DIFFICULTIES OF ELEMENTARY GEOMETRY. LONDON: IRCIIARD CLAY, PRINTER, BREAD STREET 1-1LL. THE DIFFICULTIES OF ELEMENTARY GEOMETRY, ESPECIALLY THOSE WIIICH CONCERN THE STRAIGHT LINE, THE PLANE, AND THE THEORY OF PARALLELS. BY FRANCIS WILLIAM NEWMAN, FORMERLY FELLOW OF BALL1OL COLLEGE, OXFORD, LONDON: WILLIAM BALL AND CO. 34, PATERNOSTER ROW. 1841. The reader is requested, on reading Art. 123, to peruse Appendix II., and at Artt. 55, 150, to refer to the Note at the end of the Volume. CONTENTS. PAGE INTRODUCTION................ 1 PART I. ON THE STRAIGHT LINE AND PLANE. ARTICLES 1-7. On Space generally.... 9 8-17. Volume, Area, Length. Comparison of any two lines. 12 18-22. Equality in Unlike Shapes....16 23-25. Distance, or, Least Length. Equidistance, or Parallelism of Lines and Surfaces. 19 26, 27. Laws of Rotation...23 28-32. Circles; Surfaces of Revolution. Proportional Arcs of Different Circles..... 24 33-43. Digression concerning Proportionals...27 44-58. Spheres; Poles of Sphere, and of a Circle on a Sphere. 30 59-68. Axes; STRAIGHT LINES.... 36 69-78. Cylinders; Centre of Circle... 39 79-84. Planes; their Primary Property....42 85-99. Curved Lines; Tangent, an4 Cusp, or Peak. CusPS ARE SINGULAR POINTS, 97.....44 100-111. Rectilinear Angles. (Supplements, Perpendiculars.). 53 112-119. Measurement of Rectilinear Angles.. 57 vi CONTENTS. ARTICLES PAGE 120-125. Their Periodicity, and its results... 61 126-133. Curvilinear Angles; how compared and measured. Their Orders......66 134-156. Curvature of Circles and Spheres. No PEAK IN A CIRCLE. 77ie Tangent perpendicular to the Radius. The Tangent does not meet the Circle again. A Perpendicular is the Shortest Path from a given Point to a given Straight Line, 143. Tangent Plane to a Sphere. Convexity and Concavity, 153-156.....71 157-159. STRAIGHT LINE THE SHORTEST PATH BETWEEN TWO GIVEN POINTS......81 160-162. Contact of Spheres..... 82 163-165. Intersection of Sphere and Plane.. 84 166-168. Intersection of Spheres... 8 169. Triangle of Distances... 88 170. EVENNESS OF THE PLANE.... 9 171-179. Consequent Properties of the Plane... 90 180. Parameters..... 92 PART II. ON PARALLEL STRAIGHT LINES. 181-184. Errors from Excentricity, Evanescent when the Radius increases without Limit.....93 185. The three External Angles of any Triangle are together equal to four Right Angles... 95 186-190. Chief Properties of Parallels.... 96 PART III. ON SOLID ANGLES. 192-199. Their Definition: and a Proof that their Magnitude can be measured by SphericalAreas subtending theml. 101 CONTENTS. vii Vll PART IV. ON PLANE CURVES. ARTICLES PAGE 200-206. Normals and Points of Undulation.. 106 207, 208. Deviation.....108 209-213. Length of a Curve, compared to the Sum of its Chords or Tangents..... 108 214-217. Curvature at opposite sides of the same Point in the same Curve..... 111 218. Measure of Circular Curvature....112 219. Finite Curvature......113 220, 221. The Changes of Curvature are ordinarily gradual. 114 222. Radius of Curvature.. 116 223-225. Remarks on the Evolute... 117 PART V. ON DOUBLE CURVATURE. 226, 227. On the Co-existence of two Curvatures, and the Analysis back again of the same.....120 228-230. Osculating Plane.....121 231, 232. Principal Normal..... 122 233. Second Curvature, and its Measurement... 122 234. Developable Surface..... 123 235. Osculating Sphere......124 236-239. Surface of Evolutes.....124 PART VI. ON CURVED SURFACES. 240-245. Every Curvilinear Area may be distributed into portions so small, that each may differ from a Plane Surface as little as may be required. 127 Viii CONTENTS. PART VII. SHORTEST PATH ON A SPHERE. ARTICLES PAGE 246-248. The Shortest Path, either in Space, or along a given Surface, from a given Point to a given Curve, or, between two given Curves, is Perpendicular to the Curve, or to both Curves.......131 249. On a Surface of Revolution, the Shortest Path to connect two Points in its Plane Generatrix lies along the Generatrix itself 132 250. Application of the last to the Sphere.. 134 251-255. SCHOLIUM: On the Order to be observed in composing a continuous Treatise on Elementary Geometry. 134 APPENDIX I. On Lieut.-Col. Thompson's Proof of the Theory of Parallels by help of the Equiangular Spiral... 139 APPENDIX II. On Legendres' Treatment of the Doctrine of Parallels. 141 NOTE on Articles 55, 150.. 143 INTRODUCTION. THIS book consists of extracts from one which was intended to form a continuous system of Elementary Geometry; but as the author finds no reasonable ground for hoping that any one would adopt his system as a whole, he has determined on selecting those parts which are either wholly new, or wanting in the common treatises. In this form they may be read as supplementary, by a student who has gone through Euclid; yet the endeavour has been made so to arrange them that no part shall be unintelligible to a person who may have no previous acquaintance at all with Geometry. He anticipates that objections will be made on two heads to the methods which he has employed: to the introduction of Mlotion into Geometry, and to the early use of the doctrine of Limits. It is said by many that Motion belongs solely to Mechanics, and not to Geometry; but this is a mere dogma, to which it is difficult to find reason for deferring. It is true that in Mechanics the doctrine of Motion is treated, but is treated on a perfectly different footing. In Geometry we pay no attention to Velocity, nor are we concerned with Measurements of Time, nor do we consider Motion as an effect of Force. We regard merely the successive changes of position which a body undergoes; and although we know that such changes B L-) 0 t;W INTRODUCTION. require both Time and Force, we are not concerned to estimate that Time nor that Force; but we abstract these considerations as irrelevant to the subject. Now the points which are purposely omitted in Geometry are specially discussed by Mechanics; nay, form the sole business of that branch of Mechanics which contemplates, Motion at all. Hence we are perfectly clear from the charge of intruding on the province of Mechanics. This method has been deliberately preferred, from the conviction that no definition of a geometrical figure is so vivid to the understanding, or so satisfactory in a logical point of view, as that which states how the figure is to be generated. Unless this can be done, the mind is justly in suspense, whether the definition may not have laid down something self-contradictory and absurd. It may be added that the common systems, from Euclid downwards, introduce the same thing in disguise, and cannot do without it. Geometers call it Supraposition; and in the very first theorem of the science it is employed. Yet it might seem as if Euclid had been ashamed of it; for he does not employ it afterwards, in numerous cases where it would have made his proofs clearer and more concise. If, however, it may be used once, it may be used a thousand times; and ought to be used, whenever such advantages are to be gained. If any one objects to the early use of the doctrine of Limits, it will not be as though it were illogical, but because it is imagined to belong rather to the higher Geometry. If this remark means merely to state the fact, that hitherto it has not been used in the Elements, the writer can see no reason why the beaten track should be held sacred, if a better offer itself. To him it appears that the notion of a Limit enters into the very first conceptions of Geometry, (as of a surface, a line, and a point,) and is essential to the establishment of those LAWS, on which he believes the science to rest. It is equally essential to an understanding of the doctrine of Ratio and INTRODUCTION. 3 Proportion, if incommensurable quantities are to be treated with logical accuracy. Nor does it seem to involve any difficulty comparable to those with which the Elements of Geometry abound. The Lemma concerning Proportions has been added, as appearing to him the most convenient link between the doctrines of Proportion and Magnitudes. It consists in nothing but the first and simplest problem of the Integral Calculus in disguise. He has to acknowledge his obligation to Mr. Perronet Thompson's " Geometry without Axioms," (4th edition,) for two very valuable hints, which have materially influenced the form in which the First Part of these discussions now appears; namely, First, to regard the doctrine of the Sphere as prior to that of the Plane; Secondly, to pay peculiar attention to such lines and surfaces as are capable of sliding along themselves. But in the actual execution of the plan there is here little or nothing in common with the method which Mr. T. has chosen. GENERAL PLAN OF THE FOLLOWING TREATISE. We design to discuss successively certain points which appear to be defective in the Elements of Geometry. FIRST, The doctrine of the Straight Line and Plane. In the common treatises a Straight Line is defined as " one which lies evenly between its extreme points;" but as the word evenly has not been explained, this is not more instructive than to be told that it lies strcighltly between its extreme points. A confession of the uselessness of the definition is found in the device of an axiom, that " two straight lines cannot enclose a space;" which ought to be a corollary from the definition, if the latter were adequate. Some have defined a Straight Line to be " the shortest path between two given points;" but this is not legitimate, till it have been proved that there is always some one path B 2 4 INTRODUCTION. shorter than all other paths. If any one choose to resort to a very simple experiment in proof of this, he will at once cut short all difficulties attending the doctrine both of Straight Line and of Plane. And this is, perhaps, the course which all our minds secretly follow. But it is thought right to appeal to experiment as little as possible; and perhaps the above appeal is not necessary. The method used below of explaining the term evenly is fundamentally the same with that which Professor Leslie suggested; but is much more developed. The definition of the Plane found in Simson's Euclid and elsewhere, labours under the serious fault of being redundant. A Plane (say they) is a surface, such that, if any two points whatever be joined by a straight line, this line shall lie wholly upon the surface. But how are we to know that such a surface is possible? Let us try to generate such a one. Take two straight lines, (Fig, 1.) Fig. 1. A O B, C OD, intersecting each other A\ ^ a, mOin 0. In OA, OD, /^~P ^^(or else in their prolongations OB, 0 C,) take two D/tQ M OB points, P and Q; and first, let the distances O P, 0 Q, be in all cases equal; and through P and Q pass a straight line. Then if the distance OP increase indefinitely, and again diminish indefinitely, the straight line P Q, moving with it, traces out a surface; which surface has the property, that " if any two points in it that lie in the same generatrix PQ be joined by a straight line, this line lies wholly upon the surface." Now it remains to prove that the same will be true when the points joined are neither on the same generatrix, nor in the lines A B, CD. But secondly, if to meet this difficulty we suppose OP and O Q not to be equal, but to bear some other ratio, or to vary INTRODUCTION. 5 independently, it will then be no longer manifest that the locus or surface, generated by the motion of P Q, is a single continuous sheet, and not an infinity of different surfaces. This, which needs to be proved, is assumed, and that, covertly, in the common definition. Some have put into its place the definition, that " a plane is a surface, which lies evenly between its extreme boundaries." But this is doubly objectionable, both from the vagueness of the term evenly, and from the want of proof that there is any one such surface; to say nothing of our inability to decide what is meant by " extreme boundaries." The outline may be called " the extreme boundary;" but what are the " boundaries" in an oval curve, for example? It is to meet these difficulties, about the Straight Line and Plane, that our FIRST PART is intended. To follow the methods employed in the common treatises is impossible; for they assume, from the beginning, the very properties which we want to prove. It is well, however, here to remark, that the proposition at which we are secretly driving is, that which Euclid has made his 8th; namely, that when the lengths of the three sides of a triangle are given, the shape of the triangle is hereby entirely determined. The importance of this will be clearly understood when it is remarked, that the same thing is not true of a four-sided figure; for if the lengths A B, B C, CD, DA, (Fig. 2,) were alone given, there is nothing to hinder the figure from assum- c, ing different shapes, as the diagram shows, by a change of the size of the angles. If A D any one choose to resort to experiment to establish this peculiarity of the triangle, this would be a way equally effectual with that suggested above, of cutting short our First Part. There is another defect, less fundamental, yet not unimportant, in this part of the common treatises, in their neglecting to establish any satisfactory principles to regulate 6 INTRODUCTION. the addition and subtraction of angles. It is taken for granted that if two angles, A OB, BO C, (Fig. 3,) be laid Fig. 3. down, side by side, on a plane, the angle A A 0 C may fitly be regarded as a sum of the other two. But why on a " plane " in parV ---- ticular? The very word sum implies that angles are quantities; or are resolvable into parts as small as we please, all homogeneous to each other; and conversely, that angles may be generated from the accumulation of parts indefinitely small. But nothing is laid down or proved concerning them in the Elements, as usually treated, to justify and establish this view. The subject is closely connected with that of tle Shortest Patl that joins two points on a Sphere, and that of the contact of two cones. The addition of angles may be founded on either of these doctrines, if it be judged convenient to abstain so long from every allusion to angles as quantities made up of parts. SECONDLY, The endeavour has been made to remove the celebrated difficulties embarrassing the doctrine of Parallel Straight Lines. Euclid's method of disposing of it would be honest, and so far good, if the ambiguity of the Greek term Axiom had not led to the annexing of the 12th axiom (so called) to others perfectly unlike it in kind. It might be called datoxwa, a " Postulate," or Assumption, with much propriety, and no student would demur to grant it. Yet it must tend to throw light on the philosophical basis of the science, either to demonstrate this, or to prove that no demonstration is to be looked for; neither of which seems yet to have been done, so as to satisfy geometers generally. To the writer it had always appeared that the illustration offered by Professor Playfair, of the equality of the three external angles of a triangle to four right angles, contained the germ of an unexceptionable demonstration of the same. This, accordingly, he has endeavoured here to exhibit. As the proof is concise enough, it is, if logically unimpeachable, practically deserving of acceptance. INTRODUCTION. At the same time he is so deeply convinced that every geometer secretly settles all questionings in his own mind concerning the truth of Euclid's 12th axiom, by appealing to the doctrine of proportions, as to induce him to suspect that future inquirers may succeed in obviating every objection which has been urged against Le Gendre's method. That in descending a sloping path, Fig. 4. (Fig. 4,) we make equal vertical descents, by traversing equal dis- tances along the path, is a truth of which the mind seems to possess itself before it attains to the belief that the slope may be carried so far as to descend to any required level; which latter is substantially Euclid's 12th axiom. And whether the former proposition can or cannot be established abstractedly, as by Le Gendre's triangles; in any case, the writer is persuaded that the latter should be proved by the former, and not, in the reverse order which Euclid follows, the former by the latter. THIRDLY, The method of Measuring the Solid Angle is treated; not because it has any real difficulty, but because it is rather unceremoniously slurred over in the common treatises. FOURTHLY, FIFTHLY, and SIXTHLY, Some propositions concerning Plane Curves, Double Curvature, and Curved Surfaces, have been demonstrated, which are generally assumed without proof, and are of no little importance in the higher Geometry. SEVENTHLY, The Shortest Path on a Sphere has been treated, with a view chiefly to the question concerning the Addition of Angles. The intelligent reader will probably remark of himself, that as the main difficulty of Parallel Straight Lines is identical with that of proving that no finite arc of a curve can have its curvature every where infinitely less than that of a circle; so the difficulty of demonstrating the evenness of the plane, is here virtually reduced to that of proving that no finite arc can have an infinity of cusps. 8 INTRODUCTION. Throughout, it has been endeavoured to handle every topic in such a way as to prepare the mind for that large view which must be taken in the higher mathematics; for which, naturally and necessarily, the works of a Greek geometer are wholly unfit. Especially, the undue contraction of definitions has been avoided; nor has the writer felt it requisite so to press on towards the end mainly sought, as not to tarry on collateral subjects which would tend to illustrate the matter in hand, and which in themselves are worthy of being known. Least of all can he persuade himself, that the unbending formality which characterises the Greek geometers,-the affected disdain to notice difficulties, to obviate misconceptions, to offer illustrations,-in any degree conduces to soundness of demonstration. Certainly, whatever helps the student to get vivid and distinct notions, helps towards logical reasoning, which must be in the mind, not on the paper; for the dead letter does but give hints for the mind to seize; and it is a strange feature in modern mathematics, that, as if to discourage beginners, a more repulsive and unexplanatory style is adopted in Geometry than in any other branch. Yet, with all this sacrifice for the attainment of imaginary " rigour," there is no department of exact science so full of fundamental flaws. If the reason be asked, perhaps none better will be found, than that we assiduously cultivate our Mechanics and Hydrostatics, our Algebra, our Calculus, anxiously removing their defects in successive generations, by help of the fresh light constantly poured in; while in Geometry we have set up one of the ancients for our idol, and have cramped the science in its adult state by the trammels of its infancy. DIFFICULTIES OF ELEMENTARY GEOMETRY. PART I. ON SPACE GENERALLY. 1. GEOMETRY is a particular branch of the science of Quantity, namely, that which is concerned with Space. 2. The difference in principle between this science and Mechanics or Hydrostatics, is perhaps not so great as is often supposed. In the two latter, the mathematician speaks of bars absolutely inflexible, threads wholly inextensible, balls perfectly elastic, fluids void of viscidity, and so on; although he does not expect actually to meet such things. But he seizes a few of the prominent and most influential properties of matter, and stipulates to drop the rest, at least for a while, and argue as if they had no existence. Thus the things of which he speaks are not such as are found in nature, but are imaginary limits, towards which nature only approaches more or less. Afterwards, in adapting his science to practice, he has to make allowance for the deviation, and, if possible, complete his theory by taking in the circumstances before omitted. Just so the geometer proceeds. He finds before him bodies of different material,-stone and wood, iron and silver,-but he drops all consideration of this point. They differ in weight and in colour; but this too he neglects. 10 DIFFICULTIES OF ELEMENTARY GEOMETRY. He regards solely their size and shape. Again: he sees some to be round and others square; and although on close inspection each may be found to have irregularities, being neither quite round nor quite square, he drops this circumstance also. He invents for himself shapes simpler than any found in nature, and which are mere limits more or less distant from the realities of the world. In consequence, his reasonings may possibly mislead him, when applied to practice, because what he actually encounters proves to be not precisely that of which he has been treating. As no one, without the experience of sensible Forces and sensible Fluids, could form the notion of mathematical Forces and Fluids, so neither, without the use of the senses, to give us experience of actual matter, could we arrive at the mathematical conceptions which are at the basis of Geometry. It is not without touch that we gain first the idea of Extension; as indeed also of Length, Breadth, and Thickness; of Protuberance, of Flatness, of Hollowness, of Pointedness. The earliest exercise of a baby's fingers and lips is, to assist in acquiring such notions; which gives a fair apology to those who would call geometrical notions innate, as they are probably the first that enter the mind at all. 3. We find, moreover, that those objects which resist our touch do also mutually resist each other. Hence rises the apprehension that each occupies a certain Space of its own, into which a second body cannot intrude, without displacing the former. We are farther thus enabled to neglect all consideration of the material of which bodies consist, and even the fact of substantial existence. For solid bodies we substitute the empty space which they might occupy. Consequently no absurdity is involved in speaking of two " solids" as penetrating each other, when neither has corporeal substance. 4. It is likewise allowable to imagine a solid to be transferred from one position into any other that can be described. For as by experience we learn the possibility ON SPACE GENERALLY. 11 of this in the case of numerous light and small bodies, we infer that a power may, without absurdity, be conceived, which might wield at pleasure the greatest and the heaviest. Much more, then, if we dispense with the idea of substance and weight in the body transferred, does all difficulty vanish. But, as was stated in the Introduction, it is no business of the geometer to treat on Time, (nor consequently on the Velocity of Motion,) any more than on Force. 5. The word Magnitude is employed universally to represent Geometrical Quantity, of whatever kind the quantity may be. All Quantity, and therefore Magnitude, is generally regarded as differing from Number, in being continuous. It is impossible to count, without leaving finite gaps between the numbers, as in 1, 2, 3, 4, where we proceed by units, or as in 1, 101, 102, 1'03, 0104, &c. where we proceed by hundredths of a unit: and this is Discontinuity. Whereas in weights, we conceive of every intermediate grade between one pound and two pounds; and in size, of every intermediate bulk between a cannon ball and the globe on which we stand. But the supposed difference is fictitious, and a needless source of perplexity. In the realities of life, Quantity as well as Number is discontinuous; while in theory, neither Quantity nor Number need be regarded as such. The mind which can suppose a quantity to increase from one value to another by finite increments as small as it pleases, can pass to an imaginary limit by a successive diminution of the increments, till it arrives at the idea of continuity. And in Arithmetic, we with equal ease conceive of continuous number; though we devise modes of expressing the intermediate values only so far as practical convenience dictates. Thus, " Continuity of Magnitude" is a theoretic limit invented by the mind. 6. Relative Magnitude. No magnitudes can be regarded as absolutely great nor absolutely little. There is no object so great but we can imagine its double; and of this 12 DIFFICULTIES OF ELEMENTARY GEOMETRY. latter the double again, and so on, until a magnitude is attained, such as to exceed the first in any proposed ratio. To suppose a termination of space bewilders the mind; and amazing as is the thought of space infinitely extended on all sides, yet we are incapable of conceiving a boundary beyond which space should not exist. Again: the least molecule that we can see or imagine, has opposite sides, separated by a determinate interval; and is in conception divisible into any number of parts. Hence, also, an object is conceivable, which shall be less in any required ratio than a given solid. 7. Actually to exhibit such multiples or submultiples, is sometimes an important problem with the practical geometer or mechanist. To graduate the arc of a large circle is a most delicate affair, of the highest value to astronomy. The fine screw which measures minute distances, is equally essential for accurate observations. But to theoretic geometry such matters are quite irrelevant. Appeal is made to the mind alone; diagrams are meant to assist the imagination; but expertness of manipulation is wholly needless, as far as the logical texture of the argument is concerned. Hence we should be perfectly at liberty to say:"Let the circumference of a circle be divided into 360 equal parts;" although we had not suggested by what instruments it could be done, even without gross and sensible inaccuracy. For the pure science, it suffices that no absurdity is involved in the conception. VOLUME, AREA, LENGTH. 8. By a Solid is then understood, any limited portion of space. Sometimes, however, it is convenient to attribute to it material existence, and, accordingly, to name it hard and inflexible, or to attribute to it joints, breakages, and such like. 9. The exterior boundary of a solid is called a Surface. VOLUME, AREA, LENGTH. 13 The boundary of a surface is called a Line. The extremity of a line is called a Point. 10. All three terms merely express a "limit," which the mind invents. The most obvious is the Surface, because we suppose that we see and touch the surfaces of all bodies, not being aware of that which Optics and Mechanics teach, that to the exercise of each sense some thickness is requisite in the surface which is to be seen or felt. But since the thickness may be lessened perpetually, and in any required proportion, the mind has no difficulty in imagining it wholly to vanish. Again: we conceive the diameter of a rope or string to be continually lessened, and the limit apprehended by the mind is a line. Similarly, we may suppose a solid to be perpetually diminished, till it attains a size barely appreciable to our senses. Thus, if from being as large as a cocoa-nut, it shrink successively into the size of a walnut, a pin's head, a grain of sand, we hereby readily pass to the limit, and form an idea of position independent of magnitude, in which consists the notion of a geometrical point. 11. By Volume, or Bulk, is understood the magnitude or capacity of a given solid, in comparison with that of some other, which is assumed as a standard or unit. Thus a numerical measure of volume is attained; just as when we say that a cask holds forty-two gallons; in which case the cask or jar of one gallon is the standard, or arbitrary unit. That any two solids admit of numerical comparison with each other is easy to perceive. For that all the parts of a solid are homogeneous to the whole, appears by considering, that if we repeatedly take away from the whole an exceedingly small magnitude, we may at last, as nearly as we please, exhaust the whole. And as any two solids, when placed side by side, may be regarded as one, it follows that they also are homogeneous to each other. 12. Area is the magnitude or extent of a given surface, compared with that of some other, assumed as a standard. 14 DIFFICULTIES OF ELEMENTARY GEOMETRY. It is not altogether so easy to show, in this stage, that any two areas admit of numerical comparison; or, what comes to the same thing, that all the parts of a surface are homogeneous magnitudes. Yet, by regarding a surface as entirely cut up into very small portions, we presently are able to pronounce that any one portion (A), in comparison with any other portion (B), must needs be either greater, or equal, or finally less. And if this were established, it would follow that all its parts are homogeneous. But a full proof of this is neither possible nor necessary, before the curvature of surfaces has been discussed. 13. Length is the magnitude of one line compared with that of another. It is of immediate importance to us to show that any two lines admit of numerical comparison. (Let it be observed, that although for the sake of illustration we may already speak of Length, Breadth, and Thickness, these are terms which cannot be at present employed with scientific propriety in opposition to each other. All three are at present merged in the one word, Length.) Since no magnitude is affected by change of position, neither will part of a line be hereby affected. We may then, without altering the magnitude of a line, suppose any part to be bent aside at any point; or, what is the same, we may imagine a joint to be introduced at any point, about which each portion may freely play. Now let several joints be supposed; in short, let the number of joints at every part of the line be continually increased; and neither does this imply any change of magnitude. Thus the mind approximates to the idea of a line, which is the theoretic limit of the above; namely, one which is perfectly flexible at every point. Such a line is called a Thread; and preserves the same magnitude (or length) in every position. In the place of any line under consideration, we may thus substitute a thread of equal length. Any two threads admit of direct comparison; and consequently any two lines (A, B,) are homogeneous magnitudes. A numerical VOLUME, AREA, LENGTH. 15 measure of them is obtained, by assuming one (as A) for the unit, which determines for the length of B some other number, whole, fractional, or approximate. 14. If two lines, A, B, (Fig. 5,) are equal in length, and we suppose, first, a limited number of Fig. 5. joints introduced in B; and then B to be A so applied on A as that one extremity of A i B both shall coincide (in in,) and that every /B joint moreover in B shall (as far as pos- A sible) fall on the line A; it is manifest B that by increasing perpetually the number of joints in B, the line B (which tends q more and more to become a thread,) will A B finally lie altogether along the line A. Let, m, p, q, r, s,... be the successive PB points of coincidence of the two lines; all of these points, except the first, being joints in B. It follows from the above, that the two paths which unite any two consecutive joints, (as qAr, qBr, which unite q and r,). tend more and more to perfect coincidence; so that the limit of the ratio of the two lengths, q A r: qBr, is absolutely 1: 1; when the number of joints perpetually increases in all parts of the line. 15. This conclusion, it must be observed, holds, whatever. may be the nature of the line B. It will not be vitiated, should B happen to be what we shall afterwards call a Straight Line. 16. Moreover if s is the farthest joint from m, Fig, 6. (Fig. 6,) the ratio of the line mAs: broken line mBs; (or again, of the whole line A to mBs,) has for its limit, 1:1. For mAs tends to A as r its limit, and mBs to the whole line B, while A A B and B, by hypothesis, are equal. q 17. It is convenient here to add, what is evidently comprised in the above, that if qn np q Irs be any line soever, and m n, np, pq, &c. be joined also by short lines, such as we shall hereafter S 16 DIFFICULTIES OF ELEMENTARY GEOMETRY. call Straight, then the sum of all the straight lines joining m to s has for its limit the length of the curve line m qs, if the number of points n, p, q, &c. intermediate to m and s, be perpetually increased in every part of the line. ON EQUALITY IN UNLIKE SHAPES. 18. This is perhaps the best place for bringing forward the various circumstances under which Equality is found among geometrical quantities. The three notions, Equal, Greater, Less, arise simultaneously in the mind. Each implies the others, nor is it possible to say which of the three ought to be defined first, were definition possible. But no definition of any can be given. 19. I. The simplest case of Equality is, when two objects have the same shape, as well as equal size; that is, when they are such that by a mere change of position one may be made precisely to occupy the place which was before held by the other. [Of this kind is the equality of two straight lines, or of two rectilinear angles, or of two circles, or of two curves that are equally curved the one with the other.] Magnitudes thus related are often called Identical, each being a perfect counterpart of the other. 20. II. When two magnitudes are separable into an equal number of parts, such that each of the one set has a fellow in the other set, to which it is identical. Fi. 7. Thus, let P and Q be two mag<A nitudes (Fig. 7); also, let { PA1 B.B P= A 42+ A 3+.... + An P Q=B Q =B + B2+ B3.... +;3 B B If then it be found that every A — A is identical with (and therefore equal to) its fellow B; we of Iv~ yur v/~~ r~rrruvv v ON EQUALITY IN UNLIKE SHAPES. 17 course pronounce that P = Q. For " the sums of equals are equal." [Such is the equality of two rectilinear plane areas, or of two plane-sided solids. In the Greek geometry the two cases of equality hitherto mentioned were regarded as the most rigid, and a silent effort was made to reduce all to these.] 21. III. When two magnitudes are the limits to which equal series perpetually approach. Thus, suppose that from P and Q, (Fig. 8,) are taken parts which are equal or identical; Fig.. as Al from P, B1 from Q; A where A1 = B1. Again, from 1 what remains take the equals A B A2and B2. From the remain- ders again take the equals A3/ 4 and B3, and so on. If, by re- / A peating this process continually, we can reduce both remainders to be as small as we please, then P must needs be equal to Q. By way of proof: Let p and v be the remainders after (n) subtractions; so that P = (Ai + A2 As ++ + An) + Q = (B1 + B2 + B3 + *. * * + Bn) + v Then, from the equality of each A to its fellow B, we get, P - Q = -. v [or, the difference of P & Q is equal to the difference of i & vi.] Now since y and v are each susceptible of being made as small as we please by increasing (n), much more is their difference. Hence P and Q have either no difference at all, or a difference which can be made as small as we please by increasing (n). But neither of them, nor therefore their difference, depends at all upon (n), which is the arbitrary number of subtractions. They have therefore no difference; or are equal. C 18 DIFFICULTIES OF ELEMENTARY GEOMETRY. More concisely: P is the limit of the sum of the A's, and Q is the limit of the sum of the B's. Hence P Q, because " the limits of equal sums are equal." [Such is the equality which subsists between any two plane areas, two areas on the surface of the same sphere, two solid angles, or two curve-sided solids.] 22. IV. When two magnitudes are separable into an equal, but variable, number of parts; such, that, (by increasing the number,) every ratio, of each part in the one set, to its fellow in the other set, approximates to the ratio, 1: 1, as its limit. Fig. 9. Thus, as in the second case of E-n,^^^ --- equality (Fig. 9,) let A/A B B P=Al +A2+... +A.; and e B A HA \ Q B. + B2+... + Bn i 5 B 5 P 4 Q [ B But instead of supposing (n) a fixed A\ o \B2 3B number, and the A's and B's fixed ALGA (B 2e, magnitudes, let (n) increase indefi1 A' nitely, and every A and every B diminish indefinitely. And instead of supposing every A to be precisely equal to its fellow B, let every one of the A1 A2 A 1 ratios -,,.... approximate towards as their B' B2' 2 3 1 limit. And we assert still that P = Q. The detail of proof belongs rather to Arithmetic than Geometry. It is, however,* easy to see that the ratio P A, + A2 +A- 3 + + An or B1, + B2 + B]3 +... + Bn must always be intermediate between the least and greatest A. A 2 A.2 An of the partial ratios, A-,-,.... and as none A 41 -8'2 ans non * If e is the greatest, and e the least, of the partial ratios, then A1, A2, A3... A,, are not greater than eBI, e B2, eB3 ~e ~B..B,; but some are less: consequently A1 4- A12 +... +- Ani is less than e (BI + B2 +.. * + B,); that is, P < e Q. Similarly, P > e Q, or s is less than c, but greater than e. DISTANCE. 19 of these have any finite difference from their limit 1. P neither can Q have any. But the last ratio does not 1' Q change with (n); hence it is absolutely = i. That is, P=Q. Two such equal magnitudes may be popularly said to consist of an equal infinitely great number of equal infinitely small parts. [Such are two equal lines or areas, of different curvature.] DISTANCE. 23. By Distance is understood " least length," under various circumstances. I. Between two points along a given surface. If A B are given points on a given surface, Fg. 10. (Fig. 10,) and along the surface ', numerous paths A CB, ADB, c A EB, are drawn to connect /A them, some of these paths may D be shorter than others. Yet there must exist one or more paths, than which none shorter can be found. Just as if A and B were two towns, to be joined by a road; there must needs be some limit to the possible diminution of the length of the road, unless indeed the towns were somewhere in contact. The length then of the path, than which no path joining A and B can be shorter, is called the Distance between A and B along the given surface. II. Between two points, when the path is not restricted to any particular surface. In the former case, perhaps, A EB may have been as short a path as possible; but now, (Fig. 11,) by drawing the path so as not to lie along the surface, it is conceivable that a yet F shorter may be found. If AFB is F as short as any possible, (and there E c 2 20 DIFFICULTIES OF ELEMENTARY GEOMETRY. must be some least length,) then the length of A FB is, absolutely, the Distance, (or the Distance in Space,) between A and B. Observe: We are not at present competent to assert, that there is necessarily one, and only one path, A FB, such as to be of this least length; although the mind readily persuades itself of this. But in fact, while the paths are restricted to a given surface, there may be many which have the shortest length. Thus on the globe, every meridian, joining the north and south poles, is equal to every other meridian. III. Between a point (A) and a line (D CEB) upon Fig. 12. a given surface (ig. 12). From A to the several points of the line, let there be drawn (along the sur^AC^^^ face) paths as short as possible. - If, then, of all these paths none is shorter than A C, the length of A C is the Distance (along the surface) of A from the line DB. IV. Between a point and a line, when the paths are not restricted to any surface (Fig. 13). gn The last case applies equally here, E dropping the restriction of the sur'A > eC face; in consequence of which the Distance (or least length A C,) is D probably shortened. V. Between two lines (AAA, BBB,) along a given surface (Fig. 14). From every point A in the one line, let there be drawn along the surFg'. 14'. ___ face a path as short as possible, to meet the other line. If of iA _y^ _ all these paths none be shorter al "^^ fthan a 3, the length of a j is AlW w ^ the Distance between the lines along the surface. DISTANCE. 21 Fig. 15. VI. Between two lines, when no sur- / face is given (Fig. 15). The explanation H of the last case will include this, if the a --- — surface be left out. VII. Between a point (A) and a surface (Fig. 16). If points B, C, D, on the surface, are assumed at random, and they are joined to A by paths A B, A C, AD, as short as possible; then -_ out of all such conceiv.able paths one (or more) A i=D is shorter than any other. A If A D is as short as any of them, its length is the distance of A from the surface. VIII. Between a line (A'AA') and a surface (Fig. 16). If from every point of A'AA' as short a path as possible is drawn to the surface, any one of these which proves to be as short as any of the rest expresses the distance between the line and surface. IX. Between two surfaces. In place of the line A'A A', in the last case, let a surface be introduced, and all which is there said will apply here. It is all along supposed that the points, lines, or surfaces, between which we are estimating the distances, have no part actually in common; not so much as a single point in common. Otherwise there is no shortest path, but the distance is said to vanish, or to be zero. 24. Parallelism. By this word is A Fg17 understood " Equality of Distance," 7 under several circumstances. J —I B IB B I. Parallelism of a line to line: as of AAA to BBB (Fig. 17). By K A this it is understood, that every point A in the one line lies at the same t2 DIFFICULTIES OF ELEMENTARY GEOMETRY. distance as every other point A, from the other line BBB. II. Parallelism of a surface to a surface. The same definition may be given here as in the former case, supposing only that A A A, BBB, now represent surfaces instead of lines. 25. We have as yet no way of ascertaining, whether to a given line or surface a second line or surface can be conceived such as to be parallel. But we shall very soon see examples of parallel lines and surfaces, and this will at once manifest that there is no intrinsic incongruity in that for which we have been inventing a name. Meanwhile it may be observed, that when two lines, or two surfaces, are parallel, the distance of the first from the second is the same as the distance of the second from the Fig. 18. first. In fact, if A B be a path, expressing A the distance of A from B BB, the same path BA, in an opposite direction, expresses likec wise the distance of B from A A A. Else, let B C be shorter than B A, C being a point in B \ A AAA. Then CB would be shorter than the A\ distance of every point in A AA from BBB. which is self-contradictory. It follows that the Parallelism is reciprocal; or that if A AA is everywhere equidistant from BBB, then so is BBB from A A A: provided that every point in B B B is the extremity of some shortest path drawn from A AA. III. Parallelism of a line to a surface. This means, that every point in the line lies at the same distance from the surface. But, in this case, there is no reciprocation; for not all points in the surface necessarily lie at the same distance from the line. [We might here proceed to explain the nature of Asymptotism; when two infinite lines have evanescent distance, without actually meeting. But it is of no importance to our present objects.] ROTATION. 23, LAWS OF ROTATION, 26. Let one end of a stick be thrust into the sand, and any motion given to the opposite end. Next, suppose the point to be made sharper and sharper, and that it is allowed to move as little as possible out of its place. In this way the mind passes to the conception of a body which has one point fixed; and it appears that the body may nevertheless turn on every side round this point. Secondly, let some mechanical method be used of fastening a body as nearly as can be, at two points only. Thus, it may be wedged between two walls, so as to touch each wall in but a very small part. Now in every case there is really a small surface in contact with the wall; and if we attempt to move the body, perhaps the friction so resists us that no motion can be produced. But if parts of the surface in contact are successively cut away, we shall at last be able to produce a sort of motion, even while the contact at the two sides continues in nearly the same spot of the wall. Such motion is called Rotatory. (See Fig. 19.) Many ways are conceivable of producing a more and more perfect Rotation; but it suffices here to enounce as fact, that if a continual approach be made towards an accurate fixing of two and only two points in an inflexible body, we arrive at the result, that " not all motion of the system is hereby hindered." 27. It remains to state the peculiar character of the motion which it then undergoes, wherein consist the Laws of Rotation. I. The motion of every particular point of the inflexible system is then constrained to one determinate path; which path, if the motion be continued in one direction, at length rejoins itself. The whole system has then regained its original position, and is said to have made one Revolution II. If another revolution be given to it, the same point must needs describe the very same path as before; and this, although the motion be reversed. 24w DIFFICULTIES OF ELEMENTARY GEOMETRY. III. If any revolving point be fixed, all motion is stopped. It appears to the writer, that our knowledge of these laws is as truly based upon experiment as is our knowledge that Water seeks its level. When he endeavours to assign to himself a reason, why the motion must needs be constrained, he finds himself making an inward appeal to the remembered sensation, that if an oblique pressure be applied, tending to cause motion along a new path, it is violently resisted, and cannot produce any effect, until some part of the system is crushed, or is lengthened, or slips. Easy as it is to satisfy oneself of this truth, by mechanical and experimental considerations, he has hitherto wholly failed in the attempt to prove it more abstractedly. Until, therefore, others shall have supplied a deductive proof, similar to the other demonstrations of Geometry, he will hold these laws to be experimental, and that Geometry stands on a like basis with Statics. CIRCLES. 28. Circle. In any case of rotation, the self-rejoining path (CCC C), described by any one point (C), is called a Circle. 29. Parallel Circles. Any second point D of the system may simultaneously describe another circle DDD'; and it Fig. 19. is easy to see that this must be a line parallel to the former oD circle. F or if D'C' be as short a path St/jic \' D, _, as possible from one circle to the other, then by supposing the system to revolve, D' and C' BDs~ f will describe the two circles, and the path D' C' accompanying D' expresses the shortest distance of every point D from the circle C CC. (See Art. 24.) CIRCLES. ~5 30. Sliding of the whole circle on its own ground. If a circle (CCC) be regarded as a hoop of unappreciable thickness, connected with the fixed points of the system (A and B) by inflexible lines, the rotation makes the circle turn along itself; so that while every point C is moving round, the circle as a whole does not change its position. Hence it is like itself all round: in Homer's language, rravroaF ioan, on all sides equal." 31. Surface of Revolution. If two of the circles, C C, DDD, be joined by a line CD, that revolves with the rest of the system, this line will describe a self-rejoining surface, which may popularly be regarded as a collection of Parallel Rings, indefinitely thin, since every point in the line CD describes a circle. Such a surface is called a Surface of Revolution; and if it close on all sides, so as to contain a solid, this is called a Solid of Revolution. 32. Proportional Arcs. It is manifest that any two points (C, D) in the system must complete the revolution, so as to regain their original position, simultaneously. For if C be fixed, as well as A and B, the whole is fixed. Hence any other point, as D, has its position determined by A, B, and C; and when C has regained its original place, D cannot be in any other place than that which it held at first. Let y be the point oppo- Fig. 20. site to C, in the circle of C, so that the portion (or Arc) c1, Cy is equal to the opposite A i arc y C; for there must be some middle point of the y path Cy C. Then when the system has been carried by rotation so far that C has reached 7, it may be said to have performed half a revolution. For it is evident that D will simultaneously describe half of its circle, and reach its opposite point 8; and the like may be said of every moving point in the system. 26 DIFFICULTIES OF ELEMENTARY GEOMETRY. To prove this more distinctly, we will take a larger proposition. Let C reach c, at the same moment that D reaches d; then, whatever ratio the arc Cc bears to the entire length of the circumference Cy C, such likewise is the ratio of Dd to D D. Suppose Cc, Dd, to be inflexible lines, attached to A CDB; then when the rotation brings C and D to c and d, let Cc have the position c c, and D d the position dd'. Then Cc, D d, are doubles of Cc, Dd; and, moreover, when C reaches c', D will reach d'. Similarly, if along the one circle we take any number of arcs equal to Cc, and along the other the same number of arcs equal to Dd, the points C and D will, during the rotation, describe the two sets of arcs simultaneously. If now Cc were any fraction of the circumference, say Ath, then Dd must be likewise 1th of its circumference. For by taking Cc twelve times we complete the circle; or by passnig over twelve times Cc, the point C regains its position. Therefore by passing over twelve times Dd, the point PD regains its position; which proves that twelve times Dd is equal to the circumference; or that Dd =l jth of its circumference. And the like would apply, if for 1 I th we substituted j th or 1 th, or any other submultiple; from which the mind instantly collects that the arcs described simultaneously by C and D are always Proportional. But to elucidate this conclusion the better, it is desirable to make a short digression. PROPORTIONALS. 27 DIGRESSION CONCERNING PROPORTIONALS. 33. A geometrical quantity may be supposed to vary independently, under numerous circumstances; as, when we suppose an object to become larger or smaller. But often it happens that two magnitudes vary together; as, just now, did the two arcs Cc, D d. For if by a motion of the system the length of Cc change, the length of Dd instantly changes likewise. 34. When two magnitudes thus vary together, the law connecting them may be very different under different circumstances. Sometimes, whatever increases the one diminishes the other; sometimes, on the contrary, they increase together and diminish together. Yet, even then, it may happen that their rates of increase are very different. While one doubles itself, the other may become five times as great; and while the former triples itself, the latter may become twenty times as great. But the simplest case of connected variables is, when they increase and diminish Proportionally; which is also of chief importance in the Elements of Geometry. 35. It is requisite for Proportionality, that the two variables vanish together. Thus, by taking Cc as small as we please, D d may likewise be made as small as we please; and when Cc is actually nothing, c being at C, then d is at D; or Dd is nothing. Regarding the magnitudes as increasing instead of diminishing, we may say that Cc, Dd, begin together from nothing; which is obviously necessary for Proportionality. 36. Again, when Cc increases by the Fig. 21. portion cp, (Fig. 21,) let Dd increase by c the corresponding portion d q. Then, in the case of proportional variables, since Cc is to Cp as Dd is to Dq, it follows that Cc is to ) q its increment cp as Dd to its increment dq. Suppose Cy, D, to be any other corre- / sponding values of the variables, which of 28 DIFFICULTIES OF ELEMENTARY GEOMETRY. course are (by hypothesis) proportional to Cc, D d; then we infer that the increments cp, dq, are proportional to the fixed magnitudes C, D8. Consequently, if the increment cp be a given magnitude, the magnitude of dq is instantly thereby determined, be the magnitude of C c what it may. Thus, if a new value be assumed for Cc, as in the diagram, and consequently a new value for Dd, and yet cp be assumed just as great as before, the dq likewise will be just as great as before. 37. This last property of Proportional Variables admits of being concisely expressed by saying, that they increase uniformly. For if any number of successive increments to the former be all equal, then the corresponding successive increments to the latter will be also equal to one another. 38. So much being premised, it will be easier to understand the following LEMMA, which is the converse of all this: viz. that " Magnitudes which begin together from nothing, and increase uniformly, are Proportional." PROOF. Let x and y be two such variables, which have increased together from nothing. Then x has been formed by an aggregation of small increments, every one of which may be regarded as equal to the first of them, viz. = h; in which case, by hypothesis, y will have been produced by the aggregation of the same number of increments, each equal to the first of them, or = k. Thus, x is the same multiple of h, as y is of k; and consequently x and y are proportional to h and k; (x: =y: k). This holds, however many fresh increments are added to both; so that if x' and y' are new values of the variables, these also are proportional to A and A, and consequently to x and y. That is, x: x'=y: y'. Which was to be proved. 39. The only objection to this is, that if x and y increase by finite additions, they will not receive all conceivable values intermediate to the first and last. The reply is, that the increments h and k may be in imagination lessened as much as we please; and in this way x and y may be made PROPORTIONALS. 29 to approach ever so close to any intermediate magnitude. If farther satisfaction be desired on this head, let it be supposed that x' and y' do not exactly contain h and k; but that x' contains j a certain number of times, and y over; then y' must contain k the same number of times, and v over. Consequently (x' - ) is the same multiple of h, as (y'- v) is of k; and, reasoning as above, we find that x: x'- =y: y- v. Here u and v may be called the errors incident to the 2d and 4th terms of the proportion which we are aiming to establish. But by diminishing I and k as much as we please, we may make the errors less than any proposed value; since ft is less than i, and v than k. And such diminution leaves the values of x x', y y', unchanged. Omitting, therefore, the errors as unreal, because they have no assignable fixed value, we have as before, x: x'- y: y'. In fact, in all cases, it is needless and useless to refine concerning Geometrical quantities, as though, in respect of Continuity, they required a different treatment from Numbers. Every conclusion drawn generally, by treating them as discontinuous, may unceremoniously be received as universally proved: for an error which has nzofinite value (as,u and v just now) has evidently no existence at all. 40. The Lemma may be also modified conveniently, as follows. " Let Sx, ~y, represent the new increments which x and y are just about to receive: if, then, the values of 8x and 8y are determined solely by one another, without any reference* to the values of x and y, supposing also that x * This may become yet clearer by considering the opposite case. Let S be a point inside an oval curve; x =an arc AP, y = area A SP, contained between the curve, and two straight lines. Let Pp = Ax; then area PSp = By. Now a little thought shows, that P P P if AP be made longer, (or P be taken at a more distant Ad / - point in the curve,) although Pp should have the same T, ) length assigned to it as before, yet the area PSp would ordinarily be different; so that here ay would depend not only on i x, but also on x, or on the length AP. Thus here, x does not vary proportionally to y. $0 DIFFICULTIES OF ELEMENTARY GEOMETRY. and y begin together from nothing, it follows that these last vary proportionally." For since the values of x and y do not affect those of x and Sy, a succession of increments to x, each equal to 3x, would produce a succession of increments to y, each equal to 8y, and thus x and y would increase uniformly. 41. Returning to the case of the arcs Cc, Dd, in Art. 32, it is there plain that the two increments cp, dq, are determined solely by one another, and are nowise affected by the lengths Cc, Dd, already attained. Moreover Cc, Dd, begin together from nothing. Hence we infer that they throughout " vary proportionally." 42. In like manner it appears that the surface passed over by the line CD, (Fig. 20,) which revolves with C, varies proportionally to the arc described by C. For they begin together from nothing; and so long as cp, the increment of the arc, is uniform, the corresponding increment of the surface is likewise uniform. 43. Similarly, if A be a fixed point of the system, and A C be a line revolving with C, the surface generated by A C is proportional to the arc described by C. SPHERES. 44. If A C be a stiff line, (Fig. 22,) of which one end A is.fixed, the other end C is free to play round A in all PFi. 22. directions. The locus in which C then ic, C lies is a surface. For if CC' be any path described by C, suppose CC' to be a rigid line attached to A C and A A 4 A C', and let the whole revolve about v~ v' A. Then CC' sweeping round A on all sides, traces out a surface, into any point of which the point C may evidently be brought. And as CC' may play round on all SPHERES. 31 sides of A, the surface rejoins itself, and encloses a Solid. This Solid is called a Sphere, and A its Centre. 45. Parallel Circles on the Sphere. If B be a second fixed point, (Fig. 23,) and B C an inflexible line, the place of C becomes restricted to a circle. But this circle must lie upon the Fg. 23. sphere, since A C is still the same. Change the length B C to BD and BE; D and E being still on the < - A sphere, and BD, BE, inflexible; B then D and E generate new circles on the sphere, which are parallel to the former circle described by C. It is immaterial whether the second fixed point B be within or without the sphere; or upon its surface. 46. A Spherical Surface is the Locus of all the points that lie at one particular distance from the centre. For, first, that all the points on the surface, as C, D, E, &c. lie at one and the same distance from the centre A, is manifest; because the paths A C, A D, A E, &c. may be made wholly to coincide. Next, (Fig. 24,) that a point F, which is outside the sphere, lies at a Fig. 24 greater distance from the centre than do the points on the surface, follows from this; that no path can join F E and A, without piercing the surface. Lastly, if G be a point within the sphere, we may conceive a new sphere to be described from the same centre A, by means of the line A G; so that G may be on its surface. Therefore all points on the surface CDE are such, that the shortest path connecting them with A pierces the surface of G, and consequently G is nearer to A than are C, DE,E, &c. It thus appears that every point in the surface CDE, and no point not in this surface, lies at the same distance as C from the centre A; and this is what is meant by 32 DIFFICULTIES OF ELEMENTARY GOEMETRY. calling that surface the LOCUS of all points which lie at that distance. COR. We moreover infer that a Spherical Surface consists of but one sheet. No part of the surface is overlapped, or comprised within another part. 47. Parallel Spherical Surfaces. The concentric surfaces described by C and G are obviously Parallel. Moreover, the surface G is entirely contained within the surface C; and every point in G is nearer to the surface C than is the centre A. 48. Spherical Shell. The solid intercepted between two concentric spherical surfaces is called a Shell. It is evident that, about a given sphere, a shell may be conceived to be added, such as to be less than any solid proposed. For the inner surface of the shell having a determinate magnitude, it is evident that if the thickness be perpetually diminished, the magnitude of the shell becomes evanescent. 49. Continuous Increase of the Sphere. Hence a Sphere in increasing from one size to another may be supposed to pass through all intermediate magnitudes. For the successive accessions of magnitude may be made as small as we please. Moreover, since a Sphere is readily conceivable less than any proposed solid, and another Sphere greater than the same; if the former increase till it reaches the magnitude of the latter, it must pass through a state in which it was equal to the solid in question. Or, " There is some Sphere equal to any proposed solid." 50. No Second Centre to a Sphere. The Centre of a Sphere is equidistant from all points on the surface. Conversely, if a point within a sphere be thus equidistant, it will possess all the properties of a Centre; for there would be no impropriety in conceiving the sphere to have been generated from it. But, we say, " there cannot be two such Centres." For it was shown that the centre A is more remote from the outer surface than is any other point (G) within the sphere, (Art. 47.) But it is obvious that there SPHER'ES. >cannot be two points within the sphere, each more remote from the surface than any besides itself. 51. In fact, suppose the surface CDE to be given, we may thus approximate towards one determinate centre '(Fig. 24). Take G within the body of the sphere, and through it pass a new spherical surface, parallel to the former; for we know that there may be one parallel. But it is evident that no second surface can pass through G, also parallel to CDE; for the mere principle of equidistance suffices to fix the surface of G, when the first surface CDE and the point G are given. Within the surface of G take a new point H, and it in like manner appears that through H may pass one and only one spherical surface parallel to that of G. By continuing this process, we form a series of spheres, less and less, each interior to the former, and tending continually to shrink into a single determinate point, which of course must be the centre A, This suggests another mode of stating the result of the last Article, viz. " If a spherical surface be given, the centre is determined." 52. Sliding of the Spleree on its own ground. If the surface be stiff, and inflexibly connected with the centre, which we may suppose to be fixed, rotatory motions in various directions may be given to the sphere, by which its surface will move, but only along itself; so that the sphere, as a whole, will suffer no change of place. During such sliding, every point G in the interior, (supposing the sphere to be solid,) of course moves along a concentric spherical surface, so that the centre is the only point not susceptible of motion, while the outer surface, as a whole, retains its position. Thus every one of the concentric surfaces, G, H, &c. is constrained to keep its own place, and slide along its own ground, if the outermost surface is thus constrained; and conversely. Hence it is obvious that if by any means we can secure that one surface, as CDE retains its place, we may infer that G, and that H, and every other yet more D 34 DIFFICULTIES OF ELEMENTARY GEOMETRY. interior surface, does the same, and consequently that the centre A remains fixed, since this series of surfaces may be conceived to approach as near to the centre as we please. 53. Poles of a Sphere. Let the centre A, and some other point B, be fixed, (Fig. 25,) B being either within or Fig. 25. without the sphere, or on -_ ~.the surface, but connected with it inflexibly. Then the only motion of which \4i/ the system is susceptible | A:L / jllljl is, a rotation about the X. "fixed points A and B. Of course every point C describes a circle C CC upon the surface, which slides on its own ground; and all such circles are Parallel. Now any one circle, as CCC, divides the spherical surface into two parts. On either side of CCC take a point D on the surface, and let DDD be the circle parallel to CCC. This cuts off a less portion of the sphere's surface on that side, and leaves a larger portion on the other. Within the area cut off by DDD on the opposite side from C, take a new point E, and let EEE be its circle parallel to the former circles. Within EEE, and towards the same side again, take F, and describe the parallel FFF. The process may be repeated continually, and each portion of surface intercepted is wholly interior to the preceding. Nor is there any internal area which is a limit towards which these areas converge; for within any such area a new circle might be placed. Hence the series of circles tends towards a determinate point, which we may call P, intermediate to all such possible areas. P will then have the peculiarity of turning about itself, so as not to change its position during the rotation. Again; on the other side of CCC we may suppose a succession of parallel circles, G G G, HHH, &c., and it may SPHERES. 35 similarly be proved that they tend towards a point Q on the surface, which has similar properties to P. The points P and Q are called Opposite Poles. 54. If B be on the surface, it must obviously coincide either with P, or else with Q. Thus one pole (say B or P) being given, the opposite pole Q is determined. 55. Poles of a Circle on the Sphere. Let CC'C" be any circle upon a sphere, and suppose that we do not know from what two fixed points the circle was generated. The poles P and Q may nevertheless be determined just as before. For we may conceive the circle as a line painted upon the solid sphere, which sphere is hung upon an immovable centre. Now by turning the sphere about, any point on its surface may be guided along any line soever drawn on its surface, but immovable in space. Wherefore any one point C may be made to describe the circle CC' C', without shifting the sphere's centre. As in this motion C successively takes the place C', C', &c., it is clear that:C', C", &c. also move round in the same path. Thus the circle slides along its own ground, and the rotation is constrained, just as when a fixed point B was given. We have therefore only to proceed as before, and determine the points P and Q, poles of the Circle. 56. It is easy to see that (counting distances along the surface) any two parallel circles are equidistant; as FFF from C CC. Hence that P is more distant (along the surface) from CCC, than is any other point F within the same area CP CC. 57. Since P and Q remain fixed during the revolution of the circle, they may be regarded as the fixed points from which it was generated; and they are at once poles of the Sphere, and of the Circle upon the Sphere. [Here we might proceed to explain the notions of Latitude and Longitude, and others connected with them, if a complete Treatise of Geometry were aimed at.] 58. Points that lie evenly. It has appeared, that when the points A, B, are fixed, the points P and Q turn about D 2 36 DIFFICULTIES OF ELEMENTARY GEOMETRY. themselves, or are fixed likewise. It is manifest that the very same rotation of the system would be effected if B were free, but P and A were fixed; or if B and A were free, but P and Q were fixed. In fact, if all are inflexibly connected, any two being fixed, the rest are fixed likewise. Three or more points thus related are said to lie evenly. AXES. 59. Suppose, as originally, that A and B are given points (Fig. 26). If the size of the sphere change, many new Fig. 26. pairs of poles P Q', P" Q", &c. are attainable, which will all lie evenly with A and B. Let the sphere change its magnitude by a -P? P'( ( B iH' A K A' ycontinuous motion; then b y its perpetual increase, Vi >V^- /'l t the opposite poles move away from each other, tracing out two lines, P P, Q Q', stretching to an unlimited extent both ways. If the sphere perpetually diminish, the poles move towards each other, and tend to meet in the centre A. When the sphere is such that B is on its surface, B is itself one of the poles. Hence the whole locus in which the poles lie, forms a single continuous line, passing through A and B, and susceptible of indefinite prolongation each way. All the points in this line lie evenly with A and B, so as to suffer no change of place by the rotation of the system, of which this line is called the Axis. 60. It is now manifest that to fix any two points in the axis is equivalent to fixing all its points. Also, to fix all its points offers no impediment to rotation. But if, besides the axis, a point likewise is fixed which is not in the axis, AXES. 37! then by the third Law of Rotation no farther motion is possible. 61. Axis of a Circle, or of a Surface of Revolution. All the above applies to any rotatory system soever, in which A and B are two fixed points. For it is evident that the very same axis is produced, whether A or B is made the centre of spheres; and that such spheres may be introduced arbitrarily into any rotatory system. 62. No second Axis to a Circle. If a circle be given, but the fixed points from which it was generated are not given; or, what is the same, if the axis be not given; it may be inquired whether more than one axis is conceivable, from which it might have been generated. We shall therefore show that only one is conceivable; so that, when the circle is given, the axis is determined. Let CCC be a given circle, and (since it must needs have at least one axis,) let one axis pass through A and B (Fig. 25). A sphere is conceivable, large enough to contain the circle within it, and consequently too large to pass through the circle as through a hoop. Let such a sphere have its centre placed upon the axis, and be moved along the axis until it strikes against the circle in one point C, the centre of the sphere being then at A. Then, since neither the circle nor the sphere changes its place by rotation about the axis AB, it follows that every point C of the circle lies on the surface of the sphere. Now the axis AB pierces the sphere's surface in two opposite points, P, Q, which are poles to the circle, upon the sphere, and each of which is equidistant from all parts of the circle, counting the distances along the surface. But if the circle could have any second axis, this must pierce the sphere in some other two points than P and Q. Suppose, for an instant, it pierces the sphere in a second point F, on the same side of GCC as P is, and we shall see the absurdity of it. For F would need to be equidistant from all the points C, C, C, of the circle, (counting distance along the surface,) a property which, it is evident by 38 DIFICCULTIES OF ELEMENTARY GEOMETRY. Arts. 55, 56, no point on the surface but P can possess. There is then no second axis to the Circle. 63. The Axis of a Circle is the Locus of the Points, each of which is equidistant from all points in the Circle. For, first, that any one point in the axis is thus equidistant, is manifest from the nature of rotation. But, next, we have to prove that every point which has this property is in the axis. Let A be equidistant from all points in the circle CCC, then A may be the centre of a sphere, on the surface of which the circle CCC lies. Hence, reasoning as in Art. 55, it will appear that A remains fixed, while C C revolves on its own ground; and consequently A is a point in the axis, which was to be proved. 64. Axes of the Sphere. In contrast to the circle, it is manifest that a sphere possesses an infinity of axes, all uniting in its centre. For A being the centre, B may be chosen arbitrarily, and an axis A B may be determined, piercing the sphere in opposite Poles. Thus, also, a Sphere is a Surface of Revolution. 65. That portion of a Sphere's Axis which is intercepted between opposite poles, is called a Diameter, and the half of it, between the centre and surface, a Radius. It is manifest that all the diameters of a sphere are equal, as likewise all the radii. 66. The mind here naturally guesses that the sphere is the only rotatory system which has more than one axis; which is nearly the truth. The exception is, that a Plane, of which we shall soon speak, has likewise an infinity of axes, all Parallel to each other. 67. Straight Line. It is with reference to the rotation of bodies that the word Axis is used; but when rotation is not immediately contemplated, a line, which has all its points lying evenly, is called Straight. 68. The following properties of a straight line are manifest from the mode of generating it. I. If A, B, are points within a finite solid, the straight line A B may be prolonged so as to pass out of the solid; CYLINDERS. 39 so also as to return no more into it if prolonged yet farther. This is evident from the fact, that a sphere of centre A is conceivable, large enough to envelop the whole solid, and of course having the poles P, Q, which lie evenly with A and B, exterior to the solid; which must be true equally of all larger concentric spheres. II. In the same way it appears that if a straight line lie on the surface of a finite solid, it can be prolonged so far as no longer to lie on the surface. III. Any two points in a straight line of unlimited length determine the whole. IV. Hence, also, two straight lines of unlimited length being applied together, so as to have two points in common, will entirely coincide. V. Between two given points but one straight line can be drawn. VI. A straight line has but one prolongation each way. VII. The parts of straight lines are straight. VIII. A straight line may slide along another, or along its own direction; and if inverted it will, as a whole, still occupy the same position. CYLINDERS. 69. Let the points A, B, be inflexibly connected, as before, with the circle or hoop CC'C", and an axis pass through A, B. We have before remarked, Art. 63, that any one point in the axis is at the same distance from all points in the circle. But, on the other hand, not all points in the axis lie at the same distance from the circle; else the whole axis would lie on the surface of a sphere whose centre is any one point C in the circle-an infinite straight line on the surface of a finite solid. Hence if A slide to an indefinite distance along the axis, carrying B with it, and the connexion of C C' C" with A 40 DIFFICULTIES OF ELEMENTARY GEOMETRY. and B be preserved inflexibly, the circle C' C cannot remain fixed, but must at length move also. This remark Fig. 27. being alike true, whatever point on,^.... the axis A may have reached, it fol-:~ J_ \ lows that by moving A as far as we X please, we shall cause CC'C" also to A move indefinitely. By such motion ^ C/ <GCC'C'" traces out, or generates, a cerD / tain surface, which forms a continuous sheet surrounding the axis, and prolonged indefinitely in both directions. This surface is called a Cylinder, and the circle's axis, the Axis of the Cylinder.* 70. It is well to add here, that since 1\ B X the system A CB C' is inflexible, and \ E~ E like itself in all positions, C C' " cannot F /- X remain motionless, if A has any ever so small a motion along the axis. |^^G |f ~ In fact, (since the circle is determined, when one point in it is given, and the axis is given,) all the points of the circle either move together or are at rest together; and not all its points lie in one straight line. Hence, if the circle be fixed, the whole system is immovable (Art. 60); wherefore no motion whatever can take place in A, unless the circle move with it. 71. A Cylinder is the Locus of all the Points which lie at one particular distance from the Axis. For, first, that all points C, D, on the cylinder, lie at the same distance, is manifest. Next, that a point F, which is outside, is farther from the axis than are C and D, appears by this, that every path from F to the axis must pierce the surface. Lastly, * Here, then, is an example of a line (the Axis) which is Parallel to a surface, (that of the Cylinder.) Also there is here a reciprocation found, such as as not calculated on in Art. 25, No. III. CYLINDERS. 41 if G be within the cylinder, we may suppose G to generate round the axis an inner circle, and this circle to generate an inner cylinder; whence it will follow that C is more distant from the axis than is G. Thus no point not on the surface is at the same distance from the axis as are all the points on the surface; which completes the proof of our statement. 72. The Cylinder is a Surface of Revolution. For if D D'D" be a new position of the circle, and D C* be a line drawn upon the cylinder, every point in D C lies in one of the generating circles. Hence the revolution of D C round the axis would generate the cylinder. It also thus appears that the cylinder may revolve about its axis, without changing its place as a whole; this being common to all surfaces of revolution. 73. Sliding of the Cylinder. Regarding the surface as indefinitely extended each way in the direction of the axis; if the axis slide along itself, carrying the surface with it, the cylinder, as a whole, does not change its place, but slides along itself. This appears from the circumstance that each particular circle, such as C C'", DD'D", will thus be made to slide along the surface. 74. Inversion of the Cylinder. Let the whole system be removed, and be then replaced with the axis inverted, but holding its former place. Then, since the axis is where it was, and the distance of the points on the surface of the cylinder from the axis is as before, the surface also will have regained its position as a whole. 75. After the inversion, let the axis slide up till the circle C'C" has regained its former place; and then let the cylinder revolve till the point C has regained its own place upon the circle. Let y be the point in the circumference which is opposite to C, so that y bisects the circumference, counting round from C to C again. It will * The mind readily perceives that if D C be as short as possible between the two circles, it will be a straight line. But to prove this, involves the whole difficulty of Parallel Straight Lines. The reader ought to be aware that we have not yet proved it possible to draw a straight line on a cylindrical surface. 42 DIFFICULTIES OF ELEMENTARY GEOMETRY, follow that y also has, after the inversion, regained its proper place; but the opposite halves of the circle have exactly changed places. Thus it appears that every circle is susceptible of being doubled about itself, by a half revolution about two opposite points in the circumference; so that the opposite halves coincide. 76. Centre of the Circle. Suppose that after the inversion, and after C and y have regained their own places, A and B have the new places a and P on the axis. Let also E be any point on the axis between A and B; and after the inversion, let E be found at e. Of course AE = a; but ordinarily E and E do not coincide, and A E is not equal to half of A a. Let A E increase or diminish, till it = - A a, and then it is evident that E and E will coincide, as at 0. Thus if O be connected with A, and A 0 = a 0, the point 0, like C and y, regains its own place after the inversion. This amounts to saying, that if the circle perform half a revolution about the points C and y, O turns about itself, or lies evenly between C and y, that is, between any two opposite points of the circle. Thus all the straight lines which join opposite points in the circle meet the axis, and meet it in the same point. This point (0) is called the Centre of the Circle. 77. The whole straight line C Oy is called the Diameter of the Circle, and its half, CO, the Radius. That all the diameters of a circle are equal, and all the radii are equal, is evident. 78. While A slides along the axis, and CC' C slides with it (Art. 70,) along the surface, the centre 0 accompanies CC'C" in its motion; so that every point (as 0 or E) on the axis, is centre to one, and only one, of the generating circles. PLANES. 79. If C revolve through the circumference, and carry the diameter COy with it, the two halves, C, Oy, trace PLANES. 34, out or generate one and the same surface, since, after half a revolution C and y exchange places. This surface is the LOCUS of all the diameters, and is called the Plane of the Circle. It is a species of Surface Fi. 28S of Revolution. A 80. Inversion of the Plane. After inverting the system as above, the plane, as a whole, exactly regains its own position. Thus it is like itself at both sides, and cannot be said to bend either way. 81. Concentric Circles. While O C, revolving, generates the plane, any point H in 0 C (Fig. 28,) gene- o rates a new circle, HH' H", which lies in the same plane with CCC, \ and has the same centre 0 with it. C The two circles are parallel, as in Art. 29. But they are likewise evidently parallel, or equidistant, in another point of view, viz. by counting distance along the plane surface. [For we have not yet shown, what however is the truth, that the shortest path from H to C lies along the plane.] Thus 0 is more distant from the outmost boundary, C C' C", than is any other point H, which is within the circular area. 82. No second point (besides the centre) within that area, can be equidistant from all the points of the outline CC' C". For by Art. 63, no point can be equidistant from the circle, unless it be in the axis; and the axis has no point in common with the plane, except the centre. 83. In fact, if the circle C C' ", and therefore its plane, be given, (Fig. 28,) we may thus approximate towards one determinate centre. Within the circle, take any point H, and through H pass a line upon the plane, parallel to C C'C". There can be but one such parallel through H, and of course it will be a circle, concentric with CC' C". Within it again take K, and by it determine another circle K'K', also 44 DIFFICULTIES OF ELEMENTARY GEOMETRY. parallel and concentric with the former. This process may be repeated as often as we please, and the central area be diminished as much as we please. Each new circle is determinate, if one point in it be given; and hence they converge towards a single determinate point, interior to them all, which of course is the centre. 84. Lastly, the Plane may be looked on as the LOCUS of Circles, which have the same Axis and Centre. For there are no circles having the same axis and centre with CC C, except those which lie on the plane; and there is no point in the plane which is not likewise a point in one of these circles. We here conceive of the plane, as indefinitely extended, as it evidently may be, by prolonging the line 0 C which generates it. And in this case it suffices to speak of " A Plane," without adding " Plane of Circle;" since the circle is but accidentally connected with it. CURVED LINES. 85. The nature of Straight Lines will be yet better understood, after putting them in contrast with Curved or Bent Lines. Let A C be some stiff line, (Fig. 29,) united to a point Fig. 29. B exterior to it. A and B reA A maining fixed, let A C generate round A and B a self-rejoining <D o.^ 3" Vc surface, like an umbrella round ^C-^p -^D n its stick. It will be a continuB B ous surface of revolution. Again: if at A be a joint, and A C be set in a new direction A D, so as no longer to lie on that same surface; and be fixed in the new position; then it may generate a new surface of revolution round AB. Of the two surfaces thus produced, one is interior to the other in the immediate neighbourhood of A, and will therefore be justly said to be sharper at A than the other. We CURVED LINES. 45 may also say that this surface, or indeed ordinarily each surface, has a Peak at A. If to the possible sharpness of the peak, produced by altering the direction of A C, there is any limit, let A E (Fig. 80) be that position of A C which makes the peak sharpest. If there be no limit. 3 attainable without making a part A of the surface disappear at A, this F A is equivalent to saying that a part A F of the line A C is susceptible of lying evenly with B, so as to project out beyond the surface of revolution. Then A F is a part of the axis A B, and is absolutely B B Straight. In any other case it is manifest that no portion (as AF), however small, can be cut from A E such as to be straight; or no portion of A E, counted from A, coincides with the straight line AB. Then A E is properly called Curved at A. The point A is a Peak of the surface * of rotation; and there is an evident propriety in calling A E more or less curved at A, according as it deviates more or less from the straight line AB; that is, according as the peak of the surface at A is blunter or sharper. 86. Every line may be divided into finite portions that are Straight, and finite portions that are Curved; if it be not curved throughout, nor straight throughout. For instance, if AE, which is curved at A, is not curved throughout, there must be some definite point P, along A E, at which it first begins to be straight. Then the finite portion A P is wholly curved. Next, setting out from P, we may cut off a definite portion which is wholly straight. And so on alternately. 87. Tangent, or Osculating Straight Line. A E having been brought into such a position that the peak of the If a name be needed for this surface, it is obvious to call it a BelL 46 DIFFICULTIES OF ELEMENTARY GEOMETRY. surface at A is as sharp as possible; we are justified in using the phraseology, that A E lies " as near as possible to the straight line AB at A." This is to use Distance in a new sense, yet in a sense perfectly intelligible, and not at all repugnant to its former use. It is immaterial whether A E, turning about A, be brought towards A B, or A B, turning about A, be brought towards A E, as far as proximity of the two lines at A is concerned. In either case, A B and A E are made to take, as nearly as possible, the same direction at A; that is, as nearly as the nature of the curvature at A, and the nature of the straight line A B, allow. Hence, (if the position of A E be given,) of all possible straight lines that can be drawn from A, none takes so nearly the direction of A E at A, as does the straight line A B; in other words, none lies so close to A E as does A B. This line A B is therefore said to be the Tangent, (or Rectilinear Tangent,) to the curve A E at A. In a popular sense, any two lines touch one another, when they meet one another; and this is a defect in the name Tangent. In consequence of this ambiguity we shall need to be much on our guard; for instance, two solids might "touch one another," and yet not be "in contact with one another." It is to be regretted that the word Osculator has not been used in preference to Tangent; and Osculation for Contact. But in the higher Mathematics, Osculation has unfortunately been appropriated to a yet more intimate sort of Contact. 88. It is well to remark on the sisual characteristic of lines in contact; which is the same, whether one line, as. B, be straight, or both lines be curved, (Fig. 31.) Let P be any,J^ Ipoint in the curve A E, and PN the A N' N B shortest path connecting P to the line A B. Then, when A P diminishes,,i r X~ (by taking P nearer to A,) it is A --—;;< — N presumed that P N and A N likewise CURVED LINES. 47 diminish; at least * after A P is less than a certain limit. Now if PN diminish at last far more rapidly than AP, PN attains a size too small to be discerned at all by the eye, while AP is still distinctly visible. When PN is invisible, the two lines appear actually to coincide through the part AP, AN; and this apparent coincidence is the Visual Peculiarity of lines in contact. But the pure science of Geometry recognizes the existence of PN, so long as hypothesis alleges its existence, whether it be visible, or no: hence these considerations do not affect our argument at all. 89. Any continuous por- Fig. 32. tion, AP, of a curved line, is called an ARC; and the straight line AP, joining c A B the extremities, its CHORD. 90. If A be, not the extremity of a curve, but some intermediate point in a Fig. Fig. 23. curve PA Q, it has two c\ Q arcs, A P, A Q, on oppo- \ site sides of it, to each of which we may suppose a tangent to be A A drawn, viz. A B and A C. Now three cases may happen: (I.) as in Fig. 32, the two tangents A B, A C, may be Fg. 3 opposite branches of the same \ straight line BA C: (II.) as in -A Fig. 33, AB and AC may be two different straight lines: (III.) They may, as in Fig. 34, lie along the same branch of the same straight line A B. In the first case, the curvature is said to be Continuous on each side of A. In the two latter cases, the curvature is * For after AP attains a certain length, it is p conceivable that a farther increase of A P might: __ -cause a diminution in P N, as in this diagram. B 48 DIFFICULTIES OF ELEMENTARY GEOMETRY. Broken or Discontinuous at A, and the curve has a PEAK or CUsp at A. It is evident that, in the second case, the curve at A makes an abrupt deviation from straightness, immensely greater than in the first case; while in the last the curve at A turns right back in just the opposite direction. 91. If, in the last case, we look on AP, A Q, as two separate curves, then, because they have the common tangent A B, they are said themselves to be in contact. It is indeed evident that in no other position could they lie so close together in the neighbourhood of A. 92. If AE be a curve (Fig. 35,) divided at P, Q, R.... and the chords A P, P Q, Q R.... be all drawn, it Fig. 35. appears by Arts. 14-17, that B /P / /by increasing perpetually the points of division in all parts of the curve, the sum of the chords tends towards the sum of the NC// arcs, (or, towards the whole length A E,) as its limit. Also: that each particular chord, as A P, tends to become equal in length, and coincident in position, with its arc, so as to be entirely confounded with it. It immediately follows, that if the chords A R, A Q, A P, are prolonged to r, q, p, the straight lines A r, A q, Ap, tend more and more to coincide in direction as nearly as possible with the are AE at A. Now the tangent AB, of all straight lines, coincides in direction most nearly with the curve at A. Hence, if AP be perpetually diminished in length, the straight line A Pp tends towards the position of the tangent, as its limit. 93. The review of Arts. 14-17 shows farther, that though a true geometrical curve is not made up of little straight lines, it may be looked on as a limit to which we pass by considering a path made up of straight lines, which become shorter and shorter, and bend oftener and oftener. CURVED LINES. 49 A very small arc nearly coincides with its chord; and by making an arc as small as we please, we may make it coincide, in direction and length, as nearly as we please, with its chord. (Art. 14, 15.) Now, if PN be as short a path as possible from P to the tangent A B, it is manifest, since Ap tends to assume the direction AB as its limiting position, that the two lines A P, AN, tend to confound themselves entirely, when the arc A P perpetually diminishes. If we call the length AN by the name of Tangent, in reference to the arc A P, we are now warranted to pronounce, that the Tangent, Arc, and Chord, all tend to confound their directions, when the arc perpetually diminishes; and that the limit of the two ratios (Tang.: Arc) and (Chord: Arc,) is, the ratio (1: 1). 94. That the distance P N vanishes when the length of the arc A P vanishes, is, of course, obvious. But this fact alone will not suffice to account for AP and AN tending to assume the same direction. It is farther necessary that P N should diminish much faster than A P, so that the ratio (PN: AP) must be perpetually getting less, while AP diminishes. In fact, this ratio must be susceptible of indefinite diminution, by lessening A P; but the full proof of this must be reserved until the subject of Proportional triangles has been discussed. 95. If PA Q be any curve, having no peak (Fig. 36), and ML be any two points F 3 in it, let a straight line of / indefinite length be drawn A through Ol and L; then - suppose the points ML to // move up towards each M L -- Q other, carrying the line p / with them. If A be the intermediate point in which they tend to concur, and m A be the limiting position towards which ML tends, then " z I is a Tangent to the curve at A." E 50 DIFFICULTIES OF ELEMENTARY GEOMETRY. For as the arc IMAL tends perpetually to confound itself with the chord ML, the chord ML with the tangent at nl, and the tangent at M with the tangent at A; the conclusion is evident. It fails only when there is a peak at A, which is excluded by the present hypothesis. In that case the two tangents at M and L do not tend towards the same tangent at A, nor does the arc MA L tend to confound itself with the chord ML. 96. All the above will enable the reader to appreciate the statement, that the Rectilinear Tangent "is drawn through two consecutive points of a curve." This implies that a chord is first drawn through two neighbouring points, and is prolonged each way; and next, that the two points move together, carrying the chord with them. It then tends to become a Tangent, which is the limit. But if the presence of a Peak be possible, then one of the two points must be stationary, and the other must move towards it. Thus any curve may be approximately represented by portions of its tangents, or of lines which tend to the tangents as their Fig. 37. limits. If A Pp, P Q q,,Q R r, R Ss, S T t, &c. (Fig. / 37,) be straight lines, the bent line APQRS... is arude / As ' representation of a curve; |,1~R ~ and how far it differs from a |^ /r curve, depends on thelengths FP// I P\ V of A P, PQ, &c. and their W \ AR relative directions. IA \iQ Suppose A, P, Q, R... to B^ g^p khave been originally taken in a I some particular curve, and to fix ideas, let the arcs A P, P Q, QR... be all equal. Then, as in Art. 92, if the arcs are perpetually lessened, the lines Ap, Pq, Qr,... will tend to become tangents. We may call them, "ultimately tangents to consecutive points in the curve," and we perceive that the tangents to two consecutive points inter CURVED LINES. 51 sect, as Ap, and Pq in the point P. Thus in passing from A to any other point T along the curve, we find the tangent turn about successive points in its own length, and so deviate into a new position. If Q A, Q P... be prolonged to a, P... the lines Q a, Q 3 also tend to confound themselves with the tangent at A, when the arcs are perpetually diminished; for A Q tends to Ap, as was seen in Art. 92, and so does Pq, or Q 3. The same must be true of the prolongations of P R, P S,... QR, Q S, Q T... since all the points P, Q, R, S... tend to merge themselves in A. 97. It now readily follows, that in a limited arc the number of Peaks must be limited; or, what is the same thing, that no two consecutive points of a curve can be Peaks. For instance, if A be a peak, Q, indefinitely near to it, cannot also be a peak. (Fig. 37.) For if we take P and R on opposite sides of Q, and draw 3 P Qq, QRr, straight lines, and indefinitely diminish P Q, QR, as also A Q; the two lines Q3, Qr, whose limits are the tangents at Q, do both at once approximate towards Ap, so that the opposite tangents Q(3, Qr, at length become a single straight line. Peaks or Cusps are on this account called Singular points; because a finite arc, while it contains an infinity of points which are not peaks, has but a finite number which are; and every two consecutive peaks are separated by a finite distance. This is as obvious, as that on a knife edge not every part can be a point or peak. 98. Deviation of the Tangent. While a point lM, as in Fig. 36, traverses the curve from P to Q, let its tangent move with it. Then, by Art. 96, the tangent deviates continually into new positions. But by the same article it appears that (except at a peak) the deviation is gradual, depending on the length of the arc through which M passes, and capable of being perpetually lessened and caused to vanish, by reducing the length of the arc, and causing it to vanish. Only at a peak is the deviation of a tangent abrupt, and "finite through an infinitely small arc." To explain what E 2 52 DIFFICULTIES OF ELEMENTARY GEOMETRY. may seem an absurd phraseology, consider the curve in Fig. 36, in which there is a peak at A. As M moves up from P towards A, the tangent at 1 tends more and more to assume the position AB, and when M actually reaches A, the tangent attains the position A B. But Mcannot move farther along A Q, through any arc, however small, without the tangent abruptly passing over into the position A C, or a position indefinitely near to A C. Thus an "infinitely small" motion of M, through A, produces a finite transference of position in the tangent. Fig. 38. 99. Deviations of Curves from their Tangents. A -If now we suppose two curves to be placed together, so as to have a common tangent at a,: common point, as A 0, AD have the same tangent AB at A, (Fig. 38,) a ready test presents itself, as to "which curve deviates the '/&Xl more from the tangent." For if we suppose D Dti D them simultaneously to generate bells around AB, the bell whose peak at A is exterior to B the other evidently deviates the more. But this does not enable us to pronounce anything concerning the ratio of the two deviations. It does not even suggest under what circumstances one curve might be said to deviate twice or three times, &c. as much as another. Since it is manifest, by Art. 98, that the principal deviations of curves are at their peaks, at which the tangent itself deviates abruptly, this suggests the propriety of treating on the deviations of straight lines, before considering any further those of curves. RECTILINEAR ANGLES. 53 RECTILINEAR ANGLES. 100. When two straight lines proceed from one point, the deviation of each from the direction of the Fig. 39. other is called a Rectilinear Angle, or more (l simply, an Angle. This Latin term may at first seem to mean the same thing, as its A English representative, Corner; yet in Geometry they do not mean the same. For the Corner is the bare point in which D 0 the lines meet, while the Angle (as we \ said) is the " deviation," being a relation between the direction of the one line, and c the direction of the other. B Depending thus solely on the direction of the lines, the Angle remains the same, whether they be ever so long, or ever so short. It is usual to denote the angle made by A B and A C, by saying, "the angle BA C," or, " the angle CAB," putting the letter which is at the corner between the other two. 101. A method perfectly similar to that of Art. 99, enables us to decide which of two angles is to be called the greater. For if A C, AD be two straight lines, each deviating from the third line AB, we may suppose the whole system to be inflexible, and each of them to generate a surface of revolution round the axis AB; then if (as in Fig. 39) the peak of A D is exterior to the peak of A C, we pronounce that A D deviates more from A B, than does A C. Since A C, AD cannot meet in any second point, the surface of A C forms an entire covering, wholly separating AD from AB; nor is it possible to pass from a point in AD to a point in AB, without piercing the surface. AD is then called exterior, because we regard AB as interior. But if BA be prolonged to 3, then if A 3 be looked on as interior, A C becomes exterior to the surface of AD, and A C deviates more from A 3 than does A D. 54 DIFFICULTIES OF ELEMENTARY GEOMETRY. 102. Cone.-If any straight line AC hang loosely from A (Fig. 40), and after performing every circuit soever Fig. 49. round AB, return to its original poA sition; the surface which it has traced out is called a Cone. In the pariticular case supposed above (Fig. 39), A C was inflexibly attached to AB; so c ll^ that in every position its deviation was Ca-cl the same, and the motion was a Rotation. Such a Cone is called for distinction, a B Cone of Revolution;" and as it is for the most part the only Cone spoken of in Elementary Geometry, this is generally understood, when " a Cone'" is mentioned, and when the contrary is not specified. 103. Supplement of an Angle.-Suppose now that in Fig. 39, the line A C shifts its place, and occupies that of A D. Hereby the angle which it makes with AB is increased; but the angle which it makes with A 3, the prolongation of BA, is diminished. Thus the two angles BAC, AC, stand in such a relation, that we cannot increase the former, without diminishing the latter; and of course, conversely. These two are then "Variables," mutually dependent, and they are called Supplements to one another. 104. Vertical or Opposite Supplements.-But when a particular angle, BA C, is given, two ways Fig. 41. now offer themselves of producing a / Supplement to it. For by prolong/ ing BA to 3, we get, as before, the supplement f AC; while if (Fig. 41) we instead prolong CA to y, the supplement to BA C is y A B. ^c/~ IBefore proceeding farther, we must B consider whether this involves us in any ambiguity. The following reasoning shows that the opposite supplements are absolutely equal; or are, what we called in Art. RECTILINEAR ANGLES. 55 19, identical magnitudes. Suppose the whole system of lines (remaining inflexible) to be entirely removed; and then replaced so that the angle BAC shall occupy the same place as before, but with its lines interchanged, A C being where A B was, and AB where A C was. Then the two prolongations A 3, Ay, will also have exchanged places, and, consequently, the angle CA 3 has precisely exchanged with BAy. These angles are then coinciding magnitudes, every way equal; and it is indifferent in which of the two ways the Supplement to BA C is estimated. 105. Right Angles.-Let us now imagine that the line A C originally very nearly coincided with A. B; in which position the angle BA C was very small, and, consequently, its supplement 3 A C was large. Suppose then, that BA C gradually opens, and as it increases, its supplement will diminish. If it continue to increase, the supplement will at last become very small, until it all but vanishes. In such a progress, the angle must have passed one, and only one intermediate position, in which it is EQUAL to its supplement. Thus, let A D be such as to make the angle BAD = its supplement 3 AD. In this position, each angle is said to be Right. 106. Moreover, if DA be prolonged to 8, the opposite angles B A 8, 3 A X, which are the other supplements, are equal to these by Art. 104. Thus the two intersecting lines B A 3, DA 8, produce four right angles. Each of them is said "to make right angles," "to be at right angles," or " to be Perpendicular " to the other. 107. We said, there is only one intermediate position, in which the angle is equal to its supplement. Although no one will question this, it may not be clear to some, how we know it. To remove any doubt on this head, let x be an angle, and y its supplement, and let it be remembered, that the greater x is, the less y is. Hence, if x' is another angle, and y its supplement, and x' is greater than x, then y' is less than y. 56 DIFFICULTIES OF ELEMENTARY GEOMETRY. It follows, that should x = y, x' is greater than y, and much more greater than y. Thus in no way can we have simultaneously, x = y, and x' = y'; unless x and x' were absolutely identical. Or; there is but one intermediate angle, between the least and greatest, equal to its supplement. This is generally expressed, by saying that " All Right Angles are equal." 108. Obliquity.-Every other straight line, as A C, which is not perpendicular to B A f3, is said to be Oblique to it: and of the two angles which it makes, the less (as CAB) is called Acute; while the greater (as CA 3) is called Obtuse. 109. Erecting a Perpendicular from a Straight Line.It is now evident, that if A be any given point in a given straight line BAP; a perpendicular, as A D, may be erected from A. Nevertheless, " to erect a perpendicular," is not a determinate problem: for an infinity of perpendiculars can be drawn, the LOCUS of all which is the surface of revolution generated by AD about the axis BAP. 110. On comparing Articles 79, 80, it is very manifest that the surface generated by A D is a Plane, and B A 43 the Plane's Axis. Since the Axis is thus perpendicular to the generatrix of the Plane in all its positions, the Axis is said to be " Perpendicular to the Plane." Fig.42. A 111. Dropping a Perpendicular on a Straight line.-Let A B (Fig. 42) be the straight line, which must be supposed susKn ceptible of indefinite prolongation; and let C be a given point without it. Sup^c^ —o — o pose C to generate a circle round the axis A B; this circle must have a determinate ddjaC )l _11|0 centre 0; which is a point in the axis. Join C0, and it will evidently be perpendicular to A B. It is said to be "dropt," or "let fall," from C on to A B. RECTILINEAR ANGLES. 57 " To drop a perpendicular" is a wholly determinate problem. There can be no perpendicular as C P, P being some other point in the axis. For if the system be inflexible, and slide together along the axis, so that the circle may generate a cylinder, when P takes the place which 0 had before, it appears by Articles 70, 78, that the circle must needs have some new position, as C C' C. Thus the line PC will have come into the position 0 C'; and, consequently, the angle CPA being equal to C' 0 A, is not equal to C OA; and is not a Right Angle. 112. Thus far we have succeeded in establishing between different angles the relations of Greater, Equal, and Less. But nothing has appeared as yet, by which a numerical measure of angles may be attained. It is not possible to affix any sense to the statement that one angle is double or triple of another, until we can fix on a method by which any number of angles can be added together; and conversely, by which an angle can be resolved into any number of parts, whose sum shall constitute the whole angle. Until this shall have been done, an Angle, if entitled to be called a Quantity, is yet incapable of being measured, or appreciated numerically. Now that a method of addition may not be illusory, it is requisite and sufficient, (1) that the result may be unaffected by the order in which the parts are combined; (2) that no number of resolutions and recompositions may affect it. Yet of various devices by which an unambiguous Sum of several angles may be obtained, not all are equally natural and proper, though all may be Fig. 43. equally logical. Moreover, if B A B, A C, AD, (Fig. 43,) are c three straight lines proceeding from A, and making three angles, BA D, BA C, C AD, it is by no means justifiable to say that the greatest of the angles is the Sum of the other two. For in fact, if the magnitude of the two smaller be given, this does not 58 DIFFICULTIES OF ELEMENTARY GEOMETRY. suffice to determine a single value for the third. For instance, if the angles BA C, CA D, be given, and B A revolve round CA so as to generate a Cone, we have no right to assert (what indeed is obviously untrue) that the angle BAd D remains constantly the same. 113. One method which recommends itself as at once unambiguous and natural, is, to inquire in what position of BA, the angle BA D attains its maximum: and to consider this as the genuine sum of the two constituent angles. But in our present stage we cannot have recourse to this. We must be satisfied with ultimately proving that the course which we have taken produces this very result. Fig. 44. 114. When we consider /P that the quantity to be mea/ sured, is, the deviation of 0; -~ ___1 nthe direction of one line from. 9/ \\; a~' the direction of another line, o \ the thought will instantly 1i7</ ~~ arise, whether a comparison of two angles cannot be made, by measuring them p, ~ from leg to leg. For instance, if PO Q, M L N, are two..... \ — angles, to fix ideas, (Fig. 44,),f___fj 1..measure off from their legs the lengths Op = Oq=L m \" -= L n, = one yard; and draw straight lines p q, mn. Then if p q prove to be double of m n, it might at first seem that the angle 0 must be double of the angle L. An objection to this presently discovers itself. Ifp Or, rO q, were angles having a common leg Or, and Op = O r = Oq, the line p q, which we have assumed as measuring the angle p 0 r, is not made zup of the two lines p r, r q, which are supposed to measure the smaller angles. But if p 0 q is to be regarded as a whole, made up of parts p 0 r, RECTILINEAR ANGLES. 59 r 0 q, then the measure of the whole ought to be made up of the measure of the parts. 115. This remark readily leads to a mode of obviating the difficulty. Let a plane be laid upon the lines 0 P, 0 Q, so that the plane's centre may fall on 0, and 0 P, 0 Q become two generatrices of the plane: then one circle of the plane (Art. 84) will pass through p and q.* Similarly, through m and n, pass a circle whose centre is L. And let us assume the lengths of the circular arcs p q, m n, as the measures of the angles P 0 Q, ML N. The former objection will not now apply; for if p Or, r Oq, be laid down on one plane, whose centre is 0, one circle of the same whose centre is 0, will pass t through p, r, q; and as the whole arc p q = sum of the arcs p r, r q, it is congruous that the angle p 0 q measured by the arc p q, should be regarded as the sum ofp 0 r, and r 0 q, measured by the smaller arcs. Thus far then, we are led to the principle, that angles which are to be added together should be laid down side by side on a plane, with the plane's centre for their common corner. 116. But a new difficulty may be started, which must be removed before we can acquiesce in this method of measuring the amount of deviation; namely, that an arbitrary quantityhas been introduced, in the length of the radius Op. Now if a change in this length will give different results in our measurement, the method must be abandoned as useless. To take a simple case: If (Fig. __s ig.4 45) PRQ is a circular are of Fg 4.. centre 0, in which R Q is double \ of P R; whence we infer that the angle R 0 Q may, without im- propriety, be called double of the angle P 0 R; let us inquire, whether the propriety of it will be overturned by a change in the * In future this may be expressed, " With centre 0, and radius Op, describe a circular arc pq." t Art. 84. 60 DIFFICULTIES OF ELEMENTARY GEOMETRY. radius 0 P. Take some other length Op along the line 0 P, and from centre 0 describe a circular arc p r q, cutting 0 R in r, 0 Q in q; which is possible, by what has preceded. It is now very manifest, that if R Q be bisected in S, and SO be joined, dividing the arc rq in s, we shall have rs = sq. For if the system ROS be applied on the equal and identical system S 0 Q, so as to coincide, r and s will take the places of s and q. Thus also, P R, R S, S Q, being all equal, p r and r s and s q are likewise equal; which gives, r q double of p r. We find, then, the same ratio as before, between the angles R 0Q, POR, if we measure them by p q, and p r, instead of P Q and PR. But it is at once clear that this may be generalized. For remembering that an axis of rotation passes through O, about which the circles of P Q and p q are described, we may regard P R, p r, as two Variables, depending on each other, as in Articles 33, 41; and in Art. 41, it was shown that they vary proportionally. Thus the ratio of P R to R Q, is always equal to that of p r to r q, whatever is the size of the angles at 0, and whatever the lengths of the radii 0 P, Op. 117. Not only, then, does this second objection fall to the ground, but we perceive that we entirely succeed thus in measuring the deviation, (or width of opening between the lines which form the angle,) in the only linear method.*' For no other sort of line but a circular arc would give to every elementary equal angle into which we might resolve the whole, an equal measure. As, however, by Art. 42, it appears that the Areas (or Circular Sectors) P 0 R, p Or, are proportional variables, these sectors also might be assumed as measures of the angles. For as the sector PR O is proportional to the arc * That is: "the only linear method attainable on a plane." We might suppose the angles laid down on the surface of a cone, with their common corner at the cone's vertex. But this has a double objection; (I.) It is arbitrary, what sort of cone to choose, or with how large a rotatory angle; (II.) That some angles will be so large as not to lie on the cone at all. RECTILINEAR ANGLES. 61 PR, and this arc to the angle P OR; it follows that the sector is proportional to the angle. Thus: Sector PR 0: Sector OR Q = Angle P OR: Angle RO Q. 118..If, as an arbitrary unit for angular measurement, four right angles were assumed; which are measured by the entire circumference of the circle; then since, Any Angle: Four Right Angles, (or 1,) = Arc: Circumference; we get: Angle= ( Arc ) Angle- (Circumference as a numerical valuation. By similar reasoning, we get: Sector Angie= (Whole Circular Area) The preceding articles show, that the value of these ratios or fractions is not at all affected by a change of the radius. 119. It is usual to divide the circumference into 360 equal parts, called Degrees; so that every Right Angle contains 90 degrees. But it is needless to enlarge on that which is fully explained in so many other books. It is sufficient here to remark, that, Any angle + its supplement = 180~. 120. Periodic Magnitude. We have established that angles are not only Magnitudes, but are Magnitudes resolvable into parts all homogeneous to each other and to the whole, so as to allow of numerical valuation. They have, however, a great peculiarity, distinguishing them from the other magnitudes which we have hitherto met, in their not being susceptible of indefinite increase. There is a maximum value for the angle, which it cannot pass, namely, 180~. On attaining this, the angle vanishes; and if the arc which measured the angle increase yet farther, the angle begins again to increase from nothing. We may, however, with propriety, extend the limits of angles from 0~ to 360~, in order to distinguish between the 62 DIFFICULTIES OF ELEMENTARY GEOMETRY. direction of a line A B from A to B, and the direction of the same from B to A. In Fig. 34, we may say that the tangent at A, (which turns back upon itself,) deviates through an angle of 180~. Generally, if A be the centre Fig. 46. of a circle B CD EF GB, (Fig. If o0 46,) and AC, AD, AE, AF, /~ \ \c A G, lines issuing from A in all wEi \ /\ directions, we may estimate all of 1~80' — z — ^ 0 B these with reference to the single \ /r \ j direction AB, by means of the F< \ / arcs BC, B CD, B C E, B D F, 2~G BE G, all counted round in the same direction, and some of them, perhaps, greater than 180~, or than 270~. Moreover, if CA F be a straight line, and B C (for example) = 45~, so that BDF = 225~, by assigning these two different arcs, we distinguish between the opposite directions, A C, A F; a matter which is often of importance in the higher mathematics. Thus in Mechanics, two opposite forces might act along A C and A F. But while we may thus justify the extension of angular magnitude as far as 360~, it is evident that beyond this limit the angle does not increase with the arc. If to any arc, as B C, we add 360~, the direction determined for the line A C is the very same as before; and by the " angle " we explained that only "relative direction" was meant. Herein, then, consists the periodicity of angular magnitude. If the arc by which the angle is determined begin from 0, and increase till it attain the length of 1, 2, 3, 4... circumferences, the angle at the completion of each circumference, vanishes, and then goes through the same series of magnitudes as before. 121. But it may be proper here concisely to point out the method of determining directions universally, whether they do or do not fall on one plane. Suppose a sphere, (Fig. 47,) whose centre is A, and A y, A S, to be two straight lines, whose directions, relatively to 4 B, are to be described SCHOLIUM. 63 or noted down. Let a plane whose centre is A, and axis PA Q, pass through A B, and cut Fig. 47. the sphere along B CD, which, of course, is a circle. Pass a like j| / plane through A P and A y, and a ' third through A P and A 8, cutting i ^ / the sphere along new semicircles ^ill Py C Q, PD Q. Then the direction A y, with reference to A B, is fixed by the two arcs B C, P y; also the direction of A 8 is similarly fixed by the arcs B D, P 8. And so with any other radii. In geography, B C or BD would be called Longitude, and Py or P 8 North Polar Distance. SCHOLIUM. 122. Very eminent modern geometers,* considering the periodicity which characterizes Angles, have thought themselves justified in pronouncing that the angle has "a natural unit;" and assuming this to be true, have, by a very few steps evolved conclusions, generally supposed to be attainable only by long processes of reasoning, and by help of the properties of Parallel Straight Lines. In Art. 118, we assumed "four right angles" as an angular unit, and called it " arbitrary.". Arbitrary it is; for any other angle might, with equal logical propriety, be assumed: yet at a glance we see that it does not stand on a like arbitrary footing with the assumption of a foot, a yard, or a mile, for the linear unit; inasmuch as it is the maximum value of angles, while lengths have no maximum. Yet while we are thus led to remark a difference in the two cases, it remains rather vague and uncertain what inferences may be drawn. * See Dr. Brewster's Translation of Legendre's Elements of Geometry. 64 DIFFICULTIES OF ELEMENTARY GEOMETRY. But, observe, that if instead of writing; angle = arcu circumf' we had been able to show that the angle is proportional to ( r-); we should have demonstrated all that Legendre desired. For it is clear that when the arc and the radius are given, the angle is hereby determined, without knowing any angular unit; and the above proportion would show that to give their actual length comes to the same thing as to give their numerical representatives, and conversely. Hence an angle could be determined from knowing the mere ratio of two lines, and without having previously settled on any angular unit. Angles, consequently, do not need an artificial unit at all, which circumstance was naturally, and, as the writer believes, truly accounted for, by saying that they had a natural unit in the entire circumference. 123. But can we establish in the present stage, that the angle is proportional to ra)? This obviously depends on our ability to prove that the circumference and the radius vary proportionally, which must rest on the following train of reasoning. Let R be radius of a circle, and C its circumference. Then, if R be given, C is geometrically determined, no other element whatever affecting the value of C. We are led to infer, that a mind perfect in intelligence, could deduce by some process of reasoning, the arithmetical length of C from the arithmetical length of R; and this would imply, that so long as the numerical value of R remained the same, the numerical value of C would likewise be unchanged, namely, that whether the R meant R yards, or R miles, or R furlongs, &c., accordingly, the result would be C yards, C miles, C furlongs, the linear unit being unimportant to the calculation. And this amounts to saying that the circumference must needs bear in all cases the same ratio to the radius. If any one refuse to admit the inference, it must be by alleging, that for the computation SCHOLIUM. 65 of C it might be insufficient to know the numerical length of R, without knowing farther whether it was in yards or miles, &c. But this seems opposed to the very nature of calculation; as appearing to imply, that help from the senses is needed; by which alone a mile can be distinguished to be a mile. 124. Whatever cogency this reasoning may possess, is certainly not due to our secret knowledge that the same result has been attained by other processes, according to the received methods of geometry; for it rests on a far wider principle, applicable alike to other sciences, and known as the Law of Homogeneity. In Mechanics, for instance, if homogeneous quantities be mutually dependent on one another, it is considered to be a sort of axiom, that the relation between them, or rather, the equation which expresses it, can only involve their ratios; insomuch that such an equation is called " Homogeneous in respect to them." And one might think that every geometer must be conscious, that his mind seizes with a kind of intuition on certain truths which depend solely on this principle, so as to prove fully that we do not need to deduce them by the ordinary steps, of which many are less strikingly obvious than the conclusion. Such truths are those involved in the doctrine of Similar Figures:- that if the three dimensions of a figure vary proportionally, all the linear measurements vary in the very same ratio;-the areas in the duplicate of it, (or as the squares of the lines,)-the volumes in the triplicate of it, (or as the cubes of the lines.) 125. The reasonings of Art. 123, are not really needed as a part of Legendre's argument; but they are more or less available for answering objections to it. Yet no geometer thinks it logically incumbent on him to answer objections, which, if his demonstration be perfect, must spring from ignorance. It is a condescension on his part to try to help the objector out of a difficulty into which he has plunged himself. At the same time it is hard to deny that the reasoning F 6 DIFFICULTIES OF ELEMENTARY GEOMETRY. (whether as laid down above, or as by Legendre,) involves an assumption which we would not willingly make, while on the threshold of a science which aims at perfect demonstration, namely, the possibility that the science of Geometry should exist. If it be conceded as possible, that by a mental process the circumference can be deduced from the radius, the demonstration appears complete. But if a stiff objector * protest, that for anything which has been yet proved to him, geometry cannot in the nature of things become a science of calculation, it may be very hard to answer him. CURVILINEAR ANGLES. 126. We may now return to the question which we left in Art. 99, and consider how the deviation of curves, from one another or from straight lines, is to be measured. And we are naturally led to the following process by what has been already laid down concerning the deviations of straight lines. Let A B be a common tangent to two curves, A C, A D, (Fig. 48,) just as in Fig. 38; Fr~ig. 48. and suppose a sphere of centre R? A, small enough to meet A B QR Din P, A C in Q, AD in R. Then with centre A, and radius of the sphere for radius, circular arcs P Q, P R may be described. The lengths of these arcs, when the radius is very short, give a rough or approximate measure of the two curvilinear angles B A C, * This is probably the meaning of Col. Perronet Thompson, who asserts that Legendre has confounded the determination of one quantity by others, with its calculability: an assertion, the full force of which I did not estimate, when with undue decisiveness I contested it in a Review of his work on Geometry without Axioms.-(West of England Journal, 1835.) CURVILINEAR ANGLES. 67 B A D. For, in that case, the arcs A C, A D, do not differ greatly from their chords. If the chord A Q be drawn, the rectilinear angle B A Q is accurately measured by ( iuP- ); and if the radius of the sphere be perpetually diminished, the chord and arc A Q tend more and more to coincidence. Thus the limit of the rectilinear angle B A Q is the curvilinear angle B A Q. Wherefore this last angle is measured by the limit of ( crcuf). But this is of no direct utility to us; for as the circumf. tangent lies closer to the curve than does any other straight line, it is evident that the peak of the surface generated by A C round AB is sharper than any conical peak; or no conical peak can be introduced at A interior to the peak of A C. Hence the curvilinear angle B A C is sharper than any possible rectilinear angle. (Which, it will be observed, is thus proved generally of the angle between any curve and its tangent.) It follows that( iuQ ) is a ratio which circumf. s can have no finite limit, but must vanish more and more, and tend perpetually towards zero as the radius of the sphere diminishes. 127. But while the above ratio does not help us to compare the curvilinear with the rectilinear angle, (because the former is indefinitely less,) we may probably in many cases compare two curvilinear angles with each other. For, drawing the chord A R, we have: Rect. angle BAQ: Rect. angle BAR = PQ: PR, the radius being here quite immaterial, while it is the same for both arcs. Let the radius perpetually diminish; in which case the rect. angles tend to confound themselves with the curvilinear ones; so that we get: Angle BA C: angle BAD = limit of {P Q: PR}. Now a priori it is impossible to foresee what will be the limit of the last ratio, which must differ exceedingly in different curves. It is not difficult, however, to invent Fr2 68 DIFFICULTIES OF ELEMENTARY GEOMETRY, curves in which the limit may be (1: 1) or (2: 1) or any other finite limit: or again, in which it shall be zero, that is, in which the ratio shall diminish below all limit. 128. To exemplify this, it will somewhat simplify the matter to confine ourselves to curves drawn on a plane, Let A be the centre of a plane, upon which are drawn an Fig. 49. indefinite number of circles, S whose common centre is A, <se' as in Art. 8 Let ABbe t \g a generatrix of the plane, _<g\r and AC any curve drawn HI^^Q F 1lpupon the plane, so as to be;A & p Pl> P' P B touched by A B in A; and of course A C, like A B, will cross the circles. Let PP' P".. be points in which A B cuts them, and Q Q' Q"... the corresponding points of intersection for A C. We will first show, that a new curve is conceivable, which shall have its curvilinear angle (or curvature) at A just one half that of A C. Bisect the arc B C in D, the arc P Q in R, the arc P' Q' in R', P" Q" in BR"; and so on; then the series of points D, R ', RR"...lie between the curve A C, and tangent A B. And they are indefinite in number, as are the circles which lie on the plane. By increasing the number perpetually, the points R ' "S... approach nearer and nearer to one another, and tend to form a continuous line. To use another form of speech; if the radius A P is arbitrary, and the arc PR = l P Q, the LOCUS of R is a certain curve line A R D, which lies between P C and A B. And since (PR PQ) = 1: 2, a ratio which remains constant however small A P becomes, it follows that L BAD: Z BA C = 1:2; or that the curvature of A D at A is half that of A B. 129. It is evident that in like manner a curve is conceivable, whose curvature at A shall bear any required ratio to that of A C. -CURVILINEAR ANGLES. 69 But as it is only the bending in the immediate neighbourhood of A which affects the curvilinear angle at A, and the farther parts of the curve R D may be bent aside without affecting that angle; it is by no means requisite that every arc PR should be to its fellow PQ in that required ratio. All that is needed, is that the limit of (P R to P Q) should = the ratio proposed. 130. To illustrate this simply, we will devise a new curve wholly distinct from A C,-having no portion, however short, in common with it at A,-and yet having equal curvature with it at A. For this, call the length P Q = a, where a is some fractional number, referred to some linear unit, suppose an inch. Then a2 represents the second power of a, according to algebraic notation; and is less than a, while a is less than 1. Thusifa, - a2 _; ifa a-, a2 -; if 1 2 =43 a =, a = ': if a = -1, a2 = -01; if a = -05, as = '0025, &c. Now whatever may be the length of P Q or a, which continues to diminish, always take Q S, (in the prolongation of the are P Q,) = a?; and hereby we determine a series of points S, S', "... whose locus is a curve A S E. It can never fall upon the curve A Q C; for Q S by hypothesis has always some length. Yet this length (a2) bears to QP a perpetually decreasing ratio: indeed Q: Q P = a2: a = a: 1, a ratio which becomes less than any limit, as P approaches A. Thus the approach of ES to C Q in the neighbourhood of A, is indefinitely closer than that of CQ to BP. It is immediately evident that A SE and A Q C have equal curvature at A. For P S= a + a2; P Q = a;. PS: PQ a + a2: a = 1 a: 1. But / BAE: L BA = limit of (PS: PQ) = limit of (1 + a: 1); which limit is barely (1: 1); since a is evanescent. Hence LBAE= L BAC. This teaches us that two curves which have unequal deviations, (estimated after the manner of Art. 99,) may nevertheless have equal czrvatures. The respective deviations through two equal finite arcs, however short, 70 DIFFICULTIES OF ELEMENTARY GEOMETRY. may be unequal; and yet they may approach towards equality as the limiting state, if the arcs be perpetually shortened. 131. Instead of measuring a' along the prolongation of P Q, cut off from P Q itself a length P T = a2; or what is the same, P T = QS; P T' = Q'S'; P"T" = QS", &c.... and let the locus of the points T T' T"... be a curve ATF, lying of course between AB and A Q. It is then evident, that as FT approaches A, it lies indefinitely nearer to BP than does C Q. For PT: P Q - a: a -a: 1, which is evanescent with a. Hence the curvature B AF is indefinitely less than B A C. 132. Orders of Curvature. We are led on to remark, that yet a new curve is devisable, whose curvature at A shall be indefinitely less than that of AF, which was itself indefinitely less than that of A Q. For we may suppose P U always cut from P Q, such as to be = a3, the third power of a; (thus, if a= a3 =; and if a= 1, a3 = 001;) Then P U: PT =a: a2- a: l.. LBA U: ABAT limit of a: 1; which is evanescent with A P, so that B A U is indefinitely less than L B A T. This process may be carried farther and farther by means of the powers a4, a5, &c. so that an endless series of curves is devisable, passing through A, and touched by A B, each having its curvature indefinitely less than that preceding it. There is nothing paradoxical or mysterious in this. It is only one form in which the infinite divisibility of space (in conception) is set forth. If we can suppose a perpetual and indefinite division of the line PQ as it moves towards A, we can of course equally conceive of curves tending towards A B more and more closely in their approach to A. The student who is familiarized to algebraic conceptions, will at once perceive that between the series of curves just now supposed, we can at pleasure interpolate others having a like relation to the series. Thus between the curve of a, CURVATURE OF CIRCLES AND SPHERES. 71 and the curve of as, we may interpolate a curve of 3 3 a or i/a3. Nowa a: a = V/a: 1, which is evanescent: also a: a = / a:, which again is evanescent. Hence 3 the curvature of a2 is indefinitely less than that of a; while the latter again is indefinitely less than that of a. Thus the geometrical doctrine of the Orders of Contact, is identical with the algebraic doctrine of the Orders of Infinitesimals. See Cauchy's Cours d'Analyse. 133. The most natural inference from the above, is, that Curvatures differ so enormously, and form so many new series of magnitudes not homogeneous with each other, as to render hopeless the thought of ordinarily comparing them. Yet the inference is mistaken. It may be interesting to the student, even in this stage, to be informed, that except at singular points, the curvature of any two curves soever is of the same order. Thus A C and A F in Fig. 49, although at A their curvature is so different, yet at Q and Tprobably, (or indeed at every other point but A,) have curvatures readily admitting of comparison. And in every curve of limited length, the number of points which have any other than ordinary curvature, is limited. CURVATURE OF CIRCLES AND SPHERES. 134. We have pursued the subject of Curvature into details not logically essential to the argument immediately before us, yet, perhaps, useful in helping the student to distinct ideas on the matter concerning which we are reasoning. We now resume the consideration of the particular curve, which is to us in the present stage most important; taking up the subject in reality from Art. 111. A Circle is everywhere curved: that is, no circular arc, however short, can be a straight line. For since a circle can slide along itself, the curvature or noncurvature at 72 DIFFICULTIES OF ELEMENTARY GEOMETRY. every point is the same. If then any, however small a part, were perfectly straight, the whole would be straight; and a straight line would rejoin itself. 135. Prolongation of a Circular Arc. Any circular arc, however short, admits of sliding along only a single determinate path. For let CD be any arc (Fig. 50,) and let it Fig. 50. slide into the position 7 D 8, the part D 8 c being its prolongation. We say then, there is no second prolongation as D E, such that it might equally slide into the position y D e. For, if this were possible, then the system yD 8 might revolve on y D, as an axis, until D came into the position D;.~D ~ which would imply that the part 7 D is a straight line. But we have shown that no portion of a circular are can be straight. Hence D 8 is the only prolongation of CD. 136. We infer that the sliding of CD along itself takes place by a constrained motion; and that if the prolongation be continued on and on, the are will at length complete the whole circumference, and rejoin itself at C. Moreover; Any are of a circle, however small, is thus proved sufficient to determine the whole circle. Wherefore, two circles cannot have any small arc in common. 137. No Peak in a Circle. The uniformity of curvature all round in a circle, proves that there can be no peak anywhere. For if one point were a peak, so would all be, which is contrary to Art. 97. Fig. 51. 138. The Tangent is PerpenT sC S~ dicular to the Radius. Let a A l// B:s circle have centre 0, (Fig. 51,) rP! \\ | n diameter CO 7, and let C T, C S, 0o be tangents to arcs CA, GB, on \ \p D opposite sides of C. It appears ^R\\\ 2 ~ by the last Article, that TCS S -~ ~ must be a straight line. But besides, since by Art. 75, the circle may be doubled about CURVATURE OF CIRCLES AND SPHERES. 73 its diameter Cy, so as to make the opposite halves CA y, CB y, coincide; it follows that CS and C T would coincide, and the angles 0 CS, OCT are therefore equal. Consequently, each is a right angle. 139. Three points in a Circle cannot be in the same straight line. Thus C, A, P cannot lie evenly together. Else, if the circle were doubled about itself, till P and A changed places, and C fell upon Q, Q also would lie evenly with A, P, and C; where P Q = A C. Similarly, if Q R = P A, it would follow that R lay evenly with Q, P, A, C. Again, take R S = P Q, and S will be likewise in the straight line. Continue measuring off parts, alternately equal to A P and A C, and we shall at last come round either to C exactly, or to a point beyond C. If we never light on any point twice, however often we go round the circumference; we shall determine an infinite number of points, (as lying evenly with C, A, P,) whose locus is the circumference itself. This would prove the whole circumference to be straight. But if we light again on some point, as C, then a straight line may rejoin itself. Either result is absurd. 140. The Tangent does not meet the Circle again. That no small arc of the circle at C coincides with the tangent, appears from Art. 134. Hence at C, the tangent and circle part; the tendency of the circle, even on starting at C, being towards y, in which CA and CB will at last meet; the tangent having no tendency towards any point on one side of it more than on another side. Now the arcs CA, CB, having once quitted the tangent, by reason of their tendency towards y, can never again return towards the tangent; but as their path is prolonged, must bend perpetually more and more away from it, since their curvature is all one way. Hence the tangent and circle meet in no point but C, the point of contact. Otherwise: If the tangent met the curve again as at P, take Cp = C P, at the opposite side of C; and it must evidently meet the curve likewise at p. Then P, C, p would be three 74 DIFFICULTIES OF ELEMENTARY GEOMETRY, points in the circle, lying in one straight line; which we have just proved cannot be. For this cannot be evaded by saying that P and p might be but one point, namely y, (as CPy = Cp y): for if y were a point in the tangent, then the diameter and the tangent would coincide; contrary to Art. 138. 141. Curvature of the Sphere. If the semicircle CAy revolve about the diameter COy, the arc will generate a sphere; since the surface thus generated is evidently equidistant from 0. (See also Art. 64.) Let C ay be any position of the generating semicircle. Then since the curvature of CA and of Ca, estimated at the point C, is identical, we are justified in saying, that the curvature of the spherical surface at C in every direction round C is equal. Again; since the sphere may slide on its own ground in every direction, until any point C assumes the place of any other point D; in which case the surface immediately round C would occupy the place before held by the surface immediately round D; it follows that the curvature at C = that at D. It is manifest also, that the curvature of a sphere is measured by that of its generating circle. 142. Straight line lying upon a Sphere. The line TC, which is tangent to the generating circle, is obviously perpendicular to the sphere's radius, which is the circle's radius. It is besides wholly without the sphere. For it cannot be wholly within; for no infinite straight line can be shut up within a limited solid, (Art. 68.) Nor can it pierce nor again meet the surface; for similar reasons to those urged in Art. 140. Hence it lies wholly without the sphere, and has in common with the surface only the isolated point C. 143. It hence follows that if C Tbe a given straight line of indefinite length, and 0 a point without it, the perpendicular 0 C being dropped determines C as nearer to 0 than is any other point in the line. 144. While the semicircle CAy revolves about the diameter C7, and generates the spherical surface, let the CURVATURE OF CIRCLES AND SPHERES, 75 tangent C T revolve with it. Then the locus of C T is a Plane, whose axis is Cy. This is fitly called the Tangent Plane to the Sphere at C, because every one of its generatrices is a tangent to a generating circle of the sphere at C. But for a fuller understanding of this, we must take up the matter on more general grounds. 145. If P be a point on any curved surface whatever, and lines P Q, P Q', P Q"... be drawn from it along the surface, to The_ c Ph. which P T, P T', P T".... are tangents, the locus of all these tangents is a single sheet, forming a Cone whose vertex is P. For if by varying the nature of the curves P Q, (which are subject only to the condition of being drawn from P along the surface,) we could produce two or more sheets, then the sheet which lay closer to the given surface would entirely separate the other from it, so that P T on the outer could not lie in contact with the surface at all, nor with any curve drawn on the surface, Such a cone is called the Tangent Cone at P. But the Cone is susceptible of several varieties. (1.) It is possible that the generatrix P T may move round in such a way as to be always perpendicular to some axis drawn through P. In this case the locus becomes a Plane, which is really only a variety of the Cone; though from the absence of any peak in the plane at P, we are not used to denominate a plane a sort of Cone. The axis to the Tangent Plane is then called the Normal at P. (2.) The Cone, although not a Plane, may be such, that every line PT has its prolongation P t also lying on the Cone's surface. In this case we may conceive the cone to be generated by the half revolution of the line TPt, which is fixed on a pivot P, and vibrates above and below a certain plane, while performing 76 DIFFICULTIES OF ELEMENTARY GEOMETRY. its motion (Fig. 53.) No particular name has been given to Fig. 53. this variety. (3.) The number of lines P TP T',... which are such that their prolongat tions lie upon the cone, may be finite; in which case, these form so many Ridges crossing in P. Or there may be but one ridge TPt, as along the - top of a bank. (4.) No line P T may have its prolongation P t lie upon the cone; and then there is a true Peak at P; ~ ~~..A_ -~:~this is what we generally understand by a Cone. 146. It would lead us into too long a digression to attempt to prove, what the reader will readily convince himself of,-that except at " Singular Points," a curved surface always admits one Tangent Plane and Normal. To express this otherwise: " Consecutive points on a curved surface cannot be Peaks, and consecutive lines cannot be Ridges." This will be taken up afterwards. 147. Returning to the Sphere, we now see that the plane generated by the revolution of C T round Cy (Fig. 51,) is fitly called tangent to the sphere, inasmuch as it contains the tangent at C not only to the circular arcs CA, C a, &c.... but to every possible line that can be drawn from C along the spherical surface. Also it is evident that no straight line can be drawn from C, between the plane and the sphere. For indeed this is contrary to the very nature of a tangent plane. And the sphere and plane have but one point in common, the point of contact. Consequently, a straight line T CS, which in a popular sense touches the sphere externally, at a point C, does also in a mathematical sense touch (or osculate) it; and meets it in that one point Conly. It is also perpendicular to the sphere's radius 0 C, which meets it in the point of contact. CURVATURE OF CIRCLES AND SPHERES. 77 148. It is included in the above, that a Sphere does not admit of Peaks or Ridges; for we have proved that it has but one Tangent Plane at every point of the surface. 149. Any portion of a sphere's surface is sufficient to determine the whole sphere. For in the given portion assume a point A, and upon the surface Fig. 54 -take a distance less than the least distance of A from the boundary line. With this dis- tance, determine upon the surface a circle whose pole is A. The circle has one determinate axis, as A a, which must also be an axis of the sphere. Assuming a second point B, we similarly determine B 3, a second axis of the sphere. But A a, B 3 cannot have more than one intersection, O, which is thus the single determinate centre to the spherical surface. But when the centre 0 is settled, and a point B in the surface, the whole sphere is determined. Hence there is but one spherical surface, of which the given area can form a portion. This is equivalent to saying, that " two spheres cannot have any area of their surface, however small, in common.' 150. The given area may of course slide along the sphere's surface, which is its prolongation (or extension) on all sides. But, moreover, it must slide by a certain constraint, so as never to be able to deviate from Fi.. this one surface; that is, "it has one determinate extension." For if two sheets were imagined, into ^ A either of which it might slip, as B C and B D in Fig. 55, then since D these must have a common tangent plane at B, one must have less curvature at B than has the other. Yet unless both had everywhere curvature equal to that of A B, and 78 DIFFICULTIES OF ELEMENTARY GEOMETRY, therefore mutually equal, it is evident that A B could not glide along them. 151. It is evident, farther, if in Art. 95, (Fig. 36,) we suppose P AQ to be no longer a curve line, but a spherical surface, and M, L, two points which run together, the straight line ML being prolonged will tend towards a tangent (mA 1) to the sphere, as its limit. Else, the sphere would have a peak at A; which has been proved impossible. 152. Hence we infer, that on a spherical surface, not more than two points can lie in a straight line. For if M, N, Fig. ~5 6 ~. P (Fig. 56,) were three points Fig. 56. _AM on a sphere that lay evenly, we N=^ might suppose the line P NM IPip ^^^M A y to revolve about P, so that M ~r ^^N /~ - ~ and N might run together into A ^ sA aoa point int in which case PA would touch (or osculate) the sphere in E, although it likewise meets the sphere in P; which is contrary to Art. 142. 153. Convexity and Concavity. If any solid be enclosed Fig. 57. by a surface (Fig. 57,) such T 6 that the straight liqe (A B) B joining any two points (A, B) on the surface lies wholly within the solid, it is evident that every tangent plane, or tangent cone to the surface lies outside. For if the chord B A revolve about the point B, so that A may move up towards B along any curve A C B drawn on the surface, the prolongation of B A lies entirely without the solid, and, consequently, the tangent B T to which B A tends, will also lie wholly without. Hence the curvature is everywhere turned away from the part exterior to the solid. The outer side is called Convex, (protuberant, bulging); the inner, Concave (hollow). The same names are popularly used, (and may be used CURVATURE OF CIRCLES AND SPHERES. 79{* with much propriety,) in the case of solids enclosed by planes, and which therefore possess Fig. 58. no curvature, at least, as far as has.o yet appeared concerning the plane. /7 < -~ Thus if AB C DEF (Fig. 58,) be a B(/\ / solid, fulfilling the condition that no \ straight line pq joining two points \__ / p, q, on its surface, has any point in X it exterior to the solid, then the outer surface is called Convex, and the same looked at from within, Concave. A part of a surface may be called Convex on one side, even when the whole is not. Fig. 59. Thus, let A B C D E (Fig. 59,) C be a solid, which is not wholly convex externally, inasmuch as the straight line C B lies without it. Nevertheless another portion, as B A E, may be ex- - _ _ ternally convex, tried by the following test; that if a line B n n Ep B be drawn on the surface, cutting off a certain area around A, a new surface is conceivable, which shall fill up the line B mn n Ep; and shall, with the opposite area round A, form a solid outwardly convex. If so, we are justified in calling the area A B m n Ep B A by this name. [154. If a solid is not only outwardly convex, but also free from peaks and ridges, and is everywhere curved, so that no straight line upon it can touch it in more than one point at the part of contact; it is often called oval, or Egg Shaped (see Fig. 57): although this term is sometimes confined to such solids as have a peculiar symmetry, especially those of Revolution. It is evident from the above, that " Spheres are a species of Ovals," according to this definition. Also, Spheres are externally convex. 155. Ovals touch one another externally in but one point. Let the two ovals, P Q, PR, be in external contact at P. Do DIFFICULTIES OF ELEMENTARY GEOMETtRY Then, the curvatures at P being in opposite directions, Fig. G6. the surfaces bend away from each other on all sides round P. If the two surfaces coincided near the point of contact over any small area, this must be because the curvature of one or both was there crushed by pressure. For such area could not possibly be convex on each of its sides, since convexity is the reverse of concavity. Nor can the ovals have any line in common, in the neighbourhood of P: for they must have a common tangent plane at P, which entirely separates them; and such line (if it existed) must lie along that plane. Yet if it were a generatrix of the plane, then it would be straight, and would prove the solids not to be convex in this part: or, if Fig. 61. it were curved, (Fig. 61,) then it would be P met by generatrices (Pp, Pp', &c.) of the plane in two or more points; and of such generatrices many must be external to the solids, otherwise the solids would have an area (Ppp") in contact. But if one such generatrix existed, it would prove the solids to be there externally concave; since the straight line (Pp) joining points to (P, p) in their surface, is exterior to them. Lastly, it having appeared that in the immediate neighbourhood of P, the ovals have no point in common but the isolated point P itself, it is farther plain that the surfaces meet no more. For as they continue to bend away in opposite directions they can not approach each other again. 156. It is included in the above that Spheres touch each other externally in but one point, and have no second point of the circumference in common. But as the reasoning in tje last Article will to some appear lax, to others difficult to follow, no use has been made of it in the seqzel.] CONTACT AND INTERSECTION OF SiPHERES. 81 CONTACT AND INTERSECTION OF SPHERES. 157. "A straight line is the Fg. 62. shortest path between two given points." Let the given points be A, B, (Fig. 62,) and let C be any point in the straight line A CB. We will prove that no path joining A and B can be as short as it might be, unless it passes through C. Draw C T at right angles to A CB, and let it generate a plane round the axis A CB. Also, with centres A, B, and radii A C, B C, describe two spheres. Then since the sphere of centre A is on the same side of the plane that its centre is, and the sphere of centre B is on the same side as is its centre; and the centres are on opposite sides; therefore the plane wholly divides the spheres (Art. 147,) which have thus only the point C in common. Hence any path AP Q B which does not pass through C, must pierce the spheres in separate points, as P and Q. Thus a needless length P Q is incurred; for the paths A P and B Q might be otherwise directed, (without change of form,) so as to join the points A and C, B and C; which would save the distance P Q. Thus any path, to be as short as possible, must pass through C. But C is any point in the straight line A B. Therefore no path can be as short as it might be, unless it run along the whole straight line A B. Cor. Hence the.straight line is the measure of distance between two points in Space. 158. Addition of Distances. If now A, B,, D...are points whose distances, two and two, are given, (A from B, B from C, C from D, &c.) or, what is the same thing, if the line A BCD... is elsewhere inflexible, but has joints at B, C, D,.. then the distance of the G 82 DIFFICULTIES OF ELEMENTARY GEOMETRYt first from the last is greatest, when the points A, B,.. E are ranged in order along a straight line. For, first, if C be not in the straight line A B,.' the straight line A C is shorter than the sum of the straight lines A B, B C; so that the distance A C is not so great as it might be, namely, by bringing C into the prolongation of A B; and thus the distance A C attains its maximum. Next, the same reasoning shows that A D does not attain its maximum, until D is in the proD longation of A C. And so on continually. 159. Strained Thread. If then A E be a thread, which is drawn tight, so as to pull its extremities as far apart as possible, (the length being supposed invarible,) all the points in A E will dispose themselves in a straight line. This is an experiment made inadvertently by every human creature; so simple and convincing, that it might justly be made an Experimental Law of Geometry. And if any of the reasoning above used is at all questionable as to accuracy, this would be the most preferable mode of obviating all objection. It would also greatly shorten the process of attaining our farther results; but this, in writing an entire scientific treatise, is not always an advantage; for every step which we make is perhaps of intrinsic value, and if omitted in one part, must be introduced in another. 160. External Contact of Spheres. Let two spheres be placed at a distance, and two points C, D, on their surfaces, (Fig. 64,) which we design to bring into contact, be placed in the line of their centres A, B. Then let A and B, C and D, slide with the spheres along this line, till C coincides with D. Since the tangent planes at C and D lie outside the spheres, and do not meet one another till C and,D unite, (for else, contrary to Art. 111, from their concourse would be dropt two perpendiculars to the line A B,) it follows that the spheres are wholly separated CONTACT AND INTERSECTION OF SPHERES. 83 until C runs into JD, and the two tangent planes merge into one. In this Fig. 64. position no second point is common to the surfaces of both spheres; C c (because this is the only point on which either meets the tangent plane, Art. 147;) and the point of contact lies in the straight line of the cenztres. Now I say, this is the only mode in which C and D can be in contact. For if they could remain united while B received ever so small a displacement, by the motion of the sphere B about the fixed point C, let A B be joined, and it would be shorter than A C + CB, by Art. 157, or the distance between the centres less than the sum of the radii; which would imply that the curvature of one sphere or other had been crushed in by the motion.: which is not contemplated by our hypothesis. We see, therefore, that though ordinarily one body must be fixed to another immovable body in "' at least three points which are not in the same straight line," in order to retain the latter immovably; yet in this case it suffices to fasten the sphere B to the immovable sphere A (or conversely,) by a single point C. For the curvatures, though they are not in linear or superficial contact, preclude angular motion as effectually as if they were; the curvilinear angle being less than any rectilinear angle. 161. Internal Contact of Spheres. Of two given spheres, let that whose centre is A be the larger, (Fig. 65,) and AC a radius. From CA cut off CB, equal to the radius of the smaller; and join A P, B P, where P is any point in the surface of the larger. Then since A P is less than AB + BP, while A P = A C = AB + B C; therefore BC is less than BP. Hence P is beyond the sphere whose centre is B, and radius B C. If, then, the smaller G 2 DIFFICULTIES OF ELEMENTARY GEOMETR.Y. sphere e e laced with its centre at B, it will touch Fig. 65. the other internally at C, pD, b anppled in exo oter point. Also,.of the twohe point of contact is eesow at. Hence byin the line of the centres proi A ~ —~ ___ longed. __ _ ~Now I say, this is the __r r~ ~ only position in which the h c d nt be spheres call have internal contact at C For if a third sphere, whose centre is, be applied in external contact at C with the greater of the two it is, fortiori, in o contact with the less, at C. Hence byrthe last Article, A, C, and D, are in a straight line; and so are B, C, and, in a straight line; which could not be unless A, B, and C lay evenly with one another. 162. The above is in harmony with what appeared in Arpt. 149, that it is impossible for two different spheres to have a portion of their surface, however small, in common. Also, the greater sphere has the less cune vature. 163. Intersection of Splere and P th ae. Let O be any point exterior to the straight line A B C, (Fig. 66,) and Fig. 66. QA be dropt perpendicular to A C. W VTe have shown that OA is the shortCt est path from 0 to the line, (Arts. 143. and 157). If then, P traverses the tr line A P C, the distance O P increases at first, when P starts from A. Now ~ -;? - IjA we farther assert that OP always l continues to increase, as the distance (A P) of P from A increases. For if not, we must make one of two suppositions. EITHER, "there is some portion (B C) of the line, such, that while P traverses it, the distance 0 P is invariable." Now this would imply that an entire ring of the plane, generated by B C aroulnd the axis 0 A, is a portion of the CONTACT AND INTERSECTION OF SPHERES. 85 spherical surface whose centre is 0, and radius 0 C or B: thus, in every position, B C, as it revolved, would be a straight line touching the spherical surface along its entire length. But this is impossible, by Arts. 142, 152. Again: ELSE, " there is a point B, up to which OP keeps increasing, and at which O P reaches a maximum, and afterwards decreases." If so, then j3, y, being two points at opposite sides of B, in the line A B C, and ever so near to B, the distances 0 3, 0 i are less than 0 B. Consequently j3 and y would be points zwithin the sphere of centre 0, and radius 0 B; and the straight line 3 B 7 would be a tangent to the sphere at B, and yet be inside the sphere. which again is contrary to Art. 142. Since then the length OP begins by increasing, when P starts from A, and afterwards, it never remains constant, and never diminishes, while A P continues to increase; it follows that A P and 0 P perpetually increase together. 164. If now the line A C generate a plane round the axis 4 A, (Fig. 67,) the points B, P, C will each generate a circle of this plane. Let Fg. 7. BA B', PAP', CA ', be diameters of these circles. Of course, then, 0 B = OB, \eOP = ORPP, O C = O U; or the points equidistant from A along the line CA C', are also -:-0 equidistant from 0; as B and \!A B', P and P', C and C';&c... But farther, if a sphere be./_ described with centre 0 and radius OP, it is clear that the circle of P is a circle on the sphere. Also, by the last Article, the circle of B lies within this sphere, since 0 B is less than the radius OP. This being true for every point B between A and P, it follows that the whole plane area enclosed by the circle of P lies within the sphere. On the other hand, since 0 C is greater than the radius OP, the 86 DIFFICULTIES OF ELEMENTARY GEOMETRY. circle of C is outside of the sphere: and this being true, so long as A C is greater than A P, it follows that all that part of the plane which is exterior to the circle of P, is outside the sphere. Hence the plane crosses or cuts the sphere in the circle of P; and only this circle is common to the surface of the sphere and the plane. 165. As it will soon be proved that all parts of a plane are like all other parts, the above shows generally, that "the intersection of a plane and sphere is a circle, whose axis is the common axis of the sphere and plane." 166. Intersection of Spheres. Let B, (Fig. 68,) be within or without a sphere of centre A; and let B A cut the sphere Fig. 68. in the poles C and y, of which C is avL^ ~ nearer to B than is y. It is then evident from what was proved about the contact of spheres, (Arts. 160, 161,) that if P is a point traversing the surface, BP is least when P is at C, and is greatest when P is at y. Suppose then that P moves along /\\ / the surface from C to y by as short a path as possible. On its starting from C, the distance BP begins to P~ r increase but we now farther assert, that B P perpetually increases with, CP. For, FIRST, there is no small portion (QR) of P's path,, such, that while P traverses it, the distance ____t_ o BP can remain invariable: else the sphere of centre B and radius B Q or B R would have a band of surface, of which Q R is the breadth, in common with the given sphere; namely, the surface generated by QR round the axis Cy. But this, by Arts. 160, 161, is impossible. NEXT; neither can B P attain, as at Q, a maximum value and then again decrease. CONTACT AND INTERSECTION OF SPHERES. 87 For if q, r, are two points in P's path, very close to Q, on opposite sides, then B q and B r would be both less than B Q, however short the distances q, Q r. Consequently, the sphere of centre B and radius B Q, would be touched internally by the given sphere along the whole circle which Q generates round Cy: for the band of the given sphere generated by q Q r would all lie inside the other sphere. This again is obviously opposed to Arts. 160, 161. As, therefore, during the increase of the distance CP, the other distance BP never remains constant and never diminishes, it follows that B P ever increases with C P; that is, until C P reaches its maximum; which must be when P arrives at the other pole y. Hence, if two spheres having centres A and B, are placed so near that the distance of their centres is less than the sum of their radii, they intersect in a circle, (as in the circle of Q,) whose axis is the line of the centres. For it is manifest from the above, that a portion of the surface of sphere A, which is intercepted by the circle of Q round the pole C, is interior to the sphere of centre B and radius B Q; while the rest of the surface is exterior to that sphere. 167. lTe distance of points on the sphere increases with their absolute distance. For the above reasoning holds equally, if B (Fig. 69,) coincides with C. Then it appears that G P in space increases F with CP on the surface. 168. Moreover, (Fig. 68,) the angle CBP increases with the distance CP along the sphere's surface. Wherefore the distance BP in a straight line increases with the angle CBP. When B coincides with C, (Fig. 69,) c we can say that " the chord of CP increases with the arc CP," and consequently, " increases with the angle CA P." Thus the diameter CT is evidently the longest chord in a sphere. 88 DIFFICULTIES OF ELEMENTARY GEOMETRY. 169. Triangle of Distances. When A B Q is an infexible system, (Fig. 70,) it was an experimental Law of Geometry that the point Q revolving round the fixed points A and B, Fig. 70. generates a single self-rejoining (Q aQ line, which we call a Circle. But we are now able to say more; namely, that if the system be not otherwise known as inflexible at A 1B 3 B A, B, and Q, these three points will be inflexibly connected, if the distances A B, A Q, B Q are three assigned invariable lengths. We may state the matter thus. Let A and B be points fixed and known: let Q be a point whose distance from A is known, and whose distance from B is also known. Then Q is on the spherical surface of centre A and radius A Q; and also on that of centre B and radius B Q. Hence its locus is in the circle which is the intersection of those! spheres. But this is the very circle to which it would be restricted if the system were by hypothesis inflexible. Consequently Q is laid under the very same restrictions by assigning its distances from A and B, as by connecting it inflexibly with those points. Thus if A B, B Q, QA, are three given distances, the shape of the system is determined, and (if these lines are straight,) the three angles of A, B, Q are determined. ON THE PLANE. 89 ON THE PLANE. 170. Evenness of the Plane. Let the axis B A C (Fig. 71,) have AR perpendicular to it, Fig. 71. and let A R generate a plane a round it. We have to prove that all parts of the plane are Even; that is, "there is no curvature at any part towards either side." This will have been proved, if we have shown that a straight line pressed against any part of the plane lies close against it along the whole length; or, (what amounts to the same,) thatif D, E be any two points soever on the plane, every point P in the straight line D E, or in its prolongation, is likewise on the plane. Now observe; FIRST; that every point R in the plane, lies at the same distance from B as from C, if A B = A C. This is clear by inverting the plane and its axis, so that B and C may exchange places, while A and R remain as before; which is possible by Arts. 80, 81. NEXT; a point Q on the same side of the plane as B, is nearer to B than to C. For if Q Cbe joined, it must cut the plane as at S: join B Q, BS. Then B S = CS, as before, S being on the plane, like R: also B S + S Q is longer than B Q, (Art. 157);.'. CS + SQ or CQ is longer than BQ. THIRDLY; let B and C be fixed points, and R a movable point whose position is assigned by the single condition that it shall be as far from B as from C: and the above justifies us in affirming, that the locus of R will be the plane. Now D,'E being two points on the plane, each is as far from B as from C; so that the three distances D E, EB, BD are identical with the three, DE, E C, CD. Consequently B and C are in a circle whose axis is D E 90 DIFFICULTIES OF ELEMENTARY GEOMETRY. (Art. 169). Now P being in this same axis, is equidistant from all points in the circle; hence PB = P C. It follows that P also is on the plane. And P being any point in D E, the whole line D E is on the plane; as was alleged. Fig. 72.,Kr \ _ StyI 171. New mode of generating E - the Plane. It is now manifest, that a plane can be generated from any two given lines in it, - Y" as D, Es (Fig. 72). For if a straight ruler X Y press always against both lines, the locus of X Y is the plane, if no other restriction is added to its motion. 172. It thus appears also that one and only one plane can pass through two lines, or through a line and a point, or through three points; provided in each case that not all the data lie in one and the same straight line. 173. Sliding of a Plane. Supposing D 8, EE to be at rest, the plane may be so transferred as to keep it always pressing close against D 8, E e. But by such transference, the plane (as an indefinitely extended whole,) would always occupy the very same position. Thus it slides on its own ground. 174. Axes of the Plane. During the sliding, the axis may be brought to pass through any point required of the plane: so that the plane may be said to have an axis through every point of it. Originally, we supposed the plane to be generated from some one particular axis and centre: but it now appears that there is nothing peculiar to distinguish this axis from others. 175. Perpendicular to a Plane. Since an axis is perpendicular to every generatrix of the plane, it is justly called Perpendicular to the plane itself, as was noted in Art. 110. From a given point in the plane there can be but one axis to the plane, because there can be but one axis to every circle on the plane. Hence but one perpendicular to a plane can be drawn from a given point in it; for a second perpendicular might evidently be made a second axis. ON THE PLANE. 91 176. Moreover, "from the intersection of two straight lines, can be drawn one and only one line perpendicular to them both:" namely, an axis to the plane which passes through them. For if a second common perpendicular could exist, it might be made axis to a second plane that should pass through both. 177. Perpendicular dropt on a(Plane. If A be a point (Fig. 73,) which is not on a plane B C D,. there is some shortest path A C, from A to the plane. Then the sphere of centre A and radius A C cannot intersect and go beyond the plane; therefore, neither call it coincide with the plane in more than n _ one point, as appears by Arts. 142, 147. Hence, too, we ascertain that A C is an axis to the plane, and perpendicular to it. But no second perpendicular can be dropt, as AE; else, joining CE, we should have AC and A E two perpendiculars dropped on the same straight line CE. 178. Extension of the Plane. Any small plane may be prolonged or extended indefinitely, but in a single determinate sheet. This is evident from the generation of the plane in Art. 171, or by its sliding along itself. It follows also that any small plane within a given solid, however large, may be extended so as to cut the surface in a selfrejoining line, and then pass out. 179. Intersection of Planes. Moreover, two planes which have two points in common, have in common likewise the straight line joining these two points. But they can have no other point in common without becoming one and the same. Either plane may revolve on this line, until it coincides with the other plane: and from the nature of rotation we deduce that the planes here cross each other. The common line is called their Intersection. The angle between them is called Dihedral, and may be readily measured, as in Art. 32. It is clear that if two planes have one point in common, they must intersect, viz. in a straight 92 DIFFICULTIES OF ELEMENTARRY GEOMETRY. line: for did they not cross each other, one or other would have a peak or curvature at the common point. 180. Parameters. Since any one plane (of indefinite extent,) can be made to coincide with any other, planes do not differ at all. Having everywhere no curvature, any two such surfaces are in quality perfectly alike. The same is true of Straight Lines. But Circle differs from Circle, Sphere from Sphere, Cylinder from Cylinder, by reason of the difference of radii, which occasions a difference of curvature in them. When, however, the length of the radius is given, the shape and size of the figures (though not their position,) is completely determined. With reference to this property, their radii are called Parameters to them: but Planes and Straight lines are said to have no Parameter. PART Il. ON PARALLEL STRAIGHT LINES. 181. THE term Parallax is well known in Astronomy, to indicate "the change in the apparent position of objects, caused by a change of position in the observer: " and more especially, the difference produced by the fact that the observer is on the earth's surface, instead of being at its centre. Although the name is not particularly needed in Geometry, we meet with the thing. We may regard it as an error arising from Excentricity, in the computation of angles, when the corner of the angle is regarded (for simplicity,) as though it were in the centre of a circle, although this should not be accurately true. Fig. 74. 182. Let CD be the ^.~-~ --- > arc of a circle (Fig.,/ P /\ 74,) of which A is the ~/^~ ^^~~. \ ~ real centre, and B the / \ supposed centre. This / / ' false supposition inK L v:' 4 \~ volves the notion that the angle CBD is N <M / ^]l'- j measured by the number of degrees in the \arcs CD: which we \ / y know to be true con\~-. - ^/ cerning the angle CA D. But as B is 94 DIFFICULTIES OF ELEMENTARY GEOMETRY. not the centre, it is most probably erroneous to imagine those two angles equal: and the difference between them is the error which results from assuming the arc CD as the measure of / CBD.; or, (as we may also put it,) the error arising from the excentricity of B. 183. There are two ways of diminishing the error. The more obvious is, by seeking to diminish the distance of the pointB from the true centre A. But another method equally effectual, is, by increasing the radius of tie circle, and supposing B C, BD prolonged, so as to meet the new circle in C' and D'; or (if the radius be again increased,) in C" and D"; then, I say, the angle B will at last, when the radius is of enormous magnitude, be measured with far less error by the degrees in the ar C' D', or C" D". To illustrate the meaning, and at the same time bring conviction of its truth, let B be one foot distant from the centre A, and the original radius A C = two yards. The error of taking the arc CD to measure the angle B, may be looked on as gross. But take A C', a new radius, = a thousand miles: and it is clear that one foot is so insignificant in comparison, that the error of confounding B with the centre A, must be very small indeed. If, however, we wish to make it still smaller, take A C" = a million miles: and so on, till it be as small as we choose. The same method would hold, if B were a mile, or were a thousand miles, distant from A: for we might then suppose A C' = a billion miles, A C" = a trillion miles; and so on. In short, if KLMN be any finite area enclosing the centre A, we may suppose a radius A C" so vast, that this area may be but a speck in comparison with the circle: and be the error what it may, of confounding any point in this area with the centre, that error may be reduced as small as any one chooses to demand of us, if we may increase the dimensions of the circle at pleasure. To express ourselves in the phraseology of the higher mathematics: we do not yet know how to estimate the error, when the radius is given; but so much we know a priori ON PARALLEL STRAIGHT LINES. 95 c, it is such a function of the radius, as to vanish when the radius is infinite." And if it be asked how we assure ourselves of this, the only reply is, that by the very nature of Quantity, anything that is finite, (as the amount of excentricity,) becomes insignificant and evanescent in comparison with that which is susceptible of increasing indefinitely: and if any difficulty attach itself to the subject, it is not a purely Geometrical one, but is equally found in the doctrine of Quantity and Number. 184. Like reasonings apply to the sector B CD, as well as to the arc CD. Were B the true centre, it was shown (Art. 117,) that if we assume the whole area of the circle as the measure of four right angles, the sector B CD would measure the angle BCD. This consequently will, when B is excentric, deviate from the truth only by an error which is evanescent when the radius is indefinitely increased. 185. If the above be conceded as valid argument, all difficulty on this subject is broken down: for it is now easy to prove that " the three external angles of any triangle are together equal to four right angles." Let L MN be any triangle (Fig. 75,) within which take Fig. 75. a centre 0; and describe a circle containing the triangle. Prolong / \ \ LM, MIN, NL, to u/ Bt \ meet the circumference g \ \ a \ in G, K, H. Let LY, L AL-v MO \i_~, N~, be the angles 4 &) - HL G, GCMK, KNH, (^^^ ^^-^^ I~ / in degrees, and let X, \/, v, be the circular areas intercepted by the same angles: also let a = whole area of the circle, r = area of the triangle. Now if L were the true centre, and not 0, we should 96 DIFFICULTIES OF ELEMENTARY GEOMETRY. X L~ have -: but this most probably is erroneous." a — J60 Let 8 be the error, so that we have accurately: A L~ a 600 where 8 is either (possibly) zero, or else some number positive or negative. One thing only is known about 8, by the preceding article, that if the radius of the circle is indefinitely increased, 8 diminishes beyond all limit. We similarly have: _ M0 v N 0, 360+ [' an.d - 8; a 3600) a 8600 Consequently: X + + v _L + M0 + (8 + )NO a 860' Now A + u + v = area of circle, minus area of triangle; A + +_ v a-T r or, =a- r; hence --- _= = 1 - -. a a a Also, when the circle perpetually increases, a increases ad minlz., while r remains finite. Consequently, - is evanescent, and the limit of + ' + v is. But L~, i~, N'~ do not vary with the variation of the radius; and the limit of (8 + 8' + S") is zero, because each separately tends to zero. Hence we obtain: 1 L~ + M~ + NO 1 360~ Or, L~ + iV~ + No = 3600; which was to be proved. Cor. 1. Therefore the three internal angles, together = 1800. Cor. 2. The four internal angles of a plane Quadrilateral, together = 360~. 186. This is the proposition, from which Playfair in his X By using the phrase ultimately in its well known acceptation, the details of this argument will be shortened. Thus: and so on. of this argument will be shortened. Thus: -ultimately = -- and so-on. ON PARALLEL STRAIGHT LINES. 97 Geometry deduces the whole theory of Parallel Straight Lines. But as I have introduced a new definition of the word Parallel, (as equivalent to Equidistant,) it is desirable to pursue the subject a little farther. First, it must be proved that: "If two straight lines (A B, CD,) in the same plane, are both perpendicular to a third straight line, (EF,) they are Parallel." (Fig. 76). Let B, CD intersect EF in G and H. Fig. 76. Then since the angles at G and by hypothesis c __ are right, G H is the distance of the point H " G from the line A B, and of the point G from the line CD: (Arts. 143, 157.) Let B be any other point in AB, and drop BD per- L K pendicular to C HD; then BD is the distance of B from the line CD. Also: since the quadrilateral B G H D has its four angles rD- together equal to four right angles, and of these we know three to be right, viz. G, H, and D; it follows that B, the fourth, is also right. Therefore D B is the distance of D from the line G B. And the point B was arbitrarily chosen. It appears then, that to prove the parallelism, or equidistance, of AB and CD, it is only requisite to show that GH = B D. Let K be the middle point in G B, and drop KL perpendicular to CD; then, as we showed above that the angles at B were right, so can we show that those at K are right. Now let the figure KBD L turn round KL through half a revolution, till KB, L D have come into the directions KG, L H. Then since KB = KG, B will fall on G. But B D being in the plane of KC, and ZKB D == Z KGH, it follows that B D will have the direction of G H. But L D has simultaneously taken the direction of L H. Therefore D, the intersection of B D and L D, falls upon H, the intersection of G H and L H. Thus G H, B D, coincide, and are equal. Which was to be proved. 187. " Through a given point (G) can be drawn one and HI 98 DIFFICULTIES OF ELEMENTARY GEOMETRY. only one line in a plane, parallel to a given straight line CD; " and it will be itself straight. For if we drop GH, perpendicular to CD, we have only to suppose G H to move in the plane G H D, so as to remain always perpendicular to CD, while H traces out the line C D. Then G traces out the only line on the plane which can lie at the distance G H from CD; and it is evident from the last Article that the locus of G will be none other than the straight A G B, perpendicular to E F. 188. "A straight line (G H) perpendicular to another (CHD) is also perpendicular to every line (A GB), which cuts G H, and which, being in the plane G HD, is parallel to the other (CHD)." For if AGB were not perpendicular to G H, the line which should be drawn through G in the same plane and perpendicular to G H, would be also parallel to CD. Thus in the same plane through the same point G pass two lines parallel (or equidistant) to CD; which is obviously absurd. 189. " If a point (P) moves along a sloping path (AB) towards ahorizontal line (CD) in the same plane, the vertical approach of P towards this line is proportional to the length of the slope which it has traversed." (Fig. 77.) Let P, Q be any two points in the path, and PS, Q T Pig. 77. be perpendicular to CD: drop A Q R perpendicular to PS. Then, first, PS, Q T are parallel (by P / N Art. 186,) and next, so are QR, o/ R PTS: whence Q T= = R S. There/ 7: I fore in moving from P to Q, the B point has come nearer to the line CD by the distance PR.-Take _ s T_ S _ _ p = P Q, in another part of the D t S T S 0 same slope; and similarly construct the system of lines p tsr: then it is easy to show that the triangles Q PR, qp r, are every way equal; and.'. pr = PR. Thus, if the length (P Q = p q) along the slope be in two cases the same, then the vertical descent ON PARALLEL STRAIGHT LINES. 99 (PR = p r) is also in each case the same, be the previous distance A P what it may. Thus the sloping descent (A P), and the vertical descent (A N), begin together, and increase uniformly; and consequently, (by Articles 38, 39,) the one varies proportionally to the other. Which was to be proved. 190. " Straight lines in the same plane, which are not parallel (or equidistant,) may be prolonged so far as to meet." Or, what is the same thing: " Those which, being in the same plane, will never meet, cannot but be parallel; or everywhere equidistant." For if A PB be not parallel to CD, let A N be the vertical approach made towards CD, in the descent A P; (regarding CD as horizontal, to fix ideas.) Also let A NTC be the entire perpendicular from A to CD. Since then a multiple of A N can be found, so great as to exceed A C, the same multiple of A P would assign a prolongation of A P sufficient to carry it across CD. This proposition establishes the identity of Parallelism (or equidistance,) with the notion of the same as given in Euclid, and in most other geometrical treatises: and here, therefore, we stop. 191. But as a matter of curiosity, it may be worth while to go back to Art. 185, and offer an additional thought concerning the argument there employed. We proceeded upon the concession, or established truth, that S, 8', S", were all evanescent when the radius increased indefinitely. Yet it does not appear that the knowledge of this is absolutely essential to the conclusion at which we are driving. It would be sufficient to admit, that " if 8 have a limit other than zero, yet that limit does not depend on L; and so neither the limits of 8' and 8" on MA and N. In this case, if = limit of ( 8 + 4' + s"), we know that E does not depend on L, Il, N; and we get L~ + M~ 4+ NO 1 8-360~ E consequently (L~ + M~ + N~) has a constant value in every, H 2 100 DIFFICULTIES OF ELEMENTARY GEOMETRY. triangle; and from this, Legendre shows that we can easily deduce the same results as before. It may at first appear that this is lower and safer ground: yet in fact we gain nothing by it. For if we cannot infer that 8 is evanescent for infinite values of the radius, we have nothing at all to convince us, that the limit of 8,is not dependent on L; in which case E would vary in different triangles. Hence the latter mode of stating the argument is unserviceable. PART III. ON SOLID ANGLES. 192. Trihedral and Polyhedral Angles. If several straight lines in different planes, as O A, OB, O C, &c. meet in one point O; (Figs. 78, 79,) and planes pass through each contiguous pair, when these are taken in a certain order; there is formed at 0 that which is called a Solid Angle, by reason of its analogy to plane angles. It is named, according to the number of the planes which form it, Trihedral, Tetrahedral, Polyhedral, for three, four, or more planes. Fig. 78. A C Fig. 79. A. A n \' -a~~~~~~2 193. But we may also conceive of a solid angle formed without any planes at all; as at the vertex of a Cone. Such a solid angle bears to a polyhedral angle the same relation as a curved line bears to a straight line, which is bent in many places. We now encounter a difficulty in part like that which was met in discussing plane angles, viz. an inability to estimate their relative magnitudes. But plane angles readily admitted of a direct comparison as to greater and less, which is not the case with many solid angles. For instance, if two cones be constituted (Fig. 80,) with the same vertex, upon two oval curves, as their 102 DIFFICULTIES OF ELEMENTARY GEOMETRY. directrices, which intersect each other; no immediate Fig. 80. supraposition of the vertical angles avails to establish the relation of greater and less. 194. In another view, however, the Solid Angle is more manageable than the plane angle: viz. that it is very obvious under what circumstances one solid angle is justly said to be divided into two others. In fact, if any plane pass through the vertex of the angle, (as in one of the Cones just supposed,) and divide the base into two parts; this plane also divides the solid angle into two parts, the sum of which makes up the whole solid angle. 195. It is, then, easy to show the homogeneity of any two solid angles, by a proceeding similar to that which we used in the case of any two solids of limited extent. For, any very small part of a solid angle is homogeneous to the whole. To convince ourselves of this, we have only to consider that by repeatedly taking from the whole a very small part, we may as nearly as possible exhaust the whole. This shows that we may institute a numerical comparison between the whole and such a part; which indeed may be an exact submultiple of the whole. By subdividing this part continually, we may make it differ as little as we please from being a submultiple of any second proposed solid angle. Wherefore any two proposed solid angles admit of numerical comparison. Fig. 81. 196. But as the comparison of M..iir;^ plane angles is facilitated, by proving that they are proportional to the arcs of a circle, when the vertex of the angles is at the centre; so the comparison of solid angles is more vividly apprehended, by a like use of the spherical surfaces on which they stand. ON SOLID ANGLES. 103 Let two solid angles have 0 for their common vertex. With centre 0, suppose any sphere to be described; and on the surface, let the solid angles determine the areas or bases M N P,, v 7r (Fig. 81.) The two solid angles shall be proportional to these their spherical bases. Call the two solid angles A and a, and the areas of their bases B and A; we have then to prove, that A is to a as B is to 3. 197. Now, FIRST, it shall be shown that in every case in which the base B = the base /3, it is also true that the angle A = the angle a. The possible cases of equality between B and /3 are three: (I.) B and 3 may have the same shape as well as size; or be absolutely identical. In this case the sphere may slide on its own ground, while the centre is unmoved, until B actually coincides with /3; and then A precisely coincides with a; or A = a. (II.) B and 3 may be divisible into an equal finite number of parts, as Rii 1 R R.... Rn, and pl p2 p3... pn, such that every R is identical with a p that corresponds. For this gives B = R + R +...+ Rnl Also, R, = p 13= p 2 + p + P J R2 = p2J whence B = 3. &c. &c. Now in this case, the division of the base B into (n) such parts, furnishes a corresponding division of the angle A into (n) parts, which may be called S\ 82 83... Sn; so that A = S + &2 + 83 +... + ^. Similarly a = a, + 4 2 + m3 +... 4-+ ar; since the division of the base /3 equally gives rise to a division of the other angle a into (n) parts. Moreover, since R1 is identical with pi, in shape and size, therefore 81 = <a, (by Case I. just treated,) if these are the angles which have the bases Ri and pi. Similarly we have S = 9a, 83 = 3, and so on. Wherefore A = a. 104 DIFFICULTIES OF ELEMENTARY GEOMETRY. (III.) B and 3 may be called equal, on yet a third ground; viz. when each is divisible into parts which form a converging infinite series, say, B = limit of RI + R, R, + R, + &c. ad infin. 3 = limit of pi + p2 + p3 + p + -&c. ad infi. such that R =- pi, R2 = p R3 = p3, &c. ad infin. where these last equalities imply areas which are either identical, or may become so by a mere redistribution of parts. This falls under Cases I. or II. just treated. Hence, retaining the same notation as before, we infer by those Cases, that S1 = —, = S2 2 C = =, &c. ad infin. At the same time we get: A = limit of S\ + 82 + S3 + &c. ad in. a = limit of a1 + -2 + 03 + &c. ad infin. j whence it follows that A - a. The fourth case of conceivable equality, mentioned in Art. 22, need not here be treated, because the bases B and 3 are surfaces of like curvature everywhere, so that every part may coincide by supraposition with every other part. Generally then, it has been proved that if B = 3, A = a; or, what is the same, that the magnitude of the angle (A) is determined, when the area of the base (B) is given. 198. We have further to prove, that if the area (B) varies at all, the angle (A) varies in the same ratio, the sphere being unaltered. They are magnitudes which vary together, by what has already been established. Moreover, when B becomes very small so as to be ready to vanish into nothing, A likewise is ready to vanish, and may be made as small as we please by diminishing B. Contrariwise then, if we regard A and B as increasing, we may state that they begin together from zero, and increase together. Suppose, then, any increment A B to be assigned to B; then a corresponding increment A A at once accrues to A, ON SOLID ANGLES. 105 Also the magnitude of A A is determined by the magnitude of AB, and by that alone; be the magnitude of B what it may. Hence it immediately follows, (by the doctrine of Proportion delivered in Art. 40,) that A varies proportionably to B. 199. We have thus established generally, that on the surface of a given sphere, the area inclosed by a solid angle whose vertex is at the centre, is a proper MEASURE of the Solid Angle. PART IV. CERTAIN ELEMENTARY PROPERTIES OF PLANE CURVES. 200. FROM the Evenness of the Plane, it follows that all the Chords of a plane curve lie in its plane: so therefore do all its Tangents, since they are limits to the chords. Fig. 82. 201. That side of a plane curve on which the tangent falls, (Fig. 82,) is suitably called Convex, and the opposite side Concave; in conformity with the language used in Art. 153. Fig 83. 202. Normal. By this word is understood a perpendicular to the tangent, \>-\ ~ drawn through the point of contact, and in the plane of the curve. (Fig. 83.) \ ence the radii of a circle are all Normals. 203. Undulation. If, at any point (A) in a plane curve, (Fig. 84,) the tangent changes Fig. 84. A ) its side, so that the curve, A B Duf from being convex at one side, becomes presently concave, or vice versa; the curve is A ~~\ ~ said to undulate at this point, Y- ~A if there is here no breach of the continuity; that is, if there is here no peak. The tangent cuts the curve at a point of undulation. ELEMENTARY PROPERTIES OF PLANE CURVES. 107 This may lead us to remark the incorrectness of laying down that a Tangent is a straight line which " meets a curve without cutting it; " a coarse definition, suited only to a very crude state of geometrical knowledge. 204. Points of Undulation, like peaks, are Singular points; that is, are of finite number in a finite arc. If we try to conceive them occurring consecutively, all idea both of convexity and of concavity is destroyed. Wherefore from a given curve at a given point A, there can always be cut an arc A B so short as to have neither peak nor undulation. (Fig. 84.) 205. If a connected curve have three points (A, B, C) in a straight line, this implies that Fig. 85. the curvature has not been all towards one side continuously: hence A - C there must be between the extreme points either a peak or an undu- /C G lation. (Fig. 85.) 206. If the plane curve (A B) (Fig. 86,) be destitute of singular points, and m, n,p, q.... are taken in it, between A and B; and chords A m, in n,.... B are drawn; it is evident that all the angles 1 g A m n,m,.... p qB, are pointed towards the side on which the curve is convex. For if two of these angles, A B as A mn, m n p, were turned opposite ways, a straight \A line could be drawn from a point in the arc A m, cutting the chords mn, np; and consequently, cutting the arcs of those chords. It would then meet the curve in as many as three points, (r, s, t.) But this is not possible, if the curve have neither peak nor undulation. The Rectilinear path A m i p q B may be called Convex on the side towards which the angles present themselves. 108 DIFFICULTIES OF ELEMENTARY GEOMETRY. 207. Deviation. In such a path, each straight line leaves the direction of that immediately preceding it, and deviates into a new direction: but all deviate towards the same side. If we prolong Am to I/, r n to v, np to 7r, p q to K (Fig. 87,) the successive deviations at m, n, p, q, are measured by the angles u im n, Fig. 87. w p, wpq, KqB; ^s'87' i vnp, rrpq, KqB; and the sum of all these together might,, -^ --- —-— " be called the Total A: deviation. \B - k But if again we prolong qp, Bq backward, so as to meet Aut in P and Q, between m and Ft, the angles t Pq, At Q B, measure the deviations of pq and q B from the original direction A m. We may call Z At Q B the ultimate deviation attained by the path qB. 208. That the ultimate is equal to the total deviation, readily appears from the doctrine of Parallel Straight Lines, and the propositions connected with it. In fact, considering the triangle P q Q, of which A Q B is an external angle, and PqQ an internal angle = icqB, we have at once: ZAtQB = ZLiPq + LiKqB: which expresses, that the ultimate deviation of q B, exceeds that of p q, by the amount of the deflection at q. Thus each successive deviation is added to that which before existed. On the other hand, it is so simple a principle as to bring conviction to the mind by a direct process; that when all the deviations are in one plane and towards the same side, the Ultimate deviation must be equal to the SUM of the separate deviations. It may, therefore, deserve the consideration of geometers, whether this might not be proposed in such a form, as to make it the foundation on which the doctrine of Parallels might rest. 209. Once more, suppose such a path, A B CD EF, with the angles all turned one way. (Fig. 88.) Select one corner, as C, and cut it off by the line m n. By taking either m or n ELEMENTARY PROPERTIES OF PLANE CURVES. 109 as near as we please to C, we may evidently make the difference of length between (m C + Cn) and nz n, as small as we please: yet (m C 4- Cn) is always longer than m n, while a triangle n C n exists. Hence AB m n D E F is a new path like the former, ig. 88. only shortened, and that, as - little as we please. But again, we may cut B off the corner n by a straight \ / line p q; and next, cut off the corner q by a straight - line rs; and so on continually. The resulting interior path which connects A to F, is always shorter than the exterior. Hence if A bc dF be any such interior path, (the angles at b, c, d... being all turned the same way as B, C, D...) it is shorter than A B CD F. 210. Let now A P F be any arc of a plane curve (Fig. 89,) concave towards the chord A F, and without peak: and by perpetual division of the arc Fig. 89. let chords be inscribed. Thus, c — first, take D on the arc, and join AD, Di. Next, on the arc \\ A D take C, and on D F take E; join A C, CD; DE, EF. Next; on A C take B; and so on. A Then the chord A F is shorter than AD F; ADF than ACDEF; this last than ABCDEF: and so on. Or the sum of the chords, as they thus increase in number and diminish in length, is a perpetually increasing quantity. But the limit towards which their sum tends, is, the length of the arc A PF itself. (Art. 17.) Since, then, they increase towards it, it is always "greater than their sum," which also is manifest from the circumstance that each small arc is longer than its chord, by Art. 157. 211. Again: let the tangents at A and F meet in T, on the side of the convexity AP'; and we may now show 110 DIFFICULTIES OF ELEMENTARY GEOMETRY. that * the sum ofthetangents (A T + T F) is longer than the arc A PF; the limitation being continued as before, that A PF Fig. 90. has no Singular point; that T is, neither undulation nor peak. (Fig. 90.) AnrL/^ ^ Ai ~For, a series of paths is conceivable, of which A TF A F is the first in order; that shall continually diminish in length, and shall tend towards the arc A P F as their limit. If this can be proved, then the arc will manifestly be shorter than any of them, and of course shorter than the longest of them, namely, than A TF. To exhibit the truth of the above: first suppose the angle T cut off by a straight line mnp n, which touches the curve in p. Then since m n is shorter than (m T + Tn), the path Amn F is shorter than A TF.-Next, let the corner n be cut off by a straight line q Pr, touching the curve in P; then qr being shorter than (q n + n r), the path A m q r F is shorter than Am n F. By proceeding thus, always cutting off the angles of the path by tangents, it is evident that each new path which is produced is shorter than that from which it was formed. It is likewise manifest that the curve AP F itself is the limit towards which we tend: and this is what we undertook to prove. 212. Let us now imagine two curves A P F, A QF, both Fig. 91. p concave towards the common chord AF, and on the same side c/ ^ - \ ~ of it. If one contains the other, B\ \Q^\ D that which is the interior (as A Q F) is the shorter.-For between the two we can draw a A ' rectilinear path ABC... F, * Many Geometers assume this as an Axiom. But as all our definitions.are complete, we have no pretence for any such assumption; but if it be true, it can be, and ought to be, deduced from the definitions. The proposition, moreover, makes a very cumbrous axiom, because of its being embarrassed by the limitation that the arc must be free from peaks and unidulations. ELEMENTARY PROPERTIES OF PLANE CURVES. 111 having all its corners presented towards A PF: and by the preceding articles it is evident that this path will be shorter than A PF, but longer than A QF (Fig. 91.) 213. The reader will with great ease infer, that of two plane ovals that which may be con- Fig. 92. tained within the other, (as QQ Q Q within PPP,) has the shorter outline. (Fig. 92.) 214. Besides Peaks and Undu- Q lations, there are other singular points to be noticed, depending on P an irregularity in the curvature. It was explained above, (Art. 127,) how two different curvatures may be arithmetically compared: and this same method is evidently applicable to compare the curvatures at the opposite sides of the same Fig. 93. point in one and the same curvee. %M m P n N Thus, if A PB is a curve, c (Fiq. 93,) we have to consider A B at P two different curvatures, that along PA, and that along P B. With centre P and any radius PMl = P N, describe a semicircle, cutting the tangent at P in A2 and V, and the curve in Q and R. Then the ratio (f Q: N R} will roughly show the ratio of the curvatures P Q and PR to each other, if PM is small: and if P M is perpetually diminished, the limit of the ratio {M Q: N R} expresses the ratio of the two curvatures. 215. If now the limit of {AM Q: N R Fig. 94. is the ratio {1: 1}, the opposite cur- M r N vatures are exactly equal. But no one Q I is able to predict concerning any curve soever that this must reeds be the case. In fact, it is easy to construct a curve in which it shall be otherwise. Thus, if two unequal semicircles P Q S, P R T be applied on opposite sides of a straight line P S T, which is perpendicular to lM P N, it is at once clear that the curve 112 DIFFICULTIES OF ELEMENTARY GEOMETRY. 8Q PR T has on opposite sides of the point P unequal curvatures. (For although we have as yet given no rigorous demonstration that unequal circles have unequal curvatures, it is allowable to assume it, when we are only aiming at illustration.) 216. A similarly abrupt change of curvature might happen at a point of undulation; as will be seen by reversing Fig. 95. one of the semicircles so as to s produce an undulation at P. Then, if for illustration we supQ P pose the curvature of S Q P to M N be represented by 1, and that of PR T by 3, we may with propriety say that at the point P the curvature changes suddenly from + I to - 3; or from - 1 to + 3; since it changes in T direction as well as in amount. 217. This leads us to remark that a point of undulation will always imply an abrupt and finite change of curvature, unless the curvature becomes actually zero on each side of Fig. 96. the point: in which case we may say that, in N changing from positive to negative, it passes R through zero. There is then no discontinuity. Should this be the case, the curve in the immediate neighbourhood of the undulation P appears almost straight; and although the tangent cuts it, the contact is infinitely closer than under ordinary circumstances. (Fig. 96.) (QoM 218. Measure of Circular Curvature. Let us assume, in accordance with Art. 208, that the ultimate deviation of a bending path is equal to the sum of all the separate deviations. It immediately follows, that, if A B be a circular arc, (Fig. 97,) and A Q, BQ tangents, the angle B Q Ix, which is the ultimate deviation of the arc, between A and B, is also the sum of the deviations, (or, as we may now say, the sum of the curvatures) of the whole ELEMENTARY PROPERTIES OF PLANE CURVES. 113 arc. Hence the angle B Q i, divided by the lengtl A B, is the proper measure or quantum of curvature at every point of the circular arc. This expression admits of a very simple o Fig. 97. transformation. Let OA, OB be radii of the circle. Then the angles at A and B being right, in the quadrilateral 0 A Q B, the other two angles at 0 and Q in that quad- \ rilateral must be mutually supplements; (Art. 185, Cor. 2): and therefore,/ B Q t = A < /. Next; the angle 0 is to four right angles, as the arc AB is to the circumference. Hence the fraction /- -Q )> which measures the curvature = 1 arc AB j f 4 right angles 3 1* r-cuf- = radus' if we suppose a right angle L circu1mf. j radius to be measured by one quarter of the circumference whose radius = 1.-Also, when rad. = 1, curvature = 1. Making then the standard "unit of curvature," the curvature of the circle whose radius is 1, the curvature of any other circle is measured by (r I) radius 219. Finite Curvature.-Let A B be Fig. 98. any plane curve soever, which is not a I circle, (Fig. 98,) and having its curva- ture turned all one way. Suppose the / extreme tangents A A, B Q to meet in Q, — Q, so that / B Q /t as before expresses the ultimate and total deviation through A B, or the sum of all the curvatures; which may be conceived of as angles infinite in number, but each infinitely small. Then the fraction { ar A B } expresses the average curvature in the course of the arc A B: and as each term of this ratio or * We here borrow the proposition, that the circumferences of circles are proportional to their radii. I 114 DIFFICULTIES OF ELEMENTARY GEOMETRY. fraction is finite, a finite line R must exist, such that (h) is equal to the fraction. Then the circle whose radius is R, has a curvature which is the average of that in the arc A. B. It is then impossible that this arc should have, at every point of it, curvature infinitely less than that of a circle, or, what is the same, LESS than the curvature of any circle however great: for in that case, its average curvature would also be less than that of any circle; which, we have just seen, cannot be. And this holds, however short the arc A B be supposed, so long as it is finite. In like maner it appears, that the curvature of AB cannot be, at every point, GREATER than that of any circle however small. In the above, we have supposed the curvature of A B to be turned all the same way: but every finite plane curve can be divided into a finite number of portions, alternately concave and convex. We can then pronounce generally, that the points of curves at which the curvature is infinitely greater or infinitely less than that of a circle, are isolated or Singular; their number is finite in a finite arc, and every adjoining pair of such points is separated by a finite distance. The ordinary curvature may thus be measured by that of a Circle; the cases in which this happens being, in a finite arc, infinite in number, but the exceptions finite in number. Hence the curvature of circles is the ordinary standard, and is named Finite Curvature. 220. The CHANGES of Curvature are ordinarily gradual: or the anomaly remarked on in Art. 215, can occur only at Singular points. To prove this, we must first consider how to determine for any point of a curve, the circle which shall have equal curvature with it. That some such circle exists, except at singular points, appears by the last article. If then A C P B be any curve, (Fig. 99,) take I, I, points in the curve near to C, and through C, k, 1, suppose a circle to pass, which is ordinarily possible. Next, let k, i, move ELEMENTARY PROPERTIES OF PLANE CURVES. 115 up towards C, and the circle change its form and position with them. The limiting circle, if any, to which the variable circle will tend, may be Fig.99 said to pass through thlree contiguous points of the ^ - curve; and since no circle can be made to pass through more than three given points, no circle can be imagined that shall more nearly coincide with the curve at C than this does. It will therefore be the equicurve circle itself; and this proves, that, unless C be a singular point, such a limiting circle exists. Take A, g, on the opposite sides of C; and by passing a circle through C, h, g, we may form the idea of a second equicurve circle osculating the curve along ClAg. Our immediate business is to prove, that, except at singular points, the same circle osculates along Ck 1, and along C g. Let A be a singular point, having unequal curvatures on its opposite sides. Then, I say, C, c contiguous point, cannot have a like property. This means, that if C constantly move up towards A, the two circles which osculate along Ckl and along CGg, must, at least at length, tend to become one and the same; and this, nearer than any assignable difference. FOR: when C moves up towards A, it simultaneously moves up towards g, which is always between A and C. Therefore the circle through g, i, C, and that through C, k, 1, tend to confound themselves in one; since indeed we may regard the former as either osculating at C, in the direction Cg, or (equally well) as osculating at g, in the direction g C. 221. We do but put the same under a different aspect, in saying, that if P is not a singular point, we may measure off Pn, P q, finite distances, on each side of it, such, that within n Pq the curvature receives no abrupt increment i 2 116 DIFFICULTIES OF ELEMENTARY GEOMETRY. (like that noticed in Art. 215)'; but, that if n, q move together, meet where they will within nPq, they bring equal curvatures with them, counted along n A, q B, opposite ways. Thus we may announce generally, that " Except at Singular points, the change of curvature in an indefinitely small arc is indefinitely small." 222. Supposing AB to be any curve soever, having finite Jig. 0o0. curvature at A, (Fig. 100,) a part o Am may be cut off so small, that "the change of curvature in A m may be less than any assigned amount;" as is manifest from the last article. This is equivalent to B saying, that " the arc Am shall differ from a circular are as little as may be required." Hence if tangents be drawn from A and m to meet each other in T, the parts A T, m T are nearly equal, if A m be very small; and when Am is perpetually diminished, the ratio {A T: Tmn tends to the limit {1:1}. Thus also the angles (whether curved or rectilinear) TAm, TimA, are said to be qultimately equal. Moreover, if from A and m2 normals to the curve are drawn, meeting in 0, it follows that these likewise are ultimately equal; that is, the limit of {A 0: m 0 is {1: }, if the arc Am is perpetually diminished. On this is founded the simplest method of determining the circle which shall have equal curvature with the curve at A; viz. Draw normals from contiguous points A, m, and suppose them to intersect in 0; then 0 is the centre, and OA or Om' the radius of the circle required. Also, as in Art. 218, we have 1 angle 0 rad - anre; a formula much used in arithmetic rad. arc J mn geometry. In strictness, if A m has a smallfinite length, this formula does but determine a radius which expresses the average ELEMENTARY PROPERTIES OF PLANE CURVES. 117 curvature of the arc A m; yet as the variation of curvature is very small, this is nearly the same as the curvature at A. But we must suppose the length A m perpetually diminished, and so pass to the limit of the ratio, in order to determine a radius which rigorously corresponds to the curvature at the very point A. This is called the Radius of Curvature to that point; and the centre 0 which corresponds, the Centre of Curvature. The Circle is also said to osculate the curve. From Articles 219, 220, we infer, that except at Singular points, the radius only a single value. 223. In concluding this subject, it may be well to point out how the results at which we have arrived affect the nature of the curve which is the Locu. of the centres of curvature. of curvature has a finite and Fig. 101. P Q 7 5 RL Fig. 102. 0 r 7 ] Let Q P R be any curve, and suppose the centre of vature corresponding to every point in it to be found. assemblage of these centres will make a new curve, called the Evolute to the M former; or to speak more accurately, - the locus of the centres is the Evolute. \ In Fig. 101, the portion P Q gives rise to the evolute p q, and the opposite portion P R to the evolute p r. In this figure it is supposed that the curvature is always finite, increasing from Q to P, and decreasing from P to R. But if QP R con- tains a singular point / P, at which the cur- n vature is infinitely - great, the radius of curvature is here infi- \ nitely small, or p runs up into P, as in Fig. curThe,. 103. -\ I' '1,4 :118 DIFFICULTIES OF ELEMENTARY GEOMETRY~ 102. Then the evolute and original curve have the point P in common. If the curvature at P is infziitely small, as in each of the figures 103, then the radius is here infinitely great. Consequently the evolute has two infinite branches rn, z m. Should there be also a point of undulation at P, as in the second Fig. 103, then q n and r n are at opposite sides of Q P R, and the infinite branches run off in contrary directions. If the singularity of P consist in this, that the curvature at opposite sides is unequal, //,\ there are here twio centres of curvature, as 7r O. \P and p in Fi. 104. Then the evolute consists of two finite portions, as p q, 7r r', which aare broken apart at 7- and p. A yet more entire disruption of the evolute would happen, if there were a peak at P. Fig~,~ 105. Finally, if the curvature come to 3?p a maximum at.P, as in Fig. 101, -...."-. or to a minimum, as in Fig. 105, Q R without becoming infinitely small or infinitely great, the evolute 'xV/;' turns directly back on itself at rp ~ the corresponding point p, producing there a sharp peak. 224. It is also very obvious that any radius of curvature to the original curve, as 0 L in Fig. 103, is a tangent to the evolute at 0. In fact, if AK, L are consecutive points in the curve, and o, 0 the corresponding centres of curvature, which are therefore consecutive points of the evolute, it has already appeared that 0 is likewise a point in the normal IKo, since it is the intersection of the contiguous normals K 0, L 0. Wherefore K o 0 passes through the consecutive points o, 0 of the evolute; and is consequently a tangent. On this is grounded the property which has given rise to the name Evolute; viz. that the original curve may be ELEMENTARY PROPERTIES OF PLANE CURVES. 119 looked on as produced by the unwinding of a string KO, which is ever kept stretched, and pressing against the evolute. 225. Hence also it follows, that the length of any part of the evolute, as 0 q, is equal to the difference of the radii 0 L and q Q, which belong to the points 0 and q. PART V. DOUBLE CURVATURE. 226. If we suppose a curve to be drawn on a piece of flat paper, and next that this paper is curled up in some way, as by rolling it upon any cylinder; the curve assumes a new shape, in which it may be said to have two curvatures. The one is its own, such as it had while yet on the plane, the other is the curvature which the plane itself has received, or the curvature of the cylinder, which is identical with that of the cylinder's base. Upon regarding the matter thus, we are led to inquire, conversely, whether, if any curve which is not a plane be rig. 106. given, we can resolve its curvature into that of two plane curves. And the.-Y^-1-\ ~ above suggests the method. —. --- — - —.. - of projecting the curve ".... by perpendiculars on to a plane; so as to produce a cylindrical* surface (Fig. 106.) The uncurling of this surface will exhibit upon it a plane curve, the curvature of which, joined to the curvature of the cylinder's base, constitutes that of the given curve. 227. On farther consideration it appears that this introduces an arbitrary element,-the position of the plane, -which might be such as to produce results needlessly * The word Cylinder is here used in a larger sense than in Art. 69. DOUBLE CURVATURESo complicated. Nay, if the given curve were actually plane, yet by projecting it thus on another plane, it would seem to be a curve of double curvature. It is therefore to be desired, that we estimate the curvature of the proposed line without introducing anything arbitrary. 228. Along any curve take points Fig. 107. A, B, CU D, E. o. which we call consecutive; (that is, we intend at a later stage to introduce the sup- ID position, that they approximate towards one another without limit:) and to fix ideas, let them be at equal distances, two and two, counting along the curve. Draw chords AB, BC, CD, o &c.... Then since the curve is by hypothesis not plane, we presume that the plane A B C is not the same as the plane B CD, except in singular positions of A, B, &co Yet as the points A B C can be in one straight line only in singular positions, a plane A B C is always determined by them, when the distances AB, BC perpetually diminish. Hence the plane which is the limit towards which AB C tends, may be said to pass through three consecutive points in the curve; and is called the Osculating Plane. Thus each different point, as A and P, (if these are separated by a finite distance,) has its different osculating plane; and when a pencil traces out the curve, we may suppose an osculating plane to accompany its motion, turning about into such and such successive positions. Consecutive points A, C,... yield consecutive osculating planes. 229. If the chords A B, B C be prolonged, these ultimately confound themselves with tangents. Thus the limit to the plane A B C is the same as the limit to the plane of the tangents to two consecutive points at A, B. Hence we may also describe the Osculating Plane as passing throughi two consecutive tcagents. 230. Since the arcs A B, B C may be made to differ as 122 DIFFICULTIES OF ELEMENTARY GEOMETRY. little as we please from their chords, by diminishing the distances A B, B C; the whole arc A B C may be made to differ as little as may be desired from a plane curve, by shortening it as much as we please. Bisecting, therefore, the chords A B, B C, by perpendiculars drawn in the plane of B A C, we obtain two consecutive normals, which intersect in 0. This will of course be the centre of the Osculating Circle. 231. Every point, as A, admits evidently an infinity of normals, the locus of which is the plane passing through A, perpendicular to the tangent A. This plane is called the Normal Plane. But of all the normals, that which lies in the osculating plane is the most important; and is named the Principal Normal. By the last Article it appears that two consecutive principal normals of necessity intersect each other; viz. in the centre of curvature; wherever the curvature is finite. 232. Another way of looking at the principal normal is sometimes convenient. Let A T be tangent to the curve,mg. 10s. AP; and take equal lengths A T, AP T TTT along them. Join TP and prolong it to I P lrPl P R; then let the distance A T be perpetually i — / - f; diminished; and TPR will tend more / it 1 and more towards some position A 0, as its limit. I say, this is the Principal Normal. 1\ 'kU// For since the amount of curvature in A P becomes indefinitely small, when A P is perpetually diminished, A P T tends more and more to become an isosceles triangle. But the angle TA P being infinitely small, TP R is ultimately perpendicular to A T; or becomes a normal at A. Yet this normal (A 0) is in the plane of the tangent A T, and of a consecutive point P; that is, it is in the osculating plane. Hence it is the Principal Normal. 233. It must be already manifest, that the two sorts of curvature which of necessity meet us in any curve which is not plane, are, first, the curvature as measured by the oscu DOUBLE CURVATURES. lating circle in the osculating plane; secondly, the curvature or revolving which is to be discovered in the osculating plane itself. The former differs not at all from plane curvature, and needs no farther remark here, except as it leads us to estimate the latter. If A B (Fi. 107,) = a, and the angle 0 between two consecutive principal normals = w, it appears by Art. 222, that 1ad = limit of (- ); which determines the first lkad. alio curvature. Suppose, then, that while the principal normal thus turns through angle = w, the osculating plane (revolving about that normal) turns through an angle = —. Take a length p, such that - limit of ~. Then it is clear that p a the circle whose radius is p will exhibit the second curvature at A; or the proportionate rate at which the osculating plane is revolving. 234. The locus of all the centres of curvature produces an evolute, exactly as in plane curves; but besides, the locus of the radii of curvature is a surface worth attention. Generated by the motion of this radius, it has the property that any two consecutive generatrices meet each Fig. 109 other, (viz. in the evolute,) and of course, as in a7 plane curves, these are tangents to the evolute, but principal normals to the original curve. But, in consequence of the mode of generation, it is a surface of such a nature as to be susceptible of being unfolded and spread out on a B J plane: for which reason it is named Developable. /I To see the truth of this statement, suppose // a series of straight lines a A, b AB, c BC, / d CoD, e D E, EF.. to be drawn, (Fig. 109,) / intersecting one another successively in A, B, C, D, E,...but in any planes soever. If, then, we suppose the system capable of revolving about any of these straight lines, so as to change its shape in any manner; provided only that the angles a A b, b B c, c Cd, &c.... 124 DIFFICULTIES OF ELEMENTARY GEOMETRY. retain the same magnitude, and the lengths A B B, CD, &c.... remain unaltered; it is evident that we may unfold each angle in succession so as to bring all into the plane of the first, without any tearing of the system. This, being true however small the angles a A b, b Bc, &c... become, and however short the distances A B, B C, C D, &c. will be true also when the lines a A, bB, &c. are the tangents to a curve A B.. F. Wherefore the surface which is the locus of the tangents of any curve of double curvature, is Developable: and this will apply to the osculating surface above spoken of, by considering that its generatrices are tangents to the evolute. It is manifest that the plane of two consecutive generatrices, as d C D, e D, is ultimately a Tangent Plane to the developable surface. 235. Let P 0, p o be consecutive radii of curvature of the Fig. 10o. curve P Q (Fig. 110); and from 0, o, /^ RA ^ the successive osculating planes. Then / whatever is the angle of inclination between those planes, the same must be the angle between the perpendiculars, if they meet. Suppose them to meet in C; then the angle 0 Co = the angle between successive osculating planes, or may be used to measure the second curvature of P Q at P. Since every point in 0 G may be made centre of a circle passing through three consecutive points of the curve, say P,p, q; and similarly every point in og centre of a circle passing through p, q, r, the three next points: it follows that C, the point of meeting, is equidistant from four consecutive points P, p, q, r; and may thus be regarded as centre of an Osculating Sphere. It is the point of concourse of three successive normal planes. 236. But it may be inquired whether 0 G, og are ultimately in one plane, so as to have any ultimate point of concourse C. The reply is, that except at Singular Points, DOUBLE CURVATURES. 15 the deflections of the curve Pp q r follow the law of continuity, and so therefore does the change of position in the normal plane. Wherefore the locus of 0 G is a surface, whose curvature does not change abruptly except at Singular Positions of 0 G. It follows, that ordinarily the pair of lines 0 G, og determine its tangent plane. There is, then, no incongruity in supposing them to meet at C. It is true that 0 G, og, may be parallel at Singular Points. But if through a finite arc P Q, the series of lines 0G, O'G', 0"G"... were always parallel to each other, their locus would be a Cylinder, and the arc P Q would be Plane. Hence in a curve of double curvature, the point C ordinarily exists. 237. The locus of the centre C is a curve, to which the perpendicular 0 C is a tangent: that is, the system of tangents to this curve forms the developable surface, which has for its tangent planes the normal planes of our given curve. This surface is named by French geometers, La Surface Reglee; (ruled surface?) 238. It is now not difficult to C Fig. 111. show, that, (besides the principal Evolute, which is the locus 0, as in plane curves,) an infinity of other evolutes exist, all lying on the ruled surface: each having the property which gave \ rise to the name Evolute, vizo that the original curve can be generated from it by the un- O//// winding of a string. Thus if 0 Z O C, o C, o c, o' c' are successive A/"" generatrices of the Ruled Sur- /: / face, (Fig. 111,) meeting one another in C, c '; while 0, 0, 0', o", are the centres of curvature determined by the intersections of five consecutive principal normals, at P, p, p',p", "'; take G arbitrarily in 0 C; join Gp, and 126 DIFFICULTIES OF ELEMENTARY GEOMETRY. let it cut o C in g. Next, join gp', and let it cut o" c in g'. Join g'p", and let it cut o"c' in g"; and so on. Then G g g' g is a part of an Evolute; and a string Ggg' g" p"' by unwinding will generate p" P, for the string is always perpendicular to P Q, and the circle whose radius is Gp, gp"... passes through three consecutive points of P Q. Because of this property, the Ruled Surface might be termed the Surface of Evolutes. 239. Since the curve which is the locus of the centres C C, '.... cannot have two consecutive Cusps; since also its successive deflections are equal to the second curvature of the given curve, the second curvature is subject to the same laws of continuity as is the first curvature. Therefore also it is only at Singular Points that a curve can have two Osculating Planes, any more than two tangents. In fact, since the successive angles 0 Co may be all laid down on a plane, by uncurling the Surface of Evolutes, we can thus produce a plane curve, whose deflections shall accurately represent the second curvature of our original curve. PART VI. CURVED SURFACES. 240. If P be any point on a curved surface, (Fig. 112,) and lines P Q, P Q", P Q'.. be drawn in all directions round P along the surface; it ap- pears by Art. 238, that if we take all of these lines as short as we / please, we may make every one / - \" differ as little as we please from a Q. straight line. Round P draw on the surface a a line Q Q' Q" Q"' Q rejoining itself; then the above may be otherwise stated thus: If we take the area round P as small as we please, we may attain a portion of surface differing as little as we please from a Cone, generated round P by the motion of a straight line: (See Art. 102.) Such an area then, differs to an indefinitely small amount from an area marked out on the Tangent Cone round P, (See Art. 145). Now the cases in which the tangent cone does not merge itself into a tangent plane, may be spoken of under two heads: first, when at P is a peak, as in ordinary cones; secondly, when P is (what we may call) a centre of undulation, such as is described in Art. 145, under numbers (2) and (3). 241. Consider next the case of a Cone, (Fig. 113,) (using the word in its most extended sense,) which has P for vertex, 128 DIFFICULTIES OF ELEMENTARY GEOMETRY. and Q Q' Q" Q"' Q for the directrix by which it is generated. If this directrix have any peaks, as at Q' and Q"', the lines,'ig. 3. 1P Q', Q"' will probably be.u~p ~~Ridges along the surface. For if |^ ' mA~n, n be taken in the directrix, \ at opposite sides of Q"', and the ) v\\\ \ ^^ ~distances Q m, Qnz,be diminished Ky ~J\i 8 - indefinitely; each of the planes Yt \\\..a< Q'"' m9, P Q"' n, tend to become,'": tangent planes to the cone along P Q"' But by reason of the peak at Q.' in the directrix, it is possible that the two planes may not tend to one and the same plane: in which case P Q"' will be a Ridge. But two consecutive generatrices P m, P Q"', could not be ridges; otherwise the consecutive points n, Q"', must needs be both peaks in the directrix; which is impossible. It is obvious also, that no point but P in the line P Q or P Q', &c. can be a peak to the surface; for these lines are straight. Hence both the peak P and any ridges that proceed from P, are Singular; so that in a finite curvilinear area both the peaks and the ridges are finite in number. Fig. 1. The same is true of such points as we have called centres of undulation. Q' If P be such a one, (Fig. 114,) sup(Rifc / pose R to be a second point indefinitely oI( F" near to P, and in the generatrix P Q; still considering P Q, P Q', &c. as straight lines. Take P Q" another generatrix, indefinitely near to P R Q; then the plane Q P Q" is a tangent plane to the surface along the line P R Q, and consequently the point R admits a tangent plane, which contains the tangent to every curve R R" drawn from R to meet P Q". Every point R in Q P Q', except the point P, (which admits of no such lines as R B",) has thus a tangent plane. Wherefore P is altogether isolated and unique. 242. What has been proved of the Cone, applies to the CURVED SURFACES, 129 surface in Fig. 112, which by perpetual diminution might be made to differ as little as we pleased from the Cone. Thus any given curved surface of finite dimensions,-except at a limited number of isolated points, and along a limited number of straight lines,-admits at every point of it one, and only one, Tangent Plane. 243. Every curved surface may con- Fig. 115. sequentl/ be distributed into portions M so small, that each may difer from a L plane surface as little as may be re- quired.-To fix ideas, let Fig. 115 represent any curved surface, having -- an isolated peak P, and ridges NKM, KL. Round P draw a line, cutting off an area as small as we please; and we may make it differ as little as we please from a cone. But a cone is a developable surface, and may be laid out on a plane; hence this portion, (which might indeed be neglected as evanescent,) is comparable to a plane surface. Divide the rest into three parts, N Q 3K, MIKL, and NKL R, (omitting the system of P;) then neither of these parts has any singular point such as have here been discussed. Consequently, each is resolvable into small parts, differing from the tangent planes as little as we please. 244. A curved surface is then always comparable in respect to magnitude with a plane surface; or, Any curved surface being given, a plane surface is conceivable, (say a circle or a square,) equal to it in area. See Art. 12. 245. Analogy might seem to require that we follow the subject of curvature in surfaces; and establish the very remarkable and elegant properties, which, except at Singular Points, they are known to possess. But the * In the Diff. Calc. this is assumed as an Axiom. Thus, if u = area of an ellipsoid, whose semi-axes are a, b, c, it is at once assumed that u is afunction of a, b, c, of two dimensions. K 130 DIFFICULTIES OF ELEMENTARY GEOMETRY.,writer has not succeeded in finding how to do this by any simple * and satisfactory method; and, in fact, there is little or no occasion for it, inasmuch as every step has now been made good, which was a prerequisite to the application of the Differential Calculus to curved lines or surfaces. * Monge has attempted it in his Descriptive Geometry. He appears to me to fall short of demonstrative reasoning on this subject, which yet is well worthy of being studied by those who do not aim at high mathematical attainments. PART VII. SHORTEST PATH ON A SPHERE. 246. In investigating the question, " What is the shortest path along a sphere from one point to another? " it is found convenient to establish several subordinate propositions. First: " The shortest path A P (in space) (Figq. 116,) from a point A to a curve B P C that has no peak, is at right angles to the curve." For if Q, R are points near to P, and at opposite sides of it, the distances A Q, A R are never less than A P, however near Q and R are to shortest path from A to the curve. Fig. 116. C A P; because AP is a Hence if a sphere be conceived, of centre A and radius A P, Q and R can never fall within the sphere. And since we suppose that there is no peak in the curve at P, the straight line Q R being prolonged each way, tends to become a tangent to the curve at P, when the distances P Q, P R, are perpetually diminished, (Art. 95.) But it also tends to become a tangent to the sphere at P; for if it entered the sphere at P, either Q or R must lie inside the sphere; which is not the case. Thus the curve and sphere have a common tangent at P. This tangent being perpendicular to the radius A P, it remains that the shortest path A P is justly called perpendicular to the curve at P. K 2 132 DIFFICULTIES OF ELEMENTARY GEOMETRY. Observe: If P were an extreme point of the curve (Fig. Fg. 117 117,) so as not to allow opposite points Q, R; e ~ this would be equivalent to the case of P being a peak: for the curvature would be there suddenly arrested. 247. Cor. If B C, A D are curves without peaks, no path P 0 to join them can be as A short as might be, unless it is perpendicular to both curves. (Fig. 118.) Fig. 118. 248. Secondly: "The same things are c true, if the path is constrained to a pari?^ ~ ticular surface, on which the curves B P C, A OD, lie." (Fig. 119.) / o~ WTWhen from a given point A, the path A D A P is drawn along the surface to B C, let a, /,... be points in the path. Then A P cannot be as short as possible, if a P could be shortened without shifting a; and the like may be said of 3 P, y P, &c. however near a, /3,... may be to P. But when these distances become Fig. 119. very small, the arcs aP, 3 P, y P, * 'tend to confound themselves with the chords; and by the theorem just proved, /,~,1 l Ithe chords are perpendicular to B C, when \ they are the shortest paths (in space). Here, therefore, the chords tend more %: and more to perpendicularity, as they diminish: consequently, the tangent P 7' to the curve P A, is actually perpendicular to A P C; that Fig. 120. is, the two curves A P, BP C, are perpend ' I' dicular to each other at P. II!i>i P a! When the point A is not given, but two curves B C, A D, are given it follows exactly as in the Cor. to the former article, that OP must be perpendicular to both curves, if it is as short a path as possible from one to the other. Fig. 120. 249. Thirdly: " On a Surface of Revolution, the shortest SHORTEST PATH ON A SPHERE. 133 possible path to connect two points in its plane generatrix lies along the generatrix itself." (Fig. 121.) Let GP 0 be any portion of a plane curve, which, by revolving round the axis G MVN in its own plane, generates a surface of revolution. It is first easy Fig. 121. to see, that everz circle genzercttecl is G peipendicular to the generating curve. For if P generate the circle B P C, whose centre is K1, and P be a cB / M 1c tangent to the circle, and P T a j7 tangent to the curve P 0; then /?!/ since the axis NMl1 is perpendicular 11! N to the plane of the circle, so also is N the plane NAiP that passes through 0 the axis. Now the intersection of these rectangular planes is AiP; to T which is drawn in one of them the perpendicular PR. Wherefore P R is perpendicular to the other plane VN IP; and consequently, since P T is in this last plane, the angle RP Tis right. That is to say, OP is perpendicular to B P C at the point P. Next: a path on the surface to join 0 and G, will not be as short as possible, unless the path from 0 to the circle B P C be also a minimum: for all points in that circle lie at the same distance from G. Hence by the last Article, the shortest path from 0 to G, must cut the circle B P C (and every parallel circle) at right angles. But no path joining 0 and G, except the plane generatrix OP G, can cut all the parallel circles at right angles. That this path does, we have shown; but any deviation from this must instantly produce a path more or less oblique to the circles. Wherefore no other line drawn on the surface from 0 to G, is so short as the plane generatrix O PG. Finally; if 0 and P be given points in the same plane generatrix, this line P 0 is as short as any other that can be drawn on the surface to connect 0 and P. For 0 P G 134 DIFFICULTIES OF ELEMENTARY GEOMETRY. cannot be a minimum, unless its part O P is also a minimum. Which was to be proved. 250. A Sphere is a particular sort of Solid of Revolution, its plane generatrix being a semicircle. By the theorem just proved, it appears that the shortest line which can join two points on a sphere, is, one arc of the generating semicircle which passes through them. To determine this arc, we have only to pass a plane through the centre of the sphere, and the two given points: the intersection of the plane and sphere affords the path sought for. SC IOLIUM. 251. To the establishment of Art. 249, some propositions concerning the intersections of planes are needed, which are found in the XIth book of Euclid. In all this, however, there is nothing which absolutely demands that we shall have settled any principles concerning the addition and subtraction of rectilinear angles. The comparison of angles, as regards greater and less, is effected as in Art. 101; and we may proceed from thence as far as Art. 111, and then skipping to Art. 134, continue to the end of the 1st Part. If any one choose next to treat of the Inclination of Planes, so far as to establish the simple theorems needed in Art. 249, (which is easy, especially by help of the Sphere,) he might then lay down principles for effecting the addition and subtraction of angles, in sonme respects more satisfactory than those of Articles 113-117. 252. For if A 0 B, B 0 C are any two angles to be added Fio. 122. together, (Fig. 122,)we must inquire how we can make the angle A 0 C greatest; and we JI ^C must count this to be their Suim. Sup-!)! M pose then, that we cut off OA = 0 B = 0 C, and regard A, B, C as on the surface of a sphere whose centre is 0; and that A B, SCHOLIUM. 135 B C, A C, are arcs of great circles. Then by Art. 250, A C is less than A B + B C, except when C is in the prolongation of AB; as at y. Thus the distance A C, (and consequentiy, the angle A 0 C,) is greatest, when A, B, C are all in one plane with 0, and A, B, C stand in order round the circle. And this requires that the two angles A 0 B, B 0 C, be put side by side upon a plane. 253. In composing a continuous treatise of Elementary Geometry, the writer is (on the whole) of opinion that it might be advisable to follow this order. Articles 113-133, will then be omitted from their place above; and after establishing the Evenness of the Plane, &c., as in Articles 170-180, Dihedral Angles must be treated, and a certain part of what is commonly found in Spherical Trigonometry: after which, the doctrine of the Addition and Measurement of Angles upon the basis just suggested would follow. The nearer we come to the true and natural method of a science, the more immediately do we find ourselves able to deduce our conclusions out of first principles. This necessarily draws after it a new difficulty, viz. an uncertainty as to which arrangement of subjects is most to be preferred: and if there be also a doubt, which of several Experimental Laws deserves to be made the basis of the science, there is still more room for hesitating as to the most advisable Order of a treatise. This has evidently been much felt by writers on Statics; and I am strongly conscious of the same, in regard to Geometry. But there is here some danger of a fantastic desire of an artificial consecutiveness. For however specious may be the system, which would make Geometry as nearly as possible a chain of propositions, linked each to the one before it, so as to admit of no dislocation; it may be questioned whether this can ever be an arrangement to be aimed at. On the contrary, if we desire a deep knowledge on the student's part as to what he is about, the more our proofs are drawn from first principles, the better. Yet this will give to a treatise the appearance of being ill-connected. 136 DIFFICULTIES OF ELEMENTARY GEOMETRY. 254. The doctrine of Parallel Straight Lines may be regarded as dividing Geometry into two parts. Before this doctrine is established, the whole theory of the mensuration of angles, plane, spherical, curved, and solid, can be treated satisfactorily; but we can scarcely touch the theory of linear, superficial, and solid measurements without it. As this is the critical step by which we pass to the calculation of lengths, surfaces, and volumes, so also it was shown in Art. 123, that we can prove the doctrine itself by barely assuming that Geometry is a science of calculation. It may appear strange to some, that we do not propose to prove by experinent some fundamental law on which the doctrine of Parallels may rest, since this proceeding has been vindicated above, as the basis of the science. The only reply which the writer can give, is, that while he has an internal consciousness that his conviction of the Laws of Rotation is of a mechanical origin, he has just the opposite inward persuasion concerning the relations of Parallel Straight Lines. Here the appeal seems to him to be made to the pure reason, and not to outward trial of any sort; and the remark made in Art. 124, concerning the doctrine of Homogeneity, as common to other sciences, confirms him in this view. 255. Yet, as the Evenness of the Plane has been shown to result out of the principle that Peaks are Singular points, there is much speciousness (to his mind) in the thought, that the doctrine of Parallels ought to be elicited out of the theorem that "curvature is finite," (or comparable with that of a circle) "except at singular points." He once thought that he had succeeded in demonstrating this; but at last it appeared, that there was a concealed assumption that " the total deviation is not infinitely greater than the ultimate deviation." But in case any should be disposed to pursue this investigation, it may be remarked, (1) that the doctrine of Parallels is easily established, if it can be shown that the Cylinder, defined as in Art. 69, has no longitudinal curvature. (2) Since a longitudinal SCHOLIUM. 137 section of the cylinder is clearly not a circle, it must either be a straight line, or an unknown curve which has at every point equal curvature. (3) The latter alternative is instantly disproved, if it be conceded that ordinary curvature is finite. For the (supposed) curve under examination can nowhere have its curvature so great as that of any circle: which is contrary to the concession. It may finally be noticed, that by the method of Articles 127, 214, the conception of finite curvature can be distinctly formed, without any premature assumptions concerning the circle of curvature. APPENDIX. No 1. AFTER the foregoing pages were in the printer's hands, my attention was arrested by Lieut.-Colonel Perronet Thompson's small book, on the " Equiangular Spiral," used as the foundation of the Theory of Parallels. Having before found him to supply instructive suggestions, I turned with interest to the examination of his new proof. It appears quite safe to assert, that if the Second Corollary to his Proposition A is sound, his method is logically perfect. But I am sorry that in that one most important step his proof is quite deficient, in my judgment. He is essaying to demonstrate, that if (r, o) are coordinates to the Spiral, we may make (co) as great as we please, by taking (r) as great as we please; or, what is the same, that if (r) increases without limit, (co) cannot approximate to a finite limit; or that while (o) is finite, (r) cannot be infinite. But there is a clear p]etito principii, in his proof, that (r) cannot be infinite; and moreover, it proves too much. His Second Corollary is not limited to the Equiarngular Spiral, but to a spiral which fulfills the condition named in his first Corollary. This is, virtually, the equation, dr 0cc ( (r). dco, where dp (r) = circumference of a circle, whose radius is (r); a function about which he has established nothing, except that it is finite, while (r) is finite. It is easy to prove farther, that it increases with (r). But the conclusion from such premises, in his Second Corollary, is undoubtedly too wide: for instance, if ( (r) = el', it is readily proved false. Or, again, if q (r) = a r-1+,, the conclusion is false as long as (pu) has a real positive value. It is then essential to Col. Thompson's method, first to establish that the circumference does not increase in a higher ratio than as the radius directly. On a superficial view it may seem that all that is needed, is to prove that the Equiangular Spiral cannot have an asymptote thlrough its pole; for which an easy geometrical proof is devisable. But this is not really enough; nor can I complete the proof without assuming the following 140 APPENDIX. Lemma, which is only a new Axiom. " If (a b c) are the sides, and (A B C) the opposite angles of a triangle, and, B being constant, A diminishes without limit; then the limit of the ratio of b: c, is thatfof 1: 1, even though the sides b, c, simultaneously increase ever so much." P. S. Delays, over which the author had no control, have so retarded this small volume, that Col. Thompson has brought out meanwhile four editions of his pamphlet. The fundamental fallacy above commented on is only reproduced in a new shape. He, however, distinctly permits his reader to assume, for argument's sake, that the circumference of a circle increases in a higher ratio than the radius; and strange to say, so acute a writer does not perceive that this vitiates the result. If a point recede from a centre, with a velocity varying as a power of the distance higher than unity, (say, the power 1 -+ /,) it is a mathematical certainty that the distance will become infinite in a finite time. No. II. ON LEGENDRE S TREATMENT OF THE DOCTRINE OF PARALLELS. IN the same interval,-while these pages have been passing through the press,-I have read the article on Parallels in the Penny Cyclopedia; and so for the first time I have learned the form which the celebrated Legendre finally gave to his argument concerning the sides and angles of triangles. By a method quite elementary, and by no means tedious, he shows that if one triangle has its three angles together equal to two right angles, the same is true of all triangles: and if it be denied that this holds of any triangle at all, it will follow that the magnitude of a triangle is determined, when the three angles are given. Which is absurd. This ingenious and beautiful process fails to reply to the objection which Leslie so decisively urged against Legendre's algebraical treatment of the same subject; namely, that by exchanging Angles into Sides, and Sides into Angles, a fac-simile of the argument is produced, which nevertheless is false. We know that the angles of a triangle are determined, when its three sides are given (in length); why then may not its sides be determined, when its angles are given? How can the one be a demonstrative truth, the other an axiomatic falsehood? At first sight it does certainly appear requisite to reply to this; and we are forced back on Legendre's answer, which is virtually, that " Angles can be determined from abstract numbers, without an angular unit; but Lengths cannot, without a linear unit." The truth of this I endeavoured to demonstrate above, in Art. 123, while still conscious that I was unduly assuming, that " what is geometrically determined from any data, is arithmetically computable from the same." By a very slight change in the mode of statement, it now appears to me possible to get rid of that assumption, as follows: Length and Number are essentially heterogeneous; it being wholly arbitrary, whether a particular length shall be called ten or six, or any other number. Therefore, there can be no mathematical relation between one particular length, and one particular number, more than between the same length and every number. Neither then is a length sufficient to determine a number, nor a number a length. Let this be conceded 14 2 APPENDIX. as sufficiently proved. Now the ratio,-Circumference: Radius of Circle;-is an abstract number, like every other ratio; and it must EITIER remain constant, OR it must vary with the length of the radius, there being nothing else to determinje it. But the latter alternative is inadmissible; else it would constitute an abstract relation between a certain length and a certain number. It follows then that the ratio is the same for all lengths of the radius. Q E D. And in like manner, (after a square and a cube have been defined,) it may be proved that the ratios, Area of Circle: Square on Radius; and, Volume of Sphere: Cube of Radius; are the same for all radii. I confess myself unable to see what objection attaches to this reasoning. It may perhaps be said, that it is " transcendental;" it appeals to loftier and subtler principles, than are usually invoked in elementary geometry. This is most true; but the failure of more familiar methods seems to prove that these are inadequate to untie the knot; and our sole question at present is, whether a proof be logical. While it may not be quite certain that it is incumbent on one who rests on Legendre's argument, to reply to Leslie's objection at all; it will be admitted that it were better so to exhibit that argument, as toobviate the possibility of even starting the objection. I can now hardly account to myself, why I have so clung to this embarrassment, without any necessity. The fundamental principle of Legendre makes the cases of Angles and Lengths reciprocal: " a length is not sufficient to determine an angle; and, an angle is not sufficient to determine a length: " (this does appear to me a rigorous deduction from the definition of the terms;) and I showed in the " West of England Journal," (1835,) p. 77, how from this admission we might readily demonstrate what is equivalent to Euclid's 12th Axiom. I presume no one will question its logical validity: if any deprecate the introduction of the notion of a Limit, that is a matter of mere taste * and convenience. FIRST, then: a straight line of indefinite prolongation has in itself no connexion with one length or distance, more than with another. In technical phraseology, it has no Parameter (Art. 180). Now a rectilinear angle is made up by two such lines; their position alone, irrespective of their length, deciding the magnitude of the angle. Hence no mathematical relation whatever can subsist between a certain rectilinear angle, and a certain length, more than between the same angle and all lengths. For example, an angle of 60 degrees, has no more to do with a yard, than with a mile. Reciprocally; a mile has no more to do with 50 degrees than with 60 degrees. SECONDLY: this being conceded, we say; If a straight line A C (Fig. 77,) be perpendicular to a second, C D, and make any acute angle, * Indeed, in the West of England Journal, the notion of a limit was not brought forward. APPE NDIX. 14 3 CAB, with a third, AB, in the same plane; then the third may be prolonged far enough to meet the second. For: in A B take a point P, and drop P N perpendicular to A C; then if P move on to Q, V will move on in the direction of C; or A N increases with A P. Therefore, if A P increase perpetually and beyond all limit, AN either does the same, or increases up to some length (co), which is its limit. The latter alternative is impossible; for (o) would be determined by nothing but the angle CA B; and a relationship would be established between a length and an angle. This being rejected, it follows that A P may be taken so long, that A N shall equal or exceed A C, that is, A P may be prolonged to meet and cross CD. Being unable to discover ground for hesitating about the sufficiency of the argument, I must retract the too strong remark made in page 7 above, that the proof of Euclid's 12th Axiom ought in any case to follow another proposition there named. It now appears to me that the neglect to examine with sufficient depth the nature of the straight line,-its noncurvature, and its independence of a Parameter, —has been the true cause of the difficulties encountered in the doctrine of the Plane, and in that of Parallel Straight Lines.-The above proof, being manifestly limited to the case of rectilinear angles, is free from the only other plausible objection which was urged against Legendre. NOTE ON ART. 55. Postscript.-I regret to discover that the proof advanced in Art. 55 is inadequate; as it would apply equally, if C C' C" were any selfrejoining line drawn on a sphere. The following proof is shorter, as well as more correct:" Since the circle C C' C" must have been generated from some two fixed points, let one of them be called B. Then since B is equidistant from the circle, and so also is A, the centre of the sphere, the circle may move on its own ground, even while B and A are fixed: that is to say, it might have been generated from A and B. We have then only to proceed as in Art. 53." The argument also in Art. 150 needs to be expanded, as it confounds curvature with deviation at present. But nothing depends upon that Article, and it may be omitted without mischief. R. CLAY, PRINTERt BREAD STREET HILL, LONDON.