^>.-
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES 
 
 GIFT OF 
 
 John S.Prell
 
 A 
 TREATISE 
 
 ON 
 
 FLUXIOJVS. 
 
 IN TWO VOLUMES. 
 
 COLIN MACLAURIN, A.M. 
 
 Late Professor of Mathematics in the University of Edinburgh, and Fellow of the 
 Royal Society. 
 
 SECOND EDITION. 
 
 TO WHICH IS PREFIXED 
 
 AlSl ACCOUNT OF HIS LIFE. 
 
 THE WHOLE CAREFUtLY CORRECTED AND REVISED BY 
 
 An Eminent Mathematician. 
 
 ILLUSTRATED WITH FORTY-ONE COPPERPLATES. 
 
 VOL. II. 
 
 LONDON : 
 
 PRINTED FOR WILLIAM BAYNES, 54, PATERNOSTER ROW, 
 
 AND WILLIAM DAVIS, 
 
 Editor of the Geatleman's Mathematical Companion, Author of the Complete Treatise of Land 
 Stirveying, Use of the Globes, &c. &c. 
 
 SOLD ALSO BY 
 
 Hanwell and Parker, Oxford ; Deighton, Cambridge ; Dickson, Edinburgh ; 
 and Dugdale, Dublin. 
 
 1801. 
 ^^^g^ & Comptajv Print^^I^ddl&gn^., - 
 
 Cri»7 & Mechanical Engineer. 
 
 SAN FBANOISOO, CAL.
 
 f
 
 Engineering 
 Library 
 
 TABLE OF THE PRINCIPAL CONTENTS. 
 
 ^^, 
 
 BOOK I. 
 
 V^HAPTER XII. of the method of infinitesimals, of the limits of 
 ratios, and of the general theorems which are derived from this 
 doctrine, for the resolution of geometrical and philosophical pro- 
 blems 
 Of the harmony betwixt the method of fluxions and of infinitesimals . 495 
 Some objections against the method of infinitesimals considered. . . . 498 
 The true reason why parts of' the element are to be neglected in the 
 
 method of infinitesimals 501 
 
 Of Sir Isaac Newton'* method by the limits of ratios 502 
 
 Propositions of the preceding chapters danonstraled briefly by this 
 
 method 506 
 
 Theorems concerning the centre of gravity and its motion, and their 
 use shown in resolving several problems coiicerning the collisions 
 
 of bodies 510 
 
 Of the descent of bodies that act vpon one another, of the descent 
 and ascent of their centre of gravity, and the preservation of the 
 
 vis ascenclens, or vis viva 521 
 
 Of the centre of oscillation 534 
 
 Of the motion of water issuing from a cylindric vessel 53/ 
 
 Of the motion of water issuing from any vessel , 550 
 
 Of the Catenaria, when gravity acts in parallel lines 551 
 
 General theorems concerning the trajectories, lines of sxviftest de- 
 scent, the Catenaria, SfC 563 
 
 Chap. XIII. The analysis (f the problem concerning the lines of 
 swiftest descent, when an unij'orm or variable gravity acts in pa- 
 rallel lines BTi 
 
 The sijnihctic demonstration 576 
 
 » 2 The 
 
 713S80
 
 I^r CON'TENTS. 
 
 The same, when gratity tends to a given centre , . . . 578 
 
 Another synthetic demonstration 584 
 
 Of the lines of swiftest descent amongst those of the same perimeter 
 
 in any hypothesis of gravity 588 
 
 The first general isoperimetrical problem resolved by first fiuxions, 
 
 and the resolution demonstrated synthetically 592 
 
 The problem extended further by the same method 5.97. 
 
 The second general isoperimetrical problem resolved in the same 
 
 manner 601 
 
 The property of the solid of least resistance demonstrated in this 
 
 manner 606 
 
 Ch a p. XIV. Of the ellipse considered as the section of a cylinder . 609 
 General properties of the conic sections transferred briefly from the 
 
 circle 622 
 
 Of gravitation towards spheres and spheroids 628 
 
 Supposing the density of the planets uniform, their figure is accu- 
 rately that of the oblate spheroid, which is generated by the conic 
 
 ellipse about its second axis 6sS 
 
 Of the figure of the planets and variation of gravity towards them 641 
 The gravitation at the pole and equator, or any point on the surface 
 
 of a spheroid, measured accurately by circular arks or logarithms 6^2 
 The gravitation in the axis or plane of the equator produced, mea- 
 sured accurately by the same 648 
 
 Of the figure of the earth in particular, supposing its density uniform 655 
 Of the gravity towards a spheroid, supposing the density variable. . . 660 
 Of the figure of Jupiter, and the effects of his spheroidical form upon 
 
 the motions of the satellites 682 
 
 Of the tides 686 
 
 Of other laws of attraction 6c)(S 
 
 BOOK II. 
 
 Of the Computations in the Method of Fluxions. 
 
 CHAP. I. Of the fiuxions of quantities considered abstractly as 
 represented by general characters in Algebra. 
 
 Of the import of some algebraic symbols Art. 6,9.9 
 
 I'he principles of this method adapted to algebra 700 
 
 Of the fiuxions cf powers of all hinds 707 
 
 Of thcfiuxions of products and quotients 715 
 
 Of
 
 CONTENTS. V 
 
 Of the fluxions of logarithms 717 
 
 Of second and higher fluxions 720 
 
 Chap. II. Of the notation of fluxions Art. 723 
 
 The rules of the direct method 724 
 
 The fundamental rules of the inverse method 735 
 
 Of inflnite series 745 
 
 An investigation of the binomial and vmltinomial theorems 748 
 
 Other theorems 751 
 
 Examples of their use 753 
 
 Chap. III. Of the analogy hetvcixt elliptic and hyperbolic sec- 
 tors Art. 758 
 
 Of resolving trinomials into quadratic divisors 7^5 
 
 Of reducing fluents to circular arks and logarithms when thefluxion 
 
 is expressed by rational quantities 770 
 
 Of reducing fluents to the same measures when thefluxion involves 
 
 an irrational binomial or trinomial 789 
 
 Of reducing fluents to hyperbolic and elliptic arks 798 
 
 Of reducing fliie?its of a higher kind to others of a more simple form 810 
 Chap. IV. Of the area ichcn the ordinate is expressed by a 
 
 fluent Art. 813 
 
 Of the area when the ordinate and base are both expressed by fluents 819 
 Instances xchercin. the total area, or fluent, is measured by circular 
 arks or logarithms, when it does not appear that the same fluent 
 
 can be generally reduced to those measures 822 
 
 ^Theorems derived from the method of fluxions for approximating to 
 
 the sums of progressions by areas, and conversely 828 
 
 Theorems for finding the sum of any powers, positive or negative, of 
 the terms in an arithmetical progression, and for flnding the sums 
 
 cf their logarithms 833 
 
 Of the ratio of the sum of all the unciae of a binomial of a very high 
 
 power to the vincia of the middle term 844 
 
 Of computing the area from afexo equidistant ordinates 848 
 
 Theorems derived from the method of fluxions for interpolating the 
 
 intermediate terms of a series , 850 
 
 Chap. V. Of the general rules for the resolution of problems by 
 computations, with examples 
 
 Of the rules for determining the tangents Art. 857 
 
 The greatest and least ordinates 858 
 
 The points of contrary Hexure and cuspids 86' J 
 
 The
 
 ■*> CONTEXTS. 
 
 The centre of curiaiure , §70 
 
 The caustics by reflexion and refraction 8/2 
 
 The centripetal forces g^4. 
 
 The constrtiction of the trajectory that is described by a force ivhick 
 is interselii as theffth pouer of the distance, by logarithm in cer- 
 tain cases fi/S 
 
 la these case.'; a body may recede from the centre continually, so as 
 never to rise to a certain altitude, or may approach to it for ever, 
 
 and never descend to a certain distance, 879 
 
 TJie construction in other cases 881 
 
 The rules for computing the time of descent along a given curve . . . 884 
 The time in a finite circular arch measured by the arks of conic sec- 
 tion's 886 
 
 T//« same by infnite series 887 
 
 Rules concerning the computation of motions in a medium 888 
 
 Iluksfor determining the figure (f the catenaria, and the lines of 
 
 swiftest descent 889 
 
 Hulesfor the computation of areas, solids, curvilineal arks and sur- 
 faces 890 
 
 The meridional parts in a spheroid computed by circular arks or lo- 
 garithms 895 
 
 The gravitation towards a spheroid at the pole and equator, mea- 
 sured by circular arks and logarithms, when the force towards 
 any particle is inversely as any power of the distance from it. . 9OO 
 
 Of the centres of gravity and oscillation 906 
 
 Of the proportion of the power to the weight, that a machine may 
 
 have the greatest effect 907 
 
 Of the same when the friction is considered 90S 
 
 2'he most advantageous position of a plane, which moves parallel to 
 itself with a given direction, that a stream may impel it with the 
 greatest force, when the velocities of the stream and plane are 
 
 given 910 
 
 7'he wind ought to strike the sails of a wind-mill in a greater angle 
 
 than 5.4°. 44'. 914 
 
 The most advantageous position of the sails that the wind may impel 
 a ship with the greatest force in a given direction, the velocities of 
 
 the wind and ship being given 91^ 
 
 How an ark is to he divided into any number of parts, that the 
 profluct .of an I) powers of the sines of the several parts may be a 
 
 maximuiii 921 
 
 The
 
 CONTENTS 1!U 
 
 ^The most advantagemts direction of the motion of a ship, and best 
 position of the sail, that the ship may recede from a given line or 
 
 coast with the greatest velocity 922 
 
 Of reducing equations from second tofrsffuxious, 'with examples.^ 924' 
 The construction of the elastic curxe, and of other figures, by the 
 
 rectification of the conic sections . 927 
 
 Of the vibrations of musical chords 929 
 
 Problems concerning the maxima a7id minima that are proposed 
 with limitations concerning the perimeter of the figure, its area, 
 the solid generated by this area, SfC. resolved by frst fluxions. , 931 
 
 Examples of this kind relating to the solid of least resistance 934 
 
 An example of the method of computing from the general principles 
 
 in art. 563 935 
 
 An instance of the theorems by xchich the viilue of the ordinate may 
 
 be determined from the value cf the area, by common algebra. ., . 936 
 It is. relative not absolute space and motion that are supposed in the 
 method of fluxions , 937 
 
 TREATISE
 
 TREATISE ON FLUXIONS. 
 
 BOOK I. 
 
 CHAP. XII. 
 
 Of the Method of Infinitesimals, of the Limits of Ratios, and of 
 the general Theorems which are derivedfrom this Doctrine for 
 the Resolution of geometrical and philosophicgtl Problems. 
 
 49j.JLN the account which we have given of the method of 
 fluxions, in the preceding part of this treatise, magnitudes were 
 supposed to be generated by motion; and, by comparing the in- 
 crements that were generated in any equal successive, parts of 
 the time, it was first determined whether the motion was uni- 
 form, accelerated, or retarded. When the motion was uni- 
 form, the fluxion of the magnitjide was measured by the in- 
 crement which it acquired in a given time. When the mo- 
 tion was accelerated, this increment Was- resolved into two 
 parts; that which alone would have been generated if the motion 
 had not been accelerated, but had continued uniform from the 
 beginning of the time, and that which was generated in conse- 
 quence of the continual acceleration of the motion during that 
 time. The latter part was rejected, and the former only re- 
 tained for measuring the motion at the beginning of the time. 
 And in like manner, when the motion was retarded, the quanti- 
 ty, which was found deficient inconsequence of this retardation, 
 was supplied; so that the motion at the term proposed was ac- 
 curately measured, and the ratio of the fluxions always ac- 
 curately represented. In the method of infinitesimals, the ele- 
 ment, by which any quantity increases or decreases, is supposed 
 to be infinitely small, and is generally expressed by two or 
 more terms, someof which are infinitelv less than therest, which 
 VOL.11. B * being
 
 2 Of the Method of Infinitesimals. Book I. 
 
 being neglected as of no importance, the remaining terms form 
 what is called the difference of the proposed quantity. The 
 terms that are neglected in this manner, as infinitely less than 
 the other terms of the element, are the very same which arise 
 in consequence of the acceleration, or retardation, of the gene- 
 rating motion, during the infinitely small time in which the ele- 
 ment is generated; so that the remaining terms express the ele- 
 ment that would have been produced in that time, if the gene- 
 rating motion had continued uniform. Therefore those differ^ 
 cnces are accurately in the same ratio to each other as the ge- 
 nerating motions or fluxions. And hence,though in this method 
 infinitesimal parts of the elements are neglected, the conclusions 
 are accurately true, without even an infinitely small error, and 
 agree precisely with those that are deduced by the method of 
 fluxions. Forexample, *inprop.2, whenDGr//^. 21), thein- 
 crementof the base ADof the triangleADE,issupposed to become 
 infinitely little, the trapezium DGHE (the simultaneous incre- 
 ment of the triangle) consists of two parts, the parallelogram 
 EG, and the triangle EIH; the latter of which is infinitely less 
 than the former, their ratio being thatof-i-DG to AD. There- 
 fore, according to this method, the part EIH is neglected, and 
 the remaining part, viz. the parallelogram EG, is the difference 
 of the triangle ADE. Now it was shown above (art. 93), that 
 EG is precisely that part of the increment of the triangle ADE 
 which is generated by the motion with which this triangle 
 flows, and that EIH is the part of the same increment which 
 is generated in consequence of the acceleration of tliis motion, 
 while the base by flowing uniformly acquires the augment 
 DG, whetiier DG be supposed finite or infinitely little. In 
 prop. 3, case 1 , the incrementDELMHG C/ig-'l") of the rectangle 
 AE consists of the parallelograms EG, EM, and lb; the last of 
 which 16 becomes infinitely less than EG or E^I,when DG and 
 LMtheincrementsofthe sides arc supposed infinitely small; be- 
 cause lb is to EG as LM to AL, and to EM as DG to AD; 
 therefore 16 being neglected, the sum of the parallelo- 
 grams EG and EM is the difference of the rectangle AE : 
 
 * The figures cited from Vol. I. are repeated in this Volume in plate 25, opposite 
 
 to p. 1 i. 
 
 and
 
 Ghap. XIL Of the Method of hifinitedmah. 3 
 
 and it was shown in art. 102, that the sum of EG and EM 
 is the space that would have heen generated by the motion 
 with which the rectangle AE flows continued uniformly, but 
 that \b is the part of the increment of the rectangle which is 
 generated in consequence of the acceleration of this motion, 
 in the time that AD and AL by flowing uniformly acquire 
 the augments DG and LINE The same may be observed of 
 all the other propositions wherein the fluxions of quantities are 
 determined above. 
 
 496. In general suppose, as in art. 66, that while the point P 
 (.,^'g. 2'20) describes the rightline Aawith anuniformmotion,the 
 poi n t M se ts o u t from L with a velocity th at is to the constan t vei o- 
 city of P as he to Dg. and proceeds in the right line Ee with a 
 motion continually accelerated or retarded, that LS any space 
 described by M is always to DG the space described in the same 
 time by P as I/to Dg, that ex is to J)g as the difference of the 
 velocities of M at S and L to the constant velocity of P, and 
 that LS is always to LC as I/to Lc. Then LS being always 
 expressed by LC + CS, it is manifest that (since LC is to DG 
 as Lc to Do-, or as the velocity of M at L to the velocity of P) 
 LC is what would have been described by M if its motion had 
 continued uniformly from L, and that CS arises in this expres- 
 sion in consequence of the acceleration or retardation of the 
 motion of the point M while it describes LS. But if LS and 
 DG be supposed infinitely small increments of EL and AD, cv 
 will be infinitely less than Dg; and since (/is less than ex by 
 what was shown in art. QG, it follows that c/'will be infinitely- 
 less than Lc, and CS infinitely less than LC. Therefore -when 
 the increment LSis supposed infinitely small, and its expression 
 is resolved into two parts LC, and CS, of which the former LC 
 is always in the same ratio to DG (the simultaneous increment 
 of AD while the increments vary, and the latter CS is infinite- 
 ly less than the former LC, we may conclude that the part CS 
 is that which arises in consequence of the variation of the mo- 
 tion of M while it describes LS, and is therefore to be neglect- 
 ed in measuring the motionof M atL, or the fluxion of the right 
 line EL. Thus the manner of investigating the differences, or 
 fluxions of quantities in the method of infinitesinials maybe de- 
 
 B 2 duced
 
 4 OftheMttlwdoflnJinitesimah, Book I. 
 
 duced from the principles of the method of fluxions demonstrat- 
 ed above. For instead of neglecting CS because it is infinite- 
 ly less than LC (according to the usual manner of reasoning in 
 that method)^ we may reject it^ because wemay thence conclude 
 that it is not produced in consequence of the generating motion 
 at L, but of the subsequent variations of this motion. And it 
 appears why the conclusions in the method of infinitesimals arc 
 not to be represented as if they were only near the truth, but 
 are to be held as accurately true. 
 
 497. In order to render the application of this method easy, 
 some analogous principles are admitted, as that the infinitely 
 small elements of a curve are right lines, or that a curve is a po- 
 lygon of an infinite number of sides, which being produced give 
 the tangents of the curve, and by their inclination to each other 
 measure the curvature. This is as if we should suppose that 
 when the base flows uniformly the ordinate flows with a motion 
 which is uniform for every infinitely small part of time, and in- 
 creases or decreases by infinitely small differences at the end of 
 every such time. Buthowever convenient this principle may be, 
 it must be applied with caution and art on various occasions. It 
 is usual therefore in many cases to resolve the element of the 
 curve into two or more infinitely small right lines; and some- 
 times it is necessary (if we would avoid error) to resolve it into 
 an infinite number of such right lines, which are infinitesimals of 
 the second order. In general it is apostulatimi in this method that 
 we may descend to the infinitesimals of any order whatever as 
 we find it necessary, by which means any error thatmight arise 
 in the application of it may be discovered and corrected by a 
 proper use of this method itself. This will appear by consider- 
 ins: some instances wherein it is said to lead us into error. 
 
 498 (F/g.C21). The most no ted of these is taken from the doc- 
 trine of pendulums. If we were to consider the circle ABH, whose 
 diameter AH is perpendicular to the horizon, as a polygon of an 
 infinite number of sides, and consequently the infinitely small 
 arch AB as coinciding with its chord, it would seem to follow 
 that the time of a vibration in such an arch ought to be equal to 
 tlie time of descent in its chord, which is ecjual to the time of 
 descent in the diameter HA ; whereas if the ratio of those times 
 
 be
 
 Chap. XII. Of the Method of Infinitesimals. 5 
 
 be at all assignable, it must be that of the quadrant of a circle 
 to the diameter, as may be shown from art. 408. But it is easy 
 to discover that we are not in this case to argue from infi- 
 nitesimals of the first order,since if we should suppose the same 
 arch to coincide with its tangent AT, the time of descent in it 
 would be found infinite. This difficulty however cannot be re- 
 moved (as some others) by resolving the infinitely small arch 
 AB into two infinitely small chords BD and AD, or tangents 
 BC and AC, or into any finite number of such chords or tan- 
 gents. The time in the tangent BC must be supposed the half 
 of the time in the chord BA, because BC is equal to CA, and 
 when BDA is supposed infinitely small, BC is one half of BA ; 
 the time in CA is the half of the time in BC ; consequently the 
 time in BC and CA is three fourths of the time in the chord 
 BA, or diameter HA, which is nearer to the true time in the 
 arch BDA, but is not yet equal to it. By supposing the arch 
 BDA to be continually subdivided into more and more equal 
 parts, and the tangents or chords to be drawn at each division, 
 the times in the circumscribed and inscribed figures will conti- 
 nually approach to the time in the arch, and will at length 
 agree with it when the divisions are supposed infinite in number, 
 in the same manner that the circumscribed and inscribed poly- 
 gons approach to the circumference of the circle, and are said 
 to coincide with it when the number of their sides is supposed 
 infinite. But the time in such an infinitely small arch is briefly 
 determined by considering it as coinciding with the time in the 
 arch of the cycloid of the same curvature, which was determin- 
 ed in art. 408. 
 
 ^'HQi.Fi.g. 222). When a curve is considered as a polygon of an in- 
 finite number of sides, and CE, EH are any two of those sides, if 
 CE produced meet GH the ordinate from H in T, CT is com- 
 monly supposed to be the tangent, and HT the subtense of the 
 angle of contact ; and if CL, EI parallel to the base meet the 
 ordiuatcs DE, GH in L and I, IT will be equal to LE, and 
 TH equal to the difference of LE and IH which are the first 
 dificiences of the ordinates; and hence HT the subtense of the 
 angle of contact is often supposed by authors on this method to 
 be equal to the second difference of the ordinates ; whereas it 
 
 B 3 follows.
 
 6 Of the Method of Injinitesimals. Book I. 
 
 follows, from what was shown above, that when the arch is 
 infinitely diminished, the subtense of the angle of contact is 
 equal to the half of the second difference, or second fluxion of 
 the ordinate, only. But it is obvious that there is no reason why 
 the tangent of the curve at E ought to be supposed to coincide 
 with one of those elements CE, EH, rather than tiie other; and 
 that it ought to be considered in this method as equally inclin- 
 ed to both, or rather as forming with each infinitel}' small 
 angles that difler from each other by an angle infinitely less 
 than either. Therefore let the tangent tE^ be supposed equal- 
 ly inclined to EC and EH, and meet BC, GH in t and t; then 
 the second difference of the ordinate (or the difference of LE and 
 IH) will be equal to Ct-j-Hf or 2H^, that is to twice 
 the subtense of the an2;le of contact. Thev however who 
 consider the subtense of the angle of contact as equal to the 
 second difference of the ordinate^ compensate this error by suppos- 
 ing that angle in effect to be double of what it is. But whether 
 we suppose CEand EH to be rectilineal or curvilineal elements 
 of the figure, the subtense of the angle of contact ought to be 
 supposed equal to the half o^lhesecond difference of the ordinate 
 only .See art. 234. If we would compare these subtenses at 
 different distances from the point of contact, it is better then to 
 consider the element of the curveas aninfinitely small arch of a 
 circle, unless when the curvature is of those kinds which were 
 described in art. 377 and 378, that are either less or greater 
 than the curvature of any circle. Hence when the ray of cur- 
 Vature is finite, the subtenses of the same angle of contact are in 
 the duplicate ratio of the arches ; but in the cases described in 
 those articles they follow other proportions. 
 
 500. When the value of a quantity that is required in a 
 philosophical problem becomes in certain particular oases infi- 
 nitely great, or infinitely little, the solution would not be al- 
 ways just though such magnitudes were admitted. As when it 
 is required, to find by what centripetal force a curve could be 
 described about a fixed point that is either in the curve, or is. 
 so situated that a tangent may be drawn from it to the curve, 
 the value of the force is found infinite at the centre of the forces 
 in the former case, and at the point of contact in the latter; 
 
 yet
 
 Chap. XII. Of the Method of Infiiiitedmah. 7 
 
 yet it is obvious that an infinite force could not inflect the line 
 described by a body that should proceed from either of these 
 points into a curve ; because the direction of its motion in 
 either case passes through the centre of the forces, and no force 
 how great soever that tends towards the centre could cause it 
 to change that direction. But it is to be observed that the 
 geometrical magnitude by which the force is measured is no 
 more imaginary in this than in other cases where it becomes in- 
 finite ; and philosophical problems have limitations that enter 
 not always into the general solution given by geometry. 
 
 50 1 . But to insist on no more instances : what we have chief- 
 ly in view is to show how these scruples may be obviated^ 
 which the brief manner of proceeding in the method of infini- 
 tesimals is apt to suggest to such as enter on the higher parts 
 of geometry, after having been accustomed to a more strict and 
 rigid kind of demonstration in the elementary parts. To such 
 it may seem not to be consistent with the perfect accuracy that 
 is required in geometrical demonstration, that in determining 
 the first differences, any part of the element of the variable 
 quantity should be rejected merely because it is infinitely less 
 than the rest, and that the same part should be afterwards em- 
 ployed for determining the second and higher differences, and 
 resolving some of the most important problems. Nor can we 
 suppose that their scruples will be removed, but rather confirm- 
 ed, when they come to consider what has been advanced b}' 
 some of the most celebrated writers on this method, who have 
 expressed their sentiments concerning infinitely small quantities 
 in the prcoisest terms ; while some of them deny their reality, 
 and consider them only as incomparably less than finite quanti- 
 ties, in the same manner as a grain of sand isimcomparably less 
 than the whole earth ; and others represent them, in all their 
 orders, as no less real than finite quantities. It was with a view 
 to remove any ground there might seem to be given for scruples 
 of this kind, that we followed a less concise method in the 
 preceding chapters of this treatise, and showed in art. 495 and 
 496, that a satisfactory account may be given for the more 
 brief way of reasoning that is in use in the method of infinitesi- 
 mals. AVhen we investigate the first differences, we may reject 
 B 4 the
 
 "8 Of the Limits of Ratios. Book I. 
 
 the infinitesimal parts of the element, not merely because they 
 are infinitely less than the other parts ; but because the quanti- 
 ties generated, and their mutual relations depend upon the ge- 
 nerating motions (art. ^4, 33, 42, 43), and are discovered by 
 them: and because in measuring these motions^ at any term of 
 the time, the infinitesimal parts of the element are not to be re- 
 garded, since they are not generated in consequence of those 
 motions themselves, but of their variations from that term ; as 
 was shown at length in prop. 2, and its corollaries, and in seve- 
 ral other parts of the preceding chapters. The same infinitesi- 
 mal parts of the element hovi'^ever may serve for measuring the 
 acceleration or retardation of those motions from that term, or 
 the powers which may be conceived to accelerate or retard 
 them at that term : and here the infinitely small parts of the 
 element that are of the third order are neglected for a similar 
 reason, being generated only in consequence of the variation of 
 those powers from that term of the time. In this manner we 
 presume some satisfaction may be given to the scrupulous (who 
 may be apt to demur at the usual way of reasoning in this me- 
 thod), while nothing is neglected without accounting for it: 
 and thus the harmony may appear to be more perfect betwixt 
 the method effluxions and that of infinitesimals. 
 
 502. But however safe and convenient this method ma}- be, 
 some will always scruple to admit infinitely little quantities, and 
 infinite orders of infinitesimals, into a science that boasts of the 
 most evident and accurate principles as well as of the most rigid 
 demonstrations ; and therefore we chose toestablish soextensive 
 and useful a doctrine in the preceding chapters on more unex- 
 ceptionable ^os^w/a^a. In order to avoid such suppositions. Sir 
 I&aac Newton considers the simultaneous increments of the 
 flowing quantities as finite, and then investigates theratio which 
 is the limit of the various proportions which those increments bear 
 to each other, while he supposes them to decrease together till 
 they vanish; which ratio is the samewith theratioof thefluxions 
 by what was shown in art. 66, 67, and 68. In order to discover 
 this limit, he first determines the ratio of the increments in gene- 
 rJil, and reduces it to the most simple terms, so as that (generally 
 speaking) a part at least of each term may be independent of the 
 
 value
 
 Chap. XII. Of the Limits of Ratios, 9 
 
 value of the increments themselves ; then by supposing the in- 
 crements to decrease till they vanish, the limit readily appears. 
 503. For example, let a be an invariable quantity, x a flow- 
 ing quantity, and o any increment of or ; then the simultaneous 
 increments of xx and ax will be 2xo-\'(H) and ao, which are in 
 the same ratio to each other as Q,x-\-o is to a. This ratio of 
 2r-|-o to a continually decreases while decreases, and is al- 
 ways greater than the ratio of 2jr to a while is any real incre- 
 ment, but it is manifest that it continually approaches to the ra- 
 tio of2xto« as its limit; whence it follows that the fluxion 
 of XX is to the fluxion of ax as Q.x is to a. If x be supposed to 
 flow uniformly, ax will likewise flow uniformly, but xx with a 
 motion continually accelerated : the motion with which ax 
 flows may be measured by ao, but the motion with which xx 
 flows is not to be measured by its increment 2xo-|-oo (by 
 ax. 1), but by the part 2xo only, which is generated in conse- 
 quence of that motion ; and the part 00 is to be rejected because 
 it is generated in consequence only of the acceleration of the 
 motion with which the variable square flows, while the in- 
 crement of its side is generated : and the ratio of 2x0 to ao is 
 thatof2j: to a, which was found to be the limit of the ratio of thein- 
 crements '2xo-\-oo and ao (fig. 2G0). In general, if (as in art. 
 QQ, ^c.) the point P be supposed to describe DG upon the right 
 line Aa with an unifoim motion, and M describe LS upon Ee 
 with a variable motion in the same time, the velocity of M at 
 L be to the constant velocity of P as Lc is to J)g, and I/be al- 
 ways to Tig as LS to DG ; it was shown in those articles that 
 if LS and DG (the simultaneous increments of EL and AD) be 
 supposed to decrease till they vanish, then the ratio of I/(or 
 Lc '+ cf) to J)g, or of LS to DG, will approach continually 
 to that of Lf to T)g as its limit. Therefore if the ratio be de- 
 termined, which is the limit of the various proportions in M'hich 
 I/is to Dg while the increments LS and DG decrease till they 
 vanish, this can be no other than the ratio of Lc to Dg, or of the 
 velocity of M at the term when it comes to L to the constant 
 velocity of P, that is of the fluxion of EL to the fluxion of AD. 
 If LC be to CS as Lc is to C/, then LC will be the part of LC 
 ipCS (the expressioii of LS) which arises in consequence of the 
 
 motion
 
 10 Of the Limits of Ratios. Book I. 
 
 motion of iSI at L^ and CS the part which arises in consequence 
 of the variation of the motion of M while it describes LS. 
 
 504. This hmit is discovered by any method that serves to 
 distinguish the two parts Lc and (/of Lc + rf the expression of 
 L/; or LC and CS the two parts of LC + CS the expression of 
 LS, from each other ; of which parts the former measures the 
 motion of M at h, while the latter arises from the variation of 
 the motion of M while it describes LS. We distinguished these 
 parts from each other by this property, in the preceding chap- 
 ters. But since it is the property of the part cfto decrease, and 
 at length to vanish, with the increments LS and DG, while Lc 
 remains, it appears to be a just as well as concise method of in- 
 vestigating this limit, to suppose the increments to decrease, to 
 find whatpait of the expression of 1/ decreases, and at length 
 vanishes with them, to reject this part, and retain the other Lc 
 only for measuring the velocity of M at L. It is objected 
 against Sir Isaac Newton's method of investigating this limit, 
 that he lirst supposes that there are increments (as LS and DG), 
 that when it is said let tJte increments vanish, the former supposi- 
 tion is destroyed, and yet a consequence of this supposition, /. e. 
 an expression got by virtue thereof, is retained. But the suppo- 
 sitions that are made in this method of investigating the limit 
 are not so contradictory as this objection seems to import. He 
 first supposes that there are increments generated, and represents 
 their ratio by thatof two quantities (as I/and Do)^one of which 
 (Dg) is given so as not to vary with the increments. If he had 
 afterwards supposed that no increments had beert generated, 
 this indeed had been a supposition directly contradictory to the 
 former. But when he supposes those increments to be diminish- 
 ed till they vanish, this supposition surely cannot be said to he 
 so contradictory to the former, as to hinder us from knowing 
 what was the ratio of those increments at any term of the time 
 while thev had a real existence, how this ratio varied, and to 
 what limit it approached, while the increments were continual- 
 ly diminished. On the contrary, this is a very concise and just 
 method of discovering the limit which is required. It had been 
 easv, if it had been of any use, to have supposed the generating 
 motions to have proceeded in their course: and to have substitut- 
 ed,
 
 Chap. XTL Of the Limits of Ratios. 11 
 
 cd, ill place of hisdecreasing increments, quantities that should 
 decrease so as to be always in tlie same ratio to each other as the 
 increments were while they were generated. But this was not 
 necessary, and it is to be remembered that the ratio Lc to Do-, the 
 limit of the variable ratio of Lf to Dg, is not proposed as the ra- 
 tio of increments that have vanished, but as the ratio of the velo* 
 cities with which the points M and P did set out from L and D 
 to generate real increments. 
 
 505. The ratio of Lc to T>g is likewise called ihe Jlrst or 
 prime ratio of the increments LS and DG; because though thd 
 ratio of those increments continually varies when the motion of 
 M is continually accelerated or retarded, yet the ratio of the 
 generating motions (or that of Lc to Dg) is the term or limit 
 from which the variable ratio of the increments proceeds, or 
 sets out, to increase or decrease. This ratio, strictly speaking, 
 is not the ratio of any real increments whatsoever, because any 
 increment LS partly depends on the motion of M at L, and 
 partly on the continual acceleration or retardation of its motion 
 from that term. But as the tangent of an arch is the right line 
 that limits the position of all the secants that can pass through 
 the point of contact (art. 181), though, strictly speaking, it be 
 no secant, so a ratio may limit the variable ratios of the incre- 
 ments, though it cannot be said to be the ratio of any real in- 
 crements. The ratio of the generating motions may be like- 
 wise said to be the last or ultimate ratio of the increments while 
 they are supposed to be diminished till they vanish, for a like 
 reason. 
 
 506. Most of the propositions in the preceding chapters maj- 
 be briefly demonstrated by this method. It will be sufficient to 
 giveafewexamplesCj?g;..'J8). First,let us resume the construction 
 in art. 140, where SA is invariable, SA, AP and AL are in con- 
 tinued proportion, and it is required to find the ratio of tlie flux- 
 ion of AL to th.e fluxion of AP. Because L/the increment of 
 AL is to Pp the increment of AP as DL is to SP, and the angle 
 PSD is always equal to pSA, it is manifest that if those incre- 
 ments L/ and Fp be supposed to be diminished till they vanish, 
 the angle PSD will approach to PSA, and at length coincide 
 with it, PD will become eq^ual to PL and DL to 2PL; so that 
 
 the
 
 12 Of the Centre of Gravity, Book 1. 
 
 the ultimate ratio of L/ to P/? must be that of 2PL to SP, or 
 of 2AP to SA; and the fluxion of AL must be to the fluxion 
 of AP in the same ratio. In the same manner SA, AP, AL, and 
 AM being in continued proportion, Mw the increment of AM 
 is to Vp as GM to SP; and when these increments are diminish- 
 ed till they vanish, GL becomes equal to 2LM, and GM to 
 SLM ; so that the last ratio of Mm to Vp is that of 3LM to SP, 
 or that of 3AL to SA ; and the fluxion of AM is to the fluxion 
 of AP in the same ratio. In hke manner the 8th and 9th propo- 
 sitions may be deduced. 
 
 507 . In prop. 1 4, where ADiJig. 47) is the base, DE the or- 
 dinate, DG the increment of the base, IH the simultaneous 
 increment of the ordinate, if DG be supposed to be diminished 
 till it vanish, the angle HET (contained by the chord EH and 
 tangent ET) decreases till it vanish, by art. 181; and the ulti- 
 mate ratio of DG to IH is thatof EI to IT, which is therefore the 
 ratio ofthe fluxion of the base AD to the fluxion of the ordinate. 
 The ultimate ratio ofthe arch EH to the tangent ET is a ratio 
 of equality, and the fluxion ofthe curve is to the fluxion ofthe 
 base asETto EI. In the samemanner the 15th, l6th,and 17th 
 propositions may be briefly deduced. 
 
 508. In prop. 1 8, acircle described through C,E, and K(;?o' 6 1, 
 and 62) touches the righ t line AE, because the angle ECKismade 
 equal to SEA. Therefore when P approaches to E till it coin- 
 cide with it, the ultimate ratio of the angle PKE to PCEis a 
 ratio of equalit}'^, and the ultimate ratio of the angle PCE to 
 the angle PSE is that of SE to KE, or of ST to CT; whence 
 the fluxion of the angle ACP is to the fluxion of aSP as ST is 
 toCT. 
 
 509. If the point C(y/2:. 223) be taken upon therightlineAB, 
 that joins the centres ofthe bodies A and B, so that C A be to C B as 
 the body B is to A, then C is the centre of gravity of A and B ; 
 if the point G be taken upon CE, so that GEbe to GA as the 
 sum of A and B is to the body E, then is G the centre of gra- 
 vity ofthe three bodies, A, B, and E; and in the same manner 
 the centre of gravity of any number of bodies is determined. 
 Let kn be any right line, Aa, BZ>, and Cc any parallel lines from 
 A' B, and C that meetA-y/ in a, b, and c; thenthesuni of therect- 
 
 ansfles
 
 Chap. XIT. and its Motion.' 13 
 
 angles contained by A and Aa, and by B and B6, shall be 
 equal to the rectangle contained by A-j-B and Cc when A and 
 B are on the same side of kn, but to the rectangle contained by 
 \ — B and Cc when they are on different sides of kn ; because 
 if AV and Bt? parallel to kn meet Cc in V and v, CV will be to 
 Ct) as CA to CB^ or as B to A ; and the rectangle AXCV 
 equal to B XCi). It follows that if G be the centre of gravity of 
 any number of bodies, the rectangle contained by Gg(any right 
 line from G that meets a given plane kn in g) and the sum of 
 all the bodies is equal to the aggregate of the rectangles con- 
 tained by each body, and the parallel from it terminated always 
 by kn, that is to the aggregate of AXAa, BXBi, EXE<', 6;c. 
 in collecting which any rectangle is to be considered as nega- 
 tive, or to be subducted, when the body is not on the same side of Avi 
 with G CJig.QO). Hence, cor. 6, prop. 19, maybe deduced (that 
 the surface described by any line FN/" revolving aljout the axis A^^f 
 is equalto the rectangle contained by FN/'and the line describ- 
 ed by its centre of gravity C in the same time) by applying 
 what has been shown of the bodies A,B,E,c*^c.to the elements 
 of the arch FNy, and substituting this arch itself for the sum of 
 thebodies. In the same manner itisshown thatif G(;^g-.225) be 
 the centre of gravity of any figure DBf/, kn arightline in the plane 
 of this figure parallel to X)d and given in position, G A perpen- 
 dicular to kn in A meet Mm any ordinate of this figure parallel 
 to kn in P, then the solid contained by the area DQd and the 
 perpendicular GA will be equal to the fluent of the solid con- 
 tained by the rectangle which measures the fluxion of the area 
 MBw and the perpendicular PA, by substituting the elements 
 of the area for the bodies A, B, E, <^c. and the whole area DBd 
 for the sum of the bodies. And if G be the centre of gravity of 
 a solid DBJ, of which Mm represents any section parallel to 
 J)d, let the whole solid be represented by S, the fluxion of the 
 :^olid MBm by f, and GAXS will be equal to the fluent of 
 PAX/ 
 
 510. There are several theorems concerning the centre of 
 gravity, and its motion, that are useful in the resolution of pro- 
 blems of various kinds, which we shall take this occasion to 
 "describe briefly. In any system of bodies the sum of their mo- 
 tion?
 
 U Of tU Centre of Gnivity, Book I. 
 
 tions ^vlien estimated in a given direction is equal to the motion 
 of a body that is equal to the sum of those bodies, and pro- 
 ceeds with the velocity of their common centre of gravity, if its 
 motion be reduced to the same direction f/^.2'24). Amotion that 
 is asCL, in the direction CL, reduced to any other direction Cc 
 is measured by CP, if LP be perpendicular to Cc in P. The 
 same motion reduced to the opposite direction cC is still mea- 
 sured by CP, but is then considered as negative. Let the bo- 
 dies A and B with their centre of gravity move in the same 
 time intoF, H, and L respectively ; letl^" 11//, and hi parallel 
 to Qc meet kn mf, h, and /,• and FM,HN, LP parallel to kn 
 meet Aa, B6, Cc in M, N^and P respectively ; then since the 
 sum of AXAa-fBxB6 is equal to ATBXCc, and AXF/' 
 ~fB X IVi is equal to A + B X L/, it follows that A X AM-f- 
 B X BN is equal to A + B x CP. In the same manner A X FM 
 -|-B X HN is equal to A-f- B x LP. And in the same manner 
 it appears that the aggregate of the motions of anv number 
 of bodies A, B, E, ^c. is equal to the motion of their sum 
 A-{-B-|-E, &)C. proceeding with the velocit}' of their com-' 
 mon centre of gravity, when these motions are all reduced to 
 any one direction. It follows likewise that if the motions of the 
 bodies are all uniform and rectilineal, the centre of gravity is 
 cither quiescent, or its motion is uniform and rectilineal. For 
 in this case the ratio of the right lines AM, FM, BN, HN to 
 each other being invariable, as well as the ratio of A to B, the 
 ratio of CP to LP must be invariable. 
 
 oil. As the iiirtrreo:ate of the motions of anv number of bo* 
 dies reduced to any given direction is never affected by the 
 composition or resolution of their motions, or by any actions 
 of those bodies upon one another that are mutual and equal in 
 contrary directions, or by any powers that act equally upon 
 them with opposite directions ; so the motion of the centre of 
 gravity of any sjstem of bodies is never affected by their colli- 
 sions, or when they attract or repel each other equally. In th-e 
 same manneras the motion of an3onebody continuesthesame 
 till some external force or rf^sistance cflect it, by Sir Isaac 
 Newto/i's first law of motion ; so the motion of the centre of 
 gravity of any system of bodies continues the same unless some 
 
 foreign
 
 A J) G 
 
 Fig 62. Ai-t. ^oi
 
 ni/um- ir/mi/n/ .riwii To/.j-" l'laicX\r>i'./i; r.V.II 
 
 Fill -jj. . lit .fiij Fill -J? Art 4gi Fig 4j.iit^ii7 
 
 It th 
 
 E Fii/slfAn joi. 
 
 J
 
 Chap. XII. and lis Motion. 15 
 
 foreign influence disturb it. If there was any action without 
 an equal and contrary reaction, the state of the centre of gra- 
 vity of the system would he afiected by it. And the equality 
 of these being constantly confirmed by experience, it is not with- 
 out ground that it is held to be a general law, and extended 
 by Sir Isaac Nezcton to the gravitation of bodies. It is mani- 
 fest however that it is not the sum of the absolute forces of 
 bodies, without regard to the directions of their motions, that 
 is preserved the same unalterable by their collisions, in conse- 
 quence of the equality of action and reaction* (according to 
 Sir Isaac AtaYo/i's third law of motion) ; since this is a ge- 
 neral 
 
 * When it is said that fhe quantily of ahsohilf force ts unalterable htf the colli' 
 sien of bodies, and l^ai this foUoivs so evidently from the equality of action atul 
 reaction, that to endeavour to demonstrate it would onlj render it more obscure^ 
 something else must be meant by action and reaction than has been g-enerally 
 vmderstood by these terms, and that has not been explained by those philoso- 
 phers. According to this doctrine it would seem that the equality of action and 
 reaction should take place in the collisions of such bodies only as are perfectly 
 elastic (that is of no bodies known in nature), and not even of these, uiiless we 
 measure their forces by the compound ratio of the squares of their velocities and 
 of their quantities of matter. And though this mensuration of the forces of bodies 
 was admitted, the quantity of absolute force will be found to be so far affected 
 by the collisions of bodies, that it must be less during the small time in 
 which they act upon each other than before and after the stroke ; whereas the 
 quantity of motion estimated in the same direction is preserved the same while 
 the bodies act upon each other as before and after, and never can suffer any chang(» 
 from their mutual actions. But as it might seem to be an improper digression 
 if wc should insist on this subject here, we shall only subjoin an illustration of an 
 argument which was offered some time ago, to show, that we cannot abandon the 
 old doctrine concerning the measures of the forces of bodies in motion, without 
 exchanging plain principles that have been generally received concerning the actions 
 of bodies upon the most simple and uncontested experiments, for notions that seem at 
 best to be very obscure. 
 
 Let A and B 0%". 226) be two equal bodies that are separated from each other by 
 springs interposed between them (or in any equivalent manner) in a space EFGH, 
 which in the mean time proceeds uniformly in the direction BA (in which 
 the springs act) with a velocity as 1 ; and suppose that the springs imprint on the 
 pqual bodies A and B equal velocities in opposite directions that are each as I. 
 Then the absolute velocity of A (which was as 1) will be now as 2 ; and, accord- 
 ing to the new doctrine, its force as 4 : whereas the absolute velocity and the 
 force of B (which was as 1) will be now destroyed; so that the action of tlic 
 springs adds to A a force as 3, and subducts from the equal body B a force as 1 
 only; and yet it seems mnnife^t that the action;, of the springs on these equal 
 
 bodies
 
 1$ Of the CoUhion of Bodies. Book I. 
 
 neral law, and extends to hard and soft bodies as well as to such 
 as are perfectly elastic, and the sum of the absolute motions of 
 thosecannot be said to be unalterable by ibeircollisions. It is 
 the quantity of motion estimated in the same direction that is 
 preserved the same without any change from any mutual actions 
 of bodies in consequence of the equality of action and reaction. 
 But we proceed to give some instances of the use of those 
 theorems in the resolution of problems. 
 
 .512. From these principles the effects of the collisions of bodies 
 are readily determined. ThebodiesAandB('//5-.227)beingsup- 
 posed void of elasticity, let C be their centre of gravity, and 
 tiieir velocities before the stroke be represented by AD and BD 
 respectively. Then supposing the stroke to be direct, they will 
 proceed together after it as one mass, and consequently with 
 the velocity CD of their centre of gravity. But if the bodies 
 are perfectly elastic, take CE equal to CD in an opposite di- 
 rection, and the velocities of A and B after the stroke will be 
 represented by EA and EB respectively, the change produced 
 
 bodies ought to be equal. In general, if m represent the velocity of the space 
 EFGII in the direction BA, n the velocity added to that of A and subducted from 
 that of B by the action of the springs, then the absolute velocities of A and B 
 will be represented by m-\-n and m — n respectively, the force added to A by 
 the springs will be 2ot«-{-wj, and the force taken from B will be 2mn — nn, 
 ■which differ by 2nn. Further it is allowed that the actions of bodies upon one 
 another are the same in a space that proceeds w^ith an uniform motion as if the 
 space was at rest {Ja force du choc, ou raction des corps les urn's sur les autres, depend 
 tim'qucment de leur viiesses respecti-jes. Discours sur le mouxemenf, Paris, 1726). 
 But if the space EFGH was at rest, the forces communicated by the springs to A 
 and B had been equal, and the force of each had been represented by nn. These 
 arguments are simple and obvious, and seem on that account to be the more 
 proper in treating of this question. Though there are certain effects produced by 
 the forces of bodies that are in the duplicate ratio of their velocities, we are not 
 thence to conclude that the forces themselves are in that ratio, no more than we 
 are to conclude that a force which would carry a body upwards 500 miles in a 
 minute is infinite, because it may be demonstrated (if we abstract from the resist- 
 ance of the air) that a body projected with this velocity would rise for ever, and 
 never return to the earth. And as reaction is only equal to action when both are 
 estimated in opposite directions upon the same right line, so we are never to 
 estimate the force which one body loses or acquires by that which is produced or 
 destroyed in another body in a different direction ; whence the objections against 
 the usual manner of measuring the forces of bcdict- may be resolved, and even 
 improved for to support it. 
 
 in
 
 Chap. XII. Of the CoUhion of Bodies. 17 
 
 ill their velocities in this case by the stroke being double of 
 what it wasih the former, the difference of AD and CL» 'w^ing 
 equal to the diiference of CD (or CE) and EA, and the difier- 
 cnce of CD and BD equal to the diftereuce of EB and CD. If 
 B have no motion before the stroke, then CE is to be taken 
 equal to CB, the velocity of A before the stroke being reprcr 
 sented b}'^ AB. In this case if the right line oa be to ob as A 
 is to B, and ah be bisected in e, the velocity of A before the 
 stroke will be to that of B after the stroke as half the sum of 
 A and B is to A, or as oe is to oa. And if motion be communi- 
 cated in this manner from the body A to a series of bodies in 
 geometrical progression, of which A and B are the first terms, 
 then the velocities successively communicated to those bodies 
 w^ill be in a geometrical progression, the common ratio of any 
 two subsequent terms will be that of oe to oa ; and, if it be the 
 number of bodies without including the first A, the velocity of 
 the last will be to the velocity of the first as the power of oa 
 whose exponent is n is to the same power of oe. Therefore if 
 od represent the last body in the progression, and ov the velo- 
 city communicated to it, the velocity of the first oa beino- re- 
 presented by oa, and oa be the modnlus of the system, the 
 logarithm of od will be to that of ov as the logarithm of oh is to 
 that of oe, because the logarithm of od is to that of ob as n is 
 to 1, and the logarithm of ov is. to that of oe in the same ratio. 
 
 513. Any three bodies being represented by o^r, o6, and od, 
 let the first strike the second supposed at rest before the stroke^ 
 and the second strike the third quiescent, let of he to od as oa 
 is to ob; and the velocit}'- communicated in this manner to the 
 tliird shall be to the velocity of the first as oa is to one fom-th 
 part of the sura of oa, ob, of, and od. For the velocity of the 
 first oa is to the velocity of the second ob as the sum of oa and 
 ob to 2oa ; the velocity of o6 is to that of od' as the sum or ob 
 and od to^ Qob ; consequently the velocity of the first oa is to 
 the velocity of the third od in the compound ratio of oa-^ob 
 to Q,oa and of ob-^od to Q,ob, that is (since oa, oh, of, od, are 
 proportional, so that oa is to ob as oa-\-of to cb-\-od, and oa 
 -f- ob to ob ,aS'.the , suni "of oa, oh, of, and od to ob-\-od) as the 
 sum of oa, oh, of, and od is to 4oa. Hence the velocity of oa 
 
 VOL. II. C beinsr
 
 1% Of tht ColUsidn of Bodin. Boole I. 
 
 being gvr)Sr^> t^e velocity communicated to od is inversely as 
 the sum vA oa, ob, of, and od, and is greatest when this 
 jum is least, that is, if oa and od be given, when ob and of 
 coincide with each other and with the mean proportional be- 
 twixt oa and od. Therefore the velocity communicated to od 
 is greatest when ob the body interposed betwixt o« and od is a 
 jnean proportional between them. This is one of Mr. Hui/gensi 
 theorems, from which it follows, that the more such geome- 
 trical mean proportionals are interposed betwixt oa and od, the 
 greater is the velocity communicated to od. 
 
 514. There is however a limit which the velocity commu- 
 nicated to od never amounts to (the bodies oa, od, and the ve- 
 locity of oa before the stroke being given), to which it ap- 
 proaches continually while the number of such bodies inter- 
 posed between oa and od is always increased. And this limit 
 is a velocity which is to the velocity of the first body oa before 
 the stroke in the siibduplicate ratio of oa to od. This limit is 
 not mentioned by Mr. Hiiygens, but may be determined from 
 art. 512 and 179. For while ab is continually diminished, 
 and ob approaches to oa, the last ratio of the logarithm of o6 
 to ab, or of the logarithm of oe to ae, is a ratio of equality, by 
 art. 179; consequently the logarithm of ob becomes ultimately 
 double of that of oe, and (by art. 512) the logarithm of oc? double 
 t)f that of or. Therefore if oA be a mean proportion betwixt 
 oa and od, the logarithm o^ov will become equal to the logarithm 
 of ok, but with a contrary sign ; so that ok, oa, and ov will be 
 in continued proportion : and ov the velocity of the last body 
 od will be to oa the velocity of the first oa as oa is to ok, or in 
 the subduplicate ratio of tiie first body oa to the last od. 
 
 515. The same principles will serve for determining the 
 effects of the collision, when a body strikes any number of 
 bodies at once in any directions whatever. Let the bodies first 
 be perfectly hard and void of elasticity, and the body C (fg. 228) 
 moving in the direction CD v/ith a velocity represented by CD 
 strike at once the bodies A, B, E, (Sfc. that are suj^posed at 
 rest before the stroke in directions CF, CII, CK, ^-c. in the 
 same plane with CD, and Da, Dft, De, be perpendicular to CF, 
 CH, CK iQ a, by and e respectively. Determine the point P 
 
 where
 
 Chap. XII. Of the Collision of Bodies. 19 
 
 where tlie eommon centre of gravity of the bodies C, A, B, £, 
 cVc. would be found if their centres were placed at the points 
 C^ a, b, e, Sec. respectively, by art. 509, join 3^P, and CL 
 parallel to DP shall be the direction of the body C after the 
 stroke. Let PR perpendicular to DP meet CD in 11, and DL 
 perpendicular to CD uieet CL in L; then if CLbe divided in. 
 G so that CG be to GL in the ratio compounded of that of CD 
 to Cli and that of the body C to the sum of all the bodies, the 
 velocity of C after the stroke will be represented by CG ; that 
 is, the velocity of C after the stroke will be toits velocity be- 
 fore it as CG is to CD, Let Gf, Gk, and GA-, be respectively 
 [lerpendieular to CP, CH, and CK inf h, and /c, and the ve- 
 locities of A, B, and E, after the stroke will be represented by 
 Cy, Oh, and CA-. But if we now suppose the bodies to be per- 
 fectly elastic, or the relative velocities before and after the 
 stroke be always equal when measured on the same right line, 
 produce DG till D^ be equal to 2DG, join Co-, and the body 
 C will describe Cg after the stroke in the same time that it 
 would have described a right line equal to CD before the stroke. 
 And in like manner the motions are determined when the elas- 
 ticity is imperfect, if the relative velocity after the stroke is al- 
 M ays in a given ratio to the relative velocity before it in the 
 same right line. Mr. BeimoiiiUi has deduced the computa- 
 tions of the motions in the case when the bodies are perfectly 
 elastic, and there are bodies on one side of the line of direction 
 CD that are always respectivel}^ equal to those on the other 
 side,and are impelled in directions that form equal angks with 
 CD in the same plane, from the principle that the sum of the 
 bodies multiplied by the squares of the velocities is the same 
 before and after the stroke ; which computations will be found 
 to agree with what we have shown, by supposing DP and CL 
 to fall upon CD, and restricting our supposition in other re- 
 spects so as it may agree to this case. These problems being 
 represented as of an uncommon difficulty, it may be worth 
 while tosubjointhe following construction which is still more ge- 
 neral, and is deduced from the principles. in art. 510 and 511. 
 516. Let the bodies C, A, B, E, e^'c. (>^.22g) move now in 
 the directions CD, CF, CH, CK, S^c. in one plane with 
 
 C 2 velocities
 
 20 Of the Collision of Bodies. Book I. 
 
 velocities represented by CD, Ca, Cb, Ce, ^c. and the body 
 C overtake and strike thein at once in these directions. Let 
 T be the point where the common centre of gravity of all the 
 bodies C, A, B, E, S^c. would be found if they were placed in 
 D, a, b, e, SfC. respectively ; let Ta, Tb, Te, S)-c. be perpendi- 
 cular to CF, CH, CK, &;c. in a, h, e, 8fC. and P be the point 
 where their connnon centre of gravity would be iound if the 
 bodies were placed at C, a, b, e, SfC. respectively; join TP, and 
 CL parallel to TP will be the direction of C after the stroke 
 "ulien all the bodies are supposed perfectly hard and void of 
 elasticity. Let PR perpendicular to TP meet CT in R, and 
 TL perpendicular to CT meet CL in L; let CS be to CT as 
 the body C is to the sum of all the bodies ; upon CL take CG 
 in the same ratio toCL asCTis to CS-J-CR^and CG w^ill re- 
 present the velocity of C after the stroke ; whence the veloci- 
 ties of the other bodies in their respective directions CF, CH, 
 CK, SfC. are determined as before. We omit some other 
 theorems of this kind where the directions are in different 
 planes, because they would lead us too far from our principal 
 subject. When the bodies are perfectly elastic, join DG, and 
 upon it take J)g, double of DG; but if the elasticity be imr 
 perfect, and the respective velocity after the stroke be in a 
 given ratio to the respective velocitj^ before the stroke, upon 
 DG produced take G^ to DG in that given ratio; and Cg will 
 represent the direction and velocity of C after the stroke; 
 whence it is easy to determine the velocities of the other 
 bodies. The other cases of this problem are resolved in like 
 manner from the same principles. 
 
 517, Mr. Jlin/gtns has shown that in the collisions of two 
 bodies which are perfectly elastic, the sum of the bodies multi- 
 plied by the squares of their velocities is the same after the 
 stroke as before it. It isjustly observed that this proposition is so 
 far general as to obtain in all collisions of bodies that are per- 
 fectly elastic ; but as this cannot be held an immediate conse- 
 quence of the equality of action and reaction, as was observed 
 above, and it is by some considered as a theorem of great use, 
 we shall show how it may be demonstrated when a body strikes 
 any number of bodies at once, as in art. 51d. Let l^Q, gq, 
 
 fm,
 
 <5hap. XII. Of the Collision of Bodies. 21 
 
 /m,^«,A:rbeperpendiculartoCGinQ^^,w,?iandr(/?g-.{228).Thea 
 the rectangles contained by Cm and CG, Cn and CG, Cr and 
 CG will be respectively equal to the squares oiCf Ch, and Ck. 
 If the bodies C, A, B, E be supposed to have no elasticity, their 
 velocities after the stroke will be represented by CG, Cf Ch, 
 and Ck, the velocity of C before the stroke being represented 
 by CD; because in this case no relative velocity is generated 
 by the stroke in their respective directions; and the sum of 
 A X Cm, B X Cn, E x Cr is equal to C X GQ, because the sum 
 of the motions which would be communicated to A, B, and E 
 in the direction CG is equal to the motion which C would lose 
 in the same direction by art. 511. Therefore the sum of AX 
 Cf% B X Ch\ E X Ck^ is equal to C x CG x GQ; and to 
 these if we add C X CG% the sum of all the bodies multiplied 
 by the squares of their velocities in this case would be C x CG 
 X CQ. But when the bodies are supposed to be perfectly elas- 
 tic, the velocities of A, B, and E are to be represented by 2Cf, 
 iCh, andSCA; respectively; the sum of A x 4C/% B x 4C/i* 
 and E X 4CA'^ is equal to C4- 4CG + GQ or {elem. 8, 2) C-f 
 CQ"— C X Cq"-; to which if we add C X Cg^ (or C x C^^ + 
 C X GQ"') the whole sum of the products when each body is 
 multiplied by the square of its velocity is equal to C x CD*- 
 and consequently is the same after the stroke as it was before 
 the stroke. But when the bodies are void of elasticit}'-, this sura 
 islessafter the stroke than before it in the ratio of CG + CQto 
 CD^ or of CG to CL. The same proposition is demonstrated 
 in like manner of perfectly elastic bodies in the case of art. 16. 
 And when the bodies A, B, E move before the stroke in direc- 
 tions different from those in which C acts upon them, the pro- 
 position will appear by resolving their motions into such as are 
 in those directions (which alone are affected by the stroke), and 
 6uch as arc in perpendiculars to those directions, from elem. 4:7 
 1. This proposition likewise holds when bodies of a perfect 
 elasticity strike any immoveable obstacle as well as when they 
 strike one another, or when they are constrained by any power 
 or resistance to move in directions different from those in which 
 they impel one another, as we shall show afterwards. But it is 
 manifest that it is not to be held a general principle or law of 
 
 C ci motion
 
 2'2 Of the Collision of Bodies. Book f, 
 
 motiou, since it can take place in the collisions of one sort of 
 bodies only. The solutions of some problems which have been 
 deduced from it may be obtained in a general and direct man- 
 ner from plain principles that are universally allowed,, by de- 
 termining first the motions of hard bodies which are supposed 
 to have no elasticity, and thence deducing the solutions of 
 other cases when the relative velocitiesbeforeand after the stroke 
 are equal, or in any given ratio. It will be said perhaps that 
 ^here are no such bodies known in nature. But though nobodies 
 that are perfectly elastic, or no mathematical fluid be known iu 
 nature, to investigate their motions is allowed to be an useful 
 iliquiry. It is a consequence however from the proposition we 
 have described, that while perfectly elastic bodies move in any 
 manner, if any new force act upon them that generates equal 
 velocities in the same direction in each, the excess of the sum 
 of the products of each bod}^ multiplied by the square of its ve- 
 locity, above the product of the sum of the bodies multiplied 
 by the square of the velocity of their common centre of gra- 
 vity, l? not affected by this new force or by their collisions. 
 
 5 IS. Suppose now that thebodyC(^o.'230)moving in the direc- 
 tion CD with thevelocityCDimpelsthebodiesAandBinthedi- 
 rections CF and CH; but that A and B cannot move in those 
 directions, being constrained to move in the respective direc- 
 tions C/'and C//, by planes parallel to Cy'and C// along which 
 \ve suppose them to shde without friction, or by their being fix- 
 ed to. the extremities of lines OA and UB perpeudicular to Cf 
 and CAA, and moveable about the centres O and I J, or in any 
 other equivalent manner. Suppose all those lines to be in the 
 same plane with CD, and that A and B were at rest before the 
 stroke. Let Da and Db perpendicular to CF and CH meet Cf 
 and CA in a and b respectively; draw aF perpendicuhu" to Ca, 
 and 6H {)crpendicular to C^, meeting CF and CI [ in F and Hj 
 and ¥m, iin parallel to CD meeting Da and Db iu ?« and n. 
 Let P be the common centre of gravity of the bodies C, A, and 
 B wlien their respective centres are supposed to be placed at C, 
 7n, and //, join X)P, and CL parallel to DP shall be the direction 
 of C after the stroke, the bodies being supposed to be perfectly 
 hard and void of elasticity. Let p be the common centre of 
 
 gravity
 
 Ghap. XII. Of the Collision of Bodies, 2S 
 
 gravity of C, A, and B when their respective centres are suppos- 
 ed to be placed at D, V, and H; draw pr perpendicular to DP 
 meeting CD in ?', let CS be to CD as the body C is to the suia 
 ot" all the bodies ; let DL perpendicular to CD meet CL in L, 
 join rL, and let SGparallel to rL meet CL in G,tlien CG will 
 represent the velocity of C after the stroke; and if G/ and Gh 
 respectively perpendicular to CF and CH meet C/'and Ch inj' 
 and h, then Cf and Ch will represent the velocities oif A and B 
 after the stroke. 
 
 519. \Vhen the bodies are perfectly elastic, C^ the oirection 
 and velocity of C is found as in art. 51 5, by producing DG till 
 Do- be equal to 2DG. In this case, though the m^-tion r '' ^;he cen- 
 tre of gravitj'j or the sum of the motions of the bodies l^;. the 
 direction CD, be diminished by the stroke (because of the resist- 
 ance of the planes or lines by which the bodies A and B are 
 hindered to mo se in the directions CF andCHin which C im- 
 pels them, and constrained to move in the directions Cf and 
 C^), yet the sum of the products of the bodies multiplied by 
 the squares of their velocities is tJie same after the stroke as be- 
 fore it. Foi let hf pfjipendiciilar to Ch meet CH inf, and fu 
 perpendicular to C/ meet CF in u; draw fx^uz, DQ, and gq 
 perpendicular to CL in x, z, Q, and q ; join zf, and the angle 
 Cj'z being equal to Cuz or CGf, the triangles Czf and CfGr 
 are similar, and the rectangle GCz equal to the square of C/*. 
 In the same manner the rectangle GGr is equal to the square of 
 Ch ; therelbre the sum of A X 4Cf^ and B X 4CA^ is equal to 
 the product of A X Cz+B x Cx by 4CG. But A X C2+ 
 B X Cx is the quantity of motion which C loses in the direction 
 CLwhen it communicates to A and B velocities Cy and Ch in 
 their respective directions Cj/'-and C/i, by impelling them in the 
 directions CF and CH, and therefore is equal to C xGQ by 
 art. 511. Therefore since C x GQ x 4CG is equal to C X CQ* 
 — C X C</% if we add C x Cg% the whole sum of the products 
 of the bodies multiplied by tlie squares of their velocities after 
 the stroke will be C X CD%t!iciiiime as it was before the stroke; 
 and it is manifest that this deirtonSUation is apphcable when C 
 strikes any number of bodies in any directions whatsoever. 
 
 C4 52a T
 
 24 Of the Collision of Bodies. Book I. 
 
 5C0. The demonstration of the constructions inart. 515,516, 
 and518vvill easily appear if we subjoin thatofthe firstinart. 515 
 0ig. 23 1 ). Resuming therefore the construction in that article, 
 suppose moreover that LpjL^' and Lf are perpendicular to CF^CH 
 and CK in p, q, and t ; draw aM, pm, 6N, qn, eZ, tz, and 
 W perpendicular to CD in INI, m, N, n, Z, z, and V. Let 
 the sum of the bodies C, A, B, and E be expressed by S, and 
 since P was the centre of gravity of the bodies C, A, B, and E 
 when their respective centres were supposed to be placed at C, 
 a, b and e, it follows from art. 509 that S X PV will be equal 
 to Ax«M+Ext'Z— Bx6N, and SxDV equal to Cx 
 CD -}- A X DM + B X DN + E X DZ. If we suppose the 
 bodies to be void of elasticity, or no relative velocity to be 
 generated by the collision in their respective directions, then 
 while C describes CL the bodies A, B, and E will describe right 
 lines respectively equal to Cp, Cq, and C^. Therefore if we 
 suppose CL and CH to be on one side of CD, and CF and CK 
 to be on the other side of it, C X DL -^ B x qn will be equal 
 to A X pm -f- E X tz, by art. 510, because the centre of gravi- 
 ty of the bodies had no motion in the direction perpendicular 
 to CD before the stroke, and consequently has no motion in 
 that direction after it. Let Lp meet oM in r, and pu [)arallel to 
 CD meet aM in «, then au will be to ar (or DL) as DM to 
 CD, andoM x CD equal to DL X DM, so that A x CD x pm 
 will be equal to A x CD X aM — A x DM x DL. In the 
 same manner B X CD X ^wwill be equal to B X CD X feN + 
 B X DN X DL.andE x CD x tz equal to E x CD x eZ— E 
 X DZ X DL. Prom which it follows that C X CD x DL 4-B 
 X CD X 6\ + B X DN X DL is equal to A x CD x «M— 
 A X DM X DL 4- E X CD X eZ— E x DZ x DL, or S x DV 
 X DL equal to S x CD X PV. Therefore CD is to DLas 
 DV to PV, and CL is parallel to DP. The direction of C af- 
 ter the stroke being thus detennined, suppose that CG is to CD 
 as tiie velocity of C after the stroke to its velocity before the 
 stroke. Then because the sum of the motions of the bodies 
 estimated in any given direction is not affected by the stroke, or 
 the motion of their common centre of gravity is uniform, this 
 
 sum
 
 Chap. XII. Of the Collision of Bodies. 25 
 
 sum will be equal to C x CD in the time C describes CG; and 
 C X CB + A X Cm + B X Cn + E X Cz will be to C x CD 
 as the time in which C describes CL to the time in which it 
 describes CG^, or as CL to CG. Therefore since CD x Mw is 
 equal to «M X DL, CD x Nrt to 6N X DL, and CD x Zz to 
 eZ X DL, it follows tliat A x CD X CM— A x aU x DL+ 
 BxCD xCN + B X bN x DL + E x CD x CZ— E X eZ 
 X DL is to C X CD^ as LG is to CG ; that is, S x CDX CV 
 — S'X PV X DL is to CD' as LG is to CG. But PR is per- 
 pendicular to DP by the construction, and CD to DL as PV to 
 VR, or PV X DL equal to CD X VR ; consequently S X CR 
 is to C X CD as LG it to CG. Therefore CG the velocity of 
 C after the stroke is determined by dividing CL in G, so that 
 CG may be to LG in the compound ratio of CD to CR, and 
 of Cto S the sum of all the bodies ; which was the solution given 
 in art. 515, when the bodies were supposed to have no elas- 
 ticity, so that norelative velocity of C and the other bodies was ge- 
 nerated in their respective directions.* In the samemasnnerit is 
 shown in thecase described in art. 5 16, that CSbeing to CT as C is 
 
 to 
 
 * Because the points (?,^, c, life-. (j?g-. 231,2^.2) are always in the circumference of 
 ai circle described upon the diameter CD, if we should suppose a sphere C to strike equal 
 homogeneous particles that touch it in an ark AB which is in the same plane with CD, 
 the sum of those particles be called Qj CA and CB meet the circle CaD in a and 3, 
 the point X be the centre of gravity of the ark aDb, and CX be divided in P so that 
 CP be to CX as Q is to C -|- Q, the direction of C after the stroke will be parallel to 
 DP. If AB be a semicircle bisected by CD, the point X will be the centre of the 
 circle CaD ; and the velocity which C loses by the stroke will be to its incident velo- 
 city as Q to 2C-|-Q. But because the resistance of a sphere in a fluid is notdiscovered ia 
 this manner (Newt. Princip, lib. 2, prop. 32, t5'<r. schoL), we have not insisted on 
 those cases. 
 
 These theorems are given frcm a treatise concerning the mensuration of the force 
 of bodies in motion and the effects of their collisions, written in 1728 (by way of 
 supplement to a small piece printed on this subject at Paris 1 724) that was then com- 
 municated to several persons, and intended to have been published ; wherein 1 endea- 
 vour to show, that according to those who measure the forces of bodies by the square* 
 of their velocities, equal actions generate unequal forces in equal times, and equal forces 
 in unequal times, and that the force of a body must be said to have no greater effect in 
 the direction of its motion than in other .directions, a;id that several other suppositions 
 must be admitted contrary to what has been generally agreed on. But after considering 
 ithat these would perhaps be allowed with explications by such as favour that opinion, 
 
 and
 
 2d Of the Descent of Bodies Book I. 
 
 to S, CG is to CL as CT to CS + CR Cfg. 229). In art. 518, 
 the sum of the motions (fig. 230) estimated in the direction 
 CD is not the same after the stroke as before the stroke ; and 
 while any body A acquires the velocity represented by C/'in 
 the direction C/'from the impulse of C in the direction CF, 
 we are to suppose that in generating this motion C loses a mo- 
 tion represented by A X C« in the direction QV,fu perpendicu- 
 lar to Cf inf being supposed to meet CF in u. The motion 
 A X.fu is lost by the resistance of the plane or line that con- 
 strainsthe body A to move in the direction Cf instead of CF 
 in which it is impelled bj' the body C. By reducing the mo- 
 tions A Xfu and B X A/ to the direction C D, and adding them 
 to the motions of C, A and B in that direction after the stroke, 
 (or more briefly by reducing the motions A X Cu and B X Q/to 
 
 and that it is often proposed as a definition or axiom by them, that the force of a body 
 in motion is measured by the number of springs which can produce or destroy it (though 
 the same springs act for a longer time on a greater body than a lesser, and thereby 
 generate in it a greater quantity of motion), I was unwilling to engage in a dispute 
 that was perplexed by «ach suppositions, and that after all might seem to be in a great 
 measure about words. And one of the chief designs of this chapter being to describe 
 some general principles that are of use in the resolution of problems, this seemed to be 
 » proper opportunity of publishing what was most material in that treatise. Therefore 
 I have endeavoured to show in these and the following articles, that the consideration 
 of the motions of hard bodies that have no elasticity (which are rejected for the sake 
 ©f what is called the law of continuity^ and is supposed to be general without sufficient 
 ground), is of use in order to obtain general solutions ; that the principle which Mr. 
 Uui/gens calls the conservatio vis oscendenlis (in his observations on some pieces con- 
 cerning the centre of oscillation, <?/i?r. vol. 1. p. 248, Edit, Lugd. Batav. 1724), and 
 which seems to be much the same with what is called the cofiservatio -cis vivtr of late, 
 obiaius indeed in many cases besides those he has coniidered, and may be of use in seve- 
 ji.\ inquiries concerning the motions of bodies that have no elasticity, as well as those 
 that are perfectly elastic, but is not general ; and that there is no occasion to perplex 
 the common doctrine concerning th? action and reaction of bodies, or the mensuration 
 of their force, for the sake of thi< principle when it takes place. They who hold this 
 pri.'iciple to be general confine this theory loo much to one sort of bodies, which fo- 
 any thing appears from nature have no prerogative above others. And while som? 
 insist on the preservation of the same quantity of absolute force in the universe with 
 znuch warmth against Sir Isaac Nezvion, there is nevertheless no proposition in expeii- 
 mental philosophy more evident than that in many cases force is lost or diminished in 
 the collisions of bodies from the weakness of their elasticity, whether we measure it by 
 the velocities or by the squares of the velocities. And there is ground to think that 
 it will not be generally allowed to be so easy a matter as they seem to imagine to give 
 asitiifaototy account bow tliis can bo leconcilrU with a principle so contradictory to it. 
 
 the
 
 Chap. XII. that aci upon ont another. ft 7 
 
 the same direcdou) and supposing the sum equal to C X CD, 
 wliich was the sum oF the motions in that direction before the 
 stroke, the sokition in art. 518, will appear. But we proceed 
 now to show how these theorems may be applied for deter- 
 mining the motions of bodies that descend by their gravity, 
 and at the same time impel other bodies, which will lead us to 
 consider Mr. H<;«/g>/ia'i principle concerning what he calls their 
 via ascendcns. 
 50. 1 . Let the accelerating force CA'g. ^28)and direction ofgravlty 
 be ahvavs represented by CD, and let C by its gravity impel A, 
 B and E (which we supposed at present to be avoid of gravity) 
 in the respective directions CF, CH and CK from the begin- 
 ning of its descent. If nothing hinder the bodies from giving 
 way in those directions, and if these sides of A, B and E which 
 C acts upon be planes perpendicular to CF, CH and CK (that 
 C while it descends may impel them always in the same direc- 
 tions) then C will descend in the right line CL that was de- 
 termined in art. 515 ; for CL ihedirection of C which was de- 
 termined in that article does not depend upon the incident ve- 
 locity of C, but only upon the quantities of matter in the seve- 
 ral bodies, the direction qf the motion of C, and those in which 
 itacts upon the other bodies; and when these remain the direc- 
 tion of C after the stroke is alwaj's the same. The forces that 
 accelerate the motions of C, A, B and E in their respectve di- 
 rections CL, CF, CH and CK will be to the accelerating force 
 of gravity, and the respective velocities that will be acquired 
 by them to the velocity that would be acquired in an equal 
 time by a body faihng freely m the vertical line CD by its grar- 
 vity, as CG, Cf, CJi, and CA; to CD. Let G<^be perpendicular 
 to CD in d, and the sum of the products C X CG% A X C/'% 
 B X Ck\ E X Ck' Av'ill be equal to CxCDXCd; for it was 
 shown in art. 5 17, that C X CG' + A X C/'^ -f B X CA'- -f 
 E X CA' is equai to C X CG X CQ, which (because CG is to 
 Cd as CD to CQ) is equal to C x CD X C^. But if the velo- 
 city which C acquires while it descends iiom C to G, and is 
 accelerated by Itie force CG,be represented by CG,or its square 
 by CG% the square of the velocity which it would acquire by 
 t'aJiing freely in the vertical iiom C to ^ by its gravity CD will be 
 
 represented
 
 26 Of the Descent of Bodies Book L 
 
 represented by CD X Cflf (art. 434). Therefore the sum of the, 
 products which arise by multiplying each body by the square 
 of the velocity which it acquires is equal to the product of the 
 body C (which alone is supposed to gravitate) multiplied by 
 the square of the velocity which it would acquire by falling 
 freely from C to d, or by descending freely along the inclined 
 plane CG. The same theorem holds when the directions vary 
 in which the body C acts upon A, B and E while it descends ; 
 the demonstration of which will be comprehended in a more 
 general case afterwards. 
 
 522. If thebodyCimpelAandBCAg.230)by itsgravityinthe 
 directions CFandCHfrom the beginningofitsdescent^but these 
 bodies be constrained, as in art. 518, to move in the directions 
 C/'and Qh, the direction of the motion of C and the velocities 
 that will be acquired bj' the respective bodies may be deter- 
 mined from what was shown in that article, the sides of A and 
 B upon which C acts being planes, so that C may descend in 
 the same right line CL and always impel A and B in the same 
 directions CF and CH. The velocities acquired by C,A, and 
 B at G,/'and h will be to the velocity that would be acquired 
 in an equal time by a body falling freely in the perpendicular 
 as CG, C/"and Ch to CD. ' And because" C X CG" + A X C/* 
 -|- B X CA is equal to C X CG X CQ (b}' what was shown in 
 art, 519), oi" toC x CD X Cd, Gd being perpendicular to CD 
 in d ; therefore the sum of the products of the bodies multipli- 
 ed by the squares of the velocities which they acquire is in this 
 case likewise equal to the product of C multiplied by the square 
 of the velocity which it would acquire by the same descent Cd 
 if it fell freely in the vertical CD. 
 
 523 (-F;^. 232, iV.l).Togive some examplesofthislast case. If the 
 body C impelby its gravity onebod}' Aonl}^ that is terminated by 
 a plane perpendicular to CF, and A slide along a plane paral- 
 lel to C/' without friction, let Dcr perpendicular to CF meet Cf 
 in a, rtF perpendicular to Ca meet CF in F, and Fm parallel to 
 CD meet Da in w ; upon Cm take CP to Cm as A is to C -{- A, 
 join DP, and a right line from D parallel to CF will intersect 
 CTip;nalIel to DP in G. W in this case Cfhe supposed hori- 
 zontal or perpendicular to CD, C//? will coincide ^ith Cff.and 
 
 CP
 
 Chap. XII. that act upon one anolher. 29 
 
 CP being taken upon Ca, in the same ratio to Ca as A is to 
 C + A, a right hne from D parallel to CF will intersect CL pa- 
 rallel to DP in G, so that CG will represent the direction in 
 which C will descend and the force that accelerates its motion ; 
 and CG will be described by it in the same time that it would 
 have described CD by falling freely in the vertical. If we sup- 
 pose CF (fg, 232;, N. 2.) to coincide with the vertical CD, 
 Da will in this case be perpendicular to CD, and oF being per- 
 pendicular to Crt, the points will fall upon D, and CD is to be 
 divided in G so that CG may be to DG in the compound ratio 
 of C to A and of CD to CF,or of the square of CD to the square 
 of Ca. These last are the two cases considered by Mr. Bernouilli 
 which have been lately pubhshed, Comni. Acad.Petropol. torn. 
 5; and these constructions agree with the computations which 
 he deduces by resolving the force of C into two infinite pro- 
 gressions. If the body C impel in like manner two equal bo- 
 dies A and B Cfig. 232, N. 3) in directions CF and CH that 
 form equal angles with the vertical, andfCh be one continued 
 horizontal line, CD is to be divided in G, so that CG may be 
 to GD in the compound ratio of C to the sum of the bodies A 
 and B and of the duplicate ratio of the sine of the angle FCD 
 to its cosine ; and CG will represent the force that accelerates 
 the motion of C, providing it always impel A and B in the 
 same directions from the beginning of its descent. 
 
 524. The rest remaining as in art. 323, let us now suppose the 
 bodies A andB (fig. 233) to gravitate as well as C. In this 
 case the body C while it descends will have no effect upon 
 the bodies A and B, unless the angles Dc/'and Dch exceed DCF 
 and DCH respectively. The force and direction with which C 
 descends being represented by CG, let Gf and Gh perpendi- 
 cular to CF and CH (the respective directions in which C acts 
 upon A and B) meet Cfand Ch in/and h, that Cfand Ch may 
 represent the forces by which the motions of A and B are acce- 
 lerated in the directions Cfand Ch. Let Da and Db perpen- 
 dicular to C/ and CA meet CF and CH in K and R respec- 
 tively; letyVi; perpendicular to Cfmeei CF in k, and hr perpen- 
 dicular to CA meet CH in r. Draw KM, /cm, BN, and r;i perpendi- 
 cular to CG in M, m, N, and n ; and draw Gd,fVj andhv perpen- 
 dicular
 
 50 Of the Descent of Bodies. l^ook I. 
 
 dicular to the vertical line CD in d, V^anclr. While C describes 
 CGj Adescribes Cf; and because A would have described Ca in 
 the same time by its own gravit}', the part AXfif of the force 
 which produces the motion of A is what is generated in conse- 
 quence of the action of C upon it; the force which Closes in the 
 direction CF(inwhichitacts upon A)ingcneratingthisinci'ease 
 trf the force of A in the direction C/'is AxK/c, which reduced 
 to the direction CG is AXMw. In the same manner the forc<* 
 which C loses in the direction CG by Vts action on B is BxN«. 
 Let DQ, be perpendicular to CG -in Q, and the force with 
 which C endcavoms to descend in CG in consequence of it? 
 gravity being C X CQ, it follows that C X CG + A X Mw + 
 B X Nn is e([ual to C X CQ, and C X CG" + A X CG X Mm 
 + B X CG X N/i.equal to C X CQ X CG or C X CDX Cd. But 
 the triangles Cmf/CfG being similar, INI t?i is to af as Cm to Cf 
 
 or Cf to CG, and Mm X CG is equal to Cf X af or Cp '- 
 
 CfXCa, that is (because C/is to CV as CD to Ca, and CfX 
 Ca is equal to CD X CV) to C/^-— ~ CD X CV ; and in the 
 same manner N/<xCG is equal to CJ/^- CDXCt'. There- 
 fore C X CG^ + AX C/" AXCDXCV + BxC/^^ 
 
 B X CD X Cv is equal to C X CD X Cd, or C X CG* + A X 
 Cf * 4- B X CA" equal to C X CD X Cc/ + A X CD X CV 
 + B X CD X Cv; that is (by art. 434), the sum of the pro- 
 'ducts that arise b}-^ multiplying each body by the square of the 
 velocity w'hich it acquires is equal to the sum of the products 
 when each body is multiplied by the square of the velocity 
 which itwouldhave acquired by the same perpendicular descent 
 if it had fallen freely. But it must be obsen'ed, that if any body 
 as A (for example) ascend, or the angle DC/' be obtuse, then 
 «rf is equal to Cf + Co, and the term AX CDX CVmnst be sub- 
 ducted in the latter part of the equation. The general theorena 
 therefore is, that the sum of the products of tlie bodies multi- 
 plied by the squares of their respective velocities is equal to the 
 difference of the products of those that descend multiplied by 
 the squares of the respective velocities that would have been 
 acquired by the same descents, and of the products of those that 
 ascend multiplied by the squares of the respective velocities that 
 would have been acquired by falling freely from the altitudes to 
 
 which
 
 Chap. XII. that act upon one another. 31 
 
 whi ch they have risen (Jig, Q,33,N. 2) . To give an example how the 
 motions are determined in this case, suppose that G impels the 
 equal bodies A and B in directions that form equal angles with 
 tlie vertical CD, and that those bodies move in directions Cf 
 and Ch that likewise form equal angles with the vertical greater 
 than DCF or DCH. Let Da perpendicular to C/* in a meet 
 CF in K, az perpendicular to CF meet CD in z, and KN be 
 perpendicular to CD in N ; let De perpendicular to CF meet 
 Cfin e ; divide ac inf, so thaife may be to af in the compound 
 ratio of CN to C-, and of the sum of A and B to C ; then Cf 
 will represent the force that accelerates the motion of A or B ; 
 and fG perpendicular to CF will intersect CD in G, so that 
 CG will represent the force which accelerates the motion of 
 C. Let Kk perpendicular to CF meet C/'in A-, and if C act 
 upon one body A only, ae is to be divided in f, so thatfe majr 
 be to af as A X Ck to C X Ca, and fG perpendicular to 
 CF will intersect DG parallel to CF in G. But in these con- 
 structions we suppose that the body C acts upon the bodies A 
 and B in invariable directions. 
 
 525. Tlie same theorem takes place though the sides of A and B 
 upon which C acts be not planes, and the directions vary in 
 which they are impelled by the body C ; providing C act upon 
 them from the beginning of its descent, so that there be no col- 
 lision or sudden communication of motion from one body to ano- 
 ther. Let CG^^g". 234) the direction of Cat any time meet Cy the 
 direction of A atthesame time inC, and ch the direction of B in c ; 
 let CF be the direction in which C then acts upon A, and cH 
 the direction in which it acts upon B ; and the rest of the con- 
 struction being similar to that in the preceding article, let G 
 represent the force of gravity along the inclined plane CG, g 
 the force by which the motion of C is actually accelerated in, 
 this direction, k the force of gravity along the inclined plane 
 Cf, and p the additional force by which the motion of A is ac- 
 celerated fiom the action of C, / the force of gravity along ch, 
 and q the force added to this by the action of C. Let P be 
 equal tok -^p, and Qto l-{- q. The force which the body C 
 loses in the direction CG by acting upon A in the direction CF 
 
 and.
 
 52 Of the Descent of Bodies Book T. 
 
 and generating the force A xp in tlie direction C/'is A X px — 
 
 C/ ^^ 
 
 or A X ;-> X ^^ the force which C loses in the same direction 
 
 CG by its action on B is B X «/ X-fP^' ^X qX ~; consequent- 
 
 b' C X g + A X /J X §^ + B X ^ X ;;^ is equal to C X G. If 
 
 .r, y, and z represent the fluxions of the res[)eotive spaces de- 
 scribed by C, A, and B, by their motions, x will be to j/ as CG 
 to Cf, and X to z as cG to ch. Therefore Cox -j- Apt/ -\- Bqz 
 will be equal to CaG. or Cgx -\- APy -4- ^^*^^~ equal to CxG -|- 
 Aki/ -|-B/^. But if V, n, and v represent the respective veloci- 
 ties that are acquired by C^A, and B at the points G,f, and h, 
 and I, K, L the respective velocities which the same bodies 
 would have acquired by falling freely from the same altitudes 
 from which they have descended ; then (art. 434) o-x, Py, and 
 Q:: will represent the respective fluxions of | VV, | uu, \ vv ; 
 and Gi', %, Iz will represent the fluxions of | II, | KK, and 
 ■i LL. Therefore since we suppose these velocities to begin to be 
 generated together, CVV -f- Aim -{- Bvv is equal to CII -{- 
 AKK -\- BLL, where AKK or BLL are to be subducted if A 
 or B ascend while C descends. It is obvious that if we suppose 
 the body C to act upon any number of bodies, or these to act 
 on other bodies in any directions, the theorem will still obtain 
 by collecting the sums of tlie products of all the bodies that act 
 u])on each other multiplied by the squares of their velocities. 
 526. If we suppose the bodies C, A, and B to ascend from 
 their respective places G,f, and /«wiLh the motions which they 
 have acquired, so as to be retarded by their gravity only, their 
 common centre of gravity will rise to the same level from which 
 it descended. For suppose X, Y, and Z to be the respective al- 
 titudes to which these bodies would rise in this manner, H the 
 altitude which would be described by their common centre of 
 gravity, A the altitude which it described in descending, I, K, 
 and L the respective altitudes from which the bodies dc-scend- 
 ed, and S the sum of tlie bodies ; thca by the last article CX 
 -}- AY -J- BZ will be equal to CI -f AK -f BL : and these 
 
 sums
 
 Chap. XII. that act upon one another. 33 
 
 sums are respectively equal to S X H and S x h, by art. 509 ; 
 therefore H is equal to h. These theorems extend equally to bo- 
 dies of all kinds, those that are void of elasticity, as well as those 
 tiiat have any degree of elasticity, there being no relative velo- 
 city generated by C in the directions in which it acts upon the 
 other bodies. Whereas the theorems demonstrated in art, 517 
 and 5 19, concerning the equality of the sums of the products of 
 the bodies multiplied by the squaresof their velocities compar- 
 ed together before and after their collisions, extend only to such 
 bodies as have a perfect elasticity. These last are founded on the 
 equality of the relative velocities of C, and the several bodies 
 A, B, ^c. in their respective directions before and after the 
 stroke ; but those on art. 434, and the general principle describ- 
 ed in art. 511. There may be an analogy however between 
 those theorems, that may be explained perhaps from the mo- 
 tions which are generated in bodies by the actions of springs; 
 but we are not to extend those theorems to motions of all kinds 
 for the sake of this analogy. 
 
 527. For if the body C descend from any height IC before 
 it begin to act upon the other bodies; or if there be any colli- 
 sion of the bodies while they descend, and they have no elasti- 
 city, or an imperfect one; or, in general, if there be any sud- 
 den communication of motion from one body to another, and 
 the relative velocities in their respective dh-ections be less im- 
 mediately after that action than before it; in those cases the 
 sum of the products of the bodies multiplied by the squares of 
 their velocities will be less than it would have been if the bo- 
 dies had descended freely from the same respective altitudes; 
 and if the bodies be supposed to ascend with their respective ve- 
 locitiestit any time, and their motions be retarded by their gra- 
 vity only, the common centre of gravity will not ascend to the 
 same level from which it descended. When the bodies C and 
 A (7':g. 235) that have no elasticity, or an imperfect one,suspended 
 by equal lines KC and LA from the points K and L (that are on 
 the same level, and at. a distance from each other equal to the 
 sum of the semidiameters of the bodies), after describins: the 
 arks IC and EA, strike one another,- or when any body C af- 
 ter its descent from I is loaded with a new body at C 
 
 VOL. IL D which
 
 34 Of the Descent of Bodies Book L 
 
 which it carries along with it in its ascent (as in a known expe- 
 riment made b}' Mr.GraAa;/i),itisubvioustl)attheascentoftheir 
 common centreofgravitymustbe less than its descent.To give an- 
 other simple instance: su[)po3e that the body C0ig.'236)descend.'> 
 in thever ticalC iJ^and at the same time draws any bod y Aalong the 
 horizontal line KL without friction by a line or chain CMA 
 (which we suppose to be void of gravity) that is directed by 
 the pull}' iVI, so that MA is always horizontal. First, let C draw 
 the body A as soon as it begins to descend; and the accelerat- 
 ing force being always as the absolute force directly, and the 
 matter that is to be moved inversely, the motion of each body' 
 will be accelerated by a force that is less than the accelerating 
 force of gravity in the ratio of C to C -|- A. Let CG be equal 
 to Aa, and the square of the velocity of C when it comes to G, 
 or of A when it comes to a, will be to the square of the ve- 
 locity which C would have acquired by falling freely from C 
 to G in the same ratio of C to C -}- A, the squares of the ve- 
 locities acquired by descending from the same altitude being 
 as the forces that generate them, when these forces act uniforni- 
 Iv, by art. 4S4 ; consequently the product of the sum of C and 
 A multiplied by the square of their common velocity is equal 
 lo the product of C multiplied by the square of the velocity 
 which it would have acquired by the same perpendicular de- 
 scent, if it had fallen freely from C to G; and if the bodies C 
 and A be supposed to ascend from G and a with the respective 
 motions acquired at these points, their common centre of gra- 
 . vitv will rise to the same level from which it descended in 
 this case. But let us suppose now that the body C lirst descend<> 
 from ?vl to C, and that the hue or chain AMC is not stretched 
 till it come to C, so that no motion is communicated to the 
 body A till that instant; the motion acquired by C will be 
 then divided betwixt C and A so as to produce equal velocities 
 in each ; the sum of the products of the bodies multiplied bv 
 the squares of these velocities will be less in this case than the 
 product of C multiplied by the square of the velocity which it 
 acquired bv descending freely from M to C in proportion as C 
 is kss than C 4- A; and if the bodies C and A be supposed to 
 ascend with those velocities from their respective places, the 
 
 ascent
 
 Chap. Xir. that act upon one another. S5 
 
 asceiitof theircentreof gravity will be less than itsdescent in the 
 same ratio (//g. 230). If we suppose in art. 52 1 ancl522; that the 
 body C falls from I to C before it act upon A and B, and there- 
 after descends impelling those bodies by its gravity as above, 
 let Ce be to Cf as Cd is to CD, and the sum of the products of 
 the bodies multiplied by the squares of the respective velocities 
 which they acquire when C comes to G, will be equal to the pro- 
 duct of C multi})lied by the square of that velocit}^ only which 
 it would acquire by descending freely from e to d. In the same 
 manner it may be shown, that in the case of art. 524 (Jig. 233), 
 if C fall from any altitude before it act upon A and B, and 
 thereafter descendimpelling them as in that article,and thebo- 
 flies be supposed to ascend with the respective velocities acquir- 
 ed by them fiom their respective places at any time, the ascent 
 of the centre of gravity will be less than its descent. We have 
 mentioned these instances, though they are obvious, to prevent 
 mistakes from the expressions of some celebrated authors,who 
 seem to represent this principle concerning the equality of the 
 ascent and descent of the centre of gravity as general. 
 
 528. When thebodyCC/?g-. 236) wassuppdsed'to draw thebody 
 A along KL by the hne or chainCMA froiti the beginning of its 
 descent, a greater quantity of motion was generated in C and 
 A by the uniform power of gravity acting upon C than that 
 M'hich C alone would have acquired by the same perpendicu- 
 lar descent CG, in the same proportion that the time of descent 
 in CG is prolonged in the former case above what it is in the 
 latter ; and the like may be said of those cases which were de- 
 scribed in art. 521 and 522, if regard be had to the directions 
 in which the bodies move. And as the same power acting with 
 the same direction upon the same body may be reasonably sup- 
 posed to generate a greater force in a greater time, as well as 
 a greater quantity of motion ; so there is no ground to alter 
 the usual manner of measuring the forces of bodies in motion 
 on account of the preceding theorems, or of those that folJow 
 concerning the sums of the products of the bodies "multiplied 
 by the squares of their velocities. If we were to measure the 
 forces of bodies ia motion by the product of their quantities of 
 matter, and of the squares of their velocities, the sum of the 
 D 2 forces
 
 36 Of the Descent of Bodies Book I. 
 
 forces acquired by the bodies C and A at G and a would be 
 equal to the force which C alone would acquire by the same de- 
 scent CG ; and the same force that imprinted on the body C 
 alone would cause it to ascend in the vertical from G to C, if it 
 was imprinted on the bodies Cand A at once, so as to generate 
 equal velocities in the body A from A towards a along the ho- 
 rizontal line LK, and in the body C upwards from G,. it would 
 cause the body C to rise to the same height from G to C as in 
 the other case, and at the same time cause the body A to de- 
 scribe aA equal to GC along the horizontal LK ; the force 
 which would be sufficient to produce those two effects would 
 be always the same how great soever we should suppose the bo- 
 dy A to be : and if we should likewise admit that the force 
 which causes a given body C to ascend from G to C, and de- 
 scribe a given altitude GC, is always the same without regard 
 to the time, it would thence follow that the motion of the bo- 
 dy A from A to a is an effect that ought to be held of no ac- 
 count. An observation of the same kind might be made in 
 other instances; but thereis nonecessityforperpjexingthetheo- 
 ry of motion with the consequences that follow from this doc- 
 trine concerning the mensuration of the forces of bodies ; and 
 therefore we proceed to argue from the principle in art. 511, 
 which is universally allowed. 
 
 529. Hitherto we have supposed the body C (Jig. 237)to aetim- 
 mediately bycontact on the other bodies. Let the bodies Aand 
 B be now fixed to the axis KIL at the respective distances KA 
 and LB, and the body C impinge on the inflexible lever IC 
 (that is fixed perpendicularly to the same axis) with a direction 
 and velocity represented by CD ; and supposing the figure to 
 be at rest before the stroke, let it be moveable about the axis 
 KL onlv. LetCQ perpendicular to IC meet DQ parallel to it 
 in Q ; divide CQ in N, so that CN may be to NQ as the pro- 
 duct of the body C multiplied by the square of its distance 
 from the axis of motion to the sum of the products of the 
 other bodies multiplied by the squares of their respective dis- 
 tances from the same axis; and DG parallel to CQ will intersect 
 NG parallel to IC in G, so that CG will represent the velocity 
 of C after the stroke ; and if A/ and Bk be to CN as KA 
 
 and
 
 Chap. XII. that act upon one another. S7 
 
 and LB to IC respectively, Af and Bh will represent the respec- 
 tive velocities of A and B after the stroke when there is no elas- 
 ticity. For if we suppose any line CN to represent the veloci- 
 ty of C after the stroke in the direction CQ, then (because whea 
 the figure moves about the axis KL the velocity of any point 
 A is to the velocity of any point C as KA to IC, or as Afto 
 CN) Af will represent the velocity, and A X Af the motion of 
 A. The motion which C must lose in the direction CQ by ge- 
 nerating in A this motion A X Af must be to A X Af as KA to 
 IC (by the principles of mechanics), or' to A X CN as KA* 
 to KC% and the motion which C loses in the same direction by 
 producing in B a motion B X BA is in like manner to B X BA 
 as LB to IC or to B X CN as LB^ to 1C\ And the whole 
 motion lost by C in the direction CQ being C x NQ, it fol- 
 lows that C X NQ X IC^ is equal to A xCN x KA*+B x 
 CN X LB% and that CN is to NQ as C x IC^ to A x KA* 
 + B X LB*. Therefore since CQ was divided in this ratio 
 in N, and the motion QD is not affected by the stroke, CG will 
 represent the direction and velocity of C after the stroke. When 
 C is perfectly elastic, produce DG till Dg be equal to 2DG ; 
 then Cg will show the direction and measure the velocity of C 
 after the stroke, and the respective velocities of A and B will 
 be represented by 2A/and 2BA. 
 
 530. When the bodj' C and the lever are supposed to have 
 no elasticity, the sum of the products of the bodies multiplied 
 by the squares of their velocities after the stroke is less than the 
 product of C multiplied by the square of its incident velocity 
 in the ratio of Cd to CD, Gd being perpendicular to CD j but 
 when C is perfectly elastic these are equal to each other. For 
 by what was shown in the last article C X CN X NQ is equal to 
 to A X Af- -f B X BA-, and by adding C x CG% the whole sum 
 becomes equal to the product of C by CD"" — CQ^ + QCN 
 or CD'— CQN, or (because DG or QN is to Bd as CD to 
 CQ) CD"— CD xm, that is, to C x CD x C^; which is 
 less than C x CD^ in the ratio of Cd to CD. But C x C^* + 
 A X 4A/' + B X 4Bh- is equal to C x CD* ; for let gn be 
 perpendicular to CN in n, and that sura being equal to C x Co-* 
 + C x 4QNC (by what was shown in the last article) or to 
 
 B 3 the
 
 38 Of t/ie Descent of BoJus Book I. 
 
 the product of C by C«-+^/j'' + 4QNC, it is equal (elem. Q, 
 % QN and Nn being equal) toCxCD\ Therefore if we 
 suppose LhatCacquires its incident velocity CD byfalHng from 
 any altitude cQ, and the bodies be supposed to ascend with their 
 respective velocities immediately after the coUision^ so that their 
 motions be retaided by their gravity only, their centre of gra- 
 vity will ascend to the same height from which it descended in 
 ihe latter case when C is supposed to have a perfect elasticity ; 
 but in the former case the ascent of the centre of gravity will 
 be less than its descent in the same ratio as Cd is less than CD. 
 These theorems are easily extended to the cases when several 
 bodies strike the lever IC at once, or different levers fixed to 
 the same axis with given directions and velocities ; and v*'hcn 
 the elasticity is imperfect, the ascentof the centre of gravity will 
 be always less than its descent, the motions of the bodies be- 
 ing supposed to be converted upwards after the collision. 
 
 531. Suppose now that the body C acts by its gravity only 
 upon the lever IC, and b}' means of this lever impels the whole 
 figure about the axis KL, the bodies A and B being supposed 
 to have no gravity, then the accelerating force and the direc- 
 tion of gravity being represented by CD, the force and direction 
 with which C will begin to descend will be represented by CG 
 if the body C be allowed to slide along the lever IC, but by 
 CN, if the bod}' C be fixed to the lever, the force NG being 
 destroyed in this case by the resistance of the axis. Because 
 C xCCr- + A X Af^ + B X BA^ is equal to C x CD x Cd, it 
 followj that in either case the sum of the products of the bodies 
 multiplied by the squares of their respective velocities is equal 
 to the produclof Cmultipliedby thesquareof thevelocity which 
 it would have acquired by the same perpendicular descent, if it had 
 fallen freely in the vertical CD, providing the body C act upon 
 theleverfromthebeginningof itsdescentC/Jg.SSTi^.^). Itfol- 
 lows likewise from art. 529, that if I be the centre of gravity 
 of the bodies A, B, 4fc. and a weight P act xipon the lever lA 
 at the distance IC from the axis of motion which we suppose to 
 pass through I, then the force CG with which P descends will 
 be to its gravity CD as P X IC^ to the sum of the products of 
 the bodies A, B, ^;c. multiplied by the squares of their respec- 
 tive
 
 Chap. XII. that act upon one another. 39 
 
 live distances from the axis added to P x IC^ ; and hence the 
 motion of P ma}^ be determined when it turns the figure around 
 the centre of gravity I by means of a rope PCZR that goes 
 round the axis CZR. 
 
 .532. Ifwesuppose allthebodiesCjAjandB0?g.2.'38)fixed tothe 
 axis at their respective distances CI,AK,andBLto gravitate in pa- 
 rallel lines CD, Aa, and B6; let these lines be equal to each 
 other, and represent the accelerating force of gravity ; let Cg, 
 Af, and BA represent the forces b\' which their motions are 
 actually accelerated with llieirrespective directions perpendicu- 
 lar to IC, KA, and LB, while the figure moves upon its axis 
 KL. Let DQ, am, and hk be perpendicular to those direc- 
 tions in Q, m, and k; and gd, fn, hr be perpendicular to CD, 
 Aa, and B6 in d, n, and r. If CQ be greater than Cg, but Aa 
 Jess than Aii, and Bb less than Br, then the body C loses by 
 its action on the lever IC a force C X ^C^, and thereby the bo- 
 dies A and B acquire the forces A X mjand B x kh respective- 
 ly. Hence regard being had to the lengths of the several le- 
 vers CI, KA, and BL, according to the known principles of me- 
 chanics, C X gQ x IC will be equal to A X nif x K A + B x kk 
 X LB, or (because the velocities Cg, Af, and BA are as the dis- 
 tances from the axis IC, AK, and BL) C X Cg X gQ equal to 
 AxAf X mf-i- B X BA x kh, that is, C x CQ x Cg — C x Cg^ 
 equal to A X A/^ — Ax Af x Am 4- B x BA ^ — B x BA 
 X Bk. ButCQ X Cgisequal toCD x Cd,Afx Am to AaxAu 
 and BA x BA to B6 x Br. Therefore C x C^-^^ + A x Af^ + 
 B X BA" is equal to C X CD X C^ + A x CD x'' An + B x CD 
 xBr; from which it follows (by art. 431), that when all the 
 bodies descend while the axis moves, the sum of the products 
 of the bodies multiplied by the squares of the respective veloci- 
 ties acquired by them at anytime is the same as if they had 
 fallen freely along the perpendicular altitudes from whicli they 
 have descended. But if any body as B (for example) had been 
 on the other side of the axis KL so as to have ascended while 
 the common centre of gravity of the bodies descended, AA had 
 been equal to BA-f-BA, ^ In this case the term B X CD x Br 
 iuust be subducted in the latter part of the last equation; and 
 in general the sum of the products of the bodies multiplied by 
 
 D 4 the
 
 40 Of the Centre of Oscillation. Book I, 
 
 the squares of their respective velocities is equal to the differ- 
 ence of the sum of the products of those that descend multi- 
 plied by the squares of the velocities that would have been ac- 
 quired by the same descents if ihey had fallen freely, and of 
 the sum of the products of those that ascend multiplied by the 
 squares of the respective velocities that would be acquired by 
 i'alling freely along the respective altitudes to which they have 
 arisen. In either case it follows that if the bodies be supposed 
 to ascend from their respective places at any time, and to be re- 
 tarded by their gravity only, their common centre of gravity 
 will always ascend to the same level from which it descended. 
 This principle is demonstrated in like manner when the bodies 
 C, A, B, ^"c. are supposed to act upon one another by compound 
 levers or other mechanical engines, without friction or re- 
 sistance from the ambient medium. But it will not hold if we 
 suppose any body first to impinge on the lever or engine with 
 any assignable velocity, and then to descend with it. 
 
 523. It was advanced long ago by Mr. Hui/gem* as a ge- 
 neral principle, '^ That if bodies begin to move by their gra- 
 ■^•^ vit}', their common centre of gravity can never rise higher 
 "' than where it was at the beginning of the motion." To which 
 he added as a second hypothesis, " That abstracting from the 
 ■^^ resistance ofthe air and such obviousimpediments,a compound 
 " pendulum will describe equal arks in its descent. and ascent." 
 And by these two principles he was able to determine the length 
 of a simple pendulum that should vibrate in a void in the same 
 time with a compound one in any similar arks, and to find the 
 centre of oscillation of bodies. He did not then affirm that the 
 centre of gravity of the bodies would always rise to the same 
 height from which it descended, but that it will never rise to a 
 greater height than this; which is indeed a general principle, 
 for the ascent of the centre of gravity will be always found to 
 be either equal to its descent or less than it, but never greater. 
 He seems however to go farther afterwards, and to affirm that 
 
 Horol. OfciL par. 4. Hyp. L & 2. 
 
 bodies
 
 Chap. XII. Of the Centre of Oscillation. 41 
 
 bodies always retain tlieir vis ascendens'^, as he calls it, by 
 which their centre of gravity would rise to the same level from 
 which it descended. This principle obtains indeed in all the 
 oases he has mentioned (these being called hard bodies by him 
 Mhich are supposed to have a perfect elasticity) and in many 
 others; as has been shown in the preceding articles; where we 
 have endeavoured to distinguish those cases in which this princi- 
 ple takes place from those wherein it cannot be admitted, and 
 to show at the same time that no useful conclusion in mechanics 
 is affected by the disputes concerning the mensuration of the 
 force of bodies in motion which have been objected to mathe- 
 maticians f. 
 
 534. Suppose therefore OV(/o'/238) to beequal to thelength of 
 a simple pendulum that in a void performs its vibrations in simi- 
 lar arksin the same time with the compound pendulum describ- 
 ed in art. dol, or let OV be the distance of a point in this lat- 
 ter pendulum that moves in it with the same velocity as if OV 
 was a simple pendulum suspended at O. Let S represent the 
 sum of the bodies C, A, and B,and OG be the distance of their 
 centre of gravity from the axis. While the pendulum moves, 
 let the points C, B, A, G, and V descend to c, b, a, g, and v 
 respectively; let GM be the perpendicular descent of the cen- 
 tre of gravity, and VR the perpendicular descent of the point 
 V. Then because the velocities of the points C, B, A, G, and 
 V are as their distances from the axis of oscillation, and the ve- 
 locity acquired at v is such as would cause a body to ascend 
 from R to V (by the supposition), and the altitudes to which 
 bodies would ascend by the velocities acquired at c,6, a, and v 
 are in the duplicate ratio of these velocities, it follows that C, 
 B, and A would ascend by their respective velocities at those 
 
 points to the altitudes VR X -^^i VR x — - and VR x -^. 
 And since their common centre of gravity would ascend to the 
 
 * Hicc c^nstans lex est corpora servare: vim suam ascendentem, tS" tdcirco summam 
 quadratorum vehcitatuin illorum semper manere eandem. Hoc auiem non solum obiimt 
 in ponJerthiiS pendulorum IS" percussione corporum durorum, sed in muUts quoque alns 
 mechanicis experimenUs. Cbicxv. D. ii/wj'^t'/;^ in literas D. March de rHospitalj l?''^. 
 Oper. Vol. I. p. 258, 
 
 t Aiiai^st, Qurry 9, 
 
 satne
 
 42 Of the Cmtre of Oscillation. Book I. 
 
 same altitude from which it descended^ by art. 532, it follows 
 (art. 509), that C X VR x I^% + B X VR x i-?! + A x 
 
 O V * O V ^ 
 
 VR X ~ is equal to S x GM. But the arks described by G 
 
 and V being similar, GM is to VR as OG to OV ; consequent- 
 ly SxOGxOV is equal to C X IC* + R X LB" + A X KA^ ; 
 
 and OV is found by multiplying each body by the square of its 
 distance from the axis of oscillation, and dividing the sum of 
 the products by S X OG, which is the product of the sum of the 
 bodies multiplied by the distance of their common centre of 
 gravity from the same axis. The same demonstration being ap- 
 plicable to any number of bodies, we may conclude that when 
 an}' body moves about a given axis, the distance of its centre of 
 oscillation from this axis (or the length of a simple pendulum 
 that vibrates in avoid in the same time with the body in simi- 
 lar arks) is found by computing the fluent when each particle 
 or element of the body is supposed to be multiplied by the 
 square of its distance from the axis, and dividing this fluent by 
 the product of the body multiplied by the distance of its centre 
 of giavity from the same axis. 
 
 535. If the points C, A, and B be in one plane that is per- 
 pendicular to the axis of oscillation in O, let Cc, B/ and Ak 
 be perpendicular to OG in /, / and k ; then OC" being equal 
 to OG^ -h CG"— 20G/', OB* to OG* + BG* -I- 20G/and 
 OxV* to 0G* + AG*-|-20GA; {clem. 12 and 13, 2), the point 
 i being betwixt O and G, and the points / and k on the other 
 side of G, and C X /G being equal to B X /G + A x A:G (art. 
 509), it follows that the sum of the products of the bodies mul- 
 tiplied by the squares of their distances from O, or S X OG. 
 xOV is equal to SxOG* + CxCG* 4- BxBG* -I- A x 
 AG* ; consequently S X OG X GV is equal to the sum of the 
 products when each body is multiplied by the square of its dis- 
 tance from the centre of gravity, and GV the distance of the 
 centre of oscillation from the centre of gravity is found by di- 
 viding this sum by S X OG ; whence the computation of the dis- 
 tance of the centre of oscillation from the axis in solids is in 
 some cases abridged. 
 
 536. 7q
 
 Chap. XII. Of t/ie Centre of OscUIatloii. 43 
 
 536. To give an example, let DEc/(//g.239)be theseetionofa 
 sphere through its centre G by a plane perpendicular to the axisof 
 oscillation, Dd the diameter of this circle perpendicular to the 
 axis, GE the radius perpendicular to Dd, and PFp any concen- 
 tric circle; let MP an ordinate perpendicidar to Dd at P meet 
 DEd in M, MN be perpendicular to GE in N, and the ra- 
 tio of « to 1 express that of" the circumference of a circle to its 
 radius. Let a cylindric surface be imagined to stand on the 
 circumference PF/? perpendicular to the plane DEd, and termi- 
 nated by the surface of the sphere, and its altitude being 2PM, 
 it may be expressed by Q.n X GP X GN,vvhieh being multipli- 
 ed by the square of GP (which is the distance of the particles 
 in each section of this surface perpendicular to the axis of oscil- 
 lation from the centre of gravity of the section), and the pro- 
 duct 2n X GN X GP ^ being multiplied by the fluxion of GP, 
 or (because GP"- is equal to GD^ — GN% and the fluxion of 
 GP is to the fluxion of GN as GN to GP) the product of 2/e 
 X GN^ and GE* — GN"" being multiplied by the fluxion of 
 PM, the fluent by the converse of art. 146, will be the product 
 of 2« X GN^ by i GE* — i GN*. But this fluent becomes 
 equal to rr X n X GE^ when P has described the whole radius 
 DG, and GN becomes equal to GE; and this being divided 
 by J « X GE^ X OG (which expresses the solid content 
 of the sphere multiplied by OG, by v.h at was shown in the 
 Introduction), GV the distance of the centre of oscilla- 
 tion from the centre of gravity in the s}there is found to be 
 
 GE' 
 -f Q^ or to be two flfths of a third proportional to OG and 
 
 GE. This subject havingbeen treated offuUy in the /foro/. Oscil. 
 par. 4, Acta Lipsia.-, 1714, and Method. Increm. prop. 24, and in 
 several olher pieces, we shall not insist on it further here, and 
 shallonly add, thatwhen aweightP(//o-.237. N.2)turns afiguro 
 about its centre of gravity 1 by means of a rope PCZR that goes 
 round the axis CZR as in art. 531, let V^ be the centre of oscillation 
 of the figure when C is the centre of suspension, let S denote 
 the mass or weight of the figure to be inoved, le be taken up- 
 on IC in the same ratio to IC as P is to S ; then CG the force 
 .by which the motion of P Avill be actually accelerated will be 
 
 to
 
 44 Of the Motion of Water issuing Book I. 
 
 to CD the accelerating force of gravity as el to eV. For (by 
 art. 531) CG is to GD in tiiis case as P x IC* to A x AP -f 
 B X BI% S^c. which last is equal to S X IC x fV, by art. 535; 
 consequently CG ia to GD as P x IC to S x IV, or as el to 
 IV, and CG to CD as el to eV ; which agrees with the solu- 
 tion given by the learned Mr. Daniel Bernoulli, Comment. 
 Petropolit. torn. 
 
 537. Sir Isaac Newton has considered the motion of wa- 
 ter issuing from a cylindric vessel ABDC (Jig. 24:0. N. 1) at an ori- 
 fice EF in the bottoni CD, Pn«cip./j6.C;,/?ro/). 36. His doctrine 
 on this subject may receive some illustration from the following 
 considerations.While the water issues at the orificeEF,thatwhich 
 remains in the vessel subsides at the same time; and though the 
 particles of this water descend with unequal velocities, we may 
 consider the velocity with which the surface AB descends to be 
 their mean velocity. This velocity manifestly begins from no- 
 thing (as that of any heavy body that descends by its gravi- 
 ty), and while it is accelerated is always to the velocity with 
 which the water issues at EF in the ratio of EF to AB. The 
 continual effect of the gravitation of the whole mass ol" water 
 may be considered as threefold. It accelerates, for some time 
 at least, the motion with which the water in the vessel descends; 
 it generates the excess of the motion with which the water is- 
 sues at the orifice above the motion which it would have had 
 in common with the rest of the water; and it acts on the bot- 
 tom of the vessel at the same time. Let the velocity with which 
 the water issues at EF at any term of the time be represented by 
 X, the velocity with which the surface AB subsides by V, the 
 accelerating force of gravity by g, the force which would gene- 
 rate the acceleration of V by f and the time from the begin- 
 ning of the motion by T. The gravitation of the whole mass 
 of water in the cylindric vessel ABCD may be expressed by 
 AB X AC Xg; and because the force^ is employed in generat- 
 ing the acceleration of the motion with which the water sub- 
 sides in the vessel, the force AB X AC X g—f is what we 
 are to suppose to be employed in acting upon the bottom, and 
 in generating the velocity X — V in the water that issues at 
 the orifice. Suppose that the ratio of r — 1 to 1 expresses the 
 
 propoi^i^
 
 I'i^. 23S
 
 i'iMc^Wiu;i-u.r,-/jc 
 
 ri^.23S-K.
 
 Chap. XII. from a CT/UndricFesscI. 45 
 
 proportion of the parts of this last force which produce these 
 two effects, or that r is to 1 as the force AB x AC x g—f is to 
 that part of it which we conceive to produce the velocity X — ^V 
 in the water issuing at EF; and this part will be expressed by 
 AB X AC X —-• The quantity of water which would issue at 
 
 the orifice EF in any time T with the velocity X continued 
 uniformly is expressed by EFxTxX, and the force which 
 would generate the velocity X — V in this quantity of water 
 is as the quantity of motion that would be generated in this 
 
 manner (or EF X TX X X — V) directly, and the time T in- 
 versely, that is as EF x X x X — V, or (because X is to V 
 as AB to EF, and X— V to X as AB— EF to AB) as EF 
 
 X — -- — X XX, which we are to suppose equal to AB X AC 
 X^ZZ. that represents the same force. The square of the velo- 
 city that would be acquired by the descent AC is expressed by 
 «AC Xg (art. 434), and if KC be to AC as AB X AB is to 2r 
 X EF X AB — EF, and A denote the velocity that would be ac- 
 quired by the descent KC, then AA will be to 2AC Xg as KC 
 to AC, or as AB X ABto 2r x EF x AB — EF,and consequently 
 as XX is to 2 AC Xg—f. Therefore A A will be to XX as 
 g is to g—f, and g to / as AA to AA — XX. From this it 
 follows, that if we suppose the fluxion of X (or of V which is 
 in a given ratio to X) to vanish, in order to find its greatest va- 
 lue, or the limit of all its values, by art. 242,/ the force which 
 accelerates V must vanish, g— ^must be equal to g, and X 
 to A. In general, the descent by which any velocity X of the 
 issuing water would be acquired is to KC the descent by which 
 A would be acquired as g—f to g. It appears also that the 
 fluxion of V is to the fluxion of the velocity of a body that de- 
 scends freely at the same time in the vertical asf to g, or as 
 AA— XX to AA. If the fluxions of T, X, and V be repre- 
 sented by t, X, and v respectively, then f will be represented 
 
 by I, and AA will be to A A — XX as gt io ft or gt to v, 
 
 or as AB X gt to EF x x, because v is to *^ as EF to AB, by 
 
 art.
 
 45 Of tlic Mof<u}i of JVciter issuing Book t. 
 
 art. 24, so that t is expressed by a qiicintity that is to j" in the 
 compound ratio of AA to AA — XX, and of A B to EB. By 
 AB and EF we mean the areas of the section of the cylinder 
 and of the orifice, and not their diameters. We shall suppose 
 the ratio of r to 1 to be invariable ; and because diflerent sup- 
 positions have been made concerning this ratio, we shall show 
 how the motion of the water may be computed from any of 
 them, before we enquire what this ratio i.s. 
 
 538. Let««/(^g.240,N. land2)beaneqiii]atcralhyperbola,c 
 the centre,^ the vertex, ab perpendicular to ccr meet the asymp- 
 tote in b) ah represent the velocity A, ad the veloci ty X,join cdixnA 
 produceittillitmeetthehyperbolain;? ; thcnthetimeTin which 
 the velocity of the issuing water becomes equal to X will be 
 to the time in which a body would fall from K to C by its 
 gravity in the compound ratio of the area of tlie orifice EF to 
 the area of the bottom CD or AB, and of the hyperbolic sector 
 can to the triangle cab, if the water be supposed to be supplied 
 at the surface AB so that the vessel be kept always full. For 
 let w« be perpendicular to the axis ca in m, and ad'^ will be to 
 nni^ as ca^ to cm^ or car-\-nm'^ (by the property of the hy- 
 perbola), and cm"^ to ca^ as ab^ to ab' — ad^, or as AA to 
 AA — XX. The fluxion of the sector can is to the fluxion 
 of the triangle cad (or \ Ax) as cn^ to cd^ (art. ICO), or cni' 
 to ca"^, and therefore as A A to AA — XX, or as A\^y.gt to 
 EF XX,- consequently the iluxion of the sector can is to Agt 
 as AB to 2EF, and the sector can to AgY in the same ratio by 
 art. 24. Let Z express the time in which a body falls from K 
 to C, and by its descent acquires the velocity A, then Z will be 
 
 expressed by -> and T will be to Z as canX.lJLY to ca^ 
 
 xAB. Hence if Z, Tj^A X v^p and L be proportional, and 
 
 the ratio of which L is the logarithm (the modulus being ab) be 
 that of c to d, then the velocity ab will be to the velocity ac- 
 quired at the end of the time T as c-i-d to c — d. For ex- 
 ample, if the area AB be to the area EF as 10 to 1, and T be 
 supposed equal to Z, L will be to ab as 20 to 1, c to d in a 
 greater ratio than 48J000000 to 1 j and the excess of ab above 
 
 ad
 
 Chap, XII. from a CyUndric Feisel. 47 
 
 ad will be less than the 242000000 part of ab in the time a 
 body would fall from K to C, though according- to this theory 
 ad can never become precisely equal to ab. 
 
 539. Let mk perpendicular to the asymptote meet the hyper- 
 bola in/?, join cp, and the quantity of water that issues at the 
 orifice EF in the time T will be to the quantity that would have 
 issued at EF in the same time if the velocity had been alwa3'S 
 equal to A as the hyperbolic sector cap is to the sector cati. For 
 let the quantity of water that issues at EF in the time T be re- 
 presented by Q, and its fluxion by q, then q will be expressed 
 
 by Xt X EF, which (by art. 537) is to EF X — as A A X EF 
 
 to A A — XX X AB ; so that the fluent of Xt is to half the loga- 
 rithm of the ratio of AA— XX to AA as EF to 0^ x AB, the 
 modulus being AA. Let ae be perpendicular to the asymptote 
 in e, and if the modulus be ce^ or | AA, the area aekp will be 
 the logarithm of the ratio of pk to ae, or of ce to ck or of ca 
 to cm, that is, aekp will be equal to one half of the logarithm 
 of the ratio of ca"" to art^, or of AA — XX to AA. There- 
 fore the fluent of Xt is to the area aekp, or the sector cap, as 
 2EF to g X AB. But A X T X EF expresses the quantity of wa- 
 ter that would have issued at the orifice EF in the same time 
 T with the velocity A, and A xT is to the sector can in the 
 same ratio, by the last article. Therefore Q is to A xT X EF 
 as the sector cap to ca7t. Hence the difterence of ATxEF 
 and Q is to AT x EF as the sector cp7i to ca)i, and is to the 
 quantity that would issue at the orifice EF in the time Z (in 
 which a body would fall from K to C) as EF X ncp to AB x 
 cab. Let ch be taken on the asymptote equal to 2re, and hf 
 perpendicular to ch meet the iiyperbola iny; let mn produced 
 meet the asymptote inw, and nl be perpendicular to the asymp- 
 tote in /; then cl being always less than cu or Q.ck, it follows 
 that 7icp is always less than the sector ca/ or the hyperbolic lo- 
 garithm of 2 ; and that the difterence of the quantity of water 
 whichissues at theorificeEF, and that which would have issued 
 in the same time with the velocity A, is to what would issue 
 with the velocity A in the time Z in a ratio that is always less 
 
 than
 
 48 Of the Motion of Water issuing Book I. 
 
 than that of EF X caf to AB X cab, but that continually ap- 
 proaches to this ratio as its Umit. 
 
 5^0. In the two preceding articles we supposed the vesseltobe 
 kept ahvaysfull to the altitude AC (>?g.241),and the water to be 
 always supplied at the surface AB with the velocity V with 
 which the water in the vessel subsides. If we now suppose that 
 no water is supplied, but that the upper surface AB subsides 
 while the water issues at the orifice EF, let flC be the sdtitude 
 of the water at the beginning of the motion, AC its altitude 
 after any time, and let the ratio of e to 1 be that which is com- 
 pounded of the ratio of 2r to 1 and of AB — EF to EF. Lot 
 a series of continued proportionals be formed, of which Ca and 
 CA are the first terms, and CH be the term whose place in the 
 progression is denoted by e + 1 when e is any rational number, 
 or more generally let the logarithm of CH be to the logarithm 
 of CA as e is to 1, the moduhis being Ca ,• then the velocity of 
 the water issuing at EF will be such as would be acquired by 
 a descent that is to AH in the invariable ratio of AB* to 
 EF^X~T. For let AC be represented by H and its fluxion 
 by — h (which is negative because AC decreases), let D repre- 
 sent the descent by which X would be acquired and d its flux- 
 ion, then since — h is represented by Vt, gd by Xx (art. 
 434), V is to .r as EF to AB, and g is to f as AA to AA — XX ; 
 it follows that — h is to ^ as EF^ x AA to AB^ X A A — XX, 
 or as EF* X H to AB* x H — EF^- x e x D, so that Uh 
 
 AB* 
 
 X —J- is equal to eDh — Hd ; and hence by multiplying by 
 
 H """ and finding the fluents (by the converse of art. 99 and 
 168). EFx7:iTxD is equal to AB* x CA— CH or AB* 
 X CH — CA, according as e is greater or less than unit. But 
 when e is equal to unit, then D will be found to be to CA in 
 the compound ratio of AB* to EF*, and of the logarithm of 
 CA to the modulus Ca. When e exceeds unit the velocity of 
 the water is greatest when CA is to Ca as unit is to the num- 
 ber the logarithm of which is to the logarithm of e as unit to 
 e — 1, and the velocity is such as would be acquired by a de- 
 scent that is to AC as AB* to e x EF*. 
 
 541. Let
 
 Chap. XIT. from a CyUnchic Vessel, 49 
 
 .041. Let IG Cfg. 240, N. 1) perpendicular to the area EF 
 at G meet AB in W, and be to TH in the duplicate ratio of the 
 area AB to the area EF ; and let AMEFNB be such a catar- 
 act of water that any horizontal section of it as jSIN may be 
 always inversely in the subduplirate ratio of IR its di^.tance 
 from I. Tlien supposing, with Sir Isaac Nezcton, the water 
 around this cataract to be congealed, and the water to enter 
 sdwavs into the cataract in the surfa<;e AB with the velocity 
 tliat would be acquired l>y the descent IH, the water v. ill 
 descend in the form of this cataract the sections of which 
 diminish in the same proportion as the velocity of the de- 
 scending fluid increases, and will exert no pressure on tiie 
 ambient congealed part. Thus, the water in the vessel is 
 distinguished into two parts; the gravitation of the cataract 
 generates the increase of the motion of the water that descends 
 throuc-h every section, or theexcessof that with which it issues 
 at the orifice above what it had in entering the surface AB ; 
 while the gravitation of the ambient parts is what acts upon the 
 bottom of the vesseh The ratio of these two parts is that of 
 2EFtoAB — EF. For, since the section MNis inversely in the 
 subduplicate ratio of IR, the solid AMEFNB is equal to 2EF 
 X IG — 2AB X IH (as may be easily deduced from art. 307), 
 which is to 2EFxIG as AB — EF to AB, because III is to 
 IG as EF"- to AB^. The content of the cylinder is AB x HG, 
 
 or IG X — — ; consequently tiie content of the cataract 
 
 is to that of the cylinder- as 2EF to AB-j- EF. Supposing 
 therefore with Sir Isaac Nezctoi/, that the forces which gene- 
 rate the velocity X — V in the water that issues at EF and 
 that act upon the bottom of the vessel are the same when all 
 the water is fluid, the ratio of r to 1 will be that of AB -}- EF 
 to 2EF. And if we substitute this ratio for that of r to 1 in 
 the preceding articles, A the limit of the velocities with which 
 the water issues at EF (when the vessel is always kept full 
 to the height CA) will be suih as is acquired by the descent 
 KG, if KG be toi-AC as AB^ X eEF to EFxAB^— EF% 
 or to AC as AB" to AB^— EFS that is, if KC be equal to IG. 
 The time in which any velocity X (or ad) is acquired, and 
 the quantity of water that issues in that time, will be such 
 VOL.11. E as
 
 50 Of the Motion of Wdter issuing Book I. 
 
 as were determined in art. 538 and 339, abstracting from fric- 
 tion, the resistance of the air, and the elfect of the obhque 
 motions of the particles described by Sir Isaac Nercton, by 
 which this quantity is diiuiuished C/ig. 2-tl). If we substitute 
 this value for the ratio of r to 1 in art. .540, where the water 
 was not supposed to be supplied, we shall find e to 1 as AB* 
 — EF- to EF", or e + 1 to 1 as AB"- to EF" j and if the loga- 
 rithm of CH be to the logarithm of CA as AB^ — EF' to EF% 
 the modulus being Co, the velocity of the water issuing at EF 
 will be such as would be acquired by a descent that is to Ail 
 in the invariable ratio of AB' to AB"- — 2EF\ If we had 
 supposed that the action on all parts of the area CD is the 
 same, or that the Ibrce wiiich generates the velocity X-— V 
 in the water issuing at EF is to the action on the bottom of 
 the vessel (or 1 to r — 1) as the area EF to the area AB — 
 EF, or 1 to r as EF to AB, then KC would have been to 
 I AC as AB to AB— EF, and e to 1 as sABxAB— EF to 
 EF^ Cjig. 240, N. 1). We supposed that the forces which gene- 
 rate the motion X — Vin the water that issues at EF and that 
 act upon the bottom are in the same ratio when the water that is 
 ■without the cataract AM^sEFB is congealed, and when it is 
 fluid ; but there are several ditTerences betwixt the motion of 
 the water in these two cases. In the first the vein of water is 
 no more contracted after its exit than thefigure of the cataract 
 requires; whereas in the latter case if the water issue at EF 
 through a thin plate, the vein is immediately contracted after 
 itse.vit in consequence or the oblique motions of theparticles 
 converging towards the orifice; and the area of a horizontal 
 section of it at a little distance from the oriace is found to be less 
 than the orifice in the ratio of 1 to VI nearly when AB is 
 much greater than EF; una. the quantity of wat^r that issues at 
 EF is lound to be ne4rl3' the same that would have issued in 
 the same time if the ratio ofr to 1 had been that of AB to EF 
 according to the second h3'pothesis. If wc suppose that in this 
 case the quantity of water which issues at EF answers to the 
 second hypothesis, but that the velocity answers to the first 
 whenwc substitutethesectionof thevein of water afterit is con- 
 tracted for EF, then the area of the orifice EF will be to this 
 section of the vein of water in the subduplicate ratio of 
 
 AB'
 
 Chap. Xir. • from a Cylindric Vtssel. 51 
 
 AB^ -|- AB — EF"^ to AB% which is always less than the ratio of 
 -•2 to 1, but is very near rt when EF is very small compared 
 with AB, and is a ratio of equality when AB and EF are equal. 
 But when the water issues not ut EF through a very thin plate, 
 or when the vessel is not cylindric, the motion of the water and 
 form of the vein is different. See on this subject Princip. lib. 
 %p. 329, Edit. S. 
 
 542. When the water is supposed to be supplied in acylin^ 
 dcr, so as to stand always at tlie same altitude above the orifice, 
 there is an analogy between the acceleration of the motion of 
 the water that issues at the orifice and the acceleration of a bo- 
 dy that descends by its gravity in a medium which resists in the 
 duplicate ratio of the velocity of the body, that deserves to 
 be mentioned. Let g represent the force of gravity, R the re- 
 sistance of the medium when the velocity is X, and let R be to 
 g as XX to AA ; then g — R the force by which the motion 
 of the body is actually accelerated in its descent will be to g 
 as A A — XX to A A, and A will be the greatest velocity 
 which the descending body can acquire, or (to speak more ac- 
 curately) the limit of all its possible velocities, because if X 
 be supposed equal to A, R will be equal to g, and there can be 
 no further acceleration. The fluxions of the velocity X and 
 time T being represented by x and t, g — R will be expressed 
 
 by -> and twill be to -as AA to AA — XX. Hence if 
 
 the resistance be equal to the force of gravity when the velo- 
 city is equal to that which would be acquired by the descent 
 IG (or the limit of the velocities which the descending body 
 can acquire, and the limit of the velocities with which the water 
 issues at the orifice EF be equal), then T the time in which 
 the descending body acquires any velocity X will be toT the 
 time in which the water issuing at EF'acquires the same veloci- 
 ty in the invariable ratio of AB to EF ; because we found in 
 
 art. 537, that t was to - in the compound ratio of AA to 
 
 A A— XX and of EF to AB ; so that t is to < as AB to EF, 
 
 "and r to T in the same ratio. 
 
 543. In the same manner it appears, that if S be the space 
 described by the body while itdescends insuch a medium in any 
 
 E 2 time
 
 52 Of the Motion of Water iasmmr Book I. 
 
 time r, tlien the quantity of \vater that issues at the orifice EF 
 
 EF 
 
 in a time 2^^^ ^'iU he equal to a C3'lindric column on the 
 
 EF 
 
 base EF of aheight equal to S x— , For since the times in 
 
 which the body and the water acquire equal velocities are al- 
 ways in the invariable ratio of AB to EF, it follows that S 
 the space described by the body in the time T is to the height 
 of acolumn of water on the baseEF equal to the quantity tliat 
 
 . . . EF 
 
 issues at EF in the time T or 2^ x 7-5 in the same ratio. 
 
 AB 
 
 544. The same conclusions follow from the principles de- 
 scribed above in art. 50.5 and 531, which are applied in an in- 
 genious manner to this doctrine by ^Ir, Daniel BeruouiUi, 
 Comment. Acad.Petrop. torn. 1, who seems first to have deter-^ 
 mined rightly the manner in which the motion of v/ater issuing 
 from any vessel is accelerated, when we abstract from the impe^ 
 diments above mentioned. Supposing the surface AB of thefluid 
 to subside in the vessel, and the fluxion of the time being re- 
 presented by t, and that of the altitude AC by—// as formerly, 
 the fluxion of the square of the velocity of a body that de^ 
 scends freely in the vertical will be expressed by — *lgh, the 
 fluxion of the square of the velocity V with which the mass of 
 water contained in the vessel actually descends by — C/7j (art, 
 434), and since the particle of water which issues at the orifice 
 in the time t may be represented by AB X—h, if v/e suppose 
 ABxACx — 2gh -f ABxACx'2/A equal to ABx — h 
 j< XX — VV (in consequence of what was shown in art.oCo and 
 532), it will follow that 2AC X g^fi^ equal to XX — VV, 
 which is to XX as AB^—EF^ is\o AB\ Therefore if KG 
 be to AC as AB'^ to AB* — EF", and A be the velocity which 
 would be acquired by the descent KC (so that A A maybe to 
 SAC X o-inthe same ratio), then 2AC Xg— /will be to XX as 
 QAC y.g is to AA, and g—f to g as XX to AA ; which is 
 egreeabie to what we found in art. 537 and 541, iu a different 
 jaanner. And this is conformable to what was first taught by Sir 
 Isaac Newton, thai Lhougli llie pressure upon EF is to the prcssm-e 
 
 upon
 
 Chap. XTI. from a C^Undric Fessd. 53 
 
 \ipon the base CD^ before the orifice is opened, as the area EF 
 to the area CD; yet when we suppose the water to issue at 
 EF, and to have acquired its utmost velocity, the force that 
 generates tile velocity X — ^V in the water at EF is measured 
 by the gravity of tlie cataract AMEFNB, or by a column of 
 
 AB 
 
 the fluid of an altitude equal to 2HG X - ^^ , - ~ on a base 
 
 equal to the section of the vein of water after it is contracted ; 
 that is, the quantity of motion which is generated in the water 
 issuing at EF with that uniform velocity, is equal to the motion 
 which such a column of water would acquire by falling freely 
 with its gravity in an equal time. He has not enquired into 
 the manner in which the water is accelerated from the beginnin^^^ 
 of the motion; but if we represent the content of the cataract 
 AMEFNB by C, and suppose C x^— /equal to EF x X + X^ 
 the force which generates the velocity X — V in the water is- 
 suingat EF, then, becauseCisto 2EF X HG as AB to AB-f EF, 
 X — V to X as AB — EF to AB, and AA is supposed to be to 
 aACxo- as AB* to AB"— EF% it will follow, that XX is to 
 A.A as g—fis to g, as we found above. 
 
 545. The ratio of the action on the bottom of tlie vessel to 
 the force that generates the velocity X — ^V in the water issu- 
 ing at EF (or that of r — 1 to 1), which was deduced from the 
 cataract after Sir Isaac Netaton's method in art. 541, fol- 
 lows likewise from the principle described in art. 525 or 532. 
 Let P represent the first of these two forces, F the second, and 
 P + F will be equal to AB X AC x JH/ (by what was showa 
 in art. 5.37), which is equal to ii\.B X xx— vv or -^ AB X XX X 
 
 ^— ^~ — ^y '^^^^ ^"^'^^ deduced from that principle in the last 
 
 article. But F is equal to EF ^ X X x^ (by art 537), or 
 
 EF X XX X ^H. ; therefore P +P is to F (or r to l)as^AB 
 
 X ^^^toEFx ^-or as AB + EF to 2EF; and P 
 
 to F (or r — 1 to 1) as AB — EF to 2EF, which is the same 
 ratio that was deduced from the cataract in art. 541; and in cor. 
 2 and 5, prop. 3d, Princip. lib. 2, where tlie water is supposed 
 tQ have Hcq^uired its utmost velocity. 
 
 ' Ei 546. ft
 
 54 Of the Motion of JVatcr issuing Book L 
 
 546. It must be acknowledged, liowever, that the preceding 
 theory concerning the manner in which the watur issuing at EF 
 isacceleratcd from the beginningof the motion, is not to be con- 
 sidered as accurate in all respects, being founded on the hypothe- 
 sis, that all the particles of the fluid within the cylindric vessel 
 descend with the sam.e velocity V, and that the water issuing at 
 EF acquires the velocity X — V at once, which cannot be sup- 
 posed to hold accurately. The acceleration of V is similar to 
 thatof aheavy body descendingby its gravity in a medium that 
 resists in the duplicate ratio of the velocity (the relative gravity 
 of the body in the fluid being supposed equal to g) by what was 
 shown in art. 542. And as the fluxion of the velocity of such a 
 body is the same at the beginning of the descent, as if the body 
 fell freely by the gravity g; so when the orifice EF is opened in 
 the bottom of the vessel, if V or X be supposed to begin from no- 
 thing, AA — XX must be equal to AA at the beginning of the 
 motion, and consequently/" equal to g, so that the fluxion of V 
 must be then equal to the fluxion of the velocity with which 
 the water or any other body descends freely by its gravity. 
 From which it follows, that, according to this theory, the pres- 
 sure on the bottom of the vessel is wholly taken off at the in- 
 stant of time when the water begins to issue at EF; and as this 
 conclusion cannot be admitted, we may learn from this instance 
 that this theory is not to be considered as perfectly exact. It 
 ■will be worth while however to pursue this speculation a little 
 furlher, and to show how the method described in art. 537 and 
 541 may be applied for determining the motion of water issu- 
 ing from other vessels. 
 
 547 (Fig.'24:2). Suppose now the vessel to consist of two cylin- 
 ders a6c6?,ABCD;andletfl6thesection of theupper part be greater 
 thanAB. The velocity of-the water at EF being represented by X, 
 and the velocity ifi the vessel ABCD by V, as formerly, let its 
 velocity in abed be represented by Z, and the forces by which 
 y, Z, and X are accelerated by /', p, and F respectively. Let 
 the sections AB and ab be represented by B and C, the altitudes 
 AC and ac by b and c respectively, and the aperture EF by O; 
 let the surface ACDB continued upwards intersect the plane ab 
 in LM. Then the force that acts upon the surface CD corre- 
 sponding
 
 Chap, XII. from a Ct/Ihidric FesseL 55 
 
 sponding to that which is supposed (according to this method) 
 lo generate the velocity X — V in the water issuing at EF will 
 be expressed, as in art, 537, by ;0X xX — V, or (according to 
 the ratio or" /• to 1 that was deduced from the cataract in art. 
 
 641) by XX X —^ — . In hke manner the force which ge- 
 nerates the velocity V — Z at the surface AB is OX X V — Z, 
 or (because V is to Z as C to B, and V to X as O to B) by 
 
 XXx— 7T— x-r- > and if this force be increased in the ratio 
 of ai + AB to 2AB (according to art. 541), or of C + B to 
 2B, we shall have XX X 7~' x ^^ for the action on the 
 whole surface tv/ corresponding to that which generates the velo- 
 city V — Z in the water, while it passes from the upper into the 
 lower cylinder at the surfiicc A B. But because all the particles 
 ofthewater that arein the saniesectionofthevesselaresupposed 
 to descend with equal velocities in this theory, and to contribute 
 equally to the actionsof thcfluid, we aretodiminish thisforcein 
 the ratio of AB to ab, or of B to C, that we may have the part 
 
 of it XX X —^ — Xtttt which is to be ascribed to the column 
 
 2ti CO 
 
 LCDM, Therefore since the velocity of the water in ACDB is. 
 accelerated by the force/, and its velocity in LABM by the force 
 p, we are to suppose AC x AB X JH/ + AL x AB X JIlp,or Bb 
 
 EB— OO 
 
 X g—f + Be X g~f equal to XX x ^ + XX 
 
 X 
 
 CC— RB no T.^A^o . CC-00 
 
 2B ^CC 
 
 orXXB X ~~, that is TfT X g—bf—cp 
 equal to XX X -^- — 5 consequently if KC be to LC 
 (or b-{c) as ab^ to ab^—EF' or CC to CC— 00, and A de- 
 note the velocity that would be acquired by the descent KC, 
 XX will be to AA as Xp X g~bf—cp is to Tp X g, and A 
 will be the Umit of all the values of X. The velocities X, V,and 
 Z, and their respective fluxions are in an invariable ratio, so that 
 /' will be to F as v to x, or Y to X, or O to B ; and p will be to 
 V as Z to X or O to C. Therefore XX will be to A A as 
 . E 4 g—^
 
 56 Of the Motion of IVater issuing Book I . 
 
 g — F X -TTT. ^ J + c '■^ o ' ^^ ^^ ^^ '**^ represented by H_, and 
 i^+l2. by K, XX will be to AA as gH— FK to ^r^H ; con- 
 sequently if the fluxion of X be represented by .v, and ibe flux- 
 ion of the time by t, since x may be expre.^sed by ¥t, it fol- 
 lows that ? will be expressed by -^ x——^- Hence if the 
 
 velocity A be represented by ab (Fig. 240, n. 2), and any leaser 
 velocity X by ad, and the water be always supplied at the sur- 
 face ah with the velo<;ity Z, the time in which the water issuing 
 at EF will acquire the velocity X, will be to the time of descent 
 from K to C in the compound ratio of the hyperbolic sector 
 cati to the triangle cab and of K to H, If we had supposed r to 
 1 as the area C D to the area EF (which was Sir Isaac Nezoion's 
 hypothesis in the first edition of his Principia), then KC 
 ought to have been taken in the same ratio to ^LC as 1 — • 
 
 --f— X - ~ --'is to If and A being supposed equal to the 
 
 velocity that would be acquired by the descent KC, the con- 
 struction would have been in other respects the same. 
 
 o48. When ab the uppermost section of the vessel and the 
 area of the orifice EF with the altitude LC remain, the descent 
 KC and the velocity A are the same, without any regard to 
 the ratio of LA to AC. Hence if we suppose the water to 
 be continually supplied into a cylinder LCDM at the surface 
 LM, with a velocity that is less than V in any given ratio, let 
 this ratio be that of LC or AB to ab, and if KC be to LC as 
 ab^ to tf 6^ — EF% the utmost value of X will be the velocity 
 that is required by tbe descent K('. And if the water be sup- 
 posed to be always supplied at the surface LM, without hav- 
 ing any velocity communicated to it (but what it receives frorrs 
 the water beneath, which cannot descend without it), then KC 
 wjll be equal to LC ; and the utmost velocity of the water at 
 EF will be such as v/onld be acquired by the descent LC, the 
 altitude of the water in the vessel above the orifice EF. 
 
 ^9. Tf the cylinders fluffy, ABC D (figM4,S, N.2), communicate 
 with each other only by anaperturc (f in the plane AB, and we 
 
 abstract
 
 Chap. XII. frojn any Vessel. 57 
 
 abstract from any pressure upwards upon the lower side of the 
 plane AB, the motion of the water may be determined as in art. 
 547. Thcaction on the plane CDcorrerfpondin<2; to the force that 
 generates the velocity X — V at the aperture EF will be express- 
 
 cd as before by XX x \^^ ■ • If the aperture tf he represent- 
 ed by 0, and the velocity in efhy Y, the action on the surface 
 cd corresponding to that which generates the velocity Y — Z 
 in the water issuing at ej] will be found as above (by substitute 
 
 ins: ef or for AB) to be XX X — jrpr-x — ■> which beina: 
 diminished in the ratio of CD to ab or of B to C, gives XX x 
 -~- X — X B for the part of this action that is to be ascrib- 
 ed to the gravity of the column LCDM; and the sum of these 
 being supposed equal to BZ* X Jl^-f Be X]~^, we shall have 
 
 XX to ogH-oFK, as 1 is to 1 +22^22 _ 22; 
 
 and the descent by which the utmost velocity of the water at 
 the orifice EF would be acquired^ is to H in the same ratio • 
 
 from which it follows (because F is measured by-> that this 
 
 ratio being represented by that of 1 to t7i, the fluxion of the 
 time in which the water issuing at EF acquires the velocity X, 
 
 will be represented by ~ x—^—T' and that this time may 
 be determined by a construction similar to that in ai't. 538, 
 when the vessel is supposed to be kept always full to the alti- 
 tude LC. If O be very small compared with B and C, then 
 1 is to m as 00 to 00 + 00. And when ab is equal to AB, if 
 no water be supplied into the vessel, the velocity is determin- 
 ed by the construction in art. 540, by supposing e to represent- 
 
 BB— 00 BB— ,p(j 
 
 00 
 
 5 50. When the vessel consists of any numl>er of cy lindric or pris- 
 matic parts that have the areas B,C, D, ^c. (Jig. 'IAS) for their seve- 
 ral bases, and b,c,d,SfC. for their respective altitudes, then, by 
 proceeding as in art. 547, the forces that act at the respective 
 
 surfaces
 
 53 Of the Motion of Water issuing Book I. 
 
 surfaces B, C, D, <Src. corresponding to those that are supposed 
 in this method to generate the increase of the motion of the 
 
 water at each surface will be measured by XX X '7 ■ X 
 
 00 -,x^ CC— BB 00 x^^- DD— CC OO -, „,, 
 
 ^) XAX— 2^-Xgg> AAX — ^]7-Xcc' ^"^' ^ "^ P^^^^ 
 
 of these forces, which are to be ascribed to the gravity of 
 thecolumn which insists on the lowermost base B, areexpress- 
 A u W ^BB— 00 OOB ,.^^ CC— BB OOB ^^ 
 
 cd by XX X ^^-x — , XXx -j^ X -^, XX X 
 
 PD-CC OOB p ^1 f 1- u ■ w , B— OOB ._ 
 
 • ^^^^ X ~— » ^'C- the sum or which is XX x -^ -^^ if 
 
 She the uppci most section of the vessel. But supposing F, 
 f, p, &c. to represent the forces described in art. 547, the same 
 sum is equal to B6 X ~7+ ^^ ^ i—v + 4'^'- or (supposing K 
 equal to bx^+c x- + c?x^> 4-c.) to BHg— BKF. From 
 
 ^Yhich it follows that XX x - ,^^^^r— is equal to Hg — KF; 
 and that if A represent the velocity which would be acquired 
 
 SSH 
 
 by a descent equal to then XX will be to A A as 
 
 Hg — KF to Ho- ,• so that if the water be always supplied at 
 the surface S, with the same velocity with which it subsides at 
 S, when F is supposed to vanish, or the water at EF to have 
 acquired its utmost velocity, X is equal to A. The fluxion of 
 
 the time is expressed by — x -t-t — '^ where x represents the 
 
 fluxion of X ; and consequently the time is determined as in 
 art, 538, by hyperbolic areas or logarithms. \^ hen no water 
 is supposed to be supplied into the vessel, let D be the descent 
 by which X the velocity of the water at EF would be acquir- 
 ed, c? its fluxion, — h the fluxion of H the altitude of the 
 ^vater in the vessel above the orifice, then XX being equal to 
 2gD (art. 434), or Xx to gd, the velocity with which the sur- 
 face of the water subsides, or X X-r being expressed by -^> 
 
 F being
 
 Chap. XIT, Of the Catenaria. 5g 
 
 F being expressed by-^ or "f^— > and XX X ^~^^ equal 
 to H^'' — -KF^ by what has been shown, it follows that d, the 
 fluxion of D, is to — h the fluxion of H as H — D X^~^ — 
 
 to K X -r where S always denotes the area of the uppermost 
 
 surface of the water, O the area of the orifice, H the height 
 of the water in the vessel above O, D the descent by which 
 the velocity X would he acquired, and K is supposed equal to 
 the sum of the products w^hen the altitude of each part of the 
 vessel that contains water is multiplied by the ratio of tlie ori- 
 fice to the area of the section of that part. It easily appears 
 that the same conclusions take place when an erect vessel is 
 terminated by any curvilineal surface, supposing K to represent 
 the area of a figure, whose ordinate at any point of the axis 
 is to 1 as the area of the orifice is to the section of the vessel 
 at that point : and these agree with what is deduced by the 
 learned author above mentioned, from the principle described 
 in art. 59,5 and 532. When any sections of the vessel increase 
 from any part downwards towards the orifice, this theory sup- 
 poses that there is an action of the water from below upwards, 
 while it passes from narrower into larger parts of the vessel ; 
 and in this case the motion of the water does not seem to b^ 
 so justly determined by it; see art. 52?. Several other obser- 
 vations might be made on this doctrine, but our design obliges 
 ws to proceed now to other subjects. 
 
 55 1 . There are several other principles that relate to the 
 centre of gravity of bodies, besides these we have insisted on 
 hitherto, that are also of use in the resolution of problems. 
 When two powers sustain any body or figure that is supposed 
 to gravitate, a right line from its centre of gravity perpendi- 
 cular to the horizon passes through the intersection of the right 
 hnes in which these powers act, which with the gravity of the 
 figure are in the same ratio to one another as any three right 
 hnes constituting a triangle that are parallel to the respective 
 directions of these powers. Hence the nature of the figure is 
 discovered, which is assumed by a heavy chain or perfectly 
 
 flexible
 
 60 General Observations concerning Book I. 
 
 flexible line that is suspended from any two of its points. Let 
 FEU (/ig. 244) be such a line, F its lowermost point, where 
 the tuagent FT is parallel to the horizon, ED an ordinate from 
 E to the horizontal line AD, £T the tangent at E intersect- 
 ing FTinT, and G the centre of gravity of the portion FE of 
 the hne or chain. Then the three powers are, the gravity of 
 the chain which acts in the perpendicular to the horizon, 
 and the powers at F and E which retain those extremities of 
 the chain, by acting in the tangents FT and ET, and are equal 
 to the tension of the chain at those points. Therefore by this 
 principle the perpendicular from G to the horizon passes 
 through T; and if EI parallel to AD or FT meet TG in I, 
 the weight of the part of the chain FE will be to the tension 
 of the chain at F as IT to EI, or (by prop. 14) as the fluxion 
 of the ordinate DE to the flu>don of the base AD; consequently 
 the tangent of the angle lET, in which the curve intersects a 
 parallel to the horizon at any point E, is always as the weight 
 of the portion FE of the chain that is betwixt E and the lower- 
 most point F ; the tension of the chain at any point E, is to its 
 tension at F as ET to EI (by the same principles) or as the 
 fluxion of the curve to the fluxion of the base, and is as the 
 secant of the angle lET. We shall afterwards consider this 
 subject in a more general manner. When any body or 
 number of bodies connected together are suspended in any 
 manner, their common centre of gravity descends to as low 
 a place as possible ; and hence some problems have been re- 
 solved concerning the wa.rfwa and w//«"OTa ; but of these we are 
 to treat afterwards, and proceed now to some general observa- 
 tions on the subjects of the 10th and 11th chapters, whence 
 we shall endeavour to draw some general principles that may 
 be of use in resolving philosophical problems of various kind*. 
 552. It was observed above in art. 312, that the asymptote of 
 thebranch of acurve is considered as the tangent at itsinfinitely 
 distant extremity. In prop. 26,whilePdescribes thebranch that 
 approaches to the asymptote RX (fig. 1 17), let CP and SP 
 meet RX in m and n ; and when the revolving lines CP and 
 SP become parallel to one another and to RX, their angular 
 velocities will be in the ultimate ratio of the angles PCx and
 
 Chap. XIT. the Jfighs of Contact, Sec. 6l 
 
 TSy, or of CmR, and S/iR, and consequently in the ratio of 
 Cl{ to SR; so that SQ will be to CQ as CR k to SR, and 
 CR equal to SQ. And thus the demonstration of the 26tli pro- 
 position may be abridged, the use of which has been shown bj 
 many examples in chap, x. 
 
 553. The propositions in chap, xi, concerning the curvatm'C 
 of lines and its variation may be likewise briefly demonstrated 
 from the limi ts of ratios. LctTR (Jg. 149) parallel to EB uleet the 
 curve EMH in M, the circle ERB in R, and their common 
 tangent in T, as in prop. 32 ; then supposing £T to be con- 
 tinually diminished till it vanish, the ultimate ratio of TM to 
 Til w ill be the ratio of the curvature of the line EM at E to 
 the imiform curvature of the circle ERB; and the rays of cur- 
 vature will be in the inverse ratio. ^\ hen this is a ratio of 
 equalit}^, no circle can pass between EM and ER within the an- 
 gle of contact REM, and ERB is the circle of curvature at E, 
 Because TM, ET, and TK are supposed to be in continued pro- 
 propotion (art. 366), and when ET represents the fluxion of the 
 curve, TM ultimately measures one half of the second fluxioa 
 of the ordinate, and TK ultimately coincides with EB; it fol- 
 lows that the right lines which measure the second fluxion of 
 the ordinate and the first fluxion of the curve and a EB are in 
 continued proportion, aswas shown at greater length in prop.33« 
 When we speak of the ratio of a fluxion to a fluent, we always 
 understand the ratio of the right lines that represent them. 
 
 554. Angles of contact are in the ultimate ratio of their sub- 
 tenses, when the arches, or their tangents, are supposed to be 
 equal, and to be continually diminislied till they vanish, if 
 the subtenses are inclined in equal angles to those tangents. It 
 was shown in art, 369, that RM the subtense of the angle of 
 contact MER contained by the curve EM and circle of curva- 
 ture ER was as KQ directly, and the rectangle KTQ inverse- 
 ly, ET being given. Therefore when EB is the diameter of 
 the circle of curs^ature, and BV the tangent of BK is not paral- 
 lel to ET, the angle of contact MER is as the tangent of the 
 angle BVE directl}^ and the square of the ray of curvature in- 
 versely; and when the curvature at E is given, the index of the 
 variation of curvature (according to Sir Isaac Newton's ex- 
 plication)
 
 ^2 Ge.nernl Oh^ervnfibns concerning Book L 
 
 plication) is as the angle MER (f'S- 1'^-)- ^^"hen thecurveBK 
 touch(»> che circle BQ at B, it'C a iidO be the respective centres of 
 curvature of BQ and BKatB, then KQisa.sOCdirectly,an(l the 
 )-ectangle OBC inversel}', and the angle of contact jNIEK is as 
 OB directlv, and CB* inversely; and when the arches EM and 
 •£>» are similar in this case, the angle MER is to mer in the tri- 
 plicate ratio of E6 to EB. Tlie angle of contact, for example, 
 contained by the parabola and the circle ofcurvature at its ver- 
 tex is inversely as the cube of the parameter of the axis. AVhea 
 the contact of BK and BQ is of any order denoted by n, ac- 
 cording to the explication in art. SQQ, then the angle MER in 
 similar arches is inversely as the power of the ray ofcurvature 
 the exponent of which is n + 2. 
 
 555. The rest remaining, let MN (fig. 245) perpendicular to 
 the tangent at M, and M^ perpendicular to the chord EM meet 
 the ray of curvature FC in N and d respectively; then thelast 
 ratio of EN to theray of curvature EC and of JLd to 2EC will be 
 a ratio of equality. For Eg? is to TK as EM* to E'P, and the 
 excess of Ef/ above TK toTK as TM^ to ET^ or MTK, that is, 
 asTM to TK; consequently Ed always exceeds TK by TM, 
 which excess vanishes with £T when TK coincides with EB. 
 The fluxion of Ed is equal to the fluxion of TK when M sets 
 out from E, and may serve for measuring the variation of cur- 
 vature at E, by art. 369 and 386. 
 
 556. Any arch bein^ given, the centre of its curvature is the 
 limit of the intersections of right lines that bisect perpendicu- 
 larly the sides of the rectilineal inscribed or circumscribed 
 figures when the arch (with those figures) is continually dimi- 
 nished till it vanish; and is also the limit of the intersections of 
 right lines that bisect the angles of those figures. But the in- 
 tersection of right lines perpendicular to those sides at their ex- 
 tremities will not coincide ultimately with the centre of cur- 
 
 " vature(ji4'o-.246). Lei ?/!' bisect an}' chord M^;6 perpendicularly in w, 
 and meet therayof turvature EC in r, thenCwillbethelimitof 
 all the situations of the point ;■ when the arch E^^w is supposed 
 to he diminished till it vanish ; but if ws perpendicular to Mm 
 at m meet EC in S, the ultimate ratio of ES to EC will be the 
 same with the ultimate ratio of Ewe to EM + I Mw; so that if 
 
 E//I
 
 Chap. XII. Angles of Contact, Sec. ©3 
 
 Em be to EM as m to n, the ultimate ratio of ES to EC will 
 be that of 9,m to Q.m — //, 
 
 557- SupposingasaboveET(^^o-.o4.5)tobetheiangcnlof EM 
 atEjTM thesubtense of theangleof contactparalleito EB,TK 
 
 ET* 
 
 to be always equal to — — > and FK the locus of the point K 
 
 to intersect EB in B ; it is manifest that when ET is supposed 
 to be continually diminished and at length to vanish, TK then 
 coincides with EB; and this seems to be sufficient to justify the 
 expression, when it is said that EB is the ultimate value of 
 
 FT* 
 
 —r- which is supposed to be always equal to TK. But if it 
 
 should be objected, that when ET vanishes TM likewise va- 
 nishes, the ratio of ET to TM is not assignable, and the value 
 
 of -— must therefore be then inconceiv.able or imaginaiy. 
 
 In answ'er to this we may observe first, that nothing is more 
 usual in Geometry than to determine the points of one figune 
 from those in another b}' a construction or equation, as in this 
 case any point K in FKB from the corresponding point M in 
 
 IIME by supposing TK always equal to t^vt' that the point 
 
 in the former v.hich corresponds to E in the latter can be no 
 other than B where the locus FKB intersects EB ; that EB must 
 
 ET* 
 
 either be allowed to be the ultimate value of — — -» or we must 
 
 1 M 
 
 ET* 
 only say that ^r- is equal to TK with the single exception of 
 
 the case when T falls on E : and as it has not been usual, or 
 thought necessary, to require so scrupulous an exactness, so it 
 seems unreasonable to find fault with the inventor of this me- 
 thod for making use of a convenient and conciseexpression that 
 is not liable to more exceptions than such as were allowed be- 
 fore his time. When EMH is an arch of a semicircle described 
 upon the diameter EB, FKB is an arch of the same semicircle, 
 
 ET* 
 
 and — T- is generally allowed to be always equal to TK or 
 
 EB — TM without excepting any particular case; from which 
 
 it
 
 64 Gcnernl Ohbervnt'inm coiicerning Book I. 
 
 k would follow that since Eli — TM becomes equal to EB 
 when ET and TM vanish, the ultimate value of =tt-t- is EB. 
 
 But there is no necessity for making use of exceptionable ex- 
 pressions in any part of Geometry ; and the same author lias 
 shown us how to avoid them in this case. For we may consider 
 EB as the ultimate value of TK, but only as the limit of the 
 
 values of r;;Tr when ET is continually diminished till it vanish : 
 
 and such a limit may be understood to be alwaj^s meant by 
 what is called the ultimate value of a quantity that is deter- 
 mined in this manner from others that vanish totrethcr. There 
 can be no flexure or curvature in a point, and the curvature at 
 
 E has indeed a dependence on the values of ~- when ET and 
 
 TMare real.butin so far only as the value of their limit EB has 
 a dependence on those values: for it was shown in prop.32,that 
 in order to determine the curvature at E (as it was defined in 
 art. 364), it is sufficient to ascertain the distance EB. This is 
 no more than one of those problems that frequently occur, the 
 determining the intersection of a curve with a right line given 
 in position ; and it is, generally speaking, more easy to determine 
 the point B than the intersection of FKB with any other paral- 
 lel TK. 
 
 558. WhenS(/?of.G45)is any given point in EB,IetSjVJ meet 
 the tangent ET in /, and IM will be to TM as S/ to SE, ^vhich 
 is ultimately a ratio of equality; consequently the ultimate 
 
 FT* ■ ET* 
 
 value of ~- is the same as oi -7^- and is equal to EB ; the 
 
 . , p EM* EM* 
 same is to be said ot -— or ^.-^y* 
 
 559. Thetangcntsof thcevoh;taoCI(Xfi^. 180), intercepted by 
 AEM give a con venien t scale of the rays of curvature of the latter. 
 And if these rays CE, QAI be divided in Z, z, so that CZ be al- 
 ways to EZ in the same given ratio of m to n, and the tangent 
 of the locus of Z meet ET perpendicular to CE in t, the va- 
 riation of curvature at E will be always as - X tangent E^ Z. 
 
 For
 
 Chap. XII. centripetal Forces, &c. 65 
 
 For let an arch Zx described from the centre C meet QM in Xy 
 and the last ratio of EM to Zx will be that of EC: to ZC; and 
 because Zar is ultimately equal to CQ + CZ — Q2, and 
 Qz — CZ is to QM — CE (or CQ) as CZ to CE, the last 
 ratio of 0^ to CQ is that of EZ to EC. Therefore the lust ra- 
 tio of xz to Zx is that of CQ X EZ to EM x ZC, or of n x CQ, 
 to VI X EM; consequently the last ratio of CQ to EM, or of 
 the fluxion of the ray of curvature CE to the fluxion of the 
 curve A E (which ratio measures the variation of curvature), is 
 that of m X xz to n x Zx, or of m x EZ to 71 x Et, or of 
 
 - X tang. E^Z to the radius. It is easy to show, from art. 384, 
 
 that in all figures wherein the sine of the angle contained by 
 the ordinate and curve is as a power of the ordinate whose ex- 
 ponent is any number r (as for example in the cycloid, catena- 
 ria, clastic curve, &c.), the ray of curvature EC always meets 
 the base at Z so that EZ is to EC in the invariable ratio of 
 1 to r; consequently the base being the locus of Z, the varia- 
 tion of curvature in such figures is as x co-tang, of the an- 
 gle contained by the ordinate and curve. 
 
 560. When EMH(/7a'.247)isdescribedb3''agravity that acts at 
 E in the direction EB,letEKbe the space that would bedescribed 
 by a body falling from E in the right line EB by the gravity at 
 E continued uniformly in the same time that the tangent ET 
 would be described by the motion in the trajectory at E; then 
 this time being given, the gravity at E will be measured by 
 SEK, because a force ismeasured by the motion which it %vould 
 generate in a given time, and a space 2EK would be described 
 by the motion acquired at K in the time that EK would be de- 
 cribcd by the body descending from E to K, by art. 9.5. But 
 when ET is continually diminished till it vanish, the ultimate 
 ratio of TM to EK is a ratio of equality,- and the velocity in 
 the trajectory being measured by ET, the gravity at E will be 
 in the ultimate ratio of 2TM. It is usual in enquiries of this 
 nature first to consider =:^ie.itr(6tion as uniform in the chords mE, 
 EM inscribed in the figure, or, in its tangents, and to conceive 
 the gravity to be applied at once at the angle E. T.iet RM pa- 
 
 VOL. II. E raild
 
 66 General Observations concerning Book 1. 
 
 rallel to EB meet the chord wiE produced in R and the tangent 
 at E in T, then the ultimate ratio of KM to2TM will be a ra- 
 tio of equality^ and the gravity at E will be in the ultimate 
 ratio of RM or 2TM, whether it be conceived to act at once at 
 E (as mprop. 30, lib. 5, Princip. Edit. 3), or to act continually, 
 the velocity at E being in the ultimate ratio of ET or EM. 
 Let EM the side of the inscribed figure.be bisected in L, and 
 the angle ELc? being supposed equal to MTE, let Ld meet EB 
 in d, and the triangles MTE, ELf/ being similar, Ed will ul- 
 timately coincide with JLb half the chord of curvature EB; and 
 the ultimate ratio of the rectangle R]M x Eb to EM^ will be a 
 ratio of equality; or the rectangle contained by half the chord 
 of curvature and the right line which measures the gravity 
 equal to the square of that which measures the velocity at E, as 
 in art. 464. 
 
 56 1. In like manner if we suppose wEM to be an}' arch of a 
 perfectly flexible line or chain, n to denote the section of that 
 chain at E perpendicular to its length, EK the accelerating 
 force of the gravity at E, then EK x « x EM will express the 
 absolute gravity of an uniform chain equal in length to EM of 
 a base equal to n that is acted upon by the force EK; and this 
 is ultimately equal to the absolute gravity of the portion EM 
 of the chain ; consequently the tension of wEM at E is mea- 
 
 sured by the ultimate value of EK x w x EM X ttt-, orofEK 
 
 •' KM 
 
 X « X Eb, and is equal to the weight of a chain equal in length 
 to E6of the same thickness '»vith AEB at E that is acted upon 
 by an uniform gravity equal to EK. 
 
 562. Let E(;^'g. 248) by any point in ILarightlincgiven in po- 
 sition, A a given point that is not in I L,join AE, and let ACperpen- 
 dicular to AE meet IL in C. Then if we suppose the point E to 
 move in IL, butC to remain, AE and CE will flow proportion- 
 ally ; that is, the fluxion of AE will be to the fluxion of CE as 
 AE to CE. For let AK be perpendicular to IL in K, and the 
 fluxion of AE will be to the fluxion of KE as KE to AE (b}^ 
 prop. 15), or as AE to CE; and the fluxion of CE is equal to 
 the fluxion of KE when C is supposed to remain tixed. When 
 the point A is taken any where upon an arcli described from the 
 
 centre
 
 Chap. XII. centripetal Forces, &C; 67 
 
 centre C, and AE the tangent of this arch at A meets the dia- 
 meter XL given in position in E, then the point A being sup- 
 posed to remain if"E move in the right hne CE, the fluxion of 
 AE will be always to the fluxion of CE as AE toCE. The 
 converse of which is, that when the fluxion of AE is always 
 to the fluxion of CE as AE is to CE, the point A being taken 
 any where on the circular arch, and E being supposed to move 
 in CE, then C is the centre of the arch. In general let the 
 fluxion of AE be always to the fluxion of cE as AE is to cE, 
 the points A and c being supposed to remain ; and if while the, 
 point A approaches to the right line IL till it coincide with it, 
 ihe point c approach to C as the limit of its various positions, 
 then is C the centre of the curvature of the line upon which A 
 is supposed to move at that point of it where A falls upon IL. 
 
 563. These observations lead us to some general propositions 
 relating to philosophical enquiries, which we shall represent in 
 one view, that the analogy which is between them may the 
 better appear. The first gives the property of the trajectories 
 that are described by any centripetal forces how variable so- 
 ever, these forces or their directions may be : the second gives 
 a like general property of the lines of swiftest descent : the 
 third gives the property of the lines that are described in less 
 time than any other of an equal perimeter: and the fourth gives 
 the property of the figure that'4s assumed by a flexibfe line or 
 chain in consequence of any such forces acting upon it. Let 
 AEB (fig. 249) be an arch of any of those lines, HE//, a right 
 line in the direction of the power EK that results from the 
 composition of the several forces that are supposed to act at 
 E, and let a perpendicular from O, the centre of curvature 
 at E, meet HE in C. 
 
 I. The velocity in the trajectory at E is equal to that which 
 would be acquired by a descent equal to -i- CE by an uniform 
 gravity equal to EK the force which acts at E. And if we sup- 
 pose a body to set out from E in the right line HEA with a ve- 
 locity equal to that in the trajectory at E, and its motion to be 
 accelerated or retarded by the same powers that act at E, then 
 ils velocity and distance from C will increase proportionally j 
 that is, the fluxion of the right line V, which represents its ve- 
 locity, will be to the fluxion of its distance from C as Vis to 
 
 F 2 the
 
 6d General Observations concerning Book I. 
 
 the distance CE. Or, in other words, if EN the ordinate of 
 the figure HNG measure its velocity at any point as E of H/t, 
 and NT the tangent of HNG at N meet Wh in T, the subtan- 
 gent ET will be equal to EC on the opposite side of E. 
 
 II. The velocity in the line of swiftest descent AEB at E is 
 equal to that which would be acquired by an uniform gravity 
 equal to EK, the force that acts at E, by a descent equal to 
 t CE. The curvature of this line at E is equal to the curva- 
 ture of the trajectory that would be described by a body pro- 
 jected from E in the direction of the tangent of AEB with the 
 velocity acquired in AEB at E, and that is acted upon by the 
 same force EK. And in this case likewise V and CE flow pro- 
 portionally ; or ET the subtangent of the figure HNG and EC 
 half the chord of curvature coincide with one another. 
 
 III. Whenthesumor difference of the time in which the line 
 AEB is described, and of the time in which it would be de- 
 scribed by an uniform motion with a given velocity is a ?/i2«/wi/w, 
 the line AEB will then be described in less time than any line 
 of an equal perimeter that has the same extremities A and B. 
 And it is a property of such lines that if a body set out from 
 E with the velocity u acquired at E in EH or E/a, the fluxion 
 of u will be to the fluxion of its distance from C in the com- 
 pound ratio of ?/ to CE, and of the sum or difference of 6 and 
 u to a, b and a being supposed to represent invariable velo- 
 cities. By principles analogous to this, the nature of the line 
 that is described in less time than any line that includes the 
 same area AEB with the chord AB in any hypothesis of gra- 
 vity may be discovered, and other problems of this kind con- 
 cerning isoperimetrical figures resolved. 
 
 IV. When AEB is a flexible line or chain, its tension at E 
 is equal to the weight of a chain that is in length equal to CE, 
 of an uniform thickness equal to that of AEB at E, and that is 
 acted upon by an uniform gravity equal to EK the force that 
 results from the composition of the several powers that act at E. 
 Let A be a given point in the chain AEB, Aa equal to one half 
 of the chord of the circle of curvature at A, that is in the direc- 
 tion of the force which acts on the chain at A. Let E/cbealwaya 
 to EK the force that acts at any point E as the section of the 
 
 chain
 
 Chap. XII. centripetal Forces^ &c. 69 
 
 chain at E to its section at A, and the direction of the force 
 'Ek be opposite to that of EK ; then if a body set out from A 
 with a just velocity {viz. that which would be acquired by a 
 descent equal to a A, by an uniform gravity equal to the force 
 that acts at A), and while it is made to move along the curve 
 AEB, its motion be always accelerated or retarded by the forces 
 represented by E/c, the tension of the chain at any point E will 
 be always in the duplicate ratio of the velocity acquired at E ; 
 which is the same velocity that would be acquired by the de- 
 scent CE with an uniform gravity equal to the force EA;. And 
 if abody be projected from E with this velocity in the direction 
 of the tangent of AEB, the curvature at E of the trajectory 
 that would be described by the force JLk will be one half of 
 the curvature of the chain at E. 
 
 564. The first of these follows easily from what was shown 
 above in art. 464 or 560. For the fluxion of the velocity EN 
 being in the compound ratio of the force EK and of the flux- 
 ion of the time, whiqh is as the fluxion of the distance CE 
 directly (the point C being supposed to remain fixed), and the 
 velocity inversely, it follows that the fluxion of EN is" to the 
 fluxion of CE as EK is to EN, or as EN to EC ; but the flux- 
 ion of EN is to the fluxion of CE as EN is to ET ; conse- 
 quently CE is equal to ET. But having insisted at length on 
 this subject in the last chapter, we have mentioned this theo- 
 rem here for the sake of its analogy to the rest only. 
 
 065. Let A and B (Jig. 2.50) be two given points, XL a 
 right line that bisects AB perpendicularly in K ; and it is 
 manifest, that if a body is to move from A to B in the least 
 time with a given uniform 'avotion, it must describe the right 
 line AB ; and if it is to move from A to the right line IL in 
 the shortest time, it must describe the perpendicular AK. 
 But E being an}' point upon ILJoin AE and BE; and if we 
 now suppose that the body is to describe AEB with an uni- 
 form motion, but with a velocity that is always as CE, the 
 distance of E from C a point given upon IL, then the motion 
 will not be performed in the least time when E falls upon K, 
 but when AE is perpendicular to AC. For let KR parallel to 
 AE meet AC in R, and the time in which any hne AE is 
 
 F 3 described
 
 70 Of the Figure of a Line or Chain Book I, 
 
 described will be always directly as AE, and inversely as the 
 velocity or CE; that is, the time will be as KR, since KR is 
 to KC as AE to EC, and KC is given; but KR is least 
 when it is perpendicular to AC ; consequently AE is described 
 in the least time when AE is perpendicular to AC. It fol- 
 lows, conversely, that if AE or AEB be described in the least 
 time, and the velocity be as the distance of E from some point 
 upon IL, that point must be C, where AC perpendicular to 
 AE intersects IL. And this with art. 562, suggests the gene- 
 ral property of the curvature of the lines of swiftest descent, 
 that if IL meet this line in E, and the velocity in IL be as the 
 distance from C,or, more generally, if (the point C remaining) 
 when CE increases or decreases the velocity at E begin to 
 flow in the same proportion as CE, then the ilexure of the 
 line of swiftest descent at E must be such as to have the centre 
 of its curvature in C. In this investigation of the curvature 
 of the line of swiftest descent, we conceive AE and EB not 
 to be the whole chords that form the rectilineal figure inscrib- 
 ed in it (or the whole tangents that form the circumscribed 
 figures), but their halves only, and any two such successive 
 parts to be described uniformly with the velocity pertaining to 
 their intersection E, which is ultimately the mean velocity in 
 the arch, and tiie centre of curvature to be determined by the 
 ultimate intereection of the perpendiculars AC, BC with each 
 other, or with IL that bisects the angle AEB, according to art. 
 562. But the nature of the line of swiftest descent may be 
 discovered more easily than from this property, when the gra-r 
 vity acts in parallel lines, or is directed towards a given cen- 
 tre ; and that this theory may beset in a clear light, we shall 
 treat of it and the higher problems concerning the maxima 
 and minima in a separate chapter, 
 
 5G6. The first part of the fourth theorem, that was proposed in 
 art. 363,has been already demonstrated in art. 5(i],viz. that the 
 tension of the line or chain AEB Cfg. 249), at any point E, is 
 equal to the weight of a chain of the same thickness with AEB 
 at E that is in length equal to EC and is acted upon by an 
 uniform gravity equal to EK, and consequentl}' is measured by 
 the rectangle kEC. As to the latter part, let A-;- be perpendi- 
 cular to the tangent of AEB at E in /',• let V be a right line 
 
 determined
 
 Chap. XII. that is acted on hi/ any Powers. 7^ 
 
 determined from the forces Ek, as in art 435, so as to repre- 
 sent the velocity which the body is siH)posed to acquire at E, 
 while it moves along AEB in the manner described in the 
 theorem ; then because Er is the force by which that velocity 
 is accelerated or retarded, the rectangle contained by Er and 
 the fluxion of the curve AE will measure the fluxion of |VV. 
 But because EA; is in the compound ratio of the force EK and 
 thickness of the chain at E, Er is the force by which the ten- 
 sion of the chain increases from the point E, and the rectangle 
 contained by Er and the fluxion of the curve AE will measure 
 the fluxion of the tension at E or of the rectangle ^'EC. There- 
 fore since -iVV is supposed equal to the rectangle /cEC when 
 the point E falls upon A, they will be always equal to each 
 other. Let E^" meet in Q the circle of the same curvature at E 
 with the trajectory described by the force E/c, when the body 
 is projected from E with the velocity V in the direction Er,- 
 and by what has been shown above, if EQ be bisected in b, 
 the rectangle b^k will be equal to VV, and consequently to 
 GCExE/i. Therefore E6 is equal to 2EC, or the curvature 
 of the trajectory at E is one half of the curvature of the chain 
 at E. 
 
 567. When EK (Jig. 251) is either a centripetal or 
 centrifugal force that is directed towards a given point 
 S, or from it, take SM upon SA always equal to SE, let 
 the ordinate MN of the figure a JNM be always equal to E/c, 
 and if the area aAgd be equal to 4.VV w^hen the body sets out 
 from A, or measure the tension at A, the ajea ad^^l will al- 
 w\iys measure |-VV or the tension at any point E. And if SP 
 be perpendicular to the tangent of the cattnaria AEB at E, 
 this perpendicular SP will be always inversely as the area 
 «c?NM, or inversely as the leusion at E, or inversely as VV the 
 square of the velocity acquired at E. For, since the fluxion 
 of SE is to the fluxion of the curve AE as Er is to EA', it fol- 
 lows that the fluxion of iVV is equal to the rectangle con- 
 tained by E/c and the fluxion of SE ; so that the fluxion of V 
 is to the fluxion of SE as E/c is to V or ^V to EC. But the 
 fluxion of SE is to the fluxion of SP as EC is to SP, by art. 
 384, consequently, the fluxion of V is to the fluxion of SP as 
 V to S P ; and since SP decreases while V increases, it fol- 
 
 F 4 lows
 
 7'j; Of the Figure of a Line or Chat n Book L 
 
 lows that SP is inversely as VV or arfNM. Hence an analogy 
 appears between these figures and the trajectories described by 
 centripetal or centrifugal forces : in these, SP the perpendicular 
 from the centre S upon the tangentof the trajectory is inversely 
 as the velocity of the body that describes it ; whereas in those, 
 SP IS inversely as the square of V the velocity of the body that 
 moves along the curve, when the direction of the force is 
 changed according to the fourth theorem in art. 5(13. If the 
 forceEK beinvariable,forexample,andthechain isof an uniform 
 thickness, and if Sfl vanish (that is, if the tension at A be equal 
 to the weight of a chain of the same thickness ^rith AEB at A, 
 equal in length to SA, and that is acted upon by an uniform 
 gravity equal to the force EK), SP is inversely as SE, and con- 
 sequently AEB is an equilateral hyperbola, as Mr. Herman ob- 
 serves. But AEB is not always such an hyperbola, when the 
 force towards S is uniform, as this learned author seemed to 
 think. For when the tension at A is different, SP is inversely 
 as flM or SE + Sa. In like manner when the chain is of an 
 uniform thickness, and EK is a centrifugal force that is in- 
 versely as the square of the distance from S, and the tension at 
 A is equal to the weight of a chain of the same thickness equal 
 in length to SA and that is acted upon by an uniform gravity 
 equal to the force at A, AEB is an aich of a logarithmic spiral. 
 When the force EK is centrifugal and inversely as the cube of 
 the distance from S, AEB in a similar case is an arch of a semi- 
 circle described upon the diameter SA; and when EKis as some 
 other power of the distance, AEB is in similar cases one of 
 those figures that Avere constructed in art. 392 or 395. 
 
 568. When the forceEK (Jig. 252) acts in parallel lines, the 
 sine of the angle DEP contained by the curve and the ordinate 
 that is in the direction of the force is inversely as VV, or the 
 area adNM, or the tension at £. If the force E^ be uniform, 
 let ED in the direction of the forces meet ad in D, and the 
 sine of DEP being inversely as aM or DE, DP perpendicular to 
 EP the tangent at E will be invariable. In this figure the rect- 
 angle kEC is equal to aM x EA", and EC equal to ayi or DE; 
 and the ray of curvature EO being to EC (or DE) as DE to 
 DP, EO is inversely as the square of the sine of the angle DEP. 
 
 5C)y. When
 
 HateXXMU/;. ;-.\oa\.
 
 PJaUXXMIl : ,_l.Vil 
 
 .r - S 
 
 
 1 : 
 
 A 
 
 r 1 
 
 J
 
 Chap. XII. that is acted on by any Powers. 73 
 
 569. When the force EK is perpendicular to the ciirvCj it 
 has no effect on V, so that V is in this case constant, and the 
 tension is the same in a!i parts of the figure, and EC (which in 
 this case is the ray of curvature) is inversely as the force that 
 acts at E. This force in the velaria (according to Mr. Ber- 
 nouiUi) is as the square of the sine of the angle in which the 
 ordinate intersects the curve ; so that the ray of curvature 
 must be inversely as that square, and the velaria must coincide 
 with the common catemiria, by what was observed at the end 
 of the last article. In the elastic curve the force EK is as the 
 ordinate, and the ray of curvature is inversely as the ordinate. 
 570. When a power EI (fig. 2j3) always perpendicular to 
 the curve, and a centripetal or centrifugal force EL alwaj's 
 directed towards or from a given centre S, act at once upon a 
 line or chain AEB of an uniform thickness, the former has no 
 effect upon V ; and if LZ be perpendicular to the ray of cur- 
 vature in Z, and SM Leing equal to SE, the ordinate MN be 
 always equal to EL, as in art. 567, the rectangle contained by 
 the ray of curvature EO and Ei-f-EZ will be always equal to 
 the area ^(iNjM. For complete the parallelogram ELKI, let 
 KR be perpendicular to OE in R, and the rectangle REO will 
 be equal to tho rectangle KECor iWwhich is equal toflJNM; 
 and since IR is equal to EZ, it follows, that «£?NM is equal to 
 the rectangle contained by EO and the sum or difference of 
 EI and EZ. It is manifest that EZ is to EL the force that 
 acts in the right line ES, as SP the perpendicular from S on 
 the tangent at E is to SE; and that when EL acts in parallel 
 lines, EZ is to EL as the fluxion of the base is to the fluxion of 
 the curve; in which last ci>sc this theorem agrees with what is 
 shown comment, petropol. torn. 3. The property of the ray of 
 curvature being thus discovered, the nature of the figure may 
 in some cases be defined by first fluxions, orby acommon ecjua- 
 tion, by a proper application of the inverse method of flux- 
 ions. The problems in art. .563, considered in a general man- 
 ner, depend on the curvature of lines; and therefore the gene- 
 ral solution involves the ray of curvature, or something equi- 
 valent. But there are often particular principles which serve 
 for resolving more easily particular cases of those problems, of 
 
 which
 
 74 Of the Lines of swiftest Descent, Book I. 
 
 which we gave instances in art. 441 and 551 (where the solu- 
 tion agrees with that in art. 568)^ and we shall have occasion 
 to give other instances in the following chapter relating to the 
 Jines of swiftest descent. 
 
 CHAP. XIII. 
 
 Wherein the Nature of the Lines of swiftest Descent is deter- 
 mined in any Hypothesis of Gravity, and the Problems con- 
 cerning isoperimcirical Figures, xcith others of the same Kind, 
 are resolved byfrst Fluxions, and the Solutions verified hy 
 synthetic Demonstrations. 
 
 571. JLT was shown in chap. ix. how the greatest and least 
 ordinates of figures are readily determined by the method of 
 fluxions^ where the usual rules with the corrections that are 
 necessary to render them accurate and general were demon- 
 strated. But there are problems concerning the maxima and 
 minima which are of a higher nature, that cannot be immedi- 
 ately reduced to these. It was known long ago that of all 
 equal areas the circle has the least circumference, and of all 
 equal solids the sphere is bounded by the least surface. But the 
 first problem of this kind that required a more subtile investiga- 
 tion, seems to have been resolved by Sir Isaac Newton, Schol. 
 prop. 34, lib. 2, Princip. where he gives the property of the 
 figure, that by revolving on its axis generates the solid of least 
 resistance. Afterwards Mr. Bernouilli found, that the cycloid 
 was the line of swiftest descent in the common hypothesis of 
 gravity, and determined the nature of this line in several other 
 cases and under various restrictions. The analysis of the ge^ 
 neral problem concerning figures, that amongst all those of the 
 same perimeter produce maxima and minima was given by jNIr. 
 James Bernouilli, from computations that involve second and 
 third fluxions, by resolving the element of the curve into three 
 infinitely small parts.* And several enquiries of this nature 
 
 * AnaJif^is magni .problcmaiis isopcrimetriti . Acta erud. Lips, 1701,/. 213, ^ seqq 
 
 have
 
 Chap. XIII. when Graviti/ acts in parallel Lines. 75 
 
 have been since prosecuted in like manner, but not always with 
 equal success. In pursuit of our principal design in this treatise, 
 of vindicating this doctrine from the imputation of uncertain- 
 ty or obscurity, we shall endeavour to illustrate this subject, 
 which is commonly considered as one of its most abstruse parts, 
 by proposing the resolution and composition of these problems, 
 and to determine the properties of the lines of swiftest descent 
 (whether gravity be supposed to act in parallel lines, or to be 
 directed to a given centre, and whether the perimeter of the 
 figure be supposed to be a determinate quantity, or other limi- 
 tations of this kind be added or not), and of the isoperimetri- 
 cal figares that produce other maxima and minima immediately 
 by first fluxions, without resolving the elements of the curve 
 into two or more parts, and in such a manner as may suoo-est 
 a synthetic demonstration that may serve to verify the solution. 
 The whole might be contained in a few general propositions ; 
 but it may be useful in this, as in the preceding chapters, to 
 begin with the more simple cases, and to proceed from them 
 to such as are more complex. We shall therefore first suppose 
 the gravity to act in parallel lines. 
 
 572. The following /cwwrt is to be premised. LeiKL(yig.254) 
 
 bearightlinegiveninposition,AKaperpendicularuponKLfrom 
 a given point A, E any point in this line, join AE, suppose KE 
 to be described uniformly with any given velocity a, and AE 
 to be described uniformly with any given velocit}'^ it tiiat is less 
 than a : let L be taken upon the right line KL, so that Ah 
 may be to KL as a is to a; and the difference of the times in 
 which the right lines AE and KE will be described by the 
 
 respective velocities u and a (or — — ) will be least when 
 
 }L falls upon L ; that is, when the angle KAE is such, that 
 its sine is to the radius as u is to a. For let KH and EV 
 be perpendicular on AL in H and V respectively, and AR be 
 taken upon AL equal to AE ; then HV will be to KE as KL 
 is to AL, or (by the construction) as u to a; consequently 
 HV will be described with the velocity u, in the same time 
 that KE is described with the velocity a. Therefore the ex- 
 cess of the time in which AE is described with the velocity m, 
 
 above
 
 76 Of the Lhies of szciffest Descent, Book I. 
 
 above the time in which KE is described with the velocity a, 
 
 is equal to the time in which AE HV, or AR HV, or 
 
 AH + VR is described with the velocity u; and because AH 
 is invariable, this time is least when VR vanishes; that is, when 
 E fails upon L, and the sine ot" the angle KAE is to the radius 
 as u is to a. The same appears from art. G42, according to 
 
 which — ^^ — is a minimum when its fluxion vanishes, that 
 
 tt a 
 
 is (because u and a arc supposed to be invariable), when u is to 
 a as the fluxion of AE to the fluxion of KE, or (by art. 193) 
 as KE is to AE, that is, when E falls upon L. 
 
 573. It follows, that if A-/, KL be any two parallel lines, 
 e any point upon Id, and E an}'^ point upon KL, the right line 
 f E be described with the velocity u, and eb being perpendicu- 
 lar to KLin b, bE be described with the velocity a, the difler- 
 ence of the times in which eE and bE arc thus described will 
 be least when the sine of the angle Ee6 is to the radius as n is 
 to a ; and that when it is required that this difference should 
 be a minimum, the angle Eeb does not depend on the magni- 
 tude of tb, but on the ratio of u to a only. 
 
 574. The gravity being supposed to act in parallel lines, sup- 
 pose FED (//£;•. Q.56) to be the line of swiftest descent from the 
 pointFto any given vertical HD. LetAEbeany archof this line 
 (the point E being supposed to be lower than A), KE a parallel 
 through E to the horizontal line FH, and AK perpendicular 
 to KE. Then the excess of the time of descent in the arch 
 AE above the time in which KE would be described uniformly 
 by the motion acquired at D is always a minimum, AK being 
 given. Let Ac and Ae be any other lines drawn from A to any 
 points in KE on either side of E; let the time of descent in AE 
 be expressed by T . AE; and in like manner let the times of 
 descent in Ae and Ae, and the times in which KE, Ke, and Ke 
 would be described by the motion acquired at D, be expressed 
 by prefixing T to each; then I say that T . AE— T . KE 
 will be less than T . At-— T . Ke, or T . Ae— T . Ke. To 
 demonstrate this, Ave are first to observe, that no point of the 
 line FED betwixt F and D can be lower than D; for let FiD 
 be any line that has a point z betwixt F and D lower than D, 
 
 and
 
 Chap. XIII. when Gravitif ads in parallel Lines, 77 
 
 and let zr parallel to FH meet HD in r, then zr will be de- 
 scribed in less time than ^D, and Fzr in less time than TrD^sa 
 that F2I) cannot be the line of swiftest descent from the point 
 F to the vertical HD. This being premised, let e be any point 
 betwixt K and £, and e any point on the other side of E; let 
 ed and ed be lines equal and similar to ED and similarly situat- 
 ed, so that cE maj'^ be equal to d\}, and Ee to Dd. Then 
 by the supposition the time of descent along AED is less tliau 
 the time of descent along AedJ}, and by substracting the equal 
 times of descent along ED and ed, it follows tluitT . AE is 
 less than T . Ae + T . did, or T . Ae + T . eJL, or T . At. 
 + T . KE — T . Ke. Therefore T . AE — T . KE is less 
 than T . Ac — T . Ke. Let ed meet HD in any point x, 
 and since D is the lowermost point of FED, d must be the 
 lowermost point of ed, and x nmst be above D. By the supposi- 
 tion the time of descent along AED is less than the time along 
 Ae.r, and the time in Dd being less than the time in xd, it fol- 
 lows that the time of descent along AEDd is less than along 
 Aed, and by subducting the equal times along ED and ed, it 
 follows that T . AE + T . Dd is less than T . Ae ; that is, 
 T . AE + T . Ee, or T . AE + T . Ke — T . KE, is less 
 than T . Ae. Therefore T . AE — KE is less than T . 
 Ae — T.Ke. 
 
 575. This property of the line of swiftest descent suggests 
 immediately the nature of the figure. Let AT the tangent of 
 this hue at A meet Ke in T, let the velocity acquired at A be 
 called u, and the velocity acquired at D be called a. It is 
 manifest that when AK is continually diminished till it vanish, 
 the ultimate ratio of the time of descent along A E to the time 
 in which AT would be described \vith the velocity u is a ratio 
 of equality ; and that the ultimate ratio of KE to KT, or of 
 the times in which KE and KT would be described with the 
 velocity a, is likewise a ratio of equality. Therefore, since the 
 excess of the time of descent along AE above the time in which 
 KE would be described with the velocity a is ahvays a mini- 
 mum, it follows that the difference of the times in which 
 AT and KT would be described with the respective velocities 
 u and « is a minimum, AK being given . Therefore by art. 572, 
 
 the
 
 78 Of the Lines of swiftest Descent, Book I. 
 
 the -sine of llie angle KAT is to the radius as It is to a ; and if 
 Art be the ordinate from A to FH, the fluxion of the base Ya 
 will be always to the fluxion of the curve FA as the velocity 
 at A to the velocit3'at D. And this is Xhe anal ij sin of the pro- 
 blem when the gravity acts in parallel lines. It is obvious, 
 that the line of swiftest descent from F to the vertical line HD 
 is likewise the line of swiftest descent from F to D, or betwixt 
 any two points of FED. Because u becomes equal to a when 
 E comes to D, the cuivc is therefore perpendicular to HD at D. 
 57G. It will now be eas}- to show by a syn thetic demonstration, 
 thattheline which has this property is thelineofswiltcstdescent^ 
 Suppase that FAEBD Cfig. 9,50) is a line of such a nature that 
 the sine of the angle contained by it at any point E and EQ, 
 the ordinate perpendicular to the horizon, is always as the velo- 
 city of the body that descends along it at E ; let AEB be an 
 arch of this line, Aa and B6 ordinate? perpendicular to the ho- 
 rizontal line FH, AP, and Bp parallel to FH, AT the tangent 
 at A, TEf the tangent at E, ^B the tangent at B, and TB, tr 
 parallel to FH. It appears from art. 573, that if AT, It, and 
 iB be described uniformly with the respective velocities that are 
 acquired at the points of contact A, E, and B, the excess of the 
 time in AT^B above the time in which ab would be described 
 with any given velocity a greater than that which is acquired 
 at B, will be less than if the poiiits T, t, and B were taken any 
 where else upon the parallelsTB, tr, and Bp. LetTR and tr 
 meet the curve in 0^ and h, and Sgf, \hv the tangents atg and 
 h meet AT, Tt, and tQ in S,y^ V, and r respectively ; and AS, 
 S/,/V, Vv, vB be described with the respective velocities that 
 are acquired at the respective points of contact A, g, E, A, B, 
 then the excess of the time in which the circumscribed figure 
 AS/"VuB is thus described above the time in which ah would 
 be described by the given velocity a, will be less than if the 
 points S,/, V, Vf and B were taken any where else upon the 
 right lines SX, fx, VZ, vz, and Bj) parallel to FH, by the 
 same article. By increasing in this manner the sides of the cir- 
 cmnscribed figure, and supposing each side to be described al- 
 ways with the velocity acquired at its contact with the curve, 
 the time in which the circumscribed figure would be thus de- 
 scribed
 
 Chap. XIII. when Gravity acts in parallel Lines. 79 
 scribed will approach continually to the time of descent in the 
 arch AEB, and the ultimate ratio of those times will be a ra- 
 tio of equality; and consequently the excess of the time of de- 
 scent along AEB, above the time in which ab would be de- 
 scribed with the given velocity a will be a minimum, the point 
 A with the distance between theparallelsAPand Bpbeinggiven, 
 and the velocities being always the same in the same horizontal 
 lines. Therefore since ab is given when the points A and B 
 are given, and the velocity a is given, the time in which ab 
 would be described with this velocity is given; consequently 
 AEB will be described in less time than any other line AeB 
 that passes through A and B. It appears easily that FED per- 
 pendicular to HD is the line of swiftest descent from F to HD. 
 577. When the gravity is uniform, the velocitj'at ECJig. ^56,N. 
 2) is in the subduplicateratioof the ordinate QE; sothat(bywhat 
 hasbeen shown) the fluxion of the base FQis tothefluxion of FE 
 the line of swiftest descent in the subduplicate ratio of QE to 
 HD; and this being the property of a cycloid that has FH for 
 its base, and HD for its axis, the cycloid is therefore the line 
 of swiftest desccntinthe common hypothesis of gravity. When 
 the gravity is as the power of the distance from FH whose ex- 
 ponent is any number n, and the body is supposed to descend 
 from FH, or to descend from any point A m ith the velocity 
 that would be acquired by the descent a A, then the velocity at 
 E is as the power of QE whose exponent is |- w + 4 ; and the 
 fluxion of the base FQ is to the fluxion of FEtlie line of swift- 
 est descent, as that power of QE is to the same power of HD. 
 In those cases, if EI always perpendicular to the curve meet 
 FH in I, the motion of the point I in the right line FH will 
 be uniform while the body descends along the curve, and may 
 serve to measure the time of descent. The velocity of I in the 
 right line FH will be to the velocity acquired at D, the lower- 
 most point of the line of swiftest descent, as the difference be- 
 hvixt 1 and w is to 2; and the time in which HI would be de- 
 scribed uniformly with the motion acquired at D, will be to 
 the time of descent in ED in the same ratio. Let FQ be sup- 
 posed to flow uniformly, then (by the property of the line of 
 Bwiitest descent) the fluxion of FE will be inversely as the ve- 
 locity
 
 80 Of the Lines of swiftest Descent, Book T. 
 
 locity at E, or the power of EQ whose exponent is i w+ \, 
 and (by art, 16?) the second fluxion of FE will be to the 
 fluxion of EQ in the compound ratio ofyi + 1 to 2, and of the 
 fluxion of FE to EQ; consequently (art. 384) if CE be the ray 
 of curvature at E, and CA^ be perpendicular to EQ in k, ¥.k will 
 be to EQ as (J to ;/ + 1, and CI to CE as the difference of 
 1 and It to 2. But the fluxion of HI is to the fluxion of the 
 curve DE in the ratio compounded ol that of CI to CE_, and 
 of tlie ratio of EI to EQ, or of the velocity at JJ to the veloci- 
 ty at E. Therefore when FH is described with tlie velocity 
 acquired at D, the fluxion of the time in Vi is to the fluxion of 
 the time in the line of swiftest descent as CI to CE, or the dif- 
 ference of 1 and « to 2; and the time in which the right line 
 IH is described by a motion equal to that which is acquired at 
 D, is to the time of descent along the arch ED in the line of 
 swiftest descent in the same ratio. This theorem is not extend- 
 ed to the case wherein n is equal to unit; in which AED is an 
 arch of a circle, and the point I has no motion. What was 
 shown in art. 407, concerning the motion of a body that de- 
 scends jdong a cycloid in the common hypothesis of gravity is 
 a particular case of this theorem. 
 
 578. In order to discover the nature of the line of swiftest 
 descent, when the gravity is directed towards a given centre, 
 the following lemma will be of the same use as that in art. 572 
 wasintheprecedingcase. LetAIandKL(//^.257) be circles de- 
 scribed from the same centre S ; and, the point A being given upon 
 AI, let E be any point upon KL, and SE meet Al in M; join 
 AE, suppose AE to be described with any given velocity repre- 
 sented by u, the arch AM to be described with a given veloci- 
 ty represented by h, and the diflference of the times in which 
 
 AE and AM will be thus described will be least (or — ~~~ 
 
 will be a minimum), when the sine of the angle SAE is to the 
 radius as u is to h, if the ratio of u to b be less than that of 
 SK to SA. Let SH be to SA as u is to h, and SE meet the 
 circle HNA described from the centre S in N, then HN will 
 be to AM as SH to SA or u to b; so that HN will be describ- 
 ed with the velocity u in the same time that AM is described 
 
 with
 
 Chap. XKI. zchen Gravity tends tozcards a Centre. 81 
 
 with the velocity b. Therefore the difference of the times in 
 which AE is described with the velocity u and AM with the 
 velocity b, is equal to the time in which AE — HN is de- 
 scribed with the velocity u, and is least when AE — HN is 
 least. Let AP the tangent of the circle HN/i from A meet the 
 arch KEA: in L, and Sp be perpendicular to AE in p, then the 
 fluxion of AE will be to the fluxion of KE as Sp to SE or SK; 
 hut the fluxion of KE is to the fluxion of HN as SK to SH ; 
 consequently the fluxion of AE is to the fluxion of HN as S/> 
 is to SHj and the fluxion of AE — HN is to the fluxion of 
 HN as Sp — SH or S/? — SP to SP. Therefore KE and HN 
 being supposed to increase uniformly, AE — HN decreases 
 till E come to L, where its fluxion vanishes (because Sp becomes 
 then equal SP), and thereafter it increases till AE become a 
 
 ^j^ AM 
 
 tangent to KE/c,- consequently AE — HN, or — -j-, is a 
 
 minimum when E falls upon L, in which case the sine of SAE 
 is to the radius as SP or SH to SA, that is, as u to b. Though 
 this be sufficient for our present purpose, it may be worth while 
 to observe, that if AP produced meet the circle KLA; in I, 
 AE — HN is a maximum when E comes to /, SH being less 
 than SK ; but that when SH is equal to SK, AE — HN 
 (which in this case is AE — KE) never becomes a minimum 
 or maximum, though its fluxion vanishes when AE becomes a 
 tangent of KE/c: and this is an instance of what was shown in 
 art. 261, concerning the inaccuracy of the common rule for de- 
 termining a maximum or minimum, and the correction that is 
 requisite to render it general. For the arch HN being supposed 
 to flow uniformly, the fluxion of AE — HNis as Sp — SH, 
 and it is easy to see that the fluxion of Sp — SH or the se- 
 cond fluxion of AE — HN vanishes in this case as well as its 
 first fluxion, but that its third fluxion does not vanish. 
 
 579- In the same manner when e is taken upon kl a circle de- 
 scribed from the centre S with a radius S/c greater than SA, 
 
 and Se intersects the arch AI in in, ~ -r- is a miiiimum when. 
 
 ' u b 
 
 the sine of the angle SAe or SAE is to the radius as u to h; 
 VOL. H, G conse^
 
 82 Of the Lines of swiftest Descent, Book T. 
 
 consequently — -7- is likewise a minimum in ihia case^ <E 
 
 being any right line terminated by the circles KL, kl. 
 
 580. The gravity being directed towards the centre S, let 
 FED be the line of the swiftest descent from F to any vertical 
 line SDH, AE any arch of FED, and let the rays SA, SE meet 
 the circle DLK described from the centre S in K and L. Then 
 EM the difference of SA and SE being given, the excess of the 
 time of descent in AE the arch of the line of swiftest descent 
 above the time in which the circular arch KL would be de- 
 cribed with the velocity acquired at D will be a minimum. 
 Let /rE an arch described from the centre S meet SA in A*, let Ac 
 and Ae be any lines drawn from A to this arch, let the times 
 of descent in A E, Ae, and Ac be represented by T . AE,T . Ae, and 
 TAe, and drawing Sf, Se that meet DK in / and 1, let the times 
 in which KL, K/, and Kl would be described by the velocity 
 acquired at D be represented by T . KL, T . K/, and T . Kl, I say 
 that T . AE — T. KL will be less than T. Ae — TK/ or T . Ae 
 — T.Kl, It is manifest that D must be the lowermost point 
 of FED ; for let FzD be a line that has its lowermost point at z, 
 and zr be perpendicular to SD in r, then because zr would be 
 described in less time than zD and Fcr in less time than FzD, 
 I2D cannot be the hne of swiftest descent from F to the ver- 
 tical SIL Lete^ and ed be equal and similar to ED and situ- 
 ated similarly to the rays Se, Se as ED is to SE ; so that Id and 
 Id may be each equal to LD, and IL equal to f/D, and LI to 
 l)d. Let e be betwixt k and E, and e on the other side of E, 
 then the time of descent in AED being less tiuui in Aef/Dby the 
 supposition, and the times of descent in ed and ED equal, it 
 follows that T . AE is less than T . Ae + T . Dr/ or T . Ae + 
 T . KL — T . K/; therefore T . AE — T . KL is less than 
 '1' . Ae — T . K/. Let ed meet SII in x and the time of descent 
 in AED being less than the time of descent in Aej: by the sup- 
 position, and the time in which Dd is described by the motion 
 acquired at D less than the time of descent in jrd, it follows thai; 
 the time of descent in AEDd is less than in Aed, and by sub- 
 ducting the equal times in ED and ed, it appears thatT. AE + 
 T.LM (orT.Ll, or T.K/— T.KL) is less than T.Ae; 
 
 conse-
 
 Chap* Xlir. zohen Gravitif tends totDards a Centre. .83 
 consequently T . AE — T . K L is less than T . Ae — T . Kl. 
 Therefore T.AE — T. KLis a minimum when AE is any 
 arch of the line of swiftest descent, and EM or SA — SE is 
 given. 
 
 58 1 . The nature of the line of swiftest descent, when the 
 gravity is directed to a given centre, is easily discovered from 
 this property, by art. 578. For since T . AE — T. KL is a 
 minimum, it follows that if SE meet the circle AMP described 
 from the centre S in M, and we suppose the arch AM to be de- 
 scribed with a velocity b which is to the velocity a acquired 
 at D as SA is to SD or SK, the time in which AM is thus de- 
 scribed will be equal to the time in which KLis described wdth 
 the velocity a; and if the time in which AM is thus described 
 be expressed by T . AM, then T . AE — T . AM will be hke- 
 wise a minimum, EM the difference of SA and SE being given. 
 Therefore (art. 578) it is the nature of the hne of swiftest 
 descent in this case, that if AT be the tangent at A, the sine of 
 the angle SAT will be to the radius as the velocity at A is to 
 the velocity b, w hich is itself supposed to be to the velocity 
 acquired at D as SA to SD. That is, the sine of the angle 
 SAT, in which any ray SA intersects the curve at A in the 
 line of swiftest descent is always to the radius in the ratio com- 
 pounded of the direct ratio of the velocity acquired at A to the 
 velocity at D, and of the inverse ratio of the distance SA to the 
 xlistance SD. And this is the analysis of the problem when the 
 gravity is directed towards a given centre. 
 
 58G. LetFED (Jig. '2,58) benovvalineof suchanature thatthe 
 sine of the angle contained by the curve FE and ray SE is to the 
 radius in the compound ratio of the velocity at E to the veloci- 
 ty at D and of SD to SE. Let AEB be an arch of this line; 
 let AT, Tt, and ^B be the tangents at A, E, and B: let AM, 
 KT, rt, and^B the circles described from the centre S through 
 A, T^ t, and B, and SA, ST, S^, and SB meet the circle Da 
 described from the centre S in o, m, n, and b. Let the tangents 
 AT, T^, and ^B be described with the velocities acquired at the 
 respective points of contact A, E, and B; and the excess of th^ 
 time in which AT^B will be thus described above the time in 
 ■which the circular arch ab would be described with the velo- 
 
 G 2 city
 
 84 Of the Lines of szcif test Descent. Book f. 
 
 •city acquired at T), will be less than if the points T and t were 
 taken any where else upon the arches RT and rt. For let tlie 
 velocities acquired at A and D be called u and a, and Z' be to a 
 as SA to SD ; then since the sine of the angle SAT is to the ra- 
 dius asM is to b, it follows that if AM be described with the 
 velocity b, then T . AT — T.AM will be a minimum, TM 
 being given, by art. 578 ; but AM and am are described by 
 those velocities b and a in equal times; consequently T . AT — 
 T . am is likewise a minimum. In the same manner T . TE? — 
 T . mn and T . ^B — T . nb are minima by art. 579 ; the dif- 
 ferences of the rays ST and St, and of S^ and ST?, being given. 
 Thus, by proceeding as in art. 570, it will appear synthetically 
 that the excess of the time of descent in the arch AEB above 
 the time in which ab is described with the motion acquired at 
 D is a minimum, SA — SB being given. Therefore AED is 
 described in less timethan any other line that passes through A 
 and D, the velocities being supposed equal at A, and conse- 
 quently at all other equal distances from S. 
 
 583. What C^g. 259) we have shown concerning the lines of 
 swiftestdescentwillbe found toagreewith the secondgeneral prin- 
 ciple described above in art. 563, and may be deduced from it. 
 For let PN the ordinate of the figure HNG alwa3-s represent 
 the velocity at P, or at E, SE being always equal to SP; let 
 SX be perpendicular to the tangent at E in X, and SX will be 
 to PN as SD to DG, by the last article ; consequently since the 
 fluxion of SX will be to the fluxion of PN as SX to PN and to 
 the fluxion ofSP as SX to PT, if Ec? be one half of the chord 
 of the circle of curvature which passes through S, ^d will" be 
 equal to PT by art. S84, as it ought to be according to the se- 
 cond general principle in ai-t. 563. And from this propert}' of 
 thoselinesit may be demonstrated synthetically that AEB is not 
 only the line of swiftest descent liom A to B, but from any 
 point in Aa the ray of curvature at A to any point in Bi the 
 ray of curvature at B, providing the curve HNG be concave 
 towards HD. It may be worth while to describe this method, 
 though it is not applicable when HNG is convex towards HD;' 
 not only for the further illustration of this subject, but because 
 • ^ • .-it
 
 Chap. "XIII. Of the Lines of swift dst Desciiit. S^ 
 
 it may be applied to other cases than these we have considered 
 hitherto. 
 
 584. Let AEB (fg. 0,60) be a line that is described by tire 
 evolution of aCb, as in art. 402, let M be any point in the ray 
 CE, and if the velocity that would be acquired at M be to the 
 velocity at E always in a less ratio than CM to CE, then AEB 
 is not only the line of swiftest descent from A to B, but from 
 any point in Aa to any point in Bb. For let the velocity at M 
 be to the velocity at E as any line CZ less than CM is to CE, 
 KML any line bounded by Aa and Bb in K and L, and /MA 
 a line described through M by the evolution of the same curve 
 aCb, so as to be always perpendicular to the ray CE while 
 aCb is evolved. Then the fluxion of the time in AE will be 
 to the fluxion of the time infM in the ratio compounded of 
 that of the fluxion of AE to the fluxion of/M(orof CEtoCM), 
 and of the ratio of the velocity at M to the velocity at E, or 
 of CZ to CE^ that is in the ratio of CZ to CM ; consequently 
 the fluxion of the time in AE is always less than the fluxion 
 of the time in/M, which is itself never greater than the flux- 
 ion of the time in KM ; because CE is supposed to be always 
 perpendicular iofM, and the fluxion of/M never can exceed 
 the fluxion of KM. Therefore, the fluxion of the time in AE 
 is always less than the fluxion of the lime in KM, so that the 
 time in AE must be less than the time in KM, and the time in 
 AEB less than the time in KML. 
 
 585. Supposing the gravity to be directed to the centre S, 
 and to be always of the same force at the same distance from it, 
 let SP be taken upon any given ray SH always equal to SE ; 
 and the velocity at any distance SE,or SP, being- such as would 
 be acquired by a descent from the distance SH,let PN the or- 
 dinate of tlie figure HNG represent this velocity; and NT the 
 tangent of HNG at N meet SH in T. Let Ed be taken upon 
 SE produced from E always equal to the subtangent PT, and 
 f/C perpendicular to S(/raeet EC, a perpendicular to the curve 
 AEB in C. Then, if the point C determined in this manner, be 
 always the centre of the curvature of AEB at E, and HNG be 
 concave towards HD, AEB will be the line of swiftest descent 
 from any point in the right line Aa to any point in Bb. For 
 let M be any point upon CE different from E, IMf an arch of 
 
 G r* a circle:
 
 85 Of the Lines ofszoiftest Descent. Book I, 
 
 a circle from the centre S meet SH in p and SE in l,pn tlicor^ 
 dinate dXp meet HNG in n and >\T in z, and /x perpendicular 
 to S/meet CE in x. Then the velocity at M will be to the ve- 
 locity at E Q.%pn to PN, and consecjuentiy in a less ratio than 
 vz to PN, orTp to TP, or dl to dliL, or Co: to CE, and there-r 
 fore {Cx being less than CM) in a less ratio than CM to CE. 
 From which it follows (by the last article) that AEB is describ- 
 ed in less time than anyother line drawn from any pointin Aa to 
 any point in B6, the velocities being supposed equal at equal 
 distances from S, or being such as would be acquired by a dcr 
 cent from the same distance from the centre S. The same de-r 
 monstration takes place when HNG is a right line, because ia 
 that case Cx is still less than CM. It is not applicable when 
 HNG is convex towards HD, nor is AEB in this case the line of 
 swiftest descent from Aa to B6 ; but it appears from art. 583, 
 that it is the line of swiftest descent from the point A to the 
 point B. It is obvious that the same demonstration takes place 
 when the gravity acts in parallel lines and HNG is concave 
 towards H D, by substituting a horizontal line in place of the 
 arch p^ll: but it is not applicable when HNG is a right hue, 
 in which case an arch of a circle is the line of swiftest descent 
 from A to B, but a similar concentric arch is described in the 
 same time with AEB, as may be easily demonstrated. 
 
 586. The figures constructed in art. 392 and 393, which are 
 the trajectories and catenaries (by art. 436, 437:, 438, and 569), 
 in some of the most simple cases when the centripetal or cen- 
 trifugal force is inversely as a power of the distance from the 
 centre, of an exponent greater or less than unit, are likewise the 
 lines of swiftest descent in certain analogous cases. When a 
 centripetal force is inversely as a power of the distance /( greater 
 than unit, and the velocity at any point F is to the velocity in 
 a circle at the distance SF in the subduplicate ratio of 2 to n 
 ' — 1, the line of swiftest descent from F to the vertical SDH is 
 the same that was constructed in art. 392, from the right line 
 AM Cfg' 171) by taking always the angle ASL to ASM as 2 
 to »+ 1, and SL to SA as the power of SM of the exponent 
 
 —j-r is to the same power of SM. For in these figures the sine of 
 
 the angle SAT is inversely as the power of SA of the expo- 
 nent
 
 Chap. XII r. Of the Linen of szciftcst Dcsce?if. 87 
 
 ncnt {n-\-l, and therefore in the compound ratio of the direct 
 ratio of the velocity (which is inversely as the power of SA 
 of the exponent ^n — |-) and the inverse ratio of the distance ; 
 consequently, these are the lines of swiftest descent in this case, 
 by art. 582. For example, v.hen the force is inversely as the 
 cube of the distance (or n is equal to 3), and a body descends 
 from any point F with a velocity equal to the velocity in a 
 circle at the distance SF, the line of swiftest descent AEB is 
 an arch of an equilateral hyperbola that has its centre in S. 
 When )i is equal to 2, and the velocity at F to the velocity in 
 a circle at the distance SF as \/ 2 to 1, the figure is construct- 
 ed by taking ASL equal to l-ASM and SL equal to the first of 
 two mean proportionals betwixt SM and SA. When a cen- 
 trifugal force acts upon the body tliat is inversely as a power 
 of SA of an exponent w less than an unit, and the velocity at 
 any point F is to the velocity in a circle described at the dis- 
 tance SF by a centripetal force equal to the centrifugal force 
 at F in the subduplicate ratio of 2 to 1 — n, AEB is likewise 
 one of the figures constructed in art. 392, by taking ASL to 
 ASM as 2 to 1 — w, and SL to SA as the power of SM of the 
 
 2 
 
 exponent y~ ^o the same power of SA. For example, when 
 
 the centrifugal force is constant, and the velocity at F to the 
 velocity in the circle at the distance SF as 'v/2 to 1, the line of 
 swiftest descent is an arch of a parabola that has lis focus m 
 S. W^hen the centrifugal force is as the distance from S, and 
 the velocity at F equal to the velocity in the circle, the line of 
 swiftest descent from one given point to another is an arch of 
 a circle or of a logarithmic spiral. When the centrifugal 
 force is as a power of the distance of the exponent n greatei 
 than unit, and the velocity at F is to the velocity in a circle 
 described at the distance SF with a centripetal force equal 
 to the centrifugal force at F as 2 is to w-j-l, the line of 
 swiftest descent is one of those constructed in art. 393 C/ig. 172) 
 from the circle, by taking ASL to ASM as 2 to u — 1 and SL 
 
 to SA as the power of SM of the exponent — r to the same 
 
 power of SA, And, in this case according as ?i is equal to 3, 3, 
 
 G 4 01
 
 g8 Of the Lines ofszcifiest Descent Book I. 
 
 or 5, AEB is an arch of an epicycloid described by a point in 
 the circumference of a circle that revolves on an equal circle, 
 or it is an arch of a semicircle, or of the lemniscata. It is obvi- 
 ous that these figures satisfy likewise the problem, when a body 
 wHich is acted upon by a force diliused from S that is as any 
 power of the distance, moves from a givfen point A to a given 
 point E with a given velocity, and it is required that the sum 
 of the actions shall be aminimum. For example, if the power 
 be inversely as the square of the distance, AE is an arch of a 
 semicircle described through the three points S, A, and E ; if it 
 be inversely as the pov/er of the distance of the exponent m 
 greater than unit, then AMS being a semicircle let the angle 
 ASL (Jig. 172) be to ASM as 1 to wi— 1, and SL to SAas the 
 
 power of SM of the exponent —— to the same power of SA. 
 
 587. To conclude this subject with an instance which may 
 show the extensive use of the second general theorem in art. 
 563, and how it may serve for finding the curvature of the line 
 of swiftest descent when the gravitation tends to several cen- 
 tres, let equal and uniform forces tend towards C and S as in 
 art. 491. Let the velocity CJig. 217) be such as would be ac-. 
 quired by a descent from F, E any point in the line of swiftest 
 descent, upon CE produced take Em equal to the excess of 
 CF + SF above CE + SE, and uz perpendicular to Cm shall in- 
 tersect the right line Ez, that bisects the angles CES, in z, so 
 that E2 shall be one fourth part of that chord of the circle of 
 the same curvature with the line of swiftest descent at E which 
 bisects the angle CES. 
 
 588. Suppose now that it is required to find the nature of the 
 line that of all those that pass through two given points A and 
 B (jig.QoG), and have equal perimeters, is described in the least 
 time. Let the time in which AEB is described In' a body that 
 descends along it b}' its gravity be expressed by T . AEB, and 
 the time in which the same line would be described uniformly 
 with any given velocity a by ^ . AEB, and let the ratio oi' in 
 to n be any given ratio ; then if - xT . AEB+" t . AEB be 
 
 a minimum, AEB will be described in less time than any other 
 
 line
 
 Chap. XIII. amongst those of the same Perimeter. 89 
 
 line AeB of an equal perimeter. For letT . AeB represent the 
 time of the descent along x\eB, t, AeB the time in which AeB 
 would be described uniformly with the same given velocity a ; 
 
 then by the supposition - X T . AEB+ ^, AEB is less than 
 
 ^ X T . AeB T t . AeB. But t . AEB is equal to t . AeB, these 
 being the times in which equal lines are described by equa^ 
 uniform motions. Therefore T . AEB is less than T . AeB, that 
 is, AEB is described in less time than any other line of an 
 equal perimeter that passes through A and B. It is obviousthat 
 
 - X T. AEB + t , AEB cannot be a minimum, when AEB is 
 greater than the line which is described in less time than any 
 other line whatever that passes through A and B ; and that 
 
 - X T . AEB — t, AEB cannot be a minimum^ when AEB is 
 
 n 
 
 less than that line. 
 
 589. Let the velocity at any point E be represented by m, 
 ^nd ^ be to a as m is to n ; let V be to m as a is to b -J- u, and 
 if AEB be the line that is described in less time than any other 
 whatsoever that passes through A and B^ by a velocity which 
 at any point E is represented by V, the same line AEB will be 
 described in less thne than any other of an equal perimeter that 
 passes through A and B, by a velocity which at any point E is 
 represented by u : for the fluxion of T . AEB is expressed by 
 the ratio of the fluxion of AE to u, the fluxion of t , AEB by 
 the ratio of the fluxion of AE to a ; consequently the fluxioa 
 
 of ^ X T . AEB + t . AEB by the ratio of the fluxion of AE 
 to V. And ^ X T . AEB T t . AEB is equal to the time in 
 
 which AEB is described with a velocity that is always equal, 
 to V at any point E. Therefore when this time is a minimum, 
 
 ^ X T . AEB -f t . AEB is likewise a minimum, and AEB is 
 described in less time than any other line of the same perimeter 
 \\ydt passes through A and B. 
 
 590. Suppose
 
 90 The Resolution of the general Book I. 
 
 590. Suppose the gravity to act in parallel lines, which is 
 the case considered hy Mr. BernouiUi ; and if the sine of the 
 angle FEQ be to the radius (or the fluxion of the base FQ be 
 to the fluxion of the curve FE) as V is to an invariable velo- 
 city, that is in a ratio compounded of the ratio of u the velo- 
 city at E to the sum or diflcrence of h and // and of an invari- 
 able ratio, then AEB will be described in less time than any 
 equal line that passes througli A and B, by art. 576, which 
 coincides with the equation of the curve that was found by that 
 learned author, by resolving the element of the curve into three 
 infinitely small rectilinec}] pnits and computations from second 
 fluxions. Mem. de VAcad. lioj/ale dts Sciences, 1718, prop. 4, 
 and schol. 2. 
 
 .59 1 . The same method discovers the property of those lines, 
 when the gravity is directed towards a given centre S with the 
 same facility. Let the velocity of the body that descends along 
 the curve AEB at any point E be represented by M,and 6 express 
 an invariable velocity ; then if the sine of the angle S EB (Jig.Q.5't), 
 contained at E by the curve AE and ray SE, be always to the 
 radius in a ratio compounded of that of u to the sum or difier- 
 ence of h and u, and of the ratio of an invariable line to the 
 distance SE, the line AEB will be described in less time than 
 any equal line that can be drawn from A to B, the velocities 
 being supposed equal at A in each. In general, let EC be half 
 of the chord of the circle of curvature at E, that is in the di-;. 
 rection of the force EK that acts upon the body at that point, 
 as iri art. oGS, and suppose the body to descend from E in CE 
 ■with the velocity u acquired in the curve at E, then the point 
 C -being su|)posed to remain, if the curvature of the line is such, 
 that the fluxion of u be to the fluxion of EC in the compound 
 ratio oi u to EC and of 6 + a to b, then AEB will be described 
 in less time than any equal line that passes through A and C, 
 the velocities being equal at A. The demonstration is similar 
 to that in art..3f>5. 
 
 59'i. The celebrated isoperimetrical problems maj' be treated 
 in the same manner, and rendered more general than is usual, 
 without having recourse to the fluxions of the higher orders ; 
 and the solutions of these problems (that are generally consi-» 
 
 dere4
 
 Chap.^IlI. Isoperimdrical Problems demonstrated. 01 
 dered as of a very abstruse nature) ma}' be verified by easy syn- 
 thetic demonstrations. The lemma that is required for this 
 purpose differs notmaterially from that in art. 572, which wede- 
 monstrated withouthaving recourse lo fluxions. Let KL^/Zo-.QGI) 
 be a right hne given in position, A any given point that is not ia 
 KL, A K perpend icuhn- to KL in K, E any point upon KL; and 
 let a and u represent any given or invariable hnes : then if 
 KL be to AL, or the sine of the angle KAL to the radius, as 
 u is to a, AE x a — KE x u will be vcmirnmum wdien E falls 
 upon L. For let KH and EV be perpendicular to AL in H 
 and V, and AR made equal to AE ; because KE is to HV as 
 AL to KL, or a to u, KE X u is equal to HV x a, AE x a 
 — KE X u equal to AE— HV x a, or AH+Vll x a, which 
 Is evidently least wlienVR vanishes, that is when E falls upoa 
 L. It follows from this that the point A, the distance AK, and 
 EA; the distance of the parallels KL and kl, being given, E/c be- 
 ing perpendicular to Id in k, and E and c being any points 
 upon these parallels; if «, u, and v be supposed invariable, then 
 A Ed' X a — KE y. u — ke x v will be least when the sine of 
 the angle KAE is to the radius as u to a, and at the same 
 time the sine of A:Ee to the radius as ■y to « ,• for in this case 
 AE X a — KE X u will be a minimum, and Ee x « — kc x "9 
 will be less than when the angle kEe is of any other magni- 
 tude, so that their sum will be less than if the right lines AE 
 and Ee were inflected in any other manner. 
 
 593. Itis easily demonstrated from what was shown in thelast 
 article, that kC (fig.26Q,) and CG being perpendicular to each 
 otherinC,KMNLGagivenfigureappliedonCG,Eany pointiii 
 the curve AED,EPN a parallel to kC that meets CG in P, and 
 the curve ML in N, Epn a parallel to CG that meets /cC inp, 
 upon which pn is supposed to be taken always equal to PN the 
 ordinate of the figure KMNLG,and to generate the area kmnlg 
 in this manner, then if the point A, the distance KG (or the 
 difference of A/c andDg the ordinates from the curve AED to 
 /C), and the figure KMNLG with the right line a be given, 
 the excess of AED x a above kmnlg, the area generated hy pn, 
 will be a minimum, when the sine of the angle AEp contained 
 ty the curve AE and any ordinate Ep, is to the radius as PN 
 
 the
 
 §3 The Hesolution of the general Book Ia 
 
 the corresponding ordinate of the figure KMNLG is to the in- 
 variable line a. For let TE? the tangent at E meet AT the 
 tangent at A in T, and D/ the tangent at D in i, let TRS and 
 /VX parallel to kC meet CG in R and V, and ML in S and X, 
 Trs and tvx parallel to CG meet kC in r and v, and mnl in $ 
 andx; complete the parallelograms /jr/t/,7;;;«Ojp»j/t'^t;x2:o-; and 
 the excess of ATEil '0 x. a above the figure kufoyxzg, the sum of 
 these parallelogi'auiSj will be less than if the points T, E, t, D 
 were taken any where else upon the parallels TR, EP^ i\ , 
 DG, hy the last article. Then by drawing tangents to the curve 
 AEB at the points where TR and t\ intersect it^ and parallels 
 |o kC through the points where these tangents intersect AT, 
 Tt, and tjy, and proceeding in this manner, the ultimate ra- 
 tio of tiie circumscribed figure AT^I) to the curve AED, and 
 of the area /^{/b^a:;!^ to the curvilineal area kmnlg, will be a ra- 
 tio ofequality ; consequently AED Xa — kmnlg will be a mivi- 
 mum ; the point A, with the right hues a and KG^ and the 
 figure KMLG being given. 
 
 594. It follows that the point A being given with the figure 
 KMLG, \^ pn be always equal to PN, and the sine of the 
 angle AEp be always to the radius as PN to a, then the area 
 kmnlg will be greater than any area A'wz;/g- generated in the same 
 manner from any line AED that is drawn from A to LGD of a 
 perimeter equal to AED. For since AED X a — knndg is less 
 than Ae D X a — kmvlg, by the last article; and AeD is equal to 
 AED, by the supposition ; it is manifest that kmnlg must be 
 greater than kmvlg. Therefore when the sine of the angle AEp 
 is always to the radius as PN to a, AED is the line that amongst 
 all those which are drawn from A to LGD, and have equal 
 perimeteis, produces the gix?atest area knniig. 
 . 5fj5. The rest remaining as before, let HI parallel to KG 
 at any given distance KII meet EP in Q, and hi parallel to kg 
 at an equal distance kh meet Epin q, and the points h, k, yn lie 
 in the same order from each other as H, K, M. Then qn will 
 be always equal to QN, and the area hilnm generated by the 
 ordinate qn will be a maximum or minimum according as H I and 
 KG cue on the same or different sides of MjNL, the points A 
 and D being given.. For since kmuIg is always a maximum, and 
 
 the
 
 Chap. XIII. Isoperhnetrical Problems demonstrated. t)3 
 
 the rectangle hg is given, hmnli will be a maximum when Iii 
 and kf are on the same side o^mnl, that is when HI and KG 
 are on the same side of MNL; and hmnli will be a minimum 
 when hi and kg are on different sides of mnl, that is when HI 
 and KG are on different sides of MNL. It appears therefore 
 that the area hmnU generated by the ordinate qn, equal to QN, 
 is -A maximum vvhentlie sine of the angle AEa is ahvays to the 
 radius (or the fluxion of the base hq to the fluxion of the curve 
 AE) as QN + HK is to a, or as QN — HK to a, if QN in this 
 latter case be never less than HK; because when HI and KG 
 are on the same side of MNL, PN (or jpw) is either equal to 
 QN-f-HK, or to QN — HK, QN being in this case never less 
 than QP orHK; but that the area hmnli is •^minimum, when 
 the sine of the angle AE^ is always to the radius (or the fluxion 
 of the base hq to the fluxion of the curve AE) as HK — QM 
 to a, HK being supposed to be never less than QN any ordinate 
 fromMNLto the axis HI; because when KG and HI are on dif- 
 ferent sides of MNL, HK — QN is equal to PN. The points 
 A and D with the figure HMLI are supposed to be given, and 
 the perimeter AED to be always the same. And this property of 
 the line AED, which amongst all those of aa equal perimeter that 
 pass through A and D produces the greatest or least area hmnli, 
 by taking the ordinate qn always equal to QN the correspond- 
 ing ordinate of the figure KMNLG, is' the same that Mess. 
 Bernouilli, Dr. Taylor, and others, deduced from computations 
 that involve third fluxions. It is obvious that the curve AED 
 is concave, or convex, towards AK, according as the sine of the 
 angle A'Eq increases, or decreases^ while KP increases; that is, 
 according as PN increases, or decreases, while ICP increases. 
 When HI intersects MNL in any point betwixt M and L, hi 
 intersects mnl at some poin t betwixt w and /,- and one part of 
 the figure AED produces a maximum and the other a minimum. 
 When MNL meets KG, the angle AEg- vanishes, and the curve 
 touches the ordinate Eg'; and if PN become equal to a, KEq 
 will become a right angle, and the curve perpendicular to the 
 ordinate. 
 
 596. Because the sine of the angle contained by the curve 
 and ordinate Eg is to the radius as PN to a, it follows from 
 
 ai-t.
 
 g4 The Resolution of the general Book Ij 
 
 art. 57()j that AED is likewise the line of swiftest descent when 
 the velocity in any part of the right line EPN is always mea- 
 sured by the corresponding ordinate PN, as was observed by 
 Mr. James Bernouilli; and this analogy between the lines that 
 satisfy these two problems is accounted for, from the similitude 
 of the methods by which their properties were demonstrated 
 in art. 576 and 595. 
 
 597. Suppose now that S (fg. 265) is a given point, that a 
 circle ^p/: is described from the centre S with a given radius Sg-, 
 that SA, SB, and SD meet this circle in k, p, and g, that the 
 figure HMLD being given, and SQ being always equal toSE,S» 
 is taken upon SE always equal to QN the corresponding ordinate 
 of HMLD; and that Sm and SI being taken upon SA and SD 
 respectively equal to HM and DL the corresponding ordinatesof 
 the same figure, it is required to find the nature of the line AED 
 that amongst all those which pass through the points A and D, 
 and have equal perimeters, produces in this manner thegreatest 
 or least area Smnl. It is manifest that a being an invariable 
 line, if AED X a — kmlg be a minimum,thenkmlg\y'i\\ be greater 
 than any area formed in the same manner from any figure that 
 has its perimeter equal to AED ; and that Smnl will be the 
 greatest or least of the areas terminated by S/re and SI ac- 
 cording as the point Sand the circular arch kpg arc on the same 
 or on different sides of mnl. To find when AED X a — kmlg 
 is a minimum, let AT the tangent at A meet the circle QE de- 
 scribed from the centre S in T, and ST produced meet the arch 
 ARH described from the same centre in R, and it appears from 
 art. 578, that AT X a — AR Xu is a minimum (IIQ being given) 
 when the sine of the angle SAT (in which any ray SA inter- 
 sects the curve AED) is to the radius as uio a. Let us there- 
 fore suppose AR x?^ to be ultimately equal to the area kmnp; 
 and since kmnp is ultimately to the sector SAR (or ^SA x AR) 
 as the difference or sum (according as k and m are on the same 
 or different sides of S) of the squares of Sw and S/c to thesquare 
 of SA, it will follow that u is to ISA as the difference or sum of 
 those squares is to SA^; so that u is to a, and consequently the 
 sine of the angle SAE is always to the radius, as that difference 
 or sum is to 2SAxa. Therefore in the figure AED, if SX be 
 
 perpen-
 
 Ff^.2S. 
 
 H
 
 11a«eX\LX^«,fl^. IbfM. 
 
 F^^HXl 
 
 d
 
 C^iap. XIIl. Isoperimetricai Problems demomtrated. ^5 
 perpendicular to the tangent EX at X, So^ and QN be repre- 
 sented by c and V, respectively^ the rectangle '2tf xSX will be 
 cqnal to the difterence or sum of VV and cc. The invariable 
 Quantities a and c (with another invariable line that will arise 
 in determining the figure from this property) serve to satisfy the 
 conditions of the problem^, which requires that the curve shall 
 pass through A and Y), the perimeter being supposed of the 
 same magnitude in all these figures amongst which AED pro- 
 duces the greatest or least area Slum. 
 
 598. When So- is supposed to vanish, 2fl! X SX is equal toQN*, 
 or SX is a third proportional to Ca and QN, and the area Smnl 
 U -A maximum. In this case, if HMNLD be a parabola that 
 has its vertex in S and axis in Sli, or an hyperbola of any 
 order whose ordinate QN is inversely as any power of SQ, 
 AED is one of the figures constructed in art. SQ1 or 3^S, 
 ■which we have found already to satisfy the most simple cases 
 of several problems in art. 436, 457, 438, 439, 567, and 586. 
 For example, when S/i is taken upon SE alwaj^s equal to a 
 mean proportional betwixt SE and a given line, and AED is 
 an arch of a logarithmic spiral that has its pole in S, the area 
 Sw«/ is the greatest that can be produced in the same manner 
 from an arch of an equal perimeter that passes through A and 
 D. When Sw is inversely in the subduplicate ratio of SE, and 
 AED is an equilateral hyperbola that has its centre in S, 
 S;«/// is the greatest area that can be produced in the same 
 manner, from any arch of an equal perimeter that passes 
 through A and D. When MNL is a right line thai passes 
 through S, AED is an arch of a circle, and midh likewise an 
 arch of a circle similarly situated with respect to S, whether 
 Sn^ be supposed to vanish or not ; and, in this case, it is well 
 known that tlie area 'Stintl is a maximum or minimum accord- 
 ing as mill is concave or convex towards S. 
 
 599- It appears, in the same manner, that if any given line 
 hqi meet SA, SE, and SD in h, q, and / ,• and qu be always 
 taken upon Sr/ equal to QN, it will be the property of the figure 
 AED, that amongt those of an equal perimeter which pass 
 tlirough A and D^ produces the greatest or least avcahmli, that 
 
 the
 
 9a The Resolutio7i of the general Book 1, 
 
 the sine of the angle AEX will be to the radius as the differ- 
 ence or sum of the squares of S^ + QN and Sg to 2SE x a. 
 
 600. The property of the line AED that is described by a 
 velocity -svhich at the distance SE^ or SQ^ is always measured 
 by QN, the ordinate of a given figure HMLD, and that of all 
 those which pass through A and D, and are described in the 
 same time, comprehends the greatest or least area SAED, is 
 that the sine of the angle SEX is to the radius in the com- 
 pound ratio of QN to an invariable velocity, and of the differ- 
 ence or sum of the square of SE and an invariable square to 
 the rectangle contained by SE and an invariable line. For the 
 construction being the same as in art. 597, '\^ aa x T . AED — 
 A^Dgpk be a minimum, the point A and right line HD being 
 given, the area AEDo-pA; will be greater than any area AeDgpk 
 ■which is terminated by any line AeD that is described in the 
 sametimev/ith AED; and SAEDvvillbe greater or less thanSAeD, 
 according as the point S and arch hpg are on the same or dif- 
 
 ferent sides of AED. Because — X ^^ ^ minim,um (HQ being 
 
 given) when the sine of SAT is to the radius as u is to 6, by 
 
 art. 578, let QN be equal to u, and suppose — — ultimately 
 
 AR 
 
 equal to-7-' Then because AEpA; is ultimately to the sector 
 SAR, or iSA x AR, as the difference or sum of SA* and S/c* 
 to SA- ; it follows, that b will be to 2SA, as aa is to that differ- 
 ence or sum ; and that u is to b, or the sine of the angle in 
 which any ray SA intersects the curve to the radius, in the 
 compound ratio of the same difference or sum to aa and of QN 
 to gSA. When the gravity acts in parallel lines, the nature 
 of the line AED is discovered in the same manner. 
 
 601. The following lemma leads to an easy solution of the 
 second general isoperiraetrical problem. The ^omtACjig. 264) 
 being given, the right lines AE, b, and u being given in magni- 
 tude, and AG given in position, let EKbe perpendicular to AGin 
 K, and EK X 6 4- AK x u will be a maximum, when the tangent 
 of the angle AEK is to the radius as u is to h. For let the 
 circle BEZ» described from the centre A with the given radius 
 AEmeet AG in G ; let Gg be erected perpeadicuJar to AG, so 
 
 that
 
 Chap. XIII. Tsoperimetrical Problems demonstrated. 97 
 thatGg and KE may be on different sides oFAG, and let G^ 
 be to AG as w is to 6 ; join Ag, and Ag will he given in posi- 
 tion ; let EK produced meet Ag in O^, then because KO is to 
 AK as Go- to AG or u to b, AK X u will be equal to KO x b ; 
 so that EK X 6 + AK x u will be equal to EO x b, and will 
 be a maximum when EO is greatest; and it is manifest that EO 
 is greatest, when ET the tangent of the circle at E is parallel 
 to A^, or when AE is perpendicular to Ag, that is, when AK 
 is to KE, or the tangent of the angle AEK to the radius, as 
 Gg to AG, or as u to b. In the same manner it appears, that 
 if th^ right lines Ee and v be likewise given in length, and 
 ek be perpendicular to AG in k, then ek >c b •{- AK X u 
 4- KA X V will be a maximum when the tangent of the angle 
 AEK is to the radius as u to 6, and at the same time the tanarent 
 of EcA; to the radius as x; to 6 ,• because EK x 6 + AK x u 
 is then a maximum; and, Em parallel to AG being supposed 
 to meet ek in m, em X b -\- Em X v will be greater than if 
 tlie angle Ee^ was of any other magnitude. In general, let the 
 line AEefD consist of any number of parts AE, Ee, ef^fJ) 
 whereof each is given in length ; let EK, ek,fl, DL be perpen- 
 cliculap to AL ; and upon these perpendiculars take Km, kv, 
 Ix, Jjz equal to any given lines ; then the radius being supposed 
 equal to^, let the figure of the line AEf/'D be such that the 
 tangents of the angles AEK, EeA', efl,fDL may be respectively 
 equal to the perpendiculars Km, kv, Ix, Ls ; complete the pa- 
 rallelograms AKz/r, Kkvs, klxt, Ihzy ; and the sum of the 
 rectangle LD x b added to the area Arw/i'^orj/^L (which is made 
 up of those parallelograms) will be greater than if the line 
 AEe/"D was disposed into any other figure. 
 
 602. Let AEDL (JigMQd) and ABM ILbe applied upon the same 
 axisAL,EPM parallel to DLmeet BMIinM,and thetangentof 
 the angle AEP, in which the curve AED intersects its ordinate 
 •EP, be always to the radius as PM to b. Let Aed be any 
 other line equal in length to AED, and the arch Ae being al- 
 ways equal to AE, let epm parallel to DL meet AL m p, and 
 pm always equal to PM generate the area A^mil ; then it fol- 
 lows from what was shown in the last article, that DL X b 
 + ABMIL will be always greater than (f/ X 6 + A^mil. From 
 VOL. H. H which
 
 98 The Resolution of the general Book L 
 
 which it follows^ that when dl is. equal to DLj the area ABMIL 
 is greater than ABmil. And if we suppose d to coincide with 
 D (and consequently il coincide with IL)^ and, AH being 
 Fiven upon AB, if HG parallel to AL meet ID in G, PM 
 in Q, and pin in (/, the area HBMIG generated by the ordi- 
 3iate QM will be greater or less than HBwIG generated by the 
 ordinate qm, according as HG and AL are on the same or dif- 
 ferent sides of jMNI. Therefore since QM is equal to the sum 
 or difference of PM and AH, it is the property of the line AED, 
 which of all those that pass through thepoints A and Dandhave 
 equal perimeters, produces the greatest or least area HBMIG, 
 when the ordinate QM depends upon the length of the arch 
 AE, being what is called a function of the arch AE (that is, 
 QM being always equal to the ordinate of a given figure 
 when the base is equal to AE), that the tangent of the angle 
 AEP is to the radius, or the fluxion of the base AP to the flux- 
 ion of the ordinate PE, as the sum or difference of QM and 
 an invariable line AH is to an invariable line b. And this 
 agrees with what the authors above-mentioned found by their 
 computations when carried on justly. 
 
 603. Itappearsas in art. 597 (Jig- 3556), that when S is a given 
 point, and upon SE arightline SM is always taken equal to afunc- 
 tion of the arch AE (that is, equal to the ordinate of any given 
 figure when the base is equal to the arch AE), and the area 
 SBML is the greatest or least of all those that can be thus pro- 
 duced by lines of equal perimeters that pass through A and D, 
 the tangent of the angle SED, in which any ray intersects the 
 curve, is always to the radius as the difference or sum of the 
 square of SM and an invariable square to ^SE x b. 
 
 604. The other isoperimetrical problems may be reduced to 
 these, or treated in like manner. Por example, letERparallelto 
 AK (^V.<2f)7) meet AS parallel to K DinR,and RN, SVtheordi- 
 nates of the figure ANVS be always equal to fund io7is of the 
 arches AE and A ED ; let NZ and VX be perpendicular to KA 
 produced in Z and X. Then because the area AVX is equal 
 to AS X SV — ANVS, and when A and D are given, and the 
 nrcli A ED is given in length, hs function SV being given, the 
 rectangle AS x SV is given, it follows that the area AVX is 
 
 a niaxi-
 
 flMcXWl./^a.cuf.lW.il. 
 
 K 
 
 ]> 
 
 Fa?. -J? J. 
 
 Fig. 2S2.N^2. 
 
 Fiff.232.N'°S. 
 
 C v/ 
 
 V" 
 
 / 
 
 \^/ 
 
 \ 
 
 \ 
 
 7/ \v/ 
 
 \ 
 
 \ 
 
 \ / \ 
 
 \.- 
 
 ril 
 

 
 Flalc-X.\\-l./},_„<f,|;./]T 
 
 ^
 
 "Cihap. XIII. IsopcrinietHcal Problems demonstrated, 99 
 
 s maximum when AVS is a minimum, and that AVX is a mini- 
 mum when AVS is a maximum, that is (by art, Q02), when the 
 tangent of the angle AER is to the radius as the sum or differ- 
 ence of RN and an invariable line is to an invariable line. The 
 ai-ea AVX has its ordinate ZN equal to EP the ordinate of 
 AED, and its base AZ equal to RN a. function of AE ; and the 
 line AED is the figure which is assumed by a chain perfectly 
 flexible^ and suspended from A andD, v/hen the thickness of the 
 chain at any point Eis as the fluxion of RN; because the centre of 
 gravityofsuchachain would descend to aslow a place as possible. 
 605. The restremaining as in art. 572 (Jig. 254), let usnow sup- 
 pose that m the velocity with which AE is described is not given, 
 but varies as a power of AE whose exponent is any number n. 
 And when the sine of the angle KAE is to the radius as u is to 
 
 AE— KE AE KE . . . ,. 
 
 i—nXa, — ' -^, or { < is a minimum according as n 
 
 AE 
 
 is less or greater than unit. For — will be as AEl— «> 
 
 AE 
 
 and the fluxion of — to the fluxion of AE (art. I67) as 
 
 ^ AE 
 
 1—"^ X — to AE, or as 1 — n io ii. The fluxion of AE ia 
 to the fluxion of KE as KE to AE by art. IQS, consequently 
 
 A^ KE 
 
 the fluxion of — ^ is to the fluxion of — as 1— ^ x KE to 
 
 M a 
 
 AE X -• Therefore if n be less than unit, these fluxions are 
 
 a 
 
 A 17 VI? 
 
 equal, and — — is a minim, um, or its fluxion vajiishes, Vvhen KE 
 is to AE as u is to ~ « X a. And when n is greater than unit, 
 
 AF KE . ,, AE , KE . . . ■ 
 
 «— decreases while — mcreases, and — + " is a minimum, 
 
 ff a w t* 
 
 or its fluxion vanishes, when KE is to AE, or the sine of KAE 
 is to the radius, as u is to ;;ilT X a. In these cases therefore 
 likewise the sine of the angle KAE is still as u ; and this theo- 
 lem thus extended will serve for resolving problems concerning 
 
 H S thff
 
 100 The Propertif of the Solid 15ook li^ 
 
 the maxima and minima, to which the lemma in art. 572 does 
 not reach. 
 
 606 {Fig. 266). It remains now to show how the problem con- 
 cerning the solid of least resistancemay be resolved by first flux- 
 ions. The point A being given, let the ordinate AD meet KL 
 (parallel to the axis DG) in K;, let E be any point upon KL, 
 join AE ; and the resistance of the fluid being represented by 
 a given right line AR, the resistance of the right line AK mov- 
 ing in the direction KL will be represented by AK x AR. Let 
 RM be perpendicular to A E in M, and IMN perpendicular to 
 AR in N ; and the resistance which the conical surface gene- 
 rated by AE (when the l^gure is supposed to revolve about the 
 axis DG) meets with, will be to the resistance of the annular 
 Space generated by AK as RN to AR, and therefore (AK 
 being bisected in a) that resistance will be as Da X AK X RN. 
 Because, AK being given, RN decreases while KE increases, 
 let us enquire w hen Da x AK x RN + KE x AR x a is a m.ijii-i 
 mum. The fluxion of this sum vanishes, AR (which measures 
 the direct resistance of the fluid) with AK, Dfl and a beings 
 supposed invariable, when the fluxion of KEis to the fluxion o^ 
 RN as AK x Da to AR x a. But RN being to AR as AK* 
 to AE", RN is inversely as AE% and (art. 167) the fluxion 
 of RN is to the fluxion of AE as 2RN to AE; consequently 
 the fluxion of KE is to the fluxion of AE, or (art. 193) Ap 
 to KE as 2Dfl X AK x RN to a x AE x AR ; that is, as 
 ^Da X AK^ to a X AE^ Therefore Da X AK x R 4-KE 
 X AR X a is a minimum when 2Da x AK^ X KE is equal to 
 a X AE\ 
 
 607. From this it follows, that KE parallel to the axis BH 
 being supposed to meet the ordinate A D in K, and AE the tan- 
 gent at A in E, and a being an invariable quantity, if the liue 
 FAL be of such anatitrfe that AD x AK^ X KE be always 
 equal to -i-a X- AE"", then the solid generated by FALi revolv- 
 ing about the axis BH will ineet with less resistance, when it 
 moves in a given fluid with a given velocity in the direction of 
 the axis BH. than the solid generated in the same manner by 
 any other figure whose perimeter passes through Fand L. For 
 when AK is continually diminished. Da is ultimately ecjual to 
 
 DA;
 
 Plate XKXTa JOO. T^'/.R. 
 
 B D L H
 
 piat<-.\xxRi/,w.r;./jL 
 
 \ 
 
 .B \S 
 
 41 
 
 9 
 
 i -r I I, 
 
 \J 
 
 / 
 
 y 
 
 ^'^ 
 
 
 ^~~~-~-^tz— 
 
 Fig. 26ej\'°iArt. 606 
 
 \ 
 
 J
 
 ^hap. XIV. of the Ellipse comideredy^c. 101 
 
 P A ; and it follows from the last article that the sum of the so- 
 lid which measures the resistance of the conical surface gene- 
 rated by AE about the axis BH, added to KE x AR X a will 
 be always ultimately a minimum. And because the sum of the 
 resistances of these conical surfaces is ultimately equal to the 
 resistance of the solid generated by P'AL about the same axis 
 BH; and the sum of the solids KE x AR x a is ultimately a 
 given solid &L X AR X a (because b\j, AR, and a are supposed 
 to be given), it appears that the resistance of the solid gene- 
 rated by FAL is a minimum. It is easy to see that this agrees 
 with the property of this solid, which was given by Sir Isaac 
 Newton. 
 
 608. InthesamemannerwhenaplanefigureFAL(//g.267)re- 
 volves aboutagiven centre S,thatisintheplaneofthefigurejiua 
 medium that resists in the duplicate ratio of the velocity, it is 
 the property of the line which in this case meets with the least 
 resistance, that the sine of the angle SAP contained by the ray 
 SA and tangent at A is inversely as the cube of the tangent AP, 
 SP being perpendicular to AP in P. There are several other en- 
 quiries of this nature which might be prosecuted in the same 
 manner, but we proceed to what may be of more use in philo- 
 sophy. 
 
 CHAP XIV. 
 
 Of the Ellipse considered as the Section of a Cylinder. Of the 
 Gravitation towards Bodies, zvhich results from the Gravita- 
 tion towards their Particles. Of the Figure of the Earth, 
 and the Variation of Gravity towards it. Of the cbbin<r and 
 Jlowing of the Sea, and other Enquiries of this Nature. 
 
 609. JL H E properties of the circle demonstrated by Euclid, 
 Pappus, Greory a St. Vincentio, and others, suggest analogous 
 properties of the ellipse ; which, generally speaking, are most 
 easily and briefly deduced by considering it as the oblique sec- 
 tion of a cyhnder, or as the projection of a circle by parallel 
 
 H 3 rays
 
 102 Of the Ellipse considered Book I. 
 
 rays upon a plane that is oblique to the plane of the circle. 
 For the centre of the circle by this projection gives the centre 
 of the ellipse ; any diameters of the circle that are perpendicu- 
 lar to each other with the tangents at their extremities 
 (which form the circumscribed square) and their respective ordi- 
 nates, give conjugate diameters of the ellipse with the circum-. 
 scribed parallelogram, and the ordinates of these diameters; any 
 parallel lines in the plane of the circle are projected by parallel 
 lines in the plane of the ellipse, that are to each other in the 
 same ratio as those of which they are the projections; any area 
 in the plane of the circle gives by its projection an area in the 
 plane of the ellipse, which is always to the area in the plane of 
 the circle as the transverse axis of the ellipse to its second axis, 
 the cylinder being supposed upright; and any concentric circles 
 give similar concentric ellipses. Having found this method very 
 convenient on these accounts for discovering the properties of 
 the ellipse, and particularly some that are of use in the follow- 
 ing enquiries, it may be worth while lo prosecute it a little 
 farther than the illustrious Marquis de l' Hospital has 
 done, lib. 6, sect, conic, the rather that it is not difficult to de- 
 rive the analogous properties of the hyperbola and paraboloi 
 from those of the ellipse when known. 
 
 610. Let aABb^Jig. 268) be a section of an upright cylinder 
 through its axis cC, adbea section of this cylinder perpendicular 
 to cC, ADBE a section perpendicular «o the plane aAB6, but ob^ 
 lique to the axis cC ; the former will be a circle that has its centre 
 in c; and the latter will be an ellipse that has its centre in C, 
 of which AB will be the transverse axis, and the second axis 
 DE perpendicular to AB will be equal to ab the diameter of 
 the circle. For let h be any point in the circumference of the 
 ■ circle, hp perpendicular to ab in p, //H and^F parallel to cC 
 meet the p-lane ADBE in H and P, so that H may be what we 
 call the projection of//, and P oi' p; join HP, and it will be per- 
 pendicular to the plane aABb, consequently AHPp is a'paral- 
 lelogram, and HP equal to hp. Therefore the square of HP is 
 equal to the rectangle apb which is to APJi (because ap is to 
 AP, and pb to PB, as ab to AB) as ab'' to AB'^; consequently 
 AHB is an ellipse, and its second axis DE is equal to ab. It 
 
 appears
 
 Chap. XIV. a$ the Section of a Cylinder. 105 
 
 appears in the same manner, that any right line in the plane of 
 '^he circle that is perpendicular to ab, is projected by an equal 
 right line in the plane of the ellipse perpendicuiar to AB. 
 
 611. Any parallels gh and hi (jig. 269) in the plane of the 
 circle are projected by parallel lines GH and KL in the plane 
 of the ellipse that are in the same ratio to each other as gh and 
 kl. For the planes gGYili, A'KL/ being parallel^ the sections 
 of those planes with ADBE are parallel; and because the 
 angles of the figures oGH//,/:KL/ are respectively equal to each 
 other, GH will be to gh as KL to kl. 
 
 612. It is obvious, that according as /"is a point without or 
 within the circle, its projection is without or within the ellipse. 
 Therefore the tangent of the circle at any point A is projected 
 by the tangent of the ellipse at H. Hence any right lineVRpti- 
 rallel to the tangent at H is bisected in M by the diameter that 
 passes through H, and VM, MR are the ordinates of that dia- 
 meter, being projected from vm and mr equal perpendiculars 
 to ch. Let vt the tangent of the circle at v meet ch produced 
 in ty and VT the tangent of the ellipse at V meet CH in T. 
 Then because ml^l, h\l, and /T are parallel, and cm is to cv as 
 cv (or ch) to ct; it follows that CM, CH, and CT, are in conti- 
 nued proportion. Let tv produced meet the semidiameter cl 
 perpendicular to ch in z, and because the rectangle tvz is equal 
 to the square of cv or ck, it follows that if TV meet CL the semi- 
 diameter conjugate to CH in Z, the rectangle TVZ will be equal 
 to the square of CK the semidiameter of the ellipse that is paral- 
 lel to TZ. And if HNQ parallel to CK meet CV and CL in 
 N and Q, the rectangle NHQ will be equal likewise to CK^. 
 
 613. Any right line 5- /i in the plane of the circle is to GH its 
 projection in the plane of the ellipse as the second axis of the 
 ellipse is to the diameter that is parallel to GH; because this 
 diiuneter is the projection of the diameter of the circle which 
 is parallel to gh ; and parallel lines in the plane of the circle 
 are in the same ratio as their projections in the plane of the 
 ellipse, by the last article. 
 
 614. Hence right lines j9w and^^i (Jig. 270) in the plane of 
 the circle that form equal angles with ah, or with phgany pa- 
 fallei to at), on the same or on diflererit sides of that parallel, 
 
 H 4 ar
 
 104 Of the Ellipse comidered Book 1} 
 
 are projected by right lines PM and PN that form equal angles 
 with AB the axis of the ellipse, or with PHG the projection 
 ofphg; and PM is to PN as pm to pn. Tor let mr perpendi- 
 cular to pg in rmeeljjn in f, and mr will be equal to Jr; conse- 
 quently their projections MR and FR will be equal by art.6lO, 
 and the angle MPR equal to NPR ; and because the diameter 
 parallel to PM is equal to the diameter parallel to PN^ it fol- 
 lows that PM is to PN as pm to pn, by the last article. Ifnq 
 be perpendicular to pg, and NQ to PG, PQ will be to PR as 
 pq to pr, and PQ + PR topq -\-pr as PQ to pq or as AB to DE. 
 Tlie use of this property will appear afterwards, but we will 
 first show how other properties of the ellipse are briefly deduced 
 in like manner. 
 
 615. If any line VR (fig. 271) terminated by the ellipse in 
 V and R meet any parallels GH and KL in M and N, and 
 VR, GH, and KL be projected from vr, gh, end kl in the plane 
 of the circle, GM will be to KN as gm to k7i (art. 6l 1), and 
 MH to NL as mh to id; consequently the rectangle GMH 
 will be to KNL SL^gmh to knl. In the same manner the rect- 
 angle VM R will be to VNR as vmr to v)ir. Butgrnh is equal 
 to vmr (elem. 35. 3), and knl to vnr. Therefore the rectangle 
 GMH is to KNL as VMR to VNR. 
 
 616. When he and cl (fig 272) are perpendicular semidiame- 
 ters of the circle, let hp and Iq be perpendicular to ab inp and q, 
 and let HP and QL be the projections of these ordinates in the 
 plane of the ellipse. Then HP will be equal to hp (art. 6 10), or 
 cq, and LQ equal to Iq or cp. Because CP* is to cp% and CQ" to 
 cq"-, as CA^ to c«% it follows, that CP^+ CQ"-}- cp^+ cq-, or 
 CH*+ CL% is to cp^-\- cq'^, or f«% as CA^+ ca^ to car -^ 
 consequently CH^ + CL^ is equal to CA^ + ca% or CA^ + 
 CD^; that is, the sum of the squares of any two conjugate dia- 
 meters is equal to the sum of the squaresof the axis AB and DE. 
 
 617. Any rectangle ghlk (jig. 273) in the plane of the circle 
 that is contained by right lines oneofwhichgA; is parallel and the 
 other g/i perpendicular to ah, is to its projection GHLK in the 
 plane of the ellipse (which is likewise a rectangle) as the second 
 axis DE to the transverse AB. For Wi and GH the sides of 
 those rectangles perpendicular to ab and AB are equal, by art. 
 
 610,
 
 Chap. XIV. «s the Section of a Ci/Iinder. 105 
 
 610, and gk is to GK as ab or DE to AB. Any triangle mnr in 
 the plane of" the circle is to MNR its projection in the plane of 
 the ellipse in the same ratio; for if nq parallel to ab meet mr ia 
 g, and NQ parallel to AB meet MR in Q^ rh and mg parallel 
 to ab meet nk and qk perpendicular to ab in h, I, g, and k, and 
 GHLK be the projection of ghlk, the triangles MNR and mnr 
 will be the halves of the rectangles GHLK and ghlk ; conse- 
 quently mnr will be to MNR as DE to AB.' It appears from 
 this, that any figure described in the plane of the circle is to its 
 projection in the plane of the ellipse asDE to AB; and that any 
 equal figures described in the former are projected by equal 
 figures in the latter. Thus the squares described about the circle 
 being always equals the parallelograms described about any 
 conjugate diameters of the ellipse (which are the projections of 
 those squares) are always equal. If CP and CS be taken upoa 
 any diameter FI, from C in any given ratio to CF^ the area 
 of the parallelogram contained by the tangents drawn from P 
 and S to the ellipse will be given ; and thus the property of the 
 ellipse described in thelntroduction^p. 8j is easily- demonstrated. 
 618. If the point r (fig. 274) describe the circumference of 
 the circle adbewiih an uniform motion, and Rbealwa3's the pro- 
 jection of r, the ray CR will describe equal areas aboutCinequal 
 times, and rv and RV being arches described in the same time, 
 and vt the subtense of the angle of contact in the circle parallel 
 to cr being to VT the subtense of the angle of contact in the 
 ellipse parallel to CR as ca to CR, it follows that the force di- 
 rected towards the centre C by which the ellipse could be de- 
 scribed, is to the force by which the circle arb upon the diame- 
 ter ab could be described uniformly in the same time as CR to 
 CD. The velocity in the ellipse at R is to the velocity in the 
 circle as CZ the semidiameter conjugate to CR to CD by art. 
 610. And when the force towards the centre is as the distance, 
 the periodic times in circles being equal, the timesinwhichellip- 
 ses are described are likewise equal. Thus prop. X, lib. l,Princip. 
 with its corollaries are briefly demonstrated. In like manner, 
 /being any point in the plane of the circle, let r move in the 
 circumference, so that 7/ may describe equal areas in equal times 
 about/; then if S and R be the projections in the plane of the 
 
 ellipse
 
 106 General Properties of the conic Sections Book T* 
 
 ellipse of/ and r, RS will describe equal areas in equal times 
 about S, by art. 6 17, and the force at R towards S will be to the 
 force at r towards/as SR to/r or (eg and CG parallel to the le- 
 spective tangents at r and R, being supposed to meet/r and SR 
 in g and G, y/and re being produced till they meet the circle in 
 :c and o) as GRt.ogr. But^/j being perpendicular to the tangent 
 at r inp, the force at r towards/is inversely as^" x rx; conse- 
 quently the force at R towards Sis directly as GR, and inversely 
 asfp^X-grx, or (because the rectangle grx is equal to Q.cr^, and 
 
 C R 
 
 tlierefore invariable) as — -, or (because fp is to re as/r to gr, 
 
 or as SR to GR) as •— j-; and when S is the focus of the el- 
 lipse, GR being always equal to CA, the force at R towards 
 S is inversely as the square of SR the distance from the focus, 
 as was shown in art, 446. 
 
 619. Let Aa and Bb (fig. 275) be any two diameters of the 
 ellipse that are pei-pendicular to each other, and CL die perpen- 
 dicular from C on the chord AB is always of the same length. 
 For ABflft is a rhombus, K/ and GH that bisect AB and Ba in 
 P and V are conjugate diameters, CG^ — CV* is to BV% or CV^, 
 asCGHoCK^andCGnoCP^asCG" + CKMoCK^ ButKQ 
 being perpendicular to GH in Q, CP* is to CK^as CL^toKQ*; 
 consequently CG^ is to CG^-f CK* as CL^ to KQ^; therefore 
 CG--f CK^ being invariable, and CG X KQ being likewise in- 
 variable, by art 616 and 6 17, it follows that CL is invariable 
 and always equal to the perpendicular from Con the chord that 
 joins the extremities of the transverse and second axis. Hence 
 the area of a rhombus ABab inscribed in the elUpse is as AB the 
 &icie of the figure, and is least when the figure is rectangular, 
 or when K/ and GH are the axis of the ellipse, and is greatest 
 when AB joins the extremities of the transverse and second 
 axis. 
 
 620. Upon AB (fig. 276) any diameter of an ellipse take the 
 points G and F, so that the square of GF may be equal to the 
 rectangle AFB; from G draw a right line GE that meets the el- 
 lipse in H and K, and FE(paral!el to the tangent atB)in E,then 
 HE,GE,and KEwillbe in continued proportion. For let a,b,g, 
 f, e. h, and k be the points in the plane of the circle from which 
 
 A, B,
 
 Chap. XIV. transferred hriefiy from the Circle. 107 
 
 A,B,GjF,E,H, and K are projected on the plane of the ellipse; 
 then ye will be perpendicular to cf, and the rectangle afb, or 
 cf' — c'6% equal to gf^ ; consequently ge^ is equal to ef^ + cf* 
 — c6% or ce^ — cZ>\ that is, to the square of the tangent et, or 
 to the rectangle hek ; therefore, he, ge, and ke being in con- 
 tinued proportion, HE, GE, and KE are likewise proportional. 
 It appears, in the same manner, that when G is any other 
 point upon the diameter AB, the difference of the rectangle 
 HEK and of the square of EG is to the square of the semi- 
 diameter parallel to EG, as the difference of AFB and 
 GF^ to CB% 
 
 62 1 . Let any quadrilateral figure aefb {Jig, 277) be inscribed 
 in the circle, and gm any parallel to f, one of its sides, meet 
 the other sides ah, ae, hfm g, k, I, respectively, and the circle 
 in h and m ; then gh, gk, gl, and gm will be proportional ; for 
 the angle gak being equal to efh, or gib, it follows that the 
 triangles gak and gib are similar, and the rectangle kgl equal 
 to agb, or hgm. From this it follows, that if AEFB be any 
 quadrilateral figure inscribed in the ellipse, and any right line 
 GM parallel to one of the sides EF meet the other sides AB, 
 AE, BF in G, K, L, and the ellipse in H and M, then GH, 
 GK, GL, and GM will be proportional. In this manner, many 
 other properties of the ellipse are briefly deduced, as lemma 
 C!4, 25, lib. 1, Princip. , But we shall onl}^ subjoin an instance 
 or two of the properties of the conic sections, that are briefly 
 demonstrated, by showing first that they take place in the 
 circle, and then transferring them to any conic section in 
 general, by considering it as the projection of a circle upon an 
 oblique plane, by rays that issue from a given point. 
 
 622. Letg (^"^.278) beagiven point in thcplane of thecircle, 
 c/* a right line through g that meets the circle in e andy^ et 
 and /if the tangents at e andy^ and their intersection t will he 
 always found in a right line given in position ; for, join eg, and 
 let td be perpendicular to eg in d, join ct, and it will bisect ef 
 in ra. The rectangle met is equal to ce% and the triangles cgm, 
 cf J being similar, the rectangle o^c^ is equal to met, and conse- 
 quently to ct^ ; therefore ed is given, and dt is given in posi- 
 tion. But it is obvious, that if this figure be projected upon 
 oblique plane by rays issuing from a given point V, the projec- 
 tion
 
 lOS General Properties of the conic Sections took Ti 
 
 tion of the circle will be a conic section, the right lines et 
 and ft will be projected by the tangents of the conic section, 
 the point g by a given point, and td by a right line in llic 
 plane of the conic section given in position. Therefore vvheo 
 G is a given point in the plane of any conic section, and EF 
 always passes through G, the intersection of ET and FT the 
 tangents at E and F will be always found in a right line TD 
 given in position. 
 
 623. Let the five points C, S, E, A, and B (fig. 279) be in 
 the circumference of the circle ; produce BC and ES till they 
 meet in D ; let CP and SP be drawn from C and S to any 
 point P in the circle ; let CP meet EA in N, and SP meet BA 
 in Q, then D, Q, and N will be always in a right line. Foe 
 let N« parallel to SE meet SQ in n, AP in m, and AB in r ; 
 let An meet the circle in b and SE in G, and BA meet with SE 
 in K. Because the angle AN« is equal to AEK, or APS, a 
 circle will pass through A, N, n, and P, and the angle N An (or 
 EA6) will be equal to NPw, or CPS; consequently the arch 
 Eb will be equal to CS, and C6 parallel to SE. Let BA, BS, 
 and AE meet Cb in f e., and /, and /"i will be to^^ as^e is to 
 jTC, by art. 62 1 . But KG is to KE 'as fb to^, and, because 
 BfS is a right line, KS is to KD as fe to fC. Therefore 
 KG is to KE as KS to KD ; and KG being to KE 
 as rn to rN, KS is to KD as rn to rN ; and because S, Q, and 
 n are in a right line, it follows that D, Q, and N are likewise 
 in a right line. From which it follows, by supposing this 
 figure to be projected as in the last article, that if C, S, E, B, 
 and A be five points in a conic section, and any two of the 
 light lines BC and ES intersect each other in D, CP and SP 
 be drawn to any point P in the section, CP meet EA in N, 
 and SP meet AB in Q, then D, Q, and N will be always in a 
 right line. Hence it appears that a conic section can be 
 drawn through those five points C, S, E, B, and A, by drawing 
 any line DQNfrom D meeting AEin Nand AB in Q,joining 
 SQ and CN; for their intersection P will be a point in the 
 conic section. And this is the method of describine: a conic 
 section through any fivegivenpoints(when nomore than two of 
 those points are in a right line), that was mentioned in art. 322. 
 The way of drawing a tangent to any point C of the conic sec- 
 tion^
 
 
 
 U/. 
 
 1 
 
 a 
 
 V4, 
 
 1 _ I 
 
 
 E^ i 
 
 i 
 
 h 
 
 / 
 
 f^ 
 
 ^7- 
 
 \ 
 
 1 
 
 H 
 
 ; 1 
 
 V 
 
 p" 
 
 \ 
 
 \ 
 
 /■ 
 
 3 
 
 "i 
 
 i 
 
 > 
 
 ^ 
 
 Mk 
 
 7 
 
 \ 
 
 [^ 
 
 \ 
 F 
 
 1 
 
 
 

 
 ■ all- XaX-flzjcrf. I'vUI. 
 
 ■ "m^^' 
 
 r\
 
 Chap. XIV". transferred briefly from the Circle, 199- 
 
 tion> that was described in art. 324, may be demonstrated in the 
 same manner^ by showing first that it takes place in the circle. 
 By supposing one or more of the right lines that were inscribed 
 in the conic section to be tangents,, several properties of those 
 figures may be briefly deduced from this proposition ; parti- 
 cularly that which was mentioned (as analogous to aproperty of 
 the lines of the third order) in art. 401, is the case when the 
 right lines ESD and BCD are tangents at S and C. 
 
 624. Let P, H, and K {fig. 280) be any three points in an 
 ellipse, let PM parallel to HK and KN parallel to PH meet 
 the ellipse in M and N, and a right line through H parallel to 
 MN will be the tangent at H. When the iigure is a circle, 
 the arch HM is equal to HN, MN is perpendicular to the 
 diameter through H, and consequently parallel to the tan- 
 gent at H. This property is extended to the ellipse, by art. 
 6\ 1 and 6l2, and may be demonstrated of any conic section. 
 But we proceed now to these properties of the ellipse, which 
 we had chiefly in view, because of their use in the following 
 enquiries. 
 
 625. Let PH (fg. 281) be a chord of an ellipse parallel to 
 the axis DE, LK any ordinate to this axis at V, meeting the 
 (ellipse in L and K, join DL and DK, let PM and PN parallel 
 to DL and DK meet it in M and N, let MQ and NR be per- 
 pendicular to PH in Q and R, then the sum or difll:erence of 
 PQ and PR (according as Q and R are on the same or differ- 
 ent sides of P) will be to 2DV as the chord PH to the axis DE. 
 For supposing, first, the figure to be a circle, let the semidia- 
 meter HC meet the circumference again in I ; and the arches 
 HM and HN being equal to EL and EK, and consequently 
 equal to one another, the right line MN will be bisected 
 perpendicularly by the diameter HI in X. Because 
 the arch MH is equal to LE, IX is equal to DV ; let 
 XZ be perpendicular to PH in Z, and because the angle HPI 
 is right, PZ will be to IX (or DV) as PH to IH or DE. But 
 QR is bisected in Z; therefore PR + PQ is equal to2EZ,and 
 is to 2DV as PH to DE. This is extended to the ellipse by 
 art. 6 11 and 6 14, and may be demonstrated of any conic section. 
 626. Let PJ and He {fig. 282) be perpendicular to the axis 
 JDE in d and e, describe an ellipse adde upon the axis dc simi- 
 lar
 
 110 Of the Gravitation Boole T, 
 
 lar to ADBE. Let Ik an ordinate from the internal ellipse to 
 the axis de in v meet this ellipse in / and k. Let PM and PN 
 parallel to dl and dk meet the external ellipse in M and N, MQ 
 andNRbeperpendiculartoPHinQandR^andPR + PQv/illbe 
 equal to Q.dv. For dv will be to DV'^ as de (or PH) toDE, and 
 therefore as PR + PQ to 2DV, by the last article ; consequently 
 PR + PQ is equal to idv. This may be demonstrated in the 
 usual manner from the property of the ellipse described at the 
 end of art. 6 12. 
 
 627. Let the right hne PS perpendicular to the ellipse in P 
 meet the axis DE in Sj and SZ be perpendicular to the semi- 
 diameter CP in Z, then the rectangle CPZ will be equal to the 
 square of the semiaxis CA, that is conjugate to CD. For let 
 PY parallel to AC meet CD in d and CO the semidiameter 
 conjugate to CP in Y ; and let PS meet CO in T. Then, be- 
 cause PS is to PZ as PC to PT, the rectangle CPZ is equal to 
 SPT, which (because PT is to PY as Vd to PS) is equal to 
 the rectangle c?PY, and therefore is equal to CA% by art. 6l2. 
 In the same manner if PS meet the axis AB in /, the rectangle 
 /PT will be equal to CD% and PS will be to P/as CA^ to CD*. 
 Because dC is to dS as P/to PS, <?S is to dC as C A^ to CD*. 
 
 628. Supposingthat the gravitation towards anyparticlede- 
 creasesin the same proportion that the square of the distance from 
 itincreasesJetPAEa;,PBF60?g'. 28S)besimilarconesconsisting 
 of such particles, terminated by spherical bases AEcf, BF6that 
 have theircentrein P; and the gravitation at P towards the solid 
 PAEa will be to the gravitation at P towards PBF6 as PA to 
 PB,or inthesame ratio as any homologous sidesof these similar 
 solids. For let MNw be any surface similar to AEa having lis 
 centrelikewisein P; and the gravitation towards the surface AEa 
 w-illbe to that towardsMNw in the ratio compounded of the di- 
 rect ratio of the surface AEa to MNttz (or PA"" to PM^) and of 
 the inverse ratio of PA^ to PM^, that is, in a ratio of equality ; 
 consequently, the gravitation towards the surface AEaA being 
 represented by A, the gravitation towards the solid PAEa will be 
 represented b^-A X PA, and that tov/ards the similar solidPBF^ 
 byA X PB,whichareintheratioof PAtoPB. in thesamcman- 
 ner the gravitation towards the frustum that is bounded by the 
 surfaces AEA, M Nw is represented by A x AM . It is manifest 
 
 that
 
 Chap. XIV. towards Spheres and Spheroids. 1 1 1 
 
 that though the surfaces AEa and MN/Tz be of any other form, yet 
 the ultimate ratio of the gravitations at Ptowards the conical or 
 pyramidal solids PAEa, PMNw is that of PA to PM; and that 
 if AQ and Mq be perpendicular to PH in Q and q, these forces 
 reduced to the directionPH will be ultimately in the ratioof PQ 
 to Pq. 
 
 629. The forces with which particles similarly situated with 
 respect to similar homogeneous solids gravitate towards these so- 
 lids are as their distances from any points similarly situated in 
 the solids, or as any of their homologous sides. For such solids 
 may be conceived to be resolved into similar cones, or frustums 
 of cones, that have always their vertex in the particles ; and the 
 gravitation towards these cones, or frustums, will be always in 
 the same ratio. 
 
 630(Jig.2S% N.2).A particle placed within thehollow solid that 
 isgeneratedbythetheannular space terminated by two concentric 
 circles, or similar concentric ellipses, ADBEand adbe, revolving 
 abouttheaxisAB,hasnogravity towardsthis solid. For letp be 
 any such particle, pk any right line from p that meets the in- 
 ternal circle or ellipse in any points /* and q, and the external 
 figure in x and r; then if xr be bisected in z,fq v.ill be like- 
 wise bisected in z, because the figures are similar and similar- 
 ly situated; consequently yT is equal to qr; and the gravita- 
 tions of p towards opposite frustums of the solid that have their 
 vertex in p and are terminated by the same right lines produced 
 fromp with opposite directions will be always equal, by art. 
 628, and mutually destroy each other's effect. 
 
 631. It follows from this that the gravity at any point p in 
 the semidiameter CP towards the sphere or spheroid is to the 
 gravity at P as Cp to CP ; because the gravitation towards the 
 solid generated by the annular space that is included betwixt 
 APB, apb has no effect upon a particle at p; so that the 
 gravity at p tov/ards the whole solid ADBE is equal to the gra^ 
 vity at p towards the solid adbe, which is to the gravity at P 
 towards the sohd ADBE as Cp to CP, by art. 629- 
 
 632. Let two planes PMKI and PNLI (Jg. 284) intersecting 
 each other in the right line PI include a proportion of a solid be- 
 tween them,th at consists Qfparticleswhich attract in this mannep. 
 
 M
 
 112 Of the Gravitation Book I. ^ 
 
 ]et PandG be any two points in the right line PI,GKbe always 
 parallel to PM, and tlie planes PMN, GKL perpendicular to 
 PMKI intersect the plane PNLI in the right lines PN and GL; 
 then tlie gravitation of P towards the pyramidal solid PMwwN 
 generated by the plane PMN revolving about P will be to the 
 gravitation of G towards the pyramidal solid GK/:/L generated 
 by GKL revolving about G ultimately in the ratio of PM to 
 GK, when the inclination of the planes and the equal angles 
 MPwZj KGAare supposed to be diminished till they vanish. For 
 the right lines PM and GK being always parallel, MN will be 
 ultimately to KL as PM to GK, and the angle MPN equal to 
 KGL; consequently, the angles MP/w and KGA being equal, 
 the gravitation of P towards the solid PMw2/?N will be to the 
 gravitation of G towards Gliklh as PM to GK, by art. 628. 
 
 QSo. Any sections of a spheroid made by parallel planes are 
 similarellipses.LetAB(^£;'.285)betheaxisofthesolid,GPHasec-. 
 tion of the spheroid by a plane perpendicular to the generating 
 ellipse ADBE in GH and PM be perpendicular to GH in M, 
 let KML perpendicular to AB the axis of the solid meet the 
 ellipse ADBE in K and L, and CQ be the semidiameter paral- 
 lel to GH. Then because PM is perpendicular to the plane 
 ADBE, the points K, P, and L will be in a semicircle described 
 upon the diameter KL, and the square of PM equal to the rect- 
 angle KML, which (by art. 6lo) is to the rectangle GMH 
 as CD^ to CQ"^; consequently the section GPH is an ellipse> 
 and is similar to any other section of the solid by a parallel 
 plane, the ratio of the axis GH to the other axis being that of 
 CQ to CD. From this it follows that the sections of two si- 
 milar concentric spheroids similarly situated, which are made 
 by the same plane, are similar ellipses; because they are similar 
 to the sections of the same solids by a parallel plane that passes 
 through their common centre; and these last are similar by 
 art. 122. It appears likewise that all sections of the spheroid 
 made by planes perpendicular to the circle generated by the 
 axis CD (which we may call the Equatoj' of the solid) are simi- 
 lar to the generating ellipse ADBE. 
 
 634. The gravity of any paiticle of a sphere or spheroid be- 
 ing resolved into two forces, oue perpendicular to the axis .of 
 
 the
 
 Chap. XIV. towards Spheres a)id Spheroids. IIS 
 
 the sohd, the other perpendiculai* to the plane of its equator, all 
 particles equally distant from the axis tend towards it with 
 equal forces, and ail particles at equal distances from the plane 
 of the equator gravitate equally towards this plane, whether the 
 particles be at the surface of the solid or within it. And the 
 forces with which particles at different distances from the axis 
 tend towards it are as these distances : the same is to be said of 
 the forces with which they, tend. towards the plane of the equa- 
 tor. This easily appears of the spherQ from what was shown in 
 art. 6S 1 > and is mentioned for the sake of the analogy only. Let P 
 (//g. '286) be anypointinthesurfaceof aspheroid^APDBEasec- 
 tion of the solid through its axis AB, Vf a perpendicular to AB 
 in f, Pd a perpendicular to the equator of tlie solid in d ; and 
 the gravity at P towards the solid being resolved into a force in 
 tiie direction P/"and another force in the direction Vd, the for- 
 mer will be equal to the gravity at d towards the solid, and the 
 latter equal to the gravity atf. Let adbe be a spheroid similar 
 to ADBE having the same centre C and its axis ab in the same 
 risht line AB with the axis of the external solid. The sections 
 of these spheroids by any plane that passes through the right 
 line Pdfl will be similar concentric ellipses similarly situated by 
 art. 633, and the gravity of P in the direction lyperpendiculat 
 to the axis AB that arises from the attraction of any portion or 
 slice of the external solid contained by two such planes will be 
 equal to thegravityatJin the direction <iC which arises from the 
 attraction of the slice ofthe internal solid that is contained by the 
 same planes. To demonstrate this, let PMNIG (fig. 287), Vmnlg 
 (in the next figure) be the sections ofthe external sohd by two 
 such planes, dKLd and dkld the sections ofthe internal solid by 
 the same planes ; let KL be an ordinate at V to de, the axis of 
 the internal ellipse which is in the plane of the equator ofthe 
 solid, join dK and dh, and let PM and PN be always parallel to 
 dK and dh, respectively. Let the planes PMw, PN?/, dKk, 
 </L/perpendicular to the planePMISlGmeetP/72«rginthe right 
 lines Fm, P//, dk, and dl, respectively; and let those planes re- 
 vohe about the points P and D, PM being always parallel to 
 dK and PN to dL, while V is supposed to describe the rightline 
 ed. Then the forces with which P and d are attracted towards 
 VOL. n. I the
 
 114 Of the Gravitation Book I. 
 
 the p3-raraidal solids generated by the planes PMw, PN«, dKk, 
 and dLl will be ultimately as the right lines PM, PN,<fK, and 
 dlj, respectively, by art. (33'2; and Pp being parallel to de, and 
 j\IQ and NR perpendicular to Pp in Q and R, if these forces be 
 resolved into such as act in tlie directions Pp and de, and such as 
 act in the right lines perpendicular to these, the former will be 
 as the right lines PQ, PR, dV, and dY. But PRTPQ is al- 
 ways equal to Q,dY by art. 626. Therefore the gravity of P m 
 the direction Pp arising fyom the attraction of the pyramidal so- 
 lids generated by the planes PM,vi and FiSn is ultimately eqr.ai 
 to thegravity of ^ in the direction </e arising from the attraction 
 of tha pyramidal solids generated by the planes dK/c and dLi. 
 And since this always holds, while we conceive the point V to 
 describe cd, and these phmes to describe the portions of the ex- 
 ternal and internal solids terminated byPMNlGand Pm«I^,it 
 follows that the gravity of P in the direction Pp arising from the 
 attraction of the whole portion of the external solidbounded by 
 theplanesPMNlG, P/w Jg is ultimately equal to the gravity of d 
 in the direction de that arises from the attraction of thepartof the 
 internalsolid bounded by the same planes, when the angle con- 
 tained by those planes is supposed to be diminished till it vanish. 
 ByC^g .286) conceiving other slices of the solids contained by 
 planesthatpassthroughPJI,and form equal angles v.'iththeplane 
 APDBontheothersidcj to attract the particles P and c?, it willap- 
 pear that the gravi ties of P and rf towards the axisof th e spheroid 
 wisingfromthe joint attraction of those slices will be equal. And 
 since thisholdsof all the portions of the solids contained bysuch 
 planes, it foUovvsthat the force with which P tendstowards the 
 axis ABarisiug from the attraction of the wl'.ole spheroid, isequal 
 lo the gravity of d towards the internal solid, or (by art. 631) 
 to its gravity towards the whole external solid ADBE. The 
 gravity of any particle p situated in the right line l*d toward* 
 the spheriod ADBE, is equal to its gravity towards a similar 
 concentric spheroid similarly situated that has Cj) for its semi- 
 diameter, by art. 631, and therefore its gravity in the direction" 
 perpendicular to AB is equal to the gravity ol d towards the 
 solid adbe, by what has been shown. Therefore all particle^ 
 equally disfea-nt from the axis tend U)wards it with equal forces ; 
 
 aad
 
 Chap. XIV. towards Spheres and Spheroids, 115 
 
 and because the gravity at d is to the gravity at D as Cd to CD, 
 bv art. 6Sl, it follows that the gravity of P towards the axis is 
 to the gravity at D towards the spheroid as P/"to DC. In the 
 same manner it is shown, that the gravity of P towards the 
 plane of the equator is equal to the gravity of y towards the 
 spheroid ADBE, and is to the gravity at A towards the spheroid 
 as/C or P^ to AC. 
 
 6S5. In order therefore to find the direction in which the 
 spheroid attracts any particle atP, and the force of this attfac* 
 lion, let A denote the attraction at the pole A, and D the at- 
 traction at the equator, let Fd be perpendicular to the plane of 
 the equator in d, upon dC take dQ from d towards C, so that 
 «?Q may be to dC asT) x CA to A X CD, join PQ ; the attrac- 
 tion towards the spheroid wWi tend in the direction PQ, and be 
 always measured by this right line PQ. For the gravity towards 
 the spheroid in the direction Vd being to A the gravity at the 
 pole as Pc? to AC, and the gravity at Pin the direction Pf, or 
 dC, being to D as dC to DC, by the last article ; it follows, 
 that the gravity at P in the direction Vd is to the gravity at P 
 
 in the direction P/as A X ^ to D x^> that is (by the sup- 
 position) as Vd X dC to dQ X dC or as Pc? to dQ; consequent- 
 ly the gravity at P is in the direction PQ ; and if the gravity 
 at A towards the spheroid be represented by AC, the gravity 
 at P in the direction I*d will be represented by PJ, and the gra- 
 vity at P towards the spheroid by PQ. In the same manner if 
 fq be taken upon the axis from /"tow^ards C in the same ratio to 
 /C as A X CD to D X CA, then P^' will always show the direc- 
 tion and measure the force of the gravity at any point P to- 
 wards the spheroid, supposing the gravity at D to be represent- 
 ed by DC. Let Dx perpendicular to CD represent the gravity 
 at D, join C±, and besause the gravity of any particle in the 
 semidiameter CD is as its distance from C, the gravity of the 
 column DC (the spheroid being supposed to be fluid) will be 
 measured by the triangle CD-r or ^ CD X Dx, or | CD X D. 
 In the same manner the gravity of the column AC will 1*» 
 measured by i AC X A. And as the columns CD and AC gra- 
 yi tate equally in the-sphere, so the gravity of the column CD is 
 
 I 2 greatei
 
 Il6 Of the Figure of tliePlaiiets, Book f , 
 
 greater or less than the gravity of the column AC in the s^*he- 
 roid according as it is oblate or oblong, and CD x D is greater 
 or less thanCA X A in the spheroid, according as CD is greater 
 or less than CA (because the fluid would sink in the former 
 case at D,and in thelatter at A, till its figure becomesspherical). 
 But this will appear more fully afterwards when we come to 
 determine the ratio of A to D in a given spheroid. 
 
 636. Hitherto we have supposed the particles of the spheroid 
 to be affected only by their mutual gravitation towards each 
 other. Let us now suppose any new powers to act upon all the 
 particles of the spheroid in right lines, either perpendicular to 
 the axis of the spheroid, or to the plane of its equator ; or some 
 powers to act in right lines perpendicularto the axis, and others 
 in lines parallel to it ; and let each force vary always as the dis- 
 tance of the particles from the axis, or equator, to which the 
 direction of the force is supposed perpendicular. Then the 
 spheroid being supposed to be fluid, if CA be to CD inversely 
 as the whole forces that act on equal particles at A and D, the 
 fluid will be every where in aquilibrio. To demonstrate this 
 proposition fully, we shall show, 1. That the force which re- 
 sults from the attraction of the spheroid and those extraneous 
 powers compounded together acts always in a right line perpen- 
 dicular to the surface of the spheroid. C. That the columns of 
 the fluidsustain or balance eachotherat the centreof the sphe- 
 roid. And, 3. That any particle in the spheroid is impelled 
 equally in fill directions. 
 
 637. 1 {Fig.Q.BQ). Lettheforcesthatresultfromtheattractiort 
 of the spheroid and the extraneous powers at AandD be call- 
 ed M andN; andMwillbetoNasCDtoCA,bythesupposition. 
 Because the attraction of the spheroid at Pin the direction Vd 
 is to its attraction at A as Pc? to AC, and the force of each ex- 
 traneous power at Pis supposed to be to the force of the ^ame 
 power at A in the same ratio of P^ to AC, it follows that the 
 whole force by which a particle atP tends in the direction Pc? 
 is to Mas Vdio AC. In the same manner the whole'force with 
 which a particle at P tends in the direction ly is to N as P/'or 
 dO to DC ; consequently the force with which P teiids in the 
 direction Vd is to the force with which it tends in the direction 
 
 Tf
 
 Chap. XIV. and Variation of Gravity towards them. 117 
 P/as M Xtc *° -^ '^Iic ^ ^"^^ supposing PK that meets CD 
 in K to be the direction in which a particle at P tends towards 
 the spheroid from the composition of those two forces, Prf will 
 be to dK in the same ratio ; so that dK will be to dC as N X 
 AC to M X DC, or (because N is to M as AC to DC, by thp 
 supposition) as AC^ to DC^. But if PK was supposed pet^ 
 pcndicular to the ellipse APDB at P, d K would be to dC in 
 this same ratio, by art. 627. Therefore any particle as Pat 
 the surface of the spheroid tends towards it in a right line per- 
 pendicular to its surface ; and the force M which aqts on a 
 particle at the pole A being represented by the semiaxisAC,'the 
 force wiiich acts on an equal particle at any point of the surface 
 P will be alwaj's represented by the perpendicular PK terminat- 
 ed by the plane of the equator of the solid in K. It appears 
 iikewise that any particlej^ within the spheroid in the semidia- 
 meter CP tends in the direction p/r parallel to PK, with a force 
 that is measured by the right line pk terminated by the same 
 plane in A:, because th,e forces that act on P and^ in right lines 
 perpendicular to the axis, or equator, are as the distances fronx 
 the axis, or equator, by th-e supposition, 
 
 638. In order to show, that when the spheroid is fluid, the 
 columns sustain each other at the centre, let KZ (fig.Q^Q) an4 
 J:z be perpendicular to PC in Z and %; then the force M which 
 acts at the pole A being represented by the semiaxis AC, the 
 force with which particles at P and p tend in the direction PC 
 will be represented by PZ andps respectively ; and because pz 
 is to PZ as Cp to CP, the gravity of the whole column PC in 
 the direction PC will be measured by i PZ X PC, which i;i 
 equal to |- CA^ (by art. 627), or to i CAxM. Therefore the 
 gravity of any column PC in the direction PC is equal to the 
 gravity of the column AC in the direction AC, and all the 
 columns of the fluid sustain each other at C. 
 
 639- Letp (Jig. 288) be any particle in the spheroid, Vp a co- 
 lumn from the surface to the point p, produce Cp till it meet 
 the surface in q ; upon CA take CO in the same ratio to CA as 
 Cp is to Cq, and the gravity of the column P/? in the direction 
 Tp will be equal to the gravity of the column AO in the direc- 
 
 I 3 tion
 
 1 18 Of the Figure of the Planets, Book 1 . 
 
 tion AC. First, let Vp be in the plane APDB, let PG and pg 
 be perpendicular to the plane of the equator in G and g, PL and 
 pi perpendicular to the axis AB in L and /,- and Pp being sup- 
 posed to meet AB inland DE in h, let gc and lu be perpendi- 
 cular to Vp in e and u. Then the force with which the particle 
 p tends in the right line parallel to the axis will be to M nspg 
 to AC, by the supposition ; and this force reduced to the direc- 
 tion Pp will be to M as pe to AC. The force with which p 
 tends in the duection perpendicular to the axis is to N as pi 
 to DC, and this force reduced to the direction Vp is to N as 
 pu to DC. Therefore the whole force with which p tends 
 
 in the direction P» is M x -^ + N x ^5 or M X 4?; x 
 
 AC-* J-zO AC/ 
 
 ^ + N X ^ X -75* From which it follows that the gra- 
 vity of the whole column Vp in the direction Vp is jtjtt x 
 
 ^ X hV^-hp^ + 2DC ^ ^» ^ /P -//^ that IS 2AC ^ 
 
 PG=-/^' + ^ ^ rL"-/^- But PL^ is to CA* — CL^ 
 and pi"- to CO* — C/* as CD* to CA* ; consequently PL* 
 — pt is to the difference of CA*— CL* and CO* — C/* 
 in the same ratio, and (because M is to N as DC to AC) the 
 whole gravity of the column Vp in the direction Vp will be 
 toi M X AC as CL* — C/* + CA* — CO* + C/* — CL* 
 to CA*, that is as CA* — CO* is to CA*, and consequently 
 equal to the gravity of the column AO in_lhe direction AC. 
 Therefore the particle p is pressed equally in all directions in 
 the meridian plane APDB that passes through p. In like man- 
 ner it is shown, that any other columns from the surface of the 
 spheroid to the particle p press equally upon it, and sustain 
 each other. 
 
 640. We conclude, therefore, that when the particles of a 
 fluid spheroid of an uniform density gravitate tqwards each 
 other, with forces that are inversely as the squares of their dis- 
 tances from each other, and any olher powers act on the par- 
 ticles of the fluid, either in right hnes perpendicular to the
 
 PlateXSXi XTaiib. Vrl.lL.
 
 FlateZXXll«( j,/tv,<u 
 
 ^ 
 
 r
 
 C'liap. XIV. atid Fariation of Gravity totcard^thtm. 119 
 axis that vary in the same proportion as the distances from the 
 ;i.\is, or in riglit lines perpendicular to the plane of the equator 
 that vary as their distances from it^ or when any powers acton 
 the particles of the spheroid that may be resolved into such as 
 these; then if the whole force that acts at the pole A be to the 
 whole force that acts at the circumference of the equator as 
 the radius of the equator to the semiaxis of the spheroid, the 
 fluid will be every where in (cquiUbrlo ; surfaces similar and con- 
 centric to ADBE will be the level surfaces at all depths; and 
 the forces with which equal particles at those surfaces tend to-' 
 wards the spheroid, will be measured by perpendiculars to the 
 surfaces terminated either by the plane of the equator, or by the 
 axis of the spheroid. 
 
 641. This theorem is of use in several philosophical enquiries. 
 Supposefirsta fluid spheroid ADBE(Jig. 286) of an uniform den- 
 sity to revolveonitsaxisAB; let the attraction of the spheroid at 
 the pole A be represented by A, the attraction at the circum- 
 ference of the equator by D, the centrifugal force there arising 
 from the rotation of the spheroid by V; then if CA be to CD as 
 
 D — V to A, or V be equal to the excess of D above A ^ p-> 
 
 the fluid will be every where in aquiUhrio. For in this case M 
 is equal to A, there being no centrifugal force at the pole; the 
 force N that acts on any particle in the circumference of the 
 equator is equal to D — V the excess of the attraction above 
 the centriiugal force there; and the centrifugal force, with 
 which any particle of the spheroid endeavours to recede from 
 its axis in consequence of the rotation of the spheroid, is as its 
 distance from the axis; consequently if A be to I) — V as CD 
 to CA, tlie fluid will be in (tquilibrio in all its parts, by what 
 has been shown. It appears therefore that if the earth, or any 
 other planet, was fluid, and of an uniform density, the figure 
 which it would assume in consequence of its diurnal rotation 
 would be accurately that of an oblate spheroid generated by 
 an ellipsis revolving about its second axis, as Sir Isaac New- 
 tun supposed : and v.e cannot but observe, that as no theory 
 of gravity has a foundation in nature but his only, so no 
 Other gives so simple a figuie of the planets, as vyill appear 
 
 I 4 by
 
 120 Of the Figure of the Planets, Book I, 
 
 by comparing what was demonstrated above in art, 4913, 
 This theorem is applicciblc in Hkc manner to the theory of the 
 tides. But before vye proceed to a more particular application 
 of it, we 3re first to show how the gravity towards a spheroid 
 at the pole is easily measured by a circular or hy})erbolic area, 
 according as the spheroid is oblate or oblong ; and how the gra- 
 vity towards it at the circumference of the equator, or at any 
 distance in the axis, or in the plane of the equator produced, is 
 determined from the gravity at the pole, without any new qua- 
 drature or computation. For this end we premise the follow- 
 ing lemma. 
 
 642. LetADfZa (7?g-CS9)be any section ofa solid of an uniform 
 density by aplanethatpassesthroughagivenpointP,andPC,PH 
 be right lines given in position in this plane; let ariy right line 
 PM drawn from P meet the figure ADr/a in jM and m, and a 
 circle BN6 described with the given radius PC in N; let MQ 
 and mq be always perpendicular to PC in Q and q, and NR be 
 perpendicular to PH in P^; upon RN take RK equal to 
 PQ — Vq; and let the ordinate RK always determined in this 
 manner generate the area HGgA, while PJNI revolves about P. 
 from PA to P«. Then if we suppose another plane that passes 
 through the right line PH to cut the same solid, the gravity of 
 the particle P towardsthe slice of the solid included betwixt 
 those two planes and that stands upon the base AD(/<z, when re- 
 
 duced to the direction PC, will be ultimately as - Jt ? the angle 
 
 contained by the two planes being supposed to be continually 
 diminished til! it vanish. For let anothcrrightline PSmeet the 
 figure ADda in S and s, and the circle BN6 in ri ; let MZ and 
 nr be perpendicular to PH in Z and r, the arch Mo described 
 from the centre P meet PS in 0, and l\\x, on be perpendicular 
 on the ether plane that is supposed to pass through PH 
 in X and u. Then if the point P be without the figure KDda, the 
 gravity at P towards the p3'raniidal solid PMo//a.' will be ulti- 
 mately as Isltn X —~~-^ hy art. 628, or (because M.r is as 
 
 MZ) as Mm x -^rr-' that is (because Mo is to N« as MZ 
 
 to
 
 Chap. XlV. and Variation of Gravltji lotbards them. 121 
 toNR, and the rectangle Mo x MZ to Nw x NRas PjVP to 
 PC^) as Mm x ^"^^f -? or (N« being to Rr as PC to NR) 
 as Mw X ~ ; and the gravity of P towards that solid reduced 
 
 1 T • T>/-i -11 1 /^ Kr RK X Iir -r. 
 
 to the direction PL^ will be as Qq X -j^r; or — — — But 
 
 IIK X Rr ultimately measures the fluxion of the area IIGKR. 
 Therefore the gravity of P in the direction PC, that arises Irom 
 the attraction of the whole slice of the solid which has the 
 
 figure ADc/a for its base, is ultimately' as— p^, the angle con- 
 tained by the planes which terminate the slice being continual- 
 ly diminished. When the point P is betwixt M- and m, thea 
 RK is to be taken equal to PQ — P^, and the gravity at P is 
 measured in the same manner. It follows from this lemma, 
 that, supposing the figure ADc?a to revolve about the axis PH, 
 and to generate a solid, and the direction PC to coincide with 
 
 PH, the gravity at P towards this whole solid willbe as ■ f^ -* 
 
 When the particle P is so situated with respect to the figure 
 ADda, that the perpendiculars from the points M intersect PC 
 on different sides of P, the gravity at P in the direction PC is 
 to be determined from the difference of the areas generated by 
 the ordinate RK. 
 
 643. Let aparticle atP (fig. CQO) gravitate tow^ards the sphere 
 generated by thesemicircleADBabout theaxis AB,and C being 
 the centre of the sphere, let any riglitline PM meet the semi- 
 circle in INI and m, and the circle CNH described from the cen- 
 tre P in N ; let NR be perpendicular to PC in R,-and RK be 
 always equal to Q^ when P is without the sphere or in contact 
 with it.'^ . Let CL be perpendicular to PM in L, and Mm being 
 bisected in L, LM^ will be equal to PL' — MPm or PR^ — 
 APB, a«d the fluxion of LM"- equal to the fluxion of PR"- ; so 
 that the fluxion of PR will be to the fluxion of LM as LM to 
 PR. And because KR or Qq is to 2LM as PR to PN, the 
 ^uxion of the area generated by the ordinate KR is in this case 
 
 equal
 
 122 Of the Figure of the Planets, Book I. 
 
 PG 
 
 equal to the rectangle contained by -7—- and the fluxion of 
 
 LM. Therefore the area IRK is etjual to p -> and the 
 gravity at P towards the portion of the sphere generated by the 
 segment MDm about the axis AB is as „, > and consequent- 
 ly as the cube of Mm the chord of the segment ISID/w directly, 
 and the square of PC the distance of the particle from the cen- 
 tre inversely. Hence the gravity at P towards the whole 
 sphere is as the cube of its diameter (or its quantity of matter, 
 the density being given), directly, and the square of PC in- 
 vers<:ly ; and is the same as if we should conceive the whole 
 matter in the sphere to be collected in its centre. The same i* 
 to be said of the gravity towards the aggregate of any number 
 of sucli spheres that have a common centre ; from which it fol- 
 lows, that however variable the density of a sphere may be at 
 diflevent distancesfrom the centre, providing the densisty be al- 
 ways the same at the same distance from it, the gravity of a 
 particle(that is not within the sphere) towards it will be as the 
 quantity of matter in the sphere directly, and the square of the 
 distance of the particle from its centre inversely. It appears 
 from vrhat has been shown, that the whole area IKGC is equal 
 
 to ^^[^r-r-j and that the trravity at A towards the spliere ADBB 
 
 ismeasuredbv^- — accordms; to the last article. 
 
 644. Let ADB E (fig. G9 1 , IV. 1 and 2) be now a spheroid of an 
 uniform density, generated by the semi-ellipse ADB revolving 
 about the axis AB; letAM any rightlinefrom the pole Ameetthe 
 ellipse in M and the circle CN H in N; let MQand Nil be perpen- 
 dicular to AB in (iand K; upon RX take RK always equal to 
 AQ, and let this ordinate RK generate the area AKGC while^ 
 AM revolves about A and describes the area of the semi-ellipse ' 
 ADB. Then the gravity at the pole A tov^ards the spheroid 
 
 ADBE will be measured by — r-r- (bv art. 642), and is to the 
 
 gravity at A towards a spliere described upon the diameter AB 
 
 (whicli
 
 Chnp. XIV. and Variation of Gravity towards tJiem. 123 
 (which is measured by \ AC, by the last article) as the area 
 AKGC to I CA\ 
 
 64-5. The gravity al D at the circumference of the equator 
 towards the spheroid is to the gravity at D towards a sphere 
 described upon the diameter of the equator as SCA* — AKGC 
 to ^ CA% and to the gravity at the pole A as 2CA'' — AKGC 
 
 CD 
 
 to 2AKGC X ^' For suppose the two elliplic sections 
 
 DBEA and J)bea to be perpendicular to the plane of tlie equa- 
 tor of the solid, and to intersect each other in the right line hdg 
 theircoramon tangent atD; let any rightline D/wfrom D meet 
 the ellipse in m, and the circle cnh described from the centre 
 D with the radius Dc (equal to AC) in 7i, let mq be perpendi- 
 cular to DE in q, and nr perpendicular to DA in r meet mq in 
 k; and let AA-ED be the area generated by the ordinate rk^ 
 while J))n revolves aboutDand describes the elliptic areaBBE; 
 then the gravit}^ at D towards the shce of the spheroid con- 
 tained by the planes DBEA and Yibea will be ultimately mea- 
 sured by ~^y the angle contained by those planes being 
 
 given, by art, 642. But if RK produced meet GI parallel to 
 AC in X, and the right lines AM and Dm revolve about A and 
 D so that the angle hDm be always equal to BAM ; then D« 
 being equal to AN, and the angle rJ)n to RAN, Dr will be 
 always equal to AR, and hr equal to CR or G.r. Because qm"^ 
 is to the rectangle D(/E as the rectangle AQB to QM% and the 
 triangles Dyw, MQ A being similar^ qm is to D^- as AQto QM^ it 
 follows that qm is to </E as QB to QM, and J)q to ^E as QB to 
 AQ ; consequently DE and BA are divided in the same propor- 
 tion in q and Q, so that D^ is to QB, or rk to xK, as DE to 
 AB. Therefore the bases lir and Gx being always equal, the 
 area hrk is to the area Ga:K in the same constant ratio of DE to 
 AB, and the area MED to GKAI (or CA X AB — AKGC) 
 as CD to CA. When ADBE is supposed to be a circle describ- 
 ed upon the diameter- DE, or CD is supposed equal to CA, 
 the area AKGC is equal to |. CD' (by art. 643), and GKAI 
 equal to ^ CD*. Therefore by article G42, the gravity at D 
 towards the slice of the spheroid contained by the planes DBEA 
 
 and
 
 1«24 Of the Figure of the Planets, Book I. 
 
 and D^ffltisto the gravity at D towards the slice of the sphere 
 
 ^ described upon the diameter DE that is contained by the same 
 
 , MEQ , 4CD ,1 . . 2CA^— AKGC CD , 
 
 planes, as — ^r— to --r- that is, as —. x tt- to 
 
 ■» ■' Dc d i CA CA 
 
 4CA* 
 
 ^ CD, or as 2CA^ — AKGC to — ^. The gravity at D tOr 
 
 wards the spheroid is to the gravity there towards the sphere de- 
 scribed upon the diameter DE in the same ratio; because the 
 section of the spheroid by an^' plane perpendicular to the equa- 
 tor is always an ellipse similar to DBEA, and the section of the 
 sphere described upon the diameter of the equator made by the 
 same plane is always a circle having that axis of the former 
 which is homologous to DE ibr its diameter ; and the gravity at 
 D towards the elliptic slice of the spheroid contained by any 
 tvvosuch planes, is always ultimately in the same ratio tothegra- 
 vity at D towards the circularslice ofthe sphere contained by the 
 same planes. Therefore the gravity at D towards the spheroid 
 ADBE is to the gravity at D towards the sphere described upon 
 the diameter of the equator as iCiV — AKGC to I CA^ But 
 the gravity at Dtowards this sphere istothegravit}" atA towards 
 & sphere described upon the axiSx'VB asCDtoCA; andthis latter 
 gravity is to the gravit}^ at A towards the sphcriod ADBE as 
 f CA^to AKGC by the last article; consequently the gravity at 
 D towards the spheroid is to the gravity at A towards it as 
 
 2CA^— AKGC to 2AKGC x ^. It appears likewise that the 
 
 gravity at A towards a sphere, described upon the axisAB being 
 represented by f CA according to article 043, the gravity at 
 
 A towards the spheroid will be measured by -■? and the 
 gravity at D towards it by ^-^^^Z^- X CD or CD — 
 
 AKGC CD . . 
 
 ~^CA~ ^ CA* 
 
 646. In order to measure the area AKGC, let F be the focus 
 of the generating ellipse, and because AQ is to QM (or the 
 jrectangle AQ x QM to QIVP) as All to KN, and Q:\I== is to 
 AQ X QB as CD^ to CA% it follows that AQ x QM is to AQ x 
 
 QB.
 
 Chap. XIV. and Varhilion of Graviti/ to7i;ar(is them. I5.*5 
 QB (or QM to QB) as AR x CD^ to RN x CA^ consequeuU 
 ly AQ is to QB as AR^ x CD^ to RN^ X CA*, and (becaase 
 RN" is equal to CA^ — AR") AQ, or RK, to AB as AR^, 
 
 X §5!. to CA" + gr X AR* or CA" — §^ x AR% ac- 
 cording as the spheroid is oblate or oblong; that is (if C/'lje 
 taken upon CF in the same ratio to AR as CF is to AC), as 
 
 C/" X ^A r in the fonner case, and to CA" -^ Cf^ in 
 
 the latter. In the former case, let A/and AF meet the circle 
 GNH in/ and S ; and the fluxion of C/— C/will he to the 
 liuxion of C/"as.C/" to Af' (art. 195), tliat is, as RK X CF* 
 to AB X CD-, and to the fluxion ofARasRK x CF^ to 
 2Cx\* X CD", consequently the fluxion of the area ARK will 
 be to the fluxion of 2CA X cjf~c/ as CA x CD" to CF^ ARK 
 will be to 2CA X q/Pc7 in the same ratio, and the whole 
 
 area AKGC, equal to '^ — ^^ — X cf— cs. Therefore the 
 
 gravity at A towards the sphere described upon the axis AB 
 being represented by §• AC, the gravity at A towards the sphe- 
 roid ADBE will be measured by - — ^^5 — ^ cf^^^^ ; the 
 gravity at J) towards the same spheroid (art. 645), by CD 
 CDJ CD3 X cs — CD X CF X CD' — CF* , 
 
 cfI ^ CF— cs or ^p -^ that is 
 
 by CD X ^P^xcs-CA^XCF ^ ^^^^j^^ ^^.^^^.^y at A' to the 
 
 gravity at D, as 2CA X CD X cf — cs to CD" x CS — 
 CA" X CF, or if the arch FO described from the centre A 
 meet CB in O, as CD X cf— csto the segment FCO ; be- 
 cause this segment is equal to I CD X FO — •§ CA X CF, or 
 ^ CD^XCS — ca^xcf 
 ^O 2CA . 
 
 647.WhenCAisgreaterthanCD,thatiswhenADBEC%.291, 
 JV.2), isan oblong spheroid, therestremainingasinthe last arti- 
 cle, let LC be taken upon C A equal to tjie logarithm of the ratio of 
 CD to AF, or of the subdupUcate ratio. of BF to AF, the mo- 
 dulus being AC ; and the gravity at the pole A will be to the 
 
 gravity
 
 126 Of the Figure of the Planets, Book T. 
 
 gravity at D as 2CA X CD X I.F to CA* X CF — CD^ X CL. 
 Vox in this case we Ibund that the ordinate RK was to AB as 
 Qf^ X — - to CA^ — C/*. But if CI be taken upon CA, so 
 
 as to represent the logarithm of the ratio of vca + c/ to 
 ycA — c/, the modnim being AC, the fluxion of C7 — C/* 
 will be to the fluxion of C/"as Cf^ to CA*— C/S or as RK 
 X CF* to AB X CD% and to the fluxion of AR as RK x CF^ 
 to2CA* X CD"- ; consequently the fluxion of the area ARK is 
 to the fluxion of 2CA x 'cT^Cf as CA x CDHo CF^ and the' 
 
 2CA^ X CD* 
 
 whole area AKGC is equal to ^^^ X LF. Therefore the 
 
 gravity at A towards the oblong spheroid ADBE is measured by 
 " rf"3~~~ ^ ^' ^^ gravity at D (art. 64.5), towards the same 
 
 :^heroid by CD — ^— X Lt or CD X ^^i 9 
 
 and the giavity at A to the gravity at D as 2CA x CD X LF 
 toCA^ X CF — CD* X CL. What has been shown concern- 
 ing the gravity at the pole A, agrees with what was advanced 
 long ago by Sir Isaac Nezvton and Mr. Cotes, who contented 
 themselves with an approximation in deterpiining the gravity 
 at the equator, which is exact enough when the spheroid 
 differs very little from a sphere. The approximations proposed 
 lately for this purpose, Phil. Trans. N. 438 and 445, are more 
 accurate ; and Mr. Stirling, after determining the gravity 
 at the equator by a converging series, since found that the sum 
 of the series could be assigned from the quadrature of the cir- 
 cle. It was shown in art. 645, how this gravity at the equator 
 is deduced accurately from the gravity at the pole, without 
 any new quadrature or computation. The gravity in any. 
 otherlatitudeisdetermined from whathasbeeu demonstrated by 
 art. 635 Cjig. 286), where dQ is to be taken in the same ratio to 
 dC as CD^ X CS — CA* x CF to eCD^ x of -cs, thatPQ 
 may measure the force and show the direction of the gravity 
 at P, The gravity at any distance in the axis of the spheroid, 
 or the plane of the equator produced, is likewise accurately 
 
 determined
 
 Chap. XIV. and Variation of Gravihf towards thcnu 127 
 determined tiom what has been shown by the following lemma, 
 without any new computation. 
 
 648. Let ADB,Pr/pr/^.292)be two semi-ellipses that havethe 
 same centre C and the same focusF. Let any righ L linePmM from 
 P meet the internal ellipse in m and M, and Pa: meet the exter- 
 nal ellipse in x, so that CL the perpendicular from C on P;r 
 may be to CR the perpendicular from C on PM as Cd to CD, 
 then Mw will be to Px as CA to CP. For let Vyp and AuB be 
 semicircles described upon the diameters Pja and AB; let m» 
 and MV parallel to CD meet the circle At)B in v and V, and 
 xy parallel to CD meet the circle Vyp in 7/ ; produce mv und. xy 
 till they meet Vp in q and I; and let Cr and C/ be perpendi- 
 cular to Vv and Vy in r and I respectively. Then since Pm^ is to 
 PC^ as qm" to CR% and Vx"- to PC^ as Ix"- to CL% it follows 
 
 that P»^* is to Px^ as ^, to ^^> that is, by the supposition, 
 
 as ~^ to ~7 > or (because qtii^ is to the difference of qm^ and 
 qv^ as CD' to CF% and Ix"- is to the difference of Ix* and ly* 
 as Qd^ to CF^ as the difference of qm^ and qv^ to the differ- 
 ence of Lr' and ly'^\ consequently Pm"' is to ^m^ — qm^ + qv^^ 
 or Pz)* as Pa' to Py% and Pw to Pz? as Vx to Vy. And because 
 q'V is to ^m as CA to CD, and ly to Ix as CP to Cd, so that 
 
 ^ is to ^ as 1^ to ^; and therefore as Fm to pjr, or (by 
 Trhat has been demonstrated) as Fv to Vi/, it follows that P 
 
 is to ^/ as CA to CP : therefore Cr is to CI as CA to CP It 
 
 Py 
 
 follows, that the triangles Crv, C/P are similar, and Yv (or 
 ^rv) to Fy (or 2?/) as CA to CP. But Mtn is to Vv as Pm to 
 Pt), or Vx to Pj/ ; consequently Mm is to Pa^ as Yv to Pj/-, or as 
 CA to CP. It appears from hence, that when two ellipses Fdp 
 APB have the same centre and focus, if any semidiameters 
 GE andCe of those ellipses constituteanglespCE,pCe with the 
 axts Cp, whose sines are in the same ratio as CD to Cd, these 
 semidiameters will be to each other as CP to CA. For if CE 
 
 and
 
 IG'8 Of the Figure of the Plaficfa, Book I. 
 
 and Ce be respectively parallel to Px and Pm^ CE will ht to 
 Ce as Px to Mm. 
 
 649. The ellipses Pdp, ADB that have the common centre 
 C and focus F being supposed to revolve about the axis PCp, 
 and to generate spheroids of the same densit}', the gravities at 
 P towards these solids will be in the same ratio as the quanti- 
 ties of matter contained in them, or as Cd^ X CP to CD^ X CA,- 
 For let any right line P,nlsL from P meet the internal elHpse 
 in m and M, and the circle CNH described from the centre F 
 in N, and P.t^ meet the external ellipse in x and CNH in L; 
 let 7?iq, MQ, xl, NRj and LZ be perpendicular to P^; in q, Q, 
 I, R, and Z respectively ; upon RN take RK always equal 
 to Qq, and upon ZL take Zk ahvays equal to PI, and the 
 gravity at P towards the internal solid will be to the gravi- 
 ty at P towards the external solid, as the area generated by 
 the ordinate RK to the area generated by the ordinate Zk : 
 suppose LZ to be always to NR as Cd is to CD, then Mni 
 will be always to Px as CA to CP, by the last article. But Qq 
 is to Mm as PR to PC, and PI to P.r as PZ to PC; and the 
 fluxion of the area CRKG is to the fluxion of the area CZkg 
 in the compound ratio of Q^ to PI, and of the fluxion of PR 
 to the fluxion of PZ, that is, in the compound ratio of Mm to 
 PXf and of the fluxion of PPt- to the fluxion of PZ^ (or of 
 the fluxion of NR- to the fluxion of LZ-), and consequently in 
 the compound ratio of CA to CP, and«f CD^ to Cd'- ; and the 
 areas CRKG, CZkg being in the same ratio, it follows that 
 the gravity at P towards the portion of the internal spheroid 
 that is generated by the segment AwMB is to the gravity atP 
 towards the portion of the external spheroid generated by the 
 segment Pxp, as CA x CD- to CP x Cd-, and that the gravities 
 at P towards the whole spheroids are in the same ratio; because 
 Px by revolving about P describes the semi-ellipse P^, while 
 t/iM describes the semi-ellipse ADB. 
 
 ().30. Hence the gravity towards the oblate spheroid ADBE 
 at any point P in its axis produced beyond A, is measured bv 
 
 — — — — X CF— cs, PF being supposed to meet the arch 
 
 CNH described from the centre P in S; because the gravity 
 
 at
 
 <rhap. XIV. and Variation of Gravity tozcards them. 129 
 at P towards the external solid Vdpe is measured by — —-^ — x 
 
 CF— cs (art. 646), which is to the gravit}^ at P towards ADBE 
 as CP X Cd"- to CA X CD% b}' the last article. In the same 
 manner the gravity atP towards an oblong spheroid ADBE (^of. 
 
 29 1, N".2) ismeasured by — -^^ — X LF, C L being theloganthin 
 
 of the ratio of Cd to PF, the modulus being PC {Jig. 292). 
 Because the gravity at P towards any spheroid ADBE 
 that has its centre in C and fociis in F, and is described on any 
 axisAB that isnot greater than Vp,\s as the quantity of matter 
 in that spheroid, it follows that if thedensity of the solid Vdpc 
 vary,butso as to be alwaysthe same over thesurface of any such 
 spheroid ADBE, the gravity towards Fdpe in this case will be to 
 the gravity towards it when its density is uniform, as tlie quan- 
 tity of matter contained in it in the former case to the quanti- 
 ty of matter contained in it in the latter. 
 
 65 1 . Let V(Jig-"QS) now be any point at the circumference of 
 the equator ofthe external spheroid arf6e,and the gravity at P to- 
 wards the internal spheroid ADBE will be to the gravity at P 
 towards theexternalsolidasCA X CD^toCa X C^%or as the qu an- 
 tics ofmatter in these spheroids, the generating ellipses being sup- 
 posed to have the same centre andybcws, as in art. 649- To demon- 
 strate this, PC being supposed perpendicular to the meridian 
 plane adbe, let it meet D/?E the circumference ofthe equator of 
 theinternal solidinp,* and letthesectionsPZC, pVC perpendi- 
 cular to the plane adbe intersect it in CZand CV , so that Zr and 
 VR perpendicular to Qd may be always in the same ratio to one 
 another as Ca to CA, and the elliptic sections PCZ,pCV will 
 have the same excentricity, or CD^ — CV* will be equal to 
 Ct/* — CZ*. For let rZ and RV produced meet the circles 
 dgh and DGH in g and G respectively, and Ca* — CZ* will be 
 equal to gZ X Zr+^r, which is to ha X aC+AC or Cd^ — Ca* 
 as Zr* to Ca*. In the same manner CD* — CV* is to CD*- — 
 CA* as VR* is to CA*. But C^* — Ca* is equal to CD*— 
 CA*, and Zr* is to Ca* as VR* to CA*, by the supposition ; 
 consequently Ct/* — CZ* is equal to CD- — CV*. Let the 
 sections PCZ,pCV move into the places VCz,pCVf and it fol- 
 
 VOL. IL K lows
 
 t3t) Ofth Figure of the Plane f.^, Book I,- 
 
 lows from what was shown in the last article, that the gravity 
 at P towards the slice contained by the planes PZC, PzC is ul- 
 timately to the gravity at P towards the slice contained by pVC 
 and/jrCin the compound ratio of GZ'-xCPto CV^ + Cp and of 
 the angle ZCz to Vcv, or (because the areas of sectors are in the 
 compound ratio of the squares of the rays and of the angles of 
 fhe sectors) as CZs x CPto CVv X Cp. ButbecauseZ/is always 
 to VR as Crt to CA, and consequently Cr is to CR as Cd to> 
 €D; the area CaZr is to CAVR, and CaZ to CAV, as Ca X Cd 
 to CA X CD, and CZ:; to CVv in the same ratio. Therefore 
 the gravity at P towards the slice terminated by theplanes PZC 
 and P^C is always to the gravity at P towards the slice termi-- 
 nated by pYC and piC in the invariable ratio of Ca X Cd'- to- 
 Cx4 X C\y ; and the gravity at P towards the whole external 
 solid is to the gravity at P towards the whole internal solid i» 
 the same ratio, or as the content of the former to the content 
 of the latter solid. 
 
 652. Henceto measure the gi-avity towards any oblate spheroid- 
 AUBE of an uniform density, at any distance CP in the plane 
 of its equator produced, let F be the focus of a section of the 
 solid through its axis, describe from the centre F with a radium 
 equal to the distance CP an arch intersecting the axis in a, and 
 from a as centre describe with the same radius the arch FQ 
 meeting CB in- O'; then the gravity at P towards the spheroid 
 
 *T^T^T^ -111 ji SCAxCD^ FCO . , ,. 
 
 ADBE will be measured by — ~^ — x -^ > because this 
 
 gravity is to the gi\avity at P towards the external solid adbe 
 
 (which is measured by —^^ — x -^y by article o4o), as 
 
 CA X CD' to Ca X Cd-, by the last article. The gravity at P 
 towards ADCE is to the gravity at a towards it as FCO to CP-f- 
 CF— cs, by art. 650. And if the density of the spheroid adbe 
 be supposed to vary from the surface to theceutre,but so as to be 
 always the same in the difibrent parts of the same surface gene- 
 rated by any ellipse ADB that has always the same centre and 
 focus vv'ith adby the gravity at the equator of the solid Gc/^t; will 
 be to the gravity at the pole a in the same ratio as if the densi- 
 ty of this spheroid was unitorm.- 
 
 653. The
 
 Plate XXXinJ^^.*.?^. To/, i [ 
 
 •2^2.^1. 
 
 Fu^.'2<^i.y.'2.
 
 I J- G 
 
 Plate xxxinj",;^,!,'. rh/.ii . 
 
 ia. lyj.J't. 
 
 r^
 
 tJhap. XIV. arid V'ariation of Gravity towards them. 131 
 Q53. The rest remaining as in art. 6,5 1 , suppose ihe solid not 
 to be a spheroid, or Cp to be greater or less tlian CD, but so 
 that the difference of the squares of Cp and CD be equal to 
 the difference of the squares of CP and Cd, that the sec- 
 tions DpC, dVC may be still ellipses that have the same centre 
 and focus; and if we suppose the sections PCZ.^pCV to be al- 
 ways ellipses that have PC and CZ, pC, and CV for their re- 
 spective axes, the distances of ihew foci from the centre C 
 \vill be always equal, as before; and it will appear in the same 
 manner, that the gravity at P towards the external solid will 
 be to the gravity at P towards the internal solid as Ca x Qd 
 X CP to CA X CD X Cp. 
 
 654. Let X be any point in the surface of the spheroid adbe, 
 which is supposed to be generated by an ellipse that has the 
 same centre and focus with ADBE, as formerly, and the gra- 
 vity at X towards the internal solid ADBE will be to the gra- 
 vity at X towards the external spheroid adbe either accurately 
 or nearly when the spheroids differ little from spheres, as 
 CA X CD* to Ca X Cc?*, or as the content of the external to the 
 content of the internal solid. When x is at the pole of the sphe- 
 roid, or at the circmnference of the equator, this appears from 
 art. 650 and 652> and in other cases it may be deduced from 
 the last article ; but we proceed to the application of those 
 theorems to enquiries that relate to the planetary system. 
 
 Q55. The gravity towards the spheroid ADBE (fg. 294) at 
 the pole A being represented by A, at the equator by D, and 
 the centrifugal force at D by V, as formerly ; if the density 
 of the spheroid be uniform, D will be to A as the area of the 
 segment FCO to CD X CF— cs, by art. 646, that is (by the series 
 usually given for the mensuration of circular segments and 
 arches, the proof of which we are to give in the second book), 
 b, a, and c, being supposed to represent CD, CA^ and CF re- 
 
 spectivelyas 1 +-_ + —, &cto 1 +--+—, &c. There^ 
 fore Db — ka, or Yb, will be to D6, or V will be to D, as 
 
 ratio of c to 6 is given^ the ratio of V to D may be determined 
 
 K 2 to
 
 .132 Of the Figure of the Phaicts, Book 1, 
 
 to any degree of exactness, at pleasure. When the ratio of 
 
 V to D is given, and thence the ratio of c to ^ is required, let 
 
 V be to D as 1 to m, and c^ to 6* as z to 1, then ~ +-^, 8cc. 
 
 6 Jo 
 
 will be to 1 +1Q+-355 Sec. as 1 to m ; from which it fol- 
 lows (by the methods for inversion of series) that z is equal to 
 r — ^ — , &c. This series may be continued at pleasure : but 
 when the spheroid differs little from a sphere, 2 will be nearly 
 equal to - . -^ and c* to 6* nearly as 5V to 2D + ^ — , con- 
 sequently^ in this case, the excess of the semidiameter of the 
 equator above the semiaxis is to the mean' semidiameter nearly 
 
 as5V to4D— ^-. 
 
 606. The ratio of z to 1, or cc to bb, may be discovered 
 several ways, without having observations made at the equator 
 of the spheroid. For this end the two following properties 
 of the ellipse are subjoined. Let PK perpendicular to the el- 
 lipse at any point P meet CD in K. Let the sine and co-sine 
 of the angle PKD (which is the latitude of the place P) be 
 denoted by S and K respectively, the radius being unit. Then 
 PK^ will be .to CA^ as CA^ to CA^ + CF^ x KK, or as CA* 
 to CD^ — CF"^ X SS. For Vd being perpendicular to CD in d, 
 dK will be to dC as CA^ to CD% by art. 627, and dC being 
 to CA^ — Prf* as CD^ to CA% dK^ is to CA* — P^^as CA^ 
 to CD* ; consequently CA*— PK- is to dK"- as CF^ to CA* ; 
 and since dK^ is to PK* as KK to 1, CA* — PK* is to PK* 
 as CF*xKK to CA*, and PK* to CA* as CA* to CA + CF* 
 X KK, or as CA* to CD*— CF* x SS. 
 
 657. The ray of curvature at any point P is always in the 
 triplicate ratio of the perpendicular PK. For let CG be the se- 
 midiameter conjugate to CP, and because the ray of curvature 
 at P is as the cube of CG, by art. 374, and PK is inversely as 
 PZ the perpendicular to CG in Z (art. 627) which is inversely 
 as CG, it follows that PK is as CG, and that the ray of curva- 
 ture
 
 ■Chap. XIV. and Va?'iation of Gravity towards them. 133 
 ture at P is as llie cube of PK. Hence the ray of curvature, or 
 a degree upon the meridian, at any latitude P, is in the triph- 
 cate ratio of PK, or of the force with which a body descends 
 towards the spheroid at P, by art. 637. 
 
 638. The magnitude of the earth is usually determined from 
 that of a degree upon the meridian. This however gives us only 
 the ray of curvature at that place of the meridian, or the ra- 
 dius of a sphere that would have all its degrees equal to that 
 degree ; and the centrifugal force derived from thence, and 
 from the period of the earth's revolution upon its axis, is the 
 centrifugal force at the equator of such a sphere when it is sup- 
 posed to revolve on its axis in the same time with the earth. In 
 order to derive the ratio of cc to hb in the spheroid from the ob- 
 servations made in any latitude P, let g represent the force with 
 which a body is found by observation to descend towards the 
 earth at P^t' the centrifugal forceat the equatorof a sphere that 
 has its degrees equal to the degree which we suppose to be 
 measured at P, and that revolves on its axis in the same time 
 with the spheroid ; and, the radius being supposed equal to 
 unit, let the sine of the latitude of P be S. Then CF^ will be to 
 
 CD% or cc to hb, nearly as 5v io Q,g — -^ + SSSy. For let 
 
 Vo the ray of curvature at P meet CD in K, PK be represented 
 by L, and the ratio of g to x? by that of n to unit. Then (by 
 
 the last article) Po is to the ray of curvature at A, or - > as L^ 
 
 to a^ ; V is to v as DC to Po, the times in which the spheroid 
 ADBE and the sphere of the radius Po are supposed to revolve 
 being equal ; consequently V is to t; as a* to bU. But g is 
 to A the gravity at the pole as L to a, by art. 637^ and A to 
 D — V as b to a, by art. 64 1, consequently g is to D — V as L6 
 
 to aa ; and m, or rrj equal to —r + ], or — - -f- 1, or 
 
 •^ V •' Lb\ aav 
 
 ~- + 1, that is (art. QoQ), ^^l^css "^ ^' Therefore m + 1 
 — nz — SSs is to 1 — SS^ as m to unit, or (art. Qoo) as 1 + 
 fg + ^) &c. to 1" + ^, &c. From which it follows, 
 
 K3 that
 
 134 Of the Figure of the Earth, Book I, 
 
 that when n is a large number, z is nearly to u nit, or cc to bb, 
 as 1 to "I + ^Q + SS + 5J + X' °^ (because z is nearly- 
 equal to \) as 1 to I" + SS — |., and therefore (« being 
 to 1 as g to ;;) as 5v to 2g + 5SSv — - • Hence the ratio, 
 
 of cc to bb is determined from the magnitude of a degree mea- 
 sured on the meridian in any latitude, and the length of the 
 pendulum that vibrates in a given time in the same latitude (the 
 earth being supposed of an uniform density), by computing v 
 from the former, and g from the latter. At the equator this ra- 
 tio is that of 5u to 2g — •^''j at the poles, that o^5v to 2g + -;^'- 
 
 659. The ratio of c^ to b^ (fig. 293), may be likewise disco- 
 vered from what has been demonstrated, by comparing the 
 eravity of a satellite that revolves about the spheroid in the 
 plane of its equator with the centrifugal force at D. Let Cc? be 
 any distance in the plane of the equator, and let Qa be taken 
 upon the axis so that aF may be equal to Qd ; from the centres 
 a and A describe the arches FO and Fo meeting CB in O and 0, 
 and the p^ravity at d will be to the gravity at D as FCO X CI> 
 to FCo X Cd (by art, 646), or, supposing Cdto be represented 
 
 3c* 9c* 
 
 by d,'m the compound ratio of b^ to rf% and of 1 -f j^ + —^t. 
 
 &c. to 1 + Y~- -1- -^9 &c. It appears from this, that the 
 
 gravity towards an oblate spheroid decreases in the plane 
 of its equator in a greater ratio than the square ol' the 
 distance from the centre of the spheroid increases. 
 Hence the periodic times of the satellites of Jupiter 
 ought to increase in a greater proportion than accord- 
 ing to KepIer^s law, or the sesquiplicate ratio of their 
 distances from the centre of Jupiter; but the variation from his 
 Jaw will hardly be sensible even in the nearest satellites. In like 
 manner, the gravity towards an oblate spheroid decreases in the 
 ^xisproduced ina less ratio tliantiiat in which the square of the 
 
 distance
 
 Chap. XIV. its Densitj/ being supposed uniform, 135 
 
 distance from the centre increases. For, the righ t lines aV and 
 AF being supposed to meet thearcht'S C/andCS described from 
 the centres a and A in/ and S, the gravity at a towards the 
 spheroid ADBE will be to the gravity at A towards the same 
 spheroid as CF — C/to CF — CS, that is, Ca and CA being 
 represented by e and a, in the compoimd ratio of a* to e% and 
 
 of 1 — — +, &c. to 1 — — > &c. It appears in the same 
 
 manner, that the gravity towards an oblong spheroid decreases 
 in the plane of the equator in a less ratio than that in which 
 the squares of the distances from the centre increase, but in a 
 greater ratio in the axis produced from tkepole. 
 
 660. Let N be to 1 in the compound ratio of the cube of Cd 
 to the cube of CD, and of the square of the time in which the 
 spheroid revolves on its axis to the square of the time in which 
 a satellite revolves about the spheroid in the plane of its equa- 
 tor at the distance Cd ; and let Cd be to CD as M to unit ; 
 
 45 3 
 
 then cc will be to hh nearly as 5 to 2N + rj — - • For, 
 by the last article^ the gravity at d is to the gravity at D in the 
 compound ratio of 1 to MM and of 1 -f - ^^ ? &,c. to I 
 
 4- j^ J &c. Bui the gravity at D is to V as 1 -f ^^ j &c. to 
 
 -^ + ^> 8cc. consequently the gravity at d is to V, or --rji 
 is to unit, in the compound ratio of 1 to MM and of 1 -f 
 ira' ^^- *° r + S-'* ^^- Therefore 1 -f ^, &c. 
 is equal to -^ + -^> &c. From which it follows, that 
 when we may neglect the terms of the equation that involve the 
 higher powers of z, it is tqual to -——^L ^ '^ 
 
 2N i»6NN ^ 8MMNN 
 
 &c. or z IS nearly to unit, or cc to bh, as J to^^ -i 
 
 5 14 
 
 3 
 
 ,i"0MM' ^^^ ^^^^ excess of the semidiameter of the equator 
 
 K 4 above 
 
 e
 
 336 Of the Tigure of the Earth, Book I. 
 
 above the semiaxis is to themeansemidiameteras5 to 4N + 
 
 10 3 , 
 
 -^ — -nrr. nearly. 
 
 7 MM -^ 
 
 66 1. To apply those theorems to the earth, a degree of the 
 meridian about the latitude o^ Paris is 57060 tohes according to 
 Mr. Picart ; conscvquently if the earth was a perfect sphere, 
 its radius would be 1961,5783 French feet, and a bod}' at the 
 equator of such a s|)here would describe 1430 . 4 feet in a se- 
 cond of time by the diurnal motion, the versed sine of which is 
 7.510148 hnes. Thcji because a. pendulum that vibrates in a 
 second at Paris (by the observations made lately by Mr. De 
 Mairaif) is 440.57 lines; and the space described by a body 
 that descends freely by its gravity in any time is to the length of 
 fipenduhimihiit vibrates in the same time in the duplicate ratio of 
 the semicircle to its diameter, by art. 405, it follows that in that 
 latitude a bod})- would describe by its gravity about 2172 . 9 
 in a second of time, and that v is there to g (according to 
 the notation in art. 658), as 7 . 510148 to 2172 . 9 • or as 1 to 
 289 . 3. From which it follows by art. 658 that cc is to bb 
 as 1 to 11 6, and that the excess of thesemidiameter of the equa- 
 tor above the semiaxis is about ^-jf of the mean semidiameter. 
 If the degree of the meridian near to Paris be greater thaa 
 57060 toises, the ratio of this excess to the mean semidiameter 
 "will be greater almost in the same ratio ; but though this degree be 
 57183 toises (as it is said to be found by some late observations), 
 that excess virill not be above a-j-f of the mean semidiameter. 
 By the mensuration and observations of the members of 
 the Royal Academy of Sciences nt Paris made near the polar 
 circle, u is to g there as 1 to 287 . 8 . as will appear by com- 
 paring in the same manner the degree measured by them with 
 the length of the pendalinn, which, by their observations, vi- 
 brates at Pe//o in a second of time. From which cc is to bb 
 as 1 to 1 15 . 9 • i^'^d almost the same excess of the semidiame- 
 ter of the equator above the semiaxis arises as from the obser- 
 vations ■d.tParis. This ratio jfnaybelikewisedetermined from the 
 distance and periodictimeof the moon, compared with the time 
 in which the earth revolves on its axis, and thence finding the 
 
 ratio
 
 Chap. XIV. its Density/ being supposed U7iiform. 157 
 
 ratio of N to unit, according to art. 660. By this computation 
 the difterence of the semidiameters of the earth is nearly the 
 same as by the former. And these agree nearly with Sir Isacc 
 Nezctons, according to which the semidiameter of the equa- 
 tor is to the semiaxis as 230 to 229. 
 
 662. But supposing the earth to be a spheroid, according to 
 what was demonstrated above, upon the supposition that the 
 density is uniform from the surface to the centre, if we com- 
 pute the difference of those semidiameters of the earth from 
 the lengths of pendulums that have been found to vibrate in 
 equal times in different latitudes ; or from the increase of the 
 degree of the meridian from Paris to the polar circle, as it has 
 been determined lately ; the difference of these semidiameters 
 will be found to be considerably greater than -^ of the mean 
 semidiameter. Let L and /denote the lengths of two suchjsm- 
 dulums mtiw^'o places P and p, and, the radius being unit, letS and 
 /represent the respective sines of the latitudes of P and p (that 
 is, of the angles PKD,pA;D), then cc will be to bb as LL — // 
 
 to LLSS — llff. For PK^ is to pie as i — fl^" to 1 — — , 
 
 b bb 
 
 by art. 6b6. The space described in a given time by a body 
 descending freely is as the gravity ; and it follows by art. 408, 
 that the length of a pendulum that vibrates in a given time is 
 likewise as the gravity ; consequently LL is to // as PK* to pk*- 
 
 by art. 647, or as 1 — ^ to 1 — -^ • Therefore cc is to 
 hh as LL — // to LLSS — llff. Hence if L be to / as 1 + m 
 to 1 — u, cc will be to bb nearly as 4m to SS — ff + 2m SS 
 + Q^nff 
 66o. If a degree upon the meridian at P be to a degree atp 
 
 as G to g, cc will be tobb as G ^ — g ^ to G ^ SS — g ^ ff; 
 because G is to g as the ray of curvature at P to the ray of cur- 
 vature at p, that is, as PK^ to pk^, by art, 657 ; consequently 
 
 G ^ IS to o- ^ as 1 — -fr to 1 rr > and cc to bb as 
 
 '-' t?b bo 
 
 G T_^ ^ to G ^ SS — g ' #. This rule for finding the ratio 
 
 of
 
 158 Of the Grcvitij tozvards a Spheroid, Book L 
 
 of cc I o hb (whence the ratio of bb — cc, or' aa, to bb is 
 easily computed) is accurate, and is founded on no particular 
 theory of gravity^ but on the supposition that the earth is a 
 spheroid only. When the spheroid differs little from a sphere, 
 let the degree at P be to the degree at ^; as ] + a: to 1 — x, 
 
 and cc will be to bb nearly as -^ to SS — ff -\- -~ -f t^ m 
 
 664. For example, the length of the pendulum that vibrates 
 in a second of time at Pello latit. Q&^ . 48' . is 441 . 17 by the 
 observation * of Mr. De Maiipertuis, See. The penduhint 
 that vibrates in the same time at Paris, Latit. 48° . 50' . 10" . is 
 440 . 57 lines. Suppose therefore L to be to / as 441 . 17 to 
 
 , . 30 SO J , 
 
 440 .5/, or as 1 -|- ^;^^ to 1 — ^^^53^? and by computing 
 
 from either of the rules in art. 662, cc will be to bb as 1 to 
 102 . 8. By comparing in the same manner the observations 
 made in Jamaica by Cohn Campbell, Esq. t^ and at London 
 hy Mr. Graham, cc is to bb nearly as 1 to 95 ; and by com- 
 puting from some other observations of this kind, t this ratio is 
 found still greater ; which ought to be that of 1 to 1 16, if the 
 earth was of an uniform density, by art. 66O. 
 
 665. The degree that cuts the polar circle was found to be 
 57438 foises, and the middle of the arch was in latit. 66°. 
 19' • 34". The degree measured by Mr. Picart, allowing the 
 correction mad6 lately by Mr. De Maupertuis, is of 57183 
 toises, and the middle of the arch that was measured is in 
 latit. 49'"- 21'. 24''. Suppose therefore G to ^ as 57438 to 
 
 571S3, or as 1 + ~~-^ to 1 — ^~^, and by the rules 
 
 in art. 663, cc will be to bb as 1 to 89 y. Hence, 6 is to c 
 or thcsemidiameterof the equator to the semiaxis, in the sub- 
 duplicate ratio of 89 y to 88 ^, or nearly as 178 i to 177 ^j 
 and consequent!}^ tiie difference of those semidiameters is about 
 22 miles, which if the density of the earth was uniform ought 
 to be 17 miles only. If the correction of Mr. Picart's arch 
 be not allowed, the difference of those semidiameters will be 
 considerably greater. 
 
 * Figure of the eaithj Bool 3, Ch. 6. § 6. + Phil. Trans. N. 432. J Mem. de 
 I'Acad. 173fi. 
 
 666. From
 
 Chap. XIV, Xi)hen the Derisilij is variable. 139 
 
 666. From these observations, there ;j ground to thhik that 
 the variation of the density of the internal parts of the earth is 
 considerable; and to enable us to form somejucrment of this, 
 it may be of use to enquire what proportions of the seniidame- 
 ters DC and AC, and of the gravitation at A to the gravitation, 
 at D, arise, when the density is supposed to increase or decrease 
 towards the centre ; or even when the earth is supposed to be 
 hollow with a nucleus included, according to the ingenious hy- 
 pothesis advanced long ago by Dr. IlaUry. If the density 
 was uniform, the increase o^ gravitation (by which we shall un- 
 derstand with Mr. Dc Maupertuis in what follows the force 
 with which a body actually tends downwards, or the excess of 
 the gravity above the centrifugal force) from the equator to 
 the poles ought to be in the same proportion to the mean gra- 
 vitation as the diiference of the semidiameters DC and AC to 
 the mean semidiameter ; because A is to D— V as DC to AC, 
 hy art. 641, and the excess of A above D — V to half their 
 sum, as DC — AC to | DC rf i AC. If we suppose new mat- 
 ter to be added at the centre, or the density to be increased 
 there, the attraction of this new matter will add more to the 
 gravity at the pole than at the equator, the distance being less, 
 and may account for a greater increase of gravitation from D tp 
 A than arises from the hypothesis of an uniform density, as; 
 Sii" Isaac Nezcrton has justly observed. But this will not ac- 
 count for a Q-reater difference of the semidiameters DC and 
 AC. Supposing the columns to be fluid (after Sir Isaac^s 
 manner), and to have sustained each other before the new mat- 
 ter was added at the centre, the attraction of this new matter 
 will add more to the gravity of the longer column DC than of 
 CA,- and though we suppose the centrifugal force at D to be in- 
 creased till it be in the same ratio to the whole gravity at D 
 as before, the column CD will be more than a counterpoise for 
 CA, till CD and CA come nearer to an equality, and the figure 
 nearer to a sphere. For let d represent the increment of the 
 gravity at D from the attraction of that new matter, N the in- 
 crement of the gravitation of the column AC arising from the 
 same attraction, and the increment of the centrifugal force at 
 D being represented by v, let t) be to c? as V to D, that the ra- 
 tio
 
 140 Of the Gravity towards a Spheroid. Book T, 
 
 tio of 1 to m (or Vt) to D + d) may remain the same as before; 
 tlien -- + N will represent the whole gravitation of the co- 
 lumn AC, imd ^-^^^ ^. N + c? X b—a — -| the gravita- 
 tion of DC ; but I Aa is supposed equal to -*- h x d— v-, and 
 - being equal to -^ > or (because c^ is to 5^ as 5V to !2D, 
 
 nearly, by art. 655) -^y which stands less than d X JIZ, or j^ 
 
 in the ratio of 2 to 5, nearly, it follows that the gravitation af 
 DC is now greater than that of AC ; so that these columns 
 cannot balance each other, unless the fluid subside at D and 
 rise at A. If the new matter be in the form of a sphere about 
 the centre C, it is shown in the same manner, that the column 
 AC will not balance DC ; and the same will appear after- 
 wards, when tlie new matter is supposed to be formed into a 
 spheroid similar and concentric to ADBE. 
 
 667. On the other hand, if we suppose the density to be less 
 at the centre, or some matter to be taken away there, the co- 
 lumn DC will no longer balance or sustain the column AC ; 
 and the fluid in the canal ACD will not be in cequilibrio till it 
 rise at D and subside at A ; that is, till the figure vary more 
 from a sphere than in the case when the density was supposed 
 uniform: for supposing the decrement of the gravity at D in 
 consequence of the rarefaction of the matter at the centre to be 
 represented by d, and the decrement of the gravity of the 
 whole column AC by N; let z) the decrement of the centrifu- 
 gal force be such, that V — v may be now to D — d in the same 
 
 ratio as V was to D; then -7; N will represent the gravitation 
 
 of the column AC, and „ ^ ^ — ^^ — '^ ^ ^--^ ^^^^ gravi- 
 tation of CD. But ^ being less than d x JZ^ , as in the last 
 
 article ; and — equal to — — ^ ^> because the columns were 
 
 supposed to sustain each other before the matter at the centre 
 
 was
 
 Chap. XIV. when the "Demitij is variable. 141 
 
 was taken away ; It appears tlmt the column AC is now more 
 than a counterpoise for DC. Thus the rarefaction of the matter 
 at the centre will account for a greater difference of the semi- 
 diameters DC and AC, or a greater variation from the spherical 
 figure, than the hypothesis of an uniform densit}'. But it will 
 not account for a greater increase of gravitation from the equa- 
 tor to the poles. On the contrary the increase of gravitation 
 will be less in this case than when we suppose the density uni- 
 form. For since A — D + V is to A + D — V as 6 — a to 
 b -{■ a, that is, as 5 to 8m, nearly, by art, Q55, the increase of 
 gravitation from the equator to the poles is nearly to the meaa 
 gravity (which we shall call G) as 5 to 4m, when the density 
 of the spheroid is uniform. But when the matter about the cen- 
 tre is supposed to be rarefied, as above, let c? be to G as 1 tor; 
 
 and the gravity at A being A -^ and the gravitation at 
 
 D equal to D — d — ^V + v, the difference of which is to half 
 
 their sum as A — D 1 + \—v to|A + AD 
 
 — I V + |t) ; it follows (because A — D + t; is to 2G as ft — a 
 to 6 + a or 5 to 8w, c^ to a^ as 5 to 2m, and t^ to G as 1 to 
 rm nearly), that the increase of gravitation from the equator to 
 the poles will be in this case to the mean gravitation nearly as 
 
 9 2 
 5r — 14 to 4mr — 4m + 2, or as 5 -. to 4m -| r)which 
 
 r — 1 r — 1 
 
 is a less ratio than that of 5 to 4m. And if we suppose the fluid 
 to rise at Dand subside at A, till the columns AC and DC sus- 
 tain each other, the increase of gravitation from D to A will 
 in this case be to the mean gravitation in a less ratio than before. 
 The hypothesis therefore of a greater density towards the cen- 
 tre may account for a greater increase of gravitation from the 
 equator to the poles than that of an uniform density, but not 
 for a greater increase of the degrees of the meridian : and the 
 hypothesis of a less density towards the centre may account for 
 a greater increase of the degrees of the meridian, but not for a 
 greater increase of the gravitation, supposing always (after Sir 
 Isaac Nezvton's manner) the columns DC and AC to extend 
 from the surface to the centre, and there to sustain each other. 
 
 Tliis
 
 142 Of the Graviti/ touards a Spheroid, Book t. 
 
 This is likewise the result of our computations (some of which 
 we are to subjoin),when we have supposed thedensity toincrease: 
 ordecrease continually from the surfaceof the spheroid ADBE to 
 the centre,, so as to be uniform in the different parts of any one 
 similar and concentric elliptic surface; and in several other cases. 
 And hence there seems to be some foundation for proposing it, 
 as a Querifi, Whether the internal constitution of the parts of 
 the earth;, above-mentioned, that was proposed by Dr. Halley 
 for resolving some of the phaenomena of the magnetick needle, 
 will not be found to account in a probable manner for the in- 
 crease of gravitation^ and at the same time of the degrees of 
 the meridian from the equator to the poles ; as these have 
 been determined by thebestobservalionshitherto. The grounds 
 upon which we mention this will appear better from wh at follows. 
 668. LetADBE(^g.295) beasectionof a spheroid through its 
 axisAB,Ftheybc«s,andFOan arch described from thecentre A, 
 as formerl}^, meeting CB in O ; let adbt be any similar concen- 
 tric ellipse, y its focm^fX a parallel to the axis meeting the 
 arch FO in Z, and ZV a perpendicular to the axis in V. Sup- 
 pose the density to be always the same over the surface generat- 
 ed by any ellipsis adb about the axis AB, however variable it 
 may be in different elliptic surfaces; and let e represent the den- 
 sity at the surface adbe. Then if VKbe taken upon VZ in the; 
 same ratio to VZ as e is to CD, and the ordinate VK generate 
 the area OKHC, the gravity at D towards the whole spheroid 
 
 ADBE will be measured by ^^^^- x OKIIC. For let 
 
 Imnr be another similar and concentric ellipsis^ ^ its/ofMS, xz 
 parallel to AB meet FO in z, zv be perpendicular to AB in vi 
 then (by art. C)51) the gravity at D towards the solid gene- 
 rated by the annular space bounded by adb and Imn revolv- 
 ing about the axis AB, of the density c, will be measured by 
 
 ' -^j X — — • ) which, when at is continually dimi- 
 
 nished, is ultimately equal to rrrr^ X or (by 
 
 ''CD^XCA 
 
 the supposition) to -^^^r^ — . x VK X Vv; consequently the 
 
 ijravity
 
 Clifip. XIV. its Densifi/ being supposed variable. 143 
 
 gravity at D towards the whole spheroid ADBE is measured by 
 :: — ^p— X OKHC ; and is to the gravity at D towards the 
 
 spheroid ADBE, when its density is supposed uniform, and re- 
 presented by E, as OKHC x CD to FCO x E. For example, 
 if the density in the ray CD at any point d be inversely as Cd 
 the distance from the centre, the gravity at D towards this sphe- 
 roid will be to the gravity at D towards a spheroid of an uni- 
 form density equal to that of the former at D, as CF x CO to 
 the area FCO ; because if E represent the density at D, V K 
 will be to E as VZ (or Cf) to Cd, or as CF to CD, and the area 
 OKHC X CD equal to E X CF x CO. In this case the gra- 
 vity is the same in all parts of the column DC. In the same 
 manner, when the density at d is inversely as the square of the 
 distance Cd, the gravity at D towards such a spheroid is to the 
 gravity at D towards the spheroid when its density is uniform 
 and equal to that of the former at D, as CF* X CO to FCO x 
 CD : and the gravity at any point d in the column CD is in- 
 versely as the distance Cd. 
 
 669. Inlike manner,letjf^ perpendicular to CD atj^ thejhcus 
 of adbe, be to e as Cf"- to A/% and the ordinate /A generate 
 the areaCA-oF; and the gravity at A towards the spheroid ADBE. 
 
 ''CD"' X CA 
 
 will be measured by - — ^^^3 — X CkoV. This is demonstrated 
 
 in the same manner from art 650. The gravity towards such 
 a spheroid at any point in its axis, or in the plane of its equa- 
 tor produced without the solid, may be determined iu the same 
 manner. 
 
 670. Suppose, for example, that the density in any semidia- 
 meter is as the distance from C, and the density at the surface 
 being represented by E, e will be to E as Cd to CD, and VK 
 to VZ as E X Cd to CD% or (because Cd is to CD as ZV to 
 CF) as E X ZV to CD X CF ; and VK will be to E as ZV^ 
 or AO^— AV% to CD x CF; consequently if AM perpendicu- 
 lar to AO be to AO, or CD, as E is to CF; and a para- 
 bola be described upon the axis MA that shall have its vertex. 
 in M and pass through O, OKH will be a portion of this 
 parabola; and the area OKHC will be found equal to 
 
 E X
 
 M4 Of the Figure of the Earth, Book I. 
 
 x, SCD'XCO — CD3 4-CA3 , j- , ,1 . i- • 
 
 JL X 3CDXCF ' °^" (according lo Uie notation in 
 
 art. 655), to -^ X Li^. Therefore the jrravity at D to- 
 
 wards such a splieroid will be measured by -j-< x yT^' ^ "'^ 
 
 gravity at rf is to the gravity at D in the compound ratio of Cd 
 to CD and of the density at d to the density at D, and conse- 
 quently as Cd^ to CD\ Therefore if the gravity at D be repre- 
 sented by Q, and Cd by z, the gravity at d will be represented 
 
 Ly-^) the density at c? by -^9 and the gravity of the co- 
 iumn DC will be measured by an area upon the base CD that 
 has its ordinate at any point d equal to -^j- > and this area is 
 equal to ^QEb. Any distance in the plane of the equator, as 
 Cp, greater than CD being represented by d, and Vd^7 by a, 
 
 2i'aE 2i/-|-a . .,, 
 
 the gravity at p will be measured by — ^ X - — - > as wili 
 
 appear in the same manner. 
 
 671. In the same spheroid^yA* is to be taken to E as Cf to 
 A/^ X CF ; and if L denote the logarithm of the ratio of DC to 
 AC, the modulus being AC;, the area CkoV will be found equal 
 
 ^^ ICF ^ CF* — 2ACXL j and the gravity at A towards the 
 spheroid will be measured by — ^ x c' — 2aL. The gravity 
 
 at a towards it will be to the gravity at A in the compound 
 ratio of the density at a to the density at A and of Ca to CA, 
 that is, as Ca^ to CA^. Therefore h' q denote the gravity at A, 
 
 and Ca be represented by u, the gravity at a will be ^» 
 and the density at a will be — ; consequently the gravity qF 
 
 the column AC will be measured by ^ qlLa. 
 
 672. Let V represent the centrifugal force at D, arising from 
 the rotation of the spheroid on its axis, the centrifugal force 
 
 at
 
 Chap. XIV. its Density being supposed variable. 14^ 
 
 ate? will be -^5 andj the density at rf being -^j the quantity 
 to besubdiicted from the gravity ofthe column DC, on this ac- 
 count, will be measured by an area on the base CDlh'.t has^ 
 
 the ordinate at any point cif equal to -jp'^ "and this areabeino-. 
 equal to ^ YEb, the gravitation of the column DC is f 
 E^Q — I- EbV, or (supposing V to be to Q the gravity at 
 D as I to m, as formerly) E6Q x "'~ • 
 
 673. If we now suppose (after Sir Isaac Newton's man- 
 ner) the columns DC and AC to be fluid, and to sustain each 
 other at C, we shall have bQ x -^^ equal to ~^j or b to 
 a as 7 to Q X -^- But when the spheroid differs little from 
 a sphere, Q and q may be considered as equal ; for by art. 67O 
 and 67 1 (supposing CD to be equal to I -{- x, and CA to 1 — x), 
 
 W will be to fl' as 4 X === or i -i — ? to - x cc—9^i » 
 
 which last being likewise expressed by t, those terms only will 
 be found different that involve the second and higher powers of 
 a,'. Therefore ^ is to a nearly as 3m to 3m — 4, and b — a to 
 b nearly as 4 to 3m, that is, as 4V to SQ. , And in this case the 
 excess of the semidiameter of the equator above the semiaxis is 
 greater than when the density is supposed uniform in the ra- 
 tio of 16 to 15, the ratio of V to Q being supposed the same 
 as that of V to D was before. Let Q be to V as 289. to 1:> as in 
 the earth, and CD — CA will be to CD as 4 to 3 X 289, or 
 as 1 to 216 I-,- consequently CD will be to CA as 2l6 I to 
 215 1. This hypothesis, therefore, of a density that de- 
 creases as the distance from the centre decreases, might ac- 
 count for a greater excess of the semidiametei of the equator 
 abo^■« the semiaxis, than-'that which resirivs from the supposi- 
 tion of- an uniform dens>ty', but it would not account for a 
 greater increase of gravitation from the equator to the poles. 
 For since the values of Q and q almost coincide in this case, it 
 follows that the gravitation at the equator is to the gravity at 
 VOL. II. / L th^
 
 146 Of the Figure of the Earth, Book I. 
 
 the pole as Q — V to Q, or as m — 1 to m, that is, as 288 
 to 289 i whereas in the hypothesis of an uniform density, this 
 ratio was that of 230 to 231. In Uke manner, by supposing 
 the density to decrease in the same proportion as the cube of 
 the distance from C, the ratio of DC to AC will be found to 
 be that of 226 to 225, nearly, but the increase of gravitation 
 will be less than in the former hypothesis. 
 
 674. It will be easy, from what has been shown, to measure 
 the gravity at D and A towards a spheroid, when the depths 
 from the surface being supposed to increase uniformly, the den- 
 sity increases likewise uniformly, till at the centre it become 
 any multiple of what it is at the surface ; and to determine the 
 form of the ellipsis ADBE. Let L be taken upon CD produ- 
 ced upwards, so as thatCL may be to LD as any number n to 1 ; 
 and suppose the density at any point d to be always as hd. 
 Let e denote the density at the surface, and ne will represent 
 the density at the centre. In this case, we may conceive the 
 density of the spheroid at any orb adbe, as the difference of the 
 densities of a spheroid of an uniform density ne, and of another 
 spheroid that has the density at its surface equal to «_i x e, 
 and its density decreasing downwards in the same proportion 
 as the distance from C, as in the preceding articles ; because 
 the difference of those densities at any point d will be equal to 
 
 ne — «_i X ~ X e or — j-^ Xe, or— X e, which re- 
 presents the density at d of the spheroid ADBE that we are 
 now considering; consequently the gravity at any point towards 
 this spheroid ADBE is equal to the difference of the gravities 
 towards those two spheroids at the same point. Therefore if 
 P denote the gravity at D towards a spheroid of an uniform 
 density represented by tie, and Q denote the gravity at D to- 
 wards the spheroid, whose density at D is n— 1 X e, and at any 
 other point d is asCc?; then P — Q shall denote the gravity at 
 D towards the spheroid ADBE, and (CD and Cd being repre- 
 sented by b and z as formerly) -^ — -rr the gravity at d 
 
 towards it. The density at d is represented by ?ie — n^ X j » 
 
 con-
 
 Chap. XIV. its Density being supposed variable. 147 
 
 consequently the gravity of the column CD will be measured 
 by an area upon the base CD, of an ordinate at d equal 
 
 to ^-f — ~ X ne — n — 1 x t> that is, by ^ x n+2 X P 
 
 646 ; and the value of Q is ^hae X "—^ X .-=": > t>y art. 67 0. 
 
 -^- X Q- The value of P is — - x 7ie X -j-> by art, 
 
 b^a'^ 
 
 If it is required to determine the gravity towards this sphe- 
 roid at any point p in the plane of its equator produced, de- 
 scribe from the centre F, with a radius equal to Cp, an arch 
 intersecting the axis in p, and the arch Fo with the same ra- 
 dius from the centre p intersecting the axis in ; and the gra- 
 vity at p towards the spheroid ADBE will be measured by 
 
 •200^ X CA ^ FCo 2DCa X CA SCp+C/. 
 
 jrr^ X -- X nc ^tP X ^^ \ X «- i X e. 
 
 Ci^ Cp 3Cp Cp+C/^ 
 
 675. The centrifugal force at D being represented by V, the 
 centrifugal force at d will be -^S and the density at d being 
 ne «:rr X ~) the quantity to be subducted from the gra- 
 vitation of the column DC, on account of the centrifugal force, 
 will be measured by an area upon the base DC the ordinate at 
 
 d being always equal to -^ «- 1 X -jj- ? and this area 
 
 is ch\ X 2i^ (or supposing the ratio of the centrifugal force 
 V to P — Q, the gravity at D to be represented by that of 1 to 
 jfi\ e5 X ^~^ X — • Therefore the gravitation of the co- 
 
 lumn DC, by subducting this quantity, is reduced to e6P X-^ 
 
 X — — — "T" X o ~~~ ""„ 
 
 676. Leip denote the gravity at the pole A towards a sphe- 
 roid of an uniform density represented by we, and q the gravity 
 at A towards the other spheroid, the density of which in any 
 column AC is as the distance from C; then p — q ^vill denote 
 
 L2 the
 
 148 Of the Figure of the Earth, Book I, 
 
 the gravity at A towards the spheroid ADBE (the density of 
 Avhich at any orb acIOe is supposed to be as hd), and by pro- 
 ceeding as in art. (i74, the gravity of the cohimn AC will be 
 
 found to be measured by — X „_^2 X p — -— X q. The 
 
 value of ^ is ^—f- X cf— Cb X ne, and the value of q is b^ae 
 
 X .^^ . X c^- — 2a'L, by art. 67 1. 
 
 1^77. .The supposition of the (Equilibrium of the columns DC 
 and AG ffives us )6P X -~ 5Qx ■~^, equal to an 
 
 — aq^ sT^' oiNbeing supposed equal to ^^j Z*P X ~ 
 
 — i6Q X "'^" equal to ap •—• ^aq; and 5 to a as p — N<7 
 
 to - — X P — Q X -^-3 ; -so that b — a will be to & + a^ 
 
 or (supposing Z> equal to 1 + .r^ and a to 1 — a:) x to 1, as 
 
 p — P X to p + P — 2NQ, nearly ; because Q 
 
 and q may be considered as equal, by what was observed in 
 art. 673, and m is supposed to be a large number. From this 
 it will be found (by substituting for p, V, and Q their values, 
 from art. 674 and 676, and neglecting the terms where the 
 index of x is greater than unit, and where x is divided by 
 
 va) that —^ — N X «— i X x is equal to ^~> and (substi- 
 
 tutmg for N its value jttt) ^ equal to -. x ,-= — . .„ , ■* 
 
 The same value of x is found when n is a fraction ; that is/ 
 ? being taken upon CE, when the density at the centre is less 
 than the density at D in the ratio of /C to /D, and the den- 
 sity at any point d is as Id. According as n is greater or less 
 
 thaa tinit,ar is less or greater than „— ; for x is equal to g—
 
 Chap. XIV. its Density being supposed variable. 149 
 
 15 3,7+1 X «^T rrii r- .1 ■ ^ ■. 
 
 ' — -r- X T ^ , „, I v; - 1 heretore the ratio of the centri- 
 
 , Sot 1 7;.'« -|- J4'«-p43 
 
 fugal force at D to the gravity being given, the spheroid is 
 found to differ less from a sphere, when the density increases to- 
 wards the centre in the manner we have described above, than 
 when the density is supposed uniform ; but to vary more from a 
 sphere when the density decreases towards the centre. 
 
 678. The increase of gravitation from the equatoT to the 
 
 pole is to the mean gravitation as p — ^^^ X P— -to 
 
 ■T ^ ■'-mm 
 
 p Q 
 
 ■|P + ip — Q + "l^T"' ^^^^* ^^> '"* ^^^^ compound ratio of 
 1 to m, and of 2.5 X «+i^ + 20 to 17 x Jipi^ + 28 ; or in 
 the compound ratio of 5 to Am, and of 1 -f 3 x " X " — } 
 
 ^ ■ 17««-f.31.;-f-23 
 
 to 1 . Therefore the increase of gravitation from the equator 
 to the poles is to the mean gravitation in a greater or less ra- 
 tio than that of 5 to 4m (which is the ratio when the density 
 is uniform) according as n is greater or less than unit ; that is 
 according as the density increases or decreases towards the 
 centre. And it appears from hence, and from the last article^ 
 that no supposition of this kind can account for a greater va- 
 riation from the spherical figure, and at the same time for a 
 greater increase of gravitation from the equator to the poles, 
 than the hypothesis of an uniform density ; if the columns AC 
 and DC be supposed to extend from the surface to the centre, 
 and be supposed to balance each other at C. 
 
 679. To mention some examples : if the density at the cen- 
 tre be double of what it is at the surface, or n be equal to 2, 
 the excess of DC above AC will be to the mean seinidiameter 
 as 200 to 181 m; consequently in the earth (m being equal to 
 289) the semidiameter of the equator will be to the semiaxis 
 as 262 to 261, and the gravitation at the equator to the'oravi- 
 tation at the poles as 213 to 214. If n be equal to 3, the dif- 
 ference of CD and CA will be to the mean semidiameter as 1 
 to m ; and in the earth the semidiameter of the equator to the 
 semiaxis as 289 to 288 ; in which case the gravitation at the 
 e<|uator will be to the gravitation at the poles as 20G to 207. 
 
 L 3 if
 
 150 Of the Figure of the Earth, Book I. 
 
 If the density be as the distance below the surface, or the point 
 L coincide with D, the difference of CD and AC will be to 
 the mean semidiameter as 10 to 17 w; in the earth DC will 
 be to AC as 492 to 491, and the gravitation at D to the gi'avi- 
 tationatAas 196to 197. 
 
 680. Suppose the density to be uniform from the surface 
 ADBE to the similar concentric orb adbe, and to be uniform 
 likewise iVom adbe to the centre ; and the density within the 
 orb adbe be to the density without it as 1 + t' to 1. In this 
 case the increase of gravitation from D to A will be greater 
 than in the hypothesis of an uniform density ; but supposing the 
 columns AC and DC to sustain each other at C, and DC to be 
 to dC as n to 1, then the excess of the semidiameter of the 
 equator above the semiaxis will be to the mean semidiameter 
 nearly in the compound ratio of 5 to 4m, and of «' + en^ 
 
 + eri^ 4- ee to w' + eii^ + e«* + een^ + 3e X —5 — i which 
 
 compound ratio, when e is positive, is manifestly less than that 
 of to 4m (the ratio of the difference of CD and CA to the 
 mean semidiameter when the density is supposed uniform); since 
 7ih necessarily greater than unit. This likewise holds, when 
 there are three or more such orbs, providing the density be 
 always greater within the orbs that are nearest to the centre, 
 
 681. Let us therefore now suppose the earth ADBE to be 
 hollow with a 7iuclais Imnr included ; let the convex and con- 
 cave elliptic surfaces ADBE, adbe that bound the external part 
 be similar ; and first let Imnr be a sphere. Let CD be to Cd 
 as fi to 1, the area of the sphere Imnr to the area of the sphe- 
 roid ADBE as 1 to N, the centrifugal force at D to the gra- 
 vity as 1 to m ; and the external part bounded b}- ADBE and 
 adbe being supposed of an imiform density, if we suppose the 
 columns Aa and Dd to gravitate equally, the excess of CD 
 above CA will be to the mean semidiameter nearly in the com- 
 pound ratio of on + 5 to 2a«N, and of ?i^ -t- n^ N — N to 
 
 2«* + 2w' + 2«^ — 3n — 3 — -j^ j and the increase of 
 
 gravitation from the equator to the poles will be to £lie mean 
 
 gravitation
 
 Chap. XIV. its Densiti/ being supposed variable. 151 
 
 j'ravitation nearly as 1 + o,,n,.Q» o q — ttTt X o" 
 to w. la this cage the difference of the semidiameters 
 CD and CA, and the increase of gravitation from D to A, may 
 be both greater than when the density is supposed uniform, 
 the ratio of 1 to m being supposed the same in both cases. For 
 example, let n be supposed equal to 5, N to 45, and m to 289; 
 then the semidiameter of the equator will be to the semiaxis as 
 180 I to 179 i nearly ; and the increase of gravitation from 
 the equator to the poles will be -^ of the mean gravitation. 
 \i Imnr be a spheroid(as is more probable), and y the focus of 
 a meridian section of it, let Of be to the mean semidiameter 
 of ADBE as 1 to r; and the rest remaining as in the former 
 case, the difference of CD and CA will be nearly to the mean 
 
 1 • /-iT-v » <-i A « -+- 1 n^ 3n* 
 
 betwixt CD and CA as 5 x ■■ ^— - x w^ — ^ + n — 2>7n 
 
 X „« + « + ! to 2n* + 2k' + 2«* — 3/i — 3 — ^« This 
 ratio may be computed from the same principles, when the 
 density is supposed to increase or decrease from ADBE to adbe. 
 But, because the hypothesis of the equal gravitation of the co- 
 lumns Aa and J)d, as well as of an uniform density in the dif- 
 ferent parts of every elliptic orb similar and concentric to 
 ADBE, may seem precarious, we shall not insist on the con- 
 sequences that would follow from such a constitution of the 
 internal parts of the earth, as we have here supposed. If we 
 suppose the density to be uniform in the different parts of 
 every orb adhe that is generated by an ellipse, which has always 
 the same centre and ybcMS with ADBE, but to vary in different 
 orbs ofithis kind, the gravity at any point in CD or CA may 
 be computed from the principles in art. 650 and ^59.. But the 
 conclusions deduced from this hypothesis, when the density is 
 supposed to increase towards the centre, agree no better with 
 the phaenomena than those in art. 677 ^md 6/8. By imagin- 
 ing the density to be greater in the axis than in the plane of 
 the equator at equal distances from the centre, an hypothesis 
 perhaps might be found that would account for most of the 
 
 L 4 phseno-
 
 152 Of the Figure of Jupiter, Book I. 
 
 phaenomena ; but as this may seem to be an improbable suppo- 
 sition, and it is not so easy to compute the consequences that 
 would result from it, we shall insist on this subject no further. 
 When more degrees shall be measured accurately on the me- 
 ridian, and the increase of gravitation from the equator, to- 
 wards the poles determined by a series of many exact obser- 
 vations, the various hypotheses, that may be imagined concern- 
 ing the internal constitution of the earth, may be examined 
 with more certain t3^ We have always abstracted from any 
 powers thatmay affect the gravitation, besides the mutual gra- 
 .vity of the particles and their centrifugal force. 
 . 682. The figure of the planet Jupiter is found to differ consider- 
 ably from a sphere, by the observations of Astronomers, as well as . 
 by this theory. By Dr. Poim^^'s observations, the distance of the 
 fourth satellite is to the greatestsemidiameter of Jupiter as 26, 63 
 to ] , and its periodic time to the time in which Jupiter revolves 
 on his axis as 24032,15 to 596. Therefore let N be to 1 in the 
 compound ratio of the cube of 26,63 to 1, and of the square of 
 596 to 24032,15 according to art. 66O, and N will be found 
 equal to 11,615. By continuing the series in art. 66O, one 
 step further, the excess of the semidiameter of the equator 
 above the semiaxis is to the mean semidiameter as 5 is to 4N 
 
 + ? — mT^ + ^' ^''' ^^ ^^'""^ ^'1"''^ ^"^ ^^'^^ • *''^"" 
 sequently this ratio is that of I to 9,8; and the semidiameter of 
 the equator to the semiaxis as 10,3 to 9,S, the density being 
 supposed uniform ; and this agrees with Sir Isaac Newton s 
 computation. But the difference of thosesemidiameters, accord- 
 ing to Mr. Cassini, is only -^-^ of Jupiter's semidiameler, and 
 
 by Doctor PouncVs observations is betwixt ^5^ and --^ of it. 
 Hence, according to what was shown in art. 677, the density 
 of Jupiter seems to increase towards the centre. We have ab- 
 stracted from the efiect of thegravitation of the fourth satellite 
 towards the other satellites, and towards the atmosphere of Ju- 
 piter (if there is any); but the difference betwixt this computa- 
 " tion and the observations cannotbe imputed to these. Itis near-
 
 Chap. XIV. arid of his St/stem. 153 
 
 ]y the same ratio of the semidiameters of Jupiter that is foimd 
 by computing from Dr. Pou;2(i's observations of the eiongatioa 
 of the third satelHte. 
 
 683. If we suppose the density of Jupiter to increase from the 
 surface to the centre, in the manner described in art. 674, so as 
 to become quadi'uple at the centre of what it is at the surface; 
 then, by art. 677^ CD being supposed equal to \ -{- x, and CA 
 
 210 
 
 to 1 — X, X will be nearly equal to rrr" ^J computing 
 
 from what was shown at the end of art. 674, m will be nearly 
 
 to N as w + 3. +, — - + lOx to i + 2;^ X „ + 3 + 5J^j^; 
 
 and supposing n equal to 4, m will be equal to 12 nearly, x to 
 
 rr--j and the semidiameter of the equator to the semiaxis as 
 
 13,4 to 12,4; which differs little from the mean ratio that re- 
 sults from Dr. Pound's observations. 
 
 684. Sir Isaac Nezoton has found, that the mean density 
 of Jupiter is to the mean density ol" the earth as 94 | to 400. If 
 we suppose n equal to 4, as in the last article, the density at the 
 surface of Jupiter will be to the mean density as 4 to 7, and. 
 consequentljr to the mean density of the earth as 94 i to 700. 
 The earth is therefore not only more dense than Jupiter, but there 
 is some ground to think, from whathas been shown concerning 
 those planets, that the ratio of the densities at their respective 
 surfaces is greater than the ratio of their mean densities (or that 
 of 94 I to 400), and that it approaches more towards the ratio 
 of the densities of the rays of the sun incident upon them at their 
 respective distances. 
 
 685. It cannot be expected that we should be able to disco- 
 ver, by observation, the variation of the distances of the satellites 
 from Kepler's law mentioned in art. 659-, For let z denote 
 the distance of the first satellite as it is deterjnined by this law, 
 from its periodic time, and from the distance and periodic time 
 of the fourth satellite; that is, let the square of the periodic 
 time of the fourth satellite'be to the square of the periodic time 
 of the first, as the cube of the distance of the fourth to z^; let 
 c denote the distance of ihefocus of the meridian section of Ju- 
 piter
 
 154 Of the Tides. Book I. 
 
 piterfrom the centre; and the mean distance of the first satel- 
 lite will be nearly z + ^^-^ >^ 75^ ^ which, when the density 
 
 cc 1 
 
 is uniform, exceeds z by -— only, that is, by less than -■ ' 
 
 part of Jupiter's semidiameLer; and this excess is still less when 
 n is greater than 1, or when the density' is supposed to increase 
 towards the centre. It would seem, therefore, that if there are 
 any irregularities observed in the motions ofthose satellites, or in- 
 deed in an}' of the celestial motions, they are not to be ascribed 
 to the consequences of the variation of the figure of the sun or 
 planets from that of perfect spheres, but to their gravitation to- 
 wards one another^ or to some other causes. 
 
 686. We are next to apply the proposition demonstrated 
 above from art. QSQ to art. 641 to the theory of the tides. It 
 follows from it, that if we suppose the earth to be fluid, and ab- 
 stract from its motion on its axis, and the inchnation of the 
 right lines in which its particles gravitate towards the sun or 
 moon, the figure which it would assume, in consequence of the 
 imequal gravitation of its particles towards either of those bo- 
 dies, would be accurately that of an oblong spheroid having its 
 axis directed towards that body. For (^'g.296)let ADBEbeany 
 section ofthe earth through the right lineDCEth at is supposed to 
 bedirectedtowardsthesunatS; and what was shown concerning 
 the inequality of the gravities of the earth and moon towards 
 the sun in art. 47 1 and 472. being applied to the particles ofthe 
 earth, it will appear, in the same manner, that any particle P 
 may be conceived to be affected by two forces, besides its gra- 
 vity towards the earth ; a force in the direction PC which the 
 action ofthe sun adds to the gravitation ofthe particle P; and 
 another in the direction PA", parallel to CS, by which the action 
 ofthe sun draws the particle from the plane ArfB perpendicular 
 to the right line SC at C. The former force is always as the 
 distance PC; and if V represent this force at the mean distance 
 d, then (PN and PM being perpendicular to AB and DE in N 
 
 V 
 
 and M respectively) it may be resolved into a force PN x j 
 
 perpen-
 
 Chap. XIV. Of the Tides, 155 
 
 perpendicular to the plane AfZB and a force PM x t perpendi- 
 cular to DE, which we now suppose to be the axis of this oh- 
 
 3V 
 
 long spheroid. The latter force is PN x -j • Therefore if the 
 gravity at D be represenled by D, and the gravity at A by A, 
 CA and CD by a and b, as formerly ; the particle P will gravi- 
 tate in the direction PN perpendicular to the plane A^B with a 
 
 force -J X PN by art. 634, and the whole force, with which it 
 will tend in that direction in consequ ence of its gravity and 
 
 the other two forces, will be -j j- x PN. The particle 
 
 tends in the direction PM perpendicular to the axis with a force 
 
 - J X PM. The former force is always as PN the distance 
 
 of the particle P from the plane Ac?B to which its direction is 
 perpendicular; and the latter as the distance from the axis DE. 
 Therefore by art. 640, if the whole force at A be to the whole 
 
 force at D (that is, if A + --j- be to D -j-) as b\oa; the 
 
 fluid will be every where in aquilibriQ. And any particle P will 
 tend towards the spheroid in a direction PK perpendicular to its 
 surface APDB, with a force that is always measured by the 
 right line PK terminated by the axis DE in K. 
 
 687. Let L represent the logarithm of the ratio of CAto DF, 
 or of the subduplicate ratio of 6 + c to 6 — c, the. modulus be- 
 ing b ; and by art. 647, D will be to A as 2ff6L — 2a6cto bbc 
 
 — aoL ; consequently Db will be to Aa as £6Z/L — 266c to 
 bbc — aaL. Therefore if L — c be represented by K,D6 — Aa 
 will be to D6 as 366K — ccK — ccc to 266K, or (because K is 
 
 equal to ^ + ^ + — , Sec.) as |, + ^1^' See. to 1 + ^,, 
 &c. And (because Db — Aa is equal to — ^ — x V, by the 
 
 last article) '- — -— X V ib to D6 in the same ratio. Hence if we 
 
 suppose h equal to J + J, and a equal io d — x, we shall find 
 
 that
 
 156 Of the Tides. Book 1. 
 
 that .r is to d nearly as 1.5V to 8D, or, more nearly^ as 15V is 
 to 8D — 9 y V ; and that the excess ol" CD above CA is to the 
 mean semidiameter d, as 15V to 4D — 4 -H: X V. 
 
 688. The mean force, which the solar action adds to the jjra- 
 vity ol' the moon in the quadratures, is to the gravity of the 
 moon towards the earth at her mean distance, in the duplicate 
 ratip of the periodic time in which the moon would revolve 
 about the earth in a circle at her mean distance, by her gravity 
 towards the earth only, to the periodic time of the earth about 
 the sun. By diminishing the former of these forces in the ratio 
 of the mean distance of the moon to the semidiameter of the 
 earth, and increasing the latter force in the duplicate ratio. Sir 
 Isaac Newton finds V to be to D as 1 to 38604600. There- 
 fore the ascent of the water under the equator, in consequence 
 of its unequal gravitation towards the sun, ought to be to the 
 semidiameter of the equator as 15 to 4 X 38604600; and this 
 ascent oua^ht to be about 1 foot 1 1 i^- inches; which almost co- 
 incides with that which Sir Isaac founds by computing it brief-^ 
 ly from what he had shown before concerning the figure of the 
 earth. Hededuces the lunarforce from the solar, by comparing 
 their effects in the syzygies, when they conspire together, with 
 their effects in the quadratures of the sun and moon when these 
 forces act against one other. The effect of the moon is much 
 greater than of the sun, by common experience ; and by his 
 computations, the lunar force is to the solar as 448 to 100. 
 These effects (according to observations and his theory) depend 
 upon the positions of the luminaries to one another, their dis- 
 tances from the earth, their declinations from the equator, the 
 latitudes of places, and the form and situation of the channels 
 by which the tides are propagated to them from the ocean. 
 
 689. The ascent of the water, which was determined in the 
 ]ast article, is that which would be produced under the equator 
 in consequence of the solar force, if the earth was fluid, and had 
 no diurnid rotation ; the gravitation towards the particles of the 
 earth being supposed to decrease as the squares of the distances 
 from them increase. But it does not follow, that the ascent 
 of the water which arises from the solar action will be so great, 
 if the oblong spheroid ADB£ be in a different situation, and
 
 Chap. XIV. Of the Tides. \5l 
 
 its transverse axis be not directed towards the sun ; or when the 
 whole mass (because of the constant figure of the solid parts) 
 cannot assume the figure of such a spheroid. For the difference 
 of CD and CA, that we have computed, proceeds not from 
 the action of the sun only, but in part from the excess of the 
 gravity at A above the gravity at D, which is owing to the 
 spheroidical figure, and depends upon it. If the gravity had 
 been supposed uniform in all parts of the surface, the ascent of 
 
 the water would have not been above ~ X c/, which is less than 
 
 -rrp X t? by — > or one fifth part of the whole ascent. When 
 
 the transverse axis of the oblong spheroid is directed towards the 
 sun, the solar force and the dimhiution of gravity at the extremi- 
 ty of the transverse axis conspire together to produce the ascent 
 of the water from A to D. But when DE the transverse axis 
 of this oblong spheroid constitutes an angle with the right line 
 CS, that joins the centres of the sun and earth, while the solar 
 force endeavours to raise the water in this right line, the excess 
 of the gravity at A above the gravity at D tends to raise it in 
 a different part; and if by increasing the velocity of the diur- 
 nal rotation, the transverse axis DE should become perpendicu- 
 lar to CS, these causes would act directly against one another. 
 690. Sir Isaac Newton has shown that the lunar orbit (ab- 
 stracting from its excentricity) ought to be an elliptic figure, 
 having its centre in the centre of the earth, and the shorter axis 
 directed to the sun, in consequence of the inequality of the gra- 
 vity of the moon and earth towards the sun ; and, supposing 
 it to be a perfect ellipsis, endeavours to determine the ratio of 
 the second axis to the transverse, jarop. 28, lib. 3, Princip. In 
 the same manner, if we should suppose the earth to re . olve on 
 its axis with a sufficient velocity, the particles of the sea at the 
 equator would describe figures of an elliptic form about the 
 centre of the earth, and revolve as satellites, without gravitating 
 on those beneath them; andDE the greater axis of those figures 
 being perpendicular to CS,the greatest ascent of the water would 
 beat Dand E. If we g.297) should suppose all the sections of 
 the earth perpendicular to its axis to be ellipses of this kind 
 
 similar
 
 158 Of the Tides. Book f. 
 
 similar to each other^ and the whole mass to form either an ob- 
 late spheroid, such as would be generated by the semi-ellipsis 
 ADB revolving about the second axis AB, or an oblong sphe- 
 roid, such as would be generated by DAE, about the transverse 
 axis DE ; then if the ratio of CD to CA was such, that (A 
 and D being supposed to represent the gravities at A and D, as 
 
 CA 
 
 formerly) A — ^ x D should be to SV as CA is to the mean 
 
 semidiameter, the whole force that would acton each particle 
 P, resulting from its gravity and the solar action, would be di- 
 rected precisely to the centre C, and vary in the same ratio as 
 the distance PC. For CD being represented by b, CA by a, 
 and the mean semidiameter by d, as formerly, let PN be per- 
 pendicular to DE in N, PM perpendicular to AB in M, and 
 Nj be taken uponNC in the same ratio to NC as D X a to A X ^6, 
 join Vq, and produce it till it meet AB in Q. Then, by art. 
 QS5, the gravity at P towards this spheroid will act in the di- 
 rection PQ, and be always as PQ. Because N^ is to NC as 
 Dtf to Ah, Cq is to NC, or CQ to MQ, as A5 — Da to Ab, or 
 
 (by the supposition) as -^- to A ; that is, as the force by which 
 
 the solar action endeavours to draw the water at A from the 
 plane perpendicular to CS, to the gravity at A ; or as the force 
 PA- by which the sun endeavours to draw the particle P from 
 that plane to the gravity at P in the direction PN, by art. 634. 
 Therefore CQ is to PQ as the force Pk to the gravity at P ; 
 and the force which acts at P, compounded from the gravity 
 and the force P/c, acts precisely in the direction PC, and varies 
 in the same proportion as the distance PC. The other force 
 which the solar action adds to the gravit}' is directed to C, and 
 varies likewise as PC. Therefore the whole force that in this 
 case acts on any particle P tends precisely to the centre of the 
 spheroid, and is as the distance PC, And (by article 445) any 
 particle Pin the plane of the equator issuing from any point P 
 with a just velocity, would dcscrihe the ellipse ADBE accurate* 
 ly, in the same time that a bod}' would describe a circle about 
 
 C at the distance DC by the force D + j x V, or at any 
 
 distance
 
 presented hy d + x, and ahy d — x) as d -\- -^ io d + 
 nearly ; consequently, A5 is to Da as fi? + -r^ to c? + -^ / and 
 
 Chap. XIV. Of the Tides. 159 
 
 distance CP by the whole force that acts at P : or if the earth 
 was supposed to revolve on its axis in this time, the water in the 
 canal EADB would move freely in this figure without gravi- 
 tating on the bottom of the canal. 
 
 691. The ascent of the water in this case, or the excess of 
 CD above CA, depends on the supposition that Ah — aD is to 
 0V6 as a to d, by which the whole compounded force that 
 acts on any particle of the spheroid is reduced precisely to the 
 direction PC, so as to be measured by PC. To determine this 
 ascent, and the form of the ellipsis EADB, the distance of the 
 
 focus from the centre being represented by c, A was to D as 
 
 ^ "*" 5^7? ^^' *^ 1 + Ioij» ^^' ^y ^^^' ^^^' ^^ ^^ being re- 
 
 b V rJ. t\ n<i fJ -U . ^„ ^ . 
 
 5 — "- • 5 
 
 5 + 1 
 
 Ab — Da to Da as ~ to d. Therefore sV is to D as 12ar 
 
 to 5d, or X to d as 5V to 4D ; and the whole ascent of the wa- 
 ter, or Qx, to d as 5V to 2D. This will be found to be the 
 ascent of the water likewise, when the figure is supposed to be 
 such an oblong spheroid as would be generated by a semi- 
 ellipsis EAD about the axis ED. But when we abstracted from 
 the diurnal rotation of the earth in art. 688, the ascent of the 
 
 15V 
 
 water was found to be -^ x d (the same which Sir Isaac 
 Newton has defined) which is greater than this ascent 
 ~ X rf in the ratio of 3 to 2. The transverse axis of the sphe- 
 roid DE is not in the position wehave here supposed; but when 
 the axis DE is inchned to CS in any angle, the ascent deter- 
 mined in art. 688 is to be diminished on this account. 
 
 692. Sir Isaac Newton having considered the ascent of 
 the water, and the elliptic figure of the lunar orbit (abstracting 
 from its excentricity) as simWar pha;nome7ia arising from the so- 
 lar force, let us imagine DBAE now to represent the lunar orb 
 replenished with water; and the difference of the semidiame- 
 
 ters
 
 160 Of the Tides. Book L 
 
 teis CD and CA^ according to the last article, would be to the 
 mean seniidiameter as oV to 2D, that is (in the present suppo- 
 
 sition) as 5 to 2 X 178 -q, or as 1 to 71 ; and CA would be 
 
 to CD as 70 to 71. According to pro/?. 28, lib. 3, Frincip. this 
 ratio is that of 69 to 70. This agreement seems to be acci- 
 dental; but it appears from it, that if \vc had determined the 
 ascent of the water, or the difference of CD and CA from the 
 figure which is there ascribed to the lunar orbit, it would have 
 been foinid nearly the same as in the last article. It would be 
 found however nearly equal to that which was computed in 
 art. 688, if we wei'fe to determine it from the figure which Dr. 
 Hnlhy ascribes to the lunar orbit from observations. 
 
 693. Because the earth is not fluid, and the solid parts re- 
 tain the same figure in all positions of the .sun or moon, the 
 ascent of the water will be different fiom what was determined 
 in art. 688, on this account likewise; and that the effect of 
 this may be sensible, appears from art. 689:, vvhere we found 
 that a difference of two feet only betwixt CD and CA, the se- 
 midiameters of the spheroid,gavc occasion to an assent of near 
 five inches. If the earth was a solid sphere of an uniform den- 
 sity, and (abstracting from its diurnal rotation) the water in a 
 small canal ADBE at the surface of its equator was affected by 
 the solar foice, it -will- be found (as in art. 492), that if we 
 should sttppfosp' thcAvaier iiTtke«c!anEil' to- 'assume such a figure* 
 that the.w;holc f^rce which acts on an}'- particle P resulting 
 from the gravity and solar action, should be always perpen- 
 dicular to the surfL\ce of the fluidi the difference of CD and 
 CA Avould be to the seniidiahieter CD as jV to 2D ; conse- 
 quently in this c^s.e the ascent of the water would be only -f- of 
 that which was defined in art. 688. The forces- which pro- 
 duce this phcenomenon -are very minute in comparison of the 
 gravity of the water; and circumstances, that in other enquiries 
 are safely neglected, may have a sensible effect upon it. 
 
 694. There areparticdar causes, besides these mentioned by 
 Sir- Isaac Nezcton, and in the last article, that interfere in 
 producing the various phoenomena of the tides. The inequali- 
 ty of the velocities with which bodies revolve, by the diurnal 
 
 motion
 
 Cbap. XIV. Of the Tides. l6l 
 
 motion about the axis of the earth, in different latitudes, may 
 have some effect on the motions of the sea and air ; and may 
 contribute to occasion greater tide's, than might be otherwise 
 expected from the theory, especially if their course be not far 
 from the meridian. A current that sets out directly towards 
 tlie north, ouglit, on this account, to bend its course soon after- 
 wards somewhat towards the east ; if it set out towards the 
 south, its course ought afterwards to incline towards the west ; 
 and with the change of direction it may in some cases acquire 
 greater force. Several phajnomena may perhaps be accounted 
 for from this consideration. But we are not to enter farther 
 into this enquiry in this place. 
 
 (395. If there is an ocean in Jupiter, the tides may be veiy 
 considerable when all or most of his satellites are in one right 
 line ; and it may be worth while to observe, whether the great 
 and sudden changes,thataresometimesperceived byastronomers 
 to happen on the surface of this planet, have any analogy with 
 their conjunctions and oppositions. If the other secondary 
 pla nets, as well as the moon, move on their axis so as to h ave nearly 
 the same hemisphere turned always towards their primary planet, 
 the tides in their seas (if they have any) will chiefly proceed 
 from the variation of their distances from it, and such maybe 
 sufficient ; whereas their tides would probably be too great if 
 they revolved on their axes with a greater velocity. 
 
 696. In this chapter we have endeavoured to determine ac- 
 curately some of the consequences of Sir Isaac Nezoton's 
 theory of gravity ; being persuaded that, however obscure the 
 cause of gravity may be, there is hardly any proposition in ex- 
 perimental philosophy established on better evidence, than that 
 there is such a power in Nature, and that it observes the laws 
 we have supposed. We have sometimes made use of the term 
 attraction, as a convenient expression only, and because itserved 
 to distinguish the real gravity from the apparent ; svhich last is 
 often a combination of gravity and several other powers. Sir 
 haac NezctoH has shown how to compute the attraction of 
 bodies, when the particles ave supposed to attract each other ac- 
 cording to other laws. We shall only subjoin an easy proof 
 of one proposition on this subject. Suppose that the attraction 
 
 VOL. U. M of
 
 162 Of otheT Laics of A tlr action. Book t, 
 
 of the particles of the cone PAEa (fig. 283) decreases in the 
 same proportion as the cubes of the distances from them increase; 
 and a particle atP will tend to the splierical surface MN//j(that 
 has its centre in P) with a force that is as this surface (or PM^) 
 directly, and PM^ inversely ; that is, with a force which is as 
 PM inversely, or directly as MV the ordinate of the hyperbola 
 KVI described betwixt the asymptotes PA and PH. Therefore 
 the attraction o^ xhefriishim MNmAErt will be measured by the 
 hyperbolic area MVIA bounded by the ordinates at A and M ; 
 and the attraction of the cone PMNw by the infinite hyperbo- 
 lic area that is conceived to be formed betwixt the ordinate MV 
 and asymptote PH. It follows, that if such a law could take 
 place, the particle P Avould tend towards the least portion of 
 matter in contact with it by a greater force than towards the 
 greatest body at any distance how small soever from it. The 
 same is easily shown from art 297, when the attraction of the 
 particles decreases as any powers of the distances, higher than 
 their cubes, increase. As such laws would be very improper 
 for preserving the celestial bodies in their regular courses (by 
 art. 447 and 448), so they would be very unfit for producing a 
 just force, by which their several parts might be kept together. 
 The true law of gravity is better adapted for those purposes. 
 It is the chain that holds the parts of each in a proper union, 
 that perpetuates the motions in the great system about the sun, 
 the preserves the revolutions in the lesser systems, of which it is 
 composed, nearly regular. Its inequalities, in some cases, have 
 their use, as in the tides ; and a remarkable geometrical sim- 
 plicity is often found in the conclusions that are deduced from 
 it ; of which we have had several instances, as in art. 445, 
 446, Q^3Q, Q^Q, and 690. 
 
 BOOK
 
 BOOK II. 
 
 OF THE COMPUTATIONS IN THE METHOD OF FLUXIONSs 
 
 CHAP. I. 
 
 Of the Fluxions of Quantities considered abstractly/, or as 
 represented hy general Characters in Algebra, 
 
 697. J- HE idea of a fluxion, as described in art. 10 and 11, 
 after Sir Isaac Newton, seemed to be more immediately applica- 
 ble to geometrical magnitudes, which we may very naturally 
 conceive to be generated by motion, than to quantities consider- 
 ed abstractly, or as they are expressed by general symbols in al- 
 gebra. For this reason we chiefly considered the fluxions 
 of geometrical magnitudes in the first book ; and most com- 
 monly gave demonstrations from geometry, because these are 
 often preferred, as more satisfactory than algebraic computa- 
 tions. The evidence of the method had been disputed, and ob- 
 jections had been made to the number of symbols employed in 
 it, as if they might serve to cover defects in the principles and 
 demonstrations. In order to obviate any suspicions of this kind, 
 we endeavour to describe it in a manner that might represent 
 the theorems plainly and fully, without any particular signs of 
 characters, that they might be subjected more easily to a fair 
 examination. 
 
 698. But an important part of this doctrine still remains td 
 be described. The improvements that have been made by it, 
 either in geometry or in philosoph}^ are in great measure ow- 
 ing to the facility, conciseness, and great extent of the method 
 of computation, or algebraic part. It is for the sake of these 
 
 M 2 advantages
 
 164 Of the Fluxions Book II. 
 
 advantages that so many symbols are employed in algebra, the 
 number and complication of which (together with the greater 
 care there has been taken in treating of geometry, after the ex- 
 cellent models left us by the antients), have contributed more 
 to occasion the preference that is often ascribed to geometry, in 
 respect of perspicuity and evidence, than any essential difference 
 that can be supposed to be between them. It is a general kind 
 of arithmetic ; and this is what renders its usefulness so univer- 
 sal ; nor can this be supposed to derogate from its evidence, if 
 we have no ideas more clear or distinct than those of numbers, 
 and often acquire more satisfactory and distinct knowledge 
 from compulations than from constructions. It may have been 
 employed to cover, under a complication of symbols, abstruse 
 doctrines, that could not bear the light so well in a plane geo- 
 metrical form ; but without doubt, obscurity may be avoided in 
 this art as well as in geometry, by defining clearly the import 
 and use of the sym])ols, and proceeding with care afterwards. 
 
 G99. The use of the negative sign in algebra is attended with 
 several consequences that at first sight are admitted with diffi- 
 culty, and has sometimes given occasion to notions that 
 seem to have no real foundation. It implies that the real value 
 of the quantity represented by the letter to which it is prefixed 
 is to be subtracted ; and it serves, with the positive sign, to 
 keep in view what elements or parts enter into the composition 
 of quantities, and in what manner, whether as increments or 
 decrements (that is, whether by addition or subtration), which 
 is of the greatest use in this art. In consequence of this, it 
 serves to express a quantity of an opposite quality to the posi- 
 tive, as a line in a contrary position, a motion with an opposite 
 direction, or a centrifugal force in opposition to gravity ; and 
 thus often saves the trouble of distinguishing, and demonstrating 
 separately, the various cases of propositions, and preserves their 
 analogy in view. But as the projX)rtion of lines depends on 
 their magnitude only, without regard to their position ; and 
 motions and forces are said to be equal, or unequal in any given 
 ratio, without regard to their directions ; and in general the 
 proportion of quantities relates to their magnitude only, with- 
 out determining whether they are to be considered as incre- 
 ments
 
 Chap. I. of algebraic Quantities. l65 
 
 ments or decrements; so there is no ground to imagine any 
 other proportion of — b and + a (or of — 1 and 1) than 
 that of the real magnitudes of the quantities represented by b 
 and a, whether these quantities are in any particular case to be 
 added or subtracted. It is the same thing to subtract a decre- 
 ment as to add an equal increment, or to subtract — b from 
 tf — 5 as to add + 6 to it : and because multiplying a quan- 
 tity by a negative number implies only a repeated subtraction 
 of it, the multiplying — ^ by — ■ n, is subtracting — 6 as 
 often as there are units in n; and is therefore equivalent to ad- 
 ding + 6 so many times, or the same as adding + 7ib. But if 
 we infer from this, that 1 is to — n as — b to w6, according 
 to the rule that unit is to one of the factors as the other factor 
 is to the product; there is no ground to imagine that there is 
 any mystery in this, or any other meaning than that the real 
 magnitudes represented by l,7i, b and nb are proportional. For 
 that rule relates only to the magnitude of the factors and pro- 
 duct, without determining whether any factor, or the product, 
 is to be added or subtracted. But this likewise must be de- 
 termined in algebraic computations ; and this is the proper use 
 of the rules concerning the signs, without which the opera- 
 tion could not proceed. Because a quantity to be subtracted 
 is never produced, in composition, by any repeated addition of 
 a positive, or repeated subtraction of a negative, a negative 
 square number is never produced by composition from a root. 
 Hence the \/^, or the square-root of a negative, implies an 
 imaginary quantity, and in resolution is a mark or character of 
 the impossible cases of a problem ; unless it is compensated by 
 another imaginary symbol or supposition, when the v.'hole ex- 
 pression may have a real signiiicution. Tims 1 + V~l, and 
 1 — V~\ taken separately are imaginary, but their sum is 2 ; 
 as the conditions that separately would render the solution of a 
 problem impossible, in some cases destory each other's effect 
 when conjoined. In the pursuit of general conclusions, and of 
 simple forms for representing them, expressions of this kind 
 must sometimes arise where the imaginary syuibol is compensat- 
 ed in a manner that is not always so obvious. By proper sub- 
 stitutions, however, the expression may be transformed into 
 
 •M 3 auother,
 
 1G6 i^i^^^-Fhlxiofis Book II. 
 
 another, wherein each particular term may have a real signifi* 
 cation, as well as the whole expression. The theorems that 
 are sometimes briefly discovered by the use of this symbol, 
 may be demonstrated without it by the inverse operation, or 
 someother way: and though such symbols are of some use in the 
 computations in the method of fluxions, its evidence cannot be 
 said to depend upon any arts of this kind. We have just men- 
 tioned these things without enlarging upon them, for we sup- 
 pose that the common algebra is admitted. 
 
 700, The rules for the computations in this method may be 
 deduced from art. 99. bat it may be worth while to demon- 
 strate them here briefly, from general principles that may seem 
 more immediately applicable to algebraic quantities. Any quan- 
 tities that are produced from each other by an algebraic ope- 
 jation, or whose relation is expressed by any algebraic form, 
 being supposed to increase or decrease together, some will be 
 found to increase or decrease by greater differences, or at a 
 greater rate; others by less differences, or at a less rate; and 
 while some are supposed to increase or decrease at one constant 
 rate by equal successive differences, others increase or decrease 
 by differences that are always varying. We have no occasion 
 for considering such quantities in this doctrine as generated by 
 motions, and for enquiring into the velocities of those motions, 
 or for considering the prime or ultimate ratio of their incre- 
 ments or decrements, but for ascertaining the respective rates, 
 according to which they increase or decrease, when they are 
 supposed to vary together; in order from these to discover the 
 properties of the quantities themselves. Thus by comparing 
 the velocities of points that are supposed to generate lines at 
 the same time, it appears when a line increases at a greater or 
 less rate than anoLlier, and in what proportion. The same is to 
 be said of any quantities, which, while they vary together, 
 are always in the same proportion to one another as those lines. 
 But it does not seem to be necessary to have always recourse 
 to such suppositions; thoughin treating of geometrical magni- 
 tudes, that are often conceived to be generated by motion, this 
 method of comparing the rates of their increase or decrease is 
 natural and clear, and has other advantages *. When a quanw 
 
 * Art. i74.
 
 Chap. I. of algebraic Quantities. 1 67 
 
 tity A increases by differences equal to a, 2 A increases or de- 
 creases by differences equal to 2^/, and manifestly increases or 
 decreases at a greater rate than A in the proportion of 2a to a, 
 
 or 2 to 1 : and if m and ii be invariable, — increases or de- 
 creases by differences equal to — . and therefore at a greater 
 
 or less rate than A in proportion as — is greater or less than a, 
 
 or m is greater or less than n. This seems to be easily conceiv- 
 ed, without having recourse to any other considerations than 
 the relations of the differences by which the quantities increase 
 or decrease. In order therefore to avoid the frequent repeti- 
 tion of figurative expressions in this algebraic part as much as 
 possible, we will endeavour to substitute in place of the defini- 
 tions and axioms above (art. 11 and 15), others that are ra- 
 ther of a more general import, but are perfectly consistent with 
 them, and are best explained by them ; as other principles and 
 propositions in algebra are commonly best illustrated from geo- 
 metry. 
 
 701. By thejluxions of quantities we shall therefore now un- 
 derstand, ani/ measures of their respective rates of increase or de- 
 crease, while they vary (or fioio) together. There can be no 
 difficulty in determining those measures when the quantities 
 increase or decrease by successive differences that are always in 
 the same invariable proportion to each other, as in the last arti- 
 cle. While A by increasing becomes equal to A -J- a, or by 
 decreasing equal to A — a, 2A becomes equal to 2A -^ 2a, or 
 to 2A — 2a ; and as 2 A increases or decreases at a greater 
 rate than A in the proportion of 2a to a ; so the fluxion of A be- 
 ing supposed equal to a, the fluxion of 2 A must be equal to Q.a. 
 
 In the same manner the fluxion of- X A Cor of - x Ajl^, 
 
 supposing m, n, and e to be invariable), is - X a ; and since 
 
 m may be to n in any assignable ratio, a quantily may be al- 
 ways assigned that shall increase or decrease at a greater or 
 Jess rate than A in any proportion, or that shall have its flux- 
 
 M 4 ion
 
 V. 
 
 168 Of the Fluxions Book II, 
 
 ion greater or less than the fluxion of A in any ratio. In such 
 cases the ratio of the fluxions and that of the differences by 
 which the quantities increase or decrease are the same. 
 
 702. But while A is supposed to increase at a constant rate 
 by any equal successive difterences, ifB increase or decrease by 
 differences that are always varying, B cannot be said to increase 
 or decrease at an}' one constant rate ; and it is not so obvious 
 how, the fluxion of A being supposed equal to its increment a,, 
 the variable fluxion of B is to be determined. It cannot be 
 supposed that the fluxions and differences are always in the 
 same proportion in this case ; but it is evident, however, that 
 if B increase by differences that are always greater than 
 
 the equal successive differences by which - x A increases, it 
 cannot increase at a less rate than- x A : and that it cannot 
 increase at a greater rate than - x A, while its successive dif- 
 ferences are always less than those of- X A. The fluxion of 
 A being still represented by a, the fluxion of B therefore can- 
 not be less than - X a in the former case, or greater than - x a 
 
 in the latter. The following propositions are consequences of 
 this, and will enable us to determine at what rate B increases 
 when its relation to A is known. 
 
 703. The successive values of the root A being represented 
 by A — a, A, A + a, S)C. which increase by any coaistant 
 difference a, let the corresponding values of any quantity pro- 
 duced from A by any algebraic operation (or that has a depend- 
 ance upon it so as to vary with it) be B — b, B, B -f- 6, ^c. 
 Then if the successive differences b, b, S)-c. of the latter quan- 
 tity always increase, how small soever a may be, then B can- 
 not be said to increase at so great a rate as a quantity that 
 increases uniformly by equal successive differences greater than 
 b, or at so small a rate as any quantity that increases uniformly 
 by equal successive differences less than b. In like manner, if 
 the relation of the quantities is such, that the successive differ- 
 ences
 
 Chap. I. of algebraic Quantities. 169 
 
 ences, b^ b, &c. continually decrease; then B cannot be said 
 to increase at the same rate as a quantity that increases uniform- 
 ly by equal successive differences greater than b, or less than b, 
 
 704. Therefore the fluxion of A being supposed equal to the 
 increment a, the fluxion of B cannot be greater than b or less 
 than b, when the successive differences b, b, &c. continually 
 increase; and cannot be greater than b, or less than b, whea 
 these successive differences always decrease. 
 
 705. In the samemanner if the latter quantity decrease while 
 the former increases, iiud its successive values be B + 6, B, 
 B — b, &c. then if the decrements Z», b, &c. continually in- 
 crease, B cannot be said to decrease at so great a rate as a quanti- 
 ty that decreases uniformly by equal successive differencesgreater 
 than b, or at so low a rate as a quantity that decreases uniform* 
 ly by equal successive difference less than b. Therefore, in 
 this case, the fluxion of A being supposed equal to a, the fluxion 
 of B cannot be greater than b, or less than b. And in the same 
 manner if the successive decrements b, b, &c. always decrease, 
 the fluxion of B cannot be greater than b or less than b. 
 
 706, As the fluxions of quantities are any measures of the re- 
 spective rates according to which they increase or decrease, by 
 art. 701, so it is of no importance how great or small soever 
 those measures are, if they be in the just proportion or relation 
 to each other. Therefore if the fluxions of A and B may be 
 supposed equal to a and b, respectively, they may be likewise 
 
 supposed equal to \a and \b, or to — and — • These prin- 
 ciples (as other algebraic propositions) may be illustrated from 
 geometry, as we observed in art. 700. And the propositions 
 concerning the fluxions of areas, ordinates, &c. in the first 
 book, may be demonstrated immediately from them ; but this 
 would be needless. 
 
 707. Prop. I. Tht fluxion of the root A being supposed equal to 
 a, theJiuxloH of the square A A will be equal to qA X a. 
 
 Let the successive values of the root be A — «, A, A + «> 
 and tke corresponding values of the square will be AA — 
 
 2Am
 
 170 Of the Fluxiom Book 11. 
 
 SAm + uu,AA, A A + <2Au + mt, wliich increase by the dif- 
 ferences 2Am — nil, 2A?/ + uu, &c. and because those differ- 
 ences increase, it follows from art. 704, that if the fluxion of A 
 be represented by it, the fluxion of AA cannot be represented 
 by a quantity tliat is greater than Q,Au + uu, or less than 
 SAm — uu. This being premised, suppose, as in the proposi- 
 tion, that the fluxion of A is equal to a; and if the fluxion of 
 AA be not equal to Q.Aa, let it first be greater than 2Aa in any 
 ratio, as that of 2A + o to 2 A, and consequently equal to 
 SAct 4- oa. Suppose now that u is any increment of A less than 
 o; and because a is to u as Q.Aa + oa to QAu + ou, it follows 
 (art. 706) that if the fluxion of A should be represented by u, 
 the fluxion of AA would be represented by Q,Au -\- ou, which 
 is greater than 2 Am + uu. But it was shown, from art. 
 704, that if the fluxion of A be represented by u, the fluxion of 
 AA cannot be represented by a quantity greater than 2Am + uu. 
 And these being contradictory, it follows that the fluxion of A 
 being equal to a, the fluxion of 2AA cannot be greater than 2Aa. 
 If it can be less than 2Aa, when the fluxion of A is supposed 
 equal to a, let it be lees in any ratio of 2A o to 2A, and there- 
 fore equal to 2Aa — oa. Then because a is to u as 2Aa — oa 
 is to sAm — on, which is less than QAu — uu (u being sup-i- 
 posed less than 0, as before), it follows that if the fluxion of A 
 was represented by w, the fluxion of A A would be represented 
 by a quantity less than 2Ak — uu, against what has been shown 
 from art. 704. Therefore the fluxion of A being suppose^ 
 equal to a, the fluxion of AA must be equal to Q,Aa. 
 
 708. The fluxions of A and B being supposed equal to a and 
 b, respectively, the fluxion of A + B will be a + b, the fluxion 
 of A+B^ or of A A + 2AB + BB, will be 2 X 1C\^ x 7+7 
 or 2Aa + 2B6 + 2Ba + 2 A6, by the last article. The flux- 
 ion of AA + BB is 2Aa + 2B6, by the same; consequently 
 the fluxion of 2AB is 2Ba + 2A^; and the fluxion of AB is 
 Ba + A6. Hence if P be equal toAB, and the fluxion of P be 
 p, then p will be equal to B« + Ab, and dividing by P, or AB, 
 
 we find p- = X "»"§• ^^^ — g 5 and ^r be the fluxions of Q, 
 
 then
 
 Chap. I. of algebraic Quantities, 171 
 
 then QB = A,| + -r=jorJ=^ — ^; and consequent- 
 
 T Qa Qb a Ah cB — Kb ^-^-r, « , 
 
 •^ ^~X"~T — B~"BB°^ ~"bb~- ^^"en any of the 
 quantities decrease, its fluxion is to be considered as negative. 
 
 709. If w be any integer number, and the sum of the terms 
 E«-', E«-^ F, E«-^ F% E«-* F^ S^c. continued till their 
 number be equal to n, be multiplied by E — F, the product 
 will be E'* — F«. For the terms being formed by subducting 
 continually unit from the index of E and adding it to the index 
 of F, the last term will be F"— '. The product of their sum 
 
 multiplied by E will be E« + E«-^ F + E«-^ F^ + 
 
 EF»— ' ; their sum multiplied by — F gives — E«— ' F — 
 £„_2 F* .... — EF«— ' — F« J and the sum of these two 
 products is E» — F«. 
 
 710. Supposing E to be greater than F, E" — F^ will be 
 less than wE«— ' x e — F, but greater than /iF*— ' X e — f. For 
 each of the terms E«— % E'»— ^ F, E'*— ^ F^, &)C. is greater than 
 the subsequent term in tlie same ratio that E is greater than F, 
 and E»— ' is the greatest term ; consequently the number of 
 terms being equal to n, /iE"— ' is greater than their sum ; and 
 ?iE"— ' X E — F is greater than their sum multiplied by E — F, 
 or (by the last article) greater than E'» — F". Because the 
 last term Fi*— ' is less than any preceding terra, wF«— ' x e— F 
 is less than the sum of the terms multiplied by E — F, or less 
 than E» — F», 
 
 711. When n is any integer positive number, the root A be- 
 ing supposed to increase by any equal successive differences, the 
 successive ditfercnces of the power A'* will continually increase. 
 For let A — a. A, A + a, be any successive values of the root, 
 and A — a". A", a + «" will be the corresponding values of the 
 power. But A 4- a" — A'Ms greater than wA«— 'b; as appears 
 by substituting, in the last article, A + « for E, A for F, and a 
 for E — F. In like manner wA"— 'a is greater than A" — A— o*. 
 Therefore A~rS" — A** is greater than A" — a — J^, and the 
 successive differences of tl'fe power continually increase. 
 
 712. Prop.
 
 172 Of the Fluxions Book IL 
 
 712. Prop. II. The fluxion of the root A being supposed equal 
 to a, ihefuxion of the power A" Zi)ill be waA"— '. 
 
 For if the fluxion of A« can be greater than naAr-—^, let the 
 excess he equal to any quantity r ; suppose o equal to the ex- 
 
 cess of ^ A"—' +X, above A, and consequently ^-f.o«— * = 
 
 na 
 
 A«— ' + -^ • Then na x T+T"— ' will be equal to nflA**—* +r, 
 
 the fluxion of A». Let u be any increment of A less than o ; 
 and because a is to m as na X a + o"~* to nii X a + J^—^, it fol- 
 lows (by art. 706), that if the fluxion of A be now repre- 
 sented by the increment ^<, the fluxion of A will be represented 
 "by nil X A4-o»— ' which is greater than nu X a-j-w**^*, and this 
 last is itself greater than a -}- «*— ' — A«, by art. 7 10. But 
 when the successive values of the root are A — w. A, A + m, 
 those of the power are a + r A A^, a + u^, the successive dif- 
 ferences of which continually increase ; consequently (by art. 
 704), if the fluxion of A be represented by u, the fluxion of A" 
 cannot be represented by a quantity greater than a+ «" — A", 
 or less than A'* — a + «". And these being contradictory, it 
 follows that when the fluxion of A is supposed equal to a, the 
 fluxion of A*^ cannot be greater than waA«— '. If it can be less 
 than naA»— ', let it be equal to waA" — r, or (by sup- 
 
 «— 1 
 
 posing o = A — '^ A«-*— »- ) to na X a^«-'. Then u 
 
 na 
 
 being supposed less than o, if the fluxion of A was represented 
 by u, the fluxion of A** would be represented by 7iu X a — o«— ', 
 •which is less than nu x a — »."— ' (because we suppose u to be 
 less than o) and therefore less than A« — X'^I^'^by art. 710. 
 But this is repugnant to what has been demonstrated from 
 art. 704. Therefore the fluxion of A being supposed equal to 
 a, the fluxion of A" must be equal to 72flA'*— '. 
 
 713. The
 
 Chap. I. of algebraic Quantities. 17S 
 
 713.Thefluxionsof— jorofA-, may be determined in the 
 
 same manner : but these being comprehended in the following 
 theorem, it is needless to consider them separately. We shall 
 only observe that the lemma for determining the former is, that 
 
 1 J £'i prt 
 
 when E is greater than F, — ■ — -r. or — is less than 
 
 o 'f« E" E"F'* 
 »£" — * , ,, . nE—nF i • i . , 
 
 X E— F (by art. 710), or — x e — f which is less 
 
 E"F" EF" 
 
 than "I^-~\ but greater than "-^^ X eTTf (art. 710), 
 
 and consequently greater than ~"_ T* ^^^ hence it may be 
 demonstrated, as in art. 712, that when the fluxion of A is 
 supposed equal to «, the fluxion of ^ is '^[77, the sign being 
 
 negative because — decreases while A increases. We have 
 
 supposed n to be an integer positive number in this and the last 
 article. 
 
 714. Prop. III. Thejliixion of A being supposed equal to a, (he 
 fluxion of A^i zvill ie — X ^ ~ , 
 
 First, let the exponent - be any positive fraction whatsoever , 
 
 ■m 
 m n 
 
 suppose An—K\ consequently A =: K ; and the fluxion of K 
 being supposed equal to k, maA^—^ zz wA'K"—^ by art. 7 12, 
 
 and
 
 174 Of the Fluxions Book it. 
 
 and k or the fluxion of ^^ will be eqflal to — — — r: 
 ^- = - X aA'~ • When ^ is negative^ let it be equal to 
 
 n 
 — r 
 
 — r; and suppose A r=K,or 1 = A''K ; then taking the flux- 
 
 r— I 
 
 ions (by art. 708)^ rA aK + khr — o, and k — 
 
 =: — rA = - X ,. , a. 
 
 r a n .n. 
 
 715. Prop. IV. Suppose P to he the product of any factors A, B, 
 C, Jy, E, 4 f. {or P = ABCDE, 4^-.) ht the fuxions of P, 
 A, B, C, D, E, be re^ptclixdy equal' to p, a, b, c, d, e, ^c. 
 
 cnd^ mil Be equal ^^ x '^ b ^ c '^ d '^^* 
 
 Let Q be equal to the product of all the factors of P, the 
 first A excepted ; that is, suppose P =r AQ. Suppose R equal 
 to the product of all the factors, the first two A and Bexcepted, 
 that is, let P zn ABR, or Q = BR. In the same manner let 
 R — CS, S := DT, and so on. Then, the fluxions of Q, R, 
 S, T, 8^c. being supposed respectively equal to q, r,f, t, 8$c. it 
 
 follows, from art. 708, that ^ =: ^ + ^ — (because ^ = - 
 
 r\ a h r ,, r c s \ a b c s 
 
 + mJ A + B + iT == ^^^^^^^^ R=G+?Ja + B + C+S 
 , .9 d t\ a h c d f T 
 
 = (because- ^p+^pjx + B'^c + D+TJ ^"*^ ^° *^"- 
 
 Therefore ^ is equal to the sum of the quotients when the 
 
 fluxion of each factor of P is divided by the factor itself. 
 
 716. It the factors be supposed equal to each other, and 
 their number be equal to n, then P — A", and by the last pro- 
 position
 
 Chap. I. of algebraic Quantities. 175 
 
 position p = ^ ; consequently^—'^ — naA"~^*y aswefound 
 in art. 712. 
 
 717. Prop. Y.IfV - KLMx^c. ^^^ ^^^ fluxions of the re- 
 spective quantities be expressed by the small letters, p, a, b c 
 
 m 
 
 A. J. J. ^ "ft') ^t yy ^i 
 
 &c. as formerly, then | = ^ + ^ + i - i _ i^_ - ^^. 
 
 For PKLM x S^c. - ABC x S^c. and, by art. 715, 
 
 ^ k I m Q o b c 
 
 p +K+L + M'*^'^*~A + B+C} ^'^' whence by trans- 
 
 . . p a b c k In 
 
 position- =- + - +-_-_-, ^c. 
 
 718. The fluxion of the logarithms being supposed invariable, 
 the fluxions of any quantities N and M will be in the same 
 proportion as these quantities themselves. For it is the funda- 
 mental property of the logarithms, that when they are taken 
 in any arithmetical progression, the quantities of which they are 
 the logarithms are always in a geometrical progression. There- 
 fore, the logarithms being supposed to increase by any equal 
 differences, these quantities will increase or decrease by dif- 
 ferences that increase or decretvse in the same proportion as the 
 quantities themselves. Let A — a. A, A + a, be the respec- 
 tive logarithms of N — n, N, N + n\ and B — a, B, 
 B -j- a the logarithms of M — mM,M -\- m ; then because 
 the logarithms increase by the constant difference a, n will be 
 to w as N to N + w ,• m to m as M to M + m ; and n to m 
 as N + « to M. Therefore when the quantities and their lo- 
 garithms increase together, it follows from art. 704, that if the 
 constant fluxion of the logarithm be supposed equal to its incre- 
 ment a, the fluxion of N will not be greater than n, or the flux- 
 ion of M less than m ; consequently the fluxion of N is to the 
 fluxion of N in a ratio that is not greater than that of « to m, 
 or of N + w to M. But if the fluxion of N could be to the 
 
 fluxion
 
 176 0/ t/ie Fluxions Bookll. 
 
 fluxion of M in any ratio greater than that of N to M, as in that 
 of N + M to M ; then by supposing n to be less than u, the fluxion 
 of N would be to the fluxion of M in a ratio greater than that 
 of N ■{• ?i to M. And these being contradictory, it follows 
 that the ratio of these fluxions is not greater than that of N to 
 M. In the same manner the fluxion of M is to the fluxion of 
 Nina ratio that is not greater than that of M to N. Therefore 
 the ratio of the fluxions of M and N is the same with the ratio 
 of the quantities M and N. When the quantities decrease while 
 ihe logarithms increase, the demonstration is the same. 
 
 719' Prop. VI. The fluxion of any quantity N w to the fluxion 
 of its logarithm as N is to the modulus of the logarithmic 
 system. 
 
 For the quantities and their logarithms being suppossedto in- 
 crease or decrease together, when the quantity increases or de- 
 creases at the same rate as its logarithm, it is then equal to the 
 'modulus. Suppose this quantity to be M, and since the fluxion 
 of N is to the fluxion of M as N is to M, by the last article ; 
 it follows that the fluxion of N is to the fluxion of its logarithm 
 as N is to the modulus. Hence if N := A«, e being any invari- 
 able exponent, the log. N — e x log. A, consequently, the flux- 
 ions of N and A being supposed equal to n and a respectively, 
 
 ■r- ■=. -7—, and ;i rr -r = e\ „, We msisted on this, at 
 
 N A ' A " ' 
 
 some length, in chap. (). book I. 
 
 720. When the fluxion of a quantity is variable, it may be 
 considered as a fluent; and its fluxion may be determined 
 (which is called the second fluxion of that quantity') by the pre- 
 ceding propositions. Thus we found in art. 707, that the flux- 
 ion of A being supposed equal to a, the fluxion oi AA is 2Aa ; 
 and if Abe supposed to increase at an unil'orm rate, or its flux- 
 ion a be invariable, <l\a will increase by equal successive dif- 
 ferences ; consequently ils fluxion, oi the second fluxion of AA, 
 will be equal to any of those <lifferences fart. 701), as to Q.a X 
 TT~a — 2Aa, or '2,aa. If a be variable, let its fluxion be 
 
 equal
 
 Chap. I. of algebraic (Quantities. 177 
 
 equal to z, and the fluxion of 2Aa (or second fluxion of AA) 
 will be £.aa -}• Q.Az, by art. 708. In the same manner, the 
 fluxion of A being constant, the fluxion of 7iA"~^a, or the 
 second fluxion of A", is na x."— ^ x A"~^a, oin x « — i x aa 
 A""" ; the fluxion of this, or the third fluxion of A", is n 
 X i^^TT X T^ITi X fl3A''""3. And the fluxion of A" of any or- 
 der denoted by w, is w X «— i X n — 2 X » 3, Sec. x a"'A"~'", 
 where the. factors in the coefficient are to be continued till their 
 number be equal to m. When n is any integer positive num- 
 ber, the fluxion of A", of the order «, is invariable and equal 
 to H X «— 1 X n—2 X n — 3, &ic. X a". The quantities that re- 
 present those fluxions of A" depend on a, which represents-the 
 fluxion of A. When A remains of the same value, the first 
 fluxion A" is greater or less in the same proportion as a is 
 
 supposed to be greater or less; the second fluxion of A" is in 
 the duplicate ratio of a ; and its fluxion of the order m is as 
 
 «'", If a be variable, but z the fluxion of a, or the second 
 fluxion of A, be constant, then the fourth fluxion of AA will 
 be constant and equal to 6zz ; for we found that the second 
 fluxion of AA was Qua + '2Az ; the fluxion of which is 
 4az + Q,az, or 6az ; and the fluxion of this is 6zz. In like 
 manner the sixth fluxion of A^ will be constant in .this case, 
 and equal to 90^^ 
 
 721. The second differences of any quantity B are the suc- 
 cessive differences of its first differences ; and as the fluxibn of 
 B increases when its successive differences increase, so its second 
 fluxion, or its fluxions of any higher order, increase, when .its 
 second or higher differences increase. If we arrive at diffe- 
 rences of any order that are constant, the fluxion of the same 
 order is constant, and is expressed by that difference. Thus when 
 A is supposed to increase by constant differences equal to a, and 
 It .5 fluxion is supposed equal to a, the second difference of AA 
 
 VOL. II. N ^or
 
 173 ■ . Of the "Notation Botok 11. ■ 
 
 (or "A + o' — SAA + A — a") is Q.aa, which is hkewise it» 
 second fluxion ; and the third difference of A^ is Qa, which 
 is its third fluxion. When n is any integer and positive num- 
 ber;, the fluxion of A" of tlie order n is equal to the fluxion of 
 any of its first differences of the next inferior order, or to the 
 fluxion of any of its second differences of the order n — 2, and 
 so on. For the fluxion of A + «" — A" (one of the first dif- 
 ferences of A") of the order ?? — 1 is n x n— i x„— 2, &,c. 
 
 xaT^"-"+^ A X a"^-^=?i x^^^ X „-:r-2. Sec. X «". 
 
 where the coefficients are supposed to be continued till their 
 number be n — 1, so that the last must be 2. And this we found 
 to be the fluxion of A" of the order n^ in the preceding article. lu . 
 
 the same manner, the fluxion of A + «" 2A" + A — a", (the. 
 
 second difference of A") of the order n — 2, is equal to the 
 fluxion of A-i-a" — Ji of the order « — 1 ; and consequent- 
 ly equal to the fluxion of A" of the order 71. These fluxions 
 are invariable and equal to the last or invariable differences. 
 But in other cases the fluxions of A of any order are less than 
 its subsequent differences of the same order, but greater than 
 the preceding differences, as in art. 703. 
 
 722. The preceding propositions are demonstrated briefly 
 by finding the ultimate relation of the differences of the flu- 
 ents, for this will determine their respective rates of increasing 
 or decreasing, or the relation of their fluxions. Thus, because 
 A + a* — A", the increment of A''^ is less than wa x A + a*~^. 
 but greater than naA , by art. 710; and when a is sup- 
 posed to be diminished continually till it vanish, the ultimate 
 ratio of «a x A + a""~^ to 71a A"~ is a ratio of equably: it 
 follows that the ultimate ratio of the increment A + a" — A" to 
 7iaA^~ is a ratio of equality ; and that the fluxion of A 
 
 being supposed equal to a, the fluxion of A" must he 71a A"~~ 
 
 as
 
 Chap. ll. of Fluxions. t79 
 
 as in art. 712. In the same manner the second or higher flux- 
 ions of A", or of any other fluent, are ultimately equal to 
 the corresponding diflferences of the fluent. If we suppose 
 (with Mr. Leibnitz, and those who have followed his me- 
 thod) a to be an infinitely small difference of A, and suppose 
 quantities to be equal when their difference is infinitely less 
 
 than the quantities themselves, 7m X A-\-a"~ must be suppo- 
 sed equal to /jaA"" ; and, since A + 0' — A", the difference 
 of A", cannot be greater than the former, or less than the 
 latter (art. 710), it must be supposed equal to naA"'~ * 
 
 CHAP. II. 
 
 Of the Notation of Fhixio7is,the Rules of the direct Method, and 
 the fundamental Rules of the inverse Method of Fluxions. 
 
 723. Jo>IR Isaac Newton, on some occasions,* represent- 
 ed the fluents by capital letters, and their fluxions by the 
 small letters that correspond to them. We followed this no- 
 tation in the last chapter, in demonstrating the grounds of 
 the operations. But it is convenient that the fluxions should 
 be distinguished from other algebraic expressions, and in such 
 a manner that the second and higher fluxions may be represent- 
 ed so as to preserve the original fluent in view. In his last me- 
 thod he represented the variable or flowing quantities by the 
 final letters of the alphabet, as .r, y, z ; their first fluxions by 
 the same letters pointed once, as by x, y, z', their second 
 fluxions by the same letters pointed twice, as by x, y, z; the 
 
 third fluxions by the letters pointed thrice, as by x,i/, z, and 
 so on, where the number of points serves to show the order of 
 the fluxion that is represented with respect to tlie first fluent; 
 and the diflTerence of those numbers show of what order any of 
 
 * Piincip. lib. ii. lemm. 2. 
 
 N 2 then^ 
 
 /
 
 , 180 The Rules in the Book \\, 
 
 tlicui is the fluxion of lliose that precede it, as y is the first 
 
 fluxion of I/, but the second fluxion of y. Mr. Leibnitz re- 
 presented tl\e infinitely small differences of x, y, z, by dx, dy, 
 ■dz ; their second difterences by ddx, ddy, ddz ; and their in- 
 finitesimal difierenccs of any order n, by rf x, d y, d z. The 
 symbol x, or dx, expresses the fluxion of x generally, without 
 determining whether it is to be consid.ered as positive or nega- 
 tive ; that is, whether x increases or decreases with respect to 
 the other fluents. Invariable quantities arc reprcseuted by the 
 first letters of the alphabet, as a, h, c, Sec. These have no 
 fluxions ; and, in the same manner, when any fluxion is sup- 
 posed constant, its fluxion vanishes. Sir Isaac JSezaton* has 
 comprehended most of the rwles of the direct method in one 
 general proposition ; but it is more usual to represent them 
 separately ; and it may be of use to proceed gradually from 
 tiic simple eases to those that are more complex. 
 
 724. I. When one simple fluent only enters each term of a 
 compound quantit}^, the fluxion of this quantity is found by 
 collecting the fluxions of each term, or by placing a point over 
 each fluent. Thus the fluxion of x -\- y — '-z, is x ■\- y — z; 
 the fluxion oi' ax '\- by — czis ax -{- by — qz. The fluxi- 
 on of ax, or of ax -\- bb, is ax. This rule is obvious, and fol- 
 lows from art. 701, or art. SQ, 41, and 78. 
 
 725. II, As the fluxion of xy is xy -^-yx (by art. 708 and 
 99), so the fluxion of a product of any two fluents is the sum 
 of the several products when the fluxion of each factor is mul- 
 tiplied by the other factor. Thus the fluxion of 'a-\-x X. T^ 
 
 isx X T^ — y X rt-f.v =: xb xy-r^-ya — yx. As the flux- 
 ion of az is fl^; so the fluxion of axy [& a x .Ty-f-j^x == xya 
 
 7^26. III. As the fluxion of the fraction- is -^—^ by art. 
 
 708, so the fluxion of any fraction is found by multiplying the 
 fluxion of the numerator by the denominator, substracting the 
 
 * See hi? Lemma H. to Prop. VIJI, Ub, II of Ki? Prin.ipia. 
 
 product
 
 Chap. IL direct Method of Fluxiom. 181 
 
 product of the fluxion of" the denominator multiplied by the 
 numerator, and dividing the remainder of the square of the de- 
 
 nomuiator. Thus the fluxion of — r- is — ^ — ' — 
 
 I -|- * fl;<r -j- .r 
 
 7^7. IV. As the fluxion of a" is - x.i"" x.r, by art. 
 
 714 and 7ig, so the fluxion of a power of any invariable 
 exponent is found by multiplying by the exponent, substract- 
 ing unit in the index of the power, and multiplying by the 
 fluxion of the root. Thus the respective fluxions of 2% .r^, 
 
 .r "*, <^c. are ^x^~^ x x or C .r.r, 3 x .r^~^ x 'x or S.i^ x, 
 4t X X XX or 4 0? :r, &c. In order to give this rule its full 
 extent, and to reduce fluxions to the most simple expressions, 
 we are to suppose from the common algebra, that a quantity 
 may be carried from the numerator of a fraction to its denomi- 
 nator, or from the denominator to the numerator, providing the 
 sign of its index or exponent be changed. Thus the fluxions of 
 
 I —I 1 —2, 1 _3 . - 
 
 — or X , ~ or .r — or x , are respectively 
 
 -, v> —1 — 1 • .r —2—1 
 
 — I X X X a' or — -, —2 X x x x er Qx, 
 -~3X X X xoY-~3x; and the fluxion of -1 or x~" is 
 
 — n XX X xov — ;-:-• The fluxions of surds are found 
 
 X 
 
 by expressing them as powers with fractional exponents. Thus 
 
 the fluxion of -/^ or x^is-X.rl~ x x=-^x x~^x=^, 
 
 <> 2 2xi 
 
 ■ 3 — . i 1 »_1 . ^^j;. 
 
 The fluxion of V x or x^ is- x x X x =■ ' 
 
 N 3 :
 
 182 The Rules in the Book II. 
 
 - 1 
 
 - ^ f The fluxion of v' x orr isixj;^' y'x 
 
 ^^ S\^xx 
 
 .7 
 
 1 _1 
 
 . ; and the fluxion of « or x n is 
 
 n n — 1 ^' 
 
 n x "■ 
 
 —l—tt 
 
 1 n ' 
 
 XX j< r =: — .. — p-^— : — 1— ,— , Theflux- 
 
 n — ?— . 
 
 nx « 
 
 -«— 1 a -f 
 
 ion of a + x is 7« X a + a^ X :r; and the fluxion of • 
 
 *4- 
 
 m — I . n n — 1 . m 
 
 m y. a-Y X xx x b -^^ x n y. b -{■ x xxxa-{-x 
 
 is : 
 
 ( . . . — " — M 
 
 Vdividing the nui^ierator and denominator by b-\-x , / 
 
 ■ • m 1 m 
 
 mx X b -i- X X a + X nx X a + x . 
 
 «4-i 
 
 728. V, As when jp := x x i/ x z X u, &c. or to this product 
 
 multiplied by any invariable quantity K, it follows, from art. 
 • • • . . • 
 
 V X 2/ Z U ' PX 
 
 719, that -=7 + - + 7 + - &c. or that p - ^ + 
 
 ^ ^^ ^^ &c. So the fluxion of any product divided 
 
 by the product itself is equal to the sum of the quotients, when 
 the fluxion of each factor is divided by the factor ; or the flux- 
 ion of any product is equal to the sum of the several quantities 
 that are formed, by substituting successively in that product the 
 fluxion of each factor in place of the factor itself. Thus if/; = 
 
 XJ/Z,
 
 Chap. II. direct Method of Fluxions. 183 
 
 xyz, then p —xyz +yxz + zxj/. If p = i + -* X M-7 x c + * 
 then £ :r -^+ -l- + -1__. It> ztTT^", then ^ 
 
 + -li. If p = .r + ^/J^r then £ - , ^^^ = 
 
 — , ,/ ■ _ ' w • — = — '— * ^ =;• Hence if p + -/Ty, t i 
 
 /^ 
 
 '"^^ -h^ XV 4- 1 , then ^-=7 - ^— 
 
 729. VI. As when p — *^-^^'' Sec. it follows, from art. 
 
 ^ — 5 X « X ^ 
 
 717, that£:=i+£+i-i-£-i &c. or p-^i + ££ + 
 
 p X tf z s u t X y 
 
 — — — — — — ^ &c. So when any fraction is proposed, 
 
 if we divide the fluxion of each factor of the numerator by the 
 factor itself, and from the sum of the quotients substract the se- 
 veral quotients that arise by dividing the fluxion of each factor 
 of the denominator by this factor itself, the remainder will be 
 equal to the fluxion of the fraction divided by the fraction; or 
 the remainder muItipUed by the fraction will give its fluxion. 
 
 Thus \\ p ~ - — - ^ 7 : X ; — - ; then - — rr— + ' 
 
 •T a — X D — X c — X ' p a -\- X a — x 
 
 X X , * I ■'^ ^^* I ^^-^ I 2cjr 
 
 + -r-x T 7 1 r 1 — r TT 1 ' 
 
 b -\- X o — X c -\- X c — X aa .xx bb-xx cc-xx 
 
 730. VII. Any equation of this form ^^X^^ItTx^wX 
 &,c. = o being proposed, the equation for the fluxions will be 
 ic—r Xx—'s xi— «x &c. = 0. For since x must be equal to 
 r, or to s, or to u, 8cc. x must be equal to r_, or to s, or 
 U, 8cc. 
 
 731. VIII. Let L represent the logarithm of x, the modulus 
 being equal tocr; then as L=:— - by art. 721, so the fluxion 
 
 N 4 of
 
 184 The Rules in the Book IT, 
 
 of thelogarithm of any quantity is found by dividing its fluxion 
 by the quantity itself, and multiplying by the modulus. 
 
 If p — a"^ the fluxion of the logarithm of p is fP or 2££^ 
 
 ^, ^ . ,.11 1 ' • nax may ^„ 
 
 The fluxion of the logarithm ot j«y"' is --- + ~^. U p = 
 
 7X7 X np7 ^ c + z >^ Sic. then the fluxion of the logarithm of 
 p is «/^ or (by art. 728) -^ + *^ + -^+ &c. This 
 
 likewise follows from the property of logarithms, that the lo^ 
 garithm of the product is equal to the sum of the logarithms of 
 the factors; and consequently the fluxion of the logarithm ofj3 
 equal to the sum of the fluxions of the logarithms of the fac-- 
 
 tors a + X, b + 1/, c -{■ z, that is to — -L. + .^^ + _fl. 
 
 In the same manner the fluxion of the logarithm of , is the 
 difference of the fluxions of the logarithms of x — a and a^ + a, 
 
 and therefore equal to — : — - ~__:::: — — The fluxion 
 
 X — a X -\~ a XX — xx 
 
 ot the loofanUim or xi x &c. is 1- &c. 
 
 73'2. IX. A quantiiy that has a variable exponent, asj/J^, is 
 called an exponential or percurrent * quantity ; and its fluxion is 
 
 AT— I . 
 
 1 X ?/^' X log. j/+ »y j/. For if we suppose y^ zz-ii, then 
 
 a 
 
 by the properties of logarithms (art. 157) x X log. y rr log. u. 
 And finding the fluxions by ait. 7*25 and 731, x X \og.y-{- 
 
 _^X.r — Jfi consequently the fluxion of j/^, or «— — ,xlog. 
 
 ij -f-.iL__— _ X j/^ X log. y -\- n/^— ' ?/.f In like manner the 
 
 ' Aytu I.ij'-. it. 'k * V. Eaicrfon'.^ Fhuions, p. 14, Kx. !3. & 19 
 
 fluxion
 
 chap. II. direct Method of Fluxions. 185 
 
 fluxions are found of exponential quantities of higher de- 
 grees. 
 
 733. X. The second fluxion is determined from the first flux- 
 ion, and the fluxion of any order from that of the preceding 
 order, by the same rules. It is often useful to suppose one of the 
 variable quantities to flow uniformly, or its fluxion to be con- 
 stant; in which case that quantity will have no second or liigh- 
 er fluxion, and the second or higher fluxions of quantities that 
 depend upon it will be expressed in a more simple manner. 
 Thus the fluxion of x being supposed constant, the first fluxion 
 
 of a." being nxjp , its second fluxion will be n x n— i xa* 
 X ~ , its third fluxion n X « — i X « — 2 X x' x"~ ; and its 
 fluxion of any order m will be « x n—i x ^"i x ^IIs X 8cc. x x"^ 
 X ^~'"^ where the factors in the coefficient are to be continu-: 
 ed till their number be equal to m. 
 
 734. The second or higher fluxions of quantities may be 
 found (without computing those of the preceding orders) by 
 particular theorems, as in the last example. Thus the fluxion 
 
 of xj/ is .r^ + yi" ; the second fluxion of xt/ is therefore xy + 
 
 •2^^ + xi/ ; its third fluxion is xj/ + 3xy -]- 3xi/ -{-xt/ ; and 
 in general the fluxion of Xj/ of any order denoted by m is found 
 by multiplying the fluxion of .r of the order m by j/, the fluxion 
 
 of X of the order m — 1 by 7/, the fluxion of .r of the order 
 7n — 2 hy )/, and proceeding alwa3's in this manner (diminishing 
 the order of the fluxion of a, and increasing the order of the 
 fluxion of 7/ by unit), then prefixing to the several products the 
 respective coefiicients of the binomial 1 + 1 raised to tiie power 
 m ; the last term being the product of x by the fluxion of 1/ of 
 the order m. If we suppose the fluxion of x to be constant, then 
 the two last terms will give the fluxion of xi/ of the order re- 
 quired : and if the second fluxion of .r be constant, the three 
 last terms will give that fluxion of .ry ; and so on, \T'hen the 
 fluxion of .r of any order r, and the fluxion of t/ of an}' order 
 s, ai"e supposed constant, the fluxion of xi/ of any order m (sup- 
 posing tn not to exceed r + s) is determined by this theorem. 
 
 735. In
 
 186 Of the inverse Method of Fiiaions. Book II. 
 
 735. In the inverse method, it is required to find the fluent 
 when the fluxion is given ; and the rules are derived from those 
 of the direct method ; as the rules of division and evolution in 
 algebra are deduced from those of multiplication and involu- 
 tion. As when a fluent consists of a variable and an invariable 
 part, the latter does not appear in the fluxion ; so when any flux- 
 ion is proposed, it is only the variable part of the fluent that 
 can be derived from it. If x represent any fluxion that may be 
 proposed, the variable part of the fluent will be equal to x ; for 
 supposing j/ to be any variable quantity, if a, -f ?/ could represent 
 tlie fluent of x, then x + t/ would be equal to x, and 1/ — 0, or 
 ^ would be invariable, against the supposition. But supposing 
 K to represent any invariable quantit}-, then jr-f K may gene- 
 rally represent the fluent of .r. If it be required to find such 
 a fluent of x as shall vanish when x is supposed to vanish, it 
 can be no other than x ; and if it be required that the fluent 
 should vanish when .r is equal to any given quantity a, then 
 b\^ supposing x + K to vanish when x becomes equal to a, we 
 have af + Kr=o, or Kn — a; whence the fluent is x — a. 
 
 In the same manner the fluent of — x may be generally repre- 
 sented by K — X. When a fluxion, that is proposed, coincides 
 with any of those vj^hich were deduced from their fluents in any 
 of the preceding articles, the variable part of the fluent requir- 
 ed must coincide with that which was there proposed. As di- 
 vision in algebra leads us to fractions, and evolution to surds, 
 so the inverse method of fluxions leads us often to quantities 
 that are notknowu in the common algebra, and that cannot be 
 expressed by the common algebraic symbols. In the following 
 articles we will endeavour to give some account of the pro- 
 gress that has been made in this method. 
 
 736. I. As the fluxion of ax-\-hi/-cz is ax + by-cz\ so, 
 conversely, when any aggregate of quantities is proposed, each 
 of which involves a simple fluxion that is not multiplied by any 
 flowing quantity, the variable part of the fluent is found by sub- 
 stituting in place of each fluxion its particular fluent ; or by 
 taking away the points, or other fluxionary symbols. Thus the 
 variable part of the fluent of ax-^by-cz is ax -{- bj/-cz. If 
 it is required that this fluent should vanish when x vanishes, let 
 
 7/ he
 
 Chap. II. Of the inverse Method of Fluxions. 187 
 
 y be then equal to e, and z equal to /"; and the fluent will be 
 ax + b X "p^ — ex 717. For the whole fluent may be ex- 
 pressed by ax + by - cz + K, wliere K is supposed invariable. 
 But, by the supposition, when x vanishes, this fluent vanishes, 
 and is equal to be - cf -{■ K ; whence Krr- be + c/'; and conse- 
 quently ax -i- by - cz + K is equal to ax -\- by - be - cz + cf, 
 or to ax + b X y—e -ex z~f. 
 
 737' II. As the fluxion of x is nx ^^by art. 727, so, con- 
 versely, when the fluxion proposed is the product of any power 
 of a variable quantity multiplied b^ats fluxion,with any invaria- 
 ble coefficient, the variable part of the fluent is found by adding 
 unit to the exponent of the power, dividing by the exponent 
 thus increased and by the fluxion of the root. Thus the vari- 
 
 «—!-{- 1 . 
 „_l . nx X _ 
 
 able part of the fluent of w.r .r is , — , r—x^"■: and if 
 
 f «— 1+1 Xx 
 
 it is required that the fluent should vanish when x vanishes, it 
 is then precisely .r ; but if it is to vanish when .r is equal to any 
 given quantity o, the whole fluent is x — a . In general we 
 may express it by x + K, where K may represent any inva- 
 riable quantity. In the same manner the fluent of axx is 
 
 ; + i; ,1 
 
 a X '• — + K r: - axx + K ; the fluent of ax^x is 
 
 2x 2 
 
 2 + 1. . 1 , • —2. 
 
 ax '^ _j_ K n -ax"^ + K; the fluent of££, or ax x, is 
 
 3; ^ ■ x^ 
 
 -2+1. , ^ ^+1 . 
 
 ^^ '5_-f K =— f +K; the fluent of x^r is "^ ^ 
 
 — 1 X ^ "^ ^^ 
 
 J- K =: ""^^ -f K. The fluent of an aggregate of quantities 
 
 of this kind is found by computing the fluent of each term se- 
 parately. Thus the fluent of x'^x -{- axx + bbx is i x^ + |- ax^ 
 + 66a- + K; the fluent of xxxa^x^, or of aaxx + Q.ax'-x + 
 
 r\ 3 1''^ • ■ 'ft 
 
 x' X is i a\i- + ~~ + '-J. The fluent oi x^x X x+a. 
 
 wlien
 
 188 Of the wverse Method of Fluxions. Book II. 
 
 vhen n is an integer, is found by raising x^a to the power n, 
 
 multiplying each tenn by x^^x, finding the fluent of each se- 
 parately by this nile, and collecting them into one sum. The 
 variable part of the fluent is assignable in all those cases, un- 
 
 — 1 
 less when the fluxion £ or xx is involved in one of the 
 a- 
 
 terms, of which case we are to treat afterwards. 
 
 738. III. As the fluxion oi xy is xy + yx, by art. 725, so 
 when any proposed fluxion can be resolved into two terms of 
 this form, where there are two fluxions, each of which is sepa- 
 rately multiplied by the fluent of the other, then the product of 
 the two fluents is the variable part of the fluent recpiired. Thus 
 
 the fluent of hz — uz — au — zu, or zxT^" — uxT^z is b—u 
 xl^7 + K. In the same manner, when a fluxion can be re- 
 solved into three parts in the form xyz + yxz + zxy, where 
 there are three fluxions x, y, z, and each of these is separately 
 multiplied by the product of the fluents of the other two 
 fluxions, then xyz the product of the three fluents is the vari- 
 able part of the fluent required. These theorems are easily 
 continued from art. 728. 
 
 739. IV. The fluxion of exz-\-zx, where e is supposed to . 
 be invariable, is not of the same form with any of those in 
 
 the preceding article, but by multiplying it by r , the pro- 
 duct exx z + zx is easily reduced to the fust of them, 
 lor supposing y — x , exx ;:+:-r nj/.s -f zj/, the fluent 
 of which 1? yz^ or zx ; which is therefore the fluent of 
 exz -\- zx X J'^ '. In the same manner, if eryz -\-fyxz-\-zxy 
 be multiplied by x 7/ , the fluent of the product will h6 
 of y z. And wlien the fluxion exyzu-\- fyxzu -\- gzxyu -f* 
 ■)ixyz is multiplied by .1 y z , the fluent is x y 
 z^ u ; and so on. It follows from the first of these, that when 
 
 un equation exz -f- zx — ar x is proposed, the equation 
 
 for
 
 Chap. II. Of the inverse Method of Fluxiom. .189 
 
 for the fluents is zx^ 4- K — , x ax^ •. p,nd in this 
 manner the fluents in art. 540 were found. 
 
 740. V. When i - £ 4. i! + i -f- Sec. then we may 
 
 p X y z -^ 
 
 conclude thatp is the product of r, ?/, z, Sec. and of some in- 
 variable quantity K; for this fluxional equation will arise (by 
 
 art. 728) when we suppose p r= K ay: X S^c. If £. — * -J- 
 
 p X 
 
 ^ 4- - — L — - &c. then we may conclude that p — !^.^-^.- 
 
 y ^ z s u ^ ^ su 
 
 X &c. Thus if .£ — — f^ 1^. we may conclude that 
 
 « = K X ^:-^. liii-'lk.jL.'i (e, m and n being sup- 
 
 *• + * p X * y Oi 
 
 posed invariable), then j3« = K x'^y'^. 
 
 74 1 . VI. A fluxion that is hot proposed under any of the pre- 
 ceding forms may in some cases, by a proper substitution, be 
 changed into an equal fluxion that will appear under one or 
 more of them; and thus the fluent may be discovered. The 
 
 fluxion % z y^a-\-z is not immediately comprehended 
 under any of the preceding forms when m is a fraction, or any 
 negative number. But by supposing x ■=■ a-\- z,oxz-=.x — n. 
 
 and consequently z .=: x^ and z = x — a the proposed flux- 
 
 ion is transformed into XX Y,x — a ; the fluent of which is 
 found by raising x — a to the power of the exponent n, mul- 
 tiplying each term by .r x, and computing the fluent of e,acli 
 product separately by art. 737. 
 
 742. The fluxion x x x c-h/^" being proposed; suppose 
 e -f/r" = z, and '"-±^ - r- then a« = IZZ^ , x" -^ * - 
 
 «j -4- 1 
 
 -■ ~- 1 " zz ■ •~ ~> and (by taking the fluxions) m-\-l X 
 
 m 
 X X
 
 IQO Of the inverse Method of Fluxions. Book If. 
 
 1 
 
 X .r = y^ X z — e X z. Tlicrefore the fluxion that was 
 
 r-l 
 
 proposed will be equal to " w /v X ^ — ^ X zh — 
 
 m + 1 X/ 
 
 * / f 1 
 
 IL- y^ 2 g • consequently the fluent is found by raising 
 
 2 — e to the power of the exponent r — 1, multiplying each 
 
 terni of this power by zz^, finding the fluent of each product 
 separately by art. 737, and dividing the sum of these fluents by 
 
 nO\ This fluent is assignable in finite terms when r or *" "*" \ 
 
 •^ n 
 
 is an integer (unless / be of such a value as to give occasion to 
 the exception mentioned above at the end of art. 737), and will 
 consist of as many terms as there are units in r; because this is 
 the number of terms in the power of z — e of the exponent r — 1 , 
 
 For example, the fluent of i'"x X <^-\-f^^ is assignable in alge- 
 braic terras equal in number to m-^-l, when m is any integer 
 and positive number; for in this case n:zz\ and rzzm'\-\. 
 
 The fluent o^ x^x%e-\-fx'^ is assignable in finite terms when 
 m is any odd positive number; because in this case «=2, and 
 
 r ■=. "' "^ zz ^ which is an integer when m is an odd posi- 
 
 tive number. The fluxion x^x »/ex +/rr — x ^ x X 
 ^ , r^' ; and consequently the fluent is assignable when in-\- — 
 is an integer positive number, that is when m is equal to any 
 
 fraction of this series — _, -, -, -, &c. The fluent of xx i 
 
 2 2 2 2 • 
 
 k 
 
 X e -\-fx s 1 is assignable in finite terms when s -|- 1 is any mul- 
 tiple of A:; for in this case r (or _ -}- 1 divided by _ ) is equal 
 
 to UlL, and is an integer when 5-f- 1 is a multiple of k. 
 
 743. The
 
 Chap. II. Of the hiterse Method of Fluxions. ig\ 
 
 743. The same fluxion x x x ^ +fx (multiplying 
 the first part .r"' j: by j: , and dividing the other part e-\-fx 
 by the same x , by ^vhich their product is not altered) is ex- 
 pressed by xx"* X ex "-\-f . As the value of r taken from the 
 first expression was '-~~, so its value computed from the se- 
 
 cond expression is -3;^ — * Therefore the fluent is assignable 
 in a finite number of algebraic terms^ not only when ^^ is an 
 
 integer and positive number^ but likewise when "' _ "^ is such 
 
 a ninnber. Thus the fluent of x X ^4"^" "isassignablein 
 
 a finite number of terms when k is integer and positive, whatever 
 number be reprciented by n ; for in this case ?n — o, l— — k — - 
 
 ~-—j—, ''/+ 1 -—7>k, and -i:^ rr- ^- 
 
 •744. ^\^hen the diflerence or sum of two fluents is invariable, 
 their fluxions arc equal, as we observed in art, 735. And hence 
 when the same fluxion is represented by two diflferent expres- 
 sions, as in the two preceding articles, there may be some dif- 
 ference betwixt the fluents that arc derived from them bv the 
 preceding rules; but by the addition or substraction of an inva- 
 riable quantity, they will be found to agree with one another. 
 Thus, for example, the fluent of x x «+*'l~^ is (by art. 737) 
 
 ^X o 4-x — -i-. The same fluxion is equal to x~~^ x y 
 
 ~ |— 2 
 ax -f 1 ' , and the fluent of this fluxion (by the same ar- 
 
 ticle)isi: 'l^J!ll±l :^JIi_Jll -, -' 
 
 —2 
 
 The latter fluent vanishes when x vanishes. The 
 
 former ~^, by adding the invariable quantity K, becomes
 
 IQ'i Of the inverse Method of Fhixi&ns, Book IT. 
 
 1 
 
 K -{"^nr;:; and if we suppose tliis fluent to vauisli when x va- 
 
 nlshes;, K +7T^ = o^ K = — -, and the fluent will be 
 
 ■~^+^-i-v. — — ,— . —I ===, which coincides with 
 
 the latter fluent. 
 
 745. VII, Wlien a fluent cannot be represented accurately 
 in algebraic terms, it is then to be expressed by a converging 
 sericsj or by a more simple fluent that is already known. In di- 
 vision in tlie common algebra(and in decimal arithmetic) the 
 
 fjnotient is often such a series. Let—- be the fluxion propos- 
 ed ; and if we divide a by a x by the usual method, we shall 
 find the quotient or -— —\-\- - -\ 1 + 77 &c. Hence 
 
 - ^ ; ■ — X -|" "T "^ — a^ ^ "Ts" ^^' ^^^ ^^^^ fluent of ^_^ - 
 
 XX, x'- X 
 
 is equal (by finding the fluents of the terms x, -^ —^, &c. 
 
 X^ X^ X^ 
 
 separately from art. 737) to the series ^^^ + 2J + i^^ + 4^ 
 4- &c. which may be of use for determining the fluent when 
 .1' is very small in respect of a ; because, in that case, a few 
 terms at the beginning of the series will be nearly equal to the 
 value of the whole. This series gives us the logarithm of ^^ 
 the modulus being supposed equal to a, by art. 731. For if we 
 
 suppose — = r, then - — =i-, by art. 7*28, and the fluent 
 
 A ^ a—x a — X z 
 
 of—— is equal to log. z or log. ., or to — log. a — x. 
 
 aa *■* *■* *^ 
 
 746. In the same manner — ; — — \ — -r +^ — "^ 
 
 aa-\-xx 
 0.0. X • n c ' ■*^"''^ ^^'"^ 
 
 &c. and the fluent of — 1- is the fluent of .r — TT +^ 
 
 aa-\-xx
 
 Ghap. II. Of the inverse Mtthod of Fluxions. 19S 
 
 — ^ &c. that is (by art. 737), ^ — ^» + ^ — ^s &c. Be- 
 
 cause the fluxion of the arch is to the fluxion of its tangent in 
 the duplicate ratio of the radius to the secant (by art. 195), it 
 follows that if the radius be a, the tangent x, and consequently 
 the secant equal to */aa-\-xx, the fluxion of the arch will be 
 
 equal to j~^ ; and the ark itself will be expressed by the se- 
 ries T- g + ^-^ &c.or z X 1 - 3~ + ~~i, &c. 
 This series was given by Mr. James Gregorj/ for computing 
 the arch from its tangent. Comnur. epistoL I671. Dr. Halley 
 has computed the ratio of the circumference of the circle to its 
 diameter from it, by supposing x to be the tangent of an arch 
 of 30 gr. in which case the tangent x is to the secant '/aa -{-xx 
 as 1 to 2, and consequently x to cr as 1 to -v/?; so that the arch 
 
 of 30 gr. is the product of -TT multiplied by the series 1 — . 
 
 g + — — 189 "^ 7^ ^^* ^^^ ^^^ whole circumference to th€ 
 diameter as vTS multiplied by this series to miit. This series 
 may be represented by 1 — ^ + ^ _ ^ 4. ^_L ^ 
 
 -- — — - Sec. that the law of its continuation may appear. 
 
 747. In like manner, when the roots of powers are extract- 
 ed by the usual rules in algebra, the root is often expressed 
 
 by a series of this kind. Thus : Vaa—xx = a — ,~ — g-5 — 
 
 * * „ .1 • . • **Af x^x x^:t 
 
 ^-^ &c. consequently I- >/ aa-xx -ax— — — — — .^ 
 &c. Therefore the fluent of x >/ aa—xx is (by art. 737) ax — 
 
 ? TTT-i — TTTn — ,,-0 n ^^' And if C A the radius of the circle 
 
 be represented by a, upon which CP(^'g. 298) be taken from the 
 
 centre C equal to x, CB and PM perpendicular to CA meet the 
 
 VOL. II. O circle
 
 
 CBMP 
 
 
 
 X 
 
 Va 
 
 
 • = 
 
 ax 
 
 — 
 
 ,r3 
 
 = 
 
 W XX 
 
 " 6^ 
 
 — 
 
 
 + 
 
 *5 
 
 8ai 
 
 + 
 
 x7 
 
 16aS 
 
 &C. 
 
 — 
 
 x3 
 
 3^ 
 
 + 
 
 194 Of the inverse Method of Fluxions. Book If. 
 
 circle in B and M; then the area CBMP will be expressed by 
 this series; for PM = aa-xx, the fluxion of the area CBMP 
 (art. 107) equal to PM X x — x */ aa—xx: and consequently 
 the area CBMP equal to the fluent. Let MN be perpendicu- 
 lar to CB in N, and the area BMN = CBMP — CP x PM 
 
 „ &c. 
 
 748. Because the fluxion of the arch BM is to x the fluxion 
 of its sine MN or CP, as CB to PM, that is, as a to »/ aa~xx, 
 
 the fluxion of BM is expressed by — ^I— = -}}t'EEl- 
 
 Vaa—xx a,i—xx 
 
 (dividing the series which expresses Vaa—xx by aa — xx) x + 
 ^ -I- -^ + T^ + &c. consequently the arch BM is equal 
 
 to a- + -^ + 40^ + i7^ ^c- - ^' + 23^ ^ -^ + 
 f^^ X — 4- -^ X — 4- &c. where A represents the first 
 
 term x, B the second term — , C the third tenn, and so on. 
 
 "It is useful to represent a series in this manner, that it may be 
 easily continued to any number of terms, and the fluent com- 
 puted to any degree of exactness that may be required. Let 
 the arch NS described from the centre C meet CM in S, and 
 KS will be to BM as CN to CB, that is as \/aa~xx ^o a ; con- 
 sequently, if the series which expresses BM be multiplied by 
 
 the series wnich expresses — ^ viz. ^ — -^r—^^ 
 
 rk, ^^-^ t^-^e Pi^oduct X — :^ _ ?il _ '^^ &c. will 
 
 represent the ark NS. Therefore MN— NS = ^^ + ff 
 
 3 2a* 8d* 
 
 5?f5 fly-? 
 
 , ^iL- &.C. And the area BMN is to CB X mIwNS as 
 
 x3
 
 Chap. II. Of the inverse Method of Fluxions. IQS 
 
 r^ + To^ + 6S7s ^^- *^ 3^ + TTaT + To5^ &c, or as 
 1 + ^x + ISt &c. to 1 + g + |1 &c. This ratio 
 
 by substituting ^ instead of - coincides with that which was 
 
 given in article 655, without a proof, as the ratio (fg. 294) 
 
 of the segment FCO to CD X CF— CS. 
 
 748. Sir Isaac Newton's binomial theorem is of excellent 
 use for extracting the roots of powers, or reducing a quantity to 
 aseriesof this kind ; and, having made no use of this theorem in 
 demonstrating the rules in the direct method of fluxions, we 
 may the rather give an investigation of it from art. 727 . Let it 
 
 be required to find 1 +.r , where n may represent any integer, 
 number, or fraction, whether it be positive or negative. It is 
 evident, from what is shown in the common algebra concern- 
 ing powers and their roots, that the first term of any power of 
 1-\'X is 1, and that the subsequent terms involve x, x% .r', x*, 
 
 &c. with invariable coefficients. Suppose, therefore, l-{-x ~ 
 1 + Ax + Bx' + Cx' + Bx* + &.C. where A, B, C, D, &c. 
 represent any such coefficients. By finding the fluxions (art. 
 
 -rt— 1 
 
 727) nx X l+x = Ax + 2Btx + sCj^x + 4Dx'x + 
 
 &c. and, dividing b}; 7ix, we have i ^-x :=: — -f- — 
 
 4- "-7- + — ^ + Sec. And since this equation must be 
 true, whatever the value of x may be, it follows by supposing 
 X = (or because the first terra of T+~a'"~ must be 1), that 
 - = 1, and A = n. By taking the fluxion of the last equation, 
 
 -——n—^ . ' 5Bx 6C.r.r . 12D;r»r 
 
 n— 1 X l+x X X = — - + — + . 4- &c, 
 
 ' ■ « n ijt ' 
 
 -n— 2 2B 
 
 and dividing by w — 1 X xj we have l+x rr - —- — 1. 
 
 nxn—X ~ 
 
 2 ecAT 
 
 »x>j— 1
 
 196 Of the inverst Method of Fluxitns. Book II. 
 
 ■ . -J. + nxn—i "^ ^*^* ^"^" "3' siipposmg X = (or because 
 
 the first term of any power of 1 + -r must be 1), " = 1 
 
 or Jj — n X -g— By taking the fluxions again, we find «— T x 
 
 n— 3 V • 6Ci J 240;^^ . n-3 _ 
 
 iJLx ^ X — ^-= H == gcc. and l + x ~ 
 
 • nxn — 1 nX"— I 
 
 6C , 24DAr , 2, „ .V . 6C 
 
 _|_ — - — — -\- &c. SO that — 
 
 «X«— IX"— 2 nx«— lX«— 2 nx«— IXn— 2 
 
 1, or C — n X -Y' ^ "i~ ' ^^^ ^° °^*- Therefore T+T"' 
 = 1 + nx 4- « X -^— X a- + ?i X -^— X -^— X j,'^ + &c. 
 
 And a -{- b =:a + b X «» = 1 +- 
 
 X tt" 1= (by substitut- 
 
 mg - lor X) a + + « X -ir X — r- + ;< x —r- 
 
 n — 2 a"i3 n « — 1, « — 1 „ o 
 
 X -3- ,X —^ 4- &c. = a + ?m b + n x —^ xa 6 * 
 
 -f w X — g- X —J-- X a"~ 6' + Sec. which is the binomial 
 theorem. 
 
 749- "En the same manner if we suppose a+ox+cxx-\-dxi k'c^ 
 =:^ + Ba: 4- Ci* + Dx^ &c. b}^ supposing x = a, we have 
 
 A-=tt . By faking the fluxions, and dividing by x, we shall find 
 
 a + bx + cxx &.C. X nb + 2iicx + otidx"- Sic. rr B 4- 
 
 £Cx + SDx^ + &c. and by supposing x =: o, we have B = 
 
 na"~ b. By taking the fluxions again, dividing by Q.x, and then 
 
 supposing X = 0, we shall find C rr w x ^^^ X a*~^bb "^ 
 
 na*^ c. And by proceeding in the same manner, we may in- 
 vestigate the other coefficients D, E, 8cc. in Mr. De itf o/rre's 
 theorem for raising a multinomial to any power of the index n. 
 
 Of
 
 Chap. II. Of the i7iverse Method of Fluxions. 1S7 
 
 Of this the reader will find a fuller account in the Philosoph. 
 Trans. n. 230, otMiscel. Analyt. pi 87- There are sev'eralother 
 methods by which these theorems are investigated, but we have 
 described that which is immediately suggested by the method 
 of fluxions, and will be of use afterwards in other enquiiies. 
 
 750. When any fluxion i-P is proposed, and P is any quan- 
 tity that can be expressed by any powers of x and invariable 
 quantities, the value of P can be resolved into a series by these 
 theorems ; and each term being multiplied by x, the fluent of 
 each may be found separately by art. 737, such excepted as are 
 of the same form with Aa'x~ . Thus to find the fluent of a:"^r x 
 c •fyi" , it is first transformed by supposing 2 — e 4*/^" 
 
 into —r X ~ — t''"^ (as in art. 742) r= (by the binomial theo- 
 
 rem)— x z — r — 1 x 2 e + r — Ix'-^x^ fe-&c. 
 nf . 2 . 
 
 ■ ,1-'- '+'■ 
 
 consequently the fluent is the product of L^mulfiplied by 1— 
 
 r— 1 l-X-r—l , . r— I ', r—2 ' I+r—2 
 
 xz e 4* —9- X ., ^ • Xz ee &c. 
 
 /+r— 1 '"^ - * 2 l+r—2 
 
 which (by restoring c + /a," for z, supposing / 4* r =:. s, and 
 
 7-;/ — 71 =■ py will be found equal to — ;_ x « + A'^ ^ — 
 
 nj^r ■ 
 
 T — 1 eA r — 2 *c? r — 3 eC „ > . . . 
 
 X — A — X -rr X -r— &c. where A re- 
 
 s_l yx« ^ s— 2 /;i" s— 3 ./j:»* 
 
 presents the first term, B the second, C the third, and so on. 
 This series was given long ago by Sir Isaac 'Sezcton, Com-. 
 mtr. cphtol. And in his Treatise of Quadratures he has shown 
 how to assign the fluent of !rP in a series,-when P is the product 
 of a J^ hx^ + CI'*'' See. multiplied by x'^, and \>y any multi- 
 nomial c + fx'^ + go:^" + hx"'^ 8cc. raised to a power of any 
 exponent I; or when P is equal to the product of those quanti- 
 
 O 3 ties
 
 198 Of the invetst Method of Fluxions. Book 11. 
 
 ties multiplied by k + 1 j:'» + m x^ &c. raised to a power of 
 any exponent k. De quadrat, curvar. prop. 5 & 6. 
 
 751. The follo^ving theorem is likewise of great use in this 
 doctrine. Suppose that ?/ is any quantity that can be expressed 
 by a series of this form A + Bz + Cz^ + Dz^ 4 &c. where 
 A, B, C, &c. represent invariable coefficients as usual, any of 
 which may be supposed to vanish. When z vanishes, let E be 
 
 the value of y, and let E, E, E, 6cc. be then the respective 
 
 values of y, y, y, &c. z being supposed to flow uniformly. 
 
 rp, „ Ez Ez* Ei3 Ez* 
 
 Then 3^ = E + — + r + + + 
 
 z lx2i^ 1x2x323 1x2x3x42* 
 
 &c. the law of the continuation of which series is manifest: for 
 since 3/ = A + Bz + Cz^ -\- Dz^ + &c. it follows that 
 when z = 0, A is equal toy; but (by the supposition) £ is then 
 equal toy; consequently A = E. By taking the fluxions, and 
 
 dividing by z] i: = B + sCz + SDz* + &c. and when 
 
 z 
 
 2 = Oj B is equal to ^ j that is to _ • By taking the fluxions 
 
 z z 
 
 again, and dividing by z (which is supposed invariable) ^ = 
 
 z* 
 
 2C + 6Dz + &c. let z — o, and substituting E fory, - — 
 
 z* 
 
 2C, or C ~ -^. By taking the fluxions again, and dividing by 
 
 22 
 
 . " E 
 
 z, — — QJ) ^ Sec. and by supposing z — 0, we have D = -r-. 
 
 z^ Si} 
 
 Thus it appears that y = A + Bz + Cz^ + Dz' + &c. = 
 Jt' + -T- T ^ — V — ^ T : r occ. 1 his pro- 
 
 I lX2z- 1X2X323 1x2X3X1-4 ^ 
 
 position may be likewise deduced from the binomial theorem. 
 
 Let
 
 Chap. IT. Of the inverse Method of Fluxions. 199 
 
 Let BD Cfg.Q.99), the ordinate of the figure FDM at B, be equal 
 to E, BP = z, PM = y, and this series will serve for resolving 
 the value of PM, or ?/ (some particular cases being excepted, 
 
 as when any of the coefiicients E_, -7-^ -r~> &c. become infi- 
 
 z z^ 
 
 nite), into a series, not only in such cases as were described in 
 the preceding articles, but likewise when the relation of y and 
 z is determined by an affected equation, and in many cases 
 when their relation is determined by a fluxional equation. This 
 theorem was given by Dr. Taylor, method, increm. By sup- 
 posing the fluxion of 5; to be represented by BP, or 2 = z, we 
 
 • F" P P 
 
 have y = E+fi + -2- + -6'+24r'*' ^^- (^^ '^^''^^ observ- 
 ed in art. 155):, and hence it appears at what rate the fluxion 
 of y of each order contributes to produce the increment or de- 
 
 E E E 
 
 crement of y, since ?/ — E=:E + '2- + -g-' + 2^ + &c. If 
 Bp be taken on the other side of B equal to BP, then pm— A 
 — B2 + Cz* — -Dz^ + &c. = (the same quantities being re- 
 
 P P 
 
 presented by - , -;-,. g^c. as before, or the base being supposed 
 to flow the same way) E r + 
 
 lX2z^ 
 :: 4 
 
 8cc. consequently PM + pw = 2E + 
 
 IXSXSX'^i lx2z' 
 
 :: 4- 
 2Ez . o 
 
 r- + &C. 
 
 lX2X3Xt^ 
 
 752. The area BDMP, or the fluent oiyz, is equal to the 
 
 a *. c X}' I Ezz . Ez'^z . Ez^z 
 
 fluent of Ez + -r~+ r- + + &c. that is (be- 
 
 2. Ix2z* Ix2x3z^ ^ 
 
 cause while this area is generated by the ordinate PM, the 
 
 O 4 quanti-
 
 200 O/tJiC inverse Method of Fluxions* Book II. 
 
 . . _ e" E „ . . , , ^ T^ Ez* 
 ciuantities E, t j t- > &c. are invanable) to E? H r + 
 
 * 12* lX2i 
 
 + &c. wliich theorem is not materi- 
 
 1X2X3^' lX2X3X4i* 
 
 ally different from Mr. BernouilWs Act. Erud. Lips., 1694. 
 
 Ei* F ' 
 
 In the same manner the area BDw^ = Er — r + 
 
 f.z^ . Ez 
 
 lx2z Ix2x3i' 
 Ea4 
 
 -77 &c. where a is supposed to be the same as in tb 
 
 e 
 
 lX2X3x4z' 
 
 former case; therefore the area PMw/> bounded by the ordi- 
 nates PM and pm, that are at equal distances from BD (or E), 
 
 :: $ 
 2Ei5 9 Ej: 
 
 on opposite sides, is 2E2 + n + rr + &c, and 
 
 ^^ 1x2x32^ lX2X3X4X5s* 
 
 is equal to the rectangle contained by the base Pp (or 2r) and 
 
 1 • -n . Ez . Ez .0 
 
 the series h H r, + r- + &c. 
 
 2x3x« SxSXIXaz" 
 
 753. The series for finding the number of a given logarithm 
 may be deduced by the theorem in art. 751. Let z represent 
 the logarithm of ^, the modulus being represented by M; and 
 
 since^ =: ~ by art. 73 1, it follows that?/ = ^> 3/ = ^—- 
 
 • i :. • • » ;^3 :: •• • 3 '4 
 
 • :::: ~) V rr ^^-^ =: — > V = -^^ — —5 and so on, 
 
 M* -^ M^ Mi "^ M3 M+ 
 
 When z =■ 0, then 3/ rr 1 : therefore we are to suppose E =: 1, 
 i^=i^'E=S^'^ = S^'^=S^' consequently, by 
 the theorem, 3^ = 1 + ^ + sS^ + 6-mT + ii^ •^ &c. 
 The same series is found by supposing y = l 4* -Az 4* Bz* •{• 
 Cz^ 4" Dz* 4" Sec; and therefore My rr yz r= z 4" Azz •!* 
 Bz* 2 4" Cz' 2*4" &-C, consequently by finding the fluents,My (or 
 
 M
 
 Chap. II. Of the inverse Method of Fluxions. 201 
 
 M J^ AM2 + BM2^ + CM2^ 4- &c.) = K 4*2 4- -|^ •* 
 ?|! ^ _1 1^ &c. and by comparing the terms of those two 
 values of My that involve the same powers of 2, K r: M, AM 
 = lorA = ±,BM = torB = A, CM = | or C = Ij. 
 
 and so on ; therefore y = 1 -I* -^ 4* 7^ 4" gJjT "^ ^<=- % 
 supposing z = M, y — \ ^ 1 4" t •!- | 4" 2V •!• rio 4* &c. 
 = 2, 7182818, &c. The ratio of this number to unit is that 
 which Mr. Cotes calls the ratio modularis, the modulus beins: 
 always the logarithm of this ratio in any logarithmic systeiUo 
 See art. 175- 
 
 754. In the same manner the series for finding the cosine, 
 when the arch is given, is deduced from the theorem in art.7dl. 
 Let the arch BM = z, and its cosine PM =r y, then because 
 
 V : z : : CP : CM : : V^^- : a, t :=, ±± %L =^ 
 
 and so on. When the arch BM vanishes, or z "^-o, PM rz CB 
 = a ; supposing therefore E r= a, substitute a for ?/ in those va- 
 lues of y,y,y, &c. in order to obtain E, E, E, &c. and ^ 
 
 Thereforey = E 4* j- 4* |jr 4- &c. = « — g- ^~ 
 
 2.6 
 — 72Qfl5 4* &c. If we suppose BM still equal to Zy but its right 
 
 sine MN now equal to y, the same equation ~- r: ^~^ 
 
 will express the relation of y to r, and the values of y, v, &c. 
 
 will
 
 SOS Of the inverse Method of F/iixiotis. Book II. 
 
 will be the same as in the former case : but because when BM 
 vanishes, its sine MN likewise vanishes, we are now to sup- 
 pose E rr o, and to substitute o for ?/ in the values ofy, y, y. 
 Sec. in order to obtain E, E, E, &c.; therefore, in this case, 
 
 rz — ^) h = o, &c. And y r: 
 
 a3 aS 
 
 E rr 0, — 
 
 z^ 
 ?.7 
 
 6a^ ' 120a4- 
 
 5010flfi 
 
 + See. If y represent the tangent 
 of the ark z, then (as in art, 746) -4- — flt^^^ and suppos- 
 
 ing R r:i o (because y vanishes with ::), and, proceeding as 
 
 before, we shall find y =r s + |i + y£i + i^ + &c. 
 
 \i y represent the secant of the ark z, then y : z :: y V yy—aa 
 : fl«, and supposing E rr cr, because the secant becomes equal 
 to the radius when the ark vanishes, it will be found thaty rr 
 
 ^ + 2^ + 24^ + 720^ + ^^' ^" "^^^ ^^'^^ manner ge- 
 neral theorems are found for the reversion of series, such as 
 are given by Sir Isaac Nezoton, Commerc. Epist., in his letter 
 of October I676, towards the end. We now proceed with 
 our account of the inverse method of fluxions ; but will have 
 occasion to return to the doctrine of series afterwards, and to 
 show further the use of the theorems in art. 7j1 and 753. 
 
 CHAP
 
 203 
 
 CHAP. III. 
 
 Of the Analogy hetzoixt circular Arches and Logarithms, 
 and of reducing Fluents to these, or to hyperbolic and 
 elliptic Arches, or to other Fluents of a more simple 
 Form, when they are not assignable in finite alge- 
 braic Terms. 
 
 755. vV HEN it does not appear that a fluent can be assigned 
 in a finite number of algebraic terms, we are not, therefore, 
 to have recourse immediately to an infinite series. The arches 
 of a circle, and hyperbolic areas or logarithms, cannot be 
 assigned in algebraic terms^ but have been computed with great 
 exactness by several methods. By these, with algebraic quan- 
 tities, any segments of conic sections and the arks of a parabo- 
 la are easily measured ; and when a fluent can be assigned by 
 them, this is considered as the second degree of resolution. 
 When it does not appear that a fluent can be measured by tiie 
 areas of conic sections, it may however be measured in some 
 cases by their arks ; and this may be considered as the third 
 degree of resolution. Ifitdoesnotappearthat afluent can be as- 
 signed by the arks of any conic sections (the circle included), it 
 may however be of some use to assign the fluent by an area or 
 ark of some other figure that is easily constructed or described ; 
 and it is often important that the proposed fluxion, be reduced 
 to a proper form, in order that the series for the fluent may not 
 be too complex, and that it may not converge at too slow a 
 rate. 
 
 756. The rule in art. 737 is of no use to find the fluent of 
 
 X X, or-^; for, according to that rule, the fluent is 
 
 X X f _ X ,, O 1 — 1 ^ ^ 1 
 
 == — : — --— = (because .r — x — — r: 1) - ; 
 
 I— 1 X * O ^ X 
 
 from which expression no computation of the fluent can be de- 
 duced ^
 
 £04 Of the Ajiahgy hdicixt Book II. 
 
 duced, and therefore this case was excepted. By art. 731, the 
 
 fluent of -is equal to — ^— > M being the modulus, and the 
 
 fluent being supposed to vanish when x is equal to 1, or to the 
 quantity whose logarithm vanishes. If we suppose x =: a^z, 
 
 then i^ — -^j and the fluent will be found (as in art. 745) 
 " ^-"sT + 3?^ ± 411 + 5^ ^^- &Wose;> - j~, and 
 ^art. 728) — — —- + —^ '} consequently the fluent of ^ 
 
 CTlog.p = 2M X I + g 4- — + :^ + Sec. as in art. 
 
 173. In the same manner other theorems are found for com- 
 puting logarithms. 
 
 757. The fluent of ^ is equal to AH IE (Jg.300) the area of the 
 
 equilateral hyperbola,AHbeing perpendicular from thevertexA 
 andEIfrom any point E Lo the as3'mptote OH in H and I^ suppos- 
 ing OH = 1 jandOI — a',0 being the centre of the figure; because 
 
 the ordinate EI = — — — =. — Hence the area AHIE, or 
 
 the sector AOE, is called the hyperbolic logarithm of 01, or 
 EI, the ?woc?w/ms being supposed equal to AH x OH or 1 ; and 
 such coincide with the logarithms in Napier s first tables ; 
 whereas the tabular logarithms are now equal to these multi- 
 plied by the reciprocal of the hyperbolic logarithm of 10, as 
 was more fully explained in art. 174. If the sector OAK : 
 OAE ::n: 1, and KL be perpendicular to the asymptote in L, 
 then log. OL = n X log. 01; and OL : OH :: 0I» : OH". 
 
 758. The properties of the circle and ellipse often suggest si- 
 milar properties of the h^'perbola; and reciprocally the pro- 
 perties of hyperbolic areas (which are sometimes more easily 
 discovered because of their analogy to the properties of loga- 
 rithms described in book 1, chap. 6) are of use for discovering 
 tije analogous properties of circular and elliptic areas. The 
 
 fol-
 
 Chap. III. elliptic and hj/perboUc Sectors. 205 
 
 following theorem serves to show how great this analogy is, and 
 leads us in a brief manner to various general theorems that re- 
 late to the multiplication and division of circular sectors or arks. 
 Let O (fig. 300 and 30 1 ) be the centre of the ellipse or hyperbola 
 AEK, OA either semi-axis of the ellipse^ but the semi-transverse 
 axis in the hyperbola, av the axis perpendicular to OA, OAK a 
 sector that is the same multiple of the sector OABin both figures, 
 Kk and B6 perpendicular to av in k and b; suppose OA := «_, B^ 
 =: X, and Kk = z, when the perpendiculars B6, Kk are on the 
 same side of the axis av witli OA (as they always are in the hy- 
 perbola) ; but B6 = — X or Kk = — z, when Bb, or Kk, are on 
 the other side of av in the ellipse. Then the relation of r to x 
 will be determined by the same equation in both figures. To 
 make this appear, let AOB, BOC, COD, DOE, &,c. be any 
 equal sectors in the hyperbola; and let AOB, BOC, COD, DOE, 
 &,c. be likewise any equal sectors in the elhpse ; let Bb, Cc, Dd, 
 Ef, &c. be perpendicular to av in b, c, d, c, &c. in each figure; 
 join AC, BD, CE, DF, &c. intersecting the semidiameters OB, 
 OC, OD, OE, 8cc. in M, N, P, Q, &c. respectively. Because 
 the sectors AOB, BOC, COD, DOE, &c. are equal, AC, BD, 
 CE, DP, Sec. are ordinates of the respective semi-diameters OB, 
 OC, OD, OE, &c. For the same reason OB is to OM, OC to 
 ON, OD to OP, OE to OQ, &c. always in the same ratio of 
 Bb to OA, in the same figure ; as was shown above of the ellipse 
 iintrod. p. 8, 4- §6 17), and is easiij^ extended to the hyperbo- 
 la. Let M/», Nw, Pp, Q^, &c. be perpendicular to the diame- 
 ter av in each figure in /«, n,p, q, r, &c. respectively ; and the 
 ratio of Mw to B6, of N« to Cc, of Pp to Y)d, of Qq to '£x, &c. 
 will be always the same as that of Bb to OA. Then because 
 
 AC is bisected in M, Cc -{- OA == sMr/i = SB^ x ^=£B6 
 
 X ~ * because BD is bisected in N, Dt? -f B6 = 2N« = SCc 
 
 X -• In the same manner Ec -f Cc = iVp = SDc? x-: and 
 
 SO on : therefore, since in both figures Cc =: 2B^ X OA, 
 
 Dd
 
 206 0/ the Analogy betwixt Book II. 
 
 Dd — 2Cc X - — B&, Ee = 2Dc? x - -^ Cc, and so on; it 
 
 appears that the relation of Cc to B6, of Dc? to B6, Ee to B5, 
 and, in general, the relation of YJi to B6 (the sector OAK be- 
 ing the same multiple of OAB in both figures), will be expressed 
 always by the same equation in the ellipse and hyperbola, 
 the perpendiculars B6 and K^' being on the same side of the di- 
 ameter av with OA, But if the perpendicular F/*(for example) 
 stand in the ellipse on the other side of ar, then — F/"will be 
 determined from B6 andOAin the ellipse by an equation of the 
 same form with that which serves for determining 4- Pyfrom 
 BftandOA in the hyperbola; for in this case we find in the ellipse 
 
 DJ__F/-- 2Q^=r 2Ed x^,or — F/=2Ee X j— D<^,-and 
 
 in the hyperbola + F/'zr 2Ee X Dc?. In the same manner 
 
 in the ellipse — Gg— — 2F/ x j — Ee, but + Gg = sFf X 
 
 - — Ee in the hyperbola; whence — Yf, — Gg, &c. are deter- 
 mined in the ellipse by the same equation as + F/", 4- Gg, &c. 
 in the hyperbola : and in general it appears that + KA- or z 
 is always determined from + BZ*, or x, and OA, or a, in both 
 figures by the same equation. 
 
 759. In the equilateral hyperbola, let BS and KT (fig-oOO) be 
 perpendicular to the transverse axis in S and T, VBand LK per- 
 pendicular to the asymptote meet the same axis in X and Z ; 
 let the sector OAK : OAB : : n : 1, OX = y, B6 or OS = x, 
 and Kk or OT =z z as before : then by the common property 
 of this hyperbola, BS- r= OS'^ — OA% that is BS = >/ xx-aa^ 
 and OX {-y) = OS + SX = OS + BS = x + V^^^^ in 
 the same manner KT n '/^■i.—aa, OZ := OT + TZ :=■ OT 
 + TK r: 2 4* V~2— flfl. Because the sector AOK : AOB : : 
 w : 1, it follows (art. 757), that OV« : 0H« ( : : OX" : OA«^) 
 : : OL : OH : : OZ : OA ; that is,y« : a" : : 2 -f -• »a— aa : a^
 
 Chap. III. elliptic and hyperholic Sectors. 207 
 
 w" , 
 
 or Z + '/zz—aa := ^'^J because ?/ == X + *^ xx — 7a, y-^ 
 
 zz XX — aa, or yy — Q,xy + aa :=. o ; and because z + '/%z—aa 
 
 yn 
 
 = "^^x' it follows thaty" — 2a«— ' zy' + a""" = o. Hence 
 
 the relation of z to j? is found in the hyperbola by comparing 
 the two equations y^" — 2~a"^' y" + fl'" = o, and t/j/ — 2ij/ + aa 
 = 0, and exterminating y. Therefore, by the last art. (Jig. 
 301), if the sector OAK be to OAB in the circle i\s n to 1, or 
 the ark AK =z n x AB, then the relation of ip K/t (the cosine 
 of the ark AK) to If Bh (the cosine of AB) will be determin- 
 ed by supposing If Kk z=: z,Zf Bb—x, OA r: a, and extermi- 
 nating y from the two equations 7/^" — 22a"—' y" + «^" = 
 o, and yy — <2xy + aa = o ; of which theorem Mr. De Moivre 
 has made excellent use for resolving a trinomial of the form 
 y^" — 2^y + 1 into quadratic trinomials {Miscel. Analyt.lib. 1), 
 as we shall see afterwards. 
 
 76Q. Produce SB and TK (fig. 300), till they meet the asymp- 
 tote in s and t, and K^ : OA : : Bs« ; OA" ,• that is 
 
 Z Visa: — aa — CI X 
 
 a : : x — '/xx — aa : fl" ," consequently 
 
 rj 
 
 X V XX — aa 
 
 Therefore since z + -v/^a 
 
 1 
 
 X + V XX- 
 
 - aa 
 
 a 
 
 
 , it follows (by adding those equations) 
 
 , a a + V x.v — aa -^ X */ xx — aa , . , , , 
 
 that z r: - X — — ^, 3 which (by the 
 
 binomial theorem) is equal to -^ multiplied by x" + ri x -j— 
 
 «— 1 
 
 XX«— ^Xr;r — ^ + WX — ^ X -^ X—J— XX«— *X;*-Ar— is* 
 
 + &c. Or, the radius a being supposed equal to unit, raise 
 ^'+ 1 to the power of the exponent n, multiply the terms taken 
 
 alter-
 
 SK)S Of the Analogy betwixt Book II. 
 
 alternate^, beginning with the first x** by 1, xx — 1, 
 
 2 3 
 
 XX — ], XX — 1, &c. respectively, and the sum of the pro- 
 ducts will be equal to 2. Hence if 06 the sine of the ark AB 
 (,%. 30 1 ) be represented by u, or uu=:aa — xx, then Kk or z will be 
 
 equaUotheproductof— — , multipliedbyl — n X x 1- 
 
 ,_, n — 1 « — 2 n — 3 «* , t • • i f> i i , 
 
 nx.-^~ X -sp X -— - X — — ice. Itisevidentfromwhatbasbeea 
 
 1— >« 
 and 2a "X 
 
 shown that a.- if: '•;^Ar— ^a = a X IJl^ 
 
 = 2+ V'^i3^i"4- 2— -v/^JH^r- Let O^- (the sine of the 
 ark AK) = S, or SS — aa — zz "=. (by substituting the value 
 
 ci z) , 2«^" - ^4- A/.._-^ - X- V..-.. . consequently 
 
 X -\- V XX — aa X 's/ xx — i 
 
 ^ = " " — ogn-i ~^ — ~ ^ V— 1 which (by the 
 
 "binomial theorem) is equal to — 3j multiplied by nx^—^ u 
 
 — n X -g— X -~ X X ' ?i^ + Sec. The series given by 
 
 Sir/s«<2c Newton for finding the sine of the ark AK from 
 the sine of AB, may be derived from this theorem, or from 
 ai'ticle 751. 
 
 76 1 . Let Ar and AR (fg. 300 and 301 ), the tangents of the hy- 
 perbola or circle at A intefcepted by the semidiameters OB and 
 OK,be represented by i and T; and because B& : 06: :OA:Ar,we 
 
 find in the hyperbola. r = — and ^/xx—aa^z — ■ ^ but in 
 
 *y aa — // '/aa—tt 
 
 the circle x = — ~ — and '/TT^Z = ~ * By substitut- 
 
 V aa-\-it Vaa-iftt 
 
 ing these values for x and */ xx—aa, and similar values for z 
 and ty^-^a in the first equation in art. 759, z + V^-^ = 
 
 flX
 
 Chap. III. circular Arks and Logarithms. fi09 
 
 a X ~ — —\ , which was shown to be common to both 
 
 — -,« 
 
 figures we have in the hyperbola — ° ■ = - —^ I or (be- 
 
 and T = /z x 
 
 cause — h--, =: — ^) — ^p = — !~ 
 
 g + ^— g— f . but in the circle f±Z^rl =: l±LYjzl. 
 7+T" + ^7'' 'y^'^^ -f Tf~ -/ "^7+ /7" 
 
 or — 
 
 a — IV— 1 a — f-/— 1 
 
 and 1 = a X — - .. 
 
 a + tVZri'-i-u-y/'-Zf 
 
 na'^^ t—nx -— x — - x a t' + H^c, 
 
 r= (by art. 748) a x ^j— -j i . 
 
 fl"— n X 1-^ X ««— ^ ^ * + &c. 
 
 This theorem was given by Mr. BernouiUi, Act. Lips. 1712. 
 ' 762. The same theorems are immediately deduced from the 
 inverse method of fluxions, by representing circular arks as ima- 
 ginary logarithms; for inthismanneran analogy is preserved in 
 the expressions of the fluents, as near as possible to that which is 
 betwixt their fluxions, or betwixt the equations of the circle 
 and h3rperbola. The fluxion of the hyperbolic sector OAB is 
 to the fluxion of the triangle OAr (or la't ) as BS* to Ar*, 
 or as OS^ (= OA* + BS*) to OA% and consequently as OA* 
 to OA* — Ar*, that is, as aa to aa — it; and is expressed by 
 
 -'' X — rr» the fluxion of OAK is in the same manner 
 
 2 aa—'tt ' 
 
 - a t X — ^TTTp. Therefore since OAK = w X OAB, we have 
 «i:ZTf = 7^::rt' ^3^ supposmgp =: a X — -, we have (art. 
 
 728) - = -rr + — 7 = T,^ and rrTT = pT. I" the 
 
 VOL. II. P same
 
 210 Of the Atialogy betmxt Book II. 
 
 samemanner, by supposing ^f rr o X ^^^, the iiuxioii 
 
 aa-TT 
 
 = %'j consequently f^ = ^, J = ^J ^"^ (^^■^' 7^8) p = 
 .7'^ X K where K is invariable, or a x --t^ = « K x "■—-] or 
 (because T and t vanish together, and a"K =: «) — ^ = 
 f-lL- , as in the last article. 
 
 a — t\ ' 
 
 763. In the same manner the fluxion of the circular ark AB, 
 VIZ. (art. 740 K by supposmg p = ax — =, is trans- 
 
 formed mto 7=' because (art. 728) - zz 
 
 , ^^, __ ^= + 
 
 2/?-/— I p a + ^A/-.! 
 
 / ^/ZII[ 2 a/ Hi 
 
 Therefore the circular ark is equal to 
 
 a — 1\/^\ aa-\-tt 
 
 the fluent of — ^^=5 and is expressed by — r-i — = X loff. p =: 
 2/JV/-1 ^ ^ Qy\^~^ ^•f 
 
 a a + t\/Z:i. f. • . 
 
 - ■ ■ , r ■ -z:^ X log. a X 7—7=:} where the value or »is imasrma- 
 
 ry, and is so far compensated by the imaginary symbol M '/^, 
 that the whole compound expression maybe supposed to denote 
 the circular ark ; as such imaginary symbols compensate each 
 other in the expressions of the real roots of cubic and higher 
 equations. See art. 699. In the same manner the fluxion of the 
 
 1 AT^ ««T , . r/+T\/irr 
 
 circular AK, or ™=-' by supposmg q = a x — -i 
 
 au +11 ci — 1 \/ I 
 
 is transformed into -^: and since T^ ^=^ """" * it fol- 
 
 lows that 1 — !!^5 </ rr p» X K. or « X „, "l!- — a" K x 
 
 q P a~iV—\ ' 
 
 . . > — ,1 
 
 ~ 5 or (because T and t vanish together, and conse- 
 
 a-^t'/—i 
 
 a + T\/-^i a + tV^i 
 
 quently a»K = a) — , ,p . ^ = z= 5 as m art. 76 1. 
 
 764. In
 
 Tlatc XXXJnLFa.QiP. VolJI. 
 
 
 B 
 
 E 
 
 JJ- ay 
 
 F 
 
 c 7 
 
 M 
 A 
 
 D 
 
 Fi^.^oiJt^.Q^rtj^g. 
 
 P A 
 
 O e> ^ 
 
 A j3 
 
 F^.-i^^.
 
 TlatcXXXIIIlJ°rf.2/^-. f7vy/.
 
 Chap. III. circular Arks and Logarithms. 211 
 
 764. In like manner, supposing, as above, OA — a,Bb~ x, 
 
 K/c — z, the fluxion of the ark AB (art. 747) is ' ~°^ ^ and 
 
 V aa —' XX 
 
 the fluxion of AK is ■ """^ -. . These fluxions are trans- 
 
 formed, by supposing p —x ^ Vxx — aa, and q-=.z-\- sf^^^^a 
 
 (by art. 728) into -~r=z. and — ~r Therefore since AK 
 ^ ■^ pV—x qV—x* 
 
 -=.11 Y. AB, ^ = ^ tf = p« X K, or s + '/^^^^a = 
 1 /'^ ^ 
 
 a:+ '/xx—aa'' X K r= (because when B6 or .r = a, then 2 = a> 
 
 and consequently a = «"K) a X ^^.'^^n^'' 9 as we 
 
 found in art. 759- And supposing y =. x •\- </ xx ^aa, 
 the relation of z to x will be determined by exterminating y 
 from the equations 3/*" — 2;ra"— ' 3/" + a^** + o, and ?/'■ — 
 ^zy + a* =: 0, as we demonstrated in art. 759^ without mak- 
 ing use of the imaginary sign, by showing that the relation of 
 2 to X must be determined by the same equations in the circle 
 and hyperbola. 
 
 765 . Let (Jig. 302) the circumference of the circle be represent- 
 edby C, the radius OA by 1, thearkAKby A, and consequently 
 AB by -. Let the circumference be divided into as many 
 equal parts (beginning from the point B) BC, CD, DE, EF, 
 EG, GB, as there are units in n ; then AB = -, AG = BG— - 
 AB = ^, AC = AB + BC = -£±A^ aF = i^, 
 
 AD = -^^7-^, and AE = — ~-. The cosine Kk of the 
 ark AK being represented by + z, as before, let the cosines 
 of those several arks AB, AG, AC, AF, AD, AE, &c. be repre- 
 sented by 4:0, 4:6, z^c, z^d, qif, z^f, Sec. respectively, where 
 the sign of each cosine is supposed positive or negative accord- 
 
 P 2 ing
 
 SIC 0/ resolving Trinomiaia Book II; 
 
 ing as it is on the samt side of the diameter au with OA, or not. 
 Then because those arks are in the same ratio of 1 to w to the se- 
 veral arks A, C— A, C + A, cC— A, 2C + A, SC— A, sC+A, 
 &,c. which have all the same cosine Kkzzlfz; the several, 
 relations of z to a, z to b, z to c, z to d, Sec. are found by 
 comparing successively the equation _y*" — Qzi/^ + 1 == Oj 
 with the several quadratic equations yy — 2aj/ + 1 =zo,^i/ — 
 26y + 1 = o,i/j/ — Q.CI/ + 1 —o,yy — ^dy 4-1=0, 8cc. and 
 always exterminating j/. From which it follows, that the trino- 
 mials yj/ — 'lay + \,yy — <lhy + \,yy — <lcy + \,yy — (idy 
 + 1, S;.c. are divisors of j/" — 22y" + 1, or this last is equal 
 to the product of those trinomials. 
 
 766. Upon OA take OP to represent y, join PB, PC, PD, 
 PE,&c. .and PB^ = O*^^ + B6±OP' = Oh"- + B6* + 2B6 
 X OP + op- = 1 — 2ffy + yy. In the same manner PG^ =: 
 1 — Ohy 4- yy, PC^= 1 — ^cy 4- yy, &c.; consequently PB* 
 X PG^ X PC" X PF" X PD" X &c. =/-" — 22j/" 4- 1 = 3/"'* 
 jf; 2K^ Xy« 4- 1. Let the ark AK or A be now supposed 
 
 A C 
 
 equal to the whole circnmferenceC, then r^g'.303)BA 3r - = - 
 
 riB G, andG coincides with A. In this ccsse the point K fallson 
 thepointA,4- z = KA; =r OA = 1 ; and PB" X PA" X PC" X PD' 
 X &c. rz?/^" — 2j/« + ];andPB X PA X PC X PD X &c. 
 z=. t/^ — 1 or 1 — 3^. From which it follows, that if the cir- 
 cumference he divided into as many equal parts at A, B, C, D, 
 8cc. as there are units in n, upon OxV (= 1) 3^ou take OP r: y^ 
 and from P draw right lines to the other points B, C, D, E, F, 
 &c.; then the productof all those right lines, PA X PB X PC X 
 PD X Sic, will be equal to y — 1, or 1 — 3/", that is, to OPf^ 
 — OA", or OA" — OP", according as OP is greater or less than 
 OA; which coincides with the first part of the elegant theorem 
 invented by Mr. Cotes, and described by Dr. Smith, Harmon. 
 Memurar. p. 1 14. Let the semidiamcter AO produced meet 
 the circle in a, and if n he an even number, one of the points 
 A^bcrein the circumference is supposed to be divided will fall 
 
 OQ
 
 ■ Chap. III. into quadratic Divisors. *216 
 
 oil a; consequently the divisors of 1 — j/« will be the rectangle 
 APa with the squares of the right hnes PB, PC, gcc. thatareon 
 one side of Aa; but if n be an odd number, PA (or ] — y) will 
 be a simple divisor of I — y«,and thesamesquaresof PB,PC, hcc 
 will be its other divisors. Suppose now the ark AK equal to the 
 
 semi-circumference, or to IC r/7£f.303); then BA rr - r= — rr 
 
 -| BG; and in this case K falls upon a, K/c, or Oa, = — ^ =: 1, 
 or 2: rr — 1 . From which it appears, that if the circumference 
 be divided at B, C, D, JL, &.c. into as many equal parts as 
 there are units in ?t, and one of those parts BG being bisected 
 in A,youjoin OA, and take OP upon it to represent j/, then from 
 P draw the right lines PB, PC, PD, PE, Sec. to the several di- 
 visions of the circumference ; the product of the squares of all 
 those lines PB^ X PC^ X PD^ x PE^ x &c. will be equal to 
 1/^" X2zif + 1; and consequently PB X PC X PD X PE X &;c. 
 = 1 t^i yn — OA" + OP"; which coincides with the latter 
 part of Mr. Cotes s theorem. When n is an even number, the 
 same product is that of the squares of the several right lines 
 PB, PC, &c. that are on the same side of the diameter Aff; 
 which are therefore the quadratic divisors of 1 + y in this 
 case ; but when n is an odd number, one of the divisions of the 
 circumference falls upon a; and the same squares with Pa, or 
 1 + ^, are the divisors of i+y". By supposing AK 0?g.304) equal 
 to a quadrant of the circle, or to $ C, or s = 0, it will appear 
 that the circumference being divided as before, if the ark BA 
 be taken upon BG equal to | BG, and upon OA you take OP 
 =zi/, then PB^ x PC^ x PD* x 2cc. = 1 + y"-" - 0A^» 
 -f OP''"'. The reader will find this subject treated in a differ- 
 ent manner, Epist. adamicum de Cotesii invattis, 1722. 
 
 767. In general, the circumference being divided into 
 thesamenumberof equal parts,letP (//g.300)beanypointinthe 
 plane of the circle, let OP meet the circumference in A, 
 take the ark AK = n X AB, and upon OA take OQ : 
 
 P 3 OP
 
 Q.IA- Of resolving Trinomials Book II. 
 
 OP : : OP»-l : OtV-\ join QK; and PB^ X PC^ X PD* 
 X PE^ X &c. = QK* X OA-"— ^; because \ik being sup- 
 posed equal to + z, or — z (according as KA; is on the same 
 side of azj with OA, or on a different side), then QK* = 1 — 
 9.Z X OQ + OQ- = 1 — 9.zif + y^-'K In this manner Mr. 
 Cotes's theorem was rendered more general by Mr. De Moivre. 
 Hence several other propositions relating to the circle may be 
 briefly derived. By supposing OP to concide with OA, it ap- 
 pears Lhat the productofthechoidsAB X AC X AD X AE x &,c. 
 =: AK X OA"-'. For in this case OA, OP, and OQ, being 
 equal, AB* x AC^ x AD* x AE* x Sec. ;:= AK* xOA*"— -,and 
 AB X AC X AD X &c.=:AK x OA«— ' ; which is demonstrated 
 in adifferent manner. Hospital. sect.coniq.Ub. lO,theof\ 1 andS. 
 768. The quadratic divisors of a trinomial may be likewise 
 discovered from ibe common algebra. To resolve a quantity 
 as _y*" ■ — 2-^" + 1 into its divisors, is a problem equivalent 
 to the resolution of the equation y*" — 2x7/" -}- 1 — 0. By 
 proceeding as is usual in the resolution of quadratic equation 8,7*" 
 — 2zy -{- zz = zz — 1, 1/"' — z = If. v'zz— 1 ; and the di- 
 visors are y — z + 's/as,— 1, and 3/" — z — -v/^a— 1, the pro- 
 duct of which is 1/'^'^ — 223/'* + 1. When z is less than 1, 
 Vaz 1 is imaginary, and those divisors involve imaginary ex- 
 pressions. Butwe arenotthence toconclude that otherdivisors 
 cannot be assigned in this case, which may involve real quantities 
 only. It is obvious that JZ17 X JUJ X JUJ X JH^ may be resolv- 
 ed into several different pairsof quadratic divisors, asinto^ZI^ X 
 ^—i, and JZ7 X JZj, or into ^^ X y—c,vind^—i X J^l; and 
 though the first two may involve the imaginary symbol, the 
 latter may involve no quantities but such as are real. Thus sup- 
 posing j/^^ — 2z7/* +1=0, we have ?/* =. z + \/2.z— 1 or z — 
 Vzs — 1; and the four simple divisors (by extracting the square 
 root again)arc in this casej/ -f \^z-r V-^z—l, y — "/;:+ VH^^
 
 Chap. III. into quadratic Divisors. Cli 
 
 1/+ y'z—v'ii—i, and 1/ — \/z- a/;!— 1. The product of tlie 
 first and second gives j/y — 2 — v'z;— 1 ; and the product of the 
 third and fourth gives iji/ — z+ v'zz—i, the same iuteruiediate 
 divisors from which the simple divisors were derived ; both of 
 "which involve an imaginary quantity when* is less than l. But the 
 productof the first and third gives yj/ 4- \/ z-^V ^^^ -{- '^z-\/zzA 
 X ^ + 1 rr (because the square of \/z-^\/I^i^+ y^v—v^^THY 
 
 is z + \/i^zri :j: o ^ 5; _ ^z^rrr — iz :^ i) yy + /"^ 
 
 X ?/ + \, ox yy — y'-^s+s x 3/ + 1 ; and the product of the third 
 and fourth agrees with the-e. Thus we find ?/'^ — '2zy^ + 1 
 
 zz yy + \/2z+2 X y + 1 X j/j/ — a/sz+s Xj/+ 1. And these 
 divisors may involve no imaginary quantity, though she supposed 
 negative, and less than unit. B3' continuing the resolution till 
 we have the simple divisors, and then compounding those di- 
 visors together variously, quadratic divisors may be formed in this 
 manner, some of which will have all their coefficients real when 
 z is greater than unit, and others when z is less than unit ; and 
 we are not to conclude that no real quadratic divisors can be 
 assigned, because those of one combination are imaginary. The 
 latter quadratic divisors are likewise found by resolving the 
 equation y* — 223/^+ 1 =0 in a manner somewhat different from 
 the usual way of proceeding ; for since ?/ + 1 zz 2zy", com- 
 plete the square on the first side of the equation by adding the 
 middle term If 2y% and y^~^ Q.y^ + I = + '^zy^'Z^ qif ; 
 consequently by extracting the square root, ?/* IjT ] ■=.~^ 
 VSz^iS X y ; and the quadratic divisors are y^ — -v/s^qis X ?/ + 1, 
 and j/^-|- v'Si+s Xj/+ 1, as before. By the same method the 
 
 divisors of ?/^" — Izy^ + 1 arey If V'Siqi^ xy^ + 1, which 
 may be again resolved in the same manner into divisors of infe- 
 rior dimensions; and by a continual bisection of the exponent, 
 when n is any power of the number 2, we may at length, 
 find the quadratic divisors. But this last method is not appli- 
 cable when n is any other number. 
 
 769- Supposing therefore j/^" — 22^" -f 1 rr 0, as before ; 
 and consequently y'^ — z + v'lz— 1 or z — VIi^ ; then by 
 
 P 4 evolu-
 
 Ql6 Of reducing FlutnH Book II. 
 
 evolution y = '^z^Vzz—i or v^z-a/zz— i ; consequently 
 
 n 
 
 two of the simple divisors are y — Vz+ VTt^i and y 
 
 ^z-~\/zz~\ \ and^ the quadratic divi sor arising fro m their mul- 
 
 n n 
 
 tiplication by each other being ^ry — '^ z-\-VTz'^\-\- 'V^z— a/TT^ 
 X y + 1, suppose the coefficient of the middle term to be Ix, 
 or this quadratic divisor to be yy — 9^xy + I, and 2x = 
 
 ^2-|-\/z7~r + ^z— a/II^T. By comparing this equation 
 with that which was deduced in art. "59, it will appear that, 
 when z is less than 1, if z be the cosine of a circular ark A, 
 
 then X will be the cosine of an ark equal to -, Or if we sup- 
 pose z + v'zz— 1 = p and 2 — v^IT^'i — q, and consequently 
 
 ^z — p ^ q, X ^ 2. — --L, -/^;7ri — (because pg' = 1) 
 — o J ^■*" V^A^i =p" and .r — >/xx~\^q;'\ so that 2z = 
 
 I'+V'^T^+j: — ^j(.;;f_i, the same equation that we found 
 in art. 7o9, for the cosines ; which, expanded as above, will 
 be found always to agree with those by which the relations of 
 the cosines are determined by the common methods. But 
 let us now proceed to show how fluents, or areas, are measured 
 by circular arks and logarithms ; and, first, when the ordinates 
 are expressed by rational quantities. 
 
 n , 
 
 770. Let it be required to assign the fluent of •^^— ^^ n being 
 
 any integer positive number. It was shown in art. 709, that if 
 y»— » ^ yn—T- ^ .j. yu—i o^ , , . a"^' be multiplied by y — a, 
 
 the product will be y'» — a'K Therefore ^--^ zz y^—^y +
 
 •Chap. III. to circular Ark$ and LogaritJms. 217 
 
 ay y+«y y... + a y -^- • consequent- 
 h' the fluent of i::!li^(by art. 737 and 740) is i:! + ^Z! 
 
 manner y — y a +y a . . , , q: c r= — - — • 
 Therefore the fluent of -^"-^L is ^ — 'i^Zl + .^-f."! . . , 
 
 j/-\-a n n— 1 »— 2 
 
 t: f_ X log. y + fl. 
 
 77 1 . Any integer number bei'^g represented by w, the fluent 
 
 of ■ ^ -^ is expicsscd by a circular ark, or logarithm (^with 
 
 algebraic quantities), according as n is an even or odd num- 
 ber. For it appears, as in the last article, that when n is an evea 
 positive number, if 3^"—^' — a-j/"— * ^ a+y"— g — a'y^—^ 
 Hh 8cc. be multiplied by j/^ + (i^, the product will be y — ■ 
 a", or^" + «", according as |« is an even or odd number. 
 
 Therefore J!J- -V^-S — "y^V + a^y^'-^y 
 
 -T ^ y 
 
 • consequently if A represent the ark whose tangent 
 is equal to y, the radius being equal to a (so that A =: 
 
 n . 
 
 , -> by art. 744), the fluent of - ' . - will be equal to 
 y:zL__^lrzL . <^ . . . +«-.^ X A. Whennisan 
 
 odd affirmative number, suppose it equal to ;« + 1 ; and, by what 
 
 has
 
 CIS Of reducing Fluents Book IL 
 
 has been shown, ^ - ^^ ^ — 3/'"— 'j/ — 'if'—^a-y + y^'—'^a'^i/. . , 
 + — — ■; the fluent of which (because the fluent of -— - 
 
 n — 1 n — J ' n — 3 M 
 
 — « . . « . 
 
 By supposing ^ =-, ^^^ is transformed nito — ^^:^^ x 
 
 —J the fluent of which (by what has been shown) is express- 
 ed by a circular ark or logarithm, according as n is an even 
 or odd number. By supposina: z zz a x ^~y the fluxion -^^ 
 
 is transformed into — . b}-^ art. 728; consequently the fluent is 
 
 r^ X log. a X ^^^\ and the fluent of ''^ - • beins: equal to 
 
 2M '=' i'+« Hi/ — '^^ 
 
 — ~ — M^^~~' ^^ easily follows that when n is any integer num- 
 
 ber, the fluent of - — ^ is expressed by logarithms and alge- 
 braic quantities. 
 
 772. Let ^-—. — r= o., and let it be required to find 
 
 aa-\-1bi/-\-i/y 
 
 the fluent of 6 . By supposing y + 6 = 2,, and consequently 
 iiTzz. i/ij + 2fji/ + aa zz zz ^^ aa — bb,Q = — ; -5 and the 
 
 fluent Q is expressed by a circular ark, or logarithm, accord- 
 ing as a is greater or less than b, by the last article ; and if 
 
 a =6 the fluent is ^ K, by art. 737. The fluxion 
 
 — -—-, — is transformed into —^^^^^-^3 by the same substitu- 
 
 aa-\-^bj/-\-yif siz-{-aa—oo '^ 
 
 tion ;
 
 Chap. Til. to circular Arks and Logarithms. 219 
 
 lion; and the fluent maybe found by thelast article. Or supposing 
 
 VZ+^bf^ = u, bccause ^^^,^^,^^^ is equal to - , by art. 
 728, it follows that the fluent of ,,_,.-gy^^^ is ^^ — ^Q = 
 log. ^/..+g^^-f//y _^Q^ The fluent of Jj^ is found 
 
 (when n is any integer positive number) by dividing j/** hy 7/_y+ 
 o^y _{. ««^ and continuing the operation tilltheremainder beof the 
 form Au-^^ (where A and B represent invariable coefficients), 
 multiplying each term of the quotient by y, finding the fluent 
 of each product by art. 737, and determining the fluent of 
 
 -^~r, — ^, bv this article. The fluxion ', ^u 1 is trans- 
 
 I . —a" z 
 
 formed, by supposing _y = -, into ^ ^^ _ , and the fluent 
 
 is found as before. It appears, therefore, that, the ordinate be- 
 ing expressed by a fraction, if the denominator be any quadra- 
 tic trinomial 1 + Q.hi/ -f yy, and the numerator consist of terras 
 that involve any powers ot t/ and invariablequantities; and the 
 exponents of those powers of j/ be integers; the fluent may be 
 assigned by circular arks or logarithms with algebraic quanti- 
 ties. And any fluxion Py being proposed, if P can be resolved 
 into an}^ number of fractions of this form, the fluent of Pj^ can 
 be assigned in like manner. 
 
 773. What was demonstrated in art. 715 and 7 17, or in 728 
 and 729> is often of use for resolving an ordinate into such frac- 
 tions. For example, as when p rr xi/zu X &c. it follows, that 
 
 T — ^+-+-' + &c. so if we resolve 1 + ?/" (sup- 
 posing n to be an even number) into its quadratic divisors 
 1 — 2ffy + yy, 1 — 2ij/ ~^yy> ^ — ^^1/ + M> ^^- according to 
 
 art. 765, it follows, that ^^^^^^^l = -l^^^^Zf^- + SJlilfZ^li
 
 ©20 Of reducing Fluents Bool? II, 
 
 A ^ — 1 — -I- &c. and consequently — ■ -r-^ — r^ + 
 
 + Sec. so that the fluent of ■■ ' ■ - is equal to ?/ added to the 
 fluents of the several fluxions — —^ — x - — ' ■^"" - - 
 
 ^^J/+I/I/ n' 1— 23y+_y^ 
 
 — . 8cc. which are found by the last article. 
 
 774. The same method serves for investigating briefly the 
 first four propositions of Mr. De Moitres Miscel. Anali^t. lib. I. 
 Suppose, first, n to be an even number, and since \-\-y^^zz 
 
 l_2ay4.j/_y X \—2by-ifj/j/ X \-.2cj/-\-j/y X &,C. it foUowS, 
 
 dividing l+ybyy", and each quadratic divisor by y^, that 
 li^ —\—<laij—'-\-ij-^ X l—'lby—'^ y—^ X &c. There- 
 
 fore, as when - = xyzu X &c. ^ — -— -J* -4« 
 
 - + &c. (byart. 728), so in this case 2L — Z =^ ( = 
 
 z 1 "T^y" y 
 
 or (dividing both sides by— ^;~')y^ = —±±'^^ 
 
 .2^Z::J— 4. --iIZi!^ • which is the first of those propositions. 
 
 When n is an odd number, then, besides the quadratic divisors 
 of 1 ^if (which, according to art. 766 (Jig. 304), are PB", PC*. 
 
 PD*
 
 Chap. III. to circular Arks and Logarithms. 221- 
 
 PDS &c.), there is a simple divisor Pa — 1 4-y,- and it appears 
 
 that in this case Yqp^ - T-ji;^ + 1^^^+^ ■*" i-'2by+yj/ 
 + &c. When w is an even number, the rectangle APa, or 
 l~-7/y, is one of the divisors of l—^"" (by art. 766), and the 
 other quadratic divisors PB% PC*, PD% &c. being expressed 
 by 1 — Ofly + j/y, I — <2.bi/ + j/y, &c. as before, it appears, in 
 
 .... « _ 2 2— 2«y 
 
 the same manner, that in this case-j^^-- p^+ 1_2«^^^,; + 
 
 ; ""^ --^ 1- Sec: and when n is an odd nuraber,PAor 1 — j/being 
 
 one of the divisors of 1— j/" (by art. 766), we shall find in the 
 same manner that = 1 1- , q, f r , — 1-77: + 
 
 &c. The ordinate -z=- — bein^j resolved in this manner into 
 fractions with quadratic denominators, the area or the fluent of 
 ^-=-^ (and consequently of -—-^^ when m is any integer 
 number) is reduced to circular aiks or logarithms with alge- 
 
 braic quantities, by art. 772. The ordinate TTTytT+T^ ^s 
 
 resolved into fractions of the same kind by this method, when 
 ^"is greater than 4€g, that is, when the roots oi" the quadratic 
 
 equation ^2;; _j. -^ +1=0 are real. For supposing those 
 
 ^ £■ 
 
 roots to be -R« and -r", or e + fi/n + gy2« n g x j/'^ + R« 
 X y» -f ?•»», let j/» + K" and y" + r" be resolved into their 
 respective divisors (art. 766) RR — 2Ay + yj/, &c. and rr — 
 
 « n 
 
 ^ay +yj/Ac.; and, by what has been shown, ,^;:^, — ^nj^yn 
 
 or
 
 222 Of reducing Fluents Book II. 
 
 _„ ^ ""^ X m = J- X — - + See. — — X 
 
 5RR— 2Ay 
 
 — &C. 
 
 Kli— 2Ay+y// 
 
 77j. When z is less than 1, lct7/2« — 2ry» + 1 be resolved 
 (by art. 766) into its divisors 1 — 2cry + yy, 1 — 2^3/ +3/j/> &c. 
 Then (2 being supposed invariable) it follows, as in art. 728, that 
 
 -^ ^^^^^ —T^-r h , ^Z ■ +8cc.or(mnltiplying 
 
 , *^ ^ ""y" — "^^ .V — '^ K ^-~^ 
 
 1 — 26j/— ' 4- j/— ^ X &c. it follows (as in art. 728), that 
 
 ^2«_2,y«4-l - 1-2.^+^y -T TZIoJh^ "^ ^''- ^^^ '""^ °^ 
 
 those equations eives ■■ """""" _j^--g_x =^ 3.-"+^ "i* i— ay 
 
 ^ ° /"— 2^j,'^+l 1— 2aj.+jj, 
 
 — 1-- :%+^^ 'T c^c.; and It follows, from art. 
 
 772, that the fluent of ^ „. , is assignable by cir- 
 
 cular arks and logarithms w'ith algebraic quantities. 
 
 77G. By a similar application of what was shown in art. 728, 
 if w'e suppose xx — Ax ± B :r: x—a X x—6, the fraction 
 
 T— 7-0 is resolved into fractions that shall have the simple 
 
 divisors .r — a, x — b for their respective denominators wnth in- 
 variable coefficients. Tor since xx — Ax+ B zz "^IZ X IHJ, it 
 
 follows
 
 Chap. III. to circular Arks and Logarithms. 2Q,Sv 
 
 follows (art. 728), that ■ — ^ '[„ = ~ — | — ^; and be- 
 
 ^ ■" xx—Pi.x-^D x—a ' X — o* 
 
 cause 1 — A.r— * + Br—* =1 — az— ' x 1 — bx--\ it fol- 
 
 lows that 
 
 Axx—'^Bx ax . bx 
 
 jcx—'-.^lix ax ox x? xi 
 
 — ■ . „ = \- — r, rrom these two equa- 
 
 — A;f+B X — a X — b T "■ 
 
 , X Ih—a X , 4 A — h 
 
 tions we have t—ts — rji — riT ^ — ": + "^ — Tjr X 
 
 ^-^ = (because A :=. a + b and B := ah, by the known pro- 
 
 1 !• 1 ■* 
 
 perties of equations) --_- X -— - + ■^- X ^::rj. Hence 
 
 r— -^ = -i-r X + T — X — r. Therefore ifr' — Ax" 
 
 a:,x»— A;f + li a — b x—a b—a x — * 
 
 -{-Bx—C—'^IZ X :;zr X ~o it follows, that _^3_^^,^g^_^ 
 zz — =:=-X =:(bywhathasbeenshown)rrr= — z=r- — ==. 
 
 X — a X X — b * '^ a-—b X x — a X x—^ 
 
 -f — r- — — ■ ■ - r: (by resolving each of these last fractions 
 
 b — jx X — by, X — c 
 
 in the same manner) := — + =^ — + 
 
 a — b X o — c X x—^ a—b X c—^ x x—'a 
 
 b — a X b — c X X — b b—a x c — b X x—c Of—b X a—c X x^-a 
 
 — : . The continuation of those 
 
 b — a >< b — c X X — b c — a X c-^b X ■*■ — e 
 
 theorems is manifest, the coefficient of x — b, for example, 
 being always the product of the differences b — a, b — c, &c. by 
 which the root b exceeds the other roots of the equation ^a 
 X "7^ X ~7 X &.C. r: 0, This subject is considered by Mr. 
 Leibnitz, Act. Lips. 1702, and Mr. Dc Moivre, Phil. 'Trans. 
 N. 373, &c. 
 
 777. But these fractions are briefly discovered in the follow- 
 ing manner. Suppose x^ — A.t"— ' + Bcr"— ^ — C:r"— ^ + &.c. 
 := "i^II^ X Zm X ^37 xX^ X &,c. and let this product be 
 represented by P; letQ represent the product of all the simple 
 divisors, the first x — a excepted; that is, let Q = ^7 x 7^ 
 
 X a;— </
 
 224 Of reducing Fluents Book II, 
 
 X "Z^ X &c. Suppose a, I, c, d, &c. to be unequal, and r be- 
 
 ing any integer positive number less tban «, suppose -p- or 
 
 - I- + M + JL+ &c. 
 
 a"— Ax**— ' + Ba"— ^ — &c. "" x — a x — b x — c 
 wbere L, M, N, &c. represent the invariable coefficients that 
 are to be determined. By reducing those fractions to a com- 
 mon denominator, and multiplying by P or x —a X x — b X 
 
 ^Z7 X &c. we have a:'" = LQ + MQ X ^ + NQ X ~ 
 
 + &c. Then by supposing x =r a, or x — a r=.o, we find that 
 a'' (i. e. in this case a*") is equal to LQ, or that a-' = L X a— ^ X SIH^ 
 
 X^IUX See. and L = — -""J •' In the same 
 
 manner by supposing a:=&, we find M=_ ^_-^_- ^g,^^ 
 
 The other coefficients of the fractions into which -^ is to be re- 
 solved, are expressed by similar values. 
 
 778. Because P =: Q X x—a, it follows, that p = d X 7^ 
 + xQ. ; and when :r = c, p == ;;Q, or Q (which in this case is 
 
 equal to a— J X a—c X a-d X &,C.) = - = ?2a"— ' — n—\ X 
 
 Aa»--* + ;;:i2 X Ba'i--' — &c. Therefore L = — = 
 
 In the same 
 
 7^a"— ' — ;]Z:i xAa''— " + «— 2 X JBa"— ^ — 6lc. 
 
 manner M - ,,^„-i__^zTx A6«-^ -f ;r:2xB6«-^— &c. 
 and the values of the other coefficients are similar. The rule 
 
 for
 
 Chap. III. to circular Arks and Logarithms, 22^ 
 
 for finding the coefficient in any of those fractions (as in 
 
 N \ . 1 . , X'- ■ . , - , 
 ) into which — -—J •; — T^ ■ — - — IS to be resoiv- 
 
 ed_, is, substitute c for x in the numerator x^, find the fluxion 
 of the denominator x^ — Ax''— ' + 8cc. which being divid- 
 ed by X, and c being substituted for x in the quotient^ you have 
 the denominator of the value of N. 
 
 779.LetitberequiredtoresoIve ^,.._a^.._. ^ I^^.^,. _^^^ 
 
 into fractions that shall have quadratic denominators^ where 
 r is supposed to be any integer and positive number less than Qti; 
 and the denominator (P) to be the product of the quadratic 
 divisors xx — 2ax + gg, xx — Qbx + hh, xx — 2car + kk, &.C.; 
 
 .f L — Ix M—mx 'N—rtx 
 
 suppose p-— xx^2az+gg- "^ xx—9ix+hh "^ 'xx--2cx+kk "^ 
 
 Sic, Let Q represent the product of all the quadratic divisors, 
 
 thefirst .r.r — 2a.r + o-g excepted;thatis,letP=:x:r-2aa: + o'o- xQ. 
 Then by reducing the fractions to a common denominator, 
 
 — — y^ . ■' ' r^ xx—^axA-ffP- ____ _^ xv-2ffx A- s-p- 
 
 L-/.xQ + M^.,.xQx ^_,,,;g + N-«. X Q X -^z£^ 
 4- &c. =: X'*'. Let e and /be the rocts of the equation xx — 
 2«T -^ gg — 0. Let M be the value of Q when x — e, and N 
 its value when x=:f. Then substituting e for x, xx — ^ax-{-ggj 
 and all the terms that are multiplied.by it, vanish; consequently 
 L— /e X M =: e^. In the same manner, by substituting/ for 
 
 ^,t:^xN=/'. Henc^L^5j|^_jj^= (be- 
 
 cause ef -gg)y^^ x -^ ^-^^' Because P=xx^2ax+ff^ 
 
 X Q, it follows (by taking the fluxions), that p = 2x-2a 
 
 X Q V + d X xx—iax+g^; and by substituting e for x, r- (which ia 
 VOL. n. Q "" this
 
 S26 Of resohvig Trinomials Book II. 
 
 this case is 2«e*«— "-i^ilUr X Ae'"— ^ + 2^112 x Be*"— ^ — &c.) 
 =: 2<r— .2a X M = (bccausc e + / = 2tf, and e — / = 2e — 2a), 
 <^/ X M. In the same manner N x /^ — ^rif'-^--^ — ^n-x 
 
 X A/^t-» + &c. Therefore L = ^^.,,,_. 
 
 g^/''-' 
 
 2w/ '"— '— 2^irr X A/'^«- ^ + &,c. 
 
 -- — And in like n 
 
 ■ ,., t — -^ ^ A .„ z , c — And in like manner we. 
 2«e^"— ' — 2«--i X Ad'"— "• + inc. 
 
 shall find / r: 
 
 IVl X tf-/ IN X f-e 
 fr 
 
 The values of the coef- 
 
 "M X77 N X 7:7 2/ie'"-'-2«-i X Ae^«-^ + &.c. 
 
 2/j/'"— ' — "S^i^ X A/"^«— ^ + &LC. 
 
 fifcients M and m, N and n, &c. are similar. 
 
 780. Let ggi hh, kk, &c. the last terms of the quadratic 
 divisors, be all equal to each other, and to unit. Then M =: 
 
 ee—2be-\-l X es— 2cf+l X ee—2de-\- 1 = (becaUSe €6 
 
 9.ae +1=0, and ce 4- 1 = Sflc) lae—^be X s^TITs^ x 
 
 2flf — 2c/ff X 8CC. =: e"— ' X 2fl— 23 X 2«— 2c X 2a— 2J X &C. 
 
 In the same manner, N =/'="-* X 2^ — 2! x 2^ — 2^ x 2a— 2J 
 X &c. Therefore if I represent the product of the differences 
 by which 2« the middle coefficient of the first divisor exceeds 
 oh, 1c, Q.cl, &c. the middle coefficients of the other divisors ; 
 then M = e"—' I and N =■/""-* 1. Therefore, by substituting 
 those expressions forM and N in thevalues of L and / in the last 
 
 article L = — r — z. ■■ when ?^ is greater than /• ,- or L =r 
 
 1 X «— / 
 
 «-7 — -d^^ -" when n is less than r ; that is, the difference of » 
 
 IX/-ff 
 
 pTfl //Tl 
 
 and r being represented by m, L =: rrr — - — : where the sign 
 
 of
 
 Chap. III. to circular Arks and Logarithms. 227 
 
 of I is positive or negative, according as u is greater or less than 
 
 r. In like manner / =r . — =^= when n — 1 is great- 
 
 I X ^—f 
 
 er than r, or / — : — =^ when 7i — 1 is less than 
 
 1 X e—f 
 
 r : that is, / :=. — ; — -^ — — in the former case, and / =: 
 
 1 Xe—f 
 
 — - — -' .. in the latter. Because e andf are the roots of the 
 
 A X e — 'J^ 
 
 equation xx — 2aT -f- 1=0, e = a + -v/^7Zl,/=rfl — Vaa — ly 
 
 ande—fzz 2Vaa — \, and L= ■ - -t-— ~ 
 
 _«+-/«<>— I a \^aa—l 
 
 — ^TTr — 77= ^ v'— 1. Hence if Bo, the 
 
 cosine of the ark AB (Jig. 302) be equal to a, the radius OA 
 being unit, the ark AQ be to the ark AB asm to ], and Q^ be 
 the cosine of AQ, then Ob being equal to '/i.—aa, it follows 
 (b}^ comparing the value of S determined in art. 760), that L— 
 
 -i^ , Let the ark QZ be made equal to AB, so that AZ may 
 
 be equal to ^T X AB, or ^T x AB, according as m — 1 
 is greater or less than r, and Zz be the cosine of the ark AZ ; 
 
 then I = T^rrpr, Therefore the fraction - — ^r— ^^ — = 
 
 - Qi^^^ X i_2ax+xx * "^^^^ values of the other frac- 
 tions are similar. And thus it appears how the fluent of 
 
 r' 
 
 — - — ""' '^. T, ... -, ^ is assignable by circular arks and 
 
 x"-"- — Ax"-"—' + Bx"'''—'' -&c. ^ -^ 
 
 logarithms when the denominator is the product of any qua- 
 dratic divisors. 
 
 781. If the values of L and / are to be expressed algebraical- 
 ly, then raise tr + 1 to the power of the exponent m, tlie differ- 
 
 Q 2 ence
 
 228 Of reducing Fluentu Book II, 
 
 cnce of n and r, by art. 748 ; multiply the dd, 4th, 6th, &c. 
 
 terms of this power by 1, aa — 1, aa — I ,aa — 1 , 8cc. re- 
 spectively ; and the sum of the products divided by Sa— 24 
 X 2^112^ X 2a— 2d X &c. wiU be equal to + L or — L, ac- 
 cording as n is greater or less than r. The other coefficient / 
 
 of the fraction - — is found bv multiplying tlie like terms 
 
 1 — 2ax-\-xx •' 1 ^ o 
 
 of the power ofa + 1 of the exponent n-r-\ or r — n + 1 
 (according as n — 1 is greater or less than r) by 1, aa — 1, 
 
 aa-X , aa-l , &c. respectively, dividing the sum of the pro- 
 ducts by 2<j— 23 X 2d— 2c X 2a— 2d X &,c. The quotient will be 
 equal to + /in the former case, but to — / in the latter. 
 The coefficients ]\I and m, N and n, Sec. are found in like man- 
 ner. But when n=.r, the coefficients L, M, N, &c. vanish j 
 and when w-lrrr, the other coefficients /, m, n, Sec. vanish. 
 782. When the fraction that is to be resolved in this manner 
 
 is of the form —^ where the denominator is a tri- 
 
 nomial, the coefficients of the fractions ■; — - — ; — . &c. into 
 
 ' 1 — ^ax+xxi 
 
 which it is to be resolved, may be more briefly determined from 
 art. 779, by which (substituting in this case o for t*) L rr 
 
 :— — — , , , : where, P beinar supposed equal to 
 
 Nx^Z/ Mx^-/' a rr 4 
 
 xx—2ax-^\ X Q, M is the value of Q when x =z e, and N its 
 value when x ■=. f. By taking the fluxions, as in that article, 
 
 2«— 1 . «— .1. 
 
 P = Q,nx X Q.nZX X = 2.v.V— 2av' X Q -|- Q X 
 
 ^x — 2ax-\-\, and substituting e for x, and M for Q, Q.nc^ — ' 
 — Inzfi-—^ rr 2e—2a X M =z 7^ X M ; consequently 7^ 
 X M =: 2;<e"— ' x e» — z. In the same manner J—^ x N = 
 enfn"* X /"— ;r. But e'» — 2ze" + 1 = o and e" + ^ (or
 
 Chap. III. to circular Arks and Logarithms. 229 
 
 e« + /«) = 2/, and e« — /« = 2e« — 2/ or 2/ — 2/« ; so that 
 ;i7" X M z= 7ie«— = X e"-^/'' and 7^ x N = «/«— * X e'*/". 
 Therefore L = — ; ' . -^ i^' ■ — 1 By 
 
 art. 779, / = .r "—- — ^t ^- = ~^ -^-^ 
 
 Mx^_/ N X/_e we«— 'X £»»-/' 
 
 g+A/aa— 1 a \/flfl— 1 Therefore, by comparing 
 
 a+V'aa— 1 « \/aa~l 
 
 the value of S in art. 76O, it appears that if B6 the cosine of 
 the ark AB be equal to a, AK =: n x AB, AZ =: ^T x AB, 
 
 Kk and Z2; be the cosines of AK and AZ : then / == - x 
 
 ' n 
 
 1 Oz 
 
 2f; and the fraction , ^~^. ^ iZZpllf , which co- 
 
 O* ^ 1 — 2ax-\-xx l—2ax+xx ' 
 
 incides with Mr. De Moivre's fifth proposition, lib. 1. Miscei, 
 Analyt. as it is concisely expressed, p. 42, of that treatise. 
 
 783. When the fraction proposed is of this form 
 
 jr 
 
 l,-Qlx"Jrx^'^ 
 
 r being any integer positive number less than 2/?, let the differ- 
 ence of 71 and r be represented by ?w, as above ; and L == (art. 
 
 780) •; — -~ — (because Ie«— * = M =r 7<e«— * X — ^ 
 
 I TO. (-Tp 
 
 by what was proved in the last article) - x — -■• • There- 
 
 fore if the ark AQ be to AB as m to 1, and Q^ be the cosine 
 
 Q3 of
 
 ^30 Of reducing Fluents Book II. 
 
 of AQ, L = ^; and I = ^xl^ ov _j_^=_ 
 
 according as 7i is greater or less than r ; that is, / =: - X 
 
 7IJ — I rm—i 1 ^m + j — /"'-i-', . 
 
 r~^i ^^ n X ;r-ii > O^^ supposing AZ = m^l 
 
 e J e —f 
 
 X AB, and Zz to he the cosine of AZ, / = — ^, There- 
 
 fore the fraction -i±^ = ^-^^^I__ , When 
 
 y = M, or to w — 1, the numerators of those fractions consist of 
 one term only. 
 
 784. It appears, as in art. 773, that the fraction 
 
 \.^^ i,xJr qx^ 4- '-x^ ■'- •'^''^- . •, K L M 
 "1, — ;7~; r^ — ;: is equal to + — 7 + 
 
 N 
 H J + &c. If a, h, c, d, the roots of the equation a:" 
 
 ■=— At'"~i + Bj;'*— * — tScc. r= o be all unequal, the index of x 
 in the numerator 1 + px + qx^ + &c. be less than n, and we 
 suppose the coefficients K, L, M, N, &c. respectively equal to 
 the quantities that result when we substitute successively a, by 
 c/d, &c. for X in h^±^±l^^ ^'+^o 
 
 This theorem serves for reducing briefly the fluent of 
 ' '^ '^' X ;^ to los;arithms or circular arks. 
 
 785. Suppose now that some of the factors of the denomina- 
 tor of the fraction, by which the ordinate is expressed, are 
 equal to each other ; and let it be required to find the fluent or 
 aica. For example, let it be required to find the fluent of 
 
 , ' __,, _„ X ^ , Suppose ^ _1„ ■— =: 
 
 x+a X.v-J-5 x^a Xjf-1-^ 
 
 H
 
 dhap. III. to circular Arks and Logarithms, 23 1 
 
 ji ^ L hi 
 
 — -i^OT 4- m — 1 4- =~:ot — 2 ... J- =« a. "■ -ft-.1 _i 
 
 / 
 
 =~^«~2 + &;c. where H, K, L, h, k, I, &c. are the in- 
 variable coefficients that are to be determined, and the fractions 
 of each sort are to be continued till their number be equal to 
 m and u respectively. By reducing the equation to a common 
 denominator, H x 7+i'' 4- K x ~a x 7fl» + L x "^^^ 
 X 7+J" . . . + A X T+;«» -i- k K r+7 X Tj^a"^ + / X 
 *+ A* X J+;'» + &c. =1 + px -h qx"- + rx^ + &c. By 
 supposing X + a — o, oy x =: — a, all the terms of this equa- 
 tion that involve x-{-a vanish, and we find H X r^a^=:l — ■pa 
 
 ^ qa^ — ra' + &c. orH = — 1=~ . By taking 
 
 b — a 
 
 the fluxion of the equation, dividing each term by x, and 
 then supposing x z=. — a, we have riYi X JTr^"—* + K x iTZ^^ 
 zzp — iqa^sra"- — Sec. By taking the fluxions again, divid- 
 mg by X, and supposing x zz — a, we find n x ^i X H X 
 Xr^n-2 ^ o^K X 33^«— • + 2L X JZr^n = 2^ — Qra J^ 
 &c. Thus the values of H, K, L, &,c. are easily computed 5 
 and by proceeding in like manner, and supposing xJ^^b =0, the 
 values of A, k, I, See. are determined. If the fraction proposed 
 be --— — , that is, if p, q, r, 8cc. be supposed to vanish, 
 
 then H = -i-, K =-=!.--, L = « x 2±L x -i---, * 
 
 n n-\-\ 2 w-f-1 
 
 rr-i— , A: = ~" ■., / = m X « + 1 X J --,&c.which 
 
 ' — « w + i — 2~~ "^^ 
 
 « — 3 a — b a — b 
 
 coincides with Mr. Ldbnitz's theorem, Act. Lips. 1703. If 
 
 Q 4 th«
 
 2.3a Of reducing Fluents Book il. 
 
 the fjactioii proposed be ! , and we sup- 
 
 poseitequalto 4- — ^ — 4* — ^^+ See. it will appear in 
 
 — -m m-\ m-2 ■•■ -^ 
 
 x-\-a x-\-a x-\-a 
 
 the same manner, that H — , K z= z::!!!^ — i — — 
 
 n s b — a c-^a,' 
 
 b — a Y^c — a 
 
 
 
 1 — -.^ 
 
 
 
 
 
 
 ■ - - 
 
 
 
 • ^- 
 
 oC, 
 
 rr 
 
 771 X m- 
 
 -1 
 
 + 
 
 
 i'ms 
 
 
 + 
 
 sy^s- 
 
 -1 
 
 
 2 
 
 
 
 
 — . 
 
 _2 
 
 
 
 b—a 
 
 
 
 i— <a 
 
 Xc- 
 
 -a 
 
 
 c — a 
 
 
 X 
 
 K, 
 
 &C. 
 
 
 
 
 
 
 
 
 
 5 ^ H + ^ 
 
 c — a 
 
 780. Ihe ordmate or traction ■ ;„ ^ — _, bemg 
 
 *-{-a X x + b 
 
 resolved in this manner, the fluent will be found (by art. 
 737) equal to _"!_„,_ , -^=L___^. . . _I±__j 
 
 711 — 1 X * + a m — 2 X *• + <z n — 1 x *• + A 
 
 — , &c. If the last coefficient of each sort vanish, 
 
 »— 2 X *• + -^ 
 
 the fluent will be assignable in algebraic terms : in other cases, 
 the fluent is assigned by logarithms with algebraic quantities. 
 
 787. If we suppose the fraction - ^^ + g ^ ~r — 
 
 X '{• a X X •{■ b 
 
 xJfb '" 
 
 X ■\- a 
 
 &fter Mr, De Moivre's method, Miscel. Analyt. p. 59 ; then 
 K X 7^'" 4- Ab ^ A4-B6 X a: 4- B-i-C6 xa- 4-C4-D6 
 X a:^ = 1 4* p>r 4* ^'^'^ 4^ Sec. By supposing x 4« * = o, or 
 X — — Z», K X "7^"* = 1 — p6 4* (jb'' — See. The supposi- 
 tion of X =: gives Ka™ 1^ Ab ■=. \, By taking the fluxions 
 of the equation, dividing by x, and supposing x •=:. 0, wKa"»— '
 
 Chap. III. to others of a more simple Form. 233 
 
 »i* A'i'Bb—p; and by proceeding in this manner the coeffi- 
 cients K, A, B, C, &c. may be deteraiined. If the fraction 
 
 be ^^ ,the coefficient of any term in the numerator of the 
 
 second fraction, as of F^''^ is found by raising a — b to the power 
 of the exponent 7n, rejecting as many of the terms of thispower 
 a""^, ma'"'—^ b, Sec. us there are units m r ^ 1 ; and dividing 
 
 the sum of these that remain by 6 X a — b ; for the quo- 
 tient will give •{" i*^ or — F, according as r is an even or odd 
 
 number. The fraction ^;^^ is resolved in like 
 
 i-ax X 1 — 1>^ 
 
 manner by a similar rule, and supposing it equal to 
 
 A + B.r + Co:^ . . . G i'«-' ,. b"^ . b^ 
 + ; K = , A =: 1 — 
 
 m m n 
 
 I — ax b — a I— -a 
 
 B == 6A 4* , C =: bB . &c. 
 
 b — a 2x3—^ 
 
 788. The fluxion "■ — -r- is transformed into . 
 
 e+/a" e'^fz''"- 
 
 by supposing r =■ z; because a; "^ -rz z , x'^xrzrz r 
 
 and ar'« =: z''*. In like manner the fluxion is transformed, so 
 as to become rational, when the denominator is a trinomial - 
 and the fluent ma}^ be found by the preceding articles. 
 
 789- Sh- Isaac Nezoton has given some excellent theorems 
 for reducing fluents to others of a more simple form, in the 7th, 
 8th, and llth prdjp. of his treatise DeQuadrat.Curvar. Let R 
 n= e +yx« + gx''"- + hx'^^'- + &c. and, consequently, r — 
 n/x^-^'x + 2wgr^-«— i!r + S/i/i*^'*— '* + &c. Let a =:
 
 234 Of reducing Fluents Book II. 
 
 j,m— I ;. Yyi^ B* — \x^, c = 'bx\ d = cx^, &c. and In + n 
 = p. Then ?weA + ff^ x /B + 2^ « x gC + 3^+;;r 
 X /«D + &c. =: x''^ R'+'. This theorem will appear by tak- 
 ing tlie fluxion of a:'" IV-'r', which (by art. 725) is a"*-' R' x 
 mRx -h Tfi XXR —x^^—'x R' X me + mfx" + tngx'-n + &c. 
 i4« 7'+T X ?//'r« X 7+1 X 2/<^'.r'''i + &c. — mek + p-J-w X 
 fh'i'^p-\-m X gc +, 3p 4- wi X ho + See. Let the num- 
 ber of terms in e + /i" + gx-" + &c. the value of R be re- 
 presented by q ; and if as many of the successive areas A, B, C, 
 D, &c. be known as there are units in q — 1^ the rest can be 
 computed from these, by this theorem. Thus if R be a bino- 
 nomial e + fx^'} any one of the areas. A, B, C, D, &c. being 
 given, the rest may be computed from it ; and when R is a tri- 
 nomial e + fx"- 4- gx^^, any two of those areas, as A and B, 
 are sufficient for determining the rest. 
 
 790. Let H = .r"*-'; R'+» r= a"*-'; R« x e^fx'' '^ 
 
 g x"-" »J- &c. — ek 4*/b 'i' g'c ^ hb 4" ^C'* and it fol- 
 lows that H = eA >^fB 4* g^ 4" hD 4* &c. Hence it ap- 
 pears that m and /being any numbers whatsoever, if /• and s be 
 any integer numbers, and as many of the areas A, B, C, t), &c. 
 be known as there are units in q — 1, the fluent of x'n+^«— * '^, 
 
 X e 'i'fx'^ "^^ may be computed from them. 
 
 791. Li like manner it appears, that if R rr e ^fx'^ 4- gx^^ 
 + Sec. S =: E -I- Fa'i + Gx->^ + &c. a = a"*-*; R' S^, 
 B :=: A X", c =: Br", &c. and In + n = p, kn *i> n — q, then 
 i^'R^^* S^ + ^=«zeEA+ meV + mjE +p/E + ^eF X 
 B + yncG + mgl^-\rmfY + p/F + qfl^ + 2pgE + IqeG x C 
 
 792. From
 
 Ghap. IIL to others of a more simple Form. 235 
 
 792. From these, particular theorems are easily cletluced that 
 may be of use in the resolution of problems. Let R be the bi- 
 nomial e—fx'^; and, as in art. 789, let a —x'^—^x R^ b =a jt", 
 c = B a", D = 'cx», &c. ; and M =.In +« + m. Let m and 
 / + 1 be any positive numbers whatsoever, that x^ R^+' may- 
 vanish either when x zr 0, or e — y^" = ; and let r be any 
 integer positive number ; then if A represent the fluent of a,"*— • 
 
 X X e — fx^ that is generated while x flows, and from being o 
 becomesequaltoTl'',thefluentofx'"+'"«— '.^ x.e—fp' generated 
 m the same time will be to A X — as *tt X • x — ^^— - 
 
 r ^ M-|-» M+2« 
 
 X ^t£ ^ ^^- *^ ^ ' wliere the fractions ^^ 8cc. are to be 
 continued till their number be equal to r. For in the present 
 case weA— M/B=:o, by art. 789, and B= ^ X ^, C= -"+.!! 
 
 ^ / — M ^ M+« ^ Jf ^ - M+2« ^ y-M ^ M+; 
 
 X Sr2« ^ /r^ ^"^ so on. The same theorem serves whea 
 A = x"^'x X yi" — ■?. 
 
 793. For example, let A = ~~^' '"^^^ consequently A 
 equal to the fourth part of the circumference of the circle 
 whose radius is 1, and the fluent of ■ — - generated while 
 
 V 1 — >xx 
 
 X flows, and from being becomes equal to 1, will be to A as 
 2x4xt>x8X&c'. ^^ ^ ' 'fi\^QT:& the fractions are to be conti- 
 nued till their number be equal to r. Because in this case a 
 Qixm~^x X e—fxrt = x^—'x X l—xx"^, SO that ;» = 1,
 
 236 Of reducing Flumts Book II, 
 
 / — — 1, n = 2, M = 2, e = 1, and/= 1, The fluxion 
 
 jstransfonued into 7x;-M' supposing j:= — =r, and — =^= 
 
 ^T" V 1 + zz Vl — jfx 
 
 V'l— ;rj» 
 
 x~' X 
 
 ■XX 
 
 ,2r 
 
 into ^ ■ { ,• the fluent of which is, therefore, to A the 
 
 1 +Z2; 
 
 fluent of \ v.Zi (P^ ^^^ quadrantal ark of the circle of the ra- 
 dius 1) as ^ ^ 4 ^ 6 ^ 8 ^ ^^- *° ^- If ^^^ suppose a = 
 X Vl—xx, in wnich case A is the fourth part of the area of the 
 circle whose radius is 1, then the fluent of x'^^'x '/x-^xx will be 
 
 *« A ^' 1x6x8xIq ^ ^"- '•^ '• ^^ ^Wosing ^ = ;;=^, 
 the corresponding fluent of — ^— ^ — - will be to the fluent of 
 
 f -, in the same ratio. In Kke manner other theorems of 
 
 this kind may be deduced from those in art. 789, &c. 
 
 VT—Z 
 
 794. The fluxion i-'"—':r X e+/ar'* being transformed, as 
 in art. 742, by supposing e. + /i« ~ 2, the fluent will be 
 
 measured by the areas of conic sections when - is any in- 
 teger number positive or negative, by art. 789, When — \- I 
 is any integer number, the same will appear by suppos- 
 ing 2 •=. — — , Or if / be equal to the fraction -- we 
 
 ^ e4- fxi^ 
 
 may suppose 2 =: e +y x" « in the former case, andz; =: - ^ | 
 
 in the latter. The fluxion xX^"—* x '^+■^^" 1 is transformed, 
 
 by
 
 Chap. III. to others of a more simple Form. 237 
 by supposing — ^^ = z (and consequently a" = \ ^, 
 
 nx 
 
 _^^^^-e/. .. ^/l^ 
 
 and - = =:^_== X 2) into ^L-11 xz^^ x ^=|2 . 
 
 By supposing z =z gx" + ^ f, the fluxion ar"»— ' ;. X 
 e+/x«+g.r"« ^ is transformed into- x . ^~/ _'"" 
 
 - ^"1 J and the fluent may be found in both those 
 cases by the preceding articles^ when r is any integer number. 
 
 It we suppose 7/ = ^ S_ and transform 
 
 the last fluxion from x to ?/, its expression will become rational 
 as is shown, Miscel. Analyt. p. Q5. When any of those fluxions 
 is multiplied or divided by a rational binomial E + F^, or tri- 
 nomial E + Fa" -r Gx^'^, or by any quantity that can be re- 
 solved into such binomial or trinomial factors, the fluent may 
 be measured by the areas of conic sections (that is, either by 
 algebraic quantities, or by circular arks, or logarithms, or these 
 compounded together), by the preceding articles. 
 
 795. When a fluxion is proposed that involves an irrational 
 quantity, the fluent is sometimes obtained in finite terms, or 
 compared with a circular ark or logarithm, by supposing the 
 quantity that is under the radical sign equal to a new flowino- 
 
 quantity. Thus if q =3 - — X ^+^ 
 
 E + Yx y. e-iffx eJffx 
 
 -J and we 
 
 suppose z - ^±1^ then - = ^1 Y_^ x * and 
 
 '—1 
 
 Q — Ye—zf ' ^consequently the fluent Q — 
 
 X 
 
 m X Ve—Ef e-i-Jx 
 
 m X Fc— E/ 
 
 E+F.v^ 
 
 But this is often more easily obtain- 
 
 ed
 
 238 Of reducing Flvcnts Book 11. 
 
 ed by transfonning the fluxion from the sine or cosine of an 
 ark to the tangent or secant, or to the sum or difference of the 
 secant and tangent, or by the converse operations. If we sup- 
 pose X = — ' ■ (2 being the tangent that corresponds to the 
 
 smex), then — "^ =r — - — , Hence if d = 
 
 VI-—XX 1+22 X^v/l— XX 
 
 then d = ^, and Q r= - = — ^, And if q = 
 
 j___fff____ _ .JgL- then Q is equal to the 
 
 aa-^-xxV^l — XX (ICL -\- aa — \ y. ZZ^ 
 
 ark of a circle described with the radius -—:==== that has its 
 
 tangent equal to z ox — = If we suppose a: + Vxx-^-X 
 v' 1 — ■«•*• 
 
 __ . 1 X K T f (I -^ V aa + XX __ 
 
 = z, then — ■ = -. It we suppose — ^ = — — ^t 
 
 -/x^+l z '" A' 
 
 then — ^"^ ~ -; so that the fluent is the logarithm of 
 
 X V aa^^rxx ^ 
 
 . f + ^ ee + ffx'^^ 
 Zj the modulus being 1. And by supposing ;^-^-^ rr z, 
 
 , is transformed into —-; so that the fluent is 
 
 ■— X loff. z, the morfM/ws beina: unit. 
 
 796. Supposing, as in art. 789, R = f+/i", A = x*"— 1 
 X W, and B = A3:" we found meA + M/B — a"^ R^+1 ; 
 from which it follows, that if neither »i=o, nor M=o, A and 
 B depend mutually upon each other ; but if ?;^ =0, B is as- 
 signable in finite algebraic terms; and if M r= 0, A is assignable 
 
 in such terms. If neither^ nor - be equal to o, or to an in- 
 teger number, the fluents of all the fluxions in the series 
 
 A X^"*
 
 Chap. III. to others of a more simple Form, 239 
 
 A:r^", 'Ax'\ a, Ax-'S A:c-'«, &c. (wliicli may be conti- 
 nued either way) depend upon the fluent of any one fluxion 
 
 in the series ; but when either - or - is an intecrer, or when 
 
 n n o •' 
 
 either of them vanishes, this cannot be said of the whole series. 
 Let A r= ^ "" -, where M = O;, m = — 1, and A =: 
 
 X V oc — XX 
 
 '"'"'"' but the fluent of Aa'' (= --^^L=) is the circular 
 
 ax ' V aa — XX' 
 
 ark whose sine is x, the radius being a : the fluents of 
 ^2'—% Ax—*, &c. depend upon the former, and are assignable 
 in finite algebraic terms ; but the fluents of Ax*, Ax''', &c. de- 
 pend upon the latter, and are assignable by that circular aric 
 
 with algebraic quantities. If A = —^^^r::;: m zz o, M= 1, 
 
 V aa — xx^ 
 
 A = V7a—xx, and the fluents of 'ax^, ax*, &c. are assignable 
 by algebraic quantities ; but the fluent of A.r-- (rr ~^ - ^ 
 
 X\^ aa — XX J 
 
 is the logarithm of — — ~J the modulus being unit, and 
 
 the fluents of Ax —*, Aa— -^ &c. depend upon this logarithm. 
 In like manner, if A = ■ - — , A is the logarithm of a; + 
 
 V'*^— I, and the fluents ofAx^Ax'^, &c. depend upon it; 
 but the fluents of Ar— % Aa— ■% &c. are assignable in finite 
 
 algebraic terms. If A = — — — , A = V xx-^aa, and the flu- 
 
 Vxx—\ 
 
 ents of A-r% "ax*. Sec. are assignable in finite algebraic terms ; 
 but the fluent of Act— * (=: "N is the ark whose secant 
 
 XVxx—\ ) 
 
 is x, the radius being unit, and the fluents of Aa:—*, Ax"'^, &c. 
 
 depend
 
 240 Of redtiting Fluents Book If. 
 
 depend upon it. If a = -- .— , and m be a fraction, the 
 
 fluents of all the fluxions in the series Ar+'^j A>r+*, kx+^, 
 &c. depend upon A. 
 
 797. Let R -fx^^ — e/k - a:'«-';R«, b — kx~'\ c = 
 B.TT— " = kx—^^, D = c.r— " = kx—^'\ &c. and M = //z + w + 
 
 m, as formerly ; then whenyi" = e, B =: —^ x — , C = 
 
 — — - x*^ — , D = — 5- X -^ &c. Ihereforer beino-anyin- 
 
 teger positive number, if q == kx—^^, Q : — : : ~" x 
 
 ■ ? X "1- X &c. (where these fractions are to be con- 
 tinued till their number be equal to r) : 1 . For example, let 
 
 A = 
 
 Q = — rr7-T-=^J ^'^^" Q : A : : i X I X 
 
 xVxx-\^ x''+W 
 
 - X - X &c. : 1 . these fluents are generated while - from beine: 
 
 6 8 ° X O 
 
 becomes equal to 1 . 
 
 798. After the fluents that can be accurately assigned in finite 
 terms by common algebraic expressions, and those which can be 
 reduced to circular arks and logarithms, the fluents that deserve 
 the next place are such as are assigned by hyperbolic and eUiptic 
 arks; which with the former are all comprehended under these 
 which are measured by the lines that bound the conic sections 
 (the triangle and circle being figures of this kind), as the first 
 two are measured by the areas of conic sections. The fluent of 
 
 is of the first class ; that of — -= - — -- or of -— ^^ — 
 
 -v/l + v '^x X a/I-I-a; \/l-\.xx 
 
 is of the second ; but the fluents of "* ^ ..? — :; — > 
 
 V l^xx Vx X V iJ^xx 
 
 ■r^— 1< and =^1t are of the third class, and (as far as has 
 
 ap-
 
 Chap. III. to others of a more simple Form. 241 
 
 appeared hitherto) cannot be reduced to the former. The 
 fluents of this class are sometimes required in the resolution of 
 useful problems, and our design obhges us to give some account 
 of them likewise. 
 
 799. Let AEH(^g.305)b6anequilateral hyperbola, thathasits 
 centre in S and vertex in A, AD a right line perpendicular to SA, 
 suppose SA=1, SN = :r, and let a circle described with the 
 radius SN from the centre S meet AD in M, let SE bisect the 
 angle ASM, and meet the hyperbola in E ; then the hyperbolic 
 
 ark AE shall be equal to the fluent of — ^ ^^ » For let the 
 
 ^ QV XX— 1 
 
 ark AE =: s, SEr= r, and SP be perpendicular on EP the tan- 
 gent of the hyperbola in P5 then the triangles SMA and SEP 
 will be similar, by art. 181, and ; : )■ : : SE : EP : : SM - 
 AM :: X : '^l^^x ; but SA, SE, and SM, are in continued pro- 
 portion, or r == v';^ so that r : X : : 1 : ^V x', conse- 
 quently i = ■■ • * "^ : and supposing the fluent of -.■ — : 
 
 ^V XX — 1 w XX — 1 
 
 to begin to be generated when a:r:i, and thereafter to in- 
 crease while :r increases, it will be always equal to 2iAE. If 
 Am be perpendicular to SM in m, and we now suppose Sw=:.r, 
 
 then the hyperbolic ark AE will be the fluentof — ■•• i^ : — /. t-. 
 
 (as will appear by substituting in the former fluxion ar^' for ir); 
 and EP — AE the excess of the tangent above the hyperbolic 
 
 ark AE will be the fluent of -=^^^; because EP will then 
 be equal to ' — xj and its fluxion to —- 
 
 800. Let AB be perpendicular from A the vertex of the hy- 
 perbola to the asymptote SB in B. Suppose now SB r: 1, up- 
 on BA take BLrrx, join SL, and let it meet the hyperbola in 
 E; from the centre S describe the ark AQ, intersecting SE in 
 Q; and the hyperbolic ark AE shall be equal to the fluent 
 
 VOL. H. R of
 
 2-12 Of reducing Fluents Book II. 
 
 of S^ — ~^5 because if A^, LZ, and EK, be perpendicular 
 
 to the other asymptote in b, Z, and K, respectively, S6 . SK : : 
 EK : A6(= LZ) : : SK : SZ, SZ =:BL = x, SK = ^/s6l^£ 
 rrv'r, SE*=SK* + EK"=: j: + ;; ; and the fluxion of AE be- 
 ing to the fluxion of SK as SE to SK^ it is therefore equal to 
 
 if- X Vr+^ or — ~^» The fluxion of SE or of QE is 
 
 2x X 2x\/x 
 
 r — ^."^ ^. r^by adding which to the fluxion of AE, it appears 
 
 QX'/xX\/xx+l •' ° y Lf 
 
 that AE 4- EQ is the fluent of — =~r which begins to be 
 
 '/l+xx 
 
 generated when a: = 1 (or when BL — BA), and thereafter in- 
 creases while X increases. In the same manner AE — EQ is the 
 
 fluent of "" " "" that begins to be generated when x zz 1, and 
 
 \/l+xx 
 
 thereafter increases while x decreases. 
 
 801. Suppose SA = 1, AM = x, and £AE will be the 
 
 fluent of ===r *^^^ vanishes with x ; as appears by substitut- 
 
 l + ATXli 
 
 ing in the first value of '$, in art. 799> -/i+T^in the place 
 ofT. Suppose SA = 1, Am—Xy and 2EP — sAE will be 
 
 the fluent of ===rj7 that begins to be generated when or = 1, 
 
 and thereafter increases whiles decreases. Ifwe suppose SB = 1, 
 
 SL = X, then AE + EQ will be the fluent of+rs^ 
 
 XX — lljT 
 
 that begins to be generated when xx = 2. 
 
 802. As for the fluent of -— ^=r— ^— === or of ==7, it 
 
 does not appear that it is possible to represent it by any hyper- 
 bolic arch and algebraic quantities. But by assuming an elliptic 
 ark, likewise, it may be assigned by the following construction. J 
 The rest remaining as in art. 799- Let an ellipse ARD be de- 
 scribed having its centre in S, SF the distance of the focus F 
 
 from
 
 Chap. III. to others of a more simple Form. 243 
 
 from the centre S equal to the shorter semi-axis SA, and con- 
 sequently the semi-transverse axisSD : SA : : -/"s": 1. Sup- 
 pose SA = 1, S»« = Xf take SX upon SA equal to SP (or to a 
 mean proportional betwixt SA and Sm), let the ordinate XR 
 
 meet the ellipse in R: and the fluent of _ ~^ . r::that 
 
 begins to be generated when x ■=. \, and thereafter Increases 
 while X decreases, will be equal to AR -\- AE — EP, the dif- 
 ference by which the sum of the elliptic and hyperbolic arks 
 AR and AE exceeds EP the tangent of the latter. -» For SX = 
 y^j and if RT the tangent of the ellipse at R meet SA in T, 
 
 ST = -i=, XT = ST— SX r= i=L, XR^ - o x T^^, 
 
 V x^ V x^ 
 
 RT* = XT^ + XR^ = ^:=^; and the fluxion of the el- 
 
 X ' 
 
 liptic ark AR will be to the fluxion of SX as RT to XT, that 
 is, as */ i—xx to 1 — .r, or as 1 -\- x to >/ \—xx \ consequently 
 
 (the fluxion of SX being — ^ ] the fluxion of the ark AR is 
 
 IVx J 
 
 -l4= v-l±-"— = -^ i^; and the fluent 
 
 of — "7^ (by the latter part of art. 799) equal to 
 
 -iiV X y^ V \'—xx 
 
 AR + AE — EP. If we suppose km = r, AR + AE — EP 
 
 will be the fluent of — ■ » as will appear by substitut- 
 
 ing Vx-'xx for x in the former fluxion. By supposing BL — z, 
 and SB = 1, the same diff"erence AR + AE — EP gives 
 
 the fluent of , "^ ■ -.. -.1 because if Sw — x, then x ~ 
 
 It is likewise the fluent of - — 3 ? if we sup- 
 
 pose SL=:z, and SB ;= 1, or of —-^=73, if we suppose AM 
 
 ~ z, and S A = 1 . 
 
 R 2 803. The
 
 244 Of reducing Fluents Book IL 
 
 803. The fluent of ,. . — , — . (which we found equal 
 
 'XX 
 
 to AR + AE — EP, art. 80'i) is equal to AP the ark of the 
 curve that is the locus of P (where the perpendiculars from the 
 centreSintersect the tangents of theequilateral hyperbola) which 
 is called the lemniscata. For (art. 212) the fluxion of the curve 
 AP is to the fluxion of SP as SE to EP, or as Sx^ to A?n, that 
 is (supposing SA ~ 1^ and Sm zn x), as 1 to -v/TZ^; but SP : 
 
 SA : : SA : SE^ and SE = — r=5 consequently SP zz VT^ the 
 
 -/ X 
 
 fluxion of SP is r= and the fluxion ofAPrr 
 
 ^2\/ X '2V X X Vl^xx 
 
 IfFbethefocusofthehyperbola, FH (Jig. SOG) perpendicular to 
 the tangent EP in H, then it is known th at H will be always found 
 in a circle described from the centre S with the radius SA ; 
 and if FH produced meet this circle in h, SP will be equal 
 to \ ah. From which it appears^ that the lemniscata may be 
 constructed in the following easy manner. Bisect SF (Jig. 307) 
 inf, from the centrey describe a circle with a radius equal to 
 I SA, let any right line SX meet this circle in X and x, set oft" 
 SP from 8 on the same right line always equal to the chord 'Kjx, 
 and the point P shall be in the lemniscata : and the fluents that 
 were described in art. 802 may either be represented by AR + 
 AE — EP, or by the ark AP of this curve, which is so easily con- 
 structed. 
 
 804. Let AEH be any other hyperbola, SA the semi-trans- 
 verse and SDthe semi-conjugate axis, SA = a, SD =r 3, e =: 
 
 ^^, SE= r, SP = p, AE - s, and x =z ^; then the hyper- 
 bolic ark AE shall be equal to the flnpnh nf /^ ^'^ * 
 
 ^ Q\^xx + 2^x—6I> 
 
 For let SH be the seraidiameter conjugate to SE,then SE' — 
 SH* - aa—hb- 2ea, or SH^ sirr — oea ; and SH x SP 
 
 SA X SB =: abi and rr — 2ea zz. -^- zr ax. But 
 
 i'P 
 
 SE;
 
 Vla.telQOC/ra.244yo^il
 
 Tiate-jaac/rm^Vo/n. 
 
 H
 
 Chap. III. to others of a more simple Form, 245 
 
 SE : EP : : r: '/77:^p, and ; - "'' Ic/ ^a _ 
 
 */ rr-~pp 'i'/ xx-\- 2ex — bb 
 
 Because EP = Vrr—pp ~ V"-^ — ^^ — ^ the fluxion of 
 EP is ^=5 X —~=~=z^ and EP — AE (the excess of 
 
 2x V X V xx-\-2ex--bb 
 
 the tangent above the hyperbolic ark) is the fluent of 
 bb X \/T , . f>P bb\ ^ 
 
 -i^v-j. ^/;;Ti;^ ' °' (^..pposmg z = - = -; of 
 
 — •-« a/ ax. 
 
 .,, „ — * It appears likewise that the ark AE is the 
 
 y bb+2e-i—zii ^^ 
 
 fluent of ~''^'^ -, and that EP— AE is the fluent of 
 
 ■pp V a- b^ + laepp—p"-^ 
 
 ^J ^ ==» Inlikemanner it appears that if AEB(;^g. 309) 
 be an ellipsis, S the centre,SA=«, SB=6, aa-\-bb—0,e.a, SPbe 
 perpendicular on the tangent EP in P, and SP = j9, a? = ~, 
 
 then the ark AE will be the fluent of - " " ^ "" - , or of 
 
 V 2ex — xx~^bb 
 
 " ' . . that beeins to be srenerated when p z=: a. 
 
 pp 
 
 805. In order to represent the fluent of - 
 
 or of ■ . . . , " , we must have recourse to both the hyper- 
 
 -y a^b'-Z\-'-2aep^—p'^^ •'^ 
 
 bolic and elliptic arks. The rest remaining as in the last article, 
 join AD, and let AF Qig. 308) perpendicular to AD meet DS 
 produced in F, describe an ellipse AR6 that has its focus in 
 F, centre in S, and SA for the second semiaxis ; upon SA take 
 SQ equal to SP, let QR the ordinate at Q meet the ellipse in R, 
 
 and -X AR + AE— EPshallbethefluentof ■ ^^- 
 
 '/aa-~ppX Vbii-pp 
 
 R3 or
 
 246 Of reducing Fluents Book 11, 
 
 or (supposing pp zz uz, and Q.ta — bb — aa, as above) of 
 
 o^^lT^f^^^^l' ■^°' '^ ^'^ ^^'^ ^^'"Sent of the ellipse at 
 R meet SA in T, AR =/ SQ (= SP) = p, and SF - k, then 
 
 ST zz^.,qt: - ".^-ill^ RT zr ^15^^ v/««4. il|:, . 
 
 — ^ : : RT : QT, and/ i= ^ x "!jff' _ 
 
 (because /iZ»=:<7rt,by the supposition)^^ X • — — i>b+ pp ^^ 
 
 '> Vaa—pp X Vbb-^pp* 
 
 But the fiuxion of EP — AE was found (art. 802) equal to 
 -r^^ 7=-.. Therefore i x AR + AE— EP is the 
 
 Vaa—pp X VbbJfpp "■ 
 
 fluent of ■ _-Z.' ^ — ■ or of "", \ that 
 
 Vaa—ppy^Vbb\pp^ ^V a% ^ V bb—'i.ez—>.% 
 
 begins to be generated when p and z are equal to a ; and that the 
 fluent is flniie which is generated while x decreases till it va- 
 nish, appears from art. 327. 
 
 806. The values of those fluents, as of . Cthe 
 
 M V aa — pp ^ 
 
 fluxion of the elliptic ark BR), may be computed either by re- 
 solvmg ^ aa—sA ^^^**^ ^ scncs, muUiplymg each tei-ni by 
 t. , and findino^ the fluents of the several products by art. 737, 
 which will form a series of algebraic quantities. Or we may 
 compare it with the fluent of- — _ (the circular ark of the 
 
 V aa — pp 
 
 radius a and sinep^, by resolving V'J^^f- only into a series, by 
 art. 793. Thus / = -^ x 1 + ^^ ~ ^ + &c.; 
 
 V aa — pp 
 
 consequently the elliptic quadrant ABB is to the quadrant of 
 the circle of the radius a as 1 + _ _ ^^ + jr-^ — &c. 
 
 to
 
 Chap. III. to others of amort simple Form. 24f 
 
 to 1. The same elliptic quadrant is to the quadrant of the 
 circle of a radius equal to SB the semi-transverse axis, suppos- 
 ingSB = fl, asl— i;^ — ^5^ — -2567— .&c. tol. 
 
 807. Let ^y - Q^ ^4=r = A, then a 
 
 V aa—pp X V bb-\-pp Vaa^pp 
 
 - a/^TIT;^ '^^' "" « ^^ 2W+8M 166" + ^c. 
 
 and, by art. 793, the fluent Q that is generated betwixt the 
 terms, when p zz o and ^ r= cr, is (N being supposed to repre- 
 sent the ratio of the semi-circumference of a circle to its dia- 
 meter) K6 X 1— £^ -I- ^ — ^ -I- &c. where the nu- 
 merical coefficients 1, 1,1^, &,c. are the squares of the several 
 uncia of a binomial raised to the power of the exponent — |. If 
 
 we suppose x — a — ^, and E r: a -i , then q s 
 
 bb X , - 
 
 -7=X-— -== X 1 — -r 
 
 5 or - X ■ ^ XI 
 
 "" ^\^Ex—xx o\ 
 
 Let A first denote the ark of a circle described upon the dia- 
 meter fl, whose versed sine is equal to x, and A = — : 
 
 2 V oa; — XX 
 
 = I axx X a — X ; whence m — 1 =: — |, w = |, w=: 1, 
 / = — h € = a,f — 1, M =bi -^ n + 171 = 1 ; and, by art. 
 792, when r is any integer positive number, the fluent of 
 
 r 
 
 ' A X ^ , that is generated while x increases from o till it be- 
 
 come equal totf, isAx4-X|-x|-x4-X &c. these fractions 
 being continued till their number be equal to r, and A being sup- 
 posed now to represent the semi-circumference on the diameter 
 
 Ibk. X 5xx bx"^ _ 
 
 a. Therefore, smce G = ^^^7=^1 + .2e + 8EE+T6eI+ ^^• 
 
 it follows, that Q = ^^ >^ ^ + lE + 64E^ + 256E3 + ^^^ 
 
 R4 In
 
 248 Of reducing Tlucnts Book 11. 
 
 In like manner, q z:— x — ~- x 1 — -l"^. Whence Q 
 
 may be corapfircci with an ark of a circle upon the diameter E 
 that has its versed sine equal to .r. 
 
 808.Letd - , ~^'*^ , and A = '~"^ then d 
 
 Vaa—pp X V hb-\-pp y aa — /./»' 
 
 = ^- X > + W' = T. ^f-W^%- &c. Thcr.- 
 fore the fluent Q generated betwixt the terms when p =: oand 
 
 — • ^'"^ I ~^- ' ~^^ 175fl6 „ 
 
 jp _ «, IS -T- X 2 i5Z,i + 128M ~ " 8 X 256^6 + ^^' 
 
 This fluent is the ultimate excess of the tangent EP above the 
 hyperbolic ark AE, that is, the limit of this excess while the 
 figure is produced, or (according to the usual manner of expres- 
 sion) the excess of the asymptote above the curve AE, when both 
 are supposed to be infinitely produced. By supposing aa — fp 
 
 z=. ax Q. zz — — - — 3 and the same fluentwillbefound 
 
 •* '2.x V £ — X 
 
 (by art. 792) equal to _x^— + 23^ + 4^^ + 
 
 ^ ■ ' " -. + Sec. where A denotes in the usual manner 
 
 6x8E ^ SxlOE 
 
 the first term — X ^ ~, -^ the second term ^, C the 
 
 third term, and so on. 
 
 8O9. It follows, from what was shown above, art. 792, 
 799, &c. that when r is any integer number, the fluent of 
 
 is assignable by the arks of conic sections; that is, by 
 
 right lines, when r is equal to 4, or to any multiple of 4 ; by cir- 
 cular and parabolic arks (which may be reduced to loga-- 
 rithms) with right lines, when r is any other even number ; by 
 arks of an equilateral hyperbola with right lines, when r is any 
 munber oft^e series .3, 1, 11, 15, &c. ; and by arks of the 
 
 same
 
 Chap. III. to others of a more simple Form. 24f) 
 
 same hyperbola and right lines^, with arks of an elhpsis that has 
 ks excentricity equal to the second axis, when r is any of the 
 numbers 1, 5, 9, 13, 8cc. For if we suppose z« = xx, the pro- 
 
 r 
 
 posed fluxion will be transformed into - — ■> when r = 3, 
 2 — I =l> and the fluent is found by art. 799 or 800 ; but when 
 
 *' = 1^2" — ^ — — T-> ^"^ t^>^ fluent is found by art. 802. 
 810. \jeXn (Jig.S \0) beany fraction whatsoever,and thefluent 
 
 <^* oi' .. -.—r . be required. For this end let AL be one 
 
 of the figures constructed in art. 392, where the point S, and 
 right line AE, were supposed to be given in position, SA was 
 perpendicular to AE in A, M any point upon AE ; and the ra- 
 tio of the angle ASL to ASM, and that of the logarithm of 
 the ray SL to the logarithm of SM, was always the same inva- 
 luable ratio of 7i to 1 ; that is, ASL : ASM : : n : 1, and SL to 
 SA asSM +« to SA+ «. LetSA — 1,SM = .t,SL = /•,andthe 
 ark AL n s ; consequently r rr a?+'», V = + wa;+"— 'x,and (by 
 art. 392) 's : + r : : SM : AM : : x : VT^^, or / = 
 
 nxTK rp, f an . 0^+"^ . 1 I 
 
 -- , . 1 hererore the fluent or ^ is -- X s — r x 
 
 Vxx—\ V xx—l " 
 
 AL. If Am be perpendicular to SM in m, then Sm == 
 - ; consequently, if we suppose Sm =: z, then the fluent of 
 
 ■ — will be equal to - X AL. By supposing z =: 
 
 V'l— yy and u — O — 2A-, =—=7; is transformed into - , 
 
 1— jy^l vi— s» 
 
 nm 
 
 1 w ^ ' 
 
 and the fluent is - x AL. By supposing y'^—.z^, — — ■ — 
 
 2c"--'2" 9 
 
 -IS transformed into ; , and tlie fluent is -^ x AL. 
 
 These
 
 950 Of reducing Fluents Book IL 
 
 These are the figures which we found to resolve the most simple 
 cases of problems of various kinds in the first book, art. 436> 
 569, &c. 
 
 811. LetA/,Ap,AL,APr//g.31 1), &c.be such a series of figures 
 as was described in art. 212, where each curve is supposed to 
 be always defined by the intersections of the tangents of the 
 preceding curve with the respective perpendiculars on those 
 tangents drawn from the given point S. Let AL be a figure 
 of the kind described in the last article; that is, let the angle 
 ASL be to ASM, and log. SL to log. SM always in the same 
 invariable ratio. Then A/, Ap, AP, and all the other figures 
 in the series, shall be likewise of this kind. By art. 212,^ the 
 
 angle AS/ == ASL + 2ASM, SI =x +'*^', and the fluxion of 
 Al to the fl[uxion of AL as the fluxion of S/ to the fluxion of 
 
 SL, or as «q:2 x ar"^ to n; consequently the fluxion of AL 
 
 is li- X sx^ ; and the ark A/ is assignable by s and al- 
 It 
 
 gebraic quantities, by art. 792. The same is to be saidof all the 
 other arks in the series taken alternately, that is, of the 2d, 2th, 
 6th, &c. from AL. The other curves in the series Ap, AP, &c. 
 are all assignable by AP and right lines; but the arks of any 
 twofigures that immediately succeed each other in the series, as 
 of AP and AL, cannot be compared with each other by an al- 
 gebraic equation. WhenA/)S(//g.31 l,A^.2)issupposedtobease- 
 lnicircleuponthediameterSA,/coincideswithA,andA/vanishes, 
 ALand ihesubsequent arks in the series taken alternate]y(which 
 have all a cuspid in S) are assignable by right lines ; but the 
 other arks in the series are measured by the circular ark Ap and 
 xightlines. When AL(N.3)issupposedtocoincidewith the right 
 line AM itself (or n — 1), P coincides with A, and AP vanishes, A/> 
 jsacommon parabola thathasitsfocusinS,A/and the arks in the 
 scries continued backwards, taken alternately from A/,admitof a 
 perfect rectificiilion; huttheotherarksin the same series are mea- 
 sured bv parabolic arks and right lines. Ofall the figures wherein 
 the an<^ie ASL is to the angle ASM and log. SL to log. SM in 
 tbesameinvariableratio,thcrearenoncbesides these that seem to 
 
 admit
 
 Chap. III. to others of a more zimph Form. 251 
 
 admitof a perfect rectification, or an accurate mensuration by cir- 
 cular arks or logarithms. When AL is an equilateral hyperbo- 
 la that has its centre in S (or «=|), the curves taken alternate- 
 ly from AL either way in the series,, are measured by AL and 
 right lines ;but the other curves in the series are measured by 
 AL with an elhptic ark (described above, art. 802) and right 
 lines. By supposing u—^, ^, |, &c. other series of curs'es 
 will be formed. And every series of such curves gives tvpo 
 distinct sorts of fluents, vl^hich cannot be compared with each 
 other, or with those of any difterent series of this kind. 
 
 CHAP. IV. 
 
 Of the Area, when the Ordinate and Base are expressed by Flu- 
 ents ; of computing Fluents from the Sums of Progressions, or 
 the Sums of Progressions from Fluents, and other Branches 
 of this Method. 
 
 812. JL HE base being represented by z, and the ordinate of 
 
 the figure by y, the fluxion of the area is zy. \^ y and z be 
 both assigned by quantities compounded in common algebraic 
 terms from the powers of the same variable quantity x, the 
 fluxion of the area will be expressed by such quantities mul- 
 tiplied by X. Having insisted on the fluents of such expres- 
 sions in the preceding chapters, we now proceed to enquire 
 into the area or fluent when the ordinate is itself assigned by 
 an area or fluent, or when the ordinate and base are both ex- 
 pressed by fluents : and in this case it will be sufficient if wc 
 can reduce the area of the figure to the fluents of the former 
 kind ; as to circular arks and logarithms, or to elliptic and hyper- 
 bolic arks, or, in general, to the fluents of expressions that in- 
 volve one variable quantity only in algebraic terms with its 
 fluxion. In this case we shall find that the total area (or that 
 which insists upon a certain given base) may be sometimes mea- 
 sured by circular arks or logarithms, though it may not appear 
 
 that
 
 252 Of the Area, when the Ordinate Book IT. 
 
 that in the same instances the part of the area can be assigned 
 in this manner which stands upon any segment of the base that 
 may be proposed. For example, let ADcr, BE6 (//g. 312), be con- 
 centric circles described from thcsame centre, CBbeingless thaa 
 CA;ietAG be the tangent atA, and T any point upon AG; join 
 CT intersecting the circle BE6 in V and v. Now, let the figure 
 CHKR be constructed so that the base CR may be always 
 equal to the logarithm of the ratio of CT + AT to CA, and 
 the ordinate RK always equal to the logarithm of the ratio of 
 Tiy to TV, the modulus being CA. Then the whole area 
 CHKLLRC generated by the ordinate RK, while the point V 
 describes the quadrant BVE, shall be equal to the rectangle con- 
 tained by the quadrant AFD and the ark DF whose sine is CB ; 
 but it does not appear that the part of this area CHKR, that 
 stands upon any given base CR, can be measured in this manner. 
 The fluents of this kind are sometimes required in the resolution 
 of useful problems, and the mensuration of the whole area is 
 commonly what is most valuable. But before we treat of the 
 ai'ea, when the ordinate and base are both expressed by fluents, 
 some theorems are to be premised concerning the area, when 
 the ordinate only is expressed in this manner. 
 
 813. Let A represent any area on the base x, suppose Ax =; 
 K, K;^ = h ,<'hx = M, Mx "==■ ii , &c. where K represents the 
 area when the ordinate is A, L the area when the ordinate is 
 K, M the area when the ordinate is L, and so on. LetoTA =: 
 B, ^B — Cy xc ~ i), xb = E, &c. ; and suppose B^r' — i, k'x •=. 
 /, I'x — m, nix = n. Sec. Then shall K =: x\ — B, 2L = xK 
 — k, 3M z= arL — /, 4N rr xM — m, and so on. For since 
 Ar + XA = k + B, it follows, by finding the fluents (art. 
 738), that Ax =: K + B, and K = Ax — B. Because Ki 
 -f xk — i = L + AX;; — Bi- = L + Ax—B X i = t 
 -I- K:^ = 2l, by taking the fluents xK ~ k z=: 2L. In like 
 manner, Lx + xl — / — M + Kxr — kr" = M + iLx ~ 
 ;1m, and SM ■=■ xL — /,• M;:^ + xm — m =: n + Lx^-' — Vx 
 := N + 3M>" == 4n, and 4N zr xM — m, and so on. 
 
 814. In
 
 Chap. rV. ayid Base are expressed by FIue7its. ^53 
 
 814. In the same manner that K=tA — B, it is manifest that 
 k — xB — C ; consequently 2L = a:K — k = jt^A — arB — 
 a:B + C = x"A — Q.xB + C, and 2/ = or^B — 2aC + D. 
 Hence 6M = (by the last art.) 2xL — <2,l z=. x X ;ciA— 2*'B+c 
 — x'-'E.—'ixC-fD — x'^A — 3j:^B 4- SxC — D, and Cmi ■=. 
 a,3g _ 3^^c + SxY) — E ; 24N = GxM. — 6m = x x 
 
 ;^3A — 3.*r»B +5xC—D — x3B— 3;tr*C+ 3xD — E = X*A — 4x^B 
 
 -f 6x^0 — 4xD + E. And in this manner it is mani- 
 fest, that if r denote the place of any fluent Z in the series 
 
 x'-A—rx^—' B + r X ^^ x a:'-—'C-8LC. 
 
 K, L, M^ N, &C. Z = IX2X3X4X . . . Xr " 
 
 which is the first part of prop. 1 1, De Quadrat. Curvar. When 
 «'-A— rcf-'B + r X ^^ a'--"C-&c. 
 
 X = a, then Z = 
 
 1 X 2 X3. .. Xr 
 
 815. Let» zz 7Z.J X A = fl'"A — ra'-—'xk +rx ^- X 
 
 a*"— *j:^A — &c. r= a'' a — ra''— ' b + ^ X ^- X a''— *c 
 
 &c. ; consequently j2 — a'A — ra'— ' B + r x ^- x a''-— »C 
 
 — &c. and when x =: a, Z = i^zxsx ...Xr » ^^^"*^"^^ ^ 
 the second part of the same proposition. 
 
 816. Let .TA = P, Pa =: Q, Qa = r, T^a = i, &c. and 
 the fluent of AV will be equal to xA" — ;jA«— ' P + ?* x 
 "^r x A«— ^Q — M X "^T X 'IJIir X A"--^R — &c. For, 
 by art. 738, the fluent of AVis xA^ — F, /?A«— ' xk (where F 
 is prefixed to denote the fluent of the expression that imme- 
 diately follows) n xA'^ — F, wA"— ' p = iA" — ?< A''— ' P + l\ 
 n X IZT X A«— ^ d = xA^ — «A«— ' V -\- n x tliZIi x A''—"- 
 Q — F, w X ^T X 7^ X A"— ^Q, and so on. 
 
 81 7. For example, let a = ■ — , and K the fluent of 
 
 V 1 +.VX' 
 
 A'x will be J A — B = (because b = xA = ^ ^ _ and B
 
 254 Of the Area, when the Ordinate Boole ^11, 
 
 = hP Vi^xx) x\ + »/ iz^xx ; and, because the fluents B, Q, 
 J), &c. are expressed by circular arks or logarithms with alge- 
 braic quantities,, according as A is itself a circular ark or loga- 
 rithm, the same is to be said of the fluents K, L, M, N, 8cc. 
 
 by art. 814. Let */\^.xx = z ,• then A = 1-, p r= .ta = 
 
 T. 
 
 -^ ■=■ + 5=1 and P = + ;:; ; q = Pa = + x\ and Q = + 
 t;r = Qa = T~ =2, and R =: r,- s = Ra = v, and S 
 
 3 
 
 = X. Therefore the fluent of A''^ is ivV ± ?2A«— '2; + n x 
 ^ZT X A«— ^a^ — w X ^r X ;;^ X A«— ^2 + w x "^ZTi x 
 ^;Z2 X ^Z^s X A"— ^a: + 8cc. 
 
 818. Supposing A = .yi-, B '=■ oc'i^rz yxx. Ify can be ex- 
 pressed by X, B may be expressed by a fluxion that involves an 
 invariable quantity only (viz. xj with its fluxion ; and if A and 
 B can be reduced to algebraic quantities, or to circular arks or 
 logarithms, by the preceding articles, the same is to be said of 
 K, the fluent of A^., becauseK^rxA — B, Itisobviousthat if A 
 anda^be assignable by each other, A.r or km ay be easily express- 
 ed by a fluxion that involves one variable quantity only (viz. 
 X or A) with its fluxion ; and the fluent of A;^ may, in many 
 cases, be assigned in algebraic quantities, or compared with 
 circular arks or logarithms, by the preceding articles. But be- 
 sides these more obvious cases, there are others wherein the 
 fluent of x X F,y;^ ( or of A;^ ) can be reduced to such as have 
 been considered above. 
 
 819. The base of a figure being represented by z, and the 
 
 ordinate by 3f, lets =: Vi''"— ' xE + F.i»l^', and ^ = xX'^"-* 
 X e-f/x"^» and let x = f? when E -f Fx« = (that is, let 
 (Jn — — - j ; then if 7- + s -f I + k :=: o, the area of the figure 
 
 (or the fluent of ij/) that is generated while .r by flowing 
 from becomes equal to d, shall be equal to the simultaneous 
 
 fliuents
 
 Chap. IV. and Base are expressed hi/ Fluents. 255 
 
 fluents of x^^+^'^^i X E + P>^~' and :rs»— * ; x e +/>^+' 
 multiplied by p^; that is, let Q, G, and P, represent the se- 
 veral fluents of i^'"— I X E + Fx«'""' x F. irs"— ' x e+fJF^^ 
 ixrn+tn^i ^ E+lV'^"'^ and xx""-' X ^7^'^+^ that are 
 generated while x by flowing from ^6 becomes equal to d ; and 
 
 Q = - 
 
 e 
 
 I^. i^OY, by the supposition, -^ = xX*"-' X 
 
 1 +- — I = (by the binomial theorem) a^"— »; + — X 
 xsn+n^z; + A X i=l X -f X x^'^+'n^i^ + &c. and (A) 
 ,i:ir« - 1 + m X T- + 7+^ X A- X ~ x-'^-+ &c.; 
 consequently ?/s is equal to the product of ^-^ x a:»-«+sM— i 
 
 •J-i 
 
 x E + Fa" multiplied by this series. Therefore, by art. 792, 
 if in this series you substitute d for x, and multiply the terms 
 
 respectively by 1, -^^, -^ ^;T7T7TT. ^^- ^^^ ^^^^^^^^ 
 y+s = — / ~ A-, and r + s + / = — k) by 1, ^i, d^f x 
 — ip-j-j &c. and suppose the series thence arising, viz. i + 
 
 $ /J" , i + l—l ffd'-n 
 
 — X i+ / X V+7P ^ * + ' X -2-- ^ ~ + &C. = 
 el 
 
 L, we shall have Q =: - X GL. But by substituting: in the 
 
 equation A, by which the value of y was determined, P 
 
 for y,k-i-l for k, and d for x, it is manifest that L zn ■ ,'", -. 
 
 e'^+'d^"9 
 
 GP 
 
 consequently Q = ~T^« This theorem is founded on art. 
 
 792, and is to be understood with similar limitations, particu- 
 
 larlv
 
 250 Of the Area, when the Oydinat6 Book IL 
 
 laily with those described in art. 79G. We have supposed r + s 
 -f / -f k "=1 0, or s =3 — r — / — A"; but it is easy to see that if 
 s be increased Or diminished by any integer number, this theo- 
 rem will be of use for discovering the fluent of y^l when x =: d, 
 or for reducing it to common fluents, that is, to such as involve 
 the powers of one variable (quantity compounded together in 
 common algebraic terms with the fluxion of that quantity. Sup- 
 pose, for example, that d = » X F, yX", and let e +/i'* = R, 
 
 then Q = —-. — ==r. — -p — ^ -• 
 
 nJXs + k+l JXs+i + i 
 
 820. Let X — D when e + fx^ — o ; and, the values of « 
 and y remaining the same as in the last article, let Y, Z, and q, 
 he the respective fluents ofy, z, and ^2, when e + fx"' = Oi 
 Let g and p be the simultaneous fluents of ^2rji-f.s»— i >j 
 
 e-\- fx^^ and 'xx'"'~^ X E + Ri-'''+^ Then if r + 5 + / + A 
 
 —0, as formerly, q =YZ — vkt^^Ura ' '^^'^ theorem follows 
 
 from the last, because F, y^ zz yz—V, Zy . 
 
 821. Let a = — ;.r''— ™— * X Ex«TF^""', 7 = — ix"^ 
 
 '1/ 
 
 X ex^ + f , X •=. d when E + Fjt" =: 0, and the area or fluent 
 of ya' which is generated while a: flows till from being equal to 
 d it becomes infinitely great (or the limit to which this area 
 approximates while x increases continually),i9theproductof the 
 fluents of— ;a:n+"fc— I x EIm-F'""' and - ;,r'»— "^Xf.r"+/^""^^ 
 multiplied by each other and by - ^ , ^^^ • as will appear 
 
 by substituting .r— > for x'\n art. 819^ and supposing w = rn + 
 In — ]. The fluent of y«* when e+fx^ rr is the excess of 
 the product of the corresponding values of 3^ and z above the 
 product of the simultaneous fluents of — .Vx"'— ' X t'.r"+/'^ 
 and — -^x' —m—nh—'Z >; li,jv«^i.+ multiplied by each other 
 
 Dw-fi — id 
 
 and by -r=n— > by the last article. 
 
 822. from
 
 Chap. IV. and Base are expressed hi/ Fluents. 'Z57 
 
 822. From these theorems tables might be computed of 
 fluents of this kind that may be reduced to circular arks and lo- 
 garithms ; but we shall only give a few examples of their use. 
 Suppose tliatit is required to find this area or fluent of y.*, when, 
 
 m being anv positive number, \ = and ,; zz. — — — • 
 
 that is, let it be required to find the fluent of — * X F, 
 
 •^ X^\/bh—xx 
 
 ■ _ • when X rzh. In this case, by comparing the expo- 
 nents with those in art. 819, 11 — ^, r— ^^^, / — 1 s — ~. 
 
 2 2^ 2 ' 
 
 and k rr — 1, so that r + / + s + /.• - ^~"+^+" '— l rro,as 
 the theorem requires. Because in this cased ~ '^ and ^ 
 
 V bb — XX 
 
 — " _ ; it follows, that if N denote the ratio of the cir- 
 
 cumference of a circle to its diameter, the fluent required is 
 -_-- >c F. ___f_ , if b be substituted for x in the value of this^ 
 
 2flO"» Vaa^Lxx 
 
 last fluent after it is determined. Thus if 4;=loo-. ^^~^^^ ^-^^ 
 
 X ■ 
 
 and j/ = log. a x -/—^/and H represent the ark described 
 with the radius a that has its sine equal to b, then the area re- 
 quired will be equal to ~ x H ; whence the proposition that 
 
 was advanced in art. 812 follows: for in this example ^ =: 
 
 bbv . aay' 
 
 ~7~=. , = ^^^-,m =: 1 and P = F, -JL If, the 
 
 xv'ii—xx aa — XX -v/J^HTv 
 
 same value of z remaining, we suppose y to be always equal to 
 
 the ark described with the radius a that has its tangent equal to 
 
 X, or 3, ~ — 1 — ^ then p == — ; and in this example 
 
 aa-{-xx VaaJr^x ^ 
 
 VOL. II -^ the
 
 258 Of the Area, when the Ordinate Book II. 
 
 the fluent required is — — X log. .!Lff±£ilL- ; so that the flu- 
 '2 a 
 
 cnt, whicli in the former case was the product of two circular 
 arksj is now the product of a circular ark by a logarithm. If 
 m be any integer number, the fluent required may be measured 
 by the areas of conic sections, and if m be equal to any of the 
 fractions ^, |, f, &c. it may be measured by their arks. 
 
 823. It is obvious that in other cases the fluents may be com- 
 puted from the theorem besides those wherein ?• + s + / -f. /; 
 
 r=o. Suppose, for exam ple,K — " — - and y — i. 
 
 xW bb—xx e -i-Jxx 
 
 Because y = x — -ILJL — , ^t/ zh f -f — x 
 
 f4-/r.r a.rVM— Arr xWsb-^xx 
 
 A'l • 
 
 ^ F, — i^~. The fluent of the former part is assignable in 
 
 e +fxx 
 
 algebraic terms, and the fluent of the latter (by the theorem 
 in art. 819) by circular arks and right lines. 
 
 824. In like manner it follows, from art. 822, that if i' =3 
 
 ___f___ and y = — ~—> then the area, or fluent of y« 
 x^Vx.x—bb xx+aa 
 
 that is generated while x flows till from being equal to 6 it 
 
 N x^~^x 
 become infinite, is — _ x F, — - . 
 
 825. The theorems in art. 819, Sec. are cliiefly of use for 
 reducing fluents to circular arks or logarithms, or to others of 
 a more siiiipie form (and consequently for rendering our solutions 
 of problems more simple and elegant than when we have im- 
 mediately recourse to an infinite series), when neither j/ nor z 
 can be expressed by x in algebraic terms^ But they may be of 
 some use, likewise, for finding the fluent of j/a;* when^ is as- 
 signed by X. Thus, to find the fluent of 
 
 aax 
 
 aa -{- XX ^ V" bb—xx 
 
 when X ■=:: b) suppose ri = — tj/=: , and conse- 
 
 Vbb—xx . aa + xx 
 
 quently
 
 Ghap. iV". Unci Base are expressed hy Vluenti. ^5^ 
 
 queritly^ •=. ." - ■ / By comparing these values of » and 
 
 aa-\-xx 
 
 p with their general values in art. 819> ?« — 2, ^ — i> ^ =^ |^ 
 s =r 1, /c = — 2> and r+l-\-s-^k:=:o,as the theorem re- 
 
 quires, G the fluent of -- is -j and P the fluent of 
 
 ^ Vbb—XX 4 
 
 -^ . '^— is — ■ * whence^ hy the theorem in art. 819, the 
 
 diz+ xx\^ l/aa-\-xx 
 
 fluent required is ■ - . Other exariiples might he given, 
 
 2 V aa-\-bb 
 
 if we were not obliged to hasten towards a conclusion, this Trea- 
 tise having already grown to a far greater bulk than was at first 
 intended. 
 
 826. If we assume an equation as x-^-a/^- X 7+%" = E, 
 where m, n, A, B, E, are supposed invariable, then (by art. 728) 
 
 i/'x + otA4-«b X *-}^ + OT+n X ABjy^ = 0. If we had as- 
 sumed T+XP'" X j/« = E, then wj/x + w% + ^Ij^ X Aj/^ =0, 
 where the term x'x is wanting. When a fluxional equation is 
 proposed that can be reduced to a form of this kind, then, by 
 comparing its coefficients with those of the equation of the 
 same form, the values of wz, n, A, Sec. maybe determined, and 
 the equation for the fluents discovered; as is shown more fully^ 
 Comment. Petropol. torn. 1, &c. 
 
 827. When an area or fluent is reduced to a: series by the 
 methods described in art. 745, &c. the series in some cases con- 
 verges at so slow a rate as to be of little use for finding the area. 
 Suppose FjVI/*(y?g. 313) to be an equilateralhyperbolathathasits 
 centrein Oand Oa for one of its asymptotes ; let OA =. 1, AP =r, 
 
 PM = y, and y = -— = l — x -i- x^ — x'^ + &c. whence 
 
 the area APMF =1- p^^; = x — j + ^ -j- '1* + Sec ; and 
 
 S2 if
 
 •fiGO Of computing the Sn)7is of Progressions Book II.^ 
 
 ii'AB = \, tlic area ABEF z=i— i+l._l-^-i_ 
 -■ + 5cc. rr 2 + T2" + so" "^ ^^' ^^'^''^^^ ^^ '^''^ sei'ies 
 mentioned foi- the quadrature of the lij'perbola (or for finding 
 the hyperbolic logaritliui of 2) in art. 36 1. But this series con- 
 verges so slowly, that the sum of the first 1000 terras * of it is 
 found deficient from the true value of the area in the fifth deci- 
 mal; and other examples similar to this might be brought; 
 wherein the area may be more easily computed from the in- 
 scribed polygons tlian from the series. Some further artifice is 
 therefore necessary in order to compute the area in such cases^ 
 instances of which were described in art. 36l, and others are to 
 be met with in several authors^ particularly those who have 
 treatcdof the compulation of logaritliras and mensuration of the 
 circle. The following theorems, derived from the method of 
 fluxions, may be of use for this purpose; and serve for the reso- 
 lution of many problems that are usually referred to what is call- 
 ed Sir Isaac Ne?i:tuifs differ enlial method. 
 
 828. Suppose the base AP=:r (fig. 314), the ordinate PMrry, 
 and, the base being supposed to flow uniformly,let '%^=.\. Let the 
 firstordiuateAFbe representedby «, ABn: 1, and theareaABEF 
 =:A. As Ais the area generated by the ordinate?/, so let B,C,D, 
 E,F,&c. represent the areas upon the same baseABgeneratedby 
 
 the respective ordinates^,/,^,]^", &.c. Then AF = <z r= A — 
 
 |j C V o 
 
 2 + T<J "~ 720 + 30240 ~ ^^' ^^^' ^y ^^^' "^^"^ A = a + 
 •^ + 4- + -^ + Y,5^ 4- Sec. whence we have the equa- 
 tion (Q) c = A — -^ ~ — ^ — ^ — Sec. In like 
 
 manner, c=:b — -^ — — — -^j — c^c a = k^ ^ — 
 
 ^^ Sec. :i = D — 4r — Sec. a' = E — Sec. by which lat- 
 
 * Stirling De Summat, Scrierum, p, 28; 
 
 ter
 
 Cliap. IV. and Areas from each other. 2^1 
 
 ter equations^ if we exterminate a, a, a, a, Scc. from the va- 
 
 R C 
 
 liie of rt in the equation Q, we sliall find tliat « = A — -5 + j^ 
 — ^ -\- Sec. The coefficients are continued thus: let k, I, m. 
 
 n, &,c. denote the respective coefficients of a, a, a, &c. in the 
 
 \_^ _ 1 
 
 24 ' """ T20' 
 
 equation Q ; that is, let k =. --^ I = -^ m ~ ~ 
 
 &c. ; suppose K = /c 1= ^ , L rr /cK — / — — , M zz kh — • 
 
 IK + »« =0, N = kM — /L + mK — 71 =z — — , and so on ; 
 
 then a = A — KB + LC — MD + NE — &c. where the 
 coefficients of the alternate areas J), F, H, &c. vanish. 
 8'29. As A is the fluent of 3/z, so B is the fluent of ^i, C 
 
 of j/z, E of 7/2, &.C. Therefore, since z =■ 1, and these 
 areas are generated while the ordinate PM moves from AF 
 to BE, the area B will be expressed by the excess of the last 
 ordinate BE above the first AF, C by the difterence of the 
 fluxions of the ordinates (having due regard to the signs of 
 these fluxions), E by the difference of their third fluxions, and 
 the other areas G, I, &.c. by the respective differences of their 
 fluxions of the corresponding higher orders. Therefore if « 
 represent BE — ^AF the difference of the ordinates, and (i, 5, 
 ^, &,c. the differences of their fluxions of the first, third, fifth 
 
 o„le,-., &c. then a ^ A -^ + i^ - L + J- + &c. 
 
 830. Supposing now the base Aa to be divided into the equal 
 successive parts AB, BC, CD, Sec. and each part equal to unit, 
 let the sum of the equidistant ordinates AF, BE, CK, &c. ex- 
 clusive of the last ordinate oj] be represented by S, the total area 
 AF/'« upon the base Aa by A, the excess of cr /"above AF by «, the 
 respective excesses of their first, third, fifth fluxions, 6cc. by /3, 
 d,(^,&ic. the fluxion of the base being supposed equal to I, 
 
 then it follows, from the last article, that S = i\. — .,- + ' -jg
 
 ^62, Of computing the Sum$ of Progressions Book If ^ 
 
 - 7I0 + i + &c.andA = S + f-A + J. ^ 
 — i-- + &c. which eive two of the theorems mentioned in art. 
 
 302*0 ° 
 
 352 and 553, where we had hyperboHc figures chiefly in view. 
 This proposition more generally expressed, without supposing z 
 
 or » equal to unit, is that S = — — - + — ^ . -|- 
 
 ^ ^ 2 12^ 720^3 
 
 ^ ^ -f occ. 
 
 30240is 1209^0021^ 
 
 831. The ordinate AF being still represented by a, let AR 
 and Arbe taken on opposite sides of the point A equal to each 
 other, RV and rv the ordinates at R and r terminate the area^ 
 RVrr; let 7/ represent any ordinate as PM of the figure, and, 
 the base being supposed to flow unifonnly, let A, C, E, &c. 
 represent the areas upon the base Rr that are generated by the 
 
 respective ordinates 3/, j/,j/, Sec ; then, supposing AR=:2, the 
 
 middle ordmate AF ( = «) = -- ^^ + ^. - ^^_- 
 
 r , "RrvY _ A , z^-^ 
 
 + &c. ; for, by art. 752, -— — - — =«+ r— r.- + 
 
 2X32 
 
 A z^a 
 
 z^a 
 
 ^ "' +&C. or« = — — ^::^ — ^^A Sec. In 
 
 2X3X4X5^,4 2z Oz^ 120a+ 
 
 •• C z'a /i-^ •• _ £ 
 
 like manner, a zz — — -— — — '^c- a — — — &c. ; 
 
 23 (>i* 22 
 
 whence, by exterminating a, a, Sec. from the value of a, the 
 theorem will appear. The coefticients of C, E, G, &c. are 
 
 continued thus : let the several coefiicients of a, a, &c. in the 
 
 value of -^ (derived from art. 752) be represented by k, I, 
 
 -^ z^k zH 
 
 m, n, &.C. that is, let k-zz — --d I = r, m = ^— — -> ^* 
 
 2 X 02* 4 X oz-- 6 X 7a* 
 
 - ^''^ -, &c. ; then let K = - = -i-, L = AK — 

 
 Chap. IV. afid Areas from each other. CCS 
 
 ^,M -kh — lK + ~, N = AM — /L + wK — -^j &c. 
 And the values of the coefficients K, L, j\T, N, &c. being thus- 
 computed, then a = -^ — KC + LE — MG + NI — &c. Be- 
 cause the areas C, E, &c. are the respective fluents of^^, 
 y% , &c. if the respective differences of the first, third, fifth, 
 seventh, and higher alternate fluxions of the ordinates rv and. 
 
 RV, be expressed by /3, I, ^, 6, Sec. then a = ~ -%- -f 
 
 : :- + r- &C. 
 
 720!23 30240z5 1209600;=7 
 
 832. Fromthisitfollows,thatifAF,BE,CK(y70f.3 15)&c.bease- 
 ries of equidistant ordinates upon the base Aa, of which i^F is 
 the first and oythe last; AB their common distance be equal to 
 Q,z; AR be taken backwards from A equal to z or \ AB, and 
 ar be taken forwards from a also equal to \ AB; the ordinates 
 RV and rv terminate the area RVzt; and this area being repre- 
 sented by A, the differences by which the first, third, fifth, 
 seventh, and higher alternate fluxions of rv exceed the same 
 fluxions of RV, be expressed by |3, 5, ^, 0, &c. and the sum of 
 the ordinates AF, BE, CK, &c. (including af) by S, then 
 
 __ _A_ _ z^ 7z'S _ 31 z'^ 127z'S 
 
 22 12a 720^5 30240a;S 1209600^7 
 
 ' r + &C. A = O- S + -,- + — ;- ?- 
 
 47900160i;9 fo SbOi^ 15120-j? 
 
 , 127/6 511;'°>t . Tr Ao 
 
 + r- — r— + &c. Ir we suppose AB rr i, 
 
 6048002i'' 23950080*'' 
 
 and ^ zz I, then 2 = | and S = A ~ '— 4- rr:- — —l,^ 
 
 ' ^ 24. • 3760 9o7o80 
 
 "r 1 f.Afto»snr^ *^c. ana n — ^ + "SZ Tt^ + ^ 
 
 164828800 """ """* j.x — ^ i 24. 6760 ^ 967680 
 
 &c. which are the two other theorems mentioned in art. 
 ."52 and 353, only, in order to include the term of, ar is 
 here taken forwards from a, whereaa afwixs there excluded^ 
 and ar taken the contrary way. 
 
 S 4 ^33. Tlic
 
 254 Of computing the Sums of Progressions Book 11. 
 
 833. The use of these theorems will best appear by examples. 
 First, let w, m+e, m + Q^e, m-{-3e, . . . n, be a series of num- 
 bers in arithmetical progression, where m denotes the first term, 
 c the common difference, and n the last term ; and r being any 
 number positive or negative ( — 1 excepted) S the sum of the 
 powers of tiiese numbers of the exponent r, that is, m^ + 
 
 mT"'" + ^7+2^'- 4* "'i-a^'' + . . . • + n''- — == — X ?/'• - m^ 
 
 n^' + nf re > — -■ ^ r.r — ] . r — 2 . e^ 
 
 ^ 1- -rr X //'•-'—?«'•—' — X 
 
 2 12 720 
 
 ',^7-— 3 — ^;i'-— 3 -f &c. For,supposingOPm',PM=r_y,letO?g.3 I4,N^. 
 
 1 &2) FAI/be the paraboiaor hyperbola whoseequationisy^ijr'', 
 OA =z m, Oa:=:n ; consequently Af :=im'', af z=. w'", F. i/x = 
 
 T.x'^^'x = and the area AF/a = A = » 
 
 af — AF zr fi{ =: ?i''^— m*"; ^ = ra:'"— 'i , and, supposing;^ = 
 
 2 = 1, the difference of the fluxions of af and AF is rtO'—t 
 
 — rm^'—^= /5 ; ^' — r X~T x 7Il2 x j'— ^ ^ , and ^:=zrx7^ 
 X 7^2 X w'— ^ — m"— \ In like manner, ^, 6, &c. are com- 
 
 putedj and it follows, from art. 830, that S — W = — = — — ' 
 
 n^ — m'" re 
 
 -\- — X n'—^ — m"— I — &.C. therefore S 
 
 2 12 
 
 w^+i — W+i n^' + m^ re — • „ _ 
 
 7+Txe ^ 2 ^ 12 ^ 
 
 supposing e =. I and tn = o, it follows, that the sum of the 
 
 powers of the numbeis 0, 1,2,3,4, . . . n of any integer and posi- 
 
 . w'+i n^' rn''—^ r x ~ X 7^ X ?/'■— ^ 
 
 tive exponent ris— + ^J + ITT ^q " 
 
 -J- &.C. this series being continued to as many terms as there 
 are units in 2 + ~- only, when r is an odd number : be- 
 cause when r ■=. 1, the fluxions of AF and of are equal, and 
 
 /3 = c>:
 
 Chap. IV. and Arca& from each other. 265 
 
 /5 — ; when r = 3^ 5 = o ; when r — 5, ^— o, 8cc. Thus if 
 r r: 1, S rr ^- + ^; it r = 2, b = - + ^ + _; and 
 
 if r =;: 3, S rz - + g + J"* This is the theorem given 
 
 by Islx. James Bernouilli, Ars, Conjeclcmdi, p. 97. When r 
 is -a fraction or negative number, the sum of the powers of the 
 
 same numbers (by supposmg m ~ 1) is -{ 4. 
 
 rtv—i —^r r X r— 1 x r_2 
 
 X W— ^ -1 + &c. 
 
 12 720 
 
 834. The sum S (fig. 3 1 5) of the same powers of m + e,m + 3e, 
 
 m-\-5e,m-\-1le,. . . n — e, where 2c is the common diflerence of the 
 
 terms, computed by the theorem in art. 832, by supposing OR 
 
 = m, RA :=. e =. ar, Or =. n (and computing the area RVrr 
 
 with the differences of the first, third, and higher alternate flux- 
 
 1 ■ ■ — 7'g 
 
 ions of TV and RV), is ■= X «'+i -w'+» X 
 
 ^' r+iX2e 12 
 
 77'Xr — 1 Xr— 2 Xe^ 
 
 720 
 Sir xTZT xm? X~3 xTZi 
 
 30240 ^ ^' ^ ;.'-^-m'-= + &c. 
 
 By supposing m zz e — \, the numbers are 1, 2, 3, 4, 5, ... . 
 
 T , _, w'+i nV— I 7r X~Tx~2X;i''— ^ 
 w — i, and S — ■ 4- 
 
 r+1 24 5760 
 
 31rxrZTx7Il2x733X~+ „ 1 
 
 X ?!'■— 5 + CCC. r:r 
 
 9^)7(380 7+TX2M-I 
 
 , ^ 7rx;n:Tx~2 
 
 "*" 24 X or-i 5760 X 2'--^ ■*" 
 
 835. When r is negative, let r r= — s; and if s be 
 
 greater than unit, then, by what we have shown in article 
 
 1 ' ' 
 
 833, tne sum of the proa-ression — + =, + ==-, 
 
 J , 1 J 
 
 + ;rF37' + ^^' (by supposing ~ zzo)zz -7 x cv;.-i + ^^^J 
 
 +
 
 ^Gd Of computing the Sums of Progressions Book IF; 
 ^ s£ s X 7+T X 7+2 X t^ .sxT+T x 7+2 x ~i x 7+1 
 
 X e^ — ■ &c. This series was deduced from different principles 
 by Mr. De Moivre. In like manner it follows, from the last 
 
 1 1 J I 
 
 article, that =p.^- -f ^=F=« + -^^^^-s -i. ==s j- &,c, 
 
 _ 1 se 7.S X 7+T X 7+"2 x e^ 
 
 &c. For example, if s = 2 and e zz ^, then S =: r-r-r 
 
 7 31 
 
 r^ siOffls" — i3Um'^ "^ ^^* "^^ compute the sum of the 
 progression l + r + -5+ 7^ + -^ +&c. find the sum of 
 
 J 
 
 the tenns at the beo-innina: of the series as far as -=7=71 exclu- 
 
 sively, then compute the sum of the subsequent terms by this 
 
 theorem. Thus if we add the three first only 1 + - + g- r= 
 
 1. 361 1 i &c. suppose 771 -i- J zz 4, or m zz ^, and the sum of 
 
 the following terms will be | X 1 — j^ + -jj^^ - &:c. 
 
 three terms of which series only collected and added to the 
 former number 1. 361 ] 1 &c. give 1. 64493 <Scc. for the sum of 
 the series required, true to the fifth decimal. Us — f, and ? = |, 
 
 then b zz X '^ ■ . 4- — . - — 4- ccc. 
 
 836. These theorems may serve likewise, in many cases, for 
 computing the area when the series that arises in the common 
 method (described above, art. 745, &c.) converges at too slow a 
 rate. For example, let Vwr be a common hyperbola, O the 
 centre, OR = m, Or zz n, and R/- be divided into any even 
 number of equal parts of which RA is the first and ar the last ; 
 let RA = e, and S denote the sum of the or(Hnates AF, BE, 
 . . . af that inL^st upon the base ;it the distance 2c from each 
 
 other.
 
 Chap. IV. and Arca&from tacU other, 257 
 
 'e^ 1 1 
 
 Other. Then the area RVrr — 2f S + ~ X — 
 
 i js niTn tjTt 
 
 3x2x^x7^* _i_ f , 2x2x3x4x5x31^^ 1 I 
 
 72P ~" ^ n& "^ "• 30240 ^1^ ^6* 
 
 ■ — &c. because m this case y = -5 y — -5 „ = -.. 
 
 &c. Hence, if OR — m— \,Or — n —% and RA := e r= 
 
 li o 8,8.8,8 j^c-2,2 
 -, then S rr - + ^T + 1? + TI' ^^^^ ^^^ ^ 9 + If + 
 
 2 2 ^ 2^' 1 ^ _ Jl_. 2x2x3x7<?4 1 p 
 
 'H "^ Tr> I2 ^ ^ nn ~ bl2i 720 ^ "^ ^ 
 
 =; V — --: and by so few sub-divisions of the base B.r 
 
 64xb4x64' •' 
 
 and terms of the series, the area R/-z;V, or the hyperbolic lo- 
 garithm of 2, is 0. 693146 &c. which differs from the trulh by 
 ^n unit only in the sixth decimal. 
 
 837. The logarithm of m being given, the logarithm of 
 ^ + s is assigned by this theorem, log. w-j-^ - — log. m ziz 
 
 — ^ multiplied by the series 1 — it X — -- — i -f 
 
 == w, Aa-=z, FM/the logarithmic curve havingits asymptote EZ 
 perpendicular to EA, EP=x, PM =2/, the modulus or ordinate 
 
 Ee =: 1 ; then by the nature of the figure (art. 178), j^ zr - 
 
 or a-^ = ^ ; consequently, MN being perpendicular to the 
 asymptote in N, the area Ee FMN = r — ], <^MP = EP x PM 
 
 — Ee FMN =: r X log. a: — -ar + 1, and the area AFfa =: 
 Z^ X log.^^l — m X log. m — 2; == A. And because u r= 
 af— AF = log. ^ir\^ — log. m (supposing i = 1), AF = a z= 
 
 log. m = (art. 829) T " 2 + T2 ~ 720 + 30240 ' ^<^- = 
 
 — + J X log.;.+.- --o X log. m- 1 + 12" - 720 + I02I0 
 
 — &c. Therefore ~ + 2 ^ ^^S- "^ — ^^o* ^ — I — 
 
 f? 
 12
 
 CGS Of compulijig the Sams of Progressions Book I L 
 12 + 7l^ - sllo + ^^'- = (becausei = 1 and ^ = ;^^ 
 3 „ = — - and S — =% -i 8cc.) 1 x 
 
 1 I z' I 1 z' 1 1 ^^ 
 
 — — + :T7^;^x=^=3— T3'— T:;7^x=^-s -T:?' Hence, 
 
 if we suppose w IT l^andz = ], because log. 1 = o, it follows, 
 
 Q 1 7 31 
 
 that J X log. 2=1+ "123^ — 3^o^8 + "12603^ — ^C. 
 
 And by supposing z = |^ |- x log, \ — I + -^ — 
 
 360x8x37 
 31 1 
 
 + 1260x32x343 ~ ^^' ^^ ^ similar computation, it appears 
 that if z denote the excess of the logarithm of a + c? abov^ 
 the logarithm of a, or measure the ratio of « + d to a, then d 
 the difference of the numbers may be found by dividing az by 
 
 the series i _| -}. ^_ ^ + _^___g^c. Other the- 
 orems of this kind may be derived from art. 832. 
 
 838. Let it be required to find the sum of the logarithms of 
 a series of numbers m + e, m + Se, m 4- be,m + 7^ • . • . 
 n — e, in arithmetical progression, where m -\- e denotes the least 
 term, n — e the greatest, and 2e the common difference of the 
 terms ; or, to find the logarithm of the product m-f e X m^?)e X 
 m^-be X m-\-ie X ... X 7i^, whcn all thcsc numbors are 
 supposed to be multiplied continually by one another. For this 
 end, the figure being the same as in the last article, let EA be 
 now equal to m + f, Ea rz n — f, take All from A towards E 
 equal to e, and ar the contrary way equal to AR, and still sup- 
 pose the fluxion of the base equal to 1 ; then ER r= m, Er=^)i, 
 the area RVrr — n X log. n — m X log. m — u + m :=■ A, 
 the difference of the fluxions of the ordinates rv and RV^, is 
 
 i 1— c?x — r. ^/-£i_£lA — !!? 
 
 ft m '^ rr nry ^ n^ nr ' n' 
 
 _ Z2^, &c. Therefore (by art. 832), S = — — ^ + -^ 
 3/a' ^ ^ '2e 12 720 
 
 — 1- -f 5cc. rz — ■ — ^ o 
 
 30240 26- 2e 12 
 
 X
 
 Chap. IV. and Areas from each other. 269 
 
 n m 360 n^ m3 1260 «5 ^5 1680 
 
 1 1 — 8cc. And this is the same sokition which Mr. 
 
 «'/ w' 
 
 Stirling derives from his method, pi^op. 28, De Interpol. 
 Serierum. 
 
 8.39. The terms in arithmetical progression being represent- 
 ed by m, m + f , m + 2e, m + 3e,m + 4e, . . . . n — e, where ni de- 
 notes the least term, f the common difference, and n — c the great- 
 est term; thesumof the logarithms computed by the theorem for 
 
 S,inart.830, is equalto the excess of the series - — - X log. n 
 
 — _ + — - — + 8vc. above X 
 
 e \2n 360«3 1260;j5 e 2 
 
 losf. m — - + T^ -^ — + --^, 8cc. For if 
 
 » e I2m '360mi 1260/7i5 
 
 we now suppose EA=m, and Ea — n, AF will be the first or- 
 dinate, the area AF/Y/r:« xlog. n — m xlog, m, a'=-af — AF= 
 log. n — ^log. 7n, the difference of the fluxions of cr/and AF, or /3 
 
 — « 5 = -, — — r. &,c. ; and the theorem appears 
 
 by substituting these values for A, (i, 5> &c. in the equation for 
 S, in art. 830. This coincides with the value of S derived by, 
 Mr. De Moivrc in a different manner, Suppl. ad MisceL 
 Analyl. " 
 
 840. The sura of the logarithms of the odd numbers, 3, 5, 
 7, 9, \\,...n — lis obtained expeditiously, when w is a 
 
 large number, by computing | X log. n—\— ^^ 4* -3^5^ 
 + , " — &c. and thereafter adding —^5 ^^ 
 
 1260«S ' 1680«7 ^^' ""^ ."^.v,«.v^. «v.«,xio 2 
 
 the constant logarithm .346573590 &c. Because, if we suppose, 
 
 _ _ 1 1 "» X loff. 771 f"- 1 
 
 m art. 838, e z=. l, and m =: 2, then ~ — — 3- — -nr- 
 
 » 7 31 „ , 1.7 
 
 ^ 360/7;3 1260;«5 ^^ *^^- — ^'-'a* ^ ^ 12x2 ^ 360x8 
 
 ^^ + &c. = (by art. 837) log. 2 — | X log. 2 = 
 
 1260x^2
 
 270 0/ computing the Sams of Progressions Book 11.- 
 
 -^ ^ X log. 2 ; and this quantity is to be substracted from 
 
 I" X log. n — - — — + Sec. in order to obtain the sttm of 
 
 the logarithms of 3:, 5,1, .. .n — 1, by art. 838. 
 
 841. The sura of a scries^ of which the terms are alternately 
 positive and negative, is found by computing separately the sums 
 of such as are affected with the same sign by either of the theo- 
 rems in art. 830 or 83'2, and then taking the difference of these 
 sums. But when the terms^ which are added and substracted al- 
 ternately, may be considered as the successive ordinates of the 
 same figure, the computation of the area may be avoided, and 
 the sum of the series more elegantly obtained by the following 
 theorem. Let AF represent the first positive term, aythe term 
 which when the progression is continued succeeds after the lasfe 
 negative term, ethecommondistance ofthe ordinates, Sthesum 
 of the terms that precede ctf, and let 0, 5, ^, &c. denote the 
 differences by which the alternate fluxions of cr/ exceed the re- 
 spective fluxions of AF, as formerly. Then S =z — ^^ -f 
 
 £:_ — £_£ -f £^ — Sec. For the sum of the positive terms- 
 
 4 43 4aO ^ 
 
 (by art. 830> the common distance ofthe ordinates whicfh tb'- 
 present them being 2e) is ^ — _ 4- r__ — _^ 4< 8cc. 
 
 9e 2 12 720 
 
 A__^ 
 2^ 12 
 
 and the sura of the negative terms is (art. 832) -^ — -7^ 4* 
 
 il— — &,c. the difference of which (ai being equal to af-^AF)t 
 
 720 
 
 is 
 
 2 4 48 
 
 cr alternate fluxions of fl/" vanish, and ^, d, ^, &c. represent 
 
 the first, third, arid higher alternate fluxions of AF, without 
 
 1 • .v • • .u o AF— 0/ e/3 ,e'S e'? 
 changmg their signs, then S = ^-^ — _ 4. __. — _| 
 
 80640 
 
 842. Hence if EA =: 2, AF = log. 2, Ea =: n, af— log. w, 
 and /3, J, ^, &c. denote the several fluxions of AF, the loga- 
 rithm
 
 Chap. IV. and Areas from each other. 27! 
 
 lithm of the ultimate value of the product f x|-X7-X|-Xff^ 
 
 X ... X ^^ X 2 v/T will be equal to ^°-- '^'"^' ^ ~-| 
 
 + :: + &C. 4- loo- Q J- -2. — z: — X loff. 2 — -" 
 
 ^ 43 480 ^ ^ ^ o "^ ^ 2 2 ^ 4, 
 
 ■'■ 43" ~ 4^ + ^^' - (because, by art. 837, | X log. 2 = 1 
 /3 7$ 31C „ . , 2g 8? 32^ 
 
 ''" 12 720 "^ 3G240 '' 12 ''" 720 "30243 
 
 + 8cc.=(because(3rriJ=|,^=||,&c.) 1— i + i^— ^ 
 
 + TTgQ — &c. But (by what has been shown by Dr. Wallis) 
 
 if c denote the circuraference of the radius unit, c = 8 X 
 
 8 24 48 80 „ , . , , . J .„ ,, 
 jrXTrr- Xtt-X-ttX ccc. which product continued till the 
 
 9 23 49 81 '^ 
 
 denominator of the last fraction be ^ZJi*, may be expressed by 
 
 4 16 36 64 ^P ,v 
 
 *''9''25'^49'^8r'^"*-''S^ '^"' *^«nseq^^"^^^ 
 
 _. , , . , .2468 w— 2 , 
 
 -v/ c IS the ultimate value ot TrX-rXz-Xr-x.... X — r* ^ 
 
 Q^/7; and log. ^/^ = -^- =: i _ ± + 34o- t4o + 
 -^^ — 8tc. This (which was first observed by Mr. Stirling) 
 
 serves for abridging the computation in finding the sura 
 of the logarithms of the numbers 1, 2, 3, 4, 5, ... n — 1. For 
 suppose, in art. 839, mi=e=:l, then the latter series in that ar- 
 
 tide X log. m + -r^ rr—r ■ -f &c. = — I 
 
 I 1 I „ — log. c •< r^ 
 
 + 12 — sTo + 1260 — ^^^- = "-J-^ consequently S = 
 ^ X log. n — n + ~ — 3go;r + Tik^ — &c. + ^; 
 
 or ifw-idcnote thegieatest number in the progression, then (sub- 
 stituting
 
 272 Of computing the Sums of Progressiom Book IT." 
 
 stitiUing I for e in art. 838) S =. 7i x log. 7i — 7^ — — 4* 
 
 ssk^-iolo^ + &c- +-T; ''^'^^^^ ^»-^ ^^>« rules giveri 
 
 for this case in the treatises above-mentioned. Bat if it is re- 
 
 8 24 48 
 quired to find the value of 8 Xg X 25" ^ ig" ^ , ^c. by the 
 
 tiieorem in art. 841 (that is, to compute c from Dr. Ji'allis's, 
 proposition), then, because the series for the logarithm of a/T" 
 converges at too slow a rate, when EA is supposed equal to '2^ 
 lei r be any greater even number; find the number whose loga- 
 rithm is -T rrr-: + ;^T Scc. Call this number N, and 
 
 T 2 V" ~r~ 
 
 >/" = 2>^ |X|-X|X| .. . X73j-^~N~* If?-=10, then 
 let N be the number whose logarithm is equal to Tq- — "24000 
 
 1 „ , ,_ 256 A/lO 
 
 -„7; — occ. and -/ c = 
 
 ' 2000000 " *• ~~ -313 N 
 
 843. In like manner the logarithm of the ultimate value of 
 
 21 77 
 other products of this kind may be found ; as of 3 X ^ X rr- X 
 
 l69" ^ "^9" ^ ^^' ^'^^^^'^ ^^^ denominators are the squares of 
 the odd numbers 5, Q, 13, 17,21, &c. whose common difference 
 is 4, and each numerator is less than its denominator by 4. 
 Let the ultimate value of this product be called p, and -/"/"will 
 
 be the ultimate value of-x- x — x— x .... """^ x \/~^ 
 Let r be any number in the progression, 3,7, 1 1, 15, 21, 25, &C. 
 and N the number whose logarithm is equal to — ^ + 
 
 ±- &c. then v'T" = ^Ll x2 X IxiixH . . . x II^ 
 
 *^^ N 6 9 13 17 7^» 
 
 844. The problem, concerning the ratio of the sum of all 
 the uncm of the power of a binomial to the uncia of the middle 
 term, may be resolved by article 838 or 839, with article 842, 
 or rather by the following theorem. Let r be the exponent 
 
 of
 
 Chap. IV. and Jrcasfrom each other. 273 
 
 of the power to which the binomial is to be raised when the 
 exponent isan even number, or equal to this exponent diminish-' 
 ed by uiiit when it is an odd number; and c denote the cir- 
 cumference of the circle when the radius is unit; let N de- 
 
 1 
 note the number whose looarithm is equal to — — = — 
 
 o i 4X7- -(-I 
 
 a A V ~—,i "^ 7^7rcr^^=5 — ^c. and the ratio required will be 
 
 —T' '^ r-\- \ ZU X r -(- 1 
 
 Vex TTT ^ 
 
 that of ^^ — to unit : for this ratio is equal to 1 x 3 ^ 
 
 4' 6 8 r — 2 1 . 1 „ 
 
 3- X y X g X .... X •--■ - X r, which (by art. S4'2) is 
 
 equal to the ultimate value of _^1 X ~r-^ X ^^ X —^ X 
 
 ... X V~~s, where s is supposed to represen"; a number that 
 Continually increases by the increment 2 ; and the logarithm 
 of this ultimate value is (by art. 841, supposing AF=:log. r+T^ 
 
 and./ = Iog.— )i^- i^^ + !2iLL±I _. 
 
 Tz^ -1- — ^=^ ^=-3 + &c. + l2Si.^±l - 
 
 4X;+1 ^ '24Xr+l 20Xr+l ^ ^^^- ^ 2 — 
 
 ^ 2 4 X r+ 1 'i4Xr+ 1^ 20Xr + l^^^' 
 
 — log. — - — ~ — These are always supposed to be hyperbo- 
 lic logarithms, but are converted into tabular logarithms by 
 multiplying by 0.4342944819 &c. The resolution of tliis 
 probleili derived from other principles may be found in Mr. De 
 Moivre^s Siippl. Miscel. Anah/t. p. 17, and Mr. StirJi)igs 
 Tract, de Sammat. Serier. p. 1 19. Because the other coeffici- 
 ents of the terms of a binomial (when the exponent ;• is 
 an even number) are found by multiplying the coefficient of 
 
 the middle term by -4~x X ~- X -f- X &c. these may be 
 
 likewise found by art. 838;, &c. For the use of the properties 
 
 of the terms of a binomial, when raised to a high power, see Mr. 
 
 VOL. II. T James
 
 274 Of computing the Area Book IT. 
 
 James Bcrnouill'is jIis. Conject. part 4, chap. 4, and Mr. Dc 
 
 Moivre'a Doctrine of Chances. 
 
 1 1 I 
 
 84.5. The sum of the scries — ; — =:r + 
 
 ■m'^ pi + e m-^'Ie 
 
 r- :,. • -j- • ;. — &c. is (by art. 841^ because fr/'vvith all 
 
 tit-\-3e vi-\-be 
 
 its fluxions ullinuilely vanish in this case) + —- — 
 
 12;«/« 10m//.' ItSmw 
 
 "X ft'D -}- cS:c. where Ain the usual manner denotbs the first term, 
 ■ , B the second. C the third, not includini? its sisrn. Sec. If 
 
 r rr 1, then S = -- + + r — - — + &c. 
 
 '2m 4a: m 8/;i+ 4/m^ Sm' ^^' 
 
 And hence the sum of tlie series 1 — -f +y — ;!: + i — i + 
 &c. (which is equal to the hyperbolic logarithm of 2) may be 
 easih' computed to a great number of decimal places, by firiyt 
 collecting the sun:i of the terms at the beginning of the series 
 
 (by common arithmetic) that precede --, so as that m may 
 
 be a pretty large number (ecpial to 0,5 or 27, for example), and 
 then computing the sum of the other terms by this series. If 
 
 1 113 
 
 m 2, then S = 1- -— - — — - + -— - — 5ic. whence 
 
 '2mm <2>« '2//r 2m' 
 
 the sum of the series 1 — ^ + J, — -J^ + ^ — ^^ 4- &,c. 
 
 may be computed in like manner. By supposing r =: 1 and 
 
 CZZ2, the sum of the series I — ■ ^ + ^ — y + i — tt+ tj 
 
 -V + &c. ma)' be computed by first collecting the sum of 
 
 the terms at the beginning of the series that precede -, viz. 1 
 ». -f f — -1 . . .• -^, by common arithmetic; and then 
 
 1 1 1 8 1 li 
 
 adding — + + — — — + See. This scries is 
 
 ° 2m 2inm tii^ ni' m' 
 
 equal to iof tlie circumference of the circle, the radius being 
 equid to 1, by art. 74(j j and hence the ratio of the circumfe- 
 rence
 
 Chap. iV. from a few equidistant Ordinates. 275 
 
 rence to the diameter maybe computed to many decimal places 
 with little labour. 
 
 846. The theorems in art. 830 and 832 may be applied for 
 approximating to the sum of the series that is formed by substi- 
 tuting successively any numbers in arithmetical progression in 
 
 the place ofcT in the fraction • — - . (where 
 
 x-\- a X ,r4- b X j; -f- c X cCC. 
 
 tt, b, c, &,c. represent any given numbers^j by what was shown in 
 the last chapter concerning the area, when the ordinate is equal 
 to such a fraction, &c. And in some cases this sum may be as- 
 s«igned accurately by art. SQ 1 . 
 
 Si/. If N represent the number whose hyperbolic logarithm is 
 
 fc, the sum of the series i + 1 -j- .^^ _ ^^ + ^ -. 
 j^5^g-^ + &c. ■=: r^—^\ and the sum of the series ^ -- ~ + 
 
 720 302 io' "^ 1209600 ^^' " NN—T* -^ hese appear 
 by supposing, in art. 830 and 832, the curve FM/to be the lo- 
 garithmic, Aa its asymptote, AFor RV equal iothemodulus, and 
 finding the sum of the ordinates by the conimon rule for a geo- 
 metrical progression, and putting this sum equal to the value of 
 S in those articles. 
 
 848. The base A« being divided into any number of equal 
 parts represented by n, let the area AVfa:=.Q, the sura of the 
 extreme ordinates AF -\- af — A, the sum of all the intermedi- 
 ate ordinates BE + CK -f &,c. n B, the base Aa z=. R, and the 
 same quantities be represented by /3, 1, ^, Sec. as formerly; then 
 
 the area AVfa -Q = _A_ .~I^x R — ?l1 + i^ 
 
 ^ 2/Z+2 ^ vn—\ 720«« 30240 
 
 nn-l^\ o -n. • • ^ c^ c/— AF 
 
 X — ^j &c. box supposmg, m art. 830, c——^ S + =^— ^ — * 
 
 — B4- — = V —■ — ;7-;^ V ~— — &.C. and sup- 
 
 2 K 12/j 720«3 30240/15 ^ 
 
 posing e rr R in the same theorem, — —-^ z= - r: ~ •{< 
 
 R5 R^5 , R'^ o .1 x. 
 
 1:2- ■" -7?o- "^ Tm ■" ^''- ^^'^"' '^ '''^ exterminate /2 
 
 T 2 by
 
 ■076 Of interpolating Book II, 
 
 by these two equations, the proposition will appear. It" 
 we neglect 5, t, d. Sic. AVJa = -^^^ + -^^^^-^, Sup- 
 pose n ■=■ 2, or that there are three ordinates only (in which 
 case B denotes the middle ordinate), then the area A¥fa =: 
 
 :^±f^ X R— J^ + — ^- —Sec. Ifvve-suppose7/-3, 
 
 6 4x7^20 16x30240 ^^ 
 
 or that there are four ordinates onlj^, B will represent the sum of 
 the second and third, and the area AF fa — — Z x R — — — 
 
 •^8 9x720 
 
 + '. z-^ — Sec. Bv nearlectina: 5, K, G, Sec. we shall 
 
 81x302i " ^ Q ^ > ^ 
 
 have two of the theorems given by Sir Isaac Nercton and 
 others for computing the area from equidistant ordinates, the 
 
 latter of which (viz. AFfa =z '— - — x R) is much recom- 
 mended by Mr. Cotes. 
 
 849. By exterminating 5, ^, 6, &c. successively, other the- 
 orems will be found by which the area will be more and more 
 accurately determined from the ordinates. Let there be five 
 ordinates, A the sum of the first and last, B the sum of the se- 
 cond and fourth, and C the middle ordinate : then the area 
 
 J — 90 ^ ^^ 6x16x16x30210 ^ ' 
 
 for by the rule for three ordinates ^ — '—^ — X R — 
 i_ -f ^ • By dividinac the base into two equal 
 
 4x720 16x30240 -^ ^ '■ 
 
 parts, and computing from thesame rule the area thatstands up- 
 on each part, and adding these areas together, ^ ,zz —^A- — 
 
 R^S oR'f 
 — + ^TTrTT-r — ^'^' then, byexterminatina: 
 
 32x720 ^ 2x16x16x30240 •' » 
 
 § by these two equations, the proposition appears. These the- 
 orems may be continued in like manner, and some judgment 
 formed of the accuracy of the several rules, by comparing the 
 quantities that are neglected in them. 
 
 8jO. The
 
 Chap. IV. the intermediate Terms of a Scries. -77 
 
 850. The theorems in art. 830 and 832, from which we have 
 drawn so many conclusions, may he of use for interpolating the 
 terms of a series likewise, or for finding the intermediate ordi- 
 natcs of a figure when the equidistant primary ordinates are 
 given. When the equation of the figure FM/is known, the in- 
 termediate ordinates are found without any difficulty, by sub- 
 stituting the intermediate values of the base in theequation; but 
 it is not so obvious how we are to interpolate the values of S, 
 or thesums of thoseordinates. SupposeFNs Cfg-3\7) tobethe 
 figure whose succcssiveordinates at the points A,B,C,D,&c. are al 
 ways equal to the successive sums of the ordinates of the figure 
 FlNl/'at the same points beginning with AF; that is, let AF = 
 AF, Bf = AF + BE, Ck = Bf"+ CK, Da - Ck + DL, 
 &:c. and let it be required to determine any intermediate or- 
 dinate PN of the figure VNz. Let this ordinate PN meet the 
 curve FM/in M, AF = a, PM =r y, the common distance of 
 the ordinates AB = e, the area AFMP — Q, '>/ — a — &, '^ 
 _ -J =: 5, SCO. then PN = S + 1±^ + i^ - g + 
 
 ~S_ — , -f Sec. because, if we suppose PN to 
 
 3024.0 1209o00 ' ^^ 
 
 move successively into the places of the ordinates of the 
 figure FNs at A, B, C, D, &c. its successive values will be 
 rightly determined by this theorem, by art. 830. Or if we 
 Tv'ould avoid the arpa Q in the theorem for PN, let APrr/n, and, 
 
 since :- == — + — — + — _ — ixc. It follows. 
 
 2 >» 12 720 30240 
 
 that P\T — t±l V ^+!i _L ^"""•* V R ^^~"'' V X -L 
 
 gQ2^ X ^ — Sec. A siiijilar theorem follows from art. 832 : let 
 
 AR be taken backwards from A, and Vr forwards from P, each 
 equal to i AB, R V and rv meet FM/in V and v ; let Q now denote 
 the area RVvr, and /}, 5, ^, &c. denote the differences by which 
 the first, third, fifth, and higher alternate fluxions of the ordi- 
 Hate rv, exceed the respective fluxions of RV, and AB = e, as 
 
 T 3 for-
 
 278 Of interpolating Book II, 
 
 formerly : then any ordinate PN = _1 — -I — + IlJ^ — 
 
 •^ -^ e 2x12 8x720 
 
 ^ — + &c. The ordinates al the points A, B, C, D, 
 
 &c. are called the primary ordinates of the figures FN^ or 
 TMf. If Pp = AB, /j« meet FNz in n and FAJ/'in m, then j9« 
 r:; FN •!« p«z or PN — pm, according as Pp is taken forwards 
 or backwards from P : and hence any intermediate ordinate 
 PN being known,, all other ordinates of the figure FNz that are 
 at a distance from i t equal to AB, or any multiple of AB, are easily 
 found by adding or substracting the intermediate ordinates of 
 the figure FM/". 
 
 851. LetTX and T' X'be the primary ordinates of the 
 figure FiM/' adjoining to the intermediate ordinate PN; bisect 
 TT' in X, let the ordinate a,;?/meetFM/in //, the area xi/vr—q^ 
 the ordinate at T of the figure FNz, viz. Tr — J, x\j — y, rv 
 
 =z u, then PN =/4- ^ — o—j ^ '^""-^ "^ sTtFo ^ "— J^ 
 ■ — Sec. For if RV =r a, the area RVrr rr Q, then RVyj? — 
 
 Q— /7, and PN = - — ^~fj x '« _.;j + ^^^^X « — a— &c, 
 by the last article ; and "Ir —f — —- — Tr-ro X y— « + 
 
 fx720 
 
 X 'if— a — cic by art. 830; consequently PN — /n: 
 
 7 — sT ^^ + 8"xW ^ «-i; — &c. andPN zz/4. -f 
 
 — V "^~ + &c. This series will convero;e very fast in 
 
 many cases, when PN is at a great distance from AF. 
 
 852. This leads us to some easy and simple theorems for 
 finding the intermediate terms of a series by interpolating the 
 difitbrences of the terms. First, suppose the diflerences of the 
 terms to decrease continually, so that by continuing the series 
 these differences may become less than any given quantitv, but 
 never vanish; or that the terms of the series being represented 
 by the ordinates of the figure FNz, and their diflFerences by the 
 ordinates of FM/, this latter figure has the base F/for its asymp- 
 tote.
 
 Chap. IV. the inlermediate Terms of a Series. 279 
 
 tote. In this case ttv the term of the series tliat precedes the 
 first primary term AF, at any distance A% less than AB, is 
 equal to tiie excess of the sum olthe priauuy differences AF+ 
 BE + CK + &c. above the sum of the interpolated differ- 
 ences be + ck + dl + ^c. the chstances Bb, Cc, Dd, 8cc. being 
 each equal to At, iind taken the same way from B^ C, 1), 
 &c. For in this case PS is ultimately equal to Tt; that is 
 (supposing PT' = ttA), ttv + be »>5*c/; 4* J/ + &c. is ultimately 
 equal t© AF -J- BE 4* ^^ 4" "^^"-j consequently %v :=^ AF — 
 *£" 4- BE — c/c 4- CK — dl 4- &c. 
 
 853. For example, suppose AFrrl, BE — |, CK = i, DL 
 ~ ^, &c. then the successive primary ordinates of the figure 
 
 T?x- -n 1- 3 11 25 137 147 o t . a t * -r. 
 
 FiVwillbel,^, -, 725 -60 5 -60J ^^' LetAxrriAB, 
 and because the intermediate differences be = ^, ck rz |-, c?/rz 
 f , &c. it follows, that tv — 1— 1 + ^ — t + t — f + f 
 ~ t + &c. = 2 X 4- — i + 4: — i + J- — I -5- &c. =1 (be- 
 cause log. o-i_|. + ^_i.4.±_|.4. 8cc.) 2 X 1-10,^.2. 
 And the other mtermediate terms are found b}^ adding succes- 
 sively the intermediate differences f, f , ^, &c.* 
 
 854. In like manner, if we suppose AF =: 1, BE—^, CK =: 
 f, DL =: yV> ^c* ^1' the successive primary ordinates of FM/" 
 to be the squares of 1, ~, -j, ^, \, &c. and we suppose At =: 
 4 AB, then the intermediate differences be, ck, dl, &c. will be 
 ^o> 4-g) T9":> *-^c. the ordinates AF, Bf, Qv., Da, i.'v-c. will be 
 
 * The intermediate terms of this series are determined by the learned Mr. Euler, 
 Comment. PetropoL iom. b, p. 93, by finding a fluent that expresses the terms of the 
 
 1— v" 
 series in a general manner, which in this case is F, X''* supposing n to denote 
 
 the place of the term in the scries (that is, 1 for the first term, 2 for the second, &'c. 
 and i for the term ^j^i), and 1 to be substituted for .v after the iluent is determined ; 
 
 l-VJ . 
 
 whence ^v = ^^YH X *• = 2 — 2 x log. 2. I take this opportunity to 
 
 mention, that, having occasionally shown, in 1737, the 292 and 293d pages of this 
 Treatise (after they were printed) to Mr. S/irling, he took notice that a theorem 
 similar to the first of these described in art. 352 had been coinmimicatcd to him by 
 Mr, Eu/er. 
 
 T 4 1,
 
 280 Of interpolating, &c. Boole IT. 
 
 1, 1 + I. 1 4- I + i, 1 + I + f + ^\, &c. and the or- 
 dinate TTV that stands bel"ore the first primary ordinate 
 AF, at hall' tb.e common distance of the ordinates, will be 
 
 4"" 1 4i 1 Aa 
 
 eoiial tol — -J-- — _j__ _ r s^p — A ^ 
 
 I 9 ~ 4 25 • 9 49 — 4 X 
 
 4 — 9 "^ ~Q — 25 + ^^^- Therefore, if the sum of the 
 
 1 1 1 1 1 
 
 series 1 — - + - _ -^ _}. -^ ^ _ + 8i.c. (which may be 
 
 computed easily from art. 845) be denoted by N, then xv =: 
 4 _'4N. If AF = 1, BE = \, CK = f, DL = ^V <^c. 
 and Att =i AB, then ttv— 1 — ^ +i — f + &c. which 
 is equal to the eighth part of the circumference of the radius 
 unit. 
 
 855 rTi>.3 1 8). When the terms may be continued withoutend;, 
 and their second differences decrease so as ultimately to vanish, 
 let K denote the ijltimate value of the first differences of the 
 
 terms; and ttv vviilbe equal to -— — x K added to the excess 
 
 of the sum of the primarydifferences AF + BE + CK, &c. above 
 the sum of the intermediate differences he + ck +c?/ + &o. 
 because in this case the iluxions of rv and xy ultimately vanish, 
 
 PN i§ ultimately equal to/ -1 — , 5 to K x xr rz K X ab— A^ 
 
 and consequently ttv = — —'- X K "J* AF — be + BE — ek 
 
 4- 6vc. A like theorem may be applied, when the second 
 differences of the terras continually approach to a certain 
 limit. 
 
 856. The series 1, 1 x 1, 1 XG, 1 X 2 X 3, 1 X C X 3 X 4, 
 1 x2X3X4X5j £cc. being proposed, let it be required to 
 find the term that is betwixt the two first piiraary terms at 
 equal distances from each. Thedifferences of the logarithmsof 
 the terms are log. 1, log. 2, log. 3, log. 4, log. 5, &c. and 
 the ordinates of the figure FM/' being supposed to represent 
 these logarithms, the inlermediate ordinates will be log. f, 
 log. I, log. I, log. I, &.C. Therefore the logarithm of the 
 
 term required is - + log. 1 — log. |- + log. 2 — log. I + 
 
 lo^.
 
 Fuj.3o3.XU 
 
 H 
 
 F^.QH-N''^ 
 
 V 
 
 N 
 
 Fi^. 316
 
 PlateX)CX'\l/;;?A'I,Vn 
 
 fM.aatl.XU. X, Fi^.3o8.yi.jirtJod T^gaoj). ^^ 
 
 Tig Sio.
 
 Chap. V. Of the general RuJea, kc. ^81 
 
 iQg^ 3 — log. I + &c. v.'hich is equal to ihe Icgarithin of the 
 
 ," r.2 4 6 8 n , __ 
 
 ultimate vakie ot ^- x -^- X y X g- • • x — - X v^/.-fi = 
 
 •\/r 
 
 (by what was shown in art. 842, from Dr. Waliis) log. __.-;con- 
 
 seqnently the term required is equal to half the square root of 
 the cireumfercnee of the radius 1 ;v*'hieh is agreeable to what 
 has been discovered by other methods. This subject might be 
 prosecuted further, and other instances given of the use of the 
 method of liuxions in finding the sum of a series, or interpo- 
 lating its terms; but we proceed to what is more necessary for 
 bringing this Treatise to a proper conclusion. 
 
 CHAP. V. 
 Of the general Rules for the Resolutton of Problems, 
 
 857 (Fig' 319). J[T remains that we describe bi'iefly tlie ge- 
 neralrulesthatarederived from this method for the resolution of 
 problems, andilli)stratethembyexamples. The base AP being 
 repni-sented by .r, and the ordinate PM by y, the subtancrent 
 PT (which is the right line intercepted upon the base betwixt 
 
 the ordinate and tangent) is found by computingjff . When y 
 
 increases while .rincreases, this value of PT is positive, and PT 
 is on the same side of P with PA; but when y decreases while 
 X increases, this value of PT is negative, and PT is on the other 
 side of P. If x vanish in respect of y, PT vanishes, and the 
 ordinate is the tangent; but if ^ vanish in respect of i, the 
 tangent is parallel to the base. If the curve FM be repre- 
 sented by z, then the tangent MT == ~ =: '^ ^ "^ -JL. If 
 
 MN perpendicular to the tangent MT meet the base in N, PN 
 (which is sometimes called the subnormal) zz M. These fol- 
 
 low
 
 <282 Of the general liu/cs ]5ook II. 
 
 low from art. 188, &.c. by which r, ,}, and », are in the same 
 proportion as the right hnes PT, PM, and MT; or as P^.I^PN, 
 
 andMN. For example, if y^ =rt"'--\r, then (art.7CS,/g.320) ^ 
 
 - % and PT rz ^ - mx - m x AP. Let the rai/ SE 
 
 revolve about a given centre S, and meet the curve AEB in E, 
 the arkyE be described from the centre S, SE := r, the ark of 
 the curve AE rr s, the fluxion of the circular ark^E be repre- 
 sented by x, and SP be perpendicular from S on the tangent 
 
 EP in P ; then SP zr ^, and EP - II, by art. 202. There 
 
 s s 
 
 are other theorems relating to the tangents which are of use 
 in particular enquiries, of wiiieh some were given in book I. 
 chap. 8. 
 
 858. When the first fluxion of the ordinate vanishes, if at 
 the same time its second fluxion is positive, the ordinate is then 
 a minimum, but is a maximum if its second fluxion is then ne- 
 gative; that is, it is less in the former, and greater in the lat- 
 ter case than the ordinates from the adjoining parts of that 
 branch of the cuiTe on either side. This follows from what 
 was shown at great length in chap, g, b. I., or may appear 
 thus. Let the ordinate AF r: E, AP rr r (fig. 319), and, the base 
 being supposed to flow uniforml}', the ordinate PMr=(art. 751) 
 
 E -1- ~ + f" + £i_. + &c.; let Ap be taken on the other 
 
 A.- S.v" Gx " 
 
 side of A equal to AP, then the ordinate pm = E — —1 J^ 
 
 X 
 
 E f* E.'"^ , . E r^ 
 
 -^— r--+ Sec. Supposenow E:=o,thenPM=:E^+ -r — - 
 
 E.r^ 
 ^c. and pm r:: E * + — ?cc. Therefore if the distances 
 
 AP and Ap be small enough, PM and pm will both exceed 
 the ordinate AF when e is positive; but wiU be both less than 
 
 AF
 
 Chap. V. for the Resolution of Problems. 283 
 
 AF if E be negative. But if e vanish as well asE, and e 
 does not vanish, one of the adjoining oidinatcs PM orpm shall 
 be greater than AF, and the other less than it; so that in this 
 case the ordinate is neither a maximum nor minimum. We al- 
 "jvays suppose the expression of the ordinate to be positive. 
 
 859. In general, if the first fiiixion of the ordinate, with its 
 fluxions of several subsequent orders, vanish, the ordinate is a 
 minimum o\- maximum, when the number of all those Huxions 
 that vanish is 1, 3, 5, or any odd number. The ordinate is a 
 rninimum when tlieliuxion next to those that vanish is positive; 
 but a maximum when this iiuxion is negative. This appears 
 from art. 261, or by comparing the values of PM ancljjm in the 
 last article. But if the number of all the fluxions of the ordi- 
 nate of the first and subsequent successive orders that vanish be 
 an even number, the ordinate is then neither a maximum nor 
 piiftimum. 
 
 860. When the fluxion of the ordinate y is supposed equal to 
 nothing, and an equation is thence derived for determining x, 
 if the roots of this equation are all unequal, each gives a value 
 of X that may correspond to a greatest or least ordinate. But if 
 two, or any even number of these roots be equal, the ordinate 
 that corresponds to them is neither a maximum nor miuimum. 
 If an odd number of these roots be equal, there is one maximum 
 or m«y/i/;»//?i that corresponds to these roots, andoneonly. Thus if 
 
 ^ zz x^ -{- ax'' + hx'^ + cr + d, then, supposing all the roots 
 
 of the equation a,* -f ax'^ + hx^ + ex + d -rz o to be real, if 
 the four roots are equal, there is no ordinate that is a maximum 
 or minimum; if two or three of the roots only are equal, there 
 are two ordinates that are maxima or minima ; and if all the 
 roots are unequal, there are four such ordinates. 
 
 86 1. To give a few examples of the most simple cases. Let 
 y z=. a'-x — x^, then ^ = a'^'x — 3x^x and j." ■=. — Gx.x-^. Sup- 
 
 a , 
 pose ^ zz o, and 3x^ = a' or x = Tz-F' ^^^ which case ^ =:
 
 264 Of the general Rules Book II, 
 '■^' Therefore, .^ being negative, ?/ is a maximiun wlien 
 
 V 3 
 .r — - 71-1:: , and jts greatest value is -—7=. Jfy •=. aa -{■ ibx 
 
 — rr, then ], — ibx — Ixx , and i, -- — ■ 2>; consequently 
 y is a maximum when 26 — 2x =: 0, or xrzb. li'y zz aa - — 
 Qbx 4- XX, then ^ = — 2bx +2X;^,and y =r2i'; consequent- 
 ly y is now a minimum when xzzb, if" a be greater than /;. 
 
 862. TherighthnesBFandGH (yzff 321) being perpendicular 
 to the given righ t line BG in the same planCjH a given poi n t,C any 
 point upon BF, and the figure being supposed to revolve about 
 the axis BG, let it be required to determine the position of the 
 rio-ht line HC when the conical surface described bv it is a 
 viinimum. l-et DE bisect BG perpendicularly in D, and meet 
 HC in E, and (by art. 2 16) the surface described by HC about 
 the axis BG will be as DE X EH =: (supposing GH = a, DG 
 ::r b, DE z=i X, and consequentl}^ EH^ zz bb -\- a — x" ) 
 x\/ bb-\-aa—'2-ax + xx. Supposc, therefore, j/ =: 6\r^ + a'^x^ — 
 2ax^ + x^, then^ = ^x'^'x — Qax'^'x + ^a'x'x + ^ib'^x'x, and 
 'y zz \1x'x'^ — \^axx^ + la'x'' + ^b\-^. By supposing 
 ^ zz 0, Ave have iixx— &ax + 2aa + ^bb X x ZZ ; tiie re- 
 solution of v/hicli equation gives (besides x z± u) x =: 
 
 So + Vaa—Hbb 3a \/aa—Sbb ^nr^rr ^->r^ 
 
 oxx — • IfGII^-v/VxBG, 
 
 then aa zz SZ^Z*, and these tuo values of j. become equal to 
 
 each other and to-. In this cascT- =12 X -* x — 
 4 ■ x'^ 44. 
 
 + 2«fl + — rr o, and y is neither a maximum nor minimum; 
 
 but while we suppose the point C to move from B along the 
 right line BF, the conical surface described by the right line 
 HC about the axis BG continually increases, though its fluxion 
 
 00 IT 
 
 vanishes when DE = ""j"* If GH be greater than v^2 xBG, 
 
 tlienrt«> 866, the former value of xgivesj^ positive, tin(\yii mini- 
 mum :
 
 Chap. V. for the Resolution of Problems. 285 
 
 mum ; but the latter value of j gives}' negalive, and y a max'mum; 
 
 that is, tlie value of j/ is greater when .r ~ "^ '^ ■"'-^^'' thur, 
 
 4 
 
 its atljoining values on either side. But this is not to be under- 
 stood as if the value of ij was then the greatest possible; for it 
 is obvious that, by supposing the point C to proceed in the right 
 line BF, DE X EII may exceed any given rectangle. See art. 
 239. AVhen GH is less than v'~ x BG, aa is less than 8bb, 
 and the values of x are imaginary. Examples of this kind may 
 frequently occur; and Avhat has been shown of ordinates is 
 transferred to the rays that are drawn from a given point to a 
 curve, by art. 1277. 
 
 8().j. When ^J = o, if j," be at the same time infinite in re- 
 spect of X (which is supposed constant), we cannot conclude 
 that y is then a maximum or minimum without some further 
 enquiry; for the ordinate may then pass through a point of con- 
 trary flexure or a cuspid. Let -^ — — ^^fZZfl^ then ^ zr 
 
 ~ — . , The supposition of L =. o gives a.r — xx =r o, 
 
 2a \/ ax— XX ^^ J CD f 
 
 and orrr a, or x zz o ; in both cases }' is infinite ; and it is ob- 
 vious that the curve is reflected from the ordinate, because 
 when X is supposed greater than a, or negative, the values of L 
 are imaginary. In like manner, if j,", ],", and }, vanish, and ]; be 
 infinite in respect of x, we cannot thence conclude y to be a 
 maximum ox minimum. But it maybe admitted as a rule, that 
 when ^'=■0, and, x being constant,]^ is real and finite; or when 
 any odd number of fluxions of i/ of the successive orders jj, if\ 
 ],', &.C. vanish together, and the fluxion of the next order to 
 these is real and finite in respect of ;^, we may safely conclude 
 (without any further enquiry) that;y is then a maximum ox mi- 
 nimum, according as this last fluxion is negative or positive. 
 However, when, after supposing y — o, x is determined by a 
 
 simple
 
 2S6 Of the general "Rules Book II. 
 
 simple C({aation, we may conclude 7/ to be a maximum or mini- 
 mum without farther trouble. 
 
 8G4. It was observed in art. 244, that when any quantity 
 
 N is expressed by a fraction -^ if P and Q vanish at the same 
 time, we are not thence to conclude that N =: 0. Thus, sup- 
 
 pose N = 7-=*, ana when x — a, the numerator and dt- 
 
 * a V ax 
 
 nominator of N vanish together; but if we reduce the value 
 of M to a more simple form, by dividing the numera- 
 tor and denominator by their common divisor >/T — V~7, 
 
 we shall find 1^-a X ^~+^_L- (when x = a) a xi^ 
 
 a 1/ a 
 
 t: 2a. In such cases the value ofN is found by computing 
 
 — ; because when P and Q decrease till they vanish, the ulti- 
 Q 
 
 mate ratio of P to Q is that of p to d. If P and Q vanisli 
 
 p 
 
 at the same time, then N = — • This rule was ^iven in the 
 
 Q 
 
 Anal, des Injiniment Petits,]). 145, and is sometimes of usein pre- 
 venting mistakes concerning the gl-eatest and least ordinates 
 ( as are described Mem, de VAcad. des Sciences, 17 06), as well 
 as on other occasions. The computations in enquiries of this 
 kind are sometimes abridged by art. 730. Thus ifmxx ±= nj/y 
 -i-T^iiZ:^ X j/Tj thenjqr;!; X m^ — o, andj^ -\-mx X ;; — ny 
 =: 0. 
 
 865. The 2;reatest and least ordinates are likewise discover- 
 ed, in some cases, by supposing ^ to be infinite in respect of x -, 
 \)ut it is obvious that there are several exceptions to this rule, 
 Sincethe curve may then form a continued arch that is reflected 
 from the ordinate after touching it, or may be continued on 
 tlic other side wi th a contrary flexure. See art. 262. By com- 
 paring the signs of ^ on the ditferent sides of the ordinate 
 (which in this case is a tangent to the curve), tiie latter of these 
 cases maybe distinguished from that wherein the ordinate is a 
 maximum OYmininium; and when the curve is reflected from 
 
 the
 
 Chap. V. for the Resolution of Problems. QS7 
 
 the ordinate, some of the values of^ beeoine imaginaiy on one 
 side of" that ordinate. As for the maxima and minima, which 
 were said to be of the second kind in art 240, see art. 27G. 
 
 8()6, The points of contrary flexure and reflexion are usu- 
 ally determined by supposing y ■=: o or infinite. But this rule 
 being liable to several exceptions, it was shown, in art. 263, 
 that the ordinate y passes through a point of contrarj^ flexure, 
 when, the curve being continued on both sides of the ordinate, 
 ^ is a ?7iaximum or minimum ; which (by what has been shown) 
 does not always happen when ^ =-o or infinite. Hence, if ^ 
 
 ~ 0, and 7/ be real and finite, then ?/ passes through a point of 
 contrary flexure (fig. 319). Thisappearslikewise by comparing 
 thevaluesofPMandpw inart.859. Let PMmeetthetangentat 
 
 Ex 
 
 Fin V, and pw meet it in r; then PV r= E +-r— , and pv :r 
 
 E — ~ ; but when E =:o, PM -E + ^^, + ~ 4- &c. 
 
 X XX 
 
 andy;w rz E — ^j^ r- + &c. ; consequent!}', if E be 
 
 positive, and the distances AP and Ap small enough, PM will 
 
 be greater than PV^, and^m less thanjst',- and whether e be 
 positive or negative, the arks FM and Vm shall be on difi'er- 
 cnt sides of the tangent TF^,- consequently' F will be a point 
 
 of contrary flexure: but if E likewise vanish, and E be of a real 
 value, PM andpw will be both greater or both less than the re- 
 spective perpendiculars PV andpy intercepted by the tangent^ 
 and there v> ill he no point of contrary flexure at F. In gene- 
 
 ral, if ij, I/, y. Sic. vanish, the number of these fluxions be- 
 ing odd, and tlie fluxion of the next order to them have a real 
 and finite value, then y passes through a point of contrarv flex- 
 ure; but if the number of these fluxions that vanish be even, 
 it cannot be said to pass through such a point, unless it should 
 be allowed that a double infinitely small flexure can be formed 
 
 at
 
 S8S Of the general Prides Book IL 
 
 at one point. To give one of" the most simple examples, sup- 
 pose j/ = 1 — .r'^, then y zz — Ax^x, y— — \1x x'^, j/ = — 
 ti^xx^, and y ■=: — 24.r'^. If we suppose y ~ o, then x — o; 
 
 but, because y is then likewise nothing, and y real and finite, 
 y does not pass through a point of' contrary flexure, but is in- 
 deed a maximiuii ; the truth of which might easily be shown 
 otlierwise. 
 
 8G7. The curve being supposed to be continued from the 
 ordinate PAT, ov y,on both sides, if/ be infinite, M is notthere- 
 fore always a point of contrary flexure, as j^ is not in this 
 case aKvays a maximum or minimum^ by art. 813 j, and the curve 
 jnay have its concavity turned the same way on both sides of 
 M. But these cases may behkewise distinguished by comparing 
 the signs of y on the different sides of PM, for, when these 
 signs are different, M isa point of contrary flexure: for example, 
 
 let ?/ ~ 1 — .r^^, then j/ = — -t-^=' whi eh becomes infinite 
 
 when .mo or j/= 1, and is affected with contrary signs on differ- 
 ent sides of y; consequently the ordinate passes through 
 a point of contrary flexure when x rr 0. The suppositions of 
 *y — or iniinity, and of i/ := or infinity, serve to direct us 
 where we are to search for the maxima or mimima, and for 
 points of contrary flexure, but where we are not always sure 
 to find them; for though an ordinate or a fluxion that is posi- 
 tive never becomes negative at once, but by decreasing or in- 
 creasing gradually (as w'as shown in art. 262), yet, after it has 
 decreased till it vanish, itmay thereafter increase,continuingstin 
 positive; or, after increasing till it becomes infinite, itmay 
 thereafter decrease, without changing its sign. 
 
 868. Tlie points of reflexion, or cuspids, were distinguished 
 into two kinds iti art. 268. When the curve is reflected from: 
 the ordinate PM ox y, it always forms a cuspid, unless when 
 y is infinite in respect of x, in which case likewise M is some- 
 limes a cuspid of the second kind ; and when y or y is real 
 and finite, M is always a cuspid of the second kind. If/ —o, 
 
 the
 
 Chap. V. for the Resolution of Problems. 289 
 
 the cuspid may be of either kind. But the most simple kind 
 of the cuspids of the first sort (such as are in some of the Unes 
 of the third order) are formed when y is infinite, as the most 
 simple kind of points of contrary flexure are formed where y 
 no; see art. 270 and 379. When^ is such a maximum or 
 minimum as was described in art. 865, y passes through a cus- 
 pid of the first kind. Other observations may be derived from 
 art. 269. 
 
 869. Suppose(asinart.857)ST (;/fg.,S22) perpendicular from 
 the given point Son the tangent PT in T, SP = r, the fluxion of 
 the curve equal to /, the fluxion of the ark/P described from 
 
 the centre S equal to «',• consequently ST — ^ ; and (by 
 
 s 
 
 art. 281) P is a point of contrary flexure, when the angle 
 SPT is oblique, and ST is a maximum or minimum; whence 
 rules may be deduced analogous to the former for determin- 
 ing those points. Suppose % constant, and, the fluxion of ST 
 
 being equal to "^JLLlULLl. , the points of contrary flexure 
 
 are found by supposing 'r $ .t /,' or (because V s "=■ r r »{« 
 2 a and s "s •=. r r ) '^ — T 'r\ equal to nothing or infinity ; 
 but with exceptions similar to those described in art. 866 and 
 
 867. 
 
 870. LetC(J?g.3 1 9)be the centre of the curvature at M,C6 per- 
 pendicular to PM in h, AP=.r, PM=:^, the ark FM=s, and 
 
 (by art. 382), supposing x constant, M6 ~ il r= ILiLlI, or 
 (because ,' V = ^ y) M& = 4r-j and the ray of curvature 
 
 s 
 
 CM = -7-^. For example, if ay = xx, then a'^ = o^x'x^ J} 
 
 = 2xh x» + ^* = ■"T?'" X ^* = •—-- X ^*f and Mb = 
 VOL. II. U ^ s*
 
 290 Of the general Rules Book IL 
 
 ^ =: 2y 4- la. If the ray of curvature be expressed by R, the 
 
 3' 
 
 variation of curvature (according to Sir Isaac Newton's ex- 
 plication) will be as -p. But we have insisted on this sub- 
 
 ject at length in chap. 11, b. I. 
 
 87 1. Resuiningthesuppositionsinart.869,let(^o^.322)ST=j:;/ 
 
 then the ray of curvature at P, viz. PC = — ; and^ ifCIbeper- 
 
 pendicular to SP in I, IP = ~. This was demonstrated in 
 
 art. 384, and may be briefly shown thus. Let S^ be per- 
 pendicular to /?^ the tangent at p, and the arks tn, pu, de- 
 scribed from the centre S, meet ST and SP in n and u. 
 Then the angles VCp, TSt, being equal, PC will be to ST 
 in the ultimate ratio of Pp to tn ; but IP is to PC in the 
 ultimate ratio of pu to Vp; consequently IP is to ST in 
 the ultimate ratio of pu to tn, or (because the angles SPp, 
 STtj are ultimately equal) of P« to Tw, that is, of r to 
 
 ;,. therefore IP = A and PC = IP x - = — . And by sub- 
 stituting for p the fluxion of ^, or (supposing the circle AD 
 
 s 
 
 to be described with the given radius SA from the centre S, SP 
 to meet this circle always in D, SArra, AD =:: c, and conse- 
 quently » — — j of -^^j and supposing c", z, /, or r, constant, 
 
 a ' as 
 
 various forms may be derived for expressing the ray of curva- 
 ture CP, or IP half the chord of the circle of curvature that 
 passes through S. To give one of the most simple examples, 
 let i : a : : a^ : r", as in the figures constructed in art. 393 ; 
 
 then;) = — = — —> — = ;»+ 1 x _, IP = ^ = -—-, 
 s a"^ p r p n+V 
 
 1 rt" 
 
 a^d PC = -i- X JL^. 
 w+1 ;»—' 
 
 872. The
 
 Chap. V. for the Refiolution of Problems. ^g\ 
 
 872. The rest remaining as in the last article^ suppose S to be 
 ii radiating points SP any ray incident upon the curve AP, and 
 reflected by it so as to touch the caustic at m. Then the angle 
 CPm ~ CPS; and the reflected ray Fm will be to the incident 
 
 rav SP (or r) as 1 is to — x t- + 1^ where this unit is to be 
 p 
 
 added or substracted according as the ark at P has its convexity 
 
 or concavity towards the radiating point S. For if CU be 
 
 perpendicular to Fm in R, PR bisected in q, and P/ be 
 
 taken on the reflected ray equal to the incident ray PS ; then 
 
 (art. 410) of: o-R : : ^R : qm, and P^ being equal to — , it 
 
 follows, that Pm : SP : : — ; r + ~, For example, if i : i : : 
 
 tt" : /■«, then - x- ^ n + I, and Fm : SP : : 1 : Qn + 1. 
 P 
 873. Suppose the curve AP (fig. 322, N, 2) to refract the ray 
 SP, let PM be the refracted ray, and touch the caustic in this 
 case at M. The rest of the construction remaining the same as 
 before, let Or be perpendicular to PM in r, PR = e,Fr =f3 
 PM = X, and let the constant ratio of the sine of incidence to 
 the sine of refractidn (or of CR to Cr) be that of « to 1 ; then 
 (by art. 413) PM : rU : : x : x—f: : CR X SP X Pr t 
 C/- X SR X PR : : nfr : r^ X e ; consequently x r= 
 
 -'-' £> K^Mf^fv ^/-inol tr\ ^__ 
 
 , , e being equal to ^ and /"—-:- X \/„„—\ x rr -hpp^ 
 
 nfr—er±ee1 p '' np 
 
 874. Suppose the curve AP (fig. 3Q.%N. 1 ) tobe described by any 
 centripetal forces, and the force that acts at any point P will be 
 directly as the square of the velocity at P, and inversely ias half 
 the chord of the circle of curvature that is in the direction of 
 the force : when it is directed towards a given centre S, the 
 ctrea described by the ray SP about S flows uniformly ; the ve- 
 locity at any point P is inversely as ST the perpendicular from 
 S on the tangent, and is to the velocity by which a circle could 
 be described about S at the same distance SP by the same cen- 
 
 U 2 tripetai
 
 X 
 
 29*3 Of the general Rules Book IT. 
 
 tripetal force as - to ^ - • and the force at P is as -^, 
 r J) pif ' 
 
 because the velocity is as 1, and PI =: £!_• The same force is 
 
 as-— 7, or as — ± r, the fluxion of the area (or r^) being 
 
 supposed constant. Thus if i : 3 : : a'* : r"^ the centripetal 
 force directed towards S will be inversely as the power of SP 
 
 of the exponent Q.n + 3 (because p = and-^ = n+l 
 
 ^J— j, and the velocity at P to the velocity in a circle at 
 the same distance as ^ ~ to ^ ^ 5 that is, as 1 to -v/T+T. The 
 
 r p ' 
 
 demonstration that was promised in art. 4.51 may be deduced 
 in the following manner. 
 
 875. Let AMB (^g. 323) be any figure th at canbe described by a 
 centripetal force directed towards S that is always as the power 
 of the distance SM of the exponent m. Constitute the angle 
 ASL : ASM : : m + 3 : 2 ; and, supposing SA = \, SM == x, 
 
 SL rz r, let r z=. x ^-; that is, let the angle ASL be to the 
 
 angle ASM, and the logarithm of the ray SL to the logarithm 
 of SM always in the same invariable ratio of w + 3 to 2 ; then 
 the curve ALD may be described by a centripetal force direct- 
 ed towards S that always varies as the power of the distance 
 
 4, 
 SL whose exponent is -— - — 3. For let SQ and SP be per* 
 
 pendicular to the respective tangents of AM and AL in Q and 
 P, SQ =: y, and SP •=. p. Then, by the supposition, -~r- := 
 
 €X^, where e represents an invariable quantity. By finding 
 
 1 2fx"*+' 
 
 tlie fluents -r = 2K — — > where K denotes an in- 
 
 y* m + 1 
 
 variable
 
 Ghap. V. for the Resolution of Problems. Q^S 
 
 variable quantity, according to art. 735. The triangles SMQ, 
 SLP, being similar (art. 394), it follows, that — — — r~i — 
 
 1 2K 2e 
 
 (because r^ zz. a"» + ^) _ __ __ 
 
 2K/- , and -—• = — ^^-— X Kr , 
 
 wi + 3 wi 4- 1 p r jn + 3 m-\-3' 
 
 4 
 or as the power of r of the exponent — -^ — 3 . If the rays from S 
 
 be perpendicular to the curve AMB in A andB, and to the curve 
 ALD in A and D, the angle ASD ; ASB : : w + 3 : 2, by 
 the construction. 
 
 S7 6 (Fig. 32'2). Suppose the centripetal force to be always the 
 same ateqnal distances from the centre S. Let e and V denote the 
 forces at the respective distances SA and SP, A andw the veloci- 
 ties at A and P, let SA zz a, and SP = r ; then mi = F. — 
 2Vr (by art. 43.5) ,* in determining which fluent, care must be 
 taken that u become equal to h when r zz a. When V is to 
 « as r to a, or as aa to rr, the trajectory is a conic section, by 
 art. 445 and 446 ; and when Y : e : : a^ :r^, the trajectory 
 may be constructed by the areas of conic sections, as has 
 been already shown by several authors. When V : e : : a* : r*, 
 the trajectory is constructed, in some particular cases only, by 
 the areas of conic sections (or circular arks and logarithms), 
 but is constructed in general by the arks of conic sections. la 
 this case abody may continually descend in a spiral line towards 
 the centre, and yet never descend so far as to enter within a 
 circle of a certain radius ; and a body may recede for ever from 
 the centre, so as never to arise to a certain finite altitude, but 
 revolve in a spiral that is always within a certain circle. This 
 remarkable circumstance could not take place in the trajectories 
 that are described in the former cases, which have been already 
 constructed by others ; and therefore we have chosen the con- 
 struction of this case for an example of the method of determia- 
 ing the trajectory from the law of the centripetal force. 
 
 U 3 877. Let
 
 294 Of the general Rules Book II. 
 
 877. Let A denote the velocity, and AG or GA be the direc- 
 tion of the body at any given point A. Let h be to the veloci- 
 ty with which the body would describe a circle at the same 
 distance by the same centripetal force as y'l+wm to VT"; that 
 is, let (fig. oQ-i) hh:ac::\-\- mm : 2. Let SG be perpendicular to 
 AG in G, and any ray SP from S meet the trajectory in P, 
 and the circle AX described from the centre S in X, 
 SA = a, SG r=. b, SP rr r, ST (the pependicular on the 
 tangent at P) "^^ p, the ark AX ~ c ; and the same flux- 
 ions be represented by '$ and j^, as before. Then uit ■=. P, 
 
 — ^ — rr r-T + K rr (because when r r:^ a, then uu r= lih 
 
 ae -,T l-i-mm _■, . tt- mmae\ a*-\-mmr* 
 
 = - + K =r -^ X ae, so that K = -^j ■ ^^ ■ 
 
 7 7 a 1 + mm aehbs^ . . i • • 
 
 X «e = nh X — = X :-} consequently i" : z* 
 
 _^___ _ . TV c*- \ 
 
 : : a* + wwr^ : \\.mm X bbr^, and r* : z» (— — } : • 
 a* + >n//2/-+ — 1 -)-OTOT X Z?^/''^ : "Tf^ x bbr^\ therefore 
 
 p — J, , — Ihe ratio of V\A-mm to 
 
 \fa'^-\.^,Mn X bbr^-^vimr"^ 
 
 1 is that of the velocity at A in the trajectory to the velocity 
 that would be acquired by an infinite descent to A. Yim rr o, 
 
 .c — — ■ sand the traj ec tory is an ark of a circle that passes 
 
 througb S, described upon a diameter equal to yj which is 
 
 agreeable to art. 437. 
 
 878. The trajectory is constructed by circular arks and loga-. 
 rithms (and is of that kind of spiral lines which were mention- 
 ed at the latter end of art. 343), when the body sets out from 
 A in the trajectory with a velocity that is to the velocity in a 
 circle at the same distance SA as SA is to ^/SA^ + v'sA* — SG*. 
 In this case (supposing SA* : SG* : : w : 1, or aar=inbb), 1 -^mm : 
 
 2 : : a* : «"• + Va'^^b^ : : n : n ^ i/nn—l, ;w = m
 
 Chap. V. for the Resolution of Problems. 295 
 
 ± v'nn—i, and i-\-mm X bb = — " ^ •■ zz Q.maa ; conse- 
 
 . . 4- 'ruaV'^ ^ 111- * 
 
 quently c = • Suppose, 1, that the velocity at A. 
 
 in the trajectory is to the velocity in a circle at the distance 
 
 SA as v^T to ^/ n + v/ «/j_i (in which case m ■=. n — \/«„ — i), 
 
 upon SA produced take Sk : SA : : 1 : V"^ describe the 
 
 circle AxK from the centre S, take the ark AK (on the same side 
 
 1 1 — V7t 
 of A; that AG is of A) equal to , X loa;. -i the modulus 
 
 being S/c, join SK, and it shall be the tangent of the trajectory 
 at the points. To find any other point of the trajectory, as 
 
 1 ^ ;• 
 
 P ; let SK ■=:. d, take the ark Kx -=■ — = X loo-. -—- 5 join Sar, 
 
 and upon the right line Sx take SPrzr. For, suppose the ark 
 
 Tr 11 . + dd'r V~ - , , 
 
 \s.x — y, then, by art. 731,^ = dd—r ' r — ^ (because ddx 
 
 J\-aa'r y/T , ., au +aa'r a/I^ 
 Cfl : : 1 : m) and c "=■ -^ zz , as itougrht 
 
 aa—mrr 
 
 to be. Therefore describe an equilateral hyperbola Ka^x; hav- 
 ing its centre in S and vertex in K ; let any right line Srw 
 meet the hyperbola in n and the tangent at K in r, then let 
 the circular sector SK x : SKn : : \/~: 1, and SP betaken upoa 
 Sx equal to Kr, and P shall be a point in this trajectory. 2. 
 Let the velocity at A in the trajectory be to the velocity in a 
 circle at the distance SA (fig. 325) as v'h to V;* — '/nn—i, 
 then m =. n + Vnn—^i, SK is to be taken less than SA in the 
 ratio of 1 to v'Z) the sector SKa: : SKn : : VJ: 1 ; and SP is to 
 be taken upon the ray Sjt, so that Kr : SK : : SK : SP, 
 
 879- In the first case [when the velocity at A (Jig. 324) in the 
 trajectory is to the velocity in a circle at the same distance as a 
 to \/a^ -f- v^a4_z,4], if the body set out from A with the direction 
 GA, it will perform its revolutions in a spiral always within 
 the chcle Kxz, and never can arise to the altitude SK from 
 the centre S ; because Kr (to which the distance SP is always 
 
 U 4 equal)
 
 296 OJ the general Rules Book II. 
 
 equal) cannot become equal to SK^ while the area SKw or 
 
 ark Kx are finite. The area described by the ray SP about the 
 
 centre S is always to the hyperbolic area generated by the 
 
 rJQfhtline^vnn the invariable ratio of VT to 1 ; because* : y 
 
 r'z y riWHT -harr'r V^. j ^ n • 
 
 : : rx/^ : a, — =. - — ^ = —n n ? and the fluxion 
 
 of the ai-ea Km(=:SK« — SKr) is ^^^^„,^^ * Therefore 
 if the body set out from A, with the direction AG, it will de- 
 scend in the curve APS to the centre S in the time that, by 
 proceeding in the tangent AG with its velocity at A, it would 
 describe about S a triangle equal to VT x KRN, KPt being 
 supposed equal to SA. In this figure the area SoPxK (termi- 
 nated by the curve SoP, the circular ark K.r, and right lines SK 
 and Pi:) admits of a perfect quadrature, and is to the triangle 
 SKr as v'"2~to 1. 
 
 880. In the second case^ when the velocity at A Cfg. 30.5) is to the 
 velocity in a circle at the same distance as a to ^^ a'- — v' a*— i<, 
 if the body set out from A with the direction AG, it will re- 
 volve in a spiral that always approaches to the circle Kj:, but 
 
 it never can descend to this circle ; because SP (= -j^J can^ 
 not become equal to SK in any finite time. This spiral has an 
 asymptote at a distance from S equal to -^ x V 2 , because, 
 
 by art. 877, pp = i '" i ^ hhr"^, and the ultimate value 
 
 or pp IS y. OQ =. (in this case) — — j — , 
 
 88 1 (Fig. 32G).Iaothercases,thetrajectoryma3' be constructed 
 by hyperbolic and elliptic arks, from art. 805. If the velocity at 
 A be to the velocity in a circle at the distance SA as i/ i—mm 
 to -/"sT 3.nd the direction at A be perpendicular to SA 
 (fit a ^b), then by substituting, in art. 877, — mmiovmm.
 
 Chap. V. for the Resolution of Problems. 297 
 
 'Z a'— I— mm X aarr--)nni7"^ \/aa — rr X V^flj + zwr 
 
 may be compared with -== — ^ _ , b}' supposing 6 
 
 V aa—pp X \/ hh-\-pp 
 
 =: - andp=r. The fluent of this last fluxion was found (art. 
 
 805, fig, 308) to be equal to j X AT? + AE— EP. Therefore, 
 
 when the velocity at the distance SA is less than the velocity 
 by which a circle would be described at the same distance in 
 the ratio of y'l-— »iw to -/ 2^ the trajectory may be constructed 
 in the following manner. Let SD : SA : : 1 : m, S6 : SA : : 
 */\^m \ 1 ; describe an hyperbola AEZ having SA and SD 
 for its two semi-axes, and an elUpse AlXb having SA and Sb 
 for its semi-axes ; draw E/? a tangent to the hyperbola at any 
 point E, and Sp a perpendicular to E/? ; upon SA take SQ z= 
 Sp, and let the ordinate at Q meet the ellipse in R ; then up- 
 on the circle Axz described from the centre S take the ark 
 
 Ax: - X AR + AE — . Ep : : w a/ i—mm : 1, upon the ray 
 
 S^ take SP = Sp, then P shall be a point in the trajectory. 
 In this case the velocity at A is such as could be acquired by 
 a body descending to A from some greater distance by the 
 same centripetal force. 
 
 882. When the velocity at the distance SA is to the velo- 
 city in a circle at the same distance as \/i+mm to v''2, then 
 
 * ~i~raa v 1 + mm zi • v 
 
 c = . ^ J- = (by supposmgpp - aa — rr) 
 
 V aa——rr A v aa — mmrr 
 
 3l /)aa V' 1 + mm ■, ■, . ,i • n 
 
 — =r — ^^=====: ; and, by comparing this flux- 
 
 X^aa—'pp >< V I — mm x aa-\-mmpp 
 
 ion with that in art. 805, it appears that, when m is less than 
 1, we are to take SD : SA : : Vi—mm : m, Sb : SA : : 1 : 
 \/ 1 — mm, and to proceed in the construction as in the last 
 article ; only, after Sp and Ax are deter mined, we are now to 
 take SP upon the ray Sj: equal to -/sa^-— b/^. 
 
 883. When
 
 293 Of the general Rules 
 
 883.When wis greater than l^then^bysupposingjs: 
 
 , aa ' J^fi.T */ l^mm 
 
 andconsequenUyrz: — ==;:: ' "~ 
 
 V^oa— />/ V aa—pp X a^^ ^jn— 1 Xaa + pp 
 
 therefore, in this case, we are to make SD : SA : : \^mm—i : 1, 
 S6 : SA : : m : Vmm—x, to determine Sp and Ax, as in art. 
 88 b awcl then we are to take SP (upon the ray Sr) equal to a 
 third proportional to v'sA^— s/.* and SA. If upon Sxyou take 
 SP a third proportional to Sp and SA, P will be a point in the 
 trajectory which is described by a centrifugal force directed 
 from S that is inversely as the fifth power of the distance. 
 When the direction of the body at the distance SA is oblique 
 to the ray drawn from the centre S, the trajectories may be 
 constructed in a similar manner. 
 
 884. If the curve FM (Jig. 319) be described by powers di- 
 rected in any manner whatsoever, and the force at any point 
 M, resulting from the composition of these powers, act in the 
 direction MK, and be measured by MK ; let MK be resolved 
 into the force MO in the direction of the ordinates MP (=j/), 
 and the force OK parallel to the base AP { zz x)\ then, the 
 time being supposed to flow uniformly, or the velocity at M 
 being represented by the fluxion of the curve FM, the force 
 "^10 will be measured by y\ and the force OK by x\ by art. 
 460 and 466 ; but we insisted on this, and its use, in book I. 
 chap. 11> article 4G5, &c. 
 
 885. Let abody descend along the curve FPA(/' g, 327) by its 
 s;ravitation towards S, the time of the motion be represented by^, 
 the velocity at any distance SP or r by u, the centripetal force 
 at the same distance by g, the ark FP by s ; then the motion of 
 
 the body along the curve is accelerated by the force — ^il — 
 
 s 
 
 If— — (because / — — ) ^; consequently J« — — g'r, uu r: 
 
 1 us 
 
 p — o ' and /' — " When the gravity is uni- 
 
 v^F.-'V 
 
 form.
 
 Chap. V. for the Resolution of Problems. 299 
 
 form, and acts in parallel lines, letz be the space described ia 
 a vertical line from the beginning of the descent, then uu = . 
 
 = F. 2^ =: 9.gz, i — — =, and t—V — The gravity be- 
 
 V2g^ g 
 
 ing still uniform, let {fg. 23-', N. 1) the body begin to descend 
 along the curve DMS from D, MN be perpt;ndicula. to the 
 horizontal line DA in N, the ark SM — s, MN = z, and t re- 
 present the time of descent from M to the lowermost point S; 
 
 then i = - ^ ■ . If DMS be an ark of a semi-cycloid that has 
 
 its axis perpendicular to the horizon, the diameter of the gene- 
 rating circle r::a, ASrz^^ then (by the second property of this 
 
 figure in art. 805) i ; — » :: V7 ; v'^H,", and / — ~" 
 
 If N be to 1 as the semi-circumference of a circle to tiie dia- 
 meter, N shall represent the fluent of —-" — ^ that is gene- 
 
 rated while Z- becomes equal to h ; consequently the time of 
 descent in the ark of the cycloid DMS is expressed by N x 
 
 y -^, and is to the time of descent in the axis a (viz. \^~ \ 
 g g ^ 
 
 fis N to ], as we found in art, 408. 
 
 886. But when DMS is an ark of a circle, f is a fluent of a 
 higher kind, and is not to be represented by the areas of conic 
 sections,butby their arks. LetC(j^"g.238,]V.2)be the centreofthe 
 circle, HCS the vertical diameter, MV perpendicular to HS in 
 V, HS = E, CA = F; then ; : — ^ : : CS : MV : : ^ E : 
 
 ViEE—F¥—2Fz—xz, and / = — '^ > 
 
 2\/^«= X ViEE-Ft'-Q¥z-2Z 
 
 Let this fluxion be compared with (d =) = J~ ' ^ . 
 
 the fluent of which was determined in art. 805 ; and we have 
 hh- \ EE — FF, or h = AD, 2F =: 2e r= ^^=^, and a =
 
 300 Of the general Rules Book II. 
 
 IE — F = SA. Therefore, let S be the centre, A the ver- 
 tex, and SD the asymptote of the hyperbola AE ; produce 
 HD till it meet S6 perpendicular to SA in k, take S6 = DA:, 
 and describe the ellipsis A116 ; let SQ or SP — -/^ and the 
 
 AD 
 
 fluent Q will be represented by -^ X AR + AE — EP, by 
 art. 805 ; t the time of descent from INI to S will be express- 
 ed by Q X — - — — ^, and is to the time of descent in the 
 *' AD- V^s 
 
 vertical SA as Q to ^^, 
 
 887. It follows, from art. 807> that if the semi-circumference 
 be to the diameter as N to 1, and HA : AD ■.m:: 1, then the time 
 
 HS X N 
 
 in the whole ark DMS will be represented by — ■■■ 
 
 V2^ X HA 
 
 X 1 — r — 4- TT-L — 8cc. the ultimate value of which 
 when SA is supposed to vanish, is \/^ x N. Therefore the 
 
 time of descent in the ark DMS is to this ultimate value of ^ 
 (which is said to be the time in an evanescent ark, and, by art. 
 885, is equal to the time in any ark of a cycloid that has the 
 
 diameter of the generating circle equal to | CS) as x 
 
 \ — J- jL. _: L- a. &c. to U By the sequel 
 
 of the same, art. 807, if SH : SA : : « : 1, then the whole 
 time in the ark BMS will be expressed by N ^ ~ x 
 
 I jL. J— ^ ^__ ^ &.C. and the time in DMS will be to the 
 
 4.T 64'i- 
 
 time in an infinitely small ark (or the ultimate value of t) as 
 1 + ±. .1- _L" + -^'^- + &c. to 1. When DMS 
 
 is
 
 Chap. V". for the Resolution of Problems. 301 
 
 is a quadrant, the time of descent is measured by the arks of 
 the /em/«*sc«^a, of which we gave an easy construction in article 
 803 (fig. 307). 
 
 888. If a body descend or ascend in the vertical line 2 in a 
 medium, and the resistance be represented by R, its motion is 
 
 accelerated or retarded by g± R —~ =. i^* and+^a 
 
 / 55 
 
 — e±^ ^ «*•• For example, if the resistance be as the square 
 of the velocity, and a denote the velocity when the resistance 
 is equal to the gravity, or R : g : : wm : aa, then + «« = /s 
 
 aa-\-uu . aa ^Zuu , . u aa 
 
 aa * g aa-^uu^ if + R S 
 
 ^^q~5 whence z and t may be computed from u by loo-a- 
 
 rithms or circular arks. See art. 542. When the body descends 
 along a curve line, it is accelerated by the excess of the force 
 
 ^^~ above R, which is therefore equal to — 5 and if it ascends, 
 
 the sum of these forces is equal to ^-, When a trajectory 
 
 is described in a medium, and the centripetal force is directed 
 towards S (fig.o21),\ei this force at any pointP be to the centri- 
 petal force atP by which thesame trajectory would be described 
 in a void as zto a, and (retaining the same symbols as in art. 869) 
 
 the resistance at P will be as — r, or, if the area of the figure be 
 
 supposed to flow uniformly, as % I (by art. 452), and is to the 
 centripetal force at P in the medium as pr a to 22 's p. If the 
 resistance R be in the compound ratio of the density D and 
 
 square of the velocity mm, then D is as — ^ or (because uu is as 
 
 rf) as Ar and if the curve be such as can be described ia a 
 
 void
 
 302 Of the general Rules Book if. 
 
 void by a force directed towards S that is as any power of 
 
 the distance, D will be inversely as —. If the centripetal 
 
 r 
 
 force in the medium be uniform, and act in parallel lines, and 
 ^ be an ordinate in thedirection of the force, then the resistance 
 
 -^ill be to the gravity as ^ s to 2^'^ ; and if R be as Duu, then- 
 D will be as —-• 
 
 889. Suppose FP A CJig. 327) to be thefigure wliicli is assumed by 
 achain that isperfectly flexible,and gravitates towards thegiven 
 point S. Then ST the perpendicular from S on PT the tangent 
 at P shall be inversely as F. gr, and the tension of the chain at 
 £iny point P inversely as ST, by art. 567. If FPA be the line 
 of swiftest descent from F to the lowermost point A, SA = a, 
 SP (=: /•) meet the circle AD described fi-om the centre S in D, 
 the ark AD =r c, and w be to « as the velocity at P to the ve- 
 locity acquired at A ; then c = — ■ . by art. 581 and 582. 
 
 T V rr—uu 
 
 If the gravity act in parallel lines, let VM.{— y) be an ordinate 
 in the direction of the force, FMni-, PMrsy, the ark FPi=s; 
 
 then if FTA be the catenaria, ~ will be as F. gy, by article 
 
 568. And if FPA be the line of swiftest descent, u denote the ve* 
 locity acquired at P (or u zz "V^F. 2g^ ), and a the velocity ac 
 quired at the lowermost point A, then j : i : : a : m, by art. 
 575 and 576. 
 
 890. The base AP (fig. 3 1 9) being represented by i', and the or- 
 dinate PM by 1/, if the F, i/x, be computed, and the expression 
 be made to vanish when a:=ro, according to art. 735, it will give 
 the area APMF. When the fluent is negative, it gives the area 
 on the other side of PM. For example, let 1/ =: a"*, then F.- 
 
 yx ^ F. x"'x = ' ^ ■ ■, which gives the area when m is any 
 
 positive number, or is a negative number less than 1. But when 
 
 m is
 
 t)hap. V. for the Resolution of ProhkinS. 303 
 
 m is a negative number greater tlian unit, this expression is ne^- 
 gative^and gives the area on the other side of PM (^o-.322).The 
 area generated by the ray SP about S (according to tlie symbols 
 
 in art. 869) is the fluent of ^ or of—. We have had many ex- 
 amples above of the computation of areas from those theorems. 
 There are several general theorems for computing the area de- 
 scribed above, as in art. 752, 819, 830, 832, &c. 
 
 89 1 . The solid generated by the area APMF (fig. 319) about 
 the axis AP is found by computing F. 2Ny* x, where N denotes 
 the ratio of the semi-circumference to the diameter. For exam- 
 ple, let the figure be any conic section, AP the axis, and the ge- 
 neral equation of the figure being yy=.Axx + B jr + C, the so- 
 lid generated by APMF about AP will be equal to — -^ -j. 
 
 NBjt* + 2Nc:r. Let Ap be taken on the other side of A 
 equal to AP, and pm be the ordinate atp, ihen pm^-=.Axx — • 
 Bx + C; consequently the solid generated by the area ApmF 
 
 about the axis Ap will be equal to "-^ NBjt* -f 2NC2*. 
 
 Therefore the solid generated by the area VMmp is equal to 
 
 — ^ + 4NCx. When x =. o, 1/1/ ~ C •, consequently the 
 
 cylinder generated by the rectangle PH//p(HFA being parallel 
 to Pp) is equal to 4NCjr,- and the excess of the frustum gene- 
 rated by the area PMmp above this cylinder is -^ x A:r' = 
 
 (supposing Vp — 2x = v) ^^ ; which (if PZ : Pp : : v'a 
 : 1) is I of the cone generated by the right-angled triangle 
 PZp about Vp, and is always of the same magnitude when t> 
 and A are the same. The frustum is greater or less than the 
 cylinder according as A is positive or negative ; and they are 
 equal when A rr o,- that is, when the figure is a parabola. In 
 this manner the properties of these solids described above, p. 
 
 24, are briefly demonstrated. When the value of F . QNyy^r 
 
 is
 
 304 Of the general Rules Book II. 
 
 is negative, it represents the solid that is generated b}- the area 
 on the other side of the ordinate PM. Thus if yrrj:""'", then 
 
 F. SNvvo: =: ^ = — -x — ——, Avhich expression 19 
 
 -'-' — . '2m-\- 1 2m— -I 
 
 negative when m is greater than |, and represents the limit to 
 which the solid generated by the hyperbolic area on the other 
 side of PM continually approaches whilst that area is supposed 
 to be produced. See art. 307, &c. 
 
 892« The ark FP is the fluent of s, or of '^>+j^-. For 
 
 3. X • 
 
 example, let ai/i/ zz x^, then y •=. ^, y — -Li-j s = 1 X 
 
 -/E+E, and by art.7a7,s=~tii"+K. If we suppose s^o 
 " 27«^ 
 
 when-a:=o, then K= -^^r- and s=: —^ . In like man- 
 
 27 a' 
 
 ner, if we make use of the notation in art. 869 (Jig. 322, N. 1), 
 6* •=. — —=.. Suppose, for example, «pp =: r% then s s 
 
 V rr--pp 
 
 rr a/7 . 2 , — f J , ^ ^^^. ^ I il 
 
 — =: rn xa — r\ , and (art. 727) « = 2a Xa — r 
 
 V ffr;' — rrr 
 
 = 2v'aa— ar. If wc supposc AP (fig. 329) to be a parabola, 
 S the focus, and A the vertex, then T will be always found 
 in the right line AE perpendicular to SA ; and the parabo- 
 lic ark AP = PT + log. ST + TA, the tnoduhis being SA. 
 For let SA = a, ST = p, SP = r, AP = s, and PT = q, 
 
 then pp — ar, s = ""^ = • Z — (because q = 
 
 V rr — pp V rr—ar 
 ^/rr—ar aud q = " , ) *? + ^ , . But if W = ST 
 
 + TA
 
 Chap. V. foT the Resolution of Prohhms. 305 
 
 + TA = '/■77+ */7^^^, then *. consequently 
 
 s = j -^ — 5 and % z=. q-^ log. u, the modulus being equal to a. 
 
 See art. 746 and 845, for the mensuration ot circular arks, and 
 art. 806, 807, 808, for hyperbolic and elliptic arks. 
 
 893. The surface generated by the ark s, when the figure re- 
 volves about the base (the ordinate being represented by y and 
 
 base by .r), is F. 4%s or F. 4Nj/ v^F+7% by art. 229. Thus if 
 the parabola AP (Jig. 329) revolve about the axis ASM, PM being 
 perpendicular to AS in M, PM —y — o, AT == 2 */ ar—aa, and 
 
 's — /■^■~~~ ' consequently _ysn2r -/^the surface generated 
 
 , , I * r. • 16N __ -. 16N — — 
 
 by the ark AF is — x rV ra + K = -j- X sp x st-sa% 
 
 and (if SE be a mean proportional betwixt SP and ST) this 
 surface is to the circle of the radius AE as 8 to 3. 
 
 894. Let C (fig. 330) be the centre, CD half the transverse 
 axis,andCAhalf theseoond axis of the ellipse ADB,F the focus, 
 PN perpendicular to CD, PM perpendicular to CA, and PK 
 perpendicular to the curve meet CD in K, CA r: a, CD 
 = h, CF = c, CN = X, PN =: y, and the ark AP = s ; 
 
 then NK : NC : : a^ : h\ or NK - ~, PN* = ~ X 
 
 _^^___ c^c'^t'- a ___^__ 
 
 hb—xx, VYJ- — a^ —, and PK = — X v/i4— c»x\ 
 
 But s. X : : PK : PN — y, ys rr j^ = (sup- 
 
 posing c.b::b:d-CG) "''' "^J^' -* Therefore let CA 
 
 and NP meet the circle GZE described from the centre C in E 
 and Z, and when the figure is supposed to revolve about the 
 axis CD, the surface generated by the elliptic ark AP will 
 be to the area CEZN as 4N x ac to hh ; and if DI perpen- 
 dicuhir to CD meet GZE in I, the whole surface of the spheroid 
 VOL. IJ. X will
 
 506 Of the general Rules Book II, 
 
 tvill be to the surface of the sphere of the radius CA as ^^ y, 
 
 So 
 
 CEID to 4Na«, that is, as EI + CA to <2CA. In hke manner, 
 if PK produced meet AC in /:, Ml: : MC { — y) : : b"- : a"-, and 
 
 Vk =-^ X Vir^TcTp. ■ let DP =/, and / : ; : : P/c : PM, or 
 PM x/ =. 'j/XFk; consequent!}' the fluxion of the surface get 
 nerated by the ark DP about the axis CA is — - x Va* + c^/ 
 = (if c : « : : a : e rr Cg) — — X y Veejfj/i/, the fluent of 
 
 L'm 
 
 which is - — X y Vee-\-j/j/ + 2N6 X \o^.y-\- -/ ee-\-j/y (the 
 modulus being equal to c or Cg) = 2N X CM X PA + 2N6 X 
 log. CM + P/c X g^. Hence the surface generated by the 
 elliptic quadrant DPA about the axis CA is 2N6 X 
 b 4- log. a X —-y and the surface of this spheroid is to the- 
 
 surface of a sphere of the radius CD as CD + log. ^^ ■ ■ 
 
 lo 2CD, the modulus being Cg. These constructions agree with 
 Mr. Coles's Harmon. Mensurar. p. 28 and 29, where he 
 iUustrates the transition from circular arks to logarithms (or 
 from the measures of angles to the measures of ratios), that so 
 often occurs in the resolution of the various cases of a problem, 
 from an analogous transition observed long ago by f^ieta in 
 the resolution of cubic equations ; the roots of which are in 
 some cases obtained by trisecting an ark, and in other cases bjr 
 what may be called the trisecting a ratio {/,. e. interposing two 
 mean proportionals betwixt the terms of the ratio) ; so that the 
 trigonometrical and logarithmical canon are mutually supple- 
 ments to each other. Theharmony of thoscmeasures, which was 
 so much considered by this excellent author, may be further 
 illustrated by the resolution of the two following useful pro- 
 blems relating to the spheroid. 
 
 895. In
 
 Chap. V. for ike Resolutkn ofProhlems, '307 
 
 895- I" plain sailing the meridians are supposed parallel, and 
 the degrees of longitude as well as those of" latitude are sup- 
 posed equal ; whereas the meridians intersect each other in the 
 pole, the degrees of longitude decrease in the same proportion 
 as the semi-diameters of the parallels of latitude, and the de- 
 grees of latitude (because of the oblate figure of the earth) 
 increase from the equator towards the poles. In order to cor- 
 rect some of the errors that arise in Navigation from these false 
 suppositions, a projection was invented (commonly called Mer~ 
 cator's Chart) in which the meridians are still supposed parallel, 
 and the degrees of longitude enlarged as in the former, but the 
 degrees of latitude upon the meridians are enlarged in the same 
 proportion. The arks of the meridian thus enlarged (or the me- 
 ridional parts)iive found in a sphere or spheroid by the following 
 theorems. LetthearkDHr/g.SSljiV. I),orangleDCH,bethe 
 latitude for which the meridional partsz are required, HEits sine, 
 let CT bisect the ark H6?(the complement of HD),and meet the 
 
 CD 
 
 tangent at d inT. Then, 1, in the sphere 2=:log, ^,i[iemodu- 
 
 ius being CD. 2. In the oblate spheroid, let Dh be an ark 
 whose sine eh is to EH as CF the distance of the focus from the 
 centre to CD the semi-diameter of the equator ; let Ct bisect 
 
 the ark dh, and meet dT in i ; then z = loi?. ■^t^ — ttt: K 
 
 ^ a I CD 
 
 log. -^, 3. In the oblong spheroid> let Dq (fg. 331, JV. 2) be 
 the ark whose tangent is to EH the sine of DH as CF to CD, 
 
 , , CD CF TA 
 
 and z = log. ^ + — X D^. 
 
 896. For, supposing ADB to be a meridian section through 
 the poles A and B, as in art. 894, let CA =: a, CD = b, 
 CF = c, CM = 1/, EH-w, and the elliptic ark DP=5. Then, 
 to find the meridional parts z, we are to suppose the element 
 or fluxion of the ark DP to be always enlarged in the ratio of 
 CD the radius of the equator to PM the radius of the parallel 
 
 CD 
 
 of P, that is, 4 == ^ X p;^ = (because 's : } : : PK : NK : : 
 
 X2 CH
 
 308 Oft%t general Rules Book IL 
 
 CH : CE) JL^ X ~ =(because PM : NK : : &6 : aa, and 
 
 NK : CE : : FN : EH) I x "'"'■. By what we found in art. 
 
 y bb — i/u 
 
 894^ PA = — X '/a''-\-ccyy or — X Va'^—cc^, according 
 as CD is greater or less than CA ; consequently Vk be- 
 ing to yik as CH to EH;, we have u = — ■ j^ or y = 
 
 ? and (by art. 728) — = h ti=- = - 
 
 Vb^^ccuu y u b'^+ccuu u 
 
 O* mi r ■ aab'^u , 
 
 1 heretore a = ■ — — - < that is, a = 
 
 b'^+CCUU " bb—uuysb'^-'t-ccuu 
 
 'U 
 
 hb —uu 
 
 Ihu • .1 1 • bbu b-ccu . , 
 
 m the sphere, % = rr — tt i" the ob- 
 
 late spheroid, and i = - — ^ + Tr-r— — ^" the oblons 
 
 spheroid. Suppose now dh to be the diameter of the circle d\yh, 
 join <?H and H6, then the triangles TrfC and c?H6 will be simi- 
 lar, ^T : <fC : : </H : H6 : : v/cJITeh : ^Cd+Eti, dT zz 
 
 b X ^T— 5 and the modulus being 6, the fluxion of I02:. 
 
 (orofloff. v -r--j shallbcTT-^^ — • In like manner, because 
 
 ^ O i) — u' bb— uu ' 
 
 CU T ^ /bb — <■« ii n • <- 1 CD 
 
 eh --J-) dt- h XV 27:^' the fluxion oi log. -^ (or 
 of I02:. v/7- z "^"" j is 71— -—• Therefore in the sphere z 
 — log. Tjw = log- CD — log. dT ; and in the oblate sphe- 
 
 CD CF CD 
 
 roid, i; 1:1 log. -^ — ^ X log. -^, In the oblong sphe- 
 roid^
 
 Chap. V. for the Resohdd^n of Problems. 309 
 
 3+ + CfMM 
 
 roid, Dg is the fluent of ^^" • ; consequently z = log 
 
 CD . CF T^ 
 
 5t + CD ^ ^^• 
 
 897. These logarithms are hyperbolic, or of Napier's first 
 sort; but it is easy to adapt the theorems to the tabular lo- 
 garithms, and to express the meridional parts in minutes, as is 
 usual. Thus in the sphere substract the logarithmic tangent of 
 half the complement of the latitude from the logarithm of the 
 radius (or 10.000000), and multiply the remainder by 
 791J-704467897 Sec. {viz. the number of minutes contained 
 in the radius divided by the modulus of the tables), then the pro- 
 duct shall give the meridional parts in minutes. 
 
 898. In the oblate spheroid we have this easy rule : let CF: 
 CD : : 1 : w, and u be the sine of the given latitude DCH for 
 which the meridional parts are required. The table for 
 the meridional parts being already computed in the sphere, 
 find the meridional parts in this table for the latitude whose 
 
 sine is - X u, divide these by n, substract the quotient from the 
 
 meridional parts in the same table for the given latitude DCH; 
 and theremainder shall be themeridionalpartsfor the same lati- 
 tude in the oblate spheroid. This problem is resolved by infi- 
 nite series, and a table of the meridional parts is computed for 
 the oblate spheroid wherein cc : bb : : 20, : 1000, in an in- 
 genious treatise published lately by the Reverend Mr. Mur- 
 dochf whose table may be examined by this rule; and it may 
 likewise serve for facilitating the computation, when a differ- 
 ent ratio is assumed for that of cc to hb. The greatest differ- 
 ence betwixt the meridional parts in the oblate spheroid and 
 sphere is easily computed by finding the meridional parts in the 
 sphere for the latitude whose sine is to the radius as 1 to n, and 
 dividing these by n. 
 
 899' In the oblong spheroid, to find the meridional parts for 
 the latitude whose sine is u, add to the meridional parts in the 
 
 1 
 
 5phere for the same latitude - x J)q, J)q being the ark whose 
 
 X 3 900. Let 
 
 tangent is - X u.
 
 310 Of the general Rules Book II, 
 
 900. Let PMowj: (fg. 33'2) be q pyramid of an uniform density 
 upun the rectangular baseMo?<x, and suppose that the attraction 
 of its particles is inversely as the power of the distance of any 
 exponent n less than 3. Let the attraction at the given dis- 
 tance PC (=:a) be represented by e^ the attraction at any dis- 
 tance PM (=»■) by V; then if the angles MPo, MPjt, be con-c 
 tinuallydiminished,thegravityatP towards the pyramid PMowa? 
 
 Vr 
 
 will be ultimatel V as r — x Mo x Mx. For if N«A7 be a section 
 
 of the pyramid parallel to Moux at a distance PN = PC^ the 
 gravity at P towards the pyramid shall be ultimately equal to 
 
 the fluent of Vj* x Mo x Mx, or (because N/ : Mx : : N« : Mo 
 
 QP"^^ f'r 
 
 '.-.u : r and V : e : : a» : r»*) of — x N/ X Nwj that is, to 
 
 liL_ X N/ X Na = T-- X ^^0 X Mar. 
 
 S — n j—n 
 
 901. Hence it will appear (by proceeding as in art. 642), 
 that if a portion of a solid contained by planes that intersect 
 each other in PH attract a particle at P^ PMA be one of these 
 planes, the right line PM meet the circle BNC described from 
 the centre P with the given distance PC in N, MQ be perpend* 
 ciilar to PC and NR to PH, PC = a, PM = r, PQ = Zy 
 PR =r X, the sine of the inclination of the planes to the ra- 
 dius as^ to a; and, supposing this angle to be diminished con- 
 tinually till it vanish, the ultimate value of the gravity at P to- 
 wards the slice of the solid contained by the planes in the di- 
 rection PC be represented by q, then q will be equal to the 
 
 fluent of ^=A^- or of ^i— ^^^ x r-K If PC coincide 
 S— « X aa 3 — n 
 
 with PH, then x:a ::z:r, and the gravity at P will be as the 
 fuentof— - — - X r'^—'"xx. If PH be perpendicular to PC, 
 
 the gravity at P towards the portion of the solid will be ulti- 
 mately
 
 Chap. V. for the Resolution of Problems, s\i 
 
 mately as the fluent of "^^J- X r '~^x Vaa>^xx, because in 
 this case r : z : : a : Vaa-^xx' 
 
 902. Suppose a solid to be generated by the figure PMA (ftr, 
 332,iV.2)revolvingabout the axisPC^and the gravity atPtowards 
 this sohd in the direction PC to be measured by Q, then Q = 
 
 — 3^^ " ■ X F. r """a\rj N being supposed to denote the ratio 
 
 of the semi-circumference to the diameter, as formerly. For 
 example, if PMA be a semicircle of the radius PC, then PM =:: 
 
 12PR, or r = Q^x ; and Q = F. ^"""1 ^ ~"^ = (when r be- 
 
 comes equal to 2a) ^ ^ 1.1 . If w=2, Q=: ^^l ' ,andthe 
 
 gravity is the same as if the whole matter in the sphere attract- 
 ed from the centre C, because the solid content of the sphere is 
 
 — ^* and in other cases the gravity is to this attraction as 
 
 3X2"^ to 3 — n X 5 — n> Ifw =-— 1, these are equal. If 
 71 = 0, their ratio is that of 4 to 5; and if ?« =: 1, the ratio is 
 that of 3 to 4. In different spheres, when n is given, the gra- 
 vity is as PC^ — " , because e is as PC— » . 
 
 g03. To find the attraction at the pole A (/«g.333) towards the 
 spheroid generated by the semi-ellipsis ADB about the axisAB, 
 suppose P to coincide with A, AC = a, CD =: h, CF (F 
 being the focus) — c, AR == x, AQ =: 2, AM =: r; then 
 
 AR* : : NR' : : AQ* : QMS or xx : art — xa: : : zz : -^ x 
 
 lb Si" a** 
 
 1a% — zz'. '. z '. — X 2a — ;; ; consequently z = — — , 
 
 aa a"' 4I c* ;«" 
 
 2A^ a*-x aa 
 
 and (because x '. a \\ z \ r) r zz. — — — , or ,r r= — x 
 
 ——————. 2' — " Na*""^ /^''"""^"r 
 
 If id + Vh^^^^r. Therefore Q = "^ ^^^ X 
 
 X 4 '^ V.
 
 312 Of the general Rules Book IL 
 
 X 
 
 F. = -^— — .-.jj r^ ; or, if we suppose bb zz dc, Q will be 
 
 equal to ■ X F. ———= 1 . , which 
 
 fluents are easily measured by the areas of conic sections, when 
 n is any integer number. The upper signs are for the oblong 
 spheroid. 
 
 904. To find the attraction at the pointD(j^g.334)in the equa- 
 tor of the spheroid, let P coincide with D, DBE be a section of 
 the solid perpendicular to its equator, PH or DH a tangent at 
 D, HNc a circle described from the centre D with the radius 
 Dc ( = CA) meet DM in N, MQ perpendicular to DE, and NE 
 to DH, CA zz. a, CD = b, CF zz c, as formerly, and DQ = z, 
 DM = r, DR = X. Then NR^ : DR^ : : DQ* : QM% that 
 
 IS, aa — XX : XX : '. zz : -rr X ^bz—^z :: z: -rr X 2b — % and zzz 
 
 bo bo 
 
 , aa — XX .1 / az \ iba^ >/ aa-xx 
 
 Ihaa X — —\-^ \ consequently 7- (=- ~ 
 
 a^-^rC^X^ V aa—xx' a'' -\- CCXX 
 
 mi r a^—'^ef T? T ,' , T- Sa^b^fex 
 
 Therefore a = ± x F. r^—^'x '/aa—xx — F. 1 
 
 ^ 3 n 3— « X '2aab[* 
 
 4 — » 
 
 X L^ ^ , — y which gives the ultimate value of the gra- 
 
 vity at D towards a slice of the spheroid contained by 
 two planes perpendicular to its equator that intersect each 
 other in DH, when the angle contained by the planes va- 
 nishes, by art. 901. If we suppose c zzo or a — b, the last 
 
 fluxion becomes equal to ■ ^ — x aa — jrxl— ^- zz (sup- 
 
 posing yy zz aa — jtx) ^-- X — ^ — ^ , the fluent of 
 
 3~« X 2" >/ aa— yy 
 
 which gives the ultimate value of the gravity at D towards the 
 slice of the sphere (described upon the diameter of the equator 
 
 of
 
 Chap. V. for the Resolution of Problems. 313 
 
 of the spheroid) that is contained by the same planes. Because 
 the sections of the spheroid by planes perpendicular to the 
 equator are ellipses similar to the meridian section and to one 
 another, and the secticns of the sphere by these planes are cir- 
 cles, the gravity at D towards the spheroid is to the gravity at 
 D towards the sphere described upon the diameter of the equa- 
 tor as the former to the latter fluent, that is (supposing cc ^ 
 
 aa :: m : 1), as F. a'-'^ b'~"x x — ~^^'^l ,^ to F. x x 
 
 4— n 
 
 aa — jr-rj'~2~. These flaxionary expressions are rational when 
 n is an even number ; and when n is an odd number they are 
 transformed into rational expressions by supposing .r rr 
 
 -— — ^^ — . Hence, therefore, the gravity at the equator, as well 
 
 as the gravity at the poles, is measured by circular arks or lo- 
 garithms when ;/ is any integer number less than + 3. 
 
 905. When w=:2, the gravity at the pole or equator is easily 
 computed from the first theorem in art. 901, viz. 5^ — F. 
 
 fl" — eizx ' 
 
 — - — '■—- X r^— «rr(when ?i=:2) F. efzx. For when the par- 
 ticle P, whose gravity is required, is at A, as in art. g03, z (or 
 AQ, supposing All z=. x) ■=. — 
 
 X F.— ™— ; consequently the gravity at the pole A towards 
 the spheroid is to the gravity at A towards the sphei-e of the 
 diameter A-B as hb X F. — L!' - to \aa. When the particle 
 
 aa-\-mxx * 
 
 P is at D on the circumference of the equator, suppose, as in 
 art. 904, DR = x, then DQ =: :: = 26 x Jl^^Hil and a 
 
 =: F. efzx =: 2bef X F. a' X -^^'^- ; consequentlv the 
 
 gravity at D on the circumference of the equator towards the 
 
 spheroid
 
 314 Of the general Rules Book II, 
 
 spheroid is to the gravity at D towards the sphere upon the dia-f 
 meter DE as F. abx x ~r=- — to F. x x aa—xx = (when 
 
 aa-\-mxx ^ 
 
 xz=.a)~y and these fluents give the same constnictions 
 
 by circular arks and logarithms that were described in art. 646 
 and 647. The gravity at any point P on the surface of the 
 spheroid in the direction parallel to the axis, or perpendicular to 
 it, may be computed in like manner from the theorem ^ = F, 
 efzr; but this case is reduced to the former by art. 634. When 
 the density varies, but so as to be uniform over any surface si^ 
 milar and concentric to ADBE, the gravity at any place in the 
 plane of the equator, or axis of the spheroid, may be computed 
 ■ by art. 668, Sec. The reader will find this subject treated in a 
 different manner in a late ingenious essa}', Phil. Trans. N. 449, 
 by INIr. Clairaut. It was demonstrated in art. QsQ, &c. 
 that if the density of the earth was uniform, its figure would 
 be such a spheroid as is generated by an ellipsis revolving about 
 its second axis, according to the theory of gravity; but this 
 was assumed as an hypothesis in art. 67 9^ 681, &c. where the 
 density was supposed variable. 
 
 906 {Fig. 335).Thecentresof gravity and oscillation of figures 
 are determined from art. 509 and 534. LetG be thecentre of gra- 
 vity, and O be the centre of oscillation of theplane figure F/znM 
 when it revolves about the axisF/] OG A perpendicular to Vf'm A 
 
 and to Mm in P,AP = .r,Mm=3^,GA=:2;,OArr»; then 2= L^ 
 
 F. yr 
 
 and u ~ ■ — =^— — . inus_, u -^t/ — a: , ^ _ - — — — 
 
 T.yxxr F.x"'x 
 
 r> ?)2 -I- 2. * 
 
 "•^^ X X, or G A : PA : : m + 1 : m + 2 ; and u - — — -I- 
 
 ^ !1±| X ^, or OA : PA : : m + 2 : m + 3 (Jig. 239). The centre 
 
 of oscillation of a sphere was determined^ in art. 536, from the 
 
 fluent
 
 Chap. V. for the Resolution of Frohhms. S15 
 
 fluentof 2«_y^y X aa—uy [supposing, in fig. 239 {Jig- 239) the ra- 
 llius GE — «, GN:=PM=:y, 0(jt=.z and n to 1 as the circum-f 
 
 fcrenceof the circle to its radius], which is <2.nip x — ^* 
 
 •^ a 6 » 
 
 and this fluent becomes equal to -^ when PM = GE oryrra, 
 
 which being divided by — X a^ X z (the solid content of the 
 
 sphere multiplied by the distance of its centre of gravity from 
 
 the ajcis of oscillation) gives - j< -— — m. The centre of 
 
 percussion is in a right line perpendicular to AO at O. Several 
 principles concerning the centre of gravity and its motion, that 
 iare of use in the resolution of problems, ^vere explained in art. 
 511, 526, 5oS, 544, 551, 8cc. The motion of a fluid issuing 
 from a cylindric vessel was considered in art. 537, 540, 541, &c. 
 and an example of the method by which the principle con- 
 cerning the equality of the ascent and descent of the centre of 
 gravity is applied to this enquiry {Comment. Petropol. torn. 2) 
 is described in art. 544. But the same theory has been since 
 prosecuted more fully by the learned author, and illustrated 
 by various experiments, in a particular treatise, entitled 
 JHydrodynamica . 
 
 907. In any engine the proportion of the power to the weight, 
 when they balance each other, is found by supposing the en- 
 gine to move, and reducing their velocities to the respective 
 directions in which they act; for the inverse ratio of those ve- 
 locities is that of the power to the weight, according to the 
 general principle of mechanics. But it is of use to determine 
 likewise the proportion they ought to bear to each other, that 
 when the power prevails, and the engine is in motion, it may 
 produce the greatest effect in a given time. When the power 
 prevails, the weight moves at first with an accelerated motion; 
 and when the velocity of the power is invariable, its action up- 
 on the weight decreases while the velocity of the weight in- 
 creases. Thus the action of a stream of water or air upon a 
 wheel is to be estimated from the excess of the velocity of the 
 
 fluid
 
 31& Of the gcneralUuks **lBook IT. 
 
 fluid above the velocity of the part of the engine which it strikes, 
 or their relative velocity, only. The motion of the engine ceases 
 to be accelerated when this relative velocity is so far diminished 
 that the action of the power becomes equal to the resistance of 
 the engine arising from the gravity of the matter that is elevat- 
 ed by itj and from friction; for when these balance each other, 
 the engine proceeds with the uniform motion it has acquired. 
 Let a denote the velocity of the stream, u the velocity of the 
 part of the engine which it strikes when the motion of the 
 machine is uniform, and a — u will represent their relative ve- 
 locity. Let A represent the weight which would balance 
 the force of the stream when its velocity is a, and p the weight 
 which would balance the force of the same stream if its veloci- 
 
 A X u^ 
 
 ty was only a — ii ; then p : A : : a—Z^ : a-, or p =; , 
 
 and p shall represent the action of the stream upon the wheel. 
 If we abstract from friction, and have regard to the quantity of 
 the weight only, let it be equal to qA (or be to A as 5- to 1), 
 and, because the motion of the machine is supposed uniform, 
 
 . Ax a—u a — u m . r» 
 
 p — q X A :=: , or q =: . Ihe momentum of 
 
 A )/ yc. ' ^ 
 this weight is qku :=. , which is a maximum w^hen 
 
 7/ yc * ~ ^ • 
 
 the fluxion of '^—^ vanishes, that is, when u x 7-I!7* — 
 
 ^uu X a—u :=. o,ora — 3u =■ 0. Therefore, in this case, the 
 
 machine will have the greatest effect if u = -, or the weight 
 
 A X ~" 4 A 
 qA := -—^ n: ~ ; that is, if the weight that is raised 
 
 by the engine be less than the weight which would balance 
 the power in the proportion of 4 to 9 ; and the momottum of 
 
 the weight is -r-- . 
 
 908. If
 
 Chap. V. for the Resolutioti of Problems. 317 
 
 g08. If we would likewise consider the friction arising: from 
 the motion of the weight, let 1 be to n as the weight is 
 to the resistance of the engine which would arise from this 
 friction, if the motion of the engine was such that the part of 
 it impelled by the stream moved with the given velocity a; then, 
 supposing the friction to be always in the compound ratio ofthe 
 weight and velocity, the resistance of the engine arising from 
 the same cause when the part of the wheel impelled by the 
 
 stream moves with the velocity u will be — -. Suppose, there- 
 
 n A . ngfiiu A X a — u ,, . A a — u ^ 
 
 lore, p = oA + -^ — = , then oA = - x « 
 
 and the momentum ofthe weight qAu rr — f x -""" : theflux- 
 ion of which being supposed to vanish, we shall find aa — Sau — • 
 Q,nuu = 0, or u = — - — .- , and the weidit oA r= 4A x 
 
 — — , ; that is, the machine will have the ffrealest ef- 
 
 3+'/9 + 8« ^ 
 
 feet (according to this supposition) when m : a : : 2 : 3 -f 
 Vg + Sn, and the weightistothatwhichwouldbalance the power 
 as 2 + 2 V9 + 8« to 9 -t- 4w + 3 VTlT^n. For example, if 
 
 n — -, then u—-^, and ^ A : A : : 20 ; 49 ^ consequently, 
 
 though the velocity u be less than in the former case in the 
 ratio of 6 to 7 (and therefore the action of the power on the 
 wheel be greater), yet the weight that is raised is less in the ratio 
 of 45 to 49^ and the effect of the engine is less in the ratio of 
 270 to 343. If n be very small in respect of 1, then u: a : : 
 
 1 : 3 + J-, and qA : A : : 4 + — : 9 + 4n nearly. But if we 
 
 would have likewise regard to the friction arising from the mo- 
 tion ofthe parts of the engine, as well as to that which arises 
 from the elevation of the weight, the computation will be some- 
 what
 
 ^1^ Of the general Rules Book ll» 
 
 ivhat different. Let the friction be equal to mA when the ma- 
 chine moves without any charge in such a manner that the ve- 
 locity of the part impelled by the stream is equal to a; 
 
 and the friction will be equal to ^- when this velocity is 
 
 «, where we suppose m invariable, because the machine re- 
 mains the same. When the motion of the engine is uniform^ 
 
 mjAu niAu A x "^Z^Z"- , . , 
 
 p =z qA + --— + — ^ — 3 and, supposmg the 
 
 momentum of qA to be a maximum, u will be found by re- 
 solving the equation u^ -{■ -^^ 1 — j X au^ — ^- Xaau^ 
 
 ■^, For example, if ;^ = — and w = TTT? " ^^ ^^^""^y To"* 
 qA IS about -j^, and the effect of the engine about -i- of Aff 
 or |-ths of what it would have been if there was no friction^ 
 and u was equal to -r, 
 
 909. Suppose that the given weightP (^o-.^SS) descendingby 
 Jtsgravity in the vertical line raises a given weight W by the line 
 PMW (that passes over the pully M) along the inclined plane 
 BD, the height of which BA is given; and let the position of 
 the plane BD be required, along which W will be raised in the 
 least time from the horizontal line AD to B. Let AB ■=. a, 
 BDrrjT, ^z-lime in which Wdescribes DB; the force which ac- 
 celerates the motion of W is P — - — . tt is as - „ "" ■' -. and if 
 we suppose the fluxion of this quantity to vanish, wc shall find 
 X — — — or r =: • consequently the plane Hu requir- 
 ed is that u[)on which a weight equal to 2W would be sus- 
 tained by P; or if BC be the plane upon which W would 
 sustain P, tiien BD r= <ZBC. But if the position of the plane 
 BD be given, and W being supposed variable, it be required to 
 find the ratio of \V to P when the greatest momentum is pro- 
 duced
 
 Chap* v. fot tlie Uesolution of Problems. 5ig 
 
 duced in W along the given plane BD; in this caseWought to 
 be to P as BD to BA + -/bd + ba X -/ba. 
 
 910. TheradiusCA (Jig. 337) 'dnd angle ACB being given, let 
 E be any point upon the ark AB,EM the sineof the angle ECA, 
 EN the sine of the angle ECB, ;e any positive number, and let it 
 be required to determine the point E when EM« x EN is a max- 
 imum. Upon AC produced beyond C take CD : CA : : ti — I 
 : ?i+\ ; draw DG parallel to CB, meeting the circle AB in G, 
 and if CE bisect the angle ACG, it will meet the circle in 
 the point E required. For, let ER parallel to CB meet CA 
 in R, CR =: x, ER zz y, and when y^x is a maximum (or when 
 
 its fluxion vanishes), "~ — j- — = o, by art. 728, or Jix zr 
 
 ~ • Let the tangent at E meet CA in T, CB in Z, and CG in 
 y 
 Q, and AP perpendicular to CE meet CB in K and the circle 
 
 again in H ; then RT zz'4^ z=i nx, or RT : CR : : w : 1 - 
 
 y 
 JET : EZ : : AP : PK : : PH : PK; consequently HK : KA 
 : : n^\ : «+ 1 : : CD : CA,and DH is parallel to CB. Therefore 
 H coincides with G, and the ark GA is bisected in E when ER« 
 X CR is a maximum, or (because ER is to EM and CR to 
 EN in the same invariable ratio of the radius to the sine of the 
 given angle ACB) when EM" X EN is a maximum. 
 
 911. Let a fluid that moves with the velocity and direction 
 AC strike the plane CE; and suppose that this plane moves pa- 
 rallel to itself in the direction CB. Take CD : CA •.•.\:3, dravs^ 
 DG parallel to CB meeting the circle AB in G; and if the plane 
 CE bisect the angle ACG, then the efiectof the fluid upon CE 
 will be greatest at the beginning of the motion (Jig. 337) N.'2'). 
 ButiftheplaneCEhasalready acquired amotion in the direction 
 CB, let its velocity in this direction be to the velocity of the 
 stream as Aa to AC, and let Aa be parallel to CB ; let a circle 
 described from the centre C with the distance C« meet DG in 
 g; and theeflectof the stream upon the plane CE will be greatest 
 
 in.
 
 390 Of the general Rules BooklL 
 
 Bii ibis case when the plane bisects the angle uCg. For let AP 
 and ap be perpendicular toCE in P and p, and aJi perpendicular 
 to AP in h ; tben the motion of the particles of the fluid in the 
 direction perpendicular to CE will be represented by AP, their 
 motion in the direction parallel to the plane CE by CP, the 
 motion of the plane in the former direction by A/i, and its mo- 
 lion in the latter direction by ah. The action of the fluid on 
 the plane depends on their relative velocify only, that is, on 
 the dift'erence of the motions AP and AA (whicli is equal to 
 hP:::zap), and on the sum or ditfercnce of tlic motions PC and 
 ha, which is equal to pC It ibllows, that the action of the 
 stream on the plane CE is the same in this case as when the 
 plane is at rest, and the stream strikes it with the direction and 
 force flC. Let this force aC be resolved into ap perpendicular to 
 the planeCE, and jsC parallel to it; and because the latter has no 
 effect upon the pl9,ne CE, let the force ap be resolved into the 
 force ak parallel toCB,andpA; perpendicular to it; then because 
 the force pk has no effect to impel the plane CE in the direc- 
 tion CB,aA; will measure the force with which any particleof the 
 fluid impels CE in the direction CB; and the number of parti- 
 clesincident upon theplane CEin the same time beingasrtp, the 
 effect of the stream to move the plane CE in the direction CB 
 shall be measured by ak x «p = (Emand EN being perpendicular 
 to Crt andCBln ?w and N, and consequently a/i;: cfp:: EN: CE) 
 ^/x^/-xEN _ Eyn» X EN X -^, which is a maximum when CE 
 
 CE. C£.^ ' 
 
 bisects the angle «Co^, by the last article ; because CE and Ca 
 are supposed to be given, n r: 2, and CD : CA : : 7i — 1 :n+l 
 : : 1- : 3. If Aa z=: o, g coincides wilh G, and the stream has 
 the greatest effect when CE bisects the angle ACG. 
 
 912. Let CV(/?g. 338) be perpendicular to Aa in V, and CEpro- 
 
 duced meetAainf,- takeVLrrVCx -/ 2^ ¥/"=—, join I/, and 
 
 V^ the tangent of the angle VCE(VC being radius) shall be 
 equal to L/'+V/jwhen the plane CEisinthemostadvantageous 
 position, CA the velocity and direction of the stream, and Aa 
 the velocity and direction of the plane CE being given. For 
 
 let
 
 Chap. V. for the Resolution of Problems. 321 
 
 let pq perpendicular to CV in q meet Ca in u, let oC and VC 
 produced meet J)g in z and d; then, because aC—SCu, at=z 
 3i?M = (because ag = ^crp) -|- =: ^-^t — = ^ if __ ~ 
 
 (because gd^ = Cg* — Cd' = Ca"- — ^ = ~ + 
 
 9 — 9 
 
 9Vfl' 4 
 
 I/, and ^f = L/) L/+ ^; and V^ = c^ + Va = L/ 
 q:V/. If CV = ff, Va = o then Yt = y/^"" +£2: q:2- 
 
 '^ 4 2 
 
 The negative sign is to take place when aCB is greater thaa 
 a right angle. 
 
 913. When the angle ACB is right, A {fg. 338> N. 2) co- 
 incides with V, DG is perpendicular to AD, and A^ =: Lf + 
 
 A/= V 2aa + ^ "*" "T" ^^ A« = c —o, then A^ : AC 
 : : -v/? : I, or AP the sine of ACE to the radius AC as AG 
 to 2CA, or as \/ad to V 2ca, that is, as \/2 to v's'. There- 
 fore the stream at the beginning of the motion will have the 
 greatest effect upon the plane CE, if the angle ACE be of 
 54*^. 44'. ; and this is the case which has been considered by 
 several authors : but if the plane CE has already a motion in 
 the direction CB, the stream will have the greatest effect 
 upon it if the angle ACE be greater. For example, if the 
 velocity of the plane CE in the direction CB be a third 
 
 part of the velocity of the stream, or c — ^, then A^ = 
 
 ^2aa 4- T" "*■ I ~ ^'^•^ ^'' ^^^^ tangent of the angle ACE 
 ought to be double of the radius, that is, ACE = 63°. 2fi'. 
 If c : a : : \^s : \^9, then A^ : AC ; : 2 -f a/J : 1, and ACE 
 ©ught to be of 73° 40'. If c = «, then ACE = 74"* 19'. 
 
 914. Hence the sails of a common windmill ought to be so 
 situated that the wind may strike them in a greater angle than 
 
 VOL. II. Y that
 
 322 Of the general Rules Book II. 
 
 that of 54" 44'; for this is the most advantageous angle at the 
 beginning of the motion only; and when any part of the en- 
 gine has acquired a velocity c, the effect of the wind upon that 
 part will be greatest when the tangent of the angle in which 
 
 the wind strikes it is to the radius asva + ~- + ~^^ ^' 
 
 Let the rightline Mrepresentthelcngthofoneof thesails, take 
 AC to Ab as the velocity of the wind to the velocity of the given 
 point 6 about the axis of motion, LA r: AC X \/ 2, and a being- 
 
 any point upon bh, take Af=. —— • then if the sail be so form- 
 ed that the wind shall strike it at any distance Aa from the axis 
 of motion in an angle wliose tangent is always to the radius as 
 L/'-J- A/to CA, the wind shall have the greatest eft'ect upon the 
 sail. It is true, that a celebrated author has drawn an oppo- 
 site conclusion from his computations, viz. that tlje angle in 
 which the wind strikes the sail ought to decrease as the distance 
 from the axis of motion increases ; that if c r= a, the wind ought 
 to strike the sail in an angle of 45'^; and that if the sail be in 
 one plane, it ought to be inclined to the wind at a medium in an 
 angle of about 50 degrees : but if he had reduced the equation 
 of six dimensions, by which he has determined the maximum, to 
 a biquadratic equation^ our conclusions would have agreed ; and 
 the divisor by which this reduction may be made is of no use 
 for determining themostadvantageous position of the sail when 
 the engine is in motion ;becauseitdoes not give a maximum, but a 
 7nm/m?/w that corresponds to the case when CE coincides with Ca, 
 and the stream has no effect upon the planeCE. Su ppose Aa n AC, 
 or c-=.a ; and if the angle ACE be of 45'^, CE will coincide with 
 Ca, the velocities of the plane CE and of the stream estimated 
 in the direction perpendicular toCE must be equal; so that the 
 stream will have no effect upon the plane CE in this case to pre- 
 serve or accelerate its motion; and the angle ACE mustbein. 
 creased, that the velocity o^the stream in the direction ap (in 
 which it acts upon the plane) may be greater than the veloci- 
 ty of the plane in the same direction. In the same manner it 
 
 is
 
 Chap, V. for the Resolution of Prohknts, S23 
 
 is obvious that, if Aa was equal to 2AC, and ACE of 54° 44'. 
 then the stream could have no effect upon the plane CE, and 
 the angle ACE must be increased. 
 
 915. When (fig. 339) the engine is of such a nature that the 
 whole fluid, or the same quantity of it, is always incident on the 
 plane CE in its various positions, the force by which it impels 
 
 CE in the direction CB is as ak — E?n X EN X ^^ which is a 
 
 maximum (Ca and CE being given) when CE bisects the angle 
 oCB, by art. 9 10, because in this case «= 1, CD : CV : : n — 1 
 : ;^ + 1 : : o : 2, that is, CD vanishes, and DG coincides with 
 CB. In this case, if AC and Aa, the velocities of the stream 
 and plane, be given, with CB the direction of the motion of the 
 plane, but the angle ACBbe variable, and Aa be greater than 
 I AC, the action of the fluid upon the plane will not be greatest 
 when AC is perpendicular to CEand CE to CB ; but when ACB 
 being an obtuse angle, the sine of ACV is to the radius as AC 
 to 2Aa, and the plane CE is perpendicular to AC. For let C^ 
 =: Ca, aq be perpendicular to CB in q, then ak — igq. Suppose 
 CA = «, Art =: c, AV = X, then ak — Co- + C^ = Ca +rtV" 
 =: Vaa-^-cc—^cx-Vx — c,' and whcn the fluxion of this quantity 
 
 vanishes, — -^-x =: o, aa+cc — Qcx — cc, aarzocx, 
 
 */ aaJf cc-^cx 
 
 or .r : a : : rt : 2c ; and it is easy to see from the construction that 
 in this case ACE must be a right angle. For example, if c =a 
 then .T=f a, ACV = 30 degr. ACB = 120 degr. ACE = go 
 degr. and ECB = 30. degr. 
 
 916. Suppose now that AC (fig. 338) represents the direction 
 and velocity of the wind,CB the direction in which a ship moves. 
 Art parallel to CB the velocity of the ship, CE the situation of the 
 sail, and letus abstract from her leeward wtdiX, or suppose that no 
 deflexion from the direction CB is occasioned by the obliquity of 
 the wind or sail to the course CB. Then, in order to determine 
 the most advantageous position of the sail CE (when CA, CB, 
 and Art, are given), that the wind may act with the greatest 
 foicc to impel the ship in the given direction CB, produce AC 
 
 Y 2 till
 
 324 Of the general Rules Book IF. 
 
 till AD : AC : : 4 : 3. Let DG be parallel to CB, and a circle 
 aeg described from the centre C with the distance Ctr meet DG 
 in g; then the sail CE ought to bisect the angle aCg, by art. 
 91 1 ; or let CV be perpendicular to Aa in V^ LV =: VC X Vli 
 y/r: I Ma, \t = Lf + V/', and CE produced pass through t. 
 When Aa the velocity of the ship is neglected, or when the 
 motion begins, CE ought to bisect the angle ACG ; which is 
 the case that was resolved long ago by Mr. Fatio and Mr. 
 lluygens by a biquadratic equation; and has been considered 
 more fully since by Mr. Bernouilli, Manoeuvre des l^aisseaux, 
 chap. 5. But in some cases the ratio of Art to AC is not in- 
 considerable; and supposing AC perpendicular to CB, if (for 
 example) Aa z=. \ AC, the angle ACE ought to exceed \ ACG 
 ( =: 54'^ 44' in this case) by about 9 1 degr., if we would have 
 the wind impel the ship with the greatest force in the direction 
 CB. 
 
 917. The force with which the wind impels the ship in the 
 direction CB is always measured by ah X ap; and when this 
 force and the resistance of the water become equal, the motion 
 of the ship becomes uniform. Let CK represent the uniform 
 velocity which the ship would acquire by the same wind in its 
 direction AC, if the sail was perpendicular to AC, and the force 
 in this case which sustains the motion of the ship, and balances 
 the resistance, will be measured by KB*. Therefore (the re- 
 sistance of the water being as the square of the velocity of the 
 ship) CK"" : Art"" : : KB* : ap X ak zz. (supposing Aa parallel to 
 
 CB to meet CE in aV- X -^j; consequently Aa : at -. \ CK 
 
 X \/^: KB. LetCA=a,Aa = .r,EN=?/,AV =», 
 CK:KB: : 1 :w;then Aa '.at : :7/\/'7":?wa -/T,- and, because 
 ht - Vf =P AV = V7^::=Tp X ^^^ + i?,.Aa - xzz 
 
 ^' ^ — .—- '-^ — - • Suppose a,p, and nu to be con- 
 
 stant^
 
 Chap. V. for the Rcsolutwn of Problems. 3C5 
 
 stant, and when xxsa maximum we shalf find that aa — oj/y — 
 
 • • '-^+ ■ ■ • ^^ = 0- liiis IS an equation for determm- 
 
 m \/aa — fifi 
 
 ingthe sine of the angle ECB which ought to be contained by 
 the sail and the hue of the ship's motion, in order that the velo- 
 city of the ship in this line maybe the greatest possible, a,p, and 
 m, being given. 
 918. If AC be perpendicular to CB, then p := o, and 5j/i/ + 
 
 •^ — r^ = aa. For example, let m nay's", that is, let the ve- 
 
 locity of the ship be to the velocity of the wind when the ship 
 moves in the direction of the wind, and the wind is perpendi- 
 cular to the sail as 1 to 1 4" 2-/ 2 (or nearly as 1 to 3.828) ; 
 then, if the ship sail in a direction perpendicular to that of the 
 wind, the sail oughtto be inclined to the wind in an angle of 60'', 
 or to the way of the ship in an angleof 30''. For the equation 
 fory v/hen i" is a maximum is, in this example, aa — ^yy — 
 
 '^—^- — 0, which gives w = - ; and in this case the velocity of 
 V2 2 "^ 
 
 a V y X V aa — yy __ U 
 
 the ship is — ^ "" £ - = —--, The sine of tlie angle 
 
 ECB is always less than x CB. 
 
 919. The angle ECB (fig. 340) contained by the sail CE and 
 course oftheshipCB,withACthe velocity of the wind bcinggiven, 
 the velocity of the ship is greatest when ACE is a right angle, 
 that is, when the wind is perpendicular to the sail; as is obvious, 
 and agrees with art. 917j where, if a, y, and w, be given, x be- 
 comes a maximum when p'^y- Supposing AC to be perpendicu- 
 lar to CE, x~ ^-=~ — r., and is a maximum when y oxpzza 
 
 ma-Za-^-yVif 
 9 
 
 X V^~- ; that is, of all cases wherein tlie wind is supposed to 
 
 be perpendicular to the sail, the velocity of the ship is greatest 
 
 Y 3 (providing
 
 o 
 
 26 t)f the general Rules Book II. 
 
 (providing CK be not less than -f- CA, or m be not greater 
 than 2) when the sine of the angle ECB contained by the sail 
 
 and course is to the radius as v'wot to vT> and the velocity 
 
 of the ship is gi-eater in this case than when the wind biows in 
 
 the direction of the course, and is perpendicular to the sail in the 
 3 
 
 ratio of ?» + 1 to 3 ^'^, or (supposing « = 2 — m) of 1 — 
 
 -to 1 — ^' ^. If, for example, CK : CA : : 1 : 2, the velo- 
 city of the ship in the direction CB will be greatest when the 
 
 sine of ECB, or ACV, is to the radius as 1 to v'T; that is, when 
 the angle ECB is about 39^ 3', or when the angle AC&, 
 in which the direction of the wind is inclined to the course 
 of the ship, is an angle of about 50*^ 57'. And the velocity 
 of the ship is in this case greater thanwhenthe same wind blows 
 directly in the course of the ship, and the sail is perpendicular 
 to the wind (in which case the wind is commonly thought 
 
 S3 3 
 
 to be most favourable) in the ratio of v/ss to \^2T, or of Q^/T 
 to 3; and by inclining the sail CE to the wind, so as to increase 
 the angle BCE, the velocity of the ship in the right line CB 
 will be still greater. There may be many other cases supposed 
 from art. 916, wherein a side-wind would promote the motion 
 of the vessel more than a direct wind. For example, if the ve- 
 locity of the vessel in the direction CB be to the velocity of the 
 wind as 1 to 3, and the angle ACB be only of 109'' 28', the 
 force by which the wind will promote the motion of the vessel 
 in the course CB will in this case be greater than when the 
 
 wind is direct, or the angle ACB is of 180°, in the ratio of 
 
 3 ___^ 3 
 \/ 32 to 1/21 ; the sail being supposed in both cases to have the 
 
 most advantageous position, which was determined in art. QlG. 
 
 920. AgiVenlineAC(y?of.341) beingdividedinB, therectangle 
 
 AB X BCisiimaximumvfhenAB = BC, by what is shown in the 
 
 elements of geometry. Hence it folWvvs, that, if a given line 
 
 AG
 
 Chap. V. for the Resolution of Problems. 327 
 
 AG be divided into a given number of parts AB, BC, CD, DE, 
 EF, FG, the product of the parts AB x BC x CD x DE, &c. 
 is Si maximum when they are equal to each other; because if 
 BD the sum of any two adjoining parts be divided equally in 
 C and inequally in c, BC X CD is greater than Be x cD, and 
 AB X BC X CD X DE x &c. is greater than AB x Be x cD 
 
 X DE X &c. If a given right line AG be divided in C, and 
 AC« X CG"* be a maximum, then AC : CG : : n :m; for if we 
 
 suppose AG = a, AC =: x, a" x a — x*" = y, then -^ = ^ 
 ^, and if 7/ = 0. - = , that is, AC : CG : : n : m. 
 
 a—x ^ X a — X •• 
 
 The same proposition may be derived from the former case 
 when n and m are any integer numbers : for example, AB X 
 BG* is a maximum when AB is to BG as 1 to 5; because if 
 BG be divided into five equal parts BC, CD, &c.thenAB x BG* 
 = 5XoX5x5x5xABxBCxCDxDExEFx FG, 
 jvhich is a inaximum when AB zr BC ::= CD = DE = EF = 
 FG. If AG be divided into three parts AB, BD, and DG, 
 then AB x BD« x DG»» is a maximum when AB, BD, and DG, 
 are to each other in the same proportion as the numbers l,w,and 
 m, respectively; because, wherever we suppose the point Bio 
 be, BD« X DG"* cannot he a maximum {and consequently AB 
 X BD« X DG"* is not amaximum) unless BD : DG : : n :m; 
 and wherever we suppose the point D to be, AB x BD" cannot 
 be a maximum unless AB : BD : : 1 : w. The continuation of 
 those theorems is obvious; and this brief method of resolving 
 several questions relating to maximaand minima that cannot be 
 so easily reduced to the common rules, was mentioned in a letter 
 to Martin Folkcs, Esq. Phil. Trans. No. 408. The following 
 article gives another useful instance. 
 
 921. TherudiusACC/ig-.342)andarkAFbeinggiven,letAFbe 
 divided into three parts, AE, EB, and BF, let EM, EN, and 
 BR, be the sines of the arks AE, EB, and BF; then if £M« x 
 EN X BR"»be amajwrtMw, the tangents of the arks AE,EB, and 
 
 y 4 BF;
 
 '28 Of the general Rules Book 11. 
 
 BF, shall be in the same pioportion as the numbers n, \, and m. 
 This follows trooi art. 910, because, wherever we suppose the 
 point B to be placed upon the ark FE^, EM" X EN is not a max- 
 imum {aii ^10), unless the tangent of the ark A E be to the 
 tangent of EB as ^^ to 1 ; consequently the ark AB must be 
 divided in this manner, t'uit BI\"» x EN X EM" may be a 
 maximum. In like manner, wherever we suppose the point Eto 
 be taken upon the ark AB, EN X BR*" cannot be a maximum, 
 luiless the tangent of EB be to the tangent of BF as 1 to m; 
 and the arkFE must be divided in this manner, that EM" X EN 
 xBR"^nvdyhe a maximum. Therefore if EM:« X EN x BR"» 
 te a tnaximum, the tangent of AE must be to the tangent of EB 
 as n to 1, and the tangent of EB to the tangent of BF as 1 to 
 m; that is, the ark FA mnst be divided in such a manner in B 
 and E that the tangents of AE, EB,and BF,maybe in the same 
 proportion to each other as the numbers n, 1, and m. liii—my 
 then AE •=. BF. The continuation of these theorems is like- 
 wise obvious. If a given ark be divided into any given num- 
 ber of parts whose sines are represented by a, h, c, d, e, &c. and 
 din X !>« X C' X c?s X &c. be a maximum, then the tangents of 
 the respective parts must be in the same proportion as the in- 
 dices, m, n, r, s, &,c. and (because the sine of an ark is to the 
 radius as the radius to the secant of the same ark) the product 
 of the same powers of the respective secants of those parts is a 
 maximum. 
 
 922. For an example of this, the force and direction of the 
 wind being given, let it be required to find the most advanta- 
 «-eous course of the ship and position of the sail, that the ship 
 maybe carried in a given direction, or removed from a given 
 coast or right line, as fast as possible. Let AC represent the 
 force and direction of the wind, CF the line from which the 
 ship is to be carried as fast as possible, CB the course of the ship, 
 and CE the position of the sail. Let AQ be parallel to CB, AP 
 perpendicular to CE in P, and PQ perpendicular to AQ in Q. 
 Then the force by which the wind impels the ship in the di- 
 rection CB at the beginning of the motion will be as AP X AQ
 
 Chap. V. for lite Tle^ohition of Trohlcms. 329 
 
 EN 
 
 = EM* + — ; and the velocity of the ship (supposing it to 
 be incomparably less than the velocity of the wind) shall be as 
 EM X '/en ; which, reduced to the direction B II perpendicu- 
 lar to CF, is as EM X -/en X BR ; and this last velocit}- is a 
 maximum (by the last article) when the tangents of the arks 
 AE, EB, and BF, are in the same proportion as the numbers 
 1, land 1, or 2, 1 and 2. Let the radius CErra, the tangent 
 of AF be represented by h, the tangent of AE or BF by t, and 
 the tangent of AB or FE by T. Because the arks AB-f BF 
 
 =:AF, it will easily appear that if— ax- ~",„ : and in the same 
 
 '' *■ * au-\-o i 
 
 manner, because BF -}- BE — FE, the tangent of BE (= -) 
 
 -"" ^ -^^Tt/' ^'^^^lenceT = ^__-.; consequently ?^—4Z*« 
 — 5aat + 2baa — o ; and, h being given, t and T may be 
 found by the resolution of this cubic equation. 
 
 923. If FCA be aright angle, then b is infinite, and (ltt=aa, 
 OTt:a:: I : -/s", and T : « : : a/s" : 1 ; that is, ACB = FCE= 
 54° 44' ; consequently, if the velocity of the ship may be neg- 
 lected as incomparably less than the velocity of the wind, the 
 course ought to contain an angle of 54° 44', and th^s sail an 
 angle of 35° iG' with the direction of the wind, that the ship 
 piay gain upon the wind as much as possible; and this is the 
 case resolved by Mr. Bernouil/i, Manoeuwc chs Faisseaiix, 
 p. 50, 8cc. If the course CB and position of the sail CE is re- 
 quired, that the ship may get away from the line AC as fast as 
 possible, then we are to suppose ACF to be a continued right 
 line, or b =. o, in which case tt =: 5aa, or t : a : : '/'^ : i ; 
 consequently the angle ACE ought to be of 65° 54', and ACB 
 of 1 14° 6'. If the angle ACE be given, the tangent of ECB 
 ought to be to the tangent of ECF as 2 to 1 ; and ECB is de- 
 termined by a construction similar to that in art. 910. We 
 have always supposed the sail to be a plane, and have abstracted 
 from the lec-way of the ship, but shall not enter farther into 
 
 this
 
 330 Of the general Rules Book II, 
 
 this theory at present. Mr, Renau published an ingenious 
 treatise on this subject in 1689; but some particulars in it 
 have been corrected by Mr. Hui/gens and Mr. Bernouilli. 
 Several other mechanical problems may be resolved in the 
 »amc manner as these we have considered. 
 
 9C4. In book I. chap. 13, it was show^n how many problems 
 may be immediately reduced to equations that involve first 
 fluxions only, which it has been usual to resolve first by equa- 
 tions that involve second or higher fluxions ; but as that me- 
 thod is not always applicable, we shall give some examples of 
 the method of reducing equations from second to first fluxions. 
 Suppose X constant, and if the equation involve x, y, and y, 
 but if either x or y be w'anting (of which kind are those which 
 arise most commonly in the resolution of problems), it may be 
 reduced to first fluxions, by introducing a new variable quantity 
 
 z, and supposing it equal to ^ or "-. Suppose, for example, 
 that a* -J- y* = — , let y zz. zx, and consequently y zzz x, 
 
 n 
 
 then nx- X ] +zz zzyzx, or 7ix K l+zz { — 7iyX —^^J — yz ; 
 
 therefore ^ = ■ ^^ , and (by art. 740) w'a = 1 + 22 X A, 
 y ]-^zz 
 
 or zz :=■ 1 rz ^ ; consequently x = - ^ — , where 
 
 A ;^ Vj^i»-A 
 
 A denotes an invariable quantity. 
 
 925. Let the point T (^g.343>) move in the right line Aff, and 
 the point M in the curve FM, so that the velocity of the point 
 T may be to the velocity of the point M in the invariable ratio 
 of » to 1, and the motion of M may be always in the direction 
 MT or TjNI ; and let it be required to determine the curve FM. 
 Let AP = X, PM = y, FM = s, AT = t; then ns - 't = (be- 
 cause t "=■ X — rr-^ or to ^ — X, and x is supposed constant) 
 y n
 
 Chap. V. for the Resolution of Problems. 331 
 
 ,JLI1-* Let X — zij, then zy •\- zy ■=. o, and ny ^/Tl^ — 
 
 / _ . . . 
 tlili = ±yz, and ^ = -^ whence Av" = 
 
 zy"- y '•l + zs' "^ 
 
 <•!+»» i^^ an^ AAj/-»+ 2A~j/» = 1, or 2z = ^ r= ~ti 
 
 if _ V 
 +1 
 
 ± Ay», consequently 2r=: i^- ± ky^y, and 2a: =£ 
 
 —^ + K, where K denotes an invariable quantity. 
 
 If w r: j^ then x — — — — ±— — -~ + K, and the curve is 
 
 aparabolaof the third order oflines. Ifw =rl, the curve (^g-. 344) 
 is constructed by logarithms or the equilateriil hyperbola. Let 
 NDN be such an hyperbola described betwixt the asymptotes 
 Ca and Cb, D a given point in the hyperbola, join CD, let 
 KLM perpendicular to the iisymptote Ca in L meet CD in K, 
 and let LM X 2FD be always equal to the area DJNK ; then M 
 shall be a point in the curve. 
 
 926. An equation that involves second fluxions is sometimes 
 easily reduced to first tluxions, by the common rules of the in- 
 verse method, which were described in chap. 2 ; and that the 
 solution may be general enough, when any fluxion is supposed 
 constant, a quantity compounded from it or from its powers and 
 invariable quantities ought to be added to the equation. For 
 example, let it be required to find the nature of the line 
 in which the curvature is always as the ordinate, this being a 
 figure by which several problems of different kinds are resolved. 
 Let the ray of curvature be represented by R, and because 
 
 R IT . / \ „ 3 suppose s constant, then R r= -^ • In the 
 
 s X X s X 
 
 figure required R is inversely as the ordinate^ ; consequentl}'^, 
 a being an invariable quantity, we may suppose — =: R = 4r 
 
 or
 
 33 2 . Of^ the general Rules Book II. 
 
 or Q.J/1/S = aax ; and by finding the fluents, t/j/s = aax -f K« 
 Avhere K denotes an invariable quantity, and Ks is' added 
 
 because s is supposed constant. If K = o, then s'^ : ,i^ : : a* : 
 
 • • -p. ' 
 
 y,_y» : x^ : : a+ — y : j/% and consequently .r = ^~-^'' 
 
 927. The celebrated author who first resolved this as well 
 as several other curious problems, after his account of this 
 fi'nire (which is commonly called the clastic curve), adds, Ob 
 graves causassuspicor cttrvid nostra const ructionem a nuUiussecUo- 
 nis conica scu qiiadratura seu rectijicatione peyidere, Act. Lips. 
 1694, p. 272. Butit is constructed by the rectification of theequi- 
 lateral hyperbola ; for if the base of a figure be always taken 
 equal to the perpendicular from the centre on the tangent of such 
 an hyperbola, and the ordinate equal to the excess of the tangent 
 terminated by that perpendicular above the ark intercepted be- 
 twixt the vertex of the hyperbolaand thepoint of contact, then 
 the figure shallbe the elastic curve. Let AEZ fjg. 345) be an equi- 
 lateral hvperbola that has its centre in S and vertex in A, let E 
 be any point in the hyperbola, ET a tangent at E, and SP per- 
 pendicular from the centre S to the tangent at P; upon S A take 
 SQ=:SP,andthe ordinate QM always equal to the excess of the 
 tangent EP above the ark AE of the hyperbola ; then M shall 
 be a point in the elastic curve AMB. For suppose SA — a, SQ 
 (=:SP) = y, QM=:.r, SE = r, EP = z, and the ark AE = 5, 
 
 then r = — 3 EP = 5 = Vrr-^j/y -) z — ■ ^ -^ ^ - 
 
 But s : r : : r : 2 : : a« : v^a*— _y4 and r = — ^-» consequently 
 
 
 
 5 and X 
 
 3J ^_X — ; which is the equation for the common clastic 
 
 curve. 
 
 92s. In
 
 Chap. V. for the Resolution of Problems. Sf^s 
 
 928. In general, the equation for the elastic curve was aax =z 
 yyi — Ks,- consequently X =: + i/x - - j'JCZ ! ^^,^ — -— — » 
 
 y aa — K -\-ijif X ^a -|- K —yif 
 
 and by comparing this fluxion with those described in art. 804 
 and 805, it will appear that the elastic curve may be construct- 
 ed in all cases by the rectification of the conic sections. Let 
 SArA"o;-346') be halflhe transverse axis of the hyperbola AEH,SD 
 half the second axis ; upon DS take SF. SA : : SA. SD, and S3 
 =:AF, describe the elliptic quadrant A11Z>, and, E being any 
 point in the hyperbola, SP perpendicular to the tangent EP in 
 V, upon S A take SQ — SP, and let the ordinate QR meet the 
 ellipse in R ; then, by taking QM upon QR equal to ~1 x 
 
 . SD*— SA^ 
 
 EP— AE + g^^ ^ g^ X AR, M shall be a point in the elastic 
 
 curve ; and the ray of curvature at any point M shall be 
 A D^ 
 
 equal to ^^, because in comparing those fluxions we suppose 
 aa—¥.- SD^ and «a + K = SA% or <laa = SD* + SA* 
 =■ AD% and the ray of curvature was supposed equal to 
 
 aa AD^ 
 
 929. Let SA CJg-347)he incomparably less than SD, then, be- 
 cause SD : SA : : SA : SF, we may suppose SF to vanish, AR6 to 
 be a quadrantof a circle, and EP — AE to vanish ; consequent- 
 ly QM = ~ X AR and SB = ~ x ARb - (if the ratio of m 
 to 1 denote that of the circumference to the diameter) ~ X SD ; 
 
 and the elastic curve in this case will represent the figuie 
 which a musical chord BAG assumes in its small vibrations bj 
 the converse of art. 569, the tension of the chord being every 
 where equal. Let P represent this tension, or a weight equi- 
 valent to it, n a section of the chord perpendicular to its length, 
 R the ray of curvature at any point M, and V the force by 
 ■^hich the motion of any point at M towards BC is accelerat- 
 ed.
 
 S34 Of the general Rules Book IL 
 
 ed, while the chord returns to its natural state; then, by art. 
 561, the tension would be equal to the weight of a chord of 
 the same thickness of the length R, if the gravity was equal 
 
 to V; that is, P = nRV, orV=4.--x^= (be- 
 
 cause SD is to AD nearly in a ratio of equality) - x -^rr =r 
 
 "^ X ^^. If we suppose, with Dr. Taylor, N to repre- 
 sent the weight of the chord, L its length, g the force of 
 gravity, D the length of a given pendulum, SA = a, SQ or 
 
 MN = V ; then, because N =: wL«-, or » z= — - and L = SSB, 
 
 it follows, that V =: ^ , Because V is as 7/ the distance of 
 
 M from BC, the vibrations of the chord are similar to 
 those of a pendulum; and the time in which M describes 
 MN is to the time in which the pendulum D performs 
 
 half a vibration as v |. to v -5 or as — -— to v'd ; con- 
 
 sequently the number of vibrations made by the chord, while 
 
 the pendulum vibrates once, is expressed by m X ^^; which 
 
 is Dr Taylor's theorem, and serves for determining the 
 number of vibrations made in a given time by any given chord 
 that is extended by a given weight ; or for comparing the 
 number of vibrations made by different chords in equal times, 
 upon which their tone depends. Thus if the weight P be 
 
 the same, the number of vibrations is as — ; and when 
 
 V LN" 
 
 the chords are of the same kind (or N is as L) the vibration* 
 are as - , If the chord be given, the number of vibrations is 
 as v/p. The ratio of w to 1, or of the circumference to the 
 diameter, enters the expression of the number of the vibrations 
 of the chord; because the ratio of 2SB the length of the chord 
 to the ray of curvature involves it; and there seems to be a dif- 
 ference
 
 Chap. V. for the Resolution of Problems. 335 
 
 fcrence in this respect betwixt the theorems by which the vi- 
 brations of a musical chord, and these which are produced in 
 the air by organ-pipes, or other wind-instruments, are to be de- 
 termined. 
 
 930. Because the elastic curve is defined by the equation 
 
 (art 926), 1/7/s — K.S = aax, it follows, that s = — — "-^ 
 
 ~aay 
 
 — ,/ — T-v V ^ ^ , * Let this fluxion be compared with 
 
 that in art. 805 (fig. 346), of which we found the fluent to be-^ 
 
 SA. 
 
 xAR+AE — EP; and it will appear, by supposing aa-\-K:zz 
 SA% and aa — K=:SD% that AM the ark of the elastic curve 
 
 is equal to ^ x ^ X AR + AE— EP. Therefore the 
 
 figures that have been constructed by the rectification of the 
 elastic curve maybe constructed by the rectification of the hy- 
 perbola and ellipsis ; particularly the curve along which if a 
 heavy body moved it would recede equally in equal times 
 from a given point, which Mr. Leibnitz constructed by the 
 rectification of a geometrical curve of a higher kind than 
 the conic sections, and Mr. James Bernouilli by the elastic 
 curve. Act. Lips. I694, p. 272, 277, 338, 370, &c. The fluents 
 
 r. is-is a^x a-z , z'z ^ i • i 
 
 ol — < --==> ^ — 9 and (which are 
 
 mentioned, ibid, p. 338, where it is said of the first only, that it 
 may be assigned by the rectification of the hyperbola) are all 
 assignable by the rectification of the equilateral hyperbola, and 
 of the ellipsis, whose excentricity is equal to the second axis. 
 Let AE and AR (Jig. 348) be such an hyperbpla and ellipsis, 
 
 SA=:a, and SE=s, then the F — — rrAE, and the fluent of 
 
 — ^=: = AR + AE — EP. if SP be peipendicular from 
 
 the
 
 336 Of the general Rules Book II, 
 
 the centre S on the tangent at E in P, SA—a, and SP = z, 
 
 then the F. -I^^ = AR + AE — EP, and the F. -^ ^j.. 
 
 = EP — AE, as appears from art. 799 and. 802. Fluents of 
 other forms may be assigned by the rectification of the conic 
 sections by art. 804 and 805. 
 
 931. It may be worth while to show here how the same 
 easy method which was described in chap. 13, book I. for de- 
 termining, by first fluxions only, thenatureof the lines of swift- 
 est descent, of the figures that amongst all those of equal peri- 
 meters produce wiaxma and minima, and of that which gene- 
 rates the solid of least resistance, serves with equal facility and 
 evidence for discovering the equation of the curve when other 
 limitations are added in the problem ; as when it is required to 
 find the solid, which amongst all those of equal capacities, and 
 that are bounded by equal surfaces, meets with the least resist- 
 ance in a fluid. The fundamental lemma (demonstrated in ' 
 art.572and 592)isthat, 'dAK(Jig.349) begiven,KE be perpen- 
 dicular to A K, a and 2< denote any given or invariable quantities, 
 
 AE KE\ 
 
 then AE x a — KE X m (or ) is a minimum whea 
 
 KE :AE::u: a, or a xKE-uX AE. Let the base FP == 
 X, the ordinate PA zz y, the ark G A = s, AK = y, and if AE 
 the tangent at A meet KE parallel to the base in E,*then AE 
 = s and KE = jr ,• and it follows from the lemma, that if 
 V and u represent any quantities compounded from the powers 
 of j/ (so as to be of the same value when 2/ is the same), and if 
 
 y be given, then Vs — ux, and -^ -^ are minima when 
 
 Yx — us. From this it follows (as in art. 576 and 593), 
 that if GAD be the whole curve, and DH the difference of the 
 ordinates at G and D be given, then the F. Vs — F. «.r, or the 
 
 F. -^ F. — , shall be a minimum when the nature of the 
 
 figure is defined by the equation Nxzzm. Therefore, supposing 
 
 this
 
 Chap. V. for the Resolution of Prohlcms. 337 
 
 this to be tlie equation of the curve, and DH to be given, if 
 the fluent of Vs be also given, then the F. ux shall be nmaxi- 
 miim ; or if the latter fluent be given, then the former shall be 
 
 a minimum : ant! if the fluent of— be given, the F. -^ shall 
 
 be a maximum ; or if the fluent of™ be 2;iven, the fluent of — ^ 
 shall bo a muiiinum. It appears, likewise (as in art. 59-5), that if 
 DIi witii the base FC or G II be given, and the fluent of Vsbe 
 given or invariable, then the F, ux will be a maximum or mini- 
 mum when the equation of the curve is V.r =: e+u X s, where 
 € dei;otes an invariable quantity that may be positive;, or ncgn^ 
 tive, or vanish. .'■■:> 
 
 932. Suppose, therefore, V= A + Bj/ + Cyy|- J)f + ifeVi 
 and uzz a -\- h !/-{■ c r/^ + df -r&ic. where A, B, C, &c. and a, h, 
 c, &c. denote any invariable coefficients that may be positive or 
 negative, any of which may be supposed to vanish; and the flu- 
 en t of V.S — ux, tiiat is, of s X A -TBj/ + Cj/j/ + &c. — x X 
 a -f bij + o/y + &c. shall be a minimum when the equation of 
 the figure is .r X A + Bj/ + Cj/y -!- &c. =: sxa-\-hy-rcyy-\-^c. 
 the ordinate DH being given. Therefore, if the fluent of .s- 
 X A -f B// -f Ci/i/ 4- "Sec. be also given, the fluent of a? x 
 « + ^j/ + cj/j/ ►J- &c. siuill be a maximum-; or if the latter be 
 given, the former shall be a minimum : and if the base FC or 
 GH be given with DH and the F. s x A + Bi/-\- Oyy + &c. then 
 the F. X X a + by + cyy -f »Scc. shall be a maximum or minimum 
 
 when X X A -f Bj/ -|- Cyy + &c. ~ s X e + A + Bj/ ^Cyy + &c. 
 Of which theorein it is an obvious but a particular case only, 
 that, if the nature oi'the figure be defined by the last equation, 
 and G H, DH, with the fluents of As, 'Bys, Cyys, &c. byx, cyyx, 
 (hf^X) 8cc. be all given or invariable, one only excepted, this last 
 fluent shall be either a maximum or minimum. 
 
 VOL. II. Z 933. For
 
 53S Of the general Rules Book II. 
 
 933. For example, the [)oints G and D being given, if the 
 perimetei- GAD (or tiie F. As) be also given, tlie area FGADC 
 (or the F. ?/x) is a niaxitimm or niini/mim when Ar =s X e+A+Bi/, 
 that is, wlien GAD is an ark of" a cirelc. If the surface gene- 
 rated by the ark GAD about the axis FC (or the F. By.s) be 
 given, then the solid generated by the figure FG A DC about the 
 same axis (or the F. tj/j/.i) is a /naiiniiitu or tui/iimum when Bi/x 
 = s X e'-fiyi/; and when e r: o, this is again a circle. If the 
 perimeter GAD (or the F. As) be given, the solid generated by 
 FGADC about the axis FC is a maximum, or minimum when Ax 
 = e-\-cy>j X 6', and GAD is the elastic curve, which was con- 
 stnicted by the arks of conic sections in art. 928. If the F. s'x 
 A + By be given^ then the fluent of .r X a + bij + cyy is a max i- 
 
 mum or minimum when jr X A + B// = s X c+a + Oi/ -{■ cyy. 
 And it is no more but a particular case of this theorem that ihe 
 , same equation comprehends that of the figure when the points 
 G,D,with thesnrface generated by GAD.about FC and thearca 
 FGADC, are given or invariable, and the solid generated by this 
 area about the axis FC isa maximum or miuimutn. For since, by 
 the supposition, the fluents of i- X A-\- By with the F. ax and 
 F.^j/o: are given, so that thcF. cyyx alone is variable, and the 
 fluent of F. x X a -{-by-i- ctjy is a maximum or minimum, it is ma- 
 nifest that the F. cyyx is a maximum or minimum. Nor is there 
 any occasion, in order to obtain surhec|uations, to have recourse 
 to higher fluxions, or to resolve the element of the curve into a 
 number of infinitesimal parts. Other examples may be derived 
 in the same manner at pleasure. 
 
 934. The same metiiod is extended to several other sorts of 
 problems, by art. 605. Let V" be now compounded from the 
 powers of * and y as well as iVom the powers of y and iu- 
 
 variable quantitie.^. For example, let V =~v~ + A + By + 
 
 C//-+Dy^ + &.c. where K is supposed to be eompoundeJ from 
 the powers of j/ and invariable quantities, antl u — a <^ by +
 
 Chap. V. for the Resolution of Problems. 3iJ9 
 
 cyy + See. as formerly. Then it appears, as in article 
 GOo, that \s + ux shall be a mininmm when T^ X 
 -^ + Ax + V>i/x + Cy\i' 4- Sec. = + 'sii = 
 
 + s X rt+ /yj/ + cj/j/ + &c. and by substituting 3 for u this 
 equation serves for resolving theproblems that may be proposed 
 concerningthe solid of least resistance. For, supposing the solid 
 thatisgenerated by the revolution of thefigure I^G XDC(tig.ooO) 
 to move in a fluid with a given velocity, and in the direction of 
 the axis CF, then, according to the common doctrine of the 
 action of the particles of fluids on bodies (or if the fluid be 
 rare, as Sir hauc Xetctoii supposes), the resistance of the co- 
 nical surface generated by the tangent AE will be ultimate- 
 
 AK- //^ ,',^ • • • 
 
 Iv as PA X AK X = -^ =^^ X s, and £i/i/^x — xs 
 
 AE~ > si ^^ 
 
 X A + Bj/ + Ct/i/ + Sec. = s* X a + bj/ + cj/y + Sec. is 
 the equation for the curve that generates the solid of least resist- 
 ance, when the points G and D with the fluents of As, Bj/s, 
 
 Cj/j/s, and the F. r X « -f bj/ "4* rj/j/ + 8cc. are supposed tobegiven 
 or invariable. Thus if the points G and D only are given, 
 
 the equation is 2//j/lr rr as^, as Sir Isaac Nezcto/i found. If 
 the solid of the least resistance is required amongst all the solids 
 of equal capacity, the equation is 'lyif x zz as^ + cif s^. If 
 the solids are supposed to be bounded by equal surfaces, the 
 equation for the figure which generates the solid of least resist- 
 ance is ili/i/x — By is' ii: as'^. If the solid is to have the least 
 resistance of all those that have equal capacity, and are termi- 
 nated b}' equal surfaces, the equation is C'/j/^r — Vjj/xs^ •=. «s*4- 
 cifs^ ; and in like manner the equation is fouiid, when other 
 limitations that relate to the perimeter GAD, area FGADC, &c. 
 are superadded. 
 
 93 j. Because the theorems proposed in art. bGS, and explain- 
 ed in the subsequent articles, are of more general use, it may 
 be proper to give one example of the manner oi" applying them 
 
 Z 2 for
 
 ,^^Q Of the general Rules Book IT. 
 
 for discovering the equation of the figure required. Let u the 
 Telocity acquired at any point Abe as Aa'"«'l-Bj/'» xi'y, and the 
 equationofthelineofswiftestdescentbe required. Let OA (fig. 
 3,51) the rayof curvature at A be considered;;? given in})o^itioii, 
 and, supposing the point O to remain, let A move in the right 
 line OA, and APbe always perpendicular to FC in P ; let OA 
 r= <^, FF = :r, and AP :r: ?/, then if OA meet PC in I — x : q 
 : : PI : L\ : : j/ : s, and y : q \ \ PA \\k . : x : s. But, 
 by the theorem in art. 565, OA and u increase proportionally 
 while the point A is supposed to move in the right line OA, 
 
 that IS, -^ = — = — f =^ •r — + ^-' 
 
 q u Ax^+Bj/"" X y 
 
 Hence by substituting ~4^ for x, and ~ for j/, then dividinn; 
 
 S S 
 
 hy q, and substituting for the ray of curvature q its value 
 _i 5 X being supposed constant, it follows, that i-1 = 
 
 —X J/ ' ^ ^ s" 
 
 nAx^';-v.^^r-^x _j_ V. ^ ii. Kit be required that tho 
 
 Ax"+ Bf X y • 
 
 curve shall be described in less time than any other of an equal 
 perimeter, the equation may be found by the third general 
 principle described in art. 563. 
 
 936. Tiie preceding examples may serve to show the exten- 
 sive usefulness of the method of fluxions in geometry and the 
 various parts of philosophy. In the account ^ve gave of this 
 cloctrine in the iir?t book, we supposed with Sir Isaac NeWf 
 ton quantities to be gei}erated by motion, and considered the 
 fluxion of a quantity rts the velocity or n^easure of this motion. 
 Some propositions, however, were demonstrated (as prop. 20 
 and 32) without making use of fluxions; and several other 
 theories described in this and the preceding book may be like- 
 wise established inauianncrindependent of the notion of a flux- 
 ion.
 
 Chap. V. for the Resolution of Problems . S4l 
 
 ion. Thus, it is easily demonstrated from art. 7 10, that, sup- 
 posing n to be any integer and positive number, if the area 
 uponlhebaseAPfy/g.Soa^M-xbealwa^'sequalto^r", thentheordi- 
 iiatePM orj/shalibealwayseqnal to ?a:«— '. For let o represent 
 Vp any increment of the base .r; and, because a^ and y increase 
 together, PM X Pp, or y X o, shall be less than PM/np z=z 
 X +0" — a", the simultaneous increment of the area, which (by 
 substituting x + o for E, and x for F, in art. 710) is less than 
 
 ii X 1 -f 0"— ^ X o; consequently 3/ is less than n X a:+o«— '. 
 In the sarnemanner, itappearsthatPM X Vp, ory X o, isgreater 
 
 than PAJ/xT = >r'* — x — o", which, by the same article, is 
 
 greater than no X x — o"— ' ; consequently the ordinate y is 
 
 greater than n X .r — o'— '. But if j/ be said to be greater than. 
 
 ??.i"— % suppose y =r //.r"— ' -f r, and o rr t»— ' + - ,._i 
 
 r, or j+ o« — ' rr a"—' -f -, then y rr ?/.r« — * y^ r zz n x 
 
 a -j- 0"— ', the contrary of which has been demonstrated; 
 and if y be said to be less than ?/.i" — ', suppose y n nx^—^ 
 
 1 
 
 r 
 
 r, and r= jr — a"—' — -«— ij, or x — o"— ' rr an—' — 
 
 — , then j/ rr n X a — oi , against what has been demon- 
 strated : therefore?/ = ;u"~'. 1 intended to have subjoin- 
 ed demonstrations of this kind of some other theorems ; 
 but this seems to be unnecessar}', after what has been shown 
 at so great length in the first book, and the first chapter of this 
 book, for demonstrating the evidence of this method. Some- 
 times we have spoke of infinites in this chapter in the 
 usual style of writers on this subject; but Ave took no greater 
 libcrt}" in making use of such expressions than is allowed to 
 authors in the inferior parts of these sciences, particularly to 
 
 such
 
 S42 Of the general Rules, kc. Book II. 
 
 such as treat of trigonometry, wlio, while they assign a tan- 
 gout and secant to every ark, and find that no finite tangent 
 or secant can belong to the quadrant, therefore mark it injinite 
 in their tables. In the same sense // or j/ arc in certain 
 t.*ases supposed to become infinite; but we pretend to draw 
 no conckisions conecrninginfinitcs from the use of such concise 
 and convenient expressions^ nor inferences of any kind, but 
 such as may be otherwise justified by unexceptionable evi- 
 <^ence. 
 
 937. In this doctrine^ when the velocity of amotion is de- 
 termined, it is always with relation to the veloeit)' of some 
 ©iher motion; and when we enquire at what rate the ordinat(;, 
 for example, increases or decreases, it is always in relation to 
 the base, or some other magnitude, with which it is compared. 
 It is only relative space and motion we have occasion to consi- 
 der in this method, than which no sort of quantities scein to be 
 q[iiore clearly conceived by u.s. 
 
 FINIS. 
 
 Forjht au'i Coiapldi), Pnntcv , '..'iJd'e jfect. 
 Cloth Fair.
 
 Fi^.-^ct.7lo..j\ 
 
 Fi(f.-3ii. 
 
 B D 
 
 \ 
 
 Fig. 326. Nj 
 
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 FlHicXXX\-Ui;. bf nl,i,;i ,,/ th/ f:„.i .„vr^i_ii 
 
 
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 F M 
 
 
 Fig. 330 
 
 FU].332.N'^2. 
 
 KC <
 
 nateXXXVIH. /•(' /.<• piacd,it the End of l.'l I I 
 
 1^111320. 
 
 Tujisj. 
 
 330 / 
 
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 On Noirmbcr ISth uih' be publis/iLiI, 
 THE 
 
 GENTLEMAN'S 
 
 MATHEMATia\L COMPANION 
 
 Fo?' the Year 1802 ; 
 
 CONTAINING 
 
 ANSWERS TO THE LAST YEAR'S ENIGMAS, REBUSSES, CHARADES, 
 QUERIES, AND QUESTIONS; 
 
 ALSO 
 
 NEW ENTv^xMAS, REBUSSES, CHARADES, 
 QUERIES, AND QUESTIONS, 
 
 Proposed to be answered next Year : 
 
 ■VVITII SOME 
 
 Curious Papers from the Fhilosojjhical Tranmcl/wns. 
 
 Printed for the Editor, W. Davis, Member of the Mathematical and Philoso- 
 phical Society, London ; and sold by H. D. Synv-snds, VV. Baynes, 
 and J. Wallis, Paternoster Row ; ?ip:a tlie Editor, 
 W. Davis, No. 2, Albion ITuildings, 
 Aldersgate Street. 
 
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 SURVEYING, 
 
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 AND HIS USE OF THE GLOBES, 
 
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 In a few JFeeh 'uill be published bj/ the Editor^ 
 AN ENGLISH EDITION 
 
 OF THE 
 
 Spherical Elements of TJieodosius. 
 
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