About the Work
Mathematical Principles of Natural Philosophy is Sir Isaac Newton’s foundational treatise on the laws of motion and universal gravitation. The work, originally composed in Latin, was translated into English by Andrew Motte. The edition described in the notes is the First American Edition of 1846, published by Daniel Adee of New-York, which was carefully revised and corrected, and includes a Life of the Author by N. W. Chittenden, M. A. This edition also incorporates “Newton’s System of the World” and features a portrait derived from the bust at the Royal Observatory at Greenwich. The volume is dedicated to the Teachers of the Normal School of the State of New-York, with Chittenden’s introduction framing the work as a contribution to a “new era” in education, arguing that liberties are preserved not by legal documents but by the cultivated hearts and heads of the people, and that teachers serve as the “prime Conservators of the public weal.”
The book itself is a comprehensive mathematical and physical treatise, organized into three major books. Book I concerns the motion of bodies under various forces, establishing the mathematical foundations of dynamics. Book II addresses the motion of bodies through resisting mediums, including fluids and pendulums. Book III presents the “System of the World,” applying the principles developed in the earlier books to celestial phenomena, including planetary orbits, the moon’s motion, tides, and comets. The work is characterized by its rigorous geometrical method, a deliberate choice that the American edition’s introduction defends against the encroachment of analytical techniques, citing Newton’s own preference for the geometrical approach despite his invention of fluxions (calculus).
The edition also contains substantial front matter, including a detailed biographical account of Newton’s life, from his birth at Woolsthorpe in Lincolnshire on 25 December 1642 (Old Style), through his education at Trinity College, Cambridge, his discoveries in optics and mathematics, his later career, and his death in 1727. The biography includes Pemberton’s testimony on Newton’s character, describing him as modest, open, and generous, and recounts Newton’s famous self-assessment uttered before his death: that he seemed to himself “only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me.”
Context
The Principia emerged from a period of intense scientific inquiry and was shaped by Newton’s personal intellectual journey. The biographical notes trace Newton’s early education and the development of his key ideas. Born prematurely and posthumously, Newton was initially an inattentive student at Grantham, but a physical altercation with a schoolmate spurred him to study and rise to the head of his class. His mechanical ingenuity manifested early through working models of windmills, a mechanical carriage, a water clock, and sundials. He was admitted to Trinity College, Cambridge, in 1660, where he studied Euclid, Descartes’s Analytical Geometry, Wallis’s Arithmetic of Infinites, Saunderson’s Logic, and Kepler’s Optics. Notes on Wallis contained the germ of his fluxionary calculus.
The context of the work is also defined by the scientific community of the time. Dr. Barrow, Newton’s friend and mentor, was made Lucasian Professor of Mathematics in 1663 and later resigned the chair in 1669 in favour of Newton. Newton’s method of fluxions was possessed by 1666, the same year he experimented with a triangular glass prism and concluded that light was not homogeneous but consisted of rays of different refrangibility. Forced to leave Cambridge due to the plague, he retired to Woolsthorpe, where in the autumn of 1666 he first considered that gravity might extend to the moon, finding that if planets are held in course by a power like gravity, its strength must decrease in the duplicate proportion of the increase of distance. His calculations using the common estimate of sixty miles to a degree did not answer his expectations, and he abandoned the inquiry, only to return to it later.
The publication of the Principia was facilitated by Edmund Halley, who assisted in correcting the press and whose solicitations led to the work’s publication. Halley obtained demonstrations of celestial orbit figures and pressed Newton to communicate them to the Royal Society. Newton deferred publication to investigate lunar inequalities, gravity laws, figures described by attracted bodies, motions in resisting mediums, and comets’ orbits. The author’s preface, dated Cambridge, Trinity College, May 8, 1686, acknowledges Halley’s role and begs readers’ candour. The work thus stands at the intersection of Newton’s personal discoveries, the institutional support of Cambridge and the Royal Society, and the collaborative efforts of contemporaries like Halley.
Structure of the Book
The Principia is organized into three books, each subdivided into sections that develop specific aspects of Newton’s system. The structure reflects a deliberate pedagogical and logical progression from abstract mathematical principles to concrete physical applications.
Book I — Of the Motion of Bodies establishes the foundational mathematics and mechanics. Its sections include:
- Section I: Of the method of first and last ratios of quantities, laying the mathematical groundwork for dealing with infinitesimals and limits.
- Section II: Of the Invention of Centripetal Forces, introducing the concept of forces directed toward a center.
- Section III: Of the motion of bodies in eccentric conic sections, applying the theory to elliptical, parabolic, and hyperbolic orbits.
- Section IV: Of the finding of elliptic, parabolic, and hyperbolic orbits from given data.
- Section V: How the orbits are to be found when neither focus is given.
- Section VI: How the motions are to be found in given orbits.
- Section VII: Concerning the rectilinear ascent and descent of bodies.
- Section VIII: Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force.
- Section IX: Of the motion of bodies in moveable orbits; and of the motion of the apsides.
- Section X: Of the motion of bodies in given superficies, and of the reciprocal motion of funependulous bodies.
- Section XI: Of the motions of bodies tending to each other with centripetal forces.
- Section XII: Of the attractive forces of sphærical bodies.
- Section XIII: Of the attractive forces of bodies which are not of a sphærical figure.
- Section XIV: Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any great body.
Book II — Of the Motion of Bodies in Resisting Mediums addresses the effects of resistance on motion. Its sections include:
- Section I: Of the motion of bodies that are resisted in the ratio of the velocity.
- Section II: Of the motion of bodies that are resisted in the duplicate ratio of their velocities.
- Section III: Of the motions of bodies which are resisted partly in the ratio of the velocities and partly in the duplicate ratio.
- Section IV: Of the circular motion of bodies in resisting mediums.
- Section V: Of the density and compression of fluids; and of hydrostatics.
- Section VI: Of the motion and resistance of funependulous bodies (pendulums).
- Section VII: Of the motion of fluids, and the resistance made to projected bodies.
- Section VIII: Of motion propagated through fluids.
- Section IX: Of the circular motion of fluids.
Book III — The System of the World applies the principles of the earlier books to celestial mechanics. It derives the motions of planets, comets, the moon, and the sea from the theory of gravity. The notes indicate that Book III includes propositions on the perturbations of Jupiter and Saturn, the fixity of aphelions and nodes, the diurnal motions of planets, the figure of the Earth, the precession of the equinoxes, the theory of the moon, and the forces of the sun and moon to move the sea (tides). The book also addresses the nature of comets, arguing that they descend into the planetary regions.
The work concludes with a discussion of tides, including detailed observations from ports such as Bristol and Plymouth, and an argument about the immense distances of fixed stars based on the lack of annual parallax. The final notes address comets, citing specific observers such as Kepler, Hevelius, and Flamsted, and arguing that comets’ annual parallax, curvature of paths, and the light of their heads and tails demonstrate their descent into the planetary regions.
Detailed Section-by-Section Summary
The Title Page and Dedication of the First American Edition
The 1846 First American Edition presents Newton’s masterwork in an English translation by Andrew Motte, augmented with “Newton’s System of the World” and a portrait from the Royal Observatory at Greenwich. This edition, carefully revised and corrected, includes a Life of the Author by N. W. Chittenden and is published by Daniel Adee of New-York. The dedication is addressed to the Teachers of the Normal School of the State of New-York, with Chittenden proclaiming a “new era” in education and elevating the Teacher to the highest and most responsible office, whose influence rolls up “a measureless curse or a measureless blessing” along the infinite curve. He argues that liberties are preserved not by legal documents but by the cultivated hearts and heads of the PEOPLE, making teachers the “prime Conservators of the public weal” and leaders in “Educational Reform.”
The Introduction to the American Edition
The introduction notes that despite Newton’s revered name and the work’s celebrated history, the Principia has remained generally unknown in America, previously inaccessible due to the “dead language” of Latin and the enormous price of English copies. This edition aims to be the most accurate and beautiful, quoting that the finest monument to Newton’s glory is a good edition of his works. It argues for the study of Geometrical Synthesis over Analytical encroachment, citing Newton’s own preference for the Geometrical Method despite his invention of Fluxions, and suggests students be led from Euclid or Legendre to the Principia, where Geometry is found “in varied use from the familiar to the sublime.”
The Life of Sir Isaac Newton: Early Years and Education
Newton was born on 25 December 1642 at Woolsthorpe, a posthumous and premature child. At age twelve, he was placed at the public school at Grantham, where an initial inattentiveness was cured by a kick from a boy above him, spurring him to rise to the head of the school. He demonstrated early mechanical genius, building a working model of a windmill powered by a mouse, a mechanical carriage, a water clock, and sundials, one named “Isaac’s dial.” He formed an early attachment to Miss Storey, but inadequacy of fortune prevented their union. Recalled from school in 1656 for farming, he proved distasteful to it, preferring to study. He was admitted to Trinity College, Cambridge, in 1660, where he studied Euclid, Descartes’s Analytical Geometry, Wallis’s Arithmetic of Infinites, and Kepler’s Optics, with his notes on Wallis containing the germ of his fluxionary calculus.
The Plague Years and the Dawn of Discovery
In 1665, Newton received his Bachelor of Arts degree, and in 1666 he entered upon his brilliant discoveries, possessing his Method of Fluxions and inventing the Binomial Theorem. Experimenting with a triangular glass prism, he concluded that “LIGHT WAS NOT HOMOGENEOUS, BUT CONSISTED OF RAYS, SOME OF WHICH WERE MORE REFRANGIBLE THAN OTHERS.” Forced to leave Cambridge due to the plague, he retired to Woolsthorpe. In the early autumn of 1666, seated in his garden, he considered that gravity might extend to the moon, finding that if planets are held in course by a power like gravity, its strength must decrease in the duplicate proportion of the increase of distance. However, his calculations using the common estimate of sixty miles to a degree did not answer his expectations, so he abandoned the inquiry.
Academic Rise and the Reflecting Telescope
In 1667, he was made a Junior Fellow, and in 1668 he took his Master of Arts degree and was appointed to a Senior Fellowship. That same year, he constructed the first reflecting telescope ever executed, six inches long and magnifying about forty times, which he sent to the Royal Society in 1671. In 1669, he wrote a letter of advice to Francis Aston on travelling, and communicated his work “Analysis per Equationes Numero Terminorum Infinitas” to Dr. Barrow. Dr. Barrow resigned the Lucasian Professorship in 1669 in favour of Newton, who delivered his Optical Lectures that same year.
Pemberton’s Testimony and Newton’s Final Illness
Pemberton reports that Newton’s extreme age and universal reputation had not made him rigid or elated, and that he received remarks on the Principia with utmost goodness. The biographer praises Newton’s modesty and generosity, noting he was “Full of wisdom and perfect in beauty” yet uncorrupted by pride. Newton reportedly uttered before death: “I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the sea-shore… while the great ocean of truth lay all undiscovered before me.” After residing about two years at Kensington, Newton went to London in February 1727 to preside at a Royal Society meeting, but fatigue brought a violent return of his bladder affliction. Dr. Mead and Dr. Cheselden pronounced the disease the stone, holding no hope. He suffered intensely but never groaned or complained; on Saturday morning, the 18th of March, he became insensible, continuing until Monday, the 20th, when he expired between one and two o’clock in the morning, in his eighty-fifth year.
Burial, Memorials, and Estate
His body was placed in Westminster Abbey, and in 1731 relatives erected a monument there. A medal was struck at the Tower that same year, and in 1755 Roubiliac executed a white marble statue placed in Trinity College’s ante-chamber. Newton left about thirty-two thousand pounds, divided among four nephews and four nieces of the half blood. Woolsthorpe and Sustern fell to John Newton, heir-at-law, while the Kensington estate went to Catharine, Mr. Conduit’s daughter, later Viscountess Lymington. Dr. Stukeley visited Woolsthorpe in October 1721, describing Sir Isaac’s study with shelves made from deal boxes; the apple-tree had been destroyed by wind about twenty years prior but was preserved as a chair. The house was repaired in 1798 with a tablet inscribed: “Sir Isaac Newton, son of John Newton, Lord of the Manor of Woolsthorpe, was born in this room, on the 25th of December, 1642.”
The Author’s Preface: On Mechanics and Philosophy
Newton cites Pappus on the ancients’ valuing of mechanics, noting that moderns have abandoned substantial forms and occult qualities, subjecting phenomena to mathematical laws. Mechanics is twofold: rational (accurate demonstration) and practical (manual arts), with errors lying not in the art but in artificers. Geometry is founded in mechanical practice—the art of measuring—and rational mechanics concerns motions from forces and forces producing motions. Newton’s design concerns natural powers: gravity, levity, elastic force, fluid resistance, and attractive or impulsive forces. The difficulty of philosophy is to investigate nature’s forces from motion phenomena, then demonstrate other phenomena; Book III exemplifies this through the System of the World, deriving gravity toward the sun and planets from celestial phenomena, then deducing the motions of planets, comets, moon, and sea. Newton suspects all phenomena depend on forces by which particles are mutually impelled or repelled. Edmund Halley assisted in correcting the press and schemes, and his solicitations led to publication; Newton deferred publication to investigate lunar inequalities, gravity laws, figures described by attracted bodies, motions in resisting mediums, and comets’ orbits. He begs readers’ candour, dating the preface at Cambridge, Trinity College, May 8, 1686.
The Definitions and Laws of Motion
The edition notes that the second edition enlarged Book I’s second section, improved Book II’s fluid resistance theory, and more fully deduced the moon’s theory and equinox precession in Book III; the third edition added falling-body experiments and various astronomical observations. Definition I states that quantity of matter arises from density and bulk conjointly, known by weight. Definition II states that quantity of motion arises from velocity and matter conjointly. Definition III defines vis insita as matter’s power of resisting, persevering in its present state, called vis inertia, exercised only when another force changes its condition. Definition IV defines impressed force as an action changing state, consisting in action only.
Propositions on Falling Bodies and Centripetal Force
CASE 2 concerns a parabolic figure DES, where it is found that the falling body’s velocity at C equals the velocity of uniform circular motion at interval ½SC (Prop. XXXIV). This velocity relates to that of a circle with radius SK in the subduplicate ratio of SK to ½SC, which is SK to ½CD (Cor. 6, Prop. IV), establishing that area KSk equals area SYk. PROPOSITION XXXVI (Problem XXV) determines descent times from place A, describing semicircle ADS on diameter AS and equal semicircle OKH about centre S, making sector OSK equal to area ASD. PROPOSITION XXXVII (Problem XXVI) defines ascent/descent times for a body projected from a given place G along line GS with any velocity, taking GA to ½AS in the duplicate ratio of this velocity to uniform circular velocity at interval SG; if the ratio is 2:1, a parabola is described, and if less or greater, a circle or rectangular hyperbola respectively. PROPOSITION XXXVIII (Theorem XII) states that if centripetal force is proportional to altitude or distance from centre, the times and velocities of falling bodies and spaces described are respectively proportional to the arcs, right sines, and versed sines of the arcs; Corollary 1 states that times are equal for a body falling from A to centre S and another revolving describing a quadrantal arc, and Corollary 2 states that all times are equal for bodies falling from any places to the centre. PROPOSITION XXXIX (Problem XXVII) supposes a centripetal force of any kind, granting quadratures of curvilinear figures, to find the velocity of a body ascending or descending in a right line at several places, and the time to arrive at any place; the velocity at E is as a right line whose square equals curvilinear area ABGE, and the time describing AE is as curvilinear area ABTVME. Corollary 1 lets P be the place from which a body should fall so that, urged by any known uniform centripetal force, it acquires at D a velocity equal to that which another body falling by any force has acquired at D.
Corollaries on the Motion of Bodies and the Tides
Cor. 14 states that when body S is very distant, forces NM, ML are nearly as force SK and ratio PT:ST conjunctly, and all effects scale with S’s absolute force directly and ST³ inversely. Cor. 15 states that if orbits retain figure, proportions, and inclination but alter magnitude, all effects will be similar and proportional, with linear errors as orbit diameters and angular errors the same. Cor. 16 states that forces NM, ML are as radius TP, and their periodical effects are as forces and the square of P’s periodical time conjunctly, so angular errors of P observed from T are as P’s periodical time squared directly and T’s periodical time squared inversely. Cor. 17 states that the mean force LM is to the force retaining P in its orbit about T in the duplicate ratio of periodical times. Cor. 18 supposes many fluid bodies moving round T at equal distances, contiguous, forming a fluid annulus or ring, whose parts draw nearer to T and move swifter in conjunction and opposition with S than in quadratures; the annulus’s nodes rest at syzygies, and out of syzygies they are carried backward, swiftest in quadratures. Cor. 19 supposes a spherical body T of non-fluid matter enlarged to extend as far as the annulus, with a channel cut round its circumference containing water; the water, accelerated and retarded by turns, is swifter at syzygies and slower at quadratures than the globe’s surface, ebbing and flowing like the sea, with force LM attracting water downwards at quadratures and force KL attracting it upwards in syzygies. Cor. 20 states that if the annulus becomes hard and the globe diminished, flux and reflux cease, but oscillating inclination and precession of nodes remain; the greatest angle of inclination of the annulus single is when nodes are in syzygies, and the greatest motion of decreasing inclination happens when nodes are in quadratures, with the least angle of inclination in the octants after quadratures; the case is the same for a globe without the annulus if it is a little higher or denser in the equatorial than polar regions, and though the centripetal force of the globe be increased so all parts tend downwards as our earth’s parts gravitate, the phenomena would scarce be altered, except that places of greatest and least height of water differ, as water is now sustained not by centrifugal force.
Spherical Bodies Resisted in Duplicate Ratio of Velocity
The passage opens with a proposition concerning spherical bodies moving through a resisting medium where resistance is proportional to the square of velocity. It establishes that if the times are taken directly as the first motions and inversely as the first resistances, then the bodies will lose parts of their motion proportional to the wholes. Consequently, the spaces described will be proportional to the times and the first velocities conjunctly. The proof demonstrates that the lost parts of motion are as the resistance and time conjunctly, and for these losses to be proportional to the whole motions, the resistance and time must be inversely proportional to the motion. Several corollaries follow: for equally swift bodies resisted in the duplicate ratio of their diameters, they lose proportional parts of motion while describing spaces proportional to their diameters; if resistance is in a sesquiplicate ratio, the spaces are in the sesquiplicate ratio of diameters; universally, if resistance is as any power D^n, the spaces are as D^(3-n); denser globes require augmented spaces in proportion to their density; and different mediums require spaces diminished in proportion to the greater resistance.
The Moments of Generated Quantities
Lemma II defines a “genitum” as any quantity produced by multiplication, division, or root extraction, but not by addition or subtraction. The moments of such quantities are their instantaneous increments or decrements, not finite particles. The lemma states that the moment of any genitum is equal to the moments of its generating sides, each multiplied by the index of its power and its coefficient. The cases demonstrate this: the moment of a rectangle AB is aB + bA; the moment of a content ABC is aBC + bAC + cAB; the moment of a power A^n is naA^(n-1); and similar rules apply to reciprocals and roots. Corollaries extend this to continually proportional quantities and given terms.
A General Method of Tangents
The Scholium quotes Newton’s 1672 letter to Mr. J. Collins, describing a general method of tangents. This method extends not only to curves but also to the calculation of areas, lengths, and centres of gravity, and is not limited to equations free from surds. It is interwoven with infinite series and relates to a treatise from 1671.
Absolute Forces in a Uniform Medium
Proposition VIII states that if a body moves in a uniform medium with gravity, and the spaces are divided equally, the absolute forces are in a geometrical progression. The proof employs hyperbolic areas and the lines AC, AK, KC, and AP. Corollaries provide the greatest velocity as AC and its ratio to the known resistance.
Ascent and Descent Times
Proposition IX relates the time of ascent to a circular sector and the time of descent to a hyperbolic sector, with tangents proportional to the velocities. Cases demonstrate the particle ratios, and corollaries connect the spaces, velocities, and times in resisting versus non-resisting media.
Medium Density for a Given Curve
Proposition X finds the density of a medium required for a body to move in any given curve, with resistance proportional to density and the square of velocity. The solution involves tangents, ordinates, and series expansions, with examples including the semi-circle, parabola, and hyperbola. Rules for practical application follow, including determining the hyperbola from phenomena.
Resistance in Simple and Duplicate Ratio
Section III begins with Proposition XI, which states that for a body resisted partly in the ratio of velocity and partly in the duplicate ratio, if the times are in arithmetic progression, the quantities reciprocally proportional to the velocities increased by a given quantity are in geometric progression. Proposition XII states that if the spaces are in arithmetic progression, the velocities augmented by a given quantity are in geometric progression.
Resistance from Arc Difference
From the law of resistance and the difference (Aa) of the arcs (Ca, CB), the proportion of resistance to gravity can be found. If resistance (DK) is uniform, the figure BKTa is a rectangle, and DK equals half of Aa, making resistance to gravity as half of Aa to the pendulum’s length. If resistance is as velocity, the figure is nearly an ellipse, and with OV as the exponent of resistance at the middle point O, Aa is to OV as 11 to 7, so 7/11 of Aa is to the pendulum’s length as the resistance at O is to gravity. If resistance is in the duplicate ratio of velocity, the figure is almost a parabola, making OV equal to 2/3 of Aa, so the resistance at O is to gravity as 2/3 Aa to the pendulum’s length.
Proposition XXXI, Theorem XXV
If resistance in each proportional part of the arcs is augmented or diminished in a given ratio, the difference between the descent and subsequent ascent arcs is augmented or diminished in the same ratio. This difference arises from retardation and is proportional to the retarding resistance. The rectangle under Aa and half aB is as aB and the resistance conjunctly, so Aa is as the resistance. Corollaries state that if resistance is as velocity, the difference is as the whole arc; if in duplicate ratio, the difference is in the duplicate ratio of the arc; and this extends to any ratio of velocity. If resistance is partly simple and partly duplicate, the difference is partly in the simple and partly in the duplicate ratio of the arc. Thus, the law of resistance for velocity is the same as the law of that difference for arc length.
General Scholium: Experiments on Air Resistance
The author found air resistance by suspending a wooden globe (weighing 57 7/8 ounces troy, diameter 6 7/8 London inches) by a fine thread, with the distance from hook to center of oscillation being 10 1/2 feet. A ruler was placed at a point 10 feet 1 inch from the center of suspension to observe arc lengths. He counted oscillations where the globe lost 1/8 of its motion. For initial descents of 2, 4, 8, 16, 32, or 64 inches, the globe lost 1/8 of its motion in 164, 121, 69, 35 1/2, 18 1/2, and 9 2/3 oscillations, respectively. The differences between first descent and last ascent arcs were 1/4, 1/2, 1, 2, 4, and 8 inches. Dividing by the number of oscillations, the mean differences for arcs of 3 3/4, 7 1/2, 15, 30, 60, and 120 inches were 1/656, 1/242, 1/69, 4/71, 8/37, and 16/29 parts of an inch. These differences are nearly in the duplicate ratio of the arcs for greater oscillations, but somewhat greater for lesser ones, so the resistance is nearly in the duplicate ratio of velocity when swift, and somewhat greater when slow.
Letting V represent the greatest velocity, and A, B, C be given quantities, the difference of arcs is AV + BV^2 + CV^2. Since greatest velocities are as half the arcs in a cycloid and as half the chords in a circle, the differences are nearly the same in both curves. To reduce observations to the cycloid, the greatest velocities are analogous to the numbers 1/2, 1, 2, 4, 8, 16. Using the 2nd, 4th, and 6th cases, equations were reduced to find A = 0.0000916, B = 0.0010847, and C = 0.0029558. The resistance of the globe is to its weight as 0.0000583V + 0.0007593V^2 + 0.0022169V^2 to 121 inches. Thus, in the 2nd, 4th, and 6th cases, the resistance is to the globe’s weight as 0.0030345, 0.041748, and 0.61705 to 121, respectively.
The arc described in the 6th case was 119 5/6 inches, and the center of the globe described an arc of 124 3/16 inches. The greatest velocity is nearly the same as if the globe described 62 3/16 inches in a non-resisting medium, equivalent to falling from a height of 15.278 inches. With this velocity, the globe meets a resistance to its weight as 0.61705 to 121, or 0.56752 to 121 for the part in the duplicate ratio of velocity. An hydrostatical experiment showed the wooden globe’s weight to a water globe’s weight as 55 to 97, so the resistance to a water globe is as 0.56752 to 213.4, or 1 to 376 1/2. The force of resistance uniformly continued would take away 1/3762 part of the whole velocity in the time the globe would describe 15.278 inches.
Perturbations of Jupiter and Saturn
The passage discusses how Jupiter’s orbital error, despite Saturn’s great force, is minimized by placing the lower focus of its orbit in the common center of gravity of Jupiter and the sun. The greatest error scarcely exceeds two minutes, and the mean motion error, two minutes yearly. In their conjunction, the accelerative forces of gravity (sun towards Saturn, Jupiter towards Saturn, Jupiter towards the sun) are as 16, 81, and 156609. The difference of forces (sun towards Saturn, Jupiter towards Saturn) to Jupiter’s gravity towards the sun is as 65 to 156609, or 1 to 2409. Saturn’s disturbing power is proportional to this difference, so Jupiter’s perturbation is much less than Saturn’s. Other orbits are less perturbed, except Earth’s, which is sensibly disturbed by the moon. The common center of gravity of Earth and moon moves in an ellipsis about the sun, describing areas proportional to time, while Earth revolves about this center with a menstrual motion.
The Fixity of Aphelions and Nodes
Proposition XIV, Theorem XIV states that the aphelions and nodes of planetary orbits are fixed, per Prop. XI and Prop. I, Book I. Inequalities from mutual actions of planets and comets are so small they are passed by. Corollary 1 states that fixed stars are immovable, keeping the same position to the aphelions and nodes. Corollary 2 states that stars have no sensible parallax from Earth’s annual motion and, due to immense distance, no force in our system; their contrary attractions destroy mutual actions.
The Motion of Aphelions
In the Scholium, it is noted that planets near the sun (Mercury, Venus, Earth, Mars) are small, so their aphelions and nodes are fixed except for disturbances by Jupiter, Saturn, and higher bodies. Their aphelions move a little in consequentia, in the sesquiplicate proportion of their distances from the sun. If Mars’s aphelion moves 33′ 20″ in 100 years, Earth’s, Venus’s, and Mercury’s move 17′ 40″, 10′ 53″, and 4′ 16″ respectively; these are neglected.
Finding Planetary Orbits
Proposition XV, Problem I is to find the principal diameters of planetary orbits. They are taken in the sub-sesquiplicate proportion of periodic times, then augmented in proportion of the sum of masses of sun and each planet to the first of two mean proportionals between that sum and the sun’s matter. Proposition XVI, Problem II is to find eccentricities and aphelions of planets, resolved by Prop. XVIII, Book I.
The Uniformity of Diurnal Motions
Proposition XVII, Theorem XV states that the diurnal motions of planets are uniform, and the moon’s libration arises from its diurnal motion. This is proved from the first Law of Motion and Cor. 22, Prop. LXVI, Book I. Jupiter revolves in 9h. 56′; Mars in 24h. 39′; Venus in about 23h.; Earth in 23h. 56′; the Sun in 25¼ days; the moon in 27 days, 7 hours, 43′. The lunar day is menstrual, so the same face of the moon is turned to the upper focus of its orbit, deviating a little to the lower focus (libration in longitude). Libration in latitude arises from the moon’s latitude and the inclination of its axis to the ecliptic. Mr. N. Mercator explained this theory in his Astronomy from Newton’s letters. Saturn’s utmost satellite revolves with a like motion, as M. Cassini observed; Jupiter’s utmost satellite also seems to, due to a spot.
The Axes of Planets
Proposition XVIII, Theorem XVI states that the axes of planets are less than diameters perpendicular to them. Diurnal revolution causes parts to recede from the axis and ascend about the equator, enlarging diameters there and shortening the axis. Jupiter’s diameter is shorter between poles than east to west. If Earth were not higher at the equator, seas would subside at poles and flood the equator.
The Proportion of a Planet’s Axis
Proposition XIX, Problem III is to find the proportion of a planet’s axis to its perpendicular diameter. Mr. Norwood measured 905751 feet between London and York in 1635, with latitude difference 2° 28′, determining a degree as 367196 feet, or 57300 Paris toises. M. Picart measured an arc between Amiens and Malvoisine, finding a degree as 57060 toises. M. Cassini (father) and his son measured from Collioure to Dunkirk, total 486156½ toises, 8° 31′ 11⅝” difference, giving 57061 toises per degree. These give Earth’s circumference as 123249600 and semi-diameter 19615800 Paris feet, assuming a spherical figure. In Paris latitude, a heavy body falls 15 Paris feet, 1 inch, 1¾ line in a second (2173 lines). Correcting for air weight (1/10000), it falls 2174 lines in vacuo. A body in a sidereal day (23h. 56′ 4″) revolving at 19615800 feet describes an arc of 1433.46 feet per second, with versed sine 0.05236561 feet or 7.54064 lines. The force of descent in Paris is to centrifugal force at the equator as 2174 to 7.54064.
The Earth’s Shadow and the Moon’s Theory
The passage discusses the Earth’s atmosphere, which refracts the sun’s light to a height of 35 or 40 miles, scattering it and dilating the Earth’s shadow. Consequently, in lunar eclipses, Newton adds 1 or 1.5 minutes to the shadow’s diameter as derived from parallax. The theory of the moon should be examined from phenomena in the order: syzygies, quadratures, then octants. For this work, Newton assumes mean motions at the Royal Observatory of Greenwich for the last day of December 1700 (O.S.): the sun’s mean motion is ♉ 20° 43′ 40″, its apogee ♊ 7° 44′ 30″; the moon’s mean motion is ♍ 15° 21′ 00″, its apogee ♎ 8° 20′ 00″, and its ascending node ♓ 27° 24′ 20″. The difference of meridians between Greenwich and the Royal Observatory at Paris is 0h 9′ 20″. The mean motions of the moon and its apogee are not yet obtained with sufficient accuracy.
The Force of the Sun to Move the Sea
Proposition XXXVI, Problem XVII is to find the force of the sun to move the sea. The sun’s force in the moon’s quadratures is to gravity as 1 to 638092.6; in syzygies, the force is double. Descending to Earth’s surface, these forces are diminished in proportion of 60.5 to 1. The former force on Earth’s surface is to gravity as 1 to 38604600, depressing the sea 90 degrees from the sun. The other force, twice as great, raises the sea directly under and opposed to the sun. The sum of these forces is to gravity as 1 to 12868200, and this is the total force of the sun to disturb the sea. In other positions, this force is as the versed sine of double its altitude directly, and the cube of the distance reciprocally. The corollary states that the centrifugal force of Earth’s parts (1 to 289 of gravity) raises equatorial waters 85472 Paris feet above the poles. The sun’s force (1 to 12868200) is to this centrifugal force as 1 to 44527, and will raise waters under and opposed to the sun only 1 Paris foot and 11 1/3 inches above those 90 degrees removed.
The Force of the Moon to Move the Sea
Proposition XXXVII, Problem XVIII is to find the force of the moon to move the sea. This is deduced from its proportion to the sun’s force, collected from the motions of the sea. At the mouth of the river Avon, three miles below Bristol, Samuel Sturmy’s observations show the ascent in vernal and autumnal syzygies is about 45 feet, but in quadratures only 25. Thus, L + S to L – S is as 45 to 25, or 9 to 5. At Plymouth, Samuel Colepress’s observations give a rise of about 16 feet, with syzygies exceeding quadratures by more than 7 or 8 feet. Supposing the greatest difference to be 9 feet, L + S is to L – S as 20.5 to 11.5, or 41 to 23, which agrees with the former. Newton prefers Sturmy’s observations and uses the proportion of 9 to 5.
The greatest tides are the third in order after the syzygies, falling nearly upon the forty-third hour after the new or full moon. At this port, they fall about the seventh hour after the moon’s appulse to the meridian, when the moon is distant from the sun or opposition by about 18 or 19 degrees in consequentia. The sun’s force at this distance is less in proportion of the radius to the co-sine of double this distance (37 degrees), or 10000000 to 7986355. Therefore, in place of S, we must put 0.7986355S. The force of the moon in the quadratures must be diminished on account of its declination from the equator (about 23° 13′), nearly in the duplicate proportion of the co-sine of the declination, making its force 0.8570327L. Thus, L + 0.7986355S to 0.8570327L – 0.7986355S is as 9 to 5.
The diameters of the moon’s orbit are as 69 to 70, so its distances in syzygies and quadratures are as 69 to 70. Its distances when 18.5 degrees advanced beyond the syzygies and quadratures are to its mean distance as 69.098747 and 69.197345 to 69.5. The moon’s force to move the sea is in the reciprocal triplicate proportion of its distance, so its forces in the greatest and least distances are as 0.9830427 and 1.017522 to 1. From this, 1.017522L x 0.7986355S to 0.9830427 x 0.8570327L – 0.7986355S is as 9 to 5, and S to L as 1 to 4.4815. Since the sun’s force is to gravity as 1 to 12868200, the moon’s force is to gravity as 1 to 2871400. Corollary 1 states that the sun’s force raises waters to 1 foot and 11 1/3 inches; the moon’s force raises them to 8 feet and 7/14 inches; their joint force raises them to a greater height.
The Retardation and Compound Motions of Tides
The analysis of tidal forces continues by examining how the vis insita—the inherent force of reciprocation retained by moving waters—modifies the simple lunar cycle. This retained force causes the greatest tides to occur not at the first syzygy but at the third, a phenomenon observed at ports like Falmouth and Bristol. In shallow channels, the retardation is even more pronounced, delaying the peak tides to the fourth or fifth after the syzygy, with specific propagation times noted along various coasts and out to islands such as the Cassiterides. The text explains that tidal waves travel through the ocean more slowly than the moon’s apparent course, a necessary delay that allows for simultaneous rises and falls in distant regions, such as the coasts of Brazil and New France.
The Compound Tides of Batsham and River Obstructions
When tides from different channels converge on a single port, they can compound to create new and complex motions. This is illustrated through a hypothetical division of tides and the real example of Batsham in Tunquin, where the waters stagnate entirely after the moon crosses the equator, only to resume a single daily flow and ebb, with the flood occurring at the moon’s setting. The passage also notes how river currents can obstruct the ingress of tides while promoting their egress, citing Sturmy’s observations on the River Avon as a specific case.
The Magnitude of Tides and the Forces of Sun and Moon
The scale of tides is directly related to the breadth of the sea, with a figure showing the oval shape of ocean basins to demonstrate that tides are smaller in the mid-sea than along the shores. The sun’s force is computed to raise the sea by approximately 9 inches, and from this, the moon’s force is derived as being 5⅓ times greater, a magnitude sufficient to generate the observed tides. These forces, while powerful over vast oceans, are deemed far too weak to have any measurable effect on pendulums or barometers.
The Density of the Moon and the Centre of Gravity
From these calculations, the moon’s density is determined to be greater than that of the sun and the earth. Furthermore, the common centre of gravity of the earth and moon is now determined with greater precision, refining the understanding of their mutual orbital motion.
The Immensity of the Fixed Stars
The discussion then turns to the fixed stars, arguing for their immense distances based on the complete lack of annual parallax. The text refutes the idea that their light could be lost over such vast spaces, confirming that their apparent faintness is due to distance alone, not any attenuation of light in transit.
The Descent of Comets into Planetary Regions
Finally, the passage argues that comets descend into the planetary regions, a conclusion supported by three lines of evidence: their observable annual parallax, the curvature of their paths, and the light emitted by their heads and tails. This is confirmed by the observations of specific comets recorded by astronomers such as Kepler, Hevelius, and Flamsted, whose data demonstrate that these bodies are not distant celestial phenomena but active members of the solar system, moving within the same spatial domain as the planets.
Main Arguments
The central argument of the Principia is that the phenomena of the natural world, both terrestrial and celestial, can be explained by a unified set of mathematical laws governing motion and force. Newton’s stated design, as expressed in the author’s preface, is to investigate the forces of nature from the phenomena of motion, and then to demonstrate other phenomena from those forces. The work argues that gravity is the universal force governing the motions of planets, comets, the moon, and the sea.
A key argument concerns the nature of mechanics itself. Newton distinguishes between rational mechanics, which concerns accurate demonstration, and practical mechanics, which concerns manual arts. He argues that geometry is founded in mechanical practice, and that rational mechanics concerns motions from forces and forces producing motions. The work thus elevates mechanics from a practical art to a rigorous mathematical science.
The Principia argues that the difficulty of natural philosophy lies in deriving forces from motion phenomena and then using those forces to demonstrate other phenomena. Book III exemplifies this method by deriving gravity toward the sun and planets from celestial phenomena, and then deducing the motions of planets, comets, the moon, and the sea from that gravity. Newton suspects that all phenomena depend on forces by which particles are mutually impelled or repelled, a conjecture that extends the scope of his theory to the microscopic realm.
Specific arguments within the work include the demonstration that planets are held in their orbits by a force that decreases in the duplicate proportion of the increase of distance, a finding that Newton first considered in 1666 but only later confirmed with accurate calculations. The work also argues for the immobility of the fixed stars, which keep the same position relative to the aphelions and nodes of planetary orbits, and which have no sensible parallax from the Earth’s annual motion, indicating their immense distance.
The work argues that the moon’s motion is subject to perturbations from the sun, and that the tides are caused by the combined forces of the sun and moon acting on the waters of the Earth. The detailed tidal analysis, based on observations at Bristol and Plymouth, derives the proportion of the moon’s force to the sun’s force, concluding that the moon’s force is about 4.4815 times the sun’s. The work also argues that comets descend into the planetary regions, evidenced by their annual parallax, the curvature of their paths, and the light of their heads and tails.
Key Concepts
Several specific named concepts and mathematical techniques are developed throughout the Principia. These include:
Definitions of Fundamental Quantities: Newton opens with definitions that establish the conceptual vocabulary of the work. Definition I defines quantity of matter as arising from density and bulk conjointly, known by weight. Definition II defines quantity of motion as arising from velocity and matter conjointly. Definition III introduces vis insita, or the innate force of matter, which is its power of resisting changes to its present state, also called vis inertia (inertia). Definition IV defines impressed force as an action changing the state of a body, consisting in action only.
First and Last Ratios: Section I of Book I develops the method of first and last ratios, a geometrical approach to dealing with infinitesimal quantities. This method allows Newton to reason about instantaneous rates of change and limits without explicitly using calculus, providing a rigorous foundation for his proofs.
Fluxions and Fluents: The biographical notes explain that Newton named the velocities of generation “Fluxions” and the quantities generated “Fluents.” His method of fluxions, possessed by 1666, was a precursor to calculus. The Principia itself, however, employs the geometrical method of first and last ratios, a choice defended in the American edition’s introduction.
Centripetal Force: A central concept of Book I, centripetal force is the force that draws bodies toward a center. The work invents methods for finding centripetal forces and applies them to various orbital configurations, including circular, elliptical, parabolic, and hyperbolic paths.
Resistance: Book II develops the concept of resistance, the force opposing motion through a medium. Newton analyzes resistance proportional to velocity, resistance proportional to the square of velocity, and mixed cases. He conducts pendulum experiments with a wooden globe to measure air resistance, deriving empirical laws for how resistance scales with velocity.
Genitum and Moments: Lemma II of Book II defines a “genitum” as a quantity produced by multiplication, division, or root extraction. Moments are the momentaneous increments or decrements of such quantities. The lemma states that the moment of any genitum equals the moments of the generating sides drawn into the indices of powers and coefficients, a result that anticipates the product rule and chain rule of calculus.
Vis Insita and Reciprocation: In the tidal analysis, Newton invokes the vis insita, or innate force, retained by waters, which alters tides by lessening differences between alternate tides and causing the greatest tides to occur not first but third after syzygies.
Syzygies, Quadratures, and Octants: These terms describe the positions of the moon relative to the sun and Earth. Syzygies are the points of conjunction and opposition (new and full moon), quadratures are the points 90 degrees away, and octants are the points 45 degrees away. Newton’s lunar theory examines the moon’s motion in these different positions.
Libration: The work discusses the moon’s libration, the apparent wobbling of the moon that allows us to see slightly more than half its surface over time. Libration in longitude arises from the moon’s diurnal motion and its elliptical orbit, while libration in latitude arises from the moon’s latitude and the inclination of its axis.
Precession of the Equinoxes: Book III addresses the precession of the equinoxes, the slow westward shift of the equinoctial points, which Newton attributes to the gravitational influence of the sun and moon on the Earth’s equatorial bulge.
Themes
Several deeper recurring meanings and interpretive themes emerge from the Principia, grounded in the text’s content and framing.
The Unity of Heaven and Earth: A dominant theme is the unification of celestial and terrestrial mechanics. Newton’s derivation of gravity as the force governing both planetary orbits and the fall of bodies on Earth dissolves the Aristotelian distinction between the sublunary and superlunary realms. The same laws that govern the moon’s motion also govern the tides and the pendulum’s swing. This theme is reinforced by the work’s structure, which moves from abstract mathematics (Book I) through resisting mediums (Book II) to the System of the World (Book III), demonstrating that the principles developed abstractly apply directly to the observed cosmos.
The Power of Geometrical Method: The Principia is as much a demonstration of method as it is a collection of results. Newton’s choice of geometrical synthesis over analytical calculus is presented as a deliberate preference, and the American edition’s introduction argues for the study of geometrical synthesis over analytical encroachment. The theme here is that rigorous demonstration, built on the foundations of Euclidean geometry and extended through the method of first and last ratios, can yield certain knowledge about the natural world. The work is presented as a monument to the power of this method, with the introduction quoting: “Le plus beau monument que l’on puisse elever a la gloire de Newton, c’est une bonne edition de ses ouvrages.”
The Role of Observation and Experiment: Despite its mathematical abstraction, the Principia is deeply grounded in empirical observation. The tidal analysis relies on specific observations from Bristol and Plymouth. The pendulum experiments measure air resistance with a wooden globe of known weight and diameter. The figure of the Earth is determined from arc measurements by Norwood, Picart, and the Cassinis. The theory of the moon is examined from phenomena in the order of syzygies, quadratures, and octants. This theme emphasizes that mathematical theory and empirical data are mutually reinforcing, with theory guiding interpretation and observation constraining theory.
The Limits of Human Knowledge: Newton’s famous self-assessment, quoted in the biographical notes, frames the Principia as a modest contribution to an infinitely larger ocean of truth. This theme of epistemic humility is reinforced by the work’s acknowledgment of uncertainties, such as the note that the mean motions of the moon and its apogee are not yet obtained with sufficient accuracy. The work presents itself as a provisional but powerful step toward understanding, not a final and complete system.
The Moral and Civic Purpose of Science: The dedication to the Teachers of the Normal School of the State of New-York frames the Principia within a broader civic and educational mission. Chittenden argues that liberties are preserved by the cultivated hearts and heads of the people, and that teachers are the prime conservators of the public weal. The work is thus presented not merely as a scientific treatise but as a contribution to the education of citizens and the preservation of democratic institutions. This theme connects the pursuit of scientific knowledge to the cultivation of an informed and virtuous populace.
The Character of the Scientist: The biographical notes present Newton as a model of scientific virtue: modest, open, generous, and uncorrupted by pride or ambition despite his universal reputation. His patience in the face of intense suffering during his final illness, his lack of complaint, and his continued conversation and reading until the end are presented as evidence of his character. This theme elevates the scientist as a moral exemplar, whose personal virtues are as admirable as his intellectual achievements.
The Author’s Conclusions
The Principia concludes with a series of arguments and findings that complete Newton’s system of the world. The final notes address the forces of the sun and moon to move the sea, the distances of the fixed stars, and the nature of comets.
On the tides, Newton concludes that the sun’s force raises the sea by about 9 inches, and the moon’s force is derived as 5⅓ times the sun’s, sufficient to generate the observed tides. He notes that these forces are too weak to affect pendulums or barometers. The moon’s density is calculated as greater than the sun’s and the Earth’s, and the common centre of gravity of the Earth and moon is more exactly determined.
On the fixed stars, Newton argues that their immense distances are evidenced by the lack of annual parallax, and he refutes the idea that light is lost over vast spaces. The stars are immovable, keeping the same position relative to the aphelions and nodes of planetary orbits, and their contrary attractions destroy mutual actions due to their immense distance.
On comets, Newton concludes that they descend into the planetary regions, evidenced by their annual parallax, the curvature of their paths, and the light of their heads and tails. He cites specific comets and observers, including Kepler, Hevelius, and Flamsted, to support this conclusion.
The work thus ends with a comprehensive account of the solar system: the motions of planets and their perturbations, the figure of the Earth and the precession of the equinoxes, the theory of the moon and the tides, the immobility of the fixed stars, and the nature of comets. All of these phenomena are explained by the single principle of universal gravitation, derived from the mathematical principles established in the earlier books. The Principia concludes as a complete and unified system of natural philosophy, demonstrating that the diverse phenomena of the heavens and the Earth are governed by one set of laws, discoverable through mathematical reasoning and confirmed by observation.
Significance
The Principia is one of the most significant works in the history of science, fundamentally transforming humanity’s understanding of the physical world. Its significance lies in several dimensions.
First, the work established the mathematical framework for classical mechanics. The definitions of quantity of matter, quantity of motion, and impressed force, along with the laws of motion, provided a rigorous foundation for all subsequent work in physics and engineering. The methods developed in the work, including the method of first and last ratios and the treatment of moments, advanced the mathematical tools available to scientists.
Second, the Principia unified celestial and terrestrial mechanics. By demonstrating that the same force of gravity governs the fall of an apple and the orbit of the moon, Newton dissolved the ancient distinction between the heavens and the Earth. This unification was a profound conceptual shift, paving the way for the modern view of a single, law-governed universe.
Third, the work established a methodological ideal for natural philosophy. Newton’s approach—deriving forces from phenomena and then demonstrating other phenomena from those forces—became the model for scientific inquiry. The emphasis on mathematical demonstration, combined with empirical verification, set a standard that subsequent scientists sought to emulate.
Fourth, the Principia had immediate and lasting practical applications. The improved theory of the moon’s motion aided navigation. The analysis of tides contributed to understanding coastal phenomena. The work’s treatment of resisting mediums informed the study of fluid dynamics and ballistics.
Fifth, the work’s influence extended beyond science into philosophy and culture. The image of Newton as the discoverer of universal laws, modestly playing on the seashore of truth, became a powerful cultural symbol. The Principia was seen as proof that human reason could uncover the order of the cosmos, inspiring the Enlightenment’s faith in rational inquiry.
Finally, the American edition of 1846, with its dedication to teachers and its framing of the work as a contribution to educational reform, demonstrates the Principia‘s enduring relevance as a tool for cultivating the minds of citizens. The work was not merely a scientific treatise but a monument to human achievement, offered as an inspiration and a foundation for future generations of learners. The Principia remains a testament to the power of mathematical reasoning to reveal the hidden order of nature, and its significance endures as both a landmark of scientific achievement and a model of intellectual rigor.
