COR. 1. Hence it appears, that if the time be expounded by any partAD of the asymptote, and the velocity in the beginning of the time by theordinate AB, the velocity at the end of the time will be expounded by theordinate DG ; and the whole space described by the adjacent hyperbolicarea ABGD ; and the space which any body can describe in the same timeAD, with the first velocity AB, in a non-resisting medium, by the rectangle AB X AD.COR 2. Hence the space described in a resisting medium is given, bytaking it to the space described with the uniform velocity AB in a nonresistingmedium, as the hyperbolic area ABGD to the rectangle AB X AD.COR. 3. The resistance of the medium is also given, by making it equal,in the very beginning of the motion, to an uniform centripetal force, whichcould generate, in a body falling through a non-resisting medium, the velocity AB in the time AC. For if BT be drawn touching the hyperbolain B. and meeting the asymptote in T, the right line AT will be equal toAC, and will express the time in which the first resistance, uniformly continned, may take away the whole velocity ABCOR. 4. And thence is also given the proportion of this resistance to theforce of gravity, or ay other given centripetal force.COR. 5. And, vice versa, if there is given the proportion of the resist-; nee to any given centripetal force, the time AC is also given, in which ccentripetal force equal to the resistance may generate any velocity as AB ;and thence is given the point B. through which the hyperbola, having CHCD for its asymptotes, is to be described : as also the space ABGD, which abody, by beginning its motion with that velocity AB, can describe in anytime AD. in a similar resisting medium.PROPOSITION VI. THEOREM lVrcHomogeneous and equal spherical bodies, opposed hy resistances that arein the duplicate ratio of the velocities, and moving on by their innateforce only, will, in times which are reciprocally as the velocities at thr.260 THE MATHEMATICAL PRINCIPLES |BOOK II,vbeg-in fiing, describe equal spaces, and lose parts of their velocities proportional to the wholes.To the rectangular asymptotes CD, CH describe any hyperbola B6Ee, cutting the perpendiculars AB, rib, DE, de in B, b, E, e; let theinitial velocities be expounded by the perpendiculars AB, DE, and the times by the lines Aa, Drf.Therefore as Aa is to l)d, so (by the hypothesis). is DE to AB, and so (from the nature of the hy- C "^perbola) is CA to CD ; and, by composition, so isCrt to Cd. Therefore the areas ABba, DEerf, that is, the spaces described,are equal among themselves, and the first velocities AB, DE are proportional to the last ab, de ; and therefore, by division, proportional to theparts of the velocities lost, AB ab, DE de. Q.E.D.PROPOSITION VII. THEOREM V.If spherical bodies are resisted in the duplicate ratio of their velocities,in times which are as the first motions directly, and the first resistances inversely, they will lose parts of their motions proportional to thewholes, and will describe spaces proportional to those times and thefirstvelocities conjunctIt/.For the parts of the motions lost are as the resistances and times conjunctly. Therefore, that those parts may be proportional to the wholes,the resistance and time conjunctly ought to be as the motion. Therefore thetime will be as the motion directly and the resistance inversely. Wherefore the particles of the times being taken in that ratio, the bodies willalways loso parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their first velocities.And because of the given ratio of the velocities, they will always describespaces which are as the first velocities and the times conjunctly. Q.E.D.COR. 1. Therefore if bodies equally swift are resisted in a duplicate ratio of their diameters, homogeneous globes moving with any velocitieswhatsoever, by describing spaces proportional to their diameters, will loseparts of their motions proportional to the wholes. For the motion of eacho-lobe will be as its velocity and mass conjunctly, that is, as the velocityand the cube of its diameter ; the resistance (by supposition) will be as thesquare of the diameter and the square of the velocity conjunctly ; and thetime (by this proposition) is in the former ratio directly, and in the latterinversely, that is, as the diameter directly and the velocity inversely ; andtherefore the space, which is proportional to the time and velocity is asthe diameter.COR. 2. If bodies equally swift are resisted in a sesquiplicate ratio oftheir diameters, homogeneous globes, moving with any velocities whatsoSEC. 1L] OF NATURAL PHILOSOPHY. 261ever, by describing spaces that are in a sesquiplicate ratio of the diameters,will lose parts of their motions proportional to the wholes.COR. 3. And universally, if equally swift bodies are resisted in the ratioof any power of the diameters, the spaces, in which homogeneous globes,moving with any velocity whatsoever, will lose parts of their motions proportional to the wholes, will be as the cubes of the diameters applied tothat power. Let those diameters be D and E : and if the resistances, wherethe velocities are supposed equal, are as T) n and E"; the spaces in whichthe globes, moving with any velocities whatsoever, will lose parts of theirmotions proportional to the wholes, will be as D 3 n and E 3 n. Andtherefore homogeneous globes, in describing spaces proportional to D 3 nand E 3 n, will retain their velocities in the same ratio to one another asat the beginning.COR. 4. Now if the globes are not homogeneous, the space described bythe denser globe must be augmented in the ratio of the density. For themotion, with an equal velocity, is greater in the ratio of the density, andthe time (by this Prop.) is augmented in the ratio of motion directly, andthe space described in the ratio of the time.COR. 5. And if the globes move in different mediums, the space, in amedium which, cccteris paribus, resists the most, must be diminished in theratio of the greater resistance. For the time (by this Prop.) will be diminished in the ratio of the augmented resistance, and the space in the ratio of the time.LEMMA II.The moment of any genitum is equal to the moments of each of the generatinrrsides drawn into the indices of the powers of those sides, andinto their co-efficients continually.I call any quantity a genitum which is not made by addition or subductionof divers parts, but is generated or produced in arithmetic by themultiplication, division, or extraction of the root of any terms whatsoever :in geometry by the invention of contents and sides, or of the extremes andmeans of proportionals. Quantities of this kind are products, quotients,roots, rectangles, squares, cubes, square and cubic sides, and the like.These quantities I here consider as variable and indetermined, and increasing or decreasing, as it were, by a perpetual motion or flux ; and I understand their momentaneous increments or decrements by the name of moments ; so that the increments may be esteemed as added or affirmativemoments ; and the decrements as subducted or negative ones. But takecare not to look upon finite particles as such. Finite particles are notmoments, but the very quantities generated by the moments. We are toconceive them as the just nascent principles of finite magnitudes. Nor dowe in this Lemma regard the magnitude of the moments, but their firsf262 THE MATHEMATICAL PRINCIPLES [BoOK 11proportion, as nascent. It will be the same thing, if, instead of moments,we use either the velocities of the increments and decrements (which mayalso be called the motions, mutations, and fluxions of quantities), or anyfinite quantities proportional to those velocities. The co-efficient of anygenerating side is the quantity which arises by applying the genitum toihat side.Wherefore the sense of the Lemma is, that if the moments of any quantities A, B, C, &c., increasing or decreasing by a perpetual flux, or thevelocities of the mutations which are proportional to them, be called a, 6,r, (fee., the moment or mutation of the generated rectangle AB will be B-h bA ; the moment of the generated content ABC will be aBC -f bAC 4-1 -2. .1cAB; and the moments of the generated powers A2. A 3, A4, A 2, A 2. A 3,A*, A , A 2, A * will be 2aA, 3aA2, 4aA 311 , |A *, fA* 3i A s, |/iA3, aA 2, 2aA 3, aA 2respectively; andin general, that the moment of any power A^, will be ^ aAn-^. Also,that the moment of the generated quantity A 2 B will be 2aAB 4- bA~ ; themoment of the generated quantity A 3 B 4 C2 will be 3A2 B 4 C 2 + 4/>A3A 3B 3 C 2 4-2cA 3 B C; and the moment of the generated quantity orA B 2 will be 3aA 2 B 2 2bA 3B 3; and so on. The Lemma isthus demonstrated.CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, asyet, there wanted of the sides A and B half their moments 。a and 。b, wasA 。a into B 。b, or AB a B 。b A + 。ab ; but as soon as thesides A and B are augmented by the other half moments, the rectangle becomes A 4- 4-a into B 4- 。b, or AB -f ^a B 4- 。b A -f 。ab. From thisrectangle subduct the former rectangle, and there will remain the exces.?aE -f bA. Therefore with the whole increments a and b of the sides, tinincrement aB + f>A of the rectangle is generated. Q.K.D.CASE 2. Suppose AB always equal to G, and then the moment of thecontent ABC or GC (by Case 1) will be^C + cG, that is (putting AB andaB + bA for G and *), aBC -h bAC 4- cAB. And the reasoning is thesame for contents under ever so many sides. Q.E.D.CASE 3. Suppose the sides A, B, and C, to be always equal among themselves; and the moment B + />A, of A2, that is, of the rectangle AB,will be 2aA ; and the moment aBC + bAC + cAB of A 3, that is, of thecontent ABC, will be 3aA 2. And by the same reasoning the moment ofany power Anis naAn. Q.E.DCASE 4. Therefore since -7 into A is 1, the moment of -r- drawn into A ASEC. 11.] OF NATURAL PHILOSOPHY. 263A, together with A drawn into a. will be the moment of 1, that is, nothing.Therefore the moment of -r, or of A ,is -r . And generally since A .AT- into Anis I, the moment of drawn into Antogether with intoA n A. AnnaA"! will be nothing. And, therefore, the moment of -r- or A nAwill be T^~7- Q-E.D.V .t. iCASE 5. And since A 2 into A2 is A, the moment of A1 drawn into 2A 2will be a (by Case 3) ; and, therefore, the moment of A7 will be n~r~r or^A-j#A . And, generally, putting A~^ equal to B, then Am will be equalto Bn, and therefore maAm !equal to nbBn, and maA equal to?tbB , or tibA ^ 5an<i therefore ri aA ^~ is equal to &, that is, equalto the moment of A^. Q.E.D.CASE 6. Therefore the moment of any generated quantity AmBnis themoment of Am drawn into Bn, together with the moment of Bn drawn intoA", that is, maAmB" -f- nbBn ! Am; and that whether the indicesin arid n of the powers be whole numbers or fractions, affirmative or negative. And the reasoning is the same for contents under more powers.Q.E.D.COR. 1. Henoe in quantities continually proportional, if one term isgiven, the moments of the rest of the terms will be as the same terms multiplied by the number of intervals between them and the given term. LetA, B, C, D; E, F, be continually proportional ; then if the term C is given,the moments of the rest of the terms will be among themselves as 2A,B? D, 2E, 3F.COR. 2. And if in four proportionals the two means are given, the moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.COR. 3. And if the sum or difference of two squares is given, the moments of the sides will be reciprocally as the sides.SCHOLIUM.In a letter of mine to Mr. /. Collins, dated December 10, 1672, havingdescribed a method of tangents, which I suspected to be the same withSlusius*s method, which at that time wag not made public, I subjoined thesewords This is one particular, or rather a Corollary, of a general nte264 THE MATHEMATICAL PRINCIPLES [BjOK II.thod, which extends itself, without any troublesome calculation, not ojdyto the drawing of tangents to any curve lines, whether geometrical ormechanical, or any how respecting right lines or other cnrves, but alsoto the resolving other abstrnser kinds of problems about the crookedness,areas, lengths, centres of gravity of curves, &c. ; nor is it (as Hudd^ri smethod de Maximis & Minimia) limited to equations which are freefromsurd quantities. This method I have interwoven with that other ojworking in equations, by reducing them to infinite serie?. So far thatletter. And these last words relate to a treatise I composed on that subject in the year 1671. The foundation of that general method is contained in the preceding Lemma.PROPOSITION VIII. THEOREM VI.If a body in an uniform medium, being uniformly acted upon by theforceof gravity, ascends or descends in a right line ; and the whole spacedescribed be distinguished into equal parts, and in the beginning ofeach of the parts (by adding or subducting the resisting force of themedium to or from the force of gravity, when the body ascends or descends] yon collect the absolute forces ; Isay, that those absolute forcesire in a geometrical progression.For let the force of gravity be expounded by thegiven line AC ; the force of resistance by the indefinite line AK ; the absolute force in the descent of thebody by the difference KC : the velocity of the I tody<^LKJL&i>F/ by a line AP, which shall be a mean proportional between AK and AC, and therefore in a subduplicate ratio of the resistance;the increment of the resistance made in a given particle of time by the lineolaKL, and the contemporaneous increment of the velocity by the lineolaPQ ; and with the centre C, and rectangular asymptotes CA, CH,describe any hyperbola BNS meeting the erected perpendiculars AB, KN,LO in B, N and O. Because AK is as AP2, the moment KL of the one willbe as the moment 2APQ of the other, that is, as AP X KC ; for the increment PQ of the velocity is (by Law II) proportional to the generatingforce KC. Let the ratio of KL be compounded with the ratio KN, andthe rectangle KL X KN will become as AP X KC X KN ; that is (becausethe rectangle KC X KN is given), as AP. But the ultimate ratio of thehyperbolic area KNOL to the rectangle KL X KN becomes, when thepoints K and L coincide, the ratio of equality. Therefore that hyperbolicevanescent area is as AP. Therefore the whole hyperbolic area ABOLis composed of particles KNOL which are always proportional to thevelocity AP; and therefore is itself proportional to the space describedwith that velocity. Let ,that area be now divided into equal partsSEC. IJ.J OF NATURAL PHILOSOPHY. 265as ABMI, IMNK, KNOL, (fee., and the absolute forces AC, 1C, KC, LC,(fee., will be in a geometrical progression. Q,.E.D. And by a like reasoning, in the ascent of the body, taking, on the contrary side of the pointA, the equal areas AB?m, i/nnk, knol, (fee., it will appear that the absoluteforces AC. iG, kC, 1C, (fee., are continually proportional. Therefore if allthe spaces in the ascent and descent are taken equal, all the absolute forces1C, kC, iC, AC, 1C, KC, LC, (fee., will be continually proportional. Q,.E.D.COR. 1. Hence if the space described be expounded by the hyperbolicarea ABNK, the force of gravity, the velocity of the body, and the resistance of the medium, may be expounded by the lines AC, AP, and AK respectively and vice versa.COR. 2. And the greatest velocity which the body can ever acquire inan infinite descent will be expounded by the line AC.COR. 3. Therefore if the resistance of the medium answering to anygiven velocity be known, the greatest velocity will be found, by taking itto that given velocity in a ratio subduplicate of the ratio which the forceof gravity bears to that known resistance of the medium.PROPOSITION IX. THEOREM VII.Supposing ivhat is above demonstrated, I say, that if the tangents of t-heangles of the sector of a circle, and of an hyperbola, be taken proportional to the velocities, the radius being of a fit magnitude, all the timeof the ascent to the highest place icill be as the sector of the circle, andall the time of descending from the highest place as the sector of t/iehyperbola.To the right line AC, which expresses the force of gravity, let ADdrawn perpendicular and equal. Fromthe centre D with the semi-diameterAD describe as well the cmadrant A^E -tof a circle, as the rectangular hyperbola AVZ, whose axis is AK, principalvertex A, and asymptote DC. Let Dp,DP be drawn ; and the circular sectorAtD will be as all the time of the ascent to the highest place ; and the hyperbolic sector ATD as all the time of descent from the highest place; iiBO be that the tangents Ap, AP of those sectors be as the velocities.CASE 1. Draw Dvq cutting off the moments or least particles tDv and^?, described in the same time, of the sector ADt and of the triangleAD/?. Since those particles (because of the common angle D) are in a duplicate ratio of the sides, the particle tDv will be as -^-^-, that is266 THE MATHEMATICAL PRINCIPLES [BOOK li.(because tD is given), as ^f. But joD8 is AD 3 + Ap 2, that is, AD 2 -hAD X AA-, or AD X Gk ; and (/Dp is 1 AD X pq. Therefore tDv, theBOparticle of the sector, is as ^ ,; that is, as the least decrement pq of thevelocity directly, and the force Gk which diminishes the velocity, inversely ;and therefore as the particle of time answering to the decrement of the velocity. And, by composition, the sum of all the particles tDv in the sectorAD/ will be as the sum of the particles of time answering to each of thelost particles pq of the decreasing velocity Ap, till that velocity, being diminished into nothing, vanishes; that is, the whole sector AD/ is as thewhole time of ascent to the highest place. Q.E.D.CASE 2. Draw DQV cutting off the least particles TDV and PDQ ofthe sector DAV, and of the triangle DAQ ; and these particles will be toeach other as DT2 to DP2, that is (if TX and AP are parallel), as DX 2to DA2 or TX 2 to AP 2; and, by division, as DX2 TX2 to DA2 -AP 2. But. from the nature of the hyperbola, DX2 TX2is AD 2; and, bythe supposition, AP 2 is AD X AK. Therefore the particles are to eachother as AD 2 to AD2 AD X AK ; that is, as AD to AD AK or ACto CK ; and : and therefore the particle TDV of the sector is -PQtherefore (because AC and AD are given) asCKthat is, as the incrementof the velocity directly, and as the force generating the increment inversely ; and therefore as the particle of the time answering to the increment.And, by composition, the sum of the particles of time, in which all the particles PQ of the velocity AI3 are generated, will be as the sum of the particles of the sector ATI) ; that is, the whole time will be as the wholesector. Q.E.D.COR. 1. Hence if AB be equal to afourth part of AC, the space which a bodywill describe by falling in any time willbe to the space which the body could describe, by moving uniform]} on in thesame time with its greatest velocityAC, as the area ABNK, which expresses the space described in falling tothe area ATD, which expresses thetime. For since AC is to AP as AP_ to AK, then (by Cor. 1, Lem. II, of thisBook) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thenceLK is to ^PQ as AP to JAC or AB ; and KN is to AC or AD as AB tc. II.] OF NATURAL PHILOSOPHY. 267UK ; and therefore, ex cequo, LKNO to DPQ, as AP to CK. But DPQwas to DTV as CK to AC. Therefore, ex aquo, LKNO is to DTV r,?AP to AC ; that is, as the velocity of the falling body to the greatestvelocity which the body by falling can acquire. Since, therefore, themoments LKNO and DTV of the areas ABNK and ATD are as the velocities, all the parts of those areas generated in the same time will be asthe spaces described in the same time ; and therefore the whole areas ABNKand ADT, generated from the beginning, will be as the whole spaces described from the beginning of the descent. Q.E.D.COR. 2. The same is true also of the space described in the ascent.That is to say, that all that space is to the space described in the sametime, with the uniform velocity AC, as the area ABttk is to the sector ADt.COR. 3. The velocity of the body, falling in the time ATD, is to thevelocity which it would acquire in the same time in a non-resisting space,