IMAGE EVALUATION TEST TARGET (MT-3) ^ ^ / // O % .% 1.0 I.I i;^|28 U, |3|2 12.2 iM 12.0 1.25 ■ 1.4 1.8 1.6 ^. .>^ -> ^W'^^ ^ ^ ^ o ^ / /S r Photographic Sciences Corporation 23 WEST M^IN STREET WEBSTER, N.Y. 14580 (716) 872-4503 CIHM/ICMH Microfiche Series. CIHM/ICMH Collection de microfiches. Canadian Institute for Historical Microreproductions / Institut Canadian de microreproductions histodques C Technical and Bibliographic Notes/Notes techniques et bibliographiques The Institute has attempted to obtain the best original copy available for filming. Features of this copy which may be bibliographically unique, which may alter any of the images in the reproduction, or which may significantly change the usual method of filming, are checked below. D Coloured covers/ Couverture de couleur I I Covers damaged/ D Couverture endommagie Covers restored and/or laminated/ Couverture restaur6e et/ou peliiculAe I I Cover title missing/ Le titre de couverture manque I j Coloured maps/ D n n D Cartes g6ographiques en couleur Coloured ink (i.e. other than blue or black)/ Encre de couleur (i.e. autre que bleuo ou noire) I I Coloured plates and/or illustrations/ Planch3s et/ou illustrations en couleur Bound with other material/ ReM6 avec d'autres documents Tight binding may cause shadows or distortion along interior margin/ La reliure serrde peut causer de I'ombre ou de la distortion le long de la marge intirieure Blank leaves added during restoration may appear within the text. Whenever possible, these have been omitted from filming/ II se peut que certaines pages blanches ajouties lors d'une restauration apparaissent dans le texte, mals, lorsque cela 6tait possible, ces pages n'ont pas 6tA filmias. Additional comments:/ Commentaires supplAmentaires; L'Institut a microfilm^ le meilleur exemplaire qu'il lui a *t6 possible de se procurer Les details de cet exemplaire qui sont peut-Atre uniques du point de vue bibliographique, qui peuvent modifier une image reproduite, ou qui peuvent exiger une modification dans la methods normale de filmage sont indiqu6s ci-dessous. I I Coloured pages/ y D This item is filmed at the reduction ratio checked below/ Ce document est film6 au taux de reduction indiquA ci-dessous. Pages de couleur Pages damaged/ Pages endommagies n Pages restored and/or laminated/ Pages restaur^es et/ou peiiicuiies Pages discoloured, stained or foxed/ Pages dicoiories, tachet6es ou piqudes I I Pages detached/ Pages ddtachies Showthrough/ Transparence Quality of prir Quality in^gaie de I'impression Includes supplementary materit Compreotd du materiel supplAmentsire Only edition available/ Seule Edition disponible I I Showthrough/ I I Quality of print varies/ I I Includes supplementary material/ I — I Only edition available/ Pages wholly or partially obscured by errata slips, tissues, etc., have been refilmed to ensure the best possible image/ Les pages totalement ou partiellement obscurcies par un feuiilet d'errata, une pelure, etc., ont At6 film6es A rtouveau de fa^on A obtenir la meilleure image possible. 10X 14X 18X 22X 26X 30X y 12X 16X 20X 24X 28X 32X The copy filmed here has been reproduced thanks to the generosity of: L'exemplaire film6 fut reproduit grdce d la g6n6rosit6 de: Ills iu difier ine lage Hamilton Public Library The images appearing here are the best quality possible considering the condition and legibility of the original copy and in keeping with the filming contract specifications. Hamilton Public Library Les images suivantes ont 6t6 reproduites avec le plus grand soin, compte tenu de la condition et de la nettet6 de l'exemplaire film6, et en conformitd avec les conditions du contrat de filmage. Original copies in printed paper covers are filmed beginning with the front cover and ending on the last page with a printed or illustrated impres- sion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres- sion, and ending on the last page with a printed or illustrated impression. Les exemplaires originaux dont la couverture en papier est imprimis sent film6s en commenpant par le premier plat et en terminant soit par la dernidre page qui comporte une empreinte d'impression ou d'illustration, soit par le second plat, selon le cas. Tous les autres exemplaires originaux sont film6s en commen9ant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une telle empreinte. The last recorded frame on each microfiche shall contain the symbol —^^ (meaning "CON- TINUED "). or the symbol V (meaning "END"), whichever applies. Un des symboles suivants a^jparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbols — ► signifie "A SUIVRE ", le symbols V signifie "FIN ". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent dtre film6s d des taux de reduction di.drents. Lorsque le document est trop grand pour dtre reproduit en un seul clichd, il est film6 d partir de Tangle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la nidthode. ata slure. 3 I2X 1 2 3 1 2 3 4 6 6 CENTRIFUGAL FORCE AND GRAVITATION Parts, 1. 2. 3. BT ( K U K L O S ) JVUy HARRIS. The Attniotivc Force and Ttinu'entiul Motion. The Planetary axis of rotation, and plane of the I^lcliptic. The neighbouring' Stellar Sy.stem, and Aberration of Light. The Cometary Orbit of Eevoliition. Party J. & II. The Undulatoi-y and other Theories of Light. Part III . Light, and other manifestations of Force. THE CIRCLE AND STRAIGHT LINE, The Geometrical Relationship demokstkated. The Construction of the Ciucm;. Conclusion. I -. I 11^^^^^^^'^^''''^''^ a in '^^l C ^ -OO- AND GRAVITATION fii©^*' — CEITEIFTJGAL FORCE AND GEAVITATIOK ^ LECTURE. Vo\ \ BY JOHN HARRIS. ■^♦-♦-♦^ MONTREAL : JOHN LOVELL, ST. NICHOLAS STEEET. 1873. Entered aeeoraing to Act of rarliamcnt in tlio your one thnnsdml eight liiindrc.'. ftn.l seventy-thrce, by John II.viuus, in the Office of the Minister of Agricultn.o and Statistics at Ottawa. 'nnm MOXTHEAL :— JOHN LOVBLL, Printer. PREFACE. The ' inductive system' as taught by Francis Bacon bases science upon facts. The necessity of correctly observing and of verifying with the most scrupulous care those fun- damental fiicts upon which the various divisions of natural science are (have to be) constructed, is therein inculcated and insisted upon ; the slow and uncertain progress of science when ' undemonstrated theories' and ' suppositious facts' are substituted for ' truths and realities' is pointed out and dwelt upon ; the pernicious effect of accepting conclusions resulting from reasoning not based on truth or certainty as ' scientific' is plainly shown ; and the dan- gers (and the possibly disastrous consequences) of system- ising unsound knowledge and arraying falsehood \n the garb of truth are clearly indicated. The primary purpose of this book is to ascertain whether the rules of the inductive system as laid down and taught by Francis Bacon are still recognized ; that is to say, whether the number of those who do recognise those rules is sufficient to compel their public and gene- ral recognition as constituting, collectively, that law of science which may not be set aside by any human author- .1- iv PREFACE. ity,— that law to which each indivi(hial, if educated and (Hiahfied, has the rigiit (the most important of nil riglits ■which freedom can give) to appeal in his own interest hnd that of others ; and, that law according to which his appeal must be heard, and by which the soundness or unsoundness of his claims, nuist be decided and deter- mined. With this object certain important doctrines now taught as belonging to science are herein contested on the general ground that the specified conclusions so taught as facts are false or unsound ; tlie precise nature of the error being pointed out in eacii case, and the cor- rect or sound explanation set forth. The subjects relate more particularly to what may be termed the Physics of Astronomy. Of these may be instanced : The Tlieory known as tlie (( (( (< (( ' Newtonian law of Gravitation.' ' Kepler's third law.' " " " " ' The Law of Equable Areas.' The Teaching on the subject of the tirles. The present teaching on tliese subjects is challenged, according to the rules of the inductive system, on fact and for cause shown. For example— The distance of the sun (i. e. the approxnnate distance) from the earth, as ascer- tained by parallax, is an astronomical fact j also the moon's distance from the earth, and the periodic times of the moon's orbital revolution round the earth, and of the earth's revolution round the sun, are astronomical facts. The attractive force of gravitation at the surface of the earth as measured by the space fallen through in a de- PREFACE. y finite time by an attracted body, is a fact belonging to physical science. It is asserted for reasons particularly- set forth that the theory known as the Newtonian law of gravitation is not based upon and is in-econcileablo with these facts ; and the theory (alleged to be) correctly based upon these facts and demonstrated by them to bo sound, is stated and explained. Our statements and arguments are put forward in such a form that if erroneous or inconclusive they may readily be shown to be so. If a deficiency as to a proper knowledge of the subject (or subjects) is apparent, that may be easily pointed out. If it is said that we have not in this investigation employed the analytical methods of what are called the higher branches of mathematics, we acknowledge that we have not done so j but we are not bound or called upon by the mles of the inductive system to treat the subject in such a manner, and we are decidedly of opinion that these subjects being of a funda- mental and primary character, such a method of treat- ment would be improper and unsafe. Montreal, 2S^/t June, 1S73. LECTURE. CENTRIFUGAL FORCE AND GRAVITATION. INTRODUCTORY OBSERVATIONS. Before coinmeiiciug my formiil lecture it will be proper to give a brief explanation of tlie particnlar purjiose and of the reasons which I trust will be liekl to justitV me iii coming forward as a ])ublic teaclier or lecturer on a ■scientific subject. It is my intention to mal\t' some general observations eitlier at the close of this lecture or soon afterwards, on obstacles to the progress of science, having particular reference to certain scientific difficul- ties and (piestions in controversy on scientific subjects. I should have preferred to defer all explanation until that time, and to connuenco at once upon the particular sub- ject of the lecture, but so doing would certainly expose me to the risk of a prejudice behig formed in your mhids, antagonistic (and possibly very strongly antagonistic) to my purpose as an instructor or lecturer. Under the cir- cumstances this could be only a temporary obstacle, but an unfavorable prejudice is in itself an obstacle of a kind which I am aware that a teacher should by no means despise oi underrate ; and, as the subject has already suffi- cient difficulties of its own, it is requisite for me to guard, as carefuUy as I can, from uimecessarily increasing them. The subject of my lecture (Centrifugal Force and Gravi- tation) may be considered to belong more directly and 8 INTRODUCTORY OBSERVATIONS. particularly to that division of natural science called media- nics, or meclianical philosophy ; but also, since it embraces those laws by which the motions and relative positions of the planetary bodies are determined and regulated, it enters largely into that division tenned astronomical science. The express puqiose of my lecture is to call your particular attention to and clearly explain, so for os may be necessary, the teaching on this subject at present con- sidered scientific (i. e. scientifically orthodox), in order to object, on scientific grounds and for reasons which will be particularly stated, to certain parts of tliat teaching. The fomi of the argument will be to give first that which I assert to be the correct and sound teaching on each of those parts of the subject, and then to compare these with the present authorised explanations (i.e. the teaching now recognised as scientific) in order to show and bring dis- tinctly under your consideration the particidars against which my arguments will be directed, as being erro- neous and as belonging therefore to unsound science. In making the few preliminaiy remarks which seem to be necessary I am anxious to avoid making any state- ment or assuming anything which may even appear to be objectionable or open to dispute. For the moment, it may seem that I aiu at least risking something in sucli a sense by proposing to assume that there is in the minds of a great many educated people a conventional, loose, unjustifiable and incorrect meaning attached to the word ' Science,' and therefore also to such compound temis as 'scientific teaching' and ' scientific authority.' I feel sure fhat I may safely go further than this, and still for the mo- ment only risk dissent, in assuming that this conventional and incorrect meaning is not confined to the minds of per- sons of good general education, but also finds a place in very many minds which have had the advantage of what is termed a special scientific training — in the minds of men who it may be are quite qualified (possibly much better quahfied than myself) to give a correct definition of, and INTRODUCTORY OBSERVATIONS. 9 therefore to attach a correct meaning to the e^^pression ' Science/ but who in fact go on retaining and subjecting themselves to the influence of a more or less indefinite and incorrect meaning attached to that expression. ' Science ' is not a new word — it has been in use for at least several centuries, and lias been used all that time to convey the same general idea or meaning. TKe way in which the term has been and still is used, may be des- cribed as a definite word used indefinitely; or, to amplify this a little, a word or expression which is essentially definite and discriminating used indefinitely and for the most part so loosely as to be allowed to include things not only dissimilar but even such as are opposed to each other, in the sense that truth is opposed to untruth, or as right is opposed to wrong. Let us go back only one cen- tury • and, with om- present advantages and better means of forming a con-ect judgment, reflect whether all the knowledge which was at that time considered or classed as scientific knowledge was correctly entitled to be so classed or considered. It is beyond question and dispute that opinions, judgments and conclusions were held by and formed a part of the so-called scientific knowledge ot the men of science of that day which at the present time would be unhesitatingly condemned by any educated person as bring certainly erroneous ; and this misapplica- tion of the word was not peculiar to that or to any other one period of the world's pasi; history; nor has it ceased to be still used in esseudaUy the same way, that is mis- applied ; a source of mischief and confusion resulting from a tacit agreement to deceive ourselves and to call a thing that whicli every one qualified to judge knows that in fact it is not ; namely, t • call a collection of knowledge, some of which is undoubtedly true, some of which is almost certainly or very probably true, some of which is doubtful (i.e. possibly true and possibly untrue), and some of which is certainly untrue, (because no sane sci- entifically educated man will assert that all which we i 10 INTRODUCTORY OBSERVATIONS. now call science contains no errors, no doubtful theories, or merely plausible assumptions held as possible truths until more certain knowledge can be obtained) — to call such a collection collectively by the name science, a word expressly meaning true knowledge as distinguished, not only from unsound knowledge, but also from knowledge not known to be certainly true. Tlie mistake may bo partially rectified by ado^ating some such expression as * autliorised science ' or ' classified science,' and defining this to include, together with the scientific knowledge, a cer- tain indefinite amount not certainly known to be sound; but to apply even such a modified expression correctly, a careful re-classification and separation would be necessary; because the present collection, known as science, includes also tlie ffreat antagonist of science — that evil and false knowledge whicli, assuming a systematic and apjiarently scientific form, and entering in unnoticed, mingles with and contaminates knowledge otherwise wliolesome and true ; and whicli liaving insidiously and firmly esta))lis]itMl itself in tlie educational strongiiold of civilization displays that hatred of definite and true knowledce — that orsxan- ized opposition to real progress,and that skilful and un<;eas- ing endeavor to darken, confound and destroy the human intellect which has ever been the characteristic of ' unsound science ' : It may be remarked : well, — as to all this, — there may be something in wliat you've ])een saying in a philoso- phic sense, a sort of abstract trutli perhaps ; Init it can't be practically a matter of any pa'-*^'"- lar conseijuence, and can scarcely be considered as more than a sort of quibble about the meaning of a word. Feeling sure that an inevitable controversy is impending upon this particular question — a controversy that will not be confined to a few disputants — not to any one par- ticular locality, nor even to any one nationality, Imt — a controversy which will become wide-spread,into which all those belonging to the educated world will be drawn, and INTRODUCTORY OBSERVATIONS. 11 in which each one will have to choose his side and take u part — a controversy which will become a conflict of the most uncompromising character, a verbal battle asto which even those who most love peace and detest discord, wUl agree with tliose to whom dispute and strife are less un- welcome, that it must be fought, and must be fought out, until the one party or the other is completely vanquished and subdued; — with the conviction that suclia controversy is at hand, if indeed it may not be said to have already commenced, it would be unadvisable to enter now into an argument the merits of which could not be fairly stated without occupying a good deal of time. Intend- ing, however, as already stated, to make some remarks ' u])ou obstacles to the progress of science ' at the close of this lecture I may then perha])s have something further to say upon this particular sulyect. Meanwhile I will suppose the foregoing statement or some other erpiivalent statement to have Ijcen nifide; namely, to the effect — that it is not of })ractical importance to discriminate, and to de- fine tlie difference between 'science' {i.e. sound science) and ' luisound science '; and I w ill for tlie present content myself with asserting the directly contrary, namely — that it is of practical importance so to discriminate — that it is, in an educational sense, of an importance which can be neither oventated nor overrated, to define the difference and to distinguish between 'science' and 'unsound science.' Before conmiencing my formal lecture, I )jeg to state for the benefit of those who take an interest in such sub- jects, that it will be continued to-morrow evening, as there will not remain much more than sufficient time this evening to make a connnencement and to indicate the particular form of the argument. Although I may venture to say that the statements and explanations of my lecture will be conveyed in the most simple and intelligible fonn that scientific treatment of the subject will justify, it cannot be expected that those who have 12 CENTRIFUGAL FORCE AND GRAVITATION. not some previous knowledge of the subject will be able to follow with much interest the reasoning and conclu- sions in an argument which will mainly consist in con- trasting and distinguishing between cases essentially dis- similar, but wliich to the scientifically iminstructed must appear very much alike. I will therefore now take the opportunity to express my personal thanks to those who may this evening have attended in consequence of parti- cular invitation ; the lecture itself having commenced those who remain or return will then do so knowing what they have to expect. Of the remarks with which it is intended to conclude the lecture, and which may per- haps be sufficient for an evening to themselves, due notice will be ffiven. CENTRIFUGAL FORCE AND GRAVITATION. We will first take three cases for consideration in which a body revolving round a centre (a central body or cen- tral point) is subjected to conditions differing as stated. Case 1. — A body attached to a central fixed point revolving in a circle. Case 2. — A body revolving in a circle with a definite unifonn velocity around a central body, the two bodies being subject to the influence of gravitation. Case 3. — A body retained by and subject to the influ- ence of gravitation, revolving around a central body, the distance of the moving from the central body, being detennined by the relative proportions of the velocity and the amount of gravitating force. CENTRIF.GAL FORCE AND GRAVITATION. 13 Case (1) may be conveniently subdivided into the case (a) where the revolving body is attached to the fixed central point by a non-elastic line or hnk ; and the case (b) when attaclied by an elastic line or by a line connected with a sprijig sucli as to allow of limited extension when acted upon by the centrifugal force of the revolving body, (a). Fig. 1. Let a weigiit A be attached by a string to a central point C, around f^ ^ ^^H which it revolves, the ,^- motion of A, from the y* point A, in the direction AD, would carry it to the point D, in a definite time (t). But the motion of A being restrained by the attachment to the central point, is com- pelled to take place through AE. To what distance from circle will the weight A move in the definite time (t) in which it would, if unrestrained, have moved to the point D, on the straight line AD I Let a point B on the circle AE be the point at which A will arrive in the time (t). Is AB greater or less than, or equal to AD ? If greater, then is the motion accelerated ; if less, it is retarded ; and if equal to AD, then is the motion neither accelerated nor retarded by the action of the string. As there is no ground whatever for assuming that any motive or active force can develope itself from the central point or out of the string, there is no accelerating cause ; no extraneous or additional force from which additional motive power or velocity can be derived, therefore AB cannot be greater than AD. Is the motion of A, (lea\ang out of consideration friction and so forth) retarded by the restraining action of the string ? Let FA on the straight line FAD be the direction in which the moving weight A has arrived at the point A, on the tangential straight fm arc of the point the circle A on the u CENTRIFUGAL FORCE AND GRAVITATION. line FAD. Now if the angle FAC were greater than a 1 ight angle in any degree (e. g. fAC), it must be admitted that the string could not then retard the motion of A iii tlie direction fAd. If on the contrary tlie angle DAG were greater than a right angle in any degree (e.g.) 'dAC, it must be admitted that the string would then retard the motion of A in ihn (hrection TAti. But the tangent is necessarily at every point in the circle at right angles to the radius ; where, then can any re- tarding effect commence f in what does it consist ? or ichat influence is there to which it can be attributed ? If we can suppose tlie radius AC, or the string repre- sented by it, to be of infinite length ; then there would be no restraint, and A would move in an absolutely straight line. Or if we suj>|iose the radius AC to be of indefinitely great length ; the deviation of A's motion from a straight line through any given space will be inde- finitely small, nevertheless every point and every part in the line wall represent a point and apart, in the arc of a circle, and if there be no retarding force at any one point or any one part, neither can there be at any other point or part, because thi.'t would suppose a dissimilarity in the motions 01- relative p(,sitions of the parts contrary to the condi- tions of the case under corisideration. It may be argued that (1) where there is no retardation, as in the supposed instance of an infinite radius, there is also no restraint ; but that (2) when the motion is restrained and caused to de\'iate from the straight line into the arc there is retardation, and that the force causing or resulting from such retardation is that known as centrifugal force, and is represented or measured by the tension of the string. In order to examine this objection we will now take the subdivision of the case (b), and suppose an elastic spring to form a part of the connecting line by which the weight A is attached to the central point C, (Fig 2.) The weight A is assumed to start from the point A, at the same dis- tance from the centre and to move with the same velocity CENTRIFUGAL FORCE AND GRAVITATION. 15 as before, which would cause it to arrive at D in the time (t) ; the effect will now be that the tendency of the weight A to move in the straight line will, in the firet instance, be only partially restrained by the string, because the eliistic spring, connected therewith, will be extended and allow the weight A to move outside the circle ABE. ■\Ve will suppose this extension, which will be limited to a certain short time, sufficient to allow the weight A to aiTive at the point 1, and afterwards to arrive at the point II, when the elastic o force of the spring reacting towards the '^ ^.,-^^P^^s^j^ ^ centre C, may be sup- posed to become e(|ual to the centri- fugal force of the weight, and equality l)eing established, the weight A will con- tinue to move in the J»7 greater circle of wliich the arc HK forais a part. Now, if we take any point 1, at which A arrives when the clastic spring has been partially extended, then since the weight A has moved from the point A, in the curve Al, and A is a point on the circle ABE, and the curve Al is between that circle and the tangential straight line AD, the direction of the motion of the weight A at the point 1 will be the tangent to the curve Al, and which may be represented by the line mln. Join CI, and through the point i, where CI cuts the circle, draw i\iQ tangent o. p. Evidently the angle Cln is greater than aright angle (for the angle Cip is a right angle and Cln is greater than Cip,) and consequently it must be admitted that the string CI retards the motion (i. c. diminishes the velocity) of the mov- ing weight A. When the weight A haa arrived at H, then since the centrifugal force is unable to extend the spring any further, and the reactionary force of the extended 16 CENTRIFUGAL FOKCE AND GRAVITATION. pprinc: is unable to overcome the centrifiiijiil force, tlie coiulitions of the case now become isimihir to those of the subui', ision (a), with the (Utrerence, however, that the weiirht A will now revolve in the greater circle of which UK is an arc, and will move tlierein with a velocity less than tlie velocity which it had wlien leaving the point A (if. less than the velocity wiiich would enable it to travel from A to D in the time (t). l^it the spring which fonus part of the line connecting the weight A with the central point C is now extended ; and the force employed in extending the spring is ecpiivalent to the loss in velocity of the revolving weight A. Case (2). Under the conditions of this case, the cen- trifugal and centripetal forces being e(pud (/. r. the central gravitating force being counter-balanced by the tendency of the body A to move along the tangential straight line), the motion of the body A will be restrained from deviat- ing out of the circular orbit of revolution, and the velocity of revolution will be neither accelerated nor retarded. The body A (Fig 3) will arrive^ at the point B on the circle in the same time it would otherwise have arrived at the point D on the straight line, therefore AB = AD. Case (3). The conditions be- longing to this case will be includ- ed if we suppose that the central gravitating force is in the first place more than sufficient, to restrain the moving body from increasing the distance between itself and the central body ; the superior gravitat- ing force therefore causing the revolving body A to deviate from the circle and to approach the central body. The body A (Fig 4) is moving at the point A, in the direction FAD, with the same velocity as before, and would arrive at D in the time (t). By case (2), if the gravitating force acting towards the centre was equal to CENTUIFUGAL FOKCK AND ORAVITATION. 17 th(! contrifugal ion;e, the body A would move in the arc A f B of a circle ; rc^taiii- ing always the same distan- ce from C as it had w'len leavinir the point A ; bnt the gr;i\ iiating force is now greater than equal to the centrifugal force, and conse- quently A, when it has moved through half the time (>) lias been caused by the sujierior gravitating force to ap- proach C, by the space contained between the point f on the circle and tlie point on the curve Adm. at which tlie line fC cuts that curve (as this point would be very near to d, we may denote the space for illustration as fd.) The space fd will therefore represent additional motion imparted to the body A, and which, being compounded with the motion in tlie circle, will cause A to move with increased velocity, and to arrive at the point d in the same time (I) in which it would have amved at e, on the line FAD, (Ae being the half of AD, and Ad being greater than Ae), but iri moving through the remaining half of the time {-) the body A is caused to continually approach the central body, and at the end of the time (t) A arrives at ra ; the distance Am on the curve being greater than the distance AB on the circle which is eqtial to the distance AD on the line FAD ; and the distance dm, is greater than the distance Ad, because the motion is thus far con- timially accelerated, {l. c. so long as the moving body actu- ally approaches the centre of gravitation.) The difference by which the space Am on the curve, is greater than the space AB on the circle, measures the additional motion which has been imparted to the moving body by tlie supe- rior gravitating force, and it represents the quantity of motion (or space) by which the body A has approached the centre. When the body A arrives at the point E IS CENTRlFUfSAL FORCK AND nUAVITATlON. opposito TO A (at the opposite oxtroinity of tlieiiinjor axis oi'tlio ollipsr) A lian approacluMl the central body by the d\(\vrt. ace between tlie ihstaiices AC and EC ; and if A were suppose(^ to l)e at rest (deprived of antjnlar ;notion) at each of these (Ustances from C, viz., AC and EC, then wonM the irraviratintl influence bo inversely proportionate to the irreaterand tlu' lesser distance viz., inversely as AC : EC. * I>ut at the point E tlie angular velocity of the l)ody A iMOvinjf in tlie direction FED is considerably iireaterthan at the opposite point A ; and the areal velocity is also greater, /. c. the distance to which A would move from E in the direction FED in the time (t) would be irreater than AD. Since the irravitatiniy influence is uniform at any siiven distance, and since it is contimious, the amount of force exerted beinu in^' liy or revulviu';' arounnce falling towards the central body ; and as the orbital motion proceeds, to continue falling witli accel- erated s]ieed according to the law of gravitation apjdied to falling bodies. The analysis is soTuetimes shewn I)} draw- ing a line from that })oint arrived at by the moving body in the time (t) in tlie »'ircle or orbit of revolution, (/. r. by drawing the sine of the arc) at right angles to AC, and then taking the distance of the point where the line •iO drawn cuts the radius AC from the point A ; that distance is then stated to be the space through which the moving body has fallen towards the centre whilst moving through that portion of its orbit, thus Ad', on the line AC, would be considered the sj)ace through which the moving body A had fallen towards the centre C in moving through the arc or curvilinear fraction of the orbit Ad. Similarly Ab', Ac', would be considered as spaces fallen through by the moving body in travelling from A to b, and A to c CENTKIFUGAL FOKCK AND CiRAVTATION. •31 respectiw'ly, in tlu' orbit. An jipparciitly ^«inlilar, but not. equivalenr,niO(UM>t\'xpl;inationisl)y(Ini\vinij;tlu' tangential line Al> and deriving the space moved through in the orbit tioni the assumed motion in the direction AI.) by drawing a pcrpendii idiir iVoni the assuuieil point of arrival on rli:it line AD to the arc or curve representing the orbit, the perpendicular is then considered to be the space tidlen iln\»ugh towards the centre ot" gravitation, and the niotio'.i in the orbit or arc is supposed to be com- pounucd ot'ihe pi'r[)endicular motion towards the centre and the horizontal motion in the directi(ui of the tangent. Another way in whii'h the same (last) explanation is varied, analyses die composiiion of the orbital motion by subdivision ol'tlie ciirve into shorter s['.a('es ; instead ol drawing tlie one tangential line AD from A, tangents repri'sentini; e(|ual divisional periods of time are drawn one to eacii of tl^e lVaeli(.»nal curves which constitute the divisional spaces of the orbit. Thus (Fig. ■') a), if Ad re- present tliL- whole time (>) ; then Ab— 3,- Ac^-o :Ad--Tr =(t). From the terminal points of each of these divisional curves, the tangent is drawn representing the space through which the body would pass in the next etpnd division of time ; then from the ter- minal point of this tan- gent the perpendicular is drawn cutting the curve of the ellipse ; and from which point again the next tangent is drawn. In fig. 5 (a) the perpendiculars b, c, d, would be considered the falling motions belonging to the curves, and this perpendicidar motion compounded with the tangential is supposed to produce the curvilinear motion through that iipace. Hence the perpendicular c is greater than b ; and ^ ss CENTRIFUGAL FORCE AND GRAVITATION. CC d is greater than c | also the curve 1) c, is greate than Ab ; and cd greater than b c. The erroneous assumption on which these explanations are based leads (1) to a false conclrsion as to the quantity of motion imparted to the moving body by the gravitating force, and (2) to an unsound inference as to the composition or nature of the gravitating force itself. In fig 6, the case illustra- ted is that of a body moving around a central body with a velocity such that the centri- fugal force equals the attractive force, and consequently the mov- ing body can neither approach nor recede from the central body. The case is essentially similar to that of Case 1 (a), where the moving body was attached to the central point by an inelastic string or link, and in which it was shown that tlio velocity of the moving body was neither increased nor di- minished by the guiding or restraining effect of the string. (Note. It has been explained in a previous note that an oscillatinc effect of approximation and recession takes place in the case of a body revolving around a centre of gravitation as above. The connection witli the central point is here considered as equivalent to a rigid inelastic medium, whereas the radius-vector in this case (that of gravitation) may be correctly considered as extremely elastic •, this difference, however, does not in tlie least affect the argument here put forward ; ant] it will, for the moment, suffice to observe that this elasticity is also discarded by those teacliers, the correctness of whose theories and conclusions we are here disputing.) The illustrations and explanations given in the Reference from the works of Whewell and Lardner show that in the case of the moving body revolving in a circle, the erroneous assumption leads hnmediately to the conclu- sion that the velocity of the moving body is increased CENTRIFUGAL FORCE AND GRAVITATION. 23 l)y the gravitating force, l)iit which is contrary to fact, and ic is indeed manifestly impossible that any acceleration of motion can take jilace in the manner so supposed. The unsound inference as to the nature of the gravitating force may be illustrated by Fig. 5(a) (page 21) which belongs to a case similar to that I)reviously investigated in Case (3), viz., that wherein the centrifugal force of the moving body is less than the gravitating force, and in which consenuently the movino' body is made during a part of its revolution to approach the central body, and it was shown that in sucli a case the velocity of tlie moving l)Oi]y is increiised. In Fig. o(ii), for example, the moving body will travel througli tlie distance Ad of the curve, in the same time (t) in v. hich it would have moved from A to I) In tlio circh', and Ad is greater tluui Al>. T!iis neceleration ai-ises from the actual additional motion liy wliicli the i>!Ovinii- bodv approiiches the centre, and whicli isoompoun'k'd with tlie motion in the direction of the circle. Th;' spaces Ab, be, cd, in tlie orl)ital curve repr(\-*ent spaces sue- cessively passed throutrh by tlie moving body in e([ual increments of tinu^, I.)ut the spaces are unequal l)ecause, since the motion is constantly accelerated from thi> pohit A, each successive space is greater tlian that preceding it. Now if we proc(>ed I)y tlie usual method to invLStigate the gravitatina' foi'ce to which tlie moving ))ody has been subjected whilst moving tiirouuh these successive spaces, it will be made to app(>ar that the amount of gnivitating force exerted is greater in each successive space in the same proportion that the space itseU" is greater tlun the space preceding it ; for instance, in the same Fiu. o(a) : if a tangent to the curve b c, be drawn from the extremity b, and a tangent to the curve c d, be drawn from the extremity c ; and a perpendicular to each of the tangents be drawn, then according to usual teaching these per- pendiculars represent the efiective gravitating force ex- erted throughout each space respectively, and the perpen- S4 CENTRIFUGAL FORCE AND GRAVITATION". dicular d is greater than c, and c greater than b, in tlie same proportion that the space cd is greater than the space be, and tue sp«ice be greater than the space Ah. But the gravitating force is the resultant of tlie gravi- tating influence and tlie time during v^'hich the influence is exerted ; and the time represented by each of these spaces is tlie same. It is true that since the distance of the moving body from the central body is slightly lessened, the intensity of the gravitating influence is incrr^ased proportionately to the space by which this distance is diminislied, but the increase thus arising is extremely small in amount and not nearly equivalent to the diirerence inferred from the erroneous assumption in question*. The unsound inference as to the gravitating force, and therefore as to the composition of the motion, arises out of the assumption that the revolving body moves tangcntially to the curve. In the instance last given. (Fig. da) the curve Ab and the curve be, would be, if the velocity of the moving body were sucli that the centrifugal and gravitating forces w^ere equal, the equal arcs of a circle, and of which arcs the perpendiculars b, C) and d, would be equal. Now if an additional impulse were to be given to the moving body at the commencement of the second cur\'e, and the velocity be thereby increased, then would the second be greater than the first arc, and the perpendicular of the second proportionally greater than the perpendicular of the first ; the gravitating force exerted during the progress of the moving body through each of the arcs respectively would have been the same, becaui=;e the time was the same, and the distance of the moving body from the centre of gravitation was through- * Note Tliim actual increaHC in iuteneitj', small as it is in amount ought not to be taken as a deduction I'rom tlie error, since it is, properlv considered, quite distinct. Il' the revolution is in a circular orbit, or even if the deviation is to a curve outside the circle when the intensity would in fact decrease, the same erroneous method would still lead to the supposition of an increasing gravitating force for epch successive division of equal time. CENTRIFUGAL FORCE AND GRAVITATION. 25 out the entire distance the same. Nevertheless the per- pendicular of the second would be greater than that of the first, and this difference would, by the usual method of illustration, be made to demonstrate a greater amount of gravitating force exerted. The correct explanation may be thus stated. During the progress of the moving body through the first curve its velocity has been increased by the additional motion derived from the approach of the body towards the centre, the quantity of which motion is measured by the deviation of the curve from the arc of a circle ; the length of the curve is dependent upon the velocity, because the curve represents the space moved over in the constant and definite time, and the length of the curve throughout must have been greater tlian the length of the arc would have been. The second curve commences with the increased velocity with which the first terminated, and if the distance of the moving body from the centre remained the same throughout the second cui-ve, that cui*ve would be an arc of a length dependent upon the initial velocity, which veloci y would in tliat case remain the same through- out the cun^e ; but the gravitating influence is supposed to be still superior to the centrifugal tendency, and conse- quently an additional motion of the moving body towards the centre is again compounded with the motion of the body in the orbital curve of revolution, and a further increase of velocity and a further deviation of the orbital path from tlie arc of a circle to the elliptical curve is the con- sequence ; as before, the deviation measures the motion towards the centre, and the increased length of the curve exhibits tlie increased velocity, or as it may be called the accekM-ated compound motion of orbital revolution. 36 CENTIUl'LGAL FORCE AND GRAVITATION. REFERENCES. WJictvt'Ws Mechanics, page 1-39 (Coiitrilugiil I'urce.) (107). "If u body is made to describe a circle with a iiniforni velocity it must be acted upon by a force tend- ing towards the centre of the circle," &c., «S:c. lOS. " Prop. When a body describes a circle with a iniifonn velocity, and is retained in its path by a force tending to the centre, tliis force is represented by the square of the velocity divided by the radius. — Let v be the velocity, r the railius, f the force which acts towards the centre. Let t be the small time ill which the body, not acted uj)on by the central I'orce, would des- cribe the small portion AD of the tangent ; and let D15 be the di'tk-ction l»y which tiie body is brought to B. Hence at the hniit AD= vt. BD-ilt2 . But if AE be the diameter, the triangles EAB, ABD are similar; for the angles ADB, ABE are right angles, and EAB, AJiD are equal. ilence EA : AB :: AB : BD; .... therefore EA X BD = AB X 1 AB ; and at the limit BD -ft AB- V t lu'uce •2x X lft2 = (vt) - . . . Therefore f = 1' 2 ^ ' r This explanation (or demonstration) contains the re- markable substitution (as above) of AB for AD, an assimiption apparently that AB and AD are equal. Since it is obvious on mere inspection that AB is greater than AD, the idea suggests itself that this substitution must be an oversight or misprint ; but on examination it appears to be an error of a different kind, viz., the state- ment as an axiom or fact of an unsupported assumption which is not manifest and is apparently unreasonable. It CENTUIFUGAL F014CE AND GRAVITATION. 27 tippears that the stateinont has hecu made under an impression that AD eonn>ounded vvitii DB results in AB, (viz., that A arrives at B iu the same time it would if not compounded with DB, have arrived at D). But not even a reference to such a supposition (or demonstration) could justify in tiiis place the substitution of ABfor AD, since here the object of the reasoning and the illustration is particidarly to ascertain and demonstrate the value of AB, and its relation to that of AD, and of other quanti- ties. (Observation. To avoid misunderstanding it may l)e well to observe tliat the above calculation would be in itself correct, if AB, a part of the circle, were taken in the first instance as the space moved through in the definite thne (denoted by t). But AD, representing the tangential motion Irom the point A must be made equal to AB, the arc ; and consequently DB will not be a (per- ^lendicular) straight line but a curve, and will be greater than DB shown in the figure.) To show that this unsupported assumption is not con- fined to Whewell's treatise but is at the present time a part of the recogniscdscientific teaching on the subject, we refer to Lardner's Mechanical Pliilosophy, page 147, (tig. 03) : ''Let P be the fixed point to which the string is attached. Let A be tlie ball, and let ACF be the circle in which the ball is whirled round. Let AC be the small arc of this circle moved over in a given interval of time. Startina: from A the motion of tiie ball has the direction of the tangent AD to the circle, and it would move from A to D in the given interval of time, if it were not deflected from the rectili- near course; but it is deflected into the diagonal AC, and this diagonal by the composition of forces is equivalent to two for^ "s represented by the sides AD, AB. But the motion AD, is that which the body would have in virtue 28 CENTRIFUGAL FOUCE AND GRAVITATION. of its inertia, and therefore the forec AB directed towards tlie point P is that which is impressed npon it by the tension of the string, and which, combined with the motion AD causes it to move in the diagonal AC." Here we find the same unsupported assumption in a somewhat (hflerent form. The arc AC is tal^en as re- presenting the space moved over in a given interval of time by A, which is attached by a string to the central point P ; it is then stated tliat if the motion of A were not deflected by the attachment to the central point, A would move (i. e. would have moved) in the same inter- val of time to D. But AD is less than AC, therefore the velocity (in the deflected movement to C) has been increased, and this increase in the velocity is distinctly attributed to the tension of the stnng causing a motion in the direction of the centre P, and this motion repre- sented by AB is assumed to compound itself with the motion AD and to result in AC. If it be granted for a moment that such assumption may be true, it would immediately follow that the case must be one of uniform- ly accelerated motion increasing from A throughout every divisional part of AC, and the velocity of A, when it arrives at C, must be accordingly greater than when it passed the point at A, and so continue to increase throughout the circle (i. e. throughout every part of each successive revolution). This obvious corollary is appa- rently cpiite overlooked. (Observation. To siibstantiate the correctness of that teaching now recognised as soxuid, or in other words to demonstrate the conclusion thus an-ived at by Whewell and Lardner, it would be necessaiy to adopt as a postulate, or to demonstrate in the first instance, that if a force act continously on a moving body at right angles to the direction in which the body is moving, such force, by compounding itself or its efliect with the motion of the body, produces accelerated motion in the body, (in- creased velocity.) Where is demonstration on this point to be found 1 Has any one even ventured directly and positively to assert such a proposition ? THE LAW OF GRAVITATION. The Nt'wtoiiiiiu tlieory of jfruvitation assumes tluit the inteu.sity of the attriietive tbrce (gmvitiiting intluence) varies inversely as I lie 8([uure of the (listuiice, iuereasing as the sijuare of the distance decreases, and vice versa. Is such assumption supported by the facts ? (Fig. 7.) Let C be the centre of gravitation, B a body moving in the direction I'Bd at the distance CB from C, and let the velocity (v) be such as would carry B in the time (t) to d, at the distance Bd, and let the gi'avi- tating force 1.^ et^ual to the j^ ^ _ ei J> centrifugaljfore t- ;- B will there- ^> fore move in tlie arc of a circle, and will arrive at m in the same time it would have taken to arrive at d in the tangential direction TBd. Now let us suppose the intensity of the gravitating influence acting from the centre reduced to one half; and the velocity with which B moves from the point B, in the direction Bd, also reduced to one half: it is evident, since the centrifugal force (the mass and density of the body and the radial distance remaining the same) is directly dependent u^ion and proportionate to the velocity, that by reducing tlie velo- city, the centrifugal force is also reduced to the one- half; but previously the centrifugal force and the gravi- tating force were equal, therefore they must be still equal, because the half of the one must be equal to the half of the other, and in the same time (t) the body B will have arrived at a point p in the arc of the circle at half the distance of the point m from B in the same circle. We will next consider the nature of the centrifugal force. Shice if the body B was allowed to move straight fore- 80 THE LAW (»K 015AVITATI0N. ward in the direction of the tiniiiont Dd there would be no eentriliigid force, it is evi(h'nt that the centrifiigal force is in the first phice a conseijuent of tlie conipeUed deviation from tlie straight line into the arc. of the circle, caused by the restraining or guverning influence acting upon B from the centre. And again, since in every point or pan of tiie circle the tangent is at riglit angles to the radins, and the relationship of the tangent to the arc which it touches is always the same, it is evident that the auujunt of the centrifugal force is also dependent upon and directly proportionate to the velocity or speed of the moving body ; for any delinite amonnt of space moved through by the body in a given time is productive of a definite amount of centrifugal force, and if the velocity be doubled, then doubUi the space is moved through in the same time; if (juadrupled, then four times the space is moved through in the same time, and the amount of centrifugal force developed is necessarily doubled or (pia- (Irupled accordingly. In order the more readily to appreciate the effects con- sequent upon variations in the proportions of the opposing forces respectively, and in the distance at which their influence acts upon the moving body, it is desirable to examine briefly ' the relation of the enlarged circle to the lengthened radius.' (Fig. S.) With centre A and radius Ab describe the arc be ; double the length of the radius through b, and with radius A'b, describe the arc 'be' ; double the length of the ^ radius through 'b, and with ra- dius AB describe the arc BC. We have now three arcs sub- tending tlie same angle, and conserpiently similar arcs, that is to say similar fractions of the circles to which they respec- tively belong ; and the three arcs are related to each other in such « jy=- THE LAW OF GKAVITATION. 31 wise that one half the arc 'b 'c, (i. e. 'b M or ]\I c) is eciual in length to thi.' first arc be ; and again one half the arc BC (i. e. BN or NC) is equal in lengtli to the arc be, and also ont; fourth the arc BC (i. e. BO or ON) is equal in length to the first arc be. And further it should be observed that since the arcs are similar fractions of their respective circles, that they therefore contain equal amounts of curvrture, and since the first arc is half the longtli of the Sv^co id arc, the first arc contains proportionately twice the amount of curva- ture ; {i!id again, since the second arc is half the length of the third arc, the second contains proportionally double the curvature contained by the third ; and the first arc contains four tiines the curvature contained in the third arc. Hence if one half the second arc as bi\I were to be increased to double the length without increasing tlie curvature, it would become converted into an arc ))elonir-' ing to the larger circle of wliich the third arc BC is a fraction ; and the half arc 'b M so in^^reased to double the length would be then similar and equal to half of the arc BC. Returning to Fig 7. we will now suppose the moving body to be removed to twice the distance from the centre, and let CA be the radius-vector or distance so duplicat- ed. If with the distance CA we describe an arc An, intercepted at n, by a radial line drawn through the tenninal point of the arc Bm, the arc An so described is necessarily twice the length of Bm, and contains the same amount of cuiTature, and consequently, if we bisect the arc An we then have an arc AM equal in length to the arc Bm, and containing half the curvatu'-e comtained in the arc Bm. Since the moving body is now twice the distance from the centre of gravitation, the restraining force, on the supposition that the intensity of the influence decreases in some inverse proportion to the distance, will be now less. What proportion does the decrease in the intensity of the gravitating influence bear to the increase 32 i:»E LAW OF fSRAVITATION. li ,\ of tlic et ns su|>pose the decrease of intensiiy in the gravitating inllnence to he in sim- ple inverse ])roportion to the distance, that is to decrease just "^o much as the distance incretises ; then, since tlie distance has been doubled, the intensity of the inlliieiice will be now the one half of that to which the movinu body wa» snbjecteil at its former jtlace, vi/.., at U. The velocity of the movinir body is (i»y the sup|Mjsiiion) the same as before, and it will conserpiently in the same time (t) move to a distance from A equal to the distance to which it previously moved from 15 (because there is no obstacle in either case to retard the motion). By the sup- position the intensity of the gravitating influence is the one-half iind the space moved through in thesairie time is as before : evidently therefore the deviation will be now the one half of what the deviation was before, and the curve moved through wUl still be the arc of a circle, because an arc described with tluMlistance twice as great will give the rerpiired deviation, and the point at which the moving body will now arrive will be the terminal point M, of the arc AM, equal in leng"l\ to the arc Bm, but containing only one half the C(;rvatiire con- tained in that arc; that is, only half the deviation from the tangential line. ( ;^^ = dM.) But the assumption of Newton's Theory of Gravitation is that the intensity of the gravitating influence (attractive force) varies inversely as the square of the distance -, that ii i THE LAW OF GRAVITATION. 33 is, — the distance being doubled, the intensity of the influence will be reduced to one-fourth. Now, if we test this assumption by considering what, if it were true, would actually take place under such conditions as those just now supposed, i. becomes at once apparent that the moving body could no longer move in the arc of a circle, because the intensity of the influence being only the one- fbuith, and the space moved tiirough in the same time beinu: the same as before, the deviation from the tangen- tial line (direct line of motion) will be (»nly the one fourth as great as before, cons«M|uently the point at which the moving body arrives will be the point K (Fig. 9),* half the distance between the point d, and the point M ; and tiio curve AK, very nearly equalt in length to AM, would be tiie curve rijircsi'iiling the path of tiie mov- ing body; on the assumption that the intensity of the force decreases inversely as the square of tlie dis- tance. It follows tliat, if such assumption were true, the moving body would continuously recede to an indefinite distance from tlie centre of gravitation. (The orbit of revolution would become a IF spiral or helix with the curve increasing outwards.) What are the facts whicli are supposed to support the assumption ? Taking for example the case of the eartli in its revolution round the sun ; it is known that the orbit in which the earth revolves is ellii^ti- cal, and therefore when the earth is in that part of its orbit at the least distance from the sun, tliat is, at one extremity of the major axis of tlie ellipse, called the perihelion, we have an instance wherein * Fig. 9 repeats Fig. 7. t Vmj nearly, because the motion will now be very slightly re- tarded, as the angle contained by the tangent to the curve, and the radius, will be greater tlian a right angle. S.^e case 1 (b). ?' 34 THE LAW OF GRAVITATION. the intensity of gravitating influence is unable to neutral- ise the centrifugal force ; or, in other words, wherein the proportion of the gravitating force to the velocity is hisulhcient to restrain the moving body from increasing the distance between itself and the centre of gravitation. At the nioinont of perihelion the earth is moving at right angles to the radius-vector. As the motion proceeds, the distance of the earth from the sun, in consequence of the superio ■ velocity in this part of the orbit, hicreases ; that is, the deviation from the tangential direction of motion becomes less than that recpiired by tlie arc of the circle. Now if we assume that the decrease of intensity of the gravitating intluence is in simple inverse pro- portion to the distance (i. e. varies inversely as the dis- tance), it will be apparent that when the recession of the Ciirtli liiis inen'asetl tlie distaiu'e to a certain limited extent tVoiii tlie centre of gravitation, the gravitating iiillut'iice will iM'come sutlicient (or more than sntlicient) in proportion to the velocity, to restrain the earth from receding to a greater distance. The distance of the eartli from the sun ar which the centrifugal force is in equality with the attractive force must evidently be the actual average distance from the sun. If this distanct; is cor- rectly ascertained and also the distance at perihelion (i. e. the least distance) we have then, siiue the time of the entirt^ revolution and the angular velocity through defmite fractions of the orbital revolution are known, the means of determining the velocity at the perihelion in excess of the velocity required to equalize the atti.ictive force actinur from the centre of gravitation 'ralportionately to the gravitating influence than it was at tlie lesser distance, and there- lore the earth must continue to recede from the sun indefinitely. If, on tl e contrary, we assume the decrease in the attrac- tive force to vary inversely as the distance simply, tiien • The final velocity by the law of unifonr.ly acccleratod motion. 36 THE LAW OF GRAVITATION. aB the earth recedes the alterations in the conditions which were caused by her approaching the sun are sim^ ply reversed, viz., a considerable apparent decrease in the angular velocity, and a comparatively small decrease in the area! or actual velocity takes place. This decrease both in the angular and in the areal velocity is precisely equal to the increase during the motion toward the sun ; the former by causing a proportionate decrease in the centrifugal force counterbalances the equal decrease in the intensity of the gi-avitating influence (attractive force), and the latter is the immediate cause which deter- mines and regulates the average distance of the earth (or other planet) from the sun. Tlie motion toward and recession from the sun may be compared to the vibrations of a pendulum, the alliance between these manifestations of gravitating force ])eing in fact very close. The amount of these vibrations (oscillations) causing the greater or lesser deviation from the circle known as the eccentricity of the orbit, is, how- ever, partly dependent upon the disturbance (or effect) arisina from the gravitating influence of other bodies. GRAVITY. Let a number of bodies a, a, a, &c., (Fig. 10) revolve in the same circle (i.e. at the same definite distance and i» in the same plane) around a central body, and let a number of bodies b, b, b, »S:c., also revolve in another circle at twic the distance from the cen- tral body, and let the intensity of the gravitating influence on the surface of each of the bodie? QRAVITATINa INFLUENCE. 37 a, a, a, . . . and also on the surface of each of the bodies b, b, b, \ . . and on the surface of C, be the same as on the surface of the earth, (or other definite amount of intensity may be supposed, so that it be the same in all the bodies). Now supposing the bodies contained in the inner circle, viz., a, a, a, . . . brought together and combined with the central body C into one body, as in Fig. 11, and also the bodies contained in the outer circle b, b, b, . . . brought together and combined into only three bodies : wliat alteration, if any would such re-arrange ment of these bodies occa- sion in respect to the in- tensity of the gravitating influence on the surface of the remaining bodies b, b, b and 'C ? Since the mass or bulk of these com- pounded bodies would be now so much greater than for- merly ; would the intensity of the gravitating influence on the surface of each, be therefore necessarily in- creased ? And if so, would the increase in gravi- tating influence be eraal to the increase in the mass or bulk ? To take the case in the first instance in a simpler form let the several bodies a, b, d, (Fig. 12), be piled upon the lower body C, which is formed of a number of l'.ov here l)rielly d(>scribed as (•onsisting of a number of planets of various sizes, revolving at various distances in the same (or nearly in tiie same plane, round tlu'ir conmion centre of gravitation, the sun. If we consider such a system attentively it will become cident that the addition of any considerable mass to any one of the jdaiiets would have the eflect of draw- ing toget'ier the entire system : that is, of causing all the the planets to ajtproacii the sun*. For exantple take K, (Fig. Jo) andsu])pose a coiisi(h'rable addition to the mass of that ])laiu't ; tlie increase of gr-'vitating force tliere- froin would act practi.-ally as a. uiition to the gravita- ting force of the sun. The first eflecf miglit be con- sidered to be the attraction of tliose planets on the oj)po- site side (or nearest to the opposite side) causing thv'in to approach the sim ; luit the attraction of those jdancts wouhl react on F i.sclf, causing it also to appntach ; and the united '.MCct of tlu'ir attractions woidd evick-mlv act on tliose at the other two sides, with a com- pounded influence just as tiiougli a siniile nliuence 'Koh\ It may he rciii.irkcil that, iiiiloss t^wh an inliliiii.ii wore <4 !i very f,'roiit mass, the cllwl hrrc spukoii ol' wouhl l.c nitlior a tcn- iIciK'v than .in appivciahli' tli'fct ; it nould imhcd lie an actual ciii'd, but HO oxcfodingly Minall in amount as Ut ix" inajiprocialiji. •50 THE SOLAR SYSTEM. Fig. 15. proceeded directly from the centre of the sun. So soon as any actual motion towards the sun took place, the increased velocity, and consequently increased centrifugal effect, would restore the equilibrium. An addition to the mass of the sun itself, or an increase in the intensity of the gravitating influence, would have tlie like effect ; by acting directly on the planetary bodies belonging to the solar system, causing them to approach the sun and to revolve with increased velocity. The entire system, in such case, would contract and occupy less space with a proportionate acceleration of motion amongst its members. In considering the general arrange- ment of the solar system, it will now become apparent that alterations of velocity in the orbital revolution of the various planetary members, are not dependent (or consequent) upon differences in their densities ; and not- withstanding that some are vastly greater in magnitude than others, and situated at vastly greater or lesser dis- tances from the centre of gravitation, there is not THE SOLAK SYSTEM. 51 necessarily a dirterence in their areal (actual) velocities • the difference in the distance being compensated by the .greater or lesser angular (apparent) velocity, and the dif- ference in mass or quantity of matter, producing a pro- portionate difference in the amount of centrifugal force a!Kl thereby counteracting and equalizing the greater or lesser amount of gravitating force. It is evident, how- ever, that a planet of small size revolving at a very <^reat distance from the sun, and with a consequently very small angular velocity, would be liable to excessive perturbation from the influence of other planetary bodies, and we should therefore be led to expect that the planets situated at the greatest distance from the sun would be the larfest • and this is what actual observation shows to be the fact. We may now understand the unchangeable or pennanent nature of the arrangements in respect to unifomiity of the periodic revolutions, securing each planet from those changes in its velocity of motion and relative position which miglit otherwise have been occasioned by alterations in its own internal conditions. A planet hav- ing had, at the first, a definite velocity imparted to it, and a definite distance from the sun appointed for its orbital path, cannot deviate nor be readily made to depart there- from. An outside influence, as that of another planet, will cause a vibration or oscillation in its motion, resultin"- in a more or less elliptical orbit ; but any deviation in the one direction is necessarily followed by a compensating equivalent movement in the opposite ; and the average distance from the sun remains the same. And, as already explained, changes in its own physical conditions, such as contraction or expansion of its volume, will not cause alter- ation in the velocity of its motion or of its position rela- tively to the other members of the system. THE NEWTONIAN THEORY. . V. THE LAW OF GRAVITATION, AND THE NEWTONIAN THEORY. We fincl the I'ollowiiig in2)n'?f's Manual of Astronomy, [);ige 14!> : '' Assuniin<>:, as the nieasureiiients of astrono- mers jiistity us in doing, that the moon is distant O-lOjOOO miles Irom the earth \ve can easily calculate the space throuuh which siie would fall if left to herself in the lime of a minute. Tliis space would he as Newton has demon- strated in the :3^th ])ropositi(»n of thePrincijtia, the versed sine of the arc descrilx'd iu that time ; thus su])pose the arc C'd (Fis; '2^)) to l)c that di'scrihcd in a minute l)y the moon (' ; join to E tlic centre of the earth ; draw bd jxM'pi'.idicul.ir to CIO; then will Ci), the vcrst'il sine of thearc, he tlie >[»a('c throuiih which the iiiod'.i wiiiild I'all if uiisu]t])orted, I hat is if the cciitriliiual forci! should ccasf. Now thf arc ("d is oasllv fouii/ d.iys, 7 hours, V\ min : 1 min : : ;;(')(>>: :V.) nearly^Cil : of this arc. the versed sine (*1). luiiy ]»e easily computed ; it is in fact 1C> /^'^'^'^ taking tlie nwon's distance to bo 'J 10, ()()() miles. Now, on the sup- ]»osition tluit tlie force retaining the moon in its orbit be identical with terrestrial gravity, it ought to have decreased in proportion to the s(juare of the distance ; tliat is to say, assuming the moon to be sixty times fur- ther iiom the earth's cenire than a body on the earth's s irface, or distant one semi diameter (that is, EC=GOxEG) ' I THE NEWTONIAN THEORY. 53 which is about the trutli : the space describotl Ly the moon should be to the space described by a iallii:- body near the earth's surface as OO^ to P, the time being the same. Now a body would fall through 50,400 feet in one minute near the earth's surface ; hence, as G02 • 12 or as 3000: 1:: 50,400 feet: loMeet* which is the' space through which a body would fall iu a minute at the distance of the moon. Now this exactly agrees, making allowance for our using round numbers, wkh the actual distance througji which the moon would fall were the centrifugal force to cease. The moon, therefore, is retamed in her orbit by gravity, and gravity oidy, for It would be unphilosophical to assign two causes to account for effects precisely shnilar." This example, which at first sight appears to confirm the assumption that the attractive force or influence of gravi- tation is inversely as the square of the distance— sm ex- ample indeed which is considered to stand almost in the place of an observed fact, and upon which the celebrated theory is in a great measure based,— does not, when closely exandned support the assumption that the intensity of the influence is inversely as the square of the distance, but on the contrary, it proves that such assumption is quite irrecon- cileabU- with the facts ; and it also goes far to demonstrate that, ill fad, the gravitating influence (or attractive force) varies invers(;ly as the distance simply ; that is to say,* that the gravitating influence decreases in equal propor- tion to the increase in the distance. The force ot gravi- tation at the eartli's surface causes a body falling freely to descend through l(;i foct in a second. The moon is found to be by astronomical observation sixty semi-diam- eters of the earth distaiit from the earth (/. e. about :>10,- 000 miles.) The calculation then sliows that at the dil- • Nolo. TIktc is evMo.itly here an aritluhelieul enur in the cm- pututioii ; it shuul,! ivatl 57,'JOO. Itj/.^ feet. 54 THE NEWTONIAN THEORY. ! ! tance of tlie moon a body would fall towards the eaitii through 1()2 feet in a minute. Now there are (iO seconds in a minute, and the distance of tiie moon from tlw earth is 60 times as great as tlie distance of the surface of the earth from the earth's centre. Therefore as the moon's distance : distance of the surface of earth : : attractive force at earth : attract force at moon, (or, inversely, as < UiRmotor ,>ri iliamotiTu \ 1 — o • '•'^ 2~~*'' The oversight whii-h lias vitiate. i tlie calculation quoted by Mr. Drew is the non-recognizance of the law of accelerated motion in its ap[dication to falling bodies. The absolute necessity of taking into consideration tliis law, because of its direct and important application in this case, caimot fail to be iuniie- diately observed when the attention is expressly called to it, although not obvious in the first instance. In the cal- culation and its result as given by INIr. Drew, we have : '' A ])ody would fall through 57,900 feet in one minute near the earth's surface ; hence as GO^ : l^, or as 3600 : 1 : : -57,900 feet : IG,^^ feet which is the space through which a body would fall in a minute at the distance of the moon." Let us test this result by inverting the case and assuming the force of attraction so obtained at the distance of the moon •, let us see what intensity of force the application of the tlieory would in that case give us as the proportionate force at the surface of the earth. (Note. It may be here remarked that the la,w of accele- rated motion (or velocity) in falling bodies has been veiy carefully investigated, and the results verified by numer- ous experiments of a reHable character. (The calculation just now stated, being inverted, — gives P: GO- :: Intensity at Moun's distance Intensity at Earth's surface. 1 3600 IG/g- feet in a minute. 16-,^ feet in 1 nocond 60 Now IGiV ft. in a minute, increased to GO times the intensity, would (by the law of accelerated motion) be i I ! ) THE NEWTONIAN THEORY. 65 lO-i'^ ft. in a second ; bnt since we have GO x GO, this must be iigahi increased GO times ; and, consequently we June IGfV feet in the GOth jiart of a second, as tlie measure of ettective gravitating infhience at the surface of tiic earth ; or taking the result as space fallen through in a minute^ we have 3.5G4000 feet ; that is to say, the'actual intensity of gravitation at the earth's surface is thus enonnouslv exaggerated and made to appear sixty times greater thaii it is in fact. The actual i)r(>portion (i.e. the true pro- portion) may be thus simply stated : As 1 fli^m^- ; (;(> .imn^s .. 1 second : GO seconds : which (inverselv) o-ivos the gravitative intensity at the moon's distance. Or, As 1 second : GO seconds :: 1 fli]iJ2I£!l'" ; go gianiotPi-j . -^\•]lj^^.]^ gives the distance of the )noon tiom tlie centre of the earth. 0(1 VOLl'MK AND MASS OF TIIK SUN'. THE liKI.ATlVE MASS OF THE SUX TO THAT OF THE EARTH. From Brew's Maniud of Astronomy, page 174 : ^' Lot lis (irst ascertain tlio (lellcctiou of the oartli from a tan- gent caused by the sun's attraction. Proceeding in tlie same way as in that already pointed out for ascertaining tlie deHection of the moon produced by the eartli's attraction (viz., C b hi Fig. l'O,) we shall find that tlu^ proportion l»etween the distance through which the moon will be drawn l>ythe earth, and that through which tlie earth will be drawn by the sun in the same time, will be as 1 : 2.2. Now, as we have already seen, the whole amount of attraction is in a ratio compounded of the ratios of the masses directly, and of the squares of the distance inversely ; that is, letting F stand for the attractive force of the sun measured by the versed sine of the arc which tlie earth describes in one minute of time, viz. 2.2 ; and f, for the attractive force of the earth on the moon measured by the versed sine Cl>, through which the moon would lie drawn in a minute of time, viz., unity or 1 ; also the ratio of 1> to d, for that of the dis- tances of the earth from tlie sun, and of the moon from the earth, which is as 400 : I ; and ^[ to m, for the ratio of the masses of the sun and eartli, which we desire to know; then m : ^\ ;: ^ ^\^ : F 1 )^, that is, m : M :: I X r- :-J.2 X 400-, or. m : ^\ :: 1 : :j.')-J,000. So tliattht; mass of the earth is to that of the sun as I : -'{-rijOOO ; or it would take :Jo2,000 earths to make a l)ody e(pial in hulk to the sun." It is evident that in tliis calculation an error requires correction of the same kind as that which so enor- mouslv exaggerated the result of the investigation as to the intensity of the gravitating intluence at the ) "That at so vast a distajice the sun should appear to us of the she it does, and should so powerfully inlluence our condition by its heat and light, requires us to form a very grand conception of its actual magnitude; and of the scale on which tiiose important processes are carried on within it, by which it is enabled to keep up its liberal and unceasing supply of thvse elements. As to its actual mr.gnitude we can be at no loss, knowing its distance, and the anirles under which its diameter appears to us. An object place.l at the distance of elongiiig to thai ^^ystem, the centre of the sun therefore may be correctly considered as a fixed or immovable point. KEIM.ER S TJIIKI) LAW. 59 i ji> till' iiiimiliir I'lirtli (coiiiparetl tu tlic iiiooirs), and I velocity. IJut tho elK'ctivc (trravifatitig,) inlliu'uci' is in- versely astlio:3»J : I, as the projxjrtion of the volumes (masses) at an equality of density. The cube root of tliis (piaiitity . . . \' ••Jo4!):j() . . . equals (very nearly) 71. — So that the two results agree (piireas closely as can be expected, considering the inexactitude of the measurements taken. It may be remarked that, in the foregoing com])arison of the relative masses and densities of the earth and the sun, the atmosphere of the earth is entirely left out of consideration ; but evidently the atmosphere is a ])art of the earth, considered (collec- tively) as a planet,. .. .a part of the nuiss and of the volume. . . .a part of the density. . . .and (contributing) a part of the gravitating influence. Therefore the above comparison shoidd be defined as, between the solid and li((uid parts only of the earth . . rnd the entire volume of the sun. The Third of Kepler's laws. From Drcic^s Manual of Astrononuj, page 177 : " The Si^uares of the periodic times of any of the planets are to each other as the cubes of their mean distances from the sun. The value of this law will appear when we consider that knowing the distance of any one planet, say for instance the earth from tlie sun, we can by its aid cal- culate the distance of any other." The truth of this t Divided Jiy the proportionate anirular velocity if less than at the leaner distance, and inverneiy a.« the distance niultiplifd hy flie pro- portionate a. V. if grcxiter than at the lesser distance. (>(> Ki:i'i.i:i{ s i'iiii;i> i.wv I it^ law imut be in tlic first plart' (lt'|>»'inlt'iii ii|)(iii llic arcal vt'locitv (»l all tilt' planets liciiii;' the sinnc, ln'caiisr evi- dently a (iidereiice in actual velocity, would cunse a riiini lor ea<-h delinite distance iVom the sim: and which xclucity ol'etpiilihrium cannot Uv departed from. And tliis\elocity of equili- brium must be the same actual areal velocity whatever may be the distance of the parti«'ular jdanet from the centre of gravitation ; because Ixttli the cent rifu- variation in the distance, it tollows that tl.,' p lation nf the peri(,' r tinu's and the distances is also simple ; and art (inlinuly tli'' law {/.r. the third law of Kepler ) should lie read iliu^ :- • The periodic times (if any ol'the [daiiets .nc to each other as their mean distances from the sun.' TIMJICKSTIMAK (iUAVITATK »N. 61 TIIK F(H{M OF Tin-: KAimi AM) TKhMfHSTIJIAL (iKAVITATloX. Fnm Dtrir's MutiiKtl uf Astmnomji, j»iiir<' 2-5. " This ('«'iitrit'iii,Ml Wm-v lias cuiiscd inatfcr to at'cuinn- \',\W. ill the i-ct-iuiis of the r(|Mator, so tliat the earth's e(|Uiitorial (hanieter is yrcater tliaii its polar by 2i\.'i miles. And siiu-e the attraction of irravity decreases in tlie imcrse |>ro[K)rtioii ol'the s(|Uiires ol'the distance I'roin thee, •litre of f lie attractiiiy- sphere, i( is plain that the attract ion will he weaker on the earth's snrliice near the e(|uator than at or near the poles in the proportion of the s(|iiare of h.ilf the earth's a(e or coUeetive action of uravitation t<» tiie centre; since if we assume the earth to i»e a perli'ct sphere the attra<'tive force would he everywhere on or beyond the siirtiice e(|nival(!nt to (he ellec( of an iiillu e aj>[>arently situated at (he centre, l)ut it is necessary to recollect that the central force so assumed is representadve (because compounded) of (lie sum of the forces of all the parts or fractions of which (lie earth is composed : lor evidently if the (vxterior ])ar(s of the ■\irth were removed and only a small cen(ral sj)here left remaining, tlie attractive force would l)e diminished accordiiiiily. Tin- error may be immediately and dis- tinctly seen by e.\a«j;<> crating the deviation from a sj)here and suijposing the earth to be flattened into a disc-like form (/.('. tlie (brm of a grindstone) as in Fig. IG, when (\-2 TKHRESTKIAL (;HAVITATION. ii, h would rcprt'seiit rho etjuutoruil (iiainetor and c, d the polar diiinn'ttT. Now as.suiiiinir the earth to be of siu'h form, and leaving the rotation on the axis out of thc^ ((uestion, it is at once evident tluit the attractive force oii ihe suriace at ii, anti I), would hot he less than at o, and d, hut on the contr.iry would ])e cousiderahly greater, which is rhe reverse of the coiiclusion stated in tlu' text. \a. I: Frnni J>reic\^: Mdnunl of Astnmcniy, pwjr 2:1 : " It is tuund, however, by very careful adnu-asurement, that the lenuth of a degree of latitude, nscertained in the manner described, near the ecjuator, is ditferent from the letigth of a degree m-usured near the poles ; on the e(|uator it is ii', its minimum, increasing in length as the Intitude increases. In fact a section of the earth passing through both poles wotdd not l)e a circle, l)ut, as the adnuvisure- ments show, wot.ld be an «'llips«> ; indicating that the earth is ilattened at the poles, and that it protrudes in the region of the ecpiator. The first intimation which astrononu'rs received of this fact arose from the following circum- stances. As'fonomers s<'nt in the seventeentli century by the Fri'uch govs rnment to Cayenne, for the puq)Ose of making observations on the fixed stars, found tliat their clocks, the pt'iidulums of which had been so regulated as to beat seconds in tin' latitudi' of Paris, l(Kst tim*' at the rate of two minutes tw<'nty-N 63 From Dmv'.'i Mdnual of Asfronoiiiif, pan*' -Jd : " Sir r. Newton, tiikiiiii hotli tlu'se causes into afcoiinl, (Icinoii- strated that a iTVdhirijr ilnid mass of (Mjiial dciisitv throuiiliout, would assume the form of an »'llii)S(»id [that is a fifrure of which every section jtassinir tiirouirh tlie poles would he an ellipse] whose diameters would he as •J'U) : -J-JI).'" [The rapidity <»f revolution (rotat.on) is not, here slated or referred to ; a greater or lesser raj>idity would certain- ly cause (a theoretical) alteration in the proportions of the relative diameters. Nodituht the case supposes the pre- sent rate of rotation of tlie earth]. Admitiiii; tiie demon- stration ; does it not follow that, if the earth had |)recisely the ellipsoidal form, tfu-re would he no activeor positive centrifuijfal elU'ct ' Because, to sujjpose or assume a centri- fugal effect on the surfac.e of the solid, or j»artially solid, earth; evidently implies that an alt«'ration in the form wouhl take place if the fluid condition was suhstituted for the solid; or in other words, if the parti(des of matter had the necessary nxthility. It is stated, however, that tlu^ aetual diflerence hetween the two diameters is l(>ss than Newton's calculation n'ipiired ; the j)roporti<»ns derived from measurements ofs«>ctions of the earth's circumference — ?'. c. deirreesiii various latitudes from ^ t(» .sd^ inclu- sive, — hein<;as i^'OS: iii)!>, instead of '.2'^!) : 330 Ji.s rerpiired hy the calculation. This would ffive a (an efTective) centrifuiral force limited for its originating^ influence 21 toC2!«9.-Jl— GfM)) .- 2!»!)(»() of the meati diameter of the earth, a quantity which may he sutlicient to account f(»r tfie observed facts in n'spect to the pendulum vibrations, NoTK. — Jlut it must hi; observed thai un (ussioiird cornet niedsurpmcnt of the earth's dimtuter derived ft-om mtasure- metit of the earth's sar/are, assumes the correetness of tite supposed " ratio of the diameter to the cireumferrnc.e (f the circle,'- a question at the 2)resrut time in controversy. "'I [I III! i 04 TKKUr.STIUAI^ OKAVITATION. G^ TIIK KOTATION OF THE KAHTII (OR DTTIER IM.ANKT) AS INFI.rKN('IN(J TFHKIvSTKlAL ( INTFKNAI.) (J CAVITATION. All iiitpiirciit r, iiiid !* the jdaco of one of tlio l»(»lcs. Tlic rjirtli is ii>tii(iiiiii|)arati\eiv nnintliieii- eed liv this rotatoi'N niolioii. \^-_ An arirniiient to liie tollow- ^^^ar iiii; eileet is lliere(l»re likely to siiiryest itsell' to llie stii- tleiit : (I) A ltod\- re\ olviniraronnd a eeiitre of irravitalion (oreeiilral |)oiiit ) at the rate of a thousand miles an hour iiiiisi develojte a xery eonsideraltle eeiitrifnual force. (•-') This centrifugal tor«'e oj)j)(»ses the attractive force from the centre, and must therefore (:>)'»<'•'• deductitui iVom I he a|t|>areiil i:ra\ ity or \veii>ht of the l)od\'. Hence (i|iicry), is the (apparem) weiyhl of the same l>od\ coii- sidciahly <:reater in the jtolar reuioiis than near the e(|i!a- ior ;' or is there any st'iisilde dillerence in the (ajtpareiit) weight ottlie same iiody it' removed t'roiu the one to the other situation .'' 'I'lie (|iies|ion as to t.ict is answered iii the ne':ati\e: — where is the explanation .' The i:ieal stnmhlinu hlock in this case, is the s.iiiie alr»Md\ retcired to as the prohaide oriuin ot' the error in Kepltv's iliird law; \ i/.., the neulect.to discriminate i TEKUESTIMAL (iliAVITATION. 65 between angular and areal velocity. It is true the body on the surface near the e(]uator lias an actual areal velocity in the circle of rotation of about lOUO miles an hour; never- theless the motion is correctly represented by tiie hour- liand of a watch, vvliich likewise [tasses throu<,di the circle once in 24 hours ; and if the hour-hand were extended and made about -iOOO miles in len^tli, tlu; extremity of the hour-hand would also then move with u velocity of about 1000 miles an liour. .*-',j soon as this (iict is correctly appreciated the dilliculty will ju-oltably be in a irreat mea- sure removed, because it will be understood that theaiiiin- lar velocity as observed in the honr-liand of the watch or clock is not sullicient to develope iiiiy very apju'ecialde amount of centrifugal force. The ditlicnlty may, however present itself in a diHerent form, and one in which tlm solution is not so readily apjtarent. A body revolvinet\\een tlie Acised sine of the arc throuuh w hich the l»ody moves v: a definite time, and the distance thnninh which the gravitation would move the liodv towards the centre in the snme definite time; ]»e- cause the latter measures the tbrmer and, if e(|Ual, causes no more than tlie deviation from the taiigt-ntial direction, which is re(|nired to prodiu'e tlie arc, and in that case jh«' bodv will continue to revolve ; but if the deviation is in any degree greater than re(|uisite for the versed sine of the arc. then must the Ixidy fall to the e;irtli. ]\y taking the e(|iiivalent to a thousand miles at the ecpiator cm a circh', the case mav V»e illustrated, — as in fig. li>, where the angular divisions I 2 3 4 f> (> are each equivalent to about 1000 miles ; and the body A, near the surface of the earth is supposed to be moving in the tangential direction AD, with a velocity of about 3C)h miles in 1 minute; (which would carry it through a space equal to the circumference of the earth in about 10 hours and ;j7 miinites.) Tlio •effect of terrestrial irravitation actiim freelv on h bodv TERRKSTRIAL GRAVITATION. near tlie surfjico is known to equal in efl'ect an approach Unvards tlie centre (or space fallen throngli) of 57,!)(»u feet in aniiiniTc. Tliis would be therefore the deviation from the tanj-ential line of motion caused in the body A by the attractive force; viz, (about) MSfi feet in 52SiO feet, (ecpial to about tiOO miles in 2000 miles.) The body A, would ^ conseinientlytia- velL>OOU miles in o45 minutes, and ^ *^ arrive at IJ, hiiv- ing in that same time uiiderir( lie a deviation of its motion ti(tm ilie I C tangential line AD, measured by the verse 1 sine AE, of a^out 60 ) miles. And since the distance J)F, is «'qual to the distunco AE the body A, will revolve continually in an orbit at that dis- tance. (The velocity, however, should be a little increased to e(|iialize the dillc'rence in the radial distance of CA greater than CK ; (e g, if AK=I0 niilrs; then CA : CE: : 4010: 4000 mih's.)* THE TIDKS. Dirtr's MdiDiiiJ (if Asfrononiu. Paire 7^. *' InjKj l'> jiJati; VI, let Z represent tiie moon, K the €)^ earth. Now ihe moon attracts every particle of the earth': rmd ilie wat(>r i)eirig free to move, will tend towards her at o : it will be higli tide, therefore, to those idaces situated at o and its neiuhbourhood, which have •Tlie puppositioii ff mcli iiioroa.ep is inclmlod in the nssurnjjtion above that the (liHiuiici' iih\ iiroiimil to the diHtunccAL;. I- ()S THE TIDE.S. the inoon on the meridian ; but since the quantity of water remaiiiiM the same, tlie place? at n and s, 90° dis- tant from will supply the rise at p ; with them there- fore and down the line n R s it will be low water. As the earth turns round with her diurnal motion, other places will advance towards the moon, or will have her in the meridian ; it will therefore be liigh tide to them at that time. So far the matter is clear ; but the pecu- liarity is, tliat when it is high tide at o it will be also high at q, diametrically opposite, or with those places on whose inferior meridian the moon is situated. To render our explanation of tliis fact more lucid, let us investigate the operation of attraction on three bodies, at ditlerent dis- tances from the attractive body (Fig 1'2). The etfect of a { \ h' z. ^ a •eo liody Y, operating on three others r, z, x, in the same line would be to increase their mutual distances; for r would be drawn to w, tiirough th(> space rw ; z, being further oil' from Y, would be drawn through a less spac;', in t!ie proportion Yi^ : Yz- , viz., to v, zv being less than rw ; X would be still less operated upon, and would pass through a less space towards the attracting body ; viz. xt. The result will be, that the distance of tlie two bodies r and x, from z, will l)e increased ; vw and vt, tlieir new distances being greater than zr and zx, their original distances. Let the waters on either side of the earth R, in fig. 4o, pi. vi, be considered in the same circumstances as the two bodies x and r with respect to z in fig 12. The operation of the attrac- tion of the moon z, upon them and the earth will be to raise the waters at p, and to draw the earth, as it were away from the waters at r, causing a simultaneous rising of the tides at o and ([." The explanation here given is that *he moon attracts the water on the earth's surface, at i4< THK TIOKS. ()0 tlie side next to her, and draws it tioni tlie earth towards Jicr, and in addition attracts the earth itself, and (h-aws that also from the water on the opposite side; thus accounting for the high tide at both sides ; but if such were the case the earth and moon would soon come together; because, if tiie larger body,— the earth, deviates i'rom her orbit to approach the moon, the lesser body, — th" moon, must deviate so m,ichthe more (in inverse propor- tion to the mass) to approacli the earth; and, as this approx- imation is supposed to be continuous, the result would necessarily bo, within a certain limited time, contact be- tween the earth and the moon. The hypothesis, that the water is drawn towards the moon at the side opposite to her, does not when attentively considered appear to be tenable ; no reason is shown why such an effect should be confined to the water only ; tiie air would be proportion- ately subject to the same action ; and solids on the earth's surface also ; vanations in the barometnc level and in pendulum vibrations would indicate and measure such a partial and local effect of the moon's gravitating influence. Such a cause operating in tlie slightest imaginable degree, in the maimer supposed, would have the effect of accumu- lating all the water on the earth's surface, in that part exposed to the moon's direct influence. Moreover the sun's attraction is supposed to operate in the same manner. Page 79 (Drew^s Astmwmy). " Not only is the moon an agent in producing tides, but the sun also ; in conse- quence, hov ever, of his greater distance his attraction is not so mu'.h felt ; the whole force of attraction being in compouiid proportion of the mass directly and the distance squared inversely. The force of attraction thus deduced will give the sun's attraction : the moon's : : 2 : 5." The hypothesis is negatived by the fact that the influ- ence of the solar gravitation on the earth's surface is the same by night as by day. Substances, whei-ier solid or fluid, weigh the same at midnight as at midday whereas if the hypothesis were sound there would necessarily be a difTerence, and the apparent gravity of each thing on the (. ;, ' 70 TIIK TIDAL rUKXOMKXA. |:i i'i \\ r. earth's surface would be greater at uiglit, and less b} day. OF THE TIDAL PHENOMENA a reasonable exi»lanatiou may be found in the fomi of the earth as deviating from an ellipsoid and in (con- jtmction therewith) the unevenness of the earth's surface. As already stated, assuming Sir I, Newton's calculation of the perfect planetary ellipsoid, and the correctness of the actual measurements of the polar and ecpiatorial 01 . . diameters, the dillerence of -77:",,:^; indicates a cause of iiie(|uality ; whicii would allow the rotatory motion to occasion a positive centrii'ugal ellect in the equatorial regions ; and granting any etlect, however slight, from this cause, the quantity of ellect would be in a measure (hqiendeut upon tlie conformation of the ' arth's surface (ami up(»n l(»cal ditlerences in the conformation^ greater or less in flitferent [daces according to the depression or ele- vation of the surface at those places. And moreover, the manner in whicli the etVect would manifest itself would be to some extent determined or modified by the particular ciuitiguration of the uneven surface, (i. e. the [tart of the surface covered with water; \'iz., the eleva- tions and dt'[)ressions in the bed of the ocean). It is not ditlicult to understand how an oscillatory etlect U[)on the waters of the ocean thus [troduced, — being modified, iiicreast'd or U'ssenetl, by tlie action of suci. extra causes as winds, aerial and ocean currents, the une([ual influence of light, heat, v'cc., — may result in what is known as the ebb and tl«»w of the tide (and of the other tidal [)heno- iin'im). Tlie «'X[»lanation of the double tide is to be t'oiind in the law reuulating the e([uilibrium of bodies rotatini; on a cfntral axis: when a [tart of the matter cm iqtosintr the bo[urbance ot' tor ill) aiteratitdi in the quantity of matter (or the quiiiitity of gravitatiuir etlect) on the one side, by in- creasinu' or diminishing the mass eqindly on the o[»posite »iili' 7J ANGULAR AND AREAL VELOCITIES. Apparent motion of the sun. From HcrscJters OuiUncft of Asfrinioniif, [xiges 219 to '2'^-2 ^ (:J47). "We have nlroudy seen (art. 14(i) that the sun's motion in right ascension among the stars is not uni- form. Tliis is partly accounted lor by the obli(piity of ^ the ecliptic, in conse(pU'nce of which equal variations in longitude do not correspond to efpial changes in right ascension. Jiut if we observe the place of the sun daily throughout the yer.r, by the transit and circle, and from these calculate the longitude for each day, it will still be found that, even in its own proju'r path, its apparent angular motion is far from imiform. Tlie change of longitude in twenty-four mt'an solar hours averages 0^' o\)' S". 33; but about the 3Jst December it amounts to 1° 1' 9 '. 9, and about the 1st of July is only 0* 57 J I '. -5. Such are the extreme limits, and stich the mean value of the sun's apparent angular velocity in itS annual orbit. *^ (34*^). This variation of its angular velocity is accom- panied with a corresponding change of its distance from us. The change of distance is recognized by a variation observed to take jdaee in its apparent diameter, when measured at ditlerent seasons of the year, with an instru- ment adaj)ted for that purpose, called the heliometer, or, liy calculating from the time which its disc takes to tra- verse the meridian in the transit instrument. The greatest, apparent diameter corresponds to the 1st of Jamnny, or to the greatest angular velocity, and measures 32' 30" .3 ; the least is 3L 32'. 0, and corresponds to the 1st of July ; at which epochs, as we have seen, the angular motion is also at its extreme limit either way. Now, as we cannot suppose the sun to alter its real size period- ii :':i l\ ' r 72 ANGULAR AND AUliAI- VELOCITIKS. ically, the ohservod change of its iippiirtMit si/.«> can only arise from an actual change of distance. And tlie sines or tangents of sucli small arcs being pnmortional to the arcs tliemselves*, its distances from us, at tlu- ahovc- iiamed epoch, must be in the inverse pr(»portion of tlif apparent diameters. It appears, therefore, that tlie irreatest, the mean, and the least distances of the sun from us are in the respective proportions of the iiuml)er> l.OKw!), J.OOOUO, and 0.9S321 ; and that its apparent angular velocity diminishes as the distance increases, and vice versa. ^^ ^^ (:U!»). " It follows from this, that the real orbit of tiie sun, as relened to the eartli supposed at rest, is not ii circle with the earth in the centre. Tiu> situation of the eartii within it is cxcentrk, the cxcentricitif amounting to 0.01079 of the mean distance, wliich may be regarded as our unit of measure in this encpiiry. But l)esi(les this, the form of the orbit is not circular, but elliptic.'' § (3-50). '' The mean f or tHiigiTitc of ail arcs are proportional to the arcs, whether large or .«niall. Has not .Sir John intcnilod to say that the sines or tangents of very small arcs nearly correspoml with (i.e., are nearly eipial Uj) the arcH theniselve.s ? ANliri.AU AND AkKAL VKLOCITIES. 7:{ latter, and vice versa. Heiico we are led to conclude that the angular rclociti; |is*Jii the inverse propoition, not oi the (listance simply, hut ol' its s(|uure ; so that, to coin- pare tlie daily niiition'jn lonijitnde of the sini, at one point A, of its path, 'with that at 1*, we must state the proportion tfius": — OB'-^ : OA3 : : daily motion at A : daily motion at H. And this is found to he exactly verified in '"every part of the orbit." (Art. :il;M §(:]-')l). *' Hence we de^ {^i-j'-i). '' From this it necessarily follows, that in tuuMpial times, the areas descrihed nnist hi- proporti(»nal to tlu' times. Thus, in the fiuure of alt. -iVJ, the time in which the sun moves from A to H, is to the time in w hich it moves from (J to I), as the area of the elliptic sector AOB is to the area of the sector U()C." ^^ (;}o4). "The circumstances of the sun's a[iparent annual motion may, therelbre, ]>e summed up as follows: — It is performed in an orljit lyinu in tuie plane passing through the earth's centre, called the plane of the ecliptic, and whose projection on the heavens is the great circle so-called. In this plane its motion is from west to east, or to a s[)ectator looking down on the plane of the e(diptic from tlie northern siile, in a direction the reverse of that of the hands of a watch laid face ui»per- niost. In tiiis plane, however, the actual path is not circular, Itnt elliptical ; having the earth, n(tt in its centre, but in one focus. The eccentricity of this ellijtse ANOILAU AND AKEAL VKLOCITIKS. 7.5 is 0.01(170, ill parts of a unit cf^uiil to the mean distancrf or half the hufjer lUitnnicr of the vUipsc ; i.e., ubuut oue- sixtietli [tiirt, of rliat st'iiii-dianictor ; and the motion of the sun in its rircunifcrcnco is so reijulated, that eciual areas of the )'lli[tso luv j>assoil over Ity tlie nuUus vector in e([iiiil times.'' The conchisions here arrived at — vi/., that the angular velocity varies inv«'rsely as tlie 8(|uare of tiie distance, and not in sini]tle inverse proiiortion — appears to he (is ohvinnsly) nnreasdUidde ; for example, let us supi>oso the distance of the earth fidin the sun to he suddenly reduced to the one-hall, and that the earth continues Still to travel in the orbit of revolution with the same Miocity as before. Since the earth to nuike a complete circle of revolution will now liave half the distance to travel throuirh, and travels with the same speed as before, it is reasonable to iid'er that the time occupied would be the one-half (/. c.\^2^ days). The above conclusion, or theory, would make it necessary for the earth to accom- plish the half distance in one-fourth of the time, and this without any increase in the areal velocity of motion : (/. c. in the actual sjtccd). The precise nature of the error, as herein manifested, it is not dilUcult to point out. — Sir John Herschel has omitted to observe that in reducimr the distance, the standard of comparison is likewise reduced. iNn avc subtending any deiin 'e angle is still at the lesser distance a similarjnrc, buj evide<.itly the actual length is reduced, and one degr: o a , '),2 lesser distance is therefore, althougli it is W a 'ei'.ve, not a degree of the same lenutli as the d'^gref .'iC th. greater distance. Referring again to the exau f»ifc of ic earth at the half distance from the sun, th- i.-'Sti circle will contain 'M\0 degrees, but since two of these circles would be only e(pial t(» the one circle at the greater distance, if the increased velocity be eonsidir-red iin comparison with the greater circle as the standard of comparison, the increiis.! will be oO per cent. ; but if eompitred with I 70 AXfULAU AXI> AKKAL VKL( CITIKS. the lesser circle as the .sraiidard, then it will hv loo p.-r cent., and what is true in resj)ecr to the circle also applies to every (ecpial anuiilar)jlivi8ioii of the circle ; as Ibr example, to one deiiree. In.the succeedioe section (^ :Jol) Ilcrschel, apparentlv. attriimtes the increased velocity, to motion acquired in tlie approach of tiie earth to tlu- sun.* In our previous calculation, page :3o, the increas.' (in areal velocity) iuisinir in this maimer, was shown to he — ?— ^.r alxint ''■'♦id ""v'lii I mile in :>:>(K) nuVs. ) The tiieory of "an increase (.1 velocity inversely as the scpiare of "the distance would re(iuire an increase (in areal velocitv) of -^ (or 1 mile m thmy nules. • Tal;, t-.^rthcr with th.' first svnu-i of tl,,. .«<>elin,. (,s;|-,l), the i.loa is .su;rj:,.Htr.i that H.T.rh.l wns nu]>rv>.-,;\ \,^ this Mij,|H«<,.,l circui.iHtance (or phfiioim-noi.) us I ..i„i: a r.ii.arkiil.lr lu.-l jinsicrumH lur.-ssity. rather thai, a- ivx ctlici -ati.-lttcturiK uccuuiiti'il lur or ultrilnituhli' t... a rfCugni.Ti'l law . ii' 77 TIIK KLLII'TKAL ORIUT, AND LAW (.)F VAIVjAL AKKAS. To H'st till' suiiikIik'ss of Sir I. Ni'V.toii's (Iniioiistra- ti(»n (ill tlu' PriiR'ipiii,) let us siipixtsc a pljiiu't rcvidviiii' ill ail (irltit (»t vci-y 'rrcat (•.•(•ciitricify, in wliicli the majur axis of the ellipse : minor axis :: two : one. At (about) aplielioii tlien't'ore, flic plaiii'v luoveH in a eirele ot'wjiicli tlie (liainett'r is to tlie (liaiiieter of the circle in which tlie planel moves when at (alioiif) j)erilielion, as :.' : i. Ir is evident that — supposinu the planet, in revoiviiiir throuitli- out a i^reater orbital circle at the distance of aphelion completed the revtdntion in a certain definite time, if made to revolve thronirhont a h'sser circle at 'he distance of perihelion, it would complete a rev()liition in one half the time ; or wonid make two revolutions in the same time ill which it had made one revolution at the greater di>tancc; but what is tru«' of the entire circles is also true when appli»'d to their fractional parts; therefore, if the jdaiH't moves throuirh nine de\ a])preciate lct(' iil)S('ii('(' ol'jiir iioticrd in art (••{!), {/'(/niinil tn-cr Iter fvliolr siirfiK'r would ol'coursc be (Iccisivc. Sonic coiisidcr- iitioii.s o( a coiifnirv luitiirc, however, siiiruesr llieiiiM'lve.s ill coiisecineiicf of a remark lately made l»y I'rof. Ilaiiseii, viz., that the fact of the moon turninir always the same (iice towards the carlli is in all prohahility the result (»(' an cloiiiration ot' its liLnire in ihc iliroction of a line jdin- iiiLT the t'cntrcs ol'l)oih liic bodies actinu conjointlv trifli a niin-(iii)i(it/ri/' r iif its ciiilrc of f/na'i/if /rill/ its coifrc of si/iiitH(ffi/. To the middle of the Icnutli of a stick, htaded with a heany weiLiiil at oiieeiidand a liirht one at the other, attach a strini;, and swinir it n»niid. 'J'he heavy weight will assume anil maintain a jiosirioii in the circnlatioii n| the jniiit nia'triiii:. Suppose, then, its M-Nihc maih' np of materials imt hnmojfeneoiis, and so disjiei'sed in its interior that sdiiie coiisiderahle picpdiideraiice of wt'iulil should exist exceiitrically situated : then it will he easily apprelieiion which tlio foreffoiiur liypotliesis is based evidently does not iipply 'ulhe con- -u UAlHTAlllLlTY (»F THK Ml»UX. I I s "I !ii tlitions ami fiivuinstam-i's ol'tlu' case uiult'rcoii.siiloratioii ; viz., tlic revolution oC the inooii rouiire ; all the water, and almost all the atmosphere, ht'iiig confined ILAIUTAIUI.ITY OF THE M(»ON. 8| to the one hemisphere, in eons('((iience of a superior gra- \ iliitiiiu luicf oil tliiit side furthest from the earth ; hut. the form of the (soHd) moon is not sujipost'd to (h'viate very eonsiiU'rahly fncii ft spliere, and it is therefore not apparent how tiie liypotliesis of tlic existence of the water and iiir on tiie one 8i(h' only is to he reeoncihMl with tlie eiriumstanoea of the case ; for instance the average inten- sity of gravity on the surface of the moon compan-d to tiiat of tiie earth is ahout 4 : l'> tliiit is, if the earth's gra- vity isreprcsenti'd hy lolhs. on tiie square inch, that of the moon would hf al)ouf llhs. on thi'scpiare inch. Now tlif moon's atniospln're is sujtposed to Itc proportional in (puin- tity to that ot the eartli ; it would he, therefore, ahout one fourth of the depth (or heiglitli). Since l.'ilhs. on the sf[uare inch is the weight of tiit' yreater column, acted on hy the earth's more intense influence; svhat will Im- tiir pressure caused l)y the lesser column only onc-foiirth the heig'.ith, acted on l)y the (moon's) inlluence les«; intense in the prop(irti«»n of 1: !•'».' Wlicn it is understood that, on tiie supposition of a uniform gravitating inlluence and a imifonii distril)ntion of water, together with the (|uan. tity of atmosphere conjectured, the water W((iild he suh- p'Cted to a presturj' only \\n' rnii-fiftcnifli v{' th;\\ on the surface of the earth ; and that the air at the surface of the m(»ssihleand not perhaps (violent- 1\) improltahh', The effect would he to giv«' such a con- centration as the circumstances reipiire. Kig. 21 (a) and F 82 HABITABILITY OF THE MOON'. rig. 21. Fig 21. b IIAIUTAniLITY OF TIIK MOON. S3 (b) mny sorveto convoy a gonoivl uhni oi'the orran-oniont suiiposo.! ; (a) l,..in,ir a vortical soctioii of tl,^ niooi. through tho co.itro, {i.e., by a piano ,»a.H..inff throngh tho contros ol tho nio.M.aiul oarth) an sides [u. tho oircular boundarv] ..ouhras.isf' and regulate t ids oiled, an.l tho atnmsphoro itsolf would )'•' <-abl«' of underiroinif compression, b«' subjected to a sutlicient pre isurr, its l>ulk or vobnue will bt' iliniinished and a proportionate increase take place in its density; its weight or gravity, therefore, rvlntivdij io its bulk, will have become greater. Hence, specifn- gra- vity has a direct and very close relation to density, and for any one l>ody relatively toditlerent conditions of itself, the one tenn is so <'ntirely dependent upon the other that the tt-rms, in such limiti'd relationship, may be (!onsidt'red ulmost synonomous ; in fact, in such a case, the specilic gravity measures the dfusity ; and vice versa; but if the comparison be made with anothi'r body, compost-d of a ilitlert'ut kind or variety of matter, the sanu' necessary inter-dependence no longer holds good ; bi^ause the ('([ual bulk (jf tlu' sec(»nd body niay have [and the variety (if martt-r l»einif ditli'rt'nt, it iciU aliudst certainly Iku'v) a ilillt-rent atomic weiglit ; and constMjuently altiionirh the first body may be in its most dense, and the second n its least dense condition, tlic specific urnvify of the secund may lie nevertlieless greater than tli.ii of the first. The intensity of the uravitatiiiLr influence at the surface of the earth may be measured by tlie velocity acipiired or the spact» passed thronurh in ,i definite timeltya flillinLr body. The law wliich i-dveriis and reirulates tiie iiiotioii and pro- irri'ss of a l)ody so falling is tin' law of uravitation ; and the conditions which accelerate? or retard the di'scent of II fiilliiiu body lia\t' l)e<'n iiivestigattMl with considerable attention and care. One of the facts conclusively ascer- tained is, as previously stated, that neither an increase in tlie deiisjrv of II boilv tliroimh conr •action, nor decrease GBAVITATION AND ATOMIC TIIEOKY. So ihrongh expansion of the volume, nor yet an addition to the bulk (that is, an addition to the quantity of matter eontained in the body) makes any dirterence as to rapidity in the descent of the falling body ; the i lotiou is neither accelerated nor retarded; the velocity is the same. But: \\\mt if there be a diH'erence in the atomic weigiits ))etween two bodies? will that make no ditlerence in the relative velocity of their descent]? 8u[>po8ing there is a consider- alile diiference between the atomic weights oftwo8. (•J3<1) ^^Gidncd anil ftdtlicr expcriutiiit. — Let a glass tube AB, of live or six feet in lengtii, be closed at oneend B,and supplifd with an airtiuht cap and stop cock at thi' other end A. The cap being tmscrewed, let small pit'ces of metal, cork, paper and feathers bt. put int(» it, the cap screwed on, and the stop cock closed. Let the tiil)e be rapidly iii\<'rt('d, so as fcj let the objects included fall from end to end of the tul)e. It will be found that the heavier objects, sucii as the metal, will fall with greater, and the ligliter with loss speed, as might be exiiect- «,'d. But that this difjerence of velocity in fallinu is due, not to anv dirterence in the operation of gravity. I)ut to th»' resistance of the air, is proved in the following maimer. Let the stopcock be screwed upon the plate of an air pump, the cock being open, and let the * Vj^ k B >(» tiHAVITATlON AND ATOMIC TIU'CMIY. I! III I tul)0 l>o oxliausti'tl. Lor tlio rock ihvn W clo«<(>(l, and unsi-rewtMl from tlu' pliitc. (hi rigidly invcrlinu tlm Tiil)»', it will 1h' t'oiiiul that tlio It-atluTH will be pn'cipi- tatt'd li-oiu t'ud to end na rapidly astlio inotal, mid that in sliort, all thf ulijocts will fall together with a coiinnou vt'locity.'' This is rho ('xpcriniont and such tho rosidt ohtainod, wliiili is ifciu'ralist'd, iiiid applii'd as lolhtws : — Ta!:*' 10!». •• Wiii/lit of bud, .'t priiftortiontil fn (heir ijudiitifii'i of )niittrr. — >inco the attraction ot" tii«' »'arth acts equally on all thf the coni|»(tiu'nt paits of Ixtdios, n'ld si net' tin' airuiciiatf forces |r <»f their ((luintity of matter. Hence, in the connuon atlJiirs (»f cdiuiiierce, the (piaiitities of l)odies .ire estimated by their weights. It will apjiear, heienfter, that the weiirlit of a l»ody, or the force witii which it is attracted to tin- surface, is slinliily dilferent in ditl'erent [daces iipti'i till' cirtii; l»ut this is a point which need imt be insisted on at pH'scnt. At the same place tlu' wciulits are invarialdy and exactly pro^iortional to tlie ((iianties of matter composinu: the boarti(des of matter, whate\er des( riptH»ns or \iirieti<'s of matier may be contained in the various sul>stanct's known to us, are all (»f e(pud weight ; and this is not merely an assinnption taken as a basis tor arL'ument, nor is it a theory (tllered for conside- ration, but a positive statement as of a fact nianitestly and conchisivelv established, " Jt is clear that the OKAVITATIUN AiND ATOMIC TIIEUIJV. S7 woigl.rs Of bo.lio. niM.t bo iu the cxm ,,roiM,rtion of tlu' nmi.luT of i.i.rticlt's coiiiposimr ^u^,^^^ ^,^. ^,f ^,j^. i ;;■_) Cli loriuf •; - - I'Mliiic . . ]'_»y F'ntashiuiii ;^(, fi'dii 2^ ^'")'l'<'»' .•U.7 '■•■'»'' iu;i.7 ■"^ilv'T loH Now tin' law iu (piesrion is to this ellect :— II' such Vi <^ % /] % A^, "=> ^^' /A .!> o 7 IMAGE EVALUATION TEST TARGET (MT-3) 1.0 |22 I.I IS 11, 1.8 1-2^ llll|i-4 IIIIII.6 Photographic Sciences Corporation // ^^ f/. ^ 23 WEST MAIN STREET WEBSTER, N.Y. US80 (716) 872-4503 I ^^ o^ 88 GRAVITATION AND ATOMIC THEORY. I i IN IS numbers represent the proportions in which the different elements combine with the arbitrarily fixed quantity of the starting substance, the oxygen ; they also represent the proportions in which they unite among themselves, or at any rate bear somp exceedingly simple ratio to these proportions." (Pa^e 193. Conihinaiion by vohime.) " The ultimate reason of the law in question {comlin- ntion hy volimie) is to be found in the very remarkable relation established by the hand of Nature between the specific gravity of a body in the gaseous state and its chemical equivalent ; a relation of such a kind that quantities by weight of the various gases expressed by their equivalents, or in other words, quantities by weight which combine, occupy under similar circumstances of pressure and temperature either equal volumes, or volumes bearing a simple proportion to each otlier." '^If both the specific gravity and the cheniical equivalent of a gas be known, its equivalent or combining volmne can be easily detennined, since it will be represented by the number of times the weight of an unit of volume (tho specific gravity) is contained in the weight of one chemical equivalent of the substance. In other words, the equivalent volume is found by dividing the chemical equivalent by the specific gravity." If we consider the elementary atoms of the chemist to be the elementary particles of matter, then, it is quite evident, that these results, of very numerous carefully conducted chemical experiments, entirely disagree with the deductions from the guinea taid feather expeiiment prenously detailed ; because the information furnished us by these experiments is that the weight of a body consists in the atomic weight of its elementary particles multipUed into the number of those particles ; or in other words, the atomic weight of that particular descrip- tion of matter of wh;ch the body consists multiplied uito the quantity thereof GRAVITATION AND ATOMIC THECRY. 89 Now we have seen that the physicist as represented by Dr. Lardner, has unlimited confidence in these gene- raUzations nn 1 conclusions which based upon the guinea ^nd feather experiment are considered as the exposition of an estabhshed law. Does the chemist show equal con- fidence in the atomic theory, and in those experiments upon the results of which its title and claim to confidence are founded. Fowne^s Manual of Chemistry, Page 200. '•' The theory in question (the atomic theory) lias rendered great service to chemical science, &c., &c." " At the same time, it is indispensable to draw the broadest possible line of distinction between this, which is at the best but a graceful, ingenious, and in its place, useful hypothesis, and those great general laws of chemical action which are the pure and unmixed result of inductive research." " Note. — The expression atomic weiglit is very often substituted for that of equivalent weight, aiid is, in fact, in ahnost every case to be understood as such ; it is perhaps better avoided." So that the atomic theory is not only considered in- conclusive, but it is thought proper to caution the student to look upon it with a sort of distrust, as being at best, only a graceful and ingenious hypothesis. To show how far we dissent from this teaching on the subject, we will express our belief that the atomic theory is, and has been for some time past, virtually a demonstrated theorem) and, as such, shown to be a compound fact, — the great fundamental fact upon which the svmcture of chemical science rests. It is true, it has not been, as yet, formally demonstrated | but that is, apparently, because no one has taken into consideration the possible consequences direct and u:diryct of leaving a science such as chemistrj'- without any demonstrated and acknow- ledged basis. One of the consequences is the caution given to the student, as above. There might be, how- ever, an objection to admitting the atomic theory (and the law of combining equivalents) as a demonstrated 00 GRAVITATION AXU ATOMIC THEORY. (ills liCi •i theorem, side by side with that (so called) law of natural philosophy (j^reviouely stated) based on the guinea and feather experiment; because if understood in the usual sense, one of these laws evidently to some extent con- tradicts the other. It therefore seems desirable to give a little more particular attention to the g. and f. experiment as recorded by Dr. Lardner. We wiU first consider for a moment the circumnstances under which the experiment has probably been tried. What were the expectations and what the express object of the experimenter ? The experimenter has been previously informed, perhaps, as to what the result will be, and is consequently pre- judiced or predisposed to expect and to accept such result. The express object of the experiment in the first instance was evidently to ascertain the effect of removing the resistance of the air to the descent of a foiling body j not to compare the relative velocities of descent in !>ubstances diti'ering in kind. Taking tlie lenfi:th of the glass tube at five feet and one third, and assuming that there may be soine difference in the velocities of descent, what might be expected to take place ? The space fiillen through by a weight in a second has been established as 10^ feet; lead or iron being the material generally used for the weight. The pieces of metal, therefore, might be expected to fall from end to end of the tube in about one tliird of a second ; now supposing a veiy considerable difference, and that the jueces of feather or cork were to take more than half a second (orsny even two-thirds of a second) it might be possible, but it would be by no means very easy, to observe such a difference ; and this is supposing that all the objects in a glass tube of only 4 inches bore, which has to be inverted, start fairly together and do not come into contact with the side of the tube. The experiment, as to its affording any pre- cise and reliable information about the velocities of bodies iiiUing in vacuo, seems to us scarcely worthy of consider- ' GRAVITATION AND ATOMIC THEORY. 91 ation. If those who object to the atomic theoiy and the teaching of chemistry have nothing more rehable to oppose to it than the result of such an experiment, the chemist may fairly claim demonstration, on his side, and to have the general result of his numerous experiments admitted to the place of an established fact. But can we not get at what would necessarily be the result of the G. and F. experiment, if it were to be properly tried, by deducing the result from the results of rehable experi- ments which have already been tried and recorded ? The specific gravities of the various metals have been carefully detennined. A cubic inch of (cast) iron weighs 4.17 oz. A cubic inch of (cast) lead weighs 6.37 ozs. Now if we take a cubic inch of each of these metals, and connecting the two weights by a fine hue, suspend them in an Attwood's macliine by passing the connecting line over the grooved wheel, — we can say with certainty what will happen ; viz., the lead weight which is more tlian 2 ozs. heavier than the other will descend wi':h a consider- able and a continually accelerated velocity ; and, in doing so will raise the iron weight with an equal velocity. If the two weights are now detached and allowed to fall from the same height to the ground — will they descend with an equal velocity f No doubt they will ; because in doing so the equally rapid descent of the heavier weight will represent a larger quantity of effect exactly propor- tionate to the preponderance of weight. Does this decide the question ? Dr. Lardner's deduction is that the quantity of matter or number of particles contained in the lead ^is greater than in the iron, and therefore both of them descend with the same velocity ; but, taking the atomic theory, are we thereby tauglit that an atom of lead is of precisely the same weight as an atom of iron, or of gold, or of potassium ? Does the cubic inch of lead contain a greater number of elementary particles of lead than the cubic inch of iron contains of the elementary particles of iron ? It is evident that the deduction of Lardner becomea "P V2 GRAVITATIO>f AND ATOMIC THEORY. or includes a primary definition of matter ; in other words it involves the positive statement (corollary), that if (a) and (b) represent two distinct varieties of matter, and the combining equivalent (or atomic weight) of (a) is twice that of (b), the elementary atom of (a) contains twice the quantity of primary matter {i.e. of matter in a more sim- ple and elementary form or condition) contahied by tiie elementary atom of (b). If the elementary atoms are of the same size, then that of (a) must have twice the density compared with that of (b). The important distinction herein defined is that gravity is not a property of which one variety of mattpr possesses more or less than others ; but belongs to a primary fonu or condition of matter, and that a fundamental ditTerence between all those varieties of matter known to us, is that the elementary atom of any one variety is compounded of a greater or of a lesser quantity of primary matter, than the elementary atom of any one of the other varieties. We do not say that the conclusion thus arrived at is unsound ; on the contrarv, we are strongly of opinion that such conclusion may be demonstrated and established, and with such intei-preta- tion and definition the law stated by Lardner and the atomic theory harmonize perfectly ; but it does not follow that the admission of a hasty generalization based on a rough and inconclusive experiment to stand as a part of the national science is justified because it may eventually appear that the generalization was not in fact false. We are under the impression that Dr. Lardner himself would have hesitated to accept, and might very possibly have rejected the corollary to the proposition so positively stated. ii IP CONCLUSION. 95 It has been said — by a writer whose reasoning powers cannot be lightly esteemed, and whose opinions and statements, considering the general state of knowledge and other circumstances at the time in which he wrote, are c»;rtainly entitled to respect — that 'a little knowledge is a dangerous thing ' ; the saying may be amplified, and it may be said, with no less trutli, tliat a good deal of l^nowledgo is a dangerous thing, — if that knowledge is of an uncertain, unsound and disorderly nature, containing sound and vmsound knowledge, — truth and imtruth — good and evil — mingled indiscriminately together; and it may be also said — tliat men possessed of a little knowledge, or of unsound and uncertain knowledge, are dangerous ; dangerous to themselves and to each other ; unless they be controlled by that superior knowledge whicli is sound and certain. Many persons think that human knowledge is now for in advance of what it has been at any former time. This is an opinion which may have, and very pro- bably lias l)een fretjuently, entertained at earlier periods of the Avorld's history. Jlany educated per- sons suppose tliat civilization and science have })ro- gressed so much and are now so firmly establislied, tliat no general catastrophe or even serious reverse is any longer to be feared ; a nation here and there may fall behind, and mistakes may sometimes he made ; l)ut, in a general or universal sense, civilization is safe, and must continue to progress with accelerated rapidity. Tlie grounds npon which sucli rehance is based a})pear to bo somewliat indefinite. Some put dieir trust in the ex- tended area of civilization and the general difiiision of education ; some have faith in the multitude of books; and others feel a happy assurance in the power of railways, steamboats and telegraphs, to ward ofl!' and avert any dan- gers that civilization may be exposed to. We do not wish to exaggerate the signs which appear to us to indicate particular danger, and we certainly da * 94 CONCLUSION. not wish to create any unnecessary alarm ; but wc will conclude with one word of caution, and will say to those who think, because we have lived for long in a time of calm, because the mutterings of danger occasion- ally heard have ceased, and the signs of storm, which from time to time have shown themselves, have again passed away and the heavens are still serene, that there- fore we may live on in careless security, and that pmdence and precaution are needless ; to such persons we say. . . .take care, it is unsafe. 1