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The Mathematical Principles of Natural Philosophy (1729)/Section 3

If a body be resisted partly in the ratio, and partly in the duplicate ratio of its velocity, and moves in a similar medium by its innate force only; and the times be taken in arithmetical progression: then quantities reciprocally proportional to the velocities, increased by a certain given quantity, will be in geometrical progression. Plate 3. Figure 1.

With the centre C, and the rectangular asymptotes CJDd and CH describe an Hyperbola BEe, and let AB,DE, de, be parallel to the asymptote CH. In the asymptote CD let A, G be given points: And if the time be expounded by the hyperbolic area ABED uniformly increasing; I say that the velocity may be express'd by the length DF, whose reciprocal GD together with the given line CG, compose the length CD increasing in a geometrical progression.

For let the areola DEed be the least given increment of the time, and Dd will be reciprocally as DE, and therefore directly as CD. Therefore the decrement of 1/GD, which (by Lemma 2. Book 2.) is Dd / GD² will be also as CD / GD² or (CG + GD) / GD², that is, as 1 / GD + CG / GD². Therefore the time ABED uniformly increasing by the addition of the given particles EDde, it follows that 1 / GD decreases in the same ratio with the velocity. For the decrement of the velocity is as the resistance, that is, (by the supposition) as the sum of two quantities, whereof one is as the velocity; and the other as the square of the velocity; and the decrement of 1 / GD is as the sum of the quantities 1 / GD and CG / GD², whereof the first is 1 / GD it self, and the last CG / GD² is as 1 / GD²: therefore 1 / GD is as the velocity, the decrements of both being analogous. And if the quantity GD, reciprocally proportional to 1 / GD, be augmented by the given quantity CG; the sum CD, the time ABED uniformly increasing, will increase in a geometrical progression. Q.E.D.

Corollary 1. Therefore, if, having the points A and G given, the time be expounded by the hyperbolic area ABED, the velocity may be expounded by 1 / GD the reciprocal of GD.

Corollary 2. And by taking GA to GD as the reciprocal of the velocity at the beginning, to the reciprocal of the velocity at the end of any time ABED, the point G will be found. And that point being found, the velocity may be found from any other time given.

The same things being supposed, I say, that if the spaces described are taken in arithmetical progression, the velocities augmented by a certain given quantity will be in geometrical progression. Plate 3. Figure 2.

In the asymptote CD let there be given the point R, and erecting the perpendicular RS meeting the Hyperbola in S, let the space described be expounded by the hyperbolic area RSED; and the velocity will be as the length GD, which, together with the given line CG, composes a length CD decreasing in a geometrical progression, while the space RSED increases in an arithmetical progression.

For, because the increment EDde of the space is given, the lineola Dd, which is the decrement of GD, will be reciprocally as ED, and therefore directly as CD; that is, as the sum of the same G D and the given length CG. But the decrement of the velocity, in a time reciprocally proportional thereto, in which the given particle of space DdeE is described, is as the resistance and the time conjunctly, that is, directly as the sum of two quantities, whereof one is as the velocity, the other as the square of the velocity, and inversely as the velocity; and therefore directly as the sum of two quantities, one of which is given, the other is as the velocity. Therefore the decrement both of the velocity and the line GD, is as a given quantity and a decreasing quantity conjunctly; and because the decrements are analogous, the decreasing quantities will always be analogous; viz. the velocity, and the line GD. Q.E.D.

Corollary 1. If the velocity be expounded by the length GD, the space described will be as the hyperbolic area DESR.

Corollary 2. And if the point R be assumed any how, the point G will be found, by taking GR to GD, as the velocity at the beginning to the velocity after any space RSED is described. The point G being given, the space is given from the given velocity: and the contrary.

Corollary 3. Whence since (by Proposition 11.) the velocity is given from the given time, and (by this Proposition) the space is given from the given velocity; the space will be given from the given time: and the contrary.

Supposing that a body attracted downwards by an uniform gravity ascends or descends in a right line; and that the same is resisted, partly in the ratio of its velocity, and partly in the duplicate ratio thereof: I say that, if right lines parallel to the diameters of a Circle and an Hyperbola be drawn thro the ends of the conjugate diameters, and the velocities be as some segments of those parallels drawn from a given point; the times will be as the sectors of the areas, cut off by right lines drawn from the centre to the ends of the segments; and the contrary. Plate 3. Figure 3.

Case 1. Suppose first that the body is ascending, and from the centre D, with any semidiameter DB, describe a quadrant BETF of a circle, and thro' the end B os the semidiameter DB draw the indefinite line BAP, parallel to the semidiameter DF. In that line let there be given the point A, and take the segment AP proportional to the velocity. And since one part of the resistance is as the velocity, and another part as the square of the velocity; let the whole resistance be as AP² + 2.BAP. Join DA, DP cutting the circle in E and T, and let the gravity be expounded by DA², so that the gravity shall be to the resistance in P, as DA² to AP² + 2.BAP; and the time of the whole ascent will be as the sector EDT of the circle.

For draw DVO, cutting off the moment PQ of the velocity AP, and the moment DTV of the sector DET answering to a given moment of time; and that decrement PQ of the velocity will be as the sum of the forces of gravity DA² and of resistance AP² + 2.BAP, that is, (by 12 Proposition, 2 Book Elem.) as DP². Then the area DPQ, which is proportional to PQ, is as DP², and the area DTV, which is to the area DPQ as DT² to DP², is as the given quantity DT². Therefore the area EDT decreases uniformly according to the rate of the future time, by subduction of given particles DTV, and is therefore proportional to the time of the whole ascent. Q.E.D.

Case 2. If the velocity in the ascent of the body be expounded by the length AP as before, and the resistance be made as AP² + 2.BAP, and if the force of gravity be less than can be expressed by DA²; take BD (Figure 4.) of such a length, that AB² - BD² may be proportional to the gravity, and let DF be perpendicular and equal to DB, and thro' the vertex F describe the Hyperbola FTVE, whose conjugate semidiameters are DB and DF, and which cuts DA in E, and DP, DQ in T and V; and the time of the whole ascent will be as the hyperbolic sector TDE.

For the decrement PQ of the velocity produced in a given particle of time, is as the sum of the resistance AP² + 2.BAP and of the gravity AB² - BD², that is, as BP² - BD². But the area DTV is to the area DPQ as DT² to DP²; and therefore, if GT be drawn perpendicular to DF, as GT² or GD² - DF² to BD², and a GD² to BP², and, by division, as DF² to BP² - BD². Therefore since the area DPQ is as PQ, that is, as BP² - BD²; the area DTV will be as the given quantity DF². Therefore the area EDT decreases uniformly in each of the equal particles of time, by the subduction of so many given particles DTV and therefore is proportional to the time. Q.E.D.

Case 3. Let AP be the velocity in the descent of the body, and AP² + 2.BAP the force of resistance, and BD² - AB² the force of gravity, the angle DBA being a right one. And if with the centre D, and the principal vertex B, there be described a rectangular Hyperbola BETV (Figure 5.) cutting DA, DP, and DQ produced in E, T, and V; the sector DET of this Hyperbola will be as the whole time of descent.

For the increment PQ of the velocity, and the area DPQ proportional to it, is as the excess of the gravity above the resistance, that is, as BD² - AB² - 2.BAP - AP² or BD² - BP². And the area DTV is to the area DPQ, as DT² to DP²; and therefore as GT² or GD² - BD² to BP², and as GD² to BD², and, by division, as BD² to BD² - BP². Therefore since the area DPQ is as BD² - BP², the area DTV will be as the given quantity BD². Therefore the area EDT increases uniformly in the several equal particles of time by the addition of as many given particles DTV, and therefore is proportional to the time of the descent. Q.E.D.

Corollary. If with the centre D and the semidiameter DA there be drawn thro' the vertex A an arc At similar to the arc ET, and similarly subtending the angle ADT: the velocity AP will be to the velocity, which the body in the time EDT, in a non-resisting space, can lose in its ascent, or acquire in its descent, as the area of the triangle DAP to the area of the sector DAt; and therefore is given from the time given. For the velocity in a non-resisting medium, is proportional to the time, and therefore to this sector; in a resisting medium it is as the triangle; and in both mediums, where it is least, it approaches to the ratio of equality, as the sector and triangle do.

One may demonstrate also that case in the ascent of the body, where the force os gravity is less than can be express'd by DA² or AB² + BD², and greater than can be express'd by AB² - DB², and must be express'd by AB². But I hasten to other things.

The same things being supposed, I say, that the space described in the ascent or descent, is as the difference of the area by which the time is express'd, and of some other area which is augmented or diminished in an arithmetical progression; if the forces compounded of the resistance and the gravity be taken in a geometrical progression. Plate 3. Figure 5, 6, 7.

Take AC (in the three last figures) proportional to the gravity, and AK to the resistance. But take them on the same side of the point A, if the body is descending, otherwise on the contrary. Erect Ab, which make to DB as DB² to 4.BAC: and to the rectangular asymptotes CK, CH, describe the Hyperbola bN, and erecting KN perpendicular to CK, the area AbNK will be augmented or diminished in an arithmetical progression, while the forces CK are taken in a geometrical progression. I say therefore that the distance of the body from its greatest altitude is as the excess of the area AbNK above the area DET.

For since AK is as the resistance, that is, as AP² + 2.BAP; assume any given quantity Z, and put AK equal to (AP² + 2.BAP) / Z ; then (by Lemma 2. of this Book) the moment KL of AK will be equal to (2.APQ + 2.BA.PQ) / Z or 2.BPQ / Z, and the moment KLON of the area AbNK, will be equal to 2.BPQ.LO / Z or BPQ.BD³ / (2.Z.CK.AB)

Case 1. Now if the body ascends, and the gravity be as AB² + BD², BET, (in Fig. 5.) being a circle, the line AC, which is proportional to the gravity, (AB² + BD²) / Z, and DP² or AP² + 2.BAP + AB² + BD² will be AK.Z + AC.Z or CK.Z; and therefore the area DTV will be to the area DPQ as DT² or DB² to CK.Z.

Case 2. If the body ascends, and the gravity be as AB² - BD², the line AC (in Fig. 6.) will be (AB² — BD²) / Z and DT² will be to DP² as DF² or DB² to BP² - BD² or AP² +2.BAP + AB² - BD², that is, to AK.Z + AC.Z or CK.Z. And therefore the area DTV will be to the area DPQ as DB² to CK.Z.

Case 3. And by the same reasoning, is the body descends, and therefore the gravity is as BD² - AB², and the line AC (in Fig. 7.) becomes equal to (BD² - AB²) / Z; the area DTV will be to the area DPQ as DB² to CK.Z: as above.

Since therefore these areas are always in this ratio; if for the area DTV, by which the moment of the time, always equal to itself, is expressed, there be put any determinate rectangle, as BD.m, the area DPQ, that is, ½BD.PQ, will be to BD.m as CK.Z to BD². And thence PQ.BD³ becomes equal to 2.BD.m.CK.Z, and the moment KLON of the area AbNK, sound before, becomes BP.BD.m / AB. From the area DET subduct its moment DTV or BD.m, and there will remain AP.BD.m/AB Therefore the difference of the moments, that is, the moment of the difference of the areas is equal to AP.BD.m/AB; and therefore (because of the given quantity BD.m/AB) as the velocity AP; that is, as the moment of the space which the body describes in its ascent or descent. And therefore the difference of the areas, and that space, increasing or decreasing by proportional moments, and beginning together or vanishing together, are proportional. Q.E.D.

Corollary. If the length, which arises by applying the area DET to the line BD, be called M; and another length V be taken in that ratio to the length M, which the line DA has to the line DE: the space which a body, in a resisting medium, describes in its whole ascent or descent, will be to the space, which a body, in a non-resisting medium, falling from rest can describe in the same time, as the difference of the aforesaid areas to BD.V²/AB; and therefore is given from the the time given. For the space in a non-resisting medium is in a duplicate ratio of the time, or as V²; and, because BD and AB are given, as BD.V²/AB. This area is equal to the area DA².BD.M²/DE².AB, and the moment of M is m; and therefore the moment of this area is DA².BD.2.M.m/DE².AB. But this moment is to the moment of the difference of the aforesaid areas DET and AbNK, viz,, to AP.BD.m/AB, as DA².BD.M/DE² to ½.BD.AP, or as DA²/DE² into DET to DAP; and therefore, when the areas DET and DAP are least, in the ratio of equality. Therefore the area BD.V²/AB and the difference of the areas DET and AbNK, when all these areas are least, have equal moments; and are therefore equal. Therefore since the velocities, and therefore also the spaces in both mediums described together, in the beginning of the descent, or the end of the ascent, approach to equality, and therefore are then one to another as the area BD.V²/AB, and the difference of the areas DET and AbNK; and moreover since the space, in a non-resisting medium, is perpetually as BD.V²/AB, and the space, in a resisting medium, is perpetually as the difference of the areas DET and AbNK: it necessarily follows, that the spaces, in both mediums, described in any equal times, are one to another as that area BD.V²/AB, and the difference of the areas DET and AbNK. Q.E D.

The resistance of sphærical bodies in fluids arises partly from the tenacity, partly from the attrition, and partly from the density of the medium. And that part of the resistance, which arises from the density of the fluid, is, as I said, in a duplicate ratio of the velocity, the other part, which arises from the tenacity of the fluid, is uniform, or as the moment of the time: and therefore we might now proceed to the motion of bodies, which are resisted partly by an uniform force, or in the ratio of the moments of the time, and partly in the duplicate ratio of the velocity. But it is sufficient to have cleared the way to this speculation in the 8th and 9th Proposition, foregoing, and their Corollaries. For in those Propositions, instead of the uniform resistance made to an ascending body arising from its gravity, one may substitute the uniform resistance which arises from the tenacity of the medium, when the body moves by its vis insita alone; and when the body ascends in a right line, add this uniform resistance to the force of gravity, and subduct it when the body descends in a right line. One might also go on to the motion of bodies which are resisted in part uniformly, in part in the ratio of the velocity, and in part in the duplicate ratio of the same velocity. And I have opened a way to this in the 13th and 14th Proposition, foregoing, in which the uniform resistance arising from the tenacity of the medium, may be substituted for the force of gravity, or be compounded with it as before. But I hasten to other things.


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The Mathematical Principles of Natural Philosophy (1729)/Section 2

If a body is resisted in the duplicate ratio of its velocity, and moves by its innate force only through a similar medium; and the times be taken in a geometrical progression, proceeding from less to greater terms: I say that the velocities at the beginning of each of the times are in the same geometrical progression inversely; and that the spaces are equal, which are described in each of the times.

For since the resistance of the medium is proportional to the square of the velocity, and the decrement of the velocity is proportional to the resistance; if the time be divided into innumerable equal particles, the squares of the velocities at the beginning of each of the times will be proportional to the differences of the same velocities. Let those particles of time be AK, KL, LM, &c. taken in the right line CD; and erect the perpendiculars AB, Kk, Ll, Mm, &c. meeting the Hyperbola BklmG, described with the centre C and the rectangular asymptotes CD, CH, in B, k, l, m, &c. then AB will be to Kk as CK to CA, and, by division, AB - Kk to Kk as AK to CA, and, alternately, AB - Kk to AK as Kk to CA, and therefore as AB x Kk to AB x CA. Therefore since AK and AB x CA are given, AB - Kk will be as AB x Kk; and lastly, when AB and Kk coincide, as AB ². And, by the like reasoning, Kk - Ll, Ll - Mm, &c. will be as Kk², Ll², &c. Therefore the squares of the lines AB, Kk, Ll, Mm, &c. are as their differences; and therefore, since the squares of the velocities were shewn above to be as their differences, the progression of both will be alike. This being demonstrated, it follows also that the areas described by these lines are in a like progression with the spaces described by these velocities. Therefore if the velocity at the beginning of the first time AK be expounded by the line AB, and the velocity at the beginning of the second time KL by the line Kk, and the length described in the first time by the area AKkB; all the following velocities will be expounded by the following lines Ll, Mm, &c. and the lengths described, by the areas Kl, Lm, &c. And, by composition, if the whole time be expounded by AM, the sum of its parts, the whole length described will be expounded by AMmB the sum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c. so that CA, CK, CL, CM, &c. may be in a geometrical progression; and those parts will be in the same progression, and the velocities AB, Kk, Ll, Mm, &c. will be in the same progression inversly, and the spaces described Ak, Kl, Lm, &c. will be equal. Q.E.D.

Corollary 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB; the velocity at the end of the time will be expounded by the ordinate DG; and the whole space described, by the adjacent hyperbolic area ABCD; and the space which any body can describe in the same time AD, with the first velocity AB, in a nonresisting medium, by the rectangle AB x AD.

Corollary 2. Hence the space described in a resisting medium is given, by taking it to the space described with the uniform velocity AB in a non-resisting medium, as the hyperbolic area ABGD to the rectangle AB x AD.

Corollary 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, which could generate, in a body falling thro' a non-resisting medium, the velocity AB, in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the asymptote in T; the right line AT will be equal to AC, and will express the time, in which the first resistance uniformly continued, may take away the whole velocity AB.

Corollary 4. And thence is also given the proportion of this resistance to the force of gravity, or any other given centripetal force.

Corollary 5. And vice versa, if there is given the proportion of the resistance to any given centripetal force; the time AC is also given, in which a centripetal force equal to the resistance may generate any velocity as AB; and thence is given the point B, through which the hyperbola, having CH, CD for its asymptotes, is to be described; as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any time AD, in a similar resisting medium.

Homogeneous and equal spherical bodies, oppos'd by resistances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at the beginning, describe equal spaces, and lose parts of their velocities proportional to the wholes.

To the rectangular asymptotes CD, CH describe any hyperbola BbEe, cutting the perpendiculars AB, ab, DE, de, in B, b, E, e; let the initial velocities be expounded by the perpendiculars AB, DE, and the times by the lines Aa, Dd. Therefore as Aa is to Dd, so (by the hypothesis) is DE to AB, and so (from the nature of the hyperbola) is CA to CD; and, by composition, so is Ca to Cd. Therefore the areas ABba, DEed, 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 the parts of the velocities lost, AB - ab, DE - de. Q.E.D.

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 the wholes, and will describe spaces proportional to those times and the first velocities conjunctly.

For the parts of the motions lost are as the resistances and times conjunctly. Therefore, that those parts may be be proportional to the wholes, the resistance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the resistance inversely. Wherefore the particles of the times being taken in that ratio, the bodies will always lose 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 describe spaces, which are as the first velocities and the times conjunctly. Q.E.D.

Corollary 1. Therefore if bodies equally swift are resisted in a duplicate ratio of their diameters: Homogeneous globes moving with any velocities whatsoever, by describing spaces proportional to their diameters, will lose parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mass conjunctly, that is, as the velocity and the cube of its diameter; the resistance (by supposition) will be as the square of the diameter and the square of the velocity conjunctly; and the time (by this proposition) is in the former ratio directly and in the latter inversely, that is, as the diameter directly and the velocity inversely; and therefore the space, which is proportional to the time and velocity, is as the diameter.

Corollary 2. If bodies equally swift are resisted in a sesquiplicate ratio of their diameters: Homogeneous globes, moving with any velocities whatsoever, by describing spaces that are in a sesquiplicate ratio of the diameters, will lose parts of their motions proportional to the wholes.

Corollary 3. And universally, if equally swift bodies are resisted in the ratio of 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 to that power. Let those diameters applied to that power. Let those diameters be D and E; and if the resistances, where the velocities are supposed equal, are as D" and E": the spaces in which the globes, moving with any velocities whatsoever, will lose parts of their motions proportional to the wholes, will be as D3-"and E3-". And therefore homogeneous globes, in describing spaces proportional to D3-"and E3-" will retain their velocities in the same ratio to one another as at the beginning.

Corollary 4. Now if the globes are not homogeneous, the space described by the denser globe must be augmented in the ratio of the density. For the motion, with an equal velocity, is greater in the ratio of the density, and the time (by this Proposition) is augmented in the ratio of motion directly, and the space described in the ratio of the time.

Corollary 5. And if the globes move in different mediums, the space, in a medium which, caeteris paribus, resists the most, must be diminished in the ratio of the greater resistance. For the time (by this proposition) 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 generating sides drawn into the indices of the powers of those sides, and into their coefficients continually.

I call any quantity a Genitum, which is not made addition or subduction of divers parts, but is generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever; in geometry by the invention of contents and sides, or of the extreams and means 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 esteem'd as added, or affirmative moments; and the decrements as subducted, or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first proportion as nascent. It will be the same thing, if, instead of moments, we use either the Velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities) or any finite quantities proportional to those velocities. The coefficient of any generating side is the quantity which arises by applying the Genitum to that 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 the velocities of the mutations which are proportional to them, be called a, b, c, &c. the moment or mutation of the generated rectangle AB will be aB + bA; the moment of the generated content ABC will be aBC + bAC + cAB: and the moments of the generated powers, A², A³, A4, A1/2, A3/2, A1/3, A2/3, A-1, A-2 , A-1/2 will be 2aA, 3aA², 4aA³, 1/2aA-3/2, 3/2aA1/2, 1/3aA-2/3, 2/3aA-1/3, -aA-2 , -2aA-3, -½aA-3/2 respectively. And in general, that the moment of any power An/m will be n/mA(n-m)/m. Also that the moment of the generated quantity A².B will be 2aAB + bA²; the moment os the generated quantity A³.B4.C² will be 3aA².B4.C² + 4bA³B³C² + 2cA³B4.C; and the moment of the generated quantity A³/B² or A³.B-2 will be 3aA²B-2 - 2bA³B-3; and so on. The Lemma is thus demonstrated.

Case 1. Any rectangle as AB augmented by a perpetual flux, when, as yet, there wanted of the sides A and B half their moments ½a and ½b, was A - ½a into B - ½b or AB + ½aB + ½bA + ½ab; but as soon as the sides A and B are augmented by the other half moments; the rectangle becomes A + ½a into B + ½b or AB + ½aB + ½bA + ½ab. Fr0m this rectangle subduct the former rectangle, and there will remain the excess aB + bA. Therefore with the whole increments a and b of the sides, the increment aB + bA of the rectangle is generated. Q.E.D.

Case 2. Suppose AB always equal to G, and then the moment of the content ABC or GC (by case 1.) will be gC + cG, that is, (putting AB and aB + bA for G and g) aBC + bAC + cAB. And the reasoning is the same for contents under never so many sides. Q.E.D.

Case 3. Suppose the sides A, B, and C, to be always equal among themselves; and the moment aB + bA, of A², that is, of the rectangle AB, will be 2aA; and the moment aBC + bAC + cAB of A³, that is, of the content ABC, will be 3aA². And by the same reasoning the moment of any power An is naAn-1. Q.E.D.

Case 4. Therefore since 1/A into A is 1, the moment of 1/A drawn into A, together with 1/A drawn into a, will be the moment of 1, that is, nothing. Therefore the moment of 1/A or of A-1 is -a/A². And generally, since 1/An into An is 1, the moment of 1/An drawn into An together with 1/An into naA1/An-1 will be nothing. And therefore the moment of 1/An or A-n will be -na/An+1. Q.E.D.

Case 5. And since A1/2 into A1/2 is A, the moment of A1/2 drawn into 2A1/2 will be a, (by case 3:) and therefore the moment of A1/22 will be a/2A1/2 or ½aA-1/2. And generally, putting Am/n equal to B, then Am will be equal to Bn, and therefore maAm-1 equal to nbBn-1, and maA-1 equal to nbB-1 or nbA-m/n; and therefore m/naA(m-n)/n is equal to b, that is, equal to the moment of Am/n. Q.E.D.

Case 6. Therefore the moment of any generated quantity AmBn is the moment of Am drawn into Bn, together with the moment of Bn drawn into Am, that is, maAm-1Bn + nbBn-1Am; and that whether the indices m and 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.

Corollary 1. Hence in quantities continually proportional, if one term is given, 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. Let A, 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.

Corollary 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.

Corollary 3. And if the sum or difference os two squares is given, the moments of the sides will be reciprocally as the sides.

In a letter of mine to Mr. J. Collins, dated December 10. 1572. having described a method of Tangents, which I suspected to be the same with Slusin's method, which at that time was not made publick; I subjoined these words; This is one particular, or rather a corollary, of a general method, which extends itself, without any troublesome calculation, not only to the drawing of Tangents to any Curve lines, whether Geometrical or Mechanicals or any how respecting right lines or other Curves, bttt also to the resolving other abstruser kinds of Problems about the crookedness, areas, lengths centres of gravity of Curves, &c. nor is it (as Hudden's method de Maximis & Minimis) limited to equations which are free from surd quantities. This method I have interwoven with that other of working in equations, by reducing them to infinite series. So far that letter. 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.

If a body in an uniform medium, being uniformly acted upon by the force of gravity, ascends or descends in a right line; and the whole space described be distinguished into equal parts, and in the beginning of each of the parts, (by adding or subducting the resisting force of the medium to or from the force of gravity, when the body ascends or descends) you collect the absolute forces; I say that those absolute forces are in a geometrical progression. Plate 2. Figure 1.

For let the force of gravity be expounded by the given line AC; the force of resistance by the indefinite line AK; the absolute force in the descent of the body, by the difference KC; the velocity os the body 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 lineola KL, and the contemporaneous increment of the velocity by the lineola PQ; 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 will be as the moment 2APQ of the other, that is, as AP * KC, for the increment PO of the velocity is (by law 2.) proportional to the generating force KC. Let the ratio of KL be compounded with the ratio of KN, and the rectangle KL * KN will become as AP * KC * KN; that is, (because the rectangle KC * KN is given) as AP. But the ultimate ratio of the hyperbolic area KNOL to the rectangle KL * KN becomes, when the points K and L coincide, the ratio of equality. Therefore that hyperbolic evanescent area is as AP. Therefore the whole hyperbolic area ABOL is composed of particles KNOL which are always proportional to the velocity AP; and therefore is itself proportional to the space described with that velocity. Let that area be now divided into equal parts, as ABMI, IMNK, KNOL, &c. and the absolute forces AC, IC, KC, LC, &c. 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 point A, the equal areas ABmi, imnk, knol, &c. it will appear that the absolute forces AC, iC, kC, lC, &c. are continually proportional. Therefore if all the spaces in the ascent and descent are taken equal; all the absolute forces lC, kC, IC, iC, AC, IC, KC, LC, &c. will be continually proportional. Q.E.D.

Corollary 1. Hence if the space described be expounded by the hyperbolic area 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.

Corollary 2. And the greatest velocity, which the body can ever acquire in an infinite descent, will be expounded by the line AC.

Corollary 3. Therefore if the resistance of the medium answering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio subduplicate of the ratio which the force of gravity bears to that known resistance of the medium.

Supposing what is above demonstrated, I say that if the tangents of the angles 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 time of the ascent to the highest place will be as the sector of the circle, and all the time of descending from the highest place as the sector of the hyperbola. Plate 2. Figure 2.

To the right line AC, which expresses the force of gravity, let AD be drawn perpendicular and equal. From the centre D with the semidiameter AD describe as well the quadrant AtE of a Circle; as the rectangular Hyperbola AVZ, whose axe is AX, principal vertex A, and asymptote DC. Let Dp, DP be drawn; and the circular sector AtD 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: If so be that the tangents Ap, AP of those sectors be as the velocities. (Figure 2).

Case 1. Draw Dvq cutting off the moments or least particles tDv and qDp, described in the same time, of the lector ADt and of the triangle ADp. Since those particles (because of the common angle D) are in a duplicate ratio os the sides, the particle tDv will be as qDp * tD² / pD², that is, (because tD is given) as qDp / pD². But pD² is AD² + Ap², that is, AD² + AD * Ak, or AD * Ck; and qDp is ½AD * pq. Therefore: Dv, the particle of the sector, is as pq/Ck; that is, as the least decrement pq of the velocity directly, and the force Ck, 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 sector ADtt will be as the sum of the particles of time answering to each of the lost particles pq, of the decreasing velocity Ap, till that velocity, being diminished into nothing, vanishes; that is, the whole sector ADt is as the whole time of ascent to the highest place. Q.E.D.

Case 2. Draw DQV cutting off the least particles TDV and PDQ of the sector DAV, and of the triangle DAQ; and these particles will be to each other as DT² to DP², that is, (if TX and AP are parallel) as DX² to DA² or TX² to AP²; and, by division, as DX² - TX² to DA² - AP². But, from the nature of the hyperbola, DX² - TX² is AD²; and, by the supposition, AP² is AD * AK. Therefore the particles are to each other as AD² to AD² - AD * AK; that is, as AD to AD - AK or AC to CK: and therefore the particle TDV of the sector is PDQ * AC / CK; and therefore (because AC and AD are given) as PQ / CK that is, as the increment of 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 AP are generated, will be as the sum of the particles of the sector ATD; that is, the whole time will be as the whole sector. Q.E.D.

Corollary 1. Hence is AB be equal to a fourth part of AC, the space which a body will describe by falling in any time will be to the space which the body could describe, by moving uniformly on in the same time with its greatest velocity AC, as the area ABNK, which expresses the space described in falling, to the area ATD, which expresses the time. For since AC is to AP as AP to AK, then (by corollary 1. lemma 2. of this Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence LK is to ½PQ as AP to ¼AC or AB; and KN is to AC or AD as AB to CK; and therefore, ex œquo, LKNO is to DTV as CK to ACC. Therefore, ex æquo, LKNO is to DTV as AP to AC; that is, as the velocity of the falling body to the greatest velocity which the body by falling can acquire. Since therefore the moments 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 as the spaces described in the same time; and therefore the whole areas ABNK and ADT generated from the beginning, will be as the whole spaces described from the beginning of the descent. Q.E.D.

Corollary 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 same time with the uniform velocity AC, as the area ABnk is to the sector ADt.

Corollary 3. The velocity of the body, falling in the time ATD, is to the velocity which it would acquire in the same time in a non-resisting space, as the triangle APD to the hyperbolic sector ATD. For the velocity in a non-resisting medium would be as the time ATD, and in a resisting medium is as AP, that is, as the triangle APD. And those velocities at the beginning of the descent, are equal among themselves, as well as those areas ATD, APD.

Corollary 4. By the same argument, the velocity in the ascent is to the velocity with which the body in the same time, in a non-resisting space, would lose all its motion of ascent, as the triangle ApD to the circular sector AtD; or as the right line Ap to the arc At.

Corollary 5. Therefore rhe time in which a body by falling in a resisting medium, would acquire the velocity AP, is to the time in which it would acquire its greatest velocity AC by falling in a non-resisting space, as the sector ADT to the triangle ADC: and the time in which it would lose its velocity Ap by ascending in a resisting medium, is to the time in which it would lose the same velocity by ascending in a nonresisting space, as the arc At to its tangent Ap.

Corollary 6. Hence from the given time there is given the space described in the ascent or descent. For the greatest velocity of a body descending in infinitum is given (by Corollory 2 and 3. Theorem 6. of this Book) and thence the time is given in which a body would acquire that velocity by falling in a non-resisting space. And taking the sector ADT or ADt to the triangle ADC in the ratio of the given time to the time just now found; there will be given both the velocity AP or Ap, and the area ABNK or ABnk, which is to the sector ADT, or ADt, as the space sought to the space which would, in the given time, be uniformly described with that greatest velocity sound just before.

Corollary 7. And by going backward, from the given space of ascent or descent ABnk or ABNK, there will be given the time ADt or ADT.

Suppose the uniform force of gravity to tend directly to the plane of the horizon, and the resistance to be as the density of the medium and the square of the velocity conjunctly: it is proposed to find the density of the medium in each place, which shall make the body move in any given curve line; the velocity of the body, and the resistance of the medium in each place. Plate 2. Figure 3.

Let PQ be a plane perpendicular to the plane of the scheme itself; PFHQ a curve line meeting that plane in the points P and Q; G, H, I, K four places of the body going on in this curve from F to Q; and GB, HC, ID, KE four parallel ordinates let fall from these points to the horizon, and standing on the horizontal line PQ at the points B, C, D, E; and let the distances BC, CD, DE, of the ordinates be equal among themselves. From the points G and H let the right lines GL, HN, be drawn touching the curve in G and H, and meeting the ordinates CH, DI, produced upwards, in L and N; and compleat the parallelogram HCDM. And the times, in which the body describes the arcs GH, HI, will be in a subduplicate ratio of the altitudes LH, NI, which the bodies would describe in those times, by falling from the tangents; and the velocities will be as the lengths described GH, HI directly and the times inversely. Let the times be expounded by T and t, and the velocities by GH/T and HI/t, and the decrement of the velocity produced in the time t will be expounded by GH/T - HI/t. This decrement arises from the resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity, with which it would be able to describe twice that space in the same time, as Galileo has demonstrated; that is, the velocity 2NI/t : but if the body describes the arc HI, it augments that arc only by the length HI - HN, or MI * NI / HI; and therefore generates only the velocity 2.MI * NI / (t * HI). Let this velocity be added to the beforementioned decrement, and we shall have the decrement os the velocity arising from the resistance alone, that is, GH / T - HI / t + 2.MI * NI / (t * HI). Therefore since in the same time, the action of gravity generates, in a falling body, the velocity 2.NI / t ; the resistance will be to the gravity as GH / T - HI / t + 2.MI * NI / (t * HI) to 2.NI / t, or as t * GH / T - HI + 2.MI * NI / HI to 2.NI.

Now for the abscissa's CB, CD, CE put -o, o, 2.o. For the ordinate CH put P; and for MI put any series Q.o + R.o² + S.o³ + &c. And all the terms of the series after the first, that is, R.o² + S.o³ + &c. will be NI; and the ordinates DI, EK and BG will be P - Q.o - R.o² - S.o³ - &c. P - 2.Q.o - 4.R.o² - 8.S.o³ - &c. and P + Q.o - Ro² - S.o³ - &c. respectively. And by squaring the differences of the ordinates BG - CH and CH - DI, and to the squares thence produced adding the squares of BC and CD themselves, you will have o.o + Q.Q.o.o - 2.Q.R.o³ + &c. and o.o + Q.Q.o.o + 2.Q.R.o³ + &c. the squares of the arcs GH, HI; whose roots o.√1 + Q.Q - Q.R.o.o / √1 + Q.Q, and o.√1 + Q.Q + Q.R.o.o / √1 + Q.Q are the arcs GH and HI. Moreover, if from the ordinate CH there be subducted half the sum of the ordinates BG and DI, and from the ordinate DI there be subducted half the sum of the ordinates CH and EK, there will remain R.o.o and R.o.o + 3.S.o³ the versed sines of the arcs GI and HK. And these are proportional to the lineolæ LH and NI, and therefore in the duplicate ratio of the infinitely small times T and t: and thence the ratio t / T is √(R + 3.S.o) / R or (R + frac12;S.o) / R; and t * GH / T - HI + 2.MI * NI / HI, by substituting the values of t / T, GH, HI, MI and NI just found, becomes (3.S.o.o / 2R).√1 + Q.Q to 2.R.o.o. And since 2.NI is 2.R.o.o, the resistance will be now to the gravity as (3.S.o.o / 2R).√1 + Q.Q to 2.R.o.o, that is, as 3.S.√1 + Q.Q to 4.R.R.

And the velocity will be such, that a body going off therewith from any place H, in the direction of the tangent HN, would describe, in vacuo, a Parabola, whose diameter is HC, and its latus rectum HN² / NI or (1 + Q.Q) / R.

And the resistance is as the density of the medium and the square of the velocity conjunctly; and therefore the density of the medium is as the resistance directly, and the square os the velocity inversely; that is, as 3.S.√1 + Q.Q / 4.R.R. and (1 + Q.Q) / R inversely; that is, as S / (R.√1 + Q.Q). Q.E.I.

Corollary 1. If the tangent HN be produced both ways, so as to meet any ordinate AF in T: HT / AC will be equal to √1 + Q.Q, and therefore in what has gone before may be put for √1 + Q.Q. By this means the resistance will be to the gravity as 3.S * HT to 4R.R * AC; the velocity will be as HT / (AC.√R), and the density of the medium will be as S * AC / (R * HT).

Corollary 2. And hence, if the curve line PFHQ be defined by the relation between the base or abscissa AC and the ordinate CH, as is usual; and the value of the ordinate be resolved into a converging series: The problem will be expeditiously solved by the first terms of the series; as in the following examples.

Example 1. Let the line PFHQ be a semicircle described upon the diameter PO; to find the density of the medium that shall make a projectile move in that line.

Bisect the diameter PQ in A; and call AQ, n; AC, a; CH, e; and CD, o: then DI² or AQ² - AD² = nn - aa - 2.ao - oo, or ee - 2.ao - oo; and the root being extracted by our method, will give DI = e - ao/e - oo/2e - aaoo/2e³ - ao³/2e³ - a³.o³/2.e5 - &c. Here put nn for ee + aa, and DI will become e - ao/e - nnoo/2e³ - anno³/2e5 - &c.

Such series I distinguish into successive terms after this manner: I call that the first term, in which the infinitely small quantity o is not found; the second, in which that quantity is of one dimension only; the third, in which it arises to two dimensions; the fourth, in which it is of three; and so ad infimitum. And the first term, which here is e, will always denote the length of the ordinate CH, standing at the beginning of the indefinite quantity o. The second term, which here is ao/e, will denote the difference between CH and DN; that is, the lineola MN which is cut off by compleating the parallelogram HCDM; and therefore always determines the position of the tangent HN; as, in this case, by taking MN to HM as ao/e to o, or a to e. The third term, which here is nnoo/2e³, will represent the lineola IN, which lies between the tangent and the curve; and therefore determines the angle of contact IHN, or the curvature which the curve line has in H. If that lineola IN is of a finite magnitude, it will be express'd by the third term together with those that follow in infinitum. But if that lineola be diminished in infinitum, the terms following become infinitely less than the third term, and therefore may be neglected. The fourth term determines the variation of the curvature; the fifth, the variation of the variation; and so on. Whence, by the way, appears no contemptible use of these series in the solution of problems that depend upon tangents, and the curvature of curves.

Now compare the series e - ao/e - nnoo/2e³ - anno³/2e5 - &c. with the series P - Qo - Roo - So³ - &c. and for P, Q, R and S put e, a/e, nn/2e³ and ann/2e5, and for √1 + Q.Q put √1 + aa/ee or n/e; and the density of the medium will come out as a/ne, that is, (because n is given) as a/e, or AC / CH, that is, as that length of the tangent HT, which is terminated at the semidiameter AF standing perpendicularly on PQ: and the resistance will be to the gravity as 3.a to 2.n, that is, as 3.AC to the diameter PQ of the circle; and the velocity will be as √CH. Therefore if the body goes from the place F, with a due velocity, in the direction of a line parallel to PQ, and the density of the medium in each of the places H is as the length of the tangent HT, and the resistance also in any place H is to the force of gravity as 3.AC to PQ, that body will describe the quadrant FHQ of a circle. Q.E.I.

But if the same body should go from the place P, in the direction of a line perpendicular to PO, and should begin to move in an arc of the semi-circle PFQ, we must take AC or a on the contrary side of the centre A; and therefore its sign must be changed, and we must put -a for +a. Then the density of the medium would come out as -a/e. But nature does not admit of a negative density, that is, a density which accelerates the motion of bodies; and therefore it cannot naturally come to pass, that a body by ascending from P mould describe the quadrant PF of a circle. To produce such an effect, a body ought to be accelerated by aa impelling medium, and not impeded by a resisting one.

Example 2. Let the line PFQ be a Parabola, having its axis AF perpendicular to the horizon PQ; to find the density of the medium, which will make a projectile move in that line. Figure 4.

From the nature of the Parabola, the rectangle PDQ is equal to the rectangle under the ordinate DI and some given right line: that is, if that right line be called b; PC, a; PQ, c; CH, e; and CD, o; the rectangle a + o into c - a - o or ac - aa - 2ao + co - oo is equal to the rectangle b into DI, and therefore DI is equal to (ac - aa)/b + (c - 2a)o/b - oo/b. Now the second term (c - 2a)o/b of this series is to be put for Qo, and the third term oo/b for Roo. But since there are no more terms, the coefficient S of the fourth term will vanish; and therefore the quantity S / (R.√1 + QQ), to which the density of the medium is proportional, will be nothing. Therefore, where the medium is of no density, the projectile will move in a Parabola; as Galileo hath heretofore demonstrated. Q.E.I.

Example 3. Let the line AGK be an Hyperbola, having its asymptote NX perpendicular to the horizontal plane AK; to find the density of the medium, that will make a projectile move in that line. Figure 5.

Let MX be the other asymptote, meeting the ordinate DG produced in V; and from the nature of the Hyperbola, the rectangle of XV into VG will be given. There is also given the ratio of DN to VX, and therefore the rectangle of DN into VG is given. Let that be bb: and, compleating the parallelogram DNXZ, let BN be called a; BD, o; NX, c; and let the given ratio of VZ to ZX or DN be m/n. Then DN will be equal to a - o, VG equal to bb / (a - o), VZ equal to (m/n) * (a - o), and GD or NX - VZ - VG equal to c - (m/n).a + (m/n).o - bb/(a - o). Let the term bb / (a-o) be resolved into the converging series bb/a + (bb/aa).o + (bb/a³).oo + (bb/a4).o³ + &c. and GD will become equal to c - (m/n).a - bb/a + (m/n).o - (bb/a³).o³ - (bb/4).o³ + &c. The second term (m/n).o - (bb/aa).o of this series is to be used for Q.o, the third (bb/a³).o² with its sign changed for R.o², and the fourth (bb/a4).o³ with its sign changed also for S.o³, and their coefficients m/n - bb/aa, bb/a³ and bb/a4 are to be put for Q, R and S in the former rule. Which being done, the density of the medium will come out as or, that is, if in VZ you take VT equal to VG, as 1 / XT. For aa and (m²/n²).a² - 2mbb / n + b4 / aa are the squares of XZ and ZT. But the ratio of the resistance to gravity is found to be that of 3.XT to 2.TG; and the velocity is that with which the body would describe a Parabola, whose vertex is G, diameter DG, latus rectum XT² / VG. Suppose therefore that the densities of the medium in each of the places G are reciprocally as the distances XT, and that the resistance in any place G is to the gravity as 3.XT to 2.TG; and a body let go from the place A, with a due velocity, will describe that Hyperbola AGK. Q.E.I.

Example 4. Suppose indefinitely, the line AGK to be an Hyperbola, described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the Hyperbola in G and its asymptote in V, VG may be reciprocally as any power DNn of the line ZX or DN, whose index is the number n: To find the density of the medium in which a projected body will describe this curve. Figure 5.

For BN, BD, NX put A, O, C respectively, and let VZ be to XZ or DN as d to e, and VG be equal tobb / DNn; then DN will be equal to A - O, VG = bb / (A - O)n, VZ = (d/e).A - O, and GD or NX - VZ - VG equal to C - (d/e).A + (d/e).O - bb / A - On. Let the term bb / (A - O)n be resolved into an infinite series

and GD will be equal to

The second term of this series is to be used for Q.o, the third  for R.oo, the fourth  for S.o³. And thence the density of the medium S / (R. √1 + QQ), in any place G, will be, and therefore if in VZ you take VT equal to n * VG, that density is reciprocally as XT. For A² and  are the squares of XZ and ZT. But the resistance in the same place G is to the force of gravity as 3S * XT / A to 4RR, that is, as XT to. And the velocity there, is the same wherewith the projected body would move in a Parabola, whose vertex is G, diameter GD, and latus rectum (1 + QQ) / R or. Q.E.I.

In the same manner that the density of the medium comes out to be as (S x AC) / (R x HT), in corollay 1. if the resistance is put as any power Vn of the velocity V, the density of the medium will come out to be as Figure 3.

And therefore if a curve can be found, such that the ratio of, or of S²/R4-n to (1 + QQ)n-1 may be given: the body, in an uniform medium, whose resistance is as the power Vn of the velocity V, will move in this curve. But let us return to more simple curves.

Because there can be no motion in a Parabola except in a non-resisting medium, but in the Hyperbola's here described 'tis produced by a perpetual resistance; it is evident that the line which a projectile describes in an uniformly resisting medium, approaches nearer to these Hyperbola's than to a Parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them, than these Hyperbola's here described. The difference however is not so great between the one and the other, but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful, than an Hyperbola that is more accurate, and at the same time more compounded. They may be made use of then in this manner. Figure 5.

Compleat the parallelogram XYGT, and the right line GT will touch the hyperbola in G, and therefore the density of the medium in G is reciprocally as the tangent GT, and the velocity there, as √GT² / GV² and the resistance is to the force of gravity as GT to {(2nn + 2n) / (n + 2)} * GV.

Therefore if a body projected from the place A in the direction of the right line AH, (Figure 6.) describes the Hyperbola AGK, and AH produced meets the asymptote NX in H, and AI drawn parallel to it meets the other asymptote MX in I; the density of the medium in A will be reciprocally as AH, and the velocity of the body as √AH² / AI, and the resistance there to the force of gravity as AH to {(2nn + 2n) / (n + 2)} * AI. Hence the following rules are deduced.

Rule 1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed; the lengths AH, AI, HX will remain. Therefore if those lengths, in any one case, are found, the Hyperbola may afterwards be easily determined from any given angle NAH.

Rule 2. If the angle NAH and the density of the medium at A remain the same, and the velocity with which the body is projected be changed, the length AH will continue the same; and AI will be changed in a duplicate ratio of the velocity reciprocally.

Rule 3. If the angle NAH, the velocity of the body at and the accelerative gravity remain the same, and the proportion of the resistance at A to the motive gravity be augmented in any ratio; the proportion of AH to AI will be augmented in the same ratio, the latus rectum of the abovementioned Parabola remaining the same, and also the length AH²/AI proportional to it; and therefore AH will be diminished in the same ratio, and AI will be diminished in the duplicate of that ratio. But the proportion of the resistance to the weight is augmented, when either the specific gravity is made less, the magnitude remaining equal, or when the density of the medium is made greater, or when, by diminishing the magnitude, the resistance becomes diminished in a less ratio than the weight.

Rule 4. Because the density os the medium is greater near the vertex of the Hyperbola, than it is in the place A; that a mean density may be preserv'd, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT.

Rule 5. If the lengths AH, AI are given, and the figure AGK is to be described: produce HN to X, so that HX may be to AI as n+1 to 1; and with the centre X, and the asymptotes MX, NX describe an Hyperbola thro' the point A, such that AI may be to any of the lines VG as XVn to XIn.

Rule 6. By how much the greater the number n is, so much the more accurate are these Hyperbola's in the ascent of the body from A, and less accurate in its descent to K; and the contrary. The Conic Hyperhola keeps a mean ratio between these, and is more simple than the rest. Therefore if the Hyperbola be of this kind, and you are to find the point K, where the projected body falls upon any right line AN passing thro' the point A: let AN produced meet the asymptotes MX, NX in M and N, and take NK equal to AM.

Rule 7. And hence appears an expeditious method of determining this Hyperbola from the phenomena. Let two similar and equal bodies be projected with the same velocity, in different angles HAK, hAk, (Figure 6.) and let them fall upon the plane of the horizon in K and k; and note the proportion of AK to Ak- Let it be as d to e. Then erecting a perpendicular AI of any length, assume any how the length AH or Ah, and thence graphically, or by scale and compass, collect the lengths AK, Ak (by rule 6.) If the ratio of AK to Ak be the same with that of d to e, the length of AH was rightly assumed. If not, take on the indefinite right line SM, (Figure 7.) the length SM equal to the assumed AH; and erect a perpendicular MN, equal to the difference AK/Ak - d/e of the ratio's drawn into any given right line. By the like method, from several assumed lengths AH, you may find several points N; and draw thro' them all a regular curve NNXN, cutting the right line SMMM in X. Lastly, assume AH equal to the abscissa SX, and thence find again the length AK; and the lengths, which are to the assumed length AI and this last AH, as the length AK known by experiment, to the length AK last found, will be the true lengths AI and AH, which were to be found. But these being given, there will be given also the resisting force of the medium in the place A, it being to the force of gravity as AH to 2AI. Let the density of the medium be increased by rule 4. and if the resisting force just found be increased in the same ratio, it will become still more accurate.

Rule 8. The lengths AH, HX being found; let there be now required the position of the line AH, according to which a projectile thrown with that given velocity, shall fall upon any point K. At the points A and K, (Figure 6.) erect the lines AC, KF perpendicular to the horizon; whereof let AC be drawn downwards, and be equal to AI or ½HX. With the asymptotes AK, KF, describe an Hyperbola, whose conjugate shall pass thro* the point C; and from the centre A, with the interval AH, describe a circle cutting that Hyperbola in the point H; then the projectile thrown in the direction of the right line AH will fall upon the point K. Q.E.I. For the point H, because of the given length AH, must be somewhere in the circumference of the described circle. Draw CH meeting AK and KF in E and F; and because CH, MX are parallel, and AC, AI equal, AE will be equal to AM, and therefore also equal to KN. But CE is to AE as FH to KN, and therefore CE and FH are equal. Therefore the point H falls upon the hyperbolic curve described with the asymptotes AK, KF, whose conjugate passes thro' the point C; and is therefore found in the common intersection of this hyperbolic curve and the circumference of the described circle. Q.E.D. It is to be observed that this operation is the same, whether the right line AKN be parallel to the horizon, or inclined thereto in any angle; and that from two intersections H, H, there arise two angles NAH, NAH; and that in mechanical practice it is sufficient once to describe a circle, then to apply a ruler CH, of an indeterminate length, so to the point C, that its part FH, intercepted between the circle and the right line FK, may be equal to its part CE placed between the point C and the right line AK.

What has been said of Hyperbola's may be easily applied to Parabola's. For if (Figure 8.) a Parabola be represented by XAGK, touched by a right line XV in the vertex X; and the ordinates IA, VG be as any powers XIn, XVn of the abscissa's XI, XV; draw XT, GT, AH, whereof let XT be parallel to VG, and let GT, AH touch the Parabola in G and A: and a body projected from any place A, in the direction of the right line AH, with a due velocity, will describe this Parabola, if the density of the medium in each of the places G, be reciprocally as the tangent GT. In that case the velocity in G will be the same as would cause a body, moving in a non-resisting space, to describe a Conic Parabola, having G for its vertex, VG produced downwards for its diameter, and 2GT² / (nn - n * VG) for its latus rectum. And the resisting force in G will be to the force of gravity, as GT to {(2nn - 2n) / (n - 2)} * VG. Therefore if NAK represent an horizontal line, and, both the density of the medium at A and the velocity with which the body is projected, remaining the same, the angle NAH be any how alter'd; the lengths AH, AI, HX will remain; and thence will be given the vertex X of the Parabola, and the position of the right line XI, and by taking VG to IA as XVn to XIn, there will be given all the points G of the Parabola, thro* which the projectile will pass.

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The Mathematical Principles of Natural Philosophy (1729)/Section 1

If a body is resisted in the ratio of its velocity, the motion lost by resistance is as the space gone over in its motion.

or since the motion lost in each equal particle of time is as the velocity, that is, as the particle of space gone over; thesis by composition, the motion lost in the whole time will be as the whole space gone over. Q.E.D.

Corollary. Therefore if the body, destitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and also the motion remaining after some part of the way is gone over; there will be given also the whole space which the body can describe in an infinite time. For that space will be to the space now described, as the whole motion at the beginning is to the part lost of that motion.

Lemma I. Quantities proportional to their differences are continually proportional.

Let A be to A-B as B to B-C and C to C-D, &c. and, by conversion, A will be to B as B to C and C to D &c. Q.E.D.

If a body is resisted in the ratio of its velocity, and moves, by its vis insita only, through a similar medium, and the times be taken equal; the velocities in the beginning of each of the times are in a geometrical progression, and the spaces described in each of the times are as the velocities.

Case 1. Let the time be divided into equal particles; and if at the very beginning of each particle we suppose the resistance to act with one single impulse which is as the velocity; the decrement of the velocity in each of the particles of time will be as the same velocity. Therefore the velocities are proportional to their differences, and therefore (by Lemma 1. Book 2.) continually proportional. Therefore if out of an equal number of particles there be compounded any equal portions of time, the velocities at the beginning of those times will be as terms in a continued progression, which are taken by intervals, omitting every where an equal number of intermediate terms. But the ratio's of these terms are compounded of the equal ratio's of the intermediate terms equally repeated; and therefore are equal. Therefore the velocities, being proportional to those terms, are in geometrical progression. Let those equal particles of time be diminished, and their number increased in infinitum, so that the impulse of resistance may become continual; and the velocities at the beginnings of equal times, always continually proportional, will be also in this case continually proportional. Q.E.D.

Case 2. And, by division, the differences of the velocities, that is, the parts of the velocities lost in each of the times, are as the wholes: But the spaces described in each of the times arc as the lost parts of the velocities, (by Proposition. 1. Book 2.) and therefore are also as the wholes. Q.E.D.

Corollary. Hence if to the rectangular asymptotes AC, CH, the Hyperbola BG is described, and AB, DG be drawn perpendicular to the asymptote AC, and both the velocity of the body, and the resistance of the medium, at the very beginning of the motion, be express'd by any given line AC, and after some time is elapsed, by the indefinite line DC; the time may be express'd by the area ABGD, and the space described in that time by the line AD. For if that area, by the motion of the point D, be uniformly increased in the same manner as the time, the right line DC will decrease in a geometrical ratio in the same manner as the velocity, and the parts of the right hne AC, described, in equal times, will decrease in the same ratio.

To define the motion of a body which, in a similar medium, ascends or descends in a right line, and is resisted in the ratio of its velocity, and acted upon by an uniform force of gravity.

The body ascending, let the gravity be expounded by any given rectangle BACH; and the resistance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an Hyperbola, cutting the perpendiculars DE, de, in G,g; and the body ascending will in the time DGgd describe the space EGge; in the time DGBA, the space of the whole ascent EGB; in the time ABKI, the space of descent BFK; and in the time IKki the space of descent KFfk; and the velocities of the bodies (proportional to the resistance of the medium) in these periods of time, will be ABED, ABed,o, ABFI, ABfi respectively; and the greatest velocity which the body can acquire by descending, will be BACH.

For let the rectangle BACH be resolved into innumerable rectangles Ak, Kl, Lm, Mn, &c. which shall be as the increments of the velocities produced in so many equal times; then will o, Ak, Al, Am, An, &c. be as the whole velocities, and therefore (by supposition) as the resistances of the medium in the beginning of each of the equal times. Make AC to AK, or ABHC to ABkK as the force of gravity to the resistance in the beginning of the second time; then from the force of gravity subduct the resistances, and ABHC, KkHC, LlHC, MmHC, &c. will be as the absolute forces with which the body is acted upon in the beginning of each of the times, and therefore (by Law 2) as the increments of the velocities, that is, as the rectangles Ak, Kl, Lm, A1n, &c. and therefore (by Lemma 1. Book 2.) in a geometrical progression. Therefore if the right lines Kk, Lk, Mm, Nn, &c. are produced so as to meet the Hyperbola in q, r, s, t, &c. the areas ABqK, KqrL, LrsM, MstN, &c. will be equal, and therefore analogous to the equal times and equal gravitating forces. But the area ABqK (by Corollary 3. Lemma 7 & 8. Book 1.) is to the area Bkq as Kq to ½kq, or AC to ½AK, that is as the force of gravity to the resistance in the middle of the first time. And by the like reasoning the areas qKLr, rLMs, sMNt, &c. are to the areas qklr, rlms, smnt, &c. as the gravitating forces to the resistances in the middle of the second, third, fourth time, and so on. Therefore since the equal areas BAKq, qKLr, rLMs, sMNt, &c. are analogous to the gravitating forces, the areas Bkq, qklr, rlms, smnt, &c. will be analogous to the resistances in the middle of each of the times, that is (by supposition) to the velocities, and so to the spaces described. Take the sums of the analogous quantities, and the areas Bkq, Blr, Bms, Bnt, &c. will be analogous to the whole spaces described; and also the areas ABqK, ABrL, ABsM, ABtN, &c to the times. Therefore the body, in descending, will in any time ABrL, describe the space Blr, and in the time LrtN the space rlnt. Q.E.D. And the like demonstration holds in ascending motion.

Corollary 1. Therefore the greatest velocity that the body can acquire by falling, is to the velocity acquired in any given time, as the given force of gravity which perpetually acts upon it, to the resisting force which opposes it at the end of that time.

Corollary 2. But the time being augmented in an arithmetical progression, the sum of that greatest velocity and the velocity in the ascent, and also their difference in the descent, decreases in a geometrical progression.

Corollary 3. Also the differences of the spaces, which are described in equal differences of the times, decrease in the same geometrical progression.

Corollary 4. The space described by the body is the difference of two spaces, whereof one is as the time taken from the beginning of the descent, and the other as the velocity; which [spaces] also at the beginning of the descent are equal among themselves.

Supposing tke force of gravity in any similar medium to be uniform, and to tend perpendicularly to the plane of the horizon; to define the motion of a projectile therein, which suffers resistance proportional to its velocity.

Let the projectile go from any place D in the direction of any right line DP, and let its velocity at the beginning of the motion be expounded by the length DP. From the point P let fall the perpendicular PC on the horizontal line DC, and cut DC in A, so that DA may be to AC as the resistance of the medium arising from its motion upwards at the beginning, to the force of gravity: or (which comes to the same) so that the rectangle under DA and DP may be to that under AC and CP, as the whole resistance at the beginning of the motion to the force of gravity. With the asymptotes DC, CP describe any Hyperbola GTBS cutting the perpendiculars DG, AB in G and B; compleat the parallelogram DGKC, and let its side GK cut AB in Q. Take aline N in the same ratio to QB as DC is in to CP; and from any point R of the right line DC, erect RT perpendicular to it, meeting the Hyperbola in T, and the right lines EH, GK, DP in I, t, and V; in that perpendicular take Vr equal to tGT/N, or, which is the same thing, take Rr equal to GTIE/N; and the projectile in the time DRTG will arrive at the point r, describing the curve line DraF, the locus of the point r; thence it will come to its greatest height a in the perpendicular AB; and afterwards ever approach to the asymptote PC. And its velocity in any point r will be as the tangent rL to the curve. Q.E.I.

For N is to QB as DC to CP or DR to RV, and therefore RV is equal to DR x QB / N and Rr (that is, RV-Vr, or (DR x QB - tGT)/N ) is equal to (DR x AB - RDGT) / N. Now let the time be expounded by the area RDGT, and (by Laws Corollary 2.) distinguish the motion of the body into two others, one of ascent, the other lateral. And since the resistance is as the motion, let that also be distinguished into two parts proportional and contrary to the parts of the motion: and therefore the length described by the lateral motion, will be (by Proposition 2. Book 2.) as the line DR, and the height (by Proposition 3. Book 2.) as the area DR x AB - RDGT, that is, as the line Rr. But in the very beginning of the motion the area RDGT is equal to the rectangle DR x AQ, and therefore that line Rr ( or (DR x AB - DR x AQ) / N ) will then be to DR as AB - AQ or QB to N, that is, as CP to DC; and therefore as the motion upwards to the motion lengthwise at the beginning. Since therefore Rr is always as the height, and DR always as the length, and Rr is to at the beginning, as the height to the length: it follows, that Rr is always to DR as the height to the length; and therefore that the body will move in the line DraF, which is the locus of the point r. Q.E.D.

Corollary 1. Therefore Rr is equal to DR x AB / N - RDGT / N; and therefore if RT be produced to X, so that RX may be equal to DR x AB / N, that is, if the parallelogram ACPT be compleated, and DT cutting CP in Z be drawn, and RT be produced till it meets DT in X; Xr will be equal to RDGT / N, and therefore proportional to the time.

Corollary 2. Whence if innumerable lines CR, or, which is the same, innumerable lines ZX, be taken in a geometrical progression; there will be as many lines Xr in an arithmetical progression. And hence the curve DraF is easily delineated by the Table of Logarithms.

Corollary 3. If a Parabola be constructed to the vertex D, and the diameter DG, produced downwards, and its latus rectum is to 2 DP as the whole resistance at the beginning of the motion to the gravitating force: the velocity with which the body ought to go from the place D, in the direction of the right line DP, so as in an uniform resisting medium to describe the curve DraF, will be the same as that with which it ought to go from the same place D, in the direction of the same right line DP, so as to describe a Parabola in a non-resisting medium. For the latus rectum of this Parabola, at the very beginning of the motion, is DV²/Vr; and Vr is tGT / N or DR x Tt / 2N. But a right line, which, if drawn, would touch the Hyperbola GTS in G, is parallel to DK, and therefore Tt is CK x DR / DC, and N is QB x DC / CP: And therefore Vr is equal to (DR² x CK x CP) / (2DC² x QB) that is, (because DR and DC, DV and DP are proportionals) to (DV² x CK x CP) / (2DP² x QB); and the latus rectum DV² / Vr comes out (2DP² x QB) / (CK x CP), that is, (because QB and CK, DA and AC are proportional) (2DP² x DA) / (AC x CP), and therefore is to 2DP, as DP x DA to CP x AC; that is, as the resistance to the gravity. Q.E.D.

Corollary 4. Hence if a body be projected from any place D, with a given velocity, in the direction of a right line DP given by position; and the resistance of the medium, at the beginning of the motion, be given: the curve DraF, which that body will describe, may be found. For the velocity being given, the latus rectum of the parabola is given, as is well known. And taking 2DP to that latus rectum, as the force of gravity to the resisting force, DP is also given. Then cutting DC in A, so that CP x AC may be to DP x DA in the same ratio of the gravity to the resistance, the point A will be given. And hence the curve DraF is also given.

Corollary 5. And on the contrary, if the curve DraF be given, there will be given both the velocity of the body, and the resistance of the medium in each of the places r. For the ratio of CP x AC to DP x DA being given, there is given both the resistance of the medium at the beginning of the motion, and the latus rectum of the parabola; and thence the velocity at the beginning of the motion is given also. Then from the length of the tangent rL, there is given both the velocity proportional to it, and the resistance proportional to the velocity in any place r.

Corollary 6. But since the length 2DP is to the latus rectum of the parabola as the gravity to the resistance in D; and, from the velocity augmented, the resistance is augmented in the same ratio, but the latus rectum of the parabola is augmented in the duplicate of that ratio; it is plain that the length 2DP is augmented in that simple ratio only; and is therefore always proportional to the velocity; nor will it be augmented or diminished by the change of the angle CDP, unless the velocity be also changed.

Corollary 7. Hence appears the method of determining the curve DraF, nearly, from the phenomena, and thence collecting the resistance and velocity with which the body is projected. Let two similar and equal bodies be projected with the same velocity, from the place D, in different angles CDP, CDp; and let the places F, f, where they fall upon the horizontal plane DC, be known. Then taking any length for DP or Dp, suppose the resistance in D to be to the gravity in any ratio whatsoever, and let that ratio be expounded by any length SM. Then by computation, from that assumed length DP, find the lengths DF, Df; and from the ratio Ff / DF, found by calculation, subduct the same ratio as found by experiment; and let the difference be expounded by the perpendicular MN. Repeat the same a second and a third time, by assuming always a new ratio SM of the resistance to the gravity, and collecting a new difference MN. Draw the affirmative differences on one side of the right line SM, and the negative on the other side; and through the points N, N, N draw a regular curve NNN, cutting the right line SMMM in X, and SX will be the true ratio of the resistance to the gravity, which was to be found. From this ratio the length DF is to be collected by calculation; and a length, which is to the assumed length DP, as the length DF known by experiment to the length DF just now found, will be the true length DP. This being known, you will have both the curve line DraF which the body describes, and also the velocity and resistance of the body in each place.

But yet that the resistance of bodies is in the ratio of the velocity, is more a mathematical hypothesis than a physical one. In mediums void of all tenacity, the resistances made to bodies are in the duplicate ratio of the velocities. For by the action of a swifter body, a greater motion, in proportion to a greater velocity, is communicated to the same quantity of the medium, in a less time; and in an equal time, by reason os a greater quantity of the disturbed medium, a motion is communicated in the duplicate ratio greater; and the resistance (by Law 2 and 3) is as the motion communicated. Let us therefore see what motions arise from this law of resistance.


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The Mathematical Principles of Natural Philosophy (1729)/Proposition 6: Gravitation towards every Planet

Proposition VI. Theorem VI. That all bodies gravitate towards; every Planet and that the Weights of bodies towards any the same Planet, at equal distances from the centre of the Planet, are proportional to the quantities of matter which they severally contain.

It has been, now of a long time, observed by others, that all forts of heavy bodies, (allowance being made for the inequality of retardation, which they suffer from a small power of resistance in the air) descend to the Earth from equal heights in equal times: and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal. I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The Boxes hanging by equal threads of 11 feet, made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And placing the one by the other, I observed them to play together forwards and backwards, for a long time, with equal vibrations. And therefore the quantity of matter in the gold (by corrolary 1. and 6. proposition 24. book 2.) was to the quantity of matter in the wood, as the action of the motive force (or vis matrix) upon all the gold, to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other. And the like happened in the other bodies. By these experiments, in bodies of the same weight, I could manifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been. But without all doubt, the nature of gravity towards the Planets, is the same as towards the Earth. For, should we imagine our terrestrial bodies removed to the orb of the Moon, and there, together with the Moon, deprived of all motion, to be let go, so as to fall together towards the Earth: it is certain, from what we have demonstrated before, that, in equal times, they would describe equal spaces with the Moon, and of consequence are to the Moon, in quantity of matter, as their weights to its weight. Moreover, since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the squares of their distances from Jupiter's centre; that is, equal, at equal distances. And therefore, these satellites, if supposed to fall towards Jupiter from equal heights, would describe equal spaces in equal times, in like manner as heavy bodies do on our Earth. And by thesame argument, if the circumsolar Planets were supposed to be let sail at equal distances from the Sun, they would, in their descent towards the Sun, describe equal spaces in equal times. But forces, which equally accelerate unequal bodies, must be as those bodies; that is to say, the weights of the Planets towards the Sun must be as their quantities of matter. Further, that the weights of Jupiter and of his satellites towards the Sun are proportional to the several quantities of their matter, appears from the exceeding regular motions of the satellites, (by corrolary 5. proposition 55. book 1.) For if some of those bodies were more strongly attracted to the Sun in proportion to their quantity of matter, than others; the motions of the satellites would be disturbed by that inequality of attraction (by corrolary 2. proposition 65. book 1.) If, at equal distances from the Sun, any satellite in proportion to the quantity of its matter, did gravitate towards the Sun, with a force greater than Jupiter in proportion to his, according to any given proportion, suppose of d to e; then the distance between the centres of the Sun and of the satellite's orbit would be always greater than the distance between the centres of the Sun and of Jupiter, nearly in the subduplicate of that proportion; as by some computations I have found. And if the satellite did gravitate towards the Sun with a force, lesser in the proportion of e to d, the distance of the centre of the satellite's orb from the Sun, would be less than the distance of the centre of Jupiter from the Sun, in the subduplicate of the same proportion. Therefore if, at equal distances from the Sun, the accelerative gravity of any satellite towards the Sun were greater or less than the accelerative gravity of Jupiter towards the Sun, but by one 1/2000 part of the whole gravity; the distance of the centre of the satellite's orbit from the Sun would be greater or less than the distance of Jupiter from the Sun, by one 1/2000 part of the whole distance; that is, by a fifth part of the distance of the utmost satellite from the centre of Jupiter; an excentricity of the orbit, which would be very sensible. But the orbits of the satellites are concentric to Jupiter, and therefore the accelerative gravities of Jupiter, and of all its satellites towards the Sun, are equal among themselves. And by the same argument, the weights of Saturn and of his satellites towards the Sun, at equal distances from the Sun, are as their several quantities of matter: and the weights of the Moon and of the Earth towards the Sun, are either none, or accurately proportional to the masses of matter which they contain. But some they are by corrolary 1. and 3. proposition 5.

But further, the weights of all the parts of every Planet towards any other Planer, are one to another as the matter in the several parts. For if some parts did gravitate more, others less, than for the quantity of their matter; then the whole Planet, according to the fort of parts with which it most abounds, would gravitate more or less, than in proportion to the quantity of matter in the whole. Nor is it of any moment, whether these parts are external or internal. For, if, for example, we mould imagine the terrestrial bodies with us to be raised up to the orb of the Moon, to be there compared with its body: If the weights of such bodies were to the weights of the external parts of the Moon, as the quantities of matter in the one and in the other respectively; but to the weights of the internal parts, in a greater or less proportion, then likewise the weights of those bodies would be to the weight of the whole Moon, in a greater or less proportion; against what we have shewed above.

Corollary 1. Hence the weights of bodies do not depend upon their forms and textures. For if the weights could be altered with the forms, they would be greater or less, according to the variety of forms, in equal matter; altogether against experience.

Corollary 2. Universally, all bodies about the Earth, gravitate towards the Earth; and the weights of all, ar equal distances from the Earth's centre, are as the quantities of matter which they severally contain. This is the quality of all bodies, within the reach of our experiments; and therefore, (by rule 3) to be affirmed of all bodies whatsoever. Is the æther, or any other body, were either altogether void of gravity, or were to gravitate less in proportion to its quantity of matter; then, because (according to Aristotle, Des Cartes, and others) there is no difference betwixt that and other bodies, but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter; and, on the other hand, the heaviest bodies, acquiring the £rst form of that body, might by degrees, quite lose their gravity. And therefore the weights would depend upon the forms of bodies, and with those forms might be changed, contrary to what was proved in the preceding corollary.

Corollary 3. All spaces are not equally Full. For if all spaces were equally full, then the specific gravity of the fluid which fills the region of the air, on account of the extreme density of the matter, would fall nothing short of the specific gravity of quick-silver, or gold, or any other the most dense body; and therefore, neither gold, nor any other body, could descend in air. For bodies do not descend in fluids, unless they are specifically heavier than the fluids. And if the quantity of matter in a given space, can, by any rarefaction, be diminished, what should hinder a diminution to infinity?

Corollary 4. If all the solid particles of all bodies are of the same density, nor can be rarified without pores a void space or vacuum must be granted. By bodies of the same density, I mean those, whose vires inertia are in the proportion of their bulks.

Corollary 5. The power of gravity is of a different nature from the power of magnetism. For the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet, others less; most bodies not at all. The power of magnetism, in one and the same body, may be increased and diminished; and is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet, decreases not in the duplicate, but almost in the triplicate proportion of the distance, as nearly as I could, judge from some rude observations.


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The Mathematical Principles of Natural Philosophy (1729)/Proposition 7: Gravity tending to all Bodies

Proposition VII. Theorem VII. That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.

That all the Planets mutually gravitate one towards another, we have prov'd before; as well as that the force of gravity towards every one of them, consider'd apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence (by proposition 69. book. 1. and its corollaries) it follows, that the gravity tending towards all the Planets, is proportional to the matter which they contain.

Moreover, since all the parts of any planet A gravitate towards any other planet B; and the gravity of every part is to the gravity of the whole, as the matter of the part to the matter of the whole; and (by law 5.) to every action corresponds an equal re-action: therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A; and its gravity towards any one part will be to the gravity towards the whole, as the matter of the part to the matter of the whole. Q.E.D.

Corollary 1. Therefore the force of gravity towards any whole planet, arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this. For all attraction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet, as form'd of a number of lesser planets, meeting together in one globe. For hence it would appear that the force of the whole must arise from the forces of the component parts. If it is objected, that, according to this law, all bodies with us must mutually gravitate one towards another, whereas no such gravitation any where appears: I answer, that since the gravitation towards these bodies is to the gravitation towards the whole Earth, as these bodies are to the whole Earth, the gravitation, towards them must be far less than to fall under the observation of our senses.

Corollary 2. The force of gravity towards the several equal particles of any body, is reciprocally as the square of the distance of places from the particles; as appears from corrolary 3. proposition 74. book 1.

Proposition VIII. Theorem VIII. In two spheres mutually gravitating each towards the other, if the matter in places on all sides round about and equidistant from the centres, is similar; the weight of either sphere towards the other, will be reciprocally as the square of the distance between their centres.

After I had found that the force of gravity towards a whole planet did arise from, and was compounded of the the forces of gravity towards all its parts; and towards every one part, was in the reciprocal proportion of the squares of the distances from the part: I was yet in doubt, whether that reciprocal duplicate proportion did accurately hold, or but nearly so, in the total force compounded of so many partial ones. For it might be that the proportion which accurately enough took place in greater distances, should be wide of the truth near the surface of the planet, where the distances of the particles are unequal, and their situation dissimilar. But by the help of proposition 75. and 76. book I. and their corollaries, I was at last satisfy'd of the truth of the proposition, as it now lies before us.

Corollary 1. Hence we may find and compare together the weights of bodies towards different planets. For the weights of bodies revolving in circles about planets, are (by corrolary 2. proposition 4. book I.) as the diameters of the circles directly, and the squares of their periodic times reciprocally; and their weights at the surfaces of the planets, or at any other distances from their centres, are (by this proposition) greater or less, in the reciprocal duplicate proportion of the distances. Thus from the periodic times of Venus, revolving about the Sun, in 224d. 16¼h. of the utmost circumjovial satellite revolving about Jupiter, in 16d. 161/15h; of the Hugenian satellite about Saturn in 15d. 222/3h; and of the Moon about the Earth in 27d. 7h. 43'; compared with the mean distance of Venus from the Sun, and with the greatest heliocentric elongations of the outmost circumjovial satellite from Jupiter's centre, 8'.16" of the Hugenian satellite from the centre of Saturn, 3'.4", and of the Moon from the Earth, 10'.33"; by computation I found, that the weight of equal bodies, at equal distances from the centres of the Sun, of Jupiter, of Saturn, and of the Earth, towards the Sun, Jupiter, Saturn, and the Earth, were one to another, as 1, 1/1067, 1/3021, and 1/169282 respectively. Then because as the distances are increased or diminished, the weights are diminished or increased in a duplicate ratio; the weights of equal bodies towards the Sun, Jupiter, Saturn, and the Earth, at the distances 10000, 997, 791 and 109 from their centres, that is, at their very superficies, will be as 10000, 943, 529 and 435 respectively. How much the weights of bodies are at the superficies of the Moon, will be'shewn hereafter.

Corollary 2. Hence likewise vwe discoVeiTthe quantity of matter in the several Planets. For their quantities of matter are as the forces of gravity at equal distances from their centres, that is, in the Sun, Jupiter, Saturn, and the Earth, as 1, 1/1067, 1/3021, and 1/169282 respectively. If the parallax of the Sun be taken greater or less than 10", 30"', the quantity of matter in the Earth must be augmented or diminished in the triplicate of that proportion.

Corollary 3. Hence also we find the densities of the Planets. For (by proposition 72. book I.) the weights of equal and similar bodies towards similar spheres, are, at the surfaces of those spheres, as the diameters of the spheres. And therefore the densities of dissimilar spheres are as those weights applied to the diameters of the spheres. But the true diameters of the Sun, Jupiter, Saturn, and the Earth, were one to another as 10000, 997, 791 and 109; and the weights towards the same, as 10000, 943, 529, and 435 respectively; and therefore their densities are as 100, 94½, 67 and 400. The density of the Earth, which comes out by this computation, does not depend upon the parallax of the Sun, but is determined by the parallax of the Moon, and therefore is here truly defin'd. The Sun therefore is a little denser than Jupiter, and Jupiter than Saturn, and the Earth four times denser than the Sun; for the Sun, by its great heat, is kept in a sort of a rarefy'd state. The Moon is denser than the Earth, as shall appear afterwards.

Corollary 4. The smaller the Planets are, they are, ceteris paribus, of so much the greater density. For so the powers of gravity on their several surfaces, come nearer to equality. They are likewise, ceteris paribus, of the greater density, as they are nearer to the Sun. So Jupiter is more dense than Saturn, and the Earth than Jupiter. For the Planets were to be placed at different distances from the Sun, that according to their degrees of density, they might enjoy a greater or less proportion of the Sun's heat. Our water, if it were remov'd as far as the orb of Saturn, would be converted into ice, and the orb of Mercury would quickly fly away in vapour. For the light of the Sun, to which its heat is proportional, is seven times denser in the orb of the Mercury than with us: and by the thermometer I have found, that a sevenfold heat of our summer-sun will make water boil. Nor are we to doubt, that the matter of Mercury is adapted to its heat, and is therefore more dense than the matter of our Earth; since, in a denser matter, the operations of nature require a stronger heat.

Proposition IX. Theorem IX. That the force of gravity, consider'd downwards from the surface of the planets, decreases nearly in the proportion of the distances from their centres.

If the matter of the planet were of an uniform density, this proposition would be accurately true, (by proposition 75. book I.) The error therefore can be no greater than what may arise from the inequality of the density.