave what is now published to the examination of the public, in the expectation that, ere long, the opportunity to developo and carry further the work we have ^undertaken will be forthcoming. MoMTBXAL, October, 1875. THE CIRCLE AMD STRAIGHT LINE INDEX OF CONTENTS. An investigation of the relationship by magnitudinal and quantitive geometry. ■T JOHN HARRIS. Revised^ Corrected and Reprinted, October, 1S75. J-: INDEX TO The Arc of Increasing Curvature 8 The Tiuigentiai Line 13. D 8 Theorem (Fig. 24) 10 I'robiem (Fig. 24) 11 Theorem (Fig. 25) 12 Tho Ultimate Tangent 13 Truth and Cei-tainty. Theology and Science 18 Conclusion 20 APPENDIX. The Complementary arc, and arc of increasing curvature 23 PART FOURTH (SUPPLEMENT.) The Elements of the Circle. Analytical Examination by Fig. 23 R. PAOR The Octuple Arc. The arc of 30 degrees ; and ) . the line B. b j By Fig. 24 R. The Octuple Arc, B. Q 5 By I *!!|- ^l 5; 'Th® Sextuple Arc, B. t 6 Analytical Illustrations. Figs. 4, 5, 6, and Figs. 7, 8. ... 7 Examination of the Fig. 31 8 go.. " " Fig. 32 9 o/ " " Fig. 33 10 Tho arc of 15 degrees ; and the line J. K. N 11 The half-sextuple arc of 18-,V degrees 13 The decimal circle of the duplicated versed sine 15 Anal^'sis by decimal circles 16 APPENDIX. " , Cyclometrical Table 1 19 Do. « II 22 Arc-lengths belonging to the first table 26 49 52 52 53 54 56 PAGB a 8 8 10 11 12 13 18 20 PAOB 1 5 6 7 8 9 10 11 13 15 16 M Twice 19 22 26