it
 
 FIRST THREE SECTIONS OF THE FIRST BOOK 
 
 * OF 
 
 NEWTON'S PRINCIPIA.
 
 CAMBRIDGE : 
 
 BT W. MKTCALFE AND BON, TRINITY STREET AND E03B CEESCENT.
 
 NEWTON'S PRINCIPIA, 
 
 FIRST BOOK, SECTIONS I., II., III., 
 
 NOTES AND ILLUSTRATIONS, 
 
 COLLECTION OF PROBLEMS 
 
 PRINCIPALLY INTENDED AS EXAMPLES OF NEWTON'S METHODS. 
 
 BY 
 
 PERCIVAL FROST, M.A., 
 
 FORMERLY FELLOW OF ST. JOHN'S COLLEGE ; 
 MATHEMATICAL LECTURER OF KING'S COLLEGE. 
 
 Principiit cnim cognitit, multo Jacilius extrema intelligetis. ClCEUO. 
 
 MACMILLAN AND CO. 
 
 1880.
 
 STACK 
 ANNEX 
 
 <?03 
 
 PREFACE. 
 
 IN publishing the following work my principal 
 intention is to explain difficulties which may be 
 encountered by the student on first reading the 
 Principia, and to illustrate the advantages of a 
 careful study of the methods employed by Newton, 
 by shewing the extent to which they may be 
 applied in the solution of problems. I have also 
 endeavoured to give assistance to the student who 
 is engaged in the study of the higher branches of 
 Mathematics, by representing in a geometrical form 
 several of the processes employed in the Differential 
 and Integral Calculus, and in the analytical investi- 
 gations of Dynamics. 
 
 In my version of the first section and the begin- 
 ning of the second I have adhered as closely as 
 I could to the original form ; and, in the . cases 
 in which sections have been interpolated, or the 
 form of demonstration changed, I have indicated such 
 changes and interpolations by brackets.
 
 VI PREFACE. 
 
 It is generally advisable not to deviate from 
 Newton's words in the demonstrations of the 
 Lemmas; but in many cases, I suppose purposely, 
 he expressed himself very concisely, as in Lemmas 
 iv. and x., and he was contented with simply giving 
 the enunciation of Lemma v. ; therefore in these cases 
 interpolations have been made which, I believe, are 
 in accordance with Newton's plan of demonstration. 
 
 Throughout the Problems and Theorems which 
 depend upon the sixth proposition, the variations are 
 replaced by equations. By this method of treating 
 the subject I conceive that clearer ideas of the 
 meaning of each step are obtained by the student. 
 
 In this edition I have introduced some notes on 
 the geometrical solution of some problems relating 
 to maxima and minima, and I have placed the 
 investigations of the properties of the curves, which, 
 after the conic sections, are the best examples for 
 illustrating geometrical methods, in a more pro- 
 minent position, at the end of the first section. 
 
 I have derived great assistance in the preparation 
 of my notes from the study of Whewell's Method 
 of Limits, and from several early editions of Newton, 
 especially that of Carr. 
 
 With respect to the three Laws of Motion, I may 
 remark that I have not commenced the work by 
 enunciating and making observations upon them, 
 partly because I should only have been repeating
 
 PREFACE. Til 
 
 what has been said so well by Thompson, Tait, 
 and Maxwell, whose works are in everybody's hands, 
 and partly because in the course of reading recom- 
 mended to students, for whose benefit my work 
 was especially intended, those laws will have been 
 already discussed in the elementary treatises on 
 Dynamics. 
 
 The Problems are principally selected from the 
 papers set in the Mathematical Tripos, and in the 
 course of the College Examinations, and I have 
 generally divided them into two portions, the 
 first of which contains those problems which are 
 capable of solution by more direct applications of 
 the propositions which they illustrate, and are 
 within the powers of a larger number of students 
 In both portions I have been careful to introduce 
 very few problems which are not capable of solution 
 by methods given in the work. 
 
 At the end of the work I have given hints for 
 the solution, and in many cases complete solutions, 
 of the problems ; and in doing so I am acting in 
 direct opposition to my previously expressed opinion, 
 but additional experience of fifteen years has shewn 
 me that it a satisfaction to a student who has not 
 been able to solve a problem to see a solution of 
 it; and, even when he has been successful, to 
 compare his solution with that of an older hand. 
 The principal objection to the publication of solutions
 
 Yin PREFACE. 
 
 is that they are frequently referred to prematurely; 
 but a wise student will treat them only as a dernier 
 ressort. 
 
 In solving the Problems I have noticed two errors 
 which should be corrected as follows : 
 
 XIII. 12 half the chord. . . .is the harmonic mean, &c. 
 
 XXVIII. 6 velocity in a circle whose radius is the length 
 
 of the unstretched string, &c. 
 
 Two sets of Problems have been numbered 
 XXVII. , the second is written XXVII. Us in the 
 Solutions. 
 
 I take this opportunity to express my thanks 
 to Mr. Stearn, of King's College, for his kindness 
 in correcting the errors of the press and for many 
 valuable suggestions. 
 
 PERCIVAL FROST. 
 
 CAMBRIDGE, 
 
 February, 1878.
 
 CONTENTS. 
 
 SECTION I. 
 
 ON THE METHOD OF PRIME AND .ULTIMATE RATIOS. 
 
 PAGE. 
 
 LEMMA I. . . . . . / . , lj 
 
 Variable quantities . . , . . 1 
 
 Continuity ....... 2 
 
 Equality ....... 3 
 
 Notes on the Lemma ...... 4 
 
 Limits of variable quantities ..... 6 
 
 Ultimate ratios of vanishing quantities . . .7 
 
 Orders of vanishing quantities ..... 8 
 
 Prime ratios ....... 8 
 
 Investigation of certain limits ..... 9 
 
 PROBLEMS I., II. . . . . . . 14, 15 
 
 LEMMA II., III. . . . . .' 17, 18 
 
 Notes on the Lemmas . . .-.,-' .' . . 19 
 
 Volumes of revolution ..... 19 
 
 Sectorial areas ....... 20 
 
 Surfaces of revolution . . . . . -'."' 21 
 
 Centres of gravity . . . . rj ->,' 22 
 
 General extension , . . , , -....., 22 
 
 Notes on corollaries . . -.,.- , .. . 23 
 Investigation of certain areas, volumes, &c. : 
 
 Parabolic area ..... y>^ . 23 
 
 Parabolid, volume of . . . ,25 
 
 Spherical segment, volume of ./, , i ~, ; ' " . 26 
 
 Cone, surface of . . . . . 26 
 
 Rod of variable density, mass of . - ," - I' ? . 27 
 
 Hemisphere, centre of gravity of . t . 28 
 
 PROBLEMS in., IV. , / , *-.- . 29,30
 
 X CONTENTS. 
 
 PAGE. 
 
 LEMMA IV. ...... 32 
 
 Notes on the Lemma . . . . . .33 
 
 Application of Lemma IV to find 
 
 Elliptic area ..... 34 
 
 Parabolic area . . . . . .34 
 
 Paraboloid, volume of .... 35 
 
 Paraboloid, centre of gravity of . . .36 
 
 Rod of variable density, centre of gravity and mass of . 36 
 
 Circular arc, centre of gravity of . . .37 
 
 Surface of spherical segment ... 38 
 
 Centre of gravity of spherical belt . . .39 
 
 Volume of spherical sector .... 39 
 
 Centre of gravity of spherical sector . . .39 
 
 Attraction of uniform rod .... 39 
 
 PROBLEMS V., VI. . . . . . 41, 42 
 
 LEMMA V. ...... 43 
 
 Notes on the Lemma . . . . . .44 
 
 Criteria of similarity ...... 44 
 
 Centres of similitude . . . . . .46 
 
 Similar continuous arcs, having coincident chords, have a common tangent 46 
 
 Centres of similitude of two circles . . . . .47 
 
 Conditions of similarity of two conic sections ... 47 
 
 Instruments for drawing on altered scales . . . .48 
 
 Volume of conical figure, base of any form ... 49 
 
 PROBLEMS VII. . . . . . .49 
 
 LEMMA VI. ...... 51 
 
 Tangents to curves . . . . . .51 
 
 Notes on the Lemma ...... 52 
 
 Subtangents ....... 53 
 
 Polar subtangent ...... 54 
 
 Inclination of tangent to radius vector . . . .54 
 
 SY* = SP.SZ ...... 55 
 
 Subtangent of semi-cubical parabola . . . . .55 
 
 Cardioid, inclination of tangent to radius vector . . . 56 
 
 LEMMA VH. ....... 57 
 
 Notes on the Lemma ...... 58 
 
 Subtense vanishes compared with the arc . . .59 
 
 Exterior curve greater than interior .... 59 
 
 LEMMA VIII. ...... 61 
 
 Notes on the Lemma ...... 61 
 
 LEMMA IX. ....... 63 
 
 Notes on the Lemma ...... 64 
 
 PROBLEMS VIII., IX. t , , . . 65, 66
 
 CONTENTS. Xl 
 
 PAGE. 
 
 LEMMA X. ..... .- . 67 
 
 Finite force ... . , .69 
 
 Notes on the Lemma . . . .,: ':'; *-" 
 
 Space described under action of constant force . . ..:. ' . ''. 71 
 Geometrical representations of 
 
 Space in given time, velocity variable 
 
 Momentum in given time, force variable and depending on the time 73 
 
 Kinetic energy, force depending on the position . > ' .. .> 74 
 Motion of a particle under various circumstances : 
 
 Space when velocity varies as square of time . " . 75 
 
 Space when force varies as rath power of time , . 76 
 
 Velocity from rest, force varying as distance . . 77 
 
 Time of describing given space, force varying as the distance . 77 
 
 Simple harmonic motion . . . 78 
 Path of particle acted on by a force tending to a point and varying as the 
 
 distance ...... 78 
 
 Motion in a resisting medium . . . . . .79 
 
 PROBLEMS X., XL . . . . . . 80, 81 
 
 LEMMA XI. . . . . . . .82 
 
 Scholium ....... 84 
 
 Curvature of curves . . . . . .87 
 
 Curvature of circle constant ..... 89 
 
 Curvatures of different circles vary inversely as the radii . . 89 
 
 Measure of curvature ...... 89 
 
 Circle of curvature has closer contact than other circles . . .91 
 
 Circle of curvature generally cuts the curve .... 91 
 
 Properties of evolute of a curve . . . . .92 
 
 Involute ....... 93 
 
 Diameters and chords of curvature . . . . .94 
 
 Parabola ...... 94 
 
 Ellipse ...... 95 
 
 Hyperbola ...... 96 
 
 Relation between radius of curvature and normal in any conic section . 97 
 
 Common chord of conic section and circle of curvature . . 98 
 
 Radius and chord of curvature of a curve referred to a pole . . 98 
 
 Notes on the Lemma and Scholium .... 99 
 
 Relation between sagitta and subtense .... 100 
 
 Tangents to a curve, from the same point, ultimately equal . . 101 
 
 Example of false reasoning . ... . . . 103 
 
 Parabola of curvature . . . . . , 104 
 
 Cardoid, chord of curvature through the focus .... 105 
 
 PROBLEMS XII., XIII., XIV. . , '- . _ 105, 107, 108 
 
 NOTE ON MAXIMA AND MINIMA. . - . . . 110 
 
 PROBLEMS XV. / . ' _ . ' . . . 112 
 
 CYCLOID . . ... .114 
 
 Tangent . . . . ... 114 
 
 Length of arc, relation with abscissa . . , 114, 1 '.5
 
 XIV CONTENTS. 
 
 PAGE. 
 
 Hyperbola, repulsive force from the centre .... 206 
 
 Time in an elliptic arc . . . . . 206 
 
 Orbit described under given circumstance of projection, force tending to a 
 
 point, and varying as the distance from it . . 206 
 
 Geometrical construction of the orbit ..... 207 
 Equations for determining the position and magnitude of the orbit, 
 
 Attractive force . . . . .208 
 
 Repulsive force ..... 209 
 
 Resultant of forces tending to different centres .... 209 
 Examples of orbits described under various circumstances . . 211 
 
 Variation of elements for a given small change of velocity . . 214 
 
 PROBLEMS XXVII. bis, XXVIH. .... 215, 217 
 
 SECTION in. 
 
 OX THE MOTION* OF BODIES IN CONIC SECTIONS, UNDER THE ACTION OF 
 FORCES TENDING TO A FOCCS. 
 
 PROP. XI. PROB. VI. ..... 220 
 
 PROP. Xn. PROB. VEL . . . . . 221 
 
 PROP. XIIL PROB. VIII. . . . . .222 
 
 Notes on the propositions ..... 224 
 
 PROP. XTV. THEOR. VI. . . . .225 
 
 PROP. XV. THEOR. VH. . . . . . " 225 
 
 Notes on the propositions ...... 226 
 
 Periodic time hi an ellipse ..... 226 
 
 Time in an elliptic arc ...... 226 
 
 Eccentric, true, and mean anomalies .... 227 
 
 Time in a parabolic arc ...... 228 
 
 Kepler's laws ....... 228 
 
 Deductions from Kepler's laws ..... 229 
 
 Law of gravitation ...... 229 
 
 PROP. XVI. THEOR. VIII. . . . . .230 
 
 Velocity in the diff erent conic sections .... 233 
 
 Hodograph . . . . . . .234 
 
 Hodograph of a conic section, force tending to a focus . . 234 
 
 General properties of the hodograph of a central orbit . . .235 
 
 PROP. XVTI. PROB. IX. . . . . . 237 
 
 Notes on the proposition . . . 240
 
 CONTENTS. XV 
 
 Direct investigation for the orbit described under given circumstances of 
 projection, force tending to a point and varying inversely as the square 
 
 of the distance from it ..... 241 
 
 Geometrical construction for the orbit .... 242 
 
 Equations for determining the elements of the orbit 
 
 when elliptic or hyperbolic, .... 242 
 
 when parabolic. ..... 244 
 
 when hyperbolic under repulsive force . . . 244 
 
 Examples of orbite described under various circumstances . . 244 
 
 Variation of elements for given changes of direction of motion . . 246 
 Change of eccentricity and position of apsidal line for a given small change of 
 
 velocity ....... 247 
 
 PROBLEMS XXIX., XXX. . . . . .248, 250 
 
 APPENDIX. 
 
 SECTION vn. 
 
 ON RECTILINEAR MOTION. 
 
 PROP. XXXH. and XXXVI. . . . . .253 
 
 Notes , . . . . . . 254 
 
 PROP. XXXVIII. . 255 
 
 SECTION VIII. 
 
 PROP. XL. THEOR. XLTL ..... 256 
 
 PROBLEMS XXXI. . . . . . .257 
 
 GENERAL PROBLEMS XXXII., XXXIII, XXXIV., XXXV. 259, 261, 263, 265 
 Solutions of Problems , ,268
 
 NEWTON'S FIRST BOOK 
 
 CONCERNING THE MOTION OF BODIES. 
 
 SECTION I. 
 
 ON THE METHOD OF PRIME AND ULTIMATE RATIOS- 
 LEMMA L 
 
 Quantities, and the ratio of quantities, which, in any finite 
 time, tend constantly to equality, and which, before the 
 end of that time, approach nearer to each other than by 
 any assigned difference, become ultimately equal. 
 
 If not, let them become ultimately unequal, and let their 
 ultimate difference be D. Hence [since, throughout 
 the time, they tend constantly to equality], they 
 cannot approach nearer to each other than by the 
 difference D, contrary to the hypothesis [that they 
 approach nearer than by any assigned difference. 
 Therefore, they do not become ultimately unequal, 
 that is, they become ultimately equal]. 
 
 Variable Quantities. 
 
 1. The Quantities, of which Newton treats in this Lemma, 
 are variable magnitudes, described by a supposed law of con- 
 struction, the variation of these magnitudes being due to the- 
 arbitrary progressive change of some element of the construc- 
 tion employed in the statement of the law. 
 
 When, in the progressive change of this element, it receives 
 the last value which is assigned to it in any proposition, the 
 hypothesis is said to arrive at its ultimate form, or to be 
 indefinitely extended.
 
 NEWTON. 
 
 Thus, let ABP be a semicircle, A CB its diameter, BP any 
 arc, PM the ordinate perpendicular to AGB, then, as the arc 
 BP gradually diminishes, AM is a variable magnitude, con- 
 tinually increasing, and BP is the element of the construction, 
 
 to the arbitrary change of which the variation of AM is due ; 
 and if BP may be made as small as we please, AM may be 
 made to approach to AB nearer than by any difference that can 
 be named, and the hypothesis approaches its ultimate form. 
 
 Again, if ABG be a triangle, and AB be divided into a 
 number of equal portions, Aa, ob, Jc, ..., and a series of parallelo- 
 grams be inscribed upon those bases, whose sides aa, 5$, 07, ... 
 are parallel to BC and terminated in AC, the sum of the areas 
 of the parallelograms will be a variable magnitude, defined by 
 that construction, and changing in a progressive manner, if the 
 
 number of parts into which AB is divided be continually 
 increased. In this case the number of parts is the variable 
 element of the construction. In the ultimate form of the 
 hypothesis, it will be shewn, Lemma II., that the sum of the 
 parallelograms is the area of the triangle when the number 
 is increased indefinitely. 
 
 2. The variation of a magnitude is continuous, when in the 
 passage from any one value to any other, throughout its change,
 
 LEMMA I. 3 
 
 it receives every intermediate value, without becoming infinite. 
 When this is not the case, the variation is discontinuous. 
 
 According to the hypothesis in the last illustration, the 
 number of parts into which AB is divided being exact, the 
 magnitude varies discontinuously, i.e. the sum of the areas does 
 not pass through all the intermediate values between any two 
 states of the progress. 
 
 If the hypothesis be changed, equal portions being set off 
 commencing from B, and Aa remaining over and above after 
 ba, the last of the portions for which there is room, these equal 
 portions could be made to diminish gradually, and the sum of 
 the areas would in that case vary continuously. 
 
 Tendency to Equality. 
 
 3. Quantities are ultimately equal, when they are ultimately 
 in a ratio of equality. 
 
 4. Quantities, which always remain finite, throughout the 
 change of the hypothesis by which they are described, tend 
 continually to equality, when their difference continually dimi- 
 nishes. 
 
 Thus, in fig. 1, page 2, let BQ be an arc, always in a given 
 ratio to BP, and let QN be the corresponding ordinate ; a 
 BP continually diminishes, AM and AN remain finite, and, 
 since their difference continually diminishes, they tend con- 
 tinually to equality. 
 
 5. Quantities, which may become indefinitely small, or in- 
 definitely great, as the hypothesis is indefinitely extended, tend 
 continually to equality, when the ratio of their difference to 
 either of them continually diminishes. 
 
 To illustrate this test of a tendency to equality, let us 
 
 suppose, in fig. 1, page 2, that the arc BP is double of the arc 
 
 BQ; then, since (chdP) 2 = AB.BM, and (chd) 2 = AB.BN, 
 
 .-. BM :BN:: 
 
 : (zrcBP} 2 : (zrcBQ)* : : 4 : 1 ultimately, 
 MN: BN::Z:l ultimately;
 
 4 NEWTON. 
 
 hence, we observe that BM and BN have a difference, which 
 tends continually to become 32LV, the ratio of which to either 
 is finite, so that, although both tend to become indefinitely 
 small as the hypothesis tends to its ultimate form, BM and 
 BN do not satisfy the condition requisite for a tendency to 
 equality. 
 
 Observations on the Lemma. 
 
 6. We will now proceed to examine the force of the other 
 important terms employed in the statement of the first Lemma. 
 
 The expression " in any finite time " (tempore quovis finito), 
 signifies what has been called the indefinite extension of the 
 hypothesis from some definite state to its ultimate form.* 
 
 The law of the variation of the magnitudes under considera- 
 tion is obtained by the examination of their construction while 
 the element, to which the change is due, is at a finite distance 
 from its final value, and the finite time is the supposed time 
 occupied in the passage from this definite to the ultimate state. 
 
 In the first illustration, Art. 1, it denotes the progressive 
 diminution of BP, from being a finite magnitude to the point 
 of evanescence. 
 
 In the second, the progress from any finite number of equal 
 portions to an indefinite number. 
 
 7. The expression " which constantly tend " (quae constanter 
 tendunt) signifies that, from the commencement of the finite 
 time to the limit of the extension of the hypothesis, the dif- 
 ferences continually diminish. 
 
 To illustrate this mode of expression, let BC be a quadrant 
 
 WhewelTs Doctrine of Limits.
 
 LEMMA I. 5 
 
 of a circle whose bounding radii are OB, OC, and let BDA 
 be a straight line cutting the arc BDC and the radius OC in 
 D and A, and let OP be a radius revolving from OC to OB, 
 and cutting BA in Q, E the point of bisection of the arc BD. 
 
 OP and OQ twice tend to equality, viz. from OG to OZ) 
 and from OE to OJ?, and once from equality from OD to 
 OE; it is only from OE to 05 that OP" and OQ' tend to 
 equality constantly during the progress, and it is from some 
 position between OE and OB that the finite time must be con- 
 sidered to commence. 
 
 8. " Before the end of that time " (ante finem temporis) 
 implies that, however small the given difference may be, a less 
 difference than that difference is arrived at, while the distance 
 from the ultimate state is still finite, however near to the final 
 state it may be necessary to proceed. 
 
 Thus, if, in the last figure, the angle BOD be 60, the 
 radius one inch, and the given difference jofihny or T f ^ of 
 an inch, the difference PQ will be less than the given difference, 
 if the revolving radius be 2' or 1', respectively, from the ultimate 
 position ; and so on, however small we choose the difference. 
 
 9. In the proof of the Lemma, if the ultimate difference be 
 D, the quantities cannot approach nearer than by that given 
 difference; otherwise, they would, in one part of the pro- 
 gression, have been tending from equality in order to arrive 
 ultimately at that difference, contrary to the statement of the 
 proposition in the words " ad aequalitatem constanter tendunt." 
 
 The nature of the proof, which is more difficult than may at 
 first sight appear, can be illustrated as follows, by examining 
 the effect of the omission of some of the points in the statement 
 of the Lemma. 
 
 Draw Oy, Ox at right angles, AB any straight line meeting 
 Oy in A, CED a curve touching AB in E and meeting Oy in 
 O, CD' another touching a straight line parallel to AB in (7, 
 MQPP' a common ordinate. 
 
 As OM diminishes until it becomes indefinitely small, 
 MQPP' moves up to Oy.
 
 NEWTON. 
 
 In both curves, the ordmates MQ and MP or MP' have an 
 ultimate difference CA, equal to D suppose. 
 
 Omit the word " constanter," and the curve CED is admissi- 
 ble in a representation of the approach of the quantities ; because 
 the ordinates approach, before the end of the time, nearer than 
 by any assignable difference, as at E t although the condition of 
 continual tendency to equality is not satisfied. 
 
 Omit the words " ante finem temporis," and CD' will be suf- 
 ficient ; for, in this case, they tend continually to equality, but 
 before the end of the time they do not approach nearer than by 
 any assignable difference, and they are ultimately unequal. 
 
 In the case of the dotted line ARF touching AB at A, all 
 the conditions are satisfied. QM and EH tend continually to 
 equality, and their difference may be made less than any given 
 difference before OM vanishes. 
 
 Limit of a Variable Quantity. 
 
 10. When a variable quantity tends continually to equality 
 with a certain fixed quantity, and approaches nearer to this 
 quantity than by any assignable difference, as the hypothesis 
 determining its variation is approaching its ultimate form, this 
 fixed quantity is called the Limit of the variable quantity. 
 
 The tests are : that there should be a tendency to equality ; 
 that this tendency should be continued from some finite 
 condition; and that the approach should, during the progres- 
 sion to the ultimate form, be nearer than by any assignable 
 difference. 
 
 Thus, as is mentioned in the Scholium at the end of the
 
 LEMMA I. 
 
 section, the variable quantity does not become equal to, or surpass 
 the limit, before the arrival at the ultimate form. 
 
 Limiting Ratio of Variable Quantities. 
 
 11. If two quantities continually diminish or increase, and 
 the ratio of these quantities tends continually to equality with 
 a certain fixed ratio, and may be made to differ from that ratio 
 by less than any assignable difference, as the hypothesis deter- 
 mining their variation is indefinitely extended, this fixed ratio 
 is called the limiting ratio of the varying quantities. 
 
 Ultimate Ratio of Vanishing Quantities. 
 
 12. When the ultimate form of the hypothesis brings the 
 quantities to a state of evanescence, they are called vanishing 
 quantities ; and the limiting ratio, or the limit of the ratio, is 
 the ultimate ratio of the vanishing quantities. 
 
 The expression " vanishing quantities " does not imply that 
 the quantities are indefinitely small while under examination, but 
 only that they will be so in the ultimate form ; which observa- 
 tion implies that the ratio of the vanishing quantities is not an 
 equivalent expression with the ultimate ratio of the vanishing 
 quantities, the former being taken " ante finem temporis." 
 
 " Ultimas rationes illae quibuscum quantitates evanescunt, re- 
 vera non sunt rationes quantitatum ultimarum." See Scholium, 
 at the end of the section. 
 
 Thus, let GC, FG be two straight lines intersecting AB in 
 G, Fj and draw ADE, MPQ, perpendicular to AB. 
 
 Let a, /3 be the areas AMPD, AMQE, then it is easily found 
 
 C
 
 8 NEWTON. 
 
 that a: &:: AD+MP: AE+MQ; now, let MPQ be sup- 
 posed to move up to ADE, then, in the ultimate form of the 
 hypothesis, a and ft vanish, and are called vanishing quantities 
 from this circumstance. 
 
 Also, the ultimate ratio of the vanishing quantities is 
 AD : AE. 
 
 In this case, since HP : MQ is not equal to AD : AE, the 
 ratio of the vanishing quantities, viz. AD + HP : AE-+ MQ, 
 is different from AD : AE, the ultimate ratio. 
 
 Orders of Vanishing Quantities. 
 
 13. When we have to consider various kinds of vanishing 
 quantities, it is necessary to consider their relative magnitudes, 
 and for this purpose if one of them be selected as a standard 
 of small quantities, this quantity, and all the vanishing quan- 
 tities of which the ultimate ratio to it is finite, are called 
 vanishing quantities of the first order. 
 
 If a, ft be any two vanishing quantities, and ft : a. vanish 
 in the limit, ft is said to be a vanishing quantity of a higher 
 order than a. 
 
 If a be of the first order, and ft : of be ultimately finite, 
 ft is called a vanishing quantity of the second order, and so on 
 for higher orders. 
 
 Trigonometrical functions give familiar illustrations of these 
 orders; let 6 be taken as the standard of vanishing quantities; 
 sin# tan2#, sin^0 are all of the first order, since their ratios 
 to 6 are ultimately 1, 2 and ; vers#, which is equal to 
 2sin a ^0 is of the second order, tan#- Q and 6-smd are of 
 the third order. 
 
 Quantities which become infinite in the ultimate state are 
 also classified in a similar manner according to orders. 
 
 Prime, Ratios. 
 
 14. If the order of the change in the form of the hypo- 
 thesis be reversed, or the varying quantities be tending from 
 equality, having started into existence from the commencement 
 of the time, the quantities are called nascent quantities ; and the
 
 LI:MMA i. 9 
 
 ratio with which they commence existence is called the prime 
 ratio of the nascent quantities. 
 
 (1) Limit of - , as x gradually diminishes, and ulti- 
 2 x 
 
 Application of Lemma I to the investigation of certain Limits. 
 
 (1) Limit q 
 mately vanishes. 
 
 Since the difference between and - is , this 
 
 difference continually diminishes as x gradually diminishes, and, 
 by diminishing x sufficiently, may be made less than any 
 assignable difference. 
 
 1 -4- x 
 Hence, - - will tend continually to equality with , if we 
 
 30 
 
 commence from some value of x less than 2, and the difference 
 may be made less than any assignable quantity ante finem tem- 
 poris, therefore | satisfies all the conditions of being the required 
 limit. 
 
 2 ~4- oc 
 
 (2) Limit of : , when x increases indefinitely. 
 
 t) "r o3C 
 
 Since the difference - = -y- , which continu- 
 
 O *T" O32 O O ( ~f~ OJC\ 
 
 ally diminishes as x increases, and may be made less than any 
 assignable difference ; therefore, as before, ^ satisfies all the con- 
 
 2 -f x 
 
 ditions of being a limit of . 
 O ~r oJC 
 
 (3) Tangents are. drawn to a circular arc, at its middle point, 
 and at its extremities. Shew that, when the arc diminishes, the 
 area of the triangle formed by the chord of the arc, and the twc 
 tangents at the extremities, is ultimately four times that of tht 
 triangle formed ly the three tangents. 
 
 Let C be the middle point of the arc, AB the chord, FA y 
 FB, DCE the three tangents, and the centre of the circle, 
 
 A FDE : A FAB : FC' 2 : FG*. 
 
 Now FC(FC+2CO)=FA* = FO.FG-, 
 
 .'. FC:Fa::FO:FC+2CO', 
 
 C
 
 10 
 
 therefore, since FC vanishes in the limit, FC : FG ::CO:2CO 
 and FG = 2FC, ultimately ; 
 
 .-. &FDE: &FAB:\ 1 : 4. 
 
 (4) Limit of -- , when x differs from 1 by an indefinitely 
 
 small quantity, m being any number, integral or fractional, posi- 
 tive or negative. 
 
 First, where m is a positive whole number, 
 
 which may be made to differ from m by less than any assignable 
 difference by taking x sufficiently near to unity. 
 
 Next, let m = P ^, p, <?, and r being positive whole 
 numbers, and let x=y r ; 
 
 x-l 
 
 y-\ 
 
 y r -i 
 
 y r -l 
 
 This may be made to differ from - " or m by a quantity 
 
 less than any assignable quantity by taking x, and therefore y, 
 sufficiently near to unity; hence, whether it be integral or 
 fractional, positive or negative, m is the limit required. 
 
 When we divide the numerator and denominator byy- 1, 
 y is not equal to 1, the time chosen being ante finem temporis
 
 LEM51A I. 11 
 
 while the difference is finite. See the direction in the Scholium 
 referred to above : " Cave intelligas quantitates magnitudine 
 determinatas, sed cogita semper diminuendas sine limite." 
 
 / R \ T- -* /. ... , . , , . . . 
 
 (5) Limit of - p^j -- , when n is indefinitely in- 
 
 creased, p being any positive number. 
 
 Since this sum is the arithmetic mean of the n fractions 
 
 therefore, for all positive values of p, integral or fractional, it 
 
 (1 \ P fn\ P 
 -'} and [ - 1 or 1, therefore its ultimate value lies 
 nj W 
 
 between and 1. 
 
 This being an important limit, we will investigate it first for 
 the particular case in which p is integral and positive, and then 
 generally when p is any positive quantity. 
 
 Let =l p + 2 p +...4 n"; 
 then # 
 
 If therefore we assume that 
 
 S n = An 1 * 1 + Bn p +. . .4 Ln.+ M, 
 then S = ^n + l 
 
 we obtain, by equating the coefficients, p + 1 equations for 
 determining the values of the p + 1 constants A, S } ... L, which 
 reduce the equation to an identity. 
 
 The first of these equations is 1 = (p + 1) A ; 
 
 /. 5. --i^.tif" + -&!+..., 
 p + 1 
 
 , S 1 B C M
 
 12 NEWTOX. 
 
 hence, if n be increased, since the number of the terms following 
 
 i Q 1 
 
 is finite, we may make the difference between J\ and - 
 
 diminish until it becomes less than any assignable quantity ; 
 
 therefore is the limit required. 
 
 p + i 
 
 Next, let p be any positive quantity, and let I be the limit of 
 
 in which p -f 1, /S, 7... are in descending order, and ^ I 
 
 7* 
 
 vanishes, when n is made infinitely large ; 
 ... 4 n+1' ==?/*+ l 
 
 -I -1 
 
 -+...; 
 
 n n 
 
 therefore, observing that, when n is increased indefinitely, 
 
 i+i- 
 
 IJI + llmit of 
 
 where e, e', ... vanish ultimately. Let e t be the greatest of the 
 quantities e, e', ..., and let all the terms be positive, then 
 
 (1+ e) #/ + ... is less than (1 + e, (.Brf + 1 Cn 7 +...) , 
 and, since -5 , -5 ... are each less than 1,
 
 LEMMA I. 13 
 
 which vanishes in the limit, hence 1 = (p + 1) I ultimately ; 
 
 1 
 p + 
 
 therefore . is the limit required. 
 
 p+l 
 
 COE. is evidently also the limit of the sum 
 
 , since -^ vanishes in the limit. 
 
 , 
 
 (6) If a straight line of constant length slide with its ex- 
 tremities in two straight lines, which intersect at a given angle A, 
 and BC, be be two positions of the line intersecting in P, which 
 become, ultimately coincident, find the limits of the ratios Cc : Bb 
 andPC:PB. 
 
 By hypothesis, BC* = bc*, 
 but BC* = BA*'+CA*-2BA.CAcosA, 
 
 and Jc 2 = bA* +cA'- 2bA . cA cos A ; 
 
 .: Cc : Bb:: BA + bA- 2cA cos A : CA + cA- 2BA cosA 
 ::BA- CA cos A :CA-BA cos A ultimately. 
 
 Draw CN, BM perpendicular to AB, A (7, therefore the limit 
 of the ratio Cc : Bb is BN : CM.
 
 14 NEWTON. 
 
 Again, let BQ, drawn parallel to A C, meet be in Q, 
 then PC: PB:: Cc : BQ; 
 also Cc : Bb :: BN : CM ultimately, 
 and Bb :BQ :: Ab : Ac; 
 .-. Cc:BQ:: BN.AB : CM. A C ultimately. 
 Draw AR perpendicular to BC, then BN.AB=BR.BC 
 and CM.AC=CR.BC-, 
 
 .: PC: PB::BR: CR', 
 .: PC=BR and PB=CR. 
 
 I. 
 
 1. ARE the limits of the ratios y* : x equal in any of the three 
 equations 
 
 when x is indefinitely diminished ? 
 
 2. Find the limit of * + 3 
 
 (1) when x is indefinitely diminished, 
 
 (2) when x is indefinitely increased. 
 
 3. Find the ultimate ratio of the vanishing quantities ax + bx*, 
 Ix + ax*, when x is made indefinitely small. 
 
 4. Prove that a - Ix and I -ax tend to equality as x diminishes 
 to zero, and yet have not their limits equal. 
 
 5. BA C, I Ac are two triangles, in which AB, Ab and AC, Ac 
 are coincident in direction, and BC, be intersect in P; prove that, 
 if the areas of the triangles be equal, as B, C and b, c approach, 
 each to each, P will be ultimately in-the point of bisection of BC. 
 
 6. APQ, ABC are two straight lines which are intersected by 
 two fixed lines BP, CQ, prove that, as APQ moves up to ABC, 
 PC and QB intersect in a point whose ultimate position divides BC 
 in the ratio of AB : AC. 
 
 7. Tangents are drawn to a circular arc at its middle point, 
 and at its extremities, and the three chords are drawn. Prove 
 that the triangle contained by the three tangents is ultimately 
 one-half of that contained by the three chords, when the arc is 
 indefinitely diminished.
 
 LEMMA I. 15 
 
 8. AP is a chord of a given circle, A Q a chord near AP, find 
 the position of the point of ultimate intersection of circles described 
 on AP, AQ as diameters, when AQ approaches to and ultimately 
 coincides with AP. 
 
 9. A circle passes through a fixed point, and cuts off from a 
 fixed line a chord PQ of constant length, prove that the chord 
 of ultimate intersection of two consecutive circles bisects PQ. 
 
 10. PN is an ordinate, and PT& tangent to an ellipse, cutting 
 the axis-major in N and T respectively ; A being the vortex, shew 
 that as P approaches A, NT is ultimately bisected in A. 
 
 11. APQ is a parabola, PM, QN ordinates to the axis AMN, 
 with centres M and N and radii I'M, QN two circles are drawn ; 
 prove that, when N approaches indefinitely near to M, if the two 
 circles intersect, the distance of their point of intersection from PM 
 is ultimately equal to the semi-latus rectum. What is the condition 
 that the circles may intersect ? 
 
 n. 
 
 1. What is the test of tendency to equality? If two quantities 
 diminish so that their difference diminishes, prove that they will 
 tend to or from equality according as the ratio of their rates of 
 decrease is greater or less than the ratio of the greater to the less. 
 
 2. ABC is an isosceles triangle, base BC\ P, Q are points on 
 the straight lines CA, CB such that AP is always twice BQ; 
 prove that, if PQ and AB intersect in R, and R be the ultimate 
 position of R, when AP is indefinitely diminished, 
 
 KB : A C : : A C : 2BC - A C. 
 
 3. PMP' is a double ordinate of an ellipse, whose centre is (7; 
 R is the point of ultimate intersection of the circles described on 
 PP' and the next consecutive double ordinate respectively, and RT 
 is the ordinate of R. Shew that TM : CM: : BC Z : AC 2 . What 
 is the condition that these circles may intersect ? 
 
 4. Two concentric and coaxial ellipses have the sum of the 
 squares of their axes equal; if the curves approach to coincidence 
 with each other, shew that the ratio of the distances of any one 
 of their points of intersection from the axes will be ultimately 
 equal to the inverse ratio of the squares of the axes. 
 
 5. If a triangle be inscribed in a given circle, prove that the 
 algebraic sum of the small variations of its sides, each divided by 
 the cosine of the angle opposite to it, will be equal to zero.
 
 16 NEWTON. 
 
 6. ABC, APQ are drawn to cut a circle from an external 
 point A; BV, CT are tangents at B and C to the circle, meeting 
 APQ, in U, T; shew that the ultimate ratio of PU : QT, when 
 APQ moves up to ABC, is AB' : AC'. 
 
 7. BCRA is a diameter of a circle whose centre is C, and PRQ 
 is a chord in it perpendicular to BA. PR is bisected in S, and 
 CS meets the circle in S'. If tangents at P and S' meet BA in T 
 and T', shew that when P moves up to A, AT= 4AT' ultimately. 
 
 8. If the quadrilateral ABCD be slightly displaced in its own 
 plane, so as to occupy the position abCD, and be the point of 
 intersection of DA, CB, prove that the point of ultimate inter- 
 section of ab and AB will be the foot of the perpendicular from 
 upon AB. 
 
 9. PSp, QSq are focal chords of a parabola, prove that, ulti- 
 mately, when P moves up to Q, 
 
 PQ-.pq:: SP* : Sp*. 
 
 10. The extremities of a straight line slide upon two given 
 straight lines, so that the area of the triangle formed by the three 
 straight lines is constant ; find the limiting position of the chord 
 of intersection of two consecutive positions of the circle described 
 about that triangle.
 
 LEMMA II. 17 
 
 LEMMA IL 
 
 If, in any figure AacE, bounded by the straight lines Aa, AE 
 ani the curve acE, any number of parallelograms Ab, Be, 
 Gd, fyc. be inscribed upon equal bases AB, BO, CD, fyc., 
 and having sides Bb, Cc, Dd, fyc. parallel to the side Aa 
 of the figure ; and the parallelograms aKbl, bLcm, cMdn, 
 Sfc. be completed; then, if the breadth of these parallelo- 
 grams be diminished, and the number increased indefi- 
 nitely, the ultimate ratios which the inscribed figure 
 AKbLcMdD, the circumscribed figure AalbmcndoE, and 
 the curvilinear figure AabcdE have to one another, will 
 be ratios of equality. 
 
 For the difference of the inscribed and circumscribed 
 figures is the sum of the parallelograms Kl, Lm, Mn, 
 Do, that is (since the bases of all are equal) a paral- 
 lelogram whose base is Kb, that of one of them, and 
 altitude the sum of their altitude's, that is, the paral- 
 lelogram ABla. But this parallelogram, since its 
 breadth is diminished indefinitely [as the number of 
 parallelograms is increased indefinitely] becomes less 
 than any assignable parallelogram ; therefore, by 
 Lemma I., the inscribed and circumscribed figures, 
 and, a fortiori, the curvilinear figure, which is inter- 
 mediate, become ultimately equal.
 
 18 NEWTON. 
 
 LEMMA III. 
 
 The same ultimate ratios are also ratios of equality, when 
 the breadths of the parallelograms AB, BC, CD, ... are 
 unequal, and all are diminished indefinitely. 
 
 Jt F C J) 
 
 For, let A F be equal to the greatest breadth, and the 
 parallelogram FAaf be completed. This parallelo- 
 gram will be greater than the difference between the 
 inscribed and circumscribed figures. But, when its 
 breadth is diminished indefinitely, it will become 
 less than any assignable parallelogram. [Therefore, 
 a fortiori, the difference between the inscribed and 
 circumscribed figures will become less than any 
 assignable areas. Hence, by Lemma I., the ultimate 
 ratios of the inscribed and circumscribed and the 
 curvilinear figure, which is intermediate, will be 
 ratios of equality.] 
 
 COR. 1. Hence the ultimate sum of the vanishing paral- 
 lelograms coincides [as to area] with the curvilinear 
 figure. 
 
 COR. 2. And, a fortiori, the rectilinear figure which is 
 bounded by the chords of the vanishing arcs ab, be, 
 cd, &c., ultimately coincides with the curvilinear 
 figure. 
 
 COR. 3. As also the rectilinear circumscribed figure, 
 which is bounded by the tangents at the extremities 
 of the same arcs. 
 
 COR. 4. And these ultimate figures, with respect to 
 their perimeters acE, are not rectilinear figures, but 
 curvilinear limits of rectilinear figures.
 
 LEMMA II., III. 19 
 
 Observations on the Lemmas II. and III. 
 
 15. The statements of the propositions concerning limits of 
 quantities and their ratios contain : 
 
 I. The hypothesis by which the quantities are defined. 
 
 II. The manner in which the hypothesis approaches its 
 ultimate form. 
 
 III. The ultimate property when the hypothesis is thus 
 indefinitely extended. 
 
 The strength of the proofs lies in the examination of the 
 quantities while the hypothesis is in a finite state, before arrival 
 at the ultimate form, and the deduction of properties by which 
 the relations of the quantities can be pursued accurately to the 
 ultimate state. 
 
 If in this manner we analyse the statement of Lemmas II. 
 and III., the hypothetical constructions are given in the manner 
 of describing the parallelograms; the extension of the hypo- 
 thesis towards its ultimate form is the continual increase of the 
 number of parallelograms ad infinitum ; the ultimate property is 
 the equality of the ratio of the sums of the parallelograms and 
 the curvilinear area. 
 
 In the proof of the Lemmas, the continual decrease of the 
 parallelograms Al or Af shews that the conditions of ultimate 
 equality of two quantities are all satisfied, viz., that the sums 
 of the two series of parallelograms, since they are finite, tend 
 continually to equality, and that they approach nearer to each 
 other than by any assignable difference " ante finem temporis," 
 i.e.) while the number of the parallelograms still remains finite. 
 
 Volumes of Revolution. 
 
 16. In a manner exactly similar to Lemma II. it may be 
 shewn that, if Aa be perpendicular to AE, and the whole 
 figure revolve round AE as an axis, the ultimate ratios, which 
 the sums of the volumes of the cylinders, generated respectively 
 by the rectangles Ab, Be, ... and aB, bC, ... and the volume 
 of revolution generated by the curvilinear area AEa will have 
 to each other, will be ratios of equality.
 
 20 
 
 NEWTON. 
 
 The figure represents the cylinders generated by the in- 
 scribed rectangles. 
 
 Thus the difference of the cylinders generated by Ab and 
 aB is the annulus generated by the rectangle a5, and the 
 difference of the two series of cylinders, which have all equal 
 heights AH, J5(7, .., is the sum of such annuli, and is easily 
 seen to be the cylinder generated by aB, which, since the height 
 continually diminishes, may be made less than any assignable 
 volume, hence the conditions that the two series may have the 
 same limit are satisfied, and hence also the volume of revolution, 
 which is greater than one sum and less than the other, is 
 ultimately in a ratio of equality to either sum. 
 
 The same argument applies when the revolution is only 
 through a certain angle instead of being complete, in which 
 case the cylinders are replaced by sectors of cylindrical volumes. 
 
 Sectorial Areas. 
 
 17. The Lemmas may be extended to sectorial areas. 
 JL
 
 LEMMA II., III. 
 
 21 
 
 Thus, let SABCFbe a sectorial area, and let the angle ASF 
 be divided into equal portions ASB, BSC, ... and the circular 
 arcs Ah', aBc', bCd'j ... be drawn with centre S; then, since 
 the difference of the two series of circular sectors is the sum 
 of the areas ab\ be', ..., it is equal to the difference of the 
 greatest and least of the sectors, viz. AGrSb' ; therefore the 
 two areas SAb'Bc... and SaBbC... tend continually to equality 
 as the number of angles is increased and their magnitudes 
 diminished, and the ratios which these areas have to each other 
 and to the area SABF are ultimately ratios of equality. 
 
 Similarly, as in Lemma III., if ASB : BSC, ... be unequal. 
 
 Surfaces of Revolution. 
 
 18. The following proposition is the extension of the prin- 
 ciples of the Lemmas to the determination of a method for 
 finding the area of a surface of a solid of revolution. 
 
 Let CD be a plane curve which generates a surface of revo- 
 lution by its revolution round AB, a line in its plane. 
 
 CD is divided into portions, of which PQ is one, PJf, QN 
 are perpendicular to AB ; Pp, Qq are drawn parallel to AB, and 
 each equal to PQ in length ; pm, gn are perpendicular to AB. 
 The surface generated by CD shall be the limit of the sum of 
 the cylindrical surfaces generated by such portions as Pp or Qq. 
 
 For, the cylindrical surfaces generated by Pp and Qq are 
 one less and the other greater than the surface generated by PQ 
 
 A. M j5T*w Ji 
 
 since every portion of Qq is at a greater, and every portion of 
 Pp at a less, distance from the axis than the corresponding 
 portions of PQ.
 
 22 NEWTON. 
 
 But these surfaces are respectively ^trPM.Pp and 2irQN.Qq, 
 and their difference is 2?r ( QN- PM} PQ, and the ratio of this 
 difference to the surfaces themselves is QN- PM : PM or QN, 
 which ratio is ultimately less than any given ratio. 
 
 Hence the sums of the surfaces generated by the lines corre- 
 sponding to Pp and Qq have the ratio of their difference to either 
 sum less than the greatest value of the ratio QNPM: PM, 
 which may be made less than any finite ratio. Therefore the 
 sums of the cylindrical surfaces and the curved surface, which 
 is intermediate in magnitude to these sums, are ultimately in 
 a ratio of equality. 
 
 Centre of Gramty. 
 
 19. It is easily seen that the same methods are applicable to 
 the determination of the position of the centre of gravity of any 
 body, since it is known that, if a body be divided into any 
 number of portions, the distance of the centre of gravity of the 
 body from any plane is equal to the sum of the moments of all 
 the portions divided by the sum of all the portions. 
 
 General Extension. 
 
 20. The most general extension may be stated as follows : 
 If any magnitude A be divided into a series of magnitudes 
 A^A t ...A n , each of which, when their number is increased indefi- 
 nitely, becomes indefinitely small, and two series of quantities 
 a i a *~' a aQ d &A"A can be found such that 
 
 a n > A n > b n , 
 
 and also such that each of the ratios a l b l : a t , a 2 - 5 2 : a a , ... 
 becomes less than any finite ratio when the number is increased ; 
 then a x -h a 3 +...-f a B , \ + b 9 +...+ t m and A will be ultimately 
 in a ratio of equality. For, let 1:1 be equal to the greatest 
 of the ratios a t - b^ : a,, &c. ; 
 
 is a ratio less than 1 : 1, and may therefore be made less than
 
 LEMMA II., III. 23 
 
 any assignable ratio by increasing the number. Therefore the 
 two series 0^ + 0^4... and b l + b i +... tend continually to equality, 
 and the difference may be made, before the end of the time, less 
 than any assignable magnitude ; therefore the three magnitudes 
 are ultimately in a ratio of equality. 
 
 21. COR. 1. "Omni ex parte" has not been adopted from 
 the text of Newton, because it requires limitation, for the 
 perimeters do not ultimately coincide with the perimeter of the 
 curvilinear area. 
 
 In the figure for Lemma II. the perimeter of the inscribed 
 series of parallelograms is 
 
 A+Kb + bL + Lc+...+ DA = 2AK+ 2AD, 
 
 and the limit of this perimeter is 2Aa + 2AE. 
 
 The perimeter of the other series of parallelograms, being 
 2Aa + 2AE is constant throughout the change, and has properly 
 no limit. 
 
 COR. 2. The perimeter of the figure bounded by the chords 
 ab, be, ... ultimately coincides with that of the curvilinear figure. 
 This coincidence will be discussed under Lemma V. 
 
 COR. 3. The same is true for the figure formed by the 
 tangents. 
 
 COR. 4. Instead of "propterea," as in Newton, it is advisable 
 to state, as in Whewell's Doctrine of Limits, that, if a finite 
 portion of a curve be taken, and many successive points in the 
 curve be joined so as to form a polygon, the sides of which, 
 taken in order, are chords of portions of the curves, when the 
 number of those points is increased indefinitely, the curve will 
 be the limit of the polygon. 
 
 Application to the Determination of certain Areas, Volumes, &c. 
 
 (1) Area of a parabola bounded by a diameter and an ordinate. 
 
 Let AB, BC be the bounding abscissa and ordinate. Com- 
 plete the parallelogram ABCD. 
 
 Let AD be divided into n equal portions, of which suppose 
 AM to contain r, and MN to be the (r 4 l) th ; draw MP, NQ
 
 24 NEWTON. 
 
 parallel to AB, meeting the curve in P, Q, and Pn parallel to 
 MN] the curvilinear area AGD is the limit of the sum of the 
 
 JL 
 
 series of parallelograms constructed, as PN, on the portions 
 corresponding to MN. 
 
 But parallelogram PJV : parallelogram AS CD 
 
 -.'.PM.MN'.CD.AD, 
 and, by the properties of the parabola, 
 
 PM:CD:: AM* :AD*::r*: n\ 
 
 also MN:AD:: l:n; 
 .-. PM.MN:CD.AD::r*:n 3 ; 
 
 r* 
 therefore, parallelogram PN= -, x parallelogram ABCD ; 
 
 hence, the sum of the series of parallelograms 
 
 x parallelogram 
 
 and, when the number of parallelograms is increased indefinitely, 
 
 therefore, proceeding to the ultimate form of the hypothesis, the 
 curvilinear area ACD and the parabolic area ABO will be, 
 respectively, one-third and two-thirds of the parallelogram 
 ABCD. 
 
 Note 1. If we had inscribed the series of parallelograms in 
 ABC, AB being divided into n portions, we should have arrived 
 at the result 
 
 l* + 2*+...+ (ft-l)* 
 .**
 
 LEMMA II., III. 
 
 25 
 
 for the ratio of the series of parallelograms to the parallelogram 
 ABCD, which might thus have been directly shewn to be 
 ultimately f ; but the former method is preferable, since the 
 proof of the value of the limit depends upon simpler principles. 
 
 Note 2. If BG had been divided into n equal portions, the 
 ratio of the parallelogram corresponding to PN to the parallelo- 
 gram ABGD would have been tf r 2 : w 2 , and that of area ABG 
 to parallelogram ABGD the limit of 
 
 (2) Volume of a paraboloid. 
 
 Let ASH be the area of a parabola, cut off by the axis AH 
 
 !> 
 
 JT JL 
 
 and an ordinate HK, which by its revolution round the axis 
 generates a paraboloid. 
 
 Let AH be divided into n equal portions, and on MN the 
 (r + l) th , as base, let the rectangle PRNM be inscribed. 
 Cylinder generated by PN : cylinder by AHKL 
 
 ::PM\MN:HK\AH. 
 But PM 2 : HK* :: AM : AH:: r : n, 
 
 and MN: AH:: 1 : w; 
 /. PM*.MN:HK\AH::r:n\ 
 
 Hence cylinder generated by PN= x cylinder by AHKL ; 
 therefore the sum of the cylinders inscribed is 
 
 '" x circumscribed cylinder, 
 
 and the paraboloid is the limit of the series of inscribed cylinders ; 
 hence the volume of the paraboloid is half that of the cylinder 
 on the same base and of the same altitude.
 
 26 
 
 NEWTON. 
 
 (3) Volume of a spherical segment. 
 
 Let AHK generate, by its revolution round the diameter AB, 
 the spherical segment whose height is AH. 
 
 A MJDT X 
 
 Divide AH, as before, and make the same construction ; 
 then PM* = AM.(AB-AM)=-AH.AB--,AH\ 
 
 fl ft 
 
 Volume of cylinder generated by PN=irPM' t .MN 
 
 whence, as before, the limit of the sum 
 
 = TT AH* (\AB-l AH], 
 which is the volume proposed. 
 
 COE. If AH=\AB=A C, the segment is a hemisphere whose 
 volume is TrAC* (AC- %AC) = $7rAC 3 , which is two-thirds of 
 the cylinder on the same base and of the same altitude. 
 
 (4) Area of the surface of a right cone. 
 
 As an illustration of the method of finding surfaces given 
 above, suppose AHK to be a right-angled triangle, which 
 revolves round AH, a side containing the right angle, then 
 the hypothenuse AK generates a conical surface. 
 
 Let MN be the (r + l) th portion of AH, after division into 
 
 n equal portions; MP, NQ ordinates parallel to UK] Pp, Qq 
 each equal to PQ and parallel to AH.
 
 LEMMA II., III. 27 
 
 The areas generated by Pp and Qq respectively are 
 
 2>rrPM.Pp and 2irQN.Q& 
 and PM:HK'.'.AM:AH'.:r:n, 
 
 QN: HK: : AN: AH-.ir + l : n, 
 PQ:AK::MN:AH:: l:n; 
 
 therefore the areas are -^.^irHK.AK and T -^ r -. < 2irHK.AK 
 
 n* n* 
 
 respectively ; and the conical surface is intermediate in magni- 
 tude between 
 
 and <~H-*+..- 
 
 each of which has for its limit TrHK.AK, which is therefore 
 the area of the conical surface. 
 
 Note. The reader may notice the following method of 
 obtaining the conical surface by development, although it is 
 not related to the method of limits. 
 
 If a circular sector KAK\ traced on paper, be cut out, the 
 bounding radii AK, A K' can be placed in contact, so that the 
 boundary KLK' will form a circle. 
 
 The figure so formed will be conical, AK will be the slant 
 side, and HK in the last figure will be the radius of the circular 
 base, whose length will be the arc of the sector KAK'. 
 
 Hence, the area of the conical surface is equal to that of the 
 sector KAK' = \AK. ZirHK= irHK.AK. 
 
 (5) Mass of a rod whose density varies as ih m th power of 
 the distance from one extremity. 
 
 Let AB be the rod, and let MNbe the (r-f l) th portion, when 
 its length has been divided into n equal parts; and let p.AM m 
 be the density at Jf, or the quantity of matter contained in an 
 unit of length of the rod supposed of the same substance as the 
 rod at the point M. 
 
 The quantity of matter in MN is intermediate between 
 p.AM m .MN and p.AN m .MN,
 
 28 NEWTON. 
 
 and the ratio of the difference of these to either of them is less 
 than any assignable ratio when n is indefinitely increased. 
 
 Therefore, since AM=-AB, and MN=-AB, the mass 
 of the whole rod is the limit of 
 
 / i Nth 
 
 = f j of the mass of a rod of length AB and of uniform 
 
 density equal to that of the rod AB at B. 
 
 (6) Centre of gravity of the volume of a hemisphere. 
 Let CAB be a quadrant, which by its revolution round the 
 radius CA generates the hemisphere. 
 
 A. M JT 
 
 Let MR be the rectangle which generates the r tb inscribed 
 
 cylinder, so that CM=-xCA and MN= - xCA. 
 n n 
 
 If the mass of a unit of volume be chosen as the unit of 
 mass, the mass of the cylinder generated by MB will be 
 
 ; 
 n 
 
 hence, the mass of the series of inscribed cylinders will be 
 
 and the mass of the hemisphere 
 
 Again, the moment of the mass of the cylinder generated 
 by MRj with respect to the base of the hemisphere, will be
 
 LEMMA II., III. 29 
 
 which differs from TrPM*.MN.CM by a quantity which vanishes 
 compared with it, and is therefore ultimately (-* A irOA*^ 
 
 therefore the moment of the hemisphere, with respect to its 
 base, is 
 
 (%-$TrCA\ or frCA 4 ; 
 
 hence the distance of the centre of gravity of the volume of the 
 hemisphere from (7, which is the moment with respect to the 
 base divided by the mass, is f .(LI. 
 
 in. 
 
 1. Illustrate the terms "tempore quovis finite" and "constanter 
 tendunt ad sequalitatem " employed in Lemma I. by taking the 
 case of Lemma III. as an example. 
 
 2. Shew, from the course of the proof of Lemma II., that the 
 ultimate ratio of vanishing quantities may be indefinitely small or 
 great. 
 
 3. Shew that the ratio of the area of the parabolic curve, in 
 which PM 3 QC AM, to the area of the circumscribing parallelogram, 
 of which one side is a tangent to the curve at A, is 3 : 4. 
 
 4. Shew that the volume of a right cone is one-third of the 
 cylinder on the same base and of the same altitude. 
 
 5. AUK is a parabolic area, AITihe axis, and UK an ordinate 
 perpendicular to the axis, AHKL the circumscribing rectangle. 
 Shew that the volumes generated by the revolution of ASK round 
 AH, KI/, AL, and UK are respectively i, |, |, and ^ of the 
 cylinder generated by the rectangle. 
 
 6. The volume of a spheroid is two-thirds of the circumscribing 
 cylinder. 
 
 7. Find the centre of gravity of the volume of a right cone 
 by the method of Lemma II. 
 
 8. Shew that the centre of gravity of a paraboloid of revolution 
 is distant from the vertex two-thirds of the length of the axis. 
 
 9. Find the mass of a rod whose density varies as the distance 
 from an extremity. Find also its centre of gravity, and shew that 
 it is in one of the points of trisection of the rod.
 
 30 
 
 NEWTON. 
 
 10. The limiting ratio of an hyperboloid of revolution, whose 
 axis is the transverse axis, to the circumscribing cylinder is 1 : 2 
 when the altitude is indefinitely diminished, and 1 : 3 when it is 
 indefinitely increased. 
 
 IV. 
 
 1. Prove that the areas of parabolic segments, cut off by focal 
 chords, vary as the cubes of the greatest breadths of the segments. 
 
 2. Find the mass of a circle whose density varies as the *w th 
 power of the distance from the centre. 
 
 3. Shew that the abscissa and ordinate of the centre of gravity 
 of a parabolic area, contained between a diameter AB and ordinate 
 BC, are 3 -AB and \B C respectively. 
 
 4. A number of equal squares in one plane with their centres 
 coincident are arranged consecutively, their sides making equal 
 small angles, each with the adjacent ones; prove that the limit 
 of the length of the serrated edge, when the number of squares 
 is indefinitely increased, is equal to the circumference of a circle 
 whose radius is a side of the square. 
 
 5. By supposing the axis of a parabola portioned off into suc- 
 cessive lengths in the ratio 1:3:5, &c., apply Lemma III. to find 
 the area contained by the curve and a double ordinate. 
 
 6. Find the volume generated by the revolution of an elliptic 
 disc about an axis parallel to its major axis, and at such a given 
 distance as not to intersect the disc. 
 
 7. In the curve A CD, BE is an ordinate perpendicular to AD, 
 and FC is the greatest value of BE, and ^ = sin \^n ) . 
 
 Shew that the area ABE varies as SG, where GK is the 
 ordinate equal to BE of the circle CH, whose centre is F and 
 radius FC.
 
 LEMMA II., III. 
 
 31 
 
 8. In the curve of the last problem shew that the ratio of the 
 area A CD to the triangle whose sides are AD, and the tangents 
 AT, DT at the extremities, is 8 : TT*. 
 
 9. In the curve APC, in which the relation between any 
 
 ON PM 
 
 rectangular ordinate PM and abscissa OM is 
 
 OA 
 
 log 
 
 OA 
 
 prove that the area contained between the curve, the abscissa OB, 
 and ordinate J5C, is OA(BC-AO}.
 
 32 NEWTON. 
 
 LEMMA IV. 
 
 If in two figures AacE, PprT there be inscribed (as in 
 Lemmas II., ///.) two series of parallelograms, the num- 
 ber in each series being the same, and if, when the breadths 
 are diminished indefinitely ', the ultimate ratios of the 
 parallelograms in one figure to the parallelograms in the 
 other be the same, each to each, then the two figures 
 AacE, PprT will be to one another in that same ratio. 
 
 [Since the ratio, whose antecedent is the sum of the 
 antecedents, and whose consequent is the sum of the 
 consequents of any number of given ratios, is inter- 
 mediate in magnitude -between the greatest and least 
 of the given ratios, it follows that the sum of the 
 parallelograms described in AacE is to the sum in 
 PprT in a ratio intermediate between the greatest 
 and least of the ratios of the corresponding inscribed 
 parallelograms; but the ratios of these parallelograms 
 are ultimately the same, each to each, therefore the 
 sums of all the parallelograms described in AacE, 
 PprT are ultimately in the same ratio, and so the 
 figures AacE, PprT are in that same ratio ; for, 
 by Lemma III., the former figure is to the former 
 sum and the latter figure to the latter sum in a ratio 
 of equality.] Q. E. D. 
 
 COK. Hence, if two quantities of any kind whatever be 
 divided into any, the same, number of parts, and 
 those parts, when their number is increased and 
 magnitude diminished indefinitely, assume the same 
 given ratio each to each, viz. the first to the first,
 
 LEMMA IV. 33 
 
 the second to the second, and so on in order, the 
 whole quantities will be to one another in the same 
 given ratio. For if, in the figures of this Lemma, 
 the parallelograms be taken each to each in the same 
 ratio as the parts, the sums of the parts will be always 
 as the sums of the parallelograms ; and, therefore, 
 when the number of the parts and parallelograms is 
 increased and their magnitude diminished indefi- 
 nitely, the two quantities will be in the ultimate 
 ratio of parallelogram to parallelogram, that is, (by 
 hypothesis) in the ultimate ratio of part to part. 
 
 Observations on the Lemma. 
 
 22. The general proposition contained in the Corollary may 
 be proved independently in the following manner : 
 
 Let A, B be two quantities of any kind, which can be 
 divided into the same number n of parts, viz. a,, a 2 , 3 ... B 
 and 5 t , 5 2 , b z ...b n respectively, such that, when their number is 
 increased and their magnitudes diminished ' indefinitely, they 
 have a constant ratio L : 1 each to each, so that 
 a, : J, : : L (1 -f a,) : I, 
 
 where a t , a 2 , ... vanish when n is increased indefinitely. 
 
 Then, a, + a g +.,.: J 1 + J 8 +... being a ratio which is inter- 
 mediate between the greatest and least of these ratios, each of 
 which is ultimately L : 1, we have, proceeding to the limit, 
 
 A'.Bi:L: 1; 
 that is, A and B are in the ultimate ratio of the parts. 
 
 23. The proof given in the Prinripia is as follows : " For, 
 as the parallelograms are each to each, so, componendo, is the 
 sum of all to the sum of all, and so the figure AacE to the figure 
 PprT, for, by Lemma III., the former figure is to the former sum 
 and the latter figure to the latter sum in a ratio of equality." 
 
 The proof given in the text is substituted for this, because 
 the demonstration breaks down for any finite distance from the 
 ultimate form of the hypothesis.
 
 34 
 
 NEWTON. 
 
 Application to the determination of certain Areas, Volumes, &c. 
 
 (1) Area of an ellipse. 
 
 Let ACa be the major axis of an ellipse, BC the semi-minor 
 axis, ADa the auxiliary circle, and let parallelograms be in- 
 scribed, whose sides are common ordinates to the two curves. 
 
 Let PMNR, QMNU be any two corresponding parallelo- 
 grams. The ratio of these parallelograms is PH : QM or 
 BC:AC. 
 
 Hence, area of ellipse : area of circle : : BC : AC, but area 
 of circle = TrA C' 2 ; therefore area of ellipse = IT A C.BC. 
 
 (2) Area of a sector of an ellipse, pole in the focus. 
 If S be a focus of the ellipse, and SP, SQ be joined, 
 
 &SPM : &SQM : : BC : A C, 
 and area APM : area A QM :: BC: AC, 
 hence, area ASP : area A8Q : : BC : AC, 
 but area A8Q = &SCQ + sector ACQ 
 
 .'. area, ASP =$ {SC.PM+SC.arcAQ}. 
 
 (3) Area of a parabolic curve cut off by a diameter and 
 an ordinate to the diameter. 
 
 In the following investigation it is asserted that when a 
 chord PQ is drawn to a curve from a point P, as Q moves up 
 to P, PQ assumes as its limiting position that of the tangent 
 at P, which is deducible from the idea of a tangent being in the 
 direction of the curve at the point of contact. 
 
 Let AB, BC be the diameter and ordinate; AD the tangent 
 at A-, CD parallel to AB; P, Q points near each other; 
 PM, QN and Pm, Qn parallel respectively to AD and AB. 
 
 Let QP produced meet BA in T, and complete the parallelo- 
 grams TAmS, TAnU.
 
 LEMMA IV. 35 
 
 Then, since QP is ultimately a tangent at P, AT=AM 
 ultimately, and the parallelogram PU is ultimately double of 
 
 T -A. M 
 
 the parallelogram Pn, and the complements PN, PU are equal ; 
 therefore the parallelograms PN, Pn are ultimately in the ratio 
 2 : 1. 
 
 Hence, in the curvilinear areas ABC, ACD two sets of 
 parallelograms can be inscribed which are ultimately in the ratio 
 2:1, each to each ; therefore area ABC is ultimately double of 
 area ACD, and is therefore two-thirds of ABCD. 
 
 (4) Volume of a paraboloid of revolution. 
 Let AH be the axis of the parabola APK, AHKL the 
 circumscribing rectangle. Also let PN, Pn be rectangles in- 
 scribed in the portions AEK, AKL. 
 
 Volume generated by PN^irPM\MN=ir.PM.PN. 
 Volume generated by Pn = IT QN \AM--irPM*. AM 
 
 /. vol. by PN: vol. by Pn : : PM.PN: (QN+PM}.Pn, 
 but QN+PM=%PM and P2V=2Pw, as in (3), and therefore 
 vol. by PA r =vol. by Pn ultimately; hence, by Cor., Lemma IV., 
 the volume of the paraboloid generated by AHK is half the 
 volume of the circumscribing cylinder generated by AKL.
 
 36 
 
 NEWTON. 
 
 (5) Centre of gravity of a paraboloid of revolution. 
 
 Since the volumes generated by PN and Pn are ultimately 
 equal, the moment of the volume generated by PN with respect 
 to the tangent plane at A : moment of that generated by Pn 
 
 : : AM : \Pm ultimately, i.e. : : 2 : 1 ; 
 
 hence the moment of volume generated by AHK is twice that 
 of the volume generated by AKL, and the moment of the 
 paraboloid = f moment of the cylinder 
 
 = | volume of cylinder x \AH= f volume of paraboloid x AH; 
 hence the distance of the centre of gravity of the paraboloid from 
 the vertex is two-thirds of the height of the paraboloid. 
 
 (6) Centre of gravity and mass of a rod whose density varies 
 as the distance from an extremity. 
 
 Let AB be the rod, MN a small portion of it, then the 
 density at M"oo AM. 
 
 Construct on AB as axis an isosceles triangle CAD, whose 
 base is CD, and draw PMR, QNS parallel to CD; then PR, 
 QS, CD are proportional to the densities at M, N and B; 
 therefore the mass of MN is proportional to a rectangle inter- 
 mediate to the rectangles PR, MN and QSj MN, which are 
 ultimately in a ratio of equality. 
 
 Hence the mass of MN is ultimately proportional to the mass 
 of the rectangle PR, MN, supposed of uniform density, and the 
 moment of MN, with respect to the line CD, is proportional to 
 the moment of the same rectangle, since their distance is the 
 same ; hence, by the Lemma, the moment of the whole rod 
 : the moment of the triangle with respect to CD 
 : : the mass of the rod : the mass of the triangle ;
 
 LEMMA IV. 
 
 37 
 
 therefore, the distances of the centres of gravity of the rod and 
 triangle from CD being the same, the centre of gravity of the 
 rod is at a distance \AB from B. 
 
 Also, the mass of MN being proportional to the area PEN, 
 the mass of the rod is proportional to the area of the triangle 
 ACD, and the mass of a rod of uniform density equal to that 
 at B, and of length AB, being in the same proportion to the 
 rectangle AB, CD, is therefore double of the mass of the rod. 
 
 (7) Centre of gravity of a circular arc. 
 
 Let be the centre of an uniform circular arc ABC, OB 
 the bisecting radius, aBc a tangent at B, OD parallel to ac, 
 and Aa, Cc parallel to OB. 
 
 Let QR be the side of a regular polygon described about the 
 
 arc, P the point of contact, Qq, Rr perpendicular to ac, and PM 
 to OB. Then, since OP, OB are perpendicular to QR, qr, 
 
 qr: QR::OM:OP::OM: OB-, 
 
 but, since OM, OB are the distances of the centres of gravity of 
 QR and qr from OD, and QR.OM=qr.OB, the moments of 
 QR and qr with respect to OD are in a ratio of equality, and 
 the same is true of every side of the circumscribing polygon ; 
 therefore, by Cor., Lemma IV., the moment of the arc, which is 
 ultimately that of the polygon, is equal to the moment of ac 
 
 = ac.OB= chord A C. radius OB.
 
 38 
 
 NEWTON. 
 
 Hence, the distance of the centre of gravity of the arc from 
 radius x chord 
 
 (8) Surface of a segment of a sphere. 
 
 Let AKH be the portion of a circle which generates by 
 revolution round AH the spherical segment, the centre of 
 the circle, PQ the chord of a small arc, PM t QN perpendicular 
 to AH. 
 
 Let AOCD be the rectangle circumscribing the quadrant 
 and generating the circumscribing cylinder. 
 
 Produce 1/P, NQ, HK to meet CD in p, q, k. Since PQ 
 is in its limiting position a tangent at P, PQ is ultimately 
 perpendicular to the radius OP, also pq is perpendicular to MPj 
 
 .'. PQ:pq:: OP: PM ultimately, 
 
 and the surface generated by PQ is ultimately 2TrPM.PQj 
 Art. 18, = 27r.0P.p2 == the surface generated 
 
 
 
 
 ^ 
 
 TL 
 
 
 P. 
 
 ^ 
 
 
 
 
 / 
 
 \ 
 
 
 
 / 
 
 /, 
 
 7 
 
 \ 
 
 \ 
 
 L s: ad 
 
 If Nnv SL C 
 
 The same is true for each side of the inscribed polygon when 
 the number is indefinitely increased. 
 
 Hence the surface generated by AK, or the surface of the 
 spherical segment, is equal to the surface of the circumscribed 
 cylinder cut off by the plane of the base of the segment. 
 
 COR. Hence, also, the surface of any belt of a sphere cut off 
 by two parallel planes is equal to the corresponding belt of the 
 cylindrical surface.
 
 LEMMA IV. 39 
 
 (9) Centre of gravity of a belt of the surface of a sphere con- 
 tained between parallel planes. 
 
 The moment of the belt generated by PQ with respect to the 
 plane through A, perpendicular to AH, is evidently ultimately 
 equal to that of the belt generated by pq ; therefore the moment 
 of any belt generated by K'K is equal to that of the cor- 
 responding belt generated by k'k. 
 
 Hence, the centres of gravity of the two belts are coincident, 
 viz. in the bisection of HH', that is, the distance of the centre of 
 gravity of a spherical belt, contained between parallel planes, is 
 half-way between the two planes. 
 
 (10) Volume of a spherical sector. 
 
 Let the spherical sector be generated by the revolution of the 
 sector A OP about A 0. 
 
 The volume of the spherical sector is equal to the limit of the 
 sum of a series of pyramids whose vertices are in 0, and the sum 
 of whose bases is ultimately the area of the surface of the seg- 
 ment ; also the volume of each pyramid is ^ base x altitude. 
 
 Hence, the volume of the spherical sector is one-third of the 
 area of the surface of the spherical segment x radius 
 
 = $ . ZirAD. Dp . A = ITT AM. AO i = lirA(? ver&POA. 
 
 (11) Centre of gravity of a spherical sector. 
 
 If we suppose each of the pyramids on equal bases, they may 
 be supposed collected at their centres of gravity, whose distances 
 are \AO from ultimately, and they form a mass which may 
 be distributed uniformly over the surface of a spherical segment 
 whose radius is \AO, viz. that generated. by ar, whose centre 
 of gravity will be in the bisection of am, if rm be perpendicular 
 to AH. 
 
 Therefore the distance of the centre of gravity of the spherical 
 sector from = $ ( Oa + Om) = \OA. cos^POA. 
 
 If the angle POA become a right angle, the distance of the 
 centre of gravity of the corresponding sector, which in this case 
 will become the hemisphere, will be OA, as in page 29. 
 
 (12) To find the direction and magnitude of the resultant 
 attraction of a uniform rod upon a particle, every particle of the
 
 40 
 
 NEWTON. 
 
 rod "being supposed to attract with a force which varies inversely 
 as the square of its distance from the attracted particle. 
 o 
 
 CD P <i A 
 
 Let AB be the attracting rod, the particle attracted by the 
 rod; draw 00 perpendicular to AB, join OA, OB, and let a 
 circle be described with centre and radius 00 meeting OA, 
 OB in a, b. Let OpP, OqQ be drawn cutting off the small 
 portions pq, PQ from the arc a Cb -and the rod, respectively, 
 and draw PR perpendicular to OQ. 
 
 Then PR : PQ : : 00 : OP ultimately, 
 
 and pq iPR:: Op : OP ; 
 
 .-. pq:PQ::0^: OP* , 
 
 and, if aCb be of the same density as the rod and attract 
 according to the same law, 
 
 attraction of pq on : attraction of PQ : : 
 
 pq . PQ 
 
 ultimately. 
 
 Therefore the portions PQ, pq of the rod and arc attract 
 in the same direction with forces which are ultimately equal. 
 
 Hence, by Cor., Lemma IV., the resultant attraction of the 
 rod is the same as that of the arc aCb, which, by symmetry, 
 is in the direction OD, bisecting the angle A OB. 
 
 Again, draw qn perpendicular to OD, pr to qn] then, by 
 similar triangles, pqr, qOn, 
 
 pq : qr : : Oq : On ; 
 
 pq On 
 *' 0? ' ~0q 
 
 . 
 00* 
 
 that is, the resultant attraction of pq in the direction OD is the
 
 LEMMA IV. 
 
 41 
 
 same as that of qr at the distance OC] hence the whole re- 
 sultant attraction of AB is 
 
 where //- is the attraction of a unit of mass at the unit distance. 
 
 V. 
 
 1. Shew that the area of the sector of an ellipse contained 
 between the curve and two central distances varies as the angle 
 of the corresponding sector of the auxiliary circle. 
 
 2. Prove that the volumes of two pyramids will be equal if 
 they stand on the same base, and have their vertices in the same 
 plane parallel to the base. 
 
 3. Find the volume of a paraboloid by comparison with the 
 area of a triangle whose vertex and base are those of the generating 
 parabola. 
 
 4. Find the centre of gravity of the paraboloid by reference 
 to the same triangle. 
 
 5. Find the mass of a straight rod, whose density varies as the 
 square of the distance from one extremity, by comparison with 
 a cone whose axis is the rod. 
 
 6. Shew that the orthogonal projection of any plane area on 
 another plane is the given area x the cosine of the inclination of 
 the two planes. 
 
 As a first step, prove that, pqsr being the projection of the 
 inscribed parallelogram PQSR, pqsr : PQSR : : cosAC : 1. 
 
 7. Find the volume of a hemisphere by comparing the volumes 
 generated by the quadrantal sector and the portion of the circum- 
 scribing square which is the difference between the square and the 
 quadrantal sector.
 
 42 NEWTON. 
 
 VI. 
 
 1. Find the volume of a paraboloid generated by the revolution 
 of a semi-cubical parabola, in which PM* oc AM 3 , by means of a 
 cone on the same axis. 
 
 2. Assuming that the area of a belt of a sphere cut off by two 
 parallel planes varies as the perpendicular distance between them, 
 find by the aid of Lemma IV. the area of any portion of the curve 
 of sines. 
 
 3. Prove that, if PQ be a small arc of an ellipse, and CD be 
 conjugate to CP, the limit of the sum of all the ratios PQ : CD, 
 taken over the whole perimeter of the ellipse, will be 2ir. 
 
 4. P is any point of a curve OP; OX, OF any lines drawn at 
 right angles through 0, PM, PN perpendicular to OX, OF respec- 
 tively. Prove that, if area 0PM : area OPN : : m : 1 always, and 
 the whole system revolve about OX, volumes generated by 0PM, 
 OPN will be as m : 2. 
 
 5. Prove that the surface generated by the revolution of a 
 semi-circle round its bounding diameter is to the curved surface 
 generated by the revolution of the same semi-circle round the 
 tangent at the extremity of the diameter in the ratio of the length 
 of the diameter to the length of the arc of the semi-circle. 
 
 6. Common ordinates MPP', NQQ are drawn to two ellipses 
 which have a common minor axis, and the outer of which touches 
 the directrices of the inner; shew that the area of the surface 
 generated by the revolution of PQ about the major axis bears a 
 constant ratio to the area MP'Q'N. 
 
 7. Prove that the area included between an hyperbola and the 
 tangents at the vertices of the conjugate hyperbola is equal to the 
 area included between the conjugate hyperbola and the tangents at 
 the vertices of the hyperbola.
 
 LEMMA V. 
 
 43 
 
 LEMMA V. 
 
 All the homologous sides of similar figures are proportional, 
 whether curvilinear or rectilinear, and their areas are in 
 the duplicate ratio of the homologous sides. 
 
 [Similar curvilinear figures are figures whose curved 
 boundaries are curvilinear limits of corresponding 
 portions of similar polygons. 
 
 Let SAB CD..., sabcd... be two similar polygons, of 
 which SA, AB, BC, ... are homologous to sa, db, 
 be, ... respectively. 
 
 Then ABiabiiSA: sa, 
 similarly, BC : be : : AB : ab : : SA : sa, 
 CD: cdnBC: be :: SA : sa, 
 
 therefore, componendo, 
 AB + BC + CD + ... :ab + bc + cd + ...:: SA: sa. 
 
 Now this, being true for all similar polygons, will be 
 true in the limit, when the number of the sides AB, 
 BC, ... and ab, be, ... is increased, and their lengths 
 diminished indefinitely; if, therefore, AE, ae be 
 curves which pass through the angular points A, B, ... 
 and a, b, ... of the polygons, these curves will be 
 curvilinear limits of AB + BC-\- ... and db + be + ...,
 
 44 NEWTON. 
 
 and will be the boundaries of similar curvilinear figures; 
 therefore the curved line AE : the curved line ae 
 
 \: SA: sa:: SE : se. 
 
 Again, polygon SAB C. , . : polygon sale. . . : : SA* : sa 8 , 
 and this is true in the limit ; hence, by Lemma III. 
 Cor. 2, curvilinear area SAE : curvilinear area sae 
 
 : : SA' :sa*:: AE* :ae*:: SE* : se*. 
 
 Q.E.D.] 
 
 Observations on the Lemma. 
 
 24. In order to deduce the properties of similar curves, it 
 is premised, as before mentioned under Cor. 4, Lemma III., 
 that, if a finite portion of a curve be taken, and if a polygon 
 be inscribed in the curve, the sides of which are chords taken 
 in order of portions of the curve, and the number of sides of 
 the polygons be increased indefinitely, and the magnitudes a 
 the same time diminished indefinitely, the curve will be the limit 
 of the perimeter of the polygon.* 
 
 It is not assumed that each chord is equal to the corre- 
 sponding arc ultimately; this is afterwards proved for a con- 
 tinuous curve in Lemma VII. 
 
 Criteria of Similarity. 
 
 25. From the definition of similar curve lines, that they are 
 curvilinear limits of homologous portions of similar polygons, 
 the following criteria of similarity can be deduced, all of which 
 are very convenient in practice ; namely : 
 
 (1) One curve line is similar to another when, if any 
 polygon be inscribed in one, a similar polygon can be inscribed 
 in the other. 
 
 (2) If two curves be similar, and any point S be taken 
 in the plane of one curve, another point s can be found in the 
 plane of the other, such that, any radii SP, SQ being drawn in 
 the first, radii sp, sq can be drawn in the second, inclined at 
 
 * Whewell's Doctrine of Limits.
 
 LEMMA V. 45 
 
 the same angle as the former, and such that the following 
 proportion will hold, 
 
 sp : sq : : SP : SQ. 
 
 (3) If two curves be similar, and in the plane of one 
 curve any two lines OX, Y be drawn, two other lines ox, oy 
 can be drawn in the plane of the other curve, inclined at the 
 same angle, having the property that the abscissa and ordinate 
 OM, MP of any point P in the first being taken, the abscissa 
 and ordinate om, mp of a corresponding point p in the second 
 will be proportional to the former, viz., 
 
 om :mp:: OM : MP. 
 
 And the converse propositions can also be deduced, that if 
 these proportions hold, the curves will be similar. 
 
 26. In order to illustrate test (1), let the arcs AB, db of 
 two circles have the same centre C, and let the bounding radii 
 be coincident in direction. 
 
 Let ADEB be any polygon inscribed in AB, and let CD, 
 CE cut ah in d, e ; join ad, de, eb, these are parallel to AD, 
 DE, EB respectively, and ad : de : eb : : AD : DE : EB; hence 
 adeb is similar to ADEB; and therefore the arcs ab, AB are 
 similar. 
 
 27. Test (2) may be deduced as follows : 
 
 If ABCD..., abed..., fig. p. 43, be corresponding portions 
 of similar polygons, AB, BC, ... db, be, ... being homologous 
 sides, and AS, BS, ... be drawn to any point S, construct the 
 triangle sab equiangular with SAB, and join sc, sd, .... 
 
 Then sb : SB : : ab : AB ::lc: BC, and L SBC= L sic ;
 
 46 NEWTON. 
 
 therefore SBC, sic are similar triangles; 
 
 hence sc : SC : : sb : SB : : sa : 8A ; 
 and similarly for sd, se, &c. 
 
 Hence, if two polygons be similar, and any point be taken 
 in one, another point can be found in the other, such that the 
 radii drawn to corresponding angular points will be propor- 
 tional and include the same angles. 
 
 If we now increase the number of sides indefinitely and 
 diminish their magnitude, the same property will hold with 
 respect to the curvilinear limit of the polygon. 
 
 Test (3) can be deduced from test (1) in a similar manner. 
 
 Centres of Similitude. 
 
 28. When two similar curves are so situated that a point 
 can be found, such that the radii drawn from that point, either 
 in the same or opposite directions, are in a constant ratio, such 
 a point is called a centre of similitude. 
 
 If the radii be measured in the same direction, the point 
 will be a centre of direct similitude, and of inverse similitude 
 if they be measured in opposite directions. 
 
 It is easily shewn that there can be only one centre of 
 similitude of one kind. 
 
 Properties of similar curves and application of tests of 
 Similarity. 
 
 (1) Similar conterminous arcs, which have their chords coin- 
 cident^ have a common tangent. 
 
 Let APB, Apb be similar conterminous arcs, ABb the line 
 of their chords, AQq, APp any straight lines meeting the 
 curves in Q, q and P, p respectively ; then A will evidently 
 be a centre of direct similitude for the two curves; therefore 
 A Q : Aq : : AP : Ap ; hence AP, Ap are similar portions of
 
 LEMMA V. 
 
 47 
 
 the curves, and arcAP: arcAp :: AP: Ap :: AS : Ab; there- 
 fore the arcs AP, Ap vanish simultaneously, or, when AP 
 assumes its limiting position AD for the curve APB, this is 
 also the limiting position of Ap for the curve Apb } that is, 
 the curves have a common tangent. 
 
 (2) To find the centres of direct and inverse similitude of any 
 two circles. 
 
 If one of the circles do not lie entirely within the other, 
 let >S be the intersection of two common tangents to the 
 circles which intersect in the produced line Cc joining their 
 centres, and let CQ, cq be radii to the points of contact. 
 
 Draw SpP through 8 cutting the circles in p, P, then cq 
 is parallel to CQ, and CP : cp : : CQ:cq:: CS:cS; 
 
 .'. CS: CP::cS:cp', 
 
 also GPS, cpS are each greater or each less than a right angle, 
 and CSP is common to the triagles CPS, cp8\ therefore the 
 triangles are similar, Euclid VI. 7, and the sides about the 
 angle CSP are proportional, that is, SP : Sp : : SC : Sc ; 
 therefore S is the centre of direct similitude. 
 
 Similarly, the intersection of two common tangents which 
 cross between two circles is the centre of inverse similitude. 
 
 (3) To find the condition of similarity of two conic sections. 
 Let the conic sections be placed so that their directrices
 
 48 NEWTON. 
 
 are parallel and foci coincident, and let SpP be any line 
 through the focus meeting them in p, P; draw SaAD and 
 PQ perpendicular to the directrix DQ of AP, and join SQ, 
 and let pq, parallel to PQ, meet it in j, and draw gd per- 
 pendicular to SD. 
 
 Then Sd : SD : : Sq : SQ : : Sp : SP; and, if the curves 
 be similar, Sp : SP will be a constant ratio ; therefore Sd : SD 
 is a constant ratio, and dq is a fixed straight line for all 
 positions of p ; also, since pq : Sp :: PQ : SP, pq : Sp is a 
 constant ratio ; therefore qd is the directrix of ap, and, the 
 constant ratio being the same in both, the eccentricities are 
 the same. 
 
 (4) Instruments, like the Pantagraph and the Eidograpli, for 
 copying plans on an enlarged or reduced scale are founded upon 
 the properties of similar figures ; as are also other methods of 
 copying, such as by dividing plans or pictures into squares. 
 
 The Pantagraph is an instrument for drawing a figure 
 similar to a given figure on a smaller or larger scale; one of 
 its forms is as in the figure. AD, EF, GO and AE, DQ, FG 
 are two sets of parallel bars, joined at all the angles by 
 
 compass-joints; at B is a point, which serves to fix the 
 instrument to the drawing board; at A is a point which is 
 made to pass round the figure to be reduced or enlarged; at 
 C is a hole for a pencil pressed down by a weight, and the 
 pencil traces the similar figure, altered in dimensions in the 
 ratio of SO : AS or BF: AD. 
 
 The similarity of the figure traced by the pencil is a eon- 
 sequence of continual similarity of the triangles ABD, BFC.
 
 LEMMA V. 49 
 
 By changing the positions of the pegs at F and G the figure 
 described by G may be made of the required dimensions. 
 
 For a description of the Eidograph, invented by Professor 
 Wallace, see the Transactions of the Royal Society of Edinburgh ^ 
 vol. xin. 
 
 (5) Volume of a cone whose base is a plane closed Jiyure of 
 any form. 
 
 Let V be the vertex, AB the base, VH perpendicular to the 
 base from V; let VH be divided into n equal portions, of 
 
 which HN is the (r + l) th ; and let PQ be the section through M 
 parallel to AB. 
 
 Take VPA any generating line of the cone meeting the 
 section PQ and the base AB in PA respectively, then 
 
 PM-.AH:: VM: VH; 
 
 therefore PQ is similar to AB, M, H being similarly situated 
 points; and, by Lemma V., 
 
 area PQ : area AB :: r 2 : ri\ 
 
 also MN: VH:: 1 :; 
 therefore the volume of the cylinder whose base is PQ and 
 
 r 2 
 height MN = x area AB. VH, and the volume of the cone, 
 
 by Lemma II., is one-third of the cylinder whose base is AB 
 and height VH. 
 
 VII. 
 
 1. Apply a criterion of similarity to shew that segments of 
 circles which contain equal angles are similar. 
 
 2. From the definition of an ellipse, as the locus of a point 
 the sum of whose distances from two fixed points is constant, shew 
 that ellipses are similar when the eccentricities are equal.
 
 50 NEWTON. 
 
 3. Prove that the centre of an ellipse is a centre of inverse 
 similitude of two opposite equal portions of the circumference of 
 the ellipse. 
 
 4. Employ the properties of similar figures to inscribe a square 
 in a given semicircle. 
 
 5. Construct, by means of similar figures, two circles, each 
 of which shall touch two given straight lines and pass through a 
 given point. 
 
 6. Deduce the position of the centre of gravity of a circular 
 sector from that of a circular arc ; shew that the distance from 
 
 . 2 radius x chord 
 the centre is - . - . 
 3 arc 
 
 7. If A be the vertex of a conical surface, G the centre of 
 gravity of the base, H that of the volume of the conical figure, 
 shew that AH 
 
 8. Find the centre of gravity of the surface of a right cone 
 on a circular base. Does the method apply to the surface of an 
 oblique cone?
 
 LEMMA VI. 51 
 
 LEMMA VI. 
 
 If any are ACE given in position be subtended ly a chord 
 AB, and if at any point A, in the middle of continuous 
 curvature, it be touched by the straight line AD produced 
 in both directions, then, if the points A, B approach one 
 another and ultimately coincide, the angle BAD contained 
 by the chord and tangent will diminish indefinitely and 
 ultimately vanish. 
 
 For, if that angle do not vanish, the arc ACB will 
 contain with the tangent AD an angle equal to a 
 
 rectilineal angle, and therefore the curvature at the 
 point A will not be continuous, which is contrary to 
 the hypothesis, that A was in the middle of con- 
 tinuous curvature. 
 
 Definitions of a Tangent to a Curve. 
 
 29. (1) If a straight line meet a curve in two points A, J9, 
 and if B move up to A, and ultimately coincide with A, 
 AB in its limiting position will be a tangent to the curve at 
 the point A. 
 
 If two portions of a curve EA and AB cut one another 
 at a finite angle in A, there will be two tangents AD, AD', 
 which will be the limiting positions of straight lines AB and 
 AE, when B and E move up to A along the different portions 
 EA and BA of the curve respectively. And, similarly, if there 
 be a multiple point in A, in which several branches of the cnrve 
 cut one another at finite angles. 
 
 (2) The tangent is the direction of the side of the polygon, 
 of which the curve is the curvilinear limit, when the number 
 of sides are increased indefinitely.
 
 52 NEWTON. 
 
 This is founded on the same idea of a tangent as defini- 
 tion (1). 
 
 (3) The tangent to a curve at any point is the direction of 
 the curve at that point. 
 
 In order to apply geometrical reasoning to the tangent by 
 employing this definition, we are obliged to explain the notion of 
 the direction of a curve, by taking two points very near to one 
 another, and asserting that the direction of the curve is the 
 limiting position of the line joining these points when the 
 distance becomes indefinitely small, a statement which reduces 
 this definition to the preceding. 
 
 Observations on the Lemma. 
 
 30. " Curvatura Continua," if we consider curves as the 
 curvilinear limits of polygons, requires the curves to be limits 
 of polygons whose angles continually increase as the number of 
 the sides increase, and may be made to differ from two right 
 angles by less than any assignable angle before the assumption 
 of the ultimate form of the hypothesis. 
 
 If, however, as we increase the number of sides and diminish 
 their magnitude, one of the angles remains less than two right 
 angles by any finite difference, the curvature of the curvilinear 
 limit is discontinuous, and the form is that of a pointed arch, in 
 which the two portions cut one another at a finite angle. 
 
 A curve may be of continued curvature for one portion 
 between two points, while for another its curvature changes 
 " per saltum." 
 
 Thus, if ABC be a curve forming at B a pointed arch, it 
 
 may be of continued curvature from B to A and from C to B, 
 though not from C to A. 
 
 In this case the tangents in passing from C to A assume all
 
 LEMMA VI. 53 
 
 positions intermediate to CT, Bt, and Bt', TA, but at B they 
 pass from Bt to Bt' without assuming the intermediate positions. 
 
 31. " In medio curvature continue," implies that the point 
 A in the enunciation of the Lemma is not such a point as B 
 in the last figure, but that, in passing from a point on one side 
 of A to another on the other side, the tangents pass through all 
 the intermediate positions. 
 
 The curvature is supposed to be in the same direction in 
 the figure of the Lemma, which in all curves of continuous 
 curvature is possible, if B be taken sufficiently near to A at 
 the commencement of the change in the construction. 
 
 If the point A be not " in medio curvatures continue," two 
 tangents AD, AD' may be drawn at A to the two parts of the 
 curve, and the curve BCA will make a finite angle with one of 
 the tangents AD'. 
 
 But, even in this case, the angle between the chord and 
 that tangent which belongs to the portion of the curve con- 
 sidered continually diminishes and ultimately vanishes. 
 
 The Sultangent. 
 
 32. DEF. The part of the line of abscissas intercepted be- 
 tween the tangent at any point and the foot of the ordinate 
 of that point is called the subtangent. 
 
 33. The subtangent may be employed as follows, to find a 
 tangent at any point of a curve. 
 
 Let OM, UP be the abscissa and ordinate of a point P in 
 
 a curve, and let Q be a point near P, ON, NQ its abscissa 
 and ordinate.
 
 64 NEWTON. 
 
 Let QPU meet OX the line of abscissae in U; then, if PR 
 parallel to ON meet QN'm R, 
 
 PM : MU:: QB : PR :: QN-PM : ON- OM. 
 
 Now as Q approaches to P, the limiting position of QPU la 
 that of the tangent at P, viz. tPT, and PM : MT is the limiting 
 ratio of QN-PM: ON- OM. 
 
 The Polar Siibtangent and the Inclination of the Tangent to 
 the Radius Vector, at any Point of a Spiral. 
 
 34. DEF. Let S be the pole, PT the tangent to the curve at 
 any point P, and let ST, perpendicular to SP, meet PT in T; 
 then ST is called the polar subtangent at the point P. 
 
 35. To find the inclination of the tangent at any point of 
 a curve to the radius vector. 
 
 Let Q be a point near P, QM perpendicular to SP, pro- 
 duced if necessary, QR the circular arc, centre S, meeting 
 SP in E. 
 
 Let <>Pmeet ST'm U, then 
 
 SU: SP:: QM : PM, 
 and MR : QM:: QM: SM+SR, 
 
 Q 
 
 but, when Q approaches indefinitely near to P, QM vanishes 
 compared with SM + SR ; therefore MR vanishes compared 
 with QM or PM; therefore SU:SP:: QM: PR, ultimately; 
 therefore ST: SP is the limiting ratio of QR: PR; or 
 QR: SQ~ SP.
 
 LEMMA VI. 55 
 
 Hence ST, and also the trigonometrical tangent of the angle 
 SPT between the tangent and the radius vector can be found. 
 
 Illustrations. 
 
 (1) If 8Y be the perpendicular on the tangent PY at P in 
 a curve, Y will trace out a curve, called the pedal of the original 
 curve; to shew that if YZ be a tangent to the locus of Y, SZ 
 perpendicular to it, SY*=SP.SZ. 
 
 Let P' be a point near P, SY' perpendicular on P'P, SZ 
 perpendicular on Y'Y. 
 
 Since angles SYP, SY'P are right angles, a semicircle on 
 SP will pass through Y, Y'; therefore the angles SY'Y, SPY 
 in the same segment will be equal; the right angles SZY', 
 SYP also are equal; therefore the triangles /SPY, SY'Z are 
 similar, and SZ : SY' :: SY: /SP; but, ultimately, as F moves 
 
 up to P, P'PY' becomes the tangent at P, and TYZ that at Y 
 to its locus, also SY' = SY-, 
 
 .-. SZ.SP=SY*. 
 
 (2) To find the subtanaent in the semi-cubical parabola. 
 In the semi-cubical parabola PM 2 <x OM 3 ; 
 
 ... QN* _ pjp : pj/ 3 : : ON 9 - OM 3 : OM 3 , 
 
 but QN+PM=2PM, 
 
 and OJT-f ON.OM+ OM* = 30M'\ ultimately; 
 
 .-. QN- PM: \PM\\ ON-OM: OM ultimately, 
 
 and QN- PM: PM:: ON- OM = M.T\ 
 
 therefore MTis two-thirds of OM.
 
 56 NEWTON. 
 
 (3) To find the inclination of the tangent at any point of 
 a cardioid to the radius vector. 
 
 DEF. If BqpC be a circle, whose centre is 8 and diameter 
 BC, and pm be drawn perpendicular to BG\ then, if Sp be 
 produced to P, making SP=Bm, P will trace out a cardioid 
 APS. 
 
 .Making the same construction as before, in Art. 35, 
 
 8T : SP : : QR : SP- SQ ultimately. 
 
 Let SQ meet the circle in q, and draw qn perpendicular 
 to BC, 
 
 then QR : pq : : SP : Sp ultimately, 
 
 also pq : mn :: Sp : pm , 
 
 .-. QB:mn::SP:pm ; 
 
 but mn = Bm-Bn = SP- SQ ; 
 /. QR: SP-SQ:: SP : pm ultimately; 
 
 .-. ST: 8P::Bm:pm; 
 hence LPTS=LpBm = \LPSA^ 
 
 and it follows that the cardioid cuts the axis SCA at right 
 angles, that it touches SB at #, and that it cuts the circle BDG 
 at an angle equal to half a right angle.
 
 LEMMA VII. 
 
 LEMMA VII. 
 
 If any arc, given in position, be subtended by the chord AB, 
 and at the point A, in the middle of continuous curvature,, 
 a tangent AD be drawn, and the subtense BD, then, when 
 B approaches to A and ultimately coincides with it, the 
 ulimate ratio of the arc, the chord, and the tangent to one 
 another, is a ratio of equality. 
 
 For whilst the point B approaches to the point A, let 
 AB, AD be supposed always to be produced to points 
 b and d at a finite distance, and bd be drawn parallel 
 to the subtense BD, and let the arc Acb be always 
 similar to the arc ACB, and have, therefore, ADd 
 for its tangent at A. 
 
 But, when the points B, A coincide, the angle I Ad, by 
 the preceding Lemma, will vanish, and therefore the 
 straight lines Ah, Ad, which are always finite, and 
 the arc Acb, which lies between them [and is of con- 
 tinuous curvature in one direction, if the change 
 commence when B is near enough to A~\, will coin- 
 cide ultimately, and therefore will be equal. 
 
 Hence, also, the straight lines AB, AD and the inter- 
 mediate arc A CB, which are always proportional to 
 them, will vanish together, and have an ultimate 
 ratio of equality to one another. 
 
 COR. 1. Hence if BF be drawn through B parallel to 
 the tangent, always cutting any straight line AF 
 passing through A in F, then BF will have ulti- 
 mately to the vanishing arc A GB a ratio of equality,
 
 58 
 
 NEWTON. 
 
 since, if the parallelogram A FED be completed, it 
 will always have a ratio of equality to AD. 
 
 7 
 
 COR. 2. And if through B and A be drawn many 
 straight lines BE, BD, AF, ACr cutting the tangent 
 AD and BF, parallel to it ; the ultimate ratios of all 
 the abscissae AD, AE, BF, BCr and of the chord and 
 arc AB to one another will be ratios of equality. 
 
 COR. 3. And, therefore, all these lines in every argu- 
 ment concerning ultimate ratios may be used indif- 
 ferently one for the other. 
 
 Observations on the Lemma. 
 
 36. DEF. The subtense of the angle of contact of an arc is a 
 straight line drawn from one extremity of the arc to meet, at 
 a finite angle, the tangent to the arc at the other extremity. 
 
 This subtense is the secant which defines the limited line 
 called, in the Lemma, " the tangent." 
 
 The chord is called by Newton the subtense of the arc, see 
 Lemma XI. 
 
 37. In the construction for this Lemma, BD must be a 
 subtense, i.e. inclined throughout the change of position at a 
 finite angle to the tangent, for, otherwise, the angles BAD
 
 LEMMA VII. 59 
 
 and ADB being then both small, the ultimate ratio of the 
 chord to the tangent might be any finite ratio instead of 
 being one of equality. 
 
 This is the only limitation of the motion of BD ; the figure 
 representing changes which may take place in the approach 
 towards the ultimate state of the hypothesis. 
 
 Here b, d are the distant points, that is, points at a finite 
 distance from A] BD, B'D', B"D" are consecutive positions 
 of the subtense, when B approaches towards A, and db, db', db" 
 are parallel to these, Acb', Ac"b" are the forms of Acb changed 
 so as to be always similar to the corresponding portion of ACB 
 cut off by the chord. 
 
 It should be remarked that the curve Acb is not inter- 
 mediate in magnitude to the two lines Ab, Ad, but only in 
 position; for example, Ab may be equal to Ad, if BD make 
 equal angles with the two lines, and the curve line will then 
 be greater than either Ab or Ad] but it becomes in all cases 
 less bent, until it is ultimately rectilinear; hence the three 
 Acb, Ab, Ad will be ultimately equal, the only alternative 
 being that the curve may become doubled up, as in the figure, 
 
 which is precluded by the supposition that the curvature near A 
 is continued in the same direction throughout the passage from 
 B to A. 
 
 38. The subtense ultimately vanishes compared with the arc. 
 For BD : A CB :: bd : Acb, and, since bd vanishes and Acb 
 
 remains finite in the limit, the ratio BD : ACB ultimately 
 vanishes. It will be afterwards seen that in curves of finite 
 curvature BD varies as the square of ACB ultimately. 
 
 The ultimate equality of the lines AD, AE with the chord 
 or arc, whatever be the direction of the subtense, is due to 
 the vanishing of BD, and therefore of DE with respect to AD. 
 
 39. If two curves of continuous curvature which do not inter- 
 sect have a common chord, the length of the exterior curve will be
 
 60 NEWTON. 
 
 greater than that of the interior, provided that the curvature of 
 the interior be always in the same direction. 
 
 Let AcdeBj A CDEFB any two polygons, having a common 
 side AB : be such that the first lies entirely within the second 
 
 and that neither has internal angles, the perimeter of the first 
 is less than that of the second. 
 
 For, produce Ac, cd, de to meet the perimeter of the exterior 
 in c' t d'j e' ; then A C + Cc > Ac' ; /. A CDEFB > Ac'DEFB ; 
 similarly Ac'DEFB>Acd'EFB 1 and on on; 
 /. a fortiori, ACDEFB> AcdeB. 
 
 And, since the same is true in the limit, when the nnmber 
 of sides is increased indefinitely, the curvilinear limits of the 
 polygons have the same property, and the proposition is proved.
 
 LEMMA VIII. 61 
 
 LEMMA VIII. 
 
 If two straight lines AR, BR make with the arc A CB, the 
 chord AB, and the tangent AD, the three triangles 
 RA CB, RAB and RAD, and the points A, B approach 
 one another ; then the ultimate form of the vanishing 
 triangles is one of similitude, and the ultimate ratio one 
 of equality. 
 
 For, whilst the point B is approaching the point A, let 
 AB, AD, AR be always produced to points b, d, r at 
 a finite distance, and rid be always drawn parallel to 
 RD, and let the arc Acb be always similar to the arc 
 ACB, and therefore have Dd for the tangent at A. 
 
 Then, when the points B, A coincide, the angle bAd will 
 vanish, and therefore the three triangles rAb, rAcb, 
 rAd will coincide, and will therefore in that case be 
 similar and equal. Hence also R AB, RA CB, RAD, 
 which are always similar and proportional to these, 
 will be ultimately similar and equal to one another. 
 
 COR. And hence, in every argument concerning ulti- 
 mate ratios, these triangles can be used indifferently 
 for one another. 
 
 Observations on the Lemma. 
 
 40. If RB throughout the change in the hypothesis make a 
 finite angle with RA, the three triangles rAb, rAcb, rAd will
 
 62 NEWTON. 
 
 remain always finite, and will be ultimately identical and equal. 
 But, if the angle AEB be ultimately not finite, for example, if 
 EB revolve round a fixed point R, the three triangles rAB, ... 
 will become infinite, since r will move to r and so on to an 
 infinite distance, and there will be the same kind of objection 
 
 to dealing with these infinite triangles, as to reasoning im- 
 mediately upon the relation of the triangles BAB, BAD in 
 the former case. 
 
 In this case we can at once deduce the equality of the tri- 
 angles without producing AD to a point d at a finite distance. 
 For, the ratio of the difference of RAD and RAB to BAB is 
 BD : RB, which vanishes ultimately, since RB is finite in 
 this case; hence RAB and RAD and also the curvilinear 
 triangle, which is intermediate in magnitude to them, will be 
 ultimately in a ratio of equality.
 
 LEMMA IX. 63 
 
 LEMMA IX. 
 
 If a straight line AE and curve ABC, given in position, cut 
 one another in a finite angle A, and ordinates BD, CE bs 
 drawn, inclined at another finite angle to that straight 
 line, and meeting the curve in B, C ; then, if the pointe 
 B, C move up together to the point A, the areas of the 
 curvilinear triangles ABD, A CE will be ultimately to one 
 another in the duplicate ratio of the sides. 
 
 For, as the points B, C are approaching the point A, 
 let AD, AE\>Q always produced to the points d, e at 
 a finite distance, such that Ad : Aei: AD : AH] and 
 
 let the ordinates db, ec be drawn parallel to DB, 
 EC meeting the chords AB, A C produced in b, c. 
 
 Then [since Ab : AB ::Ad: AD :: Ae : AE : : Ac:AC t 
 and therefore Ab : Ac : : AB : AC~] a curve Abe can 
 be supposed to be drawn always similar to ABC, 
 while B and C move up to A. 
 
 Let the straight line Ag be drawn touching both curves 
 at A, and cutting the ordinates DB, EC, db, ec in 
 f,0,f, 9 . 
 
 [Now areas ABD, Abd, by Lemma V., are always in the 
 duplicate ratio of AD, Ad, and areas ACE, Ace in the 
 duplicate ratio of AE, Ae, and AD : Ad :: AE : Ae ; 
 therefore ABD : Abd :: ACE: Ace, 
 and ABD : ACE:: Abd : Ace.] 
 
 If, then, the points B and C move up to A and ultimately 
 coincide with it, the angle cAg will ultimately vanish,
 
 64 NEWTON. 
 
 and the curvilinear areas Abd, Ace will coincide with 
 the rectilinear triangles Afd, Age, and therefore will 
 be ultimately in the duplicate ratio A d, Ae. 
 But ABD, ACE are proportional to Aid, Ace always, 
 also, AD, AE are proportional to Ad, Ae ; therefore 
 also areas ABD, A CE are ultimately in the duplicate 
 ratio of AD, AE. 
 
 Observations on the Lemma. 
 
 41. By a finite angle is to be understood an angle less than 
 two right angles, and neither indefinitely small nor indefinitely 
 near to two right angles. 
 
 The angles between AD and the curve and between AD 
 produced and BD are different finite angles, because otherwise 
 BD would not meet the curve. 
 
 42. If the angle DAF be greater than a right angle, the 
 figure may assume a form in which AD will lie below ABC', 
 in this case DB, EC, ... must be produced to meet the tangent, 
 and the argument may proceed in the same manner as before. 
 
 43. It is not necessary that d and e be fixed, but only that 
 they remain at a finite distance from A, and that the proportion 
 be retained ; and the first part of this observation applies to 
 d in the previous Lemmas. 
 
 The student, by reference to Arts. 37 and 40, will be able to 
 exhibit the change in the figure which will correspond to a 
 change of the position of B and C in the progress towards the 
 ultimate position. 
 
 44. When the angle CACr vanishes, the curvilinear areas 
 Abd, Ace coincide with the rectilinear triangles Afd, Age, and 
 so are in the duplicate ratio of Ad : Ae. But if the angle 
 DAF be not finite, those triangles will not themselves be finite, 
 and the object aimed at by producing to a finite distance will 
 not be attained. 
 
 The fact is, that the triangle Adb is made up of the triangle 
 4df and the curvilinear triangle Afb, of which the latter is 
 indefinitely small ultimately, and the former is finite ; therefore,
 
 LEMMA IX. 65 
 
 in the Lemma, Afb vanishes compared with Adf; but this will 
 not be so if Adf be indefinitely small, the ratio of the triangles 
 AFB, AGO must, therefore, be found by another process, and 
 it will be found, by referring to Lemma XL, that the ratio will 
 be ultimately that of the cubes of the arcs if the curvature of 
 the curve at A be finite. 
 
 VIII. 
 
 1. RQq is n, common subtense to two curves PQ, Pq, which 
 have a common tangent PR at P. When RQ,q approaches to P, 
 RQ, and Rq ultimately vanish; will the ratio RQ, : Rq be ulti- 
 mately a ratio of equality ? 
 
 2. If PY, a tangent to an ellipse at P, meet the auxiliary circle 
 in F, and S T be perpendicular to the tangent at Y, ST will vary 
 inversely as HP. 
 
 3. If a subtense BD be drawn to meet the tangent at A at 
 a finite angle a, which remains constant as B moves up to A, and 
 DB meet the normal at A in C, shew that the ultimate ratio of 
 EC to AB will be sec a. 
 
 4. In the curve in which the abscissa varies as the cube of 
 the ordinate, shew that the subtangent is three times the abscissa. 
 
 5. Prove that the extremity of the polar subtangent from the 
 focus of a conic section is always in a fixed straight line. 
 
 6. AB is a diameter of a circle, P a point contiguous to A, 
 and the tangent at P meets BA produced in T; prove that ulti- 
 mately the difference of BA, BP will be equal to one-half of TA. 
 
 7. In any curve, if Q be the intersection of perpendiculars to 
 two consecutive radii vectores through their extremities, and SY 
 be the perpendicular from the pole S on the tangent at P, prove 
 that ultimately SP' = SY.SQ. 
 
 PQ, pq are parallel chords of an ellipse whose centre is 
 that, if 
 ultimately eqm 
 
 shew that, if p move up to P, the areas CPp, CQq will be 
 iqual. 
 
 9. From a point in the circumference of a vertical circle a chord 
 and tangent are drawn, the one terminating at the lowest point, 
 and the other in the vertical diameter produced; compare the 
 velocities acquired by a heavy body in falling down the chord 
 and tangent when they are indefinitely diminished. 
 
 10. A point moves so that the product of its distances from two 
 fixed points is constant ; shew that the normal to its path divides 
 the angle between the two radii into two whose sines are pro- 
 portional to the radii.
 
 66 NEWTON. 
 
 IX. 
 
 1. On the radii vectores of a curve as diameters circles are 
 described ; find their envelope. 
 
 2. If the intercept PQ between two curves of their common 
 radius vector OPQ be constant, and the normals at P and Q 
 intersect in N, ON will be at right angles to OPQ. 
 
 3. A right angle slides on any oval curve, so that the sides 
 containing the right angle always touch the curve ; shew that the 
 angle one tangent makes with the tangent to the locus of the 
 vertex is equal to that which the other tangent makes with 
 the chord of contact. 
 
 Hence shew that, if the oval be an ellipse, the locus of the 
 vertex will be a circle concentric with the ellipse. 
 
 4. A point moves so that the rectangle, whose sides are equal 
 to the distances of the point from a given point and a given 
 straight line, is equal to the square described on the perpendicular 
 from the given point on the given line. Find the position of 
 the point at which the tangent to the curve passes through the 
 fixed point. 
 
 5. Two points A, B describe two curves according to any 
 finite and continuous law. If A', B' be the consecutive positions of 
 A, , and ABC, A'B'C' be similar triangles, then the corre- 
 sponding sides of the two triangles will ultimately intersect in the 
 
 4 ^ t. v. . AA'.BC BB'.CA CC'.AB 
 points P, Q, R, such that ^ = Rp = p^ . 
 
 6. If SP* = AB.PM, where PM is perpendicular to a fixed 
 straight line, prove that the locus of the centre of the circle cir- 
 cumscribing the triangle formed by the tangent, the radius vector, 
 and the polar sub tangent, will be a straight line. 
 
 7. In the figure on page 30 let FB' be taken equal to AB, 
 and let the corresponding ordinate to the curve be B'E ; prove 
 that the subtangent at E' varies inversely as that at E. 
 
 8. In the hyperbolic spiral, in which the radius vector varies 
 inversely as the spiral angle, prove that the subtangent is constant. 
 
 9. In the spiral of Archimedes, in which the radius vector 
 varies directly as the angle, prove that if a circle be described, 
 of which a radius is the radius vector of the sjiral, the polar 
 subtangent will be equal to the arc of the circle subtended by 
 the spiral angle.
 
 LEMMA X. 
 
 67 
 
 LEMMA X. 
 
 The spaces tvhich a body describes [_from resf] under the 
 action of any finite force, whether that force be constant 
 or else continually increase or continually diminish, are 
 in the very beginning of the motion in the duplicate ratio 
 of the times. 
 
 [Let the times be represented by lines measured from A, 
 along AK, and the velocities generated at the end 
 of those times by lines drawn perpendicular to AK. 
 Suppose the time represented by AK to be divided 
 into a number of equal intervals, represented by AB, 
 
 BO, CD, ..., let Bb, Cc, Dd, ...Kk represent the ve- 
 locities generated in the times AB, AC, ...^^respec- 
 tively, and let Abed be the curve line which always 
 passes through the extremities of these ordinates. 
 Complete the parallelograms Ab, Be, Cd, .... 
 In the interval of time denoted by CD, the velocity con- 
 tinually changes from that represented by Cc to that 
 represented by Dd, and therefore CD being taken 
 small enough, the space described in that time is 
 intermediate between the spaces represented by the 
 parallelograms DC and Cd; therefore the spaces 
 described in the times AD, AK are represented by 
 areas which are intermediate between the sums of 
 the parallelograms inscribed in, and circumscribed 
 about, the curvilinear areas ADd, AKk respectively.
 
 68 NEWTON. 
 
 Therefore, by Lemma II., the number of intervals being 
 increased, and their magnitudes diminished indefi- 
 nitely, the spaces described in the times AD, AK 
 are proportional to the curvilinear areas ADk, AKk. 
 
 Now the force being finite, the ratio of the velocity to 
 the time is finite ; therefore Kk : AK is a finite ratio, 
 however small the time is taken; hence, if AT be 
 the tangent to the curve line at A, "meeting Kk in T, 
 KT : AK will be a finite ratio ; therefore the angle 
 TAK will be finite, or AK will meet the curve at a 
 finite angle. 
 
 Hence, by Lemma IX., if AD, AK be indefinitely 
 diminished, area ADd : area AKk : : AD* : AK* ; 
 therefore, in the beginning of the motion, the spaces 
 described are proportional to the squares of the times 
 of describing them. Q. E. D.] 
 
 COR. 1. And hence it is easily deduced that the errors 
 of bodies describing similar parts of similar figures 
 in proportional times, which are generated by any 
 equal forces acting similarly upon the bodies, and 
 which are measured by the distances of the bodies 
 from those points of the similar figures, to which the 
 same bodies would have arrived in the same propor- 
 tional times without the action of the disturbing 
 forces, are approximately as the squares of the times 
 in which they are generated. 
 
 COR. 2. But the errors which are generated by pro- 
 portional forces, acting similarly at similar portions 
 of similar figures, are approximately as the forces 
 and the square of the times conjointly. 
 
 COR 3. The same is to be understood of the spaces 
 which bodies describe under the action of different 
 forces. These are, in the beginning of the motion, 
 conjointly, as the forces and the squares of the times. 
 
 COR. 4. Consequently, in the beginning of the motion 
 the forces are as the spaces described directly, and 
 the squares of the times inversely.
 
 LEMMA X. 69 
 
 COR. 5. And the squares of the times are as the spaces 
 described directly and the forces inversely. 
 
 The proof given in the original Latin is as follows : 
 Exponantur tempora per lineas AD, AE, et veloci- 
 tates genitae per ordinatas DB, EC', et spatia, his 
 velocitatibus descripta, erunt ut areae ABD, ACE his 
 ordinatis descriptse, hoc est, ipso rnotus initio (per 
 Lemma IX.) in duplicata ratione temporum AD, AE. 
 
 Q.E.D. 
 
 45. This proof has been amplified in order to exhibit in 
 what manner the description of areas, by the flux of the ordi- 
 nates, corresponds to that of spaces by the velocities represented 
 by the ordinates ; also to shew the propriety of the application 
 of the ninth Lemma by reference to the definition of finite force 
 which may be stated as follows : A force is finite when the ratio 
 of the velocity generated in any time to the time in which it is 
 generated, is finite, however small the time be taken. 
 
 Observations on the Lemma. 
 
 46. In the proof of this Lemma, time is represented by the 
 length of a straight line, and a distance traversed by a body is 
 represented by an area. 
 
 If the length of a straight line be always proportional to the 
 period of time elapsed, the straight line will be a proper repre- 
 sentation of the time. Thus a length of n inches has the same 
 ratio to one inch which an interval of n seconds has to one 
 second ; and on this scale the length n inches is a proper repre- 
 sentation of n seconds. 
 
 If an area be always in the same ratio to the unit of area 
 that the length of a straight line is to the unit of length, the area 
 will be a proper representation of the length of the straight line. 
 
 Thus, if Ab be one foot, AB, n feet, Ac one inch, and AC, 
 t inches: complete the parallelograms ABDC, Abdc, and Bc t 
 then ABCD will contain nt such areas as Abdc. 
 
 If now a particle move with a uniform velocity of n feet 
 a second, and A C represent t seconds, on the scale of one inch to
 
 70 NEWTON. 
 
 a second, the parallelogram Be will represent the space travelled 
 over in the first second, since it contains n times the parallelo- 
 
 c t. 
 
 gram Abdc, and ABDG will represent the space travelled over 
 in t seconds. 
 
 There will be no difficulty in the representation of a period 
 of time by a line, or of a distance by an area, if the student 
 bear in mind that periods of time and lengths of lines, although 
 existing absolutely, are only estimated by their ratios to certain 
 standard periods, and standard lengths, and they are therefore 
 determined whenever these ratios are given, either directly in 
 numbers or by the comparison of any magnitudes whatever of 
 the same kind. 
 
 47. COR. 1,2. If bodies describe orbits under the action 
 of certain forces, and small forces, extraneous to those under the 
 action of which the orbits are described, be supposed to act upon 
 the bodies, the orbits will be disturbed slightly, and the errors 
 spoken of are the linear disturbances of the bodies, at any time, 
 from the positions which they would have occupied at that time, 
 if the extraneous forces had not acted. 
 
 Thus, in calculating the motion of the Moon considered as 
 moving under the attraction of the Sun and Earth, it is conve- 
 nient to estimate the motion which she would have, if subjected 
 to the attraction of the Earth alone, and then to calculate what 
 would be the disturbing effect of the Sun upon this orbit. 
 
 48. If AB be a portion of an orbit described by a body in 
 any time, A C the portion of the orbit described when a disturb- 
 ing force is introduced, BO is " quam proxime" the space which 
 would have been described in the same time from rest by the 
 action of the disturbing force alone. When the time is taken 
 small, but not indefinitely small, the expression in the statement
 
 LEMMA X. 71 
 
 of the corollaries, "approximately," is necessary for two reasons; 
 for, in the first place, the position of the body in space is not 
 the same at the end of any interval in the lapse of the time 
 as if the body had moved from rest under the action of the 
 disturbing force alone, and therefore the magnitude of the force 
 is not generally the same either in direction or magnitude ; and, 
 in the second place, since the force is not generally uniform, the 
 variation according to the duplicate ratio of the times is not 
 exact, except in the limit. 
 
 But, when the times are taken very small, the variation of 
 direction and magnitude of the force may be neglected, as an 
 approximation to the true state of the case. 
 
 49. Application of the method of Lemma X to determine 
 the space described in a finite time from rest by a particle under 
 the action of a constant force. 
 
 Let f be the measure of the acceleration caused by the 
 constant force, so that at the time t the velocity V=ft. 
 
 Since the velocity varies as the time, the curve Ak in the 
 figure of the Lemma is a straight line, dD : AD being constant. 
 
 Therefore the space which is described in the time , re- 
 presented by AK, is represented by the area of the triangle 
 AKk or \Kk.AK. The space described in time t from rest 
 is therefore \Vt = %ft\ 
 
 50. General geometrical representation of the space described 
 by a body when it moves with a variable velocity for a finite 
 time. 
 
 PROP. If a curve be found, such that the ordinate at each 
 point represents the velocity corresponding to a time represented 
 by the abscissa, then the space described by the body will be 
 represented by the area bounded by the curve, the line of 
 abscissae, and the ordinates corresponding to the commencement 
 and end of the time of motion. 
 
 Let OA, OB represent the times at the commencement and 
 end of the interval during which the motion of the body is to 
 be examined. Let OM be any other time, and let A C, MP, BD^ 
 perpendicular to OAB, represent the velocities at the ends of
 
 72 
 
 NEWTON. 
 
 the times represented by OA, OM, OB CPD the curve which 
 passes through the extremities of all such ordinates as HP. 
 
 Let AB be divided into any number of small portions, such 
 as MN] and let NQ be the ordinate corresponding to ON. 
 Complete the parallelograms PMNq, QNMp, and suppose cor- 
 responding parallelograms to be constructed on all the bases 
 corresponding to MN. 
 
 The body during the time represented by MN moves with 
 a velocity, which, if MN be taken small enough, will be inter- 
 mediate in magnitude between the velocities represented by PM 
 and QN, and the space described during that time will be 
 intermediate in magnitude between the spaces which would have 
 been described with uniform velocity represented by PM and 
 QN, or between the spaces represented by the areas PA 7 , QM. 
 
 Hence the whole space described in the interval of time 
 represented by AB is greater than that represented by the 
 inscribed series and less than that by the circumscribed series 
 of parallelograms, and each of these is, by Lemma II., ulti- 
 mately equal to the area A CDB, when the number of portions 
 into which AB is divided is indefinitely increased, and their 
 magnitudes diminished ; therefore the proposition is proved. 
 
 51. COR. 1. Since the area PMNQ is ultimately equal to 
 the rectangle PM.MN, it follows that the measure of the velocity 
 at any time is the limit of the quotient of the space described after 
 that time by the time of describing it. 
 
 52. COE. 2. Let MR represent the unit of time, and com- 
 plete the parallelogram PMRr ; then the area PMRr represents
 
 LEMMA X. 73 
 
 the space which would be described in an unit of time with a 
 velocity represented by PM\ whence it follows that the velocity 
 of a body at any instant may be measured by the space which it 
 would describe if it moved with that velocity unchanged for an 
 unit of time. 
 
 Measures of Variable Force, Kinetic Energy, Work of a Force. 
 
 53. When a particle of mass m is moving in a straight line 
 under the action of an uniform force F, if V, v be the velocities 
 at the beginning and end of the interval of time t, and s be 
 the space described in that time, the following equations will 
 hold : m (v - V) = Ft and \m (v' - F 2 ) = Fs. 
 
 These equations represent respectively that : 
 
 (1) The increase of momentum in a given time is equal to the 
 whole force which has acted during that time. 
 
 (2) Half the increase of vis viva, or the increase of the kinetic 
 energy in a given space is equal to the work of the force in that 
 space. 
 
 If F be a variable force, and F^ F^ be its least and greatest 
 values during the time t, m (v - F) will be greater than Ff and 
 less than F 2 t, each of which will become Ft ultimately when t 
 is indefinitely diminished ; and similarly for \m (u 2 F 2 ). 
 
 Hence we obtain two measures of variable force in the form 
 of the two limits : 
 
 (1) The quotient of the increase of the momentum by the time, 
 when the time is diminished indefinitely. 
 
 (2) The quotient of the increase of the kinetic energy by the 
 space through which the force has acted, when that space is 
 diminished indefinitely. 
 
 54. In the velocity curve, Art. 50, the velocity Qq is added 
 in the time MN, the measure of the acceleration at the time OM 
 is therefore the limit of the ratio Qq : Pq, or the trigonometrical 
 tangent of the angle which the tangent at P to the velocity curve 
 makes with the line of abscissae. 
 
 55. Geometrical representation of the momentum generated 
 
 L
 
 74 NEWTON. 
 
 by a finite and variable force acting for a finite time upon a 
 particle moving in the direction of the action of the force. 
 
 In the figure of p. 72, let OA, OB represent the times at 
 the commencement and end of the interval during which the 
 action of the force is considered. 
 
 Let AB be divided into any number of small portions, such 
 as MN, and let PM, QN, perpendiculars to AB, represent the 
 forces acting on the particle at the times OM, ON respectively, 
 and let parallelograms be constructed and the curve drawn as 
 in Art. 50. 
 
 The momentum generated in the time MN, if MN be taken 
 small enough, will be intermediate between the momenta re- 
 presented by the parallelograms PN and QM ; therefore, by 
 Lemma II., the whole increase of momentum is represented 
 by the area A CDB bounded by the curve, the line of abscissae, 
 and the ordinates at the commencement and end of the finite 
 interval of time represented by AB. 
 
 56. As in Arts. 51, 52, the measure of force given in (1) 
 Art. 53 can be deduced ; also that the force at any instant may 
 be measured by the momentum which would be generated if 
 the force were to continue unchanged for an unit of time. 
 
 57. Geometrical representation of the "kinetic energy generated 
 "by a force which acts upon a particle moving in the direction 
 of the forceps action through a finite space. 
 
 Let OAB be the line of motion of the particle, and when 
 it arrives at M let PM perpendicular to OAB represent the 
 force, and let the construction be made as before. 
 
 The increase of kinetic energy in the passage from M to 
 N is intermediate between the work done by the forces re- 
 presented by PM and QN, i.e. it is represented by an area 
 which is intermediate between PN and QM; therefore, by 
 Lemma II, the increase of kinetic energy or the work of the 
 force during the motion from A to B is represented by the 
 ssx&ACDB. 
 
 -58. The measure of force given in (2), Art. 53, is deducible 
 as before, since PM.MN^&rea. PMNq ultimately.
 
 LEMMA X. 
 
 75 
 
 59. In rectilinear motion of a particle under the action of 
 any variable force, the sum of the kinetic and potential energies 
 is constant. 
 
 If the motion of the particle be considered only within the 
 limits A, B, the area PMBD represents the whole work which 
 the force will be able to do as the particle moves from M 
 to the end of its path ; this work is called the Potential Energy, 
 and since the kinetic energy at M is represented by the area 
 CAMP, it follows that throughout the motion the sum of 
 the kinetic and potential energies is constant. 
 
 Application to the determination of the motion of a particle 
 under various circumstances. 
 
 (1) To find the space travelled over in a given time by a 
 "body moving with a velocity which varies as the square of the 
 time from the beginning of the motion. 
 
 Let AB represent the time, and let BG perpendicular to AB 
 represent the velocity at the end of that time. 
 
 Let AB be divided into any number of equal portions of 
 which MN is one, and let MP, NQ represent the velocities at 
 the ends of the times represented by AM, AN. 
 
 Then, since MP:NQ:BC:: AW : AN" 2 : AB 2 , a parabola 
 can be described touching AB and passing through P, Q, C 
 and the extremities of all ordinates by which velocities are 
 represented. 
 
 Hence the space described in the time represented by AB 
 is represented by the parabolic area AB G or ^AB.BG. 
 
 And if p be the velocity at end of 1", p? will be that at
 
 76 NEWTON. 
 
 the end of <" ; therefore ^pt*. t = %pt s will be the space described 
 in the time t. 
 
 NOTE. The following method of representing the space 
 serves to illustrate Art. 46. 
 
 Join AC, and let pM) qN be the ordinates, and suppose 
 the figure to revolve round AB, pM generates a circle whose 
 area cc p M* cc A M* therefore this circle may be taken to 
 represent the velocity at the time corresponding to AM, and 
 the solid generated by pqNM represents the space described in 
 time MN. The whole space is therefore represented by the 
 cone generated by ABC) or ^AB.trBC 2 ) which gives the same 
 result as before. 
 
 (2) To find the space described from rest at any time by a 
 particle under the action of a force ichose accelerating effect 
 varies as the m^ power of the time. 
 
 This problem is more simply solved by applying directly 
 the method of summation, since in order to find the area of 
 the curve, constructed as in Lemma X., we should eventually 
 be obliged to have recourse to that method. 
 
 Let the time t be divided into n equal intervals, and let the 
 acceleration by the force at the time t be pt m j hence, at the com- 
 
 (rt\ m 
 J , 
 
 and, if the force be continued uniform during this interval, the 
 velocity generated will be p ( j . - , and if the same arrange- 
 ment be made during each interval, the whole velocity generated 
 
 will be - ',!,.! ^ n pt* 1 ; hence, when the number of 
 
 intervals is increased indefinitely, it follows, by the reasoning 
 
 of Lemma II.. that the velocity at the time t=~ . 
 
 ?w+ 1 
 
 In the same manner, if the velocity at the commencement of 
 each interval were continued uniform during the interval, the 
 space described could be shewn to be
 
 LEMMA X. 77 
 
 whence, proceeding to the limit, the space described in the 
 time t = -, ^ 
 
 (3) To find the velocity acquired from rest, when a body is 
 acted on by an attractive force whose accelerating effect vanes 
 as the distance from a fixed point. 
 
 Let S be the fixed point, A the point from which the motion 
 commences, and let AB, perpendicular to SA, represent the 
 accelerating effect of the force at A. Join SB, and let MP, per- 
 
 ff 
 
 JIT 
 
 pendicular to SA, meet SB in P; then, since PM: BA\\ SM: SA, 
 PM represents the accelerating effect of the force at M, and the 
 square of the velocity acquired at M is represented, Art. 57, by 
 twice the area BAPM or SA.AB- SM.MP. 
 
 With centre S and radius 8 A describe a circle AQR, and 
 let MPQ, NR be ordinates at Q,R', then, if fj.D be the measure 
 of the accelerating effect of the force at a distance D, (vel.) a 
 at M = n (SA>- SM*} ; therefore the velocity at M= V(/"*) QM. 
 
 (4) Time of describing a given space from rest under the 
 action of a force varying as the distance from a fixed point. 
 The time of describing MN is ultimately, when MN is in- 
 N QR 1 
 
 j c i j- u j 
 indefinitely diminished, 
 
 x circular 
 
 measure of QSR; therefore, if ^ be the time from ^4 to M t 
 t V(A*) will b e tae circular measure of ASQ. 
 
 Let &4 = a, then the distance from S at the time t=a cos {< V(/*}j 
 
 and the velocity = a V(/*) sin 
 
 hence, when t
 
 78 NEWTON. 
 
 the particle will come to rest at the point A' on the opposite 
 side of Sj where SA' = SA, and, the time of oscillation from 
 
 rest to rest, being -r , will be independent of the distance 
 
 from which the motion commences. 
 
 (5) Simple harmonic motion. 
 
 DEF. The motion of a particle oscillating under the action 
 of a force tending to a fixed point, and varying as the distance 
 from it, is called simple harmonic motion. 
 
 From the preceding propositions the following construction 
 for simple harmonic motion, which may also be taken as a 
 definition, is obtained. 
 
 When a point Q moves uniformly in a circle, and an ordinate 
 QM is drawn from its position at any instant to any diameter 
 AA, the motion of M, the foot of the ordinate, is simple 
 harmonic motion.* 
 
 DEF. The amplitude of a simple harmonic motion is the 
 range SA or SA on each side of the centre. 
 
 The period is the time which elapses from any instant until 
 the moving point again moves in the same direction through 
 the same position. 
 
 (6) A particle is subject to the action of a force, whose accele- 
 rating effect varies as the distance from a faced point, in the 
 direction of which it acts, the particle is projected from a given 
 point in a direction perpendicular to the direction of the force at 
 that point, to find the path described by the particle. 
 
 Let the force tend to (7, and let A be the point of projection, 
 P the position of the particle at any time. 
 
 Let CB, perpendicular to CA, be the distance in which a 
 particle would be reduced to rest, if projected from C with the 
 velocity of projection ; so that if V be the velocity of projec- 
 tion, and pCP be the accelerating efiect of the force at P, 
 F* = ^05* by (3). 
 
 * Thomson's and Tait's Natural Philosophy, Art. 53.
 
 LEMMA X. 
 
 79 
 
 Describe circles Bb, Aa having the common centre C, and 
 draw CpP' cutting the circles in p and P', and draw pn perpen- 
 dicular to CB, andpm, P'M to CA. 
 
 Referring to (4) supra, it will be seen that two particles start- 
 ing respectively one from rest at A and the other with the 
 velocity of projection at (7, under the action of the same force, 
 would arrive simultaneously at M and n, since the time in both 
 cases is proportional to the angle P' CA. 
 
 But the particle in the proposed problem is acted on at P by 
 a force which is represented by PC, whose accelerating effect 
 parallel to AC and CB is represented by HC and PM, there- 
 fore the acceleration in AC is the same as that of the particle 
 supposed to move in A C from rest, and the retardation parallel 
 to BO the same as that of the particle in CB, projected 
 from C, therefore P is in the intersection of np and MP ', and 
 PM : P'M ::pm: FM ::Cp: CF ::CB:CA; therefore the re- 
 quired path of the particle is an ellipse whose semi-axes are 
 CA and CB. 
 
 COR. 1. Area ACP& area ACP'<x LACFcc time from A 
 to P, hence the area swept out by the radius vector is propor- 
 tional to the time. 
 
 COR. 2. The square of the velocity at P is the sum of the 
 squares the velocities of the particles at M and n=( J L.PM' 2 +/j,.pri t 
 ^jM.CD*, where CD is the semi-diameter conjugate to CP. 
 
 (7) The space described by a body moving in a medium, in 
 which the resistance varies as the velocity, when no other force 
 acts on the body, varies as the velocity destroyed.
 
 80 
 
 NEWTON. 
 
 Let the time AK be divided into equal intervals AB, BC, 
 CD, ... ; and let Aa, Bb', ... be the velocities at the beginning 
 of the intervals, the space in time AK is represented by the 
 area a'AKk'. 
 
 JL Jt C J) 
 
 Suppose the force of resistance to be constant throughout the 
 intervals of time AB, BG, ..., and equal to the amount at the 
 commencement of each, and let Aa, Bb, ... be the measures of 
 the retarding effect of those forces, then the velocity destroyed 
 s represented by the limit of the sum of the parallelograms 
 aB, bO, ... or the area aAKk] hence the space described and 
 the velocity destroyed vary respectively as the areas a'AKk 
 and aAKk ; and, since the resistance varies as the velocity, the 
 ratios Aa : Aa, Bb' : Bb, &c., are all equal ; therefore, by 
 Lemma IV., the areas a'AKk', aAKk are in a constant ratio ; 
 hence the space described varies as the velocity destroyed. 
 
 X. 
 
 1. If the square of the velocity of a body be proportional to 
 the space described from, rest, prove that the accelerating force is 
 constant. 
 
 2. At what point of the proof of Lemma X. is it assumed 
 that the body starts from rest? 
 
 3. State the proposition by which Lemma X. is replaced, when 
 the body, instead of starting from rest, commences its motion with 
 a given velocity. 
 
 4. If a body move from rest under the action of a force which 
 varies as the square of the time from the beginning of the motion, 
 shew that the velocity at any time will vary as the cube of the 
 time, and the space described as the fourth power of the time.
 
 LEMMA X. 81 
 
 5. If the velocity after a time t from rest be equal to a (2t + *), 
 what will be the shape of the curve in the figure, and the space 
 described in any time ? 
 
 6. If the square of the velocity of a moving point vary as the 
 time, find the space which will be described in a given time; and 
 shew that the acceleration will vary inversely as the velocity. 
 
 7. If the curve employed in the proof of the Lemma be an 
 arc of a parabola, the axis of which is perpendicular to the straight 
 line on which the time is measured, prove that the accelerating 
 effect of the force will vary as the distance from the axis of the 
 parabola. 
 
 XI. 
 
 1. If in the velocity curve of Lemma X. there should occur 
 a point where the two parts of the curve cut one another at a 
 finite angle, what would be the interpretation of this singularity ? 
 Explain also what a point of inflexion would imply. 
 
 2. A particle is placed in the line joining two centres of 
 attracting force, the accelerating effect of each of which varies as 
 the distance, find the time in which the particle oscillates. 
 
 3. When a body moves from rest at A under the action of a force 
 which varies as the square of the distance from S (= p . SM* at Jf), 
 the square of the velocity at M = $p (SA 3 - SM 3 ). 
 
 4. If a body be acted on from rest by a repulsive force which 
 varies as the distance from a fixed point, find the velocity when the 
 body arrives at any position. 
 
 5. Two points move from rest in such a manner that the ratio 
 of the times in which the same uniform acceleration would generate 
 their respective velocities at those times is constant. Shew that 
 their respective accelerations, at any time bearing that ratio, are 
 equal. 
 
 6. Two forces reside at S, one attractive and whose accelerating 
 effect on a particle varies as the distance from S, and the other 
 constant and repulsive ; prove that, if a particle be placed at S, 
 it will move until it be brought to rest at a point which is double 
 the distance from S at which it would rest in equilibrium under the 
 action of the forces. 
 
 7. A particle moves from rest at A under the action of a force 
 tending to S, and varying as the distance from S, and in its path 
 towards S it strikes another particle of equal mass at rest at B ; 
 prove that, if the particles be perfectly elastic, they will meet again 
 on the opposite side of S at a distance equal to SB, and continue 
 to impinge at B and B' for ever. 
 
 M
 
 82 
 
 NEWTON. 
 
 LEMMA XL 
 
 The vanishing subtenses of the angle of contact, in all curves 
 which have finite curvature at the point of contact, are 
 ultimately in the duplicate ratio of the chords of the con- 
 terminous arcs. 
 
 Case 1. Let AB be the arc of a curve, AD its tangent at 
 A, BD the subtense of the angle of contact, BAD, per- 
 pendicular to the tangent, AB the chord of the arc. 
 
 Draw AG, EG perpendicular to the tangent AD and 
 the chord AB respectively, meeting in G ; then let 
 
 the points Z>, B, G move towards the points d, I, g, 
 and let / be the point of ultimate intersection of the 
 lines BG, AG, when the points B, D move up to A. 
 
 It is evident that the distance GI may be made less 
 than any assigned distance by diminishing AB. 
 
 But, since the angles ABD and GAB are equal, and 
 also the right angles BDA, ABG, the triangles ABD, 
 GAB are similar; therefore BD : AB : : AB : AG, 
 or BD.AG = AB\ and, similarly, bd.Ag = Afr-, 
 .'. AB 9 : AV = BD.AG : Id.Ag ; 
 
 therefore the ratio AB* : Atf is a ratio compounded 
 of the ratios of BD : b d and A G : Ag.
 
 LEMMA XI. 83 
 
 Bat, since GI may be made less than any assigned 
 length, the ratio AG : Ag may be made to differ from 
 a ratio of equality less than by any assigned dif- 
 ference ; therefore the ratio A& : Atf may be made 
 to differ from the ratio BD : bd less than by any 
 assigned difference. 
 
 Hence, by Lemma I., the ultimate ratio AB* : Ab* is the 
 same as the ultimate ratio of BD : bd. Q. E. D. 
 
 Case 2. Let now the subtenses BD, bd' be inclined at 
 any given angle to the tangent; then, by similar 
 triangles D'BD, d'bd', BD' :bd' :: BD : bd, but ulti- 
 mately BD : bd :: AB* : Ab* ; therefore ultimately 
 BD :bd' :: AB* : Ab\ Q.E.D. 
 
 Case 3. And, al though the angle D be not a given 
 angle, if BD converge to a given point, or be drawn 
 according to any other [fixed] law [by which the 
 angle D remains finite, since BD' is a subtense], still 
 the angles D', d', constructed by this law common 
 to both, will continually approach to equality and 
 become nearer than by any assigned difference, and 
 will be therefore ultimately equal, by Lemma I., 
 and hence BD, bd' will be ultimately in the same 
 ratio as before. Q.E.D. 
 
 COR. 1. Hence, since the tangents AD, Ad, the arcs 
 AB, Ab and their sines BC, be become ultimately 
 equal to the chords AB, Ab, their squares also will 
 be ultimately as the subtenses BD } bd. 
 
 COR. 2. The squares of the same lines also will be 
 ultimately as the sagittse of the arcs, which bisect 
 the chords, and converge to a given point; for those 
 sagitta3 are as the subtenses BD, bd. 
 
 COR. 3. And therefore the sagittse will be ultimately 
 in the duplicate ratio of the times in which a body 
 describes the arcs with a given velocity. 
 
 COR. 4. The rectilinear triangles ADB, Adb are ulti- 
 mately in the triplicate ratio of the sides AD, Ad y
 
 84 NEWTON. 
 
 and in the sesqui plicate ratio of the sides DB, db ; 
 since these triangles are in the ratio compounded of 
 AD : Ad and BD : bd. So also the triangles ABC, 
 Abe will be ultimately in the triplicate ratio of the 
 sides BC, be. The sesquiplicate ratio may be re- 
 garded as the subduplicate of the triplicate, or as 
 compounded of the simple and the subduplicate 
 ratios. 
 
 COR. 5. And, since DB, db are ultimately parallel and 
 in the duplicate ratio of AD, Ad [therefore, this 
 being a property of a parabola,] at every point at 
 which a curve has finite curvature an arc of a parabola 
 can be drawn which will ultimately coincide with the 
 curve ; and the curvilinear areas ADB, Adb will be 
 ultimately two-thirds of the rectilinear triangles 
 ADB, Adb', and the segments AB, Ab the third 
 parts of the same triangles. And hence these areas 
 and these segments will be in the triplicate ratio as 
 well of the tangents AD, Ad as of the chords and 
 arcs AB, Ab. 
 
 SCHOLIUM. 
 
 But, in all these propositions, we suppose the angle of 
 contact to be neither infinitely greater nor infinitely 
 less than the angles of contact which circles have 
 with their tangents ; that is, that the curvature at 
 the point A is neither infinitely great nor infinitely 
 small; in other words, that the distance AI is of 
 finite magnitude. 
 
 For DB might be taken proportional to AD 3 , in which 
 case no circle could be drawn through the point A 
 between the tangent AD and the curve AB, and the 
 angle of contact would be infinitely less than that 
 of any circle. 
 
 And, similarly, if different curves be drawn in which 
 DB varies successively as AD*, AIf,AD*, &c., a series 
 of angles of contact will be presented which may be 
 continued to an infinite number, of which each will
 
 LEMMA XI. 85 
 
 be infinitely less than the preceding. And if curves 
 be drawn in which DB varies as AD*, AD$, AD , 
 ADI, AD*, &c., another infinite series of angles of 
 contact will be obtained, of which the first will be 
 of the same kind as in the circle, the second infinitely 
 greater, and each infinitely greater than the pre- 
 ceding. But, moreover, between any two of these 
 angles an infinite series of other angles of contact 
 can be inserted, of which each may be infinitely 
 greater or infinitely less than any preceding; for 
 example, if between the limits AD 2 and AD 3 there 
 be inserted ADV,ADV t AD*, AD\ AD$, AD$,ADV, 
 A D, AD'*, &G. And, again, between any two angles 
 of this series there can be inserted a new series of 
 intermediate angles differing from one another by 
 infinite intervals. Nor does the nature of the case 
 admit any limit. 
 
 The propositions which have been demonstrated con- 
 cerning curved lines and the included areas are easily 
 applied to curved surfaces and solid contents. 
 
 These Lemmas have been premised for the sake of 
 escaping from the tedious demonstrations by the 
 method of reductio ad absurdum, employed by the old 
 geometers. The demonstrations are certainty ren- 
 dered more concise by the method of indivisibles ; 
 but, as there is a harshness in the hypothesis of indi- 
 visibles, and on that account it is considered to be 
 an imperfect geometrical method, it has been pre- 
 ferred to make the demonstrations of the following 
 propositions depend on the ultimate sums and ratios 
 of vanishing quantities and on the prime sums and 
 ratios of nascent quantities, i.e. on the limits of sums 
 and ratios ; and therefore to premise demonstrations 
 of those limits as concise as possible. By these 
 demonstrations the same results are deducible as by 
 the method of indivisibles ; and we may employ the 
 principles which have been established with greater 
 safety. Consequently, if, in what follows, quantities
 
 86 NEWTON. 
 
 should be treated of as if they consisted of particles 
 [indefinitely small parts], or small curve lines should 
 be employed as straight lines, it would not be in- 
 tended to convey the idea of indivisible, but of 
 vanishing divisible quantities, not that of sums and 
 ratios of determinate parts, but of the limits of sums 
 and ratios ; and it must be remembered that the force 
 of such demonstrations rests on the method exhibited 
 in the preceding Lemmas. 
 
 An objection is made, that there can be no ultimate 
 proportion of vanishing quantities ; inasmuch as 
 before they have vanished the proportion is not 
 ultimate, and when they have vanished it does not 
 exist. But by the same argument it could be main- 
 tained that there could be no ultimate velocity of a 
 body arriving at a certain position at which its 
 motion ceases ; for that this velocity, before the body 
 arrives at that position, is not the ultimate velocity ; 
 and that, when it arrives there, there is no velocity. 
 And the answer is easy : that, by the ultimate velo- 
 city is to be understood that, when the body is 
 moving, neither before it reaches the last position 
 and the motion ceases nor after it has reached it, 
 but at the instant at which it arrives ; i. e. the very 
 velocity with which it arrives at the last position and 
 with which the motion ceases. 
 
 And, similarly, by the ultimate ratio of vanishing 
 quantities is to be understood the ratio of the quan- 
 tities, not before they vanish nor after, but with which 
 they vanish. Likewise, also, the prime ratio of nas- 
 cent quantities is the ratio with which they begin to 
 exist. And a prime or ultimate sum is that with ivhich 
 it begins to be increased or ceases to be diminished. 
 
 There is a limit which the velocity can attain at the 
 end of the motion, but cannot surpass. This is the 
 ultimate velocity. And the like can be stated of 
 the limit of all quantities and proportions com- 
 mencing or ceasing to exist. And, since this limit
 
 LEMMA XI. 87 
 
 is certain and definite, to determine it is strictly a 
 geometrical problem. And all geometrical propo- 
 sitions may be legitimately employed in determining 
 and demonstrating other propositions which are 
 themselves geometrical. 
 
 It may also be argued that, if the ultimate ratios of 
 vanishing quantities be given, the ultimate magni- 
 tudes will also be given, and thus every quantity 
 will consist of indivisibles, contrary to what Euclui 
 has demonstrated of incommensurable quantities, in 
 his tenth book of the Elements. 
 
 But this objection rests on a false hypothesis. Those 
 ultimate ratios with which quantities vanish are not 
 actually ratios of ultimate quantities, but limits to 
 which the ratios of quantities decreasing without 
 limit are continually approaching ; and which they 
 can approach nearer than by any given difference, 
 but which they can never surpass, nor reach before 
 the quantities are indefinitely diminished 
 
 The argument will be understood more clearly in the 
 case of infinitely great quantities. If two quantities, 
 of which the difference is given, be increased infi- 
 nitely, their ultimate ratio will be given, namely, a 
 ratio of equality, yet in this case the ultimate or 
 greatest quantities of which that is the ratio will 
 not be given. 
 
 In what follows, therefore, if at any time, for the sake 
 of facility of conception, the expressions indefinitely 
 small, or vanishing, or ultimate be used concerning 
 quantities, care must be taken not to understand 
 thereby quantities determinate in magnitude, but to 
 conceive them in all cases quantities to be diminished 
 without limit. 
 
 Curvature of Curves. 
 
 60. The curvature of a curve at any point is greater or less 
 as the amount of deflection from the tangent at that point, in 
 the immediate neighbourhood of the point, is greater or less.
 
 88 NEWTON. 
 
 Two curves will have the same curvature at two points, taken 
 one in each, if at equal distances from the points of contact, in 
 the immediate neighbourhood of the points, they have the same 
 deflection from the tangents at those points. 
 
 61. An exact geometrical test of equality of curvature may 
 be obtained as follows : 
 
 If AB, ab be two curves which have the same curvature at 
 A, a respectively, draw the tangents AO, ac and take AC=ac. 
 
 A C ft- & 
 
 Draw subtenses BG, be inclined at equal angles to the tangents. 
 
 If BG and be were equal, for all equal values of AC, ac, the 
 curves would be equal and similar, if BG : be be ultimately 
 a ratio of equality, when AC, ac are taken indefinitely small, 
 the curves will have the same deflection from the tangents in the 
 immediate neighbourhood of A, a, or the curves will have the 
 same curvature at those points. 
 
 If the chords AB, ab be drawn, it will be an immediate con- 
 sequence that the ultimate ratio of the angles BA C, bac will be 
 a ratio of equality. These angles are called the angles of contact. 
 
 Hence, curves will have the same curvature at two points, 
 one in each, if, equal tangents being drawn at those points, 
 and subtenses inclined at any equal angles to the tangents, the 
 limiting ratio of the subtenses be a ratio of equality, or if the 
 limiting ratio of the angles of contact be a ratio of equality. 
 
 62. The curvature of one curve will be infinitely greater or 
 infinitely less than that of another if the limiting ratio of the 
 subtense of the first to that of the second be infinitely great 
 or infinitely small. 
 
 63. The ratio of the curvature ot one curve to that of 
 another at two points, or of the curvature of the same curve at 
 two different points, is the limiting ratio of the subtenses drawn 
 from the extremities of equal tangents and inclined at equal 
 angles to the tangents.
 
 LEMMA XT. 
 
 64. The curvature of a curve is said to be finite, at any 
 point, when the ratio of the curvature at that point to that of 
 any circle whose radius is finite, is a finite ratio. 
 
 65. The curvature of a circle is the same at every point. 
 
 Let A, a be any two points on a circle, AC, ac equal tan- 
 gents at A, a, Oft, cb subtenses perpendicular to the tangents, 
 OD, Od perpendicular to the subtenses produced; therefore 
 CD = cd, each being equal to the radius, and BD = bd] hence 
 BC=lc always, and therefore ultimately, when the arcs are 
 indefinitely diminished, BC : be is a ratio of equality ; therefore 
 
 the circle has the same curvature at any two points. 
 
 66. In different circles the curvatures vary inversely as the 
 radii. 
 
 In the last figure, produce CB to the circumference in E. 
 Then, AC*=CB. CE] also, if A'C' be a tangent to another circle, 
 and A'C 1 be taken equal to A (7, and the same construction be 
 made, A'C'^^C'B'.C'E' ; therefore CB.CE=C'B'.C'E\ and 
 CB-.C'B' :: C'E 1 : CE; and when AC, AC' are indefinitely 
 diminished, CE=%AO ; therefore CB : C'B' :: A'O' : AO, ulti- 
 mately, or the curvatures are inversely proportional to the radii. 
 
 Measure of Curvature. 
 
 67. The curvature of a circle is the same at every point ; 
 the curvatures of different circles vary inversely as the diameters 
 
 N
 
 90 NEWTON. 
 
 of the circles ; and a circle can be constructed of any degree ? 
 finite curvature by varying the magnitude of the diameter. 
 
 Hence, a circle can always be found whose curvature at any 
 point is equal to that of a curve at a fixed point. 
 
 The curvature of a curve at any point is therefore completely 
 determined when the diameter of the circle is found, which has 
 the same curvature as the curve at the given point. 
 
 The diameter of the circle, which has the same curvature as 
 the curve at a given point, is called the diameter of curvature of 
 the curve at that point. 
 
 The chord of the circle, drawn in any direction, is called the 
 chord of curvature in that direction. 
 
 The circle itself is called the circle of curvature, and is the 
 circle which has the same tangent as the curve at any point, and 
 also the same curvature. 
 
 68. Any other curve might have been chosen to establish a 
 standard measure of finite curvature; but, since no curve but 
 the circle has the same curvature at every point, it would then 
 have been necessary, after selecting the curve, to specify the 
 point, the curvature at which might be made the measure of 
 curvature. 
 
 Thus, if the standard curve were a parabola, we must choose 
 the curvature of the parabola at the vertex or at the extremity 
 of the latus rectum or at some determinate point, by which to 
 obtain the measure. 
 
 The inconvenience is obvious. 
 
 General Properties of the Circle of Curvature. 
 
 69. If a circle be drawn touching a curve at a given point, 
 and cutting it at a second point, as the second point approaches 
 indefinitely near the point of contact, the circle will assume a 
 limiting magnitude, and will evidently satisfy the condition of 
 having the same curvature as the curve at that point. 
 
 70. Since a tangent at any point is the limiting position 
 of a side, terminated in that point, of a polygon inscribed in 
 the curve, when the number of sides is increased indefinitely,
 
 LEMMA XI. 91 
 
 so the circle of curvature at any point is the limiting circle 
 which passes through three consecutive angular points of the 
 polygon, one of which coincides with the point. 
 
 71. No circle can be drawn whose circumference lies between 
 a curve and its circle of curvature, in the neighbourhood of the 
 point at which the circle of curvature is drawn. 
 
 For, let AQ be the arc of the curve, Aq of the circle of 
 curvature ; and let, if possible, another circle be drawn, of which 
 the arc AS lies between the curve and circle, and having there- 
 fore the same tangent A R at A ; and let R Q, the subtense per- 
 pendicular to the tangent, cut the circles in S, q. 
 
 Then SR : qR will be ultimately in the inverse ratio of the 
 diameters of the circles; therefore SR will be ultimately unequal 
 to qR; but, since qR and QR are ultimately in a ratio of 
 equality, ififi, which is intermediate in magnitude, will be ulti- 
 mately equal to either, which is absurd ; therefore no circle, &c. 
 
 This proposition corresponds to Euclid III., Prop. XVI. 
 
 72. The circle of curvature generally cuts the curve. 
 
 For the curvature of the curve at different points taken along 
 the curve continually increases or continually diminishes, until 
 it arrives at a maximum or minimum value. 
 
 If therefore the circle of curvature be drawn at any point, 
 on the side on which the curvature is increasing, as we proceed 
 from the point, the curve will lie within the circle, and on the 
 other side, on which the curvature is diminishing, the curve will 
 lie without the circle; which proves the proposition for the 
 general position of the point. 
 
 For the particular case, in which the point is at a position 
 of maximum or minimum curvature, as at the extremities of the 
 axes of an ellipse, if the curvature be a maximum the curvature 
 at adjacent points on either side will be less than that of the
 
 92 NEWTON. 
 
 circle of curvature at the point under consideration ; therefore 
 the circle will lie entirely within the curve on both sides near 
 the point of maximum curvature; and, similarly, it will lie 
 without the curve at points of minimum curvature. 
 
 We can illustrate this by reference to the polygon inscribed 
 in the curve ; see the figure in the following page. 
 
 If, in the curve, equal chords AB, BC, CD, DE r .. be placed 
 in order, generally the angles ABC, BCD, CDE r .. will increase 
 or decrease, commencing from any point, which property of the 
 polygon will have in the curvilinear limit, when the chords are 
 diminished indefinitely, the corresponding property, that the 
 curvature decreases or increases continually. 
 
 Suppose the angles are increasing from Bj make the angles 
 CBA, CDE' equal to the angle BCD, and BA, DE' equal to 
 BC, CD...] then a circle through B, C, D will pass also through 
 A and E', and these points will be on opposite sides of the 
 perimeter of the polygon, whence, if we proceed to the limit, 
 the circle of curvature at a point in the middle of increasing 
 curvature will cut the curve. 
 
 If the angles ABC and DEF be each less than the angles 
 BCD, CDE, supposed equal, the curvature will decrease and 
 then increase, and the circle about BCD will pass through E, 
 and BA, EF will lie within the circle, and, proceeding to the 
 limit, the circle of curvature will lie without the curve, near 
 the point of minimum curvature, 
 
 Evolute of a Curve. 
 
 73. DEF. If the circles of curvature be drawn at every 
 point of a curve, the centres of those circles will lie in a curve 
 which is called the evolute of the proposed curve. 
 
 Properties of the Evolute. 
 
 74. The extremity of a string unwrapped from the evolute of 
 a curve traces out the curve. 
 
 Let ABODE be any equilateral polygon, and let a'a, Vb, c 'c 
 dd be drawn perpendicular to the sides from the middle points
 
 LEMMA XI. 93 
 
 a', &', &c., these intersect in the angular points abed... of another 
 polygon. 
 
 If a string were wrapped round a abed... the extremity a' 
 would as the string was unwrapped pass through the points 
 a'b'c'd". 
 
 Let now the number of sides of the polygon be increased and 
 the magnitude diminished indefinitely. 
 
 The points a'b'c... will be ultimately in the curve which is 
 the limit of the polygon, and since a, &, c... are the centres 
 of the circles described about ABC, BCD,... , a, &, c,... will be 
 ultimately the centres of the circles of curvature at a'b'c'... , and 
 
 the curve, which is the limit of the polygon abed..., will be the 
 evolute of the curve a'b'c... , and the property proved for the 
 polygons will be true for the limits of the polygons, therefore 
 the extremity of the string unwrapped from the evolute will 
 trace the curve of which it is the evolute. This property gives 
 rise to the name of evolute. 
 
 DEF. The curves formed by the unwrapping of a string 
 from a curve are called involutes. 
 
 75. The tangent to the evolute of a curve is a normal to the 
 curve. 
 
 Since b'b is ultimately the tangent to the evolute and is 
 perpendicular to BC, which is ultimately the tangent to the 
 curve a'b'c... , therefore the tangent to the evolute is a normal 
 to the curve.
 
 94 NEWTON. 
 
 Propositions on Diameters and Chords of Curvature. 
 
 76. If a subtense le drawn from the extremity of an arc 
 of finite curvature, in any direction, the chord of curvature 
 parallel to that direction will be the limit of the third pro- 
 portional to the subtense and the arc. 
 
 Let PQ, Pq be arcs of a curve and its circle of curvature 
 at P, let PR be the common tangent, and RQq the direction 
 of a common subtense, meeting the circle in U. 
 
 Draw the chord PV parallel to E Q. Then, since Rq.R U=PIF, 
 RU is the third proportional to Rq and PR. 
 
 But, ultimately, when PQ is indefinitely diminished, RU=PV, 
 and PR = PQ, by Lemma VII. also, Rq = RQ\)j the property 
 of the circle of curvature. 
 
 Therefore PV is the limit of the third proportional to RQ 
 and PQ. 
 
 COR. The diameter of curvature is the limit of the third pro- 
 portional to the subtense perpendicular to the tangent and the arc. 
 
 77. The two chords of curvature at any point of a parabola 
 drawn through the focus, and in the direction of the diameter, are 
 each equal to four times the focal distance of that point. 
 
 Let AP be a parabola, P any point, RQ a subtense parallel 
 to the diameter PMx, QM the ordinate at Q, S the focus. 
 Then, by a property of the parabola, QM* = SP.PM-, there- 
 fore 4&P is a third proportional to PM and QM, i.e. to RQ 
 and PR. 
 
 Hence, 4/SP is the limit of the third proportional to the
 
 LEMMA XI. 95 
 
 subtense QR and the arc PQ, and is therefore equal to the 
 chord of curvature at P in direction of the diameter. 
 
 L y \ 
 
 And, since PS, PM are equally inclined to the tangent at P, 
 the chords in those directions are equal ; therefore the chord of 
 curvature through S is four times the focal distance SP. 
 
 78. One-fourth of the diameter of curvature at any point 
 of a parabola is a third proportional to the perpendicular from 
 the focus on the tangent at that point, and the focal distance of 
 that point. 
 
 For, draw SY, QR perpendicular to PR, and let PI be the 
 diameter of curvature at P. 
 
 Then PI. QR = P# 2 = PB" ultimately, = 4P. QE ; 
 
 .-. PI: ISP:: QR : QR :: SP: SY; 
 
 since the triangles SYP, QRR are similar; therefore |P7 is 
 a third proportional to SY and SP. 
 
 79. The chord of curvature at any point of an ellipse drawn 
 through the centre of the ellipse is a third proportional to the 
 diameter through that point and the diameter conjugate to it. 
 
 Let P be any point in an ellipse, PGG the diameter, DCD' 
 conjugate to it, Q any point near P, QR a subtense parallel 
 to CP, QM an ordinate parallel to DC, PV the chord of curva- 
 ture drawn through C. 
 
 Then PV. QR = PQ* = QM* ultimately, 
 
 and QM* : PM.MG ::CD*: CP 2 ; 
 .-. PV.QR : QR.MG :: CD 2 : CP 2 ultimately;
 
 96 
 
 NEWTON. 
 
 .-. PF:2CP:: CD 3 : CP* ultimately; 
 
 .-. PV.GP: CP* :: 2CZ7 1 : OP 2 , 
 
 and PF.CP=2OD 2 ; 
 
 or PF is a third proportional to PG and DCD'. 
 
 80. !7^e cfor d of curvature at any point through tne j ocus is 
 a third proportional to the major axis, and the diameter parallel 
 to the tangent at that point. 
 
 Draw the focal distance SP cutting the diameter DCD' in E, 
 let PF' be the chord of curvature through 5, and draw the 
 subtense QR' parallel to SP. 
 
 Then PF' : PF: : QR : QR ultimately 
 
 : : CP : PE by similar triangles ; 
 .-. PV'.PE=PV.CP=2CD*; 
 .'. PF' is a third proportional to 2PE and DCD', 
 and 2PE is equal to the major axis. 
 Similarly for the other focus H. 
 
 81. The diameter of curvature at any point is a third pro- 
 portional to twice the perpendicular from the point on the diameter 
 parallel to the tangent and that diameter. 
 
 Draw QR' perpendicular to the tangent, and PF perpen- 
 dicular to DCD', and let PI be the diameter of curvature. 
 PI:PV:: QR : QR" :: CP: PF; 
 
 .-. P7.P/7=PF.<7P=2OZ)*; 
 /. PI is a third proportional to 2PFand DCD'.
 
 LEMMA XI. 
 
 97 
 
 82. Since the chord of curvature in any direction varies 
 inversely as the subtense QR drawn in that direction, it is easily 
 seen that, if PL be the portion of the chord intercepted between 
 P and DCD 1 , the chord of curvature at P in the direction PL 
 will be the third proportional to 2PL and DCD'. 
 
 83. The propositions concerning the chords and diameter 
 of curvature of an ellipse may be proved in the same words for 
 the hyperbola, employing the following figure. 
 
 84. The radius of curvature at any point of a conic section 
 is to the normal in the duplicate ratio of the normal to the semi- 
 fatus rectum. 
 
 Let PK be the normal, PO the radius of curvature at P, 
 L the semi-latus rectum, 
 (i) For the parabola, 
 
 PO:2SP:: SP: SY:: SY : SA; 
 .-. PO-.2SY:: SP : SA : : 1SP.SA : U; 
 but PK=2SY; .'. PK* = SP.SA-, .: PO : PK : : PK 1 : L\ 
 (ii) For the ellipse or hyperbola, 
 
 PO.PF= CD' and PK.PF=BC*; 
 
 .-. PO : PK:-. CD* : BC* :: AC* : PF'-, 
 
 but PF.PK=C*=--L.AC; .: AC : PF:: PK : L; 
 
 .-. PO: PK::PK*: L\ 
 

 
 NEWTON. 
 
 85. To find the chord common to a conic section and the 
 circle, of curvature at any point. 
 
 If a circle intersect a conic section in four points, as PQ UR^ 
 and these points be joined in pairs by two lines, these lines will 
 
 be equally inclined to the axis of the conic section. Thus, in 
 the conic section, PQ, E U are equally inclined to the axis. 
 
 For, if UR, QP intersect in 0, OR.OU= OP.OQ, hence 
 the diameters of the ellipse parallel to UR, QP will be equal, 
 and therefore equally inclined to the axis. 
 
 Let Q and R move up to and ultimately coincide with P, 
 then the intersecting circle becomes the circle of curvature at P, 
 and PQ is in the direction PT of the tangent, ultimately, and 
 RU assumes the position of the chord common to the conic 
 section and the circle of curvature at P. Hence, if PV be 
 drawn at an equal inclination with PT to the axis, PV will be 
 the common chord required. 
 
 And, if VI be drawn perpendicular to PF", meeting the 
 normal at P in /, PI will be the diameter of curvature at P. 
 
 86. To find the radius of curvature of a curve defined by 
 the relation between the radius vector and the perpendicular from 
 the pole on the tangent. 
 
 Let PY", PP' Y' be the directions of consecutive sides of a 
 polygon inscribed in a curve, SY, SY' perpendiculars on these
 
 LEMMA XI. 99 
 
 sides; draw PO, P'O perpendicular to the same sides, inter- 
 
 secting in 0, and P'U perpendicular to SP, and let SY, PY' 
 intersect in W. 
 
 A semicircle on SP as diameter passes through Y and Y' ; 
 
 .-. L YPW=Z. Y8Y' = LPOP', and L WYP=L OP'P', 
 
 therefore the triangles POP', WPY are similar; 
 
 .-. PO'.PF ::PW: YW, 
 
 also PP':SP::PU:PY, 
 
 by similar triangles P'UP, SY'P, and PW=PY' ultimately; 
 .% PO : SP:: PIT: YW:: SP ~ SP' : SY ~ SY' ultimately. 
 Also, if PV be the chord of curvature through S, 
 
 PV-.2PO:: SY: SP', 
 .'. PV: 2SY:: SP ~ SP' : SY ^ SY 1 ultimately. 
 
 Observations on the Lemma. 
 
 87. In the proof of Lemma XI., AI is the limit of the 
 third proportional to BD and AS, hence it is the diameter of 
 curvature of the curve at A. 
 
 88. For an example of a law according to which, in Case 3, 
 the directions of the subtenses may be determined, we may 
 suppose that they always pass through a point given in position 
 at a finite distance from A, or that they always touch a given 
 curve; but it must be observed that the case in which they
 
 100 NEWTON. 
 
 touch a curve which has the same tangent AD at A is excluded, 
 since in this case the angles D', d' do not in the limit remain 
 finite, a property required in the name subtense. 
 
 89. DEF. If a line be drawn from the middle point of an 
 arc of a curve, making a finite angle with the chord, the part 
 intercepted between the chord and the arc is called the sagitta 
 of the arc. 
 
 90. The sagitta of an arc is ultimately one quarter of the 
 subtense drawn at the extremity of the arc parallel to the sagitta. 
 
 Let the sagitta FE bisect the arc AB in E, and be pro- 
 duced to the tangent at A in G, and let BD be a subtense 
 parallel to FE. 
 
 9- 
 
 Then EG : BD :: AE* : AB* ultimately ; . 
 
 also BD:FG::AD:AG::AB:AE ultimately ; 
 .. BD = 2FG = EG', hence FE=EG = %BD ultimately. 
 
 91. COR. 5. The parabola mentioned in this corollary is a 
 parabola of curvature at that point ; for, since DB is taken in 
 any given direction, the proportion BD : bd :: AD 1 : Ad' 2 proves 
 that the curve is ultimately in the form of a parabola, and that, 
 therefore, the line through A drawn in the given direction is the 
 corresponding diameter of the parabola of curvature. 
 
 Hence the axis of the parabola may be taken in any as- 
 signed direction. 
 
 If the subtenses be perpendicular to the tangent, the parabola 
 of curvature will be the parabola whose curvature at the vertex 
 will determine the curvature of the curve, since the axis will be 
 perpendicular to the tangent, and if 4/1 Z7, in the figure page 104, 
 be the third proportional to the subtense and arc, the limiting 
 position of U will be the focus of the parabola. 
 
 By means of this corollary the proposition alluded to under 
 Lemma IX., Art. 44, is established ; viz. that the ratio of the
 
 LEMMA XI. 101 
 
 areas which takes place of the duplicate ratio, obtained in that 
 Lemma, is the triplicate ratio of the same lines, when the line, 
 AE, instead of cutting the tangent at a finite angle, coincides 
 with the tangent. 
 
 92. Scholium. Let AB, AC be two curves, having a 
 common tangent AD at A, and let subtenses DB, DBG of the 
 
 angles of contact be drawn from D at any point in the tangent 
 in the same direction, and let BD cc AD m , CD oc AD" in the 
 curves AB, AC respectively. Draw dbc a common ordinate 
 from a fixed point d, parallel to DBC. Then . ' 
 
 AD m : Ad m :: BD : Id, 
 
 and AD n : Ad n :: CD : cd, 
 
 and if m be greater than n, =n + r suppose, 
 
 AD\AD r : Ad n .Ad T :: BD : Id', 
 .: CD.AD r : cd .Ad r :: BD : Id 
 
 ::BD.AD r : ld.AD r -, 
 .'. CD : BD :: cd .Ad r :: Id .AD", 
 
 and since b, c, d are fixed, and AD vanishes in the limit, there- 
 fore CD is indefinitely greater than BD ; also, since the angles 
 of contact BAD, CAD are ultimately proportional to BD, CD, 
 it follows that, if in two curves the subtenses vary according 
 to different powers of the arcs or tangents, the angle of contact 
 of that curve in which the index of the power is the least will 
 be infinitely greater than the angle of contact of the other. 
 
 Illustrations. 
 
 (1) Two tangents AT, BT are drawn at the extremities of 
 an arc AB, to prove that AT is ultimately equal to BT, when AB 
 is indefinitely diminished.
 
 102 
 
 NEWTON. 
 
 Draw TGUV in any direction making a finite angle with the 
 tangents, and meeting the circles of curvature at A and B in UV. 
 
 Then since the circle of curvature at A is the limit of the circle 
 which passes through C and has the tangent AT at A t and 
 similarly for that at B, we have ultimately 
 
 TA* : TB*:: TO.TU: TO. TV, 
 and TU= TV ultimately ; /. TA=TB ultimately. 
 
 COR. If BD be any subtense of the arc AB, 
 
 AT+TB=AB = AD ultimately ; 
 therefore AD will be ultimately bisected by the tangent BT. 
 
 (2) If BT be a tangent at 5, AB, BC equal chords of a 
 curve of finite curvature^ drawn from B^ and AB be produced 
 to c, making Bc = AB, and Cc be joined meeting BT in T,cT 
 will ultimately be equal to CTj when the arcs AB, GB are 
 diminished indefinitely. 
 
 Let A U be drawn parallel to CT, meeting the tangent at B 
 in Z7, and let two circles touch UBT at B and pass one through 
 
 A and the other through C, and let BV, BV be chords of these 
 circles drawn parallel to AU or CT, then AU.BV=AB\ and
 
 LEMMA XI. 103 
 
 CT.BV' = BC*; but BV=BV ultimately, since the two circles 
 are each ultimately the circle of curvature at B and AB=BC, 
 therefore A U= CT ultimately. 
 
 Through B draw RBR 1 parallel to AC, meeting AU in R' 
 and Cc in R, then R'U=RT, therefore 2R T is the difference 
 between AU and C T, hence RT ultimately vanishes compared 
 with CT, and since CR = Re, therefore CT= Tc ultimately. 
 
 (3) If) from the point of contact of a curve with its tangent* 
 equal distances be measured along the curve and tangent, the line 
 joining their extremities will ultimately be parallel to the normal 
 at the point of contact. 
 
 In the last figure, let, BC, BT be equal distances, measured 
 along the arc and the tangent ; join CT, let the tangent at (7 
 meet ^Tin D, produce BT to ^making DF=DC, take BE= 
 the chord BC, and join EC, TC, and FG. 
 
 Since the arc BC is intermediate in magnitude between 
 BD + DC and BC, therefore, BT being equal to arc BC, the 
 point T lies always between E and F. But the triangles BCE, 
 DCF being both isosceles, each of the angles EEC, BFC will 
 ultimately be a right angle, therefore the angle BTC, which is 
 less than BEG and greater than BFC, will also ultimately be 
 a right angle. 
 
 Hence CT will ultimately be parallel to the normal at B. 
 
 NOTE. In order to shew the danger of falling into an error 
 by a careless employment of the propositions proved in the 
 first section, the following fallacious proof may be noticed of 
 the above proposition. 
 
 In the figure page 102, join BC, then BT: CB will be 
 ultimately a ratio of equality, by Lemma VII ; therefore CBT 
 being an isosceles triangle ultimately, CT will be perpendicular 
 to the line bisecting the angle CBT, and therefore to the 
 tangent BT, since .Z?Tand BC will ultimately coincide with the 
 bisecting line. 
 
 The fact is that Lemma VII. only allows us to assert that 
 BT and the chord BC differ by a quantity Tt, which vanishes 
 compared with either of them, and therefore Tt may cc BC*; 
 but, by Lemma XI, CTcc BC 2 ; hence Tt : CT may possibly
 
 104 NEWTON. 
 
 be a finite ratio, or C T may be ultimately inclined at any finite 
 angle to B T, at least as far as the reasoning given in the above 
 proof is concerned. 
 
 (4) To construct for the focus of the parabola of curvature 
 whose axis is in a given direction. 
 
 C D 
 
 Let AB be a curve of finite curvature, BD, Id subtenses 
 parallel to AE the given direction. Draw A U perpendicular to 
 AD, and AS making angle UAS = UAE; then since ^^is a 
 diameter of the parabola by Art. 91, AS is in the direction of 
 the focus. 
 
 Also, if kAS be taken a third proportional to BD and AD, 
 the limiting position of S will be the focus of the parabola. 
 
 (5) To find the locus of the focus of the parabola of curvature, 
 when its axis changes its direction. 
 
 Let BO be perpendicular to AD, and A 7 be chosen so that 
 4AU.BC = AC*, then the limiting position of Z7is the focus of 
 the parabola whose curvature at the vertex is the same as that 
 of the curve at A ; also, if S be the focus of the parabola whose 
 axis is parallel to DB, 4.AS.DB = AD' = AC 3 , ultimately; 
 therefore AU: AS:: BD: BC, and / SAU = L DBG', hence 
 if we join SU, the triangles SAU, CBD will be similar, and 
 LASU=/.BCD = * right angle ; therefore the locus of S is 
 a circle on A U as diameter. 
 
 (6) ABC is an arc of finite curvature, and is divided so that 
 AB : BC :: m : n, a constant ratio; join AB, AC, BC, and 
 shew that, ultimately, &ABC : segment ABC :: '6mn : (m -f n)*.
 
 LEMMA XL 105 
 
 For, by Cor. 5, Lemma XL 
 
 seg AB : seg ASC :: AB 3 : ABC 3 :: m* : (m + n? 
 
 seg BG : seg ABO :: n 3 : (m + n) 3 ; 
 .-. seg AB + seg BC : seg ABO :: m 9 + n 3 : (m + n)*, 
 
 and A A5<7 = seg AB<7 - seg AB - seg 5(7; 
 .-. A ABO : seg ABO:: 3 (wi'n + win 2 ) : (m + rc) 3 
 :: Bmn : (m -f- w)*. 
 
 (7) To ^nc? $e cAorc? of curvature, at any point of the cardioid, 
 through the focus. 
 
 It is easily seen from p. 56 (3), that SY being perpendicular 
 to PT, the triangles PSY,pBm, and OBp are similar; 
 
 /. 8Y: SP:: Bm : Bp :: J?^ : 56 T ; 
 
 .-. 8Y* : /S'P' 2 :: SP : ^(7, since 5w = 8P, 
 
 .-. SY*BC = SP 3 , and (SY 2 - SY' 2 } BO = SP 3 - SP' 3 ; 
 
 /. SP~SP' : SY^SY' :: 2SY.BO : 3 ^P 2 ultimately; 
 
 .'. by Art. 86, chord of curvature : 2SY:: 2SP: 3F; 
 
 therefore the chord of curvature through S = SP. 
 
 XII. 
 
 1. Prove that the focal distance of the point in the parabola at 
 which the curvature is one-eighth of that at the vertex is equal to 
 the latus rectum.
 
 106 NEWTON. 
 
 2. Prove that the diameter of curvature at the vertex of the 
 major axis of an ellipse is equal to the latus rectum : and shew 
 that the ratio of the curvature at the extremities of the axes is that 
 of the cubes of the axes. 
 
 3. Shew that at no point of an ellipse will the circle of 
 curvature pass through the centre, if the eccentricity be less 
 than Vi- 
 
 4. Find for what point of au ellipse the circle of curvature 
 passes through the other extremity of the diameter at that point, 
 shew that the distance of this point from the centre is the side of 
 the square of which AB is the diagonal. 
 
 5. In a rectangular hyperbola, the diameter of curvature at any 
 point, and the chords of curvature through the focus and centre are 
 in geometrical progression. 
 
 6. Prove that at a point P in an ellipse for which the minor 
 axis is a mean proportional between the radius of curvature and 
 the normal, PC = AC - BG. Shew that this is impossible unless 
 AC^. 2BO. 
 
 7. If the radius of curvature for an ellipse at P be twice the 
 normal, prove that CP = CS. 
 
 If moreover AC=2BC, prove that CP == 3PM. 
 
 8. If the circle of curvature at a point P of a parabola pass 
 through the other extremity of the focal chord through P, and the 
 tangent at P meet the axis in T, prove that the triangle PST will 
 be equilateral. 
 
 9. Prove that the distance of the centre of curvature, at any 
 point of a parabola, from the directrix is three times that of the 
 point. 
 
 10. If the circle of curvature at a point on a parabola touch 
 the directrix, the focal distance of the point will be & of the latus 
 rectum. 
 
 11. PQ is a normal at a point P of a rectangular hyperbola, 
 meeting the curve again in Q, prove that PQ is equal to the 
 diameter of curvature at P. 
 
 12. Prove that the portion of the normal intercepted between 
 the line joining the extremities of the two chords of curvature through 
 
 2B C 1 
 the foci of an ellipse, and the point of contact P, is pjf , .
 
 LEMMA XI. 107 
 
 13. A fixed hyperbola is touched by a concentric ellipse. If 
 the curvatures at the point of contact be equal, the area of the 
 ellipse will be constant. 
 
 14. Shew that the directrices of all parabolas touching a curve 
 of finite curvature at any given point, and having the same curvature 
 at that point as the curve, pass through a fixed point. 
 
 xni. 
 
 1. Prove that the chord of curvature through the vertex A 
 of a parabola : 2PT : : 2PY: AP, T being the intersection of the 
 tangents at P and A. 
 
 2. Apply the property that the radius of curvature at any point 
 of an ellipse is to the normal in the duplicate ratio of the normal to 
 the semi-latus rectum, to shew that the radius of curvature at the 
 extremity of the major axis is equal to the semi-latus-rectum. 
 
 3. Assuming only that a curve has a subnormal of constant 
 length, prove geometrically that its radius of curvature varies as 
 the cube of its normal. 
 
 4. If Pp be any chord of an ellipse, PT, pT tangents at P 
 and p, shew that the curvatures at P and p are as the cubes of pT 
 and PT. 
 
 5. Shew that the sum of the chords of curvature through a 
 focus of an ellipse at the extremities of conjugate diameters is 
 constant. Also, if p, a be the radii of curvature at those points, 
 prove that p* + ^ is constant. 
 
 6. Prove that the chords of curvature through any two points 
 on an ellipse in the direction of the line joining them are in the 
 same ratio~'as the squares on the diameters parallel to the tangents 
 at the points. 
 
 7. Prove that the distances of the centre of curvature at any 
 point of an ellipse and of that point from the minor axis are in the 
 duplicate ratio of the distances of the point and the directrix from 
 the same axis. 
 
 8. An hyperbola touches an ellipse, having a pair of conjugate 
 diameters of the ellipse for its asymptotes. Prove that the curves 
 have the same curvature at the point of contact. 
 
 9. Shew that, if D be the diameter of an ellipse parallel to the 
 langent at a point P, whose eccentric angle is 0, the length of the 
 chord common to the ellipse and circle of curvature at P will be 
 D sin 2>
 
 108 NEWTON. 
 
 1 0. Determine a parabola of curvature in magnitude and position 
 for any point in a circle, when the subtenses are inclined at 45 to 
 the tangent. 
 
 11. If #, y be the coordinates of a point P of a curve OP, 
 passing through the origin 0, the diameter of curvature at will 
 
 3? + y* 
 be -. ultimately, a being the inclination of the tangent 
 
 at to the line of abscissae. Hence shew that, if the equation of 
 a curve, referred to rectangular areas, be y 2 + 2ay - 2ax = 0, the 
 radius of curvature at the origin will be 2 V2 . a. 
 
 12. A circle is a circle of curvature, at a fixed point in the 
 circumference, to an ellipse, one focus of which lies on the circle, 
 shew that the locus of the other focus is also a circle. 
 
 13. Prove that the chord of curvature at any point P of an 
 ellipse in any direction PQ is half the harmonic mean between the 
 two tangents drawn from P to the confocal conic that touches PQ, 
 the tangents being reckoned positive when drawn towards the 
 interior of the ellipse, 
 
 XIV. 
 
 1. If AEB be the chord, AD the tangent, and BD the subtense, 
 for an arc ACB of finite curvature at A, find the limit of the ratio 
 area A CBE : area A CBD, as B approaches A. 
 
 2. An arc of continuous curvature PQR is bisected in Q, PT is 
 the tangent at P ; prove that, ultimately, as R approaches P, the 
 angle HPT is bisected by PQ. 
 
 3. If AB be an arc of finite curvature bisected in C, and T be 
 a point in the tangent at A, at a finite distance from A, prove that 
 the angle BTC will be ultimately three times the angle CTA, when 
 B moves up to A, 
 
 4. Two curves touch one another, and both are on the same 
 side of the common tangent. If in the plane of the curves this 
 tangent revolve about the point of contact, or if it move parallel to 
 itself, the prime ratio of the nascent chords in the former case will 
 be the duplicate of their prime ratio in the latter case. 
 
 5. A CB is a small arc of finite curvature ; prove that the mean 
 of the distances of every point of the arc from the chord AB is 
 equal to f of the distance of the middle point of the arc from the 
 chord, and that the mean of the distances of every point of the arc 
 from the tangent at either extremity of the arc is equal to f of the 
 distance of the middle point of the arc from the same tangent.
 
 LEMMA XL 109 
 
 6. When on an arc of continuous curvature there is no point 
 where the curvature is a maximum or minimum, the circles of 
 curvature at the extremities of the arc cannot intersect. 
 
 7. If 8 be any point in the plane of a curve, P any point on 
 the curve, Fthe corresponding point on the pedal for which S is 
 the pole, V the point where PS cuts the circle of curvature at P, 
 V the corresponding point for the pedal, then 4&P.S F'= PF. TV. 
 
 8. The radius of curvature in a curve increases uniformly with 
 its inclination to a fixed radius. Prove that the area between the 
 curve, its evolute, and the two radii of curvature of lengths a, J, 
 which contain an angle $, is (a 2 + ab + J 2 ) #. 
 
 9. A curve is such that the radius vector makes half the angle 
 with the normal that it does with a fixed line; find the chord of 
 curvature through the pole. 
 
 10. In a segment of an arc of finite curvature a pentagon is 
 inscribed, one side of which is the chord of the arc, and the remaining 
 sides are equal. Shew that the limiting ratio of the areas of the 
 pentagon and segment, when the chord moves up towards the 
 tangent at one extremity, is 15 : 16. 
 
 11. APQ, is a curve of continued and finite curvature, P and Q 
 are two points in it, whose abscissae along the normal at A are 
 always in the ratio m : 1, and from , C, two points in the normal, 
 straight lines BPb, CPc, BQb', CQc' are drawn to meet the tangent 
 at A. Shew that, when P and Q move up to A, the areas 01 
 the triangles bPc, b'Qc' are ultimately in the ratio m* : 1. 
 
 12. AB is an arc of finite curvature at A, and a point P is 
 taken such that AP : PB is in the constant ratio of m : n. Tangents 
 at A and B intersect the tangent at P in T and R, and AB is 
 joined. Prove that the ultimate ratio of the area ATRB to the 
 segment APB, as B moves up to A, is 3 (m 2 + mn + n z ) : 2 (m + n) a . 
 
 13. The tangent to a curve at a point B meets the normal at 
 a point A in T, C is the centre of curvature at A, and a point 
 on AC; prove that, in the limit, when B moves up to A, the 
 difference of OA and OB bears to AT the ratio OC : OA. 
 
 14. is a point within a closed oval curve, P any point on the 
 curve, QPQ' a straight line drawn in a given direction, such that 
 QP = PQ? = PO; prove that, as P moves round the curve, Q, Q ; 
 trace out two closed loops, the sum of whose areas is twice the area 
 of the original curve.
 
 110 NEWTON. 
 
 NOTE ON MAXIMA AND MINIMA. 
 
 93. When a variable magnitude changes its value in con- 
 sequence of the change of some element of its construction, 
 the law of its variation can be graphically represented by the 
 form of a curve in which the ordinate and abscissa of every 
 point represent respectively the corresponding values of the 
 variable magnitude and of the element on which it depends. 
 
 Examples of this mode of representation have been given 
 in Arts. 55 and 57, in which the time or space is the element 
 upon which depends the velocity or kinetic energy, which are 
 the variable magnitudes respectively considered. 
 
 94. This graphic representation enables us to obtain a 
 property of any maximum or minimum value of a variable 
 magnitude which is applicable to the solution of a variety 
 of problems. 
 
 For, let Ox be the line of abscissae and B a point in the 
 auxiliary curve at which the tangent RBS to the curve is 
 parallel to Ox, and let the abscissa OA represent the corre- 
 sponding value of the element, then the ordinate AB is a 
 maximum or minimum according as the portion of the curve 
 PBQ in the neighbourhood of B is concave or convex to 
 the line Ox. 
 
 Let a chord PQ be drawn parallel to the tangent RBS, 
 the two points P and Q one on each side of B have equal 
 ordinates 1/P, NQ, which, as PQ moves up to and continues 
 parallel to the tangent, become nearer and nearer and are 
 ultimately equal to the maximum or minimum value, while 
 the difference between the corresponding abscissae ultimately 
 vanishes. 
 
 Hence is derived the following theorem : 
 
 If a variable magnitude have a maximum or minimum value 
 there icill be two values of the element of construction, one greater 
 and the other less than the critical value^ which will give equal 
 values of the variable magnitude.
 
 LEMMA XI. Ill 
 
 95 Stationary value of a magnitude. 
 
 Let the equal ordinates MP, NQ be produced to meet the 
 tangent in R and S, then by Lemma XI., PR and QS vanish 
 compared with AM or AN^ and the ratio of the rates of 
 increase of the ordinate to that of the abscissa, which is gene- 
 rally finite, vanishes for the critical case of a maximum or 
 minimum ; on this account the magnitude is said to have a 
 stationary value. 
 
 One or two examples are sufficient to shew the application 
 of this method. 
 
 96. To find at what point on the bank of an oval pond a 
 person must land in order to pass from a given point on the 
 pond to a given point on the bank in the shortest possible time, 
 having given the ratio of his rates by land and by water. 
 
 Let A, B be the two given points, P the point at which he 
 must land, and let nv, v be the velocities by water and along 
 the bank. On opposite sides of P there are two points Q, R at 
 which if he land the time to B will be the same, in AR 
 take AM= A Q, then MR in water and QR on land are 
 described in the same time, therefore n.QR = MR, which is 
 true, however near Q and R may be to P; therefore cos< = w, 
 where (j> is the angle between AP and the tangent at P; whence, 
 when the exact form of the oval is given, the position of P 
 can be found. 
 
 97. To find the chord of an oval, which, drawn through a 
 given point, cuts off a maximum or minimum segment. 
 
 Through the fixed point A it is possible to draw two chords 
 PA Q and pAq, one on each side of the required chord, for 
 which the areas cut off are exactly equal; take away the 
 common part, and the remainders PAp, QAq are equal ; there- 
 fore, ultimately, when the angle between them vanishes, 
 PA.pA = QA.qA, and the chord which cuts off a maximum 
 or minimum area must be bisected by the fixed point. 
 
 98. If a triangle of constant shape be described about a given 
 triangle, prove that when the area is a maximum the normals to
 
 112 NEWTON. 
 
 the sides of the circumscribed triangle at the angular points of the 
 given triangle will meet in a point. 
 
 Let ABG be the given triangle, a/3y, a.'j3'y' two positions 
 of the circumscribing triangle whose areas are equal, the 
 triangle of maximum area being intermediate in position. 
 . Since the angles at a, a' are equal, the points a, a' lie in 
 the same segment of a circle whose base in j(7, and the angles 
 a(7a', ctBa! are equal. Hence the triangles a(7a', $(7/3', ft A/3' t 
 yAy, &c., are ultimately proportional to Gd\ <7/3 2 , .... 
 
 But the sum of the areas a(7a', ft A ft', 7^7' are ultimately 
 equal to the sum of ft Oft', yAy, aBa, 
 
 .'. aC* - 30* + &A* - yA' + yB* - aB 2 = 0. 
 Let the normals at A and C meet in JV; 
 
 .-. *C 2 -ftC 2 = aN 2 -ftN*, 
 ft A 3 - yA* = JV - yN* ; 
 .-. aB 2 - yft 2 = aN 2 - 7^ = aD* - yD'\ 
 if ND be perpendicular to ay ; 
 
 /. aB-yB=QLD-yD; .: BD = Q, 
 which proves the proposition. 
 
 XV. 
 
 1. In an arc AS of continuous curvature n points P v P 2 , . . 
 are taken so that the polygon AP^P Z . . B has a maximum area ; 
 prove that, when the arc AB is indefinitely diminished, the arcs 
 AP V P,P 2 , . . are all equal. 
 
 2. Find the greatest rectangle which can be inscribed in a 
 triangle, one side of which is on a side of the triangle. 
 
 3. Prove that the diagonals of the greatest rectangle which can 
 be inscribed in an ellipse, having its sides parallel to the axes, are 
 the equi-conjugate diameters. 
 
 4. Prove that the parallelograms of smallest area which can be 
 described about a given ellipse are those which have their sides 
 parallel to conjugate diameters. 
 
 5. A point is taken on the major axis AA' of an ellipse 
 produced, and a line is drawn through cutting the ellipse in the 
 points P and P. Prove that when the area of the quadrilateral 
 APPA' is a maximum the projection of PP' upon AA is equal to 
 the semi-axis-major.
 
 LEMMA XI. 113 
 
 6. Prove that the quadrilateral of maximum area that can be 
 formed with four straight lines AB, BC, CD, DA, of given lengths 
 is such that a circle can be described about it. Hence prove that 
 the curve of given length which on a given chord encloses a 
 maximum area is an arc of a circle. 
 
 7. From a point Ton the exterior of two oval curves tangents 
 TP, TQ are drawn to the inner; shew that, when the arc PQ is a 
 minimum or maximum, the radii of curvature at P and Q are in 
 the ratio TP sec a : TQ sec/3, where a, /3 are the angles which TP, 
 TQ respectively make with the normal at T. 
 
 8. Find the ultimate intersection of the chords common to an 
 ellipse and two consecutive circles of curvature, and shew that when 
 the common chord attains its maximum length for a given ellipse, 
 it cuts the ellipse at angles whose tangents are as 1 : 3. 
 
 9. A triangle inscribed in a closed oval curve moves so that two 
 of its sides cut off constant areas. Prove that when the area cut 
 off by the third side is stationary the three lines formed by joining 
 each angular point of the triangle to the intersection of tangents 
 at the other two points are concurrent. 
 
 10. Any two normal chords of an ellipse at right angles to each 
 other cut off equal areas from the curve. Hence find the position 
 of the normal chord which cuts off the minimum area. 
 
 11. An endless string just reaches round the circumference of 
 an oval, and when it is cut at any point it is unwrapped until it 
 becomes a tangent at the point of section ; shew that the involute 
 so formed will have a maximum or minimum length if the point 
 of section be chosen so that the length of the oval shall be equal 
 to the circumference of the circle of curvature at that point.
 
 114 NEWTON. 
 
 DIGRESSION 
 ON THE PROPERTIES OF CERTAIN CURVES. 
 
 THE CYCLOID. 
 
 99. DBF. If, in one plane, a circle roll along a straight 
 line, any point on its circumference will describe a curve called 
 a Cycloid. 
 
 Let C, D be the points where the tracing point P meets the 
 straight line, on which it rolls ; A the point where it is furthest 
 from CD, AB being the corresponding diameter of the circle. 
 
 The rolling circle is called the generating circle, AB is called 
 the axis, A the vertex, CD the base, and C, D the cusps. 
 
 100. Let EPS be the generating circle in any position, then, 
 since the points of the base and circle come successively in 
 contact without slipping, CS=arc PS, CB and BD are each 
 half of the circumference of the circle, and S = &rc HP. 
 
 101. To draw a tangent to a cycloid. 
 
 Let the generating circle be in the position EPS, then, con- 
 sidering a circle as the limit of a regular polygon of a large 
 number of sides, it will roll by turning about the point of con- 
 tact, which will be at rest for an instant, being an angular point 
 of the polygon ; therefore for an instant P will move per- 
 pendicular to SP, or in the direction PE of the supplemental 
 chord, which will therefore be the tangent to the cycloid at P. 
 
 If A QB be the circle on AB as diameter, PQM an ordi- 
 nate perpendicular to AB, the tangent at P will be parallel to 
 the chord QA. 
 
 102. To find the length of the arc of a cycloid. 
 
 Let EPS be the position of the generating circle corre- 
 sponding to the point P in the cycloid, PE being the tangent 
 at P.
 
 THE CYCLOID. 
 
 115 
 
 When the circle has turned through any angle POp the 
 centre will have moved through a distance equal to Pp, 
 and the motion of the generating point will be the resultant 
 
 V C 
 
 of Pp due to the rotation, and pP = Pp parallel to the base 
 due to the translation of the centre of the circle ; and PP 
 will ultimately coincide with PR. Draw pn perpendicular to 
 PR, then, since Pp = Pp, PP' = 2Pn = 2 (RP- Rp] ultimately. 
 Hence the arc of the cycloid measured from the vertex increases 
 twice as fast as the chord of the generating circle, which is a 
 tangent to the cycloid, and they vanish simultaneously, therefore 
 the arc of the cycloid is double of the chord of the generating 
 circle, or referring to the circle on the axis AB as diameter, 
 the arc AP is double of the corresponding chord A Q. 
 
 103. To find the relation between the arc and abscissa. 
 Let AM be the abscissa of the point P, then 
 AM-.AQ:: AQ : AB-, 
 
 104. To shew that the evolute of a given cycloid is an equal 
 cycloid, and that the radius of curvature of a cycloid is twice the 
 normal. 
 
 Let APG be half the given cycloid, AB the axis, A the 
 vertex, and BO the base. Produce AB to 0', making BC' equal 
 to AB, and complete the rectangle BCBC\ and let the semi- 
 cycloid C'P'C be generated by a circle, whose diameter is equal 
 to that of the generating circle of the given cycloid, rolling on 
 C'B' ; C being the vertex, CB' the axis of this cycloid. 
 
 Let SPR, SPR' be two positions of the respective gene- 
 rating circles, having their diameters RS } SR' in the same
 
 116 
 
 NEWTON. 
 
 straight line, P, P being the corresponding points of the 
 cycloids; join SP, PR and JSP', PR. 
 
 By the mode of generation, arc SP=SC, and arc SPB=BC', 
 
 .'. L PSR = L PSR ; and PSP is a straight line. 
 Also, arcP5=arcPS; .-. chd.P'S=chd.PS; 
 
 .-. PSP=2P'S=P'C r thecycloidalarc; 
 also PSP touches the cycloid C'PCat P'; 
 therefore, a string fixed at (7', and wrapped over the arc of 
 the semicycloid, will, when unwrapped, have its extremity in 
 the arc of the given cycloid ; hence, the evolute of a semi- 
 cycloid is an equal semicycloid, and the radius of curvature at 
 P is 2PS or twice the normal. If another equal semicycloid be 
 described by the circle rolling on B'C' produced, the extremity 
 of the string wrapped on this curve will trace out the remainder 
 of the given cycloid. 
 
 Thus a pendulum may be made to oscillate in a given 
 cycloid. 
 
 105. To find the area of the cycloid. 
 
 Let P, P be two points very near each other in a cycloid,
 
 THE CYCLOID. 
 
 117 
 
 Q, C$ corresponding points in the generating circle, p, p' in the 
 evolute, Rj R' the intersections of the base with normals Pp } 
 
 Pp'j T, 8 the intersections of SQ f and Pp' with PQ. Then 
 pR = PR = BQ, and &p'PS=&p'RR ultimately =4A BQT- 
 therefore trapezium PUR'S = 3 &BQT ultimately, and the same 
 being true for all the inscribed triangles and trapeziums, whose 
 sums are ultimately the areas of the semicircle and semicycloid, 
 therefore, by Cor., Lemma IV., the area of the cycloid is three 
 times that of the generating circle. 
 
 106. The following method of finding the area of a cycloid 
 is independent of the properties of the evolute. 
 
 In the figure of Art. 1 04 let P' be any point in the cycloid 
 CPC', PS the chord of the generating circle which touches 
 the cycloid, and let Q' be a point in the cycloid near P', then 
 the arc PQ' ultimately coincides with PS. Let Q'N 1 , Q'N 
 be the complements of the parallelogram whose diagonal is 
 jF/S, and sides parallel and perpendicular to the base, these are 
 equal ultimately ; therefore, by Lemma IV., the cycloidal area 
 CNP = circular segment SPN'. 
 
 The exterior portion CB'G' is equal to the area of the 
 semicircle, and the whole parallelogram BCB' 0' is the rectangle
 
 118 
 
 NEWTON. 
 
 under the diameter and semi-circumference of the generating 
 circle, and is equal to four times the area of the semicircle; 
 therefore the cycloidal area CC'J? is three times the area of the 
 semicircle. 
 
 107. All cycloids are similar. 
 
 Let two cycloids APG, Ape be placed so that their vertices 
 are the same, and their axes coincident in direction, and describe 
 
 circles on the axes AB, Ab as diameters. Draw AqQ cutting 
 the circles in q, Q. Then, since the segments Aq, AQ are 
 similar, arc Aq : arc A Q : : Aq : A Q ; and, if mqp, MQP be 
 ordinates to the cycloids, arcs Aq, AQ = qp, QP respectively ; 
 therefore qp : QP::Aq:AQ, and ApP is a straight line. 
 Also Ap : AP:: Aq : AQ :: Ab : AB, a constant ratio; hence 
 the cycloids satisfy the condition of similarity, and in this 
 position of the cycloids the point A is a centre of direct 
 similitude. 
 
 108. To construct a cycloid which shall have its vertex at a 
 given point, its base parallel to a given straight line, and which 
 shall pass through a given point. 
 
 Let A be the given vertex, AB perpendicular to the given 
 line, P the given point. In AB take any point b, and with 
 the generating circle, whose diameter is Ab, describe a cycloid 
 Ape, join AP intersecting this cycloid in^>. 
 
 Take AB a fourth proportional to Ap, AP, and Ab ; then 
 AB will be the diameter of the generating circle of the required 
 cycloid ; for, since Ap : AP : : Ab : AB, and all cycloids are 
 similar, P is a point in the cycloid whose axis is AB.
 
 THE CYCLOID. 
 
 119 
 
 109. A particle slides down the, smooth arc of a cycloid, 
 whose axis is vertical, and vertex downwards, to find the time 
 of an oscillation. 
 
 Let AB be the vertical axis of the cycloidal arc APL, L the 
 point from \\hich the particle begins to move, PQ a small arc of 
 
 its path, LR, PM, QN perpendicular to AB] and take Al, Ap, 
 Aq on the tangent at A respectively equal to AL, AP, A Q. 
 
 Suppose a point to move from I to A in the same time as 
 the particle moves on the cycloid from L to A, their velocities 
 being always equal at equal distances from A. 
 
 Let v be the velocity at P or p, and T the time of falling 
 from B to A, so that v' = ^gBM and 2AB = gT*-, therefore 
 tfT* = AB.RM= kAB.AR - AB. AM= AU- AP\ Art. 103, 
 = AF-A/. 
 
 Describe a circle with centre A and radius Al, and draw the 
 ordinates pt, qu, then AP Ap*=ptf, and^ = VjT; and if T be 
 the time from P to Q, PQ =pq = vr ultimately, hence 
 
 tu : Al : : pq : pt : : r : T; 
 therefore, if a point move in the circle from I with uniform 
 
 Al 
 velocity -^ , the point moving in IA will always be in the 
 
 foot of the ordinate and the motion in IA or LA will therefore 
 be a simple harmonic motion, by (5) page 78. 
 
 The time from L to A is the time of describing the quadrant 
 
 \ -rrAl with velocity , = \irT= |TT
 
 120 NEWTON. 
 
 The length of the string which, by the contrivance of Art. 104, 
 makes a particle oscillate in this cycloid is 2AB=l suppose; 
 therefore the time of the oscillation of a cycloidal pendulum 
 
 of length I from rest to rest = TT . /- . 
 
 V 9 
 
 ATt 
 
 9 
 
 110. We can shew that the motion on the cycloid is a 
 
 simple harmonic motion by the first definition, (5) page 78 ; for, 
 
 referring to the figure, page 115, since the tangent at P is 
 
 parallel to AQ, the acceleration along the curve at P iis 
 
 A.O A.P 
 
 varies as -^ ""! bv ( 4 ) P a & e 77 tne 
 
 time from L to A is obtained. 
 
 111. To find the time of a very small oscillation of a simple 
 pendulum suspended from a point. 
 
 A simple pendulum is an imaginary pendulum consisting of 
 a heavy particle called the 505, suspended from a point by means 
 of a rod or string without weight. 
 
 In this case the pendulum describes the small arc of a circle 
 which may be considered the same as a cycloidal arc, the axis 
 of which is half the length I of the pendulum, therefore the 
 
 time of oscillation from rest to rest is TT . /- . 
 
 V 9 
 
 112. To count the number of oscillations made by a given 
 pendulum in any long time. 
 
 In consequence of the liability to error in counting a very 
 great number of oscillations, since in the case of a seconds pen- 
 dulum there would be 3600 oscillations for each hour, it becomes 
 necessary to adopt some contrivance for diminishing the labour. 
 For this purpose the pendulum is made to oscillate nearly in the 
 same time as that of a clock; it is then placed in front of 
 that of the clock, so that when they are simultaneously near 
 their lowest positions the bob of the pendulum and a cross 
 marked on the pendulum of the clock may be in the field of 
 view of a fixed telescope.
 
 THE CYCLOID. 121 
 
 Suppose that after n oscillations of the given pendulum 
 they are again in coincidence close to the same position ; if 
 there be m such coincidences in the whole time of observation, 
 the number of oscillations in that time will be mn ; thus the 
 only labour has been to count the n oscillations, and to estimate 
 the number of the coincidences before the last one observed. 
 
 113. To measure the accelerating effect of gravity by means 
 of a pendulum. 
 
 Let g be the measure of this effect or the velocity generated 
 by the force of gravity in a second. 
 
 Let I be the length of a simple pendulum which makes n 
 
 oscillations in m hours, then = number of seconds in one 
 
 n 
 
 oscillation =ir . /-; therefore g= . . 2 ^, in whatever unit 
 
 of length I is estimated. 
 
 This would be a very exact method of determining g, if we 
 could form a simple pendulum ; but it is impossible to do this, 
 and it is only by calculations of a nature too difficult to be 
 explained here that it can be shewn how to deduce the length of 
 the simple pendulum, which would oscillate in the same time as 
 a pendulum of a more complicated structure. 
 
 114. The seconds pendulum at any place is the simple pen- 
 dulum which at the mean level of the sea at that place would 
 oscillate in one second. 
 
 If L be the length of the seconds pendulum, I the length 
 of a pendulum making n oscillations in m hours, 
 
 ? 3600m , 1L _ rfl 
 
 ~' and *Vr * '' 'WS- 
 
 115. To determine the height of a mountain by means of a 
 seconds pendulum^ the force of gravity at any point being supposed 
 to vary inversely as the square of the distance from the centre of 
 the earth. 
 
 Let L be the length of a seconds pendulum, x the height 
 of the mountain above the mean level of the sea, a the radius 
 
 R
 
 122 NEWTON. 
 
 of the earth, all expressed in feet ; and let n be the number of 
 seconds lost in 24 hours by the pendulum at the top of the 
 mountain. 
 
 If g be the measure of the accelerating effect of gravity at 
 
 the mean level of the sea, then *- ^ will be its value at 
 
 the top of the mountain, and the time of oscillation at the top 
 
 11 u I( L fa + x\*} a + x , . /L 
 
 will be TT . / -I I I > , or seconds, since TT . / = 1 ; 
 N \9\ a J ) a V 
 
 hence, writing N for 24x60x60, (N-n)- = N, and 
 x N n r? a a, 
 
 a = 4000 x 1760 x 3 and N= 24 x 60 x 60, therefore the height 
 of the mountain will be 244'4n + '0027n 2 ; thus, if n = 10, the 
 height will be 2444'7 feet. 
 
 NOTE. The attraction of the mountain would make a sensible 
 variation from the law of the inverse square, this law being true 
 only if the earth consisted of homogeneous spherical strata. 
 
 116. To find the number of seconds lost in a day, in con- 
 sequence, of a slight error in the length of the seconds pendulum ; 
 and conversely. 
 
 Let N be the number of seconds in a day, L the length of 
 the seconds pendulum, L + \ that of the incorrect pendulum, 
 Nn the number of its oscillations in a day; 
 
 ,, L X 2n 
 
 ~ '* = 
 
 THE EPICYCLOID AND HYPOCYCLOID. 
 
 117. DEF. The curve traced out by a point on the cir- 
 cumference of a circle, which rolls upon that of a fixed circle, 
 is called an Epicycloid if the concavities of the two circles be 
 in opposite directions, a Bypocycloid if the concavities be in 
 the same direction.
 
 THE EPICYCLOID AND HYPO-CYCLOID. 
 
 123 
 
 118. To shew that the evolute of an epicycloid is a similar 
 epicycloid. 
 
 Let FA be the fixed circle, APE the rolling circle in any 
 position, P the generating point, CAE a line drawn through 
 
 the point of contact, meeting the rolling circle in A, E] and 
 let GPF be the epicycloid, of which PA and PE will be a 
 normal and tangent. 
 
 Draw the chord EQ parallel to PA and join CQ meeting 
 PA produced in 0. Since EQ is parallel to AO, 
 
 CO: CQ:: CA : CE; 
 
 therefore and Q describe similar figures. But Q, being the 
 other extremity of the diameter through P, will describe an 
 epicycloid similar and equal to GPF, being at its cusp when 
 P is at G the greatest distance from C. 
 
 Draw Oa parallel to QA and therefore perpendicular to P0 1 
 meeting CA in a, then generates an epicycloid fF by the 
 rolling of a circle AOa, whose diameter is Aa, on a fixed 
 circle of radius Ca.
 
 124 NEWTON. 
 
 Also PO the normal to GF is perpendicular to aO and is 
 therefore a tangent tofF, hence fF is the evolute of the given 
 epicycloid and is a similar epicycloid. 
 
 Let a, b be the radii of the fixed and rolling circles for the 
 given epicycloid, then 
 
 Aa: CA:: OQ: CQ::AE: CE :: 25 : a + 25; 
 therefore Aa : AE : : a : a + 2&, and if a = co , Aa = AE, and 
 AFj af become straight lines, whence the evolute of a cycloid 
 is an equal cycloid. 
 
 119. Since AO : PA : : A : EQ : : CA : CE, therefore 
 PO : PA : : 2 (a + b] : a + 25, which gives PO the radius of 
 curvature at P of the given epicycloid; this will be found 
 independently of the evolute in Art. 121 below. 
 
 120. To find the length of any arc of the epicycloid. 
 
 By the properties of the evolute, see the last figure, the 
 
 arc OF of the evolute =OP=2AP. J^^, and the arc of the 
 epicycloid generated by Q, measured from Q to the highest 
 point, =OF^^-=2AP. 5L* ; therefore the arc GP from 
 
 the highest point G of the epicycloid GPF= 2EP. a -^ . 
 
 121. To find the radius of curvature at any point of an 
 epicycloid.
 
 THE EPICYCLOID AND HYPOCYCLOID. 125 
 
 Let AB, BC be consecutive sides of a fixed regular polygon 
 of m sides, AB, Be sides of another regular polygon of n sides 
 equal to those of the former, on the outside of which it rolls, 
 in a position in which two sides are coincident. 
 
 Let P be any angular point of the rolling polygon ; P will 
 generate a figure composed of a series of circular arcs such 
 as PP', P' being the position of P when Be coincides with BC. 
 Produce P4, P'B to meet in 0. 
 
 Then APB=-, and LPBP' = LcBC= 4 J 
 n m n 
 
 .'. PO: PB::sin27r (- + -}: sinirf- + - }. 
 \m n) \m n) 
 
 When the number of sides is indefinitely increased, the 
 polygons ultimately become circles, the curve traced out by P 
 becomes an epicycloid, and PO the radius of curvature at P. 
 
 If a, b be the radii of the fixed and rolling circles m.AB= Sira 
 and n.AB=2'jrbj ultimately ; therefore m : n : : a : 5; 
 
 .-. POiPAn 2 (- + -}:- + -:: 2 (a + b}:a + 2b: 
 \m nj m n 
 
 therefore the radius of curvature is 2PA. --, where PA 
 
 is the part of the normal intercepted between the generating 
 point and the point of contact. 
 
 If a = co , or the fixed circle become a straight line, the 
 epicycloid will become a cycloid, and the radius of curvature 
 will be twice the normal, as in Art. 104. 
 
 122. To find the area of an epicycloid. 
 
 In the last figure, area APP'B= A PAB+ sector PBP'-j now 
 
 sector PBP' = ^PB\27r(-+-} and ^PAB^^PB 2 sin- : 
 \m n) n 1 
 
 .'. area APP'B= APAB l + ultimately ; 
 
 hence, by Lemma IV. Cor., the area of the segment of the
 
 126 NEWTON. 
 
 epicycloid included between two normals and the fixed circle 
 
 is ( 3 H J x the corresponding segment of the rolling circle. 
 
 Compare Art. 105. 
 
 123. The corresponding properties of the hypocycloid may 
 be proved in a similar manner; and the results obtained will 
 be the same as for the epicycloid, if in the latter the sign 
 of b be changed. 
 
 Thus, if the diameter of the fixed be double that of the rolling 
 circle, the L/pocycloid will become a straight line, which agrees 
 with the result of Art. 121, since a + 2& = 0, and therefore the 
 radius of curvature at every point will be infinite. 
 
 THE EQUIANGULAR SPIRAL. 
 
 124. DEF. The equiangular spiral is a curve which cuts 
 all the radii drawn from a fixed point at a constant angle. 
 
 125. If a series of radii 8 A, SB, SO, ... be drawn inclined 
 at equal angles, and AB, BG, CD, ... make equal angles SAB, 
 SBC t ... with these radii respectively, the curvilinear limit 
 
 of the polygon ABCD ..., when the equal angles A SB, 
 BSC t ... are indefinitely diminished, will be an equiangular 
 spiral. 
 
 126. To find the length of an arc of an equiangular spiral 
 contained between two radii. 
 
 Let a be the constant angle SAB, and let SL be the n th
 
 THE EQUIANGULAR SPIRAL. 
 
 127 
 
 radius from 8A ; then, since the triangles ASB, BSC, ... are 
 similar, 8A : SB : : SB : SG ... . 
 
 Let 8B=\.8A, then BC=\.AB, CD=X\AB...FL=\ n -\AB-, 
 
 .-. AB+BC+...+ FL:AB:: 1 + X+...+X"- 1 : 1::1-X": 1-X 
 :: SA-\*.SA : SA-SB:: SA-SL: SA-SB, 
 but AB cosa = 8A - SB cozASB= SA - SB ultimately, and 
 AB+BC+... is ultimately the arc of the spiral; therefore 
 arc AL= (SA-SL) sec a. 
 
 127. To find the area of an equiangular spiral bounded ly 
 two radii. 
 
 Employing the same construction as above, 
 
 :: 1-X 8 ": 1-X*:: SA'-SL 2 : 8A*-8B>, 
 
 .: SA 2 - SB 2 = 4 A A SB x cot a, ultimately ; 
 /. area A8L = (8A*- SU] tana. 
 
 128. To find the radius and chord of curvature through the 
 pole at any point of an equiangular spiral. 
 
 Let SP t SQ be radii drawn to two points P and Q, near to 
 
 one another, let PR, QR, tangents to the spiral at P and <?, 
 intersect in R, and let the normals PO, QO intersect in O' t 
 join OR, SR.
 
 128 NEWTON. 
 
 Then, since angles SQR, 8PR are equal to two right angles, 
 and each of the angles OQR, OPR is a right angle, the circle 
 which passes through P, R, and Q will also pass through S 
 and 0, and OR will be its diameter ; therefore L OSR is a 
 right angle. Hence, proceeding to the limit, O is the centre 
 of the circle of curvature at P, and OSP is a right angle. 
 Therefore if a be the angle of the spiral, OP= SP cosecoc will 
 be the radius of curvature, and 2SP the chord of curvature 
 through the pole. 
 
 129. The following is an illustration of Art. 86. 
 If PV be the chord of curvature through 8 t 
 
 SY'-SY: SP'-SP::2SY:PV; 
 but in the equiangular spiral ST : SY' :: SP: SP' ; 
 
 .-. 8Y'-8Y:SP'-8P::8Y: SP; whence PV= 2SP. 
 
 THE CATENARY. 
 
 130. DBF. The Catenary is the curve in which a uniform 
 and perfectly flexible string, of which the extremities are sus- 
 pended at two points, would hang under the action of gravity, 
 supposed to be a constant force acting in parallel lines. 
 
 The directrix is a horizontal straight line whose depth below 
 the lowest point is equal to the length of string whose weight is 
 equal to the tension at the lowest point. 
 
 The axis is the vertical through the lowest point. 
 
 131. The tension at any point of the catenary is equal to the 
 weight of the string which, if suspended from that point, would 
 extend to the directrix. 
 
 Let A be the lowest point of a uniform and perfectly flexible 
 string hanging from two points under the action of gravity, 
 P any other point, AO the length of string whose weight is 
 equal to the tension of the string at A. Take a point B in OA, 
 and let OM, BO drawn horizontally meet a vertical PM in 
 M and C. 
 
 If a string pass round smooth pegs at APCB, it is evident 
 that there will be a position of equilibrium whatever be the
 
 THE CATENARY. 
 
 129 
 
 length of the string, or the position of BC, and for some length 
 and some position of BC the tangent at A will be horizontal. 
 
 T O 
 
 Also, since BDC will hang symmetrically, the tensions of 
 the string at B and C will be equal, and BDG may be removed 
 and replaced by equal lengths BO, CM of the string, without 
 disturbing the equilibrium of AP, therefore the tension of the 
 catenary at P is equal to the weight of a string of length PM. 
 
 132. The proposition of the preceding article may be proved 
 by considering the catenary as the limit of the polygon formed 
 by a series of equal rods of the same substance jointed freely 
 at the extremities and suspended from two fixed points, when 
 the length of the rods is indefinitely diminished. 
 
 The equilibrium will be undisturbed if each rod be replaced 
 by two weights at the extremities, each equal to half that of 
 the rod, connected by a rod without weight. 
 
 Let AB, BC be two consecutive positions of the rods, 
 
 S
 
 130 NEWTON. 
 
 weights equal to those of the rods being placed at A, B, Oj let 
 AM be vertical and BM horizontal, and produce CB to meet 
 AM in D ; draw DN perpendicular to AB. 
 
 The forces which keep B in equilibrium act in the directions 
 of the sides of the triangle ABD, and are proportional to them. 
 
 Therefore the difference of the tensions of AB and BC is 
 to the weight of the rod AB as AB- ED : AD, that is, ulti- 
 mately, as AN: AD or AM: AB; hence the difference of the 
 tensions is the weight of a rod of length A M. 
 
 Therefore, proceeding to the limit, the difference of tensions 
 at any two points of the catenary is equal to the weight of 
 string, which is equal in length to the vertical depth of one 
 point below the other, whence the truth of the proposition. 
 
 133. P is a point in a catenary, PM perpendicular to the 
 directrix, PT a tangent at P, M U perpendicular to PT; to 
 shew that PU is equal to the arc measured from the lowest point , 
 and that MU is constant. 
 
 Let PT, fig. for Art. 131, meet the direction OM in T, and 
 let A be the axis, then since the arc AP supposed to become 
 rigid is in equilibrium mider the action of the tensions at 
 A and P and the weight, and these forces are in the directions 
 of the sides of the triangle TPM, 
 
 .-. AP: AO:PM::PM:MT: TP :: PU : MU : PM, 
 by similar triangles TPM, MPU-, 
 
 /. PU=AP and MU=AO. 
 
 134. To draw a tangent to a catenary at any point. 
 "With centre M and radius equal to A describe a circle, and 
 
 draw PU touching this circle in U] then, since MU, which is 
 perpendicular to PU, is equal to A 0, PU will be the tangent 
 at P. 
 
 135. If a rectangular hyperbola be described, having centre 
 and semi-transverse axis OA, the ordinate of the hyperbola 
 will be equal to the arc of the catenary. 
 
 For, let A E be the hyperbola, therefore 
 RX* = ON*- OA*=PM 2 - UM* = PU*- /. BN=PU=AP.
 
 THE LEMNISCATE. 
 
 131 
 
 136. To find the radius and vertical chord of curvature of 
 a catenary. 
 
 Let PQ be a small arc of a catenary, RSPT, QS tangents at 
 Pand Q, PM, ^ ordinates, TOM the directrix. 
 
 T O M. N 
 
 Since QRS is a triangle of the forces acting upon P$, 
 
 tension at P : weight of PQ :: RS: QR, 
 
 /. PM :PQ::RS:QR:: %PQ : QR, ultimately; 
 
 therefore 2PM is the vertical chord of curvature, and PG, the 
 
 part of the normal intercepted between the point P and the 
 
 directrix is equal to the radius of curvature at P. 
 
 Also PG : PM :: PT: TM : : tension at P : tension at A 
 : : PM : A 0, therefore the radius of curvature is a third pro- 
 portional to A and PM. 
 
 THE LEMNISCATE. 
 
 137. PEF. The Lemniscate is the locus of tne feet of the 
 perpendiculars drawn from the centre of a rectangular hyperbola 
 upon the tangent. 
 
 138. To find the inclination to the tangent at any point of 
 the radius from the centre of the lemniscate. 
 
 Let CY be perpendicular on PT the tangent at the point P
 
 132 
 
 NEWTON. 
 
 in the hyperbola, then CY.CP=PF.CD = AC*, since A C =SG 
 and CP^CD'm the rectangular hyperbola. 
 
 Draw the ordinate PN, then CT.CM=AC*=CY.CP; 
 .-. CY: CT:: CM: CP-, 
 
 and CMP, CYTare right angles; therefore LPCM=L TOY. 
 
 Draw CZ perpendicular on the tangent at Yto the lemniscate ; 
 then ZCY and YCP are similar triangles, see page 55 ; 
 
 .-. L ZYC= L CPY= complement of twice L YCA. 
 
 139. To find the perpendicular on the tangent at any point 
 of the lemniscate. 
 
 CZ.CP=CY*, and CY.CP=AC*; 
 .-. CZ: CY:: CY* : AC*-, 
 = CY\ 
 
 140. To find the chord of curvature through the centre^ and the 
 radius of curvature at any point of the lemniscate. 
 Let YV be the chord of curvature ; 
 
 .-, YV: 2CZ:: CY- CY' : CZ-CZ', ultimately, Art. 86, 
 and 
 
 .% CY-CY' : CZ-CZ' :: AC* : ZCY* :: CY: 3CZ-, 
 
 /. FF=CF, or the chord of curvature through the centre 
 ia two'thirds of the radius vector,
 
 THE LEMNISCATE. 133 
 
 Also, the radius of curvature : ^ YV 
 
 ::CY:CZ::AC*: CY* ::CP: CY, 
 
 hence the radius of curvature is ^CP, or of the radius at 
 the corresponding point of the hyperbola. 
 
 141. Poles of the lemniscate. 
 
 Let S, H\)& the foci of the hyperbola, *, h the middle points 
 of CS and CH] s, h are called the poles of the lemniscate. 
 
 <P 
 
 Draw SY' } EZ perpendicular to the tangent to the hyper- 
 bola at Pj and let SY' meet the auxiliary circle again in Z', 
 and join sY', sZ', sF, AY, and JiZ. 
 
 Since Cs = sS, the perpendicular from s on YY' bisects it ; 
 therefore sY' = sY, similarly hY=hZ=sZ'. 
 
 Now B0.8B = l8G* = AC* = 8Y'.8Z'i 
 therefore a circle can be drawn circumscribing CsY'Z'; there- 
 fore z.Y'sZ' = /.Y'CZ'; also A Y'sZ' = %& Y'CZ', since the 
 altitude of Y'CZ' is double of that of Y'sZ'; 
 
 therefore sY. 7iY=^CA*) which is the property of the poles of 
 the lemniscate. 
 
 For this proof I am indebted to Prof. Tait. 
 
 XVI. 
 
 1. If a line move parallel to the base of a cycloid, find the 
 limit of the ratio of the segment of the cycloid to the corresponding 
 segment of the generating circle, when the line becomes indefinitely 
 near to the tangent at the vertex.
 
 134 NEWTON. 
 
 2. A balloon was found to be sailing steadily before the wind 
 at an invariable elevation above the earth. A seconds pendulum 
 suspended to the car was observed to make 2997 oscillations in 
 50 minutes ; shew that the height of the balloon was 4 miles and 
 7 yards nearly, the radius of the earth being 4000 miles. 
 
 3. If a particle be made to oscillate in a cycloid on a smooth 
 inclined plane, whose inclination to the horizon is 30, and the 
 base of the cycloid be horizontal, find the radius of the generating 
 circle in order that the particle may perform a complete oscillation 
 in n seconds. 
 
 4. If P be a point in a cycloid, and the corresponding position 
 of the centre of the generating circle, shew that PO will touch 
 another cycloid of half the dimensions. 
 
 5. Shew that the limit of the whole length of an epicycloid 
 or hypocycloid, corresponding to a complete revolution of the 
 generating round the fixed circle, is eight times the radius of 
 the latter, when that of the former is indefinitely diminished. 
 
 6. Prove that the epicycloid of one cusp is the pedal of a circle 
 referred to a point in its circumference. 
 
 7. Shew that the evolute of an equiangular spiral is a similar 
 spiral, and that the extremities of the diameters of curvature lie 
 in a similar spiral. 
 
 8. An equiangular spiral rolls along a straight line, shew that 
 its pole describes a straight line. 
 
 9. Prove that, if a catenary roll on a fixed straight line, its 
 directrix will always pass through a fixed point. 
 
 10. If S T be drawn perpendicular to the tangent to a lemniscate 
 at a point P, and SA be the greatest value of SP, prove that 
 SP 3 = SA\SY; S being the centre. 
 
 xvn. 
 
 1. From the consideration that the diameter of curvature is the 
 limit of the third proportional to the subtense perpendicular to the 
 tangent and the arc, prove that the radius of curvature of a cycloid 
 at any point is twice the normal cut off by the base. 
 
 2. On the normal to a cycloid a constant length is measured 
 both inwards and outwards; find the area included between, the 
 loci of the points so obtained.
 
 THE LEMNISCATE. 135 
 
 3. P, Q are consecutive points on an epicycloid of two cusps ; 
 from p, q, the corresponding points of contact of the rolling with 
 the fixed circle, pm, qn are drawn perpendicular to the cusp-line ; 
 prove that the elementary area PQpq is twice the elementary area 
 pmnq. Heace find the area of the epicycloid and of its evolute. 
 
 4. Prove that the diameter through the point of a rolling circle 
 which generates an epicycloid always touches another epicycloid 
 generated by a circle oi half the dimensions. 
 
 5. A hypocycloid of n cusps has at any point a tangent drawn, 
 prove that the length of the tangent, intercepted between the gene- 
 rating circle and the point of contact, is to the arc measured from 
 the point to the vertex of the branch in which the point is taken, 
 as n : 2(n-l). 
 
 6. A bead slides on a hypocycloid being acted on by a force 
 which varies as the distance from the centre of the hypocycloid and 
 tending to it ; prove that the time of oscillation will be independent 
 of the arc of oscillation. 
 
 7. If, along the several normals to an epicycloid, a system of 
 particles move from the curve under the action of a force, tending 
 to the centre of the fixed circle, and varying as the distance, prove 
 that they will all arrive at the fixed circle at the same instant. 
 
 8. A plane curve rolls along a straight line, shew that the 
 radius of curvature of the path of any point, fixed with respect to 
 
 A2 
 
 the curve, is = , r being the distance of the fixed point from 
 
 r -p sm< ' 
 
 the point of contact, $ the angle between this line and the fixed 
 line, and p the radius of curvature of the curve at the point of 
 
 contact. 
 
 9. In an equiangular spiral, which is its own evolute, the area 
 included between the curve and PQ, the radius of curvature at P 
 touching the evolute in Q, is P<2 3 tana, where a is the angle of 
 the spiral, and PQ is supposed not to cut the curve between P 
 and Q. 
 
 10. Prove, by the method of Lemma IV., that the area included 
 between a catenary, the axis, the directrix, and the ordinate at any 
 point P is twice the area of the triangle formed by the axis, the 
 tangent at the vertex, and the straight line drawn perpendicular to 
 the tangent at P from the point of intersection of the axis and 
 directrix.
 
 SECTION II. 
 CENTRIPETAL FORCES. 
 
 PROP. I. THEOREM I. 
 
 When a body revolves in an orbit, subject to the ^ action of 
 forces tending to a fixed point, the areas which it de- 
 scribes by radii drawn to the fixed centre of force, are in 
 one fixed plane, and are proportional to the times of 
 describing them. 
 
 Let the time be divided into equal parts, and in the 
 first interval let the body describe the straight line 
 
 AB with uniform velocity, being acted on by no 
 force. In the second interval it would, if no force 
 acted, proceed to c in AB produced, describing Be 
 equal to AB ; so that the equal areas ASB, BSc de- 
 scribed by radii AS, BS, cS drawn to the centre S, 
 would be completed in equal intervals. 
 
 But, when the body arrives at B, let a centripetal 
 force tending to S act upon it by a single instanta-
 
 PROP. I. THEOREM I. 137 
 
 neous impulse, and cause the body to deviate from 
 the direction Be, and to proceed in the direction BC. 
 
 Let cC be drawn parallel to BS, meeting B C in C, then, 
 at the end of the second interval, the body will be 
 found at C, in the same plane with the triangle 
 ASB, in which Be and cC are drawn. Join SO] 
 and the triangle SBC, between parallels SB, Cc, 
 will be equal to the triangle SBc, and therefore 
 also to the triangle SAB. 
 
 In like manner, if the centripetal force act upon the 
 body successively at C, D, E, &c., causing the body 
 to describe in the successive intervals of time the 
 straight lines CD, DE, EF, &c., these will all lie 
 in the same plane ; and the triangle SOD will be 
 equal to the triangle SBC, and SDE to SCD, and 
 SEF to SDE. 
 
 Therefore equal areas are described in the same fixed 
 plane in equal intervals; and, componendo, the 
 sums of any number of areas SADS, SAFS, are to 
 each other as the times of describing them. 
 
 Let now the number of these triangles be increased, 
 and their breadth diminished indefinitely ; then their 
 perimeter ADF will be ultimately a curved line ; and 
 the instantaneous forces will become ultimately a 
 centripetal force, by the action of which the body is 
 continually deflected from the tangent to this curve, 
 and which will act continuously; and the areas 
 SADS, SAFS, being always proportional to the times 
 of describing them, will be so in this case. Q.E.D. 
 
 COR. 1. The velocity of a body attracted towards a 
 fixed centre in a non-resisting medium is recipro- 
 cally proportional to the perpendicular dropped 
 from that centre upon the tangent to the orbit. 
 
 For the velocities at the points A, B, C, D, E are as 
 the bases AB, BO, CD, DE, EF of equal triangles, 
 and, since the triangles are equal, these bases are 
 reciprocally proportional to the perpendiculars from
 
 138 NEWTON. 
 
 S let fall upon them. [And the same is true in the 
 limit, in which case the bases are in the direction 
 of tangents to the curvilinear limit, therefore the 
 velocity, &c.] 
 
 COR. 2. If on chords AB, EC of two arcs described in 
 equal successive times in a non-resisting medium by 
 the same body the parallelogram ABCV be com- 
 pleted, and the diagonal BV of this parallelogram 
 be produced in both directions in that position which 
 it assumes ultimately when those arcs are diminished 
 indefinitely, it will pass through the centre of force. 
 
 COE. 3. If, on AB, EC and on DE, EF chords of arcs 
 described in a non-resisting medium in equal times, 
 the parallelograms ABCV, DEFZ be completed, 
 the forces at B and E will be to one another in the 
 ultimate ratio of the diagonals B V, EZ, when the 
 arcs are indefinitely diminished. 
 
 For the velocities of the body represented by EC, EF 
 in the polygon are compounded of the velocities 
 represented by Be, BV and Ef, EZ; and those re- 
 presented by BV, EZ, which are equal to cC,fF, in 
 the demonstration of the proposition were generated 
 by the impulses of the centripetal force at B and E, 
 and are thus proportional to those impulses. [And 
 the same is true in the limit, in which case the ulti- 
 mate ratio of the impulses at any two points is the 
 ratio of the continuous forces at those points]. 
 
 COR. 4. The forces by which any bodies moving in 
 non-resisting media are deflected from rectilinear 
 motion into curved orbits, are to one another as 
 those sagittse of arcs described in equal times, which 
 converge to the centre of force and bisect the chords, 
 when those arcs are indefinitely diminished. 
 For the diagonals of the parallelograms ABCV, DEFZ 
 bisect each other, and these sagittse are halves of the 
 diagonals B V, EZ when the arcs are indefinitely 
 diminished. [And the same will be true whether 
 ABC and DEFbe parts of the same or of different
 
 PilOP. I. THEOREM I. 139 
 
 orbits described by bodies of equal mass, if the arcs 
 be described in equal times]. 
 
 COK. 5. And therefore the accelerating effects of the 
 same forces are to that of the force of gravity as 
 those sagittse are to vertical sagittas of the parabolic 
 arcs which projectiles describe in the same time. 
 
 COR. 6. All the same conclusions are true by the 
 Second Law of Motion, when the planes, in which 
 the bodies move together with the centres of force 
 which are situated in those planes, are not at rest, 
 but are moving uniformly and parallel to themselves. 
 
 The statement of the proposition in the original Latin is, 
 " Areas, quas corpora in gyros acta radiis ad immobile 
 centrum virium ductis describunt, et in planis imrno- 
 bilibus consistere, et esse temporibus proportionates." 
 
 Observations on the Proposition. 
 
 142. In all cases of motion of bodies it is of great importance 
 for the student to distinguish between the forces themselves 
 under the action of which the bodies may be moving, and the 
 effects which these forces produce. 
 
 It is only by an examination of the motion of a body that 
 we are able to infer that it is, or is not, acted on by any force ; 
 if we find that the body is moving with uniform velocity in a 
 straight line, we infer that it is, during such motion, acted upon 
 by no force, or that the forces which are acting upon it are in 
 equilibrium ; if we find that there is any change of direction or 
 velocity, gradual or abrupt, we infer that the body is moving 
 tinder the action of some force or forces; if the change be 
 gradual, we infer that such forces are finite, by which we mean 
 that the forces require a finite time to produce a finite change 
 whether of direction or velocity ; if, on the contrary, the change 
 be abrupt, we infer that the forces are what are called impulsive, 
 that is, such as produce a finite change in an instant. 
 
 Since then, in order to make any inference with respect to 
 the forces supposed to act, a clear conception of the motion of
 
 140 NEWTON. 
 
 a body must be first attained, it becomes necessary for the 
 student to be able to describe the motion of a particle of matter 
 as he would that of a point, independently of the causes of such 
 motion. 
 
 In doing this he must give a geometrical description of the 
 line traced by the point either in a plane or in space, and then 
 he must describe the rate, uniform or variable, with which this 
 line is traversed. 
 
 He may then proceed to attribute any change of direction or 
 velocity to the action of forces upon the particle whose motion 
 he has been examining. 
 
 143. In accordance with this method of separating the geo- 
 metry of the motion from the causes of the deviations, the first 
 proposition would be stated in such a manner as the following : 
 
 " When a point moves in a curve, in such a manner that the 
 accelerations at every point are in the direction of a fixed point, 
 the areas, which it describes by radii drawn to the fixed point to 
 which the accelerations tend, are in one fixed plane, and are 
 proportional to the times of describing them." 
 
 And, generally, if the words force and body, employed by 
 Newton, be replaced by acceleration and point, the resulting 
 statements will be in accordance with this geometrical method 
 of description. It will then be easy to use such terms in the 
 proofs as will not imply, in the manner of expression, the action 
 of force ; thus, instead of saying " let a centripetal force tending 
 to S act upon the body by a single instantaneous impulse," 
 we may use the words " let a finite velocity be communicated 
 to the point in the direction of S." 
 
 144. It should be carefully observed that, before proceeding 
 to the limit, it is proved that any polygonal areas SADS, 
 SAFSj are proportional to the times of description of their 
 perimeters; so that ultimately these areas become finite curvi* 
 linear areas, described infinite times. 
 
 145. In proceeding to the ultimate state of the hypothesisi 
 it is concluded readily from Lemmas II. and III. that the 
 curvilinear areas are the limits of the polygons ; but a greater
 
 PROP, I. THEOREM I. 141 
 
 difficulty arises in the transition from the discontinuous motion 
 under the action of instantaneous impulsive forces to the con- 
 tinuous motion under the action of a continuous force tending 
 to S. For, in the curvilinear path of the body which is the 
 limit of the perimeter of the polygon, the direction of the motion 
 at the angular points of the polygon is different, and also the 
 deflection from the direction of motion is twice as great in the 
 polygon as it is in the curve. 
 
 Now, although we may assume that the curvilinear limit of 
 the perimeter of the polygon may be described under the action 
 of some force, is that force the same which is the limit of the 
 series of impulses? 
 
 The centripetal force supposed to act with a simple in- 
 stantaneous impulse, " impulsu unico et magno," is supposed 
 to generate a finite velocity at once, which effect a finite force 
 cannot produce. 
 
 If, instead of this imaginary impulse, we suppose a force 
 finite, but very great, and acting for a very short time, the 
 effect upon the figure would be to round off the angular points 
 of the polygon. 
 
 The transition from the impulses to the continuous force, in 
 the ultimate form of the hypothesis, must be considered as 
 axiomatic, like the ultimate equality of the ratio of the finite 
 arc to the perimeter of the inscribed polygon. 
 
 146. We can, however, shew that if the curvilinear limit of 
 the polygon be described under the action of some continuous 
 force tending to S, the effect of this force, estimated by the 
 quantity of motion generated in the interval between the im- 
 pulses, will be ultimately the same as that generated by the 
 impulse. 
 
 Consider first the geometrical properties of the limit of the 
 polygonal perimeter. Let BT, CU be tangents at B, C to the 
 curvilinear limit, and let Cc intersect BT'm T, fig. page 136. 
 
 Now, since Cc ultimately vanishes compared with Be, BG 
 and Be or AB and BG are ultimately in a ratio of equality, 
 and Cc is ultimately bisected by BT, by (2) page 102; also, 
 CU=BU=UT ultimately, by (I) page 102.
 
 142 NEWTON. 
 
 Consider next the effects produced by the different kinds of 
 force which act in the two cases. 
 
 In the polygonal path, the impulsive force at B generates 
 a velocity with which the body describes Cc in the time t in 
 which AB or BG is described, the measure of the effect of the 
 
 Cc 
 impulse is therefore the velocity . 
 
 In the curvilinear path, the deflection from the direction B I 
 at B, in the same time t, is TC, by means of the continuous 
 action of finite forces, and if we suppose the force ultimately 
 uniform in magnitude and direction, the measure of the ac- 
 
 2 TO 
 celerating effect of the force will be ^~ , and the velocity 
 
 2TC Cc 
 
 generated in that time will be -, .= . 
 
 t t 
 
 Hence the effects of the finite and impulsive forces, measured 
 by the quantity of motion produced, are the same. 
 
 147. We can also shew that a continuous force, which gene- 
 rates the same quantity of motion as the impulse at B in the 
 time from B to C, would cause the body on arriving at C to 
 move in the direction of the tangent to the curvilinear limit of 
 the perimeter. 
 
 For the velocity due to the action of the finite force at the 
 
 2 TC 
 end of time t being ultimately in the direction TC, and 
 
 BT 2 TTT 
 that in the direction BT being = ; therefore TC, UT 
 
 represent the velocities in those directions ; therefore UC is the 
 direction of motion at (7, that is, the body moves in the direction 
 of the tangent at C. 
 
 148. COR. 1. The corollary may be proved directly from 
 the proposition, for the proportionality of the areas to the times 
 of describing them will be true if the force suddenly cease to act, 
 in which case the body will proceed in the direction of the tangent. 
 
 Let V be the velocity at the point A, A SB the curvilinear 
 area described in any time T, AT=V.T the space described 
 if the force cease to act. Job ST and draw SY perpendicular
 
 PROP. I. THEOREM I. 
 
 143 
 
 to AT, then area ASB = triangle SAT=$V.TxSY, also 
 area ASBcc T-, therefore F varies inversely as SY. 
 
 Again, if h be twice the area described in the unit of time 
 
 employed in estimating the accelerating effect of the force tend- 
 ing to S and the velocity of the body, 
 
 2.area4=Ar; .-. h=V.SY. 
 
 By the use of this area the proportions employed in subse- 
 quent propositions by Newton may be converted into equations, 
 for the convenience of calculation. 
 
 If bodies move in curves for which the areas, described in 
 
 the same time, are not equal, Foe ~y.. 
 
 149. COR. 4. The statement in this corollary requires modi- 
 fication, for, unless the forces be considered only with reference 
 to their accelerating effects, or unless the bodies be supposed of 
 equal mass, the forces will not be proportional to the sagittae. 
 
 150. COR. 5. The object of this corollary is to determine 
 the numerical measure of the central force which governs the 
 motion of a body, when the circumstances of the motion are 
 known ; for it supplies us with the ratio of this force to the force 
 of gravity on the same body at any place, the measure of which 
 can be determined by experiment. 
 
 Applications of ike Proposition. 
 
 151. PROP. When the force, instead of tending to a fixed 
 point, acts in parallel lines, the property of the motion enunciated
 
 144 NEWTON. 
 
 in teh proposition may be replaced by tte property that the 
 resolved part of (he space described perpendicular to the direc- 
 tion of the force is proportional to the times. 
 
 This is immediately deducible from the second law of motion, 
 since there is no force in the direction perpendicular to that of 
 he forces, and the velocity in that direction is uniform. 
 
 That this is the result of the properties in the proposition 
 may be shewn by removing the centre of force to an infinite 
 distance. 
 
 S 
 
 Let 8 be the centre of force, AMN perpendicular to SB, the 
 area ABCS is proportional to the time of describing A C, and 
 the areas AMNS and ABCS are ultimately equal when S is 
 removed to an infinite distance in BMS, hence the triangle ASN 
 is proportional to the time, and therefore the base AN, which 
 varies as the triangle ASN, is also proportional to the time, 
 and therefore, since CN is ultimately perpendicular to AN, 
 the proposition is proved. 
 
 152. PROP. If a body describe a curvilinear orbit about a 
 force tending constantly to a fixed point, the area described in a 
 given time will "be unaltered, if the force be suddenly increased 
 ar diminished, of if the body be acted on at any moment by an 
 impulsive force tending to that point. 
 
 For, if in the polygon the impulse at any point B be in- 
 creased or diminished by any force tending to or from S, the 
 only effect will be to remove the vertex C of the triangle SBC to
 
 PROP. I. THEOREM I. 145 
 
 some other point in the line cG parallel to 8, hence the area 
 will be unaltered, and the argument which establishes the 
 equality of polygonal areas in a given time will proceed as 
 before. Hence in the limit the curvilinear areas described in 
 a given time will be unaltered. 
 
 If the new force introduced at S be impulsive, the angle 
 ABC will remain less than two right angles when we proceed 
 to the limit, and the two parts of the curve will cut one another 
 at a finite angle. 
 
 Hence, in any calculation made upon supposition of such 
 changes of force, the value of h, Art. 148, will be the same 
 before and after the change of the force. 
 
 Apses. 
 
 153. DBF. In any orbit described under the action of a 
 force tending to a fixed centre, a point at which the direction 
 of the motion is perpendicular to the central distance is called 
 an apse, the distance from the centre is called an apsidal 
 distance, and the angle between consecutive apsidal distances 
 is called an apsidal angle. 
 
 Thus, in the ellipse about the centre, the four extremities of 
 the axes are apses; there are two different apsidal distances, 
 and every apsidal angle is a right angle. 
 
 In the ellipse about a focus, the apses are at the greatest and 
 least distances, and the apsidal angle is two right angles. 
 
 154. In a central orbit described under the action of forces 
 tending to a fixed point, each apsidal distance will divide the orbit 
 symmetrically, if the forces be always equal at equal distances. 
 
 It is easily shewn that, in any orbit described by a body 
 under the action of forces tending to a fixed point, the forces 
 depending only upon the distance, if a second body be projected 
 at any point with the velocity of the first in the opposite direc- 
 tion, it will proceed to describe the same orbit in the reverse 
 direction, under the action of the same forces. 
 
 For, let ABC be a portion of the polygon whose limit is 
 the curvilinear path of the body, and produce AB to c, and 
 CB to a, making Be = AB, and Ba = CB. 
 
 U
 
 146 NEWTON. 
 
 The impulse at B is measured by cG when the body de- 
 scribes ABC, and if the motion be reversed, the same impulse 
 at B would cause the body to move in BA, with the velocity 
 which it had in AB, since aA = cC. And the same is true 
 
 throughout the polygonal path, hence the assertion is true for 
 the whole path, described under the action of impulses which 
 are always the same at the same points, and therefore, proceed- 
 ing to the limit, the statement made for any orbit is proved. 
 
 Hence, since the forces are equal at equal distances on 
 both sides of the apse, the path of the body from an apse 
 being similar and equal to the path which would be described 
 if the velocity were reversed at the apse, is similar to the path 
 described in approaching the apse; whence the proposition is 
 established. 
 
 155. There are only two different apsidal distances, and 
 all apsidal angles are equal. 
 
 For, after passing a second apse, the curve being symme- 
 trical on both sides, a third apse will be in such a position that 
 the apsidal distance is the same as for the first apse, and all the 
 apsidal angles are shewn similarly to be equal. 
 
 156. COR. Hence a central orbit can never re-enter itself 
 unless the ratio of the apsidal angle to a right angle be com- 
 mensurable, and if it be so, the curve will always re-enter. 
 
 Illustrations. 
 
 (1) If a body describe an ellipse under the action of a force 
 tending to one of the foci, the square of the velocity varies inversely
 
 PROP. I. THEOREM I. 147 
 
 as the distance from that focus, and directly as the distance from 
 the other. 
 
 For BC* : SY* :: HZ: SY :: HP: P; 
 
 (2) The velocity is greatest when the lody is at the extremity 
 of the major axis which is nearer to the focus to which the force 
 tends, and least at the other extremity. 
 
 For SY'is the least in the first and greatest in the second 
 position. 
 
 (3) The velocity at an extremity of the minor axis is a 
 geometric mean between the greatest and least velocities. 
 
 For at this point HZ=BC, and at the extremities of the 
 major axis the values of HZ are Sa and SA, and BC* = SA.Sa. 
 
 (4) In the equiangular spiral described under the action of a 
 force tending to the focus , the velocity cc -~_. 
 
 For, Foc SP. 
 
 (5) If the force tend to the centre of the elliptic orbit described 
 by a body, the time between the extremities of conjugate diameters 
 will be constant. 
 
 For the area PCD is constant. 
 
 (6) The velocity at any point of an ellipse about a force tend- 
 ing to a focus is compounded of two uniform velocities, one 
 
 perpendicular to the radius vector, and the other perpendicular to the 
 major axis. 
 
 Let S be the centre of force, SY, EZ perpendiculars on the 
 tangent at P, join SP, CZ. Then HZ, ZC parallel to PS, and
 
 148 NEWTON. 
 
 CH are perpendicular to the three directions; therefore the 
 velocity represented by HZ in magnitude is the resultant of 
 the two represented by CZ and EC', but the velocity perpen- 
 dicular to HZ= - S y=p - HZ '-> therefore the velocities perpen- 
 dicular to EC and CZ are ^ ae and 75 a. 
 
 XVIII. 
 
 1. If different bodies be projected with the same velocity from 
 a given point, all being attracted by forces tending to one fixed 
 point, shew that the areas described by the lines drawn from the 
 fixed point to the bodies will be proportional to the sines of the 
 angles of projection. 
 
 2. When a body describes a curvilinear orbit under the action 
 of a force tending to a fixed point, will the direction of motion or 
 the curvature of the orbit at any point be changed, if the force at 
 the point receive a finite change ? 
 
 3. A body moves in a parabola about a centre of force in the 
 vertex, shew that the time of moving from any point to the vertex 
 varies as the cube of the distance of the point from the axis of the 
 parabola. 
 
 4. In a parabolic orbit described round a force tending to the 
 focus, shew that the velocity varies inversely as the normal at any 
 point. Shew also that the sum of the squares of the velocities ta 
 the extremities of a focal chord is constant. 
 
 5. If the velocity at any point of an ellipse described about 
 the centre can be equal to the difference of the greatest and least 
 velocities, the major axis cannot be less than double of the minor. 
 
 6. If an ellipse be described under the action of a force tending 
 to the centre, shew that the velocity will vary directly as the 
 diameter conjugate to that which passes through the body; also 
 that the sum of the squares of the velocities at the extremities of 
 conjugate diameters will be constant. 
 
 7. In an ellipse described round a force tending to the focus, 
 compare the intervals of time between the extremities of the same 
 latus rectum, when AC =208. 
 
 8. In the ellipse described about the focus S, ASH A being the 
 major axis, time in AB : time in BA' : : TT - 2e : IT + 2e.
 
 PROP. I. THEOREM I. 149 
 
 9. If the velocities at three points in an ellipse described by 
 a particle, the acceleration of which tends to either of the foci, be 
 in arithmetical progression, prove that the velocities at the opposite 
 extremities of the diameters passing through these points will be 
 in harmonica! progression. 
 
 10. If v 2 be the velocities at the extremities of a diameter 
 of an ellipse described about the focus, and u the velocity at either 
 of those points when it is described about the centre, prove that 
 u (v l + 2 ) will be constant. 
 
 11. In a central orbit, the velocity of the foot of the perpen- 
 dicular from the centre of force on the tangent varies inversely as 
 the length of the chord of curvature through the centre of force. 
 
 12. A particle is describing a parabola about its focus S; if P 
 and Q be two points of its path, shew that its velocity at Q will 
 be compounded of the velocity at P and a velocity which will be 
 constant if the angle PSQ be constant. 
 
 XIX. 
 
 1. A body describes a parabola about a centre of force in the 
 focus ; shew that its velocity at any point may be resolved into 
 two equal constant velocities, respectively perpendicular to the axis 
 and to the focal distance of the point. 
 
 2. A body describes an ellipse under the action of a central 
 force tending to one of the foci ; shew that the sum of the velocities 
 at the extremities of any chord parallel to the major axis varies 
 inversely as the diameter parallel to the direction of motion at 
 those points. 
 
 3. A body moves in an ellipse under the action of a force 
 tending to the centre ; shew that the component of the velocity at 
 any point perpendicular to either focal distance is constant ; and 
 that the sum of the squares of the velocities, at the extremities 
 of any pair of semi-conjugate diameters, resolved in any given 
 direction is constant. 
 
 4. In an ellipse described about a focus, the time of moving 
 from the greatest focal distance to the extremity of the minor axis 
 is m times that from the extremity of the minor axis to the least 
 focal distance ; find the eccentricity, and shew that, if there be 
 a small error in m, the corresponding error in the eccentricity will 
 vary inversely as (1 + m} z . 
 
 5. If the velocity of a body in a given elliptic orbit be the same 
 at a certain point, whether it describe the orbit in a time t about
 
 150 NEWTON. 
 
 one focus, or in a time t' about the other, ^prove that, 2a being the 
 
 major axis, the focal distances will be , and -- . 
 
 6. A body describes a parabola about the focus; if the seg- 
 ments PS, Sp of the focal chord PSp be in the ratio n : 1, prove 
 that the time in pA : time in AP : : 3n + 1 : n 2 (n + 3). 
 
 7. If Fbe perpendicular to the tangent to a curve at P, and 
 P and Fboth move as if under the action of a central force tending 
 to S, prove that the radius of curvature at P will vary as SY. 
 
 8. If P, Q be any two points in an ellipse described by a 
 particle under the action of a force tending to the centre, prove 
 that the velocity acquired in passing from P to Q will be in the 
 direction QP', where P' is the other extremity of the diameter 
 through P. 
 
 9. Two points P, P' are moving in the same ellipse, in the 
 same directions, with accelerations tending to the centre C\ shew 
 that the relative velocity of one with regard to the other is parallel 
 and proportional to CT, where T is the point of intersection of the 
 tangents at P and P'. If the points move in opposite directions, 
 what will be their relative velocity ? 
 
 10. Two particles revolve in the same direction in an oval 
 orbit round a centre of force S, which divides the axis unequally, 
 starting simultaneously from the extremities of a chord PQ, drawn 
 through S. Prove that, when they first arrive in positions JR, T 
 respectively, such that the angle ST is a minimum, the time from 
 R to the next apse will be an arithmetic mean between the times 
 from P to the next apse and to Q from the last apse. 
 
 11. Two equal particles are attached to the extremities of a 
 string of length 21, and lie in a smooth horizontal plane with the 
 string stretched ; if the middle point of the string be drawn with 
 uniform velocity v in a direction perpendicular to the nitial direc- 
 tion of the string, shew that the path of each particle will be a 
 
 cycloid, and that the particles will meet after a time -5- . 
 
 12. If the velocity in a central orbit can be resolved into two 
 constant components, one perpendicular to the radius vector, and 
 the other to a fixed straight line, shew that the curve must be 
 a conic. 
 
 13. The velocity in a cardioid described about a force tending 
 to the pole varies in the inverse sesquiplicate ratio of the distance. 
 
 14. The velocity in the lemniscate varies inversely as the cube 
 of the central distance, when a particle moves in the curve round 
 a force tending to the centre.
 
 PROP. II. THEOEEM II. 
 
 151 
 
 PROP. II. THEOREM II. 
 
 Every body, which moves in any curve line described in a 
 plane, and describes areas proportional to the times of 
 describing them about a point either fixed or moving 
 uniformly in a, straight line, by radii drawn to that 
 point, is acted on by a centripetal force tending to the 
 same point. 
 
 Case 1. Let the time be divided into equal intervals, 
 and in the first interval let the body describe AB 
 with uniform velocity, being acted on by no force ; 
 in the second interval it would, if no force acted, pro- 
 ceed to c in AB produced, describing Be equal to AB\ 
 and the triangles ASB, BSc would be equal. But 
 
 when the body arrives at B, let a force, acting upon 
 it by a single impulse, cause the body to describe 
 BC in the second interval of time, so that the tri- 
 angle BSC is equal to the triangle ASB, and there- 
 fore also to the triangle BSc ; therefore BSC and 
 BSc are between the same parallels, hence BS is
 
 152 NEWTON. 
 
 parallel to cC, and therefore US was the direction of 
 the impulse at B. 
 
 Similarly, if at C, Z>, ... the body be acted on by im- 
 pulses causing it to move in the sides CD, DE, ... of 
 a polygon, in the successive intervals, making the 
 triangles CSD, DSE, ... equal to ASB and BSC, the 
 impulses can be shewn to have been in the directions 
 CS, DS, .... Hence, if any polygonal areas be de- 
 scribed proportional to the times of describing them, 
 the impulses at the angular points will all tend to S. 
 
 The same will be true if the number of intervals be 
 increased and their length diminished indefinitely, 
 in which case the series of impulses will approximate 
 to a continuous force tending to S, and the polygons 
 to curvilinear areas, as their limits. Hence the pro- 
 position is true for a fixed centre. 
 
 Case 2. The proposition will also be true if S be a 
 point which moves uniformly in a straight line, for, 
 by the second law of motion, the relative motion will 
 be the same, whether we suppose the plane to be at 
 rest, or that it moves together with the body which 
 revolves and the point $, uniformly in one direction. 
 
 COR. 1. In non-resisting media, if the areas be not 
 proportional to the times, the forces will not tend 
 to the point to which the radii are drawn, but will 
 deviate in consequentid, i.e. in that direction towards 
 which the motion takes place, if the description of 
 areas be accelerated ; but if it be retarded, the devi- 
 ation will be in antecedently. 
 
 COR. 2. And also in resisting media, if the description 
 of areas be accelerated, the directions of the forces 
 will deviate from the point to which the radii are 
 drawn in that direction towards which the motion 
 takes place. 
 
 SCHOLIUM. 
 
 A body may be acted on by a centripetal force com- 
 pounded of several forces. In this case, the meaning
 
 PROP. II. THEOREM II. 153 
 
 of the proposition is, that that force, which is the 
 resultant of all, tends to S. Moreover, if any force 
 act continually in a line perpendicular to the plane 
 of the areas described, this force will cause the body 
 to deviate from the plane of its motion, but will 
 neither increase nor diminish the amount of area 
 described, and therefore must be neglected in the 
 composition of the forces. 
 
 Observations on the Proposition. 
 
 157. The description of an area round a point in motion 
 may be explained by the following construction for the relative 
 orbit, in the case of motion about a point which is itself moving 
 uniformly in a straight line. 
 
 Let SS' be the line in which S moves uniformly, and let the 
 body move from A to B in the same time that S moves from 8 
 to S', and let P, <r be simultaneous positions of the body and of S. 
 
 A. 
 
 If PP' be drawn equal and parallel to a- 8, and the same 
 construction be made for every point in the path of the body, 
 the curve AP'B\ which is the locus of P', will be the orbit which 
 the body would appear to describe to an observer at S, who 
 referred all the motion to the body ; for SP will be equal and 
 parallel to o-P, and therefore the distance of the body, and the 
 direction in which it is seen, will be the same in the two cases. 
 
 If <), Q' be corresponding points near P and P', and the force 
 at a- be supposed to act impulsively, the relative motion round <r 
 will be unaltered if we apply to both P and <r velocities equal to 
 
 x
 
 154 NEWTON. 
 
 that of ff and in a contrary direction, but in this case a- will 
 be reduced to rest and the velocity of P will be the velocity 
 relative to o-. Take PQ and <r<r', which are described in the 
 same time, to represent the velocities of P and <r, and let Qq be 
 equal and parallel to o-V, then Pq represents the velocity of P 
 relative to <r; and, since Q'q = Sff a<r = P'P, P' Q is equal 
 and parallel to Pq, and therefore the velocity in the orbit Aff 
 about S at rest is equal to the relative velocity about S in 
 motion. 
 
 158. COR. 1. Reverting to the polygonal area, if the tri- 
 
 C' 
 
 angle SBC' be greater than the triangle SAB, the impulse at B 
 will not be in the direction BS, but BU, parallel to c(7', that is, 
 if the areas be not proportional to the times but be in an 
 increasing ratio, the direction of the force will deviate towards 
 the direction in which the description of areas is accelerated; 
 and vice versd, when the description is retarded. 
 
 159. COR. 2. The effect of a resisting medium is to retard 
 the motion, or, supposing it the limit of a series of impulses, we 
 must conceive an impulse at B, in the case of the polygon, in the 
 direction BA ; if therefore the description of areas be accelerated, 
 the impulse applied at B in the direction BU' must act still 
 further in consequenttd than that in BU, in order that, with the 
 impulse corresponding to the resistance of the medium, it may 
 produce a resultant impulse in the direction of BU. The effect 
 of the resistance alone is to retard the description of areas. 
 
 If the force act in consequentid, the resultant of this force
 
 PROP. II. THEOREM II. 155 
 
 and the resistance of the medium may act in the direction B8, 
 and the proportionality of the areas to the times be preserved. 
 
 160. PROP. Let ABODE be any plane curve, 8 any point 
 in the plane, to shew that, generally, the curve can be described 
 under the action of a force tending to or from 8, with finite velo- 
 city, the velocity at any given point being any given velocity. 
 
 For arcs AB, BO, ... can be measured from any point A, 
 along the curve, such that the areas SAB, SBC,... are all equal, 
 
 and of any magnitude. Also a body can be made, by some force 
 to move along the curve with finite velocity, so as to describe the 
 arcs AB, BO, ... in equal times, unless the tangent to one of 
 the arcs, as DE, pass through S, in which case, if the arcs be 
 indefinitely diminished, DE, AB will not be finite ultimately. 
 
 Hence by Prop. II. a body can move with finite velocity 
 under the action of some force tending to or from 8, generally. 
 
 161. NOTE 1. Since in making the motion of the body such 
 that it shall describe equal areas in equal times we are only con- 
 cerned with the ratio of the velocities, the velocity at any point 
 A may be any given velocity. 
 
 162. NOTE 2. Or if we please we may suppose the force at 
 any point any given force ; for, in the case of the polygon, the 
 velocity generated by the impulse at B is to the velocity in AB 
 as cCto Be, hence the impulse at B may be of any magnitude 
 if we choose the velocity in AB properly. 
 
 163. NOTE 3. The ratio of the velocities will be the same 
 at two given points, for all forces tending to a given centre, 
 under the action of which the curve can be described.
 
 156 NEWTON. 
 
 164. NOTE 4. Hence a body can move throughout any 
 ellipse under the action of a centripetal force tending to the 
 centre or focus, the force depending only on the distance, since 
 in these cases the curve is symmetrical on opposite sides of any 
 apse ; or about any point within the ellipse, if the forces do not 
 depend only on the distance, since no point within an ellipse lies 
 on any tangent. 
 
 165. NOTE 5. In the case of an oval, S being an external 
 point, a body can move with finite velocity under the action of 
 a force tending to the point S, in the portion which is concave to 
 S, and from S, in that which is convex to S, but not from one 
 portion to the other. 
 
 XX. 
 
 1. If an ellipse be described so that the sum of the areas 
 swept out by radii drawn to the vertices is proportional to the 
 times of describing them, prove that the resultant acceleration 
 will tend to the centre. 
 
 2. A body is moving in a parabola, and the time from the 
 vertex to any point varies as the cube of the ordinate ; shew that 
 this motion could be caused by the action of a central force. 
 
 3. A body was moving in a circle, and it was observed that the 
 time of describing any arc from a fixed point varied as the sum of 
 the arc and the perpendicular distance from one extremity of the 
 arc on the diameter through the other ; shew that the body was 
 acted on by a central force. 
 
 4. A heavy particle falls from the cusp to the vertex of a 
 cycloid, whose axis is vertical ; shew that a particle could describe 
 the cycloid in the same manner under the action of a constant force 
 directed to a certain moving point. 
 
 5. From the centre of a planet a perpendicular is let fall upon 
 the plane of the ecliptic ; prove that the foot of this perpendicular 
 will move as if it were a particle acted on by a force tending to the 
 Bun's centre.
 
 PROP. III. THEOREM III. 157 
 
 PROP. III. THEOREM III. 
 
 Every body, which describes areas proportional to the times 
 of describing them by radii drawn to the centre of another 
 body which is moving in any manner whatever, is acted on 
 by a force compounded of a centripetal force tending to 
 that other body, and of the whole accelerating force which 
 acts upon that other body. 
 
 Let the first body be L, the second T, T moves under 
 the action of some force P, L under the action of 
 
 another force F. At every instant apply to both 
 bodies the force P in the contrary direction to that 
 in which it acts, as represented by the dotted arrows. 
 
 L will continue to describe about T, as before, areas 
 proportional to the times of describing them, and 
 since there is now no force acting on T, T is at rest 
 or moves uniformly in a straight line. 
 
 Therefore, by Theorem II. , the resultant of the force F 
 and the force P applied to L tends to T. 
 
 Hence Fis compounded of a centripetal force tending to 
 T, and of a force equal to that which acts on T. Q.E.D. 
 
 COR. 1. Hence, if a body L describe areas proportional 
 to the times of describing them by radii drawn to 
 another body T] and from the whole force which 
 acts upon L, whether a single force or compounded of 
 several forces, be taken away the whole accelerating 
 force which acts upon the other body T\ the whole
 
 158 NEWTON. 
 
 remaining force, which acts upon L, will tend to the 
 other body T as a centre. 
 
 COR. 2. And if these areas be very nearly proportional 
 to the times of describing them, the remaining force 
 will tend to the other body very nearly. 
 
 COR. 3. And, conversely, if the remaining force tend 
 very nearly to the other body T, the areas will be 
 very nearly proportional to the times. 
 
 COR. 4. If the body L describe areas which are very 
 far from being proportional to the times of describing 
 them, by radii drawn to another body T, and that 
 other body T be at rest, or move uniformly in a 
 straight line, then either there will be no centripetal 
 force tending to that other body T, or such centri- 
 petal force will be compounded with the action of 
 other very powerful forces, and the whole force com- 
 pounded of all the forces, if there be many, may^ be 
 directed towards some other centre fixed or moving. 
 
 The same will hold, when the other body moves in any 
 manner whatever, if the centripetal force spoken of 
 be understood to be that which remains after taking 
 away the whole force acting upon the other body T. 
 
 SCHOLIUM. 
 
 Since the equable description of areas is a guide to the 
 centre to which that force tends, by which a body is 
 principally acted on, and by which it is deflected 
 from rectilinear motion, and retained in its orbit, we 
 may, in what follows, employ the equable description 
 of areas as a guide to the centre, about which all 
 curvilinear motion in free space takes place. 
 
 Illustration. 
 
 166. As an illustration of the last propositions and their 
 corollaries, we may state some of the observed facts in the 
 motion of the Moon, Earth, and Sun, and make the deductions 
 corresponding to them.
 
 PROP. Ill THEOREM III. 
 
 159 
 
 Suppose the Moon's orbit relative to the Earth to be nearly 
 circular, and let ABCD be this orbit, E the Earth. 
 
 (1) The areas described by the radii drawn from the Moon 
 to the Earth are nearly proportional to the times of describing ; 
 hence the resultant force on the Moon tends nearly to E. 
 
 (2) If ES the line joining the centres of the Earth and Sun 
 meet the Moon's relative orbit about the Earth in A, (7, and 
 DEB be perpendicular to DS, the description of areas will be 
 accelerated as the Moon moves from D to A and from B to (7, 
 and retarded from A to B and from C to D ; hence the direction 
 of the resultant force on the Moon in the positions M^ M^ 
 J/ 3 , Jf 4 , will be in the directions of the arrows slightly inclined 
 to the radii drawn to E. 
 
 From these observed facts, we see that when the force, under 
 the action of which E moves, is applied to the Moon in the 
 contrary direction, the remaining force tends in the directions 
 of the arrows. 
 
 By the supposition that the Earth and Moon are acted on 
 by forces tending to the sun, whose distance compared with EM 
 is very great, and that the differences of the forces on these 
 bodies are not very great, the accelerating effect of the force on 
 the Moon in DAB being greater than that on the Earth, and in 
 BCD less, the circumstances of the description of areas in the 
 motion of the Moon are accounted for.
 
 160 
 
 NEWTON. 
 
 PROP. IV. THEOREM IV. 
 
 The centripetal forces on equal bodies, which describe dif- 
 ferent circles with uniform velocity, tend to the centres of 
 the circles, and are to each other as the squares of arcs 
 described in the same time, divided by the radii of the 
 circles. 
 
 The bodies move uniformly, therefore the arcs described 
 are proportional to the times of describing them ; and 
 the sectors of circles are proportional to the arcs on 
 which they stand, therefore the areas described by 
 radii drawn to the centres are proportional to the 
 times of describing them ; hence, by Prop. II., the 
 forces tend to the centres of the circles. 
 
 Again, let AB, ab be small arcs described in equal times, 
 
 AD, ad tangents at A, a; ACSG, acsg diameters 
 through A, a. Join AB, ab, and draw BC, be per- 
 pendicular to A Gr, ag. 
 
 When the arcs AB, ab are indefinitely diminished, since 
 AC, ac are sagittae of the double of arcs AB, ab 
 described in equal times, they are ultimately, by 
 Prop. L, Cor. 4, as the forces at A and a. 
 
 But AC.AG = (cMABf and ac.ag =
 
 PROP. IV. THEUKEM IV. 161 
 
 .*. force at A : force at a :: AC : ac ultimately, 
 .. (cbdABf (didab)* (sicAB)* (area*)" . 
 
 ~~ *- ^~ : ~~ ' Lem - VIL 
 
 Take AE, ae two arcs described in any equal finite 
 ti fiies, then AE: ae:: AB : ab, since the bodies move 
 uniformly, and this is also true in the limit ; 
 
 .*. force at A : force at a :: 7-73 : . O.E.D. 
 
 AS as 
 
 COR. 1. Since these arcs are proportional to the velo- 
 cities of the bodies, the centripetal forces will be in 
 the ratio compounded of the duplicate ratio of the 
 velocities directly, and the simple ratio of the radii 
 inversely. 
 
 That is, if F, v be the velocities in the two circles, R, r 
 the radii, JT,/the centripetal forces, AE: ae::V:vj 
 
 ..-.. 
 
 COR. 2. And since the circumferences of the circles are 
 described in their periodic times, the velocities are in 
 the ratio compounded of the ratio of the radii directly 
 and the ratio of the periodic times inversely ; hence 
 the centripetal forces are in the ratio compounded 
 of the ratio of the radii directly, and of the ratio of 
 the squares of the periodic times inversely. 
 
 If P, p be the periodic times in the two circles re- 
 spectively, 
 
 2>rrr R r 
 
 R 
 
 COR. 3. Hence, if the periodic times be equal, and there- 
 fore the velocities proportional to the radii, the cen 
 tripetal forces will be as the radii ; and conversely.
 
 ] 62 NEWTON. 
 
 IfP=j, then V\ v:: R:r; 
 
 ,.*:,::? .*,*,, 
 
 COR. 4. Also if the periodic times be in the subduplicate 
 ratio of the radii, the centripetal forces will be equal. 
 That is, if P 2 : / : : R : r, then F=f, by Cor. 2. 
 
 COR. 5. If the periodic times be as the radii, and 
 therefore the velocities equal, the centripetal forces 
 will be reciprocally as the radii ; and conversely. 
 
 COR. 6. If the periodic times be in the sesquiplicate 
 ratio of the radii, and therefore the velocities recipro- 
 cally in the subduplicate ratio of the radii, the cen- 
 tripetal forces will be reciprocally as the squares of 
 the radii ; and conversely. 
 
 That is, 
 
 COR. 7. And, generally, if the periodic times vary as 
 any power R" of the radius R, and, therefore, the velo- 
 city vary inversely as the power 7T' 1 , the centripetal 
 force will vary inversely as 7T 1 ' 1 ; and conversely. 
 
 COR. 8. All the same proportions can be proved con- 
 cerning the times, velocities, and forces, by which 
 bodies describe similar parts of any figures whatever, 
 which are similar and have centres of force similarly 
 situated, if the demonstrations be applied to those 
 cases, uniform description of areas being substituted for 
 uniform velocity, and distances of the bodies from the 
 centres of force for radii of the circles. 
 
 Let AE, ae be similar arcs of similar curves described 
 by bodies about forces tending to similarly situated 
 points S, s; and let AB, abbe small arcs described
 
 PROP. IV. THEOREM IV. 163 
 
 in equal times ; BD, Id subtenses parallel to SA, sa 
 A V y av chords of curvature at A, a, so that 
 AV: av::AS: as. 
 
 J5 
 
 
 
 D ^ 
 
 A 
 
 T ^r 
 
 Then, force at A : force at a : : DB : db } ultimately, 
 
 AB* ab* AB* ab* ... 
 : : -r-~ : : : - n -r- : , ultimately ; 
 A V av SA sa ' J ' 
 
 and if V 9 v be the velocities at A, a since AB, ab are 
 described in equal times, AB : ab :: V: v, ultimately; 
 
 F 2 tf 
 
 .'. force at A : force at a :: -^-r : , as Cor. 1. 
 
 8 A sa 1 
 
 Again, if AB, ab be small similar arcs described in times 
 T, t, instead of being arcs described in equal times, 
 and P, p be the times of describing similar finite 
 arcs AE y ae t 
 
 T:P:: QXQQ.ASB : area ASE :: areaasb : axeaase \\t\p\ 
 
 therefore, when AB, ab are indefinitely diminished, 
 T: tr.P:p. 
 
 ::::: ^, as Co, 2. 
 
 COR. 9. It follows also from the same proposition, that 
 the arc which a body, moving with uniform velocity 
 in a circle under the action of a given centripetal force, 
 describes in any time, is a mean proportional be- 
 tween the diameter of the circle, and the space 
 through which the body would fall from rest under 
 the action of the same force and in the same time.
 
 164 NEWTON. 
 
 For, let AL be the space described from rest in the same 
 time as the arc AJE, then since, if BD be perpendi- 
 cular to the tangent at A, BD will be ultimately the 
 space described by the body, under the action of the 
 force at A, in the time in which the body describes 
 the arc AB, and the times are proportional to the arcs ; 
 
 .-. AL-.BDi: AE': AB* ; 
 .-. AL.Aa : BD.AG :: AE* : AB*', 
 and BD.AG = (cMABJ t = (aTcAB) i , ultimately; 
 therefore AL.AG = AE\ or AL : AE :: AE : AG. 
 
 Q. E. D. 
 
 SCHOLIUM. 
 
 The case of the sixth Corollary holds for the heavenly 
 bodies, and on that account the motion of bodies acted 
 upon by a centripetal force, which decreases in the 
 duplicate ratio of the distance from the centre offeree, 
 is treated of more fully in the following section. 
 
 Moreover, by the aid of the preceding proposition and 
 its corollaries, the proportion of a centripetal force 
 to any known force, such as gravity, can be obtained. 
 For, if a body revolve in a circle concentric with 
 the earth by the action of its own gravity, this 
 gravity is its centripetal force. 
 
 But, from the falling of heavy bodies, by Cor. 9, both 
 the time of one revolution and the arcs described in 
 any given time are determined. 
 
 And by propositions of this kind Huygens, in his ex- 
 cellent tract, De Horologio Oscillatorio, compared the 
 force of gravity with the centrifugal force of re- 
 volving bodies. 
 
 The preceding results may be proved in this manner. 
 In any circle let a regular polygon be supposed to 
 be described of any number of sides. And if a body 
 moving with a given velocity along the sides of the 
 polygon be reflected by the circle at each of its 
 angular points, the force with which it impinges on
 
 PROP. IV. THEOREM IV. 165 
 
 the circle at each of the reflections will be propor- 
 tional to the velocity ; and therefore the sum of the 
 forces, in a given time, will vary as the velocity and 
 the number of the reflections conjointly. But if the 
 number of sides of the polygon be given, the velo- 
 city will vary as the space described in a given 
 time, and the number of reflections in a given time 
 will vary, in different circles, inversely as the radii of 
 the circles, and, in the same circle, directly as the 
 velocity. Hence, the sum of the forces exerted in a 
 given time varies as the space described in that time 
 increased or diminished in the ratio of that space to 
 the radius of the circle ; that is, as the square of 
 that space divided by the radius, and therefore, if 
 the number of sides be diminished indefinitely so 
 that the polygon coincides with the circle, the sum 
 of the forces varies as the square of the arc described 
 in the given time divided by the radius. 
 This is the centrifugal force by which the body presses 
 against the circle, and to this the opposite force is 
 equal, by which the circle continually repels the 
 body towards the centre. 
 
 Symbolical representation of Areas, Lines, &c. 
 
 167. In the statement of the proposition the words " arcuum 
 quadrata applicata ad radios," in the text of Newton, is rendered 
 the squares of arcs divided by the radii. Such expressions as 
 
 A ^ may be regarded as representations of lines (e.g. this 
 
 expression denotes AC) whose lengths are determined by such 
 constructions as the following : 
 
 To A G apply a rectangle whose area is that of the square on 
 AB, and let AC be the side adjacent to AQ", AC is thus 
 obtained by applying the square on AB to A G. The propriety 
 
 of the symbol 4? employed to represent a line AC, assumed 
 
 A (JT 
 
 from algebra, is obvious, since the number of units of area in 
 the square on AB and in the rectangle whose sides are AG,
 
 166 NEWTON. 
 
 A C are the same ; hence, if m, w, r be the number of units of 
 
 length in these lines, m* = nxr and r = . 
 
 7i 
 
 AH 2 
 
 168. If symbols of this kind, viz. -j-^ , be used in the same 
 
 manner as a fraction, we may either treat them numerically, 
 considering AB* to represent the number of units of area con- 
 tained in the square on AB, and AG as the number of units of 
 length in A r, and thus apply the rules of Arithmetical Algebra ; 
 or we may look upon AB i as the absolute representation of an 
 
 AB* 
 area, and AG as that of a line, in which case -r^, would have 
 
 no meaning except by interpretation. In this interpretation we 
 are guided by the principles upon which Symbolical Algebra is 
 applied to any science, the laws of operation by symbols being 
 the same in Arithmetical and Symbolical Algebra, and the 
 symbols being interpreted so that these laws are not contra- 
 dicted. Thus if, in the application to Geometry, the symbol A 
 be supposed to denote an area equal to that of a rectangle whose 
 sides are represented by a and &, the assumption that A = ab 
 or la will imply that ab = 5a, hence the laws remain the same 
 
 as in Arithmetical Algebra, and = 5; so that the interpretation 
 
 is legitimate, that, if a rectangle be applied to a, whose area is A, 
 
 ^ 
 
 will denote the other side of the rectangle. 
 
 Observations on the Proposition. 
 
 169. In the statement of the proposition the word ' equal ' 
 has been inserted before ' bodies ' in order to make the theorem 
 correct, whether we suppose the centripetal force to be estimated 
 with reference to the momentum or the velocity generated. 
 
 It would, perhaps, be better to state the proposition as 
 follows : " The resultant of the forces, under the action of which 
 bodies describe different circles with uniform velocity, are centri- 
 petal and tend to the centres of the circles, and their accelerating 
 effect are to each other, &c.," for it is not known, prior to the 
 proof, that the forces are centripetal.
 
 PROP. IV. THEOREM IY. 167 
 
 170. CORS. 1 and 9. The first corollary asserts that the 
 centripetal forces on bodies moving in different circles vary as 
 F* 
 -g~ , but the ninth shews that the accelerating effects of the 
 
 centripetal forces are in each circle equal to -^- . 
 
 For, if V be the velocity, F the accelerating effect of the 
 force in any circle, Tthe time of describing any arc, FT will be 
 the length of the arc, \FT* will be the space through which 
 the body would move under ihe action of the same force con- 
 tinued constant, in the same time in which the arc is described, 
 
 : VT:: VT: 2R- .-. V* = FR. 
 
 171. Scholium. In uniform circular motion the centripetal 
 force is employed in counteracting the tendency of the body to 
 move in a straight line, which it would do, according to the first 
 law of motion, with the uniform velocity which it has at any 
 point of the circle, if the centripetal force were suddenly to cease 
 to act. This tendency to recede is called a centrifugal force 
 improperly; for the effect of a force being to accelerate or 
 retard the motion of a body, or to alter its direction, if the 
 tendency could properly be termed a force and the centripetal 
 force which counteracts it were removed, it would accelerate or 
 retard the motion of the body, or alter its direction, which it 
 does not. 
 
 The only sense in which the term centrifugal force can be 
 used with propriety as a force may be obtained by the con- 
 sideration of relative equilibrium, in which case, if the same 
 centripetal force acted on the body, the centrifugal force would 
 keep it in equilibrium, supposing the body were at rest as 
 it would appear to be to an observer moving with it. 
 
 Thus, if a body be supported on the surface of the earth, 
 since the body describes a circle about the axis of the earth 
 with uniform velocity, the pressure of the support and the 
 attraction of the earth must have a resultant, whose direction 
 will pass through the centre of this circle, and whose magnitude 
 will be such as would cause the body to describe it; this re- 
 sultant and the centrifugal force will be in statical equilibrium.
 
 168 NEWTON. 
 
 172. In this case of circular motion the force is exerted 
 not in accelerating or retarding the motion, but in changing its 
 direction. 
 
 Thus, referring to the figure of Prop. I., if the direction of 
 the impulse at B bisect the angle ABC, the triangle CBc will 
 be isosceles, and BC=Bc = AB" therefore the velocities in BG 
 and AB will be equal, and the effect of the impulse has been to 
 change the direction without altering the velocity of the body. 
 
 Hence, the regular polygon inscribed in a circle, centre $, can 
 be described with uniform velocity under the action of impulses 
 tending to the centre ; and, by similar triangles SBCj CBc t 
 
 Cc : BC :: BC : BS. 
 
 And if V be the uniform velocity in the polygon, T the 
 
 V**/* 
 time in a side BC, BC= V. T; therefore Cc= ^~ . 
 
 If now the number of sides be indefinitely increased, Cc will 
 be ultimately twice the space through which the body will be 
 drawn from the tangent by the continuous force, see Art. 146; 
 
 Cc F 2 
 therefore - = ^~ will be the measure of the accelerating effect 
 
 of the centripetal force tending to the centre of the circle. 
 
 Illustrations of Circular ^lotion. 
 
 (1) A small body is attached by an inelastic string to a 
 point on a smooth horizontal table^ to determine the tension of the 
 string when the body describes a circle. 
 
 If the body be set in motion by a blow perpendicular to the 
 string, the string will remain constantly stretched, and the only 
 force which acts on the body in the horizontal plane being in the 
 direction of the fixed point, the areas described round this point 
 will be proportional to the time, and the body will move in a 
 circle with uniform velocity. 
 
 Let v be the velocity of projection, and I the length of the 
 string, then the accelerating effect of the tension of the string 
 
 is j ; that is, j is the velocity which would be generated in an
 
 PROP. IV. THEOREM IV. 169 
 
 unit of time from rest by the action of this tension continued 
 constant, therefore the tension of the string : the weight of the 
 
 body ::^-:^. 
 
 Ex. If a velocity of two feet a second be communicated 
 perpendicular to a string whose length is a yard, 
 v 2 : Ig :: 4: 3x32 :: 1 : 24, 
 
 hence the tension is th of the weight, and the time of 
 
 O -7 / // 
 
 revolution is evidently seconds = = 9-4", nearly. 
 
 (2) If a particle be attached by a string of given length to a 
 point in a rough horizontal plane, and a given velocity be communi- 
 cated to it, perpendicular to the string supposed tight, find the 
 tension of the string at any time, the time in which it will be 
 reduced to rest, and the whole arc described. 
 
 Let Fbe the velocity of projection, Z the length of the string 
 in feet, v the velocity at any time t. Since the particle describes 
 a small arc ultimately with uniform velocity the accelerating 
 
 effect of the tension at the time t is -j . Again, if p, be the 
 
 coefficient of friction, the retarding effect of friction is fig, which 
 is constant, hence the velocity destroyed in the time t since 
 friction is the only force acting in the direction of the tangent 
 is figt, and v = V figt. 
 
 Therefore the particle comes to rest in seconds after 
 
 p W 
 
 describing the arc - feet. 
 
 *W 
 The tension of the string at the time t : the weight of the 
 
 particle : : j : g : : - ~~!^ : g ; therefore the tension cc ( 1\ 
 
 oc the square of the time which will elapse before the particle 
 comes to rest. 
 
 (3) Supposing that the Moon describes a circle with uniform 
 velocity about the centre of the Earth as its centre, to find the ratio 
 of the centripetal acceleration of the, Moon's motion to gravity at 
 the EartJis surface.
 
 170 
 
 NEWTON. 
 
 Let n= number of seconds in the Moon's periodic time, 
 R = the radius of the Moon's orbit in feet ; therefore the velocity 
 
 of the Moon is and -== . ( ) is the measure of the acce- 
 
 n H \ n I 
 
 lerating effect of the force exerted on the Moon, and the measure 
 of the same for gravity at the Earth's surface = 32.2 ; hence, 
 the ratio required is kifR : 32.2n*. 
 
 (4) A body is suspended by a string from a fixed point, and 
 being drawn out of the vertical is projected horizontally so as to 
 describe a horizontal circle with uniform velocity. Find the 
 velocity and the tension of the string. 
 
 Let A be the point of suspension, BC the radius of the circle 
 described ; therefore, the circle being described uniformly, the 
 resultant force on the body tends to the centre B, and the 
 
 F* 
 
 measure of the accelerating effect of this resultant force is -^r 
 
 in the direction CB. Let T, Tfbe the tension of the string and 
 the weight of the body, acting in CA and parallel to AB 
 respectively, therefore T: W:: CA: AB; 
 
 F* a 
 
 also, -^j : g :: CB : AB, Art. 171, /. F* = ^ 
 
 and, if CD be perpendicular to AC, BC 1 = AB. BD-, and the 
 velocity will be that due to falling through the space \BD. 
 
 XXI. 
 
 1. If the cube of the velocity, in circles uniformly described, be 
 inversely proportional to the periodic time, shew that the law of 
 force will vary inversely as the square of the radii.
 
 PROP. IV. THEOREM IV. 171 
 
 2. Compare the areas described in the same time by the 
 planets, supposed to move in circular orbits about the Sun in the 
 centre exerting a force which varies inversely as the square of the 
 distance. 
 
 3. If the forces by which particles describe circles with uniform 
 velocity vary as the distance, shew that the times of revolution will 
 be the same for all. 
 
 4. If the velocity of the Earth's motion were so altered that 
 bodies would have no weight at the equator, find approximately the 
 alteration in the length of a day, assuming that, before the altera- 
 tion, the centrifugal force on a body at the equator was to its 
 weight : : 1 : 288. 
 
 5. A particle moves uniformly on a smooth horizontal table, being 
 attached to a fixed point by a string, one yard long, and it makes 
 three revolutions in a second. Compare the tension of the string 
 with the weight of the particle. 
 
 6. A body moves in a circular groove under the action of a 
 force to the centre, and the pressure on the groove is double the 
 given force on the body to the centre, find the velocity of the body. 
 
 7. If a locomotive be passing a curve at the rate of twenty-four 
 miles an hour, and the radius of the curve be of a mile, prove 
 that the resultant of the forces which retain it on the line, viz. of 
 the action of the rails on the flanges of the wheels, and the horizontal 
 part of the forces which act perpendicular to the inclined road-way, 
 will be iko of the weight of the locomotive, nearly. 
 
 8. If a body be attached by an extensible string to a fixed 
 point in a smooth horizontal table, find the velocity with which the 
 body must move in order to keep the string constantly stretched 
 to double its length. 
 
 If W be the weight of the body, and n Wbe the weight which if 
 suspended at the extremity of the string would just double its length, 
 I the length of the string, shew that the square of the required 
 velocity = Inlg. 
 
 9. A man stands at the North Pole and whirls 24lbs. troy 
 weight on a smooth horizontal plane by a string a yard long at the 
 rate of 100 turns a minute; he finds that the difference of the 
 forces which he has to exert according as he whirls it one way or 
 the opposite is roughly 39 grains ; find the period of the rotation of 
 the earth. 
 
 10. Two equal bodies lie on a rough horizontal table, and are 
 connected by a string which passes through a small ring on the 
 table; if the string be stretched, find the greatest velocity with 
 which one of the bodies can be projected in a direction perpendicular 
 to its portion of the string without moving the other body.
 
 172 
 
 NEWTON. 
 
 PROP. V. PROBLEM I. 
 
 Having given the velocity with which a body is moving at 
 any three points of a given orbit, described by it under 
 the action of forces tending to a common centre, to find 
 that centre. 
 
 Let the three straight lines PT, TQV, VR, touch the 
 given orbit in the points P, Q, R respectively, and 
 let them meet in T and F. 
 
 T 
 
 Draw PA, QB, EC perpendicular to the tangents, and 
 inversely proportional to the velocities of the body 
 at the points P, Q, R. Through A, B, C draw AD, 
 DBE, CE at right angles to PA, QB, EC meeting in 
 D and E. Join TD, VE; TD and F# produced, if 
 necessary, shall meet in S the required centre offeree. 
 
 For, the perpendiculars SX, SY, let fall from S on the 
 tangents PT, TQ V, are inversely proportional to the 
 velocities at P, Q (Prop. i. Cor. 1), and are therefore 
 directly as the perpendiculars AP 9 BQ, or as the
 
 PROP. V. PROBLEM I. 173 
 
 perpendiculars DM, DN on the tangents. Join XY, 
 MN, then, since SXiSY:: DM: DN and the angles 
 XSY, MDN are equal, therefore the triangles SXY, 
 MDN are similar; therefore SX-.DM:: XY: MN 
 \\XT\MT, and the angles SXT, DMT are right 
 angles ; therefore, 8, D, T are in the same straight 
 line. Similarly S, E, V are in the same straight 
 line, and therefore the centre is the point of 
 intersection of TD, VE. Q. E. D. 
 
 xxn. 
 
 1. If AB, EC, CD, the three sides of a rectangle, be the 
 directions of the motion of a body at three points of a central orbit, 
 and the velocities be proportional to these sides respectively, prove 
 that the centre of force will be in the intersection of the diagonals 
 of the rectangle. 
 
 2. If the velocities at three points of a central orbit be respec- 
 tively proportional to the opposite sides of the triangle formed 
 by joining the points, and have their directions parallel to the same 
 sides, prove that the centre of force will be the centre of gravity of 
 the triangle. 
 
 3. Three tangents are drawn to a given orbit, described by a 
 particle under the action of a central force, one of them being parallel 
 to the external bisector of the angle between the other two. If the 
 velocity at the point of contact of this tangent be a mean propor- 
 tional between those at the points of contact of the other two, prove 
 that the centre of the force will lie on the circumference of a 
 certain circle. 
 
 4. If the velocities be inversely proportional to the sides of the 
 triangle formed by the tangents at the three points, the centre of 
 force will be the point of concourse of the straight lines joining each 
 an angular point of this triangle to the intersection of the tangents 
 to its circumscribing circle at the ends of the opposite side. 
 
 5. If the velocity of a particle describing an ellipse under the 
 action of a centre of force vary as the diameter parallel to the 
 direction of its motion directly, and as its distance from one of 
 the axes inversely, prove that the centre of force will be at an 
 infinite distance.
 
 174 
 
 NEWTON. 
 
 PROP. VI. THEOREM V. 
 
 If a lody revolve about a fixed centre of force, in any orbit 
 whatever, in a non-resisting medium, and if, at the ex- 
 tremity of a very small arc, commencing from any point 
 in the orbit, a subtense of the angle of contact at that point 
 be drawn parallel to the radius from that point to the 
 centre of force, then the force at that point tending to the 
 centre will be ultimately as the subtense directly and the 
 square of the time of describing the arc inversely. 
 
 Let PQ be the small arc, PS the radius drawn from P 
 to S, the centre of force. RQ the subtense of the 
 
 angle of contact at P, parallel to PS. T the time of 
 describing PQ. F the accelerating effect of the 
 force at P. 
 
 Then, when the body leaves P, it would, if not acted on 
 by the central force, move in the direction PR, and if 
 the force F continued constant in magnitude and 
 direction throughout the time T, QR would be the 
 space through which it would have been drawn by F 
 
 2O7? OK 
 in that time; therefore ultimately, F= --X.-ac ^' 
 
 COR. 1. Draw ^perpendicular to SP, and let h = twice
 
 PROP. VI. THEOREM V. 175 
 
 the area described in an unit of time. Then area 
 PSQ = \hi, Prop, i., also, since triangle PSQ 
 = SP. QT, and aiezPSQ = triangle PSQ, ultimately, 
 Lemma VIII., therefore hT = SP.QT, ultimately 
 
 /~\ T) o L* /~\ jy 
 
 hence, ultimately, JP=2 ~ = . ~ 
 
 J ' fTi'i O p* f\ fjl ' * 
 
 COR. 2. Draw/ST perpendicular on PR. Then, &PSQ 
 
 .'. hT= SY. PR = 8Y. PQ, ultimately ; 
 hence, ultimately, P=2 
 
 COR. 3. If the orbit have finite curvature at P, and PV 
 be the chord of the circle of curvature whose direction 
 passes through S, PV.QR = PQ*, ultimately; 
 
 F- W 
 ~ SY'.PV 
 
 COR. 4. If Fbe the velocity at P, then F=^, and 
 
 2QX 2QR (PQ\* 
 
 1 = ~ > ultimatel 75 
 
 that is, the velocity at any point of a central orbit 
 at which the curvature is finite is that which would 
 be acquired by a body moving from rest under the 
 action of the central force at that point continued 
 constant, after passing through a space equal to a 
 quarter of the chord of curvature at that point drawn 
 in the direction of the centre of force. 
 COR. 5. Hence, if the form of any curve be given, and 
 the position of any point 8, towards which a centri- 
 petal force is continually directed, the law of the 
 centripetal force can be found, by which a body will 
 be deflected from its direction of motion, so as to 
 remain in the curve. Examples of this investiga* 
 tion will be given in the following problems.
 
 176 NEWTON. 
 
 Observations on the Proposition. 
 
 173. In Newton's enunciation of the proposition, the sagitta 
 of the arc, which bisects the chord and is drawn in the direction 
 of the centre of force, is employed instead of the subtense used 
 in the text, but these are ultimately proportional by Art. 90. 
 
 The variations by which Newton expresses the results of the 
 first three corollaries are replaced by equations, in order to 
 facilitate the comparison of the motion of bodies in different 
 orbits and the forces acting upon them. 
 
 174. The figure employed in proof of the proposition is 
 drawn upon supposition that the force is attractive, the orbit 
 being concave to the centre of force ; the same proof will apply 
 also to the case of a repulsive force, if the curve be drawn in 
 the direction of the dotted line PQ' and the same construction 
 be made. 
 
 The exception, however, should be made, that the method fails 
 in the particular positions in which the body is at the points of 
 contact of tangents drawn from the centre of force to the curve ; 
 in such cases QR does not ultimately meet the tangent at a finite 
 angle or is not a subtense ; the result of the proposition is there- 
 fore not demonstrated for these particular positions. A further 
 discussion of the case is given on the next proposition. 
 
 175. In the proof it is assumed that the body moves ulti- 
 mately in the same manner as if the force P remained constant 
 in magnitude and direction, in which case the body would 
 describe a parabola, whose axis is parallel to PS, and which is 
 evidently the parabola which has at P the same curvature as 
 the curve. By this consideration the proposition contained in 
 Cor. 4 can be readily proved. For, since the body moves in 
 a parabola under the action of a constant force in parallel lines, 
 the velocity at P is that acquired by falling from the directrix 
 under the action of the force at P, continued constant, i.e. 
 through a space equal to the distance of the focus of the 
 parabola, which is equal to a quarter of the chord of curvature 
 at P, drawn through S.
 
 PROP. VI. THEOREM V. 177 
 
 176. The supposition that the force at P continued constant 
 in magnitude and direction, causes the body to move in a curve 
 which is ultimately coincident with the path of the body, may be 
 justified by considering that if PQ' be the arc of the parabola 
 described on this supposition in the same time as the arc PQ 
 actually described, the error Q'Q is due to the change in the 
 magnitude of the forces and the direction of their action in the 
 two cases ; now, the greatest difference of magnitude varies as the 
 difference of SP and SQ ultimately, and the ratio of the error 
 from this cause to Q'R vanishes ultimately ; also, since L PSQ 
 vanishes ultimately, the ratio of the error, arising from the change 
 of direction, to Q'R vanishes ; therefore, Q'Q : Q'R vanishes, and 
 the curves may be considered ultimately coincident. 
 
 177. It is evident that the results of the Proposition and of 
 the fourth corollary are true of the resultant of any forces, under 
 the action of which any plane orbit is described, for this resultant 
 may be supposed ultimately constant in direction and magnitude, 
 in which case the curve described is a parabola. Hence, as in 
 Art. 175, if F be the accelerating effect of the resultant of the 
 forces, QR the subtense parallel to the direction of the resultant, 
 
 V = *F.?, and ^= 2 limit -P- 
 
 Homogeneity. 
 
 178. COR. 1, 2. In the expressions for F obtained in these 
 corollaries, it is of great importance to observe the dimensions 
 of the symbols. Thus TiT represents an area and h is of two 
 dimensions in linear space and of - 1 in time ; therefore h*. QR 
 is of five in space, and of -2 in time, and SP*.QT* of four 
 
 dimensions in space ; hence, $ p *y r Is of one dimension in 
 
 space and of 2 in time, and represents either twice the space 
 through which a force would draw a body in an unit of time, or 
 the velocity generated by the force in an unit of time, either of 
 which may be taken as the measure of the accelerating effect of 
 the force ; moreover, this unit is the same by which the magni- 
 tude of h is determined. 
 
 AA
 
 178 NEWTON. 
 
 Hence, if the actual areas, lines, &c., be represented by the 
 symbols, and not the number of units, as mentioned in Art. 168, 
 every term of an equation or of a sum or difference must be 
 homogeneous, or of the same number of dimensions, both in space 
 and time; for example, PQ + V.T representing a line, V must 
 be of 1 dimensions in time. 
 
 Tangential and Normal Forces. 
 
 179. To find the accelerating effect of the components of the 
 forces, under the action of which a body describes any plane curve, 
 taken in the directions of the normal and tangent at any point. 
 
 Let PQ be a small arc of the curve described under the 
 action of any forces, T, N the measures of the accelerating effect 
 of these forces, in the direction of the tangent and perpendicular 
 to it. Then, if V be the velocity at P, t the time of describing 
 PQ, the forces may be supposed ultimately to remain constant ; 
 therefore, if QR be perpendicular to PR, we shall have 
 ultimately QR = %N.t\ and PR=V.t + \T.t*=V.t since the 
 ratio of T.t* : Vt vanishes ultimately; hence, if p be the radius 
 
 PR* 2V V 
 
 of curvature at P, 2p = _ = j=- ultimately ; therefore will 
 
 be the measure of the normal acceleration estimated towards 
 the centre of curvature. 
 
 Again, if V be the velocity at Q, V will be ultimately the 
 component of the velocity in the direction PR ; therefore, by 
 
 Art. 53, we obtain two measures of the tangential acceleration, 
 
 yiv _y* y _y 
 
 the limits of 
 
 180. To find the velocity at any point of an orbit described 
 under the action of any forces in one plane. 
 
 Let AB be any arc of an orbit, F, v the velocities at A and 
 B, and suppose the arc AB divided into a large number of small 
 portions, of which PQ is one, v r , v r+l velocities at P and Q, T the 
 accelerating effect of the tangential component of the forces at P, 
 
 w m - Vr 2 = 2T.PQ ultimately, 
 and v 8 - F* is obtained by taking the limit of the sum of the
 
 PROP. VI. THEOREM V. 
 
 179 
 
 magnitudes 2T.PQ corresponding to the different arcs when 
 their number is indefinitely increased. 
 
 That this is rigidly correct may be shewn by considering that 
 v r +* v r 2 : 2 T.PQ is ultimately a ratio of equality ; therefore, by 
 Cor., Lemma IV., or Art. 22, the limiting ratio of the sums is 
 also a ratio of equality. 
 
 In the case of a central force, whose accelerating effect is F, 
 T=FcosRPS; 
 
 ... v * r+i _ Vr = 2F.PQ cos EPS = 2F (SP- SQ) ultimately, 
 whence v* F 2 , if F depend only on the distance. 
 
 Radial and Transversal Forces. 
 
 181. To find the accelerating effect of the components of 
 force, under the action of which a body describes any plane curve, 
 taken in the direction of a radius vector drawn from a fixed 
 point, and perpendicular to it. 
 
 Let PQ be a small arc described in the time T; QRU, 
 PU parallel and perpendicular to SP', P, Q the measures of the 
 accelerating effects of the components in PS and PUj PR a 
 
 tangent at P. If V be the velocity at P, make PT=V.T, 
 draw TN perpendicular to SP, and let Qq be the arc of a 
 circle, centre S. 
 
 Since the forces may be considered ultimately constant in 
 
 magnitude and direction, ^P.T = Nn = No L + jj ultimately.
 
 180 NEWTON. 
 
 Let h be twice the area which would be described in an 
 unit of time by radii from S, if the transverse force Q ceased 
 
 to act, t\ienQn.SP=TN.SP=h.T; therefore |#- = -|P 
 
 ultimately ; and if P be the measure of the accelerating effect 
 of a force, under the action of which the body would move 
 in PS, so that its distance from S would be always equal to 
 that of the body in PQ at the same time, ^P'.T^^Nq ulti- 
 
 7i 2 
 mately ; therefore P P' -f -^- 3 . 
 
 Again, if at Q h' correspond to A, h' h, the increase of #, 
 will be due to the increase of velocity in direction PU, which 
 is equal to Q. T ultimately; therefore (h 1 - h} T= Q.T*.SP 
 
 ultimately ; hence Q --, ultimately. 
 
 Angular Velocity. 
 
 182. DEF. Angular velocity of a point moving about a fixed 
 point is the rate at which angles are described by radii drawn 
 to the fixed point. 
 
 Uniform angular velocity is measured by the angle described 
 in an unit of time. 
 
 Variable angular velocity is measured by the angle which 
 would be described by a radius in an unit of time, if moving with 
 uniform angular velocity equal to the angular velocity at the 
 time under consideration; this is the limit of the angle, described 
 in a time T, divided by T, when T is indefinitely diminished. 
 
 183. To find the angular velocity in a central orbit. 
 
 Let PQ be a small arc described in the time T, draw QN 
 perpendicular to SP, then h.T= twice the area PSQ= QN.SP 
 ultimately ; and, if the angles be supposed estimated in circular 
 
 measure, L PSQ = ~ = -j^ ultimately ; therefore the angular 
 
 P>QS) T 
 
 velocity, which is L ~ ultimately, = -^ .
 
 PEOP. VI. THEOREM V. 181 
 
 184. To find the angular velocity of the perpendicular on the 
 tangent from the centre of force. 
 
 Draw SY perpendicular on the tangent PY", and let PV be 
 the chord of curvature through S. 
 
 The angle described by SY in the time T is equal to the 
 angle between the tangents at P and (), or to twice the angle 
 PVQ', therefore angular velocity of SY : angular velocity of 
 SP::2LPVQ:^PSQ::2SQ:QV ultimately; hence the 
 
 angular velocity of SY = py sp - 
 
 Illustrations. 
 
 (1) To find the tension of a string by which a body is attached 
 to the centre of a vertical circle in which it revolves. 
 
 Let P be the position of the body at any time, (?P, GA 
 radii drawn to P and the lowest point, and let v, u be the 
 velocities at P and A. Draw PM perpendicular to GA. Then 
 
 u* v* = 2g.AM and -~-r is the accelerating effect of the forces 
 C.4 
 
 in the direction P(7, viz. the tension of the string and the com- 
 ponent of the weight of the body. Let T be the tension of 
 the string and m the mass of the body ;
 
 182 
 
 NEWTON. 
 
 therefore the tension of the string : the weight of the body 
 :: M* - 2g.CA + Sg.CM : g.CA. 
 
 NOTE 1. In order that the complete circle may be described, 
 since the string must be stretched at the highest point where 
 CA must be written for CM, u*=or>5g.CA, and if the 
 circle be just described, the tension at the lowest point will be 
 six times the weight. 
 
 NOTE 2. If the body oscillate, the extent of the oscillation 
 will be given by the consideration that at the extremity P of 
 the arc of oscillation there will be no velocity, therefore 
 u 2 = 2g.AM', and AM' is less than AC, otherwise the string 
 would not be stretched, so that the tension at A : the weight 
 ::2AM'+AC:AG. 
 
 (2) Find the, force, under the action of which a body may 
 describe the equiangular spiral uniformly. 
 
 The velocity being constant, there is only a normal force 
 
 measured by (vel.) 2 -r- radius of curvature = <cp , Art. 128. 
 
 (3) Find the force tending to the pole of the cardioidj under 
 the action of which the curve is described. 
 
 Since PV= |SP, and (vel.)* 
 
 
 see page 105, 
 
 07 S -DQ I 
 
 therefore the accelerating effect of the force is ' p4 oc -^^ . 
 
 (4) Two equal rings P, Q slide on a string which passes round 
 two fixed pegs A, B in a smooth horizontal plane ; the rings are 
 brought together, and then projected with equal velocities, so as to 
 keep the string stretched symmetrically. Shew that the tension of 
 he string varies inversely as the distance AP. 
 
 J) A 
 
 C
 
 PROP. VI. THEOREM V. 
 
 183 
 
 The figure represents the position of the system at any time. 
 Let CR bisect AB and PQ, and let DE be drawn parallel to 
 CR, so that EP=PA, then EPR = AP+PR is constant; 
 therefore DE is fixed, and P moves in a parabola whose focus 
 is A and directrix DE. 
 
 Also, the tensions of the string in PA, PQ being equal, and 
 equally inclined to the tangent to P's path, the resultant of these 
 tensions, which are the only forces acting in the plane of the 
 curve, acts in the normal, hence the rings move with uniform 
 velocity equal to the velocity of projection F, and if T be the 
 measure of the accelerating effect of the tension, PGr the normal, 
 
 F 2 
 p the radius of curvature, 2TcosAPG= , and 2p cosAPG 
 
 = chord of curvature through A = PA ; therefore 
 
 F 2 1 
 T= IPA* PA' 
 
 (5) A body revolves in a smooth circular tube under the action 
 of a force tending to any point in the circumference, and varying 
 as the distance from that point. Find the pressure on the tube, and 
 the point where there is no pressure, the motion commencing from, 
 a given point. 
 
 Take A the centre of force, C that of the circle ; let B be the 
 point of starting, PQ a small arc, BD, PM, QN ordinates to the 
 
 diameter through the centre of force, Am, Qn perpendicular on 
 OP] let }L.PA be the measure of the accelerating effect of the 
 force at P; therefore p.mA, p.Pm are those of the tangential 
 and normal forces, =p.PM and p. AM respectively.
 
 184 
 
 NEWTON. 
 
 (vel.) 2 at -(vel.) 2 at P=2fi.PM.PQ=2fi.CP.MN ultimately, 
 see Art. 179, whence, taking the limit of the summation for all 
 the small arcs in BP, (vel.)* at P=2p.CP.DM. 
 
 'vel ) 2 at P 
 Also, - 4fp = fj,.AM+t\iG accelerating effect of the 
 
 pressure of the tube, the upper or lower sign being taken 
 according as the pressure is from or towards (7; therefore the 
 pressure on the tube has for the measure of its accelerating effect 
 
 /i (AM- 2DM) = (BAM- 2AD] ; 
 
 hence the pressure is outwards from B until AM= $AD, at 
 which point there is no pressure, and inwards from that point to 
 the corresponding one on the opposite side, having its greatest 
 value at A t and the outward pressure at B is half the inward 
 pressure at A. 
 
 (6) If in a smooth elliptic tube a particle be placed at any 
 point, and be acted on by two forces which tend to the foci and 
 vary inversely as the square of the distances from those points , 
 shew that the pressure at any point will vary as the curvature. 
 
 Let be the point of starting, PQ a small arc described by 
 the body, QT, Q U perpendiculars on SP t HP. 
 
 Take ^, -^p, -R, as the measures of the accelerating 
 
 effects of the forces, and of the pressure of tube outwards. 
 
 Then, employing the usual letters for the lines of the figure, 
 
 the accelerating effect of the tangential component of force 
 
 to 8 is 
 
 j^ PT_n(SP-SQ)_ n 
 
 SP*'PQ SP.SQ.PQ~ PQ.SQ PQ.SP tey ' 
 
 and similarly for the force tending to H;
 
 PROP. VI. THEOREM V. 185 
 
 Also, . p + -*, if p be the radius 
 
 of curvature at P, and 2 P . ** = PF = 
 
 ' p AC.HP + AC.SP HP SP^HO 
 
 _ 2/4' 2/4 p /* ffl'SO flHO\ 
 
 ~ Hd + &o~Ac"Ac = (-Ho + ~so) : 
 
 which 18 constant ; therefore R varies as the curvature. 
 
 XXIII. 
 
 1. A body is attached to a point by a thread, and is projected so 
 as to describe a vertical circle, prove that, if Z 7 ,, T z be the tensions 
 of the string at the extremities of any diameter, the arithmetic mean 
 between T t , T 2 is independent of the position of the diameter, and 
 that TV~>TI is six times the component of the weight in the direction 
 of the diameter. 
 
 2. A string of given length I is capable of sustaining a weight W. 
 One end is fixed, and a given weight P less than W, attached to the 
 other end, oscillates in a vertical plane, find the greatest arc through 
 which the weight can oscillate without breaking the string. 
 
 3. A ring slides on a string hanging over two pegs in the same 
 horizontal line, find the tension of the string at the lowest point, if 
 the ring begin to fall from the point in the horizontal line through 
 the pegs, the string being stretched. 
 
 4. AB is the vertical axis of a cycloid, A the highest point, 
 AM, AN are the abcissse of points at which a body begins to slide 
 down the arc of the cycloid, and at which it leaves the curve; prove 
 that JVis the middle point of MB. 
 
 5. If in a central orbit the direction of motion change uniformly, 
 prove that the normal force will vary as the radius of curvature. 
 
 BB
 
 186 NEWTOX. 
 
 6. Given the Sun's motion in longttuie at apogee and perigee 
 to be 57' 10" and 61' 10" ; find the eccentricity of the Earth's orbit, 
 supposed to be an ellipse about the Sun in one of the foci. 
 
 7. Prove that the angular velocity of a projectile about the 
 focus of its path varies inversely as its distance from the focus. 
 
 8. A particle, constrained to move on an equiangular spiral, is 
 attracted to the pole by a force proportional to the distance, prove 
 that, at whatever point the particle be placed at rest, the times of 
 describing a given angle about the centre of force will be the same. 
 
 9. A body slides down a smooth cycloidal arc, whose axis is 
 vertical and vertex downwards, find the pressure at any point of the 
 cycloid, and shew that, if it fall from the highest point, the pressure 
 at the lowest point will be twice the weight of the body. 
 
 10. Find the law of force, tending to the centre, under the 
 action of which a lemniscate can be described. 
 
 XXIV. 
 
 1. Two straight lines AB and BC are united at B ; A B revolves 
 about A, and BC about B with the same uniform angular velocity ; 
 shew that the acceleration on C tends to A and varies as CA. 
 
 2. A particle describes an ellipse, the centre of force being 
 situated at any point within the figure. Shew that at the point 
 where the true angular velocity is equal to the mean angular 
 velocity, the radius vector is a mean proportional between the 
 
 3. A particle begins to move from any point of a smooth 
 parabolic tube, being attracted to the focus by a force which varies 
 inversely as the square of the distance ; find the greatest pressure. 
 
 4. If SY be the perpendicular on the tangent at a point P of 
 an orbit, described about a centre of force S, prove that the 
 acceleration at P will be equal to the product of the velocities of P 
 and T divided by ST. 
 
 5. A smooth cone is placed with its axis vertical and vertex 
 upwards, shew that there is a certain portion of the surface upon 
 which a particle can describe a circle, if properly projected and 
 acted on by gravity and by a force tending to the vertex and 
 varying as the distance.
 
 PROP. VI. THEOREM V. 187 
 
 6. Shew that the force required for the description of an ellipse 
 about the vertex A varies as ^ -y-a > where PN'vs, the perpendicular 
 on the axis. 
 
 7. If a particle describe an ellipse under the action of a force 
 tending to any fixed point 0, the force will vary as ^ p 2 ~, 3 , where 
 
 P is the position of the particle, PP the chord through 0, and 
 T)D' the diameter parallel to this chord. 
 
 8. Shew that in the elliptic orbit described under the action of 
 a force tending to a focus, the angular velocity round the other 
 focus varies inversely as the square of the diameter parallel to the 
 direction of motion. 
 
 9. A particle moves in a circular tube, under the action of a 
 force which tends to a point in the tube, and whose accelerating 
 effect varies as the distance, shew that, if the particle begin to move 
 from a point at a distance from the centre of force equal to the 
 radius, there will be no pressure on the tube at an angular distance 
 from the centre of force equal to cos" 1 ! . 
 
 10. A particle moves in a smooth elliptic groove, under the 
 action of two forces tending to the foci and varying inversely as the 
 squares of the distances, the forces being equal at equal distances. 
 Prove that, if the velocity at the extremity of the axis major be to 
 that at the extremity of the axis minor as AC to EC, then the 
 velocity at any point will vary inversely as the normal ; find the 
 pressure on the tube. 
 
 11. Determine the relation between /z and X and the velocity of 
 projection, in order that an ellipse -may be described under the 
 
 action of forces ^ , pz to the foci and X. CP to the centre, acting 
 simultaneously. 
 
 12 A particle is attached to a point C by a string, and is 
 attracted by a force which tends to a point 8, and varies inversely as 
 the square of the distance from 8. Find the least velocity with 
 which the particle can be projected from a point in CS, or Cb pro- 
 duced, so as to describe a complete circle. If CS be less than the 
 length of the string, prove that the tension will be a maximum at 
 a point D, where 82) is perpendicular to CS, and that if CS be half 
 the length of the string, the two minimum and the maximum 
 tensions will be as 0, 4 and
 
 188 NEWTON. 
 
 PROP. VII. PROBLEM II. 
 
 A body moves in the circumference of a circk, to find the law 
 of the centripetal force, tending to any given point in the 
 plane of the circle. 
 
 Let AP Fbe the circumference of the circle, $the given 
 point to which the centripetal force tends, P V the 
 
 chord of the circle drawn through S from P, the 
 position of the body at any time, and VGA the 
 diameter through F. Join PA, and draw SY 
 perpendicular to PYj the tangent to the curve at P. 
 
 By Prop. vi. Cor 3, if F be the measure of the accele- 
 
 2A 1 
 rating effect of the centripetal force, F = , 
 
 and, since the angles SPY, VAP are equal, and also 
 the right angles PYS, APF, the triangles SPY, 
 VAP are similar, and SY : SP : : P V : FA ; 
 
 2/f.VA* 
 ~ SP*.PV*' 
 
 therefore, since h and VA are given, F varies 
 inversely as SP\PV\
 
 PROP. VII. PROBLEM II. 189 
 
 COR. 1. Hence, if the given point S to which the 
 centripetal force tends, be situated on the circum- 
 ference of the circle, V will coincide with 8, and F 
 vary inversely as SP*. 
 
 COR. 2. The force, under the action of which a body 
 P revolves in a circle APTV, is to the force, under 
 the action of which the same body P can revolve in 
 the same circle in the same periodic time about any 
 other centre of force R, as RP\SP to SG\ SG being 
 a straight line drawn from the first centre S, parallel 
 to the distance HP of the body from the second 
 centre offeree R, to meet PG t a tangent to the circle. 
 
 For, by the construction of this proposition, since the 
 periodic times are the same, the areas described in 
 
 a given time are the same ; therefore, h is the same 
 for both centres, hence, if PRT be the chord through 
 R, the force to S : the force to R : : RP*.PT 3 
 /SjP.PF 8 ; but, by similar triangles TPV, GSP, 
 PT - P V: : SP : SG ; therefore force to S : force to R 
 : : RP'.SP* : SP\SG 3 : : EP'8P : 
 
 COR. 3. The force, under the action of which a body 
 P revolves in any orbit about a centre of force 8, is 
 to the force, under the action of which the same 
 body P can revolve in the same orbit in the same 
 periodic time about any other centre of force R, as
 
 190 NEWTON. 
 
 RP*.SPto SG- 3 , SG being the straight line drawn from 
 the first centre offeree S, parallel to RP the distance 
 of P from the second centre of force R, to meet PG 
 the tangent to the orbit. 
 
 For, in each case, the body may be supposed for a 
 short time to be moving in the circle of curvature, 
 and the forces are the same as those which would 
 retain the body in the circular orbit; therefore, 
 since the areas described in a given time are equal, 
 the ratio of the forces is RP\SP : SG*. 
 
 Observations on the Proposition. 
 
 185. In the figure employed in the proposition, the force is 
 supposed to be attractive, but the investigation of the law 
 of force applies also to the case in which the centre of force 
 
 8 is exterior to the circle, in which case the force is repulsive 
 through the arc 5(7, which is convex to the centre of force, and 
 contained between the tangents drawn from 8 to the circle. 
 
 It is important, however, to observe that this problem is to 
 find what would be the law of force tending to 8, under the 
 action of which a body would be moving, supposing that it 
 could move in the circle, or any portion of the circle, under the 
 action of such a force, but it does not assert the possibility of 
 such a motion, which is considered in Art. 165.
 
 PROP. VII. PROBLEM II. 191 
 
 111 fact, the complete description of a circle AB C, under the 
 sole action of a central force tending to an external point S, is 
 impossible, because, as the body approaches the point B, the 
 component of the velocity perpendicular to SB remains finite 
 however near the body approaches B, and since there is no 
 force to generate a velocity in the opposite direction, the body 
 must proceed to describe an arc SZJon the opposite side. SB 
 would be a tangent to both curves, because the velocity in 
 direction BS becomes larger than any finite quantity, as the body 
 approaches B, and therefore the angle between BS and the 
 direction of motion is indefinitely small at B. 
 
 That a finite velocity in the direction perpendicular to SB 
 could remain up to B, may be shewn by producing SB to T 
 in the tangent PY at P; then the -component of the velocity 
 
 7 OV" 7, I 
 
 at P perpendicular to SB is -^ . ^ = ^ = ^ , when the 
 body arrives at a point very near to B. 
 
 186. The force at a point indefinitely near to B cannot be 
 properly determined by the method of Prop. VI., because the 
 lines parallel to the direction of the force from which the mea- 
 
 sures of the force are obtained are not subtenses, or sagittas, 
 since they are in this case not inclined at a finite angle to the 
 tangent.
 
 192 NEWTON. 
 
 But it can be seen in another manner from the polygon of 
 Prop. I, that the force is infinitely great, when the distance from 
 B becomes infinitely small. 
 
 Thus, if CDEF be a portion of the polygon whose limit 
 touches the radius from S between D and E, the angle between 
 DE and DS or ES may be made as small as we please compared 
 with the angle between CD and DE, hence the velocity 
 generated by the impulse in the directions DS and SE will 
 become infinitely great compared with the velocities in CD 
 and EF. In the figure, the impulses at D and E, whose 
 directions are denoted by the arrows, have corresponding to them 
 in the limit the forces on opposite sides of the tangent, which 
 are attractive and repulsive respectively. 
 
 187. COR. 1. For the reasons given above, a limitation 
 should be made, viz., when P is at a finite distance from S. In 
 
 . 
 this case PV= SP and F= -gp- , R being the radius of the 
 
 circle. 
 
 We may also observe here that the possibility of a description 
 of a circle is not asserted, but only the law of force required 
 in case of description of any portion of the circle. The complete 
 description of the single circle is, in fact, impossible, for, under 
 the action of the force obtained, the body would pass to the other 
 side of the tangent on arriving at $, then proceed to describe 
 another equal circle, and, on arriving again at $, return into the 
 original circle. 
 
 188. COR. 2. The orbit being the same, and also the 
 periodic times about S and H being equal, the value of , in 
 the two cases, is the same; also, the force tending to S for 
 the orbit being of the same magnitude at P as that under the 
 action of which the circle of curvature would be described, and 
 SY, PV being the same in the orbit and the circle, h is also 
 the same, Prop. VI. Cor. 3 ; and, similarly, h is the same in the 
 circle and orbit described about R ; therefore it is the same in 
 the circle described about S and R as centres of force, and hence 
 Cor. 2 applies.
 
 PROP. VII. PROBLEM II. 193 
 
 Absolute Force. 
 
 189. If the force upon a body placed at any distance from 
 the point S vary inversely as the wth power of that distance, 
 the magnitude of the force, or its ratio to any given force, 
 as that of gravity, will be determined when the distance SP is 
 given. The measure of the accelerating effect of the force is 
 
 written -^ , where p the constant part of this measure is an 
 algebraical symbol of n + 1 dimensions in linear space. If the 
 unit of space = a, n is the measure of the accelerating effect of 
 
 the force on a body at an unit of distance, and p is called the 
 Absolute Force, being the measure of the accelerating effect of 
 the force at an unit of distance x the nth power of that unit. 
 The absolute force is not the measure of the accelerating effect 
 of any force, unless the symbols be treated numerically, in which 
 case fju is twice the number of units of space through which a 
 constant force, equal to the force at an unit of distance, would 
 draw a body from rest in an unit of time. 
 
 Law of Force in a Circular Orbit. 
 
 190. The law of force may be expressed in terms of the 
 distance SP, for SD, Sd being the greatest and least distances 
 of the body from 8, SD.Sd = SP.SV-, see figure, page 188. 
 
 + or according as S is within or without the circle ; 
 
 '' Jf -(8P r ~8I>.Sd)*' 
 If 8 be on the circumference Sd = 0, therefore F 
 
 If 8 be exterior to the circle, SD.Sd = SB\ and the lower 
 
 2h*AV ? . 
 sign must be taken ; therefore JJ = Tp* _ g 
 
 Velocity in the Circular Orbit. 
 
 191. To find the velocity in the circular orbit described 
 under the action of a force tending to any point in the plane of 
 
 the orbit. 
 
 CC
 
 194 NEWTON. 
 
 h h SP h VA I 
 
 The velocity at P= ^= ^. ^= ^. -^cc -^-^ 
 
 CoR. If S be in the circumference of the circle, and - 
 be the accelerating effect of the force, n = WSA* ; 
 
 hence the velocity at P = -'gpr = (j ^p 
 Or, we may employ the result of Prop. VI., Cor. 4, 
 V*- F PV_J^_ SP. . r _/M* J.* J_ 
 
 ' sf "SP"' 2 ' "W'SP* &P*' 
 
 Periodic Time. 
 
 192. To ^n^ <Ae periodic time in a circular orbit described 
 tinder the action of a force tending to a point in the circumference. 
 
 Let P be the periodic time, It the radius of the circle, and 
 
 let ^ 5 be the measure of the accelerating effect of the force at 
 
 P, then h. P= twice the area of the circle = 
 
 and p = 2^M^ = 8h*fi* ; .-. P = 
 
 193. 7b compare the periodic times in the same circle when 
 described under the action of a force tending to a point in the 
 circumference, and a force tending to the centre, of the same 
 magnitude as that of the first force at a distance equal to the radius 
 of the circle. 
 
 Let P* be the periodic time, and V the uniform velocity in 
 the circle in the second case, V* = ~ . R ; /. F= ^ , 
 
 and P'. 7 .. 
 
 At* 2V2 
 
 Illustrations. 
 
 (1) When the force in a circular orbit tends to a point within 
 the circle, to find the pointat which the true angular velocity is 
 equal to the mean angular velocity.
 
 PROP. VII. PROBLEM II. 195 
 
 The true angular velocity = -^p , the mean = ~=2ir. 
 
 therefore at the required point SP= 22, or the perpendicular from 
 the required point upon the line joining S to the centre of the 
 circle bisects OS. 
 
 (2) A body describes a circle under the action of a force 
 tending to a point within Y, the measure of whose accelerating 
 effect at the greatest and least distances SD and Sd are the radius 
 and twice the diameter respectively , the unit of time being a second; 
 find the number of seconds in passing from D to d. 
 
 TO q 
 
 and the number of seconds from Dtod= f- = - 
 
 h 4 
 
 XXV. 
 
 1. If p. be the absolute force in a circular orbit described under 
 the action of a force tending to a point in the circumference, prove 
 that the time in a quadrant commencing from the extremity of the 
 diameter through the centre of force will be (T + 2) .S 3 (/*)"*. 
 
 In what unit of time is the result expressed ? 
 
 2. A point describes a circle, with an acceleration tending to any 
 point within the circle. Prove that, if three points be taken at 
 which its velocities are in harmonical progression, the velocities at 
 the other extremities of the diameters, passing through those points, 
 will also be in harmonical progression. 
 
 3. In the case of a centre of force S within a circle, if two points 
 Z, If be taken, such that LS, MS make equal angles with the 
 diameter through S, and on the same side of it, then the forces at 
 Z and M will be to each other in the inverse ratio of the squares on 
 OL and OH. 
 
 4. The sum of the reciprocals of the velocities at the extremities 
 of any diameter is independent of the position of the centre of force, 
 and varies as the periodic time. 
 
 5. Prove that, when a circular orbit is described about an in- 
 ternal point, the sum of the square roots of the accelerations at the 
 extremities of any chord passing through that point varies inversely 
 as the square root of the length of the chord.
 
 196 NEWTON. 
 
 6. Prove that, if the law offeree tending to 8, a point without a 
 circle, be the law of force under which part of the circle can be de- 
 scribed, the body will move near B as if acted on by a force tending 
 to B and varying inversely as the cube of the distance from B. 
 
 7. OE is a radius perpendicular to the diameter through 8 in a 
 circular orbit about a central force tending to a point S within the 
 circle, SB an ordinate, perpendicular to OS, shew that, if the force 
 at B be an arithmetic mean between the forces at the greatest and 
 least distances, OE* --= SB . SE 2 . 
 
 8. Prove that, if a circle be described about a force tending to a 
 point in the circumference, and PQ be a chord parallel to the dia- 
 meter through that point, the times of describing equal small arcs 
 near P and Q will differ by a quantity which varies as PQ. 
 
 9. "When a particle is describing a circle under the action of a 
 central force, shew that at every instant the angular velocities 
 about all points in the circumference are the same. 
 
 10. The period in an orbit described under the action of a central 
 
 force, whose accelerating effect is pr" is given to be \a m -f- ^i, a be- 
 ing a line and X a number, find n. 
 
 11. Apply the proposition contained in Cor. 3, to prove that if 
 in an elliptic orbit described under the action of a force tending 
 to the centre, the force vary as the distance from the centre, then 
 the force tending to the focus will vary inversely as the square of 
 the focal distance. 
 
 12. Deduce, by Cor. 3, the law of force, when a parabola is 
 described under the action of a force tending to the focus, from the 
 constant force parallel to the axis, under the action of which the 
 same parabola may be described. 
 
 13. Shew, by the method of projections, that the centripetal 
 force at any point P tending to a fixed point in the axis major 
 of an ellipse under which the ellipse can be described, varies as 
 
 . OP, POQ being the chord of the ellipse through 0.
 
 PROP. VIII. PROBLEM III. 
 
 197 
 
 PEOP. VIII. PROBLEM III. 
 
 A body moves in a semicircle PQA under the action of a 
 force tending to a point S so distant that the lines PS, QS 
 drawn from the body to that point may be considered 
 parallel; to find the law of force. 
 
 Let CA be a semidiameter of the semicircle drawn from 
 the centre perpendicular to the direction in which the 
 force acts, cutting PS t QS in M and N, and join CP. 
 
 Let PRZ^Q the tangent at P, ZQT perpendicular to 
 PMS, meeting PRZ in Z, and let SNQ meet PRZ 
 in R. 
 
 Then the force at P= 
 
 ultimately, if the arc 
 
 .- 
 
 PQ be indefinitely diminished, and SP may be con- 
 sidered constant ; also, by Euclid in. 36, 
 
 and, since RQ is parallel to PT, and the triangles 
 PZT, CPM are similar, 
 
 RP: QT:: ZP : ZT :: CP : 
 
 A ~QR = RP '~QR 
 
 ultimately ,- 
 
 2PM 
 
 hence force at P = -
 
 198 NEWTON. 
 
 Aliter. 
 
 In fig. page 190 draw OE a semidiameter perpen- 
 dicular to SDj and let the distance SP cut the 
 circle in F", and OE in M, then, by the pre- 
 
 Q IV T>* 
 
 ceding proposition, F = ^p* py> > an d, # & ^ e 
 
 very distant, the ratio PM : SM or SO will vanish ; 
 therefore, SP = SO ultimately, and PV is ulti- 
 mately perpendicular to OE and equal to 2PM ; 
 
 __##*_ 1 
 ' * PM*' 
 
 SCHOLIUM. 
 
 A body moves in an ellipse, hyperbola or parabola, 
 under the action of a force tending to a point so 
 situated and so distant that the lines drawn from 
 the body to that point may be considered parallel, 
 and perpendicular to the major axis of the ellipse, 
 the axis of the parabola or the transverse axis of 
 the hyperbola. To shew that the force varies 
 inversely as the cube of the ordinates. 
 
 Let AMGr be the axis to which the direction of the 
 forces may be considered perpendicular, PM, PGr 
 
 the ordinate and normal, PO the diameter of 
 curvature, and PFthe chord of curvature in direc- 
 tion PA^.
 
 PROP. VIII. PROBLEM III. 190 
 
 ThcnP- 
 
 SY\PV~ SP\P 
 
 PC? pg 3 
 
 * ^* 
 
 .P F PM 3 .PO PM 3 ' 
 since POoc P# 3 , Art. 84. 
 
 Observations on the Proposition. 
 
 194. It lias been shewn in Art. 151, that the equable de- 
 scription of areas may, in the case of forces acting in parallel 
 lines, be replaced by the uniformity of the resolved part of the 
 velocity in the direction perpendicular to that of the forces. In 
 the proof given in the text, when S is removed to an infinite dis- 
 tance, h and SP are both infinite magnitudes, but the expression 
 
 ^p is finite, for area SPQ described in the time T is ultimately 
 
 equal to area SMN, whose base is equal to u T, u being the com- 
 ponent of the velocity perpendicular to the direction of the 
 
 forces; therefore hT=uT.SP ultimately, and ^p^ =w 8 , hence 
 the acceleration due to the force, when a body describes the 
 
 2 Tpa 
 
 semicircle, is -^-^ . 
 
 195. The accelerating effect of the force, acting in parallel 
 lines, may be obtained directly from the proposition of Art. 151, 
 as follows. 
 
 Let u be the constant component of the velocity F, perpen- 
 dicular to the direction of the force, and let ^be the accelerating 
 
 2F 2 F* 
 effect of the force, therefore F= - = --J 
 
 also V.u-nCP: PM; 
 
 Extension of Scholium. 
 
 196. When a lody describes any curve under the action of a 
 force tending to a point S, so distant that the lines drawn from S
 
 200 NEWTON. 
 
 to the body may "be considered parallel ; to find the law of force 
 and the velocity at any point. 
 
 Let AP be any curve, AM G the line to which the forces are 
 perpendicular, PM, PG the ordinate and normal at the point P, 
 PV the chord of curvature in the direction of the force, PO the 
 diameter of curvature. 
 
 Let F be the accelerating effect of the force at P, u the 
 component of the velocity Fin the direction AMG] 
 
 .-. V:u::PG:PM, 
 also PV-.PO:: PM-.PG; 
 
 PO 2u*.PG* 
 
 _ _ 
 
 ~~ PV~ PM\PO' PV~ PO.PM*' 
 
 PG 
 
 and the velocity = u. .,. 
 
 Illustrations. 
 
 (1) A cycloid is described by a particle, under the action of a 
 force acting in a direction parallel to the axis ; find the accelera- 
 tion and the velocity at any point. 
 
 In the cycloid P0 = 4P, and PM.AB = PGP, AB being 
 the length of the axis ; 
 
 ""' F== ~~PW~ ' PO = spur Po 3 ' 
 
 and the velocity at P=u. -^-. =u. -^ cc -^ . 
 
 (2) A particle moves in a catenary under the action of forces 
 acting in vertical lines ; find the accelerating effect of the force 
 and the velocity at any point. 
 
 Let AM be the directrix, AB the ordinate at the lowest 
 point. 
 
 Then PG : PM:: PM : AB and P0 = 2P#; 
 u\PM 
 
 PG PM 
 
 and the velocity at P= w. -, = u. - or PM.
 
 PROP. VIII. PROBLEM III. 201 
 
 XXVI. 
 
 1. A body is moving in a semicircle under the action of a force 
 tending to a point, so distant that the lines drawn from the body 
 to that point may be considered parallel ; if the centre of force be 
 transferred to the centre of the circle, when the direction of the 
 body's motion is perpendicular to that of the force, its magnitude 
 at that point being unaltered, prove that the body will continue to 
 move in the circle. 
 
 2. If a cycloid be described under the action of forces in the 
 direction of the base, the force at any point will vary inversely as 
 AM.IfQ-, AM, MQ being the abscissa and ordinate of the cor- 
 responding point of the generating circle. 
 
 3. A catenary is described under the action of a horizontal 
 force, prove that the force varies as the distance from the directrix 
 directly, and the cube of the arc from the lowest point inversely. 
 
 4. If the same parabola be described by particles when the 
 force tends to the focus, and when it is parallel to the axis, the 
 velocities will be equal at the points at which the forces are equal. 
 
 5. A parabola having its vertex at A and its Axis coincident 
 with AB the diameter of a semicircle, is described so as to cut the 
 semicircle in P ; prove that, if a body move in the semicircle under 
 the action of a force perpendicular to AB, the time of moving from 
 A to P will vary as the difference between AB and the latus rectum. 
 Prove also, that if a second body move from A to P in the parabola 
 in the same time under the action of a force perpendicular to its 
 axis, and the velocities in the two curves at P be equal, the latus 
 rectum of the parabola will be \AB. 
 
 DD
 
 202 NEWTON. 
 
 PKOP. IX. PROBLEM IV. 
 
 If a "body revolve in an equiangular spiral, required the 
 law of centripetal force tending to the pok of the spiral. 
 
 Draw SY from S, the pole of the spiral, perpendicular 
 to the tangent PF, and let PV be the chord of 
 curvature at P, whose direction passes through $; 
 then F\ the measure of the accelerating effect of 
 
 2h* 
 the force tending to the pole, is ~y , y\ but, if 
 
 a be the angle of the spiral, SY= SP sin a and 
 P V=2SP, Art. 128; 
 
 F- * * c 
 
 ~ 3 ~' 3 
 
 197. To find the velocity of a body describing an equiangular 
 spiral under the action of a force tending to the pole. 
 
 If -^p, be the accelerating effect of the force tending to S t 
 
 198. To find the time of describing any arc of the equi- 
 angular spiral. 
 
 Let AL be any arc, SA, SL bounding radii, P the time of 
 describing the arc. Then area &!.= (SA*~ Z/)tana, Art. 127 ; 
 
 2 x area SAL SA* - SL* A SA* ~ SU 
 
 .*. P= 7 = , ---- tana= -. 
 
 n 2n 2/4* cos a 
 
 199. In any orbit, described under the action of a force tending 
 to any point $, when the angle between the tangent PY and the 
 radius SP is a maximum or minimum^ the velocity is equal to 
 the velocity in a circle at the same distance about the same force 
 in the centre. 
 
 For, the curve, near this point, may be considered an equi- 
 angular spiral ultimately, since the angle is constant for a short 
 
 2SP, and V*=F.SP
 
 PROP. IX. PBOBLEM IV. 203 
 
 xxvn. 
 
 1. In different equiangular spirals, described under the action 
 of forces tending to the poles which are equal at equal distances, 
 shew that the angular velocity varies at any point as the force and 
 the perpendicular on the tangent conjointly. 
 
 2. The angular velocity of the perpendicular on the tangent ia 
 equal to that of the radius. 
 
 3. The velocity of approach towards the focus, called the para- 
 centric velocity, varies inversely as the distance. 
 
 4. A body is describing a circle, whose radius is a, with uniform 
 velocity, under the action of a force, whose accelerating effect at 
 
 any distance r is p . Prove that, if the direction of its motion be 
 deflected inwards through any angle ft without altering the velocity, 
 the body will arrive at the centre of force after a time 2 h ^ . . 
 
 5. Deduce from the time in an equiangular spiral the time of 
 passing from one point to another, when a body moves along a 
 straight line with a velocity which varies inversely as the distance 
 from a fixed point in that line. 
 
 6. A body describes an equiangular spiral in a resisting 
 medium with uniform angular velocity under the action of a 
 force tending to the pole ; prove that the force to the pole varies 
 as the distance and the resistance as the velocity. 
 
 7. Two particles of equal mass m, and at a distance 2 apart, 
 are projected simultaneously with velocity V in the same direction 
 perpendicular to the line joining them, the only force acting is a 
 mutual force of attraction varying inversely as the cube of the 
 distance between the particles, and equal at the distance 20 to mf. 
 
 Prove that, if after a time -/ft . - ^ 2 ~_ ^ \ one of ^ particles be 
 
 stopped and kept at rest, the other will proceed to describe an 
 equiangular spiral about it as pole. 
 
 8. Three particles A, B, C start from rest and move with 
 uniform velocities, A always directing its course towards B, 
 B towards C, and C towards A. Prove that if their velocities 
 be proportional to b'*c, c*a, cfl, where a, b, c are the initial distances 
 of B from C, C from A, and A from B respectively, they will 
 describe similar equiangular spirals, with a common pole.
 
 204 
 
 NEWTON. 
 
 PROP. X. PROBLEM V. 
 
 If a "body "be revolving in an ellipse, to find the law of 
 centripetal force tending to the centre of the ellipse. 
 
 Let OA, CB be the semiaxes of the ellipse, P the 
 position of the body at any time, PCG, DCD 
 conjugate diameters, Q a point near P, QT y PF 
 perpendiculars from Q and P on PC, DD ' draw 
 QU an ordinate to PCG, QR a subtense parallel 
 to CP. 
 
 Then jp= 
 
 ultimately. 
 
 But, by similar triangles QTU, PFC, 
 
 QT^ __ PF 3 QU* _ CD* 
 
 QU* - CP 3 > and PU.UG ~ CP* ; 
 QT* PF\CI? AC\BC* 
 
 ' 
 
 CP* CP* 
 
 ultimately, and PU=QR-, 
 QT* AC'.BC* 
 
 therefore the force is proportional to the distance 
 from the centre.
 
 PKOP. X. PROBLEM V. 205 
 
 Aliter. 
 
 Let CY be perpendicular on the tangent at P, and 
 PV be the chord of curvature at P which passes 
 
 through the centre = -^, Art. 79. 
 
 COR. 1. And conversely, if the force be as the 
 distance, a body will revolve in an ellipse having 
 its Centre in the centre of force, or in a circle, 
 which is a particular kind of ellipse. 
 
 COR. 2. And the periodic times will be the same in 
 all ellipses described by bodies about the same 
 centre of force. 
 For the periodic time in any ellipse 
 
 _2 x area of ellipse _ 2irA C. BC 
 ~h~ ~h ' 
 
 and the forces, at different distances in the same 
 or different ellipses, vary as the distance ; therefore 
 
 AC* BC*~* * S ^ e same in different ellipses, 
 therefore the periodic times in different ellipses 
 
 is the same, and =- 7 -. 
 \V 
 
 SCHOLIUM. 
 
 If the centre of an ellipse be supposed at an infinite 
 distance, the ellipse will become a parabola, and 
 the body will move in this parabola ; and the force, 
 now tending to a centre at an infinite distance, 
 will be constant and act in parallel lines. This 
 theorem is due to Galileo. And, if the parabola 
 be changed into an hyperbola, by the change of 
 inclination of the plane cutting the cone, the body 
 will move in this hyperbola under the action of a 
 repulsive force tending from the centre.
 
 206 NEWTON. 
 
 200. To find the velocity in the elliptic orbit under the action 
 of a force tending to the centre, the measure of whose accelerating 
 effect is fAX distance. 
 
 The vebeity at P- -*? 
 
 Aliter. 
 (Vel.) a at P=F.~= fji.CP. ^ ; .-. vel. at P= Jp.OD. 
 
 201. If a hyperbolic orbit be described under the action of 
 a repulsive force tending from the centre, the force will vary as the 
 distance, and the velocity at any point as the diameter of the 
 conjugate hyperbola parallel to the tangent at the point. 
 
 This may be proved exactly as in the case of the ellipse, 
 employing the proper figure. 
 
 202. To find the time in any arc of an elliptic orbit about a 
 force tending to the centre. 
 
 If Pbe any point of the orbit, Q the corresponding point in the 
 auxiliary circle, time in AP cc area A CPcc area A CQ cc LACQ ; 
 therefore time in AP : periodic time:: <j> : 2ir, if < be the 
 
 2ir 
 circular measure of LAC Q, and periodic time =-, ; therefore 
 
 -f- 
 
 v/* 
 
 time in AP= -- . 
 
 203. If, at a given point, the velocity of a body be knoum t 
 and the direction of its motion; to determine the curve which 
 the body will describe under the action of a given centripetal 
 force., which varies as the distance from the point to which it 
 tends. 
 
 Let Pt be the direction of motion at P, V the velocity at P, 
 ft . CP the measure of the accelerating effect of the force tending 
 to G. On PC produced, if necessary, take PV equal to four 
 times the space through which a body must move from rest, 
 under the action of the force at P continued constant, in order 
 to acquire the given velocity 7j so that V* =
 
 PROP. X. PROBLEM V. 207 
 
 Draw CD parallel to Pt, a mean proportional to CP and 
 F, and let an ellipse be constructed with OP, CD as semi- 
 conjugate diameters, then PV is the chord of curvature at P 
 through C. 
 
 MA. 
 
 In this ellipse let a body revolve under the action of a 
 force tending to 0, whose magnitude at P is that of the given 
 force, see Arts. 160, 162, then, when it arrives at the point P, 
 it will be moving in the direction Pfc, also the square of the 
 velocity at P= p,.CD' t = ^.CP.^PV^V\ or the velocity at P, 
 in the constructed ellipse, is F. Hence the body revolving 
 in this ellipse is under the same circumstances as the proposed 
 body, in all respects which can influence the motion of a body ; 
 therefore the proposed body will describe the ellipse constructed 
 as above. 
 
 A direct solution of the problem, which is solved syntheti- 
 cally in this Article, is given in pages 78 and 79. 
 
 204. Geometrical construction for the position and magnitude 
 of the axes of the elliptic orbit, described by a body about the centre, 
 when the velocity at a given point is known t and also the direction 
 of motion. 
 
 Produce CP to R, making PR a third proportional to CP and 
 
 CD; bisect CR in Z7, and draw UC perpendicular to CR t 
 
 meeting the tangent at P in 0, and with centre describe a 
 
 circle passing through C, R, and cutting the tangent in 2"and f ; 
 
 .-. PT. Pt = CP. PR = CD 2 j
 
 208 NEWTON. 
 
 Let TC intersect the ellipse in A, A, and draw PM parallel 
 to the diameter conjugate to A CA' ; 
 
 then PT* : CD* :: TA. TA' : CA* 
 
 r.CT*-CA*:CA*; 
 .'. PT* : PT.Pt :: CT*-CT.CM\ CT.CM; 
 
 .'. PT : Pt :: NT : CM-, 
 
 hence CT is parallel to P3/, and CT, Ct are in the directions of 
 conjugate diameters; but TCt is a right angle, therefore CT, 
 Ct being in the direction of perpendicular conjugate diameters, 
 are the directions of the axes of the ellipse, and if PM^ Pm be 
 perpendiculars from P upon these directions, the semiaxes are 
 mean proportionals between CJ/, CT, and CTH, Ct. Q.E.P. 
 
 205. Equations for determining the position and dimensions 
 of the orbit. 
 
 Let fi.R be the measure of the accelerating effect of the force 
 at the distance CP=R, Fthe velocity, a the angle between CP 
 and the direction of motion at the given point P. Let a, b be 
 the semiaxes of the ellipse, v: the angle which CP makes with 
 the major axis. 
 
 Then V* = p.GD t and CD 1 + CP 8 = a* + b* ; 
 
 V* 
 
 .: a* + b* = + R* (ll 
 
 P> 
 
 Also V.R since = & = *Jp.ab ; 
 
 ab = V ' R 8inct (<2\ 
 
 and, by the properties of the ellipse, 
 
 rn TM 
 
 * *' 1 (3). 
 
 The equations (1), (2), and (3) determine a, 5, and r, whence 
 the magnitude and position of the ellipse is determined. 
 
 We can obtain an equation for w, immediately in terms of 
 the data, as follows : 
 
 by (3),
 
 PROP. X. PROBLEM V. 
 
 ^ + y = cosec'a (l +^ , by (t) and (2), 
 * uIP 
 
 m ^v^ w 04XJ. UJ 
 
 '^ = r^ = 
 
 cosV sin* 
 
 cosec 2 a ( 1 + ^-J 2 
 .'. cot 2or = - tan a ( cot*a - 1 + cosec s a. 
 
 ~ 
 
 = cot 2a + cosec 2a.p ................. (4); 
 
 whence ts is known immediately from the initial circumstances of 
 the motion. 
 
 206. If the force be repulsive, the equations for determining 
 a, &, w will be 
 
 ...................... (1), 
 
 VR sin a 
 
 and 5 cos'-or -^- ain*cr = 1 .......... . ..... (3). 
 
 The direction and magnitude of the axes of the hyperbola 
 may be determined geometrically, by observing that the 
 asymptotes are the diagonals of the parallelograms of which the 
 conjugate semi-diameters are sides, and that the axes bisect the 
 angles between the asymptotes. 
 
 207. When a particle is acted on ly any number of forces, 
 'which tend to different centres^ and vary as the distances from those 
 centres t to find the resultant attraction. 
 
 El
 
 210 NEWTON. 
 
 Let /i. R, /A'. R be the magnitudes of two of the forces at the 
 distance R, A, B the centres to which they tend, P the position 
 of a particle acted on by the forces. 
 
 Let O be the centre of gravity of two particles at A and B 
 whose masses are in the ratio of fi to /*', join PA, PB, PQ. 
 
 The components of the force fJ-.PA, in the directions PG, 
 GA, are p. PG and p. GA, and those of the force /*'. PB, in the 
 directions PG, GB, are fi'.PG, and p'.GB, but p..GA = p. GB, 
 therefore the resultant of the forces tending to A and B is 
 (ft + fj.') PG, which is a single force of magnitude (/* -f //) -K, at 
 the distance R, tending to the centre of gravity of masses /*, /*' 
 placed at A and B. 
 
 Let //'.B be the magnitude of a force at the distance R, 
 tending to C, the resultant attraction is that of a force tending 
 to the centre of gravity H of particles at C and G, whose masses 
 are in the ratio /u." : /A + /*', which varies as the distance from H, 
 and whose magnitude at the distance R is (/A + /H' + /A") 72. 
 
 And generally, the resultant of any number of forces is a 
 single force, tending to the centre of gravity of a system of 
 particles, placed at the different centres, whose masses are 
 proportional to the magnitudes of the forces at the unit distance, 
 and whose magnitude at any distance is the sum of those of the 
 forces at the same distance. 
 
 208. COB. 1. If every particle of a solid of any form attract 
 with a force which varies as the mass of the particle and the 
 distance conjointly, the resultant attraction of the solid upon 
 any body will be the same as that of the whole mass of the solid
 
 PROP. X. PROBLEM V. 
 
 211 
 
 collected into its centre of gravity and attracting according to the 
 same law. 
 
 209. COR. 2. If any of the forces be repulsive, as that 
 whose centre is B, G will lie in AB or BA produced, according 
 as fi is greater or less than /*, and the resultant of the forces, 
 tending to A and from B, will be (/*-/&) PQ from (?, or 
 (p,- p!}PG towards G. 
 
 Illustrations. 
 
 (1) A body revolves in. a circular orbit about a force which 
 varies as the distance, and tends to the centre of the circle, and the 
 centre of force is suddenly transferred to a point in the radius 
 which at the moment of change passes through the body ; to find 
 the subsequent motion of the body. 
 
 Since the force varies as the distance, and is attractive, the 
 orbit will be an ellipse. And, since the force is a finite force, 
 the body will move in the same direction as before, at the 
 moment of the change. Also, the velocity will, for the same 
 reason, be unaltered at that moment. 
 
 Let CA be the radius passing through the body at the 
 moment of change, CB perpendicular to CA, p.GA the force 
 at distance CA t V the velocity in the circle. 
 
 Then F* = fj,.CA.CA = /*.(L4 2 ; and if S, the new point to 
 which the force tends, be in CA, let AB' be the ellipse described, 
 SA will be one of the semi-axes of the ellipse, sinee A is an
 
 212 
 
 NEWTON. 
 
 apse, and, SB being the other, if a body revolved in this ellipse 
 round S, p. 8B" would be the square of the velocity at -4, that 
 is, /*. SB" = p. CA*, and therefore 8B=GA=CB] hence the 
 magnitude and position of the two semi-axes SA and SB' are 
 known, and the ellipse is completely determined. 
 
 The ellipse lies without the circle at A, because, the velocity 
 being unaltered, the force has been diminished in the ratio of 
 SA : CA, and therefore the curvature diminished in that ratio. 
 
 If S had been in AC produced, as at /JT, the force would 
 have been increased, and the orbit AB" would be within the 
 circle near A. 
 
 The greatest distance from CA which the body reaches is in 
 all cases the same for this law of force, because the component of 
 the force perpendicular to CA is the same at the same distance 
 from CA in whatever curve the body moves ; therefore, in each 
 orbit, the velocity being the same at -4, the velocity perpen- 
 dicular to AC is destroyed by the force at the same distance 
 from AC. 
 
 (2) A body is describing a circle about a force which varies as 
 the distance and tends to the centre ; if the centre to which the 
 force tends be suddenly transferred to a point in the circumference^ 
 at an angular distance of 60 from the position of the particle at 
 any time, to determine the orbit described. 
 
 The orbit is an ellipse, since the force is attractive. 
 
 Let P be the position of the body at the instant the centre of 
 force is transferred from (7, the centre of the circle, to 5, where 
 8CP is an equilateral triangle.
 
 PROP. X. PROBLEM V. 
 
 213 
 
 The velocity at P is *Jp.CP= Jp.SP; and, since it is un- 
 altered by the change of the centre of force, the semi-diameter 
 conjugate to SP is equal to SP. 
 
 Draw DSD' perpendicular to CP, meeting it in F, and take 
 SD = SD' = SP. Construct an ellipse having SP, SD as equal 
 conjugate semi-diameters ; 4, SB the semi-axes bisect the angles 
 PSD, PSD.' The ellipse so described will be the orbit required. 
 
 Prove the following construction : 
 
 On CP as diameter describe a circle cutting SD in B' t A' 
 SA, SB' are the lengths of the semi-axes. 
 
 Explain why the orbit is exterior to the circle. 
 
 (3) Two bodies whose masses are m, m' revolve in an ellipse 
 under the action of a force tending to the centre; shew that, if 
 they be at one time at the extremities of two conjugate diameters 
 they will always be so, and in this case find the locus of their 
 centre of gravity. 
 
 Let P, D be their positions at any time, CP, CD being 
 semi-conjugate diameters. Let the ordinates MP, ND, meet 
 the auxiliary circle in Q and R. 
 
 Since the angles A CQ, A CR are always proportional to the 
 times, RCQ will always be a right angle; therefore the bodies 
 will always be at the extremities of conjugate diameters. 
 
 A W C It 
 
 Let GH be the ordinate of their centre of gravity. 
 Join RQ and produce HG to meet RQ in JOT; 
 
 .-. KB: GE= QM : PM, a constant ratio, 
 
 also, RK:KQ = DG: GP, ; 
 
 hence CK is constant, or the locus of K is a circle, and the 
 locus of G is an ellipse, whose axes are proportional to those 
 of APD. 
 
 Shew that the semi-major axis : CA :: (m f -H!*)* : m + m'.
 
 214 NEWTON. 
 
 (4) A body is composed of matter which attracts with a 
 force varying as the distance / shew that, however a particle be 
 projected, unless it strike the body, it will describe its orbit in 
 the same periodic time. 
 
 This is obvious immediately from Art. 208, relating to the 
 resultant of attracting forces. 
 
 (5) A body moves in an ellipse under the action of a force 
 varying as the distance ; if the velocity at any point be slightly 
 increased in the ratio 1 + n : 1, find the consequent changes in the 
 axes of the ellipse. 
 
 If, when the change takes place, the body be at the end of one 
 of the equal conjugate diameters, shew that the eccentricity will be 
 unaltered, and that the apse line will regrede through a small angle t 
 
 whose circular measure is _ . 
 
 When V is changed to (1+n) V, CD is changed to 
 (1 + n) CD ; let the corresponding changes of a, b and CT be 
 oa, /S5 and 7 ; a, /3, 7, and n being so small that we may neglect 
 their squares. Then by the equations of Art. 205, 
 
 .-. aa' + ^'^n. CD\ 
 
 Again (1 -f a)a. (1 + @}b = (1 + n) CD.R&ma.= (t +n)a&; 
 .-. a + = n, and a(a 8 - CD*} = /3 ( Cl? - 2 ), 
 a ff n 
 
 In the particular case 2^ !l = a t + J t , .*. a = /3 = n, hence, 
 a and b being altered in the same proportion, the eccentricity 
 will be unaltered. 
 
 R* H? 
 
 Also, cos* (si + 7) + r^ sin 2 (r + 7) = 1 + n 
 
 and 5 cos*r + -^ sinV = 1 ; 
 'I?
 
 PROP. X. PROBLEM V. 215 
 
 and, since the axes bisect the angles between equal conjugate 
 diameters, aJ = .KVm2t3-, therefore 7, being expressed in circular 
 
 nab 
 measure, = t 
 
 (6) In any position of a particle describing an ellipse, under 
 the action of a force tending to the centre, the centre of force is 
 suddenly transferred to the focus, find the axes of the new orbit 
 and shew that its major-axis bisects the angle between the focal 
 distance and the major-axis of the given ellipse. 
 
 Employing the equations of Art. 205, if a, fi be the semi- 
 axes of the new orbit, P the position of particle when the centre 
 s transferred to S, since the semi-diameter conjugate to SP in 
 the new orbit will be equal to CD, 
 
 and SY* : BC* :: SP: HP:: SP* . CD"; 
 
 .-. a&=CD.SY=b.SP; 
 
 ... ( a _ ) = 4 (a 2 - V] SP\ and a 2 - & = 2aeSP t 
 .'. a 2 = a(i+e) P,and/3 2 = a (l-e) 8P. 
 
 SP 2 SP* . 
 
 Also r cosOT + --surG7 = l, 
 
 a l ~ 6 ^ = (1 - e) cosV + (1 4 e) sinV = 1 - e cos 2*7 ; 
 
 therefore 2cr = zP&4, or the major-axis of the new orbit 
 bisects the angle between PS and the major-axis of the 
 original orbit. 
 
 NOTE. By the construction of Art. 204, since PR is a third 
 proportional to SP and CD, and therefore is equal to HP, the 
 circle which determines T and t passes through H, and the arcs 
 HT, TR are equal, that is, ST bisects the angle PSA. 
 
 XXVII. 
 
 1 . Shew that the Telocity in an ellipse about the centre is the 
 game at the points whose conjugate diameters are equal as that in 
 a circle at the same distance.
 
 216 NEWTON. 
 
 2. A body is revolving in a circle under the action of a force 
 tending to the centre, the law of force at different distances being 
 that the force varies as the distance ; find the orbits described when 
 the circumstances are changed at any point as follows : 
 
 i. The force is increased in the ratio of 1 : n. 
 ii. The velocity is increased in the ratio 1 : n. 
 
 iii. The force becomes repulsive, remaining of the same mag- 
 nitude. 
 
 iv. The direction is changed by an impulse in the direction of 
 the centre, measured by the velocity equal to that in the circle. 
 
 3. If a body be projected from an apse, with a velocity double of 
 that in a circle at the same distance, find the position and magnitude 
 of the axes of its orbit. 
 
 4. A particle is revolving in a circle acted on by a force which 
 varies as the distance ; the centre of force is suddenly transferred 
 to the opposite extremity of the diameter through the particle 
 and becomes repulsive ; shew that the eccentricity of the hyperbolic 
 orbit = V5. 
 
 5. A body is moving under the action of a force tending to a 
 fixed centre, and varying as the distance. The force suddenly 
 ceases, and after an interval commences to act again. Prove that 
 the radii of curvature of the orbit at the points where the body 
 ceases and recommences to be attracted are equal. 
 
 6. A body moves in an ellipse about a centre of force in the 
 centre, and its velocity is observed when it arrives at its greatest 
 distance, and again after a lapse of one-third of its periodic time. 
 If these velocities be in the ratio of 2 : 3, prove that the eccentricity 
 of the ellipse will be V*. 
 
 7. The particles of which a rectangular parallelepiped is com- 
 posed attract with a force which varies as the distance, and a body is 
 projected so as to describe a curve on one of the faces supposed 
 smooth ; find the periodic time. 
 
 8. An elastic ball, moving in an ellipse about the centre, on 
 arriving at the extremity of the minor axis strikes directly another 
 ball at rest ; find the orbits described by both bodies. 
 
 9. A body is projected in a direction making an angle cos" 1 -- 
 
 with the distance from a point to which a force tends, varying as 
 the distance from it, and the velocity = V x velocity in the circle at 
 the same distance ; prove that one axis is double of the other and 
 that the inclination of the major axis to the distance is J cos 1 "!-
 
 PROP. X. PROBLEM V. 217 
 
 10. From points in a line CA between C and A particles are 
 projected at right angles to CA with velocities proportional to their 
 distances from A, C being a centre to which the force tends, and 
 the force varying as the distance ; find the ellipse of greatest area 
 which is described. 
 
 11. Two particles are projected in parallel directions from two 
 points in a straight line passing through a centre of force, the 
 acceleration towards which varies as the distance, with velocities 
 proportional to their distances from that centre. Prove that all 
 tangents to the path of the inner cut off, from that of the outer, 
 arcs described in equal times. 
 
 12. An hyperbola and its conjugate are described by particles 
 round a force in the centre. They are at an apse at the same 
 instant ; shew that they will always be at the extremities of con- 
 jugate diameters. Also if v, v' be their velocities, o z -v'* = ft (a 2 - 5 2 ). 
 
 13. An ellipse and an hyperbola have the same centre and foci. 
 They are described by particles, under the action of forces in the 
 centre of equal intensity. If a, a 1 be their semi-transverse axes, 
 the square of the velocity of each body at a point where the curves 
 cut will be n a~ - a' 2 ). 
 
 14. If any number of particles be moving in an ellipse about a 
 force in the centre, and the force suddenly cease to act, shew that, 
 
 after the lapse of ^ of the period of a complete revolution, 
 all the particles will be in a similar, concentric, and similarly 
 situated ellipse. 
 
 15. A particle is describing an ellipse under the action of a 
 force tending to the centre. Prove that its angular velocity about 
 a focus is inversely proportional to its distance from that focus. 
 
 xxvin. 
 
 1. CX, C'Fare straight lines inclined at any angle, and a force 
 tends to C, and varies as the distance from C. If from various 
 points in C T different particles are projected parallel to CX at the 
 same moment, and with the same velocity, they will all arrive at 
 CX at the same time and place ; and they will also do so, it the 
 force cease to act for any interval of time. 
 
 2 A number of particles move in hyperbolas, under the action 
 of the same repulsive force from their common centre. Shew that, 
 if the transverse axes coincide, and the particles start from the 
 vertex at the same instant, they will always he in a straight line