m^ r ®k§^jpi»^, PRESENTED TO THE WOBNJELL TTNIVER8ITT, 1870, The Hon. William Kelly Of Rhinebeck.- arV17959 Tracts, Cornell University Library 3 1924 031 245 560 olin,anx Cornell University Jbrary The original of tliis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 245560 TRACTS, MATHEMATICAL AND PHYSICAL. TRACTS, MATHEMATICAL AND PHYSICAL. HENEY LOED BEOUGHAM, LL.D., F.E.S., MEMBER OF THE NATIONAL INSTITUTE OF FRANCE ; KOTAI. ACADEMY OF NAPLKS. CHANCELLOR OF THE UlSriVEKSITY OF EDINBUEGH. ITirokn mil (ilasgnfa : EICHAKD GEIFFIN AND COMPANY. 1860. © LONBOK: PHINTBD BY W. CLOWES AND BOKS, STAMFORD STREET AND CHARIKG CROSS. TO THE UNIVERSITY OF EDINBURGH, Ci)e0e Cracts, BEGUN WHILE ITS PUPIL, FINISHED WHEN ITS HEAD, ARE INSCRIBED BY THE AUTHOE, IN GEATEFDL KEMEMBEANCE OF BENEFITS GONFEEEED OF OLD AND HONOUES OF LATE BESTOWED. PREFACE. These Tracts were written at different times between 1796 and 1858. The first was inserted in the ' Philosophical Transactions,' with two other Papers on Light omitted in this collection. These three belong to the years 1794, 5, 6, and 7, when the author was a student at the University under Pro- fessors Playfair and Eobison. He could have wished to insert an exercise which he gave in while at the class of the former in 1794, which Mr. Playfair was in the habit of show- ing, as having had the good fortune to hit upon the Binomial Theorem, but only by induction, as its author said in answer to the Professor's question, by what means he had arrived at it. He made inquiry some years ago, and found that the Pro- fessor's papers had not been preserved. But he cannot pass over this reminiscence of the University, nor a circumstance which upon the Professor's expression of an opinion respect- ing his pupil's good fortune, at once fixed his inclination for mathematical studies. TUi PBEFACB. The Third Tract was believed to be required for elucidating D'Alembert's extension of the Integral Calculus, there being no distinct account anywhere of the history of that important step, nor indeed any very clear statement of its nature and limits. The Eleventh and Twelfth Tracts may possibly prove use- ful to students of the Principia ; at all events, they give the analytical treatment of the fundamental truths in the system — handled by Newton synthetically and with extreme concise- ness, and therefore elliptically. The Fourth Tract, on the Greek Geometry, it is hoped may have a tendency to encourage the study of the Ancient Analysis in conjunction with the modern, from which it is too often severed. The authority of M. Chasles is referred to in Note II. to this Tract, in favour of close attention to the Ancient Analysis. That he is far from undervaluing the modem is manifest ; indeed, his work on the Higher Geometry sufSeiently proves this ; and he occupies the chair of Pro- fessor of that science, the first appointed since its establishment — an inestimable benefit bestowed upon mathematical science by the government of France. Let us hope that this our University will receive the same benefit from the government of our own country ; a hope which may appear well grounded when we recollect that of its three most important members, one has been representative of Cambridge and pupil of PEBPACB. IX Stewart, another an alumnus of this University and pupil of Playfair, and a third our present Lord Eeotor, selected, not from any connexion whatever with our body, but as a testi- mony to his talents and learning. No alteration has been made in any of these Tracts in preparing them for this work, except changing the fluxional for the differential notation. But the author has very carefully gone through all the analytical processes, in order to make sure that no error or oversight had occurred in investigations conducted at different times and in various circumstances. Hardly any were found, except typographical ones iu former publications. CONTENTS. PAGE Introductory Eemarks 1 I. — General Theorems, chiefly Pokisms in the Higher Geo- metry 7 n. — Kepler's Problem 24 III. — Dynamical Principle — Calculus op Partial Diepee- ENCES — Problem op Three Bodies 33 IV. — Greek Geometry — Ancient Analysis — Porisms . . 57 V. — ^Paradoxes imputed to the Integral Calculus ... 86 VI. — Architecture op Cells op Bees 103 VII. — Experiments and Investigations on Light and Colours . 122 Vin. — Inquiries Analytical and Experimental on Light . . 166 - IX — On Forces op Attraction to Several Centres . . . 191 X. — ^Meteoric Stones 207 XI. — Central Forces, and Law op the Universe Analytically Investigated 227 Xn.— Attraction op Bodies ; or Spherical and Nonspheeical SuEPACBS Analytically treated 261 Xm. — SiE Isaac Newton. — Grantham Address .... 275 Notes 297 EBEATA. Page i, line 5 from bottom— /or neglecting read neglectod. „ 73, note— /or Grandfather read Great-giaudfathcr, and for father read grandfather. 90, notei line 1 — after lettre insert a. 96, line 7 from bottom— /or e'tait read 6taut. 96, line 6 from bottom— /or or read ou. 197, line 4 from bottom— /or throught read tliroughout. 226, last line — for subject investigates read subsequent investigations. 265, line 6— for beyond read outside and witliout. 298, line 4— /w cardiordeal read canlioide. 300, last line— /or in read m. INTEODUCTOEY EEMAEKS. It is not correct — it is the very reverse of the truth — to represent the practical applications of science as the only real, and, as it were, tangible profit derived from scientific dis- coveries or philosophical pursuits in general. There cannot be a greater oversight or greater confusion of ideas than that in which such a notion has its origin. It is near akin to the fallacy which represents profitable or productive labour as only that kind of labour by which some substantial or material thing is produced or fashioned. The labour which of all others most benefits a community, the superior order of labour which governs, defends, and improves a state, is by this fallacy excluded from the title of productive, merely because, instead of bestowing additional value on one mass or parcel of , a . nation's capital, it gives additional value to the whole of its property, and gives it that quality of security without which all other value would be worthless. So they who deny the importance of mere scientific contemplation, and exclude from the uses of science the pure and real pleasure of discovering, and of learning, and of surveying its truths, forget how many of the enjoyments derived fronj what are called the practical applications of the sciences, resolve them- selves into gratifications of a merely contemplative kind . Thus, the steam engine is confessed to be the most useful application of machinery and of chemistry to the arts. Would it not be so if steam navigation were its only result, and if no one used 2 INTBODUCTOET KEMAEKS. a steam-boat but for excursions of curiosity or of amusement ? Would it not be so if steam-engines had never been used but in the fine arts ? So a microscope is a useful practical appli- cation of optical science as well as a telescope — and a tele- scope would be so, although it were only used in examining distant views for our amusement, or in showing us the real figures of the planets, and were of no use in navigation or in war. The mere pleasure, then, of tracing relations, and of contemplating general laws in the material, the moral, and the political world, is the direct and legitimate value of science ; and all scientific truths are important for this reason, whether they ever lend any aid to the common arts of life or no. In like manner the mental gratification afforded by the scientific contemplations of Natural Eeligion are of great value, inde- pendent of their much higher virtue in elevating the mind, mending the heart, and improving the life, — towards which important object, indeed, all contemplations of science more or less directly tend, — and in this higher sense all the pleasures of science are justly considered as its Practical Uses. If it be a pleasure to gratify curiosity, to know what we were ignorant of, to have our feelings of wonder called forth, how pure a delight of this very kind does Natural Science hold out to its students ! Eecollect some of the extraordinary dis- coveries of Mechanical Philosophy. How wonderful are the laws that regulate the motions of fluids ! Is there anything in all the idle books of tales and horrors more truly astonish- ing than the fact, that a few pounds of water may, by mere pressure, without any machinery — bj'^ merely being placed in a particular way, produce an irresistible force? What can be more strange, than that an ounce weight should balance hundreds of pounds, by the intervention of a few bars of thin iron ? Observe the extraordinary truths which Optical Science discloses. Can anything surprise us more, than to find that the colour of white is a mixture of all others — that red, and blue, and green, and all the rest, merely by being blended in certain proportions, form what we had fancied rather to be no colour at all, than all colours together? INTKODUCTOKY EEMAEKS. 3 Chemistry is not behind in its wonders. That the diamond should be made of the same material with coal ; that water should be chiefly composed of an inflammable substance ; that acids should be, for the most part, formed of diiferent kinds of air, and that one of those acids, whose strength can dissolve almost any of the metals, should consist of the self-same in- gredients with the common air we breathe ; that salts should be of a metallic nature, and composed, in great part, of metals, fluid like quicksilver, but lighter than water, and which, without any heating, take fire upon being exposed to the air, and by burning, form the substance so abounding in saltpetre and in the ashes of burnt wood : these, surely, are things to excite the wonder of any reflecting mind — nay, of any one but little accustomed to reflect. And yet these are trifling when compared to the prodigies which Astronomy opens to our view: the enormous masses of the heavenly bodies ; their immense distances ; their countless numbers, and their motions, whose swiftness mocks the uttermost efforts of the imagination. Akin to this pleasure of contemplating new and extraordi- nary truths, is the gratification of a more learned curiosity, by traciiig resemblances and relations between things, which, to common apprehension, seem widely different. Mathemati- cal science to thinking minds affords this pleasure in a high degree. It is agreeable to know that the three angles of every triangle, whatever be its size, howsoever its sides may be inclined to each other, are always, of necessity, when taken together, the same in amount : that any regular kind of figure whatever, upon the one side of a right-angled triangle, is equal to the two figures of the same kind upon the two other sides, whatever be the size of the triangle : that the properties of an oval curve are extremely similar to those of a curve which appears the least like it of any, consisting of two branches of infinite extent, with their backs turned to each other. To trace such unexpected resemblances is, indeed, the object of all philosophy; and experimental science, ia par- ticular, is occupied with such investigations, giving us general B 2 INTEODtrCTOET EBMARKS. views, and enabling us to explain the appearances of nature, that' is, to show how one appearance is connected with another. But we are now considering only the gratification derived from lea,rning these things. It is surely a satisfaction, for instance, to know that the same thing, or motion, or what- ever it is, which causes the sensation of heat, causes also fluidity, and expands bodies in all directions ; — that electricity, the light which is seen on the back of a cat when slightly rubbed on a frosty evening, is the very same matter with the lightning of the clouds ; — that plants breathe like ourselves, but differently by day and by night ; — that the air which burns in our lamps enables a balloon to mount, and causes the globules of the dust of plants to rise, float through the air, and continue their race — in a word, is the immediate cause of vegetation. Nothing can at first view appear less like, or less likely to be caused by the same thing, than the processes of burning and of breathing, — the rust of metals and burning, — an acid and rust, — the influence of a plant on the air it grows in by night, and of an animal on the same air at any time, nay, and of a body burning in that air ; and yet all these are the same operation. It is an undeniable fact, that the very same thing which makes the fire burn, makes metals rust, forms acids, and enables plants and animals to breathe ; that these operations, so unlike to common eyes, when examined by the light of science are the same, — the rusting of metals, — the formation of acids, — the burning of inflam- mable bodies, — the breathing of animals, - and the growth of plants by night. To know this is a positive gratification. Is it not pleasing to find the same substance in various situations extremely unlike each other ;— to meet with fixed air as the produce of burning, of breathing, and of vegetation ;— to find that it is the choke-damp of mines, the bad air in the grotto at Naples, the cause of death in neglecting brewers' vats, and of the brisk and acid flavour of Seltzer and other mineral springs ? Nothing can be less like than the working of a vast steam-engine, of the old construction, and the crawling of a fly upon the window. Yet we find that these two opera- INTEODDOTORT EEMAEKS. tions are performed by the same means, the weight of the atmosphere, and that a searhorse climbs the ice-hills by no other power. Can anything be more sti'ange to contemplate ? Is there in all the fairy tales that ever were fancied anything more calculated to arrest the attention and to occupy and to gratify the mind, than this most unexpected resemblance between things so unlike to the eyes of ordinary beholders ? What more pleasing occupation than to see uncovered and bared before our eyes the very instrument and the process by which Nature works ? Then we raise our views to the struc- ture of the heavens ; and are again gratified with tracing accu- rate but most unexpected resemblances. Is it not in the highest degree interesting to find, that the power which keeps this earth in its shape, and in its path, wheeling upon its axis and round the sun, extends over all the other worlds that compose the universe, and gives to each its proper place and motion ; that this same power keeps the moon in her path round our earth, and our earth in its path round the sun, and each planet in its path ; that the same power causes the tides upon our globe, and the peculiar form of the globe itself; and that, after all, it is the same power which makes a stone fall to the ground ? To learn these things, and to reflect upon them, occupies the faculties, fiUs the mind, and produces certain as well as pure gratification. But if the knowledge of the doctrines unfolded by science is pleasing, so is the being able to trace the steps by which those doctrines are investigated, and their truth demon- strated : indeed you cannot be said, in any sense of the word, to have learnt them, or to know them, if you have not so studied them as to perceive how they are proved. Without this you never can expect to remember them long, or to understand them accurately ; and that would of itself be reason enough for examining closely the grounds they rest on. But there is the highest gratification of all, in being able to see distinctly those grounds, so as to be satisfied that a be- lief in the doctrines is well founded. Hence to follow a demonstration of a great mathematical truth — to perceive INTEODtrCTOEY KEMAEKS. how clearly and how inevitably one step succeeds another, and how the whole steps lead to the conclusion — to observe how certainly and unerringly the reasoning goes on from things perfectly self-evident, and by the smallest addition at each step, every one being as easily taken after the one before as the first step of all was, and yet the result being something not only far from self-evident, but so general and strange, that you can hardly believe it to be true, and are only convinced of it by going over the whole reasoning — this operation of the understanding, to those who so exercise themselves, always affords the highest delight. The contemplation of experi- mental inquiries, and the examination of reasoning founded upon the facts which our experiments and observations dis- close, is another fruitful source of enjoyment, and no other means can be devised for either imprinting the results upon our memory, or enabling us really to enjoy the whole pleasures of science. They who found the study of some branches dry and tedious at the first, have generally become more and more interested as they went on ; each difficulty overcome gives an additional relish to the pursuit, and makes us feel, as it were, that we have by our work and labour established a right of property in the subject. ( 7 ) I. GENERAL THEOEEMS, CHIEFLY POEISMS, IN THE HIGHER GEOMETRY.* The following are a few propositions that have occurred to me in the course of a considerable degree of attention which I have happened to hestow on that interesting, though difficult branch of speculative mathematics, the higher geometry. They are all in some degree connected ; the greater part refer to the conic hyperbola, as related to a variety of other curves. Almost the whole are of that kind called porisms, whose nature and origin is now well known; and, if that mathematician to whom we owe the first distinct and popular account of this formerly mysterious, but most interesting subject,! should chance to peruse these pages, he will find in them additional proofs of the accuracy which characterizes his inquiry into the discovery of this singularly-beautiful species of proposition. Though each of the truths which I have here enunciated is of a very general and extensive nature, yet they are all dis- covered by the application of certain principles or properties still more general ; and are thus only cases of propositions still more extensive. Into a detail of these I cannot at present enter : they compose a system of general methods, by which the discovery of propositions is effected with certainty and ease; and they are, very probably, in the doctrine of curve lines, what the ancients appear to have prized so much ia plain geometry ; though unfortunately all that remains to * From Phil. Trans., 1798, part ii. t See Mr. Playfair's Paper in vol. iii. of the 'Edinburgh Transactions.' O GENERAL THEOKEMS, CHIEFLY POEISMS, US of that treasure is the knowledge of its high value. I have not added the demonstrations, which are all purely geometrical, granting the methods of tangents and quadra- tures : I have given an example in the abridged synthesis of what I consider as one of the most intricate. It is un- necessary to apologise any further for the conciseness of this tract. Let it be remembered, that were each proposition followed by its analysis and composition, and the corollaries, scholia, limitations, and problems, immediately suggested by it, without any trouble on the reader's part, the whole would form a large volume, in the style of the ancient geometers ; containing the investigation of a series of connected truths, in one branch of the mathematics, all arising from varying the combinations of certain data enumerated in a general enunciation.* As a collection of curious general truths, of a nature, so far as I know, hitherto unknown, I am persuaded that this paper, with all its defects, may not be unacceptable to those who feel pleasure in contemplating the varied and beautiful relations between abstract quantities, the wonderful and ex- tensive analogies which every step of our progress in the higher parts of geometry opens to our view. Prop. 1. Porism. Fig. 10. — A conic hyperbola being given, a point may be found, such, that every straight line drawn from it to the curve, shall cut, in a given ratio, that part of a straight line passing through a given point which is intercepted between a point in the -3} curve not given, but which may be found, and the ordinate to the point where the first-mentioned line meets the curve. — Let X be the point to be found, n a the line passing through the given point n, and m any point whatever in the curve ; join X M, and draw the ordinate m p ; then a o is to c P in a given ratio. * See the celebrated aoooimt of ancient geometrical works, in tlie seventh book of Pappus. IK THE HIGHER GBOMETKT. » Corol. This property suggests a very simple and accurate metliod of describing a conic hyperbola, and then finding its centre, asymptotes, and axes ; or, any of these being given, of finding the curve and the remaining parts. Peop. 2. Forism. — A conic hyperbola being given, a point may be found, such, that if from it there be drawn straight lines to all the intersections of the given curve, with an infinite number of parabolas, or hyperbolas, of .any given order whatever, lying between straight lines, of which one passes through a given point, and the other may be found ; the straight lines so drawn, from the point found, shall be tangents to the parabolas, or hyperbolas. — This is in fact two propositions ; there being a construction for the case of para- bolas, and another for that of hyperbolas. Peop. 3. Porism. — If, through any point whatever of a given ellipse, a straight line be drawn parallel to the con- jugate axis, and cutting the ellipse in another point ; and if at the first point a perpendicular be drawn to the parallel ; a point may be found, such, that if from it there be drawn straight lines, to the innumerable intersections of the ellipse with all the parabolas of orders not given, but which may be found, lying between the lines drawn at right angles to each other, the lines so drawn from the point found, shall be normals to the parabolas at their intersections with the ellipse. Prop. 4. Porism. — A conic hyperbola being given, if through any point of it a straight line be drawn parallel to the trans- verse axis, and cutting the opposite hyperbolas, a point may be found, such, that if from it there be drawn straight lines, to the innumerable intersections of the given curve with all the hyperbolas of orders to be found, lying between straight lines which may be found, the straight lines so drawn shall be normals to the hyperbolas at the points of section. Scholium. The last two propositions give an instance of the many curious and elegant analogies between the hyperbola and ellipse ; failing however when, by equating the axes, we change the ellipse into a circle. 10 GENEBAL THEOEBMS, CHIEFLY POEISMS, Prop. 5. Local Theorem. Fig. 11. — If from a given point A, a straight line de moye parallel to itself, and another cs, from a given point c, move along with it round c ; and a point i move along a b, from H, the middle point of ab, with a velocity equal to half the velocity of d E ; then, if A p he always taken a third pro- portional to AS and b c, and through P, with asymptotes d' e' and A b, a conic hyperhola be described ; also with focus i and axis A b, a conic parabola be de- scribed ; then the radius vector from c to m, the intersection of the two curves, moving round c, shall describe a given ellipse. Prop. 6. Theorem. — A common logarithmic being given, and a point without it, a parabola, hyperbola, and ellipse may be described, which shall intersect the logarithmic and each other in the same points ; the parabola shall cut the logarith- mic orthogonally ; and if straight lines be drawn from the given point to the common intersections of the four curves, these lines shall be normals to the logarithmic. Prop. 7. Porism. — Two points in a circle being given, but not in one diameter, another circle may be described, such, that if from any point of it to the given points straight lines be drawn, and a line touching the given circle, the tangent shall be a mean proportional between the lines so inflected. Or, more generally, the square of the tangent shall have a given ratio to the rectangle under the inflected lines. Prop. 8. Porism. Fig. 12. — Two straight lines ab, ap, not parallel, being given in position, a conic parabola mh may be found, such, that if, from any point of it m, a perpendicular m p be drawn to the one of the given lines nearest the curve, and this perpendicular be produced till it meets the other line in B ; and if from b a line be drawn to a given point c ; then m p shall have to p b together with c B, a given ratio. IN THE HIGHER GEOMETRY. 11 Scholium. This is a case of a more general enunciation, which gives rise to an infinite variety of the most curious porisms. Prop. 9. Porism. Fig. 13. — A conic hyperbola being given, a point may be found, from which if straight lines be drawn to the intersections of the given ciirve y with innumerable parabolas, or hyper- bolas, of any given order whatever lying between perpendiculars which meet in a given point, the lines so drawn shall cut, in a given ratio, all the areas of the parabolas or hyperbolas contained by the / peripheries and co-ordinates to points of it, found by the innumerable intersections of another conic hyperbola, which may be found. — This comprehends evidently two propositions ; one for the case of parabolas, the other for that of hyperbolas. In the former it is thus expressed with a figure. Let e m be the given hyperbola ; B a, A c, the per- pendiculars meeting in a given point A : a point x may be found, such, that if x m be drawn to any intersection M of e m with any parabola a m n, of any given order whatever, and lying between A B and A c, x m shall out, in a given ratio, the area a m jsr p, contained by A m n and a p, p w, co-ordinates to the conic hyperbola f n, which is to be found ; thus, the area arm shall be to the area r m n p in a given ratio. Prop. 10. Porism. — A conic hyperbola being given, a point may be found, such, that if from it there be drawn straight lines, to the innumerable intersections of the given curve with all the straight lines drawn through a given point in one of the given asymptotes, the first- mentioned lines shall cut, in a given ratio, the areas of all the triangles whose bases and altitudes are the co-ordinates to a second conic hyperbola, which may be found, at the points where it cuts the lines drawn from the given point. Prop. 11. Porism. — A conic hyperbola being given, a straight line may be found, such, that if another move along it in a given angle, and pass through the intersections of the 12 GBNEEAL THEOEEMS, CHIEFLY POEISMS, curve with all the parabolas, or hyperbolas, of any given order whatever, lying between straight lines to be found, the moving line shall cut, in a given ratio, the areas of the curves described, contained by the peripheries and co-ordinates to another conic hyperbola, that may be found, at the points where this cuts the curves described. .* Peop. 12. Porism. — A conic hyperbola being given, a straight line may be found, along which if another move in a given angle, and pass through any point whatever of the hyperbola, and if fhis point of section be joined with another that may be found, the moving line shall out, in a given ratio, the triangles whose bases and altitudes are tbe co- ordinates to a conic hyperbola, which may be found, at the points where it meets the lines drawn from the point found. Scholium. I proceed to give one or two examples, wherein areas are cut in a given ratio, not by straight lines, but by curves. PfiOP. 13. Porism. Fig. 14. — A conic hyperbola being given, if through any of its innumerable intersections with all the parabolas of any order, lying between straight lines, of which one is an asymptote, and the other may be found ; an hyperbola of any order be described, except the conic, from a given origin in the given asymptote perpen- -K dicular to the axis of the parabolas, the hyperbola thus described shall cut, in a given ratio, an area, of the parabolas, which may be always found. If from G, as origin, in a b, one of l m's asymptotes, there be described an hypeibola i c', of any order whatever, except the first, and passing through m, a point where l m cuts any of the parabolas a m, of any order whatever, dravra from a a point to be found, and lying between ab and Ac, an area acd may be always found (that is, for every case of a m and i c'), which shall be constantly cut by i c', in the given ratio of m : n ■ that is, the area amn : nmdc : : m : n. I omit the analysis, which leads to the following construction and composition. B ,.14. I J \ j^ / v~ ■— S p X " G ■v^^ G — -c K IN THE HIGHEE GEOMETRY. 13 Constr. Let to + n be the order of the parabolas, and p + q that of the hyperbolas. Find ^ a 4th proportional to m-\- n, q — p and m-\-2n; divide G B in A, so that A K : A G : : q '■ p + ^^^° r -. q-.-. M (ij — p) V TO + 2ft i.e., the conBtaiit rectangle or space to which at . sii is equal. 14 geiteeaij theoebms, chiefly .poeisms, Par. L M X (m + n) , . .... AG : q — p; consequently a c D = r muiu- plied by ( — ■ ^ — - +p 1 and diminislied by — — g--*-® ii. 4- 4. • Par. LM X (m + n) ^ X A N X ; theretore, transposing -, r — X g -p MX (q -p) m ■ m ■\-2n q . AG n q — p \ . ,, M + N — X , + p IS equal to a c d 4 X a n x 2ft 1 -^ J ^ M q-p ; and par. l m will be equal to M + N O . A g\ M ACD H X AN X X M q — p / q — p , — ^^ — -, that is, m + ra q — p \ , N / o X — 1— +1^ X (m + n) m + 2(1 1 / M , , X \1— P) X ACD+g'.ANXAG M + N m + 2ft ^ -^ Now it was before demonstrated, that the parameter of lm g . AG is equal to ap x (mp + p+ V-P + 'v • W — W \ rpj^jg j^ V TO + 2m y M , . X {q —P) X ACD+g.ANX AG M + N therefore equal to " , ~ li.- 1 • 1, j.1. 1, (" + ") X (? - p) , , M multiplying both by — — + p, we have X (g -i')+P 7ft + 2ra -^ M + N X (g — i?) X ACD + q . AN X AG = APX(MrX (^3 + (« + «) X (q -p)'^ m -\-2n Tj+q.KCi). Prom these equals take g' . AG X an, and there remains M , . ,, /(« + «') X ((/ — p") X (g — i') X ACD equal to AP x pm x -^ — ■ — - — -^ — — M + N V ™ + 2ft rsr THE HIGHER GBOMBTET. 15 )M + q . AG X (ap — an) ; or, dividing by q —p, M + N XACD = APX T-TT + -^- X PM + — i— X AG X \OT + 2«y \q -pj q-p (AP — AN). JSfow, -— X AP X PM IS equal to the area m -{- 2n A p M ; therefore the area A p m together with —^ — x ap . pm, q -p Q p and X AG X (ap — an), orAPM with — - — x ap . p m q-p q-p X AG X (AN — ap), or APM + — X AP . PM — q-p q-p X rect. PT, is equal to x acd. Now ic' is an q-p M + N hyperbola of the order p + q; therefore its area is x p -q P rect. G H . M H. But a is greater than p ; therefore — - — is p-q ,. , p X GH . HM . ^_ , , ^, negative, and ■ is the area m h k c : and the area q-p P NT kg' is equal to —^ — x gt x tn: therefore mnth is equal P to (mHKC' — NT kg'), or to —^ — X (GH . MH — GT - Tn). q — p From these equals take the common rectangle a t, and there p remains the area mpn, equal to — ^—— x ap x mp — q-p q-p X P T ; whieb was before demonstrated to be, together with M APM, equal to .acd. Therefore mpn, together with M -f- N If APM, that is, the area a m n, is equal to . a o d : con- ' ' M M + S sequently amn : acd :: m:m + N; and (dividendo) a m n : N m D c : : m : N. An area has therefore been found, which the hyperbola ic' always cuts in a given ratio. Therefore, a conic hyperbola being given, &c. Q. e. d. 16 GENEBAL THBOEEMS, CHIEFLY POEISMS, Scholium. This proposition points out, in a very striking manner, the connexion between all parabolas and hyperbolas, and their common connexion with the conic hyperbola. The demonstration here given is much abridged ; and, to avoid circumlocution, algebraic symbols, and even ideas, have been introduced; but by attending to the several steps, any one will easily perceive that it may be translated into geometrical language, and conducted on purely geometrical principles, if any numbers be substituted for m, n, p, and q; or if theSe letters be made representatives oi lines, and if conciseness be less rigidly studied. Prop. 14. Theorem.— A common logarithmic being given ; if from a given point, as origin, a parabola, or hyperbola,' of any order whatever be described, cutting in a given ratio a given area of the logarithmic ; the point where this curve meets the logarithmic is always situated in a conic hyperbola, which may be found. Scholium. This proposition is, properly speaking, neither a porism, a theorem, nor a problem. It is not a theorem, be- cause something is left to be found, or, as Pappus expresses it, there is a deficiency in the hypothesis : neither is it a porism ; for the theorem, from which the deficiency dis- tinguishes it, is not local. Prop. ,15. Porism. Fig. 16. — A conic hyperbola being given ; two points may be found, from which if straight lines IP Fi.g.«. ^® inflected, to the innumerable intersec- tions of the given curve with parabolas or hyperbolas, of any given order whatever, described between given straight lines; and if co-ordinates be drawn to the inter- sections of these curves with another conic hyperbola, which may be found ; the lines inflected shall always cut off' areas that have to one another a given ratio, from the areas con- tained by the co-ordinates. — Let x and y be the points found • H D the given hyperbola, f is the one to be found ; a d c one of the curves lying between a b and A g, intersecting h d and f e • join XT), YD; then the area AYD:XDCBina given ratio. IN THE HIGHER GEOMETRY. 17 Prop. 16. Porism. Pig. 16. — If between two straigM lines making a right angle, an infinite number of parabolas of any order whatever be described ; a conic parabola may be drawn, such, that if tangents be drawn to it at its intersec- tions with the given curves, these tan- gents shall always cut, in a given ratio, the areas contained by the given curves, the curve found, and the axis of the given curves. — Let amn be one of the given parabolas ; d m o the parabola found, and t m its tangent at m : a t m shall have to t m r a given ratio. Prop. 17. Porism. — A parabola of any order being given; two straight lines may be found, between which if innu- merable hyperbolas of any order be described ; the areas cut off by the hyperbolas and the given parabola at their inter- sections, shall be divided, in a given ratio, by the tangents to the given curve at the intersections; and conversely, if the hyperbolas be given, a parabola may be found, &c. Prop. 18. Porism. — A parabola of any order (m -f- n) being given, another of an order (m -J- 2n) may be found, such, that the rectangle under its ordinate and a given line, shall have always a given ratio to the area (of the given curve) whose abscissa bears to that of the curve found a given ratio. Example. Let to = 1, n = 1, and let the given ratios be those of equality ; the proposition is this : a conic parabola being given, a semi-cubic one may be found, such, that the rectangle under its ordinate and a given line, shall be always equal to the area of the given conic parabola, at equal abscissae. Hcholium. A similar general proposition may be enunciated and exemplified, with respect to hyperbolas ; and as these are only cases of a proposition applying to all curves whatever, I shall take this opportunity of introducing a very simple, and I think perfectly conclusive demonstration, of the 28th lemma, " Principia," Book i., " that no oval can be squared." It is well known, that the demonstration which Sir Isaac Newton gives of this lemma is not a little intricate; and, c 18 GENBEAL THBOEEMS, OHIEFLT POKISMS, wJietlier from this difficulty, or from some real imperfection, or from a very natural wish not to believe that the most celebrated desideratum in geometry must for ever remain a desideratum, certain it is, that many have been inclined to call in question the conclusiveness of that proof. Let AMC be any curve whatever (fig. 17), and d a given line ; take in a & a part a-p, having to a p a given ratio, and erect a perpendicular pm, such, that the rectangle pm . "o shall have to the area apm a given ratio; it is evident that m will describe a curve amc, which can never cut the axis, unless in a. Now because j) m is pro- A P M portional to , or to apm, pm will always increase ad infinitum, if amc is infinite ; but if amc stops or returns into itself, that is, if it is an oval, jj m is a maximum at 6, the point of a 6 corresponding to b in a b ; consequently the curve a mo stops short, and is irrational. Therefore p m, its ordinate, has not a finite relation to ap, its abscissa ; but ap has a given ratio to A P ; therefore p m has not a finite relation to a p, and apm has a given ratio to ^ jn ; therefore it has not a finite relation to a p, that is, A p m cannot be found in finite terms of ap, or is incommensurate with A P ; therefore the curve a m b cannot be squared. Now a m b is any oval ; therefore no oval can be squared. By an argu- ment of precisely the same kind, it may be proved, that the rectification, also, of every oval is impossible. Therefore, &C. Q. E. D. I shall subjoin three problems, that occurred during the consideration of the foregoing .propositions. The first is an example of the application of the porisms to the solution of problems. The second gives, besides, a new method of re- solving one of the most celebrated ever proposed, Kepler's problem ; and the last exhibits a curve before unknown, at least to me, as possessing the singular property of a constant tangent. IN THE HIGHER GEOMETRY. 19 Prop. 19. Problem. Fig. 18. — A common logaritlimic being given; to describe a conic hyperbola, such, that if from its intersection with the given curve a straight line be drawn to a given point, it shall cut a given area of the logarithmic in a given ratio. The analysis leads to this construction. Let bme be the logarithmic, g its modula; A B the ordinate at its origia a ; let c be the given point ; a n o b the given area ; m : n the given ratio : draw B Q parallel to A n ; find D a 4th proportional to m, the reotartgle bq . oq, and M + N. From A D out off a part a l, equal to A c together with twice G; at l make lh perpendicular to ad, and between the asymptotes al, hl, with a parameter, or constant rectangle, twice (d + 2 . A B . g) describe a conic hyperbola ; it is the curve required. Prop. 20. Problem. Fig. 19. — To draw, through the focus of a given ellipse, a straight line that shall cut the area of the ellipse in a given ratio. — Const. Let AB be the transverse axis, ef the semi-conjugate ; e, of conse- quence, the centre ; o and l the foci. On A B describe a semicircle. Divide the quadrant ak in o in the given ratio of m to u, in which the area is to be cut, and dessciibe the cycloid G M R, such, that the ordinate p m may be always a 4th propor- tional to the arc o Q, the rectangle a b x 2 f e, and the line c l ; this cycloid shall cut the ellipse in m, so that, if m c be joined, the area A c m shall be to c M b : : m : jst. Demmstr. Let ap = a;, pm = y, ac = c, ab = a, and 2ef = & ; then, by the nature of the cycloid g m e, — p m : o Q : : 2 FE X AB : CL, and QO=AO-AQ = Dy const. ' ^ X (ak - aq) ; also, CL = AB - 2 AC, since ac = lb. There- c 2 20 QENIiBAIi THEOEEMS, CHIBFLT POEISMS, fore, — p M : x ak — aq :: abx 2ef : ab — 2ac; M + N or — V : X arc 90° — arc vers. sin. x :: ah : a — 2 c; M + N ^ therefore - y (a - 2 c) or + 2/ (2 c - a) = a 6 x I X arc . . lis ^ "1" . 90° — arc v. s. x ), and by transposition a 6 X arc . v. s. a; + y (2 c — a) = — '- — X arc 90°. To these equals add 2y {x — x) = 0, and multiply by — 1 ; then will ab x arc v. s. M X + (2 X — a) y — 2 1/ (x — c) = X ab arc 90°, of which ,, , , . , ,, . a6 X arc v. s. x the 4th parts are also equal ; theretore -^ ]- ' (2x — a)y y , , ab m i ; — '-2- -^-(x — c) = -r X X arc 90°. Now be- 4 2^ ' 4 M+N 6* b cause A F B is an ellipse, y'^ = ~-^ x {ax — a?), and y = - , , ,, ,, f a 6 X arc V. s. a; 2 a; - a V (a 37 — a;^) ; therefore -| — X - V (aoo -a?) -'^(x-g) ^%- X X arc 90°. Mul- ct^ ^2^ ^ 4 M + N tiply both numerator and denominator of the first and last , ^1 .,. 6 a' 2x — a b terms by a ; then will - X -r X arc v. s. a; H X - 01 4 4 a 4 {ax-x^)-^(x-o) = - X ^ X ~— X arc 90°. Now ^ '2^ a4M + N the differential of an arc whose versed sine is x and radius — , A is equal to -— T7 —, which is also the differential of the 2jJ (ax — ar) X h / n^ arc whose sine is V- and radius unity; therefore - X I — X Ci a \ 4 arc rtf THE HIGHER GEOMBTET. 21 ,x 2x — a , \ V I smV- H J — X V(ax -x')) - ^ (a; - c) is equal to -, a M X 7 X X arc 90° ; and, by the quadrature of the circle, "'. ■ r ^ 2x — a ,,i „. . -^ X arc sm. V ^ -I j — X */ {a x — x% is the area whose abscissa is x ; consequently the semicircle's area is — X arc 90°- But the areas of ellipses are to the corresponding areas of the circles described on their transverse axes, as the conjugate to the transverse : therefore - X I — X arc sin. a \4 . X 2x — a . \ .. V — 1 J — X fj [ax — x^)\ is the area whose abscissa is a ' 4 X, of a semi-ellipse, whose axes are a and 6 ; and consequently 6 a* - X -J- X arc 90° is the area of the semi-ellipse. Therefore the area a p m — - fa; — c) is equal to of a m p b. But y /■ \ PM ^ . pm'' . ,, , . , - Qa; — c) = -— X (A p — A c) = -— X P c, is the triangle M c p M ; consequently, a p m — c p m, or a c m, is equal to X AMPB; and ACM : AMFB :: M : m + n; or (dividendo) ACM : CMFB :: m : n; and the area of the ellipse is cut in a given ratio by the line drawn through the focus, q. e. d. Of this solution it may be remarked, that it does not assume as a postulate the description of the cycloid; but gives a simple construction of that curve, flowing from a curious property, by which it is related to a given circle. This cycloid, too, gives, by its intersection with the ellipse, the point required, directly, and not by a subsequent construc- tion, as Sir Isaac Newton's does. I was induced to give the demonstration, from a conviction that it is a good instance of the superiority of modem over ancient analysis ; and in itself '^perhaps no inelegant specimen of algebraic demonstration. 22 GENEEAI, THEOREMS, CHIEFLY POKISMS, Prop. 21. Problem. Kg. 20. — To find tlie curve whose tangent is always of the same magnitude. Analysis. Let mn be the curve required, a b the given axis, s m a tangent at any point m, and let a ^ he the ' given magnitude ; then, SM.g. =SP.g.+PM.g'. = d'; -: therefore, (^ a; = — X Va^ — y^- In order to integrate r y a^dy dy this equation, divide — a/ a* — v" into its two parts, 7= y y 'Jot y and ; to find the integral of the former. "'~f 1 + a^dy ^ a^dy /J a" - y^ = - ax y ^/a'-y' 2/ a + V"^^' a X differeiitialof \ J / ady c? dy yWa' - y y a + ^/ a" -y^ y therefore the integral of a^dy a + »J a^ -y^ ; y is — a X hyp. log. y .\/ a^ - y" a+ 'J a' -y^ ^ ^^^ ^^^ integral of the other part, ~^ ^ , y V a^-y^ dy is + V a^ — 2/* ; therefore the integral of the aggregate — V a' _ j,«, IS ^ (jB _ y ZrV^ - a X h. 1. a + V a'- y\ y IN THE HIGHBE GEOMETET. 23 y V ffiS _ y« _|_ a X h. 1. J ' * filial equation to the curve as required. Q. E. i. I shall throw together, in a few corollaries, the most re- markable things that have occurred to me concerning this curve.* Carol. 1. The subtangent of this curve is \/ (a^ — y^). Carol. 2. In order to draw a tangent to it, from a given point without it ; from this point as pole, with radius equal to a, and the curve's axis as directrix, describe a concoid of Nicomedes : to its intersections with the given curve draw straight lines from the given point; these will touch the curve. Coral. 3. This curve may be described, organically, by drawing one end of a given flexible line or thread along a straight line, while the other end is urged by a weight to- wards the same straight line. It is consequently the curve of traction to a straight line. Carol. 4. In order to describe this curve from its equation ; change the one given above, by transferring the axes of its eo-ordinates : it becomes (y being = p' m and x — A p'), y = V («* - a^) + « X h. 1. °^ -jr ; which may be used with ease, by changing the hyberbolic into the tabular logarithm. Thus, then, the com^mon logarithmic has its sub- tangent constant; the conic parabola, its subnormal; the circle, its normal ; and the curve which I have described in this proposition, its tangent.f * There are other properties of this curve noted in Tract V. of this volume. t This Tract was printed in Phil. Trans, for 1798, part 2. The fluxional notation has alone been altered to the differential.— The schol., p. 17, is subject to doubt from the leminscata and other similar curves. See note I. at end of this volume.— The subject of Porisms is treated of in Note II. ( 24 ) II. KEPLEE'S PROBLEM. Kepler was led, after tlie discovery of tlie law which bears his name, to the celebrated problem which also bears it. Having proved that the squares of the periodic times are as the cubes of the distances, he wished to discover a method of finding the true place of a planet at a given time — one of the most important and general problems in astronomy. By a short and easy process of reasoning, he reduced this question to the solution of a transcendental problem ; — ^to draw from a given eccentric point, in the transverse of an ellipse (or the diameter of a circle) a straight line, which shall cut the area of the curve in a given ratio ; or, in the language of astro- nomers, " from the given mean anomaly, to find the anomaly of the eccentric." This most important problem is evidently transcendental ; for, in the first place, the curve in question is not quadrable in algebraic terms ; and, in the next place, admitting that it were, the solution cannot be obtained in finite terms. As the general question, for all trajectories, is of vast importance ; and as the paper of Mr. Ivory, in the 'Edinburgh Trans- actions,' contains a most successful application of the utmost resources of algebraic skill to the most important case of it, I shall premise a few remarks upon the problem, when enunciated in different cases. Let D'' be the given area of any curve, whch is the tra- jectory of a planet or other body, or which is to be cut in the eefleb's fboblem. 25 given ratio of m to n. Let x and y, as usual, be the abscissa and ordinate, and c the eccentricity of the given point, through which the radius vector is to be drawn, if the equation is taken from the centre ; or, if it is taken from the vertex, let c be the distance of the given point from that vertex, as the focal distance in the case of the planets or comets (supposing the comets to revolve round the sun in parabolas or eccentric ellipses, having the sun in the focus), then, it may easily be found, that the following differential equation 2 j ydx -\- y (c — x) = — '■ , if resolved for the case of any given curve, gives a solution of the problem for that curve. Instead of fydx, there must be substituted the general expression for the area found by integration ; and y must then be expressed through the whole equation in terms of x, or x in terms of y : There will result an equation to x, or to y, which, when re- solved, gives a solution of the problem. Now, it is manifest, that one or both of two difficulties or impossibilities may occur in this investigation of the value of X. It may be impossible to exhibit f y d x in finite terms ; and it may be impossible, even after finding f y d x,to resolve the equation that results from substituting the value of f y d x in the general equation above given. Thus, if the given curve is not quadrable, the equation can never be resolved; but, although the curve is quadrable, it does not follow that the equation can be resolved. In the case of the circle and ellipse, both these difficulties must of course occur. The value oi f ydx in the circle being */ ax — of (where a and b are the transverse and conjugate), neither of which differentials can be integrated in finite terms, the general equations become indefinite or unintegrable. The lemniscata (a curve of the fourth order) is quadrable in algebraic terms : but the resolution of our general equation cannot, in this case, be performed in finite terms ; it leads to 26 KBPLER's PaOBLEM. an equation of the sixth order, veiy complicated and difficult.* But, if the given point is in the centre or punotum duplex of the curve, the equation is a cubic one, wanting the second term, and of course, easily resolved. It pften happens, too, that the problem may be resolved, in general, for a curve ; but that, in one particular part of the axis, Ihe solution becomes impossible. As this is rather a singular circumstance, we shall attend a little more minutely to it. Let it be required to resolve the problem for the case of comets, supposing those bodies to move in parobolic orbits. /^ The general equation for x becomes xij x + ^0 /\f x — —r- x : a cubic wanting the second term, and easily resolved. m-\-n But, in certain cases, viz., when c, the distance of the given point from the vertex, is less than 3 D x * / -: — ; ^ the ^ 'V 4 a (m + b)' problem cannot be resolved; for, in this case, the cube of one-third of the co-efficient of x is less than the square of half the last term, which is the well-known irreducible case of * The equation is of the following form, a being the lemnisoata's semi- diameter : — -|-6c(l - o)k» -|-(9c« (l-l-a2 - 2a) - qs)** jgi - 6 a2 (1 - a2) a;3 = ^ n ' -f ("s a* - 9 c2 a2 (1 + a2 - 2 a) ) ^2 -|-6ca*(l - d)x ■ a cuboBuhic having all its terms («' -|- A a* + B s* -f ic' -|- D aj^ ^ E a? -f F = 0), in which A, 0, and E vanish when the centre of motion (or of the radii vectores) is in the punctum duplex, and then the equation to « is a;^ -|- B K* -f Da;2 + P = 0, reducible to the cubic z^ + Az -^ = 0. So that the problem is soluble, except when the eccentricity is such that ( — ) is less than ( -^ ]. the irreducible case of Cardan's rule. 1 \ m + ny keplee's peoblem. 27 Cardan's rule. In tiis case, therefore, the problem of the comet is reduced to infinite series, or to the arithmetic of sines. If the given point is in the vertex of the curve, that is, hn the perihelion, the problem is always resolvable, being reduced to the simple extraction of a cube root; and this is the case of comets which fall into the sun. The resolvable case of the lemniscata is in the same circum- stances, as may easily be seen by inspecting its equation. In substituting for j y d x, its value in our general equation, we may either give it in terms of oo, that is, of the abscissa ; or in terms of x y, that is, of the circumscribing rectangle ; and neglect any further substitution. Thence arises a dif- ferent and more elegant solution of the problem, by the inter- section of curve lines ; for we obtain an equation to a new curve, which cuts the former in the point required. Thus, by such a process in the case of the comet, we obtain the 6mD^ ^ . , , , ^ equation y = -. :—, -— ^ to a conic hyperbola. JB or ^ ■^ (m -t- k) (a; -f- 3 c) ^^ brevity's sake, put = ^, it will cut the given tra- jectory in the point required : If the given point is in the perihelion, then the perpendicular must be raised at the vertex of the parabola. The solution here given by a locus, is evidently general, and has no impossible case. But there are some instances in which such solutions, although perhaps the only practi- cable ones, are nevertheless attended with an impossible case. Let us take that of the lemniscata. Instead of the irresoluble equation of the sixth order, we obtain, by the lasi^mentioned method, a cubic equation of this form, y = — -^ — ^r— | ; 28 Kepler's «peoblem. to a curve of the third order, called, if I rightly remember, by Sir Isaac Newton, in his " Enumeratio Linearum Tertii Ordinis," a paraholism of the hyperbola. Now, although this is extremely simple, in comparison of the complex equation given by the, direct method first mentioned, it has manifestly' one impossible case, viz., when <^ is equal to ax /v/ q' "'^ when the given area is to two-thirds of the square of the diameter of the curve, as m + « to m : In this case, no para- bolism of the hyperbola can be drawn, which will intersect the given curve in the point required ; and this is an impos- sibility affecting every possible value of c ; that is, every position of the given point, in this particular magnitude of the given area. But this circumstance makes no difference on the resolution of the problem by the direct method. Thus, when the eccentricity vanishes, or the given point is in the punctum duplex, the solution is derived from a cubic equation equally resolvable when 4> — ^ \/ q ^S when <^ is either < or > a X y/| The method of resolving this interesting problem by loci, is the source of an immense variety of the most curious pro- positions concerning the properties and mutual relations of curve lines ; and, more especially, leads us to the discovery of various porisms, which we otherwise should never have found out. In order to generalize and extend these, it is necessary that, instead of considering merely the case of Kepler's problem, where an area is cut by a straight line, we should consider also the far more difficult problem of cutting the area of one curve by another curve, in a given ratio ; and then the problem may be extended to the section, not of one curvilinear area, but of an infinite number of areas, contained between two given lines, or of the areas of all the curves of a particular kind which can be drawn between those gLven lines. It is easy to perceive, that the same keplek's problem. 29 resolution before adverted to, will not apply to those more complicated problems. But the reader will find a variety of examples of this species of proposition in the ' Philoso- phical Transactions of the Eoyal Society of London for 1798.,' which were investigated chiefly in the manner above de- scribed.* It is evident that the application of such problems to physics does not proceed so far ; for we have never yet discovered an example of a central force acting in a curvi- linear direction. The solutions now described, of Kepler's problem, and of several problems of a more general sort, are of a theoretical nature. They exhibit the mode of expressing by curve lines, or imaginary relations of known quantities, the relation required of the quantities given ; they rather vary the diffi- culty, or simplify the relation, than remove the impediments to practical measurement. If it be required to exhibit the anomaly of the eccentric, we may indeed adopt the solution given by Sir Isaac Newton (Principia., lib. i. prop. 31, and Schol.), or that hinted at by Kepler himself. The Newtonian solution proceeds upon the description of a cycloid, and an easy construction, by which the point required is found in the intersection of a straight line with the given trajectory. In the tract referred to, a solution is given more directly, by the intersection of a species of a cycloid of easy description, with the given curves, without any subsequent construction. But these solutions, though more pleasing and beautiful in theory, are useless, when it is required to exhibit a value of the abscissa corresponding to the anomaly of the eccentric, or its supplement, in such a manner that a comparison may be made of this line with some known measure of length. It becomes necessary, in this case, to find a numerical value of the quantity in question. Now, this can only be done by a series ; and the two great objects in finding such a series are, first, to give one which may be regulated by a simple law ; and, secondly, to give one which may converge rapidly : so * The Paper is given in this volume : it is tlie First Tract. 30 keplbe's pboblem. that its denominators rapidly increasing, the quantities may soon become so small, as not to deserve attention in our com- putations. The approximation given by Mr. Ivory in his paper in the ' Edinburgh Transactions,'* deserves the first place among those of which we are in possession, whether we consider its simplicity, universality, or accuracy. The series is of easy management, applies to the most eccentric orbits, as well as to those approaching nearer to the circle, and to all degrees of eccentricity in the given point, the centre of forces. It has the benefit, too, of a most rapid convergence. He first gives a very simple and elegant geometrical method of approximation, by an application of the rectangular case of the general problem de inclinationibus of the ancient geo- meters. But as this is by no means satisfactory to the practical calculator (for reasons before assigned), he proceeds next to the algebraic solution. He begins with investigating the series for the eccentric anomaly when the mean anomaly is a right angle. It con- verges quickly, and the terms err alternately, by defect and excess, the difference growing continually less and less. He then proceeds to the investigation of a similar series, found in the same manner, for the other cases of the mean anomaly. I should in vain attempt to give the reader a more minute idea of this solution, without a detail as full as the paper now before us, and shall only note an erratum that has crept into the twelfth article. After putting tan. A = e X COS. — , m X sec. 45°, he infers that sin. -~ = tan. — -^— 2 2 di A X 45°; it should be sin. -^ = tan. —- x sin. 45°. He next gives two examples of the application of his method to geometric problems, concerning the circle. The one, is to bisect a given semicircular area by a chord from a * Vol. V. p. 111. 1802. keplee's problem. 31 given point in the circumference. The results of the series which he gives for the eccentric anomaly are as follows : — Eccent. anom. = 47° 4' (first value, and less than the truth). „ = 47° 40' 14" (second value, and greater than the truth). = 47° 39' 12" (third value, and less than the truth). From this example, may be perceived the excellence of the method ; for, whereas the first two terms differ by nearly 36', the second and third differ only by 1' 2" ; or, in other words, while, by the two first trials, we come to a space of above half a degree, in some part of which the point required is to be found ; by the second and third trials, we obtain a space of aboiit the sixtieth part of a degree, in some'' part of which lies the result. By the third term of the series, then, we obtain a solution not more than 31" distant from the truth, and this in circumstances the least favourable. The other example is a solution of the problem — •" to draw from a point in the circumference two chords which shall trisect the circular area." Here the Eccent. anom. = 30° 33' (first value less). = 304° 4' 11" (second greater). Euler's solution (^Analysis, Inf. XI. 22) differs litte more than 30" from this solution, given by Mr. Ivory's second term. This specimen will sufficiently show the superior excel- lency of Mr. Ivory's method. Former analysts have only resolved the case within the eccentricity is small : his solu • tion extends to comets as well as planets. For the planets, his rules apply with peculiar accuracy and ease ; and his series converges with extreme rapidity ; so much so, that we may consider the approximation of one term sufficient for practice. He has given a table of the values of the errors (or differences) for the different planets computed in this way. He adds an exemplification for the famous comet of 1682, supposed to be the same which reappeared in 1759. His first approximation 32 Kepler's problem. for the anomaly of the eccentric, reckoned from the aphelion (16 days 4 hours and 44' from its perihelial passage), is 173° 61', and too small. The second approximation is 173° 64' 36", exceeding the real eccentric anomaly from the perihelion by only a few seconds. The application of the author's last correction, deduced from the comparison of the parabolic and elliptic trajectories, to the finding of the heliocentric place, and also the helio- centric distance (or radius vector of the cometio orbit), con- cludes this paper. I have been the more gratified by a perusal of this last branch of Mr. Ivory's inquiry, because the speculations had formerly occurred to me in a similar form. The introduction of the parabola, which admits of quadrature, and of definite solution, so far as regards Kepler's problem, has always appeared to me the surest method of rectifying the computations of the heliocentric places and distances of comets, or of their perihelial eccentric anomalies and radii veotores, during the small perihelial part of their trajectories which we are permitted to contemplate. In that part, the eccentric ellipse and the parabola nearly coincide ; and, after all, we are not perfectly certain that those singular bodies do not move in orbits strictly parabolic* * This Papei appeared in the Second Number of the 'Edinburgh Eeview,' January, 3803. ( 33 .) III. DYNAMICAL PRINCIPLE.— CALCULUS OF PARTIAL DIFFERENCES.— PROBLEM OF THREE BODIES. The pleasures of a purely scientific life have often been described ; and they have been celebrated with very heartfelt envy by those whose vocations precluded or interrupted such enjoyments, as well as commended by those whose more fortunate lot gave them the experience of what they praised ; but it may be doubted, if such representations can ever apply to any pursuits so justly as to the study of the mathematics. In other branches of science the student is dependent upon many circumstances over which he has little control. He must often rely on the reports of others for his facts ; he must frequently commit to their agency much of his inquiries ; his research may lead him to depend upon climate, or weather, or the qualities of matter, which he must take as he finds it ; where all other things are auspicious, he may be without the means of making experiments, of placing nature in circum- stances by which he would extort her secrets ; add to all this the necessarily imperfect nature of inductive evidence, which always leaves it doubtful if one generalisation of facts shall not be afterwards superseded by another, as exceptions arise to the rule first discovered. But the geometrician relies entirely on himself ; he is absolute master of his materials ; his whole investigations are conducted at his own good pleasure, and under his own absolute and undivided control. He seeks the aid of no assistant, requires the use of no apparatus, hardly wants any books ; and with the fullest reliance on the perfect instruments of his operations, and on the altogether certain nature of his results, he is quite assured that the truths which he has found out, though they 34 DYNAMICAL PEINCIPLB. may lay the foundation of further discoveiy, can never by possibility be disproved, nor his reasonings upon them shaken, by all the progress that the science can make to the very end of time. The life of the Geometrician, then, may well be supposed an uninterrupted calm ; and the gratification which he de- rives from his researches is of a pure and also of a lively kind, whether he contemplates the truths discovered by others, with the demonstrative evidence on which they rest, or carries the science further, and himself adds to the number of the interesting truths before known. He may be often stopped in his researches by the difficulties that beset his path ; he may be frustrated in his attempts to discover relations depending on complicated data which he cannot unravel or reconcile ; but his study is wholly independent of accident; his reliance. is on his own powers; doubt and con- testation and uncertainty he never can know ; a stranger to all controversy, above all mystery, he possesses his mind in unruffled peace ; bound by no authority, regardless of all con- sequences as of all opposition, he is entire master of his con- clusions as of his operations ; and feels even perfectly indifferent to the acceptance or rejection of his doctrines, because he confidently looks forward to their universal and immediate admission the moment they are comprehended. It is to be further borne in mind, that from the labours of the Geometrician are derived the most important assistance to the researches of other philosophers, and to the perfection of the most useful arts. This consideration resolves itself into two : one is the pleasure of contemplation, and consequently is an addition to the gratificatien of exactly the same kind, derived immediately from the contemplation of pure mathematical truth ; much, indeed, of the mixed mathematics is also purely mathematical investigation, built upon premises derived from induction. The other gratification is of a wholly different description ; it is connected merely with the promotion of arts subservient to the ordinary enjoyments of life. This is only a secondary and mixed use of science to the philosopher; OALOULUS OP PABTIAIi DIFPEBENOBS. 35 the main pleasure bestowed by it is the gratification which, by a law of our nature, we derive from contemplating scientific truth, when indulging in the general views which it gives, marking the unexpected relations of things seemingly un- connected, tracing the resemblance, perhaps identity, of things the most unlike, noting the diversity of those appa- rently similar. This is the true and primary object of scientific investigation. This it is which gives the pleasure of science to the mind. The secular benefits, so to speak, the practical uses derived from it, are wholly independent of this, and are only an incidental, adventitious, secondary advantage. (See Introductory Eemarks to this volume.) It is an illustration of the happiness derived from mathe- matical studies, that they possess two qualities in the highest degree, not perhaps unconnected with one another. They occupy the attention, entirely abstracting it from all other con- siderations ; and they produce a calm agreeable temper of mind. Their abstracting and absorbing power is very remarkable, and is known to all geometricians. Every one has found how much more swiftly time passes when spent in such investigations, than in any other occupation either of the senses or even of the mind. Sir Isaac Newton is related to have very frequently forgotten the season of meals, and left his food awaiting for hours his arrival from his study. A story is told of his being entirely shut up and disappearing, as it were eclipsed, and then shining forth grasping the great torch which he carried through the study of the heavens ; he had invented the Pluxional Calculus. I know not if there be any foundation for the anecdote; but that he cbntinually remained engaged with his researches through the night is certain, and that he then took no keep of time is undeniable. It does not require the same depth of understanding to expe- rience the effects of such pursuits in producing complete abstraction ; every geometrician is aware of them in his own case. The sun goes dowii unperceived, and the night wanes afterwards till he again rises upon our labours. They who have experienced an incurable wound in some D 2 36 .DYNAMICAL PRINCIPLE. prodigious mental afHiotion, have confessed, that nothing but mathematical researches could withdraw their attention from their situation. Instances are well known of a habit of drinking being cured by the like means ; an inveterate taste for play has, within my own observation, been found to give way before the revival of an early love of analytical studies. This is possibly a cause of the other tendency which has been mentioned, the calming of the mind. Simson (the restorer of the Greek geometry) tells us how he would fly from the con- flicts of metaphysical and theological science, to that of neces- sary truth, and how in those calm retreats he ever " found himself refreshed with rest." Greater tranquillity is possessed by none than by geometricians. Even under severe privations this is observed. The greatest of them all, certainly the greatest after Newton, was an example. Euler lost his sight after a long expectation of this calamity, which he bore with perfectly equal mind ; both in the dreadful prospect and the actual bereavement, his temper continued as cheerful as before; his mind, fertile in resources of every kind, supplied the want of sight by ingenious mechanical devices, and by a memory more powerful even than before.* He furnishes an * My late learned and esteemed friend, Mr. Gougli, of Kendal, was another example of studies being pursued under the same severe depriva- tion — but he had never known the advantages of sight, having lost his eyes when an infant, and never had any distinct recollection of light. He was an accomplished mathematician of the old school, and what is more singular, a most skilful botanist. His prodigious memory resembled Euler 3, and the exquisite acuteness of his smell and touch supplied in a great measure the want of sight. He would describe surfaces as covered with undulations which to others appeared smooth and even polished. His ready sagacity in naming any plant submitted to his examination was truly wonderful. I had not only the pleasure of his acquaintance, but I have many particulars respecting his rare endowments, from another eminent mathematician, who unites the learning of the older with that of the modern school, my learned friend and neighbour, Mr. Slee, of Tirrel. A detailed account of Mr. Gough's case, by Mr. Slee and Professor Whewell (a pupil of his), would be most curious and instructive. Euler's memory was such, that he could repeat the jEneid, noting tlie words tliat begin and end each page. Mr. Gough also was an excellent classical scholar. CALCULUS OF PAKTIAL DIFFBEENCES. 37 instance to another purpose. Thoiiglitless and superficial observers have charged this science with a tendency to render the feelings obtuse. Any pursuit of a very engrossing or absorbing kind may produce this temporary effect ; and it has been supposed that men occasionally abstracted from other contemplations, are particularly dull of temper. But no one ever had more warm or kindly feelings than Euler, whose chief delight was in the cheerful society of his grandchildren, to his last hour ; and whose chief relaxation from his severer studies was found in teaching these little ones. It has been alleged, and certainly has been somewhat found by experience to be true, that the habit of contemplating necessary truth, and the familiarity with the demonstrative evidence on which it rests, has a tendency to unfit the mind for accurately weighing the inferior kind of proof which we can alone obtain in the other sciences. Once finding that the certainty to which the geometrician is accustomed cannot be attained, he is apt either to reject all testimony, or to become credulous by confounding different degrees of evidence, re- garding them all as nearly equal from their immeasurable inferiority to his own species of proof — much as great sove- reigns confound together various ranks of common persons, on whom they look down as all belonging to a different species from their own. In this observation there is, no doubt, much of truth ; but we must be careful not to extend its scope too far, so as that it should admit of no exceptions. D'Alembert affords one of the most remarkable of these ; as far as physical science went, Laplace afforded another ; in several other branches he was, perhaps, no exception to the rule. Whatever of peace and comfort he enjoyed, D'Alembert owed to geometry, and confessed his obligations. "Whatever he suffered from vexation of any sort, he could fairly charge upon the temporary interruption of his mathematical pursuits. Both portions of his history, therefore, enforce the doctrine which I have laid down. His ' Traits de Dynamique ' at once placed him in the 38 DTNAMICAL PRINCIPLE. highest rank of geometricians. The theory is deduced with perfect precision, and with as great clearness and simplicity as the subject allows, from a principle which he first laid down and explained, though it be deducible from the equality of action and re-action, a physical rather than a mathematical truth, and derived from universal induction, not from abstract reasoning a priori. The Principle is this (' Dyn.' part 2, chap. i.). If there are several bodies acting on each other, as by being connected through inflexible rods, or by mutual attraction, or in any other way that may be conceived ; suppose an external force is impressed upon those bodies, they will move not in the direction of that force as they would were they all uncon- nected and free, but in another direction ; then the force acting on the bodies may be decomposed into two, one acting in the direction which they actually take, or moving the bodies without at all interfering with their mutual action, the other in such direction as that the forces destroy each other and are wholly extinguished ; being such, that if none other had been impressed upon the sj'stem, it would have remained at rest.* This pirinoiple reduces all the problems of dynamics to statical problems, and is of great fertility, as well as of admirable service in both assisting our investigations and simplifying them. It is, indeed, deducible from the simplest, principles, and especially from the equality of action and re- action ; but though any one might naturally enough have thus hit upon it, how vast a distance lies between the mere principle and its application to such problems, for example, as to find the locus or velocity of a body sliding or moving * Lagrange's statement of the principle is the most concise, but I ques- tion if it is the clearest, of all that have been given. " If there be im- pressed upon several bodies, motions which they are compelled to change by their mutual actions, we may regard these motions as composed of the motions wliich the bodies will actually have, and of otlier motions which are destroyed ; from whence it follows, that the bodies, if animated by those motions only, must be in equilibrio.' ' (' Me'o. An.' vol. i. p. 239, Ed. 1811.) It is not easy to give a general statement of the principle, and I am by no means wedded to the one given in the text. (See note III.) CALCTTLUS or PAETIAL DIFFBEENCES. 39 freely along a revolving rod, at the extremity of whicli rod a fixed body moves round in a given plane — a locus whicL. the calculus founded on the Principle shows to be in certain cases the logarithmic spiral.* No one can doubt that the Principle of D'Alembert was in- volved in many of the solutions of dynamical problems before given. But then each solution rested on its own grounds, and these varied with the different cases ; their demonstrations were not traced to and connected with one fundamental prin- ciple. He alone and fi.rst established this connection, and ex- tended the Principle over the whole field of dynamical inquiry. The ' Traite ' contains, further (part 1, chap, ii.), a new demonstration of the parallelogram of forces. The reason of the author's preference of this over the common demonstration is not at all satisfactory. His proof consists in supposing the body to move on a plane sliding in two grooves parallel to one side of the parallelogram, and at the same time carried along in the direction of the other side. This is not one whit more strict and rigorous than the ordinary supposition of the body moving along a ruler parallel to one side, while the ruler at the same time moves along a line parallel to the other side. Indeed I should rather prefer this demonstration to D'Alembert's. The ' Traite de Dynamique ' appeared in 1743 ; and in the following year its fundamental principle was applied by the author to the important and difficult subject of the equili- brium and motion of fluids, the portion of the ' Principia ' which its illustrious author had left in the least perfect state. Pressed by the difficulty of the inquiry, which is one of the most important in Hydrodynamics, the motion of a fluid through an orifice in a given vessel, and despairing of the data afford- . ,„ ydx^ i'Dydy^ . ., . , * The general equation is d^y = "—^ + ■ ^^ + d ~2 ^ which y is the distance of the moving body D from the fixed point, or the length of the rod, at the end of which is the body A, describing an arch of a circle, and X, that arch. The velocity of D is likewise found in terms of the same quantity. 40 DYNAMICAL PEINCIPLE. ■ ing the means of a strict and direct solution, Newton had reoonrse to assumptions marked by the most refined ingenuity, but admitted to be gratuitous and to be unauthorised by the facts. The celebrated Cataract is of this description. He supposes ('Principia,' lib. ii. prop. 36), that a body of ice shaped like the vessel, comes in contact with the upper surface of the liquid and melts immediately on touching it, so as to keep the level of the fluid always the same, and that a cataract is thus formed, of which the upper surface is that of the fluid, and the lower that of the orifice. His first investi- gation assumed the issuing column to be cylindrical, but he afterwards found that the lateral pressure and motion gave it the form of a truncated cone which he called a vein ; and his correction of the former result was a matter of much con- troversy among mathematicians. Daniel Bernoulli at first maintained it to be erroneous against Eiccati and others ; but he afterwards acquiesced in Newton's view. He, however, always resisted the hypothesis of the cataract, as indeed did most other inquirers. Newton's assumptions, in other parts of this very difficult inquiry, have been deemed liable to the same objections ; as where he leaves the purely speculative hypothesis of perfectly uncompressed and distinct particles, and treats of the interior and minute portions of fluids, as similar to those which we know. (Lib. ii. prop. 37, 38, 39.) It must, however, be admitted, as D'Alembert has observed (' Encyc' v. 889, and ' Eesistance des Fluides,' xvii.) that " those who attacked the Newtonian theory on this subject had no greater success than its illustrious author ; some having, after resorting to hypotheses which the experiments refuted, abandoned their doctrines as equally unsatisfactory, and others confessing their systems groundless, and substi- uting calculations for principles." Such was the state of the science when D'Alembert hfippily applied his Dynamical principle to the pressure and motion of fluids, and found that it served excellently for a guide, both in regard to non-elastic and elastic fluids. In fact, the par- ticles of these being related to one another by a cohesion OAIOULUS OF PAETIAL DITPEEBNCBS. 41 which prevents them not from obeying an external impulse, it is manifest that the principle may he applied. Thus, if a fluid contained in a vessel of any shape be conceived divided into layers perpendicular to the direction of its motion, and if V represent generally the velocity of the layers of fluid at any instant, and d v the small increment of that velocity, which may be either positive or negative, and will be different for the different layers, v ± dv will express the velocity of each layer as it takes the place of that immediately below it ; then if a velocity + dv alone were communicated to each layer, the fluid would remain at rest. (' Traitd de Tluides,' liv. ii. chap. 1, theor. 2). Thus the velocity of each part of the layer being taken in the vertical direction is the same, and this velocity being that of the whole layer itself, must be inversely as its horizontal section, in order that its motion may not interfere with that of the other layers, and may not disturb the equilibrium. This, then, is precisely the general dynamical principle already explained applied to the motion of fluids, and it is impossible to deny that the author is thus enabled to demonstrate directly many propositions which had never before been satisfactorily investigated. It is equally undeniable that much remained after all his efforts incapable of a complete solution, partly owing to the inherent diflS.cul- ties of the subject from our ignorance of the internal structure and motions of fluids, and partly owing to the imperfect state in which all our progress in analytical science still has left us, the differential equations to which our inquiries lead having, in very many cases, been found to resist all the resources of the integral calculus. This remark applies with still greater force to his next work. In 1752, he published his Essay on a new theory of the Eesi stance of Fluids. The great merit of this admirable work is that it makes no assumption, save one to which none can object, because it is involved in every view which can well be taken of the nature of a fluid ; namely, that it is a body composed of very minute particles, separate from each other, and capable of free motions in all directions. He 42 dtnamicaij peinoiplb. applies the general dynamical principle i,o the consideration of resistance in all its views and relations, and he applies the calculus to the sohition of the various problems with infinite skill. It is in this work that he makes the most use of that refinement in the integral calculus of which we shall pre- sently have occasion to speak more at large, as having first been applied by D'Alembert to physical investigation, if it was not his own invention. But the interval between 1744 and 1752 was not passed without other important contribu- tions to physical and analytical science. In 1746, he gave his Memoir on the general theory of Winds, which was crowned by the Eoyal Academy of Berlin. The foundation of this able and interesting inquiry is the influence of the sun and moon upon the atmosphere, the aerial tides, as it were, which the gravitation towards these bodies produces; for he dis- misses all other causes of aerial currents as too little depend- ing upon any definite operation, or too much depending upon various circumstances that furnish no precise data, to be capable of analytical investigation. The Memoir consists of three parts. In the first he calculates the oscillations caused by the two heavenly bodies supposing them at rest, or the earth at rest in respect of them. In the second, he investigates their operation on the supposition of their motion. In the third, he endeavours to trace the effects produced upon the oscillations by terrestrial objects. The paper is closed with ramarks upon the effects of temperature. The whole inquiry is conducted with reference to the general dynamical prin- ciple which he had so happily applied to the equilibrium and pressure of fluids, in his first work upon that difficult subject. In treating of Hydrodynamics, D'Alembert had found the ordinary calculus insufficient, and was under the necessity of making an important addition to its processes and its powers^ already so much extended by the great improvements which Euler had introduced. This was rendered still more neces- sary when, in 1746, he came to treat of the winds, and in the following year when he handled the very difficult subject of the vibration of cords, hitherto most imperfectly investigated CAicTJLtrs or partial dipeeeences. 43 by mathematicians.* In all these inquiries the diiferential equations which resulted from a geometrical examination of the conditions of any problem, proved to be of so diiEcult integration that they appeared to set at defiance the utmost resources of the calculus. \V hen a close and rigorous inspec- tion showed no daylight, when experiments of substitution and transformation failed, the only resource which seemed to remain was finding factors which might, by multiplying each side of the equation, complete the differential, and so make it integrable either entirely, or by circular arches, or by logarithms, or by series. D'Alembert, in all pro- bability, drew his new method of treating the subject from the consideration that, in the process of differentiation we successively assume one quantity only to be variable and the rest constant, and we differentiate with reference to that one variable ; so that x dy -{■ y d x\s, the differential of x y, a rectangle, and x y d z + x z dy + y z d x the differential of X y z, a parallelepiped, and so of second differences, d^ z being (when z - x^) = (m* — ni) a:""^ dx^ + m x"''^ d^ x. He pro- bably conceived from hence that by reversing the operation and partially integrating, that is, integrating as if one only of the variables were such, and the others were constant, he might succeed in going a certain length, and then discover the residue by supposing an unknown function of the variable which had been assumed constant, to be added, and after- * Taylor (' Methodus Incrementorum ') had solved the problem of the vibrating cord's movement, but upon three assumptions — that it departs very little from the axis or from a straight line, that all its points come to the axis at the same moment, and that it is of a uniform thickness in its whole length. D'Alembert's solution only requires the last and the first supposition, rejecting the second. The first, indeed, is near the truth, and it is absolutely necessary to render the problem soluble at all. The third has been rejected by both Euler and Daniel Bernoulli, in several cases investigated by them. D'Alembert's solution led to an equation of -7-| 1 = a^ (y^) ™ which t is the time of the vibration, x and y the co-ordinates of the curve formed by the vibration. 44 DYNAMIOAI, PEmOIPLE. wards ascertaining that function by attending to the other conditions of the question. This method is called that of partial differences. Lacroix justly observes that it would be more correct to say partial differentials ; and a necessary part of it consisted of the equations of conditions, which other geometri- cians unfolded more fully than the inventor of the calculus himself; that is to say, statements of the relation which must subsist between the variables or rather the differentials of these variables, in order that there may be a possibility of finding the integral by the method of partial differences. It appears that Fontaine, a geometrician of the greatest genius, gave the earliest intimation on this important subject ; for the function of one or both variables which is multiplied by d X being called M, and that function of one or both which is multiplied hjdy being called N, the canon or criterion of integrability is that dM. _ d'N _ dy dx ' and we certainly find this clearly given in a paper of Fon- taine's read before the Academy, 19th November, 1738. It is the third theorem of that paper. Clairaut laid down the same rule in a Memoir which he presented in 1739 ; but be admits in that Memoir his having seen Fontaine's paper. He expounds the subject more largely in his far fuller and far abler paper of 1740 ; and there he says that Fontaine showed his theorem to the Academy the day this second paper of Clairaut's was read — erroneously, for Fontaine had shown it in November, 1738 ; and had said that it was then new at Paris, and was sent from thence to Euler and Bernoulli. The probability is, that Clairaut had discovered it independent of Fontaine, as Euler certainly had done ; and both of them handled it much more successfully than Fontaine. 'D'Alem- bert, in his demonstrations, 1760, of the theorems on the integral calculus, given by him without any defaonstration in the volume for 1767, and in the scholium to the twenty-first theorem, affirms distinctly that he had communicated to Clairaut a portion of the demonstration, forming a corollary CALCULUS OF PARTIAL DIFPEEENOES. 45 to the proposition, and from which, he says that Clairaut derived his equation of condition to differentials involving three variables. It is possible ; but as this never was men- tioned in Clairaut's lifetime, although there existed a sharp controversy between these two great men on other matters, and especially as the equation of conditions respecting two variables might very easily have led to the train of reasoning by which this extension of the criterion was found out, the probability is, that Clairaut's discovery was in all respects his own. The extreme importance of this criterion to the method of partial differences, only invented, .or at least applied, some years later, is obvious. Take a simple case in a differential equation of the first order, — ■ dz = (2aacy — y'^')dx + (ax^ — Zxy'^) dy, where W = 2axy — j^, N = ffia^ — 3 xy'^. For the criterion -rr- — 2ax — Qy. dy ^ = 2ax-Zy^ dx dM. d'S gives us d^ = J^' which shows that the equation Wdx + 'RdyiB a complete differential, and may be integrated. Thus integrate (a «* — 3 xy"^) d y , as if X were constant, and add X (a function of x, or a constant), as necessary to complete the integral, and we have ax'^y — xy^ + X = Z ; now differentiate, supposing y constant, and we have —- = {2axy -y^) + (because of the criterion) = 2axy — y^, consequently - — = o, and X = C, a const Accordingly, z = aa?y~-x'f + Q; 46 DYNAMICAL PEINOIPLE. and so it is, for differentiating in the ordinary way, x and y being both, variable, we have dz = 2axy dx + aafdy — 2ixy^dy — y^dx = (2axy — y^')dx -\- {aai? — Zxy^^dy ; which was the equation given to be integrated. To take another instance in which , the differential co- dx efficient of the qiiantity added is not = o or X constant. Let dz — y^dx + ^x^dx + Ixydy, in which, by inspection, the solution is easy — z = xy' -f- a;' + C. Here M = ?/^ + 3 a;', N = 2 a;y, dM ^ rfN and — — = 2?/ = -; — . dy dx Qo z = xy'^ -\-'K, and differentiating with respect to x, dz , dX , — = y^+ -— = y^ + 3x\ dx " dx ^ Hence X = a;' + C, and z = xy'^ + x^ + G, the integral of the equation proposed. It must, however, be observed of the criterion, that an equa- tion may be integrable which does not answer the condition dM. _ d'N dy dx ' It may be possible to separate the variables and obtain 'X. d X = Y d y, as by transformation ; or to find a factor, which, multiplying the equation, shall render it integrable, by bringing it within that condition. The latter process is the most hopeful ; and it is generally affirmed that such a factor, F, may always be found for every equation of the first order involving only two variables. However, this is only true in theory : we cannot resolve the general equation by any such means ; for that gives us \dy dx J dx dy CALOULTTS OF PAETIAL DIFPEEENCBS. 47 an expression as impossible to disentangle, it may safely be asserted, as any for the resolution of which its aid might be wanted. It is only in a few instances of the values of these functions (M and N) that we can succeed in finding F. It is quite unaccountable * that Clairaut should, in reference to his equation, which is substantially the same with the above, describe it as " d'une grande utilite, pour trouver n " (that isF). It is here to be observed, that not only Fontaine had, apparently, first of all the geometricians, given the criterion of integrability, but he had also given the notation which was afterwards adopted for the calculus of Partial Differences.

S COS. 7 cos. ( m jv + 6 cos. ( \. m]v — i; cos. OALCULTJS or PARTIAL DIPFEEBNCES. 55 ( 2 m J r, terms obtained by the first or trial integration, whinli he had fully explained in his first Memoir to be the more correct mode of proceeding (' M6m.' 1745, p. 352) ; and the consequence of this is to give the multiplier, on ■which depends the progression of the apogee, a difierent value from what it was found to have in the former process. It is never to be fo7-gotten that the original investigation was accurate as far as it went ; but by further extending the approximation a more correct value of m was obtained, in consequence of which the expression for the motion of Ihe apogee became double that which had been calculated before. It should be observed, in closing the subject of the Problem of Three Bodies, that Euler no sooner heard of Clairaut's final discovery, than he confirmed it by his own investigation of tie subject, as did D'Alembert. But in the meantime, Mat- thew Stewart had undertaken to assail this question by the mere help of the ancient geometry, and had marYellously suc- cepded in reconciling the Newtonian theory with observation. Father Walmisley, a young English priest of the Benedictine order, also gave an analytical solution of the difficulty in 1749. The other great problem, the investigation of which occu- pied D'Alembert, was the Precession of the equinoxes and the Nutation of the earth's axis, according to the theory of gravi- tation. Sir Isaac Newton, in the xxxix. prop, of the thii-d book, had given an indirect solution of the Problem concern- ing the Precession ; the Nutation had only been by his un- rivalled sagacity conjectured a priori, and was proved by the observations of Bradley. The solution of the Precession had not proved satisfactory; and objections were taken to the hypotheses on which it rested, that the accumulation of matter at the equator might be regarded as a belt of moons, that its movement might be reckoned in the proportion of its mass to that of the earth, and that the proportion of the terrestrial axes is that of 229 to 230 ; that the earth is homogeneous, and that the action of the sun and moon ad mare movendum, are as 56 DTNAMIOAli PRINCIPLE. one to four and a half nearly, and in the same ratio ad equi- nootia movenda. Certainly the three last suppositions have since Newton's time been displaced by more accurate observa- tions ; the axes being found, to be as 298 to 299, the earth not homogeneous, and the actions of the sun and moon on the tides more nearly as one to three. But it has often been observed, and tmly observed, that when D'Alembert came to discuss the subject, it would have been more becoming in him to assign his reasons for denying the other hypothesis on which the Newtonian investigation rests, than simply to have pronounced it groundless. However, it is certain that he first gave a direct and satisfactory solution of this great problem ; and that he investigated the Nutation with perfect success, showing it to be such that if it subsisted alone (i.e., if there were no precessional motion) the pole of the equinoctial would describe among the stars a minute ellipse, having its longer axis about 18'' and its shorter about 13", the longer being directed towards the pole of the ecliptic, and the shorter of course at right angles to it. He also discovered in his investigations that the Precession is itself subject to a varia- tion, being in a revolution of the nodes, sometimes accelerated, sometimes retarded, according to a law which he discovered, giving the equation of correction. It was in 1749 that he gave this admirable investigation ; and in 1755 he followed it up with another first attempted by him, namely, the variation which might occur to the former results, if the earth, instead of being a sphere oblate at the poles, were an elliptic spheroid, whose axes were different. He added an investigation of the Precession on the supposition of the form being any other curve approaching the circle. This is an investigation of as great difficulty perhaps as ever engaged the attention of analysts. It remains to add that Euler, in 1750, entered on the same inquiries concerning Precession and Nutation ; and with his wonted candour, he declared that he had read D'Alembert's memoir before he began the investigation.* * This Tract is from ' Lives of the Philosophers '—Life of D'Alembert. ( 57 ) IV. GREEK GEOMETRY.— ANCIENT ANALYSIS.— PORISMS. The wonderful progress that has been made in the pure mathematics since the application of algebra to geometry, begun by Vieta in the sixteenth, completed by Des Cartes in the seventeenth century, and e.specially the still more marvelloiis extension of analytical science by Newton and his followers, since the invention of the Calculus, has, for the last hundred years and more, cast into the shade the methods of investigation which preceded those now in such ■general use, and so well adapted to afford facilities unknown ■while mathematicians only possessed a less perfect instrument of investigation. It is nevertheless to be observed that the older method possessed qualities of extraordinary value. It enabled us to investigate some kinds of propositions to which algebraic reasoning is little applicable ; it always had an elegance peculiarly its own ; it exhibited at each step the course which the reasoning followed, instead of concealing that course till the result came out ; it exercised the faculties more severely, because it was less mechanical than the opera- tions of the analyst. That it afforded evidence of a higher character, more rigorous in its nature than that on which algebraic reasoning rests, cannot with any correctness be affirmed ; both are equally strict : indeed, if each be mathe- matical in its nature, and consist of a series of identical pro- positions arising one out of another, neither can be less perfect than the other, for of certainty there can be no degrees. Nevertheless it must be a matter of regret — and here the great master and author of modem mathematics has joined in expressing it — that so much less attention is now paid to 58 GEEBK GEOMETRY. the Ancient Geometry than its beauty and. clearness deserve ; and if he could justly make this complaint a century and a half ago, when the old method had but recently, and only in part, fallen into neglect and disuse, how much more are such regrets natural in our day, when the very name of the Ancient Analysis has almost ceased to be known, and the beauties of the Greek Geometry are entirely veiled from the mathematician's eyes ! It becomes, for this reason, necessary that the life of Simson, the great restorer of that geometry, should be prefaced by some remarks upon the nature of the science, in order that, in giving an account of his works, we may say his discoveries, it may not appear that we are record- ing the services of a great man to some science different from the mathematical. The analysis of the Greek geometers was a method of investigation of pecxiliar elegance, and of no inconsiderable! power. It consisted in supposing the thing as already done, the problem solved, or the truth of the theorem established ; and from thence it reasoned until something was found, some point reached, by pursuing steps each one of which led to the next, and by only assuming things which were already known, having been ascertained by former discoveries. The thing thus found, the point reached, was the discovery of something which could by known methods be performed, or of something which, if not self-evident, was already by former discovery proved to be true ; and in the one case a construe^ tion was thus found by which the problem was solved, in the other a proof was obtained that the theorem was true, because in both oases the ultimate point had been reached by strictly legitimate reasoning, from the assumption that the problem had been solved, or the assumption that the theorem was true. Thus, if it were required from a given point in a straight line given by position, to draw a straight line which should be cut by a given circle in segments, whose rectangle was equal to that of the segments of the diameter perpendicular to the given line — ^the thing is supposed to be done; and the equality of the rectangle gives -a, proportion between the ANCIENT ANALYSIS. — POEISMS. 59 segments of the two lines, such that, joining the point sup- posed to be found, but not found, with the extremity of the diameter, the angle of that line with the line sought but not found, is shown by similar triangles to be a light angle, «.e., the angle in a semircircle. Therefore the point through which the line must be drawn is the point at which the perpendicular cuts the given circle. Then, suppose the point given through which the line is to be drawn, if we find that the curve in which the other points are situate is a circle, we have a local theorem, affirming that, if lines be drawn through any point to a line perpendicular to the diameter, the rectangle made bj' the segments of all the lines cutting the perpendicular is constant ; and this theorem would be demon- strated by supposing the thing true, and thus reasoning till v^e find that the angle in a semicircle is a right angle, a known truth. Lastly, suppose we change the hypothesis, and leave out the position of the point as given, and inquire after the point in the given straight line from which a line being drawn through a point to be found in the circle, the segments ■will contain a rectangle equal to the rectangle under the perpendicular segments — we find that one point answers this condition, but also that the problem becomes indeterminate : for every line drawn through that point to every point in the given straight line has segments, whose rectangle is equal to that under the segments of the perpendicular. The enuncia- tion of this truth, of this possibility of finding such a point in the circle, is a Porism. The Greek geometers of the more modern school, or lower age, defined a Porism to be a pro- position differing from a local theorem by a defect or defalca- tion in the hypothesis ; and accordingly we find that this porism is derived from the local theorem formerly given, by leaving out part of the hypothesis. But we shall afterwards have occasion to observe that this is an illogical and imperfect definition, not coextensive with the thing defined ; the above proposition, however, answers every definition of a Porism. The demonstration of the theorem or of the construction obtained by investigation in. this manner of proceeding, is 60 GREEK GEOMETKT. called synthesis, or composition, in opposition to the analysis, or tlie process of investigation : and it is frequently said that Plato imported the whole system in the visits which he made like Thales of Miletus and Pythagoras, to study under the Egyptian geometers, and afterwards to converse with Theodoras at Cyrene, and the Pythagorean School in Italy. But it can hardly be supposed that all the preceding geometers had worked their problems and theorems at random ; that Thales and Pythagoras with their disciples, a century and a half before Plato, and Hippocrates, half a century before his time, had no knowledge of the analytical method, and pursued no systematic plan in their researches, devoted as their age was to geometrical studies. Plato may have improved and further systematised the method, as he was no doubt deeply impressed with the paramount import- ance of geometry, and even inscribed upon the gates of the Lyceum a prohibition against any one entering who was ignorant of it. The same spirit of exaggeration which ascribes to him the analytical method, has also given rise to the notion that he was. the discoverer of the Conic Sections ; a notion which is without any truth and without the least pro- bability. Of the works written by the Greek geometers some have come down to us ; some of the most valuable, as the ' Ele- ments ' and ' Data ' of Euclid, and the ' Conies ' of ApoUonius. Others are lost; but, happily. Pappus, a mathematician of some merit, who flourished in the Alexandrian school about the end of the fourth century, has left a valuable account of the geometrical writings of the elder Greeks. His work is of a miscellaneous nature, as its name, ' Mathematical Collec- tions,' implies ; and excepting a few passages, it has never been published in the original Greek. Commandini, of Ur- bino, made a translation of the whole six books then dis- covered ; the first has never been found, but half the second being in the Savilian library at Oxford, was translated by Wallis a century later. Commandini's translation, with his learned commentary, was not printed before his death, but ANCIENT ANALYSIS. — POEISMS. 61 the Duke of Drbino (Francesco Maria) caused it to be pub- lished in 1588, at Pisa, and a second edition was published at Venice the next year : a fact most honourable to that learned and accomplished age, when we recollect how many years Newton's immortal work was published before it reached a second edition, and that in the seventeenth and eighteenth centuries. The two first books of Pappus appear to have been purely arithmetical, so that their loss is little to be lamented. The eighth is on mechanics, and the other five are geometrical. The most interesting portion is the seventh ; the introduction of which, addressed to his son as a guide of his geometrical studies, contains a full enumeration of the works written by the Greek geometers, and an account of the particular subjects which each treated, in some instances giving a summary of the propositions themselves with more or less obscurity, but always with great brevity. Among them was a work which excited great interest, and for a long time baffled the conjec- tures of mathematicians, Euclid's three books of ' Porisms :' of these we shall afterwards have occasion to speak more fully. His ' Loci ad Superficiem,' apparently treating of curves of double curvature, is another, the loss of which was greatly lamented, the more because Pappus has given no account of its contents. This he had done in the case of the 'Loci Plani' of .ApoUonius. Euclid's four books on conic sections are also lost ; but of ApoUonius's eight books on the same subject, the most important of the whole series, the ' Elements ' excepted, four were preserved, and three more were discovered in the seventeenth century. His Inclinations, his Tactions or Tangencies, his Sections of Space and of Eatio, and his Determinate Section, however curious, are of less importance ; all of them are lost. For many years Commandini's publication of the ' Collec- tions ' and his commentary did not lead to any attempt at restoring the lost works from the general account given by Pappus. Albert Girard, in 1634, informs us in a note to an edition of Stevinus, that he had restored Euclid's ' Porisms,' a 62 GSEEK GEOMETRY. thing eminently unlikely, as he never published any part of his restoration, and it was not found after his decease. In 1637, Format restored the ' Loci Plani ' of Apollonius, but in a manner so little according to the ancient analysis, that we cannot be said to approach by means of his labours the lost book on this subject. In 1615, De la Hire, a lover and a successful cultivator of the ancient method, published his Conic Sections, but synthetically treated ; he added after- wards other works on epicycloids and conchoids, treated on the analytical plan. L'H6pital, at the end of the seventeenth century, published an excellent treatise on Conies, but purely algebraical. At the beginning of the eighteenth century, Viviani and Grandi applied themselves to the ancient geometry ; and the former gave a conjectural restoration (Divinatio) of Aristiseus's ' Loci Solidi,' the curves of the second or Conic order. But all these attempts were exceed- ingly unsuccessful, and the world was left in the dark, for thei most part, on the highly interesting subject of the Greek Geometry. "We shall presently see that both Format and Halley, its most successful students, had made but an incon- siderable progress in the most difficult branches. How entirely the academicians of France were either care- less of those matters, or ignorant, or both, appears by the ' Encyclopedic ;' the mathematical department of which was under no less a geometrician than D'Alembert. The definition there given of analysis, makes it synonymous with algebra : and yet mention is made of the ancient writers on analysis, and of the introduction to the seventh book of Pappus, with only this remark, , that those authors differ much from the modern analysts. But the article ' Arithmetic ' (vol. i. p. 677), demonstrates this ignorance completely; and that Pappus's celebrated introduction had been referred to by one who never read it. We there find it said, that Plato is supposed to have invented the ancient analysis ; that Euclid, Apollonius, and others, including Pappus himself, studied it, but that we are quite ignorant of what it was : only that it is by some conceived to have resembled our algebra, or else Archimedes ANCIENT ANALYSIS. — POEISMS. 63 could never liave made his great geometrical discoveries. It is, certainly, quite incredible that such a name as D'Alembert's should he found affixed to this statement, which the mere reading of any one page of Pappus's books must have shown to be wholly erroneous; and our wonder is the greater, inasmuch as Simson's admirable restoration of ApoUonius's ' Loci Plani ' had been published five years before the ' Encyclopedie ' ap- peared. Again, in the ' Encyclopedie,' the word Analysis, as mean- ing the Greek method, and not algebra, is not even to be found. Nor do the words synthesis, or composition, inclina- tions, tactions or tangenoies, occur at all ; and though Porisms are mentioned, it is only to show the same ignorance of the subject ; for that word is said to be synonymous with ' lemma,' because it is sometimes used by Pappus in the sense of subsidiary proposition.* V^hen Clairault wrote his in- estimable work on curves of double curvature, he made no reference whatever to Euclid's ' Loci ad Superficiem ;' much less did he handle the subject after the same manner; he deals, indeed, with matters beyond the reach of the Greek Geometry. Such was the state of this science when Eobert Simson first applied to it his genius, equally vigorous and undaunted, with the taste which he had early imbibed for the beauty, the simplicity, and the closeness of the ancient analysis. He was appointed professor in 1711, and taught with ex- traordinary success ; but his genius was bent to the diligent investigation of truth, in the science of which he was so great a master. The ancient geometry, that of the Greeks of which I have spoken, early fixed his attention and occupied his mind by its extraordinary elegance, by the lucid clearness with which its investigations are conducted, by the exercise which it affords to the reasoning faculties, and above all, by the absolute rigour of its demonstrations. He never under- valued modern analysis ; it is a great mistake to represent * Euclid uses the word Corollary in his Elements. — See Note II. 64 GEEEK GEOMETET. him as either disliking its process, or insensible to its vast importance for the solution of questions which the Greek analysis is wholly incapable of reaching. But he considered it as only to be used in its proper sphere ; and that sphere he held to exclude whatever of geometrical investigation can be, with convenience and elegance, carried on by purely geo- metrical methods. The application of algebra to geometry, it would be ridiculous to suppose that either he or his celebrated pupil Matthew Stewart disliked or undervalued. That appli- cation forms the most valuable service which modern analysis has rendered to science. But they did object, and most reason- ably and consistently, to the introduction of algebraic reason- ing wherever the investigation could, though less easily, yet far more satisfactorily, be performed geometrically. They saw, too, that in many instances the algebraic solution leads to constructions of the most complex, clumsy, unmanageable kind, and therefore must be, in all these instances, reckoned more difficult, and even more prolix than the geometrical, from the former being confined to the expression of all the relations of space and position, by magnitudes, by quantity and number (even after the arithmetic of sines had been introduced), while the latter could avail itself of circles and angles directly. They would have equally objected to carry- ing geometrical reasoning into the iields peculiarly appro- priate to modem analysis ; and if one of them, Stewart, did endeavour to investigate by the ancient geometry physical problems supposed to be placed beyond its reach — as the sun's distance, in which he failed, and Kepler's problem, in which he marvellously succeeded, that of dividing the elliptical area in a given ratio by a straight line drawn from one focus — this is to be taken only as an homage to the undervalued potency of the Greek analysis, or at most, as a feat of geometrical force, and by no means as an indication of any wish to substitute so imperfect, however beautiful, an instrument, for the more powerful, though more ordinary one of the calculus which " alone can work great marvels." At the same time, and with all the necessary confession of the ANCIENT ANALYSIS. POEISMB. 65 merits of the modern raethod, it is certain that those geo- metricians would have regarded the course taken by some of its votaries in more recent times as exceptionable, whether with a view to clearness or to good taste : a course to the full as objectionable as would be the banishing of algebraical and substituting of geometrical symbols in the investigations of the higher geometry. La Place's great work, the ' Mecanique Celeste,' and La Grange's ' Mecanique Analytique,' have treated of the whole science of dynamics and of physical astronomy, comprehending all the doctrine of trajectories, dealing with geometrical ideas throughout, and ideas so purely geometrical that the algebraic symbols, as far as their works are concerned, have no possible meaning apart from lines, angles, surfaces ; and yet in their whole compass they have not one single diagram of any kind. Surely, ij _ * we may ask if j-»/ dx^ -\- dy^, 2jf'^y\ "^^^ possibly dy dx d^—^j bear any other meaning than the tangent and the radius of curvature of a curve line : that is, a straight line touching a curve, and a circle whose curvature is that of another curve where they meet ; any meaning, at least, which can make it material that they should ever be seen on the page of the analyst. These expressions are utterly without sense, except in reference to geometrical considerations ; for although x and y are so general that they express any numbers, any lines, nay, any ideas, any rewards or punishments, any thoughts of the mind, it is manifest that the square of the differential of a thought, or the differential of the differential of a reward or punishment, has no meaning ; and so of everything else but of the very tangent or the osculating circle's radius : conse- quently the generality of the symbols is wholly useless ; the particular case of two lines being the only thing to which * Or ^^""^ + '^•"'^ da?o ydxj 66 GEEEK GBOMETEY. the expressions can possibly be meant to apply. Why, then, all geometrical symbols should be so carefully avoided when we are really treating of geometrical examples and geometrical ideas, and of these alone, seems hard to understand. As the exclusive lovers of modem analysis have frequently and very erroneously suspected the ancients of possessing some such instrument, and concealing the use of it by giving their demonstrations synthetically after reaching their con- clusions analytically, so some lovers of ancient analysis have supposed that Sir Isaac Newton obtained his solutions by algebraic investigations, and then covered them with a, synthetic dress. Among others. Dr. Simson leant to this opinion respecting the ' Principia.' He used to say that, lie knew this from Halley, by whose urgent advice Sir Isa^p was induced to adopt the synthetic form of demonstratioji, after having discovered the truths analytically. Machin is known to have held the same language ; he said that the ' Principia ' was algebra in disguise. Assuredly, the pro- bability of this is far greater than that of the ancients having possessed and kept secret the analytical process of modern times. In the preface to his 'Loci Plani,' Dr. Simson fully refutes this notion respecting the ancients : a notion which, among others, no less a writer than Wallis had strongly maintained.* That he did not undervalue algebra and the calculus to * Algebra Prasf. " Hanc Grsecos olim habuisse non est quod dubite- mus ; sed studio oelatam, nee temere propalandam. Ejus effeotus (utut clam oelatse) satis conspicui apud ArcMmedem, Apollonium, aliosque." It is strange that any one of ordinary reflection should have overlooked the utter impossibility of all the geometricians in ancient times keeping the secret of an art which must, if it existed, have been universally known in the mathematical schools, and at a time when every man of the least learning, or even of the most ordinary education, was taught geometry'. Montucla touches on this subject, but not with his wonted accuracy, (1. 166). Indeed, he seems here to confound ancient with modem analysis, although no one has more accurately described and illustrated the ancient method, (I. 164, 275). He adopts the erroneous notion of Plato havitfg discovered this method ; but he does not fall into the other error of ascrib- ing to him the discovery of Conic Sections, {ib. 168). ANCIENT ANALYSIS. — POEISMS. 67 wtich it has given rise, appears from many circtimstances — among others, from what has already been stated ; it appears also from this, that in many of his manuscripts there are found algebraical formulas for propositions which he had investigated geometrically. Maclaurin consulted him on the preparation of his admirable work, the 'Fluxions,' and re- ceived from him copious suggestions and assistance. Indeed, he adopted from him the celebrated demonstration of the fluxion (or differential) of a rectangle.* But Simson's whole mind, when left to its natural bent, was given to the beauties of the Greek Geometry ; and he had not been many months settled in his academical situation when he began to follow the advice which Halley had given him, as both calculated, he said, to promote his own reputation, and to confer a lasting benefit upon the science cultivated by them both with an equal devotion. It is even certain that the obscure and most difficult subject of Porisms very early occupied his thoughts, and was the field of his researches, though to the end of his life he never had made such progress in the investigation as satisfied himself. Before 1715, three years after he began his course of teaching, he was deeply engaged in this inquiry ; but he only regarded it as one branch of the great and dark subject which Halley had recommended to his care. After he had completely examined, corrected, and published, with most important additions, the Conies of Apollonius, which happily remain entire, but which, as we have seen, had been most inelegantly and indeed algebraically given by De la Hire, L'HSpital, and others, to restore the lost books was his great desire, and formed the ' grand achievement which he set before his eyes. We have already shown how scanty the light was by which his steps in this path must be guided. The introduction to the Seventh book of Pappus contained the whole that had reached our times to let us know the contents of the lost works. Some of the summaries which that valuable discourse * Book i. chap. ii. prop. 3. F 2 68 GEEEK GBOMETET. contains are sufficiently explicit, as those of the Loci Plani and the Determinate Section. Accordingly, former geo- metricians had succeeded in restoring the Loci Plani, or those propositions which treat of loci to the circle and rectilinear figures. They had, indeed, proceeded in a very unsatisfactory manner. Sohooten, a Dutch mathematician of great industry and no taste, had given purely algebraic solutions and demon- strations. Fermat, one of the greatest mathematicians of the seventeenth century, had proceeded more according to the geometrical rules of the ancients ; hut he had kept to general solutions, and neither he nor Schooten had given the different cases, according as the data in each proposition were varied ; so that their works were nearly useless in the solution of problems, the great purpose of Apollonius, as of all the authors of the tottoq avaXuofi^vov — the thirty-three ancient books. As for the analysis, it was given by neither, unless, indeed, Schooteu's algebra is to be so termed. Fermat's de- monstrations were all synthetical. His treatise, thougli written as early as 1629, was only published among his col- lected works in 1670. Schooten's was published among his ' Exercitationes Mathematicse ' in 1657. Of the field thus left open. Dr. Simson took possession, and he most successfully cultivated every comer of it. Nothing is left without the most full discussion ; all the cases of each proposition are thoroughly investigated. Many new truths of great import- ance are added to those which had been unfolded by the Greek philosopher. The whole is given with the perfect precision and the pure elegance of the ancient analysis ; and the universal assent of the scientific world has even confessed that there is every reason to consider the restored work as greatly superior to the lost original. The history of this excellent treatise shows in a striking manner the cautious and modest nature of its author. He had completed it in 1738 ; but, unsatisfied with it, he kept it by him for eight years. He could not bring himself to think that he had given the " ipsissimre propositiones of Apollonius in the very order and spirit of the original work." He was ANCIENT ANAIiTSIS. — POEISMS. 69 taen persuaded to let the book appear, and it was published in. 1746. His former scruples and alarms recurred; he stopped the publication ; he bought up the copies that had been sold ; he kept them three years longer by him ; and it was only in 1749 that the work really appeared. Thus had a geometrician complied with the rule prescribed by Horace for those who have no standard by which to estimate with exact- ness the merit of their writings. In the meantime he had extended his researches into other parts of the subject. Among the rest he had restored and greatly extended the work on Determinate Section, or the various propositions respecting the properties of the squares and rectangles of segments of lines passing through given points. There is no doubt that the prolixity, however elegant, with which the ancients treated this subject, is somewhat out of proportion to its importance ; and as it is peculiarly adapted to the algebraical method, presenting, indeed, little difficulty, to the analyst, the loss of the Per- gsean treatise is the less to be deplored, and its restoration was the less to be desired. ApoUonius had even thought it expedient to give a double set of solutions ; one by straight lines, the other by semicircles. Dr. Simson's restoration is most full, certainly, and contains many and large additions of his own. It fills above three hundred quarto pages. His predecessors had been Snellius, whose attempt, published in 1608, was universally allowed to be a failure ; and Anderson, a professor of Aberdeen, whose work, in 1612, was much better, but confined to a small part only of the subject. About the time that Dr. Simson finally published the Loci Plani, he began his great labour of giving a correct and full edition of the Elements. The manner in which this has been accomplished by him is well known. The utmost care was bestowed on the revision of the text ; no pains were spared in collating editions ; all commentaries were consulted ; and the elegance and perfect method of the original has been so admirably preserved, that no rival has ever yet risen up to dispute with Simson's Euclid the possession of the schools. 70 GEEEK GEOMETRY. The time bestowed on tliis useful work was no less than nine years. It only was published in 1758. To the second edition, in 1762, he added a similarly correct edition of the Data, comprising several very valuable original propositions of his own, of Mr. Stewart, and of Lord Stanhope, together with two excellent problems to illustrate the use of the Data in solutions. We thus find Dr. Simson employed in these various works which he successively gave to the world, elaborated with infinite care, and of which the fame and the use will remain as long as the mathematics are cultivated ; some of them delighting students who pursue the science for the mere speculative love of contemplating abstract truths, and the gratification of following the rigorous proofs peculiar to that science ; some for the instruction of men in the elements, which are to form the foundation of their practical applica- tions of geometry. But all the while his mind never could be wholly weaned from the speculation which had in his earliest days riveted his attention by its curious and sin- gular nature, and fired his youthful ambition by its diffi- culty, and its having vanquished all his predecessors in their efforts to master it. We have seen that as early as 1715 at the latest, probably much earlier, the obscure subject of Porisms had engaged his thoughts ; and soon after, his mind was so entirely absorbed by it that he could apply to no other investigation. The extreme imperfection of the text of Pappus ; the dubious nature of his description ; his rejection of the definition which appeared intelligible ; his substituting nothing in its place except an account so general that it really conveyed no precise information ; the hiatus in the account which he subjoins of Euclid's three books, so that even with the help of the lemmas related to these propositions of the lost work, no clear or steady light could be descried to guide the inquirer — for the first porism of the first book alone remained entire, the general porism being given wholly trun- cated (mancum et imperfectum) — all seemed to present ob- stacles wholly insurmountable ; and after various attempts for ANCIENT ANALYSIS. — POEISMS. 71 years he was fain to conchide with Halley that the mystery belonged to the number of those which can never be pene- trated. He lost his rest in the anxiety of this inquiry ; sleep forsooJc his couch ; his appetite was gone ; his health was wholly shaken ; he was compelled to give over the pursuit ; he was "obliged," he says, "to resolve steadily that he never more should touch the subject, and as often as it came upon him he drove it away from his thoughts." * It happened, however, about the month of April, 1722, that while walking on the banks of the Clyde with some friends, he had fallen behind the company; and musing alone, the rejected topic found access to his thoughts. After some time a sudden light broke in upon him ; it seemed at length as if he could descry something of a path, slippery, tangled, interrupted, but still practicable, and leading at least in the direction towards the object of his research. He eagerly drew a figure on the stump of a neighbouring tree with a piece of chalk ; he felt assured that he had now the means of solving the great problem ; and although he after- wards tells us that he then had not a sufficiently clear notion of the subject (eo tempore Porismatum naturam non satis com- pertam habebam),t yet he accomplished enough to make him communicate a paper upon the discovery to the Eoyal Society, the first work he ever published (Phil. Trans, for 1723). He was wont in after life to show the spot on which the tree, long since decayed, had stood. If peradventure it had been preserved, the frequent lover of Greek Geometry would have been seen making his pilgrimage to a spot consecrated by such touching recollections. The graphic pen of Montucla, which gave such interest to the story of the first observa- tion of the transit of Venus by Horrox in Lancashire, and to the Torricellian experiment, J is alone wanting to clothe this passage in colours as vivid and as unfading. * "Firmiter animum induxi hi80 nunquam in posterum investigare. Unde qnoties menti ocourrebant, toties eas arcebam." — (Op. Eel. 320. Praef. ad Porismata.) t Op. Kel. 320. i Hist, de Math. vol. i. 72 GEEEK GBOMETET. This great geometrician continued at all the intervals of his other labours intently to investigate the subject on which he thus first threw a steady light. His first care upon having made this discovery was to extend the particular propositions until he had obtained the general one. A note among his memoranda appears to have been made, according to his custom, of marking the date at which he succeeded in any of his investigations.* — " Hodie hsec de porismatis inveni, E. S., 23 April, 1722." Another note, 27th April, 1722, shows that he had then obtained the general proposition; he afterwards communicated this to Maclaurin when he passed through Glasgow on his way to France ; and he, on his return, communicated to Dr. Simson without demon- stration a proposition concerning conies derived from what he had shown him — a proposition which led his friend to insert some important investigations in his Conic Sections. In 1723 the publication of his paper took place in the ' Philosophical Transactions;' it is extremely short, and does not appear to contain all that the author had communicated ; for we find this sentence inserted before the last portion of the paper : — " His adjecit clarrissimus professor propositiones duas sequentes libri primi Porismatum Euclidis, a se quoque restitutas." The paper contains the first general proposition and its ten cases, and then the second with its cases. No general descrip- tion or definition is given of Porisms ; and it is plain that his mind was not then finally made up on this obscure subject, although he had obtained a clear view of it generally. At what time his knowledge of the whole became matured we are not informed ; but we know that his own nature was * In one there is this note upon the solution of a problem of tactions, — " Feb 9, 1734 : — Post horam primam ante meridiem ;" and much later in life we find the same particularity in marking the time of discovery. His birthday was October 14, and having solved a problem on that day 1764, he says— 14 Octobr. 1764. „,,,,...„.. 14 Octobr. 1687. Deo Opt. Max. benignissmio Servaton Laus et gloria. 77 (scil. Anno Jitatis.) ANCIENT ANALYSIS. POEISMS. 73 nice and difficult on the subject of his own works ; that he never was satisiied with what he had accomplished; and he probably went on making constant additions and improve- ments to his work. Often urged to publish, he as constantly- refused ; indeed, he would say that he had done nothing, or next to nothing, which was in a state to appear before the world ; and moreover, he very early began to apprehend a decay of his faculties, from observing his recollection of recent things to fail, as is very usual with all men ; for as early as 1751, we find him giving this as a reason for de- clining to undertake a work on Lord Stanhope's recommenda- tion, when he was only in his sixty-fifth year. Thus, though he at first used to say he had nothing ready for publication, he afterwards added, that he was too old to complete his work satisfactorily. In his earlier days he used occasionally to affect a kind of odd mystery on the subject, and when one of his pupils (Dr. Traill) submitted to him some propositions, which he regarded as porisms. Dr. Simson would neither admit nor deny that they were such, but said with some pleasantry, " They are propositions." One of them, however, he has given in his work as a porism, and with a compliment- ary reference to its ingenious and learned author. Thus his life wore away without completing this great work, at least without putting it in such a condition as satis- fied himself. It was left among his MSS., and by the judi- oiouis munificence of a noble geometrician, the liberal friend of scientific men, as well as a successful cultivator of science. Earl Stanhope,* it was, after his death, published, with his restoration of ApoUonius' treatise De Sectione deter- minate, a short paper on Logarithms, and another on the Method of Limits geometrically demonstrated, the whole forming a very handsome quarto volume ; of which the Porisms occupy nearly one-half, or 277 pages. This work is certainly the master-piece of its distinguished * Grandfather of the present Earl, whose father also was a successful cultivator of natural science, mechanical' especially. 74 GREEK GEOMETET. author. The extreme difBculty of the subject was increased by the corruptions of the text that remains in the only ' passage of the Greek geometers which has reached us, the only few sentences in which any mention whatever is made of Porisms. This passage is contained in the preface: or introduction to the Seventh book of Pappus, which we have already had occasion to cite. But this was by far the least of the difficulties which met the inquirer after the hidden treasure, the restorer of lost science, though Albert Girard thought or said, in 1635, that he had restored the Porisms of Euclid. As we have seen, no trace of his labours is left ; and it seems extremely unlikely that he should have really performed such a feat and given no proofs of it. Halley, the most learned and able of Dr. Simson's pre-' decessors, had tried the subject, and tried it in vain. He thus records his failure : — " Hactenus Porismatum desoriptio nee mihi intellecta nee lectori profutura." These are his words, in a preface to a translation which he published of Pappus's Seventh book, much superior in execution to that of Comman- dini. But this eminent geometrician was much more honest' than some, and much more safe and free from mistake than others who touched upon the subject that occupied all students of the ancient analysis. He was far from pretending, like Girardus, to have discovered that of which all were in quest. But neither did he blunder like Pemberton, whom we find, the very year of Simson's first publication, actually saying in his paper on the Eainbow— " For the greater brevity I shall deliver them (his propositions) in the form of porisms, as, in my opinion, the ancients called all propositions treated by analysis only" (Philosophical Transactions, 1723, p. 148); and, truth to say, his investigation is not very like ancient analysis either. The notion of D' Alembert, somewhat later, has been alluded to already ; he imagined porism to be synonymous with lemma, misled by an equivocal use of the word in some passages of ancient authors, if indeed he had ever studied any of the writers on the Greek Geometry, which, from what I have stated before, seems exceedingly doubtfuL But the ANCIENT ANALYSIS. — POEISMS. 75 most extraordinary, and indeed inexcusable ignorance of the subject is to be seen in some who, long after Simson's paper had been published, were still in the dark; and though that paper did not fully explain the matter, it yet ought to have prevented such errors as these fell into. Thus Castillon, in 1761, showed that he conceived porisms to be merely the constructions of Euclid's Data. If this were so, there might have been some truth in his boast of having solved all the Porisms of Euclid ; and he might have been able to perform his promise of soon publishing a restoration of those lost books. It is remarkable enough that before Halley's attempts and their failure, candidly acknowledged by himself, Fermat had made a far nearer approach to a solution of the difficulty than any other of Simson's predecessors. That great geo- metrician, after fully admitting the difficulty of the subject, and asserting * that, in modern times, porisms were known hardly even by name, announces somewhat too confidently, if not somewhat vaingloriously, that the light had at length davsTjed upon him,"j" and that he should soon give a full restoration of the whole three lost books of Euclid. Now the light had but broke in by a small chink, as a mere faint glimmering, and this restoration was quite impossible, inas- much as there remained no account of what those books con- tained, excepting a very small portion obscurely mentioned in the preface of Pappus, and the lemmas given in the course of the Seventh book, and given as subservient to the resolution of porismatic questions. Nevertheless, Eermat gave a demon- stration of five propositions, " in order," he saj's, " to show what a porism is, and to what purposes it is subservient." These propositions are, indeed, porisms, though their several * " latentata ac velut disperata Porismatmu EucMdsea doctrina. — Geo- metrici (sevi recentioris) nee vel de nomine cognoverunt, aut quod esset solummodo sunt suspicati." — (Var. Opera, p. 166.) t "Nobis in tenebris dudum caeoutientibus tandem se (Natura Poris- matum) olara ad videndum obtulit, et pura per noctem luce refulsit." — (Bpist. ib.) 76 . GEEEK GBOMETET. entinciations are not given in the true porismatic form. Thus, in the most remarkable of them, the fifth, he gives the con- struction as part of the enunciation. So far, hovsrever, a con- siderable step was made ; but -when he comes to show in what manner he discovered the nature of his porisms, and how he defines them, it is plain that he is entirely misled by the erroneous definition justly censured in the passage of Pappus already referred to. He tells us that his propositions answer the definition ; he adds that it reveals the whole nature of porisms ; he says that by no other account but the one con- tained in the definition, could we ever have arrived at a knowledge of the hidden value ; * and he shows how, in his fifth proposition, the porism flows from a locus, or rather he confounds porisms with loci, saying porisms generally are loci, and so he treats his own fifth proposition as a locus ; and yet the locus to a circle which he states as that from which his proposition flows has no connexion with it, according to Dr. Simson's just remark (' Opera Eeliqua,' p. 346). That the definition on which he relies is truly imperfect, appears from this : there could be no algebraical porism, were every porism connected with a local theorem. But an abundant variety of geometrical porisms can be referred to, which have no possible connexion with loci. Thus, it has never been denied that most of the Propositions in the Higher Geometry, which I investigated in 1797, were porisms, yet many of them were wholly unconnected with loci ; as that affirming the possibility of describing an hyperbola which should cut in a given ratio all the areas of the parabolas lying between given straight lines. t Here the locus has nothing to do with the solution, as if the proposition were a kind of a local theorem : it is only the line dividing the curvilineal areas, and it divides innu- merable such areas. Professor Playfair, who had thoroughly investigated the whole subject, never in considering this proposition doubted for a moment its being most strictly a porism. * Var. Op. p. 118. t Phil. Trans. 1798, p. 111. Tract I. of this volume. ANCIENT ANALYSIS. — POEISMS. 77 Therefore, although Fermat must be allowed to have made a considerahle step, he was unacquainted with the true nature of the porism ; and instead of making good his boast that he could restore the lost books, he never even attempted to restore the investigation of the first proposition, the only one that remains entire. A better proof can hardly be given of the difficulty of the whole subject.* Indeed it must be confessed that Pappus's account of it, our only source of knowledge, is exceedingly obscure, all but the panegyric which in a somewhat tantalizing manner, he pronounces upon it. " CoUectio," says he, " curiosissima multarum rerum spectantium ad resolutionem difficiliorum et generaliorum problematum" (lib. vii. Proem). His definition already cited is, as he himself admits, \ery inaccurate ; because the connexion with a locus is not necessary to the porismatic nature, although it will very often exist, inasmuch as each point in the curve having the same relation to certain lines, its description will, in most cases, furnish the solution of a problem, whence a porism may be deduced. Nor does Pappus, while admitting the inaccuracy of the definition, give us one of his own. Perhaps we may accurately enough define a porism to be the enunciation of the possibility of finding that case in which a determinate problem becomes indeter- minate, and admits of an infinity of solutions, all of which are given by the statement of the case. For it appears essential to the nature of a porism that it should have some connexion with an indeterminate problem and its solution. I apprehend that the poristic case is always one in which the data become such that a transition is made from the determinate to the indeterminate, from the problem * The respect due to the great name of Fermat, a venerable magistrate and most able geometrician, is not to be questioned. He was, indeed, one of the first mathematicians of the age in which he flourished, along with the Eobervals, the Harriots, the Descartes. How near he approached the differential calculus is well known. His correspondence with Eoberval, Gassendi, Pascal, and others, occupies ninety folio pages of his posthumous works, and contains many most ingenious, original, and profound observa- tions on various branches of science. 78 GREEK GBOMETET. being capable of one or two solutions, to its being capable of an infinite number. Thus it would be no porism to affirm that an ellipse being given, two lines may be found at right angles to each other, cutting the curve, and being in a pro- portion to each other which may be found : the two lines are the perpendiculars at the centre, and are of course the two axes of the ellipse ; and though this enunciation is in the outward form of a porism, the proposition is no more a porism than any ordinary problem ; as that a circle being giveUj a a point may be found from whence all the lines drawn to the circumference are eqiial, which is merely the finding of the centre. But suppose there be given the problem to inflect two lines from two given points to the circumference of an ellipse, the sum of which lines shall be equal to a given line, the solution will give four lines, two on each side of the transverse axis. But in one case there will be innumerable lines which answer the conditions, namely, when the two points are in the axis, and so situated that the distance of each of them from the farthest extremity of the axis is equal to the given line, the points being the foci of the ellipse. It is, then, a porism to affirm that an ellipse being given, two points may be found such that if from them be inflected lines to any point whatever of the curve, their sum shall be equal to a straight line which may be found ; and so of the Cassinian curve, in which the rectangle under the inflected lines is given. In like manner if it be sought in an ellipse to inflect from two given points in a given straight line, two lines to a point in the curve, so that the tangent to that point shall, with the two points and the ordinate, cut the given line in harmonical ratio ; this, which is only capable of one solution in ordinary cases, becomes capable of an infinite number when the two points are in the axis, and when the ellipse cuts it ; for in that case every tangent that can be drawn, and every ordinate, cut the given line harmonically with the curve itself.* ' The ellipse has this curious property, which I do not find noticed by Maclaurin in his Latin Treatise on Curve Lines appended to the Algebra, AUCIENT ANALYSIS. — POEISMS. 79 Dr. Simson's definition is such that it connects itself with an indeterminate case of some problem solved; hut it is defective, in appearance rather than in reality, from seeming to confine itself to one class of porisms. This appearance arises from using the word " given " (data or datum) in two different senses, both as describing the hypothesis and as affirming the possibility of finding the construction so as to answer the conditions. This double use of the word, indeed, runs through the book, and though purely classical, is yet very inconvenient; for it would be much more distinct to make one class of things those which are assuredly data, and the other, things which may be found. Nevertheless, as his definition makes all the innumerable things not given have the same relation to those which are given, this should seem to be a limitation of the definition not necessary to the poristic nature. Pappus's definition, or rather that which he says the ancients gave, and which is not exposed to the objection taken by him to the modern one, is really no definition at all ; it is only that a porism is something between a theorem and a problem, and in which, instead of anything being proposed to be done, or to be proved, something is proposed to be investigated. This is erroneous, and contrary to the rules of and dealing a good deal witli Harmonical proportions. If from any point whatever out of the ellipse there be drawn a sti-aight Une in any direction whatever cutting the ellipse, the line is cut harmonically by the tangent, the ordinate, and the chords of the two arcs intercepted between the point of contact of the tangent and the axis. The tangent, sine, and chords are always an harmonical pencil, and consequently cut in the Harmonical ratio all lines drawn in all directions, from the given point. This applies to all ellipses upon the same axis, (all having the same subtangent,) and of course to the circle. The ellipse, therefore, might be called the Sar- monical Curve, did not another of the 12th order rather merit that name, which has it-s axis divided harmonically by the tangent, the normal, the ordinate, and a given point in the axis. Its differential equation is il yV fly *y jT* 2 dip + da.^ = - — - — , which is reducible, and its integral is an equation X of the 12th order. There is also another Harmonical Curve, a transcen- dental one, in which chords vibrate isochronously. 80 GEEEK GEOMBTKT. logic from its generality ; it is, as the lawyers say, void for uncertainty. The modern one ohjeoted to by Pappus is not uncertain ; it is quite accurate as far as it goes ; but it is too confined, and errs against the rules of logic by not being coextensive with the thing proposed to be defined. The difficulty of the subject has been sufficiently shown from the extreme conciseness and the many omissions, the almost studied obscurity, of the only account of it which remains ; and to this must certainly be added the corruption of the Greek text. The success which attended Dr. Simson's labours in restoring the lost work, as far as that was possible, and, at any rate, in giving a full elucidation of the nature of porisms, now, for the first time, disclosed to mathematicians, is, on account of those great difficulties by which his pre- decessors had been baffled, the more to be admired^ But there is one thing yet more justly a matter of wonder, when we contrast his proceedings with- theirs. The greater part pf his life, a life exclusively devoted to mathematical study, had been passed in these researches. He had very early become possessed of the whole mystery, from other eyes so long con- cealed. He had obtained a number of the most curious solu- tions of problems connected with porisme, and was constantly adding to his store of porisms and of lemmas subservient to their investigation. For many years before his death, his work had attained, certainly the form, if not the size, in which we now possess it. Yet he never could so far satisfy himself with what has abundantly satisfied every one else, as to make it public, and he left it unpublished among his papers when he died. Nothing can be more unlike those who freely boasted of having discovered the secret, and promised to restore the whole of Euclid's lost books. It is as certain that the secret was never revealed to them as it is that neither they nor any man could restore the books. But how speedily would the Castillons, the Girards, even the Fermats, have given their works to the world had they become possessed of such a treasure as Dr. Simson had found ! Yet though ready for the press, and with its preface composed, and its title AlfCIENT ANALYSIS. — POBISMS. 81 given in minute particularity, he never could think that he had so far elaborated and finished it as to warrant him in finally resolving on its publication. There needs no panegyric of this most admirable perform- ance. Its great merit is best e.stimated by the view which has been taken of the extraordinary difficulties overcome by it. The difficulty of some investigations— the singular beauty of the propositions, a beauty peculiar to the porism from the wonderfully general relations which it discloses — the sim- plicity of the combinations — the perfect elegance of the demonstrations — render this a treatise in which the lovers of geometrical science must ever find the purest delight. Beside the general discussions in the preface, and in a long and valuable scholium after the sixth proposition, and an example of algebraical porisms. Dr. Simson has given in all ninety-one propositions. Of these, four are problems, ten are loci, forty-three are theorems, and the remaining thirty-four are porisms, including four suggested hj Matthew Stewart, and the five of Fermat improved and generalized ; there are, besides, four lemmas and one porism suggested by Dr. Traill, when studying under the professor. There may thus be said to be in all ninety-eight propositions. The four lemmas are propositions ancillary to the author's own investigations ; for many of his theorems are the lemmas preserved by Pappus as ancillary to the porisms of Euclid. In all these investigations the strictness of the Greek geometry is preserved almost to an excess ; and there cannot well be given a more remarkable illustration of its extreme rigour than the very outset of this great work presents. The porism is, that a point may be found in any given circle through which all the lines drawn cutting its circumference aiid meeting a given straight line shall have their segments within and without the circle in the same ratio. This, though a beautiful proposition, is one very easily demonstrated, and is, indeed, a corollary to some of those in the ' Elements.' But Dr. Simson prefixes a lemma : that the line drawn to the right angle of a triangle froni the middle point of the G 82 GREEK GEOMETRY. hypotemise, is equal to half that hypotenuse. Now this follows, if the segment containing the right angle he a semicircle, and it might be thought that this should be assumed only as a manifest corollary from the proposition, or as the plain converse of the proposition, that the angle in a semicircle is a right angle, but rather as identical with that proposition; for if we say the semicircle is a right-angled segment, we also say that the right-angled segment is a semi- circle. But then it might be supposed that two semicircles could stand on one base ; or, which is the same thing, that two perpendiculars could be drawn from one point to the same line ; and as these propositions had not been in the elements (though the one follows from the definition of the circle, and the other from the theorem that the three angles of a triangle are equal to two right angles), and as it might be supposed that two or more circles, like two or more ellipses, might be drawn on the same axis, therefore the lemma is demonstrated by a construction into which the centre does not enter. Again, in applying this lemma to the porism (the proportion of the segments given by similar triangles), a right angle is drawn at the point of the circum- ference, to which a line is drawn from the extremity of a perpendicular to the given line ; and this, though it proves that perpendicular to pass through the centre, unless two semicircles could stand on the same diameter, is not held sufficient ; but the analysis is continued by help of the lemma to show that the perpendicular to the given line passes through the centre of the given circle, and that therefore the point is found. It is probable that the author began his work with a simple case, and gave it a peculiarly rigorous investigation in order to explain, as he immediately after does clearly in the scholium already referred to, the nature of the porism, and to illustrate the erroneous definitions of later times (veoTtpiKoi) of which Pappus complains as illogical. Of porisms, examples have been now given both in plain geometry, in solid, and in the higher : that is, in their con- nexion both with straight lines and circles, with conic sec- ANCIENT ANAIiTSIS. — P0EISM8. 83 tions, and with curves of the third and higher orders. Of an algebraical porism it is easy to give examples from problems becoming indeterminate ; but these propositions may likewise arise from a change in the conditions of determinate problems. Thus, if we seek for a number, such that its multiple by the sum of two quantities shall be equal to its multiple by the difference of these quantities, together with twice its multiple by a third given quantity, we have the equation (a+6) a; = (a — 6) a; 4-2 c x and 26 a; = 2 c a; ; in which it is evident that if c = 6, any number whatever will answer the conditions, and thus we have this porism : Two numbers being given a third may be found, such that the multiple of any number whatever by the sum of the given numbers, shall be equal to its multiple by their differences, together with half its multi- ple by the number to be found. That number is in the ratio of 4 : 3 to the lesser given number. There are many porisms also in dynamics. One relates to the centre of gravity which is the porismatio case of a problem. The porism may be thus enunciated : — Any number of points being given, a point may be found such, that if any straight line whatever be drawn through it, the sum of the perpendiculars to it from the points on one side wUl be equal to the sum of the perpendiculars from the points on the other side. That point is consequently the centre of gravity : for the system is in equilibrium by the proposition. Another is famous in the history of the mixed mathematics. Sir Isaac Uewton, by a train of most profound and ingenious investiga- tion, reduced the problem of finding a comet's place from three observations (a problem of such difB.culty, that he says of it, "hocce problema longe difBcilimum omnimodo aggres- sus,"*) to the drawing a straight line through four lines given by position, and which shall be cut by them in three segments having given ratios to each other. Now his solution of this problem, the corollary to the twenty-seventh lemma of the first book, has a porismatic case, that is, a case in which * Principia, lib. iii. prop. xli. Q 2 84 GREEK GEOMETKT. any line that can be drawn through the given lines will be cut by them in the same proportions, like the lines drawn through three harnoonicals in the porism already given of the harmonical curve. To this Newton had not adverted, nor to the unfortunate circumstance that the case of comets is actually the case in which the problem thus becomes capable of an infinite number of solutions. The error was only dis- covered after 1739, when it was found that the comet of that year was thrown on the wrong side of the sun by the Newtonian method. This enormous discrepancy of the theory with observation, led to a full consideration of the subject, and to a discovery of the porismatic case.* * The remarkable circumstance of tlie case of the comet's motion, for which Sir I. Newton's solution was intended, proving to he the porismatic case of the construction, has been mentioned in the text. It has been sometimes considered as singular, that this did not occur to himself, the more especially as he evidently had observed two cases in which the problem became indeterminate — namely, when the lines were parallel, and when they all met in one point, for he excepts those cases in express terms (Prin. lib. 1. Lem. xxvii.). It may be observed, that such oversights could very rarely happen to the ancient geometers, because they most carefully examined each variation in the data, and so gave to their solutions such a fulness as exhausted the subject. The commentators on the Principia (Le Seur and Jacquier) make no mention of the omission. The circumstance of the porismatic case was not discovered till ten years after their publication, when F. Boscovioh found it out, in 1749. But it is very extraordinary that Montucla appears to have been unaware of the matter, although the first edition of his work did not appear till 1758. Nor is the least reference made to it in the second edition, which was published the year he died (1799). There are other omissions in both editions, and also in the continuation. He appears well to have understood the ancient method, and to have read and examined some of the most celebrated works. upon it. He had given due praise to Simson in his first edition ; and to Lord Stanhope, who sent him the ' Opera Keliqua ;' and we find in the second edition a full note upon the subject, ii. 277. In the continuation — iii. 11, and seq., we have further indications of the attention which he had bestowed upon the ancient geometry ; but it is remarkable that though Matthew Stewart's Tracts, published in 1761, were known to him, he was wholly unac- quainted with the ' Propositiones Geometripse,' which appeared soon after, and with the General Theorems which had been published fifteen years ANCIENT ANALYSIS. POBISMS. 85 before. Nor does lie appear to have even seen Professor Playfair's admi- rable paper upon Porisms in the Edinburgh Transactions, 1794, the war having probably impeded the intercourse of the two countries. Had he seen this, he must have been brought acquainted with the history of the porism relatuig to the comet's place, for it is there fully given. It must be added, that Montucla's mathematical pursuits had for many years been interrupted by the duties of the places which he held under the government, until the Eevolution (Pref. 1111 ; and although the loss of those employments restored him to his studies, it is probable that he rather applied himself to the continuation of the History, the bringing it down from the period to which the first volume extended, than to supply omissions in those volumes, considerable as are the additions which he made to them. The third and fourth volumes were not published till after his death, which happened when only a third part of the former had been printed. Lalande undertook the revision of the rest, and how great soever his merits may have been as a practical astronomer, as an author, and a teacher of astronomy, he had none of the mathematical acquirements which could fit him for superintending the publication of Montucla's work. He had some assistance from a very eminent mathematician, Laoroix, and the notes given by him are, as might be expected, excellent. But we are not distinctly informed of the additions, if any, which he made to the text, while there appears considerable reason to suppose that Lalande sometimes interfered with it. Certain it is, that many things would have been suppressed, and others added, had Montucla survived to finish the work of correcting and pubUshing. There is no reason to think that the eminent analyst referred to (Lacroix), would have supplied Montucla's omissions regarding the poristic case in the Priacipia, or regarding the writers on the ancient analysis ; for on this subject he was much better informed, in all probability, than Lacroix, and the omission in the Principia comes less witliin the scope of modern than ancient geometry.* * This tract is taken from ' Lives of Philosophers,' — Life of Simson. ( 86 ) V. SUE CEETAINS PAEADOXES EEELS OU SUPPOSES, PEIN- OIPALEMENT DANS LE CALCUL INTEaRAL. Ii'examen des paradoxes, dont I'existenee a eie frequemment supposes, est d'une grande importance, parce que si la supposi- tion a ^te sans fondement, la doctrine est delivree de la charge d'inconsequence ; et si les difficultes ne recoivent point de solution satisfaisante, nous pouvons nous assurer que Ton est arrive a quelque verite nouvelle, ou a quelque limitation im- portante des propositions gen^ralement admises. On trouvera pourtant que ce chapitre (qui pourra etre appele GeoTnetria paradoxos), examine a fond, contient moins d'articles que Ton n'aurait d'abord soup9onne. II y a peu de geometres, si ce n'est Euler, qui aient plus contribue de suggestions dans ce genre que I'illustre d'Alem- bert, et Ton se propose d'en considerer quelques-unes, une surtout qui parait avoir beaucoup engage son attention, vu qu'apres I'avoir discutee dans un Memoire assez connu {Memoires de Berlin, 1747), il j revient dans ses Opuscules (vol. IV, Me- moire XXIII). Cependant c'est une chose incontestable qu'il ne traite pas le sujet avec son exactitude accoutumee, parais- sant plus d^sireux de decouvrir des paradoxes que de les expliquer ou de les resoudre. Plus d'une fois, en considerant une certaine courbe, il dit, " Voila le calcul en defaut." Ce que nous trouverons tout a I'heure n'^tre point dans une des matieres mentionnees, et dont, dans 1' autre, sa solution ne satisfait aucunement, si m^me elle n'est pas manifestement erronee. La courbe pourtant dont il parle merite bien d'etre pleinement examinee, et, dans ses rapports de dynamique, elle INTBGEAL CAIiCULUS. 87 parait offrir plus d'un paradoxe qui avait echappe k ce grand geometre, parce qu'il ne I'aTait pas consideree mecaniquement. L' equation g^nerale de la courbe est .3. ^ _2 y' -\- x' = a" ; en prenant a = 1, eomme le prend d'Alembert, y = (l - -* II prend comme I'origine A;AP = a?, PN=y, AC = 1, nous donne y = [l - (1 - x)*]^; ainsi I'arc egale J ^ dy' + dx- = J dx (1 - x)-^ = _|(l_a;)^ + | I la constante etant ~ „); mais il suppose que I'integrale est et faisant 1 — a; = C P, il tire A]sr=:|(i-cp^), g et conclut que parce que lorsque CP = 0, Tare AE = -, ainsi 2 C P dtant negatif et (- CYy =.+ C P^ AEE' devrait 6tre egal a AN, ce qui evidemmentne peut pas etre; car AEE' > AN, et ainsi, dit-U, " Le calcul est en defaut." 88 PAEADOXES. Mais tout vient de ce que Ton a pris I'equation de C, et que pourtant on a pris A pour rorigine des x. Si nous prenons A comme I'origine des x et de Tequation, nous avons 2/ = (l - x^)^, par .consequent et ainsi et an = |ap^ + |, aee' = |ap'* + | en supposant avee d'Alembert que C P' = C P. Mais quand m^me nous prenons C pour I'origine et faisons P positif et C P' n^gatif, si C P = a; et P M = ?/, nous trouvons E E' + A R, c'est-a-dire AEE'> AAN. Cela paratt clair et manifeste, si nous prenons I'origine qui est beaucoup plus commode que I'autre pour I'investigation des propri^tes de la courbe. L'^quation etant / t cT" -r / \ >. Kg. 2. E ~- \^p' A / P^ \ c c soit A le centre de la courbe : AB = AE = a; et prenez les Taleurs positives de x entre A et E, les negatives INTEGEAL CALCULUS. 89 entre A et B. Le paradoxe suppose est que A P etaut egal a A F, on trouve Tare B M egal a Tare E M', parce que — A P* = + AP^. Or, voyons quel est Tare lorsque A est rorigine ; alors dy = ^ — T — , par consequent Tare egale dx ■ ^ ^ = -a" x" + C, x" ^ 3 J. 5. et TU que Tare = 0, lorsque a? = 0, = 0, et - a" a;" repre- 3 sente Tare Au point E, ou lorsque x — a, Tare = - a, au point P', mettant on a Tare AF=^etAP=-| 3 W a = -a o g et M A egal aussi a - o, a cause de I'egalite de et + (^] et-(- EM'a = ^a = BMa, et enfin EaB = 2 .EM' a. Ainsi nous avons EM'aM = ^«+3^ 3 tandis que E M' a n'eat que - a. Par consequent, EM'aM >EM'a, 90 PAEADOXES. comme il doit I'Stre, et le paradoxe cesse. Ainsi il paraib manifeste que prenant a = 1, comme le fait d'Alembert dans I'equatiou de la courbe,* M'a = Ma = |, M'a = Ma = |,EM'a = BMa = | 15 EaB = 3, Em'aM = — , o le defaut du calcul n'existe pas. Si pourtant on pretend encore que la brancte B M ou que la section enti^re BMa est negative malgr^ I'incontestable egalite de (+ a? V et (— x^J , alors nous avons un argument de la meme espeee que celui que soutient d'Alembert comme preuve d'un autre paradoxe allegue dans le YIII" volume des Opuscules (Mem. L VIII). II trouve difficile a comprendre comment pre- nant A par I'origine des x au cercle ^S- A M B, diamfetre AC = a, la valeur radicale de A M etant = + /J a x, \a, negative sera A B lorsque A M est la positive, et non pas A M' dans le sens directement contraire a A M, et apres avoir demontre que cette negative ne peut pas etre A P, il conclut que — /J ax est A M aussi Men que + ija x. Mais il parait veritablement que tout ce raisonnement est fonde sur erreur, et que . bien qu'il ne peut pas exister un AM' parce qu'il n'y a point de cercle au-deli de A, plus qu'il ne peut y avoir de A P ; toute- fois, que A M represente — nja x autant que + ija x, et que regarder A B comme — tj ax est une erreur. Effectivement A B est trouve, comme Test A M, par V AP" + PB» ou V^^Ty^ = 'J~ax, et quoique, lorsque Ton prend le diametre pour I'axe A B = AM (d'oii vient I'erreur), si toute autre ligne est prise pour * On voit que la lettre dans la figure n'a aucun rapport avec cette lettre dans I'^quation. INTBGKAL OALCtTLUS. 91 I'axe, A M et A B sont parfaitement inegaux, eomme a M < a B si a n C est I'axe. Cependant si le paradoxe existait du tout, U s'appliquerait autant au cas de qu'au cas de AM = ± ^J ax. E.e.4. Sa valeur negative ne serait pas, salon d'Alembert, dans la direction a b, tout directement opposee a a M, mais dans la di- rection a B. On pent faire remarquer en passant que cette discussion suggere une propriete de la parabole conique dans son rapport avec le cercle, et fait voir que cette propriete n'appartient qu'a une branche de la courbe AM = Vax'et PM' = AM, si M' est dans la parabole dont le para- metre egale A C = a. Et ce rapport des deux courbes continue jusqu'a ce que x (de la parabole) = a, c'est-a-dire jusqu'a C ou y = a = C C, lei done nous avons la valeur negative de A M' et de P M' ; P P' = P M', et ils sont directement op- poses. Mais A M' et A P', comme A M et A m, ne sont pas directement opposes ; chacun d'eux doit etre trouve par un procede a^paxi, et I'un n'est pas le negatif de I'autre, +A/ax-\-cc^ est la valeur de tous les deux. On voit aussi dans cette propriete de la parabole son rapport avec rhyperbole, comme de la parabole avec le cercle, a cette difference pres que ce rapport s'etend par tout le cours des deux courbes, au lieu que le rapport de la parabole avec le cercle est borne a la portion dont I'abscisse n'excede pas le parametre. On doit de plus faire observer que mSme a I'epoque bien ant^- rieure de I'Bncyclopedie (1754), d'Alembert avait eu des opinions partieulieres sur les quantites negatives {voir I'article Courbes), et sa controverse avec Euler sur les logarithmes des quiantites negatives est assez connue. 92 PARADOXES. Maintenant on peut faire remarquer que quand mime nous pourrions conceder rexistence du paradoxe que d'A.lembert suppose sur la courbe ■ y" + OS'' = a^, la solution qu'il donne n'est aucuuement admissible. L'un des defauts du calcul, dit-il, peut etre explique par la supposition que la branche C B (Fig. 2) est situee au-dela de B, comme B D, par quoi, dit-il, il y aurait continuation de la brancbe aB, comme s'U croyait qu'il n'y eut aucune continuation en B C. Mais contre cette supposition s'elevent deux objections deci- siyes. Premierement, I'equation donne aux y entre A et B des valeurs egales et opposees des deux c&tes du A B, au point B, 2/ = 0, et au-dela de B, comme par B d, portion de I'axe qui repond a B D, y ne peut pas exister, vu que a; = > a, et que le radical devient t/ — 1. Mais secondement, U n'y a pas possibilite qu'une courbe algebrique comme Test celle-ci s'arrete tout court, ce que, par cette supposition, elle devrait faire au point D, tandis que la difficulte qui principalement fait recourir k I'hypotbese, la discontinuation supposee de la branche a B au point B n'est reellement, excepte que la courbe a un point de rebroussement (ou une cuspide) au point B. Si le celebre g^ometre eut examine la courbe entiere * au lieu de se borner k une de ses portions, il aurait trouv6 qu'elle est une ligne a E C B, k quatre cuspides, en rentrante en elle-meme ; et il aurait certainement abandonne sa tbeorie et aussi sa supposition du paradoxe et du defaut du calcul. Mais c'est certain aussi qu'il aurait trouve d'autres paradoxes que Ton doit infiniment regretter qu'il n'ait pas examines, et dont la solution ou I'explioation parait assez difficile, pour ne pas dire impossible. lis out rapport avec les rechercbes de dynamique * Nul doute qu'il donne la figure de la courbe entifere dans la planche ; mais il ne parle du tout que des deux branches E a, a B, et sa notion que la courbe s'arrgte tout court k B avait la mSme application a la branche- E a qui devait etre censee s'arreter tout court au point a; et il ne propose pas que cette branche B a soit continuee de I'autre cote de I'axe a. Ainsi il parait certain qu'il n'avait pas form^ les deux branches E C, B 0, et il se peut que la figure fut tracee apres qu'il eut fini sa description.. INTEGEAL CALCULUS. 93 plutot qu'avec I'analyse pure, et nous nous proposons de les considerer d'abord, et de finir aveo quelques autres matieres touchant la courbe, independantes de celles renfermeea dans la discussion de dynaniique. Supposons maintenaut qu'un corps ou une particule fasse une revolution dans cette courbe eomme orbite, le centre de la courbe etant celui de la force centripete. Cette force etant r proportionnelle a ; (r = rayon vecteur ; P = perpen- 2P^ .E diculaire sur la tangente ; R - rayon de courbure), Ton a la sous-tangente la tangente PT=^ = ^ dy 1 / J 2.\ et 1/2 2\ MT = o^ U"^ - x^), . ^ AT . PM X 1 /■ i i\}. AO = — --; — = a' x' {,a' - x^J^' et :& ^ 3 . a' X' {a^ - x^')" = 3P; par consequent, la force centrale/est proportionnelle a r i{a^ - x^y ^x^y^ 4. ±/ s 2^2 " ' ± JL / 2 2\2' 6(2" x'' \a^ — x'^ J 6 a' a?" \a' — x' ) -J-, et siffl = ],/ (a' - r'f '■' (1 - r^Y telle est I'expression de la force en fonction de la distance 94 PABADOXES. Cette force est repulsive par toute I'orbite, car P et E. etant des c6tes opposes de I'axe doivent avoir des signes differents, T et ainsi I'expression doit etre toujours negative. Mais ^' -tr • Jaj voici un resultat de 1' equation. La force devient iafinie lorsque a; = 0, c'est-a-dire au point a de I'orbite, et aussi lors- que X = a, c'est-a-dire au point B de I'orbite, et elle est infinie aux deux autres points E et C. Si Ton fait le cuspide (point double) C le centre de force aU lieu de A, on trouve I'expression de la force (mettant a = 1) {(.^+[i-a-.itf)'f [3 i"]8 1 ' 1 - (i - x^y _ a;^ (i - x^y\ (i - x^y et ici comme dans I'autre cas, la valeur de la force est infinie pour les deux valeurs de a;, a; = 1 et a; = 0, et qui est assez remarquable ; elle devient infinie au poiat B dans la portion de I'orbite C B oil la force est attractive aussi bien que dans la portion aBoii elle est repulsive, ou dans toutes les quatrd branches lorsque C, au lieu de A, est le centre de force. Meme resultat si Ton prend comme centre de force les points E et B. Ainsi il est manifeste que dans tous les cas la valeur de la force devient infinie lorsque le mobile arrive a I'un des points de rebroussement. Avant de discuter ce resultat, il sera bon de faire observer que la meme chose arrive dans le cas des autres orbites, et que toutes les difficultes que Ton eprouve dans la courbe dont nous sommes occup^s se recontrent dans ces autres trajectoireSf Par exemple dans la lemniscate y = a; (1 - x'f, ' dont la sous-tangeute est a; (1 - a;^) 2 x-3 a;' (2 _ 4a;' - 3 xYl ■ 1 _ 2 a;"' ~ I 2^^~Z''S^ ' '■ ^^ (2 + 4a^'-5a:')^ lx-6x INTEGEAL OALOTJLUS. 95 ainsi f = (2a^'-3) (2 -a;')" ■' ' a; (2 - 3 xy ' par consfequent, / est inflni soit que a; = 0, soit que x = \/ -. Mais I'analogie avec notre courbe parait plus complete si I'on prend le centre de force a I'une des extremites de I'axe ; car alors le mobile tournant dans I'orbite passe par le mUieu de I'axe, d'un cote a I'autre de cet axe, et a ce poiat la force est infinie. Meme chose dans la ligne que Newton appelle Para- bola nodata (Enumeratio Lin. tertii ordinis, IV. 13). II n'en donne pas I'equation, mais on peut la deduire de I'equation generale ; elle est 1 y = x{a- xy, J 1 . , 2 a; (a - «) qui nous donne pour la sous-taneente —r — , Za — ox (4 a a; — 5 a;') (a — x)^ pour la perpendiculaire [{2a-3xy + 4:(a-x)f rayon de courbure - [(2a - 3 a;)' + 4 (a - a;)P 2 a; et r etant egal aa-V^+l — a;, nous avons 2 (a + 1 - xy f = a; (4a- 5a;)' (a -a;)' La lemniscate a, comme on sait, la figure d'un buit de cbiffre. La parabola nodata se compose d'un ovale et deux branches infinies, sans asymptotes. II y a deux difficultes qui principalement se presentent dans cette discussion. La premiere est la transition du corps mobile de I'une des branches de notre courbe a I'autre, une discon- tinuite complete existant a ce que Ton a souvent pr^tendu. 96 PABADOXES. La seoonde difficult^ est la valeur infinie de I'expression pour la force a certains points de I'orbite. Sur la premiere de ces difficultes, et en partie sur la seconde aussi, la consideration de la parabola nodata et des courbes de cette forme paralt repandre de la lumiere. Car si Ton prend |)Our centre de force un point de I'axe, A', hors de I'ovale, la force repulsive fera passer le mobile de a par B, m (Kg. 6) jusqu'au point A oil cette force devienb attractive ; et en cbangeant de position de I'un des c6tes de I'axe a I'autre, le corps passe par A, ou la force devient infinie. Or on peut supposer que la ligne AB, I'axe de I'ovale, decroit inde- finiment jusqu'a ce qu'elle s'evanouit ; et alors, comme I'a remarque Newton lui-meme, I'ovale devient une cuspide (point de rebroussement). Ainsi cela pourra arriver dans le cas de cbacune des quatre cuspides de notre courbe. Toutes ont pu. Stre des ovales dont les axes s'etaient evanouis ; mais a I'instant d'evanouissement de I'axe, et lorsque I'ovale fut presque eteint et reduit aux dimensions les plus petites, pour ne pas dire infinitesimales, le corps avait ete pousse par la force d'abord repulsive, puis a I'extremite de I'axe de I'ovale attractive, et la valeur infinie de la force avait existe au point A reuni au point B apres, I'extinctidn de cette force ayant ete infinie a tons ces deux points avant I'extinction de I'ovale, Sur la seconde difficulte, il y a un exemple. plus faniilier dans le cas du cercle, loraqu'il est I'orbite d'un mobile, et que le centre de force est dans la circonf6rence ; car alors cette force devient infinie ( I'expression ^tait — au lieu de — ) au pas- sage du corps par le centre : or r = ; mais a I'autre extremity du diametre elle ne Test pas comme elle est dans la parahdht nodata. Un ami tres-savant dans la geom^trie avait pens6 que I'explication de I'infini au passage du corps de I'un a I'auti'e c&te de I'axe se trouve dans ce que la force finie ne peut INTEGRAL CALCULTJS. 97 aucunement le faire passer d'une tranche de la courbe, et qu'il doit s'^loigner k I'infini, plutSt que de prendre I'autre branche ; mais I'exemple de la lemniscate paraft repousser cette notion, aussi Men que celui de la parabola nodata, et meme du cercle ; car dans tous ces cas, le corps continue son mouvement sans aucune interruption en passant par le point oil la force devient infiuie. L'analogie des forces qui agissent en raison inverse de la distance vient nous frapper dans cette discussion. On pent pourtant remarquer que lorsque la gravitation est suppos^e d'agir avec une force inflnie, vu que la distance n'existe plus, il est question du centre du globe, ou toute la masse est supposee r^unie, et aussi il y a toujours le rayon du globe entre le corps qui gravite et le centre de force. Que devrait-on dire de la force magnetique, soit que cette force est, comme I'a supposee Newton, I'inverse cube de la distance,' soit I'inverse carre comme Ton pense aujourd'hui ? Dans I'un ou I'autre cas au point de contact la force devient infinie, et pourtant les pheno- mfenes nenous declarent aucune force infinie. M^me remarque peut se faire sur toute force ou influence quelconque venant d'un centre et propag^e a la circonference, de force ou d'in- fluenee. Peut-ltre faut-il admettre la theorie de Boscowich, qui suppose une force repulsive plus pres des corps, et croissant en raison inverse de la distance, et ainsi contrebalan^ant ou rerapla9ant la force d'attraction ; et les speculations sur I'im- possibilite d'un contact complet ont du rapport avec la propo- sition de I'infini, en tant que Ton pourrait deduire cette im- possibUite de la non-ezistenee dans la nature d'une force distrayatite (divellante). Mais il y a une plus grande difficulte que celle que nous avons consid^ree dans I'expression de I'infini. Les cas que nous venons de considerer ont rapport avec des points de I'orbite, la ou elle passe d'un c6te de I'axe a I'autre et que la tangente devient nuUe ou infinie. Mais que dire d'une valeur infinie aux autres points, comme dans la lemniscate au point oil X = a/ - a, et dans la parabola nodata, a a; = - a ? Ce- n 98 PAEADOXBS. pendant ce n'est pas a ees valeurs de x que les courbea sont le plus eloignees de I'axe et que leurs tangentes sont infinies ; au contraire, c'est la ou a; = 7 a dans la lemniseate, ou au milieu 2 de I'axe de 1' ovale, et la oii » = - a dans \& parabola nodata. Si o Ton n'^tait pas assure que le proc^de pour obtenir la valeur de la force centrale est de toute exactitude, par la conformite de ses resultats aux lois les plus connues de dynamique, par- ticulierement k la raison inverse de la distance des foyers des sections coniques, on serait tente de soup9onner quelque para- logisme en observant le resultat des memea precedes dans le cas que Ton vient de traiter. Pourtant, au lieu de dire para- doxe avec I'illustre geometre dont nous avons ose tant parler, il vaut mieux de soup9onner quelque erreur dans I'appHcation des precedes du calcul, quelque confusion telle que Ton peut remarquer dans ses raisonnements, confusion, c'est-a-dire des valeurs algebriques et geometriques, a ce qui regarde le signe negatif, et ainsi cela sera non pas le calcul en defaut, mais • ceux qui I'appliquent. Les propriet^s gen^rales et geometriques de la courbe qui nous a occup6 d'un autre point de vue, sont assez curieuses pour meriter une discussion plus suivie. 1. Ce qui nous frappe d'abord, c'est rexception que parait aj outer cette courbe aux autres exceptions au celeb re lemma (XXVIII) de Newton, portant qu'aucun ovale n'est susceptible ni de quadrature ni de rectification. D'Alembert a note sa rectification, qui ne peut pas 6tre douteuse, vu que 1 ay d X tj dy^ -^ dx^ x^ dont I'int^grale est -^a^ x^ + G; et vu que a; = 0, Tare = ; ainsi = 0. Mais la quadrature aussi est possible ; car INTEGBAl CALCULUS. 99 \ :i/dx = \ dx Va' — a; V ^ , ou (si nous mettons x = z^) et I'aire ^'•[h^l---(S)-l.l-u^l 4-9i + C; et C = si Ton prend I'aire depuis A a ; et I'aire entiere A a B, 3 X 6tant = a, est — n . a^. o2i On dira peut-efcre que lorsque Newton a enonce rimpossi- bUite, il s'est servi de Texpression figura ovalis, et qu'il a pu vouloir se borner aux courbes d'une courbure continue, comnie le cercle et I'ellipse. Pouitant Topimon universelle porte qu'n avait regard a toute courbe rehtrant en elle-"meme ; et cette opinion est appuyte par la consideration qu'en donnant les cas d'exception k son proposition, il se borne aux cas des courbes qui ont un arc infini avec leur ovale. Mais aussi il est certain que la demonstration de sa proposition s'applique aux courbes telles que celle qui nous occupe h, present. Car on pent prendre le centre pour le pivot sur lequel tourne la regie qui est supposee. Encore on n'a jamais pretendu que la lemnis- cate flit exclue de la proposition, toute carrable qu'elle soit, quoique non rectifiable. 2. La courbe est une ^picycloide engendree par le roulement d'un cercle dont le diametre est un quart du diametre du cerele ext^rieur. Si le rayon de ce cercle = a, I'equation de la courbe etant 2. 2 2, y" -\- x" — a", a le rayon du cercle roulant est -. H 2 100 PARADOXES. ^ 8 2 3. Si I'oii deorit une ellipse sur I'axe de la courbe y^ + x* = a", at que la somme des axes de I'ellipse = a, elle touchera la courbe. 4. La courbe a quelque ressemblance avec la ddveloppee de I'ellipse ; mais elle ne Test pas ; car requation de cette de?e- loppee differe de notre equation. Elle est . y* + a^ a^^ = (1 - a^Y, les axes de I'ellipse etant 1 et a. Mon savant ami M. Eouth a examind la question, n'ayant doute que notre equation ne soit celle de quelque developgee ; et il trouve que dans un cas _2. 2 ^ 2 2/" 4- a' a;" = a" est la d^velopp^e d'une ellipse, notamment de celle dont I'equation est ^ "*" a» (1 - d-y Lorsque a > 1 ou < 1, la courbe est la d^veloppee de quelque ellipse. Mais dans les cas qu'elle ne le soit pas, eUe est fre- quemment la developpee d'un ovale de quelque espece differente de I'ellipse. Lorsque a = 1, le precede manque completement, et Ton ne peut avoir aucune developpee. Dans plusieurs livres el^mentaires, on remarque la developpee de I'ellipse representee sous la forme de notre courbe ; mais elle est com- pletement differente dans le fond. 5. La perpendiculaire h. la tangente du centre, de la courbe (a etant = 1) est a; 1^1 — x'^ P et le rayon de courbure 3 . a' (l-a^O'- AinsiE = 3P. 6. Si la tangente est prolongee jusqu'a ce qu'elle rencontre les axes perpendiculaires de la courbe, cette tangente ainsi prolongee est toujours egale a I'axe, c'est-a-dire a a. 7. De cette propriety de la tangente prolongee constante, resultent des consequences asaez remarquables. Entre autres on peut noter celle-ci : Si un point est pousse sur une ligne donnee entre deux perpendiculaires, avec une vitesse uniforms, tandis que cette ligne est pouss^e sur I'une des deux perpendi- culaires avec une vitesse inversement proportionnelle a la dis- INTEGRAL CAIiCUIitTS. 101 tance de son entremite de la perpendiculaire, le point mouvant deerit la courbe y^ + a;" + a', les axes etant chacun = a. c Soit E N la ligne, M le point, A B un des axes. Si le mouve- ment de M sur E N est uniforme, et que N est pousse avec la v^locite xIcti ^ deerit la courbe. Encore prenez D pour le centre instantane de rotation de E N ; la perpendiculaire D M, de D sur E N, coupe E N en M, qui est dans la courbe ; le mouvement de rotation de la ligne etait combine avec le mouve- ment en ligne directe du point.* Si le point M reste sans mouvement sur E N, tandis que E N est poussee sur A B et A C, M deerit une ellipse, que devient un. cercle si M. est au milieu de E N. 8. La propriete de la tangente prolong^e constante mene natureUement a la comparaison de notre courbe avec une autre que j'avais decrite il y a soixante ans dans les Phil. Trans. (1798, part. II), comme ayant une tangente constante, et par consequent la sous-tangentef dx yj-y^^^'^-y^^ a etant la longueur de la tangente. L'equation difRirentielle dx - nous donne pour integrale dx = ^ ^Ja^-y\ y ± >J a^ - y"" -\- a .log. y a ± ^ a' —y^ * Gette proposition s'est presentee b. mon illustre confrere M. Chaslea, qui a eu la bonte de me la oommuniquer. t Voii' Art. 1 de ce volume. 102 PARADOXES. Et la courbe est de la formule (fig. 8) CMn, ayant une cuspide k 0, et efcant convexe k I'axe A B ; notre courbe aux quatre cuspides est CmN, ayant la tangente prolongee ST constante, = A C = A B ; tandis que la logarithmique C M w a la tangente MB = A C, et a = A C. L'arc de celle-ci S = a . log. w + C, et comma C . M « y = y et a;'' = 0, C = I'aire = tj .ydx = sdy aJ a^ — y^, ainsi la courbe a ce rapport avec le cercle, que °Mm' etant un cercle dont le rayon = B M = ffl, I'aire de la courbe, A C M P est egale a I'aire du cercle P M 6 B. Ce rapport avec le cercle n'existe par dans I'autre courbe CmN; non plus que cette autre propri^te de la logarithmique, qui la lie avec la tractrice de la ligne droite.* * This tract is the Mem. read June 1857, before the National Institute. The volume of Mem. is not yet published, but only the Compte Rendu. ( 103 ) VI. AROHITECTUEE OP CELLS OF BEES.* ■QroiQUE peu de sujets aient occupe davantage les Baturalistes de tous les siecles, et meme les geomfetres depuis le temps d'Aristote et de Pappus, que I'abeille, ses habitudes, et son architecture, on ne pent pas nier qu'avec un grand progrfes et des verites importantes, des erreurs ne se soient glissees assez remarquables pour m Writer une explication. Aussi est-il certain qu'un peu d'attention suffit pour dissiper les erreurs que la negligence ou les pr^juges ont fait nattre chez les geometres egalement, et chez les naturalistes, tandis que tous les deux s'etant arret^s tout court ont manqu6 faire des observations interessantes qui se presentent en relevant les erreurs. De ces erreurs I'une est entomologique, I'autre geometrique. L'avance- ment de nos connaissances sur ce sujet est d'un interSt, et meme d'une importance sous plus d'un point de vue, qui justifie quelques details. I. Dans les transactions de la Societe Wernerienne (vol. II, p. 260), le Dr. Barclay, celebre anatomiste d'Edimbourg, a annonce une decouverte que les naturalistes ont cite I'un apres I'autre, comme constatee sans en examiner les preuves ; on peut-etre trompes par les mfemes apparences qui avaient egare M. le docteur. II se pent qu'ils fuient disposes de I'accepter d'autant plus que nous devons a un autre anatomiste, un grand physiologiste (le celebre J. Hunter), la plus im- * This tract is the memoir read May, 1858, hefore the National Institute, " Becherches Analytiques et Exp&imentales." The volume of Mem. has not yet been publislied, hut only the Compte Bendu. 104 STETJOTUEE OP BEES CELLS. portante des decouvertes en cette branclie de science. La proposition dont il s'agit porte que chaque alveole, taut pour. ses parois hexagones que pour son fond oa base pyramidale, est, double, de maniere qu'elle est separee et independante. des alveoles qui I'entourent, et formee d'elle-meme ; que ses parois de cire sont attaches aux parois des autres alveoles par une substance agglutinante ; et que si cette substance est detraite, chaque alveole peut ^tre entierement separee des autres. Le Dr. Barclay pretend aussi que les alveoles des guepes sont doubles, et que leur substance agglutinante est plus facilement^ detruite que ne Test celle des alveoles d'abeilles. ■ II parait presque impossible de croire a cette structure apres les observations des naturalistes, surtout de Reaumur et de Huber, sur la maniere dont I'abeille travaille. Elle ne peut pas., s'insinuer entre les deux plaques de cire pour les polir ; car elles sont parfaitement et egalement polies. La substance agglutinante n'existe pas dans la cire. Mais avant tout, Tin-, speotion des gateaux de cire prouve que si les alveoles n'onu jamais servi pour faire eclore des oeufs, et pour I'^ducation des vers et des chrysalides, on ne voit aucune trace de parois doubles. Celles dans lesquelles les larves ont et^ transformees en chrysalides presentent I'apparence qui a egare le Dr. B. ; et Ton remarque que son Memoire etait accompagne d'un gateau de cire vieille, dont les alveoles avaient entretenu plusieurs successions d'insectes. Mais venons aux phenonfemes de plus pres. Un gateau fut choisi dont une portion n'avait jamais servi ni pour amasser, ni pour engendrer, et dont I'autre portion- avait re9U une seule couvee. La premiere portion etait parfaitement blanche ; la seconde avait une Mgere teinte jaun&tre, ou une nuance brune tres-legere; et dans plusieurs endroits, on voyait.' de ces raies rouges, observees par Huber, et qu'il prouve Itre. une matiere veg6tale cueillie des arbres, et surtout du peuplier. j Le gateau avait ete fait au mois d'aotit, et fut pris quatre semaines plus tard. Etant plonge dans I'alcohol, peu ou points de changement fut produit avant que I'alcohol f^t echauffe ; ; et alors la cire s'est fondue tout de suite ; la partie blanche fut STETTCTTJKE OF BEEs' CELLS. 105 entieremeut dissoute, sans qu'aucune trace des alveoles restat ; et la partie jaunatre ne se fondit pas entierement. De cette partie il decoula de la cire fondue, mais le gateau gardait sa forme et ses dimensions a peu pres, bien que la chaleur con- tinuat. Lorsque I'alcohol bouillait, cette portion du gateau dans laquelle les abeilles avaient et^ produites se separait en morceaux, mais il fallait remuer pour aider la separation, et pour faire fondre toute la cire. Lorsqu'un gSteau plus vieux, et qui egalement avait produit des abeilles, fut mis dans Talcohol moins bouillant, la separation et la fonte de la cire deman- derent plus de temps ; mais enfin en le remuant toute la cire se fondait, excepter cette. petite portion que I'alcohol ne prend pas, et qui restait dans la forme de petites globules ; mais toutes les alveoles sont restees dans le.ur forme, cbacune s^par^e des autres, et pas une seule ne fut composee de cire, mais toutes de sole, de cette sole, c'est-a-dire, que forme le ver avant sa transmutation en nymphe ou chrysalide, et dont il tapisse I'interieure de I'alv^ole de cire. Avec de I'eau bouillante on pent op^rer de meme, mais plus lentement. Avec I'esprit de terebinthe, la fonte de la cire est tres-rapide, seulement on ne pent pas voir par cette forme de I'experience dans quelle partie de I'alveole la cire existe. L'acide sulfurique pent faire precipiter ou fondre la cire sans la dissoudre autrement qu'en tres-petite quantite, et les alveoles restent. L'experience fut repetee avec un gateau dans lequel plusieurs couvees avaient ete produites. Les al- veoles furent moins larges, leurs parois plus epaisses, et leur couleur, une nuance brune foncee, 9a et 1^ presque noire. Maintenant, examinons les alveoles separees par ce precede. Chacune fut formee d'un prisme hexagone termine par nn pyramide de trois rhombes tJgaux ; en un mot, cbacune fut exactement k la matiere pres une alveole comme celles de cire ; mais formee de materiaux entierement differents. Les parois et la base furent composes d'une pellicule extremeraent mince et semi-transparente qui ressemblait a la feuille de battant d'or, mais absolument sans ride. Les plus vieilles garderent la forme plus exactement ; de sorte que leurs angles et leurs plans furent aussi bien deflnies que le sont ceux de cire dans le gateau neuf. 106 STEXJCTUEE OF BEBS' CELLS. Mais ce n'etait point la une seule pelliciile, comme celles qui n'avaient servi qu'a un seul ver ou insecte ; au contraire, ces alveoles avaient plus d'lme pellicule, I'une du dedans de I'autre ; et ces pellicules pouvaient etre separ^es au nombre de cinq ou six, toutes provenantes de la meme alveole, et toutes gardant les formes hexagones et rhopiboidales ; mais la sixieme avait des rhoHibes beaucoup moins marques ; et s'il fut jusqu'a une neuvieme ou dixifeme, la base devenait plut6t spherique que pyramidale, et etait tres-peu profonde. Les parois hexagones de toutes les alveoles gardaient cette forme ; seulement les derniers (c'est-a-dire les interieurs) avaient un plus petit- diametre. Dans les angles il y avait un peu de la matiere rouge, mais beaucoup plus dans le fond, ou partie pyramidale^ Cette base dans les alveoles internes paraissait presque rem- plie de rouge. La bouohe de I'hexagone a toujours un bord compose principaiement de cette matiere. J'ai trouve impossible de dissoudre, ou de quelque maniere que ce fut d'affeeter la pellicule, soit en la macerant dans ralcobol, dans I'esprit de terebinthe, ou de tout autre r^actif, meme bouillant, exceptor que la matiere rouge apres une longue maceration 6tait depositee, et donnait un teint jaunatre a la liqueur. L'exactitude avee laquelle la pellicule tapisse la cire de I'alveole est tres-remarquable. II n'y a pas le moindre ride, ni intervalle. Tout est convert, et avec une pellicule de la meme epaisseur partout, exceptez que la matiere rouge aux angles fait Yarier un peu I'epaisseur de la pellicule a ces angles. Tout I'int^rieur de I'alveole forme un tapis uni, sans aucune couture, et sans aucun ciment. Car apres avoir soup9onne que la matiere rouge aux angles pourrait servir de ciment, cette notion a ete de suite' contredite par I'inspection de ces parties angulaires qui n'avaient jamais eu de couche de rouge, et de celles dont la matiere rouge avait ete grattee et enlevee. Aussi j'ai trouve que la matiere rouge etait exactement sur les mSmes portions de la pellicule. Car en deeoupant un hexagone conte- nant plusieurs pellicules, de manifere a etendre tous les six c&tes (comme k la figure 1), on voyait que cette matiere etait repartie STEUOTUEE OF BKES CELLS. 107 dans toutes ces pellicules de la m^me fa9on. La fig. 2 fait voir la distribution dans les angles de la base ; et la il n'y a pas de \ 1 Kg. 1. Fig. 2. difi'6rence entre les pellicules successives par rapport a cette matiere, excepter que, etant retrecie dans celles qui sont le plus eloignees de la cire, la matiere rouge occupe une plus grande proportion, et la partie sans rouge une plus petite, la somme totale du rouge etant le meme dans toutes les pellicules. Une pellicule de la meme substance, aussi transparente mais bien plus epaisse, tapisse I'alveole de la reiae abeille. La matiere rouge est plus egalement repandue sur sa surface en nuages et raies, vu qu'il n'y a point d'angles qu'elle doit doubler. La pellicule de cette alveole royale prend la forme de la cire ; mais ce qui est trfes-remarquable, c'est qu'elle n'est pas toujours sur I'interieur de la cire. Quelquefois elle est enferm^e dans la cire, dont une coucbe est meme plus epaisse que les parois de cire, et j'en ai examine qui avait une epaisseur beaucoup plus grande. On peut constater que dans les alveoles ordinaires, la cire n'est pas platree sur la pellicule. On a examine de pres les plus vieux gateaux ; et jamais Ton n'a trouve un seul ex- ample de pellicule entre deux couches de cire, excepte dans I'alveole royale. Aussi on a vu clairement qxie dans les plus vieux gateaux, qui donnent plus de neuf ou dix suites de pelli- cules dans les alveoles ordinaires, I'alveole de la reine seule n'avait qu'une pellicule. La manifere de former ces pellicules et de tapisser I'alveole me- rite beaucoup plus d'attention qu'elle n'a jusqu'a present refue. L'opinion generale paralt Stre qu'elles sont fabriqu6es en tissue. M. Daubenton (Encyclop. 1751, vol. I, p. 21) decrit le precede de tisser comme op^re en mettant des fils trfes-fins et tres-pres, I'un de I'autre, qui se croisent. Huber semble Stre de cette 108 8TEUCTUEE OP BEES' CELLS. opinion, et que lever tapisse a la fois qii'il forme la toile, et non pas qu'il fait la toile d'abord et puis I'applique aux parois. II parait presque impossible de crbire que la toile est faite par cette operation en meme temps qu'elle est appliqu^e. Car la largeur de I'alveole varie des le commencement de la partie pyramidale a chaque point, et bien que le ver n'eut a tisser qu'autour de la mSme circonference et sans avoir le moindre aide pour le regler, cependant il devrait faire la toile si exactement adaptee a la cir- conference, qu'en I'appliquant il n'y en aurait ni de trop, iii de trop peu, et sans aucune ride. C'est certain qu'une telle opera- tion surpasse infinim'ent tout ce que fait jamais I'insecte parfait. Avec toutes les resources de notre science et de notre meca^ nique, on pent afBrmer hardiment qu'il nous serait impossible de tisser un sac de toile de largeur variaifte a tons les points, et pourtant si exacte dans ses proportions qu'etant decoupe ou fendu, il tapisserait les murs sans la moindre ride, et sans aucune intervalle.* La difficulte est nioius grande si le ver tapisse au moment d'appUquer, et qu'a chaque instant il place la toile qu'il vient de fabriquer. Mais c'est plus probable qu'il n'y a pas de tissage du tout. Certainement la plus puissante loupe ne fait voir aucune filature. Apparemment une matifere glutineuse est repandue par le ver sur les parois; et toute difficile que soit cette operation aussi, elle Test beaucoup moindre que I'autre, vu que le ver a les parois pour le guider. II n'est pas douteux pourtant que le resultat soit extraordinaire ; car non-seulement il y a une egale ^paissear par toute la pellicule, mais le ver en • J'ai mesure et oaloule la difference de la surface des trols portions du tuyau de I'alv&le. La partie pyramidale, la partie voisine, composes d'une portion de la pyramide et d'une partie de I'hexagone, et la partie de I'hexagone seule. En supposant toutes les trois portions de la mSme hauteur, les surfaces sont comme 3-03, 5-05, et 4-04 (lignes carrees) respeotivement. Ainsi en filaut le tissu le ver devrait tisser exactement dans ces proportions ; et en Slant les deux premieres parties il devrait changer k chaque instant la vitesse de son travail, vu que le contour, ou circonference de la surface ne reste par la meme, mais change & chaque instant, et que le ver devrait tisser en suivant cette cu-conference. Oette circonference varie depuis le fond pyramidale de zero a douze lignes sur la sm'fatse oi-dessus notee. STEUCTUEE OF BEES' CELLS. 109 faisant le contour pour platrer, doit s'arreter exactement au point d'ou. il est parti, vu qu'il n'y a pas le moindre vestige de la jonetion des deux c8tds ; pas la plus petite difference d'epais- seur. 11 parait presque certain que la pellicule est donee de differentes qualites, selon qu'elle est nouvellement faite ou le contraire. On expliquerait dif&cilement le phenomene des vers la fabriquant toujours avant de devenir chrysalides. Car le premier ver avait deja tapisse la cire ; et s'il n'avait besoin que de se proteger lui-mSme, ou sa chrysalide de la cire, le second ver qui naitrait dans la m§me alveole serait protege par la mSme pellicule ; et ainsi des neuf ou dix autres successeurs ; et pourtant tous doivent faire une pellicule chacun, meme en diminuant I'espace, et a la fin presque la remplissant. II ne nous est aucunement permis de dire que void une des meprises que fait I'instinct quelquefois, parce que ces meprises sont toujours accidentelles ; par exemple, lorsque la mouche trompee par I'odeur d'un fleur et croyant que c'est de la charogne, y pond ses oeufs. Mais chez les abeilles c'est une meprise con- tinuelle et reguliere, s'il en est une ; car elles preferent toujours deposer les oeufs dans une alveole oil une couvee a ete .^levee, et ou par consequent il y a une ou plusieurs pellicules de laissees aussi parfaites que pourraient 6tre une pellicule nouvellement faite. L'instinct de I'insecte etant surtout d'economiser des mat^riaux et du travail, il le porte d'abord a preferer le vieux gateau pour ne pas faire des alveoles de la cire vierge ; mais comment alors le mSme instinct ne le porterait-il pas a profiter des pellicules qu'il trouve dans les alveoles ? Mais au lieu de cela le ver prodigue son materiel et son travail a faire une pellicule neuve pour lui-meme et pour sa chrysalide. Tin instinct qui manque aussi souvent qu'il reussit ne pent aucunement etre compare a ces meprises ou fautes accidentelles. Ainsi il parait impossible de douter que la pellicule fralche nouvdlement faite possede quelque qualite ndcessaire pour I'entretien de la chrysalide. Ceux que les alveoles de soie des abeilles avaient egare jusqu'a croire que les parois de cire sont doubles, sont tombes 110 STETJOTURE OF BBEs' CELLS. dans la meme erreur a propos de la structure des guepes. lis font observer m^me que la duplication est plus facile a voir ■dans le gfiteau guepe que dans le gateau abeille, a cause disent- ils que la inati^re agglutinante est moins adh6rente. J'ai soigneusement examine ces structures, et il n'y a pas le moindre doute que I'alveole brune, faite de la limaille de bois, est doublee d'un papier blanc, tres-fin, soit file, soit platr^ ; et on pent le separer facilement lorsqu'il reste humide, niais aussi quoique plus difBcilement si on ne pent jamais fendre le parois de maniere a en faire deux de memes materiaux. Si on le tranche ou fende au milieu, on trouve d'un cote vine plaque brune, de I'autre une plaque brune d'lm. cote et blancbe de I'autre, nommement le cote double du papier blanc. La guepe etant beaucoup moins econome des materiaux de sa construc- tion que I'abeille, yu qu'ils soijt plus facile a trouver que n'est la cire a produire, n'6conomise que I'espace et le travail en formant I'alveole brune. Les parois done peuvent etre construites par le melange de la limaille de bois agglutinee avec quelque liquide savetee par I'insecte lui-mSme. Mais la pellicule blanche est evidemment une secretion entierement, soit par le ver en devenant chrysalide, soit par la gu§pe elle-meme avant de pondre I'oeuf qui produit le ver. Ce papier est tres-fin ; il est demi-transparent, et on a trouve qu'il est capable de recevoir I'encre sans barbouiller, comme s'il eut ete colle ou lave expres. On sait combien les guepes ont anticipe depuis vingt siecles nos fabricauts de papier ; mais pour papier blanc et lavi, je ne I'avais jamais entendu dire. II. Les erreurs qu'on vient de raarquer, et qui ont conduit en les exposant a des nouvelles observations sur I'economie de I'insecte, ont ete soutenues, et en partie anticipees par des autres erreurs dont I'origine fut le desir de chasser les doctrines etablies depuis bien longtemps sur la merveilleuse operation de I'instinct de I'insecte. Plusieurs philosophes ont prfetendu demontrer, les uns que I'abeille n'est pas la veritable architecte des alveoles qui- sont produites, disent-ils machinalement par les proprietes et les niouvements de la matiere ; les autres que STBUCTUEE OF BEES' CELLS. Ill rinsecte aurait pu travailler bien plus artistement. Ces erreurs, qui proviennent de g^ometrie mal entendue, autant que de negligence dans les observations sur I'insecte, bien examinees nous conduisent a la conclusion, non pas seulement qu'il n'j a aucun fondement pour les objections elevees, mais que les operations de I'insect sont encore plus etonnantes que Ton avait ci-devant suppose. La theorie de Buffon parait la plus insoutenable, pour ne pas dire la plus absurde, egarement ; et ceux qui se rappellent le controverse qu'il avait, malheureusement pour sa reputation, engage contre Clairaut, verront encore une preuve que le grand historiographe des animaux aurait bien fait de ne toucber jamais le domaine du geometre. Ayant cru percevoir des hexagones dans les boules de savon (ce qui n'est qu'une illusion optique occasionnee par les lignes de contour qui se croisent, sans qu'il y ait un seul bexagone de forme), il suppose que la cire etant d'abord disposee en cylindres, ces cylindres par leur pression mutuelle s'applatissent et ferment des tuyaux hexa- gones. Mais pour ne rien dire sur I'omission totale d'ex- plicatiou de la base pyramidale, meme la theorie ne prouve aucunement la formation hexagone, vu qu'aucun cylindre n'a jamais existe, Huber ayant prouve que I'abeille travaiHe d'une toute autre maniere ; et puis si Ton suppose toute la cire formee en cylindres, la pression manque qui est le fondement de I'hypothese. Supposons mSme que la gravite de la partie saperieure du gateau la fait pressor sur la partie inferieure, les alveoles seront dans toutes les parties de grandeur differente, eon- trairement aux phenomenes ; et qu'arriverait-il si le gateau fut forme horizon tale et non pas verticale ? Alors point de pression ; et pourtant les alveoles dans ce cas-la ont exactement la m§me figure. On ne doit pas s'^tonner que Daubenton, dans son admirable article dans I'Encyclopedie cite plus haut, ne fasse aucune mention de la theorie de son maitre et patron, avec qui il n'avait pas encore a cette epoque eu les differends qui seuls ont terni la memoire de Buffon, pour son traitement de cet eminent savant et admirable homme. Mais une erreur d'une autre espfece a ete commise par des auteurs, tous de quelque 112 STEUCTTJEE OF BEBS' CELLS. rs^putation comme geometres, et dont I'un fut meme assez distingu6, des auteurs bien au-dessus de Buffbn dans les sciences severes. Nous commencons par celui du plus grand m^rite, et qui jusqu'a present a m cense d' avoir raison, son erreur ayant ete rejetee sur les observations supposees defectueuses d'un autre et tres-celebre philosopbe. Le grand pas qu'avaient fait les connaissances sur I'archi- tecture de I'abeille depuis les observations de Pappus sur la forme hexagone, etait I'examination de la base ; et le fameux Maraldi avait trouve que pour que ces bases s'accommodassent sans perdre de I'espace, elles devaient etre toutes rhomboidales, formees de trois rhombes i^gaux. Puis il a mesur6 les angles de ces rhombes ; et il trouva que I'un 6tait de 109° 28', I'autre de 70° 32'.* La raison de cette proportion a ecbappe a ee geometre distingue et naturaliste encore plus eminent. Mais plus tard E^aumur, avec sa sagacite si connue, a soupgonne que la proportion observee par son predecesseur devait Stre celle qui donnait dans la construction de I'alveole le minimum de travail et de materiel ; et il proposait a M. Koenig (digne 61eve des Bernoullis) le probleme de determiner les angles du rhombe qui couperont le prisme hexagone de maniere a former la figure composee d'une pyramide, et des portions triangulaires du hexagone, avec le minimum de surface. M. Koenig, ne sachant pas la mesure de Maraldi, ni mime la conjecture de Reaumur, donna sa solution, et faisait les angles de 109° 26' et 70° 34'. Lorsqu'il a appris la th^orie de E^aumur et, la mesure de Maraldi, il croyait comme Eeaumur et tons ceux a qui il avait fait part de ses conjectures, que I'abeille approche de pres mais pas exactement de la solution du probleme du maxima et minima. Mais le fait est que I'abeille a raison, et que ce fut M. Koenig qui 6tait tomb6 en erreur. M'etant assure que les angles sont ceux qu'a mesure Maraldi, et que Koenig etait tombe dans I'erreur par les tables de sinus * Maraldi donne les angles comme 70° et 110° dans une partie de son m€moire, mais k ce qu'il semble approximativement ; car plus tard il donne 70° 32', et 109° 28' exactement : il parait s'etre trompe par avoir regarde le premier passage plus que le second. STEUCTUEB OF BEES CELLS. 113 ou des logarithmes, il m'a paru k propos de conduire Tinvesti- gation par chercher ]a longueur des cotes des rhombes, ou des autres lignes qui y ont rapport ; ce qui aurait le grand avantage d'^viter les erreurs en caleulant les angles. Car non-seulement il est beaueoup plus facile de mesurer une petite ligne qu'un petit angle, mais il est evident que si la mesure des angles est exacte, la perpendiculaire d'un des angles des rhombes sur le cot^ oppose — c'est-a-dire, la largeur du rhombe — doit etre egale au cote de -I'hexagone ; et ainsi la mesure que seule il faUait faire, serait de constater I'^galite ou I'inegalite de ces deux lignes droites. Nons pourrons resoudre le probleme en cberehant ou la valeur de la perpendiculaire GrG^=y, ou la valeur du cote du e J< \^ "<\ D Z H Fig. 3. rhombe KT) = x, qui donnera la surface du rhombe avec cells du trapeze 2.EFZD c'est-a-dire la surface entiere G-EFHA, le tiers de la surface de I'alveole, un minimum. Prenons x pour le variable independant, et les rectangles O Z, P Z 6tant donnes et invariables, il faut chercher la valeur de X qui donne la somme du rhombe, des triangles APD, E O D un minimum. Soit P D = S, le c6te de I'hexagone ; par la propriete de cette figure, AE = VS.S, et AB = — - — . Partaut BD = V4a;'-3SS etle triangle ADB ■ V 3SV4a;"-3S' ' 8 114 8TEUCTUEE OF BEES' CELLS. et le rhombe A D E G = V3-SV^a;' 3S^ ^ ^^.^ AF = ^x'- B% par consequent le triangle APD = , et la surface du rhombe avec celle des triangles APD, E D = V3.8 V4a;« - 3 S« + S ^ x^ - S», dont le differentiel, + — ^^=r doit = O pour trouver le minimum,, et cela nous donne X = -^-^= AD. Mais le rhombe AD EG-aussi = G-G'X AD; 2V2 , rhombe - V4£^-3S^_3S' ^ par consequent G G- = — . _ = Vo • ^3 ^ = o^ ~ "■ Ainsi le minimum est lorsque la perpendiculaire G G\ ou la largeur des rhombes, est egale au cote de I'hexagone. Mais pour trouver les angles du rhombe, il faut considerer que les deus triangles E D, S E D sont rectangulaires, et comme S D = D, les angles DEO, DES sont egaux ; ainsi prenant DE pour rayon, nous avons EO pour le cosinus de OED ; o c a et comme ED = ^, et OE = :^, Tangle OED est 2^/2 2 V2 celui dont le cosinus est \ du rayon. Si celui-ci est 1,000,000, celui-la est 333,000 ; et dans la table de sinus naturels, le nombre le plus proehe de 333,333 est 333,258, qui repond h, Tangle 70° 32'. Ainsi c'est Tangle aigu du rhombe, et Tobtus est par consequent 109° 28'. EfFectivement Tegalite des angles OED, que fait le rhombe avec le ctXk, du prisme hexagone, est Tangle de 120°, que font les rhombes par leur inclinaison Tun a Tautre, determine tout le reste, y compris les angles du rhombe DES=DEO, etDS = DO, suffit a tout determiner ; et la comparaison des deux lignes D S, STEUOTUEB OF BBES' CELLS. 115 D est tout ce qu'il faut sans mesurer ni mSme calculer des angles, exceptez que Ton a celui de I'hexagone.* Cherchons maintenant par un procede semblable, les propor- tions des lignes et des angles, qui nous donnent un autre minimum, celui des angles dihedraux de I'alveole. Ceci est tr^s- important ; car ces angles sont la partie de la structure qui de- mande le travail le plus difBcile, et qui exige aussi la plus grande consommation de cire. Les parois sont plus epais aux angles parce que la solidite depend plus des angles que des autres parties des parois. Or, la longuer de Tangle dihedrale de toute I'alveole est = 3AH + 3DZ + 6AD + 3AG,ou3AH + 3 DZ+9 AD ; on a done a diff^rencier 9 a; + 3 (A H — ^/x^ — S*). Ainsi Sdx :;:::^^^^ = O, nous donne la valeur de x, cote du Vx^ - &' 3 S rhombe = ^r — ^, comme dans le probleme pour la surface. Ainsi e'est la mgme proportion des cotes et des angles qui donne le minimum de ce travail si fin et si dispendieux de cire, c'est-a-dire la fabrication des angles, qui donne aussi le minimum des surfaces. Les geometres ont emis deux opinions opposees sur la dif- ference entre le resultat de Koenig et la mesure de Maraldi. L'une est celle du justement c^lebre Maclaurin,! qui avait resolu le probleme par la geometrie elementaire sans recoiirir au calcul ; et trouvant les deux angles a quelques secondes * Le rapport merite attention du rhomhe avec le triangle rectaDgiilaire, bien connu dea geometres, dont les carre's des cote's sont dans la proportion de 1, 2, et 3. — Aussi le rhombe a un rapport remarqnable aveo la courbe Agnesieme (La Versiera, ou Lutin'), dont la Signora Agnesi a donne une construction tr6s-elegante dans son ouvrage (* Instit. AnaKt.,' vol. I, p. 381) : son equation est Y = —. — X -; — — .d'ouronvoitqu'elledoit^treliee avec le rhombe. Effectivement si le circle g'enerateur de la courbe est de'crit sur I'un des diametres du rhombe, aveo un rayon du quart de ce diametre, et la courbe a pour asymptote la tangente du circle, les ordonnes ont une proportion donnfe aux cosinus de Tangle obtiis, 109° 28', ou aux sinus de Tangle aigu, 70° 32'. t Trans. Phil, de Londres, 1742-3, p. 569. I 2 116 STETJOTTJEB OP BEES' CELLS. pres le meme qua donne Maraldi, il a impute I'erreur de Koenig aux tables de sinus dont il s'etait servi. Mais ayant remarque que Maraldi parle approximativemenfc de 70° et 110° dans un passage de son memoire, Maclaurin impute la diversite a un accident ou a la diiBculte de mesurer ces petites quantites. L'autre opinion, ou plutfit doute, est du P. Boscowich, qui, penchant a croire que la mesure des angles sur une si petite echelle fut trop difficile pour etre exacte, soup^onne Maraldi d'avoir commence par calculer ou ddcliner leur grandeur d'une supposition qu'il eut faite de Tangle d'inclinaison des rhombes, 120°, et d'avoir fini par donner sa supposition comme le re- sultat de son mesurage. Cette opinion a ete adoptee tres- facilement par M. Castillon de Berlin et M. L'Huiller de Greneve, dans leurs memoires (Mem. de Berlin, 1781) ; et ils eroient I'avoir confirmee par certaines mesures qu'ils donnent. Or, il n'est pas permis d'accuser Maraldi d'avoir donne comme le resultat de son mesurage, ce qui n'etait qu'une conjecture, d'autant moins que n'ayant aucunement considere la question d'un minimum, il ne pouvait pas avoir un pr^juge pour une theorie favorite. Puis, a ce qui concerne les mesures de M. Castillon, elles ne valent rien, n'etant que deux a ce qui regarde la question disput^e, et dont I'une plutot soutient le calcul de Maraldi, ne faisant pas une difl^erence plus de celle entre 4-144 et 4-168, qui n'est effeetivement rien. Mais ces deux geometres out soulev6 d'autres difiicult^s sur la structure des alveoles. lis ont revoque en doute le but principal de la construction, en niant que c'est pour economiser les materiaux et le travail, et pretendent que si c'etait la I'objet, une epargne bieu plus considerable aurait pu gtre gagnee en adoptant ce qu'ils appellent le minimum miuimorum. lis afferment que I'economie actuelle ne passe pas -jV de la cire, et qu'avec une autre proportion de la profondeur a la largeur de I'alveole, I'epargne aurait ete beaucoup plus grande. Mais il est certain qu'ils se trompent sur tons les deux points. 1. II n'est pas vrai de dire que I'epargne est d'un -^V, a moins que Ton impute dans la comparaison toute la cire des parois ; cette comparaison tourne uniquement sur la difference entre la STEUCTtTEB OF BBES' CELLS. 117 base rliomboidale et le prisme hexagone. La cire des parois ne peut pas entrer dans le calcul. S'il fut question entre deux especes de toiture d'lme maison de bois de sapin, de determiner laquelle ferait le plus d'^conomie de bois, on ne mettrait jamais en ligne de eompte les raurs, pour savoir si I'economie serait d'un cinquieme de toutela d^pense. Cela ferait le calcul rouler sur la hauteur du batiment. De fait I'economie de cire et de travail est d'un ^ an lieu du -5I1- Mais ce qui fait plus inexacte et meme absurde I'importation des parois dans le calcul, c'est la difference marquee qui existe de I'epaisseur de diiferentes parties de I'alveole. Le fond, la partie pyramidale est bieu plus epaisse que les parois. J'ai tres-souvent pese des morceaux d'eteudue ^gale des rhombes et triangles, et des parois adja- centes ; et j'ai trouv^ que ceux-la avaient un poids de trois a deux en comparaison de ceux-ci. H y a plus de variation entre les giteaux en ce qui regard la difference d'^paisseur qu'il n'y en a eu ce qui regarde I'epaisseur des rhombes, mais si on est sur que la difference existe c'est assez pour detruire le calcul de M. L'Huiller. Si la proportion est de trois k deux, I'epargne monte au i sur la partie la plus epaisse, et par consequent a ■jV au lieu de -j-V sur la totalite, en emportant m^me, contre toute exactitude, les parois dans le calcul. 2. La question du minimum minimorum, dont M. L'Huiller cite un cas, depend d'un probleme, dont il n'a pas donne une solution generale. II s'agit de trouver la proportion de la hauteur a la largeur de I'alveole qui fasse la plus petite surface possible avec un contenu donne. Soit S=c6te de I'hexagone; M S = D S = perpendiculaire sur le cote oppose du rhombe d'un de ees angles ; y = A H, cote vertical du prisme ; 2A A = contenu de I'alveole. Partant, nous avons y — 5— ,-5 — ^j ; 3mS^ SW4m^-3 rhombe = — ; les triangles APD, DOE= ^== : 2V3-OT' 2V3— n." la surface d'un tiers de I'alveole = S^( ^ — ) + 2 S y 118 STEUOTUEE OF BEEs' CELLS. = S^ ( - ] + - — 7- — r;- ; et en differenciant et egalant a zero on aura fc> = I — 1, et ^3 V3(3 m - V*™" - 3)^^ 1 par consequent S : y : : 2 VS — m'' : 3 m — ^ 4 m' — 3. C'est le resultat generale pour toutes les constructions ; et dans le cas du minimum, quand D 8 = D O, ou m = 1, on a la propor- tion de S : y : : a/2 : 1. Done la construction de I'alveole sur ce prineipe donnerait la largeur et la profondeur comme ^3 : 1, ou comme 2 \/2 : 1 pour les deux largeurs de I'alveole.' II y a une omission remarquable et fatale au resultat dans le calcul de M. L'Huiller, sur le cas particulier de m = l. Mais avant d'en parler, il faut faire observer que la construction qui ferait I'alveole presque trois fois plus large qu'elle n'est haute, ou profonde, serait entierement incompatible avec cbacun des objets auxquels I'alveole doit servir. Par exemple, quoiqu'il serait possible d'y mettre les oeufs, les vers ne pourraient pas etre Aleves, ni meme exister. Encore la provision pour les insectes, et le miel lui-mSme, ne pourrait 6tre amasse et garde qu'en tres-petite quantite. M. L'Huiller eonvient qu'il faut faire le sacrifice de I'epargne qu'il pretend resulterait de cette nouvelle construction, dout il ne nie pas que les inconveniants plus que contrebalancent I'avantage qu'il suppose de I'epargne. Mais rien ne pent 6tre plus contraire a tout prineipe que la conclusion qu'il deduit, que parce que, pour cette raison, I'eco- nomie de materiel est soumise aux objets principaux de toute la construction, cette ^conomie n'entre pas de tout dans le plan et dans I'operation. Cette balance entre dans toutes les questions de maonmum et mimimum appliques aux operations naturelles. Mais meme, en g^ometrie nous avons la meme chose. S'il est question de trouver la proportion des deux cotes d'un rectangle, qui contenant une dtendue donnee de surface, aurait ses cotes les plus courts, on sait que les cotes doivent etre egaux. Mais personne ne dirait que la largeur de la figure n'entrat pas du tout dans notre consideration, quoique STRTJCTUEB or bees' cells. 119 le but principal fut de determiner la largeur, et qu'a ce but on avait sacrifie la largeur. Mais jusqu'ici nous avons regarde le raisonnement de M. L'Huiller comme si sa solution du probleme du minimum minimorum eut ete exacte ; au contraire, il n'a pas meme pos6 la veritable question. II a fait omission d'une partie de la surface, meme tres-importante, la plaque hexagone qui boucbe ou ferme le tuyau ; omission difBcile k expliquer, excepte en croyant qu'il fut egar6 par la question qu'avait soumis k Koenig, M. Reaumur. Mais dans cette question, la plaque hexagone ne po'uvait pas entrer ; etant construite, son expression aurait disparu de I'equation differentielle, dont s'est servi Koenig pour la solution. C'est tout autre cbose dans la question du mi- nimum minimorum qui fait la eomparaison entre toutes les alveoles. La plaque hexagone est une partie aussi essentieUe que toutes les autres, au moins dans ces alveoles qui gardent les provisions et le miel ; probablement aussi dans celles qui entretiennent les vers, et qui sent I'habitation des cbrysaHdes. Les vers surtout sont toujours converts. Quand meme il fut constate que la necessite de la couverture ou bouchon n'existe pas dans les alveoles qui servent a I'entretien des cbrysalides, comme eUe est de toute necessite dans les autres ; il faudrait avoir deux especes d'alveoles ; et ainsi la solution du probleme ne serait bonne que pour cette espece qui n'eut point de bouchon. Mais tout porte a croire qu'il n'y a qu'une espece ; car toute alveole est employee indifferemment a toutes les operations, et a tons les besoins de I'insecte. Voyons done quelle devait etre la solution du probleme. EUe est la meme que celle qu'on a donn6 plus haut jusqu'a un certain point ; et puis a I'expression differentielle il faut ajouter q /q as la valeur de la plaque Hexagone = . Le resultat est de nous donner la proportion de 8:y :: 2^/3— m" :3m — \/4 m' — 3 + V'5 VS — m^ pour toutes les proportions des lignes et angles ; et dans le cas de I'abeille actuelle ou TO = 1, le minimum minimorum est, lorsque le c6t6 de I'bexagone 120 STEUCTUKE OP BEEs' CELLS. est h, la largeur du prisme dans la proportion de 2 A, V 2 + V 3 = on de 2-82 k 3-14 a pen pres. Cette forme n'est pas aussi incompatible qae celle qui resulte de I'autre solu- tion, vu que la largeur de I'alveole, quoique plus grande que sa profondeur, ne I'excede pas dans la meme proportion. Pourtant la forme ne pourrait jamais convenir aux usages et aux necessites de I'insecte ; et meme il n'y aurait pas une economie de materiaux et de travail. Au contraire, on trouve en faisant la comparaison que le montant de surface dans une alveole ainsi construite est a celui d'une alveole de la construction actuellement pratiquee, dans la proportion de Sky, de 1-387 a 5, et de 56, 52 a 49, 64. Ainsi il y a perte et non pas gain par la proportion du mimimum minimorum. Cette perte est de ^ ou -I- ^ peu pres sur une seule alveole ; mais sur le gateau entier elle est assez grande. Mais ce n'est pas 1^ la seule omission qu'ont font dans leurs conseils k I'abeiUe les Academiciens de Berlin. S'ils avaient fait atten- tion k la difference en fait de travail aussi bien que de materiaux, de la fabrication des angles, ils auraient trouv«5 qu'il y a non-seulement comme on vient de faire voir un minimum en comparant les alveoles de la m6me profondeur, mais qu'U y a aussi un minimum k ce qui regarde la sur- face. La meme espece d'investigation qui nous a conduit a I'un fait voir aussi I'autre. Si la comparaison est in- stitute entre les alveoles du m^me contour on trouve la proportion du e6t^ k la largeur du prisme qui donne la plus grande epargne d'angles, dans le cas de m = 1 (largeur du rhombe = c6te de I'hexagone) est celle de 1 : ^2 + 1. II y a le meme resultat si au lieu de la limite par la supposition du contenu donne, on prenne la surface du cote du prisme bexagone comme donne — limite qui n'est pas possible par la solution de I'autre probleme de minimum minimorum pour la surface.* La longueur des angles dibedraux est 28, 92. Dans I'alveole construite selon la proportion de2av'2-)-v'3 * II va sans dire qu'une limite est abaolument neoessaire ; sans cela le plus court prisme serait celui qui ferait la plus grande Epargne de surface et d'angles dihedraux. STEUCTUEE OF BEHS' CELLS. 121 (minimum minimorum de surface) la longueur est 47, 76 ; at dans I'alveole actuelle (1387 : 5) 48 : 05. Ainsi il y a ^pargne d'augle dihedral dans ces deux cas en comparaison avee I'alveole actueUe, surtout dans celle de 1 : V 2 + 1. Mais les objections qu'on a pleinement indiquees font impossible de faire des alveoles de cette forme. La construction ne serait pas si liors de proportion de la largeur a la hauteur que celle qu'ont propose les academiciens de Berlin, de largeur k pea pres trois fois plus grande que la hauteur ; et elle n'aurait pas cause une augmentation de surface. Mais pourtant elle aurait' ete en contradiction avec le but principal d'elever les insectes et de garder les provisions et le miel. On ne pent pas douter de Timportance de tout ce qui de- montre que les abeilles ont r^solu le probleme, et que leur architecture est plus exacte sous tous les rapports qu'aucune autre que Ton pourrait imaginer, si Ton reflechit que c'est le ehef-d'oeuvre de toutes les operations instinctives. II est impossible de dire comme Virgile quandil a chante les mceurs de I'abeille, " In tenui labor" sans ajouter " at tenuis non gloria.'" Car il n'est pas permis de penser avee Descartes* que les animaux sont des maehihesi An contraire, I'hypothese, ou plutot la doctrine Newtoniennef parait plus fondee — que ce qu'on appelle instinct est Taction constante de Dieu ; et que ces speculations tendent k sa gloire, au moins a I'explication et a rillustration raisonnee de ses oeuvres et ses desseins. J * 'Tract, de Me'thode,' 36. Maia voir ses Lettres; Epist., pars I, ch. 27. t ' Optics,' lib. iii. ; Qu. 31. ' Prinoipia,' lib. iii. ' Sch. Gen.' X M. L'Huiller parait 6tre peu instruit sur I'histoire de ce fameux probleme. II dit (p. 280) que la solution du Pere Boscowich est d'aooord avec celle de Maclaurin. — ' Phil. Trans.,' 1748. Mais c'est certain qu'il n'a jamais vu le me'moire de Maclaiirin ; car il affirme que tous ceux qu'il nomme, y compris le Pere B.; aussi Men que Koenig, avaient ete d'aceord k regarder la question comme incapable de solution excepter par le calcuL Meme s'arroge-t-il le merite d'avoir le premier donne une solution par la geometric ordinaire, quoiqu'il n'y A pas de doute qui Maolauiin I'avait donne pres de quarante ans avant lui, et donne pour preuve de la force de la geometrie aucienne de laquelle il etait un admirateur zele. ( 122 ) VII. - EXPERIMENTS AND INVESTIGATIONS ON LIGHT AND COLOUES. The optical inquiries of which I am about to give an account, were conducted at this place in the months of November and December 1848, and continued in autumn 1849 at Brougham, where the sun proved of course much less favourable than in Provence : they were further prosecuted in October. I had thus an opportunity of carefully reconsidering the conclu- sions at which I had originally arrived ; of subjecting them first to analytical investigation, and afterwards to repetition and variation of the experiments ; and of conferring with my brethren of the Eoyal Society and of the National Institute. The climate of Provence is singularly adapted to such studies. I find, by my journal of 1848, that during forty-six days which I spent in those experiments, from 8 a.m. to 3 p.m., I scarcely ever was interrupted by a cloud, although it was November and December.* I have since had the great benefit of a most excellent set of instruments made by M. SoLEiL of Paris, whose great ingenuity and profound knowledge of optical subjects can only be exceeded by his admirable workmanship. I ought however to observe, that although his heliostate is of great convenience in some expe- riments, it yet is subject (as all heliostates must be) to the imperfection of losing light by reflexion, and consequently I * Of seventy-eight days of winter in 1849, 1 had here only five of cloudy weather. Of sixty-one days of summer at Brougham, I had but three or four of clear weather ; one of these fortunately happened whilst Sir D. Brewster was with me, and he saw the more important experiments. 03Sr LIGHT AKD COLOITES. 123 have generally been obliged to encounter the inconvenience of the motion of the sun's image, especially when I had to work with small pencils of light. This inconvenience is materially lessened by using horizontal prisms and plates. Although I have made mention of the apparatus of great delicacy which I employed, it must be observed that this is only required for experiments of a kind to depend upon nice measurements. All the principles which I have to state as the result of my experiments in this paper, can be made with the most simple apparatus, and vyithout any difficulty or expense, as will presently appear. It is perhaps imnecessary to make an apology for the form of definitions and propositions into which my statement is thrown. This is adopted for the purpose of making the narrative shorter and more distinct, and of subjecting my doctrines to a fuller scrutiny. I must further premise that I purposely avoid all arguments and suggestions upon the two rival theories — the Newtonian or Atomic, and the TJndulatory. The conclusions at which I have arrived are wholly inde- pendent, as it appears to me, of that controversy. I cau- tiously avoid giving any opinion upon it ; and instead of belonging to the sect of undulationists or anti-undulationists, I incline to agree with my learned and eminent colleague M. BiOT, who considers himself as a " Rieniste," and neither " ondulationiste " nor " anti-ondulationiste.'' Chateau Eleanor-Louise (^Provence),* 1st November, 1849. DEFnsriTioNS. 1. Flexion is the bending of the rays of light out of their course in passing near bodies. This has been sometimes termed diffraction, but fleocion is the better word. * In experiments at this place, in winter, I found one great advantage, namely, the more horizontal direction of. the rays. In summer they are so nearly vertical, that a mirror must be used to obtain a long beam or pencil, which is often required in these experiments, and so the loss of light countervails the greater strength of the summer sun's light. 124 EXPBBIMENTS ASD INVESTIOATIONS 2. Flexion is of two kinds — inflexion, or the bending towards the body ; deflexion, or the bending from the body. 3. Flexibility, deflexibility, inflexibility, express the disposition of the homogeneous or colour-making rays to be bent, de- flected, inflected by bodies near which they pass. Although there is always presumed to be a flexion and a separation of the most flexible rays from the least flexible (the red from the violet for example) when they pass by bodies, yet the compound rays are not so presumed to be decomposed when reflected by bodies. This is probably owing, to the successive inflexions and deflexions before and after reflexion, correcting each other and making the whole beam continue parallel and undecomposed instead of be- coming divergent and being decomposed. Peoposition I. The flexion of any pencil or beam, whether of white or of homogeneous light, is in some constant proportion to the breadth of the coloured fringes formed by the rays after passing by the bending body. Those fringes are not three^ btit a very great number, continually decreasing as they recede from the bending body, in deflexion, where only one body is acting ; and they are real images of the luminous body by whose light they are formed. Exp. 1. If an edge be placed in a beam or in a pencil of white light, fringes are formed outside the shadow of the edge and parallel to it, by deflexion. They are seen distinctly to be coloured, the red being furthest from the shadow, the violet nearest, the green in the middle between the red and the violet. The best way to observe this is to receive the light on an instru- ment composed of two vertical and two horizontal plates, each moving by a screw so as to increase or lessen the distance between the opposite edges. a, a' are (fig. 1) the vertical, 6, 6' the horizontal edges, s, s are the screws ; and these may be fitted ON LIGHT AND COLOUBB. 125 Jig-a. with micrometers, so as to measure very minute distances of the edges by graduated scales B B', B' C. For the purpose of the present proposition the aperture only needs be con- sidered, of about a quarter of an inch square. The Hght passing through this aperture is received on a chart placed first one foot, and then several feet from the instrument. The fringes are increased in breadth by inclining the chart till it is horizontal, or nearly so, when the fringes parallel to b, b' are to be examined, and holding it inclined laterally when the fringes parallel to a, a' are to be examined. It is also convenient to let the white light beyond the fringes pass through ; and for this purpose, a", b" being the figure of the instrument (fig. 2), and the light received on the chart, a hole may be made in its centre o p q, through which the greater portion of the white light may be suffered to pass. The fringes are plainly seen to run parallel to the edges forming them; as o^ parallel to 6" and p q parallel to a". The reddish is farthest from the shadow, the bluish nearest that shadow ; also the fringe nearest the shadow is the broadest, the rest decrease as they recede from the shadow into the white light of the disc. Sometimes it is convenient to receive the fringes on a ground glass plate, and to place the eye behind it. They are thus rendered more perceptible. When the edges are placed in homogeneous light, they are all of the colour which passes by any edge ; and two diversities are here to be noted carefully. First, the fringes made by the red light are broader than those made by any of the other rays, and the violet are the narrowest, the intermediate fringes being of intermediate breadths. Second, the fringes made by the red are farthest from the direct rays, the violet nearest those rays, Ihe inter- mediate at intermediate distances. This is plainly shown in the following experiment. Exp. 2. In fig. 3, C represents the image of the aperture when the rays of the prismatic spectrum are ]?;g.3. 126 EXPBEmENTS AND INVESTIGATIONS made to pass ttrough it. But instead of making the fringes by a single edge deflecting, and so casting them in the spec- trum, I approach the opposite edges, so that both acting together on the light, the fringes are seen in the shadow and surrounding the spectrum. These fringes are no longer parallel to the shadows of the edges as they were in the white light, but incline towards the most refrangible and least flexible rays, and away from the least refrangible and most flexible. Thus the red part r of the fringes is nearest the shadow of the edge a'\ the orange, o, next; then yellow, y; green, g ; blue, & ; indigo, i ; and violet, v. Moreover, the fringe r u is both inclined in this manner, so that its axis is inclined, and also its breadth increases gradually from v to r. This is a complete refutation of the notion entertained by some that Sir I. Newton's experiment of measuring the breadths in different coloured lights and finding the red broadest, the violet narrowest, explains the colours of the fringes made in white light as if these were only owing to the different breadths of the fringes formed by the different rays. The present experiment clearly proves, that not only the fringes are broadest in the least refrangible rays, but those rays are bent most out of their course, because both the axis of the fringes is inclined, and also their breadths are various. Exp. 3. Though called by Geimaldi, the discoverer, the three fringes, as well as by Newton and others who followed him, they are seen to be almost innumerable, if viewed through a prism to refract away the scattered light that obscures them. I stated this fact many years ago.* Exp. 4. That the fringes are images may be at once per- ceived, not when formed in the light disc as in some of the foregoing experiments, but when formed in the shadow. Thus when the opposite edges are moved so near one another as to form fringes bordering the luminous body's image, they are formed like the disc they surround. When you view a * Philosopliioal Transactions, 1797, part n. ON LIGHT AND COLOTJBS. 127 candle througli the interval of the opposite edges, you per- ceive that the fringes are images of its flame, with the wick, and that they move as the flame moves to and fro. "When you observe the half-moon in like manner, you perceive that the side of the fringes answering to the rectilinear side of the moon, are rectilinear, and the other side circular ; and when the full moon is thus viewed, the fringes on both sides are circular. The circular disc of the moon is, indeed, drawn or elongated as well as coloured. It is, that is to say, the fringe or image which is exactly a spectrum by flexion. Like the prismatic spectrum, it is oblong, not circular, and it is coloured ; only that its colours are much less vivid than those of the prismatic spectrum. Proposition II. The rays of light, when inflected by bodies near which they pass, are thrown into a condition or state which disposes them to be on one of their sides more easily deflected than they were before the first flexion ; and disposes them on the other side to be less easily deflected : and when deflected by bodies, they are thrown into a condition or state which disposes them on one side to be more easily inflected, and on the other side to be less easily inflected than they were before the first flexion. Let E A (fig. 4) be a ray of light whose opposite sides are E A, E' A', and let A be a bend- ing edge near which the ray passes, the side E' A' acquires by A's inflexion, a disposition to be more easily deflected by another body placed between A and the chart C, and the side E A acquires a disposition to be less easily deflected than before its first flexion ; and in like manner E' A' acquires a disposition to be more easily inflected, and E A a dispo- sition to be less easily inflected by a body placed between A and C. Exp. 1. Place A' (fig. 5) in any position between A and v r. 128 EXPERIMENTS AND INVESTIGATIONS the image made on C by A's influence, as at A' or A", or close to A at A'". If it is placed on the same side of the ray with A, no difference whatever can be perceived to be made on the breadth of rv, or on its dis- tance -uE' from the direct ray EE'. In like manner the image by deflexion r' v' is not affected at all, either in its breadth, or in its removal from EE' by any object, a, a', placed on the same side with A of the deflected ray A v'. Y g But (flg. 6) place B anywhere between A and i! r on the side of the ray opposite to A, and the breadth oi rv is increased, and also its distance from the direct ray E E', as v' r' ; and in like manner (flg. 7) the deflected rays Av, Ar are both more separated, making a broader !"%•'• image at r" v", and are further -^ removed from E E' by B's in- flexion. Exp. 2. If you bend the rays either by a single edge, or by the joint action of two edges, it makes not the least difference either in the breadth or in the distance from the direct rays of the images, or in the distension or elongation of the lumi- nous body's disc, whether the bending body is a perfectly sharp edge (which in regard to the rays of light is a surface, though a narrow one), or is a plane (that is, a broader sur- face), or is a curve surface of a very small, or of a very large radius of curvature. In fig. 8, ae is an instrument composed of four pieces of different forms, but all in a perfectly straight line ; a 6 is an extremely sharp edge ; 6 c a flat surface ; c d a, cylindrical or circular surface of a great radius of curvature; de one of a small radius of curvature. But all these pieces are so placed that B S y is a tangent to ed, dc, and is a continuation of •y ;6 K, that is, of 6, 6 a. So the light passing by the whole abode, passes by one straight line E K, uniting or joining ON LIGHT AND COLOTJES. 129 the four surfaces. It is found that the image or fringe 1 1', made hj abode (or E S y B K), is of the same breadth and in the Kg.S. a 1 same position throughout its whole length. So if directly op- posite to this edge another straight edge is placed, and acts to- getherwith abode on the light passing, the breadth of the fringe I is increased, and its distance is increased from the direct rays, but it has the exact same breadth from I to I' ; its portion I' q answering to a &, g P answering to 6 c, P answering to c d, and I answering to d e, are of the same breadth, provided care be taken that the second edge is exactly parallel to the edge E K. And this experiment may be made with the second edge behind abode, as in Exp. . 1 of this proposition ; also it may be usefully varied by having the second edge composed of four surfaces like the first, only it becomes the more necessary to see that this compound edge is accurately made and kept quite parallel to the first, any deviation, how- ever minute, greatly affecting the result. When care is thus used the fringes are as in r v, v' r', quite the same in breadth and in position through their whole length ; and not the least difference is to be discerned in them, whether made by a second edge, which is one sharp edge, or by a compound second edge, similar to ah ode. Hence I conclude that the beam passing by the compound edge, or compound edges, is exactly as much distended by the different flexibility of the rays, and is exactly as much bent from its direct course when the flexion is performed by a sharp edge, by a plane surface, by a very flat cylinder, or by a very convex cylinder ; and therefore that all the action of the body on the rays is exercised by one line, or one particle, K 130 EXPERIMENTS ANB INTESTIGATIONS and not first by one and then by otliers in succession ; and this clearly proves that after a first flexion takes place, no other flexion is made by the body on the same side of the rays. This is easily shown. For a plane surface is a series or succession of edges in- finitely near each other ; and a curve surface in like manner is a succession of infinitely small and near plane surfaces or edges. Let a h (fig. 9) be the section of such a curve surface. The particle P coming first near enough the ^' ' ray E E' to bend it, then the next particle is only further distant from E E', the unbent ray, than the particle P by the versed sine of the infinitely small arch P. But is not at all further distant than P from the ray bent by P into qr, and yet we see that ^ produces no effect whatever on the ray after P has once bent it. No more do any of the other particles within whose spheres of flexion the ray bent by P passes. The deflected ray q r' no doubt is somewhat more distant from than the incident ray was from P, \mt not so far as to be beyond O's sphere of deflexion ; for acts so as to make the other fringes at greater distances than the first. Consequently O could act on the first fringe made by P as much as P can in making the second, third, and other fringes ; and if this be true of a curve surface, it is still more so of a plane surface ; all whose particles are clearly equidistant from the ray's ori- ginal path, and the particles after the first are in consequence of that first particle's fiexion nearer the bent ray, at least in the case of inflexion. But it is to be observed, moreover, that in the experiment with two opposite edges, inflexion enters as well as deflection, and consequently this demonstration, founded on the exact equality of the fringes made by compound double edges, appears to be conclusive. For it must be ob- served that this experiment of the different edges and surfaces, plane and curve, having precisely the same action, is identical with the former experiment of two edges being placed one be- hind the other, and the second producing no effect if placed on ON LIGHT AND OOLOUES. 131 the same side of the ray with the first edge. These two edges are exactly like two successive particles of the same surface near to which the rays pass. Consequently the two experiments are not similar but identical ; and thus the known fact of the edge and the back of a razor making the same fringes, proves the polarization of the rays on one side. Thus the proposition is proved as to polarization. Exp. 3. The proposition is further demonstrated, as regards disposition, in the clearest manner by observing the effect of two bodies, as edges, whether placed directly opposite to each other while the rays pass between them so near as to be bent, or placed one behiad the other but on opposite sides of the rays. Suppose the edges directly opposite one to the other, and suppose there is no disposition of the rays to be more easily bent by the one edge in consequence of the other edge's action. Then the breadth and distension and removal of the fringes caused by the two edges acting jointly, would be in proportion to the sum of the two separate actions. Suppose that one edge deflects and the other inflects, and suppose that inflection and deflexion are equal at equal distances, following the same law; then the force exerted by each edge being equal to d, that exerted by both must be equal to 2 d. But instead of this we find it equal to bd, or 6 d, which must be owing to the action of the two introducing a new power, or inducing a new disposition on the rays beyond what the action of one did. If, however, we would take the forces more correctly (fig. 10), let A and B be the two edges, and let their spheres of flexion be equal, A C ( = a) being A's sphere of inflexion and B's sphere of deflexion ; B C ( = a) being A's sphere of deflexion and B's sphere of inflexion ; and let the flexion in each case be inversely as the mth power of the distance. Let K 2 132 EXPEEIMENIS AND INVESTIGATIONS C P = a;, P M = y, the force acting on a ray at the distance a = X from A and a — x from B. Then if B is removed and only A acts, y = j^—. If B also acts, y' ^ j^-^^ + {a — a-)"' Now the loci of y and y' are different curves, one similar to a conic hyperbola, the other similar to a cubic ; but of some such form when m = 1, as S S' and T T'. It is evident that the proportion oi y \ y' can never be the same at any two points, and consequently that the breadths of the fringes made by the action of one can never bear the same proportion to the breadths of those made by the action of both, unless we introduce some other power as an element in the equation, some power whereby from both values, y and y\ x may dis- appear, else any given proportion of y : y' can only exist at some one value of x. Thus suppose (which the fact is) y : j^' : : 1 : 5 or 1 : 6, say : : 1 : 6, this proportion could only hold when (5™ - 1) a (4™ -\) a X = • 7 or = r , liy -.y' ::1 : 5. 5" + 1 4" + 1 When m = 2, the force being inversely as the square of the distance, then x = -^ and x = — = a, are the values at 3 a/6+1 which alone y : y' :: 1 : 5 and 1 : 6 respectively. But this is wholly inconsistent with all the experiments ; for all of these give nearly the same proportion oi y : y' without regard to the distance, consequently the new element must be introduced to reconcile this fact. Thus we can easily suppose the conditions, disposition and polarization (I use the latter term merely because the effect of the first edge resembles polarization, and I use it without giving any opinion as to its identity), to satisfy the equation by intro- ducing into the value of y some function of o — a;. But that ON LIGHT AND OOLOUES. 133 tte joint action of the two edges never can account for the difference produced on the fringes, is manifest from hence, that whatever value we give to m, we find the proportion of y' : y when a; = only that of double, whereas 5 or 6 times is the fact. The same reasoning holds in the case of the spheres of flexion being of different extent; and there are other arguments arising from the analysis on this head, which it would be superfluous to go through, because what is delivered above enables any one to pursue the siibjeot, The demonstration also holds if we suppose the deflective force to act as — of the .distance, while that of inflexion acts as — . n m But I have taken m = w as simpler, and also as more probably the fact. I have said that the rays become less easily inflected and deflected ; but it is plain that on the polarized side they are not inflected or deflected at all. Their disposition on the opposite side is a matter of degree ; their polarization is absolute and their flexion null. Peoposition III. The rays disposed on one side by the first flexion are polarized on that side by the second flexion, and the rays polarized on the other side by the first flexion are depolarized and disposed on that side by the second flexion. This proposition is proved by carefully applying the first experiment of Prop. II. ; but great care is required in this experiment, because when three edges are used consecutively, , the third edge often appears to act on rays previously acted on by both the other two, when it is only acting on those previously acted on by one or other of those two. Thus when edge A has inflected and edge B afterwards deflects the rays disposed by A, a third edge may, when applied on the side opposite to B, seem to increase the flexion, and yet on re- moving A altogether we may find the same effect continue, which proves that the only action exercised had been by B and C, and that had not acted on rays previously bent by 134 EXPBEIMENTS Am) rNTBSTIGATIONS botb. A and B, which the experiment of course requires to prove the proposition. I was for a long while kept in great uncertainty by this circumstance, whether the third edge ever acted at all. That it never acted on the side of the ray on which the second edge acted, I plainly saw; but I fre- quently changed my opinion whether or not it acted on the opposite side, that is, on the same side with the first edge. Nor could I confidently determine this important point until I had the benefit of an instrument which I contrived for the purpose, and which, executed by M. Soleil, enabled me satisfactorily to perform the experimentum crucis as follows : — In fig. X. A B is a beam, on a groove (of which the sides are graduated) three uprights are placed, the one, B, fixed, the M 1 M.r d '^~ E B I*S a B \ \ c i Pig. X. other two, C and D, moving in the groove of A B. On each of the uprights is a broad sharp-edged plate, moving up and down the upright by a rack and pinion, so that both the plates F Gr could be approached as near as possible to each other, and so could F be approached to the plate E on the fixed upright B ; while also each of the three plates could be brought as near the rays that passed as was required ; and so could each be brought as near the opposite edge of the neigh- bouring plate. It is quite necessary that this instrument should be heavy in order to give it solidity : it is equally necessary that the rack and pinion movement should be just and also easy; for the object is to fix the plates at will, so ON LIGHT AND OOLOUES. 135 that their position in respect of the rays may he easily changed, and when once adjusted may be immovable until the observer desires to change their position. The light was passed under the plate E and acted upon by a b, its lower edge. The second plate F was then raised on C so as to act on the side of the rays opposite to a b, by its upper edge cd. The fringes inflected by a 6 were thus deflected by c d, in virtue of the disposition given to the side next c d. Then the third plate G, on its stand U, was moved so that it could be brought to act by its lower edge ef, which was approached to the rays deflected by cd, and placed on their opposite side. The action was observed by examining the fringes on the chart M. Those which had been as o, made by the joint action of the two first edges E P, were seen to move upwards to p as the third edge Gr came near the rays ; and p was both broader than o, and further removed from the direct rays KE'. In order to make quite sure that this change in the size and position of o had not been occasioned by the mere action of two plates, as E and Gr or F and G, it was quite necessary to remove first E, by drawing it up the stand B. If the fringe p then vanished, complete proof was afforded that E had acted as well as G. Then F was removed, and if p vanished, proof was aiforded that F acted as well as E and Gr. A very convenient variation of the experiment was also tried and was found satisfactory. When the joint action of P and Gr gave a fringe, as at q, E being, removed up the stand B, then E was gently moved down that stand, and as it approached the pencil, which was on its way to F and Gr, you plainly perceived the fringe enlarged and removed from q to p. These experiments were there- fore quite crucial, and demonstrated that all the edges had concurred to form the fringe at p, the first and third in- flecting, the second deflecting. The same experiments were made on the fringes formed by the deflexion of the first edge and the inflexion of the second, and the deflexion of the third. It is thus perfectly clear that the rays bent by the first 136 EXPERIMENTS AND INYESTIGATIONS edge and disposed on their side opposite to that edge, are bent in the other direction by the second edge acting on that opposite side, and are afterwards again bent in the direction of the first bending by the action of the third edge upon the side which was opposite the second edge and nearest the first edge. But this side is the one polarized bj' the first edge, and therefore that side is depolarized by the action of the second edge. Hence it is proved that the rays polarized by one flexion are depolarized by a second ; and as it is proyed by repeated experiments that no body placed on the same side of the rays with any of the bending bodies, whether the first or the second or the third, exercises any action on those rays, it is thus manifest that any one flexion having disposed, a second polarizes the disposed side ; and that any one flexion having polarized, a second flexion depolarizes and disposes the polarized side. Exp. 3. Another test may be applied to this subject. In- stead of a rectilinear edge, I made use of edges formed into a curve, as in fig. 12, where C is such an edge, and then the Mg.iz. figure made is gh, corresponding to the curve f. i ^ — ^ 1 e b. The first edge in the last experiment being P_-] formed like C, instead of a straight-lined edge, ' * we can at once perceive that it has acted on the rays as well as the second and third edges, because these being straight-lined, never could give the comb-like shape gh to the fringes. This completely confirmed the other observations, and made the inference irresistible. Peoposition IV. The disposition communicated by the flexion to the rays is alternative ; and after inflexion they cannot be again inflected on either side ; nor after deflexion can they be deflected. But they may be deflected after inflexion and inflected after de- flexion, by another body acting upon the sides disposed, and not by its acting upon the sides polarized. This is gathered from the experiments in proof of the second and third propositions. ON LIGHT AND OOLOTJBS. 137 Proposition V. The disposition impressed upon tlie rays, whether to be easily deflected or easily inflected by a second bending body, is strongest nearest the first bending body, and decreases as the distance between the two bodies increases. Fig. 11. Let A B = a be the distance between the first bending body and a given point, more or less arbi- E,a-.11 trarily assumed ; P the second body ; AP = a;; VM = y, the force exerted by the second body at P ; C = the chart ; P M = ?/ is in some inverse proportion to A P, but not as or ^^, because it is not infinite at A, but of an assign- able value there ; therefore y = -. r- ; and the curve which is the locus of P has an asymptote at B, when X = — a. The fringes being received on the chart at C, it might be supposed that the difference in their breadth, by which I measure the force, or y, is owing to P ap- proaching the chart C, in proportion as it recedes from A, and thus making the divergence less in the same proportion ; but the experiments are wholly at variance with this sup- position. Exp. 1. The following table is the result of one such experiment. The first column contains the distances hori- zontally of P from A, being the sines of the angle made by the rays with the vertical edges ; the second column contains the real distance of the second from the first edge, the secant of that angle ; the third column gives the breadths of the fringes at the distances given in the preceding columns ; the 138 EXPEEIMENTS AND IirVESTIGATIONS fourth gives the value of y, supposing MN were a conic hyperbola. 20 65 85 195 35 85 107i 240 Real value of y. 3* li li oi Hyperbolic value. 3J ^ The unit here is -^th of an inch. It is plain that this agrees nearly with the conic hyper- bola, but in no respect with a straight line ; and upon calculating what effect the approach of P to C would have had, nothing could be more at variance with these numbers. But Exp. 2. All doubt on this head is removed by making P the fixed point, and moving the first edge A nearer or further from it. In this experiment, the disturbing cause, arising from the varying distance from the chart, is entirely re- moved ; and it is uniformly found that the decrease in the force varies notwithstanding with the increase of the dis- tance. I have here only given the measures by way of illustration, and not in order to prove what the locus of y (or P) is, or, in other words, what the value of m is. Exp. 3. When one plate with a rectilinear edge is placed in the rays, and a second such plate is placed at any distance between it and the chart, the fringes are of equal breadth throughout their length, and all equally removed from the direct rays, each point of the second edge being at the same distance from the corresponding point of the first. But let the second plate be placed at an angle with the first, and the fringes are very different. It is better to let the second be parallel to the chart, and to incline the first ; for thus the different points of the fringes are at the same distance from the edge which bends the disposed rays. In fig. 13, B is the second plate, parallel to the chart C ; A is the first plate ; ON LIGHT AND COLOUES. 139 all the points of B, from D to E, are equidistant from C ; therefore nothing can be ascribed to the divergence of the bent rays. B bends the rays disposed by A at „ i^ different distances D D' and E E' from ' the point of disposition. The fringe | %--0. is now of various breadths from dd' to e, the broadest part being that answering to the smallest distance of D, the point of flexion, from D' the point of disposition ; the narrowest part, e, answering to EE', or the greatest distance of the point of flexion from the point of disposition. Moreover, the whole fringe is now inclined ; it is in the form of a curve from dd' to e, and the broad part dd', formed by the flexion nearest the disposition, is furthest removed from the direct rays ; the narrowest part, e, is nearest these direct rays. It is thus quite clear that the flexion by B is in some in- verse proportion to the distance at which the rays are bent by B from the point where they were disposed by A. I repeatedly examined the curve de, and found it certainly to be the conic hyperbola. Therefore m = 1, and the equation to the force of disposition is y = — . In order to ascertain the value of m, I was not satisfied with ordinary admeasurements, but had an instrument made of great accuracy and even delicacy. It consisted of two plates, A and B (Plate VI.), with sharp rectilinear edges, one. A, horizontal, the other, B, moving vertically on a pivot, and both nicely graduated. The angle at which the second plate was vertically inclined to the first, was likewise ascertained by a vertical graduated quadrant E. Moreover the edges moved also horizontally, and their angle with each other was measured by a horizontal graduated quadrant K. There was a fine micrometer E to ascertain the distances of the two edges from each other, and another to measure the breadth of the fringes on the chart. The observations made with this instrument gave me undoubted assurance that the 140 EXPBEIMENTS AND INVESTIGATIONS equation to the curve M N in fig. 11 is ?/ a; = a, a conic hyper- bola, and that the disposing force is inversely as the distance at which the flexion of the rays bent and disposed takes place. Scholium. — It is clear that the extraordinary property V7e have nov7 been examining has no connexion with the different breadths of the pencils at different distances from the point of the first flexion, owing to the divergence caused ,by that flexion. By the same kind of analysis, which we shall use in demonstrating the 6th Proposition, it may be shown, — first, that the divergence of the rays alone would give a different result, the fringes made by an inflexion following a deflexion and those made by a deflexion following an inflexion; secondly, that in no case would the equation to the disposing .force be the conic hyperbola, even where that fringe de- creased with the increase of the distance ; thirdly, even where the effect of increasing the distance is such as the dispersion would lead to expect, the rate of decrease of the fringes is very much greater in fact than that calculation would lead to, five or six times as great in many cases ; and lastly, that instead of the law of decrease being uniform, it would, if caused by the dispersion, vary at different distances from the two edges.* Nothing therefore can be more manifest than that the phenomena in question depend upon a peculiar pro- perty of the rays, which makes them change in their dis- position with the length of the space through which they have travelled. It should seem that light riiay be compared, when bent and thereby disposed, to a body in its nascent state, which, as we find by constant experience, has properties different from those which it has afterwards ; and I have therefore con- trived some experiments for the purpose of ascertaining whether or not light at the moment of its production (by * I have given demonstrations of these propositions in a memoir pre- sented to the National Institute, hut I am reluctant to load the present paper with them. ON LIGHT AND COLOUES. 141 artificial means) has properties other than those which it possesses after it has been some time produced. This 'will form the subject of a future inquiry. I would suggest, how- ever, at present that the electric fluid ought to be examined with a view to find whether or not it has any property ana- logous to disposition, that is, whether it becomes more difficultly attracted at some distance from its evolution, as light is more difBcultly bent at a distance from the point of its being disposed. On heat a like experiment may be made. The thermometer would no doubt stand at a different height at different distances from the source of the heat ; but the ques- tion is if it will not reach its full height, whatever that may be, more quickly near its source than far from it. This experiment ought above all to be made on radiant heat, in which I confidently expect a property will be found similar to the disposition of light. It is also plain that we may expect strong analogies in magnetism and electro-magnetism. — I throw out these things because my time for such inves- tigations may not be sufficiently extended to let me under- take them with success. Peoposition VI. The figures made by the inflexion of the second body acting upon the rays deflected by the first, must, according to the calculus applied to the case, be broader than those made by the second body deflecting those rays inflected by the first. In fig. 14, let Av' be the violet rays and Ar' the red, inflected by A and deflected by B. Let A )• be the red and Av the violet deflected by A and inflected by B. The action of B must inflect Ar, Av into a broader fringe F, than the action of B deflects A v', Ar' into the fringe/. Let B r- = a be the distance at which B acts on A r ; rv = d be the divergence of the red and violet ; c be the distance of 142 EXPEEIMBNTS AND INVESTIGATIONS the two bent pencils, and v' r' the divergence of the inflected pencil, equal also to d, because we may take the different inflexibility to be as the different deflexibility. B acts on the T V red of A r i; as — ; on the violet as -, ;r- ; and so on A v' as a" (o + dy T . . : on A r^ as . It is evident tliat {a + d+cY' (a+2d+c)'" the action in bending Ar, Kv, or the fringe made by that action, is to the fringe made by the action on Ar', Au', as — _ : — • : and ulti- ar (a-j-af)*" (a + 2^ + 0)™ (a + d + c)"' mately the two actions (or sets of fringes) are (supposing a = 1 and d also = 1, for simplifying the expression) as 2"' X ?■ (3 + cy (2 + c)" - i; (3 + c)" (2 + c)™ to 2'"r(2 + c)" -2'"v{a + cy". Now the former of these expressions must always be greater than the latter, because (3 + c)" > 1, and also (3 + c)" — 1 > (2 4- c)" — 1 ; and this whatever be the value of m and of c, and whatever proportion we allow of r to v, the flexibilities. But it is also manifest that the excess of the first expression above the second will be greater if the flexi- bility of the red exceed that of the violet, or if r is greater than V, as 2 v. Hence we conclude ; first, that in mixed or white light the fringes inflected by B after deflexion by A are greater than those deflected by B after inflexion by A ; secondly, that they are also greater in homogeneous light ; thirdly, that the excess of the inflected fringes over the deflected is greater in mixed than in homogeneous light. The action of flexion after disposition is so much greater than that of simple flexion, that I have only taken into the calculation the compound flexion. But the most accurate analysis is that which makes the two fringes as D H — . to D + {a + dy {a + 2d + cy {a + d + o)"' D being the breadth of the fringes on the chart by simple flexion in case the rays had passed on without disposition ON LIGHT AND COLOUBS. 143 and without a second flexion. If it be carefully kept in mind T T that D is much less than — , ov even — — , and that or {a + 2d + c)" d is still less than D, then it will always be certain that the first quantity is larger than the second. Cor. — It is a corollary to this proposition that the dif- ference of the two sets of fringes is increased by the dis- position conuminicated by the rays in passing by the first body. For the excess of the value of r over that of v being increased, the difierence between the two expressions is increased. Peoposition YII. When one body only acts upon the rays, it must, by deflexion, form them into fringes or images decreasing as the distance from the bending body increases. But when the rays deflected and disposed by one body are afterwards in- flected by a second body, the fringes will increase as they recede from the direct rays. Also when the fringes made by the inflexion of one body, and which increase with the distance from the direct rays, are deflected by a second body, the effect of the disposition and of the distances is such as to correct the effect of the first flexion, and the fringes by de- flection of the second body are made to decrease as they recede from the direct rays. In flg. 15, A P is the pencil inflected by A and forming the first and narrower fringe p ; A.r is the pencil inflected nearer to A and forming the broader fringe r. Such are the relative breadths, because they are inversely as some power of the distance at which A acts on them. But if B afterwards actst it is shown by the same reasoning which was applied to the last proposition that r will be less than p ; and so in like manner will r' be made less than o', though o' was greater than r' until B's action, and the effects of disposition with 144; EXPEBIMENTS AND INVESTIGATIONS the greater proximity of the smaller fringe, altered the pro- portions. Peoposition VIII. It is proved by experiment that the inflexion of the second hody makes broader fringes or images than its deflexion after the inflexion of the first body ; and also that the inflecto- deflexion fringes decrease, and the deflecto-inflexion fringes increase with the distance from the direct rays. Exp. 1. It must be observed that when we examine the- fringes (or images) made by the second edge deflecting the rays which the first had inflected, we can see the efi'ects of the disposition communicated to the rays at a much greater distance of the second edge from the first, than we can perceive the efi'ects of that disposition upon the infiexion by the second edge of the rays deflected by the first. Indeed w© only lose the fringes thus made by deflexion, in con- sequence of their becoming so minute as to be imperceptible to our senses. But it is otherwise with the fringes or images made by the second edge inflecting the rays which the first had deflected. These can only be seen when the second edge is near the first, because the rays cannot pass on so as to form the images on the chart., if the second is distant from the first. The pencils diverge both by the deflexion and by the inflexion of the first edge. But we can always, when the inflected rays pass too far from the second edge, bring this so near them as to act on them, whereas we in so doing intercept the deflected rays. However, after this is explained, we find no difficulty in examining the efi'ects of the infiexion by the second edge, only we must place it near the first, and thus 'we have two sets of fringes, one ex- tending into the shadow of the first edge at an inch distance between the two edges ; but at an inch and three-fourths, nay, at two inches, or even more, this experiment can well be made. Exp. 2. At these distances I examined repeatedly the comparative breadths of the two sets. In fig. 16, ab is the ON LIGHT AND COLOUES. 145 Rff.ie. white disc, on each side of which are fringes ; those on the one side, ic, cd, are by the inflexion of the second edge ; those on the opposite side, af, fe, are by the de- flexion of that second edge. I repeat- edly measured these sets of fringes, and at various distances from the second edge ; and I always found them much broader on the side of the second edge than on the opposite side. Thus ab being the breadth of 5, 6 c was 3, and c d 4i, while, on the opposite side, af " "" was = 1 and fe onlj' f or \. The fringes by inflexion of the second edge also uniformly increased as they receded from a b, the direct rays, whereas the opposite fringes as constantly decreased. Exp. 3. If however the distance between the two edges be reduced, it is observed that the disparity between the two sets of fringes decreases, and they become gradually nearly equal ; and when the edges are quite opposite each other there is no difference observable in the two sets. Each ray is disposed and polarized alike and affected alike by the two edges, and no difference can be perceived between the two sets. Exp. 4. The experiments also agree entirely with the calculus in respect of the relative values of r and v affecting the result. It appears that the fringes by the second edge's inflexion are broader than those by that edge's deflexion, whether we use white or homogeneous light. In the latter, however, the difference is not so considerable. This I have repeatedly tried and made others try, whose sight was better than my own. I may take the liberty of mention- ing my friend Lord DouEO, who has, I believe, heredi- tarily, great acuteness of vision. Proposition IX. The joint action of two bodies situated similarly with respect to the rays which pass between them so near as to be affected by both bodies, must, whatever be the law of their L 146 EXPERIMENTS AND INTESTIGATIONS action, provided it be inversely as some power of the distance, produce fringes or images which increase with the distance from the direct rays. Let (fig. 17) A and B be the two bodies, and A C = C B = a be their spheres of flexion, so that A inflects and B deflects through A 0, and A deflects and B inflects through C B. Let C P = a;, P M = ?/. The force y, exerted by the joint action of A and B on any ray passing between them at P, is equal to -; r^ + ^ -, supposing deflexion and inflexion to follow different laws. To find the minimum value of y^ -, take its difierential dy = 0; therefore we have — m(a + cc)~"'~^dx •{-n(a — x)'"~^dx = 0, or m(a — 0;)°+' = n{a + xy+K If m = K (as there is every reason for supposing), then a — X = a -^ X, or X — ; and therefore, whatever be the value of m (that is whatever be the law of the force), the minimum value of y is at the point C where A's deflexion begins. The curve S S', which is the locus of M, comes nearest the axis at C, and recedes from that axis constantly between C and B. Hence it is plain that the fringes must increase (they being in proportion to the united action of A and B) from to B ; and in like manner must those made by B's deflexion and A's inflexion increase constantly from C to A ; and this is true whatever be the law of the bending force, provided it is in some inverse ratio to the distance. Peoposition X. It is proved by experiment that the fringes or images increase as the distance increases from the direct rays. Exp. 1. Eepeated observations and measurements satisfy us of this fact. We may either receive the images on a chart at various distances from the double edge instrument, approach- ing the edges until the fringes appear, or we may receive them on a plate of ground glass held between the sun and the , ON LIGHT ANB COLOUES. 147 eye. We may thus measure them with a micrometer ; but no such nicety is required, because their increase in breadth is manifest. The only doubt is with respect to their relative breadth when the edges are not very near and just when they begin to form fringes. Sometimes it should seem that these very narrow fringes decrease instead of increasing. How- ever, it is not probable that this should be found true, at least when care is taken to place the two edges exactly opposite each other; because if it were true that at this greater distance of A from B (fig. 17) they decreased, then there must be a minimum value of P M between and B, and between C and A ; and consequently the law of flexion must vary in the different distances of A and B from the rays P, a supposition at variance it should seem with the law of con- tinuity. Exp. 2. The truth of this proposition is rendered more apparent by exposing the two edges to the rays forming the prismatic spectrum. The increase is thus rendered manifest. K.the fringes are received on a ground glass plate, you can perceive twelve or thirteen on each side of the image by the direct rays. It is also worth while to make similar observa- tions on artificial lights, and on the moon's light. The pro- position receives additional support from these. But care must always be taken in such observations, which require the eye to be placed near the edges, that we are not misled by the efiiect of the small aperture in reversing the action of the edges. Thus when viewing the moon or a candle through the interval of two edges, one being in advance of the other, we have the coloured images (or fringes) cast on the wrong side. But if we are only making the experiment required to illustrate this proposition, the edges being to be kept directly opposite, no confusion can arise. It is to be noted that the increase of breadth in the fringes is not very rapid in any of these experiments ; nor are we led by the calculus to expect it. Thus suppose m = 1, we find ( because y - — ^ j at the point C, when a; = 0, the breadth L 2 148 EXPEBIMENTS AND INVESTIGATIONS 2 a should be proportional to — . Take x = — -, and the breadth a 10 is as -j— , or the breadth of the one fringe is to the other only as 200 to 198 or 100 : 99. We need not wonder therefore if there is only a gradual increase of breadth from C to B and from C to A. The increase is more rapid between x = — and B than between C and — . Thus between the value of X - —- and -- the increase is as 4 : 5. But from -r- to —- 4 2 2 4 the increase is as 7 : 12 ; and this too agrees exactly with the experiments ; for as the edges are approached the increase of the fringes becomes more apparent. Peoposition XI. The phenomena described in the foregoing propositions are wholly unconnected with interference, and incapable of being referred to it. 1. When the fringes in the shadow are formed by what is supposed to be interference, there are also formed other fringes outside the shadow and in the white light. If the rays passing on one side the bending body (as a pin or needle) are stopped, the internal fringes on the opposite side of the shadow are no longer seen. But no effect whatever is produced on the external fringes. These continue as long as the rays passing on the same side of the body on which they are forried, continue to pass. The external fringes have many other properties which wholly distinguish them from the internal or interference fringes. 2. Interference is said to be in proportion to the different lengths of the interfering rays, and not to operate unless those lengths are somewhat near an equality. In my experi- ments the second body may be placed a foot and a half away from the first, and the fringes by disposition are still formed. ON LIGHT AND COLOTTES. 149 though much narrower than when the bending bodies are more near to one another. 3. The breadth of the interference fringes is said to be in some inverse proportion to the difference in length of the interfering rays. It is commonly said to be inversely as that difference. In fig. 20, A is the first and B the second edge. By inter- ference the fringe at C should be broadest and at D narrowest, be- cause AC— BO = AO is less than A D - B D = A P ; and so as you recede from D, the fringes should become broader and broader, because the two rays become more nearly equal. But the very reverse is notoriously the case, the breadth of the fringes decreasing with their distance from the direct rays. 4. In the case of the fringes formed by the second body inflecting and the first deflecting, there can be no interference at all ; for the whole action is on one and the same pencil or beam. A deflects and then B inflects the same ray; and when a third edge is placed on the opposite side to B, it only deflects' the same ray, which is thus twice bent further from the direct rays, the last bending increasing that distance. o. Let A be the first and B the second edge as before (fig. 20). Suppose B to be moveable, and find the equation to the disposing force at different distances of the two edges, we shall find this to be w = — == , a beins; = A E, 6 = E D, and A B = a;. But all the experiments show it to be y = — , a wholly different curve. Again, let B be fixed, or the distance of the two edges be 150 EXPBBIMBNTS AND INVESTIGATIONS constant, we shall get the equation (a being = AE, 6 = BE, and EC = x) y = , also a wholly dif- ferent curve from the conic hyperbola, which all experiments give. Therefore the conclusion from the whole is that the phenomena have no reference to interference. Having delivered the doctrines resulting from these experi- ments, I have some few particulars to add, both as illustrating and confirming the foregoing propositions, as removing one or two difficulties which have occurred to others until they were met by facts, and also as showing the tendency of the results at which we have arrived. 1 . It may have been observed that in all those propositions I have taken for granted the inflexion of the rays by the body first acting upon them as well as their deflexion by that body, and have reasoned on that supposition. It is, however, not to be denied that we cannot easily perceive the fringes made by the single inflexion, as we can without any difiiculty perceive those made by the single deflexion, and fully de- scribed in Proposition I. Sir I. Newton even assumes that no fringes are made within the shadow. I here purposely keep out of view the fringes made in the shadow of a hair or other small body, because the principle of interference there comes into play. However, I will now state the grounds of my assuming inflexion and separation of the rays by their different flexibility, when only a single body acts on them. In the/iVsf place, the first body does act in some way; for the second only acts after the first, and if the first be removed the fringes made in its shadow by the second at once vanish. Secondly, these fringes made by the second depend upon its proximity to the first. Thirdly, the following experiment seems decisive. Place, instead of a straight edge, one of the form in fig. 18, and then apply at some distance from it, the second edge, as in the former experiments. You find that the fringes assume the form, somewhat like a small-tooth comb, ON LIGHT AND OOLOUES. 151 of a b. If the second edge is furnished with a similar curve surface the form is more complete, as in c d. But the straight cU==^i edge being used after the first flexion of the curved one, clearly shows that the first edge bends as well as the second, indeed more than the second, for the side of the figure answering to that curved edge is most curved. Fourthly, the whole experiments with two edges directly opposite each other negative the idea of there being no inflexion ; indeed they seem to prove the inflexion equal to the deflexion. The phenomena under Proposition X. can in no way be re- conciled to the supposition of the flrst edge not inflecting the rays.* . 2. We must ever keep in view the difference between the fringes or images described by Sir I. Newton and measured by him, as made by the rays passing on each side of a hair, and the fringes or images which are made without the inter- ference of rays passing on both sides. It is clear that the rays which form those fringes with their dark intervals do not proceed after passing the hair in straight lines. Sir I. Newton's measures | prove this ; for at half a foot from the hair he found the first fringe x^th of an inch broad, and the second fringe ^^^ ; and at nine feet distance the former were ■Jy, the latter -^, instead of between -l and -jL, and the latter less than tV> and so of all the other measures in the table, each being invariably about one-third what it ought to be if the rays moved in straight lines ; and this also explains why the fringes do not run into one another, or encroach on the * If you hold a body between the eye and a light, as that of a candle, and approach it to the rays, you see the flame drawn towards the body ; and a beginning of images or fringes is perceived on that side. f Optics, B. iii. obs. 3. 152 EXPERIMENTS AND INVESTIGATIONS dark intervals in the case of the hair, as they must do if the rays moved in straight lines. But the case of the fringes or images which we have been examining and reasoning upon is wholly diiferent. I have measured the breadths of those formed by disposition and polarization, and found that they are broad in proportion to the distance from the bending edge of the chart on which they are received ; and vary from the results given by similar triangles in so trifling a degree, that it can arise only from error in measurement. Thus in an average of five trials, at the relative distances of 41 and 73 inches, the disc was 6f at the shorter, and 10^ at the longer distance ; the fringe next it 3/5 at the shorter, and 5-f^ at the longer distance, whereas the proportions by similar triangles would have been 9^ and 5^, so that the difference is small, and is by excess, and not, as in the hair experiment, by defect. *' Had the difference been as in Sir I. Newton's experiment, instead of 10-i- and S/^-, it would have 3-^ and 1|-|-. In another measurement at 101 and 158 inches respectively, the disc was 15-1-, the fringe 8| instead of 14f- and 9^ respectively. But by Sir I. Newton's proportions these should have been 34|- and ■^\. It is plain that if the measures had been taken with the micrometer instruments, which had not been then furnished, there would have been no deviation. I have since tried the experiment, not as above, on the fringes formed by the double-edged instrument, but on those formed by one edge at a distance behind the other, and have found no reason to doubt that the rays follow a rectilinear course. It may further be observed, that in the fringes or images by disposition and polarization, the dark intervals disappear at short distances from the point of flexion, and that the fringes run into one another, so that we find the red mixed with the blue and violet. This is one reason why I often experimented with the prismatic rays. 3. It follows from the property of light, which I have termed disposition, on one side the ray, and polarization on the opposite side, superinduced by flexion, that those two ON LIGHT AND COLOTJHS. 153 sides only, being affected, the other two at right angles to these are not at all affected by the flexion which has disposed and polarized the two former. Consequently, although an edge placed parallel to the disposing edge and opposite to it acts powerfully on the disposed light, yet an edge placed at right angles to the former edge or across the rays, does not affect them any more than it would rays which had not been subjected to the previous action of a first edge. Thus (fig. 19) :^ Kg.19 Iig.19. Jf IZD if a 6 c tZ be the section of the ray, an edge parallel to a b, after the ray has been disposed, will affect the ray greatly, pro- vided it had been disposed by an edge also parallel to a h. The sides a b and c d, however, are alone affected ; and there- fore the second edge, if placed parallel to a d ot b c, will not at all bend the ray more or make images (or fringes) more powerfully than it would do if no previous flexion and dis- position had taken place. Let us see how this is in fact : efg h is the distended disc after flexion, by passing through the aperture of the two-edged instrument (Plate XII.). It is slightly tinged with red at the two ends fg and e h, beyond which, and in the shadow of the edges, are the usual fringes or coloured images by flexion and disposition, c, c, the edges being parallel to eh,fg. Place another edge at some distance from the two, as 3 or 4 inches, and parallel to these two, but in the light, and you will see in the disc a succession of nar- row fringes, parallel to the edges, and in front of the third edge's shadow. These fringes are on the white disc, and their colours are very bright, much more so than the colours of those fringes described in Proposition I., and which are fringes made by deflexion without any disposition. But whether this superior brightness is owing to the glare of the 154: BXPEEIMENTS AND INVESTIGATIONS disc's light being diminislied by the flexion of the first two edges, or not, for the present I stop not to inquire. This is certain, that if the third edge be placed across the beam, and at right angles to the two iirst edges, you no longer have the small fringes. They are not formed in the direction hg, parallel to the edges as now placed. If the double edges are changed, and are placed in the direction h' g', you again have the bright fringes ; but then, if the third edge is now placed parallel to K e', you cease to have them. Care must, however, be taken in this experiment not to mistake for these bright fringes the ordinary deflexion fringes made by one flexion ' without disposition, as described in Proposition I. For these may be perceived, and even somewhat more distinctly in the disc than in the full light of the white pencil or beam. Now are these bright fringes only the flexion fringes, that is fringes by simple flexion without disposition ? To ascer- tain this I made these experiments. Exp. 1. If they are the common fringes, and only enlarged by the greater divergence of the rays after flexion, and more bright by the dimness of the distended disc, then it will follow that the greater the distension, and the greater the divergence of the rays, the broader will be the bright fringes in question. I repeatedly have tried the thing by this test, and I uniformly find that increasing the divergence, by ap- proaching the edges of the instrument, has no effect whatever in increasing the breadth of the fringes in question. Exp. 2. If these fringes are not connected with disposition, it will follow that the distance of the edge which forms them from the double-edged instrument cannot affect them. But I have distinctly ascertained that their breadth does depend on that distance, and in order to remove all doubt as to the distance between the chart and the third edge which forms them, I allowed that edge to remain fixed, and varied its distance from the other two by bringing the double-edge instrument nearer the third edge. The breadths of the bright fringes varied, most remarkably, being in some inverse power of that distance. Thus, to take one measurement as an ON LIGHT AND COLOTTRB. 155 example of the rest, at 4 feet from the third edge the chart was fixed and the third edge kept constantly at that distance from it. Then the double-edge instrument was placed suc- cessiTely at 14|-, at 9 and at 4|- eighths of an inch from the third edge. The breadths were respectively 2, 3f and 4^ twentieths of an inch. In some experiments these measures approached more nearly the hyperbolic values of y, but I give the experiment now only for the important and indeed decisive evidence which it affords, that these fringes are caused by disposition, and are wholly different from those formed without previous flexion. Exp. 3. If the greater breadth of these fringes is owing to dispersion, then they should be formed more in the rays of the prismatic spectrum than in white light, or even in light bent by flexion. Yet we find it more difScult to trace fringes across the prismatic spectrum than in white light, and more difficult across the spectrum when' there is divergence, than when formed parallel to its sides when there is no divergence. There are fringes formed, but of the narrow kind, which are described in Prop. I. Exp. 4. I have tried the effect on the fringes in question of the curvilinear edge described in the first article of these observations, and the effect of which is represented in fig. 18. It is certain that at a distance from the double-edge instru- ment the third edge seems only to form fringes rectilinear, or of its own form. But when placed very near, as half an inch from the instrument, plainly there is a curvilinear form given to the fringes in question ; and this is most easily perceived, when, by moving the third edge towards the side of the pencil, you form the smaller fringes so as to be drawn across or along the greater ones made by the two first edges. I think, without pursuing this subject further, it must be admitted that these fringes in light, which is bent and dis- posed, lend an important confirmation to the doctrine of disposition. It is clear that the rays are affected only on two of their four sides, or a & and o d, if these are parallel to the bending body's edge, and not at all on the sides c & and d a ; 156 EXPEEIMBNTS AND INTESTIGATIONS that, on the other hand, c 6 and d a are affected when the edges are placed parallel to these two sides of the rays ; and thus the connection of the fringes in question, with the preceding action of which disposed and polarized, is clearly proved. 4. It is an obvious extension and variation of this experi- ment both to apply edges parallel to the first and disposing edges, and also to apply edges at right angles to their direction ; and important results follow from this experiment. But until a more minute examination of the phenomena with accurate admeasurements can be had, I prefer not entering on this subject further than to say, that the extreme difficulty of obtaining fringes or images at once from the edges parallel to the first two, and from edges at right angles to these, indicates an action not always at right angles to the bending body, but whether conical or not I have not hitherto been able to ascertain. That the first body only disposes and polarizes in one direction is certain. But it seems difficult to explain the effect of the first two edges in preventing the fringes or images from being made by the second at right angles to those formed by the first two edges, if no lateral action exists. One can suppose the approaching of those two first edges to make the fringes narrower and narrower than those which the second two edges form when placed at right angles to the first. But this is by no means all that happens. There is hardly any set of fringes at all formed at right angles to the first set (parallel to the first two edges) when the first two are approached so near each other as greatly to distend the disc. 6. I reserve for future inquiry also the opinion held by Sir I. Newton, that the different homogeneous rays are acted upon by bodies at different distances, this action extending furthest over the least refrangible rays. He inferred this from the greater breadth of the fringes in those rays. It is in my apprehension, though I once held a different opinion,* not impossible to account for the difference of the * PhilosopMcal Transactions, 1797. ON LIGHT AND COLOUES. 157 breadth of the fringes by the different flexibility of the rays ; and the reasoning in one of the foregoing propositions shows how this inquiry may be conducted. But one thing is certain, and probably Sir I. Newton had made the experi- ment and grounded his opinion upon the result. If you place a screen, with a narrow slit in the prismatic spectrum's rays, parallel to the rectilinear sides, and then place a second prism at right angles to the first and between the screen and the chart, you will see the image of the slit drawn on one side, the violet being furthest drawn, the red least drawn ; but you will find no difference in the breadth of the image cast by the slit. Flexion, however, operates in a different manner, because it acts on rays, which, though of the same flexibility, are at different distances from the body. 6. The internal fringes in the shadow (said by interference) deserve to be examined much more minutely than they ever have been ; and I have made many experiments on these, by which an action of the rays on one another is, I think, sufficiently proved: I shall here content myself with only stating such results as bear on the question of interference affecting my own other experiments. First. I observe that when one side of a needle or pin is grooved so as to be partly curvilinear, the other side remaining straight, we have in- ternal fringes of the form in fig. 21. Secondly. It is not at all necessary the pin or other body forming them should be of very small diameter, although it is certain that the breadth of the fringes is inversely as the diameter. I have obtained them easily from a body one-quarter or one-third of an inch in diameter, but they must be received at a con- siderable distance from the body. Thirdly, and this is veiy material as to interference at all affecting my experiments, although certainly the internal fringes vanish when the rays are stopped coming from the opposite side of the object, the external fringes are not in the smallest degree affected, unless you stop the light coming on their own side ; stopping the opposite rays has no effect whatever. Thus, stopping the 158 BXPBEIMBNTS AND INVESTIGATIONS light on the side a (fig. 21), the fringes // vanish, but not the external fringes c. This at once proves there is no inter- ference in forming the external ones. Lastly. I may observe, that the law of disposition and polarization in some sort, though with modifica- Ixg-S-f. tion, affects the internal fringes as well as the external. It is a curious fact connected with polarization by inflexion, and which indeed is only to be ac- counted for by that affection of light, that nothing else pre- vents the rays from circulating round bodies exposed to them, at least bodies of moderate diameter. If the successive particles of the surface inflected, one particle acting after the other, the rays must necessarily come round to the very point of the first flexion. We should thus see a candle placed at A (fig. 22) when the eye was placed at B, because the rays would be inflected all round ; and even in parts of the earth where the sea -.22. is smooth, nothing but the small curva- ture of the surface could prevent us from seeing the sun many hours after night had begun by placing the eye close to the ground. This, however, in bodies of a small diameter, must inevitably happen. The polarization of the rays alone pre- vents it, by making it impossible they should be more than once inflected on their side which was next the bending body, therefore they go on straight on to C. But for polarization they must move round the body. 7. It must not be lightly supposed, that because such inquiries as we have been engaged in are on phenomena of a minute description and relate to very small distances, there- fore they are unimportant. Their results lead to the con- stitution of light, and its motion, and its action, and the relations between light and all bodies. I purposely abstain from pursuing the principles which I have ventured to explain into their consequences, and reserve for another ON LIGHT AND COLOTJES. 159 occasion some more general inquiries founded upon what goes before. This course is dictated by the manifest expediency of first expounding the fundamental principles, and I there- fore begin by respectfully submitting these to the considera- tion of the learned in such matters. In the meantime, however, I will mention one inference to be drawn from the foregoing propositions of some interest. As it is clear that the disposition varies vrith the distance, and is inversely as that distance, and as this forms an inherent and essential property of the light itself, what is the result ? Plainly this, that the motion of light is quite uniform after flexion, and apparently before also. The flexion produces acceleration but only for an instant. If ss is the space through which the ray moves after entering the sphere of flexion, and v the velocity before it enters that sphere ; it moves after entering with a velocity = /^ if -{- 'L d z, Z being the law of the bending force. Then this is greater than v ; consequently there is an acceleration, though not very great ; but because y = — , if s is the space, t the time, the force of acceleration is ^7- X ; ; but y = — shows that s is tds r X as t, else ?/ = — would be impossible ; therefore the accele- ■5 tds — sdt ^ , ., . ■■ ratma; force -— x r = Oj ^^id so it is shown there is ° as r no acceleration after the ray leaves the sphere of flexion. 160 BXPBEIMBNTS AND INVESTIGATIONS Description of the Insteuments. PLATE XII. t:J==l: Is the instrument witli two plates or edges. A, B, hori- zontal, D, 0, vertical; the former moved by the screw E, which has also a micrometer for the distances on the scale G ; the latter, in like manner, moved by F, connected with micro- meter and scale H. ON LIGHT AND COLOURS. 161 PLATE Xm. Is the instrument with four surfaces. AD, ad are two parallel plates, moving horizontally by a rack and pinion E. Each plate has an edge composed of four surfaces ; A, a, a sharp edge or very narrow surface ; B, 6, a flat surface ; C, c, a cylindrical surface of large radius of curvature, and so flat ; D, d, one of small radius, and so very convex : this is re- presented on the figure by A' B' C D' beside the other. Care is to be taken that A B C D and abc dhe a perfectly straight line, made up of the sharp edge, the plane surface and the tangents to the two cylinders. H is a jilate with a sharp and straight edge, op, which can be brought by its handle F to come opposite to the compound edge abed, when it is desired to try the flexion by the latter, without another flexion by an opposite compound edge, but only with a flexion by a lecti- linear simple edge. 162 EXPERIMENTa AND INVESTIGATIONS PLATE XIV. Is the instrument by which is tried the experimentum cruds on the action of the third edge, and also the experiments on the distances of the edges as affecting the disposing force. G is the groove in which the uprights H, I, K move. There is a scale graduated, F, by which the relative distances can always be determined of the plates A, C and B. A moves np and down upon H, B upon I, and C upon K ; each, plate is moved up and down by rack and pinion D. The uprights also move along the groove G by rack and pinion E. ON LIGHT AND COLOTTES. 163 PLATE XV. 2. Zi A- S € T S Si ■ ■ ■ ..l-.l-l. J... I I.... 9 lO TLULJJ3Xt-%S IG 2.2,3 4: s e 7 a B ibaiia.iai4iisie Is the instrument for ascertaining more nicely the effects of distance on disposition. A is a plate with graduated edge ; it moves vertically on a pivot, and its angle with the horizontal line is measured by the quadrant E. A also moves hori- zontally, and its horizontal angle is measured by the quadrant K. B is another plate with graduated edge, moving in a groove D, by rack and pinion H, and along a graduated beam I. F is a fine micrometer, by which the distance of A above B, when A is horizontal, can always be measured to the greatest nicety by the circle P and the scale G. M 2 164 EXPERIMENTS AND INVESTIGATIONS PLATE XVI. " a "' i 't-i-p'i' Is an instrument also for measuring the effect of ^^ difference entre M' P' et MP ne serait memo alors plus que xoVb- cent. Or une difference meme plus grande que celle-ci ne produit aucun effet sensible sur les franges, comme je I'ai bien des fois constats dans des experiences avec le micrometre. Done la preuve que Ton vient de donner est sous tous les rapports concluante, et auoune 170 INaUIEIBS ANALYTICAL AKD erreur ne peut s'y Jntroduire par le defaut de parallelisme des bords. II est bien pourtant de regarder la flamme par una partie des bords pas trop pres de Tangle, s'ils ont une incli- naison entre eux, parce que, bien que M P — M' P' est toujours,, — quel que soit A P, — la m^me, pourvu que P P' soit quantite constante, cependant la proportion de M P a M' P' varie avec la valeur de A P ; at quelquea experiences m'ont fait soup-f Qonner qua cette proportion varianta, quand la difference reste Constanta, pourra influer sur les phenomenes.* ^.^^**^^"'" j';*^^^*''-...-.,, * La cinquifeme figure donne I'exp^rienoe avec la flamme. P = la flamme ; B = les bords ; P = le priame ; B B = les bandes. Mais I'artiste qui les a dessiudes parait les avoir repr&ent&s un peu trop larges sur la partie supe'rieure. BXPEEIMBNTAL ON LIGHT. 171 7° Si nous Axons conjointement deux lames de verre colore, I'une rouge et I'autre bleue, examinant ainsi les bandes formees de la lumiere qui les traverse par deux bords places derriere, et que nous regardions aussi le disque distendu par Taction des bords, nous trouvons la partie rouge de ce disque plus distendue que la partie bleue, et les bandes rouges plus larges et plus separees les unes des autres. 8° L'augmentation de la distance en mime temps que de la largeur dans les bandes rouges formees par un bord seul ou par les deux, parait evidente de ce que si elles n'etaient qu'^galement eloignees de I'ombre du corps ou de I'axe de spectre, elles seraient paralleles a I'ombre ou a I'axe ; et par consequent il j aurait entre chaque bande et les bandes avoi- sinantes un intervalle croissant du rouge vers le violet, comme dans la sixieme figure, ou EiV est le spectre, e^ rv sont les bandes. Or il n'y a rien de la sorte k voir, examinez les phenomenes comme vous voudrez. Les intervalles obscurs, on les voit toujours diminuer de largeur du rouge vers le violet ; et au violet ces intervalles sont si miuces, qu'a peine peut-on les *j<^g tracer. 9° La meme diversite des rayons bomogenes, je I'ai trouvee dans tous les autres cas des bandes, soit de celles qtii sont formees par la flexion seule, soit de celles formees dans les experiences avec des speculums, ou des surfaces striees par la flexion combinee avec la reflexion. Lorsque ces bandes sont formees par la lumiere blanche, elles ont toutes les couleurs, et elles sont paralleles entre elles et a I'axe du pinceau ou de la flamme ; mais, fornixes par les rayons du spectre, elles sont toujours plus larges dans les rayons les moins refrangibles, et ont une inclinaison sensible du rouge vers le violet. Ainsi les bandes d'une surface striee exposee a la lumiere blanche etant, comme dans la septi^me figure, le rouge plus loin, le bleu plus pres de la flamme, — ces mSmes bandes regar- K*.?. dees par la reflexion des rayons du prisme sont, comme dans la huitieme figure, plus larges et plus eloignees n U 172 INQUIKIES ANALYTICAL AND de I'image EV deja flamme, dans leur portion rouge r, et s'approchent entre elles et de la flamme vers la. por- tion violette y. De meme les bandes formees par un miroir plane assez mince, et qui sont egales entre elles et paralleles aux bords du miroir, si elles sont formees dans la lumiere blanche ou dans la lumiere homogene (mais de m^me espece en plagant le miroir a travers le spectre), deviennent entiferement diiFe- rentes si le miroir est place perpendiculairement au prisme et parallfele au spectre ; car alors elles sont plus larges dans les rouges, et plus distantes des bords du miroir. Ceci est a observer meme quand on se sert d'un miroir dont les bords Sont inclines a un petit angle, comme de 5°, bien que les bandes qui repondent a la portion des bords vers Tangle soient dilatees et eloignees dans une courbe hyperbolique, si elles sont formees de lumiere blanche, ou que le miroir se trouve place en travers du spectre. Pourtant, si la partie mince du miroir est placee dans les rayons violets, et les autres parties paralleles a I'axe du spectre, la partie rouge des bandes paratt un peu plus large et plus distante de I'ombre que la partie violette, la difference de flexibilite des rayons rouges etant plus considerable que I'effet produit par le peu de largeur du miroir. 10° Done, il n'y a aucun doute sur cette propriete de la lumifere. Les rayons de differente espece sont non-seulement disposes en bandes de largeur differente par la force de flexion mais ils sont flechis differemment ; les angles de deflexion difi'ferent dans les differents rayons, etant plus grandb dans lea moins refrangibles, plus petits dans les plus refrangibles ; en un mot, leur deflexibilite est en raison inverse de leur refran- gibilite. Suivant le calcul ci-dessus donne de la proportion de 3 a 2, et supposant la deflexion moyenne telle que I'a donnee Newton (au moins telle qu'on la pent deduire de ses mesures), 3' 32", alors cat angle pour les rayons rouges sera de 4' 14", pour les violets de 2' 49". Ceci a rapport a la deflexion par un bord ou un autre corps seal. Les angles (c'est tout simple) sont beaucoup plus grands si deux bords agissent ; EXPERIMENTAL ON LIGHT. 173 mais il ii'y a pas lieu de croire que la proportion des angles est differente. Si la force flechissante varie com me — ^ (J = dis- tance du corps aux rayons), et si Taction sur les rouges est a 3 2 Taction sur les violets com me 3 a 2, elle sera comme — k — ; par consequent la distance ne signifie rien, bien que la dif- ference de TefTet produit dans les rayons diiF^rents, sur la largeur des bandes et leur separation entre elles, sera plus grande plus la distance des rayons aux bords est petite, cette difference 6tant comme -r—. Ainsi, cette difference est beau- d" coup plus facile a remarquer lorsque les deux bords agissent ; et lorsqu'il n'y a qu'un seul bord, la difference est plus re- marquable dans les bandes les plus pres de Tombre. Nous avons fait observer que Texperience newtonienne sur les largeurs des bandes ne conclut rien a cause de la propriete de lumifere que nous venons de decrire, et qui avait echappe a Tillustre philosopbe. It est probable que son erreur venait de ce qu'il avait aper9u i I'inspection simple que les bandes etaient plus larges dans les rayons rouges, et que, satisfait de cela, il n'appHquait ses mesures qu'^ eonstater la proportion des largeurs. Mais il y a une autre portion de ses observa- tions qui ne paratt pas appuyee par les phenomenes, je vieux dire la description des intervalles obscurs ou noirs lorsque les bandes sont formees par la lumifere blanche. II faut cer- tainement la plus grande hesitation, meme en osant exprimer un doute sur les recits d'un observateur si achev6. Cependant on peut concevoir que son attention n'ait pas ^te dirigee si rigoureusement au sujet du troisifeme livre qu'aux autres por- tions de son grand ouvrage. La preuve en est qu'il n'a pas remarqu^ les bandes internes ou de Tombre du tout, bien que Grriraaldi, qu'il cite, en ait fait mention. La raison est pro- bablement qu'il iivait fait ses experiences avec un cheveu ; et les bandes internes ne sont facilement observees qtt'avec un corps un peu plus large. Une aiguille -^Y de diam^tre les 174 INQUIRIES ANALTTICAL AND forme, mais pas si bien qu'utie aiguille un peu plus large. Le cheveu dont s'est servi Newton n'avait que tttt" de largeur. Notts venous de voir aussi que ses mesures etaient peu con- cluantes sur les bandes du spectre, parce qu'il n'avait pas remarque leur different eloignement. Ne serait-il pas possible qu'il se fut trompe sur les intervalles noirs en regardant comme un espace obscur ou mSme noir le teint plus fonce des bandes la oil le rouge de I'une touche au violet de I'autre? Que sais- je ? Mais si Ton se donne la peine de regarder de prfes et avec grande attention ces bandes fortnees dans la lumiere blanche, on sera convaincu que les couleurs se fondent, que le violet d'une bande se mele avec le rouge de la bande voisine, et que ce qui d'abord avait paru ligne noire n'est que la confusion de ces deux couleurs. On a fait I'experience avec toutes les mesures et toutes les proportions des observations nevytd* niennes : mSme grandeur de trou, -^^ de pouce anglais ; mSme distance de la fen^tre et du tableau au cheveu, 12 pieds I'une, 6 pouces I'autre ; et m^me largeur de cheveu. Les bandes ont ite examinees k toute inclinaison du tableau, de la verticale k I'horizontale ; elles ont ete re9ues sur le verre depoli, et; I'oeil place derrifere le verre pour les recevoir directement, examinees avec une loupe ou k I'oeil nu, et par plusieurs obser- vateurs ; et bien que d'abord it ait paru qu'il y eub un inter- valle noir, une ligne qui separat les bandes, une inspection plus attentive et serupuleuse a toujours fait voir que les bandes se fondaient I'une dans I'autre au point de leur rapprochement, la violet ou bleu de I'une se rallant par un espace trfes-petit avec le rouge de I'autre. Lorsque le verre depoli est place tres-prfes du corps, comme a moins d'un quart de pouce, on a plus de difficulte a apercevoir la fusion des bandes. Pour- tant, si trfes-prfes du corps elles sont s^parees, on ne pent pas facilement comprendre comment elles ne se croisent pas totale- ment et ne s'entrecoupent pas a une distance plus considerable. II est evident que rien ne prouve que les observations n'ont pas 6t& faites tr^s-prfes du corps, parce que les mesures de Newton etaient reprises a une distance de 6 pouces et de 9 pieds. Les lignes grises et noires au centre de 1' ombre ne peuvent EXPERIMENTAL ON LIGHT. 175 jamais 6tres confondues avec les tandes, et la separation des bandes par ces lignes-la est complete. Lorsque les bandes sont formees par la lumiere homogene, sans nul doute les intervalles noirs paraissent plus certains d'eiister, et il semble que lorsqu'il n'y a qu'une couleur elles doivent etre s^parees, a cause de la non-existence des autres eouleurs dans la bande. Cependant on doit faire observer que Newton ne donne que la plus petit difference entre les dis- tances des bandes rouges, par exemple, et des bandes de toutes cfluleurs formees par la lumifere blanche. L'une est de -^, I'autre de -^V de pouce (difference de tiVt)- H faut aussi faire remarquer que les bandes rouges, pqr exemple, examinees de prfes et sur un verre depoli, I'oeil der- riere paraissent avoir les autres eouleurs aussi. Le rouge d.ojnine, mais il j a du vert et du bleu ; bien que re9ues sur le tableau, elles paraissent toutes rouges : cela vient evidemment de la presence de lumifere blanche dispersee sans avoir pass^ par le prisme, mais aussi de la presence de lumiere imparfaite- ment separee par la refraction. Cependant, comme les rayons autres que les rouges, par exemple, doivent ^tre fl^chis aux endroits differents de ceux ou tombent les rouges, il parait que ces endroits-la doivent etre occupes par les autres eouleurs, bien qu'ils paraissent noirs. La m^me chose arrive avec le spectre prismatique lui-m^me. Faites un trou trfes-petit dans un ecran, et laissez passer les rayons homogfenes par ce trou et tomber sur le tableau. Der- ri^re le trou placez un second prisme, vous verrez un petit spectre ayant le rouge, par exemple, plus abondant et k sa place, mais ayant aussi du jaune et du vert et du bleu a I'autre extremite. Lorsque c'est le bleu ou violet qui passe par le trou le petit spectre a du vert et du rouge plus clairement que n'a de vert et de bleu le petit spectre form^ par les rayons rouges. Lorsque Ton examine les eouleurs du spectre prismatique prfes du prisme, il n'y a que du blanc, excepts aux borda, qui sont colores seulement d'une mince ligne de rouge d'un cote et de bleu de I'autre. Ces bords augmentent jusqu'4, ce que les eouleurs remplissent I'espace blanc dans la maniere decrite 176 INQUIRIES AHALYTICAL AND par Newton (^Opt., Hv. I, part II, prop, viii) ; mais, a moiiis que Tangle r^fractant du prisme ne soit tres-grand, comme de 68° a 70°, le blanc continue a quelque distance du prisme. Comme cela, selon Newton, vient du melange des diverses couleurs partant des differentes parties du prisme, il s'ensuit qu'un corps opaque, place de maniere h, interceptor une partie des rayons avant leur passage h. travers la ligne parallMe h. I'ase du prisme, fera paraitre des couleurs immediatement derrifere ce corps-la, mais non pas si le corps est place verticalement a I'axe du prisme. Apparemment c'est pour cette raison que les bords flecHssants places dans le blanc parallelement a I'axe dji spectre, et perpendiculaires a I'axe du prisme, forment des bandes de meme espfece et mSme eouleur que si les bords etaient places dans les rayons blancs non refraetes, pourvu que les bandes soient re9ues et examinees pres des bords, et dans I'espace du spectre qui continue a 6tre blanc. Eegardees plus loin, elles deviennent colorees avec les teintes du spectre. Mais on ne comprend par trop comment sur Texplication newtonienne les rayons, une fois disposes par Taction dea bords en bandes de couleurs tout k fait independantes de celles dont on suppose que la fusion produit le blanc du spectre, et 6tant devenus piuceaux de ces couleurs independantes, pour- raient plus tard, et a une distance plus grande, devenir meles ayee les couleurs qui avaient forme le blanc ; car les bords et les bandes qu'ils forment sont perpendiculaires aux rayons qui proviennent du prisme. Les bords (fig. 9, a, V) forment des bandes de couleurs entre g et /, differentes de celles cf et dg, qui s'entrem^lent avant et jusqu'a E ; au-dela de E, la fusion de cf, dg jijg, 9 ^ cesse. Mais comment est-ce que leur separation dans ce sens-la agit ou influe sur leur separation par a, b, dans un sens entiere- ment different ? Si a, h, etaient places en travers, de manifere k interceptor c / ou dg, nul doute que Teffet produit ne fftt de faire des couleurs dans la partie blanche du spectre. Mais cet effet serait produit de suite, passe a, h, et non pas a E BXtEBIMENTAL ON LIGHT. 177 seulement. Les rayons, ce semble, etant blancs a leur passage par les bords a, h, sont disposes en bandes par Taction de a, b, qui leur fait prendre line direction a un angle horizontal a a, b, c'est-a-dire que les bords decomposent le blanc en rouge, vert, bleu, par action laterale et horizontale ; et pourtant, par Taction verticale du prisme, ces mSmes couleurs sont changees passe E. - Supposons qu'au lieu des bords a, b, un prisme fut place verti- calement, il devrait former un spectre avec le rouge le plus pres de a, b, le violet de Tautre c6t6 ou a Tautre bout du spectre, si toutefois Tangle refractant du prisme est tourne vers a, b. Done, si la mime chose arrive a ce spectre qui arrive aux bandes, il s'ensuivrait que le rouge devrait 6tre change, au moins teint des couleurs qui, melees ensemble entre a et f, d et g, sont separees passe E, ce qui evidemment n' arrive pas. Cependant la grande diversite de Taction de flexion et de refraction doit toujours nous etre presente, et il n'y a rien dans ces phenomenes de plus remarquable. Lorsque la lumifere bomogene passe par les bords, parallelement et non pas diver- gente, elle est disposee en bandes non-seulement a distances diflferentes de Taxe du spectre, mais de largeur diverse. Le trait ou pinceau est distendu. Lorsque la lumifere est refractee par un second prisme place verticalement au premier ou parallele a Taxe du spectre, il est r^fracte aux diverses distances de Taxe, le violet le plus eloign^, le rouge le plus pres. En cela il y a grande ressemblance avec les phenomenes de flexion si ce n'est que les rayons les moins r^fract^s sont le plus fleehis. Mais la cesse Tanalogie des deux operations ; car il n'y a pas dans la refraction par le second prisme la plus petite distension ou dilatation du pinceau, comme il pourrait y avoir si le second prisme etait place horizontalement ou a travers le spectre ; car alors. Men que les rayons, tous de la meme couleur, ne puissent pas gtre distendus, cependant un trait compose de plusieurs coulexirs pourrait Stre distendu. Mais dans la flexion c'est different. Les bords places parallelement a Taxe du spectre ferment des bandes autant que s'ils etaient places a travers le spectre, et les pinceaux sont distendus lateralement, quand m^me les rayons qui les composent sont exacteraent de N 178 INQTJIEIES ANALYTICAL AND mSme couleur. It est vrai que les bandes sont plus largea lorsque les bords sont places parallfeleraent au prisme at en travers du spectre, a cause de la diff^rente flexion des rayons differents ; mais cette augmentation relative n'est pas trfes- considerable, paree que les rayons pres des bords (oranges, par exemple) ue sont pas autaut flechis que les rouges plus loin, cette difference etant une compensation de la plus grande distance de ceux-ci ; et ainsi la dispersion est plus petite qu'elle ne serait, k cause de la proximite des uns et de la distance des autres. Ces deux proprietes, — la differente distension (ou dispersion) des differents rayons indiquee par la differente largeur des bandes, et la differente flexibilite des rayons indiquee par la differente distance des bandes, — voyons comment on peut les expliquer, et si elles sont independantes I'une de I'autre, ou si elles peuvent ^tre ramenees au mSme principe. Newton, pour I'explication de la premifere propriete (la seule qu'il ait remarquee), a donne I'hypothese que Taction des corps s'etend plus loin sur les rayons tnoina refrangibles ; et il paralt penser qu'a la meme distance Taction est la meme, mais que cette action, plus prfes sur les una, 6gale Taction plus loin sur les autres. Cela explique eertainement la difference de lar- geur, mais non pas le different eloignement des bandes. Pour expliquer cela, il faut que Taction ne soit pas seuleraent egale k une plus grande distance, mais qu'elle soit plus forte a la meme distance. II y a pourtant une objection a faire k la theorie newto- nienne. C'est que Taction ne cesse pas avec la premiere bande ; il y en a une suite d' autres, toutes produites par la continua- tion de la meme action, diminuant avec la distance. Ainsi la theorie n'a pas d' application, a moins qu'on n'ajoute une bypo- these encore, savoir : Qu'il y a une suite de spheres d'action, chacune r^pondant k une bande, et que dans cbaque sphere Taction s'etend plus loin sur les rayons les moins refrangibles. Mais il faut ajouter encore un hypothfese, ce me semble, pour expliquer le plus grand eloignement de bandes formees par ceux-ci. II faut que la sphere, ou plut&t les spheres, d'action ESPEKIMENTAL ON LIGHT. 179 commeucent, pour les diverges couleurs, a differentes distances du corps flecliissant. En faveur de cette hypothese, ou pour- rait faire remarquer la bande ou ligne assez brilliante de blanc toucbant a I'ombre, et entre Tombre et la premiere bande coloree. Cette ligne blanche a toujours paru difficile a eipli- quer ; mais je ne suis pas d'avis que I'explication depende uniquement de ce que Ton vient de dire sur le commencement des spbferes d'aetion. II faut se rappeler que la deflexion commence la o^ I'iQflexion cesse. Ainsi les rayons qui passent le plus pr^s du corps sont exposes a toutes les deux actions, et ne peuvent pas ^tre decomposes plus qu'ils ne le sont en passant par deux prismes dont les angles refractants sont places en sens inverse. Dans ce cas-li, il n'y a que le blanc qui sorte ; ou si la lumiere est homogfene, il n'y a pas de change- raent daas le cours des rayons. Meme chose pour Taction des bords. Une troisieme hypothese me parait meriter notre attention, d'autant plus qu'elle pourrait peut-etre fournir I'explication de toutes les deux proprietes, et que les regies de philosopher defendent la multiplication de causes ou de prin- cipes. II se peut que la proportion de Taction, a la distance du corps, varie dans les difi'^rents rayons ; que le long du a * spectre dans Tequation y = — {y — force flectrice ; x = dis- tance du bord ou corps ; z = Taxe, ou plutSt les portions successives de Taxe du spectre; z = AP, a; = Pg; AB = Taxe) (fig. 10). Ainsi z varie dans les couleurs, ou le long de Taxe, et differe pour tous les rayons de Textrime rouge a Textreme violet. No\}s avons done une equation exponentielle, mais ^'^ pen eompliquee. * Nous avons fait observer que Thypothfese newtonienne n'ex- pHque aucunement la distance variaiite des bandes selon la refrangibilite des rayons. Aussi faut-il convenir qu'elle ne peut du tout expliquer les coulem-s prismatiques des bandes formees par la lumiere blanche. Supposons que Tunique difference des rayons fut que Taction du corps flechissant N 2 180 INQtrlBIES ANALYTICAL AND s'etendtt plus loin sur les rouges et moins sur les autres succes- sivement; le resultat serait que les rayons rouges seraient. disposes sur un espace, comma bande, E. V, et plus large de o E que les espaces qu'occupeiit les autres ; les oranges seraient disposes sur I'espace V o ; les jaunes, sur I'espace Y j ; et ainsi des autres, ^'E\ n " ^ de manifere que la seule partie qui serait d'une couleur simple et unique, c'est o E ; tandis que toutes les autres seraient teintes d'un melange de couleurs, jo, rouge et orange; N'j, rouge, orange et jaune ; h V, rouge, orange, jaune et vert ; il, ces couleurs avee le bleu ; vi, ces couleurs avee I'indigo ; et Y v, toutes les couleurs ou blanc. Eien ne peut etre plus different de I'apparence des bandes ; les teints saillants sorit rouge, vert et bleu. Or, selon la theorie, le vert serait mMe avee le rouge, I'orange et le jaune, et le bleu avee toutes ces couleurs ; et, finalement, I'espace qui devait gtre violet serait blanc. La diff^rente etendue de Taction n'explique done pas du tout les couleurs des bandes. Eien ne les esplique, que la diiFerente flexion des differents rayons, de manifere k faire occuper aux couleurs des places difieremment eloignees de I'orobre. Mais cette dtfierente flexion donne I'explicatiou trea-facilement. II n'y a que la difi'^reute largeur des bandes de couleurs differentes qui donne le moindre embarras, et cela n'est pas considerable. II s'ensuivrait de cette difference que la partie rouge du spectre de flexion (c'est-a-dire de la bande formee par la lumiere blanche) devrait etre plus large que les autres parties, et la violette la. plus mince de toutes. Mais la rouge et I'orange se confondent, et font une partie mat^rielle de la bande ; le violet, I'indigo et le bleu de m^me paraissent bleus ; le vert et le jaune passent pour verts; et ainsi les couleurs paraissent plut6t rouge, vert et bleu, qu'en plus grand nombre. Nous avons parle, mais pen, des bandes internes. Evidem- ment le^ rayons qui les ferment viennent des cot^s opposes du corps flecbissant, et se croisent ou au moius se rencontrent a un point plus ou moins distant du corps, selon que ce corps est plus oU moins mince. Qu'ils se croisent ou se toucbent, parait EXPERIMENTAL ON LIGHT. 1§1 d'abord par I'exp^rience fondamentale d'interftrence, 1' obstruc- tion des rayons d'un cote et I'effacement des bandes du c6te oppose. Mais cela parait aussi, si les bandes aont examinees prfes du corps, en les recevant sur un verre d^poli, I'oeil place derriere. II n'y a point de bandes si le verre n'est pas a une eertaine distance ; plus pr^s, il n'y a que I'ombre parfaitement noire. La determination de ce point, ou commencent a paraitre les bandes internes, nous donne le moyen de calculer Tangle d'inflexion, le demi-diametre de I'aiguille ou un autre corps mince etant le sinus de cot angle, et la distance du corps son cosinua. Je n'ai jamais pu iixer cet angle a moiiis de 20' ; ce qui demontre combien la force d'inflexion differe de celle de deflexion, Tangle de deflexion etant, com me nous Tavons vu plus haut, six fois moins grand. Des observations m'ont fait conclure que Tangle d'inflexion des rayons rouges est de 27' a 30', et il est probable que cet angle pour les violets est de 18' k 20' au moins, si la proportion de 3' k 2' se conserve pour les bandes internes comme pour les extemes. Pour pouvoir decider la question, savoir, si les ph^nomenes de flexion (diffraction) peuvent ^tre ramenes au principe de I'interference, il faut considerer que, selon ce principe, Teffet produit est en raison inverse de la difference entre les longueurs des rayons interferents ; je ne dis pas dans la proportion simple inverse, mais dans une proportion inverse quelconque. Soient 3^ B^ r Eg.ia. c^ B K / ^V \ \ S 1 J t c £ B, D (fig. 12), les deux bords ; B E, D E, les rayons interferents a E ; mettez AB = a, CD = 6, AC=c; soit C E = a; ; j/ = Teffet de Tinterference. Nous avons cet effet-li en raison inverse de D E — BE, c'est-a-dire 1 ^ " ( V 6' + a^ - ^/"(M-~^+^." 182 INQUIEIES AifALTTICAL AND Or, cette courbe doit avoir une asymptote, quelle que soft la valeur des constantes de I'equation, et quelle que soit la valeur de m ; c'est-a-dire dans toutes les positions des hordes, et quel que soit I'ordre de la courbe. C'est-i-dire que, quelle que soit la loi d'interference, pourvu que rinterference agisse en raison inverse quelconque k la difference des longueurs des rayons, on pent toujours trouver un point S, auquel D E = Be ou V 6^ + a;^ = V (c + a;)" + «^. Ce point-li se trouve oil 6» _ c» - a« X = 2c Done la valeur de y augmente entre A et ce point S, ou elle devient infinie. Done les bandes doivent augtnenter en largeur et en eloignement Tune de I'autre, des le point A vers le point S. Mais au contraire elles dimiiment en largeur et en distance. La plus large est la plus prfes de A ; les autres diminuent constamment jusqu'a ce qu'elles disparaissent ; et la oii il y a des intervalles entre elles, comme dans la lumifere homogfene, ces intervalles sont plus grands entre les bandes le plus prfes de A, et vont en diminuant vers S, toute comme les bandes elles-memes. Plus le bord B est pres de D dans le sens A ou J)d, et plus eloigne sera le point S. Cela paratt non-seulement par la valeur de x ci-dessus, mais aussi par la raison geom^trique de la solution du probleme de trouver le point ou deux lignes inflechies de deux points sur une troisieme ligne sont 6gales. Ainsi, plus les bords sont pres I'unde I'autre dans le sens d'D,et plus les bandes devraient 8tre minces et rapprochees I'une de I'autre ; ce qui est diam^- tralement contraire aux ph^nomfenes. Jusqu'ici nous n'avons regard^ que le cours de la courbe de A a S, en le comparant avec les ph^nomenes de ce c6t^- la de A B. Maintenant considerons la courbe du c6t6 oppos^ de A vers F. Elle approche de I'axe durant une portion de son cours, et ne commence a s'en Eloigner qu'a M, la oil il y a un point de rebroussement. Done, entre A et F (I'abscisse pour le point de rebroussement), les ordonnees diminuent, et ne commencent k augmenter que passe F. Pour trouver F, EXI-BBIMBNTAL ON LIGHT. 1^3 il faut trouver y en termes de x dans I'equation — ^ = 0. ^ dor Mais reparation devient embarrassante, meme accablante, le denominateur de la premiere differentielle de y ayant 16 facteurs multiplies par la racine carr^e d'une fonction de 4 facteurs, et puis une quantite de 30 facteurs a aouatraire ; et le numerateur est mSine plus complique, et puis le tout doit etre diff^rencie pour avoir d'^y. Mais on pent trouver la valeur approximative de — a; ; et si Ton prend les propor- tions de A C, A B et D C, les distances auxquelles rexp6rieace se fait commodement, a = 80; h = 90, et c = 1°"; le point S est a 849jdeA; etPC =9.9, ou AP = 8.9; ear a jt = - 8, 10.039' '" 10.049' 1 . . 1 y = TFTKi::- ^ ^ = -9-5. y = a; = _ 9.9, y = — — -; a; = - 10, y = 10.05' ■ '" 10.04' '" 10,049' 1 -'2'^ = 10:044'"*^^ = -'^'2' = 10.026- Done il est clair que si les ph^nomenes ^taient causes par I'interference, les bandes devraient diminuer tant en largeiir qu'en distance I'une de I'autre, de A jusqu'a F ; car y va en diminuant entre ces deux points. Mais au contraire les bandes augmentent en largeur et en distance, non-seulement passe P, mais suT toute la route de A a P. II faut faire remarquer que ces phenomenes sont tous ob- serves, et en effet ne peuvent etre observes qu'assez pres des lignes A B, CD. Car d'aprfes les proportions ci-dessus, et qui sont celles des experiences qu'on a reellement faites, les bandes d'un cote (celles de deflexion apres rinflexion) ne sont visibles que dans un espace de 3 a 4 mill. ; et les bandes augmentent de I'autre c6te, celles d'inflexion apres deflexion, dans un espace plus considerable, mais seulement de 6 a 7 miU. Mais les premieres, qui devraient augmenter jusqu'a 8, vont constamment en diminuant jusqu'a ce qu'elles cessent d'etre visibles, tandis que les secondes, qui devraient 184 INQUIBIES ANALyilCAL AND diminuer jusqu'a un certain point F, vont en augmentant toujours, et ne diminuent jamais. II faut aussi faire attention a ce que Ton a pris la courbe dans la supposition que m = 1, ou que Taction d'interference est en raisou inverse simple de la diiKrence des longueurs ; mais le raisonnement est le meme, quelle que soit la valeur qu'on donne a m. Soit Faction en raison inverse des carr^s ou ro = 2, ou des racines carrees, ou m = -J, on trouvera que la courbe est de la merae forme en ce qui regarde cette portion dont il est question. Les distances des points S, F sont les memes. Les courbes sont d'ordres difFerents, et leura autrea branches varient de beaucoup de colles de la courbe que Ton vient d' examiner. Mais en ce qui regarde la branche dont il s'agit, il n'y a pas de difference.* Si Ton regarde les bandes internes ou de I'ombre, le prin- cipe d'interference est difficile a appliquer, mais I'application n'est pas impossible. Soit a le diamfetre de I'aiguille ; 6, \ distance du tableau ou les bandes sont refues ; x, la dis- tance de I'extr^mite du diametre a, vers son centre; I'^qua- tion est y = t r^_;: . ; et ici, eomme ^ {^{a-xf+W-^V + x'Y dans I'autre caa, nous avons une asymptote, savoir, quand X = — , et les ordonnees augmentent de a; = jusqu'a x = — ; et les phenomenes s'accordent avec la theorie a un certain point, car les bandes augmentent tres-faiblement * Si TO = 1, la courbe est du huitifeme ordre. Si TO = — 1, elle est du quatrifeme ordre. Si m = 5, elle est du sixieme ordre. Si TO = ^, elle est du douzifeme ordre. Mais la forme ne varie pas beaucoup. II va sans dire que lorsque' TO = - 1, il n'y a pas d'asymptote. Si la proportion est, non pas de la difference des rayons, mais de leur carre, hypotbfese presque impossible, la courbe est une hyperbole conique, y = — ; et un porisme V' — a^ — (^ — 2cx assez curieux se rapports & cette propriete de la courbe. •-XPERIMIlNTAIi ON LIGHT. 185 jusqu'a I'axe de I'ombre, et si peu que plusieurs observateurs ont affirm^ qu'elles sont toutes de la m@me largeur. Au centre de I'axe, pourtant, il y a un espace gris, manifeatement plus large que les bandes ; et il y a deux intervalles d'un noir fence entre cet espace gris et les bandes colorees. Ces deux in- tervalles noirs sont aussi plus larges que les bandes, et que les autres intervalles noirs. Mais la theorie indique une aug- mentation de largeur beaucoup plus considerable. Prenez m = I ; a = Jg-° 6 = 2|, 6, 7^, 15, 75, 100 successive- ment, nous aureus la proportion de la valeur de y lorsque sc — 0, et lorsque x = 2\ du demi-diamfetre (ou de -Jj-)) c'est-a-dire trfes-pres du centre, comrae 1 : 12. Ainsi, les bandes pr^s du centre doivent etre 12 fois plus larges qu'a Textremit^ de I'ombre. Mais, meme en comptant les bandes noires et grises centrales, elles ne sont jamais pr^s du double. Si m = 2, ou plus, la difference est beaucoup plus grande. Meme en prenant m = ^ ou | (racine carree ou cube), la disproportion est beaucoup trop grande. Si m = ^, elle est encore considerable, comme 47286 : 872C3. Done, sans etre impossible, il est difficile de ramener les bandes internes au principe d'interf^rence. Pour ce qui regarde les bandes ex- terieures, eela devient impossible. II y a lA, certainement op- position complete des phenomfenes a la tbeorie. La theorie ou hypothese de M. Fresnel, dont j'ai parle dans mon Memoire de Tann^e passee, est d'une grande importance, je veux dire la proposition que les phdnom^nes de flexion (dif- fraction selon lui) " dependent uniquement de la largeur de I'ouverture par laquelle la lumiere est introduite." {Mim. de VInsi., 1821, 1822, p. 372.) Si cela est vrai, toute I'in- fluence des corps flechissants disparalt, et tout est reduit a la question de la largeur de I'ouverture. La preuve la plus directe du contraire est aussi la moius facile h obtenir par les experiences ; mais on peut I'avoir. Cast la mesure de I'ouverture lorsque les bords sont places direetement vis-a-vis I'un de I'autre. La largeur des bandes n'est pas en raison inverse de la largeur de I'ouverture. — La seconde preuve est de placer les bords I'lin apres I'autre sur 186 INQUIEIBS ANAIYTIOAL AND une ligne rigoureusement horizontale, et parallele aux rayona pareourant horizontalement la chambre. Les bandes et leurs distances des rayons directs, qui ne sont que de -^ de mill, et meme moins, si les bords sont distants I'un de I'autre bori- zontalement de 10 cent., ont la largeur et I'^loignement de 2 mill, lorsque la distance borizontale des bords est d'un cent. ; et les bandes ont la largeur et I'^loignement de 10 cent., lorsque les bords ne sont que -^ mill, distants I'un de I'autre. Mais la distance verticale des bords I'un de I'autre reste la mSme, elle est d'un mill. : c'est-4, dire I'ouverture reste la meme, tandis que la distance borizontale variee a entifere- ment change la largeur et I'eloignement des bandes ; d^monr stration conclusive que la largeur de Touverture ne decide pas de celle des bandes et de leur eloignemeut des rayons directs'.- Pourtant cette experience exige le paralMlisme rigoureux de la ligne ou barre sur laqueUe les bords sont places. Ainsi je donne encore une preuve qui parait decisive sans que rexaefr parallflisme soit necessaire ; et par consequent cette troisi^me experience est facile a faire. Placez les bords dans un pinceau, n'importe de quelle in- clinaison ni a quel angle les bords le rencontrent ; ils feront des bandes plus ou moins larges, plus ou moins eloign^es des rayons directs, en proportion inverse de la distance des bords I'un derrifere I'autre dans le sens du pinceau. Soit cette dis- tance de 10 cent., et faites que le bord le plus prfes du trou qui admet la lumiere dans la cbambre se porte de plus en plus dans; le pinceau, jusqu'^ ce que le passage des rayons entre les deux bords soit ferm^, et qu'il n'en passe plus. Eemarquez bien les bandes avant qu'elles disparaissent, et vous verrez qu'a cette distance des bords ces bandes u'atteignent jamais qu'une largeur trfes-petite, m#me lorsqu'elles sont ^vanouissantes. Eapprochez les bords, et retirez du pinceau celui qui est le plus prfes du trou, jusqu'i ce que les rayons puissent passer entre les deux bords (ces bords sont maintenant a une distance I'un de I'autre d'un cent., selon le cours du pinceau) ; et vous verrez que quand m^me la distance verticale des bords n'est pas trfes-petite,' il y a des bandes considerables. Faites entrer le bord dans le EXPERIMENTAL . ON LIGHT. 187 pineeau jusqu'^ ce que lea bandes soient d'un mill, ou mill, et demi de largeur, et que la distance des borda selon le cours du pineeau ne soit que d'un cent, ou | de cent. ; puia placez le second bord a une distance de 10 cent, ou de 20 cent., faites- le entrer dans le pineeau, et vous trouverez que quand mSme le bord est place dans le pineeau de manifere a faire interceptor tous les rayons, au moment de I'eTanouisaement des bandes elles ne sont jamais de la largeur dont elles etaient lorsque les bords furent places I'un pres de I'autre, pas meme du dixieme de cette largeur. Done, a des distances meme pen considerables des bords I'un derriere I'autre, il n'y a pas de petitesse d'ouverture (ou distance verticale de ces bords) qui puisse former des bandes tant soit pen larges. — J'ai vu cenx qui penchaient du c6t6 de I'hypothfese de I'ouverture ^tre convaincus tout de suite de leur erreur, en voyant qu'a plusieurs assez petites distances des bords I'un derriere I'autre, on pent varier a I'infini leur diatance verticale, c'est-^-dire I'ouverture, sans qu'aueune diminution de cette ouverture puisse augmenter la largeur ni I'^loignement des bandes eonsiderablement. Qu'il me soit permis, avant de conclure, de faire observer que Newton, dans un passage remarquable de son troisieme livre, paralt mais assez obscurement, s'Stre doute d'une pro- priete des rayons telle que je I'ai decrite sous le nom de dis- position dana mon Memoire de 1849.* En parlant dea deux bords ou trancbants de couteau, il dit que le couteau le plus pres de ehaque rayon determine le cours que prendra ce rayon, et que I'autre augmente la flexion. Or I'autre, c'est le bord oppose ; et ceci me parait approcher de tres-prfes de ma theorie. SUPPLEMENT. Dans mon dernier M6moire,| en donnant les preuves de la differente flexibilite des rayons homogenes, je me suis * Un resume des experiences et des conclusions qu'on en a tiroes, a 4i6 lu plus tard k la Socie'te royale. (Voir ' Phil. Trans.' 1850, part II.) t II precede ce Supplement. 188 INQtriRIES ANALYTICAL AND expritne avee quelque hesitation sur le cours rectiligne, que j'attribuais aux bandes fornixes par la lumifere du spectre prismatique. Les experiences avee deux bords trfes-pres I'un de I'autre, places dans les rayons parallfelement k I'axe du spectre, ont parii donner des bandes rectilignes, ou a tres- peu prfes. La distance des bords dans ces experiences ^tait en general de -J de millimetre, et rarement de moins de -jV- J'^^i fait depuis des experiences avee un r&eau de lignes gravies sur verre, et distantes entre elles de ^'-j- et aussi de Vb de millimetre ; et j'ai 6te surpris de trouver qu'a la premiere de ces distances, mais bien plus a la secoude, les bandes ^taient courb^es, et d'une assez grande courbure. Cette forme de I'exptirience est assez commode, parce que, quoique deux de ces lignes gravies, seules, donnent des bandes peu distinctes, six ou plus rendent les bandes tres-brillantes, en faisant tomber sur le tableau (ou sur la ratine lorsqu'on regarde le ph^nomfene par voie de vision directe) les bandes forraees par plusieurs lignes, ou paires de lignes, du reseau. La fig. 1 donne les bandes formees sur le tableau, lorsque le reseau est place dans les rayons du spectre prismatique. mg.s. La fig. 2 donne les bandes vues par vision directe, le rdseau etant plac6 prfes de I'oeil et derrifere le prisme, et le prisme plac6 entre le reseau et un assez petit disque de lumiere solaire jetee sur un tableau blanc, pres du petit trou par lequel la lumifere entre dans la chambre obscure. Soit A B, fig. 3, I'axe du spectre, A = violet, B = rouge. Si la force flectrice (ou I'influence, quelle qu'elle puisse 6tre, par laquelle les bandes sont formdes) augmente en raison directe de la. distance de P a A, cette force agissante en lignes paralleles (et perpendiculairement aux bords fl^chissauts), EXPERIMENTAL ON LIGHT. 189 OS P M = y, A P = a;, nous arons I'equation y = — , ligiie droite, et cfMC est rectiligne. Evideinment done si cfMC est curviligne, la force PM (fig. 4) n'est pas en raison simple de A P, c'est-a-dire en raison simple inverse de la refrangibilite ; at I'equation de dMC est y = — , et celle de cC W C est y' = — r-. Mais nous avous m pris la refrangibilite comme une fonction du sinus de refraction soustrait de la constante A B. Si Ton prend la refrangibilite en raison du sinus de refraction, toute proportion inverse de la refrangibilite donne une courbe hyperbolique qui ne peut etre d' accord avec les phenomenes, excepte en supposant le centre de I'hjperbole assez 6loigne de I'origine du spectre (le rouge) ; et quoique dans ce cas la ligne serait a peu pres droite, la force ne serait pas en raison inverse de la refrangibilite. c'est-^-dire du sinus de Tangle de refraction, mais de la difference entre ce sinus et une autre ligne. Mais si Ton doit faire cette supposition, on pourrait egalement sup- poser la proportion directe de la difference en partant du violet, ce qui donnerait une ligne rigoureusement droite. Mes mesures de la distance des courbes a Taxe du spectre, c'est-a-dire des lignes P M, P M', B C, B C, ni'ont donne lieu a supposer que » = 3, et que la courbe est parabolique. Elles ne s'accordent pas avec une courbe hyperbolique, AB = 18, B C = 13, B C = 1 5. II faut pourtant faire observer que la courbe y = — ne parait pas d'accord avec la courbure des m s ( Q 2C I 7/1 I bandes. Car le rayon de courbure etant ^^—7 — ~ — —, ce rayon parait etre plus grand pour d'M'C que pour dMC, excepte 3 tres-pr^s de d et S. Car, egalant les quantites — ^ et 190 INQUIKIES ANALYTICAL. 3 (qx* + m")^ — — — pour trouver le point M ou M', ou P, ou les deux courbes ont la meme courbure, on trouve a? = 7^ a peu pres. Dans mon dernier Memoire (qui precede ce supplement), j'ai suggere que I'hypothfese d'une force flectrice variant avec a la difference de refrangibilite, y = ~T {x = distance du rayon z au point du bord flechissant, z — distance du point a I'extremit^ rouge du spectre), pourrait expliquer la difference de la largeur des bandes form^ea par les rayons de diff^rentes couleurs ; et qu'ainsi la diff^rente flexibility expliquerait les deux phenomfenes, le different ^loignement des bandes et leur diff^rente largeur. ' jj J Soit A', fig. 5, un point des bords, sa distance de I'extrfeme rouge = z. A P = a;, la distance du rayon au point A, x' = la distance d'un rayon plus floigne du point A. Si la flexion est en raison inverse de la distance x, x', la difference B C des sinus des angles de flexion APB, AP'C, donnera la largeur B c a de la bande. Mais nous supposons que la force y = ~~j^. Ainsi, plus z est grand, et plus B 0, difference des sinus des angles de flexion, est grand. Soit A P = 2, A P' = 3 ; et pr^s de a a Textr^me rouge du spectre z = 2, y = "7= et y' = ~7^, et V 2 V 3 B D = -^^ z^ X a. Mais plus loin du rouge, z = 3; et V 6 ?/ = —=z et y' = —^ et BC = -^-^^ ^ — a --^ =? — — < ^^-^^ := , et la bande violette est moins large que la bande rouge.* " THa tract is from ' Mem. de I'lnstitut ' for 1854. ( 191 ) IX. ON FORCES OF ATTRACTION TO SEVERAL CENTRES. FORCES INTEESELY AS THE DISTANCE. 1. It is to be lamented that Sir I. Newton did not treat the prohlem of forces directed to more fixed points than one, as to two such points, either in the same or different planes from the body acted on. This is the fundamental point in considering disturbing forces when the centres are not fixed, which makes the problem more complicated and difScult. It is, however, sufficiently so even where the centres are fixed. 2. That the subject must have attracted his attention there can be no doubt. He had gone so much into the more difficult inquiries respecting disturbing forces that he must have fully considered the somewhat simpler, what may be termed the fundamental, case of fixed centres. Indeed, a paper communicated to the Eoyal Society in 1769 {Phil. Tram. p. 74) contains a demonstration by W. Jones, an intimate friend of Newton, of a proposition on this subject, which Machin had immediately after Sir Isaac's death given to the translator of the Principia. Machin had observed on the want of some investigation of the motion of forces directed to two centres, as required to explain the motions of planet and satellite, which gravitate to diiferent centres, in a word the problem of the Three Bodies. The proposition of Machin and Jones goes but a very little way to supply the defect complained of. It is confined to the case of the line joining the two centres being in different planes from the line of 192 ON FOKCEB OF AITKACTION projection ; it is that the triangle formed by the radii vectores and the line joiniag the two centres or fixed points, describes equal solids in equal times round that line ; and the demon- stration is similar to that of the first proposition, of equal areas in equal times when a single force is directed to one centre. It seems reasonable to conclude, that Xewton had, upon full consideration, found the full investigation of the subject beyond the powers of the calculus as it then existed. It is at least certain that, though he might have mastered it, he never could have delivered his results synthetically as in the Prinoipia. 3. The solutions on disturbing forces generally consider one force as acting in the one direction, that of the radius vector, and another in a line perpendicular to > that radius vector. Thus Clairaut (^Mem. Acad. 1748, p. 435) gives these eqaaXiQus rcPv -\- 2drdv = Ildx' rdv^ — the resultant at that point TO SEVERAL CENTRES, 195 P bisects a a! and is P c, and produced, P M cutting the axis. From hence may he seen how complicated would be the analysis, how next to impossible the geometrical con- struction of the locus of P, by referring the lines P M to S S' as an axis. We know indeed that one of the forces —^ r or —J- acting towards S or S', the locus of P is an ellipse ; but it would not follow that if both forces acted the same curve would be the locus. That the force would be differ- ent is certain, because it would be as P c, and not as either P a or P a'. But it may be said that the curve also would be different. Let us, however, suppose the case of the curve, whatever . it be, cutting the axis S S' produced at I and 2', points equally distant from S and S', so that 81 = 8'!'; also that the angle and the initial velocity of projection from S and I' is the same, and further that the attraction as the mass is the same from S and S', or that the mass of the body in S and in S' is the same ; then it 2 196 ON FORCES OF ATTEACTION seems impossible to avoid the conclusion that an ellipse, • and the same ellipse, must be described; because one of the forces alone acting from S, as — , would give the ellipse passing through S and S' ; and the other force alone acting from S, as -vi would give the ellipse passing through the same points S and S' ; and the initial velocity * and angle of projection would prevent any difference in the length of the conjugate axis ; and in the middle point answering to the centre C, the equality of r and q and of P a and Pa' would make the diagonal Pc coincide with the conjugate axis. But a further combination of forces may be sup- posed in this case ; two forces acting towards the points /• (7, S and S' and in the proportion of r and q, or — and — . •^ mm How will this addition affect the locus of P ? It should seem, for a reason similar to that before given, that the curve would remain the same ; for the two new forces T q ^ ^ — and — , acting in r or g or P S and P S' respectively, their resultant must, if there were none other acting, pass through the middle point C, between S and S' ; and as we know that a force acting from that point, and in proportion to the distance from that point, causes the body to move in an ellipse whose centre is that point, and r -{- q being con- stant, the ellipse must have the same axis and coincide with the ellipse produced by the combination of the forces mm -^ and -r-. 7. This had appeared to be a necessary consequence of the conditions stated, but not as at all proving the ve- locity to be the same in the ellipse, when described by * The condition of Legeudre (mentioned in page 193), that F = V^ + o' is supposed to hold ; for otherwise the centrifugal force would not be sufficient to balance the centripetal. ''to seveeal centres. 197 one force -j- or -— •, or wten described by the combined action of both, or when described by the combined action of — and m ^;;;7, Or of —5-, — J-, ^i^' ^'^d -^ ; becausc in all those cases the — , or of -J, — , — , and ^ m r (f m m velocity will be different, and particularly the action of 771 V TTh Q — J- H with — =- H — — will occasion a different velocity in r m (f m each point from that occasioned ^7 -j + — j • Thus to take the velocity at one point answering to 0. If II a and IT a' be taken as -r- and — j-, the diagonal lie' is the force of — ^ and 7fh f CI — r combined, n C is the resultant of — and — combined (f mm, (supposing m = 1). Therefore the velocity in II will be as n c' + II C, when all the forces act, and only as IT c' when the two former act alone, and as 11 C when the two latter act alone.* But the curves appear to be the same in each case. 8. These consequences seeming to follow from a con- sideration of the conditions stated, but without a full and rigorous investigation, it was very satisfactory to find that Lagrange had arrived at the same conclusion in one case of his solution of the problem of two fixed centres {Meo. Anal, part ii. sect. 7, chap. 3). That solution is marked throught with the stamp of his great genius. Buler had, in the Berlin Memoirs for 1.760, treated the case of the inverse square of the distance and the centres and orbit being in the same plane. Lagrange's solution is general for * The difference in velocity ia easily obtained, in comparing the effect of one force and of the combined forces, from the equation iP = 2 (/ X half ^p ,T chord osculating circle, the chord being = , p = perpendicular to the tangent, and E = radius of curvature. 198 ON FORCES OF ATTRACTION the force iDeiiig as any function of the distance, and of x, y, z, being the co-ordinates. Pressed by the great difficulties of the problem, and the impossibility of a general solution, he first confines himself to the inverse square of the distance (p. 97), and a general integration being still impossible, even after obtaining a differential equation with the variables separated, he makes a supposition which enables him to obtain two particular integrals (p. 99), and this gives for the orbit an ellipse in the one case and an hyperbola in the other, with the foci in the two centres of force ; and it follows, he observes, from the investigation, that the same conic section which is described in virtue of a force to one focus, acting inversely as the square of the distance, or to the centre and acting in the direct ratio of the distance, may be still described in virtue of three such forces (" trois forces pareilles"*), tending to two foci and to the "centre." He adds: " Ce qui est trfes remarquable" (p. 101). It having appeared to many persons that a portion of the demonstration was not so rigorous as might be desired, M. Serret has very ably and satisfactorily supplied the defect {Mec. An. tom. ii. note iii. p. 329, ed. 1865), but he arrives at the same result. There is also given a very important generalization of Lagrange's solution, and of Legendre's theorem already men- tioned, by M. Ossian Bonnet {Ihid. note iv.). 9. The same reason already given proves that if, instead of two points not in the trajectory we take two in it, as 1 and r, and refer the forces to those two, and make the forces — and —J in lli! and I' n' respectively, and the angle of projection and initial force the same, the same circle will be described by the body ; and that if two other forces * It is plain that " pareilles " does not mean of the same kind as -j and V ; for he resolves the force to the centre into two acting to the foci, 1' TO SEVBEAL OENTEES. 199 also act on it, as 2 n' and 2' n' f or -^ and — ) the same \ m mj circle will be described by the joint action of the forces. This is even a more remarkable consequence than the other; because the forces acting to the centre would of course give a uniform motion, and those acting to the points in the circumference an accelerated motion, and the forces combined will give an accelerated motion. At the middle point n, the velocity will be, if only the forces »» , m , aJ m .„ ^ „ r - ff . —r and — r- act, as -r— ^ ; it the forces — and — also act, it r" (f. 2 & s, as the arcs a b, bp, wHch arcs 'beiiig all as the times, the areas are proportional to those times of describing them, and therefore S and s are the centres of the deflecting^ forces. Then, drawing the tangents AC, a c, and completing the parallelograms DC, dc, the diagonals of which coincide with the evanescent arcs AB, ab, we have the centripetal forces in A and a, as the versed sines AD, ad But because ABP and o 6 p are right angles (by the property of the circle), the triangles ADB, ABB, and a(?6, apb, are respectively similar vg.5. to one another. W herefore AD:AB::AB:AP and A D = -r^TT ; and in like manner ad= — , or, as the evanescent arcs AP ap . .-, ., , , ■, . T. AB« , , ab' coincide with the chords, A D = arc -r-^=T- and ad = arc — . AP ap Now these are the properties of any arcs described in equal times ; and the diameters are in the proportion of the radii ; therefore the centripetal forces are directly as the squares of the arcs, and inversely as the radii. It is difficult to imagine a proposition more fruitful in con- sequences than this ; and therefore it has been demonstrat^ijl with adequate fulness. In the first place, the arcs described being as the velocities, if P, / are the centripetal forces, and V, v the velocities, and E, r the radii, F : f : : Y^ : v" ; and also : : r : E ; or F : / : : LAW OP THE ITNIVBESB. ■ 231 =r- : -. Now as in the circle V and E, v and r are both E r constant quantities, the centripetal force is itself constant, which retains a body by deflecting it towards the centre of the circle. Secondly. The times in which the whole circles are described (called the periodic times) are as the total circumferences or peripheries ; T : t : : P : ^. But the peripheries are as the P V radii or : : E ; r. Therefore T : ^ : : E : r ; also \ : v.: f^:-, E r therefore inversely as the radii, or T : i : : ^^^ : -, and V^ : V V E^ j^ . V v' 1^ '■'■ i^'- — r- But the centripetal forces F : f : : r=-- : — ; sub- stitutiQg for the ratio of V* : u', its equal the ratio of f^^'- -i^, E r F :/:: = :— ; or the centripetal forces are directly as the distances and inversely as the squares of the periodic times ; the forces being as the distances if the times are equal ; and the times being equal if the forces are as the distances. — It also follows that if the periodic times are as the distances, then F : /■ : : -Ri : -z- ; that is, : : ^^ : -, or inversely as the distances. — In like manner if the periodic times are in pro- portion to any power n, of the distance, or T : i : : E" : r", we ^hall have T^ : ^^ : : E'° : )•«" and F : / : : ^ : -^ ; that is • ■ -p2„_i '■ ;;5;ri; ^^'^ conversely if the centripetal force is in XL / the inverse ratio of the (2 k — 1 )"" power of the distance, the periodic time is as the m"" power of that distance. — Likewise, E r as the velocities of the bodies in their orbits or V :■!)::-=■: -, -i- t 232 • OBNTEAL FOECES ; r^ ^ , ,r ^ r 1 if we make T : i : : E" : r-", then V : ■« : : ^5- : — , or : : :^— : ' E" r" E" ' 1 3 —3-. Thus, suppose n is equal to - we have for the velocities Y : V :: : — =, or they are in the inverse suhduplicate VE V proportion of the distances ; and for the centripetal forces we have F : / : : =p5Zi ■ sTi-- ^i '■ —'> or the attraction to the centre is inversely as the square of the distance. Now if n = |, T : f : : E^ : j-^, or T' : «« : : E' : r^ in other words the squares of the periodic times are as the cubes of the distances from the centre, which is the law discovered by EJepler from observation actually to prevail in the case of the planets. And as he also showed from observation that they- describe equal areas in equal times by their radii vectores drawn to the sun, it follows from the fundamental proposi- tion, first, that they are deflected from the tangents of their orbits by a power tending towards the sun ; and then it follows, secondli/, from the last deduction respecting it, (namely, the proportion of F :/ : : :™ : -5,) that this central force acts inversely as the squares of the distances, always supposing the bodies to move in circular orbits, to which our demonr stration has hitherto been confined.* The extension, however, of the same important proposition to the motion of bodies in other curves is easily made, that is to the motion of bodies in different parts of the same curve or in curves which are similar. For in evanescent portions of the same curve, the osculating circle, or circle which has the same curvature at any point, coincides with the curve at that point; and if a line is drawn to the extremity of that * This sesquiplicate ' proportion only holds true on the ' supposition of the bodies all moving without exerting any action on each other. - ' LAW OP THE TOIVEESE. 233 circle's diameter, A M B and a mh may be considered as triangles ; and as they are right angled at M and m, A M* is equal to A P x A B and am^ to ap x ah\ and where the Fig, 4. curvature is the same as in corresponding points of similar curves, those squares are proportional to the lines A P, or a ^ ; or those versed sines of the arcs A M and a m are pro- portional to the squares of the small arcs. Hence if the distances of two hodies from their respective centres of force be D, d, the deflecting force in any points A and a, being as the versed sines, those forces are as A M" : a w' ; and from hence follows generally in all curves, that which has been demonstrated respecting motion in circular orbits. The planets then and their satellites being known by Kep- ler's laws to move in ellipticE^l orbits, and to describe roxmd the sun in one focus areas proportional to the times by their radii vectores drawn to that focus, and it being further found by those laws that the squares of their periodic times are as the cubes of the mean distances from the focus, they are by these propositions of Sir Isaac Newton which we have been considering, shown to be deflected from the tangent of their orbit, and retained in their paths, by a force acting inversely as the squares of the distances from the centre of motion. But another important corollary is also derived from the same proposition. If the projectile or tangential force in the direction A T ceases (next figure), the body, instead of moving in any arc A N, is drawn by the same centripetal force in the straight line A S. Let A w be the part of A S, 234 CENTRAL FOECES ; through which the body falls by the force of gravity, in the same time that it would take to describe the arc A N. Let A M be the infinitely small arc described in an instant ; and A P its versed sine. It was before shown, in the corollaries to the first proposition, that the centripetal force in A is as A P, and the body would move by that force through A P, in the same time in which it describes the arc A M. Now the force of gravity being one which operates like the centripetal force at every instant, and uniformly accelerates the descend- ing body, the spaces fallen through will be as the squares of the times. Therefore, if A « is the space through which the body falls in the same time that it would describe A N, A P l\.S. is to A « as the square of the time taken to describe A M to the square of the time of describing A N, or as A M* : A N", the motion being uniform in the circular arc. But A M, the nascent arc, is equal to its chord, and A M B being a right angled triangle as well asAPM, AB:AM::AM:AP AM' and A P = . Substituting this in the former proportion, ^^' '.n::KW:KW, or A re : A N« : : ^-i AB r. we have . ^ AB A M^ that is : : 1 : A B. Therefore A N^ = A « x A B, or the are described, is a mean proportional between the, dia- meter of the orbit, and the space through which the body would fall by gravity alone, in the same time in which it describes the arc. LAW OP THE UNIVERSE. 235 Now let A M N B represent the orbit of the moon ; A N the arc described by her in a minute. Her whole periodic time is found tflf be 27 days 7 hours and 43 minutes, or 39,343 minutes; consequently AN : 2 ANB :: 1 : 89,343. But the mean distance of the moon from the earth is about 30 diameters of the earth, and the diameter of her orbit, 60 of those diameters ; and a great circle of the earth being about 131,630,572 feet, the circumference of the moon's orbit must be 60 times that length, or 7,897,834,320, which being divided by 39,343 (the number of minutes in her periodic time), gives for the arc A N described in one minute 200,743, of which the square is 40,297,752,049, or AN*, which (by the proposition last demonstrated) being divided by the diameter A B gives A «. But the diameter being to the orbit as 1 : 3.14J 59 nearly, it is equal to about 2,513,960,866. There- fore A n = 16.02958, or 16 feet, and about the third of an inch. But the force which deflects the moon from the tan- gent of her orbit, has been shown to act inversely as the square of the distance ; therefore she would move 60 x 60 times the same space in a minute at the surface of the earth. But if she moved through so much in a minute, she would in a second move through so much less in the proportion of the squares of those two times, as has been before shovm. Where- fore she would in a second move through a space equal to 16j'^ nearly (16.02958). But it is found by experiments frequently made, and among others by that of the pendulum,* that a body falls about this space in one second upon the surface of the earth. Therefore the force which deflects the moon from the tangent of her orbit, is of the same amount, and acts in the same direction, and follows the same propor- tions to the time that gravity does. But if the moon is drawn by any other force, she must also be drawn by gravity ; • It is foiaad that a pendulum, vibrating seconds, is about the length of 3 feet 83 inches in this latitude ; and the space through which a body falls in a second is to half this length as the square of the circumference of a circle to that of the diameter, or as 9.8695 : 1, and that is the proper- 236 CKNTBAL FOECES; and as that other force makes her move towards the earth 16 feet I inch, and gravity would make her move as mnch, her motion would therefore be 32 feet | inch in a second at the earth's surface, or as much in a minute in her orbit ; and her velocity in her orbit would therefore be double of what it is, or the lunar month would be less than 13 days and 16 hours. It is, therefore, impossible that she can be drawn by any other force, except her gravity, towards the earth.* Such is the important conclusion to which we are led from this proposition, that the centripetal forces are as the squares of the arcs described directly, and as the distances inversely. This conclusion was the discovery of the great law of the universe. The fruit of the consequences of this proposition is the ascertaining the laws of curvilinear motion generally. The versed sine of the half of any evanescent arc (or sagitta of the arc) of a curve in which a body revolves, was proved to be as the centripetal force, and as the square of the times ; or as P X T'- Therefore the force F is directly as the versed sine, and inversely as the square of the time. From this it follows that the central force may be measured in several ways. The arc being QC, we are to measure the central force in its middle point P. Then the areas being as the times ; twice the triangle S P Q, or Q L x S P is as T in the last expression ; and, therefore, Q E being parallel to L P, the O T? central force at P is as t--=- — =^-k5- So if S Y be the perpen- S P^ X L Q^ ^ ^ dicular upon the tangent P Y, because P E and the arc P Q, evanescent, coincide, twice the triangle S P Q is equal to S Y O E X Q P ; and the central force in P is as ^—^ — ^, ^„ . Lastly, SY^ X QP if the revolution be in a circle, or in a curve having at P the same curvature with a circle whose chord passes from that * The proposition may be demonstrated by means of tbe Prop. XXSYI. of Book I. of the Prinoipia, as well as by means of the proposition of which we have now been tracing the consequences (Prop. IV.). But in truth the latter theorem gives a construction of the former problem (Prop. XXXVI.), and from it may be deduced both that and Prop. XXXV. lAW OF THE TTNrVEESB. 237 point tlirotigli S to V, then the measure of the central force will be TT^^ — ^?r^r?. — ^By finding the value of those solids in any given curve, we , can determine the centripetal force in terms of the radius vector S P ; that is, we can find the pro- portion which the force must bear to the distance, in order to retain the body in the given orbit or trajectory; and con- versely, the force being given, we can determine the tra- jectory's form. This proposition, then, with its corollaries, is the founda- tion of all the doctrine of centripetal forces, whether direct or inverse ; that is, whether we regard the method of finding, from the given orbit, the force and its proportion to the .distance, or the method of finding the orbit from the given force. We must, therefore, state it more in detail, and in the analytical manner, Sir Isaac Newton having delivered it syn- thetically, geometrically, and with the utmost brevity. 238 OENTEAL FOEOES; It may be reduced to five kinds of fomnilse. 1. If the central force in two similar orbits be called P and /, the times T and t, the versed sines of half the arcs S and s, — o „ a then F :/::_:— ; and generally F is as ■^. 2. But draw S P to any given point of the orbit in the middle of an infinitely small arc Q 0. Let T P touch the curve in P, draw the perpendicular SY from the centre of forces S to P T produced, draw S Q infinitely near S P, and Q R parallel to S P, Q o and R o parallel to the co-ordinates S M, MP. Then P being the middle of the arc, twice the triangle S P Q is proportional to the time in which C Q is described. Therefore QPxSYorQLxPSis proportional 0- to the time ; and Q E is the versed sine of —p-, therefore F as A S O T? '' ' ' 7=;^ becomes F as t^— =-— ; and if S M = a;, M P = !/, and because the similar triangles Q E o and S M P give Q E = • — ^-rj — , and because A M being the first differential of S M, oQ is its second differential (negatively), therefore QR = — (P X X ij x' 4- y^ — (taken with reference ix) dt constant), and F is as f^^±^ But L Q» = QF - L P and X X It Q" X (x" + f) L P is the differential of S P or ^/^x^ + y\ Therefore L Q'' = y^id {xdy-ydxf " \ yj -d»a;Va^ + ; = — 5—-^, and F is as - x' + y^ x^-\ xy i'l But as the differential of the time (L Q x P S) may be made constant, Q R will represent the centripetal force ; and LAW OF THE UNIVEE8B. 239 that force itself will therefore be as —,* taken X with reference to dt constant. 3. The rectangle S Y x Q P heiug equal to Q L x S P and S Y = ^-— ==?, we have F as ^dx' + df ' S Y» X Q P= QR QE (^ydx — xdyY K^i)' QR QP' 4. Because P = -prr^ rr^=r, and -;:-=- is equal to the chord S Y* X Q P Q R P V of the circle, which has the same curvature with Q P in P, and whose centre is K (and because QP'' = QExPV by the nature of the circle and the equality of the evanescent QP'' Q R arc Q P with its sine, and thus P V = 7=rp-, — therefore -=ppj ), F is as ^-rrj — 5"^==:. In like manner if the velocity, - pv/ SY'^XPV which is inversely as S Y, be called v, F is as ^^^- Now the chord of the osculating circle is to twice the perpendicular S Y as the differential of S P to the differential of the per- pendicular ; and calling S P the radius vector r, and S Y = jj, T^-TT 2pdr 1 T-, . dp ,, _,. we have P V = —^ — , and 1 is as „ .^ , ; and also F is as dp 2p'dr v^ d J) -. In these formulas, substituting for p and r their 2 dr * Of these expressions, although I have sometimes found this, which was fiist given by Herrman, serviceable, I generally prefer the two, which are in truth one, given under the next heads. But the expression first given 3" is without integration an useful one. 240 CENTEAl FORCES ; Yaluesiii terms of x and y, we obtain a mean of estimating the force as proportioned to r, which is v ic* + y^. 5. The last article affords, perhaps, the most ohvious me- thods of arriving at central forces, both directly and inversely. Although the quantities become involved and embarassing in the above general expressions for all curves, yet in any given curve the substitutions can more easily be made. A chief recommendation of these expressions is, that they involve no second differentials, nor any but the first powers of any differentials. But it may be proper to add other formulas which have been given, and one of which, at least, is more convenient than any of the rest. One expression for the centrifugal force (and one some- . d ^ times erroneously given for the centripetal)* is jp^, s being the length of the curve and E the radius of curvature. This gives the ready means of working if that radius is known. But its general expression involves second differentials, the ds' usual formula for it being „ /d j/\; consequently we \dxj d v must first find -r^ = X (a function of x^, and then there are dx ^ ' only first differentials. di' Another for this radius of curvature is 'JidJ'yJ^i^xf and this is used by Laplace ; and another is , which, with dp other valuable formulas, is to be obtained from Maclaurin's Fluxions. But the formula generally ascribed to John Ber- * This error appears to have arisen from taking the case where the radius of curvature and radius vector coincide, that is, the case of the circle, in which the centrifugal and centripetal forces are the same. — See Mrs. Somerville's truly admirable work on the Meo. Cel., where the error manifestly arises from this circumstance. LAW OF THE tTNIVEBSB. 241 nouUi (Mem. Acad, des Sciences, 1710), is, perhaps, the T most elegant of any, F = - — — ■ ; and this results from 2 r dr substituting 2 E for its value —z — , in the equation to F, deduced ahove from Newton's formula, namely, P — ■ „ , . 2p dr But the proposition is so important, that it may be well to prove it, and to show that it is almost in terms involved in the third corollary to Prop. VI. Book I. of the Principia. — By that corollary F = ^ (C being the osculating circle's chord which passes through the centre of forces). But draw- ing S Y, the perpendicular to the tangent, and P C F through the centre of the circle, which is, therefore, parallel to Y S, and joining V F, we have V P : P F :: S Y : SP or C : 2E ::p : r and C = '—, which substituted for C in the above equa- T tion, gives F = 2 P» . E' In all these cases p is to be found first, and the expres- sion for it (because, pp. 286, 287, TP : PM :: TS : S Y an TS = yl^^jdl, and PT = f >Jdf + dx-) is ^^ = SY dy dy / r R 242 CBNTEAl FORCES ; y ydx — xdy Jdy^ + d^ /s/'dy^^ dec" Also SP = Vx^ + f' Then the radius of curvature E = ,^, J L- (^ being ■— daf X dji. ^ dx in terms of w, and having no differential in it when the sub- stitution for dy is made). Therefore, the expression for the . ■ , ■, n ■, J a? + y^ X dx' X c?X . , . , centripetal force becomes — — -z — — , in which, when y and dy are put in terms of x, as both numerator and denominator will be multiplied by da?, there will be no differential, and the force may be found in terms of the radical — that is, of r, though often complicated with x also. It is generally advisable, having the equation of the curve, to find p, r, and E, first by some of the above formulas, and then substitute those values, or dp and dr, in either of the expressions for F, — ^ or — ^. 2p^dr 2p^B, To take an example in the parabola, where S being the focus, and S = a, y" = 4: a X, and T M = 2 a?, and ;> = Y S r dr V- V (a + x) a; r = SF = a + x, and E = -^ = 2(a + x) _ dp a -\- X ; we have therefore P as ■ E a + X (a(c( + j;))|- = 2 (a + x) / a + x _ a + x V a ^2a(a + xy~ LAW or THE UNIVEBSE. 243 ^ — 7 r^ = ^ r\c< — zttt;' o^j bscause S (the parameter) is 2 a (a + a;)' 2 . O S . S P" ^ -^ ^ constant, inversely as the square of the distance: And the other formula F as „ , sives the same result for the law if dr of force, or jg^,-* Again, in the ellipse, if a he half the transverse axis, and 6 half the conjugate, and r the radius vector, we have p = b -, and dp = —^ -; therefore the formula ^/ circle a = the radius = r = p; hence -^-^ hecomes — , which. 2 a - r V ^ (2 a - »-)|- dp , a b dr a ,-, n ■ ■ -r-rr- Decomes r = 7-—;, or the torce is m- p\d7' V^/rXrixdr * »" versely as the square of the distance. Lastly, as the equations are the same for the hyperbola, with only the difference of the signs, the value of the force is also inversely as r^, or the square of the distance. In the r ^ a 3-^ hecomes — , p' E a being constant, the force is everywhere the same. But if the centre of forces is not that of the circle, but a point in the circumference, the force is as — . r Eespecting centrifugal forces it may be enough to add, that if V is the velocity and r the radius, the centrifugal force ■»' /. in a circle, is as — . Also if E be the radius of curvature, r f for any curve is = -^. When a body moves in a circle by a centripetal force directed to the centre, the centrifugal force * This result coiBcidea witli the synthetical solution of Sir Isaac Newton in Prop. XIII. E 2 244 CBNTEAL POKOES ; is equal and opposite to the centripetal. Also the velocity in uniform motion, like that in a circle, heing as—, the space divided by the time, and the arc being as the radius r, f is as ; or as — ;. If two bodies moving in different circles have r . r f the same centrifugal force, then the times are as a/t. — It is to the justly celebrated Huygens that we owe the first investigation of centrifugal forces. The above propositions, except the second, are abridged from his treatise.* i. First, where the centre of force is the centre of the tra- jectory. In exemplifying the use of the formulas we have shown the proportion of the force to the distance in the conic sections generally, their foci being the centres of forces. Let us now see more in detail what the proportion is for the circle. Let S be the centre of forces and K of the circle, P T a tangent, SY a perpendicular to it, KM and MP co-ordi- nates, S K = &, K = a, P M = ?/, and M.K = x. Then,' by ST X KP similar triangles, T K P and T S Y, we have S Y = — — - — , or (because the sub-tangent M T = -^, and a' = a;' -f- y') a' + bx /2a^ +2bx or ( J ) ; also SY = ^ a' + 2b x + b\ and * Horologium Oscillatoriiim, ed. 1673, p. 159, App. LAW OF THE UNIVERSE. 245 because by the property of the circle S x S B or (a + 6) (a - 6) = a« - 6» = PS X S V; therefore Now by the formula already stated as Bernoulli's, but really Sir Isaac Newton's, the centripetal force in P is as S P „ „ — —, E being the radius of curvature, and in the circle b 1 X -tfc that is constant being = a, the semidiameter ; therefore the „ . A/a' + 2ba: + b'' 8 x a' J a' + 2b x + b^ lorce IS as — --— - — zn—^r, or as --?!— — i-— — ^ — ■ — , a (2 ct' + 2 6 a;)" ' (2a'' + 26a;/ ^, ^ . BO^x SP BO'x SP' *^"* ^^ i2^-2bxr' °^ ^' (2a- + 26.)-xSP" °' "^ BO' ^ 2a«+ 2 5a; 2.a^+26a; — But - (2a^+26a;y xPS^ SP ^ a' + 2 b x + b' SP^ BO' = P V. Therefore the central force is as ^p^ ^^ ; or (because B' is constant) the central force is inversely as the square of the distance and the cube of the chord jointly. Of consequence, where S is in the centre of the circle and b = 0, the force is constant, S P becoming the radius and P V the diameter ; and if S is in the circumference of the circle as at B, or o = b, then the chord and radius vector coinciding, the force is inversely as the fifth power of the distance, and is also inversely as the fifth power of the cosine of the angle P S 0. By a similar process it is shown that in an ellipse the force directed to the centre is as the distance. Indeed, a property of the ellipse renders this proof very easy. For if S Y is the perpendicular to the tangent TP, and NP (the normal) 246 CENTEAL FOEOES ; parallel to S Y, and S A the semi-conjugate axis ; S A is a mean proportional between S Y and P N, and therefore S Y = AS^ pTj^ ; also the radius of curvature of the ellipse is (like that of all conic sections) equal to — p^, P being the parameter. Therefore we have to substitute these values for S Y and the radius of curvature, K, in the expression for the central force, , and we have :r-i7Trr TTSe = l-^-ai X SP; so RxSY-' 4.PN A S° 4 AS' — p5— X p^ P^ that, neglecting the constant , the centripetal force is as the distance directly. From hence it follows, conversely, that if the centripetal force is as the distance, the orbit is elliptical or circular, for by reversing the steps of the last demonstration we arrive at an equation to the ellipse ; or, in case of the two axes being equal, to the circle. It also follows that if bodies revolve in circular or elliptical orbits round the same centre, the centre of the figures being the centre of forces, and the force being as the distance, the periodic time of all the bodies will be the same, and the spaces through which they move, however dif- fering in length from each other, will all be described in the same time. This proposition, which sometimes has appeared paradoxical to those who did not sufSoiently reflect on the subject, is quite evident from considering that the force and velocity being increased in proportion to the distance, and LAW OF THE UNIVEESE. 247 the lengths of similar curvilinear and concentric figures being in some proportion, and that always the same, to the radii, the lengths are to each other as those radii, and consequently the velocity of the whole movement is increased in the same proportion with the space moved through. Hence the times taken for performing the whole motion must be the same. Thus, if V and v are the velocities, R and r the radii, S and s the lines described in the times T and t, by two such bodies round a common centre, V : u : : R : »-, and S : s : : E : r ; and because V = •=- and v = — > tjt : — : : E : r, and S : s : : T E : tr; orE:r::TE:t?"; and therefore T = t. Hence if gravity were the same towards the sun that it is between the surface and centre of each planet, or if the sun were moved but a very little to one side, so as to be in the centre of the ellipse, the whole planets would revolve round him in the same time, and Saturn and Uranus would, like Mercury, com- plete their vast courses in about three of our lunar months instead of 30 and 80 years— a velocity in the case of Uranus equal to 75,000 miles in a second, or nearly one-third that of light. It also follows from this proposition that, if such a law of attraction prevailed, all bodies descending in a straight line to the centre would reach it in the same time from what- ever distance they fell, because the elliptic orbit being inde- finitely stretched out in length and narrowed till it became a straight line, bodies would move or -fribrate in equal times ' through that line. This is the law of gravity at all points within the earth's surface. Another consequence of this proposition is, that if the centre of the ellipse be supposed to be removed to an infinite distance, and the figure to become a parabola, the centripetal force being directed to a point infinitely remote, becomes constant and equable; a proposition discovered first by Galileo. 248 CENTRAL FOBCEB : Next, where the centre of force is in the focus. If P A be a conic section whose parameter is D, S Y the perpen-" dicular to the tangent T P, P E the radius of curvature at P ; then S Y : S P : : i D : P N (the normal), and S Y = ^'^^ ; 4PN= also P E D' Substitute these values of S Y and P E (p and E) in the expression formerly given for the central SP force -, and we have D» . S P^ 4PN^ or ^ ^— , 8 P N= D2 which is (D being invariable) as the inverse square of the distance. Therefore any body moviag in any of the conic sections by a force directed to the focus, is attracted by a centripetal force inversely as the square of the distance from that focus. This demonstration, therefore, is quite general in its application to all the conic sections. It follows that if a body is impelled in a straight line with any velocity whatever, from an instantaneous force, and is at the same time constantly acted upon by a centripetal force which is inversely as the square of the distance from the centre, the path which the body describes will be one or other of the conic sections. For if we take the expression D . S P and work backwards, multiplying the numerator and deno- minator both by S P, and then multiplying the denominator LAW OP THE UNIVBESE. 249 8 DV P N= by ^-y-j — p-^, ■we obtain the expressions for the value of S Y, the perpendicular, and for E, the radius of curvature. But no curves can have the same value of S Y and R, except the conic sections ; because there are no other curves of the second order, and those values give quadratic equations between the co-ordinates. By pui'suing another course of the same kind algebraically, we obtain an equation to the conic sections generally, accord- ing as certain constants in it bear one or other proportion to one another, The perpendicular S Y and the radius of cur- vature are given in terms of the normal; and either one 3 or the other will give the equation. Thus E = -^^ ., dx^xd(^£ 4 P N^ 4 „3 J = jy, = -pa J 3 X {da^ + d fy which gives B' d x^ = 4y'x((Pydx — dPxdi/) an equation to the co-ordinates. Now whether this be resolvable or not, it proves that only one description of curves, of one order, can be such as to have the property in question. The former operation of going back from the expression of the central force, proves that the conic sections answer this condition. Therefore no other curves can be the trajectories of bodies moviag by a centri- petal force inversely as the square of the distance.* This truth, therefore, of the necessary connexion between motion in a conic section and a centripetal force inversely as the square of the distance from the focus, is fully established by rigorous demonstration of various kinds. * The equation may be resolved and integrated ; there results, in the first instance, the equation dx = '— and tlierefore the- integral v'2o2/2 - D2 is this quadratic, c^x^ --icy^ - 2cCx + 0^ + J)^ = 0. 250 CBNTEAL FOECES; If we now compare the motion of different bodies in con- centric orbits of the same conic sections, we shall find that the areas which, in a given time, their radii vectores describe round the same focus, are to one another in the subdiiplicate ratio of the parameters of those curves. From this it follows, that in the ellipse whose conjugate axis is a mean propor- tional between its transverse axis and parameter, the whole time taken to revolve (or the periodic time :■ being in the pro- portion of the area (that is in the proportion of the rectangle of the axes) directly, and in the subduplicate ratio of the parameter inversely, is in the sesquiplicate ratio of the trans- verse axis, and equal to the periodic time in a circle whose diameter is that axis. It is also easy to show from the formula already given respecting the perpendicular to the tangent, that the velocities of bodies moving in similar conic sections round the same focus, are in the compound ratio of the perpendiculars inversely and the square roots of the parameters * directly. Hence in the parabola a very simple expression obtains for the velocity. For the square of the perpendicular being as the distance from the focus by the nature of the curve (the former being (f -f- a x, and the latter a + x), the velocity is inversely as the square root of that distance. In the ellipse and hyperbola where the square of the perpendicular varies differently in proportion to the dis- tance, the law of the velocity varies differently also. The square of the perpendicular in the ellipse (A being the trans- verse axis and B the conjugate, and r the radius vector) is B^ X »• B^ X »" -T ; in the hyperbola, , or those squares of the A — r A-j-r T T perpendicular vary as and — , in those curves re- spectively, B^ being constant. Hence the velocities of bodies moving in the former curve vary in a greater ratio than that * By parameter is always to be understood, unless otherwise mentioned, the principal parameter, or the parameter to the principal diameter. LAW OF THE rNIVBRSE. 251 of the inverse subduplicate of the distance, or — =, and in a ij r smaller ratio in the latter curve, while in the parabola -t= V r is their exact measure. To these useful propositions, Demoivre added a theorem of great beauty and simplicity respecting motion in the ellipse. The velocity in any point P is to the velocity in T, the point where the conjugate axis cuts the curve, as the square root of the line joining the former point P and the more distant focus, is to the square root of the line joining P and the nearer focus. It follows from these propositions that in the ellipse, the conjugate axis being a mean proportional between the transverse and the parameter, and the periodic time being as the area, that is as the rectangle of the axes directly, and the square root of the parameter inversely, t being that time, a and 6 the axes, and p the parameter, t — — :=, and _ Vp 6* = ap; therefore ab = a a/ ap = // a? X /J p; and t = v' a^ and f — c? ; or the squares of the periodic times are as the cubes of the mean distances. So that all Kepler's three laws have now been demonstrated, cJ priori, as mathematical truths ; first, the areas proportional to the times if the force is centripetal — second, the elliptical orbit, — and third, the ses- quiplicate ratio of the times and distances, if the force is inversely as the squares of the distances, or in other words if the force is gravity. Again, if we have the velocity in a given point, the law of the centripetal force, the absolute quantity of that force in the point, and the direction of the projectile or centrifugal force, we can find the orbit. The velocity in the conic sec- tion being to that in a circle at the given distance D as m to n, and the perpendicular to the tangent being p, the lesser axis will be , and the greater axis ■-—: -, the 252 CENTRAL FORCES ; signs being reversed in the denominator of each quantity for the case of the hj'perbola. Hence the very important con- clusion that the length of the greater axis does not depend at all upon the direction of the tangential or projectile force, but only upon its quantity, the direction influencing the length of the lesser axis alone. Lastly, it may be observed, that as these latter propositions give a measure of the velocity in terms of the radius vector and perpendicular to the tangent for each of the conic sec^ tions, we are enabled by knowing that velocity in any given case where the centripetal force is inversely as the square of the distance, and the absolute amount of that force is given, as well as the direction of the projectile force and the point of the projection, to determine the parameters and foci of the curve, and also which of the conic sections is the one de- scribed with that force. For it will be a parabola, an hy- perbola, or an ellipse, according as the expression obtained forj)^ (the square of the perpendicular to the tangent) is as the radius vector, or in a greater proportion, or in a less pro- portion. This is the problem above referred to, which John Bernoulli had entirely overlooked, when he charged Sir Isaac Newton with having left unproved the important theorem respecting motion in a conic section, which is clearly, involved in its solution. Before leaving this proposition, it is right to observe that the two last of its corollaries give one of those sagacious anticipations of future discovery which it is in vain to look for anywhere but in the writings of Newton. He says, that, by pursuing the methods indicated in the investigation, we may determine the variations impressed upon curvilinear motion by the action of disturbing, or, what he terms, foreign? forces ; for the changes introduced by these in some places, he says, may be found, and those in the intermediate places supplied, by the analogy of the series. This was reserved for Lagrange and Laplace, whose immortal labours have reduced the theory of disturbed motion to almost as great LAW OF THE TTNIVEESE. 253 certainty as that of untroubled motion round a point by virtue of forces directed thither.* We have thus seen how important in determining all the questions, both direct and inverse, relating to the centripetal force, are the perpendicular to the tangent and the radius of curvature. Indeed it must evidently be so, when we con- sider, first, that the curvature of any orbit depends upon the action of the central force, and that the circle coinciding with the curve at each point, beside being of well-known proper- ties, is the curve in which at all its points the central force must be the same ; and, secondly, that the perpendicular to the tangent forms one side of a triangle similar to the triangle of which the differential of the radius vector is a side ; the other side of the former triangle being the radius vector, the proportion of which to the force itself is the material point in all such inquiries. The difSculty of solving all these problems arises from the difficulty of obtaining simple ex- pressions for those two lines, the perpendicular p and the radius of curvature E. The radius vector r being always- V a;' -f- 2/* interposes little embarrassment ; but the other two lines can seldom be concisely and simply expressed. In some cases the value of F, the force, by i^r and dp may be more convenient than in others ; because p may involve the investigation in less difficulty than E ; besides that p^ enters into the expression which has no differentials. But in the greater number of instances, especially where the curve is given, the formula —^ wUl be found most easily dealt with. p -tt ii. We are next to consider the motion of bodies in conic sections which are given, and ascending or descending in straight lines under the influence of gravity; that is, the velocities and the times of their reaching given points, or * Laplace (Me'o. Cel. lib. xv. ch. i.) refers to this remarkable passage as the germ of Lagrange's investigationa m the Berlin Memoires for 1786. 254 CBNTEAL forces; their places at given times. This branch of the subject, therefore, divides itself into two parts ; the one relating to motion in the conic sections, the other to the motion of bodies ascending or descending nnder the influence of gravitation. In order to find the place of a revolving body in its trajectory at any given time, we have to find a point such that the area cut off by the radius vector to that point shall be of a given amount ; for that area is proportional to the time. Thus, suppose the body moves in a parabola, and that its radius vector completes in any time a certain space, say in half a year moves through a space making an area equal to the square of D ; in order to ascertain its position in any given day of that half year, we have to cut off, by a line drawn from the centre of forces, an area which shall bear to D^ the same proportion that the given time bears to the half year, say 3 to w/', or we have to cut off a section ASP = 3 — : D^, A P being the parabola and S the focus. This will be m. done if A B be taken equal to three times A S, and B being drawn perpendicular to A B, between B 0, B A asymptotes, a rectangular hyperbola is drawTi, H P, whose semi-axis or semi-parameter is to D in the proportion, of 6 to wi ; it will cut the parabolic trajectory in the point P, required. For calling A M = a; and P M = y and A S = a ; then A B = 3 « and y X (a; -f- 3 a) = half the square of the hyperbola's semi-axis, which axis being equal to , y (x 4- S a) = — — - = — —, LAW OF THE TJOTVERSE. 255 1 \ 3D= _ . 2 1 ^ 3D' "" + 2 '^ J = "^- ^^^^®^°^« 3 ^2/ - 2 (^ - «) 2/ = ^;^- 2 2 11 and-, AM x PM = -a;y; and- {x-a)y = - SM . PM = 3 j)s S M P ; therefore the sector A S P = — ~ : so that the radius mr from the focus S cuts off the given area, and therefore P is 3 the point where the comet or other body will be found in — ; parts of the time. K the point is to be found by computation, we can easily 18 a® D' find the value of y by a cubic equation, y' + 3 a'y = — , and making BL = y, LP parallel to AM, cuts AP in the point P required. Sir Isaac Newton gives a very elegant solution geometrically by bisecting A S in Gr, and taking the perpendicular G E to the given area as 3 to 4 A S, or to S B, and then describing a circle with the radius E S ; it cuts the parabola in P, the point required. This solution is infinitely preferable to ours by the hyperbola, except that the demon- stration is not so easy, and the algebraical demonstration far from simple. It is further to be observed, that the place being given, either of these solutions enables us to find the time. Thus, 3 J)2 in the cubic equation, we have only to find ■ — ^. It is equal to ^-^ : and as W is the given integer, or period of e.g. 6 a' half a year, the body comes to the point P in a time which bears to D® the proportion of unity to — ,o a • 256 CENTEAi forces; iii. The next object of research is to generalise the pre- ceding investigations of trajectories from given forces, and of motion in given trajectories, applying the inquiry to all kinds of centripetal force, and all trajectories, instead of con- fining it to the conic sections, and to a force inversely as the square of the distance. We formerly gave the manner of finding the force from the trajectory in general terms, and showed how, by means of various differential expressions, this process was facilitated. It must, however, be remarked, that the inverse problem of finding the trajectory from the force, is not so satisfactorily solved by means of those expressions. For example, the most general one at which we arrived or — ~ — ^^ -~ 'I {^y a X — {x — a)dyf Q being put = ^ n, or the force inversely as the square of the distance, presents an equation in which it may be pro- nounced impossible to separate the variables so as to inte- d v grate, at least while d X, the differential of — , remains in dx so unmanageable a form ; for then the whole equation is dj^ydx — d^ xdy C TT? — J -, , , so = — , and thus from 2{ydx-{x-a)dyy (^y^ j^ Q^ _ ^y^^ ■ hence no equation to the curve could be found. It cannot be doubted that Sir Isaac Newton, the discoverer of the calculus, had applied all its resources to these solutions, and as the expressions for the central force, whether or — —, or '2,p^ .r p^dr d!' X »/ x^ -{- if' - (in some respects the simplest of all, being X taken in respect oi dt constant, and which is integrable in LAW OP THE UNITBESB. 257 the case of the inverse squares of the distances, and gives the general equation to the conic sections with singular elegance), are all derivable from the Sixth Proposition of the First Book, it is eminently probable that Newton had first tried for a general solution by those means, and only had recourse to the one which he has given in the Forty-first Proposition when he found those methods unmanageable. This would naturally confirm him in his plan of preferring geometrical methods ; though it is to be observed that this investigation, as well as the inverse problem for the case of rectilinear motion in the preceding section, is conducted more analyti- cally than the greater part of the Prinoipia, the reasoning of the demonstration conducting to the solution and not follow- ing it synthetically. A is the height from which a body must fall to acquire the velocity at any point D, which the given body moving in the trajectory V I K (sought by the investigation) has at the corresponding point I ; D I, B K, being circular arcs from the centre C, and CI = C D and C K = C E. It is shown previously that, if two bodies Vhose masses are as their weights descend with equal velocity from A, and being acted on by the same centripetal force, one moves in V I K and the other in A V C, they will at any corresponding points have the same velocity, that is at equal distances from the centre C. So that, if at any point D, D 6 or D F be as the velocity at D 253 . CBNTEAL FOECBS. of the body moving in AVC, D?; or DF will also represelit the velocity at 1 of the body moving in V I K. Then take D F = 2/ as the centripetal force in D or I (that is, as any power of the distance D C, or a — «, V C being a, and C D, x) V D F L will he fydz. Describe the circle V X Y with V as radius. Let VX = «, and YX will be dz, and N K = . a Then 1 K being as the time, and d t being constant, that triangle, or , is constant, and K N is as a constant quantity divided by I C, or as — . If we take — to V A V L B (proportioned to the furce at any one point V and therefore given), as KN to IK, therefore thin will in all points be the proportion ; and the squares will be proportional, or j ydx-. *^' : : I K\ or K N^ + I N^ to K N^ ; and therefore fydx so %:%::lW,ordx^:^'-^. Therefore ^ = a;' sir a' a ^'^^ J 14.- 1 • 1 '^'^^ /i. • XT. .; and multiplying by x, (twice the '«\/fydx--. X' i TnTr> Qdx . x^dz sector 1 U K ) = — .^ =. Again a dz : : : a' : r': ^ ^ x^dz cff- (^ Qdx . , and adz = X — 5 - — j X — =^- ^r- = twice the sector Y C X. Hence results this construction. Describe the curve abZ, such that (D 6 = m) its equation shall be ;■ 2\/fy_dx-—^, and the curve acx such that (D c = severally being expressed in terms of x, for this is necessary in order to eliminate y from the equations to these curves ; and then it is necessary to integrate these expressions ; for else the angle VOX, and the curve VIK, are only obtained in differential equations. Hence Sir Isaac Newton makes the quadrature of curves, that is, first the integration oi fydx, to eliminate y, and then the integration of the equations result- ing in terms of wand x, 4> and x respectively, the assumptions or conditions of his enunciation. — The inconvenience of this method of solving the problem gave rise to the investigations of Hermann and Bernoulli. The equation of the former, involving, however, the second differential of the co-ordinate, is to the rectangular co-ordinates ; that of the latter is a polar equation, in terms of the radius vector and angle atthe centre of forces. To illustrate the difficulty with which this method of quadratures is applied, in practice — take the case of the centripetal force being inversely as the cube of the distance; s 2 260 OBNTEAL FOEOES. then y = J-^ and the curve B L F is quadrahle. If we seek the circle V X Y by rectangular co-ordinates X O, V, we find the equation to obtain V = D in terms of x, is of the form' 1 aVD Qa^dx 2a^^fydx-^ Qa'dx (c being the constant introduced by integrating f y d x). Now there is no possibility of integrating these two quantities otherwise than by sines, and we thus obtain, nor can we do more, the following equation to D in terms of x ; ^ , . a-B V2.Q.a» ^fi + 2W V — a' arc sm. = — X arc cos. — ' . « V/i + 2Q^ V2c.x And if we get D from this, in terms of cos. x, we have then to obtain P G by similar triangles, and from IPC being rights angled and 10 = a;, to obtain P I, in order to have the curve VIK. But if we proceed otherwise, and instead of working by quadratures, take v the velocity of the bodj' at I, or in the c straight line at D, and make — the area described in a second, and 6 the angle V C I, we obtain as a polar equation to V I K, (?e = — - (x being in this case both C D X \^ 4:X^V^ ~ c^ and the radius vector). Then, to apply this general equation to the case of the centripetal force being as — , let the force at the distance 1 be put equal to unity, and supposing the velocity of projection to be that acquired in falling from an infinite height, the equation to the trajectory becomes G d OC C CD ' de = - — , and integrating, = X log. — . ( 261 ) XII. ATTRACTION OP BODIES; OP SPHERICAL AND NON- SPHERICAL SURFACES ANALYTICALLY TREATED. i. The attraction exerted by spherical surfaces and by hollow spheres is first to be considered. If P be a particle situated anywhere within A B D C, and we conceiTe two lines AD, B C, in- finitely near each other drawn through P to the surface, and if these lines revolve round a P S, which passes from the middle points a and h, of tbe small arcs DC, and AB, through P, there will two opposite cones be de- scribed ; and the attraction of the- small circles D 0, A B upon P, will be in the lines from each point of those circles to P, of which lines C P, DP, are two fromi one fcirle, and A P, B P, two from the other circle. Now, this attraction of the circle CD is to that of the circle A B, as the circle C D to the circle A B, or as C D' to A B" (the diameters), arid by similar triangles C D' : A B : : P C* : P A'. But by hypothesis, the attraction of C D is to that of A B as A P" : P C^ ; therefore the attraction of DC is to the opposite attraction of A B as A P', to P C», and also as P C to A P», or as A P^ X P 0'' to A P^^ X P C', and therefore those attractions are equal ; and being opposite they destroy one another. In like manner, any particle of the spherical surface on one side of P, acting in the direction of a P, is equal as well as opposite to the attraction of another particle acting on the opposite side, and so the whole action of every one particle is destroyed by 262 ATTEAOTION OF BODIES. tke opposite action of some other particle : and P is not at all attracted by any part of the spherical surface ; or the sum of all the attractions upon P is equal to nothing. So of a hollow sphere ; for every such sphere may be considered as composed of innumerable concentric spherical surfaces, to each of which the foregoing reasoning applies; and conse- quently to their sum. We may here stop to observe upon a remarkable inference which may be drawn from this theorem. Suppose that in the centre of any planet, as of the earth, there is a large vacant spherical space, or that the globe is a hollow sphere ; if any particle or mass of matter is at any moment of time in any point of this hollow sphere, it must, as far as the globe is con- cerned, remain for ever at rest there, and suffer no attraction from the globe itself. Then the force of any other heavenly body, as the moon, will attract it, and so will the force of the sun. Suppose these two bodies in opposition, it will be drawn to the side of the sun with a force equal to the dif- ference of their attractions, and this force will vary with the relative position (coniiguration) of the three bodies; but from the greater attraction of the sun, the particle, or bodj% will always be on the side of the hollow globe next to the sun. Now the earth's attraction will exert no influ-. ence over the internal body, even when in contact with the internal surface of the hollow sphere ; for the theorem which we have just demonstrated is quite general, and applies to particles wherever situated within the sphere. Therefore, although the earth moves round its axis, the body will always continue moving so as to shift its place every instant and retain its position towards the sun. In like manner, if any quantity of movable particles, thrown off, for example, by the rotatory motion of the earth, are in the hollow, they will not be attracted by the earth, but only towards the .sun, and will all accumulate towards the side of the hollow sphere next the sun. So of any fluid, whether water or melted matter in the hollow, provided it do not wholly fiU. up the space, the whole of it will be accumulated towards the sun. Suppose it only ATTEAOTION OP BODIES. 263 enough to fill half the hollow space ; it will all be acciimiv lated on one side, and that side the one next the sun ; conse- quently the axis of rotation will be changed and will not pass through the centre, or even near it, and -will constantly be alter- ing its position. Hence we may be assured that there is no such hollow in the globe filled with melted matter, or any hol- low at all, inasmuch as there could no hollow exist without such accumulations, in consequence of particles of the internal spherical surface being constantly thrown off by the rotatory motion of the earth.* '- If A H K B be a spherical section (or great circle), P E K aftd, P I L lines from the particle P, and infinitely near each other, SD, SE perpendiculars from the centre, and Ig per- pendicular to the diameter ; then, by the similar triangles PIE, PpD, we find that the curve surface bounded by IH, * The argument is here succinctly and popularly stated respect- ing the supposition of a liollow in the centre of the earth, and several steps are omitted. One of these may be mentioned in case it should appear to have been overlooked. Suppose a mass m detached from the hollow sphere M, and impelled at the same time with that sphere by an initial projectile force, — then its tendency would be to describe an elliptic orbit round the sun, the centre of forces, and if it were detached from the earth it would describe an ellipse, and be a small planet. But as "the accelerating force acting upon it would be different from that acting on the earth, the one being as — =-5 — , and the other as — =-r— (D being the distance and S the mass of the sun), it is manifest that, ajoner or later, its motion bemg slower than that of the hollow sphere, if in be placed in the inside, it must come in contact with the interior ofcumference of the sphere, and either librate, or, if fluid, coincide with it; as assumed in the text. Where parts of the spherical shell come off by' the "centrifugal force, of course no such step in the reasoning is wanted ; nor is it necessary to add that neither those parts nor any other within the hollow shell can Ijave any rotatory motion. 264 ATTBACTION OF BODIES. and formed by tiie revolution of I H K L I round the diameter A B, and which is proportional to I H x I ?, is as = =— - • and if the attraction upon the particle P is as the surface directly, and the square of the distance inversely, or :g-=j, that attraction will be as =; =77;. But if the force actinar in Pp X PS ° the line P I is resolved into one acting in P S and another P(7 acting in S D, the force upon P wi]l be as — , or (because of Pp the similar triangles P I Q, P S^) as -p-^. The attraction, therefore, of the infinitely small curvilinear surface formed P 1 by the revolution of I H is as rr =r-rr„ or as =7-^ ; that is ■^ Pp X Po^ PS* inversely as the square of the distance from the centre of the sphere. And the same may be shown of the surface formed by the revolution of K L, and so of everj' part of the spherical surface. Therefore the whole attraction of the spherical sur- face will be in the same inverse duplicate ratio. In like manner, because the attraction of a homogeneous sphere is the attraction of all its particles, and the mSiss of these is as the cube of the sphere's diameter D, if a particle, be placed at a distance from it in any given ratio to the diameter, as m .~D, and the attraction be inversely as the square of that distance, it will be directly as D", and also as -j-=:--, and therefore will be in the simple proportion of D, mrlr the diameter. Hence if two similar solids are composed of equally dense matter, and have an attraction inversely as the square of the distance, their attraction on any particle simi- larly placed with respect to them will be as their diameters. Thus, also, a particle placed within a hollow spheroid, or in a solid, produced by the revolution of an ellipsis, will not be ATTRACTION OF BODIES. 265 attracted at all by the portion of the solid between it and the surface, but will be attracted towards the centre by a force proportioned to its distance from that centre. It follows from these propositions, first, that any particle placed within a sphere or spheroid, not being affected by the portion of the sphere or spheroid beyond it, and being attracted by the rest of the sphere, or spheroid in the ratio of the diameter, the centripetal force within the solid is directly as the distance from the centre ; — secondly, that a homogeneous sphere, being an infinite number of hollow spaces taken together, its attraction upon any particle placed without it is directly as the sphere, and inversely as the square of the distance; — thirdly, that spheres attract one another with forces proportional to their masses directly, and the squares of the distances from their centres in- versely ;— fourthly, that the attraction is in every case as if the whole mass were placed in the central point; — - fifthly, that though the spheres be not homogeneous, yet if the density of each varies so that it is the same at equal distances from the centre of each, the spheres will attract one another with forces inversely as the squares of the distances of their centres. The law of attraction, however, of the particles of the spheres being changed from the inverse duplicate ratio of the distances to the simple law of the distances directly, the attractions acting towards the centres will be as the distances, and whether the spheres are homogeneous or vary in density according to any law connecting the force with the distance from the centre, the attraction on a particle without will be the same as if the whole mass w^ere placed in the centre ; and the attraction upon a particle within will be the same as if the whole of the body comprised within the spherical surface in which the particle is situated were collected in the centre. From these theorems it follows, that where bodies move round a sphere and on the outside of its surface, what was formerly demonstrated of eccentric motion in conic sections, the focus being the centre of forces, applies to this case of 266 ATTEAOTION OF BODIES. the attraction being in tlie whole particles of the sphere ; and where the bodies move within the spherical surface, what was demonstrated of concentric motion in those curves, or where the centre of the curve is that of [the attracting forces, applies to the case of the sphere's centre being that of attraction. For in the former case the centripetal force decreases as the square of the distance increases ; and in the latter case thatforce increases as the distance increases. Thus it is to be observed, that in the two cases of attraction decreasing inversely as the squares of the central distance (the case of gravitation beyond the surface of bodies), and of attraction increasing directly with the central distance (the case of gravitation within the surface), the same law of attraction prevails with respect to the corpuscular action of the spheres as regulates the mutual action of those spheres and their motions in revolution. But this identity of the law of attrao- tion is confined to these two cases. Having laid down the law of attraction for these more remarkable cases, instead of going through others where the operation of attraction is far more complicated, Sir Isaac Newton gives a general method for determining the attraction whatever be the proportions between the force and the dis- tance. This method is marked by all the geometrical ele- gance of the author's other solutions ; and though it depends upon quadratures, it is not liable to the objections in practice which we before found to lie against a similar method applied to the finding of orbits and forces ; for the results are easily enough obtained, and in convenient forms. If A E B is the sphere whose attraction upon the point P it is required to determine, whatever be the proportion according to which that attraction varies with the distance," and only supposing equal particles of A E B to have equal attractive forces ; then from any point E describe the circle E F, and another e/ infinitely near, and draw 'ED, ed ordi- nates to the diameter AB. The sphere is composed of small concentric hollow spheres E e/F ; and its whole attraction is equal to the sum of their attractions. Now that attraction ATTRACTION OF BODIES. 267 of E«/F IS proportional to its surface miiltiplied by F/, and the angle D E r teing equal to D P E (because P E r is a right PE X Dd angle by. the property of the circle), therefore E r = — =--f; — , and if we call PE, or PP = r, ED = 3/, and DF = a;, DcZwill be d X, and E r rdx ; and the ring generated by the revo- lution of r E is equal to r E x E D, or r'E x y ; therefore this ring is equal to rdx, or the attraction proportional to the' whole ring Eg will be proportional to the sum of all the rectangles PD x Drf, or (^a — cr)dx; that is, to the integral of tliis quantity, or to ; wliicli by the property o£ the circle is equai to ^. Therefore the attraction of the Solid Ee/F will be as f x F/ if the force of a particle F/ on P be given ; if not, it will be as ?/' X F/ x / that force. PS X dx Now dx : F/ :: ?• : P S, and therefore F/ = , and the V^X^Sxdxxf ajfltraotion of E e/F is as ; or taking / = r" (as any power of the distance PE), then the attraction of Ee/F is as YS.r"-'fdx. Take D N (= u) equal to 268 ATTBACTION OF BODIES. PS.r"~'y', and let BD = z, and the curve BNA will be described, and the differential area N Dc?n will be ndz - (by construction) VB .r'"~^ y^ dx ; consequently udz will be the attractive force of the differential solid Ee/F; and fudz will be that of the whole body or sphere ABB, therefore the area ANB = fudz is equal to the whole attraction of the sphere. Having reduced the solution to the quadrature of A N B, Sir Isaac Newton proceeds to show how that area may be found. He confines himself to geometrical methods ; and the solution, although extremely elegant, is not by any means so short and compendious as the algebraical process gives. Let us first then find the equation to the curve A N B by referring it to the rectangular coordinates D N, AD. Calling these y and x respectively^ and making PA = 6, A S (the sphere's f radius) = a and P S, or a + 6, for conciseness, = •^. Then 'D'E? = 2ax-x'; PE= ^ {b+xy+2ax-x^= jV+2(a + l)x {,a+h){2ax-x') = ^b^+fx ; and D N =y = (by construction) -^ i^rp the attractive force of the particles being supposed as the -th 71 power of the distance, or inversely as (b^+fx)^. This equation to the curve makes it always of the order , If then the force is inversely as the distance, A N B is a conic hyperbola ; if inversely as the square, it is a curve of the fifth order ; and if directly as the distance, it is a conic parabola ; if inversely as the cube, the curve is a cubic hyperbola. The area may next be determined. For this purpose we h^Yefydx= r/(i^ -.fl^_ Let 2 (a/+ b') = h, this ATTBACTION OF BODIES. 269 1 h t integral will be found to he — x 5 X (b' +fx) 4:(a-j- by 6 ~ n ' •/ j 5-» - 1—- X (2 « + 6)' (6^ +/^)"° - '^^'j'-^ "'^' + C ; and 1 — ra — B the constant U is — ^ — ■ x z h ^^ —— j8-n Y rjij^jg ij^ every case gives an easy and a finite 8-n expression, excepting the three cases oi n = — l,n = 3, and ?i = 5, in which cases it is to be found by logarithms, or by hyperbolic areas. To find the attraction of the whole sphere, when a; = 2 a, we have -— — — — x ( (2 a + J)'-" - iiLx(2a + 6)--(i^+^" + J!:l+ ^°-" - 1— M 5 — n 5 — nl— n (2 a + 6)' — ) for the whole area A N B, or the whole attraction. If P is at the surface, or AP = 6 = 0, and w = 2, then the expression becomes as a, that is, as the distance from the centre directly. We may also perceive from the form of the expression, that if n is any number greater than 3, so that « — 3 = — m, the terms 6^-" become inverted, and h is in their denominator thus : 7- r-r-. Hence if m > 3 and A P (1 — m) ft" =? J = 0, or the particle is in contact with the sphere, the expression involves an infinite quantity, and becomes infinite. The construction of Sir Isaac Newton by hyperbolic areas leads to the same result for the case of n = 3, being one of those three where the above formula fails. At the origin of the abscissae we obtain, by that construction, an infinite area ; and this law of attraction, where the force decreases in any higher ratio than the square of the distance, is applicable to the contact of all bodies of whatever form, the addition of any 270 ATTRACTION OP BODIES. other matter to the spherical bodies having manifestly no eifeot in lessening the attraction. By similar methods we find the attraction of any portion or , segment of a sphere upon a particle placed in the centre, or upon a particle placed in any other part of the axis. Thus in the ca.se of the particle being in the centre S, and the particles of the segment E B G attracting with forces as the -th power of the distance S or S I, the curve A N B will by its area express the attraction of the spherical segment, if 10' (x — of — c' D N or 2/ be taken = 5-=^ = — —^- — , S being put = c, a U (x — a) and A D = a;, and AS = a, as before ; ^ ydx may be found as , - •, . , ,. (x — aydx—c'dx _„ before by integrating -^^ — ■ — — . Ihe fluent is ^^^ c'^- ^— + C; and C = -j— — ; and 3 — ?i 1 — n n— 4ft + 3 the whole attraction of the segment upon the particle at the 1 ~ r^ (a — c)" ji = 2 the attraction is as -^ , or as B' directly, and as S B inversely ; and if c = 0, or the attraction is taken at the centre, it is equal to a ; and if the attraction is as the dis- tance, or n = 1, then the- attractive force of the .segment is ii. Attractions of non-spherical bodies. The attractions of two similar bodies upon two similar particles similarly situ- ated with respect to them, if those attractions are as the same power of the distances -, are to one another as the masses directly, and the n* power of the distances inversely, or the ■ft* power of the homologous sides of the bodies ; and because centre S is equal to • — , f- -r — — . Thus, if ATTRACTION OF BODIES. 271 the masses are as the cubes of these sides, S and s, the attrac- tions are as S^ . s" : s^ . S", or as s""' : S"-^ Therefore, if n = I, the attraction is as S' : s*; if the proportion is that of the inverse square of the distance, the attraction is as S : s ; if that of the cube, the attraction is as ] : 1, or equal ; if as the biquadrate, the attraction is as « : S; and so on : and thus the law of the attractive force may be ascertained from finding the action of bodies upon particles similarly placed. Let us now consider the attraction of any body, of what form soever, attracting with force proportioned to the distance towards a particle situated beyond it. Any two of its particles A B attract P, with forces as A x A P and B x B P, and if G is their common centre of gravity, their joint attraction is as (A + B) X G P, because B P, being resolved into B G and GP, and AP into AP and GP, and (by the property of the centre of gravity) GP X A = A P X G, therefore the forces in the line A P destroy each other, and there remain only P G X B and P G X A to draw P, that is (A + B) X P G ; and the same may be shown of any other par- ticles C and the centre G' of gravity, of A, C, and B, the attraction of the three being (A + B + G) X G' P. Therefore the whole body, whatever be its form, attracts P in the line P S, S being the body's centre of gravity, and with a force proportional to tlie whole mass of the body multiplied by the distance P S. But as the mutual attractions of spherical bodies, the attraction of whose particles is as their distance from one another, are as the distances between the centres of those bodies, the attraction of the whole body A B C is the same with that of a sphere of equal mass whose centre is in S, the body's centre of gravity. In like manner it may be demonstrated that the attraction of several bodies A, B,C, towards any particle P, is directed to their common centre of gravity S, and is equal to that of a sphere placed there, and of a mass equal to the 272 ATTKAOTION OF BODIES. sum of the whole bodies A, B, C ; and the attracted body will revolve in an ellipse with a force directed towards its centre as if all the attracting bodies were formed into one globe and placed in that centre. But if we would find the attraction of bodies whose particles act according to any power n of the distance, we must, to simplify the question, suppose these to be symmetrical, that is, formed by the revolution of some plane upon its axis. Let A M C H G be the solid, M G- the diameter of its extreme circle of revolution next to the particle P ; draw P M and p m to any part of the circle, and infinitely near each other, and take PD = PM, andPo=Pm; Bd will be equal to oM (dn being infinitely near DN), and the ring formed by the revo- lution of M m round A B will be as the rectangle A M x M m. or (because of the triangles A P M, m o M, being similar, and Dd = om)PM X Jid, or PD X T>d. Let DN betaken =y = force vnth which any particle attracts at the distance P D = P M = a;, that is as of ; and if A P = 6, the force of any h y particle of the ring is as — , and the attraction of the ring, h y described by M m, is as — x D i^ X P D, or as hydw, and the a? ATTEACTION OF BODIES. 273 whole attraction of the circle whose radius is A M, being the sum of all the rings, will be as bfydx, or the area of the curve L N I, which is found by substituting for y its value in X, that is x"- This fluent or area is therefore =bfx"dx Sa;"+' — J"+^ = — r-7 + C ; and C = — . Also, making P & = P E in order to have the whole area of L N I, which measures the attraction of the whole circle whose radius is F A, we have (x being = P 6 = c) for that attraction. Then w+1b+2 taking D N' in the same proportion to the circle D E in which D N is to the circle A E, or as equal to the attraction of the circle D E, we have the curve E N T, whose area is equal to the attraction of the solid L H C P. To find an equation to this curve, then, and from thence to obtain its area, we must know the law by which D E in- creases, that is, the proportion of D E to A D ; in other words, the figure of the section AE E C B, whose revolution generates the solid. Thus if the given solid be a spheroid, we find that its at- traction for P is to that of a sphere whose diameter is equal to the spheroid's shorter axis, as — '— — to z--^„, A ^ d^ + A" — a 3 (T and a being the two semi-axes of the ellipsoid, d the distance of the particle attracted, and L a constant conic area which may be found in each case ; the force of attraction being sup- posed inversely as the squares of the distances. But if the particle is within the spheroid, the attraction is as the dis- tance from the centre, according to what we have already seen. Laplace's general formula for the attraction of a spherical surface, or layer, on a particle situated (as any particle must he) in its axis, is "" ff'^/xf'^f ^' ^^ which / is the distance of the particle from the point where the ring cuts 274 ATTRACTION OF BODIES. the sphere, r its distance from the centre of the sphere, or the distance of the ring from that centre, du consequently the thickness of the ring, ir the semicircle whose radius is unity, and F the function of / representing the attracting force. The whole attraction of the sphere, therefore, is the integral taken from f=r — uiof=r-\-u, and the expression be- comes "^JLllil + Jfdf X f d/F with (r + m) - (r - u), substituted for/, when / results from this integration. Then let F = — or the attraction be that of gravitation ; the ex- . , 2',T .udu C „, . Cdf iTT.udu f pression becomes \fafx -^ = X "hr r J J f I" 2 1 2Tt . udu (r + w) — Cr — m") 1-Kudu X - - = X - ^^ —^ ' = X - M=— 27rM'rfMX-; and the coefficient of dr, taking the r differential with r as the variable, is ■\ , — ; consequently the attraction is inversely as the square of the distance of the particle from the centre of the sphere, and is the same as if the whole sphere were in the centre.* * SKc. Cel. liv. ii. ch. 2. The expression is here developed ; but it coincides with the analysis in § 11.' ■ This Tract and the last are both taken from the ' Analytical View of the Principia,' Lib. I. ( 275 ) XIII. ADDRESS DELIVBEED ON THE OPENING OP THE NEWTON MONUMENT AT GRANTHAM, Sept. 21, 1858. To record tlie names, and preserve the memory of those whose great achievements in science, in arts, or in arms have con- ferred benefits and lustre upon our kind, has in all ages been regarded as a duty and felt as a gratification by wise and reflecting men. The desire of inspiring an ambition to emulate such examples, generally mingles itseK with these sentiments ; but they cease not to operate even in the rare instances of transcendent merit, where matchless genius excludes all possibility of imitation, and nothing remains but wonder in those who contemplate its triumphs at a distance that forbids all attempts to approach. We are this day as- sembled to commemorate him of whom the consent of nations has declared, that he is chargeable with nothing like a follower's exaggeration or local partiality, who pronounces the name of jN ewton as that of the greatest genius ever bestowed by the bounty of Providence, for instructing mankind on the frame of the universe, and the laws by which it is governed. " Qui genus humanum mgenio superavit, et omnes Eestinxit ; stellas exortus uti setlieriua sol." — (Jmc.) " In genius ■who iiurpassed mankind as far As does the mid-day sun the midnight star."— {Dryden.) But though scaling these lofty heights be hopeless, yet is there some use and much gratification in contemplating by what steps he ascended. Tracing his course of action may T 2 276 SIB ISAAC NEWTON. help others to gain the lower eminences lying within their reach ; while admiration excited and curiosity satisfied are frames of mind both wholesome and pleasing. Nothing new, it is true, can he given in narrative, hardly anything in reflection, less still perhaps in comment or illustration ; but it is well to assemble in one view various parts Of the vast subject, with the surrounding circumstances whether acci- dental or intrinsic, and to mark in passing the misconceptions raised by individual ignorance, or national prejudice, which the historian of science occasionally finds crossing his path. The remark is common and is obvious, that the genius of Newton did not manifest itself at a very early age. His faculties were not, like those of some great and many ordinary individuals, precociously developed. Among the former, Clairaut stands pre-eminent, who, at thirteen years of age, presented to the Koyal Academy a memoir of great originality upon a difiicult subject in the higher geometry ; and at eighteen, published his celebrated work on Curves of Double Curvature, composed during the two preceding years. Pascal, too, at sixteen, wrote an excellent treatise on Conic Sections. That Newton cannot be ranked in this respect with those extraordinary persons, is owing to the accidents which pre- vented him from entering upon mathematical study before his eighteenth year ; and then a much greater marvel was wrought than even the Clairauts and the Pascals displayed. His earliest history is involved in some obscurity ; and the most celebrated of men has in this particular been compared to the most celebrated of rivers,* as if the course of both in its feebler state had been concealed from mortal eyes. We have it, however, well ascertained that within four years, between the age of 18 and 22, he had begun to study mathematical science, and had taken his place among its greatest masters ; learnt for the first time the elements of gesmetry and analysis, and discovered a calculus which entirely changed the face of the science ; effecting a complete revolution in that and in ■" The Nile. GEAlfTHAM ADDRESS. 277 every branch of philosopliy connected with it. Before 1661 he had not read Euclid ; in 1665 he had committed to writing the method of Fluxions. At 25 years of age he had discovered the law of gravitation, and laid the foundations of Celestial Dynamics, the science created by him. Before ten years had elapsed, he added to his discoveries that of the fundamental properties of Light. — So brilliant a course of discovery, in so short a time changing and reconstructing Analytical, Astro- nomical, and Optical science, almost defies belief. The state- ment could only be deemed possible by an appeal to the incontestable evidence that proves it strictly true.* ' By a rare felicity these doctrines gained the universal assent of mankind as soon as they were clearly understood ; and their originality has never been seriously called in ques- tion. Some doubts having been raised respecting his inven- ting the calculus, doubts raised in consequence of his so long * The birth "of Newton was 25th Dee., 1G42. (0. S.) or .^th Jan., 1643, (N. S.) In 1661, 5th June, he was entered of Cambridge, and matricu- lated 8th July. Before that time he had applied himself in a desultory way to parts of practical meclianics, as the movement of machines, and to dialling. As soon as he^arrived at Cambridge he began to read ' Euclid,' and threw the bools: down as containing demonstrations of what lie deemed too manifest to require proof. It is, therefore, probable that he had before meditated upon the position and proportion of Unes, perhaps of angles. Upon laying aside ' EucKd,' he took up ' Descartes' Geometiy,' then Kepler's Optics, which he speedily mastered, as he did a book on Logic, showing the College Tutor that he had anticipated his lessons. In 1663 and '64 he worked upon Series and the Properties of Curves. In summer, 1664, he investigated the quadrature of the hyperbolic area by the Method of Series which he had conti-ived. A paper in his handwriting dated 20th May, 1665, gives the method of Fluxions, and its application to the finding of tangents, and the radius of curvature. So that at this time the direct method at least was invented. Another paper also in his hand- writing, Oct., 1666, gives its application to equations involving surds. The Optical Lectures in 1669, '70 and '71, give the doctrine of Different fflefrangibiUty.— In 1665 he formed the opinion of gravitation extending to the heavenly bodies, but was prevented from drawing the conclusion definitively, by the imperfect estimate of a degree as 60 miles, to which alone he had access. After 1670, when Picard showed it to be 69 J miles he resumed his demonstration, and found it exact. 278 SIB ISAAC NEWTON. withholding the publication of his method, no sooner was inquiry instituted than the evidence produced proved so decisive, that all men in all countries acknowledged him to have been by several years the earliest inventor, and Leibnitz, at the utmost, the first publisher ; the only questions raised being, first, whether or not he had borrowed from Newton, and next, whether as second inventor he could have any merit at all ; both which questions have long since been decided in favour of Leibnitz.* But undeniable though it be that Newton made the gre£|,t steps of this progress, and made them without any anticipa- tion or participation by others, it is equally certain that there had been approaches in former times, by preceding phil9- sophers, to the same discoveries. Cavalleri, by his ' Geometry of Indivisibles,' (1635,) Eoberval, by his 'Method of Tan- gents,' (1637,) had both given solutions which Descartes could not attempt; and it is remarkable that Cavalleri regarded curves as polygons, surfaces as composed of lines, whilst Eoberval viewed geometrical quantities as generated by motion ; so that the one approached to the differential calculus, the other to fluxions: and Fermat, in the interval between them, came still nearer the great discovery by his determination of maxima and minima, and his drawing of tangents. More recently Schooten had made public similar methods invented by Hudde ; and what is material, treating the subject algebraically, while those just now mentioned had rather dealt with it geometrically.f It is thus easy to per- * Leibnitzfirst published Ms method in 1684; but he had communicated it to Newton in 1677, eleven years after the fluxional process had been employed, and been described in writing by its author. t Cavalleri s ' Exercitationes Geometricse ' in 1647, as his ' Geometria Indivisibilis ' in 1635, showed how near he had come to the calculija.'%f Format, however, must be allowed to have made the nearest approach ; insomuch that Laplace and Lagrange have both regarded him as its in- ventor. He proceeds upon the position that when a Co-ordinate is a maximum or minimum, the equation, formed on increasing it by an infinitely small quantity, gives a value in which that small quantity vanishes. He thus finds the subtangent. But perhaps his most renjark- GEAKTHAM ADDRESS. 279 oeive how near au approach had been made to the calculus before the great event of its final discovery. There had in like manner been approaches made to the law of gravitation, and the dynamical system of the universe. Galileo's important propositions on motion, especially on curvilinear motion, and Kepler's laws upon the elliptical form of the planetary orbits, the proportion of the areas to the times, and of the periodic times to the mean distances, and Huygens's theorems on centrifugal force, had been followed by still nearer approaches to the doctrine of attrac- tion. Borelli had distinctly ascribed the motion of satellites to their being drawn towards the principal planets, and thus prevented from being carried off by the centrifugal force.* Even the composition of white light, and the different action of bodies upon its component parts, had been vaguely con- jectured by Ant. de Dominis, Archbishop of Spalatro, at the beginning, and more precisely in the middle of the seventeenth century by Marcus (Kronland of Prague), unknown to New- ton, who only refers to the Archbishop's work; while the Treatise of Huygens on light, Grimaldi's observations on colours by inflexion as well as on the elongation of the image in the prismatic spectrum, had been brought to his attention, although much less near to his own great discovery than Marcus's experiment. f able approacli to the calculus is the rule given to suppress all terms in which the square or the cube of the small quantity is found, because, it is said, those powers are infinitely small in comparison of the first power of the quantity. Thus calling that quantity e (or as we should say d x), he considers e^ and e' (dx^ and da?) as to be entirely rejected. — Hudde's letter to Schooten, 1658. Descartes' Geom. I. 507. * Galileo's problem on the motion of bodies by gravity acting uniformly in parallel lines could have been no novelty to Newton ; and Huygens's -Explanation of centrifugal tendency by the comparison of a stone's ten- Sency to fly off when whirled round in a sling, is as correctly as possible that now received. But his theorems had been investigated by Newton several years before, as appears from a letter of Huygens himself, t The Archbishop's explanation in 1611 of the rainbow, and his experi- ment to illustrate it by a thin glass globe filled with water and giving 280 SIE ISAAC NEWTON. But all this only shows that the discoveries of iN'ewton, great and rapid as were the steps by which they advanced our knowledge, yet obeyed the law of continuity, or rather of gradual progress, which governs all human approaches towards perfection. The limited nature of man's faculties precludes the possibility of his ever reaching at once the utmost excellence of which they are capable. Survey the ■whole circle of the sciences, and trace the history of our pro- gress in each, you find this to be the universal rule. In chemical philosophy the dreams of the alchemists prepared the way for the more rational though erroneous theory of Stahl : and it was by repeated improvements that his errors, so long prevalent, were at length exploded, giving place to the sound doctrine which is now established. The great discoveries of Black and Priestley on heat and aeriform fluids, had been preceded by the happy conjectures of Newton, and the experiments of others. Nay Voltaire* had well nigh colours by refraction, is remarkable ; but far less so than Marcus's in 1648 on the ' Iris Trigonia,' as he calls the spectrum, and his observation of the colours not changing by a second refraction, so nearly approaching Newton's ' Bxperimentum Cruois.' It is best to mention this, because writers on the history of science have so often stated that nothing hke a trace of the Newtonian doctrine of light can be found in the works of former observers. There is no appearance whatever of Newton having known Marcus's work. * In his Prize Memoir we find (among many great errors chiefly arisiag from fanciful hypotheses) such passages as this, being an observation on one of his own experiments, ' II y a certainement du feu dans ces deux liqueurs, sans quoi elles ne seraient point fluides ;' and again, in speaking of the connexion between heat and permanent or gaseous elasticity, ' N'est ce pas que I'air n'a plus alors la quantity de feu neoessaire pour faire jouer toutes ses parties, et pour le degager de I'atmosphfere eugourdie qui le renferme.' The experiments which he made on the temperature of Hquids mixed together, led him to remark the temperature of the mixture as different from what might have been expected, regard being had to that of the separate liquids. Again, speaking of his experiments on the calci- nation of metals, ' II est tr6s possible que I'augmentation du poids soit venue de la matifere re'pandue dans I'atmosphfere ; done dans toutes les autres operations par lesquelles les matieres calcinees acquierent du poids GEANTHAM ADDRESS. 281 discovered both the absorption of beat, the constitution of the atmosphere, and the oxidation of metals, and by a few more trials might have ascertained it. Cuvier had been preceded by inquirers who took sound views of fossil osteology: among whom the truly original genius of Hunter fills the foremost place. The inductive system of Bacon, had been, at least in its practice, known to his predecessors. Observations and even experiments were not unknown to the ancient philosophers, though mingled with gross errors : in early times, almost in the dark ages, experi- mental inquiries had been carried on with success by Friar Bacon, and that method actually recommended in a treatise, as it was two centuries later, by Leonardo da Vinci ; and at the latter end of the next century Gilbert examined the whole subject of magnetic action entirely by experiment. So that Lord Bacon's claim to be regarded as the father of modem philosophy rests upon the important, the truly invaluable, step of reducing to a system the method of investigation adopted by those eminent men, generalizing it, and extending its application to all matters of contingent truth, exploding the errors, the absurd dogmas, and fantastic subtleties of the ancient schools — above all, confining the subject of our inquiry, and the manner of conducting it, within the limits which our faculties prescribe.* cette augmentation pourrait aussi leur Stre venue de la rnSme cause, et non de la matiere ignee.' lie had been experimenting with a view to try if heat had any weight. (^Acad. des Sciences, 1737, Prix. IV. p. 169.) * Friar Bacon's ' Opus Majus ' was composed about the middle of the 13th century, certainly before 1267 ; and it contains, among other matters connected with experimental inquiry, a treatise expressly setting forth the a,dvantages of that mode of philosophising. His aversion to the Aristote- lian errors, and his departure from the whole philosophy of the times, was probably at the bottom of the charges of heresy under which he suffered cruel persecution for so many years. — Gilberts Treatise, ' De Magneto et Corporibus Magneticis,' was published in 1600. It is entirely founded on .experiments and observations, and is called by Lord Bacon "A painful and experimental work." Newton, who never alludes to Bacon, has been by some supposed not to have been acquainted with his writings. Sir D, 282 BIE ISAAC . NEWTON. Nor is this great law of Gradual Progress confined to the physical sciences ; in the moral it equally governs. Before the foundations of political economy were laid by Hume and Smith, a great step had been made by the French philosophers, disciples of Quesnay ; but a nearer approach to sound princi- ples had signalized the labours -of Goumay, and those labours had been shared and his doctrines patronized by Turgot when Chief Minister. Again, in constitutional policy, see by what slow degress, from its first rude elements — the attendance of feudal tenants at their lord's court, and the summons of burghers to grant supplies of money — the great discovery of modem times in the science of practical politics has been effected, the Eepresentative scheme, which enables states of any extent to enjoy popular government, and allows Mixed Monarchy to be established, combining freedom with order-^ a plan pronounced by the statesmen and writers of antiquity to be of hardly possible formation, and wholly impossible con- tinuance.* — The globe itself as well as the science of its inhabitants, has been explored according to the law which forbids a sudden and rapid leaping forward, and decrees that each successive step, prepared by the last, shall facilitate the next. Even Columbus followed several successful discoverers on a smaller scale ; and is by some believed to have had, unknown to him, a predecessor in the great exploit by which Brewater and others have peremptorily denied that his mode of inquiry was either suggested, or at all influenced by those writings. It is certain that neither he, nor indeed any one but Bacon himself, ever followed in detail the rules prescribed in the ' Novum Organum.' * The opinion of Tacitus on this subject is well known. "Cunctas nationes et urbes populus, aut primores, aut singuli regunt. Delecta (some editions add consociata) ex his et constituta rei publicse forma laudari faoilius quam evenire ; vel si eveuit, hand diutuma, esse potest." (Ann; IV. 33.) Cicero, in his Treatise ' De EepublicS,' giving liis opinion that the best form of government is that ' extribus generibus, regali, optimatum, et populari, modice confusa," does not in terms declare it to be chimerical ; yet he distinctly says in the same Treatise (II. 23) that liberty cannot exist under a king. Liberty, he says, consists "non in eo ut justo utamur domino sed ut nullo." GEANTHAM ADDEESS. 283 he pierced the night of ages, and unfolded a new world to the eyes of the old. The arts afford no exception to the general law. Demos- thenes had eminent forerunners, Pericles the last of them. Homer must have had predecessors of great merit, though doubtless as far surpassed by him as Fra Bertolomeo and Pietro Perugino were by Michael Angelo and Piaphael. Dante owed much to Virgil ; he may be allowed to have owed, through his Latin Mentor, not a little to the old Grecian ; and Milton had both the Orators and the Poets of the ancient world, for his predecessors and his masters. The art of war itself is no exception to the rule. The plan of bringing an overpowering force to bear on a given point had been tried eccasionally before Frederic II. reduced it to a system, and the Wellingtons and Napoleons of our own day made it the foun- dation of their strategy, as it had also been previously the mainspring of our naval tactics. It has oftentimes been held that the invention of Logarithms stands alone in the history of science, as having been preceded by no step leading towards the discovery. There is, however, great inaccuracy in this statement; for not only was the doctrine of infinitesimals familiar to its illustrious author, and the relation of geometrical to arithmetical series well known ; but he had himself struck out several methods of great in- genuity and utility, (as that known by the name of ' Napier's Bones,') — methods that are now forgotten, eclipsed as they were by the consummation which has immortalized his name.* — So the inventive powers of Watt, preceded as he was by Worcester and Newcomen, but more materially by Gauss and Papin, had been exercised on some admirable con- . ' ' The Khabdologia,' was only published in 1617, the year he died ; but Napier had long before the invention of logarithms used the contri- vances there described. His ' Oanon Mirificus ' was only published by him in 1611 : but it appears from a letter of Kepler that the invention was at least as early as 1594. The story of Longomontanus having anti- cipated him is a mere fable ; but Kepler believed that one Byrge had at least come near the invention, and he had done much certainly upon natural sines. (Bpist, Leips. 1718.) 284 SIE ISAAC NEWTON. trivances, now forgotten, before lie made the step which created the engine anew, not only the Parallel Motion, possibly a corollary to the proposition on circular motion in the ' Principia,' but the Separate Condensation, and above all the Governor, perhaps the most exquisite of mechanical inven- tions ; and now we have those here present who apply the like principle to the diffusion of knowledge, aware as they must be, that its expansion has the same happy effect naturally of preventing mischief from its excess, which the skill of the great mechanist gave artificially to steam, thus rendering his engine as safe as it is powerful. The grand difference, then, between one discovery or invention and another is in degree rather than in kind ; the degree in which a person while he outstrips those whom he comes after, also lives as it were before his age. Nor can any doubt exist that in this respect Newton stands at the head of all who have extended the bounds of knowledge. The sciences of Dynamics and of Optics are especially to be regarded in this point of view ; but the former in particular ; and the completeness of the system which he unfolded, its having been at the first elaborated and given in perfection,^ — its having, however, now stood the test of time, and survived, nay gained by the most rigorous scrutiny, can be predicated of this system alone, at least in the same degree. That the calculus, and those parts of dynamics which are purely mathematical, should thus endure for ever, is a matter of course. But his system of the universe rests partly upon contingent truths, and might have yielded to new experiments, and more ex- tended observation. Nay, at times it has been thought to fail, and further investigation was deemed requisite to ascertain if any error had been introduced ; if any circumstance had escaped the notice of the great founder. The most memor- able instance of this kind is the dirorepanoy supposed to have been found between the theory and the fact in the motion of the lunar apsides, which about the middle of the last century occupied the three first analysts of the age.* The error was * D'Alembert, Clairaut, Buler. GEANTHAM ADDRESS. 285 discovered by themselves to have been their own in the pro- cess of their investigation ; and this, like all the other doubts that were ever momentarily entertained, only led in each instance to new and more brilliant triumphs of the system. The prodigious superiority in this cardinal point of the Newtonian, to other discoveries, appears manifest upon ex- amining almost any of the chapters in the history of science. Successive improvements have by extending our views con- stantly displaced the system that appeared firmly established. To take a familiar instance, how little remains of Lavoisier's doctrine of combustion and acidification except the negative positions, the subversion of the system of Stahl ! The sub- stance having most eminently the properties of an acid, (chlorine,) is found to have no oxygen at all,* while many substances abounding in oxygen, including alkalis themselves, have no acid property whatever ; and without the access of oxygenous or of any other gas, heat and flame are produced in excess. The doctrines of free trade had not long been pro- mulgated by Smith, before Bentham demonstrated that his exception of usury was groundless ; and his theory has been repeatedly proved erroneous on colonial establishments, as well as his exception to it on the navigation laws ; while the imperfection of his views on the nature of rent is undeniable, as well as on the principle of population. In these, and such instances as these, it would not be easy to find in the original doctrines the means of correcting subsequent errors, or the germs of extended discovery. But even if philosophers finally adopt the undulatory theory of light instead of the atomic, it must be borne in mind that Newton gave the first elements of it by the well-known proposition in the eighth section of the second book of the ' Principia,' the scholium to that section also indicating his expectation that it would be applied to optical science ;t while M. Biot has shown how the doctrine * Eeoent inquiries are said to have shaken if not displaced Davy's theory of chlorine. . + The 47th prop. lib. II, has not been disputed except as to the suffici- ency of the demonstration, which Euler questioned, but without adding the 286 SIE ISAAC NEWTON. of fits of reflection and transmission tallies with polarization, if not with undulation also. But the most marvellous attribute of Newton's discoveries is that in which they stand out prominent among all the other feats of scientific research, stamped with the peculiarity of his intellectual character ; they were, their great author lived before his age, anticipating in part what was long after wholly accomplished ; and thus unfolding some things which at the time could be but imperfectlj', others not at all comprehended ; and not rarely pointing out the path and affording the means of treading it to the ascertainment of truths then veiled in darkness. He not only enlarged the actual dominion of knowledge, penetrating to regions never before explored, and taking with a firm hand undisputed possession ; but he showed how the bounds of the visible horizon might be yet further extended, and enabled his successors to occupy what he could only descry ; as the illustrious discoverer of the new world made the inhabitants of the old cast their eyes , over lands and seas far distant from those he had traversed;' lands and seas of which they could form to themselves no conception, any more than they had been able to com- prehend the course by which he led them on his grand enterprise. In this achievement, and in the qualities which alone made it possible — inexhaustible fertility of resources, patience unsubdued, close meditation that would suffer no distraction, steady determination to pursue paths that seemed all but hopeless, and unflinching courage to declare the truths they led to how far soever removed from ordinary apprehen- sion — in these characteristics of high and original genius we proof of its insufficiency, or communicating hia own process. Cramer has done both, and his demonstration is given by Leseur and Jaoquier, II. 364, together with another upon Newton's principle, but supplying the defects, by the able and learned commentators. The adherents, too, of the undu- latory theory have always explicitly admitted the connexion between the Newtonian experiments and their doctrine. — See particularly Mr. Airy's • very able Tracts — Thus, " Newton's rings have served in a great degree for the foundation of all the theories." S. 72, (p. 311, Edit. 1831.) GBAUTHAM ADDEESS. 287 • may be permitted to compare the career of those gi-eat men. But Columbus did not invent the mariner's compass, as Newton did the instrument which guided his course and enabled him to make his discoveries, and his successors to extend them by closely following his directions in using it. Nor did the compass suffice to the great navigator without making any observations ; though he dared to steer without a chart ; while it is certain that by the philosopher's instrument his dis- coveries were extended over the whole system of the universe, determining the masses, the forms, and the motions of all its parts, by the mere inspection of abstract calculations and formulas analytically deduced.* The two great improvements in this instrument which have been made, the Calculus of Variations by Euler and Lagrange, the method of Partial Differences by D'Alembert, we have every reason to believe were known, at least in part, to Newton himself. His having solved an isoperimetrical problem (finding the line whose revolution forms the solid of least resistance) shows clearly that he must have made the co- ordinates of the generating curve vary, and his construction agrees exactly with the equation given by that calculus. f * The investigation of the masses and figures of the planets from their motions by Newton — the discovery by Laplace of peculiarities in those motions never before suspected, a discovery made from the mere inspection of algebi-aical equations — without leaving their study — are as if Columbus had never left his cabin. t The differential equation of the curve deduced by help of the calculus of variations is of this form : — y d'f _ (dx^ + d y^)^ dx Which may be reconciled with the equation in the commentary to the Schol. of Prop. XXXIV. lib. II. — If p = j^, the equation becomes y = -i — ^ ' . T. Simpson, in his general solution of isoperimetrical p3 problems (' Tracts,' 1757), gives a method which leads precisely to the above result derived from the calculus of variations, see p. 104. See, too, Emerson's ' ITluxions,' where we see his near approach to the calculus. 288 SIK ISAAC NEWTON. That lie must have tried the process of integrating by parts in attempting to generalize the inverse problem of central forces before he had recourse to the geometrical approximation which he has given, and also when he sought the means of ascertaining the comet's path (which he has termed by far the most difficult of problems), is eminently probable, when we consider how naturally that method flows from the ordinary process for differentiating compound quantities by supposing each variable in succession constant ; in short, differentiatijig by parts. As to the calculus of variations having substan- tially been known to him no doubt can be entertained. Again, in estimating the ellipticity of the earth, he pro- ceeded upon the assumption of a proposition of which he gave no demonstration (any more than he had done of the isoperi- metrical problem) that the ratio of the centrifugal force to gravitation determines the ellipticity. Half a century later, that which no one before knew to be true, which many probably considered to be erroneous, was examined by one of his most distingnished followers, Maclaurin, and demonstrated most satisfactorily. Newton had not failed to perceive the necessary effects of gravitation in producing other phenomena beside the regular motion of the planets and their satellites, in their course round their several centres of attraction. One of these phe- nomena, wholly unsuspected before the discovery of the general law, is the alternate movement to and fro of the earth's axis, in consequence of the solar (and also of the lunar) at- traction combined with the earth's motion. This Libration, or Nutation, distinctly announced by him as the result of the theory, was not found by actual observation to exist till sixty years and upwards had elapsed, when Bradley proved the fact.* * The Nutation, and by name, is given in Prin. Lib. m. prop. 21, the demonstration being referred to as in Lib. I. prop. 66, cor. 20. Clairaut, ' Princ. de Du Chatelet,' tom. II. p. 72, 73, refers to the same proposition. P. Walmsley, ' Phil. Trans.' 1746, has an excellent paper on Precession and Nutation, treated Geometrically. It is stated in Montucla, IV. 216, GRANTHAM ADDEESS. 289 The great discoveries wMch have been made hj Lagrange and Laplace upon the results of disturbing forces, have estab- lished the law of periodical variation of orbits, which secures the stability of the system by prescribing a maximum and a minimum amount of deviation ; and this is not a contingent but a necessary truth, deduced by rigorous demonstration as the inevitable result of undoubted data in point of fact — the eccen- tricities of the orbits, the directions of the motions, and the movement in one plane of a certain position. That wonderful proposition of Newton,* which with its corollaries may be said to give the whole doctrine of disturbing forces, has been little more than applied and extended by the labours of succeeding geometricians. Indeed, Laplace, struck with wonder at one of Newton's comprehensive general statements on disturbing forces in another proposition, f has not hesitated to assert, that it contains the germ of Lagrange's celebrated inquiry, exactly a century after the ' Principia ' was given to the world. J The wonderful powers of generalization, combined with the boldness of never shrinking from a conclusion that seemed the legitimate result of his investigations, how new and even startling soever it might appear, was strikingly shown in that memorable inference which he drew from optical pheno- mena, that the diamond is 'an unctuous substance coagu- lated ;' subsequent discoveries having proved both that such substances are carbonaceous, and that the diamond is that Eoemer had given some conjectural explanation of the phenomena of what he temied vacillation; but no date is assigned — Koemer died in 1710. In the same passage it is said that before Bradley's discovery, Newton had " suspected the nutation." He had deduced it from the propositions above referred to, and was considered so to have done by Clairaut. Bradley's paper was published in the 'Phil. Trans.' 1747; and it is not a little singular that he makes no mention at all of Newton. * Lib. I. Prop. LXVI. + The XVIIth's two last Corollaries. t ' Mem de Berlin,' 1786, p. 253, is the memoir referred to by Laplace. The memoir is by Duval le Koi, but adopted by Lagrange as a supplement to his two memoirs, 1782 and 1784. U 290 SIE ISAAC NEWTON. crystallized carbon ; and the foundations of mechanical chemistry were laid by him with the boldest induction and most felicitous anticipations of what has since been effected.* The solution of the inverse problem of disturbing forces has led Le Verrier and Adams to the discovery of a new planet, merely by deductions from the manner in which the motions of an old one are affected, and its orbit has been so calculated that observers could find it — nay its disc as measured by them only varies one twelve-hundredth part of a degree from the amount given by the theory. Moreover, when Newton gave his estimate of the earth's density, he wrote a century before Maskelyne, by measuring the force of gravitation in the Scotch mountains, 1772, gave the proportion to water as 4-716 to 1 — and many years after by experiment with mechanical apparatus Cavendish, 1798, corrected this to 5'48, and Baily more re- cently, 1842, to 6-66, Newton having given the proportion as between 5 and 6 times. In these instances he only showed the way and anticipated the result of future inquiry by his followers. But the oblate figure of the earth affords an example of the same kind, with this difference that here he has himself perfected the discovery, and nearly completed the demonstration. Prom the mutual gravitation of the particles which form its mass, combined with its motion round its axis, he deduced the proposition that it must be flattened at the poles ; and he calculated the proportion of its polar to its equatorial diameter. By a most refined process he gave this * ' Optics,' Book II. prop. 10. — It might not be wholly ■without groimd if we conceived him also to have concluded, on optical grounds, that water has some relation to inflammable substances ; for he plainly says that it has a middle nature between unctuous substances and others ; and this he deduces from its refractive powers, though he gives other reasons in con- finnation. — In the celebrated 31st Query, Book III. (p. 355), he plainly considers rusting, inflammation, and respiration, as all occasioned by the acid vapours in, which he says the atmosphere abounds. In another place he treats of electricity as existing independent of its production or evolu- tion by friction. — Black always spoke of that Query with wonder, for the variety of original views which it presents on almost every branch of chemical science. GRANTHAM ADDRESS. 291 proportion upon the supposition of the mass being homogene- ous. That the proportion is different in consequence of the mass being heterogeneous does not in the least affect the soundness of his conclusion. Accurate measurements of a degree of latitude in the equatorial and polar regions, with experiments on the force of gravitation in those regions, by the different lengths of a pendulum vibrating seconds, have shown that the excess of the equatorial diameter is about eleven miles less than he had deduced it from the theory ; and thus that the globe is not homogeneous : but on the assumption of a fluid mass, the ground of his hydrostatical investigation, his proportion of 229 to 230 remains unshaken, and is precisely the one adopted and reasoned from by Laplace, after all the improvements and all the discoveries of later times. — Surely at this we may well stand amazed, if not awe-struck.* — A century of study, of improvement, of dis- covery has passed away ; and we find Laplace, master of all the new resources of the calculus, and occupying the heights to which the labours of Euler, Clairaut, D'Alembert, and Lagrange have enabled us to ascend, adopting the Xewtonian fraction of one two-hundred-and-thirtieth, as the accurate solution of this speculative problem. New admeasurements have been undertaken upon a vast scale, patronised by the munificence of rival governments ; new experiments have been performed with improved apparatus of exquisite delicacy ; new observations have been accumulated, with glasses far exceeding any powers possessed by the resources of optics in the days of him to whom the science of optics, as well as dynamics, owes its origin ; the theory and the fact have thus been compared and reconciled together in more perfect * The wholly erroneous measurement of an arc by the two first Cassinis, (Dominic and James,) was supposed to prove the shortening of the degree towards the poles, in opposition to the Newtonian theory. But all doubt on the subject was set at rest by the admeasurement in Peru in 1735, and in 1 Lapland in 1736, and in France more recently. But the error of Dominic and James Cassini was also corrected by the Oassini de Thury, who found that it had arisen from an imperfect measure employed. U 2 292 SIK ISAAC NEWTON. harmony ; but that theory has remained unimproved, and the great principle of gravitation, with its most sublime results, now stands in the attitude, and of the dimensions, and with the symmetry, which both the law and its application received at once from the mighty hand of its immortal author. But the contemplation of Newton's discoveries raises other feelings than wonder at his matchless genius. The light with which it shines is not more dazzling than useful. The diffi- culties of his course, and his expedients, alike copious and refined, for surmounting them, exercise the faculties of the wise, while commanding their admiration ; but the results of his investigations, often abstruse, are truths so grand and comprehensive, yet so plain, that they both captivate and instruct the simple. The gratitude, too, which they in- spire, and the veneration with which they encircle his name, far from tending to obstruct future improvement, only pro- claim his disciples the zealous, because rational, followers of one whose example both encouraged and enabled his successors to make further progress. How unlike the blind devotion to a master which for so many ages of the modern world para- lysed the energies of the human mind ! — " Had we still paid that homage to a name Which only God and nature justly claim, The western seas had been our utmost bound, And poets still might dream the sun was drown'd, And all the stars that shine in southern skies Had been admired by none but savage eyes." — {Dryden.) Nor let it be imagined that the feelings of wonder excited by contemplating the achievements of this great man are in any degree whatever the result of national partiality, nor confined to the country which glories in having given him birth. The language which expresses her veneration is equalled, perhaps exceeded, by that in which other nations give utterance to theirs ; not merely by the general voice, but by the well-considered and well informed judgment of the masters of science. Leibnitz, when asked at the royal table in Berlin his opinion of Newton, said that " taking mathe- GRANTHAM ADDRESS. 293 maticians from the beginning of the world to the time M'hen Newton lived, what he had done was much the better half." — ■ " The ' Principia' will ever remain a monument of the pro- found genius which revealed to us the greatest law of the universe," * are the words of Laplace. " That work stands pre-eminent above all the other productions of the human mind." f " The discovery of that simple and general law, by the greatness and the variety of the objects which it embraces, confers honour upon the intellect of man." | — Lagrange, we are told by Delambre, was wont to describe Xewton as the greatest genius that ever existed ; but to add how fortunate he was also, " because there can only once be found a System of the Universe to establish."§ — " Never," says the father of the Institute of France, one filling a high place among the most eminent of its members — " Never," says M. Biot, " was the supremacy of intellect so justly established and so fully con- fessed."|| "In mathematical and in experimental science Twthout an equal and without an example ; combining the genius for both in its highest degree." If The ' Principia ' he terms the greatest work ever produced by the mind of man, adding in the words of Halley that a nearer approach to the Divine nature has not been permitted to mortals.** — " In first giving to the world Newton's method of fluxions," says Fontenelle, " Leibnitz did like Prometheus — he stole fire from Heaven to teach men the secret." || — " Does Newton," L'H&pital asked, " sleep and wake like other men ? I figure him to myself as of a celestial kind, wholly severed from mortality." To so renowned a benefactor of the world, thus exalted to the loftiest place by the common consent of all men, one whose life without the intermission of an hour was passed in the * ' Syst. du Monde,' V. 5. t lb. V. 5. t lb. rv. 5. § ' Mem. de L'Instit.' 1812, p. XLIV. II ' Joum. de Sav.' 1852, p. 135. f ' Journ. de Sav.' 1852, p. 279. ** lb. 1855, p. 552. " Nee fas est propius mortali attiagere divos.' ft ' Aoad. des Sciences,' 1727. 294 SIR ISAAC NEWTON." searcli after truths the most important, and at whose hands the human race had only received good, never evil, those nations have raised no memorial which erected statues to the tyrants and conquerors, the scourges of mankind ; whose lives were passed not in the pursuit of truth but the practice of false- hood; across whose lips, if truth ever chanced to stray towards some selfish end, it surely failed to obtain belief; who, to slake their insane thirst of power, or of preeminence, trampled on all the rights, and squandered the blood of their fellow-creatures ; whose course, like the lightning, blasted while it dazzled ; and who, "reversing the noble regret of the Eoman Emperor, deemed the day lost that saw the sun go down upon their forbearance, no victim deceived, or betrayed, or oppressed. That the worshippers of such pestilent genius should consecrate no outward symbol of the admiration they, freely confessed, to the memory of the most illustrious of men, is not matter of wonder. But that his own countrymen, justly proud of having lived in his time, should have left this duty to their successors, after a century and a half of professed veneration and lip homage, may well be deemed strange. The inscription upon the Cathedral, masterpiece of his cele- brated friend's architecture, may possibly be applied in defence of this neglect. " If you seek for a monument, look around."* If you seek for a monument lift up your eyes to the heavens which show forth his fame. Nor when we re- collect the Greek orator's exclamation, " The whole earth is the monument of illustrious men," f can we stop short of declaring that the whole universe is Newton's. Yet in raising the Statue which preserves his likeness near the plaee of his birth, on the spot where his prodigious faculties were unfolded and trained, we at once gratify our honest pride as citizens of the same state, and humbly testifiy our grateful sense of the Divine goodness which deigned to bestow upon our race one so marvellously gifted to comprehend the works * " Si monumentum quseris, circumspice." (On Wren in St. Paul's.) t Pericles. (Thuc. H. 43.) GKANTHAM ADDRESS. 295 of Infinite Wisdom, and so piously resolved to make all his study of them the source of religious contemplations, both philosophic and sublime. Besides tlie remarkable solution of T. Simpson, p. 53, his Tracts contain other singular anticipations. A very learned person (Mr Jerwood of Exeter) has pointed out a distinct anticipation of Lagrange's celebrated formula on the stability of the System. Nothing can be more delight- ful than contemplating the signal success of self-taught men, as Emerson was nearly — T. Simpson altogether — and both in humble circumstaijces. NOTES. Note L, pp. 18, 23. The demonstration of the XXVIIIth Lemma, Principia, lib. I., has been generally admitted to be inconclusive ; there being many curves which can be squared and rectified returning into themselves, and not falling within the exception in the Lemma, of curves having an oval, with infinite branches. Thus the whole of the figures whose equation is are quadrable when m is an even number ; for f ydx = f n a;""' (a" — a;")™ d x is integrable, because the power of x without the radical sign is one less than the power within ; and yet the curve can have no asymptote, because there is no divisor ; while it is plain that the — root of a" — x" is impossible when either m + a; or — a; is greater than a, n and m being both whole even numbers. Therefore the curve returns into itself; and, as y = both when x = and when a; = + a, or — a ; therefore the curve consists of two ovals touching at the origin. These are quadrable ; for the integral y d x is G - ——--(a'- - X") . - . n{m+ I) 298 NOTES. The curve considered at length in Tract V. is another instance of the failure of the XXVIIIth Lemma; for that line continues through the cusps and returns into itself, though not, strictly speaking, an oval. — The cardiordeal also. The demonstration given, instead of Sir I. Newton's, in Tract I., is not exposed to the other objections which have been made to the Newtonian demonstration ; but it is equally liable to the objection now urged from the consideration of the equation to the class of curves whereof the lemniscata is one, and from the case of the curve described in Tract V., where the rectification is possible, as well as the quadrature. Perhaps we should extend the exception in the Lemma to curves which consist of two or more ovals touching each other, and to curves having cusps though without any infinite branches. Note II., pp. 23, 74. In the Encyc. xiii. p. 126, D'Alembert states Porism to be synonymous with Lemnia in the ancient writers, but he adds that lemma is the only word used in modem times. His definition is not inaccurate as applied to lemma, a proposition of which we have need in order to pass to another more, im- portant ; and on this he grounds his notion of porism, from TTopof, passage. Under the word Poristique in another part of the Encyc, he gives a different definition of porism. Some authors, he says, call by this name the description " de la " maniere de determiner par quels moyens, et de combien,,de " diff&entes fajons un probleme pent 6tre resolu." Nothing can be deduced from the Greek for passage, because the word itopiafia is plainly not derived from Tropos, but from ■TTopi^io, which rather sanctions the opinion connecting porism with corollary, than the opinion in the text of a transition from determinate to indeterminate. That the ancients some- times used the word as synonymous with corollary there can be no doubt. The subject has been handled incidentally by one of the most eminent geometricians of our day, M. Chasles, with his NOTES. 299 wonted learning and perspicacity, in his celebrated treatise, Geometrie Superieure, Introd. p. xxi., and there is good reason to hope that a more full discussion will accompany his work lately announced as in preparation, the restoration of Euclid's three books (ies trois Livres de Porismes d'EucMe retablis pour la premiere fois) M. Chasles puts in the front of his title that the restoration is effected after the notice and lemmas of Pappus, and conformably to the view of Simson, touching the form of the enunciation of the propositions. Nevertheless, his theory differs in some particulars from that of Simson. It has been highly gratifying to find that this great geometri- cian refers with approval to a porism in the First Tract (prop, vii.) which he considers to throw light upon one of the kinds of porisms described by Pappus as belonging to Euclid's Third Book. Perhaps he will cast an eye upon an illustra- tion of the views entertained on this subject in Tract III. It seems essential to the formation of a porism that there should be a transition from determinate to indeterminate, a change in the data which makes the problem indeterminate, and so capable of innumerable solutions. Take very simple and elementary cases. — Suppose the problem is to find in the diameter of a circle produced, a point such that the line drawn from it to a given point in the circumference shall have its square equal to the rectangle under the diameter produced, and ^he portion of it between the circle and the point to be found. 'Call the diameter a, the portion without the circle d, and the line cutting the circumference /, then /' = (a -|- d') d. Let y be the ordinate, and x the abscesse, to the given point in the circumference, then also /2 = (^d+xy + ax -x'' = ^-{- 2dx + a.x; and d = — ^^— . But if the point in the circumference is such that 2(ax-x'') , ax , 2(ax-x') d + x = -^ ; then -r— + x = -^ , and a — 2x a — 2x a — 2x 2ax — 2x' = 2ax — 2x^; or the point in the diameter pro- duced is found whatever be the point in the circumference ; '300 NOTES. and also every point in the diameter produced gives a line cutting the circumference, and whose square is equal to the rectangle of the segments of the diameter. So the data may be such as to render the solution im- possible, and a change of these data making the solution indeterminate, a porism results. Thus, let it be required to draw from a given point in the diameter produced a line cut- ting the circumference, such that its square shall be equal to the rectangle contained by the whole line and that portion of it between the point and the ordinate to the point where the circumference is cut; then there is no such point of the diameter beyond the circle, because the square of the line drawn to cut the circumference must always be less than the rectangle under the segments of the diameter ; hntf being as before = (f -{- rdx + ax, and being also = to (a + d) (d + w) we have